let. - vol. 48
(2002)
{t. - no. 12
STROJNIŠKI
VESTNIK 12
JOURNAL OF MECHANICAL ENGINEERING
strani - pages 643 - 726
ISSN 0039-2480 . StrojV . STJVAX
cena 800 SIT
4.
o
CO
CM
-------------U)
CO
------------------(_>
N.
*-
1 d2tej k
+-----eg— +-----e------
P0 ij j dxi P0 ij dxpxk
for ij,k = 1,2. The vorticity equation (21) expresses time-dependent vorticity transport by viscous diffusion, convection, bouyancy forces, while the elasticity and deformation of the fluid acts as a highly nonlinear production term, making the nonlinearity of the equation even more severe, when compared to Newtonian viscous fluid flow.
By applying the curl operator to the vorticity definition:
Vx = rotv in divv = 0, robna integrala na desni strani en. (38) preoblikujemo, kar se kaže v končni obliki singularne robne integralske predstavitve kinematike prostorskega toka:
One of the most important issues in the numerical modeling of incompressible flow phenomena is to obtain a divergence-free final solution both for the velocity field and for the vorticity conservation. In the case of equations (36) or (38) it can be easily shown that solutions where none of the divergences are zero are permitted [9]. Thus, it is possible to conclude that Eq. (38) does not generally represent the kinematics of incompressible fluid flow. By using additional compatibility and restriction conditions for the velocity and vorticiy fields, <3 = rot v and div v=0, boundary integrals on the right-hand side of Eq. (38) are rewritten, resulting in the final singular boundary integral statement of the kinematics of spatial flow:
(^)v(^,tF) + a\ \F (Vu* -n ) vdtdG = a\ tF (Vu* xn )x vdtdG + a\ tF \ 3 x Vu* J dtdW + J vF-u*-dW (40)
ali tudi v zgoščenem simbolnem zapisu za krožno kombinacijo indeksov ijkij = 12312:
or in the compact symbol notation for the cyclic combination of: ijkij = 12312:
c^)v^,tF) + a\XviudtdG = al\t;
du* du* vk -------ni---------nk
1 { Čxk dxi
dtdG
du du
vj ------nj--------ni
1 <9x dx
dtdG + a\ \F coj-----dtdW-a\ \ F a>-----dtdW + \
JW lt JW lt k JW
(41)
v,- p^u^dW
-a\ \tF
JGJtF -1 j[dxi j dxj i\ WtF-1 j dxk WtF-1 k dxj
Kinematika ravninskega toka je podana z: The kinematics of planar fluid motion is given by:
c^)v^,tF) + a\G\tF vi^dtdG = a\G\tF vudtdG-ae^JtF audtdW + jW vi,F -1uF -1dW (42)
dn
Enačba (41) je popolnoma ustrezna kontinuitetni oziroma omejitveni enačbi in definiciji vrtinčnosti ter podaja kinematiko nestisljive tekočine v integralski obliki. Hitrostni robni pogoji so vključeni v robnih integralih, medtem ko je z območnim integralom zajet vpliv vrtinčnega polja na razvoj hitrostnega polja. Zadnji območni integral upošteva vpliv začetnih hitrostnih pogojev nepravega prehodnega pojava. V en. (42) se pojavljata normalni in tangentni odvod osnovne rešitve du*/dn in du*/dt, kar je pomembna razlika proti en. (39), kjer sta osnovna rešitev in njen normalni odvod, u* in du*/dn. Pri izračunu robnih vrednosti funkcij polja moramo uporabiti normalno oziroma tangentno obliko vektorske enačbe (40), [7].
Robne vrednosti vrtinčnosti so v integralski obliki zajete v območnem integralu. Izračun robnih vrednosti vrtinčnosti zaradi zapisa nesingularnega implicitnega sistema terja tangentno obliko vektorske enačbe (40).
dxj
Equation (41) is equivalent to the continuity equation, also recognized as the restriction equation and the vorticity definition expressing the kinematics of general incompressible fluid flow in the integral form. Velocity boundary conditions are incorporated into boundary integrals, while in domain integrals the influence of the vorticity field on the developing velocity field is given. The last domain integral takes into account the influence of the initial velocity conditions of false transient phenomena. In Eq. (42) the normal and tangential derivatives of the fundamental solution,5u*/cn and5u*^, are employed an important difference in comparison to Eq. (39), where the fundamental solution and the normal derivative of the fundamental solution, u* and 3u*/3n, are used. To compute the boundary values of the field functions, the normal or tangential form of vector Eq (40), [7], is required.
The boundary vorticity values are expressed in integral form within the domain integral. When the unknows are the boundary vorticities one has to use the tangential component of vector Eq. (40) because of the nonsingular implicit system of equations
c(4)n(4)xv(4,tF) + n(4)xaj jtF (Vu* -n)vdtdG = n (L)xaf \F (Vu*xn)xvdtdG + n (^)xa\ y l&xVu*\dtdW + n ()xa\ vF-1uF -1dW
4.2 Integralska predstavitev kinetike 4.2 Integral representation of flow kinetics
(43)
Za zapis kinetike toka viskoelastične tekočine Considering the kinetics of viscoelastic fluid
v integralski obliki moramo upoštevati parabolično flow in an integral representation one has to take into
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difuzivno-konvektivno naravo prenosne enačbe vrtinčnosti. Z uporabo linearnega paraboličnega difuzivnega diferencialnega operatorja:
L[-]
prenosno enačbo vrtinčnosti (33) zapišemo v obliki nehomogene parabolične difuzivne enačbe:
account the parabolic diffusion-convection character of the vorticity transport equation. Since only the linear parabolic diffusion differential operator is employed, i.e.:
dxjdxj dt
(44)
the vorticity equation (33) can be formulated as a nonhomogenous parabolic diffusion equation as follows:
\ ] d2 a dco L[a> + b = v—--------— + b = 0
dxdx dt
(45)
z naslednjim pripadajočim integralskim stavkom, zapisanim za časovni korak Dt = tF - tF-1:
with the following corresponding integral representation written in a time-increment form for a time step Dt = tF - tF-:
(č)co(č,t + v\ co-----dtdG = V\ —ukdtdG+\ bu*dtdW+ \ co u* dW
W \b f) 0JGJ 0JG-L JW-L JW F-1F -1
-1 dn
(46)
kjer je u* difuzivna osnovna rešitev, podana z en. (35). Območni integral nehomogenega nelinearnega prispevka b:
dvm 1
b =-^ + — eijgj dxj p0
vsebuje konvekcijo, vzgonske in elastične učinke, tako da velja naslednji zapis kinetike vrtinčnosti v integralski obliki:
where u* is the parabolic diffusion fundamental solution given by Eq. (35). The domain integral of the non-homogenous nonlinear contribution b, represented as:
5V
dp 1
dxi p0 dxidxk
(47)
includes the convection, the bouyancy force effects, and the viscoelastic effects. Thus the final integral statement reads as:
( ) ( ) r rtF du* r rtF dco r rtF r rtF du*
nffltu + yj co-----dtdG = vA —u*dtdG-\ covnu*dtdG + covj-----dtdW
JG Jt JG Jt JG h JW h
F-1 dn
+eij — i \ nigjpu*dtdG-eij — [ \ gjp-----dtdW + eij — [ [tF
54 dx
8x
*dtdG
(48)
1 f t F dzejk du* W f * W
-e— —--------dtd+\ coF1uF1d
JW JtF-1 JW
/VW tF -1 dxk dxi
Iz en. (48) je razvidna popolna podobnost med Eq. (48) shows the analogy between the
prenosom vrtinčnosti v viskoelastični tekočini in vorticity transport in the viscoelastic fluid and the
prenosom vrtinčnosti v toku newtonske viskozne vorticity transport in viscous, Newtonian motion,
tekočine z izjemo dodatnega elastičnega prispevka, with the only difference in the extra viscoelastic
ki deluje kot močno nelinearni izvirni člen. contribution acting as a highly nonlinear source term.
Singularno robno integralsko predstavitev By applying a similar procedure to the heat-
toplotne prenosne enačbe izpeljemo enako kakor izpeljavo transport equation, one derives the following integral
vrtinčne enačbe, tako da velja integralski stavek: statement:
c(^)T(^,tF) + a0jJtF T—dtdG = a0jJtF —u^dtdG-jJF Tvnu*dtdG
dn G tf-1 dn
+ { [tF Tvj—dtdW+\ TF1u*F1dW JW hF1 JW
(49)
4.3 Integralska predstavitev tlačne enačbe
4.3 Integral representation of the pressure equation
Tlačna en. (29) je eliptična Poissonova enačba, The pressure Eq. (29) is recognized as an
tako da uporabimo linearni eliptični Laplacev elliptic Poisson equation, thus employing the linear
diferencialni operator: elliptic Laplace differential operator:
L[]
kar se izraža v zapisu:
g2 ()
dxidxi
the following can be stated:
Lp + b = ^2p + b = 0
[] dxM
(50) (51)
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medtem ko je pripadajoča singularna robna integralska tlačna predstavitev podana s stavkom:
while the corresponding singular integral pressure representation is given by:
du*
dp
() p ($) +\Gp—dG = lfn u*dG+lbu*dW
(52)
kjer je u* Laplaceova osnovna rešitev, npr. rešitev enačbe:
where u* is the Laplace fundamental solution, being the solution of the equation:
dxdx
+ S(š,s
(53)
Za ravninsko geometrijo velja rešitev:
Z izenačitvijo navidezno prostorninskih sil:
2n
For planar geometry the following solution is suitable:
(54)
By equating pseudo-body forces with the expression:
Ofi 0(1+/ZT) idealnega plina.
0.06, 0.07 and 0.08. The mass density is given by the equation of state r = r0(1+bT).
Sl. 2. Naravna konvekcija nenewtonske viskoelastične tekočine modelirana z zgornje konvektivnim Maxwellovim reoloskim modelom (l1=0,07); vektorji hitrosti (levo), tokovnice (sredina) in izolinije
vrtinčnosti (desno)
Fig. 2. Natural convection of non-Newtonian viscoelastic fluid modeled with upper-convected Maxwell
rheological model (l1=0.07); velocity vectors (left), streamlines (middle), and vorticity lines (right)
Preglednica 1. Primerjava vrednosti Nu števila med newtonsko tekočino in različnimi modeli nenewtonske
tekočine
Table 1. Comparison of Nu number for Newtonian fluid flow and different models of non-Newtonian fluid
flow
Davis MRE - A MRE - B
l1 0 0,6 0,7 0,8 0,6 0,7 0,8
Nu 1,117 1,136 1,134 1,132 1,176 1,162 /
Geometrija problema in robni pogoji so prikazani na sliki 1. Računska mreža je sestavljena iz 80 robnih elementov in 400 notranjih celic, kar pomeni mrežo 20x20 celic z razmerjem 6 med najdaljšim in najkrajšim elementom. Vse izračune smo izvedli kot ustaljene, pri čemer smo prehodni pojav simulirali z zelo velikim časovnim korakom (At = 1016). Podsprostitev smo definirali s podsprostitvenim parametrom J, ki je znašal v primeru simulacije newtonskega toka J = 0,001 in v primeru izračuna nenewtonskih tokov J = 0,0001. Konvergenčni kriterij je vedno predstavljala napaka v velikosti = 106.
Slika 2 prikazuje vektorsko polje hitrosti, tokovnice in izočrte vrtinčnosti za zgornje konvektivni Maxwellov reološki model (B) pri vrednosti parametra 1 = 0,07. Slike 3, 4, 5 in 6 prikazujejo hitrostne profile v vzdolž vodoravne črte in hitrostne profile v vzdolž navpične črte skozi geometrijsko središče kotanje. Izvedena je primerjava newtonske primerjalne rešitve MRE na gostoti mreže 32x32, ki se odlično ujema z rešitvijo Davisa ([2] in [3]), z nenewtonskimi rešitvami MRE na že omenjeni računski mreži 20x20.
Na slikah 3 in 4 je nenewtonska tekočina modelirana kot navidezlinearna (A), na slikah 5 in 6 pa so prikazani rezultati zgornje konvektivnega Maxwellovega modela viskoelastične tekočine (B). Opazimo razmeroma velik vpliv elastičnosti tekočine
The geometry and boundary conditions of the problem are shown in Fig. 1. The computational mesh is composed of 80 boundary elements and 400 internal cells, i.e. 20x20 cells with a ratio of 6 between the longest and the shortest element. All the simulations were performed as steady with the transient phenomenon simulated using an extremely large time step (At = 1016). The underrelaxation parameter J (in the case of a temperature computation defined by Ti+1 = JTi+1 + (1- J)T and analogous for other field functions) was set to J = 0.001 in the case of Newtonian flow and J = 0.0001 in case of non-Newtonian flows. The convergence criterion was selected as s = 10-6.
Fig. 2 shows the velocity vectors, the streamlines, and the vorticity lines for the upper-convected Maxwell rheological model (B) for the relaxation-time parameter being fixed at l = 0.07. Figs. 3, 4, 5, and 6 show velocity profiles v along a horizontal line and velocity profiles v along a vertical line, through the geometric center of the cavity. The comparison between a BEM reference solution obtained on mesh density 32x32, which is in excellent agreement with the Davis solution ([2] and [3]), and BEM non-Newtonian solutions obtained on mesh density 20x20 is performed.
On Figs. 3 and 4 the non-Newtonian fluid is modeled as quasilinear (A), while in Figs. 5 and 6 the upper-convected Maxwell model (B) is used. It is easy to see a relatively strong influence of the
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Sl. 3. Nenewtonska tekočina kot Maxwellova
navidezlinearna viskoelastična tekočina (A).
Hitrostni profili v vzdolž navpične črte skozi
geometrijsko središče kotanje.
Fig. 3. Non-Newtonian fluid as Maxwell
quasilinear viscoelastic fluid (A). Velocity profiles
v along a vertical line through the center of
cavity.
(0 : 1=0; 6 : 1=0,06; 7 : 1=0,07 in/and 8 :
X=0,08)
Sl. 5. Nenewtonska tekočina kot Maxwellova
zgornje konvektivna viskoelastična tekočina (B).
Hitrostni profili v vzdolž navpične črte skozi
geometrijsko središče kotanje.
Fig. 5. Non-Newtonian fluid as Maxwell upper-
convected viscoelastic fluid (B). Velocity profiles
v along a vertical line through the center of
cavity.
(0 :X1=0; 6 :X1=0,06 in/and 7 :X1=0,07)
v primeru zgornje konvektivnega reološkega modela. V reološkem modelu (12) se ob Stokesovem odvodu dodatnega napetostnega tenzorja oziroma tenzorja gostote toka gibalne količine pojavi dodaten člen, ki opravlja prenos gibalne količine na način, ki ga lahko enačimo z raztezno - obračalnim mehanizmom v prenosni enačbi vrtinčnosti (21).
V preglednici 1 je prikazana primerjava vrednosti povprečnega Nusseltovega števila Nu za
i -0-.2 &A -O-i G.S-
Sl. 4. Nenewtonska tekočina kot Maxwellova
navidezlinearna viskoelastična tekočina (A).
Hitrostni profili v vzdolž vodoravne črte skozi
geometrijsko središče kotanje.
Fig. 4. Non-Newtonian fluid as Maxwell
quasilinear viscoelastic fluid (A). Velocity profiles
v along a horizontal line through the center of
cavity.
(0 : 1=0; 6 : 1=0,06; 7 : 1=0,07 in/and 8 :
X=0,08)
Sl. 6. Nenewtonska tekočina kot Maxwellova
zgornje konvektivna viskoelastična tekočina (B).
Hitrostni profili v vzdolž vodoravne črte skozi
geometrijsko središče kotanje.
Fig. 6. Non-Newtonian fluid as Maxwell upper-
convected viscoelastic fluid (B). Velocity profiles
v along a horizontal line through the center of
cavity.
(0 :X1=0; 6 :X1=0,06 in/and 7 :X1=0,07)
viscoelasticity in the case of the upper-convected rheological model. For the rheological model (12), not only the Stokes derivative of the extra-stress tensor is acting, but also an extra term is employed, adding the momentum transport similar to the twisting and stretching mechanism of exchange in the vorticity transport equation (21).
Table 1 shows the comparison of the values of the average Nusselt number Nu for the Newtonian
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newtonsko tekočino in za različne vrednosti napetostnega relaksacijskega časa (1) pri izbranih Maxwellovih reoloških modelih nenewtonske tekočine.
6.2 Tok v kanalu oblike Z
Kot drugi numerični primer smo preučili tok viskoelastične tekočine v zavitem kanalu oblike Z. Uporabljeni reološki model je Maxwellov zgornje konvektivni model (B). Izbran testni primer je zapletena kombinacija vstopno - izstopnega problema, ostrih robov in dodatnih nelinearnosti zaradi modeliranja toka nenewtonske tekočine. Slika 7 prikazuje geometrijsko obliko kanala z robnimi pogoji. Pri vstopu v kanal smo predpisali razviti profil laminarnega toka s povprečno hitrostjo 1,0 m/s (vin =1,0 m/ s). Pri izstopu smo definirali izstopne robne pogoje. Trdne stene zaznamujejo brezzdrsni robni pogoji neprepustnega roba.
C&jQ)
"\ 1^= &f?+6y
-&
(3,-2)
fluid and for different values of the relaxation time (A) in the case of the Maxwell non-Newtonian rheological models.
6.2 Flow in Z channel
As a second numerical example, the flow of a viscoelastic fluid in a bent channel with a Z shape is examined. The Maxwell upper-convected rheological model (B) was selected. Flow in a bent channel represents a complicated combination of the inlet-outlet problem, sharp edges, and extra nonlinearities as result of viscoelastic non-Newtonian fluid flow. The geometry of the channel with the boundary conditions prescribed is shown in Fig. 7. At the inlet to the channel, a laminar parabolic velocity profile with an average velocity of 1.0 m/s (vin =1.0 m/s) was prescribed. At the outlet, outlet velocity conditions were given. At the solid walls a no-slip velocity condition was specified.
Sl. 7. Kanal oblike Z. Geometrija in robni pogoji Fig. 7. Flow in Z channel. Geometry and boundary conditions
o-u
izstapfwlkl (?,-S)
Sl. 8. Računska mreža pri modeliranju toka viskoelastične tekočine v kanalu oblike Z Fig. 8. Computational mesh for simulation of viscoelastic fluid in Z channel
Sl. 9. Tok nenewtonske viskoelastične tekočine v kanalu oblike Z modelirane z zgornje konvektivnim
Maxwellovim reoloskim modelom (1=0,06); tokovnice (levo) in izočrte vrtinčnosti (desno)
Fig. 9. Non-Newtonian viscoelastic fluid flow in Z channel. Upper-convected Maxwell rheological model
with a1=0.06); streamlines (left) and vorticity lines (right)
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Sl. 10. Primerjava tlačnih polj v kanalu oblike Z. 0 : 1=0 (desno) in 6 : 1=0,06 (levo) Fig. 10. Comparison of pressure fields in Z channel. 0 : k1=0 (right) and 6 : I1=0.06 (left)
Sl. 11. Profil hitrosti v vzdolž navpičnice skozi
središčno točko x geometrije kanala
Fig. 11. Velocity profile v along a vertical line
through the center of the channel
(0 :X1=0 in/and 6 :X1=0,06)
Enakomerno računsko mrežo je sestavljalo 200 robnih elementov in 900 notranjih celic. Prikazana je na sliki 8.
Reynoldsovo število in Weissenbergovo število sta definirani kot:
Re =
We =
kjer sta v in A1 kinematična viskoznost tekočine in napetostni sprostitveni čas ter je (v t) karakteristična hitrost (povprečna izstopna). Karakteristična linearna izmera L pomeni polovico višine izstopne odprtine. V skladu z navedenimi definicijami je tok v kanalu blizu Stokesovemu z vrednostjo Reynoldsovega števila Re = 0,7. Vrednost Weissenbergovega števila, odločilnega za konvergenco numeričnega algoritma, je znašala We = 0,12. Vse izračune smo ponovno izvedli kot ustaljene, pri čemer smo prehodni pojav simulirali z zelo velikim časovnim korakom (Dt = 1016). Podsprostitev smo definirali s podsprostitvenim parametrom J, ki je znašal v primeru simulacije newtonskega toka J = 0,01 in v primeru nenewtonskega toka J = 0,001. Konvergenčni kriterij je bila kot vedno napaka v velikosti * = 10-6.
Sliki 11 in 12 prikazujeta profile hitrosti v in v vzdolž navpičnice skozi središčno točko geometrijske
Sl. 12. Profil hitrosti v vzdolž navpičnice skozi središčno točko y geometrije kanala
Fig. 12. Velocity profile v along a vertical line
through the center of the channel
(0 :X1=0 in/and 6 :X1=0,06)
A uniform computational mesh consists of 200 boundary elements and 900 internal cells. It is shown in Fig. 8.
The Reynolds and Weissenberg numbers were defined according to convention as:
(59) (60)
v out L
v
L
where v and k1 are the fluid kinematic viscosity and the relaxation time, respectively, and (vout) is a characteristic velocity (average at the outlet). A characteristic linear dimension L is defined as half of the outlet opening. Throughout the channel, creeping flow was assumed with Re = 0.7. The value of the Weissenberg number, which is crucial for achieving the stability of the numerical alghoritm according to the literature, was chosen to be We = 0.12. All the simulations were again performed as steady with a large time step (Dt = 1016). Underrelaxation was defined with the underrelaxation parameter J set to J = 0.01 in the case of Newtonian flow and J = 0.001 in the case of non-Newtonian flow. The convergence criterion was selected as s= 10-6.
Figs. 11 and 12 show the velocity profiles v and v along a vertical line through the center of the
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oblike kanala. Primerjamo tok newtonske tekočine (21 = 0) s tokom viskoelastične tekočine (Maxwellov zgornje konvektivni model) pri vrednosti (1 = 0,06). Slika 10 kaže primerjavo tlačnih polj omenjenih tokov.
6.3 Tok v simetričnem kanalu z nenadno zožitvijo 4:1
Predstavljeni numerični algoritem smo testirali tudi na vstopno-izstopnem problemu toka viskoelastične tekočine v simetričnem kanalu z nenadno zožitvijo 4:1, katerega geometrijska oblika je predstavljena na sliki 13.
channel. A comparison of the Newtonian fluid flow (l1 = 0) and the non-Newtonian viscoelastic fluid flow (Maxwell upper-convected model) at (l1 = 0.06) is shown. Fig. 10 presents the pressure fields in both cases.
6.3 Flow in a 4:1 planar sudden-contracted channel
The developed numerical alghoritm was tested for the test case of analysing the inflow-outflow viscoelastic fluid problem in a planar channel with a 4:1 sudden contraction. A detailed presentation of the flow geometry is given in fig. 13.
Sl. 13. Kanal z nenadno 4:1 zožitvijo Fig. 13. Channel with a 4:1 abrupt contraction
Sl. 14. Kanal z nenadno 4:1 zožitvijo. Vektorji
hitrosti (zgoraj) in vrtinčno polje (spodaj)
Fig. 14. Channel with a 4:1 abrupt contraction.
Velocity vectors (above) and vorticity field
(below)
Oba dela kanala (vstopni in izstopni) merita v dolžino L1 = L = 10H2, kar zadošča za popolno razvitje hitrostnega profila. Tokovne razmere v kanalu podobne geometrijske oblike so podrobno predstavljene v [6]. Izmera H je določena kot polovica višine izstopnega kanala (H2 = 0,125). Reynoldsovo število in Weissenbergovo število sta definirani enako kakor v prejšnjem primeru. Tok v kanalu je zelo blizu Stokesovemu z vrednostjo Reynoldsovega števila Re
Sl. 15. Kanal z nenadno 4:1 zožitvijo. Izočrte
hitrosti v smeri koordinatne osi x (zgoraj) in izočrte
hitrosti v smeri koordinatne osi y (spodaj)
Fig. 15. Channel with a 4:1 abrupt contraction.
Velocity lines in coordinate direction x (above)
and velocity lines in coordinate direction y (below)
The inlet and outlet lengths were both L1 = L = 10H2 to ensure fully developed flows in these regions. The flow field in a channel of similar geometry is described in detail in [6]. The dimension H2 is defined as the half-width of the downstream channel (H2 = 0.125). The Reynolds and Weissenberg numbers are defined as in the previous case. Throughout the channel, almost creeping flow was assumed with Re = 0.001. The
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[kerget L., Po`arnik M.: MRE za dinamiko viskoelasti~ne - BEM for the Dynamics of a Viscoelastic
= 0,001. Vrednosti Weisenbergovega števila smo spreminjali od We = 6,4*10-3U = 0,1) do We = 51,2*10-3 (21 = 0,8), kar je podobno navedenemu v [1].
Za numerično modeliranje smo uporabili diskretizacijo, sestavljeno iz 560 neenakih notranjih celic, zgoščenih okoli ostrih vogalov pri zožitvi kanala. Pri vstopu v kanal smo predpisali razviti parabolični profil laminarnega toka s povprečno hitrostjo 0,002 m/s (vin =0,002 m/s). Ta vrednost ustreza izstopnemu paraboličnemu profilu z (vout = 0,008 m/ s). Na trdnih stenah smo predpisali brezzdrsne hitrostne robne pogoje. Na začetku modeliranja (t = 0) je bilo računsko območje popolnoma napolnjeno z mirujočo tekočino. Ustaljeno stanje smo modelirali s časovnim korakom (Dt = 1016), medtem ko je vrednost podsprostitvenega parametra znašala J = 0,01. Konvergenčni kriterij smo nastavili na *= 10-6.
Sliki 14 in 15 prikazujeta vektorsko polje hitrosti, polje vrtinčnosti ter izočrte hitrosti v koordinatnih smereh x in y v ustaljenem stanju viskoelastičnega toka 1 = 0,8 za Re = 0,001 in We = 51,2x10-3.
7 SKLEP
V prispevku je predstavljena metoda robnih elementov za modeliranje toka viskoelastične tekočine. Različni Maxwellovi modeli prikazujejo široko uporabnost MRE. Kljub dodatnim nelinearnim izvirnim členom razvita shema ohranja vse prednosti modeliranja nestisljivih viskoznih tokov z MRE [5]. Kot testni primer rabijo naravna konvekcija viskoelastične tekočine v zaprti kotanji, tok v kanalu oblike Z in tok v kanalu z nenadno zožitvijo 4 : 1. Na podlagi rezultatov ugotavljamo, da je razvita shema stabilna in natančna.
Weissenberg number varied from We = 6.4x10-3 (2= 0.1) to We = 51.2x10^= 0.8), similar to
[1].
The computational mesh used for the simulations consisted of 560 non-uniform internal cells refined around the sharp corner of the inlet to the exit channel. At the inlet to the main channel the laminar parabolic velocity profile with an average velocity of 0.002 m/s (vin =0.002 m/s) was prescribed corresponding to the outlet parabolic velocity profile with (vout = 0.008 m / s). At the solid walls a no-slip velocity condition was specified. At the beginning of the simulation (t = 0) the computational domain was filled with quiescent fluid. The steady state was modeled with the time step (Dt = 1016), and with the underrelaxation parameter set to J = 0.01. The convergence criterion was selected as s= 10-6.
Figs. 14 and 15 show the velocity vector field, the vorticity field and the velocity isolines in both coordinate directions x and y in the steady state of the viscoelastic non-Newtonian upper-convected Maxwell fluid flow 1 = 0.8 for Re = 0.001 and W e = 51.2x10-3.
7 CONCLUSIONS
The boundary-domain integral approach to the solution of viscoelastic fluid motion problems is presented Different Maxwell fluid models are used to show the applicability of the proposed BEM model. All the attractive features of the BEM model, based on the application of different fundamental solutions already established in viscous fluid dynamics, are preserved [5]. The numerical scheme is verified using test cases of a viscoelastic fluid’s natural convection flow, viscoelastic flow through the Z channel, and a 4 : 1 abrupt-contraction-channel viscoelastic flow. The computational results show that the scheme is stable and accurate.
8 LITERATURA 8 REFERENCES
[1] Aboubacar, M., M.F Webster (2001) A cell-vertex finite volume/element method on triangles for abrupt
contraction viscoelastic flows. J. Non-Newtonian Fluid Mech., 98, 83-106. [2] Davis, G.D.V., I.P. Jones (1983) Natural convection in a square cavity: A comparison exercise. Int. Jou. for
Num. Meth. in Fluids., 3, 227-248. [3] Davis, G.D.V. (1983) Natural convection of air in a square cavity: A bench mark numerical solution. Int. Jou.
for Num. Meth. in Fluids., 3, 249-264. [4] Dou, H., N.P Thien (1999) The flow of an Oldroyd-B fluid past a cylinder in a channel: adaptive viscosity
vorticity (DAVSS-co) formulation; J. Non-Newtonian Fluid Mech., 87, 47-73. [5] Hriberšek, M., L. Škerget (1999) Fast boundary-domain integral algorithm for computation of incomprešible
fluid flow problems. Int. J. Num. Meth. Fluids., 31, 891-907. [6] Oliveira, PJ., FT Pinho (1999) Plane contraction flows of upper convected Maxwell and Phan-Thien-
Tanner fluids as predicted by a finite-volume method; J. Non-Newtonian Fluid Mech., 88, 63-88. [7] Škerget, L., A. Alujevič, CA. Brebbia, G. Kuhn (1989) Natural and forced convection simulation using the
velocity-vorticity approach. Topics in Boundary Element Research., 5(4), 49-86.
