VSEBINA – CONTENTS IZVIRNI ZNANSTVENI ^LANKI – ORIGINAL SCIENTIFIC ARTICLES A mathematical model for the stationary process of rolling of tubes on a continuous mill Matemati~ni model procesa kontinuirnega valjanja cevi Yu. G. Gulyayev, Ye. I. Shyfrin, I. Mamuzi}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 The temperature dependence of the parameters of non-linear stress-strain relations for carbon-epoxy composites Temperaturna odvisnost parametrov nelinearne odvisnosti napetost-deformacija za kompozite ogljikovo vlakno-epoksi T. Kroupa, R. Zem~ík, J. Klepá~ek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 Numerical optimization of the method of cooling of a massive casting of ductile cast-iron Numeri~na optimizacija postopka hlajenja pri masivnem ulivanju duktilne `elezove litine F. Kavicka, B. Sekanina, J. Stetina, K. Stransky, V. Gontarev, J. Dobrovska . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 Electrical conductivity of sintered LSM ceramics Elektri~na prevodnost sintrane LSM-keramike M. Marin{ek . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Modelling the characteristics of an inverted magnetron using neural networks Modeliranje karakteristike invertnega magnetrona z nevronskimi sistemi I. Beli~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Carnian bauxites at Muljava in central Slovenia Karnijski boksiti na obmo~ju Muljave v osrednji Sloveniji S. Dozet, M. Godec. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 STROKOVNI ^LANKI – PROFESSIONAL ARTICLES An investigation of the economics of using welded layers for some parts of worm presses for the extraction of oil from sunflower seeds Raziskave uporabnosti navarjenih plasti za dele vija~nih stiskalnic za ekstrakcijo olja son~nic V. Maru{i}, M. Kljajin, S. Maru{i} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 DOGODKI – EVENTS Slovensko dru{tvo za materiale (SDM) popularizira {tudij in raziskave materialov Slovenian Society for Materials (SDM) encouraging youngs for study and research of materials M. Torkar: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 ISSN 1580-2949 UDK 669+666+678+53 MTAEC9, 43(2)61–109(2009) MATER. TEHNOL. LETNIK VOLUME 43 [TEV. NO. 2 STR. P. 61–109 LJUBLJANA SLOVENIJA MAR.–APR. 2009 YU. G. GULYAYEV ET AL.: A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS ... A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS OF ROLLING OF TUBES ON A CONTINUOUS MILL MATEMATI^NI MODEL PROCESA KONTINUIRNEGA VALJANJA CEVI Yu. G. Gulyayev1, Ye. I. Shyfrin2, Ilija Mamuzi}3 1National Metallurgical Academie of Ukraine, Dnipropetrovsk, Ukraine 2Tube MetallurgicCompany, Russia 3University of Zagreb, Faculty of Metallurgy Sisak, Croatia mamuzicsimet.hr Prejem rokopisa – received: 2008-09-23; sprejem za objavo – accepted for publication: 2008-10-23 A mathematical model has been developed for the calculation of process parameters in continuous lengthwise plugless tube rolling. Examples of concrete calculations of rolling parameters, their comparison with experimental data and the results obtained with the application of other calculation procedures are given. Key words: tubes, plugless rolling, mathematical model Razvit je bil matemati~ni model za izra~un parametrov procesa neprekinjenega valjanja cevi brez notranjega trna. Dani so konkretni primeri izra~unov parametrov valjanja, rezultati pa so primerjani z eksperimentalnimi podatki in z izra~uni po drugih postopkih. Klju~ne besede: cevi, valjanje brez trna, matemati~ni model 1 INTRODUCTION The prospects of enhancement of the production efficiency at numerous tube rolling units are closely linked with the possibility of a reliable prediction of the forming parameters at the final stage of plastic deformation in the plugless tube reducing or sizing processes. In this connection, the problem of develop- ment of a universal mathematical model applicable in studying the process of lengthwise plugless rolling in the tube rolling mills equipped with the roll drives of different types is of a high interest. 2 STATE OF THE ISSUE AND THE AIM OF INVESTIGATION The analysis of the relevant references shows that the problem of determination of kinematical, deformational and power-and-force parameters of the continuous plugless lengthwise tube rolling process was solved up to now by consecutive analysis of forming in each indivi- dual stand. Solutions based on integration of the defor- mational parameters in all N stands of the continuous mill into a common system of equations are proposed, also 1,2. In the development of mathematical models of the continuous rolling process, e.g. in 1,2 two assumptions were made. Firstly, the mean angle of the neutral section θni (Figure 1) is defined for the condition of coincidence of the roll and the mother tube speeds within the section of the deformation zone exit in the i-th stand, though it would be logical to choose some section between the entry and exit of the deformation zone. Secondly, for the determination of the effective roll diameter Dki, the approximate formula is used: Dki = Dui – Di cosθni (where Dui, Di – are the ideal roll diameter and the mean tube diameter after rolling in the i-th stand respectively) that introduces an error because in reality Dki = Dui – 2rθi (θni) cosθni (where Dui is the ideal roll dia- meter; rθi (θni) is the pass radius at θ = θni, see Figure 2). Materiali in tehnologije / Materials and technology 42 (2009) 2, 63–67 63 UDK 62-462:621.774.35:519.673 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 43(2)63(2009) Figure 1: Scheme for the