[kerget L., Po`arnik M.: MRE za dinamiko viskoelasti~ne - BEM for the Dynamics of a Viscoelastic
[8] Škerget, L., M. Hriberšek, G. Kuhn (1999) Computational fluid dynamics by boundary-domain integral
method. Int. J. Numer. Meth. Engng., 46, 1291-1311. [9] Wu, J.C. (1982) Problems of general viscous flow. Developments in BEM. Elsevier Appl. Sci. Publ, 2(2). [10] Wrobel, L.C. (2002) The boundary element method. Vol. 1., Applications in thermo-fluids and acoustics.
Wiley.
Naslov avtorjev: prof.dr. Leopold Škerget dr. Matej Požarnik Fakulteta za strojništvo Univerza v Mariboru Smetanova 17 2000 Maribor leo@uni-mb.si matej.pozarnik@uni-mb.si
Author’s Address: Prof.Dr. Leopold Škerget Dr. Matej Požarnik Faculty of Mechnical Eng. University of Maribor Smetanova 17 2000 Maribor, Slovenia leo@uni-mb.si matej.pozarnik@uni-mb.si
Prejeto: Received:
20.12.2002
Sprejeto: Accepted:
31.1.2003
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© Strojni{ki vestnik 48(2002)12,663-676 © Journal of Mechanical Engineering 48(2002)12,663-676
ISSN 0039-2480 ISSN 0039-2480
UDK 66.063:66.069.82 UDC 66.063:66.069.82
Izvirni znanstveni ~lanek (1.01) Original scientific paper (1.01)
Metode prepoznave poplavnega stanja pri aeraciji v posodi s turbinskim me{alom
Flooding-Recognition Methods in a Turbine-Stirred Vessel
Andrej Bomba~ - Iztok @un
V prispevku so obravnavane nekatere metode, s katerimi lahko na različne načine zaznamo poplavno stanje pri dispergiranju plina v kapljevino v posodi z enojnim Rushtonovim mesalom. To so metode merjenja splošnih veličin, npr najmanjša moč mešala in največji prirastek plinaste faze ter metoda zaznave lokalnih karakteristik faznega stika. Preskus je potekal pri dispergiranju zraka v vodo ter delno pri dispergiranju zraka v vodno raztopino karboksi-metil-celuloze (KMC) različnih koncentracij. Prikazani sta medsebojna primerjava rezultatov ter primerjava z rezultati kriterijev drugih avtorjev. © 2002 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: mešala turbinska, stanja poplavna, metode prepoznavanja, dispergiranje zraka)
This article paper presents some appropriate methods for flooding detection by the dispersion of gases into liquids in a stirred vessel equipped with a single Rushton turbine. These methods are based on measurements of global properties, such as mixing-power minimum and gas-holdup maximum, and a method based on the local interfacial characteristics of two-phase systems. The experiments were performed by dispersing air into water and air partly into water-carboxy-methyl cellulose (CMC) solutions of different concentrations. A comparison of the results of the different methods is shown, as well as a comparison with the results from criteria found in the literature. © 2002 Journal of Mechanical Engineering. All rights reserved. (Keywords: turbine-stirred vessel, flooding, recognition methods, air dispersion)
0 UVOD
Pri dispergiranju plina v kapljevino v posodi z mešalom nastajajo na lopaticah mešala plinske votline, ki so osnovni mehanizem dispergiranja. V odvisnosti od količine dovedenega plina (q) in vrtilne frekvence mešala (n) ter od prenosnih lastnosti kapljevine nastajajo na lopaticah mešala različne strukture plinskih votlin. Z večanjem pretoka zraka pri stalni vrtilni frekvenci mešala lahko nastanejo naslednje strukture plinskih votlin [4]: struktura vrtinčno oprijemajočih se plinskih votlin (VC), struktura z eno veliko plinsko votlino (1L), struktura z dvema velikima plinskima votlinama (2L), struktura s tremi velikimi plinskimi votlinami (S33), struktura s tremi majhnimi in tremi velikimi plinskimi votlinami (L33) ter struktura raztrganih plinskih votlin (RC). Po viru [9] poteka dispergiranje na industrijskih napravah večinoma v režimu velikih plinskih votlin, tako da je pomembno poznavanje hidrodinamičnega režima, pri katerem lahko napovemo poplavno stanje. Po splošni definiciji poplavnega stanja, ki jo zasledimo v literaturi, je poplavno stanje predstavljeno kot prehod v neučinkovito delovanje mešala pri dispergiranju plina. Tako se z vidika struktur plinskih voltin pojavi poplavno stanje takrat, ko je na
0 INTRODUCTION
In the dispersing of air into water using stirrers in a mixing vessel, gas-filled cavities are formed behind each blade of the impeller, which represents the basic mechanism of dispersal. Depending on the air flow rate (q) and impeller rotational frequency (n), as well as liquid transport properties, various different gas-filled cavity structures are formed. By increasing the air flow rate at a constant impeller speed the following structures [4] can be formed: a vortex-clinging (VC) structure, a structure with one large cavity (1L), a structure with two large cavities (2L), a small ‘3-3’ (S33) structure, a large ‘3-3’ (L33) structure and ragged cavities (RC). According to Smith [9], dispersal with industrial-scale reactors mostly takes place in the large cavity regime, so the prediction of hydrodynamic regimes at which flooding occurs is of great importance. The general definition of flooding is often described in the literature as a transition to the unsatisfactory operation of an impeller in a gas-liquid system. Flooding occurred when an RC structure was detected in a particular location. Otherwise, flooding corresponded to the
gfin^OtJJlMISCSD
02-12
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Bomba~ A. - @un I.: Metode prepoznave poplavnega stanja - Flooding-Recognition Methods
opazovanem mestu zabeležena struktura raztrganih plinskih votlin, poplavno stanje pa je označeno z ustreznim hidrodinamičnim režimom (n, q).
Prispevek obravnava tri različne najpogosteje uporabljene eksperimentalne metode zaznave poplavnega stanja mešala pri dispergiranju zraka v posodi z Rushtonovim mešalom v vodi in v navidezplastični tekočini (karakteristike faznega stika). To so: metoda največjega prirastka deleža plinaste faze ([8] in [12]), metoda najmanjše moči mešala [7] in metoda zaznave karakteristik faznega stika ([6], [2], [4] in [5]). Iz medsebojne primerjave rezultatov teh metod je ugotovljeno, da je pri enostopenjskem mešalu poplavno stanje močprepoznati z vsemi tremi metodami, rezultati se dobro ujemajo tudi z napovedanimi vrednostmi kriterijev drugih avtorjev.
1 POSKUS
1.1 Poskusna naprava
Poskus je bil izveden v pokončni valjasti posodi notranjega premera T = 450 mm z zaobljenimi robovi in ravnim dnom. V posodi, narejeni iz pleksi stekla, so bili osno simetrično nameščeni štirje motilniki toka, uporabljeno je bilo Rushtonovo mešalo s šestimi lopaticami. Shematski prikaz je podan na sliki 1. Uporabljeni so bili: demineralizirana voda, vodna raztopina KMC ter stisnjen zrak iz pnevmatskega voda pri sobni temperaturi. Podrobnejši opis geometrijskih parametrov mešalne posode in mešala je prikazan v delih [3] in [4].
n )
T )
¦ |Q O O flD
(D
&-
Hg)
Q-
10
? ?
-€>
a
hydrodynamic regimes marked with nF and qF, where the impeller was no longer capable of dispersing all of the introduced air.
The purpose of this article is to present three different experimental methods of impeller-flooding recognition by air dispersing into water and air dispersing into a pseudoplastic fluid (interfacial characteristics) in a stirred vessel with a single Rushton turbine. These methods are: maximal gas holdup ([8] and [12]), minimal mixing power [7] and local detection of interfacial characteristics ([6], [2], [4] and [5]). Based on a comparison of the results of the given methods with single-impeller stirring it was found that the flooding state was easily recognized with all three methods. The results were in good agreement with the predicted data from the literature.
1 EXPERIMENT
1.1 Experimental setup
An experiment was performed in a cylindrical flat-bottomed Perspex vessel of diameter 450 mm with rounded edges and four baffles mounted perpendicularly to the vessel wall. A Rushton disk impeller with six blades was used. Demineralized water, a water solution of CMC and compressed air at room temperature were used in all experiments. The geometric details of the vessel and the impeller can be found elsewhere [3] and [4].
1 frekvenčni krmilnik, frequency controller
2 izmenični elektromotor, AC power drive
3 merilnik vrtljajev, tachometer
4 dinamometer, torque transducer
5 uporovna sonda, resistivity probe
6 ojačevalnik, amplifier
7 rotameter
8 tlačno zaznavalo, pressure sensor
9 PC + National 12bit AD pretvornik / converter 10 vir istosmerne napetosti, DC supply
(x) merjene veličine, measured quantities
&
8
o
komprimiran zrak
Sl. 1. Merilna proga Fig. 1. Experimental setup
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3
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Bomba~ A. - @un I.: Metode prepoznave poplavnega stanja - Flooding-Recognition Methods
1.2 Merilna proga
Merilna proga je prikazana na sliki 1. Pogonski elektromotor je bil krmiljen s frekvenčnim krmilnikom, vrtilna frekvenca mešala je bila merjena s števnikom impulzov, ki deluje na načelu odboja lastnega signala z razredom točnosti 0,1. Vrtilni moment gredi je bil merjen z dinamometrom razreda točnosti 1, prostorninski pretok zraka z rotametrom razreda točnosti 2, z upoštevanjem poprave tlaka. Za omogočanje meritev večjih pretokov zraka sta bila vzporedno vezana dva rotametra. Celotni prirastek plinaste faze (a) je bil merjen po načelu, opisanem v delu [8], prikazano na sliki 1. Zaznavalo gladine je bil nameščen na 2/3 radija posode.
Za obratovalne pogoje preskusne naprave so pri danih omejitvenih pogojih in točnosti merilne opreme izračunane tudi relativne napake naslednjih veličin [10]: največja merilna napaka momenta 16,1% (pri najnižjih vrtljajih), merilna napaka merjenja vrtilne frekvence 0,5%, relativna napaka merjenja temperature 0,5 %, relativna napaka merjenja tlaka s tlačnim zaznavalom 2%, relativna napaka Froudovega števila 1%, relativna napaka pretočnega števila 1,1% in relativna napaka globalnega deleža plinaste faze, manjša od 5%.
1.3 Metode zaznave poplavnega stanja
Za vse v nadaljevanju opisane metode je veljal enoten kriterij izvajanja meritev, po katerem so bile iskane veličine izmerjene pri nespremenljivi vrtilni frekvenci mešala (ni = konst) in postopnim majhnim povečevanjem pretoka zraka (q1, q2, q3,...qk-1, qk). Pri vsaki nastavitvi (ni, q), potem ko se je popolnoma razvil tok, so bile izmerjene višina gladine v posodi (H ), močmešala pri dispergiranju (P ) ter strukturna funkcija (M i). Pri nastavitvi na višjo vrtilno frekvenco mešala n2, je bil postopek ponovljen.
1.3.1 Metoda zmanjšanja moči
Moč je bila izračunana iz izmerkov vrtilne frekvence in zaviralnega momenta mešalne osi. Na splošno se z večanjem pretoka plina pri dispergiranju moč mešala zmanjšuje in doseže najnižjo vrednost tik pred nastankom poplavnega stanja. Pri nastanku poplavnega stanja se moč mešala poveča, (sl. 2), kar so v svojih delih opisali ([7] ter [6]). Tako lahko poplavno stanje prepoznamo iz odvisnosti razmerja moči p (P /P), ki se ob nastanku poplavnega stanja izrazito poveča. Tedaj plin mešalo obteka, mešalo pa zajema skoraj v celoti kapljevino, ki rotira znotraj dvigajočega se obročnega dvofaznega toka. Posledično se to kaže s povečanjem moči mešala. Takšen hidrodinamičen režim, pri katerem je zaznano poplavno stanje »označimo« kot (qF, n), določeno posredno pri tisti vrednosti (pi)f, ko je bilo zadoščeno pogoju:
1.2 Measurements
The experimental setup is shown in Figure 1. The AC motor drive was controlled by a frequency controller, the impeller speed was measured with an infrared reflection counter of accuracy class 0.1. The impeller torque was measured using an in-line, precisely calibrated, HBM torque transducer of accuracy class 1. The volumetric air flow rates were measured by calibrated rotameters of accuracy class 2 , corrected for actual pressure. To enable measurements of larger air flow rates two rotameters were connected in parallel. The measurements of gas holdup (a) were based on the change of the liquid height in the g vessel using the method described in work of [8], see Figure 1. A level sensor was located at 2/3 of the tank radius.
The relative errors of the measured quantities were calculated for the working conditions of the experimental setup as follows [10]: maximum relative error of the torque at the lowest impeller speed was 16.1%, the relative error of the rotational speed was 0.5%, the relative error of temperature was 0.5%, the relative error of pressure was 2%, the relative error of the Froude number was 1%, the relative error of the Flow number was 1.1% and the relative error of the global gas holdup was less than 5%.
1.3 Flooding-recognition methods
For all the experimental methods described below the properties being investigated were always measured using the same principle. At a constant rotational impeller speed (ni = const) the surface level (H i), the gassing mixing power (P ) and the structural function (M ) with successive small increases in air flow rate (q1, q2, q3,...qk1, qk) were taken. At each setting of («, q) after a fully-developed regime was achieved, H i and P i were measured or M i recorded. After increasing the impeller speed to n2, the procedure was repeated.
1.3.1 Minimum mixing power
The mixing power was calculated based on the rotational impeller speed and the mixing torque. In general, increasing the gas flow rate during dispersion reduces the mixing power and achieves the lowest value just before flooding occurs. When the impeller is flooded the mixing power increases, as shown in Figure 2, and described elsewhere [7], [6]. With this principle the power ratio curve described as p = P /P can be used for flooding recognition. In such a regime the gas remains undistributed, the impeller is filled with liquid rotating inside an annular two-phase flow, which rises around the mixing shaft towards the free surface. Consequently, an increase in the mixing power takes place. The flooding regime (<7F, n) was determined indirectly by such a value of (pi )F that fits the given condition:
stran 665
Bomba~ A ods
Pg/P
XXx X x
X X
X X XXxXxx XxxxX
n
Sl. 2. Odvisnost razmerja moči od vrtilne frekvence mešala pri nastanku poplavnega stanja Fig. 2. Power ratio vs. impeller rotational frequency
Fli-Fli-
kjer so bile predhodno izmerjene vrednosti razmerij moči (p 1, p2, p 3 ,...pr.. p k-1, pk), pri čemer je cF = 2.
1.3.2 Celotni delež plinaste faze
Celotni deležplinaste faze je bil podan z razmerjem med prirastkom prostornine zaradi vnesene plinaste faze in celotno prostornino dvofaznega sistema, kar je lahko za posodo nespremenljivega prereza prevedeno v razmerje višin:
>c
where (p1,
1p2, p3,---p
k-1
(1), , pk) were previously meas-
ured power ratios, and cF = 2.
1.3.2 Maximum gas holdup
Global gas holdup was defined by the ratio between the gas phase and the liquid phase volume that can be transformed into a ratio of heights as follows:
H-H
H
(2),
kjer pomenijo: Hg - višino gladine dvofaznega sistema pri dispergiranju in H– višino gladine vode pri mešanju brez dovajanja zraka. S povečevanjem vnosa plina v posodo se viša tudi gladina dvofaznega sistema do neke mejne vrednosti, ki je povzročena z nastankom poplavnega stanja. Pri prehodu iz dispergirnega v poplavno stanje se pojavi opazno znižanje gladine, kakor je prikazano na sliki 3. Do te spremembe pride zaradi izrazito nehomogene porazdelitve plinaste faze po prostornini kapljevine, tako v kapljevini pod mešalom ni obtekanja dvofaznega toka. Za določitev hidrodinamičnega režima poplavnega stanja (qF, n) je bil uporabljen kriterij, ko je bilo pri vrednosti (a)F zadoščeno pogoju:
where Hg means the height of the two-phase free surface during dispersion and H is the liquid height in single-phase mixing. Increasing the air flow rate during dispersion increases the gas holdup correspondingly, until flooding occurs. In the flooding transition a remarkable reduction of gas holdup can be seen, as depicted in Figure 3. This change can be explained by the extremely non-uniform distribution of gas in the liquid bulk, where, especially below the impeller, there was no two-phase circulation. The flooding regime (qF,nF) was determined indirectly by such a value of (ai)F that fits the given condition:
Fli vrednosti celotnih deležev plinaste faze (a , a ,...
Fl
>c
(3),
a,... agk-1, a g,k) so bile predhodno izmerjene. 1.3.3 Metoda karakteristik faznega stika
Mehanizem dispergiranja je tesno povezan s strukturami plinskih votlin na lopaticah mešala [4]. S povečevanjem pretoka zraka se povprečna gostota izstopajočega dvofaznega toka iz mešala manjša, kakor tudi moč mešala, vse dokler se ne pojavi poplavno stanje [13]. Za lokalno zaznavo plinskih votlin za lopaticami mešala je bila že
where the values (a 1, a 2,... aI, ... a ,k-1, a k) were previously measured. * *
1.3.3 Interfacial characteristic method
The dispersing mechanism is closely connected with the different structures of gas-filled cavities that are present at the impeller blades, as described in [4]. With increasing the air flow rate the average density of the discharge two-phase flow decreases (as well as the impeller power) until flooding occurs [13]. For local detection of the gas-filled cavity behind the impeller blade a previously
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Bo gnition Methods
X X
x x x
x -x - x
a
n
Sl. 3. Odvisnost celotnega deleža plinaste faze od vrtilne frekvence mešala Fig. 3. Global gas holdup vs. impeller rotational frequency
poprej razvita preskusna metoda karakteristik faznega stika. Velika razlika v električni upornosti vode in zraka je osnovna lastnost, ki jo z uporovno sondo zaznamo in pomeni odziv sonde v obliki električne napetosti, oziroma ustrezno strukturno funkcijo:
Mp(x,t)
1, x je v fazi / is in 0, x ni v fazi / is not
p = {L, G, S}
kjer so v področju dvofaznega toka na mestu x ter v času t mogoča tri stanja p: L kapljevita faza , G plinasta faza in S stična površina.
Frekvenčna analiza strukturne funkcije M z diskretno Fourierjevo preslikavo omogoča prikaz značilnih frekvenc pojavljajoče se plinaste faze. Fourierjevi koeficienti so bili določeni iz:
developed experimental method based on local interfacial characteristics was used. A remarkable difference in the electrical resistance of the water and the air is a basic property, which can be detected using a resistivity probe and presented as a probe voltage response or corresponding structural function Mp defined as:
phase p in phase p
where in a two-phase flow field three states p are possible at a particular point x at any time t: the liquid phase L, the gas phase G or the phase interface S.
A frequency analysis of the structural function Mp with a discrete Fourier transformation enabled the presentation of the significant frequencies of an appearing gas phase. The Fourier coefficients Xk were obtained from:
(4)
N-1 - j2pik
Xk=AtYjMp(tk)e^r
(5),
kjer pomeni Dt časovni korak med vzorcema diskretne funkcije. Med Fourierjevimi koeficienti Xk, ki se ujemajo s frekvenco k/(NDt), so veljavni le koeficienti med k = 0 to k = N/2–1. Kriterij prepoznave struktur plinskih votlin je podrobneje opisan v delu [4], ter omogoča pri mešanju z Rushtonovim mešalom prepoznavo naslednjih struktur: vrtinčno oprijemajoča struktura plinskih votlin (VC), struktura z eno veliko plinsko votlino (1L), struktura z dvema velikima plinskima votlinama (2L), struktura s tremi velikimi plinskimi votlinami (S33), struktura s tremi majhnimi in tremi velikimi plinskimi votlinami (L33) ter struktura raztrganih plinskih votlin (RC). Značilni vzorci struktur v frekvenčnem prostoru so prikazani na sliki 4. Kakor je videti iz razvoja struktur, je zadnja značilna struktura, struktura raztrganih plinskih votlin, ki pripada poplavnemu stanju. Po viru [11] se stabilna struktura L33 vrne v strukturo šestih simetričnih oprijemajočih se plinskih voltin, pri višjih vrtilnih frekvencah mešala pa se prehod naredi prek strukture s šestimi velikimi plinskimi votlinami enake velikosti, ki so opisane kot silovito vibrirajoče votline oziroma raztrgane votline [7].
where Dt denotes the time interval between successive instants t. Among the Fourier coefficients X that correspond to the frequency k/(NDt), only the coefficients from k = 0 to k = N/2 – 1 are meaningful. The criterion for gas-filled cavity-structure recognition was described in detail in [4], as were the recognized structures: a vortex-clinging (VC) structure, a structure with one large cavity (1L), a structure with two large cavities (2L), a small ‘3-3’ (S33) structure, a large ‘3-3’ (L33) structure and ragged cavities (RC). A significant pattern of the structures in a frequency domain is depicted in Figure 4. The last evident structure, as can be seen from the evolution of the frequency domain, is the ragged cavities structure, and corresponds to the flooding state. According to the literature [11] the stable L33 structure reverts to six symmetrical clinging cavities; at higher impeller speeds the transition occurs through a regime with six large cavities of identical size, which are described as violently vibrating, i.e. ragged cavities [7].
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Bom s
K,^>
A.. A
VV
_J \^
J V
J V.
naraščajoč pretok zraka increasing gas flow
AN'VAtv
Sl. 4. Razvoj struktur plinskih votlin ob postopnem večanju pretoka zraka. Poplavno stanje je določeno s pojavom strukture raztrganih plinskih votlin. Fig. 4. Gas-filled cavity structure development with incremental increase of gas flow. Flooding is determined with the appearance of ragged cavity structure.
2 ANALIZA REZULTATOV IN PRIMERJAVA
2.1 Nastanek poplavnega stanja pri dispergiranju zraka v vodi
2.1.1 Metoda celotnega deleža plinaste faze
Nastanek poplavnega staja je opažen pri tisti vrednosti pretočnega števila, pri kateri se pojavi opazno znižanje gladine dvofaznega sistema, oziroma celotnega deleža plinaste faze. Na podlagi spremembe gladine lahko prepoznamo poplavna stanja le za vrednosti Fr od 0,05 do 0,6. Na sliki 5 so prikazane vrednosti a pri Fr = 0,05 do 0,13 v odvisnosti od Fl. Vidno je enakomerno zvečanje deleža plina pri povečevanju vnosa zraka, sledi močno zmanjšanje vrednosti v točki poplavnega stanja. Pri nadaljnjem večanju pretoka zraka se zaradi večjih mehurčkov in večje količine plina v stebru plina okoli mešalne osi gladina lahko znova rahlo zviša, kar pa je z vidika učinkovitosti zaradi zrazito nehomogene porazdelitve plinaste faze neučinkovita operacija. Na sliki 6 so prikazane vrednosti a pri Fr = 0,2 do 0,6 v odvisnosti od Fl. Tudi tu so lepo vidna zmanjšanja vrednosti v točkah poplavnega stanja. Pri večjih vrednostih Fr>0,7 je vnesena količina zrakaže tako velika, da se pojavlja močno nihanje gladine s ‘pljuskanjem’ vode, kar je za odbiranje višine gladine nezanesljivo. Tudi
2 RESULTS ANALYSIS AND COMPARISON 2.1 Flooding in air dispersing into water
2.1.1 Maximum gas holdup method
In this method the flooding is detected by the particular flow number, where there is a rapid decrease of surface level or gas holdup. Based on the free-surface level change only flooding states at Froude numbers between 0.05 and 0.6 can be recognized. In Figure 5 the dependence of ag on Fl at given Fr = 0.05 to 0.13 is depicted. A reasonable increase of ag can be seen, with its dependence on the increasing flow number, followed by a rapid reduction of ag in the flooding regime. With further enlargement of the air input the surface level can be a little higher than that in the flooding state, probably due to large air bubbles that rise upwards to the surface around the mixing shaft like a slug flow. Such an extreme non-homogeneous distribution can be described from the efficiency aspect as an unsatisfactory operation. Depicted in Figure 6 is the dependence of ag on Fl at Fr = 0.05 to 0.13. Again a decrease of ag can be seen in the flooding state. At higher values, at Fr>0.7, a waving of the free surface occurred so the relevant readings were no longer possible. Otherwise,
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a
0,014 n-------
0,012 -
0,010 -
0,008 -
0,006 -
0,004 -
0,002 -
0,000--------
0,040 0,045 0,050 0,055 0,060 0,065 0,070 0,075 0,080 0,085 0,090
Fl
Sl. 5. Prirastek celotnega deleža plinaste faze v odvisnosti od Froudovega in pretočnega števila Fig. 5. Global gas holdup vs. Froude and Flow number
Fr 0,05 Fr 0,07 Fr0,09 Fr0,13
a
0,10 0,08 0,06 0,04 0,02
0,00
*r%
«fe
«
Fr 0,2 Fr 0,3 Fr 0,4 Fr 0,5 Fr 0,6
0,10 0,14 0,18 0,22 0,26 0,30 0,34 0,38
Fl
Sl. 6. Prirastek celotnega deleža plinaste faze v odvisnosti od Froudovega in pretočnega števila Fig. 6. Global gas holdup vs. Froude and Flow number
sicer pri nastanku poplavnega stanja v mešalniku težko zaznamo gladino dvofaznega sistema, kar se kaže tudi z večjo napako izmerka.
V literaturi [11] in [7] so razmejitvene črte različnih struktur dvofaznih tokov običajno prikazane v tokovni mapi, diagramu, izraženem z brezizmernimi števili Fr in Fl. V diagramu 7 so vnesene vrednosti, ki ustrezajo prepoznavnim poplavnim stanjem. Razmejitvena črta poplavnega stanja je podana z ustrezno poenostavitveno premico, katere koeficienti so prikazani v preglednici 1.
readings of Hg in all flooding states were difficult to detect precisely, which was reflected in a higher relative error of ag.
Usually the delineation line between the dispersing and flooding regimes is depicted in a flow regime map expressed with dimensionless Fr and Fl numbers, such as the one found in the literature [11] and [7]. In Figure 7 all the recognized flooding regimes (nF, qF ) given by this method are indirectly shown through Fr an Fl numbers. Based on this data the flooding line is approximated by Equation 6, coefficients k1 and k2, and the regression coefficient r; all are given in Table 1.
Preglednica 1. Pripadajoči koeficienti k enačbi [6] Table 1: Corresponding coefficients to Eq. [6]
metoda method
k1
k2
r
global gas holdup"***" -0,54 2,1 0,96
najmanjši porast moči minimal mixing power
značilnica faznega stika interfacial characteristics
0,05 1,5 0,97
0,33 1,4 0,96
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2.1.2 Metoda zmanjšanja moči
S to metodo so potekale meritve poplavnega stanja pri vrednostih Fr od 0,05 do 1,13. Na slikah 8 in 9 so podane tipiČne krivulje razmerja moči P/P0, pri katerem je v poplavnem stanju opazno zvečanje razmerja moči. Pri manjših vrednostih Fr (0,05, 0,06, 0,07 in 0,09 na sliki
1,00
0,80
2.1.2 Minimum mixing power
According to this method the measurements were taken at Fr numbers from 0.05 to 1.13. Typical curves of the power ratio P/P0 are depicted in Figures 8 and 9, where a steep increase of the power ratio during the flooding state can be seen. On the other
Fr
0,60 0,40 0,20
0,00
dispergiranje
dispersing
poplavno stanje
flooding
0,00
0,10
0,20
0,30
0,40
0,50
Fl
Sl. 7. Točke poplavnega stanja določene z metodo celotnega deleža plinaste faze v tokovnem diagramu z
mejo poplavnega stanja Fig. 7. Flooding points determined by global gas holdup method on a flow chart with approximated
flooding limit
P/P
1,0 1,0 0,9 0,9 0,8 0,8 0,7 0,7 0,6
J^A
^v?.^
^Fr 0,05 ^Fr 0,06
^Fr 0,07
^Fr 0,09
0,03
0,04
0,05
0,06 Fl
0,07
0,08
0,09
Sl. 8. Odvisnost razmerja moči od pretočnega števila pri različnih vrednosti Froudovega števila Fig. 8. Power ratio vs. Flow number at different Froude number values
1,0 0,9 0,8
Pg/P 0,7
->*>*»<
0,6 0,5 0,4 0,3
-^Fr 0,13 -^Fr 0,2 -^Fr 0,3 -^Fr 0,4 -^Fr 0,5
l
0,00
0,05
0,10
0,15 Fl
0,20
0,25
0,30
Sl. 9. Odvisnost razmerja moči od pretočnega števila pri različnih vrednosti Froudovega števila Fig. 9. Power ratio vs. Flow number at different Froude number values
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8) metoda ni dala pomembnih rezultatov. Razlog je lahko v premajhni moči mešala, saj so izmerki vrtilnega momenta mešala enakega velikostnega razreda kakor odstopki merilnika vrtilnega momenta. V diagramu, prikazanem na sliki 11, so podane vse z metodo zmanjšanja moči prepoznane točke poplavnega stanja. Podana je tudi ustrezna razmejitvena premica, katere koeficienta k1 in k2 in poenostavitveni koeficient r so prikazani v preglednici 1.
2.1.3 Metoda karakteristik faznega stika
Iz postopnega razvoja značilnih vzorcev struktur plinskih votlin v frekvenčnem prostoru (sl. 4) za vsako nastavitev parametrov hidrodinamičnega režima (n=konst, q), je lepo razviden nastanek strukture raztrganih plinskih j votlin, ki ustreza poplavnemu stanju Režimi (n, q), prepoznani kot poplavno stanje, so bili v obliki brezizmernih števil Fl in Fr vneseni v diagram na sliki 12. Izmerjene vrednosti so predstavljene z ustrezno razmejitveno premico, katere koeficienti so prikazani v preglednici 1.
2.1.4 Medsebojna primerjava rezultatov različnih metod
Pri meritvah so se glede na uporabljeno metodo in merilna oprema pojavile nekatere omejitve. Tako z metodo prirastka deleža plinaste faze ni bilo mogoče določiti poplavnega stanja predvsem pri večjih pretokih zraka, medtem ko z metodo zmanjšanja moči predvsem pri nižjih pretočnih številih. Z metodo karakteristik faznega stika je bilo mogoče prepoznati poplavno stanje v celotnem območju merjenih režimov. Za prepoznavna poplavna stanja, določena z vsemi tremi metodami, so bile izdelane ustrezne razmejitvene premice, podane z enačbo:
hand, this method did not give any relevant results for smaller Fr numbers (0.05, 0.06, 0.07 and 0.09 in Figure 8), which can be explained with the very small values of the impeller power and are of the same scale as the dynamometer's accuracy class. All the recognized flooding regimes are depicted in Figure 11, together with the corresponding correlated delineation line; the appropriate coefficients k1 and k2 and the regression coefficient r are given in Table 1.
2.1.3 Interfacial characteristic method
From the coherent development of the structure patterns in a frequency domain, as shown in Figure 4, the appearance of the ragged cavities structure can be found for each hydrodynamic regime set (n=const, q). The corresponding flooding regimes (n, q) were collected and are shown in the form of dimensionless Fr and Fl numbers in a flow regime map, see Figure 12, as well as a flooding line approximated in the form of eq. 2. The corresponding coefficients are given in Table 1.
2.1.4 Comparison of results of different methods
According to the methods used and the measuring instrumentation, some restrictions appeared. Flooding could not be recognized at higher air flow rates with the global-gas-holdup method and at lower gas flow rates with the minimum-mixing-power method. Using the interfacial characteristics method gave relevant recognition of the flooding in all the measured regimes. For all the recognized flooding states with the given methods, the corresponding regression lines were approximated by the equation:
Fr = k + k-Fl
2F
(6).
Vrednosti posameznih koeficientov so Individual values of the coefficients are given in Table prikazane v preglednici 1. 1.
Pg/P
0,50
0,45
0,40
0,35
0,30
0,25
0,20
0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80
Fl
Sl. 10. Odvisnost razmerja moči od pretočnega števila pri različnih vrednosti Froudovega števila Fig. 10. Power ratio vs. Flow number at different Froude number values
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1,20 1,00 0,80 Fr 0,60 0,40 0,20 0,00
dispergiranje dispersing
poplavno stanje flooding state
0,00 0,10 0,20 0,30 0,40 0,50 0,60 0,70 0,80 0,90 1,00
Fl
Sl. 11. Točke poplavnega stanja določene z metodo razmerja moči v tokovnem diagramu z mejo
poplavnega stanja Fig. 11. Flooding points determined by power ratio method on a flow chart with approximated flooding
1,20 1,00 0,80 Fr 0,60 0,40 0,20
limit
0,00
0,00
0,20
0,40
0,60
0,80
1,00
Fl
Sl. 12. Točke poplavnega stanja določene z metodo karakteristik struktur faznega stika v tokovnem
diagramu z mejo poplavnega stanja Fig. 12. Flooding points determined by local characteristics method on a flow chart with approximated
flooding limit
Primerjava rezultatov kaže, da so vse tri metode primerne za določitev poplavnega stanja mešala. Pri nižjih vrednosti Fr in Fl je medsebojno razhajanje manjše med metodami celotnega prirastka deleža plinaste faze in karakteristikami faznega stika. Glede na strmino pripadajočih poenostavitvenih premic je boljše ujemanje med metodama karakteristik faznega stika in najmanjše moči mešanja.
Pri dispergiranju plina z enim mešalom se lokalne spremembe na lopaticah mešala in v neposredni okolici mešala kažejo enoznačno tudi na celotni ravni. Pri poplavnem stanju mešala se torej
A comparison of the results showed that all three methods are suitable for flooding detection. The smallest differences can be seen at the smaller values of Fr and Fl between the global-gas-holdup method and the interfacial characteristics method. According to the trend, better agreement can be found between the minimum mixing-power and the interfacial characteristics methods.
In gas dispersing with one impeller, the local changes at the impeller blades and in the close vicinity of the impeller affect the global level directly. In impeller flooding the global change is expressed
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celotna sprememba izraža kot zmanjšanje celotnega deleža plina oziroma kot povečanje moči mešala. Poraja se vprašanje, ali je mogoče zaznati poplavno stanje pri dispergiranju z dvema mešaloma ali tremi mešali s katerokoli celotno metodo, kar ostaja odprto vprašanje za nadaljnje raziskave.
2.1.5 Primerjava rezultatov z napovedanimi po virih iz literature
Rezultati naših raziskav so bili primerjani tudi z rezultati kriterijev drugih avtorjev, kar je prikazano na sliki 13. Primerjani so le kriteriji, ki obsegajo tudi geometrijska razmerja in velikostni razred reaktorja kakor v naši raziskavi pri dispergiranju z enim Rushtonovim mešalom. Po [12], kriterij izhaja iz meritev prirastka deleža plinaste faze, je meja poplavnega stanja v reaktorju pri D/T = 1/3 definirana z vrtilno frekvneco mešala ter pretokom plina v obliki:
through a reduction of the gas holdup or an increase of the minimum mixing power. The question appears, whether it is possible to detect flooding by dispersing with dual or triple impellers using any of the mentioned global methods? This remains open for further research.
2.1.5 Comparison of the results with those quoted from the literature
The results of our experiments were compared with those based on the correlations of other researchers, as can be seen in Figure 13. Only the criteria including the geometrical ratios between the impeller and the tank diameter and reactor-size scale were taken for a comparison with the experimental results during dispersion with a single Rushton turbine. According to [12], these criteria are derived from global-gas-holdup measurements, a flooding regime at a D/T ratio equal to 1/3 is defined with impeller speed and gas flow rate in the form:
nF = 2,694q 0,283-T –1,207
(7).
Nienow [7] je postavil povezavo, ki zajema tudi Nienow [7] postulated a correlation including
različna geometrijska razmerja med mešalom in posodo (1/ different geometrical ratios (1/3<104;
Re = 2,01 *104; c) - f = 52,8 Hz, Re = 2,01 *104) Fig. 6. Velocity diagrams in x-y plane at the diffuser outlet: a) - f = 30 Hz, Re = 1.17x104; b) - f = 30 Hz,
Re = 2.01 x104; c) - f = 52.8 Hz, Re = 2.01 *104)
vrtilna frekvenca difuzorja in s tem razmerje N (sl. 5b, 5c). To potrdijo tudi hitrostni diagrami v ravnini x-y na izstopu iz difuzorja (sl. 6).
2.2 Potek časovno povprečenih srednjih vrednosti radialnih komponent hitrosti («)
V vrtečem se difuzorju se kot posledica radialnega tlačnega gradienta en. (1) pojavi radialna komponenta hitrosti zraka v smeri proti vrteči se steni difuzorja. Največje radialne komponente hitrosti se po pričakovanju pojavijo v območju največjih obodnih hitrosti, saj so le-te glavni vzrok za nastanek radialnih komponent (centrifugalna sila).
Na sliki 7 so prikazani diagrami radialnih komponent hitrosti v različnih delovnih razmerah difuzorja na njegovem izstopu.
Diagrama na sliki 7 kažeta, da se največje vrednosti radialne komponente hitrosti pojavijo tik ob steni difuzorja na oddaljenosti r/R * 0,99. Največje vrednosti obodne hitrosti se pojavijo sicer neposredno na vrteči se steni sami (r/R = 1), vendar zaradi stene tam radialne komponente ni. Poleg tega je razvidna odvisnost poteka radialnih komponent hitrosti od frekvence vrtenja difuzorja ter (vzdolžnega) Reynoldsovega števila, torej od razmerja N. Pri višji frekvenci vrtenja difuzorja in nespremenjeni vrednosti Reynoldsovega števila (višji N) se pojavijo večje vrednosti radialnih komponent hitrosti, kar lahko pričakujemo, saj so v tem primeru večje tudi obodne komponente hitrosti zraka. Po drugi strani pa se pri večjih vrednostih Reynoldsovega števila in nespremenjeni vrtilni frekvenci difuzorja (nižji N) pojavijo nižje vrednosti radialnih komponent hitrosti zraka; z naraščanjem radialne oddaljenosti od stene difuzorja pa vrednosti radialnih hitrosti pri manjših vrednostih N postopoma prerastejo tiste pri večjih vrednostih N.
the same value of the Reynolds number the values the rotating frequency and thus the swirl rate, N, is reduced (Figs. 5b and 5c). This is also confirmed by velocity diagrams in the x-y plane at the diffuser outlet (Fig. 6).
2.2 Distribution of time-averaged mean radial velocity components (u)
The radial velocity component in the airflow oriented towards the rotating diffuser wall appears in the rotating diffuser as a consequence of the radial pressure gradient (Eq.1). The peak radial velocities coincide with the region of the peak tangential velocities, as one would expect, since the tangential velocities are the main reason for the appearance of the radial velocities (centrifugal force).
Fig. 7 shows the diagrams of radial velocity components at the diffuser outlet at different operating conditions of the diffuser.
It can be clearly seen from the diagrams in Fig. 7 that the peak value of the radial velocity component occurs close to the wall in a rotating region of the airflow - at a relative distance of about r/R * 0.99. The highest values of the tangential velocity take place directly on the diffuser wall (r/R = 1), but there is no radial component because of the wall. Apart from that, the radial velocity distribution depends on the rotating frequency of the diffuser and on the (axial) Reynolds number of the airflow, i.e. on the swirl rate N. A higher rotating frequency leads to higher radial velocity values at fixed values of Re (higher N); this situation was anticipated because the peak tangential velocity values are also higher in this case. On the other hand, higher Re values at a fixed rotating frequency (lower N) produce lower peak radial velocity component values; the values of the radial velocity at lower N eventually exceed those at higher N when the radial distance from the diffuser wall is increased.
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a) b)
Sl. 7. Potek časovno povprečenih radialnih komponent toka na izstopu iz difuzorja:
a) - f = 52,8 Hz; b) - f = 30 Hz
Fig. 7. Distribution of the time-averaged mean radial velocity components at the diffuser outlet:
a) - f = 52.8 Hz; b) - f = 30 Hz
Sl. 8. Potek časovno povprečenih vzdolžnih komponent toka na izstopu iz difuzorja: a) f = 52,8 Hz; b) f = 30 Hz Fig. 8. Distribution of the time-averaged mean axial velocity components at the diffuser outlet:
a) f = 52.8 Hz; b) f = 30 Hz
2.3 Potek časovno povprečenih srednjih vrednosti vzdolžnih komponent hitrosti (ua)
Na prečnih prerezih, ki so bliže izstopu iz difuzorja in kjer so obodne komponente hitrosti izrazitejše, se pojavi lokalni vrh vrednosti vzdolžnih hitrosti tik ob vrteči se steni difuzorja (sl. 8).
Povečanje vzdolžnih komponent tik ob steni ni bilo opaženo ne v primeru vrteče se ravne cevi [6], prav tako pa tudi ne v primeru vrtinčnega toka v mirujočem difuzorju [4]. Eden izmed vzrokov tega pojava leži v delovanju centrifugalne sile, ki potiska tok zraka proti steni difuzorja in s tem proti izstopu iz difuzorja (stena difuzorja se namreč razširja v smeri proti njegovemu izstopu). Drugi vzrok pa je čelna stena difuzorja oz. debelina stene difuzorja na njegovem izstopu; med vrtenjem difuzorja se zrak ob tej steni zaradi delovanja centrifugalne sile pomika v radialno-obodni smeri navzven v zunanji prostor [7], na njegovo mesto pa priteka nov zračni tok, ki ga čelna stena difuzorja “črpa” iz notranjosti difuzorja oziroma iz mejne plasti ob notranji steni difuzorja. Zaradi navedenih vzrokov ni med eksperimentom nikoli prišlo do odcepljanja plasti zraka tik ob notranji steni difuzorja (povratni tok), kljub temu, da je bil kot razširitve difuzorja večji od 14°.
2.3 Distribution of time-averaged mean axial velocity components (ua)
There is a local peak of the axial velocities in the near-wall region in the transverse sections that are closer to the diffuser outlet and where the swirl is more intensive due to a higher wall tangential velocity (Fig. 8).
The increase in axial velocity component values near the diffuser wall has not been observed, neither in the case of the axially rotating straight pipes [6] nor in the case of the stationary diffuser with swirl flow [4]. One of the reasons for this phenomenon is in the action of the centrifugal force, which propels the airflow towards the diffuser wall and thus towards the diffuser outlet (since the diffuser cone diverges in the direction of its outlet). The second reason is the diffuser endplate - the thickness of the wall at the diffuser outlet. When the diffuser rotates, due to the centrifugal forces the air attached to this endplate is pushed out of the diffuser in the radially tangential direction [7]. Thus, the additional airflow is sucked in from the inner side of the diffuser cone (from the boundary layer at the inner diffuser wall) by the endplate. As consequence of the described phenomena, flow separation from the rotating diffuser wall (the so-called reverse flow) was never observed during the experiments, although the diffuser cone angle exceeded 14°.
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Iz slike 8 je tudi razvidno, da je potek vzdolžnih komponent hitrosti zraka ob steni difuzorja podoben poteku radialnih komponent hitrosti v istem območju difuzorja (sl. 7).
2.4 Rezultati numeričnih modelov
Numerični izračun toka zraka v vrtečem se difuzorju je potekal z uporabo CFD programskega paketa CFX-TASCflow, verzija 2.10. Pri tem so bili uporabljeni trije turbulenčni modeli, ki so že vsebovani v omenjenem programskem paketu: model ^, model k-a, SST in model Reynoldsovih napetosti. Vstopni in robni pogoji so bili za vse tri modele podani z eksperimentalnimi meritvami, s katerimi so bile določene časovno povprečene hitrostne komponente toka zraka na vstopu v difuzor. Na sliki 9 je prikazano območje okrog difuzorja (okolica, robni pogoji), ki je bilo uporabljeno pri modeliranju.
Rezultati numeričnih modelov in primerjava z eksperimentalno dobljenimi vrednostmi so prikazani na sliki 10.
It is obvious from Fig. 8 that the axial velocity distribution in the near-wall region of the airflow in the rotating diffuser is analogous to the distribution of radial velocities in the same region of the diffuser (Fig. 7).
2.4 Results of the numerical models
Numerical calculation of the airflow inside the rotating diffuser was carried out using a CFD program package: CFX-TASCflow, version 2.10. Three different turbulent closure models were used, which were already included in the aforementioned program package: the k-e model, the k-w SST model and the Reynolds stresses model. The inlet and the boundary conditions for all three models were determined with experimental measurements, which were used for the calculation of the time-averaged velocity components of the airflow at the diffuser inlet. A region around the diffuser (surrounding, boundary conditions) used for modelling is shown in Fig. 9.
The results of the numerical models and the comparison with the experimentally acquired values are shown in Fig.10.
Sl. 9. Shematski prikaz okolice difuzorja, ki je bila upoštevana pri numeričnem izračunu tokovnega polja
v difuzorju (mere so v milimetrih) Fig. 9. Schematic view of the diffuser surrounding used for the numerical calculation of the diffuser flow
field (all dimensions in millimetres)
I
1.6 1.4
1
ae
6L 6.4 K-2 ft -63 -64
4 eksperiment/experiment eksperiment (smer)/
* '¦-¦^**
™ experiment (trend)
......-k -a> SST _.._ ..model Raynolds. nap./ Reynolds stress. model
7*---------T-m?*^ ' /A\ r * #
6 1 &2 D 3 D 4 'Ji Dt """SV------ff^^Tjf
r,'R
Sl. 10. Primerjava vrednosti povprečenih aksialnih hitrosti na izstopu iz difuzorja - koordinatna os x
( f = 30 Hz, Re = 1,17*104) Fig. 10. Comparison of values of averaged axial velocities at the diffuser outlet - coordinate axis x
( f = 30 Hz, Re = 1.17*104)
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Vsi turbulentni modeli napovejo povečanje vzdolžnih komponent hitrosti (sl. 10) tik ob steni vrtečega se difuzorja, kjer te vrednosti dosežejo velikost 1,1 m/s (model k-co SST) do 1,4 m/s (model k-z), torej so velikostnega razreda izmerjenih vrednosti. Vzdolžne hitrosti se pri preskusu zmanjšajo na vrednost ~ 0 m/s že na oddaljenosti r/R = 0,95, medtem ko je pri turbulentnih modelih to zmanjšanje bolj položno. Najhitreje se zmanjšajo vrednosti vzdolžne hitrosti pri rezultatih modela Reynoldsovih napetosti, najpočasneje pa pri modelu k-z. Tudi majhno območje povratnega toka, ki se pojavi pri preskusu na oddaljenosti r/R * 0,92 , napovedo turbulentni modeli pri nižjih vrednostih r/R (razen modela k-co, ki povratnega toka na tem delu osi x ne napove).
Zaradi širšega pasu velikih vrednosti vzdolžnih komponent hitrosti zraka ob steni difuzorja model k-z tudi ne napove dovolj velikih vzdolžnih hitrosti v bližini vzdolžne osi difuzorja. Tu sta eksperimentalnim vrednostim precej bliže ostala dva modela, še posebej model Reynoldsovih napetosti.
3 SKLEPI
V prispevku je bila izvedena analiza kinematike toka zraka v vrtečem se vzdolžnem difuzorju s kotom razširitve 18° z namenom ugotoviti vpliv vrteče se stene na tok v difuzorju. Analiza je bila opravljena z meritvami hitrostnega polja z metodo LDA ter z numeričnim modeliranjem toka v difuzorju.
Rezultati so potrdili pojav dveh tipov toka v vrtečem se difuzorju: vrteči se tok zraka ob steni in nevrteči tok v vzdolžni osi difuzorja; to pomeni, da vpliv vrtenja stene ne seže globoko v notranjost difuzorja. Profili obodne komponente hitrosti so zato vbočeni. V območju r/R > 0,9 se pojavi ekstrem vrednosti vseh treh komponent hitrosti, predvsem na tistih prečnih prerezih difuzorja, ki so bliže izstopu. Rezultati preskusa so poleg tega pokazali tudi odvisnost poteka komponent hitrosti zraka v vrtečem se difuzorju od vrtilne frekvence difuzorja in Reynoldsovega števila.
V splošnem so najbližje rezultatom preskusnih meritev rezultati modela Reynoldsovih napetosti, kar potrjujejo tudi rezultati modeliranja toka znotraj vrtečih se cevi [8]. To izhaja iz same narave tega turbulenčnega modela, saj upošteva precej manj poenostavitev kot dvoenačbni modeli. Posledica tega je daljši čas računanja oziroma večji računski napor, ki ga model Reynoldsovih napetosti zahteva.
All three turbulent closure models predict an increase in the axial velocity components (Fig. 10), directly at the rotating diffuser wall, where the axial velocities reach values from 1.1 m/s (model^SST) up to 1.4 m/s (model k-s) - the predicted values are therefore in the region of the measured values. The measured axial velocities during the experiment drop to ~ 0 m/s at a distance of r/R = 0.95, whereas the drop in the axial velocity values predicted by the turbulent closure models is not so steep. The fastest decrease in the axial velocity values is predicted by the Reynolds stresses model, and the slowest by the k-s model. Apart from that, a small region of the reverse flow, which was observed at a distance r/R « 0.92 during the experiment, was predicted at lower values of r/R by the models (with the exception of the^model, which does not predict the reverse flow on this part of the x-axis).
Because of the broader band of high values of the axial velocity components of the airflow near the diffuser wall the k-z model fails to predict values of the axial velocities that are high enough at the diffuser longitudinal axis; the results of other two models in this region of the diffuser are thus closer to the experimental ones, especially in the case of the Reynolds stresses model.
3 CONCLUSIONS
An analysis of the flow kinematics in an axially rotating diffuser with an 18° cone angle was carried out in order to investigate the influence of the rotating diffuser wall on the airflow. This was achieved by the use of an LDA system and by numerical modeling of the airflow inside the diffuser.
The results confirm two general types of flow inside the rotating diffuser: one in the outer rotating region and the second within the inner non-rotating region. This means that the influence of the rotating wall on the airflow does not reach deep inside the diffuser. The tangential velocity profiles are therefore concave. There is a maximum of all three velocity components in the region r/R > 0.9 for the diffuser transverse sections that are close to the diffuser outlet. In addition, the results of the experiment showed the dependence of the distribution of airflow velocity components on the rotating frequency of the diffuser and on the Reynolds number.
The results of the Reynolds’ stresses model are generally closest to the results of the experimental measurements, which is also confirmed by the results of the flow modelling inside the rotating straight pipes [8]. This comes from the nature of this turbulent closure model, since the model itself incorporates far fewer simplifications in comparison to the two-equation closure models. The consequence of this is the longer calculation time (greater calculation effort) required by the Reynolds stresses model.
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Bajcar T., [irok B., Trenc F., Jo{t D.: Analiza kinematike - An Analysis of the Flow Kinematics
4 LITERATURA 4 REFERENCES
[1] Kovats, A. (1964) Design and performance of centrifugal and axial flow pumps and compressors, Pergamon
Press Ltd., Oxford. [2] Gupta, A. K., D.G. Lilley, N. Syred (1984) Swirl flows, Abacus Press, Tunbridge Wells. [3] Imao, S. M. Itoh, T Harada T (1996) Turbulent characteristics of the flow in an axially rotating pipe, Int. J.
Heat and Fluid Flow, Vol. 17, No. 5, 444-451. [4] Clausen, P. D., S.G. Koh, D.H. Wood (1993) Measurements of a swirling turbulent boundary layer developing
in a conical diffuser, Experimental Thermal and Fluid Science, 6, 39-48. [5] Cannon, J. N, W.M. Kays (1969) Heat transfer to a fluid flowing inside a pipe rotating about its longitudinal
axis, ASME Journal of Heat Transfer, Vol. 91, No. 1, 135-139. [6] Murakami, M., K. Kikuyama (1980) Turbulent flow in axially rotating pipes, Journal of Fluids Engineering,
Vol. 102, 97-103. [7] Schlichting, H., K. Gersten (2000) Boundary-layer theory, Springer, Berlin. [8] Hirai, S., T Takagi, M. Matsumoto (1988) Predictions of the laminarization phenomena in an axially rotating
pipe flow, Journal of Fluids Engineering, Vol.110, 424-430.
Naslova avtorjev: mag. Tom Bajcar
profdr. Brane Širok profdr. Ferdinand Trenc Fakulteta za strojništvo Univerza v Ljubljani Aškerčeva 6 1000 Ljubljana tom.bajcar@fs.uni-lj.si brane.sirok@fs.uni-lj.si ferdinand.trenc@fs.uni-lj.si
dr. Dragica Jošt Turboinštitut Rovšnikova 7 1000 Ljubljana dragica.jost@turboinstitut.si
Authors’ Addresses:Mag. Tom Bajcar
Prof. Dr. Brane Širok
Prof. Dr. Ferdinand Trenc
Faculty of Mechanical Eng.
University of Ljubljana
Aškerčeva 6
1000 Ljubljana, Slovenia
tom.bajcar@fs.uni-lj.si
brane.sirok@fs.uni-lj.si
ferdinand.trenc@fs.uni-lj.si
Dr. Dragica Jošt
Turboinštitut
Rovšnikova 7
1000 Ljubljana, Slovenia
dragica.jost@turboinstitut.si
Prejeto: Received:
20.12.2002
Sprejeto: Accepted:
31.1.2003
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© Strojni{ki vestnik 48(2002)12,687-695 © Journal of Mechanical Engineering 48(2002)12,687-695
ISSN 0039-2480 ISSN 0039-2480
UDK 532.57:519.61/.64 UDC 532.57:519.61/.64
Izvirni znanstveni ~lanek (1.01) Original scientific paper (1.01)
Siskov model toka nenewtonskih teko~in z metodo kon~nih prostornin
The Sisko Model For Non-Newtonian Fluid Flow Using The Finite-Volume Method
Marjan Deli} - Jure Marn - Zoran @uni~
Analizirali smo primernost metode končnih prostornin za izračun tokovnih razmer v nestisljivi viskozni nenewtonski tekočini ob uporabi Siskovega modela. Prav tako smo analizirali natančnost numeričnih rezultatov v odvisnosti od gostote mreže. Metodo smo testirali na primeru gnanega toka v kotanji in primeru toka v kanalu z nenadno zožitvijo. Numerično dobljene rezultate smo primerjali z vrednostmi iz literature. Za reološki model smo uporabili modelne parametre, dobljene na podlagi preskusa s cevnim viskozimetrom. © 2002 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: tekočine nenewtonske, modeli toka, metode končnih prostornin, modeli reološki)
We have analyzed the suitability of the finite-volume method for calculating incompressible, viscous, non-Newtonian fluid flow where the Sisko model was used. In addition, convergence criteria are presented and the convergence depending on the mesh size was analyzed. The method was tested for the driven-cavity case and flow in a channel with a sudden contraction. The numerical solution was compared with the results available in the open literature. For the rheological model, parameters obtained from an experiment with a capillary viscometer were used.
© 2002 Journal of Mechanical Engineering. All rights reserved. (Keywords: non-newtonian fluids, flow models, control-volume methods, rheological models)
0 UVOD
Nenewtonske tekočine so navzoče v vsakdanji inženirski praksi. Zaradi spremenljive viskoznosti je treba gibalne enačbe obravnavati brez običajnih poenostavitev. Viskoznost je lahko v splošnem odvisna od številnih parametrov, v tem prispevku pa obravnavamo posplošene newtonske tekočine, pri katerih je viskoznost odvisna od deformacijske hitrosti. Na področju posplošenih newtonskih tekočin obstaja več reoloških modelov z dvema, tremi ali štirimi parametri, npr. [5] do [7] in [9]. V pričujočem delu smo se osredotočili na Siskov triparametrični model.
Tokovne razmere v nestisljivi viskozni posplošeni newtonski tekočini smo obravnavali s sistemom parcialnih diferencialnih enačb osnovnih zakonov ohranitve, ki smo ga reševali z diskretno metodo končnih prostornin, opisano v [4] in povzeto po [1]. V skladu z opisanim načinom reševanja smo kontinuitetno in gibalno enačbo povezali z uporabo načela umetne stisljivosti. Konvektivne tokove smo računali po metodi karakteristik, difuzijske tokove in izvirne člene pa s končnimi razlikami (protivetrnimi in osrednjimi končnimi razlikami). Za napredovanje po
0 INTRODUCTION
Non-Newtonian fluids are very common in everyday engineering practice. Momentum equations can no longer be simplified with the usual simplifications due to variable viscosity. Viscosity is, in general, dependent on various parameters, whereas this paper deals with generalised Newtonian fluids, for which viscosity depends on the trace of the shear-rate tensor. There are several different rheological models, with two, three or four parameters among the generalised Newtonian fluids, see for example, [5] to [7] and [9]. In this study the main focus was on the Sisko three-parametric model.