determination of the value of the effective diameter Dk Slika 1: Shema za dolo~itev efektivnega premera Dk In accord with the model consisting of 2 N equations proposed1, the effective roll diameter Dki cannot be greater than the ideal roll diameter Dui and smaller than the roll diameter at the swell Dbi. This distorts the real picture of the rolling process kinematics, namely, when the rolls slip on the mother tube surface two conditions are met: Dki < Dbi or Dki > Dui. The model proposed 2 is free from this shortcoming but it is a system of 3N equations that when being solved at N > 16 is connected with considerable difficulties because of the great number of unknowns to be determined. This work is aimed at the verification and simplifi- cation of the mathematical models proposed 1,2 and to the assessment of the verification results on the basis of comparison of calculated and experimental data and it is, for this reason, of scientific and practical interest. 3 PROBLEM STATEMENT The following values have to be calculated: – angular roll rotation velocity nBi in each i-th mill stand (for the mill with individual roll drives); – angular velocities of rotation of the main (NΓ) and auxiliary (NB) motors (for the mill with differen- tial-group roll drives); – ideal roll diameters Dui (for the mill with group roll drives). These values ensure that tubes of required size (Dt·St, mm) are rolled from the mother tube of given size ((D0 × S0) mm) at a specified rolling speed V0/(m/s) in the first stand of the multiple-stand mill. Initial data for the calculation are as follows: – the total diameter and wall reduction (or just dia- meter reduction), i.e. initial mother tube dimensions D0 × S0 (or just D0) and final tube dimensions Dt × St; – the distribution of partial mother tube diameter reductions mi (%) among the mill stands of total number of N; – the value of external friction fi; – the mother tube rolling speed V0/(m/s) in the first mill stand (the problem can also be stated for V0 as the value to be determined); – the gear ratios ηΓi, ηΒi from the motors to the rolls in the lines of the main and auxiliary drives (for the mills with differential-group roll drives); – the absence of backward pull in the first mill stand (Z31 = 0) and of front pull in the last mill stand (ZnN = 0); – the number of rolls Nb forming passes in the mill stands. 4 PHYSICAL MODEL OF THE PROCESS No mother tube forming occurs in interstand spaces and the wall thickness Sj at the exit from the stand of ordinal number j = i – 1 is equal to the wall thickness S0i at the entry to the stand of ordinal number i. The deformation resistance Kfj of the mother tube material at the exit from the stand of ordinal number j is equal to the deformation resistance Kf0i of the mother tube material at the entry to the stand of ordinal number i. It follows that the coefficient of front plastic pull Znj for the stand of ordinal number j is equal to coefficient of backward plastic push Z3i for the stand of ordinal number i. The area Fki of the contact surface of the mother tube with one roll in the stand of ordinal number i is equal to the area of a rectangle with sides L D D D i i j i i i i = ⋅ ⋅ ⋅ −β ε β ( ) sin u 2 (1) with B Di i j= ⋅β (2) where Dj and Di are the mean mother tube diameters at the entry to and at the exit from the deformation zone in the stand of ordinal number i; ε i im/ % = 100 ; β i iN = π b The area Fi+ of the zone of forward creep at the surface of contact between one roll and the mother tube in the deformation zone of the stand of ordinal number i is defined as the surface of a rectangle with sides Li + = Li (3) and Li + = θni Dj (4) where θni is the neutral section angle characterizing the position of the neutral line differentiating the zone of forward creep and the zone of backward creep on the surface of contact between the mother tube and the roll in the deformation zone (Figure 1). In a real process, the magnitude of angle θni is a function of the angle α characterizing the position of a concrete diametrical section of the deformation zone relative to the diametrical section of the mother tube exit from the reduction zone. In accord with the assumption4, the magnitude of angle θni is assumed to be equal to some quantity averaged over the contact surface length. It will be regarded that θni is the value of the neutral angle in the "neutral" diametrical section of the deforma- tion zone where the extension is equal to the mean extension in the i-th stand. The axial velocity VMn of metal and axial component of the roll surface velocity VBn in the "neutral" diametrical section are given with VMni = V0 µ Σi cp (5) V n A D i i i i Bn B t cp = π θ ξ (6) where µ Σi S D S S D S S D Sj j j i i i cp = −− + − 2 0 0 0( ) ( ) ( ) is the total elongation from the mill entry to the "neutral" diametrical section of the i-th stand; YU. G. GULYAYEV ET AL.: A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS ... 64 Materiali in tehnologije / Materials and technology 43 (2009) 2, 63–67 A it cp is the mean value of the guiding cosine of the contact friction stresses 1; Dθi = Dui – 2rθi cos θ is the varying of the roll pass dia- meter across the pass perimeter (Figure 2); ( )r O A R e R e ri i i i i i θ θ θ = = − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ k o o o o o 1 1 2 2cos cos ⎥ ⎥ is the varying across the pass perimeter value of the pass diameter; R O A h i i i i i o o k k = = + − − ( sin ( sin λ λ ψ λ ψ) 2 1 2 2 1 is the pass generatrix radius; e O A h i i i i o k o k = = − − ( ) ( sin λ λ ψ) 2 1 2 1 is the pass generatrix eccen- tricity; λ ki i i b h = is the pass ovality; ψ = −( )N N b b 2 2 is the pass shape index; bi, hi are the pass width and the pass height correspond- ingly; ξ = 6·104 is coefficient of quantity dimension reduction (s·mm· r min ·m) The angle θni is defined as root of the transcendental equation VMni – VBni = 0 (7) Taking in account that in a physical sense 0 ≤ θni ≤ βi, the condition for the determination of the neutral angle assumes the following form (in symbols of Math- CAD programming language) θ β β β n arc cos if if if i i i i i i i Q Q Q Q = ≤ ≤ < > cos cos 1 0 1 i (8) The quantity Qi in (8) is defined as the root of equation 2 0 0 0 0S V D S S D S S D Sj j j i i i ( ) ( ) ( ) − − + − − πn Ai iB t cp ξ · · ( )D Q R e R Q e Q Ri i i i i i i i i u o o o o o − − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ − − ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎧ ⎨ ⎪ 2 1 1 2 2 ⎩⎪ ⎫ ⎬ ⎪ ⎭⎪ = 0 (9) Note that as distinct from the conditions used1,2, the condition (8) reflects the relation of the neutral angle θni value with the roll design parameters (R0i, e0i, λki, bi, hi). Taking into consideration relationships (5) and (6), the effective roll diameter Dki can be defined by the following equation: [ ] D S V D S S D S S D S n A i j j j i i i i i k B t cp = − − + − 2 0 0 0 0( ) ( ) ( ) ξ π (10) For Dbi < Dki < Dui, on the contact surface of each roll appears a forward creep zone with the area equal of Fi+ = θni DjLi in accord with (3), (4) and the backward creep zone with the area equal to Fi– = (βi – θni) DjLi. For Dbi > Dki, the backward creep zone extends over the entire contact surface area and the "forward roll slippage" takes place on the metal surface. If Dui < Dki, the forward creep zone extends over the entire contact surface area and "backward roll slippage" takes place on the metal surface. The magnitude of neutral angle θni must meet the condition of force equilibrium of the metal volume in the geometrical deformation zone of the i-th stand 1 that can be expressed as: θ βn if if if i i i i i i X X X X = ⋅ ≤ ≤ > < 0 1 0 0 1 1 (11) where X F F A f A Z S D S Z S D i i i i i i i i i i j j j= = ⋅ − + − − −+ k p cp t cp n n1 2 1 ( ) ( ( ) ( )[ ] S f A L S D n S D D Z j i i i j j i i j i ) t cp t cp2 1+ − ⎧ ⎨ ⎪⎪ ⎩ ⎪ ⎪ ⎫ ⎬ ⎪⎪ ⎭ ⎪ ⎪ is the coefficient of forward creep in the i-th mill stand calculated for the equilibrium of forces in the volume of the metal in the geometrical deformation zone of the i-th stand; A it cp is the mean, over the contact surface, value of the guiding cosine for normal contact stresses; fi is coefficient of external friction; nti = 1 + 0,36 fi is coefficient accounting for the effect of the contact friction stresses upon normal contact stresses3; Zπ is coefficient of forward plastic pull; ( )Z Z Z i i j cp n n 3 3 = + 2 is the mean value of the plastic pull coefficient in the i-th stand. Materiali in tehnologije / Materials and technology 43 (2009) 2, 63–67 65 YU. G. GULYAYEV ET AL.: A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS ... Figure 2: Scheme for the determination of the value of the variable across the pass perimeter of the roll diameter Dθ Slika 2: Shema za dolo~itev spremembe premera valjev Dθ na obodu vtika 5 MATHEMATICAL MODEL Equate right parts of equations (8) and (11) and use the equation of relation between the change of the mean wall thickness and force conditions of the mother tube deformation in each i-th mill stand 4,5 to obtain the mathematical model of the continuous mother tube rolling process in N stands of the mill as a system of 2N equations: θ β β β n arc cos if if if i i i i i i i Q Q Q Q = ≤ ≤ < > cos cos 1 0 1 i = = ⋅ ≤ ≤ > < β i i i i i X X X X if if if 0 1 0 0 1 1 (12) S S Z T T Z T Ti j i i i i i i i − + ⋅ − + − − − − 1 2 1 1 2 1 2 ϕ ( ) ( ) ( ) ( ) ( ) ( ) cp cp ⎧ ⎨ ⎩ * + ⋅ − + − − − − ⎡ ⎣ ⎢ ⎢ 1 2 2 1 1 2 1 2 ϕ i i i i i i i Z T T Z T T ( ) ( ) ( ) ( ) ( ) ( ) cp cp ⎤ ⎦ ⎥ ⎥ ⎫ ⎬ ⎪ ⎭⎪ 2 (13) where ϕ i i i j j D S D S = − − ln ; T S D S Di j j i i K = + ⎛ ⎝ ⎜⎜ ⎞ ⎠ ⎟⎟ ; ( ) ( )Z Z Zi j icp n n= + 1 2 ; K = 1.57 for Nb = 2; K = 1.20 for Nb = 3; i = 1, 2, … , N–1, N Distinct from the known solution2, the mathematical model includes 2N and not 3N equations that simplifies the search of solution and makes it possible to analyze the rolling process in stretch-reducing mills with N ≤ 25 stands. Depending on the type of the mill drive, the problem of determination of the rolling parameters with the use of the system of equations (12)-(13) can be formulated in different ways. For the mills with individual drives, it is necessary to determine 2N values of nBi (where i = 1, 2, … , N) and Si (where i = 0, 1, 2, … , N–1) for the specified values of St, V0 and Zni (where i = 1, 2, … , N–1). For the mills with differential-group drives, it is necessary to find 2(N–1) values of the quantities Si and Zni (where i = 1, 2, … , N–1) and the values of NΓ and NB for the specified values of S0, St, V0. For the mills with individual drives, it is necessary to determine N–1 value 66 Materiali in tehnologije / Materials and technology 43 (2009) 2, 63–67 YU. G. GULYAYEV ET AL.: A MATHEMATICAL MODEL FOR THE STATIONARY PROCESS ... Table 1: Parameters of rolling a (57 × 11.6) mm tube from a (117 × 14.8) mm mother tube Tabela 1: Parametri valjanja cevi (57 × 11.6) mm iz cevi (117 × 14.8) mm i Di/mm mi/% λki Si/mm Zni θni/° Pi/kN Mi/(kN·m) E B B B B B 1 115.25 1.50 1.037 14.77 14.86 14.85 0.327 0.361 53.7 56.8 93 92 -14.5 -16.6 2 112.59 2.30 1.024 14.82 14.82 14.81 0.568 0.606 51.3 53.2 74 69 -10.0 -10.6 3 109.67 2.60 1.032 15.55 14.68 14.64 0.721 0.756 46.9 48.2 49 44 -5.3 -5.2 4 106.49 2.90 1.028 14.35 14.47 14.40 0.729 0.766 19.9 19.1 37 32 3.0 2.7 5 102.97 3.30 1.038 14.07 14.22 14.13 0.730 0.761 17.7 14.0 38 32 3.8 3.9 6 99.47 3.40 1.034 13.75 13.97 13.86 0.729 0.755 17.00 14.4 37 33 3.8 3.9 7 96.09 3.40 1.036 13.50 13.73 13.60 0.724 0.742 16.6 13.5 37 33 3.8 4.2 8 92.82 3.40 1.035 13.28 13.49 13.35 0.715 0.743 16.3 17.2 37 33 3.9 3.4 9 89.57 3.50 1.037 13.00 13.26 13.10 0.705 0.721 16.7 14.1 37 34 4.0 4.2 10 86.44 3.50 1.036 12.70 13.03 12.87 0.693 0.711 17.2 15.4 38 35 3.9 4.0 11 83.41 3.50 1.037 12.49 12.82 12.65 0.679 0.693 17.5 16.1 38 36 3.9 4.0 12 80.49 3.50 1.036 12.41 12.62 12.44 0.665 0.676 18.2 17.3 39 37 3.8 3.9 13 77.68 3.50 1.036 12.31 12.42 12.25 0.650 0.659 18.8 18.2 40 38 3.7 3.7 14 74.96 3.50 1.036 12.18 12.24 12.06 0.635 0.642 19.4 18.9 41 39 3.7 3.7 15 72.33 3.50 1.036 11.95 12.07 11.89 0.618 0.623 19.9 19.4 42 41 3.6 3.6 16 69.80 3.50 1.036 11.78 11.91 11.74 0.595 0.597 19.7 19.2 43 42 3.8 3.8 17 67.36 3.50 1.036 11.69 11.77 11.59 0.568 0.564 20.1 19.6 45 44 3.8 3.9 18 65.00 3.50 1.036 11.58 11.64 11.47 0.532 0.518 20.1 19.2 47 47 4.0 4.4 19 62.73 3.50 1.036 11.53 11.53 11.39 0.459 0.414 18.2 16.1 52 54 5.3 6.3 20 60.53 3.50 1.036 11.40 11.49 11.41 0.267 0.078 12.6 6.5 63 72 9.2 13.8 21 59.02 2.50 1.020 11.50 11.53 11.44 0.036 -0.140 14.3 17.4 71 87 9.4 9.3 22 57.96 1.80 1.018 11.58 11.58 11.58 0.001 -0.105 25.7 29.0 74 85 3.0 1.3 23 57.60 0.61 1.000 11.60 16.00 11.60 0 0 28.4 34.3 51 54 1.0 -1.6 NOTES: A = calculation by the procedure proposed in 1 B = calculation by the procedure proposed in this work E = experimental data                                                                       !   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THE TEMPERATURE DEPENDENCE OF THE PARAMETERS OF NON-LINEAR STRESS-STRAIN RELATIONS FOR CARBON-EPOXY COMPOSITES TEMPERATURNA ODVISNOST PARAMETROV NELINEARNE ODVISNOSTI NAPETOST-DEFORMACIJA ZA KOMPOZITE OGLJIKOVO VLAKNO-EPOKSI Tomá{ Kroupa, Robert Zem~ík, Jan Klepá~ek University of West Bohemia in Pilsen, Department of Mechanics, Univerzitní 22, 306 14, Plze, Czech Republic kroupakme.zcu.cz Prejem rokopisa – received: 2008-09-19; sprejem za objavo – accepted for publication: 2008-11-26 This work focuses on the identification of the parameters of stress-strain relations for a unidirectional, continuous-fiber carbon-epoxy composite under tensile loading at various temperatures. Simple tensile tests of thin strips with various fiber orientations were performed. The identification of the parameters for the chosen non-linear stress-strain relations is obtained at each temperature for which the experiment is performed and the strength is determined. The failure analysis for the determination of the first failure with the use of Puck’s action-plane concept is performed, and the tensile and shear strength are investigated. The identification process with the use of a combination of the mathematical optimization method and a finite-element analysis is described with the necessary details. The temperature dependence of the parameters is also investigated. Key words: composite, non-linear, carbon, epoxy, tensile, FEA, temperature Cilj dela je bil identifikacija parametrov odvisnosti napetost-deformacija za enosmerne neprekinjene kompozite ogljikovo vlakno-epoksi pri natezni obremenitvi in pri razli~ni temperaturi. Izvr{eni so bili natezni preizkusi tankih trakov z razli~no orientacijo vlaken. Za vsako temperaturo, pri kateri je bil preizkus izvr{en, so bili identificirani parametri nelinearne odvisnosti napetost- deformacija, dolo~ene pa so bile tudi trdnosti. Analiza preloma s ciljem, da se dolo~i za~etek preloma, je bila izvr{ena z uporabo Puckovega koncepta o ploskvi delovanja in raziskani sta bili natezna in stri`na trdnost. Opisan je proces matemati~ne identifikacije in analize po metodi kon~nih elementov s potrebnimi detajli. Raziskana je bila tudi temperaturna odvisnost parametrov. Klju~ne besede: kompozit, nelinearnost, ogljik, epoksi, natezen, FEA, temperatura 1 INTRODUCTION The aim was to investigate the temperature depen- dence of the parameters of non-linear stress-strain relations for a unidirectional carbon-epoxy composite with the use of a finite-element (FE) analysis. The elasticity parameters and strengths were found from the comparison of the FE analysis and experimental results for simple tension tests of thin carbon-epoxy strips with dimensions 150 mm × 14.5 mm × 1.08 mm. Specimens with three fiber directions were used. The fiber directions formed angles of 0°, 45° and 90° with the direction of the loading force. The tensile tests were performed at 25 °C, 50 °C, 75 °C and 100 °C. Two types of non-linear stress-strain relations are presented. Their capabilities for prediction of the behavior of the composite material loaded with simple tension at various temperatures were investigated. The strengths of the material were investigated with the use of Puck’s failure criterion 5. 