The fluid conditions for an incompressible, generalised Newtonian fluid were treated as a system of partial differential equations of fundamental conservation laws, solved with a discrete, finite-volume method, as described in [4] and [1]. According to the described method the continuity and momentum equations were coupled with an artificial compressibility term. The convective fluxes were calculated using the method of characteristics, whereas the diffusive fluxes and source terms were computed using finite differences (upwind and central finite deferences). For time marching
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času smo uporabili izrecno metodo Runge-Kutta četrtega reda. Natančnost in konvergenco metode smo preskusili na primeru gnanega toka v kotanji in primeru toka v kanalu z nenadno zožitvijo. Za reološki model smo uporabili modelne parametre, dobljene na podlagi preskusa s cevnim viskozimetrom ([8] in [3]).
1 DEFINICIJA PROBLEMA
Ravninski tok viskozne nestisljive tekočine lahko opišemo z gibalnima enačbama in kontinuitetno enačbo, ki jih ob zanemaritvi prostorninskih sil zapišemo v obliki:
3V:
the Runge-Kutta fourth-order explicit method was used. The accuracy and convergence of the method was tested with the driven-cavity test case and flow in a channel with a sudden contraction. The model parameters of the fluid were obtained from an experiment with a capillary viscosimeter ([8] and [3]).
1 PROBLEM DEFINITION
The planar flow of a viscous, incompressible fluid can be described with two momentum equations and one continuity equation, which with neglected volume forces take the following forms:
;"'— '—'
dp dr-ji
i), i1: i).il:
(1) (2).
Reševanje enačb ohranitve gibalne količine, zapisanih za osnovne fizikalne spremenljivke za tok nestisljive tekočine, je težavno, ker tlak ni termodinamična veličina. To pomanjkljivost odpravimo z vključitvijo dodatnega člena, časovnega odvoda, v kontinuitetno enačbo. Problem nihanja numerične rešitve odpravimo z navidezno stisljivostjo:
kjer je/?umetna stisljivost.
Z vpeljavo brezrazsežnih spremenljivk, kjer sta L in v„, značilna dolžina in hitrost,
Solving the Navier-Stokes equations for incompressible fluid written with a primitive set of variables is difficult since the pressure is not a thermodynamic quantity. This drawback can be overcome with the introduction of an additional time derivative in the continuity equation. The problem of the oscillations of the numerical solution can be resolved with artificial compressibility:
(3),
where the coefficient/? is the artificial compressibility. The introduction of nondimensional variables, where L and v„ are the characteristic length and the characteristic velocity, respectively:
prevedemo enačbi (1) in (2) v naslednjo obliko:
This results in the transformation of Equations (1) and (2) into:
%+th^
3X7 8XT:
MS+g>)
V zgornjih enačbah se pojavljajo same brezrazsežne spremenljivke, zato bomo zaradi primernosti pisanje zvezdice (*) opustili.
Sistem enačb za ravninski tok nestisljive nenewtonske tekočine ob upoštevanju umetne stisljivosti lahko zapišemo v konzervativni obliki kot eno samo vektorsko enačbo ([1] in [10]):
These equations employ only nondimensional variables and we can, for convenience, skip the writing of (*).
The system of non-dimensional equations for the planar flow of an incompressible, non-Newtonian fluid with artificial compressibility can be written in conservative form as a single vector equation ([1] and [10]):
LQ 0E_ OG 6Erix DG,nfi dt 8x dy 0x dy
(4),
kjer so: Q =
where:
P/P
E=< vl+p
G =
! °
Gtu —
¦¦/¦
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in
Z vpeljavo lokalnih koordinat (x, h ), katerih povezavo s kartezijevim koordinatnim sistemom podamo z:
0__&i 0_ dji 0^ Ox Ox SL 8x 8rj ''
prevedemo enačbo (4) v obliko
and:
as be dg
8t dL dt]
With the introduction of local coordinates (x, h) connected to the Cartesian coordinate system through the relation:
3y dy dL dy Br)
Equation (4) transforms into:
kjer so:
as
where:
a?
(5),
Pri reoloskih modelih posplosenih newtonskih tekočin je t podan z izrazom t =-m (g) g, kjer lahko tenzor deformacijske hitrosti in deformacijsko hitrost zapišemo kot:
7 = W+ (VC kjer smo z operatorjem : označili sled tenzorja.
2 REOLOŠKI MODEL - SISKOV MODEL
V okviru sodelave s Premogovnikom Velenje smo opravili meritve viskoznosti na cevnem viskozimetru, prikazanem na sliki 1 [7]. Bistvena sestavna dela viskozimetra sta zalogovnik in kapilarna cev, skozi katero se pri različnih tlakih pretaka zmes, katere viskoznost določamo.
Meritve smo opravljali na treh različnih mešanicah elektrofiltrskega pepela (ostanek iz termoelektrarne Šoštanj) in vode.
stisnjen zrak compressed air
dL Or) Or] dL Rheological models of generalised Newtonian fluids have t defined with the expression t = -m (g) g, where the shear-stress tensor and the shear rate are"written as:
i = \j\ (i ¦ %
and the operator : means the trace of a tensor. 2 RHEOLOGICAL MODELS - SISKO MODEL
The setup of the viscosimeter is shown in Figure 1 [7], within the framework of a cooperation with Premogovnik Velenje. Measurements of viscosity using the capillary viscosimeter were performed. The essential parts of the viscosimeter are the container and the capillary pipe. The measured mixture flows through the pipe under various pressures.
The measurements were performed using three different mixtures of electro-filter ash (the product of the Šoštanj thermal powerplant) and water.
zalogovnik reservoir
----- kapilarna cev
capillary pipe
tlakomer
pressure indicator
+ &
tehtnica
scale "
Sl. 1. Vodoravni kapilarni viskozimeter Fig. 1. Horizontal capilary viscosimeter
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Preglednica 1. Masni deleži pepela in vode v zmeseh in gostota zmesi
Table 1. Mass fraction of ash and water in the mixtures and the density of the mixtures
w.nx!ilw.i,.JS ¦ JUU [%] p [ ky/m11
B 50 40 l.'iU 1 14CD 1524
V preglednici 1 so podani masni deleži vode in pepela ter gostote izbranih zmesi.
Kakor smo že zapisali, smo izbrane zmesi modelirali s Siskovim zakonom tečenja kot reološkim modelom posplošenih newtonskih tekočin. Zapišemo ga v obliki:
Table 1 shows the mass fractions of the water and ash and the densities of the chosen mixtures.
The chosen mixtures were modelled using the Sisko model as a rheological model of generalized Newtonian fluids. It can be written in the following forms:
-(iti) i
oziroma
- (|U_+m-y_,j7
or
{i (7) — ju_+ m-y"-1
(6)
(7).
Triparametrski model je sestavljen iz klasičnega potenčnega zakona tečenja z asimptotično vrednostjo dinamične viskoznosti M.M, m in n so vrednosti, podane za vsako tekočino posebej, in se določijo na temelju meritev viskoznosti. Če je„ = 0, n = 1 in m =M, preide model v Newtonov zakon tečenja.
Na temelju meritev so ob uporabi metode najmanjšh kvadratov [8] izpeljani parametri Siskovega modela, in sicer: .za zmes A
The three-parameter model consists of the well-known power-law model with an added asymptotical coefficient of dynamic viscosity,// . The valuesM , m and n are specified for each fluid "separately, and"are obtained from experimental viscosity measurements. If one specifies M = 0, n = 1 and m = » one gets an expression for a Newtonian fluid.
Based on the measurements performed, and with the use of the least-squares method [8], the following Sisko model parameters are obtained: .for mixture A
.za zmes B
.za zmes C
ft (7) = 0.001339 + 0.00005971-eTT1
.for mixture B (i, (7) = 0.000G28 + 0.00012471-6171
.for mixture C [i, (7) = 0.000008 + 0.00G18171-1713
(8)
(9)
(10).
3 REZULTATI
3 RESULTS
Model smo preskusili na primeru gnanega toka v kotanji. Gnana kotanja kot značilen dvorazsežni primer je pravokotnik s tremi mirujočimi stranicami in eno pomično stranico, ki se giblje z nespremenljivo hitrostjo. V našem primeru je to kvadrat z brezrazsežno dolžino stranice 1 in brezrazsežno hitrostjo pomika zgornje plošče 1. Geometrijska oblika kotanje je prikazana na sliki 2. Prekinjeni črti po sredini kotanje označujeta osrednji črti (navpično in vodoravno) po katerih smo primerjali hitrostne profile. Dodatna pričakovana težava tega primera je nezveznost v stičiščih mirujočih in gibajoče se stranice.
The model was tested with the driven-cavity flow. The driven cavity, as a typical two-dimensional example, is rectangular with three fixed walls and one moving wall that moves with constant velocity. In the selected case the cavity has a square form with a nondimensional side length of 1 and an nondimensional velocity of the top wall 1. The geometry is depicted in Figure 2. The dashed lines through the centre of the cavity show the locations (vertical and horizontal) where the velocity profiles were compared. An additional, but expected, difficulty with this case was the discontinuity in the top corners of the cavity, where the moving and fixed walls connect.
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Sl. 2. Geometrijska oblika in robni pogoji: gnana kotanja Fig. 2. Geometry and boundary conditions: driven cavity
V prvem delu rezultatov je prikazana konvergenca rezultatov v odvisnosti od gostote mreže. Rezultate smo primerjali z vrednostmi iz literature. Kot primerjalne vrednosti smo uporabili najnižje negativne vrednosti vodoravne in navpične komponente hitrosti na osrednjih črtah. Odstopanje od primerjalnih vrednosti smo definirali z izrazoma [4]:
In the first part of the results the convergence of the results due to mesh refinement is shown. The results were compared to the benchmark solution. The maximum negative values of the horizontal and vertical components of the velocity on the centre lines were used for comparison. Deviation from the benchmark solution was defined with the following expression [4]:
ErrJ| \"UtcJ\
Konvergenca rezultatov za različne gostote The convergence of the results for different
mreže je prikazana v preglednici 1. Rezultati so mesh sizes is shown in Table 1. The results were
primerjani z referenčnimi vrednostmi za newtonsko compared to reference values for a Newtonian fluid
tekočino za Re = 400. with a Reynolds number value Re=400.
Preglednica 2. Odstopanje rezultatov v odvisnosti od gostote mreže (Siskov model: M = 0, m = 0,0025, n = 1) Table 2. Deviation of the results for different mesh sizes (Sisko model: na = 0, m = 0.0025, n = 1)
A-, 7, \-/A
11x11 44.17 38.60
21x21 20.63 19.71
41x41 5.50 5.19
81 x81 1.02 0.80
129 X 129 0.28 0.19
Pri redkejših enakomernih mrežah zaradi velikih gradientov ob trdnih stenah (predvsem zgornji) pride do večjih odstopanj od primerjalnih vrednosti. Z zgoščevanjem mreže se rezultati približujejo referenčnim vrednostim. Z najgostejšo mrežo (129x129) dobimo manj ko 0,3 % odstopanja od referenčnih vrednosti za izbrane parametre. Potek hitrostnih profilov v odvisnosti od gostote mreže je prikazan na sliki 3.
Na slikah 4, 5 in 6 so prikazani rezultati izračuna tokovnih razmer gnanega toka v kotanji z uporabo Siskovega modela.
Iz primerjave oblik profilov z rezultati iz literature za newtonske [2] tekočine je glede na gradiente hitrosti moč sklepati, da je zahtevnost izračuna za najredkejšo zmes (in s tem najmanj viskozno tekočino) primerljiva z izračunom newtonske tekočine za Reynoldsovo število Re=1000.
When using coarse equidistant meshes the differences from reference values are larger due to high velocity gradients near the walls (especially the top wall). With a refinement of the mesh the results approach reference values. With the finest mesh (129x 129) a less-than-0.3% difference from the reference values for the chosen non-Newtonian model parameters was obtained. The velocity profiles for the different mesh densities are shown in Figure 3.
In Figures 4, 5 and 6 the results are shown for the calculation of the flow field in a driven cavity for model parameters of the Sisko model obtained from the experiment with a capilary viscosimeter.
A comparison of the calculated velocity profiles with the benchmark solution for Newtonian [2] fluids, taking into account velocity gradients, shows that the computation with the least dense mixture (meaning the least viscous fluid) corresponds to the computation of a Newtonian fluid with a Reynolds number value Re=1000.
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Sl. 3. Profil vodoravne hitrosti v skozi navpično sredisčnico in navpične hitrosti v skozi vodoravno
sredisčnico za različne gostote mreže (Sisko model: M = 0, m = 0,0025, n = 1) Fig. 3. Horizontal velocity profile v through the vertical centreline and the vertical velocity profile v through the horizontal centreline for different mesh sizes (Sisko model: Ha = 0, m = 0.0025, n=1) y
Sl. 4. Profil vodoravne hitrosti v skozi navpično sredisčnico in navpične hitrosti v skozi vodoravno
sredisčnico za zmes A (129x129 vozlišč)
Fig. 4. Horizontal velocity profile v through the vertical centreline and the vertical velocity profile v
through the horizontal centreline for mixture A (129x129 nodes.)
Sl. 5. Profil vodoravne hitrosti v skozi navpično sredisčnico in navpične hitrosti v skozi vodoravno
sredisčnico za zmes B (129x129 vozlišč)
Fig. 5. Horizontal velocity profile v through the vertical centreline and the vertical velocity profile v
through the horizontal centreline for mixture B (129x129 nodes.)
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Sl. 6 Profil vodoravne hitrosti v skozi navpično središčnico in navpične hitrosti v skozi vodoravno
središčnico za zmes C (129x129 vozlišč)
Fig. 6. Horizontal velocity profile v through the vertical centreline and the vertical velocity profile v
through the horizontal centreline for mixture C (129x129 nodes.)
Model smo preskusili tudi na standardnem testnem primeru toka v kanalu z nenadno zožitvijo. Robni pogoji in geometrijska oblika so prikazani na sliki 7. Oba dela kanala merita 10 enot. Prvi del kanala je visok 4 enote, nato pa se kanal nenadno zoži na 1 enoto. Za izračun smo izdelali tri neenakomerne mreže, in sicer z 1500 (40x30 + 10x30), 6000 (80x60 + 20x60) in 13500 (120x90 + 30x90) končnimi prostorninami.
Za reološki model smo pri izračunih uporabili Siskov model za zmes A (enačba 8). Na stenah smo predpisali v , v = 0, na spodnji steni simetrijske robne pogoje (3v /dyy = v = 0), na iztoku prosti iztok (dv x/dx = v = 0) in na vtoku parabolični profil s povprečno hitrostjo v = 0,0393. (Ob uporabi newtonske dinamične vkoznosti/= 0,01339 bi primer ustrezal Re=400, računano na izstopni prerez iz kanala).
Na sliki 8 je prikazana primerjava hitrostnih profilov na polovici dolžine kanala za vse tri gostote mreže. Po pričakovanju se profili hitrosti slabše ujemajo na mestu zožitve, kjer rezultati, dobljeni z najredkejšo mrežo, še dokaj odstopajo; rezultati, dobljeni z gostejšima mrežama, pa se že dokaj dobro ujemajo. Z zgoščevanjem mreže na področju razvijanja hitrostne mejne plasti rastejo gradienti hitrosti ob trdnih stenah in zaradi tega se hitrosti zmanjšujejo proti srednjici kanala.
The model was also tested on a standard test case of flow in a channel with a sudden contraction. The boundary conditions and geometry are shown in Figure 7. The inlet and outlet channels have a length of 10 units. The inlet channel has a height of 4 units and contracts to an outlet height of 1 unit. Three, non-equidistant meshes were generated: 1500 (40x30 + 10x30), 6000 (80x60 + 20x60) and 13500 (120x90 + 30x90) finite volumes.
We used the Sisko rheological model with parameters for mixture A (Equation 8). We prescribed v , v = 0, on the walls, the symmetry boundary condition (dv x/dy = v = 0) on the bottom edge, the developed profile (dv x/dx = v = 0) at the outlet and the parabolic velocity profile with mean value v = 0.0393 at the inlet. (Using the Newtonian dynamic viscosity m = 0.01339 and the outlet channel height, the equivalent Reynolds number would be Re=400).
In Figure 8 is the comparison of the velocity profiles on the contraction cross-section for three mesh densities. As expected, the profiles disagree in this area, especially for the coarsest mesh, whereas the two finest meshes show reasonably good agreement. With finer meshes in the area of the development of the boundary layer the velocity gradients near the wall increase and the velocity near the centreline of the channel decreases.
Sl. 7. Geometrijska oblika in robni pogoji: nenadna zožitev (4:1) Fig. 7. Geometry and boundary conditions: sudden contraction (4:1)
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Sl. 8. Primerjava profilov hitrosti na 1/2 dolžine kanala Fig. 8. Comparison of velocity profiles on 1/2 the channel length
Sl. 9. Tlačno polje v kanalu Fig. 9. Pressure field in the channel
—^>
Sl. 10. Izolinije hitrosti v smeri koordinatne osi x (vx) Fig. 10. Constant x component velocity contours (vx)
Sl. 11. Tokovnice (zožitev 4:1) Fig. 11. Streamlines (contraction 4:1)
Na sliki 9 je prikazano tlačno polje v kanalu, izračunano z najgostejšo mrežo.
Na sliki 10 so prikazane izolinije hitrosti v smeri koordinatne osi x (najgostejša mreža).
Na sliki 11 so prikazane tokovnice v kanalu (najgostejša mreža).
4 SKLEP
Metodo končnih prostornin smo uporabili za računanje tokovnih razmer v laminarni viskozni
Figure 9 shows the pressure field in the channel computed using the finest mesh.
Figure 10 shows the constant x component velocity contours (the finest mesh).
Figure 11 shows the streamlines in the channel (the finest mesh).
4 CONCLUSIONS
The finite-volume method was used to compute the flow conditions for a laminar, viscous,
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nestisljivi nenewtonski tekočini, pri čemer smo kot reološki model uporabili Siskov model. Formulacija je testirana na primeru gnanega toka v kotanji. Prikazan je vpliv natančnosti rezultatov v odvisnosti od gostote mreže. Prikazani rezultati za primer gnanega toka v kotanji, dobljeni z najgostejšo mrežo (129x 129 vozlišč) se zelo dobro ujemajo s primerjalnimi vrednostmi v literaturi. Modelni parametri Siskovega modela so določeni na temelju preskusa s cevnim viskozimetrom za tri različne gostote zmesi pepela in vode.
incompressible non-Newtonian fluid. The Sisko rheological model was used for capturing non-linear fluid properties. The proposed formulation was tested on driven-cavity flow. The influence of the mesh density on the accuracy of the numerical results was shown. The results obtained on the finest mesh (129x 129 nodes) agree well with the values published in the literature. The computational examples conclude with three computations based on the model parameters of the Sisko model, obtained from an experiment with a capillary viscosimeter for three different mixtures of ash and water.
5 LITERATURA 5 REFERENCES
[1] Drikakis, D., PA. Govatsos, D.E. Papantonis (1994) A characteristic-based method for incompressible
flows, Int. J. for Numerical Methods in Fluids 19, 667-685. [2] Ghia, U., K.N. Ghia, C.T Shin (1982) High Re solutions for incompressible flow using the Navier-Stokes
equations and a multigrid method, Journal of computational physics 48, 387-411. [3] Kanduti, D., J. Marn (2002) Eksperimentalna določitev toka zmesi pepela in vode, Zbornik Kuhljevih
dnevov 2002, 129-136. [4] Marn, J., M. Delič, Z. Žunič (2001) Non-Newtonian fluid flow analysis with finite difference and finite
volume numerical models, Applied Rheology 11, 325-335. [5] Škerget, L. (1994) Mehanika tekočin, Tehniška fakulteta v Mariboru in Fakulteta za strojništvo v Ljubljani. [6] Tanner, R.I. (1985) Engineering rheology, Oxford University Press, New York. [7] Ternik, P. (2002) Modeliranje toka nenewtonskih disperznih sistemov, magistrsko delo, Univerza v Mariboru,
2002a [8] Ternik, P. (2002) Neobjavljene analize eksperimentov v viskozimetru na vzorcih Premogovnika Velenje,
Univerza v Mariboru, 2002b. [9] Turian, R.M., TW. Ma, F.L.G Hsu, M.DJ. Sung (1998) Flow of concentrated non-Newtonian slurries: 1.
friction losses in laminar, turbulent and transition flow through straight pipe, Int. J. Multiphase Flow 24,
No. 2, 225-242. [10] Žunič, Z. (1997) Numerična obravnava problema aerodinamike vozil, Magistrsko delo, Univerza v Mariboru.
Naslov avtorjev: mag. Marjan Delič prof.dr. Jure Marn mag. Zoran Žunič Fakulteta za strojništvo Univerza v Mariboru Smetanova 17 2000 Maribor marjan.delic@uni-mb.si jure.marn@uni-mb.si zoran.zunic@uni-mb.si
Author’s Address: Mag. Marjan Delič Prof.Dr. Jure Marn Mag. Zoran Žunič Faculty of Mechanical Eng. University of Maribor Smetanova 17 2000 Maribor marjan.delic@uni-mb.si jure.marn@uni-mb.si zoran.zunic@uni-mb.si
Prejeto: Received:
20.12.2002
Sprejeto: Accepted:
31.1.2003
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© Strojni{ki vestnik 48(2002)12,696-706 © Journal of Mechanical Engineering 48(2002)12,696-706
ISSN 0039-2480 ISSN 0039-2480
UDK 532.5:536.22 UDC 532.5:536.22
Izvirni znanstveni ~lanek (1.01) Original scientific paper (1.01)
Vpliv Prandtlovega {tevila na turbulentni prenos toplote ob ravni steni
The Influence of Prandtl Number on Near-Wall Turbulent Heat Transfer
Robert Bergant - Iztok Tiselj
Za opis turbulentnega prenosa toplote iz stene na tekočino je pri nizkih vrednosti Reynoldsovih in Prandtlovih številih mogoče uporabiti neposredno numerično simulacijo (NNS-DNS), ki opiše vse krajevne in časovne skale pojava. Vpliv Reynoldsovega števila na turbulentni prenos toplote (hitrosti, temperature, fluktuacije itn.) je razmeroma majhen, medtem ko je vpliv Prandtlovega števila veliko večji. Pri naših simulacijah toka v kanalu smo pri Reynoldsovem številu Re « 5000 analizirali tri različna Prandtlova števila, in sicer Pr = 0,025, Pr = 1 in Pr = 5,4.
Ločljivost NNS turbulentnega prenosa gibalne količine je premo sorazmerna z Re3/4 v vseh smereh koordinatnega sistema. Pri upoštevanju prenosa toplote, za tekočine s Prandtlovim številom, večjim od ena, velja, da je ločljivost premo sorazmerna z Re3/4Pr1/2. Pri Re = 5260 in Pr = 5,4 smo opravili tri numerične simulacije pri različnih ločljivostih. Vse tri simulacije so NNS za hitrostno polje, samo simulacija z največjo ločljivostjo je tudi NNS za temperaturno polje. Rezultati so pokazali, da je mogoče temperaturno polje zelo natančno napovedati tudi z nekoliko slabšo ločljivostjo od teoretično zahtevane. © 2002 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: tok ob steni, tok turbulentni, prenos toplote, števila Prandtl, simuliranje numerično)
For describing the heat transfer from a wall to a fluid at low Reynolds and Prandtl numbers we can use a direct mumerical simulation (DNS), which describes all the length and time scales of the phenomenon. The Reynolds number has a weak influence on the turbulent heat transfer (velocities, temperatures, RMS-fluctuations...), whereas the increasing Prandtl number has a stronger influence. In our flow simulations in the channel, three different Prandtl numbers, i.e. Pr = 0.025, Pr = 1 and Pr = 5.4, at a Reynolds number Re * 5000 were analyzed.
The resolution of the DNS for turbulent momentum transfer is proportional to Re3/4 in all directions. When considering heat transfer in fluids for a Prandtl number higher than one, the resolution is proportional to Re3/4Pr1/2. Three different numerical simulations at different resolutions were performed at Re = 5260 and Pr = 5.4. All three simulations are a DNS for the velocity field, whereas only the simulation at the highest resolution is also a DNS for the thermal field. The results showed that the thermal field could be accurately described with a lower resolution than theoretically required. © 2002 Journal of Mechanical Engineering. All rights reserved. (Keywords: near-wall flow, turbulent flow, heat transfer, Prandtl numbers, direct numerical simulation)
0 UVOD
V zadnjih 15 letih se je neposredna numerična simulacija (NNS) uveljavila kot pomembno orodje pri razumevanju mehanizma prenosa toplote v turbulentni mejni plasti. NNS pomeni natančno reševanje Navier-Stokesovih enačb brez dodatnih turbulentnih modelov. Prve NNS hitrostnega polja pri nižjih vrednostih Reynoldsovih številih sta opravila Kim in Moin [5], ki sta raziskovala hitrostno polje ter opazovala turbulentne strukture v kanalu (Re = 5700). Pozneje sta Kim in Moin [6] k enačbam hitrostnega polja dodala še energijsko enačbo za prenos toplote. Tako sta med dvema stenama preučevala tudi
0 INTRODUCTION
Over the past 15 years direct numerical simulation (DNS) has become an important research tool in understanding near-wall turbulent heat transfer. A DNS means a precise solving of the Navier-Stokes equations without any extra turbulent models. The first DNS of a velocity field at a low Reynolds number was performed by Kim and Moin [5], who investigated the velocity field and observed turbulent structures in the channel (Re = 5700). Later, they [6] also added an energy equation to the equations of velocity field for the heat-transfer calculations. When considering heat transfer, the
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Bergant R. - Tiselj I.: Vpliv Prandtlovega {tevila - The Influence of Prandtl Number
temperaturno polje ter opazovala koherentne strukture, ki so se pojavljale v bližini sten. Uporabila sta nekoliko nenavaden postopek, saj sta predpostavila enakomerno gretje tekočine po celotni prostornini, steni pa sta predstavljali toplotni ponor. Tudi Kasagi [2] je raziskoval temperaturno polje, vendar je za geometrijsko obliko vzel kanal, ki je bil gret z zgornjo in spodnjo steno. Vse te simulacije so potekale pri nizkih vrednostih Reynoldsovih in Prandtlovih števil. Pozneje so raziskovalci (Kawamura [4], Na in Hanraty [7]) mejo Prandtlovega števila dvignili do deset. Kawamura [4] je raziskoval vpliv Reynoldsovega (do Re = 14000) in Prandtlovega števila (do Pr = 5). V območju tik ob steni je ugotovil majhen vpliv Reynoldsovega števila in veliko večji vpliv Prandtlovega števila na statistiko turbulentnega prenosa toplote (hitrostni profili, fluktuacije hitrosti, turbulentni toplotni tokovi).
Prispevek je vsebinsko razdeljen na dva dela. V prvem delu je posebna pozornost namenjena vplivu Prandtlovega števila na prenos toplote. NNS so bile izvedene pri Reynoldsovem številu Re = 4580 in Re = 5260 ter treh različnih Prandtlovih številih: Pr = 0,025, Pr = 1 in Pr = 5,4 (glej preglednico 1).
Drugi del je namenjen preučevanju ločljivosti pri Re = 5260 in Pr = 5,4. Analizirane so bile računske mreže s tremi različnimi ločljivostmi. Teoretično naj bi bila ločljivost NNS pri Prandtlovih številih, večjih od ena, sorazmerna kvadratnemu korenu Prandtlovega števila v vseh smereh koordinatnega sistema v primerjavi z ločljivostjo za Pr = 1 [8]. Zahtevo sta v svojih NNS upoštevala Kawamura [3] in Tiselj [9]. Pri Pr = 5,4 to pomeni, da moramo število točk računske mreže, ki je zadostno za opis hitrostnega polja pri Re= 5260, povečati približno za faktor 5, 4 v vsaki smeri. Za prvo simulacijo je bila tako vzeta dovolj gosta mreža, ki zmore opisati najmanjše skale hitrostnega in temperaturnega polja (NNS).
Da je zahtevana gostitev mreže pri Pr > 1 nekoliko preostra, sta nakazala že Na in Hanratty [7], ki sta pri Pr = 10 uporabila večjo ločljivost le v smeri pravokotno na steno. Tako je v drugi simulaciji v tem prispevku izboljšana ločljivost v smeri pravokotno na steno, tretja simulacija pa je opravljena z najmanjšo ločljivostjo, ki je zadostna le za NNS hitrostnega polja. Zadnji dve simulaciji tako izpolnjujeta pogoj za NNS hitrostnega polja, na pa tudi za NNS temperaturnega polja.
1 ENAČBE IN NUMERIČNI POSTOPEK
Pri numeričnih simulacijah turbulentnega prenosa toplote sta bili uporabljeni dve različni geometrijski obliki neskončnega kanala. Zaprt kanal (sl. 1-levo), omejen s spodnjo in zgornjo steno, grejemo z nespremenljivim toplotnim tokom, vmes pa zaradi razlike tlakov teče tok tekočine. Pri odprtem kanalu (sl. 1-desno) grejemo spodnjo steno, nad katero teče tok tekočine s prosto površino. Uporabili smo brezrazsežne
temperature field between two walls and the coherent structures near the walls were studied. But this was an unusual approach, because uniform volumetric heat generation was assumed, where the walls represented a heat sink. Kasagi et al [2] also investigated the thermal field, but in this case the channel was heated by the top and bottom walls. All these simulations were performed for low Reynolds and Prandtl numbers. Later, Kawamura et al [3], and Na and Hanratty [7] raised the limit of the Prandtl number to ten, while Kawamura et al [4] studied the influence of Reynolds numbers (up to Re = 14000) and Prandtl numbers (up to Pr = 5). They found a weak influence of the Reynolds number and a stronger influence of the Prandtl number near the wall for turbulent heat transfer (velocity profiles, velocity fluctuations, turbulent heat fluxes).
The content of this article is divided into two parts. In the first part we investigate the influence of the Prandtl number on the heat transfer. DNSs were performed at Reynolds numbers Re = 4580 and Re = 5260, and at three different Prandtl numbers: Pr = 0.025, Pr = 1 and Pr = 5.4 (see Table 1).
The resolution requirements at Re = 5260 and Pr = 5.4 are studied in the second part of the paper, where three different resolutions were analyzed. Theoretically, the resolution for Prandtl numbers higher than one should be proportional to the square root of the Prandtl number in all three directions. This requirement was taken into account in the simulations of Kawamura et al [3] and Tiselj et al [9]. Therefore, at Pr = 5.4 the grid should be improved by approximately a factor of 5.4 in all directions. In the first simulation sufficient grid was taken so as to be capable of describing the smallest scales of the velocity and thermal fields (DNS).
The grid-refinement study of Na and Hanratty at the Prandtl number Pr = 10 shows that a finer grid is not required in the streamwise and spanwise directions, but it is necessary in the wall-normal direction. This finding was considered in the second simulation, where a finer grid was applied only in the wall-normal direction, whereas the third simulation was performed with the lowest resolution, which was only sufficient for the DNS of the velocity field. Therefore, the last two simulations are sufficient for the DNS of the velocity field and insufficient for the DNS of the thermal field.
1 EQUATIONS AND NUMERICAL PROCEDURE
Two different geometries for the numerical simulations of turbulent heat transfer in the infinite channel and the flume were used. Channel (Fig.1 -left) is bounded by top and bottom walls, which are heated with a constant heat source, whereas the pressure gradient drives the fluid flows between them. In the flume (Fig. 1 - right), the fluid flows above the heated bottom wall, whereas the top surface is free.
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Sl. 1. Geometrija toka tekočine: levo) zaprt kanal, desno) odprt kanal Fig. 1. Flow geometry: left) channel, right) flume
Navier-Stokesove enačbe, normalizirane s polovično višino kanala h v zaprtem kanalu oziroma z višino h v odprtem kanalu, disipativno hitrostjo ut = tw /Tp in disipacijsko temperaturo T=q ( utPfcf). Pri tem je t w strižna napetost na steni in je enaka tw = -jj.(du / dy)w. Takšen postopek lahko najdemo tudi v objavah Kasagija [2] ali Kawamure [3]:
The dimensionless Navier-Stokes equations, normalized by the flume height h (or the channel half height h), the friction velocity ut = tw /p , and the friction temperature T = q /(„ Pfcpf), were used. t stands for the wall shear t stress, defined as tw = -jj.(du / dy)w. Such scaling can be found in the papers of Kasagi et al [2] or Kawamura et al [3]:
du+
-v-
v-u+=0
1
( u+u+ ) + —V2u+-Vp + lx
v
— = -v.(ue)+
Člena lx (enotski vektor v smeri vzdolž toka) in u+ /uB se v en. (2) in (3) pojavita zaradi numerične sheme, ki zahteva periodične robne pogoje v vzdolžni (x) in prečni smeri (z). Ret je Reynoldsovo število disipacije in je določeno kot:
Ret
kjer je h polovica višine kanala. Ne smemo ga mešati z običajnim Reynoldsovim številom, ki je v kanalu definirano kot Re = uB -2h/v . Iz Reynoldsovega števila disipacije dobimo običajno Reynoldsovo število, če Ret pomnožimo z dvakratno vrednostjo povprečne hitrosti uB v kanalu. Reynoldsovo število Re = 4580 ustreza Ret = 150.
Na osnovi Reynoldsovega števila disipacije priredimo brezrazsežne stenske enote, ki so označene z zgornjim indeksom +. Višina kanala je po definiciji enaka dvakratniku Reynoldsovega števila disipacije. Smisel stenskih enot je v tem, da lahko v območju ob steni med seboj primerjamo turbulentne tokove z različnimi Reynoldsovimi števili. Brezrazsežna temperaturna razlika je določena kot:
0+(x,y,z,t)
(1) (2) (3).
1 2 + u
RetPrV^ +i
The terms lx (unit vector in the streamwise direction) and u+x / u+B appear in equations (2) and (3) due to the numerical scheme, which requires periodic boundary conditions in the streamwise and spanwise directions. Ret is the friction Reynolds number and is defined as:
uth n
(4),
where h is the channel half height. It should not be confused with the usual Reynolds number in the channel, which is defined as Re = uB-2h/v. The usual Reynolds number in the channel can be obtained from the friction Reynolds number multiplied by the double bulk velocity uB . The Reynolds number Re = 4580 corresponds to Ret = 150.
The dimensionless wall units, denoted by the superscript +, are based on the friction Reynolds number. By definition, the height of the channel is equal to two times the friction Reynolds number. The meaning of the wall units is in the comparison of the turbulent flows near the wall at different Reynolds numbers. The dimensionless wall-temperature difference is defined as:
(Tw)-T(x,y,z,t)
(5).
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Na prosti površini tekočine sta robna pogoja za komponenti hitrosti, ki sta vzporedni s steno, enaka du/dz = 0 in dw/dz = 0, hitrost, pravokotno na steno, pa je enaka v . =0. Takšen robni pogoj ni
fizikalen, ker ne dopušča površinskih valov. Vendar so preskusi Hetsronija (1997, 1999) [12] pokazali, da so površinski valovi pri majhnih Reynoldsovih številih skoraj neznatni in ne vplivajo na obnašanje tekočine tik ob steni. Na stiku stene in tekočine so vse tri komponente hitrosti enake nič. Poleg robnega pogoja hitrosti moramo upoštevati še temperaturni robni pogoj. Prosto površino obravnavamo kot adiabatno: dq+dy = 0 , na stiku stene in tekočine pa sta mogoča dva različna temperaturna robna pogoja. Prvi, ki ga v tem prispevku ne obravnavamo, je robni pogoj nespremenljive temperature [11], drugi pa robni pogoj nespremenljivega toplotnega toka, za katerega velja:
The boundary conditions for the velocity components on the top, free surface, parallel to the wall, are du/dz = 0 and dw/dz = 0, whereas the wall-normal velocity is vfr fa =0. The velocity boundary condition at the free surfacels not physical since it does not allow surface waves. However, the experiments of Hetsroni et. al. (1997, 1999) [12] show that this is an acceptable approximation at low Reynolds numbers, where the surface waves are negligible and do not affect the near-wall behavior. The velocity components at the interface of the wall and the fluid are set to zero. Besides the velocity boundary conditions, the thermal boundary conditions have to be considered. The free surface is treated as an adiabatic surface, dq+/dy = 0 , whereas two different thermal boundary conditions can be applied at the wall-fluid interface. The first, which is not presented in this paper, is the isothermal boundary condition [11], and the second isoflux boundary condition is described as:
dq + dy+
= 0
(6).
Pri tem je brezrazsežna temperatura q na stiku stene in tekočine povprečena po času in koordinatah vzdolž in prečno na tok enaka nič:
The dimensionless temperature q at the heated wall, averaged by time and the coordinates in the streamwise and spanwise directions is zero:
q+( y =1)=0
(7).
Na vstopu v kanal moramo zagotoviti polno razvit turbulentni tok, ki ga dosežemo s periodičnimi robnimi pogoji v smeri vzdolž (x) in prečno (z) na smer toka. To pomeni, da stanje, ki ga dobimo na izhodu iz kanala, preslikamo na vstop, stanje na levi strani pa na desno stran. Pri tem je treba poskrbeti, da sta dolžina in širina kanala dovolj veliki za mešanje tekočine [11].
Kakor je razvidno iz enačb (1) do (3), je temperatura pasivni skalar, ki ne vpliva na turbulenco. To pomeni, da je vzgon zanemarjen in da lastnosti tekočine (viskoznost, toplotna prevodnost itn.) niso odvisne od temperature. Dobljeni rezultati so točni le za sisteme, pri katerih ni velikih temperaturnih razlik, medtem ko je pri večjih temperaturnih razlikah potrebna večja previdnost. Takšen približek so že uporabili Kasagi [2], Kawamura [3] in Tiselj [11].
Za reševanje enačb je bila uporabljena spektralna shema, ki uporablja Fourierjeve vrste v smereh x in z ter polinome Čebiševa v smeri y. Kontinuitetno, gibalno in energijsko enačbo rešujemo z računalniškim programom, ki temelji na delu Gavrilakisa [1].
Vse numerične simulacije so bile izvedene pri Ret = 150 in Ret = 170,8 ter Pr = 0,025, 1 in 5,4. Geometrijska oblika toka tekočine je bila v primeru Ret = 150, pri katerem je bil uporabljen zaprt kanal, enaka 2356 x 942 x 300 stenskih enot v smereh x, z in y. V primeru Ret = 170,8 je bil uporabljen odprti kanal, pri katerem je bila geometrijska oblika enaka
Fully-developed turbulent flow must be ensured in the channel entrance. This is achieved with periodic boundary conditions in the streamwise (x) and spanwise (y) directions. It means that the fields at the channel exit are mapped to the channel entrance, and the situation on the left-hand side to the right-hand side, respectively. The length and the width of the channel should ensure sufficient mixing of the fluid [11].
As can be seen from Eqs. (1) to (3) the temperature is assumed to be a passive scalar. This assumption introduces two approximations: neglected buoyancy and neglected temperature dependence of the fluid properties (viscosity, heat conductivity, etc.). The results are thus very accurate, but only for the systems where the temperature differences are not too large; and some caution is required for the systems where the temperature differences are not negligible. Such an approximation was used by Kasagi et al [2], Kawamura et al [3], and Tiselj et al [11].
The equations are solved with a pseudo-spectral scheme using a Fourier series in the x and z directions, and Chebyshev polynomials in the wall-normal y direction. The numerical procedure and the code of Gavrilakis et al[1] are used to solve the continuity, momentum and energy equations.
All the numerical simulations were performed for Ret = 150 and Ret = 170.8 at Pr = 0.025, 1 and 5.4. At Ret = 150, where a channel was used, the computational domain was 2356 x 942 x 300 wall units in the x, z and y directions, respectively. At Ret = 170.8, where a flume was used, the computational
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2146 x 537 x 171 stenskih enot v smereh x, z in y. Rezultate so začeli povprečiti potem, ko je bil dosežen polno razvit turbulentni tok, kar pomeni, da se tok statistično gledano ni več spreminjal.
Simulacije lahko glede na vrednosti Prandtlovih števil razdelimo na tri glavne skupine (preglednica 1). V prvem delu so prikazane NNS za Pr = 0,025, kjer je bila uporabljena ločljivost, zadostna za hitrostno in temperaturno polje. Ko je Prandtlovo število manjše od 1, so najmanjše krajevne skale temperaturnega polja večje od najmanjših krajevnih skal temperaturnega polja. Drugi del vsebuje NNS za Pr = 1, kjer je bila izbrana ločljivost zadostna za hitrostno in temperaturno polje. Tretji del vsebuje tri različne NNS za Pr = 5,4. Ločljivost prve študije NNS je bila zadostna za hitrostno in temperaturno polje, druga študija je imela približno 2-krat manjšo ločljivost v vzdolžni (x) in prečni smeri (z), tretja študija pa je imela v primerjavi s prvo približno 2-krat manjšo ločljivost v vseh treh smereh. Zadnjih dveh numeričnih simulacij pravzaprav ne smemo več imenovati NNS, ker ne popišeta najmanjših skal temperaturnega polja, ki se pojavijo pri visokih valovnih številih. Takšne simulacije sva poimenovala "navidez" NNS.
domain was 2146 x 537 x 171 wall units in the x, z and y directions, respectively. The results were averaged after the fully-developed turbulent flow was achieved, which means that the flow did not change from the statistical point of view.
The simulations can be divided into three main parts, according to the Prandtl numbers (see table 1). In first part the DNS at Pr = 0.025 is shown, where the applied resolution was sufficient for the velocity and thermal fields. If the Prandtl number is less than 1, the smallest length scales of the temperature field are larger than the smallest length scales of the velocity field. The second part involves a DNS at Pr = 1, where the chosen resolution was sufficient for the velocity and thermal fields. The third part involves three different DNSs at Pr = 5.4. The first DNS resolution was sufficient for the velocity and thermal fields, the second resolution had an approximately two-times smaller resolution in the streamwise (x) and spanwise (z) directions, and the third resolution had a two-times smaller resolution in all three directions. Strictly speaking, the last two numerical simulations cannot be called DNS due to the smallest scales of the thermal field at high wave-number modes. Such simulations were named “quasi” DNSs.
Preglednica 1. Izračuni pri različnih Pr in mrežah
Table 1. Computational conditions at different Pr and grids
geometrija geometry
zaprt kanal channel
odprt kanal flume
Ret
150 0,025 128x128x97 0,09
170,8 1
Pr mreža
grid
170,8 5,4 170,8 5,4
170,8 5,4
128x72x65
256x128x129 0,0256
128x72x129
128x72x65
Dt+
0,0512
0,0427
0,0683
Dx
18,41
Dz
7,36
Dy
čas povprečenja averaging time
0,08-4,9 4500
16,77 7,46 0,10-4,19 3074
8,38 4,19 0,05-2,10 2560 16,77 7,46 0,05-2,10 2562
16,77 7.46 0,10-4,19 4099
2 REZULTATI
2.1 Vpliv Prandtlovega števila
Prvi sklop rezultatov so NNS, kjer smo analizirali vpliv naraščanja Prandtlovega števila na temperaturne profile, fluktuacije temperatur in turbulentne toplotne tokove pri treh različnih Prandtlovih številih: Pr = 0,025, Pr = 1 in Pr = 5,4.
Na sliki 2a so prikazani povprečni brezrazsežni temperaturni profili za tri različne vrednosti Prandtlovih števil. Brezdimenzijsko temperaturo q dobimo s povprečenjem po ravninah, vzporednih z greto steno in po času (~ 10000 časovnih korakov). Poudariti je treba, da je 9 negativna brezrazsežna temperaturna razlika, kar pomeni, da se največje brezrazsežne temperaturne razlike med tekočino in greto steno pojavijo v sredini zaprtega kanala oziroma na prosti površini odprtega kanala. Večje ko je Prandtlovo število, večja je brezrazsežna temperaturna razlika med steno in površino tekočine. Slika 2b
2 RESULTS 2.1 Prandtl number influence
The first part of the results presents the influence of the Prandtl number on the temperature profiles, temperature fluctuations and turbulent heat fluxes at Pr = 0.025, Pr =1 and Pr = 5.4.
Fig. 2a shows the average dimensionless temperature profiles at three different Prandtl numbers. The dimensionless temperature q is averaged in the planes parallel to the heated wall and in time (~ 10000 time steps). It should be emphasized that the temperature q is a negative dimensionless difference. This means that the maximum temperature differences appear in the middle of the channel or on the top free surface in the flume. A higher Prandtl number means a higher dimensionless temperature difference between the wall and the middle of the fluid in the channel or on the top surface in the flume, respectively. Fig. 2b shows the temperature
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a) b)
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0,1
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Sl. 2. a) Povprečni brezrazsežni temperaturni profili, b) fluktuacije temperatur Fig. 2. a) Dimensionless profiles of mean temperature, b) temperature fluctuations
prikazuje fluktuacije temperatur. Fluktuacije temperatur RMS_6 dobimo tako, da trenutne temperature odštejemo od povprečnih, nato pa kvadrate razlik povprečimo po ravninah, vzporednih s steno, in času. Večje ko je Prandtlovo število, večje so fluktuacije. Pri preučevanem robnem pogoju temperature (robni pogoj nespremenljivega toplotnega toka) so fluktuacije opazne že ob steni in ostanejo konstantne v območju laminarne podplasti. Največjo vrednost dosežejo v vmesni plasti, med laminarno in turbulentno podplastjo, nato pa se zmanjšujejo do sredine zaprtega kanala oziroma do proste površine odprtega kanala.
Slika 3a prikazuje turbulentni vzdolžni toplotni tok ffu' v odvisnosti od razdalje od stene za različna Prandtlova števila: Pr = 0,025, Pr = 1 in Pr = 5,4. Izračunamo ga tako, da zmnožek fluktuacij temperature 6 in fluktuacij komponente hitrosti u v smeri vzdolž kanala (x) povprečimo po ravninah, vzporednih s steno, in po času. Vidimo, da večje Prandtlovo število pomeni večjo največjo vrednost toplotnega toka in manjšo razdaljo do stene. Zelo podobne ugotovitve sledijo za turbulentni toplotni tok v smeri pravokotno na steno ffv (sl. 3b).
100
10
0,1
0,01
fluctuations. The temperature fluctuations RMS_6 are obtained as the root mean square difference of the instantaneous and averaged temperatures averaged in planes parallel with the heated wall and in time. The temperature fluctuations already appear near the wall and remain constant through the viscous sublayer for the applied thermal boundary condition (isoflux boundary condition). In the buffer sublayer, between the turbulent and viscous sublayers, the maximum is reached; afterwards the RMS_Q is decreasing towards the middle of the channel or the top free surface of the flume.
Fig. 3a shows the profiles of the turbulent axial heat fluxes ffu versus the dimensionless distance from the wall at different Prandtl numbers: Pr = 0.025, Pr = 1 and Pr = 5.4. It is calculated as a product of the temperature fluctuations and the streamwise (x) velocity fluctuations averaged by planes parallel with the wall and by time. It is seen that the higher Prandtl number means a higher maximum value of heat flux and a smaller distance to the wall. Very similar conclusions were found for the turbulent wall-normal heat flux #v (Fig. 3b).
0,8
0,6
0,4
0,2
b)
50 y+ 100 150 200 0 50 y+ 100 150
Sl. 3. Turbulentni toplotni tok: a) vzdolžno, b) v smeri pravokotno na steno Fig. 3. Profiles of turbulent heat flux: a) axial, b) wall normal
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2.2 Vpliv ločljivosti pri Pr = 5,4
Drugi sklop rezultatov predstavljajo numerične simulacije pri Prandtlovem številu Pr = 5,4 in treh različnih računskih mrežah: 256x128x129, 128x72x129 in 128x72x65. Zaradi Prandtlovega števila, večjega od ena, samo ločljivost prve mreže ustreza teoretičnim zahtevam NNS za hitrostno in temperaturno polje, preostali dve mreži zadostujeta le za hitrostni polji.
Rezultati na sliki 4a kažejo, da različno število mrežnih točk ne vpliva na povprečne temperaturne profile. Razlike, ki se pojavijo, so kvečjemu istega reda velikosti kot negotovosti zaradi statistične obravnave rezultatov (-0,5%). Podobno lahko povzamemo za temperaturne fluktuacije na sliki 4b. V tem primeru je statistična napaka manjša od 2%.
Pri turbulentnih toplotnih tokovih (sl. 5) so razlike nekoliko večje kakor v prejšnjih primerih, vendar so ocenjene napake še vedno znotraj statistične negotovosti, razen odstopanja vzdolžnega turbulentnega toplotnega toka pri y+ > 100 na najbolj grobi mreži (sl. 5a).
60
40
20
0 0, 01
a)
128x72x65 — 128x72x129 — 256x128x129
2.2 Resolution at Pr = 5.4
The second part of the results presents numerical simulations at the Prandtl number Pr = 5.4, and three different computational grids: 256x128x129, 128x72x129 in 128x72x65. Because the Prandtl number is bigger than one, only the resolution of the first grid corresponds to the theoretical requirements of the DNS for the velocity and thermal field, while the other grids are sufficient only for the DNS of the velocity fields.
The results in Fig. 4 show that the number of grid points does not influence the mean temperature profiles. All the differences that appear are of the same order of magnitude as the uncertainties. A similar conclusion can be made for the temperature fluctuations (Fig. 4b), where the statistical uncertainties are less than 2%.
The differences are slightly larger in the turbulent heat fluxes (Fig. 5), but the estimated errors are still within the statistical uncertainties, except the turbulent axial heat flux deviations of the smaller grid at y+ > 100 (Fig. 5a).
10 8
6 4 2
6 4
0, 1
1
y+ 10
100
1000
0 0, 01
b)
//\
— 128x72x129 \
— 256x128x129
- -128x72x65 \
^
0, 1
1
y+
10
100
1000
Sl. 4. a) Povprečni brezrazsežni temperaturni profili, b) fluktuacije temperatur pri Pr = 5,4 in treh
različnih ločljivostih Fig. 4. a) Dimensionless profiles of mean temperatures, b) temperature fluctuations at Pr = 5.4 and three
different resolutions
100
10
1
0 ,1
a)
128x72x65
128x72x129
256x128x129
1
0 ,5
b)
* 128x72x65
-128x72x129
-256x128x129
40
80
y+
120
160
0 200 0
40
y+ 80 120
160
200
Sl. 5. Turbulentni toplotni tok: a) vzdolžno, b) v smeri pravokotno na steno Fig. 5. Turbulent heat fluxes: a) axial, b) wall normal
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Pri preučevanju ločljivosti računske mreže se uporabljajo spektri, ki prikazujejo, kako so v toku zastopane različne krajevne skale. Spekter fluktuacij temperature (komponente hitrosti) izračunamo z avtokorelacijsko funkcijo temperature (komponente hitrosti). Tako npr. avtokorelacijsko funkcijo temperature vzdolž kanala (x) pri izbrani razdalji od stene y0 dobimo kot:
NN 21
22
In a study of the resolution requirements the spectra are usually used to show how different length scales are represented. The spectrum of the temperature (velocity component) fluctuations is derived from an auto-correlation function of the temperature (velocity component). For example, the auto-correlation function of the temperature in the streamwise direction at a given distance from the wall y0 is:
R(x1)= X H0(xi,y0,zj)-0(xi+x1,y0,zj)
(8).
Če avtokorelacijske funkcije preslikamo po Fourier-ju in rezultat ustrezno normiramo [6], dobimo spekter temperature pri izbrani oddaljenosti od stene y0 v smeri vzdolž toka x. Na podoben način pridemo do spektrov v prečni smeri z.
Turbulentno gibanje sestavljajo vrtinci različnih izmer. Geometrijska oblika sistema povzroča največje vrtince, najmanjši pa so določeni z viskoznimi silami. Za najmanjše vrtince je značilna velik raztros kinetične energije turbulentnega gibanja v toploto. Viskozna strižna napetost opravi deformacijsko delo, ki poveča notranjo energijo tekočine na račun kinetične energije turbulence. Povedano drugače, večji vrtinci, ki jih v diagramih spektrov opisujejo nižje vrednosti valovnih števil, difundirajo v manjše vrtince, ki jih opisujejo višje vrednosti valovnih števil. Energija manjših vrtincev je manjša od energije večjih vrtincev, zato se spektri z večanjem valovnih števil zmanjšujejo. Primerjave spektrov fluktuacij hitrosti (na slikah so spektri za najbolj pomembno komponento hitrosti u ) in temperatur kažejo pričakovane rezultate: hitrostni spektri se hitreje zmanjšujejo proti nič kakor temperaturni spektri pri Pr > 1. Približno enako stopnjo zmanjševanja dobimo za Pr = 1. Večja vrednost Prandtlovega števila pomeni počasnejši razpad nižjih valovnih števil v višja, kar zahteva večjo ločljivost za opis vseh najmanjših temperaturnih skal.
To potrjuje slika 6, ki prikazuje spektre fluktuacij temperatur in hitrosti vzdolžno in prečno na smer toka pri razdalji y+=3,7 od stene. Manjše razlike so vidne v spektrih fluktuacij hitrosti (6a, 6b), medtem ko so te razlike večje v spektrih fluktuacij temperatur (6c, 6d). Največje razlike med numeričnimi simulacijami različnih ločljivosti so v območju visokih valovnih števil, kjer obe manjši ločljivosti (128x72x65 in 128x72x129) kažeta večjo moč kakor NNS največje ločljivosti (256x128x129), kar kaže na nekoliko prepočasno dušenje turbulentnih fluktuacij. Kljub temu v obravnavanih spektrih ni zaslediti kopičenja pri visokih vrednostih valovnih številih. Kopičenje (spekter pri visokih valovnih številih začne naraščati) se pojavi takrat, ko najmanjše skale niso dovolj natančno modelirane in viskozni (temperaturni) disipaciji na viskozni (konduktivni) skali ne uspe spremeniti vse turbulentne kinetične energije v
If a Fourier transformation over the autocorrelation function is made, and the result is appropriately normalized [6], the temperature spectrum at a given distance from the wall y0 in the streamwise direction x is obtained. Spectra in the spanwise direction z are obtained in a similar way.
Turbulent motion consists of vortices of different dimensions. The largest vortices are defined by the flow geometry, whereas the smallest one is defined by the viscous forces. A high dissipation of the turbulent kinetic energy into the heat is typical for the smallest vortices. Viscous shear stress makes deformation work, which transforms the turbulent kinetic energy into the internal energy of the fluid. In other words, larger vortices represented by lower wave-number modes diffuse into smaller vortices represented by higher wave-number modes in the spectrum diagrams. The energy of the smaller vortices is smaller than the energy of the larger vortices, therefore the spectra decrease with increasing wave-number modes. The comparisons of the velocity (the most important velocity spectra for the velocity component ux are shown in the figures) and the temperature spectra show the expected results: the velocity spectra decay faster to zero than the temperature spectra at Pr > 1. Approximately the same decay rate is obtained for Pr = 1. A higher Prandtl number means slower decays of low wave-number modes into the high wave-number modes, and thus requires a more detailed resolution to capture all the significant thermal scales.
This is confirmed in Fig. 6, which shows the streamwise and spanwise spectra of the temperature and the velocity fluctuations at y+=3.7 from the wall. Minor differences are seen in the velocity-fluctuation spectra (Figs. 6a and 6b), whereas these differences are larger in the temperature-fluctuation spectra (Figs. 6c and 6d). The main differences between the numerical simulations at different resolutions are observed at higher wave-number modes, where both lower resolutions (128x72x65 in 128x72x129) show stronger modes than the well-resolved DNS at the higher resolution. This indicates too-slow damping of the turbulent fluctuations. However, there are no “pile-up” phenomena seen at high wave-number modes. The pile-up phenomenon (the spectrum starts to grow at high wave-number modes) appears when the smallest scales are not properly modeled and viscous (temperature) dissipation and viscous (conductive) scale cannot
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1,0E+ 0 0n
1,0E -01
1,0E -02
1,0E -03 :
1,0E -04
1,0E -05
1,0E -06
1,0E -07—
0,001
1,0E+ 0 01—
1,0E -01
1,0E -02
1,0E -03 I
1,0E -04
1,0E -05
1,0E -06
1,0E -07 — 0,001
Sl. 6. Spektri vzdolž (a, c) in prečno na kanal (b, d) y+ = 3.7 od grete stene: a, b) temperatura, c, d) hitrost
vzdolž toka Fig. 6. Spectra in streamwise (a, c) and spanwise (b, d) directions at y+ = 3.7 from the heated wall: a, b)
temperature, c, d) streamwise velocity
notranjo energijo. To pomeni, da moramo povečati change all the turbulent kinetic energy into internal
ločljivost računske mreže, če hočemo zajeti najmanjše energy. It means that the resolution should be increased
(Kolmogorovove) skale. in order to capture the smallest (Kolmogorov) scales.