2 NON-LINEAR STRESS-STRAIN RELATIONS Several types of stress-strain relations exist. A linear stress-strain relation is the simplest way to describe the behavior of a composite material 2. Unfortunately, it cannot describe the non-linear slope of the curves obtained from the tensile tests. The next type is the non-linear stress-strain relation proposed in 1, which takes into account a non-linear relationship between the shear stress and the strain only, and is generalized in 3. This relation can be written in a material axes coordination system L (longitudinal – fiber direction), T (transverse direction) for the state of plane stress in the form ε ε γ σ σ L T LT L⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ S S S S S 11 12 21 22 66 0 0 0 0 T LT L T LT 2τ σ σ σ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ + ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ S S S 111 222 66666 0 0 0 0 0 0 ⎥ ⎥ ⎥ ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ σ σ τ L T LT (1) where S E11 1= L (2) S E22 1= T (3) S E12 = − ν LT L (4) Materiali in tehnologije / Materials and technology 43 (2009) 2, 69–72 69 UDK 678:546.26:551.52 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 43(2)69(2009) S E21 = − ν TL T (5) S G66 1= LT (6) and ν νTL LT T L = E E (7) The strains are expressed as polynomial functions of the stresses and the relation contains 7 independent parameters that can be sorted as linear parameters (Young’s moduli) EL, ET, GLT, Poisson’s ratio νLT and the non-linear parameters S111, S222 and S6666. The orders of the polynomials are predetermined and suitable for the tests performed at normal temperatures. The whole relation (R1) has to be inverted for proper use in the FE software. The Newton iteration method is used in the work to find the roots of the equation f S 0( , , )σ σ τL T LT == −σ ε (8) where S is the stress-strain matrix, σ is the stress vector and ε is the strain vector. The stress-strain relation, which takes into conside- ration the non-linear behavior of the composite material and where the stresses are explicit functions of strains, can be expressed in the form σ σ τ ν ν ν ν ν ν ν L T LT LT TL LT LT TL LT ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ = − − − E E E L L L 1 1 1 0 LT TL LT TL 0 0 LT L ν ν ν ε ET G 1 0− ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⎥ ε γ T LT ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ (9) where Ramberg-Osgood-based equations provide expressions for the tangent lamina stiffnesses E E n n L L L L L L = − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + 0 1 1 1 ε ε0 (10) E E E n n T T T T T T T = + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + 0 0 1 1 1 ε σ 0 (11) G G G n n LT LT LT LT LT LT LT = − ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ + 0 0 1 1 1 γ τ 0 (12) and the relation between the Poisson’s ratios is consi- dered as ν νTL LT T L = E E 0 0 (13) The relation (9) includes 10 independent parameters. These parameters are the initial moduli EL 0 , ET 0 , G LT 0 , the asymptotic stress and strain levels σ 0T , τ 0 LT and ε 0 L , the shape parameters nL, nT, nLT and the Poisson’s ratio νLT. Figure 1 shows the influence of the shape parameter nL for a material with EL 0 = 100 GPa and ε0L = 1. Figure 2 shows the influence of the shape parameter nLT for the material with G LT 0 = 5 GPa and τ 0LT = 50 MPa. 3 IDENTIFICATION OF THE ELASTICITY PARAMETERS The identification of the elasticity parameters is the first step in the identification process. The method used for the identification of the elasticity parameters, which combines the material axis and the off-axis tensile tests, the FE analysis and the mathematical optimization method will be described in the following paragraph. Non-linear stress-strain relations have to be imple- mented into the FE software first. The number of elements in the FE models for the calculation of the force-displacement diagrams has to be reduced as much as possible. The reduction of the number of elements is necessary for the reduction of the time consumption during the optimization cycles. The next step is to propose the residual function that represents the difference between the numerical and the experimental T. KROUPA ET AL.: THE TEMPERATURE DEPENDENCE OF THE PARAMETERS ... 70 Materiali in tehnologije / Materials and technology 43 (2009) 2, 69–72 Figure 2: The influence of the shape parameter nLT Slika 2: Vpliv parametrov oblike nLT Figure 1: The influence of the shape parameter nL Slika 1: Vpliv parametrov oblike nL results at the given optimization step, and which is minimized. The function is proposed as [ ] [ ] r F p F p F p p p = −∑ ∑ exp exp ( , ) ( , ) max ( , ) θ θ θθ num 2 (14) where θ is the fiber angle, r(θ) is the residual of the force-displacement diagram for the fiber angle θ, which had values 0°, 15°, 30°, 45°, 60°, 75° and 90°; p is the displacement where the residual is calculated; Fnum(θ, p) is the calculated force corresponding to the displace- ment p of the strip with the fiber angle θ; Fexp(θ, p) is the experimentally measured force corresponding to the displacement p of the strip with the fiber angle θ and [ ]max ( , )exp p F pθ is the maximum force measured as a response of the strip with a fiber angle θ, used as a normalization coefficient. It should be noted that the Poisson’s ratio νLT has to be identified with a special, separate test and, therefore, it will not be investigated in this paper, and its value is taken as νLT = 0.28. The tests, probably a biaxial test able to precisely describe the potential strain dependence of the Poisson’s νLT ratio, and a further analysis of the influence of the temperature on the Poisson’s ratio νLT will be performed in the near future. The flow chart of the whole automated process of the identification of the elasticity parameters is shown in Figure 3. 