3 SKLEP 3 CONCLUSION
V prvem delu je bila opisana neposredna In the first part of the paper the DNS of the numerična simulacija (NNS) polno razvitega fully-developed turbulent flow at Reynolds number turbulentnega toka pri Reynoldsovem številu Re = 4580 Re = 4580 (Ret = 150) and Prandtl number Pr = 0.025, (Ret = 150) in Prandtlovem številu Pr = 0,025 ter Re = and at Re = 5260 (Ret = 170.8) and Prandtl numbers Pr 5260 (Ret = 170,8) in Prandtlovih številih Pr = 1 in Pr = 1, 5.4, were performed. The results show that the =5,4. Rezultati kažejo, da Prandtlovo število močno Prandtl number has a strong influence on the turbulent vpliva na statistiko turbulentnega polja. Rezultati statistics. The obtained results can be used for opravljenih NNS lahko rabijo pri razvoju turbulentih developing turbulent models for flows in a more modelov za tokove v bolj zapletenih geometrijskih complex geometry and at higher Reynolds numbers. oblikah in pri višjih vrednostih Reynoldsovih števil. In the second part the numerical simulations
V drugem delu smo preučevali numerične of three different resolutions at Ret = 170.8 and Pr = simulacije treh različnih ločljivosti pri Ret = 170,8 in 5.4 were performed. Theoretically, the resolution for Pr = 5,4. Ločljivost naj bi bila teoretično sorazmerna the DNS should be proportional to the square root of kvadratnemu korenu Prandtlovega števila [8], vendar the Prandtl number [8]; however, results show that rezultati kažejo, da je vsaj pri Pr = 5,4 ta zahteva this requirement is too stringent in the case of Pr = nekoliko preostra. Naši rezultati (profili povprečnih 5.4. Our results at Pr = 5.4 (mean temperature profiles,
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128x72xC129
-256x128x129
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Bergant R. - Tiselj I.: Vpliv Prandtlovega {tevila - The Influence of Prandtl Number
hitrosti, fluktuacije, toplotni pretoki) pri najslabši ločljivosti kažejo, da je ločljivost, ki zadošča za simulacijo NNS hitrostnega polja dovolj velika tudi za simulacijo temperaturnega polja pri Pr = 5,4. Majhne razlike so vidne v spektrih fluktuacij temperatur v območju višjih valovnih števil, kjer obe računski mreži z nižjima ločljivostma kažeta počasnejšo spremembo turbulentne kinetične energije v notranjo kot NNS največje ločljivosti.
fluctuations, heat fluxes) for the lowest resolution show that the resolution, which is sufficient for the DNS of the velocity field, is also sufficient for the simulation of the thermal field at Pr = 5.4. Small differences in the temperature spectra are seen at high wave-number modes, where both low resolutions show a slower conversion of the turbulent kinetic energy into internal energy compared with the DNS of the highest resolution.
4 SPREMENLJIVKE 4 NOMENCLATURE
spekter
polovična višina kanala
enotski vektor v smeri x (1,0,0)
valovno število
vzdolžna in prečna dolžina kanala
tlak
Prandtlovo število
toplotni tok iz stene na tekočino
Reynoldsovo število
Reynoldsovo število raztrosa
avtokorelacijska funkcija
čas
temperatura
smer vzdolž, prečno, pravokotno na kanal
komponente hitrosti v smereh x, y in z
raztrosna hitrost
toplotna difuzivnost
brezrazsežna temperaturna razlika
toplotna prevodnost
kinematična viskoznost
gostota
tekočina
stena
disipacija
normalizirano z ut, Tt, v
E
h r
1x
k
L1, L3
p
Pr
qw Re Ret R t T
x, y, z u, v, w
ut
a
q
l
g
r
()f
()w
()t
()+
spectrum
channel half height
unit vector in x direction (1,0,0)
wave number
streamwise and spanwise length of box
pressure
Prandtl number
wall-to-fluid heat flux
Reynolds number
friction Reynolds number
auto-correlation function
time
temperature
streamwise, spanwise, wall normal distance
velocity components in x, y and z directions
dissipative velocity
thermal diffusivity
dimensionless temperature difference
thermal conductivity
kinematic viscosity
density
fluid
wall
dissipation
normalized by ut, Tt, v
5 LITERATURA 5 REFERENCES
[1] Gavrilakis, S., H.M. Tsai, P.R. Voke, D.C. Leslie (1986) Direct and large eddy simulation of turbulence, Notes
on Numerical Fluid Mechanics Vol. 15, ed. U. Schumann, R. Friedrich, Vieweg, Braunschweig, D.B.R., 105. [2] Kasagi, N., Y. Tomita, A. Kuroda (1992) Direct numerical simulation of passive scalar field in a turbulent
channel flow, Jornal of Heat Transfer -Transactions of ASME, Vol. 114, 598-606. [3] Kawamura, H., K. Ohsaka, H. Abe, K. Yamamoto (1998) DNS of turbulent heat transfer in channel flow with
low to medium-high Prandtl number fluid, International Journal of Heat and Fluid Flow, Vol. 19, 482-491. [4] Kawamura, H., H. Abe, Y. Matsuo (1999) DNS of turbulent heat transfer in channel flow with respect to
Reynolds and Prandtl number effects, International Journal of Heat and Fluid Flow, Vol. 20, 196-207. [5] Kim, J., P. Moin, R.D. Moser (1987) Turbulence statistics in fully developed channel flow at low Reynolds
number, J. Fluid Mech., Vol. 130, 133-166. [6] Kim, J., P. Moin (1989) Transport of passive scalars in a turbulent channel flow, Turbulent shear flows VI,
Springer-Verlag, Berlin.
Bergant R. - Tiselj I.: Vpliv Prandtlovega {tevila - The Influence of Prandtl Number
[7] Na, Y. , T.J. Hanratty (2000) Limiting behavior of turbulent scalar transport close to a wall, International
Journal of Heat and Mass Transfer, Vol. 43, 1749-1758. [8] Tennekes, H., J.L. Lumley (1972) A first course in turbulence, MIT Press, Cambridge, MA. [9] Tiselj, I., E. Pogrebnyak, C. Li, A. Mosyak, G. Hetsroni (2001) Effect of wall boundary condition on scalar
transfer in a fully developed turbulent flume, Physics of Fluids, 13 (4), 1028-1039. [10] Kader, B.A. (1981) Temperature and concentration profiles in fully turbulent boundary layers, Int. J. Heat
Mass Transf., 24, 1541. [11] Tiselj, I., R. Bergant, B. Mavko, I. Bajsic, G. Hetsroni (2001) DNS of turbulent heat transfer in channel flow
with heat conduction in the solid wall, J. Heat Transf., 123, 849-857.
Naslov avtorjev:Robert Bergant Doc.dr. Iztok Tiselj Institut “Jožef Stefan“ Odsek za reaktorsko tehniko Jamova 39 1000 Ljubljana, Slovenija robert.bergant@ijs.si iztok.tiselj@ijs.si Authors’ Address: Robert Bergant Doc.Dr. Iztok Tiselj “Jožef Stefan“ Institute Reactor Engineering Division Jamova 39 1000 Ljubljana, Slovenia robert.bergant@ijs.si iztok.tiselj@ijs.si
Prejeto: 20.12.2002 Received: Sprejeto: 31.1.2003 Accepted:
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© Strojni{ki vestnik 48(2002)12,707-718 © Journal of Mechanical Engineering 48(2002)12,707-718
ISSN 0039-2480 ISSN 0039-2480
UDK 621.924.9:531.76 UDC 621.924.9:531.76
Izvirni znanstveni ~lanek (1.01) Original scientific paper (1.01)
Mehanski in elektronski merilnik hitrosti delcev v stroju za peskanje
A Mechanical and Electronic Measurement System for Particle Velocity Measurements in a Shotblasting Machine
Ale{ Hribernik - Gorazd Bombek - Ivan Marko~i~
Osnovni problem merjenja hitrosti zrnc peska v peskalnem stroju je zelo velika abrazivnost peska in velika koncentracija prašnih delcev v zaprti testni napravi. Uporaba brezstičnih optičnih merilnih metod je zato pogosto nemogoča. Stične merilne metode so omejene na uporabo merilnikov, ki so zaščiteni z robustnimi oklepi. Kot alternativo robustnemu mehanskemu merilniku hitrosti zrnc peska smo razvili elektronski merilni sistem, ki ga odlikujeta dobra vodljivost vzdolž obeh prečnih osi curka peska in preprosta zgradba merilnega zaznavala, ki z uporabo cenenih elementov dovoljuje pogosto zamenjavo in ne potrebuje robustnega oklepa. V prispevku so predstavljeni razvoj, delovanje in uporaba mehanskega in elektronskega merilnega sistema. Prikazani so rezultati meritev in podana je primerjava obeh metod z razpravo in sklepi. © 2002 Strojniški vestnik. Vse pravice pridržane. (Ključne besede: stroji za peskanje, meritve hitrosti, hitrosti delcev, merilniki mehanski, merilniki elektronski)
Two of the main problems for particle velocity measurement in a shotblasting machine are very abrasive shotblasting particles and a high concentration of dust within the closed testing chamber. The application of noncontacting optical methods is usually impossible; and robust shields protecting the sensors from abrasion have to be used for contacting velocity measurements. An electronic measurement system has been developed as an alternative to the robust, mechanical measurement device. This alternative electronic system has a simple construction and uses low-cost elements that can be replaced very quickly and, therefore, no robust shields are necessary. This enables very good mobility and positioning of the sensor along both axes in a cross-section plane of a particle stream. The development, operation and application of this mechanical and electronic particle velocity measurement system are described in this paper. Experimental results are presented, discussed and both methods are compared. © 2002 Journal of Mechanical Engineering. All rights reserved.
(Keywords: shotblasting machines, velocity measurements, particle velocity, mechanical measurnig devices, electronic measuring systems)
0 UVOD
Učinkovitost peskalnih naprav merimo s količino materiala, ki ga s postopkom odstranimo z nadzorne površine na enoto porabljene energije. Učinek peskanja je neposredno odvisen od hitrosti zrnc peska ob udarcu v nadzorno površino, saj odnašanje materiala zagotovi le dovolj velika kinetična energija zrnc. Zaradi tega je poznavanje hitrosti zrnc peska ob trku s peskano površino izrednega pomena za pravilno izbiro parametrov peskanja in za dosego kar največje učinkovitosti peskanja.
Pri merjenju hitrosti peska v peskalnem stroju smo se podali na razmeroma neraziskano področje. V literaturi ni zaslediti podobnih primerov, obstajajo pa določene podobnosti z načinom merjenja hitrosti izstrelkov v balistiki [1]. Ker so
0 INTRODUCTION
The efficiency of a shotblasting machine is measured by the quantity of material removed from a surface by a certain amount of energy. The effect of shotblasting depends on the velocity of the particles hitting the surface, since only those particles with enough kinetic energy can remove the material. A knowledge of particle velocity is, therefore, essential when selecting the optimum operation parameters of a shotblasting machine, thus ensuring the highest possible shotblasting efficiency.
Particle velocity measurements in shotblasting machines have been insufficiently investigated so far. No similar examples can be found in the literature; however, some similarities exist with velocity measurements in ballistics [1]. The application of noncontacting optical methods, commonly used for particle velocity measurements in sprays [2], has been impossible due to the
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bile koncentracije prahu v preskuševalni napravi izredno velike, je bila uporaba brezstičnih optičnih metod [2] nemogoča. Zato smo razvili dve stični metodi: mehanski merilnik hitrosti z vrtečima obročema in elektronski merilni sistem z mikrofonskimi zaznavali, ki ju v nadaljevanju predstavljamo.
1 MEHANSKI MERILNIK HITOSTI Z VRTEČIMA SE PLOŠČAMA
Mehanski merilnik je prikazan na sliki 1. Sestavljata ga dve plošči na skupni gredi, ki jo poganja elektromotor. Pred ploščama je postavljena stena, ob katero udarja pesek iz peskalnega stroja. Na steni je izvrtina s premerom 10 mm, skozi katero pesek neovirano nadaljuje pot do prednje plošče, ki se vrti z izbrano vrtilno frekvenco. Tudi na prednji plošči je izvrtina s premerom 10 mm, ki se stožčasto (pod kotom 45°) širi od prednje proti zadnji strani. V trenutku, ko se izvrtini prekrijeta, lahko pesek nadaljuje pot proti zadnji plošči in udari ob papirnato tarčo, nalepljeno nanj. Kot j razberemo s tarče in je kot med projekcijo izvrtine na prvi plošči in zadetki na tarči (slika 1). Hitrost zrnc peska pa nato izračunamo z izrazom:
pri čemer je: l - razdalja med ploščama, co - kotna hitrost plošč.
2 ELEKTRONSKI MERILNIK HITROSTI DELCEV Z MIKROFONSKIMI ZAZNAVALI
Elektronski merilnik (sl. 2a) sestavlja nosilo, na katerem sta dve mikrofonski zaznavali. Nosilo je prek prečne konzole pritrjeno na koordinatni podajalni sistem, ki omogoča navpično in vodoravno postavitev zaznaval ter izbiro strmine nosila mikrofonskih zaznaval. Mikrofonsko zaznavalo sestavlja jeklen okrov z
high concentration of dust within the closed testing chamber. Two particle velocity measurement systems have, therefore, been developed: a mechanical measuring device using the principle of rotating discs, and an electronic system applying microphone sensors. This paper presents both systems.
1 MECHANICAL MEASURING DEVICE USING THE ROTATING DISCS PRINCIPLE
The measuring device is shown in Fig. 1. The discs on the common shaft are driven by an electric motor. The particles from the shotblasting turbine hit the screen that protects both discs. A 10-mm bore is made on the screen. The particles can continue their way unhindered through this bore to the front disc, which rotates at a known rotation speed. A 10-mm bore is made on the front disc, which is conically opened to the rear disc side. When both bores coincide the particles continue their way to the rear disc and hit the paper target that is fixed on it. The angle j between the hits on the target and the projection of the front disc bore is then measured and used to calculate the velocity of the particles with the following equation:
(1),
a 180
j n
where: l is the distance between the discs, and co is the angular velocity of the discs.
2 AN ELECTRONIC PARTICLE VELOCITY MEASUREMENT SYSTEM WITH MICROPHONE SENSORS
Two specially designed microphone sensors are used, which are mounted on a longitudinal carrier (Fig 2a). The sensor’s carrier is attached by a transverse cantilever to the coordinate tender system, which enables horizontal and vertical positioning of the sensors, and selection of the sensor carrier’s inclination. A capacitor-type microphone encapsulated in a metal hous-
j
Sl. 1. Mehanski merilnik hitrosti zrnc peska Fig. 1. Mechanical device for particle velocity measurements
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t2 ti
4%+
----------------------->
t
a) nosilo mikrofonskih zaznaval b) električni signal
a) carrier of mircrophone sensors b) electric signals
Sl. 2. Elektronski merilnik hitrosti zrnc Fig. 2. Electronic particle velocity measurement system
membrano, pod katero je nameščen ing and placed under the metal membrane forms the
kondenzatorski mikrofon. Zaznavalo S1 je microphone sensor. The sensor S1 is impressed directly
zatisnjeno neposredno v jekleno cev nosila, into the front opening of the sensor’s carrier, while the
zaznavalo S2 pa je bočno pritrjeno na nosilo. S2 sensor is mounted on the upper side of the carrier.
Udarec na površino jeklene membrane povzroči The vibration of the membrane resulting from the im-
nihanje membrane in se prenese na mikrofon, kar pact of particles is transmitted to the microphone which
povzroči naglo spremembo signala. Pri uporabi signals any sudden changes (a sudden change of mi-
elektronskega merilnika izkoriščamo pulzno crophone’s electric signal occurs). The electric measur-
delovanje peskalne turbine. V nasprotju z običajno ing system makes use of a pulsatile shotblasting tur-
centrifugalno radialno črpalko, ki jo tekočina polni bine operation. Only partial admission is used by the
po celotnem notranjem obodu, poteka polnjenje shotblasting turbine, in contrast to the radial pump where
kanalov med lopaticami peskalne turbine le na admission takes place over the whole inner circumfer-
določenem delu notranjega oboda. Zato izmetavajo ence. The turbine blades eject the particles at the se-
lopatice pesek le v izbranem delu zunanjega oboda, lected position on the outer circumference, directly into
neposredno v peskalno napravo. Pri tem je tok peska the shotblasting machine. The particle flow is, there-
v neki opazovani točki izrazito ponavljajoč, prihaja fore, pulsatile: it rises in waves of high and low particle
v obliki zgoščenih paketov zrnc peska s frekvenco, concentrations with a frequency that is equal to the
ki je enaka zmnožku vrtilne frekvence rotorja turbine product of the rotor’s rotational speed and the number
in števila lopatic turbine. Tak je tudi signal, ki ga of blades. The signals from both sensors, S1 and S2
zaznata mikrofonski zaznavali (sl. 2b). Zaradi (Fig. 2b), which are phased over a particular time inter-
vzdolžnega odmika med zaznavali sta signala val are similar to the particle flow This time interval (time
časovno premaknjena. Pri tem je čas zakasnitve delay) is inversely proportional to the longitudinal dis-
sorazmeren razmerju hitrosti zrnc in razdalje med tance between the sensors, and it is proportioned ac-
zaznavali. Torej je mogoče hitrost zrnc peska cording to the velocity of the particles. The particle
preprosto izračunati z izrazom: velocity can, therefore, be calculated as:
w = 7 (2).
2.1 Zgradba mikrofonskega zaznavala 2.1 Microphone sensor
Mikrofonsko zaznavalo (sl. 3) sestavlja A microphone sensor is shown in Fig. 3. It is
teflonska puša, v katero je privita jeklena membrana. made of a plastic tube, into the top of which a metal
Tik pod membrano je nameščen kondenzatorski membrane is screwed. A capacitor-type microphone
mikrofon, ki se tesno prilega teflonski puši. Med fastened in the plastic tube is placed under the mem-
membrano in teflonsko pušo je vstavljena ploščica iz brane. A washer made of soft silicone rubber is placed
mehke silikonske gume. Teflonska puša je zatisnjena between the plastic tube and the metal membrane. This
v jekleno pušo, ki je okrov zaznavala in prek katere je plastic tube is impressed into the metal housing, which
zaznavalo pritrjeno na nosilo. is then fixed to the sensor’s carrier (Fig. 2a).
Razvoj mikrofonskega zaznavala je potekal The development of this microphone sensor was
v več korakih. Osnovno vodilo razvoja je bilo performed over a number of successive steps. The basic
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gumijasta podložka rubber washer
okrov housing
membrana membrane
mikrofon microphone
plastična puša plastic tube
Sl. 3. Mikrofonsko zaznavalo Fig. 3. Microphone sensor
doseči čim močnejši signal na enoto obremenitve membrane ob hkratnem dobrem dušenju lastnih nihanj in nihanj, ki so posledica udarcev v bok nosila in pomenijo motnjo. Preskusili smo tri tipe zaznaval A, B in C. Zaznavalo tipa A je sestavljeno iz jeklene puše, v katero je zatisnjen mikrofon, vstavljen v teflonski obroč. Membrana nad mikrofonom debeline 4 mm je pritrjena na jekleno pušo. Na sliki 4 je prikazan odziv zaznavala na udarec, ki ga povzroči trk jeklene kroglice mase 1,5 g s hitrostjo 0,3 m/s (učinek trka ustreza trku povprečno velikega zrnca peska s hitrostjo 60 m/s). Ugotovimo lahko, da je sprememba signala
task was to increase the sensitivity of the microphone sensor (as high an output signal as possible per unit of membrane load), and to reduce the oscillations caused by the resonance in the system’s natural frequency domain or induced by the side impact of the particles. Three different types of sensor – Types A, B and C – were tested. The type-A sensor was made up of a metal housing covered with a metal membrane. The microphone was placed into this housing and fixed by a plastic ring. The membrane was 4-mm thick. The impulse response for the type-A sensor is shown in Fig. 4. The impulse input force was applied to the middle of the membrane by the impact ball. The ball’s mass was 1.5 g, and the speed
1,75 1,50 1,25 1,00 0,75 0,50
fw ^
0 12 34 5
čas - time ms
Sl. 4. Odziv zaznavala na udarec (A - membrana zatisnjena v jekleno pus o, B - membrana zatisnjena v
silikonsko pušo) Fig. 4. Impulse response of sensor (A - membrane impressed directly into the metal housing, B - membrane impressed into the plastic tube)
1,30 1,25 1,20 1,15
Az
012345
čas - time ms
Sl. 5. Odziv zaznavala na bočni udarec (B - membrana zatisnjena v silikonsko pušo brez podloge;
C - membrana zatisnjena v silikonsko pušo s podlogo)
Fig. 5. Sensor response on the side impact (B - membrane impressed into the plastic tube without rubber
washer, C - membrane impressed into the plastic tube with rubber washer)
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majhna, reda velikosti 50 mV, kar je le 4% srednje vrednosti signala in ne zadošča za izvedbo meritev. V primeru zaznavala tipa B je amplituda odziva neprimerno večja. Največja sprememba signala doseže kar 60 % srednje vrednosti signala. Tako veliko amplitudo odziva smo dosegli s spremenjenim vpetjem membrane. Da bi kar se da zmanjšali togost vpetja, smo membrano zatisnili v silikonsko pušo, ki je bila vstavljena v jeklen okrov zaznavala, kakor to prikazuje slika 3. Slaba stran tega vpetja pa so razmeroma velike motnje, ki jih povzročajo udarci ob bok zaznavala (sl. 5). Te nam je uspelo zmanjšati tako, da smo med jekleno membrano in teflonsko pušo vstavili mehko silikonsko gumo (senzor tipa C - sl. 3), s katero nam je uspelo povečati dušenje sistema in zmanjšati motnje (sl. 5). S tem smo sicer zmanjšali amplitudo odziva sistema (sl. 6), ki pa je še vedno dovolj velika (20 do 30 % srednje vrednosti signala), da lahko zaznavalo uporabimo za meritve hitrosti. Jeklena membrana ima dvojno nalogo. Ena je, da zaščiti mikrofon pred neposrednim vplivom peska, druga pa je, da ob udarcu peska odda zvok, ki ga zazna mikrofon. Pri tem je zvočni tlak neposredno odvisen od debeline membrane. Ker pa je membrana med delovanjem izpostavljena intenzivni abraziji (hitrost odnašanja materiala 1,5 mm/uro), je treba najti kompromis med amplitudo odziva in dobo trajanja zaznavala. Preskusili smo tri membrane debelin 2 mm, 3,5 mm in 5 mm. Vpliv debeline membrane na odziv sistema prikazuje slika 6. S povečevanjem debeline amplituda odziva slabi in znaša pri debelini 5 mm le še 10 % povprečne vrednosti signala. Zato smo se odločili za membrano debeline 3,5 mm. Ta zagotavlja vsaj 90 minutno dobo trajanja zaznavala in ima tudi pri začetni debelini amplitudo odziva enako 20 % povprečne vrednosti signala.
1,50 1,25 1,00 0,75
of the ball was 0.3 m/s (the effect of the ball’s impact is similar to the impact of an average shotblasting particle with a velocity of 60 m/s). The change of the signal was low, as shown in Fig. 4. It did not exceed 50 m V, which is less than 4% of the mean magnitude of the signal. This is far too small for the application of a type-A sensor for particle velocity measurements. The change of the signal was much higher for the type-B sensor (Fig. 4). The maximal signal change was 60 % of the mean signal magnitude. This was achieved by modifying the membrane fixture. In order to reduce the stiffness of the membrane’s fixture, the membrane was screwed into the plastic tube, which was then fixed into the metal housing (see Fig. 3). The weakness of this type of sensor is its high sensitivity to side impacts (undesired disturbances, Fig. 5). This problem was solved by the application of a soft silicone-rubber washer placed between the metal membrane and the plastic tube (Fig. 3). The damping of the membrane’s vibrations was increased and the sensitivity to side impacts was reduced by the application of a soft rubber washer. Unfortunately, this modification reduced the overall sensor sensitivity as well (Fig. 6). However, the signal alteration remained high enough (20–30 % of the mean signal magnitude) to carry out the particle velocity measurement. The metal membrane placed in front of the microphone performs a dual role. First, to protect the microphone from the direct impact of abrasive particles. Second, as a source of sound that is sensed by the microphone. The intensity of the sound pressure depends on the membrane thickness. Since the amount of material removed from the membrane surface during measuring is very high (approximately 1.5 mm/hour), a compromise had to be found between the signal intensity and the sensor lifetime. Three membranes with thickness of 2 mm, 3.5 mm and 5 mm were, therefore, tested. The influence of the membrane thickness on the sensor-impulse response is shown in Fig. 6. The intensity of the signal change was reduced by increasing the membrane thickness. A signal from the 5-mm membrane was too low and, since the lifetime of a 2-mm-membrane sensor would be less than 45 minutes, a 3.5-mm membrane was chosen. The lifetime of this membrane is at least 90 minutes, and the change of the signal with each impulse impact is 20 % of the mean signal magnitude of the initial membrane thickness.
C_2
C_3,5
C_5
0 0,5 1 1,5
čas - time ms
2,5
Sl. 6. Vpliv debeline membrane na odziv zaznavala tipa C Fig. 6. The influence of the membrane thickness on the response of type C sensor
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2.2 Sistem za zbiranje in analizo podatkov
Uporaba mikrofonskih zaznaval zahteva ustrezno opremo tako za zbiranje podatkov kakor tudi za njihovo obdelavo. Uporabili smo računalniško podprt sistem, ki ga sestavlja prenosni osebni računalnik (Celeron 1000 MHz, 256 MB RAM) z večnamensko kartico DaqCARD 6062 (format PCMCIA). Fizično smo električne povezave izvedli tako, da smo uporabili sistem za pripravo signalov SCXI. Večfunkcijska kartica, sistem za pripravo signalov in programska oprema so izdelek podjetja National Instruments.
Analogne vhodne signale smo pripeljali v modul SCXI-1140. Signale smo vezali diferencialno. Za napajanje mikrofonov smo uporabili modul SCXI-1124. Povezavo smo izvedli tako, da smo izenačili potenciale vseh negativnih potencialov, analogne zemlje in potenciala okrova. Poleg tega smo na maso vezali tudi kovinsko ogrodje in tako dobili Faradejevo kletko ter tako zmanjšali motnje, ki so posledica elektrostatičnega delovanja zrnc peska, ki udarjajo ob stene preskusne komore.
Za zbiranje podatkov in njihovo obdelavo smo uporabili program LabVIEW. Ta program omogoča nadzor nad delovanjem večnamenske kartice (zajemanje električnih signalov in napajanje mikrofonov), pa tudi shranjevanjem in obdelavo podatkov.
3 MERITVE IN REZULTATI
Meritve smo izvajali v sredini curka peska na oddaljenosti 650 mm od vrha lopatic rotorja turbine. Os merilnika smo postavili vzporedno s smerjo zrnc peska. Zato smo za vsako posamezno turbino pred začetkom meritev posneli t.i. sliko curka. V ta namen smo v testno komoro na oddaljenosti 1100 mm od turbine postavili zaslon, na katerem je bila pritrjena tarča. Pred zaslon smo postavili odklonsko mrežico. Med peskanjem zaslona so na tarči ostala mesta v senci odklonske mrežice nepoškodovana. Na podlagi navpičnega odmika med točko na odklonski mrežici in njej pripadajočo senco ter razdaljo med točko in zaslonom smo nato ugotavljali kot poti zrnc proti vzdolžni smeri. Analizirali smo delovanje štirih različnih turbin. Za vsako smo izvedli večje število meritev z mehanskim in elektronskim merilnikom hitrosti v eni sami točki curka. Pri tem smo meritve ponavljali v isti točki vsaj trikrat, točko meritve pa smo izbrali tako, da so bile razmere (smer curka, oddaljenost merilnika od turbine itn.) za vse turbine približno enake. V nadaljevanju bomo predstavili rezultate, izmerjene s prototipno izvedbo turbine z osmimi naprej ukrivljenimi lopaticami (G 300U) pri vrtilni frekvenci rotorja 50 Hz.
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2.2 Data acquisition and data-analysis system
Data acquisition and a data-analysis system are required for the particle velocity measurements using microphone sensors (electronic system). A computer-aided measuring system was used, which incorporates a personal computer (Celeron 1000 MHz, 256 MB RAM) and a multifunction card (DaqCARD 6062, format PMCIA). The electric signals were conditioned by an SCXI data-conditioning system. The multifunction card, data-acquisition system and application software are all products from National Instruments.
The differential analog input signals were fed to the SCXI - 1140 module. The microphones were supplied with a constant DC voltage source from the SCXI – 1124 module. The sensor wiring was made by a common negative potential, which was connected to the analog ground with the SCXI chassis and the sensor’s carrier construction. A Faraday cage was formed in this way, and the disturbances caused by the large amount of electrostatic noise (resulting from particles hitting the walls of the testing chamber) were reduced.
LabVIEW software was used to build the computer applications for the data acquisition and the data analyses. These applications are used to control the operation of the multifunction card (data acquisition, DC voltage output for microphones) and for data logging and post-processing of the data.
3 MEASUREMENTS AND RESULTS
The measurements were carried out in the centre of the particles’ stream, 650 mm from the tip of the shotblasting turbine’s blade. The longitudinal sensor axis was set parallel to the particle trajectories. A so-called stream image was, therefore, determined first showing the particle trajectories. A paper target was fixed on the vertical screen 1100 mm away from the turbine. An inclination mesh was placed in front of the screen and the turbine was allowed to run for a short period. The target in the shadow of the inclination mesh remained undamaged. The flow angle of particles within the stream intersected using the inclination mesh was determined based on the differences between the coordinates of the particular point on the mesh and the coordinates of its matching point on the target. The operation of four different turbines was analysed. Several measurements were performed using both methods, all in just one central point of the particle stream. The flow conditions were similar for all turbines. Each measurement was repeated at least three times. The results for the prototype G300U turbine with eight forward curved blades measured at 3000 revolutions per minute (50 Hz) are presented in more in detail.