4 IDENTIFICATION OF THE STRENGTHS Once the elasticity parameters are identified, the identification of the strengths can be performed. The failure criterion used in the strengths identification process is Puck’s action-plane concept. For more about the criterion, see 4–6. Once the tensile test is performed, only the tensile XT, YT and the shear SL strengths can be identified. The identification of the strengths is per- formed with the use of a minimized function, which is proposed as the sum of the errors between the average of the measured ultimate forces and the calculated ultimate force for each fiber angle [ ]r F Fu= −∑ exp ( ) ( )θ θ θ num u (15) where θ is the fiber angle, F uexp ( )θ is the average of the measured ultimate forces for the given fiber angle θ and F unum ( )θ is the calculated ultimate force for the given fiber angle θ. The flow chart of the identification process of the strengths is shown in Figure 4. Let us briefly describe the flow chart in Figure 4. At the start of the identification process the initial dis- placement increment duinit is prescribed. The FE analysis is the next step. Subsequently, the failure analysis is performed and the failure indices (FIs) are calculated. These indices represent the level of the load with respect to the strength of the material. If the FI reaches one, the material is damaged. Once the FIs are calculated, the decision process that results in the displacement increment that is applied in the next step is performed. The constant kc is used to improve the convergence speed. Once the matrix-failure index FIM or the fiber-failure index FIF reaches 1 (with the toleration εFI) the identification process ends. 5 RESULTS The identified elasticity parameters using the rela- tionship (1) are shown in Table 1. The decreasing tendency of the linear parameters and the increasing tendency of the non-linear part of the parameters with increasing temperature are evident. The exceptions are the shear modulus, which increases with the temperature 100 °C, and the Young’s modulus for the fiber direction EL, which remains constant. T. KROUPA ET AL.: THE TEMPERATURE DEPENDENCE OF THE PARAMETERS ... Materiali in tehnologije / Materials and technology 43 (2009) 2, 69–72 71 Figure 3: Identification process of the elasticity parameters Slika 3: Proces identifikacije parametrov elasti~nosti Figure 4: Identification process of the strengths Slika 4: Proces identifikacije trdnosti Table 1: Elasticity parameters used in (1) Tabela 1: Parametri elasti~nosti uporabljeni v (1) Parameter 25 °C 50 °C 75 °C 100 °C EL/GPa 111.58 111.69 111.47 111.59 ET/GPa 8.48 7.30 6.50 4.77 GLT/GPa 3.98 4.10 3.83 8.69 (S111·10–22)/Pa–2 -4.66 -4.67 -4.65 -4.49 (S222·10–19)/Pa–2 3.46 3.78 5.79 68.47 (S666·10–26)/Pa–3 12.75 21.97 58.93 442.27 Table 2 shows the identified parameters that were obtained by using relation (9). Table 2: Elasticity parameters used in (9) Tabela 2: Parametri elasti~nosti uporabljeni v (9) Parameter 25 °C 50 °C 75 °C 100 °C EL 0 /GPa 106.26 107.71 106.19 107.51 ET 0 /GPa 8.25 7.15 6.38 4.35 GLT 0 /GPa 3.44 3.27 3.00 2.34 εL 0 1.44 0.72 1.49 0.78 σT 0 /MPa 176.52 147.00 129.52 33.64 τLT 0 /MPa 56.66 38.80 31.48 17.28 nL 0.56 0.65 0.56 0.64 nT 1.38 1.61 1.35 1.33 nLT 1.44 1.83 1.53 1.89 The initial Young’s modulus in the fiber direction is constant, while the rest of the initial moduli show a decreasing tendency. The asymptotic strain level does not show any dependence on the temperature. The asymptotic stress levels decrease with the increasing temperature. The values of the shape parameters are oscillating. Figure 5 shows the temperature dependence of the residual (14). The increase of the residuals of the relation (1) with the increase of the temperature is obvious and the better capability of relation (9) is evident. Table 3: Strength identified with the use of (1) Tabela 3: Trdnosti dolo~ene z uporabo (1) Strength 25 °C 50 °C 75 °C 100 °C XT/MPa 1937.2 1937.0 1937.1 1937.0 YT/MPa 36.7 38.2 32.8 16.9 SL/MPa 58.4 49.3 39.2 73.6 Table 4: Strengths identified with the use of (9) Tabela 4: Trdnosti izra~unane z uporabo (9) Strength 25 °C 50 °C 75 °C 100 °C X T/MPa 1937.0 1937.0 1937.1 1937.0 Y T/MPa 36.4 38.2 32.8 16.9 SL/MPa 61.0 49.3 39.2 73.6 The temperature independence of the strength XT is visible from Tables 3 and 4. Also, the decreasing tendency of the strength Y T and ST is evident, except for the high value of the shear strength for 100 °C. The differences between the sums of the errors between the ultimate forces are negligible. 6 CONCLUSION The capabilities of two types of non-linear stress-strain relations were investigated. The better suitability of the stress-strain relation based on the Ramberg-Osgood equations was proven. The influence of the viscoelasticity was neglected in the work. The Puck’s failure criterion was used for the failure analysis and the prediction of the strengths. It provides acceptable results for temperatures up to 75 °C. Acknowledgements The work has been supported by the research project of the Ministry of Education of the Czech Republic no. MSM 4977751303 and the project of the Grant Agency of the Czech Republic GACR no. 101/07/P059. 7 REFERENCES 1 H. T. Hahn, W. S. Tsai, Nonlinear elastic behavior of unidirectional composite laminae, Journal of Composite Materials, 7 (1973), 102–118 2 J. M. Berthelot, Composite materials, Springer, New York 2004, p. 639 3 F. Hassani, M. Shokrieh, L. Lessard, A fully non-linear 3-D con- stitutive relationship for the stress analysis of a pin loaded composite laminate, Composites Science and Technology, vol. 62 (2002) 3, 429–439 4 T. Kroupa, V. La{, Off-axis behavior of unidirectional FRP com- posite, Mater. Technol. 42 (2007) 3, 125–129 5 A. Puck, H. Schürmann, Failure analysis of FRP laminates by means of physically based phenomenological models. Composites Science and Technology, vol. 58 (1998) 7, 1045–1067 6 V. La{, R. Zem~ík, Progressive Damage of Unidirectional Com- posite Panels, Journal of Composite Materials 42 (2008), 25–44 T. KROUPA ET AL.: THE TEMPERATURE DEPENDENCE OF THE PARAMETERS ... 72 Materiali in tehnologije / Materials and technology 43 (2009) 2, 69–72 Figure 5: Sum of the residuals calculated with the use of (14) (× – (1), o – (9)) Slika 5: Vsota rezidualov, izra~unana s uporabo (14) (× – (1), o – (9)) F. KAVICKA ET AL.: NUMERICAL OPTIMIZATION OF THE METHOD OF COOLING ... NUMERICAL OPTIMIZATION OF THE METHOD OF COOLING OF A MASSIVE CASTING OF DUCTILE CAST-IRON NUMERI^NA OPTIMIZACIJA POSTOPKA HLAJENJA PRI MASIVNEM ULIVANJU DUKTILNE @ELEZOVE LITINE Frantisek Kavicka1, Bohumil Sekanina1, Josef Stetina1, Karel Stransky1, Vasilij Gontarev2, Jana Dobrovska3 1Brno University of Technology, Technicka 2, 616 69 Brno, Czech Republic 2University of Ljubljana, A{ker~eva 12, 1000 Ljubljana, Slovenia 3Technical University of Ostrava, Tr.17. listopadu, Ostrava, Czech Republic kavickafme.vutbr.cz Prejem rokopisa – received: 2009-01-19; sprejem za objavo – accepted for publication: 2009-01-27 An original application of ANSYS simulating the forming of the temperature field of a massive casting from ductile cast-iron during the application various methods of its cooling using steel chills. The numerical model managed to optimize more than one method of cooling but, in addition to that, provided serious results for the successive model of structural and chemical heterogeneity, and so it also contributes to influencing the as solidified microstructure. The file containing the acquired results from both models, as well as from their organic unification, brings new and, simultaneously, remarkable findings of causal relationships between the structural and chemical heterogeneity and the local solidification time in any point of the casting. Therefore the determined relations enable the prediction of the local density of the spheroids of graphite in dependence on the local solidification time. The calculated temperature field of a two-ton (500 × 500 × 1000) mm casting of ductile cast-iron with various methods of cooling has successfully been compared with temperatures obtained experimentally. This has created a tool for the optimization of the microstructure with an even distribution of the spheroids of graphite in such a way so as to minimize the occurrence of degenerated shapes of graphite, which happens to be one of the conditions for achieving good mechanical properties of castings of ductile cast-iron. Key-words: ductile cast-iron, massive casting, cooling, temperature field, numerical model Uporaba originalnega programa ANSYS omogo~a simulacijo tvorbe temperaturnega polja v masivnem lito`eleznem ulitku pri razli~nih metodah hlajenja z jeklenimi hladilnimi telesi. Z numeri~nim modelom se lahko optimira ve~ metod hlajenja, dodatno pa je mogo~ izra~un pomembnih rezultatov za modeliranje strukturne in kemijske heterogenosti, kar tudi prispeva k vplivu na lito mikrostrukturo. Dobljeni rezultati obeh modelov omogo~ajo pomembne ugotovitve o odnosih med strukturno in kemijsko heterogenostjo ter lokalnim ~asom strjevanja v poljubni to~ki ulitka in napoved lokalne gostote krogli~astega grafita v odvisnosti od lokalnega ~asa strjevanja. Izra~unano temperaturno polje dvotonskega ulitka duktilne `elezove litine z izmerami (500 × 500 × 1000) mm je bilo v skladu z eksperimentalnimi rezultati pri razli~nih metodah hlajenja. Na podlagi teh ugotovitev je bilo izdelano orodje za optimizacijo mikrostrukture z enakomerno porazdelitvijo krogli~astega grafita, s ~imer se zmanj{a pojav degeneriranih oblik grafita in se ustvarijo pogoji za dosego dobrih mehanskih lastnosti ulitkov duktilne `elezove litine. Klju~ne besede: temperaturno polje, strjevanje, ohlajanje, numeri~na optimizacija, strukturna in kemijska heterogenost 1 INTRODUCTION Solidification and cooling of a classically (i.e. gravitationally) cast casting and the simultaneous heating of the mould is, from the viewpoint of thermokinetics, a case of three-dimensional (3D) transient heat and mass transfer in a system consisting of the casting, mould and ambient. If the mass transfer is neglected and – from the three basic types of heat transfer – conduction is considered as the decisive, then the problem can be reduced to the solving of the Fourier equation. Here, the used 3D model of the temperature field of the system is based on the numerical finite-element method. The simulation of the release of the latent heats of phase or microstructural changes is carried out by introducing the thermodynamic enthalpy function. It enables the evaluation of the temperature field within the actual casting, chills and mould at any point in time within the process of solidification and cooling using contour lines (i.e. so-called iso-lines and iso-zones) or tempera- ture-time curves for any nodal point of the system. It is possible to use all sophisticated sub-programs of ANSYS, such as automatic mesh generation, pre-pro- cessing and post-processing. Having one’s own numerical model available makes it possible to integrate one’s own idea of the optimal course of solidification and cooling of the object under investigation in accordance with the latest findings and experiences of a specific operation. This is the main part of the role of the technological worker, irreplaceable by any computer technology whatsoever. Computer technology, despite its perfection, is