Hribernik A., Bombek G., Marko~i~ I.: Mehanski in elektronski - A Mechanical and Electronic
3.1 Rezultati meritev hitrosti z mehanskim merilnikom
Merilnik smo postavili nad ravnino osi turbine, v višino, pri kateri je bila smer zrnc +6,250 nasproti vodoravnici. Sredina zaslonke se je ujemala s sredino curka. Izvedli smo tri zaporedne meritve. Vsaka meritev je bila sestavljena iz dveh faz. V prvi fazi smo obstreljevali tarčo med mirovanjem plošč. Tako smo določili projekcijo izvrtine prednje plošče na tarči, prilepljeni na zadnji obroč. Nato smo z elektromotorjem plošči zavrteli in pri ustaljenih vrtljajih obstreljevali tarčo v kratkem presledku. Rezultate meritve, t.j. položaj zadetkov na tarči, smo določili z odbiranjem, pri čemer je bila delitev lestvice na tarči 1°. Rezultat so zbrani v preglednici 1.
Kakor je razvidno iz preglednice 1, je ujemanje izmerjenih srednjih hitrosti zrnc zelo dobro; odstopki ne presežejo 1,3 m/s. Vendar je merilna negotovost precej večja. Vzrok za to je precej širok pas zadetkov tarče v območju ±100. Tak raztros je delno posledica širine odprtine na prednjem obroču (0=10 mm), ki tudi pri mirovanju plošč povzroči raztros v območju ±50. Dodaten prispevek k raztrosu pa prinese neenakost hitrosti posameznih zrnc peska. Delno je ta neenakost opazna že v samem toku zrnc pred merilnikom, dodatno pa k njej prispevajo medsebojni trki zrnc, ki jih povzročajo od merilnika odbita zrnca na vstopu skozi zaslonko in pri prehodu skozi odprtino prednje plošče.
3.1 Results of the mechanical measuring device
The measuring axis of the mechanical measuring device was placed above the zero plane defined by the turbine axis, and the particle velocity was inclined by 6.25 degrees from the horizontal at this particular position. The orifice (front bore) was in line with the centre of the particle stream. Three successive measurements were carried out. Each measurement was performed in two stages. In the first stage the disks and the target were at a standstill, and they were briefly exposed to the particles. The projection of the front disk bore on the target was determined in this way. In the second stage the discs were driven by an electric motor. The target was briefly exposed to the particles again at a constant rotational speed. The result of the measurement, i.e. the angle between the hits into the target when held still and the hits into the rotating target, was read from the scale that was printed on the table with a 1 degree increment. The results are shown in Table 1.
The measured particle velocities agree well, and as can be seen from Table 1 the deviation from the average value is less than 1.3 m/s. However, the measurement uncertainty is much higher due to the large scatter of the hits on the target. The scatter of the hits was ± 10 degrees. One of the reasons for this scatter is the wide bore (0=10 mm) on the front disc, which resulted in ± 5 degrees scatter of hits onto the target held still (first step of the velocity measurement). The unequal speed of the particles within the stream expanded the scatter even more when the disc and target were rotated. The particles already had a differ-ent velocity ahead of the measuring plane. These speed differences within the measuring device increase due to the collisions between the particles when passing the orifice in the screen and in the front disc.
Preglednica 1. Rezultati meritve hitrosti zrnc peska z mehanskim merilnikom za turbino G300U Table 1. Results of the particle velocity measurement of the G300U shotblasting turbine using a mechanical measuring device
Meritev Measurement
Vrtilna hitrost
Rotational speed min-1
7650
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Angle
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155
153
155
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Hitrost zrnc
Particle velocity
m/s
73,4
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74,4
Standardna merilna negotovost
Standard measurement uncertainty
m/s
+3,1________________
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3.2 Rezultati meritev hitrosti z elektronskim merilnikom
Ker imajo mikrofoni frekvenčni odziv prilagojen zvoku, ki ga lahko zaznava človeško uho (do 17 kHz), smo se odločili za približno 2-krat višjo frekvenco zajemanja - 40000 Hz. Signal, zajet s to frekvenco, lahko kasneje analiziramo in prepoznamo frekvence do 20000 Hz (po Nyquistovem teoremu) [3]. Odločili smo se, da bomo zajeli 200000 merilnih točk na kanal, kar ustreza času 5 sekund. Meritev smo izvedli tako, da smo ob zagonu turbine
3.2 Results of the electronic measurement system
The frequency response of the applied standard microphones is adjusted to the audible sound of the human ear (up to 17 kHz). Therefore, the signals were acquired using a 40-kHz acquisition frequency. According to the Nyquist theorem [3], the discrete signal acquired using this frequency can be post-analysed and frequencies up to 20 kHz can be recognised. The amount of acquired data was 200,000 per channel. This corresponds to a period of 5 seconds. A screen placed in front of the
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elektronski merilnik zastrli z zastorom, postavljenim 300 mm pred prvo mikrofonsko zaznavalo. Ko so se razmere v turbini ustalile, smo zaslon umaknili in zajeli signale.
Slika 7 prikazuje zajeta neobdelana signala. Zaradi velike gostote zapisa je slika nepregledna, vidi pa se trenutek, ko je pesek začel udarjati po mikrofonih.
Da bi lahko zajeta signala uporabili za določitev hitrosti zrnc peska, smo v programskem okolju LabVIEW izdelali poseben algoritem. Pri tem smo uporabili standardna programska orodja, ki jih programsko okolje ponuja. Z algoritmom obdelamo signal v dveh korakih. V prvem koraku s frekvenčno analizo z uporabo hitre Fourierjevo preslikavo (HFP - FFT) [4] določimo diskretni frekvenčni spekter diskretnega (digitaliziranega) signala. Frekvenčni spekter pokaže, katere frekvence in s kakšno amplitudo se pojavljajo v signalu. Frekvenčni spekter signala prvega zaznavala j e prikazan na sliki 8. Opazimo, da obstajata dve frekvenčni območji večje intenzivnosti. Frekvenčno območje 3500 Hz do 7000 Hz je posledica lastnih nihanj sistema. Bolj zanimivo je območje med 0 in 1000 Hz, ki je prikazano na sliki 9. Ugotovimo, da se prva izrazita amplituda
sensors was used to cover the sensors when the turbine was started up and until steady-state operation was achieved. At that moment the screen was removed.
The raw, acquired signals are shown in Fig. 7. The diagram lacks clarity because of the high density of the recorded data, although the moment when the first particles hit the membrane of the sensor is clearly recognisable.
In order to use the signals for particle velocity determination a special algorithm was made in the LabVIEW programming language. Standard LabVIEW program routines were used. The signals were processed in two steps using the algorithm. In the first the discrete Fourier frequency spectrum from the digitised input signal was determined by the application of the fast Fourier transform algorithm [4]. The frequency spectrum shows significant frequency components. The frequency spectrum of the signal from the first sensor (S1) is shown in Fig. 8. There are two frequency domains of higher intensity. The frequency domain between 3500 and 7000 Hz corresponds to the natural frequencies of the coordinate tender system and sensors. More important is the domain between 0 and 1000 Hz, shown in Fig. 9. The first significant frequency is 50 Hz, and this corresponds to the rota-
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Sl. 7. Signala z zaznaval S1 in S2 posneta s frekvenco 40.000 Hz Fig. 7. Signals from sensors S1 and S2 recorded by 40,000 Hz
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Sl. 8. Frekvenčni spekter signala z zaznavala S1 Fig. 8. Frequency spectre of a signal from sensor S1
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5,00E-06 ,
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Sl. 9. Frekvenčni spekter v območju 0-1000Hz - signal z zaznavala S1 Fig. 9. Frequency spectre of 0-1000 Hz domain - signal from sensor S1
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1,1 1 --*-- ¦— lr*-4______________________ S1_filt
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Sl. 10. Primerjava osnovnih in filtriranih signalov z zaznaval S1 in S2 Fig. 10. Comparison of non-filtered and filtered signals from sensors S1 and S2
pojavi pri 50 Hz, kar ustreza vrtilni frekvenci turbine. Z enako ali višjo amplitudo se nato pojavijo tudi vse njene višje harmonske. Najizrazitejša amplituda se pojavi pri 400 Hz in ustreza frekvenci vzpodbude, ki je zmnožek vrtilne frekvence turbine in števila lopatic (50 Hz, 8 lopatic). Veliko število izrazitih nižjih in višjih harmonskih napoveduje, da je delovanje prototipne turbine še precej neubrano.
Frekvenčni analizi sledi naslednji korak, to je filtriranje signala. Namen filtriranja je, da iz osnovnega signala izluščimo tisti del, ki je posledica vzbujanja sistema z osnovno frekvenco 400 Hz, s katero lopatice turbine izmetavajo pesek. Uporabili smo ozkopasovni filter z območjem 350 do 450 Hz. Zaradi nadaljnje obdelave mora biti filtriran signal čim bolj gladek prehod v območje zavrnitve pa čim bolj strm. To smo dosegli z uporabo digitalnega Čebišovovega filtra II [5]. Ta ima ravno karakteristiko v frekvenčnem območju prepuščanja signala in izredno strm, odsekan prehod v območje zavrnitve. Na sliki 10 sta prikazana osnovna in filtrirana signala z zaznaval S1 in S2. V obeh primerih je rezultat filtriranja razmeroma gladka krivulja z dovolj izrazitimi ekstremi, ki nam jih je uspelo lokalizirati z uporabo algoritma za iskanje dolov in vrhov diskretnega signala. Ker lokalni doli filtriranega signala ustrezajo naglim spremembam osnovnega
tional frequency of the turbine. All higher harmonics are presented with the same or higher intensity. The most significant frequency is 400 Hz, and this corresponds to the excitation frequency that is the product of the number of turbine blades and turbine rotational frequency (8 blades, 50 Hz). The very large number of distinctive higher and lower harmonics show evidence of a poorly tuned turbine.
The next step was signal filtering. The intention was to isolate that part of the signal which corresponds to the excitation frequency (400 Hz), at which the blades eject the particles, from the raw signal. A bandpass filter was used with a bandpass between 350 Hz and 450 Hz. The Chebyshev II design digital filter was applied [5]. These filters are maximally flat in the passband and give a sharp transition between the passband and the stop band. The filtered signal is, therefore, smooth, which is necessary for subsequent signal processing. The filtered and non-filtered signals from sensors S1 an S2 are presented in Fig. 10. The filtered curves are smooth with distinctive extremes. The extremes are localised by the algorithm for the determination of the maximum and minimum values of the discrete signal. The local minimum of the filtered signal corresponds to the rapid change of the non-filtered signal (caused by the impact of the particles on the membrane) as can be seen in Fig. 10. The time delay between the impacts on the first and
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Sl. 11. Diagram izračunanih hitrosti v merilnih točkah 1 do 800 Fig. 11. Diagram of calculated particle velocities of measurements 1 to 800
signala (padec napetosti - slika 10), ki jih povzročajo udarci peska ob membrano zaznavala, smo časovni premik med udarci v prvo in drugo zaznavalo določili na podlagi časovnih razlik med lokalnimi doli obeh filtriranih signalov. Časovni premik smo nato uporabili v izrazu (2) in izračunali hitrost potovanja zrnc peska med zaznavali 1 in 2. Končni rezultat obdelave obeh signalov v območju dolžine 2 s prikazuje slika 11. Razvidno je, da je raztros hitrosti precej velik. Povprečna vrednost izračunane hitrosti je 82,6 m/s, standardna deviacija pa znaša 3,87 m/s. Zaradi velikega števila meritev je standardna merilna negotovost majhna in je 0,15 m/s. V primerjavi z izmerjeno hitrostjo z mehanskim merilnikom je hitrost večja za 11% (8,2 m/s). Enako velike razlike so se pokazale tudi pri meritvah drugih turbin, ki se od obravnavane razlikujejo po obliki in številu lopatic. Zato lahko sklepamo, da na nastanek teh razlik ne vpliva netočnost merilne metode, pač pa so posledica različne zasnove obeh merilnikov. Mehanski merilnik izmeri srednjo hitrost zrnc, medtem ko v primeru mikrofonskega merilnika izmerimo hitrost najhitrejših delcev, ki prvi udarijo ob membrano in povzročijo njeno nihanje in s tem naglo spremembo signala. Poskusili smo najti tudi pojasnilo za velik raztros izmerjenih hitrosti. Nanj gotovo vpliva različna zrnavost peska (0 = 0,05 do 1 mm), kar povzroča različno intenzivne udarce ob membrano zaznavala s tem pa bolj in manj izrazite spremembe osnovnega signala in premik lokalnih dolov v filtriranem signalu. Dodatno prinese k večjemu raztrosu tudi neubrano delovanje turbine, ki je posledica nenatančnosti pri izdelavi (litju) ležišč lopatic v rotorju, montaže lopatic in predvsem različne obrabe lopatic. Da bi določili stopnjo neubranosti, smo analizirali filtriran signal s prvega zaznavala. Ugotavljali smo dolžino premika med zaporednimi udarci ob membrano. Vsaki od osmih lopatic smo poiskali zakasnitev glede na prejšnjo lopatico. To smo naredili tako, da smo matriko, ki je vsebovala
second sensor can, therefore, be determined from the local phase between both filtered signals. This time interval is then used in equation (2), and the particle velocity between the sensors S1 and S2 is calculated. The final result of processing a 2-seconds period of the raw signals is shown in Fig. 11. Velocity scatter is relatively high. The average particle velocity is 82.6 m/ s and the RMS (Root Mean Square) is 3.87 m/s. The standard measurement uncertainty is 0.15 m/s. This value is low due to the large number of measurements. The average measured velocity is 8.2 m/s, i.e. 11 % higher than the particle velocity measured by the mechanical device. It is interesting that similar differences were observed for other turbines as well, although they have different designs and a different number and curvature of blades. It may be assumed, therefore, that this difference is not the result of a measurement error, but is conditioned by the different measurement approaches applied in both velocity measurement methods. The average particle velocity is measured by a mechanical device, whilst the maximum particle speed is measured by an electronic system that reacts to the first hit of the membranes caused by the fastest particles. An explanation for the high velocity scatter was also found. The composition of shotblasting particles is not uniform. The particle diameter may vary from 0.05 mm to 1 mm. This causes a very high variation in the particle’s impact force intensity and, therefore, causes less distinctive magnitude changes in the measured signal and some phase shift of the filtered signal. The next reason for the large scatter of the measured velocities is a poorly tuned, prototype turbine. This is primarily due to different degrees of turbine-blade wear and non-optimised matching of the cast blades with the grooves of the cast rotor ring. An attempt was made, therefore, to establish the degree of non-tunableness in the shotblasting turbine’s operation. The filtered signal from the first sensor was analysed. The time delays between the successive membrane hits were determined first. The results were then split into eight rows, each of them corresponds
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-•-D18 ^D21
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zavrtitev rotorja - rotor revolution Fig. 12. Time delays between the particle ejection from individual blades
zakasnitve med udarci v prvi mikrofon, razdelili na 8 stolpcev. Rezultat prikazuje slika 12. Opazimo, da je povprečna dolžina premika 2,53 ms, kar dobro ustreza frekvenci vzbujanja 400 Hz. Močno izstopata krivulji A65 in A76. To pomeni da, izmetavanje peska z lopatice 6 prehiteva, zaradi česar se skrajša korak med udarci z lopatic 5 in 6 ter podaljša korak med udarci z lopatic 6 in 7. Podobno, vendar precej manj, odstopata tudi krivulji A12 in A32, kar kaže na prehitevanje izmetavanja peska z lopatice 2. Ugotovitve, ki smo jih dobili z analizo signala, so se potrdile pri pregledu rotorja. Izkazalo se je, da je obraba dveh simetrično postavljenih lopatic precej večja od obrabe preostalih lopatic. Povečana obraba je bila posledica različnih materialov, iz katerih so bile preskusno izdelane lopatice v prototipni turbini.
4 SKLEPI
V prispevku sta prikazana dva merilnika, ki omogočata merjenje hitrosti delcev v stroju za peskanje. Merilnik z rotirajočima ploščama je robusten in omogoča merjenje hitrosti delcev le v majhnem številu točk. Njegova največja prednost je, da potrebuje malo dodatne opreme (usmernik, merilnik frekvence), pomanjkljivost pa, da zaradi svojih izmer in teže ni primeren za merjenje hitrostnega profila. Elektronski merilnik je primernejši za merjenje hitrosti v večjem številu točk. Njegova prednost so predvsem manjše izmere in teža zaznavalskega dela, kar omogoča premikanje po merilni ravnini pa tudi po kotu. Dodatna prednost elektronskega merilnika je, da lahko analiziramo delovanje turbine in v določeni meri sklepamo tudi o obrabi lopatic in drugih nepravilnostih. Povzamemo torej lahko, da je elektronsko zaznavalo perspektivnejše, verjetno pa se bo zaradi primerjave rezultatov še nekaj časa uporabljalo hkrati z mehanskim merilnikom.
Obstaja kar nekaj možnosti za izboljšave, predvsem pri elektronskem merilniku. Z optimiranjem
to one of eight turbine blades and represents the time delay between the particles’ ejection of the observed blade and its forerunner blade. The results are shown in Figure 12. The average delay is 2.53 ms and corresponds well to the excitation frequency of 400 Hz. Curves D65 and D76 deviate the most from the average level. Blade 6 ejects the particles too soon, and the delay between the ejection of blades 5 and 6 is reduced. On the other hand, the delay between the ejection of blades 6 and 7 is increased. The similar, but less distinctive, deviation can be observed for curves D21 and D32, and shows evidence of blade 2 malfunctioning. These findings were confirmed by an inspection of the rotor. It was found that two of the symmetrically mounted blades were worn out much more than the other six blades, which are made from a different, more abrasion-resistant, alloy.
4 CONCLUSIONS
This paper presents two different methods for particle velocity measurement in a shotblasting machine. The mechanical device with two rotating disks is robust and can only be applied for velocity measurements in a few locations within a testing chamber. Its advantage over the electronic system is that almost no extra equipment is necessary except a rotational speed controller and an electronic tachometer. Its main disadvantage is the lack of mobility. It cannot, therefore, be used for the velocity profile measurements of a particle stream. Measurements using the electronic system can be performed in almost any location within a testing chamber. Its sensors are small and light and can be easily positioned in a measuring plane and inclined at a specified angle. Its additional advantage over the mechanical measuring device is that it can be used to analyse the operation of the turbine and make conclusions about turbine-blade wear and other similar malfunctions.
There are still some possible modifications for the electronic system. The construction of the micro-
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konstrukcije mikrofonskega zaznavala (debelina jeklene membrane, vpetje membrane, dušenje motenj) lahko povečamo občutljivost elektronskega merilnika in se v idealnem primeru izognemo uporabi filtrov, kar bi zelo poenostavilo algoritem za štetje udarcev in izračun hitrosti, s tem pa tudi povečalo točnost rezultatov.
phone sensor can be optimised (membrane thickness, membrane support, disturbances damping) for higher sensitivity of the sensor. In an ideal case, signal filtering would be unnecessary. This would simplify the algorithm for the signal analysing and particle velocity calculation, and the accuracy of the algorithm would also increase.
5 LITERATURA 5 REFERENCES
[1] Marinkovič, J. (1950) Unutrašnja balistika, Izdavačko poduzeče Narodne republike Srbije, Beograd.
[2] Jamakawa, M., S. Isshiki, J. Lee, K. Nishida (2001) 3-D PIV analysis of structural behavior of D.I: gasoline
spray, SAE Paper 2001-01-3669. [3] Merjenje zvoka in oktavna analiza, National Instruments, NI DAN 1999, Ljubljana. [4] Cooley, J.W., J.W. Tukey (1965) An algorithm for the machine calculation of complex Fourier series,
Mathematics of Computation, vol. 19, 297-301. [5] LabVIEW Measurements manual, National Instruments, 2000.
Naslova avtorjev: dr. Aleš Hribernik mag. Gorazd Bombek Univerza v Mariboru Fakulteta za strojništvo Smetanova 17 2000 Maribor ales.hribernik@uni-mb.si gorazd.bombek@uni-mb.si Author’s Addresses: Dr. Aleš Hribernik Mag. Gorazd Bombek Faculty of Mechanical Eng. University of Maribor Smetanova 17 2000 Maribor, Slovenia ales.hribernik@uni-mb.si gorazd.bombek@uni-mb.si
Ivan Markočič Gostol-TST Tolmin Ivan Markočič Gostol-TST Tolmin
Prejeto: Received:
20.12.2002
Sprejeto: Accepted:
31.1.2003
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© Strojni{ki vestnik 48(2002)12,719 ISSN 0039-2480 Strokovna literatura
© Journal of Mechanical Engineering 48(2002)12,719
ISSN 0039-2480 Professional Literature
Strokovna literatura
Professional Literature Ocene knjig
Klaus-Jurgen Matthes/Erhardt Richter: Schweisstechnik
Zal.: Fachbuchverlag Leipzig, Carl Hanser Verlag,
Miinchen, Wien.
Obseg: format 17 x 24 cm, 471 strani, 508 slik, 130
preglednic, 164 lit.pod.
V predloženi knjigi sta urednika v dvanajstih poglavjih podala izčrpen pregled današnjega stanja v varilni tehniki kovinskih materialov. Posamezna poglavja je obdelalo pet različnih avtorjev, ki pa pri delu niso bili najbolje usklajeni. To se kaže tako pri različnih ravneh strokovnih razlag kakor tudi pri neuravnoteženem obsegu, ki ga obsegajo nekatera poglavja. Kratek pregled vsebine posameznih poglavij.
1. Osnove z razmejitvijo izdelovalnih tehnik po standardih DIN 8580 in načelno shemo razmejitve varilnih postopkov po standardu DIN 1910.
2. Varivost z osnovnimi kriteriji, ki odločajo o primernosti osnovnega in dodajnega materiala ter vrste zvarnih spojev na izbiro varilnega postopka in varilnih parametrov.
3. Obločni postopki varjenja so podani izredno temeljito in obenem pregledno. Ob opisu fizikalnih osnov obločnega varjenja, pregledu osnovnih vrst izvorov varilnega toka in ob predstavitvi najpogostejših načinov obločnega varjenja je zbrana še obilica tehnoloških napotkov.
4. Plamenska tehnika je predstavljena s podrobno razlago značilnosti kisik-acetilenskega plamena, opisom osnovne, dodatne in pomožne opreme, ki jo uporabljajo pri tej tehniki varjenja in z obilico koristnih tehnoloških napotkov, ki jih zlepa ne najdemo v novejših učbenikih podobne vsebine.
5. Postopki uporovnega varjenja so glede na pomen in obseg uporabe teh postopkov v današnji industrijski praksi ter v primerjavi z obsegom predstavljenih vsebin prejšnjih poglavij, predstavljeni izredno skopo.
6. Indukcijsko varjenje je predstavljeno na vsega treh straneh, podobno skopo kakor pred tem postopki uporovnega varjenja.
7. Varjenje z velikimi gostotami energije, predstavljata varjenje z elektronskim snopom in lasersko varjenje. Za oba varilna postopka so podane fizikalne osnovne tehnične značilnosti in prikazani so posnetki neposredne uporabe naprav in izvedenih zvarov.
8. Postopki varjenja s kinetično energijo so zastopani z opisom različnih načinov varjenja: s trenjem, ultrazvočno varjenje, varjenje v hladnem s stiskanjem in eksplozijsko varjenje.
9. Varjenje s stiskanjem prek togega telesa, ki je ali ni dodatno ogrevano. Gre za opis različnih načinov mikro varjenja, ki jih uporabljajo predvsem v elektroindustriji.
10. Alumoterično varjenje je avtor razmejil na talilno varjenje in varjenje s stiskanjem, oboje pa podkrepil s skromno razlago in preprostimi skicami mogočih izvedb.
11. Difuzijsko varjenje je predstavljeno s preprosto skico, ponazorjenim varilnim krogom, tremi različnimi izvedbami spojev in z razlago osnovnih značilnosti tega postopka varjenja.
12. Navarjanje je predstavljeno s podrobno razlago tehnoloških posebnosti, ki so odvisne od namena, obsega in števila izvedb, vrste osnovnega in dodajnega materiala ter izbranega varilnega postopka.
V trinajstem poglavju je podan pregled uporabljene literature, ločeno za posamezne tehnike varjenja, na koncu je še stvarno kazalo, ki je označeno kot štirinajsto poglavje.
Knjigo bodo s pridom uporabljali tehnologi in konstrukterji v industrijski praksi ter študentje univerzitetnega in visokega strokovnega študija, če jim le nemščina ni povsem tuj jezik.
I. Polajnar
© Strojni{ki vestnik 48(2002)12,720 © Journal of Mechanical Engineering 48(2002)12,720
ISSN 0039-2480 ISSN 0039-2480
Osebne vesti Personal Events
Osebne vesti
Personal Events
Doktorati, magisteriji, diplome
DOKTORATI
Na Fakulteti za strojništvo Univerze v Ljubljani so z uspehom zagovarjali svoje doktorske disertacije, in sicer:
dne 16. decembra 2002: mag. Franc Cimerman, z naslovom: “Odzivnost dotikalnih temperaturnih zaznaval”;
dne 17. decembra 2002: mag. Jure Knez, z naslovom: “Zanesljivost delovanja strojev na osnovi meritev vibracij”;
dne 20. decembra 2002: mag. Primož Rus, z naslovom: “Paralelno reševanje fizikalno sklopljenih sistemov” in mag. Ted Prodan, z naslovom: “Vpliv hidrostatičnega tlaka in temperature na strižni modul časovno odvisnih materialov”.
Na Fakulteti za strojništvo Univerze v Mariboru je z uspehom zagovarjal svojo doktorsko disertacijo, in sicer:
dne 16. decembra 2002: mag. Gorazd Lojen, z naslovom: “Sinteza tiksotropnih mešanic”.
S tem so navedeni kandidati dosegli akademsko stopnjo doktorja tehničnih znanosti.
MAGISTERIJI
Na Fakulteti za strojništvo Univerze v Ljubljani so z uspehom zagovarjali svoja magistrska dela, in sicer:
dne 3. decembra 2002: Matej Supej, z naslovom: “Povezava prostega volumna z mehanskimi lastnostmi viskoelastičnih materialov”;
dne 23. decembra 2002: Damjan Klobčar, z naslovom: “Matematično modeliranje navarjanja” in Robert Ivančič, z naslovom: “Reparaturno varjenje orodij”.
Na Fakulteti za strojništvo Univerze v Mariboru je dne 6 decembra 2002 Simon Kovačič z
uspehom zagovarjal svoje magistrsko delo z naslovom: “Model razširitve CNC krmilnika za posebne postopke frezanja zahtevnih oblik”.
S tem so navedeni kandidati dosegli akademsko stopnjo magistra tehničnih znanosti.
DIPLOMIRALI SO
Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv univerzitetni diplomirani inženir strojništva:
dne 19. decembra 2002: Boštjan JARC, Aljoša ROZMAN, Jože JENKOLE, Dalibor PERUŠKO.
Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv univerzitetni diplomirani inženir strojništva:
dne 19. decembra 2002: Tomaž ČERNEC, Bojan GOJKOŠEK, Davorin RIŽNAR.
*
Na Fakulteti za strojništvo Univerze v Ljubljani so pridobili naziv diplomirani inženir strojništva:
dne 12. decembra 2002: Rober AVSENIK, Peter BOSCAROL, Simon KRANJC, Damjan MIHALIČ, Robert ŠPERNJAK, Ivan UMEK;
dne 13. decembra 2002: Andrej BENSA, Tomaž KALAN, Borut KUTNJAK, Viktor ROJŠEK, Anže ROVANŠEK, Miha TRČEK;
dne 16. decembra 2002: Sebastjan BOGATAJ, Domen KLEMENC, Tadej NADRIH, Anton PERC.
Na Fakulteti za strojništvo Univerze v Mariboru so pridobili naziv diplomirani inženir strojništva:
dne 19. decembra 2002: Boštjan CAFUTA, Leon GAJSER, Matej KUHAR, Milan KUŠER, Janko PLAJNŠEK, Marjan REJC, Simon VRŠIČ.
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© Strojni{ki vestnik 48(2002)12,721 © Journal of Mechanical Engineering 48(2002)12,721
Mese~nik Published monthly
ISSN 0039-2480 ISSN 0039-2480
Recenzenti letnika 2002
Reviewers of 2002 Volume
Tuji recenzenti / International reviewers: prof. dr. Matija Fajdiga
prof. dr. Francesco d’Auria, University of Pisa doc. dr. Karl Gotlih
prof. dr. Ivan Catton, University of California dr. Jože Jurkovič
prof. dr. Adolf Frank, TU Graz doc. dr. Zoran Kariž
prof. dr. Bernard Frankovič, University of Rijeka doc. dr. Breda Kegl
prof. dr. Glenn R. Heppler, University of Waterloo prof. dr. Janez Kopač
prof. dr. Roman Klasinc, TU Graz dr. Mitja Kožuh
prof. dr. Heinz-Bernd Matthias, TU Wien prof. dr. Jurij Krope
prof. dr. Herbert Schulz, TU Darmstadt prof. dr. Jure Marn
prof. dr. Dragica Noe
prof. dr. Paul White, University of Southampton prof. dr. Peter Novak
prof. dr. Luiz C. Wrobel, Brunel University prof. dr. Andrej Polajnar
prof. dr. Felix Ziegler, TU Berlin prof. dr. Ivan Prebil doc. dr. Andrej Predin
Domači recenzenti / National reviewers:
prof dr. Andro Alujevič dr. Zlatko Rek
prof. dr. Ivan Anžel prof. dr. Marko Starbek
prof. dr. Jože Balič mag. Jože Stropnik
doc. dr. Anton Bergant prof. dr. Leopold Škerget
prof. dr. Ivan Bratko prof. dr. Matija Tuma
prof. dr. Franci Čuš prof. dr. Iztok Žun
stran 721
bcšd02
© Strojni{ki vestnik 48(2002)12,722-724 © Journal of Mechanical Engineering 48(2002)12,722-724
Mese~nik Published monthly
ISSN 0039-2480 ISSN 0039-2480
Vsebina 2002
Contents 2002
Uvodnik Editorial
Kopač, J.: Moderni obdelovalni stroji - nujne Kopač, J.: Modern Machine Tools - A Urgent
investicije za Slovenijo 120 Investments for Slovenia
Alujevič, A.: Petnajst let Kuhljevih dnevov (1987- Alujevič, A.: Fifteen years of “Kuhelj’s Memorial
2002) 570 Days” (1987-2002)
Alujevič, A.: Josef Ressel - 175 let patenta ladijskega Alujevič, A.: Josef Ressel - 175 Years of the Ship-vijaka 644 Screw Patent
Razprave
Ačko, B.: Eksperimentalno ovrednotenje analitičnega preračuna deformacije krogle zaradi tipalne sile pri kalibraciji 2
Ačko, B., Šostar, A.: Prilagoditev modela vrednotenja merilne negotovosti pri kalibraciji dolžinskih etalonov na temelju avtomatizacije meritve 9
Kovač, I., Klein, A: Naprava in postopek za kalibriranje členkastih koordinatnih merilnih naprav 17
Sokovič, M., Kosec, L., Dobrzahski, L.A.: Raziskave difuzije skozi stik PVD prekrito orodje iz kermeta/ obdelovanec 33
Mesaric, M., Kosel, F.: Optimiranje dinamične uravnoteženosti krilca 41
Požarnik, M., Škerget, L.: Robno območna integralska metoda za numerično modeliranje lebdečih slojev 58
Župerl, U., Čuš, F: Model za analizo in optimiranje vpenjalnih priprav 73
Omrčen, D., Nemec, B.: Meritev gibanja kolena z industrijskim robotom - avtomatska kompenzacija gravitacije prijemala 87
Starbek, M., Grum, J., Kušar, J.: Realni pretočni časi operacij in uspešnost sistema NKP 98
Kopač, J.: Rezalne sile in njihov vpliv na gospodarnost obdelave 121
Ekinovič, S., Dolinšek, S., Kopač, J., Godec, M.: Prehod iz običajne v zelo hitro obdelavo in analiza oblikovanja odrezkov 133
Jurkovič, J., Šali, S.: Identifikacija dinamičnih lastnosti
sestavljenih nosilnih strojnih delov 143
Pavletič, D., Sokovič, M.: Šest sigm: zahtevna pobuda kakovosti 158
Pogačnik, M., Štefančič, J., Kopač, J.: Informacijska podpora pospeševanju postopkov (projektov) 169
Dolinšek, S., Bavec, C, Mihelič, A., Prodan, I.: Upravljanje tehnologije - ključ konkurenčnosti 178
Kato, Y, Cerkvenik, B., Minakami, A., Yoshizawa, Y.: Primernost reakcije magnezijevega oksida z vodo za uporabo v kemičnem hralnilniku toplote 194
Martinis, V., Matijaševič, B., Tukovič, Ž.: Računsko reševanje inverznega problema oblikovanja nadzvočne šobe 210
Papers
Ačko, B.: The Experimental Validation of an Analytical
Calculation of Sphere’s Deformation that Results
from Probing Force During Calibration Ačko, B., Šostar, A.: Modification of the Model for
Measurement Evaluation in a Gauge-Block
Calibration Based on Measurement Automation Kovač, I., Klein, A.: Apparatus and a Procedure to
Calibrate Coordinate Measuring Arms Sokovič, M., Kosec, L., Dobrzahski, LA.: An
Investigation of the Diffusion Across a PVD-
Coated Cerment Tool/Workpiece Interface Mesaric, M., Kosel, F: Optimization of the Dynamic
Balance of an Aileron Požarnik, M., Škerget, L.: Boundary Domain Integral
Method for Numerical Modeling of Fluidized
Beds Župerl, U., Čuš, F: A Model for Analysing and
Optimazing Fixtures Omrčen, D., Nemec, B.: Measuring knee movement
using an industrial robot - gravity compensation
for the automatic gripper Starbek, M., Grum, J., Kušar, J.: Realistic Lead Times
of Operations and Efficiency of the PPC System Kopač, J.: Cutting Forces and Their Influence on the
Economics of Machining Ekinovič, S., Dolinšek S., Kopač, J., Godec, M.: The
Transition from the Conventional to the High-Speed
Cutting Region and a Chip-Formation Analysis Jurkovič, J., Šali, S.: The Dynamic Properties of
Composite Supporting Parts for Machines Pavletič, D., Sokovič, M.: Six Sigma: A Complex
Quality Inititative Pogačnik, M., Štefančič, J., Kopač, J.: The
Acceleration of Processes (Projects) Through
Information Technology Support Dolinšek, S., Bavec, C, Mihelič, A., Prodan, I.: The
Management of Technology - The Key to
Competitiveness Kato, Y, Cerkvenik, B., Minakami, A., Yoshizawa, Y:
The Suitability of Magnesium Oxide and Water
for Use in Chemical Heat Storage Martinis, V., Matijaševič, B., Tukovič, Ž.: A Numerical
Solution to the Inverse Problem of Supersonic-Nozzle Design
I SsrirKitafcflMlSilrSD
| ^BSfiTTMlliC | stran 722
Vsebina 2002 - Contents 2002
Uran, S., Šafarič, R., Winther, T.: Naprava s Uran, S., Šafarič, R, Winther, T.: A Pneumatic Active-
pnevmatično aktivno površino: Razvoj Surface Device: Prototype Design and
prototipa in Ljapunovova analiza stabilnosti 218 Lyapunov Stability Analysis
Papotnik, A: Projektna naloga kot strategija vzgojno- Papotnik, A.: The Project Task as a Strategy for
izobraževalnega dela 234 Education-Training Activities
Marn, J., Žunič, Z., Ramak, M., Ternik, P: Tokovne Marn, J., Žunič, Z., Ramak, M., Ternik, P.: Flow
razmere v avtomobilski lakirnici: numerična in Conditions in an Automotive Spray-Paint
eksperimentalna analiza 244 Chamber: Numerical and Experimental Analyses
Glodež, S., Flašker, J., Jelaska, D., Kramberger, J.: Glodež, S., Flašker, J., Jelaska, D., Kramberger, J.: A
Računski model za preračun upogibne trdnosti Computational Model for Calculating the
zobnikov 257 Bending-Load Capacity of Gears
Uran, S., Šafarič, R.: Naprava s pnevmatično aktivno Uran, S., Šafarič, R.: A Pneumatic Active-Surface
površino: Tehnike krmiljenja lege togih objektov Device: Open- and Closed-loop Control-
s povratno zvezo in brez nje 267 Positioning Techniques for Rigid Objects
Nastran, M., Krušič, V., Boltežar, M.: Prispevek k Nastran, M., Krušič, V., Boltežar, M.: A Contribution
obvladovanju neuravnoteženosti krempljastih to the Unbalance Control of Claw Poles for
polov alternatorjev 283 Automotive Alternators
Jakšič, N, Boltežar, M.: Prispevek k parameterski Jakšič, N, Boltežar, M.: Parameter Identification
identifikaciji dinamičnih sistemov z eno for Single-Degree-of-Freedom Dynamic
prostostno stopnjo 302 Systems
Glavnik, A., Hriberšek, M., Škerget, L.: Prostorska Glavnik, A., Hriberšek, M., Škerget, L.: A Three-
porazdelitev faktorja ugodja na temelju Dimensional Factor-of-Comfort Distribution Based
numeričnega modeliranja naravne konvekcije 318 on Numerical Modeling of Natural Convection
Uran, S., Šafarič, R.: Naprava s pnevmatično aktivno Uran, S., Šafarič, R.: A Pneumatic Active-Surface
površino: Tehnike krmiljenja gibljivih predmetov 332 Device: Control Techniques for Flexible Objects
Drole, M., Sekavčnik, M., Tuma, M.: Energijska bilanca Drole, M., Sekavčnik, M., Tuma, M.: The Energy
občine Tolmin 355 Balance of the Tolmin Municipality
Širok, B., Dular, M., Novak, M., Hočevar, M., Stoffel, Širok, B., Dular, M., Novak, M., Hočevar, M., Stoffel,
B., Ludwig, G., Bachert, B.: Vpliv kavitacijskih B., Ludwig, G., Bachert, B.: The Influence of
struktur na erozijo na simetričnem krilu v Cavitation Structures on the Erosion of a
kavitacijskem predoru 368 Symmetrical Hydrofoil in a Cavitation Tunnel
Pandža, K., Buchmeister, B., Polajnar, A., Palčič, I.: Pandža, K., Buchmeister, B., Polajnar, A., Palčič, I.:
Proizvodna strategija, podprta s teorijo An Operations Strategy Supported with
proizvodnih virov: študij primera v podjetju Resource-Based Theory: A Case Study at the
Primat 379 Primat Company
Leš, M., Aberšek, B.: Gradnja uporabniškega Leš, M., Aberšek, B.: Building a User Interface
vmesnika na temelju programskega paketa Based on MATLAB for Control System
MATLAB za študij sistemov 395 Studies
Bešter, T: Rekonstrukcija odvaljnega bata zračne vzmeti 404 Bešter, T: Reconstruction of an Air-Spring Piston
Zupančič, J., Marn, J.: Sinteza orodij za analizo človeške Zupančič, J., Marn, J.: The Synthesis of Human-Error
napake po metodi analize spoznavne zanesljivosti Analysis Using the Cognitive Reliability and
in napak ter analize drevesa odpovedi 418 Error Analysis Method and Fault-Tree Analysis
Leskovar, M., Mavko, B.: Izviren kombiniran Leskovar, M., Mavko, B.: An Original Combined
večfazni model mešalne faze eksplozije Multiphase Model of the Steam-Explosion
pare 438 Premixing Phase
Leskovar, M., Mavko, B.: Simuliranje izotermnega Leskovar, M., Mavko, B.: Simulation of the
QUEOS preskusa mešalne faze eksplozije pare Isothermal QUEOS Steam-Explosion Premixing
Q08 449 Experiment Q08
Stojkovič, V., Mikulič, D.: Vpliv računskega polmera Stojkovič, V, Mikulič, D.: The Impact of a Fixed
obračanja goseničnega vozila na potrebno moč Kinematic Turning Radius of a Tracked Vehicle
motorja pri obračanju 459 on the Engine Power required in a Turn
Rek, Z., Perpar, M., Žun, I.: Analiza prenosa toplote v Rek, Z., Perpar, M., Žun, I.: A Heat-Transfer Analysis
postopku sintranja feritov 472 of the Ferrite Sintering Process
Horvat, A., Catton, I.: Analiza vezanega prenosa Horvat, A., Catton, I.: An Analysis of Conjugate Heat
toplote v hladilniku elektronskega čipa 482 Transfer in the Heat Sink of an Electronic Chip
Mrkič, M., Culafič, Z.: Nekateri vidiki terenskih Mrkič, M., Culafič, Z.: Some Aspects of the Research preskusov Peltonovih turbin v HE Carried out on the Power Generation Units at
“Peručica” 491 the Peručica Hydroelectric Power Plant
| gfin=i(gurMini5nLn 0212 stran 723 I^BSSIfTMlGC
Strojni{ki vestnik - Journal of Mechanical Engineering
Brožek, M.: Struženje navarov z orodji podjetja
Walter 501
Pristovnik, A., Krope, J., Črepinšek-Lipuš, L.: Simuliranje nastajanja gruč dispergiranih delcev pod vplivom zunanjega magnetnega polja 520
Mrkič, M.: Analiza parametrov reverzibilne Črpalne
francisove turbine 528
Stritih, U, Studen, S., Brenčič, M., Lapanje, A.: Analiza shranjevanja toplote v vodonosnikih - možnost uporabe v Sloveniji 541
Neslušan, M., Czan, A., Župerl, U.: Analiza porazdelitve toplote pri brušenju titanove zlitine VT 9 in njena povezava do zaostalih napetosti 557
Mejak, G.: Vogalna singularnost torzije kompozitne palice 571
Kovač, M., Simonovski, I., Cizelj, L.: Vpliv zrnate strukture na elasto-plastilčni odziv polikristalnega skupka 580
Kegl, M.: Optimiranje oblike konstrukcij: tristranični projektni element 591
Vohar, B., Gotlih, K., Flašker, J.: Optimiranje pogonskega mehanizma stiskalnice za globoki vlek 601
Bremec, B., Kosel, F.: Določanje kritične obremenitve krožnih kolobarjev v elasto-plastičnem območju 613
Slavic, J., Boltežar, M.: Izboljšana razpoznava dušenja z uporabo zvezne valčne transformacije 621
Škerget, L., Požarnik, M.: Metoda robnih elementov za dinamiko viskoelastične Maxwellove tekočine 645
Bombač, A., Žun, I.: Metode prepoznave poplavnega
stanja pri aeraciji v posodi s turbinskim mešalom 663
Bajcar, T., Širok, B., Trenc, F., Jošt, D.: Analiza
kinematike toka v rotirajočem difuzorju 677
Delič, M., Marn, J., Žunič, Z.: Siskov model toka nenewtonskih tekočin z metodo končnih prostornin 687
Bergant, R., Tiselj, I.: Vpliv Prandtlovega števila na
turbulentni prenos toplote ob ravni steni 696
Hribernik, A., Bombek, G., Markočič, I.: Mehanski in elektronski merilnik hitrosti delcev v stroju za peskanje 707
Poročila
Strokovna literatura
Osebne vesti
Navodila avtorjem
Brožek, M.: The Turning of Overlays Using Tools Produced by the Company Walter
Pristovnik, A., Krope, J., Črepinšek-Lipuš, L.: A Simulation of the Cluster-Formation Process in a Dispersion of Fine Particles Under the Influence of an External Magnetic Field
Mrkič, M.: An Analysis of the Parameters of Reversible Francis-Type Pump Turbines
Stritih, U, Studen, S., Brenčič, M., Lapanje, A.: The Analysis of Thermal Energy Storage in Aquifers - the Possibility of Application in Slovenia
Neslušan, M., Czan, A, Župerl, U.: Analysis of the Heat Distribution when Grinding of a VT 9 Titanium Alloy and its Relation to Residual Stresses
Mejak, G.: The Corner Singularity of Composite Bars in Torsion
Kovač, M., Simonovski, I., Cizelj, L.: The Effect of Grain Structure on the Elastic-Plastic Response of a Polycrystalline Aggregate
Kegl, M.: Structural Shape Optimization: A Trilateral Design Element
Vohar, B., Gotlih, K, Flašker, J.: Optimization of Link-Drive Mechanism for Deep Drawing Mechanical Press
Bremec, B., Kosel, F: Determination of the Buckling Loads of Circular Annular Plates in the Elastic-Plastic Region
Slavic, J., Boltežar, M.: Enhanced identification of damping using continuous wavelet transform
Škerget, L., Požarnik, M.: The Boundary-Element Method for the Dynamics of a Viscoelastic Maxwell Fluid
Bombač, A., Žun, I.: Flooding-Recognition Methods in a Turbine-Stirred Vessel
Bajcar, T, Širok, B., Trenc, F, Jošt, D.: An Analysis of the Flow Kinematics in a Rotating Diffuser
Delič, M., Marn, J., Žunič, Z.: The Sisko Model For Non-Newtonian Fluid Flow Using The Finite-Volume Method
Bergant, R, Tiselj, I.: The Influence of Prandtl Number on Near-Wall Turbulent Heat Transfer
Hribernik A., Bombek G, Markočič, I.: A Mechanical and Electronic Measurement System for Particle Velocity Measurements in a Shotblasting Machine
Reports
Professional Literature
Personal Events
Instructions for Authors
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© Strojni{ki vestnik 48(2002)12,725-726 ISSN 0039-2480 Navodila avtorjem
Navodila avtorjem
Instructions for Authors
Članki morajo vsebovati:
- naslov, povzetek, besedilo članka in podnaslove slik v slovenskem in angleškem jeziku,
- dvojezične preglednice in slike (diagrami, risbe ali fotografije),
- seznam literature in
- podatke o avtorjih.
Strojniški vestnik izhaja od leta 1992 v dveh jezikih, tj. v slovenščini in angleščini, zato je obvezen prevod v angleščino. Obe besedili morata biti strokovno in jezikovno med seboj usklajeni. Članki naj bodo kratki in naj obsegajo približno 8 tipkanih strani. Izjemoma so strokovni članki, na željo avtorja, lahko tudi samo v slovenščini, vsebovati pa morajo angleški povzetek.
Vsebina članka
Članek naj bo napisan v naslednji obliki:
- Naslov, ki primerno opisuje vsebino članka.
- Povzetek, ki naj bo skrajšana oblika članka in naj ne presega 250 besed. Povzetek mora vsebovati osnove, jedro in cilje raziskave, uporabljeno metodologijo dela,povzetek rezulatov in osnovne sklepe.
- Uvod, v katerem naj bo pregled novejšega stanja in zadostne informacije za razumevanje ter pregled rezultatov dela, predstavljenih v članku.
- Teorija.
- Eksperimentalni del, ki naj vsebuje podatke o postavitvi preskusa in metode, uporabljene pri pridobitvi rezultatov.
- Rezultati, ki naj bodo jasno prikazani, po potrebi v obliki slik in preglednic.
- Razprava, v kateri naj bodo prikazane povezave in posplošitve, uporabljene za pridobitev rezultatov. Prikazana naj bo tudi pomembnost rezultatov in primerjava s poprej objavljenimi deli. (Zaradi narave posameznih raziskav so lahko rezultati in razprava, za jasnost in preprostejše bralčevo razumevanje, združeni v eno poglavje.)
- Sklepi, v katerih naj bo prikazan en ali več sklepov, ki izhajajo iz rezultatov in razprave.
- Literatura, ki mora biti v besedilu oštevilčena zaporedno in označena z oglatimi oklepaji [1] ter na koncu članka zbrana v seznamu literature. Vse opombe naj bodo označene z uporabo dvignjene številke1.
Oblika članka
Besedilo naj bo pisano na listih formata A4, z dvojnim presledkom med vrstami in s 3 cm širokim robom, da je dovolj prostora za popravke lektorjev. Najbolje je, da pripravite besedilo v urejevalnilku Microsoft Word. Hkrati dostavite odtis članka na papirju, vključno z vsemi slikami in preglednicami ter identično kopijo v elektronski obliki. Prosimo, da ne uporabljate urejevalnika LaTeX, saj program, s katerim pripravljamo Strojniški vestnik, ne uporablja njegovega formata. V urejevalniku LaTeX oblikujte grafe, preglednice in enačbe in jih stiskajte na kakovostnem laserskem tiskalniku, da jih bomo lahko presneli.
Enačbe naj bodo v besedilu postavljene v ločene vrstice in na desnem robu označene s tekočo številko v okroglih oklepajih
Enote in okrajšave
V besedilu, preglednicah in slikah uporabljajte le standardne označbe in okraj Save SI. Simbole fizikalnih veličin v besedilu pišite poševno (kurzivno), (npr. v, T, n itn.). Simbole enot, ki sestojijo iz črk, pa pokončno (npr. ms1, K, min, mm itn.).
Vse okrajšave naj bodo, ko se prvič pojavijo, napisane v celoti v slovenskem jeziku, npr. časovno spremenljiva geometrija (ČSG).
© Journal of Mechanical Engineering 48(2002)12,725-726
ISSN 0039-2480 Instructions for Authors
Papers submitted for publication should comprise:
- Title, Abstract, Main Body of Text and Figure Captions in Slovene and English,
- Bilingual Tables and Figures (graphs, drawings or photographs),
- List of references and
- Information about the authors.
Since 1992, the Journal of Mechanical Engineering has been published bilingually, in Slovenian and English. The two texts must be compatible both in terms of technical content and language. Papers should be as short as possible and should on average comprise 8 typed pages. In exceptional cases, at the request of the authors, speciality papers may be written only in Slovene, but must include an English abstract.
The format of the paper
The paper should be written in the following format:
- A Title, which adequately describes the content of the paper.
- An Abstract, which should be viewed as a miniversion of the paper and should not exceed 250 words. The Abstract should state the principal objectives and the scope of the investigation, the methodology employed, summarize the results and state the principal conclusions.
- An Introduction, which should provide a review of recent literature and sufficient background information to allow the results of the paper to be understood and evaluated.
- A Theory
- An Experimental section, which should provide details of the experimental set-up and the methods used for obtaining the results.
- A Results section, which should clearly and concisely present the data using figures and tables where appropriate.
- A Discussion section, which should describe the relationships and generalisations shown by the results and discuss the significance of the results making comparisons with previously published work. (Because of the nature of some studies it may be appropriate to combine the Results and Discussion sections into a single section to improve the clarity and make it easier for the reader.)
- Conclusions, which should present one or more conclusions that have been drawn from the results and subsequent discussion.
- References, which must be numbered consecutively in the text using square brackets [1] and collected together in a reference list at the end of the paper. Any footnotes should be indicated by the use of a superscript1.
The layout of the text
Texts should be written in A4 format, with double spacing and margins of 3 cm to provide editors with space to write in their corrections. Microsoft Word for Windows is the preferred format for submission. One hard copy, including all figures, tables and illustrations and an identical electronic version of the manuscript must be submitted simultaneously.
Please do not use a LaTeX text editor, since this is not compatible with the publishing procedure of the Journal of Mechanical Engineering. Graphs, tables and equations in LaTeX may be supplied in good quality hard-copy format, so that they can be copied for inclusion in the Journal.
Equations should be on a separate line in the main body of the text and marked on the right-hand side of the page with numbers in round brackets.
Units and abbreviations
Only standard SI symbols and abbreviations should be used in the text, tables and figures. Symbols for physical quantities in the text should be written in Italics (e.g. v, T, n , etc.). Symbols for units that consist of letters should be in plain text (e.g. ms-1, K, min, mm, etc.).
All abbreviations should be spelt out in full on first appearance, e.g., variable time geometry (VTG).
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Strojni{ki vestnik - Journal of Mechanical Engineering
Slike
Slike morajo biti zaporedno oštevilčene in označene, v besedilu in podnaslovu, kot sl. 1, sl. 2 itn. Posnete naj bodo v kateremkoli od razširjenih formatov, npr. BMP, JPG, GIF. Za pripravo diagramov in risb priporočamo CDR format (CorelDraw), saj so slike v njem vektorske in jih lahko pri končni obdelavi preprosto povečujemo ali pomanjšujemo.
Pri označevanju osi v diagramih, kadar je le mogoče, uporabite označbe veličin (npr. t, v, m itn.), da ni potrebno dvojezično označevanje. V diagramih z več krivuljami, mora biti vsaka krivulja označena. Pomen oznake mora biti pojasnjen v podnapisu slike.
Vse označbe na slikah morajo biti dvojezične.
Za vse slike po fotografskih posnetkih je treba priložiti izvirne fotografije ali kakovostno narejen posnetek. V izjemnih primerih so lahko slike tudi barvne.
Preglednice
Preglednice morajo biti zaporedno oštevilčene in označene, v besedilu in podnaslovu, kot preglednica 1, preglednica 2 itn. V preglednicah ne uporabljajte izpisanih imen veličin, ampak samo ustrezne simbole, da se izognemo dvojezični podvojitvi imen. K fizikalnim veličinam, npr. t (pisano poševno), pripišite enote (pisano pokončno) v novo vrsto brez oklepajev.
Vsi podnaslovi preglednic morajo biti dvojezični.
Seznam literature
Vsa literatura mora biti navedena v seznamu na
koncu članka v prikazani obliki po vrsti za revije, zbornike in
knjige:
[1] Tarng, Y.S., Y.S. Wang (1994) A new adaptive controler for constant turning force. Int J Adv Manuf Technol 9(1994) London, pp. 211-216.
[2] Čuš, F., J. Balič (1996) Rationale Gestaltung der organisatorischen Ablaufe im Werkzeugwesen. Proceedings of International Conference on Computer Integration Manufacturing Zakopane, 14.-17. maj 1996.
[3] Oertli, PC. (1977) Praktische Wirtschaftskybernetik. Carl Hanser Verlag Minchen.
Podatki o avtorjih
Članku priložite tudi podatke o avtorjih: imena, nazive, popolne poštne naslove, številke telefona in faksa ter naslove elektronske pošte.
Sprejem člankov in avtorske pravice
Uredništvo Strojniškega vestnika si pridržuje pravico do odločanja o sprejemu članka za objavo, strokovno oceno recenzentov in morebitnem predlogu za krajšanje ali izpopolnitev ter terminološke in jezikovne korekture.
Avtor mora predložiti pisno izjavo, da je besedilo njegovo izvirno delo in ni bilo v dani obliki še nikjer objavljeno. Z objavo preidejo avtorske pravice na Strojniški vestnik. Pri morebitnih kasnejših objavah mora biti SV naveden kot vir.
Rokopisi člankov ostanejo v arhivu SV.
Vsa nadaljnja pojasnila daje:
Uredništvo
STROJNIŠKEGA VESTNIKA
p.p. 197/IV
1001 Ljubljana
Telefon: (01) 4771-757
Telefaks: (01) 2518-567
E-mail: strojniski.vestnik@fs.uni-lj.si
Figures
Figures must be cited in consecutive numerical order in the text and referred to in both the text and the caption as Fig. 1, Fig. 2, etc. Figures may be saved in any common format, e.g. BMP, GIF, JPG. However, the use of CDR format (CorelDraw) is recommended for graphs and line drawings, since vector images can be easily reduced or enlarged during final processing of the paper.
When labelling axes, physical quantities, e.g. t, v, m, etc. should be used whenever possible to minimise the need to label the axes in two languages. Multi-curve graphs should have individual curves marked with a symbol, the meaning of the symbol should be explained in the figure caption.
All figure captions must be bilingual.
Good quality black-and-white photographs or scanned images should be supplied for illustrations. In certain circumstances, colour figures may be considered.
Tables
Tables must be cited in consecutive numerical order in the text and referred to in both the text and the caption as Table 1, Table 2, etc. The use of names for quantities in tables should be avoided if possible: corresponding symbols are preferred to minimise the need to use both Slovenian and English names. In addition to the physical quantity, e.g. t (in Italics), units (normal text), should be added in new line without brackets.
All table captions must be bilingual.
The list of references
References should be collected at the end of the
paper in the following styles for journals, proceedings and
books, respectively:
[1] Tarng, Y.S., Y.S. Wang (1994) A new adaptive controler for constant turning force. Int J Adv Manuf Technol 9(1994) London, pp. 211-216.
[2] Čuš, F., J. Balič (1996) Rationale Gestaltung der organisatorischen Ablaufe im Werkzeugwesen. Proceedings of International Conference on Computer Integration Manufacturing Zakopane, 14.-17. maj 1996.
[3] Oertli, PC. (1977) Praktische Wirtschaftskybernetik. Carl Hanser Verlag Munchen.
Author information
The following information about the authors should be enclosed with the paper: names, complete postal addresses, telephone and fax numbers and E-mail addresses.
Acceptance of papers and copyright
The Editorial Committee of the Journal of Mechanical Engineering reserves the right to decide whether a paper is acceptable for publication, obtain professional reviews for submitted papers, and if necessary, require changes to the content, length or language.
Authors must also enclose a written statement that the paper is original unpublished work, and not under consideration for publication elsewhere. On publication, copyright for the paper shall pass to the Journal of Mechanical Engineering. The JME must be stated as a source in all later publications.
Papers will be kept in the archives of the JME.
You can obtain further information from:
Editorial Board of the
JOURNAL OF MECHANICAL ENGINEERING
P.O.Box 197/IV
1001 Ljubljana, Slovenia
Telephone: +386 (0)1 4771-757
Fax: +386 (0)1 2518-567
E-mail: strojniski.vestnik@fs.uni-lj.si
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