only a tool enabling real-time prediction of his/her technical thinking and decisions. Materiali in tehnologije / Materials and technology 43 (2009) 2, 73–78 73 UDK 669.1:621.74 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 43(2)73(2009) The quality of a massive casting of cast iron with spheroidal graphite is determined by all the parameters and factors that affect the metallographic process and also others. This means the factors from sorting, melting in, modification and inoculation, casting, solidification and cooling inside the mold and heat treatment. The heterogeneous pouring field is formed in such a way that the process of casting always takes a longer period of time. The different parts of the melt have trajectories of differing lengths within the mold and lose different amounts of heat. Therefore, the pouring times must be short enough in order to reduce the hetero- geneity of the temperature field as much as possible. Otherwise there is the danger, especially with massive castings—even with modularly evened out pouring—of a temporally different course of eutectic solidification with a varying density of spheroids of graphite in its individual parts. The results are unequal mechanical properties along the section, especially a ductility decrease. The cooling rate of during solidification and cooling in the mold is a significant quantity influencing the forming of the microstructure. It works not only on the morphology of the graphite but also on the segregation of elements in austenite and its transformation. The increasing rate of cooling increases the number of spheroids and improves the nodular character of the spheroids of graphite. It shortens the distances between the eutectic grains (i.e. cells) reducing the segregation on their boundaries. Simultaneously, it decreases the hetero- geneity of the pouring temperature field, it minimizes the heat convection of liquid ductile cast-iron, attempting to prevent the forming of chunky graphite inside large cross-sections. The rate of cooling is not an isolated process. Furthermore, the oxygen balance and eutectic temperature influence the density of the spheroids with mutual interaction. This paper deals with the simulation of solidification and cooling of a massive casting, with various ways of accelerated cooling using steel chills in order to reduce the heterogeneity of the pouring temperature field and to increase the rate of cooling of the casting. The results of the simulation are compared with experimentally measured temperatures. It seems that numerically controlled cooling enables the optimization of the technology of pouring of massive ductile cast-iron castings with spheroidal graphite. 2 NUMERICAL MODEL OF THE TEMPERATURE FIELD The numerical model of the temperature field must observe two main goals: directed solidification as the basic condition for the healthiness of a casting and the optimization of the technology of pouring while optimizing the utility properties of the product. The main goal achieved—in terms of the economics—is the saving of liquid metal, molding materials, the saving of energy and the already mentioned optimization of pouring and also the improvement of the properties of the cast product. The solidification and cooling of a classically cast (i.e. gravitationally poured) casting and the simultaneous heating of the mold is, from the viewpoint of thermo- kinetcs, a case of 3D transient heat and mass transfer. In systems comprising the casting, the mold and ambient, all three kinds of heat transfer take place. Since these problems cannot be solved analytically—even with the second-order partial differential Fourier equation (1) (where mass transfer is neglected and conduction is considered as the most important of the three kinds of heat transfer)—it is necessary to engage numerical methods. Equation (1) describes the transient tempera- ture field in a mold. Its properties k, c and are con- sidered to be constant. ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ T k c T x T y T zτ ρ = ⋅ + + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ 2 2 2 2 2 2 (1) The Fourier equation for a casting must be adapted so as to describe the temperature field of a casting in all its three phases: in the melt, in the mushy zone and in the solid phase. Here it is necessary to introduce the specific volume enthalpy hv = cT, which is dependent on temperature. The thermodynamic enthalpy function includes the latent heat of phase or structural changes. The equation then takes on the form ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ h x k T x y k T y z k T z v τ = ⎛ ⎝⎜ ⎞ ⎠⎟ + ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ + ⎛⎝⎜ ⎞ ⎠⎟ (2) The specific heat capacity c, density and heat conductivity k are also functions of temperature. The temperature of the general nodal point of the casting is obtained from the enthalpy-temperature dependence, which must be known for the relevant ductile cast-iron (Figure 1). The software pack ANSYS had been chosen F. KAVICKA ET AL.: NUMERICAL OPTIMIZATION OF THE METHOD OF COOLING ... 74 Materiali in tehnologije / Materials and technology 43 (2009) 2, 73–78 Figure 1: Enthalpy as a function of temperature Slika 1: Odvisnost entalpije od temperature for this investigation because it enables the application of the most convenient method of numerical simulation of the release of latent heat of phase and structural changes using the thermodynamic enthalpy function. The program also considers the non-linearity of the task, i.e.: • The dependence of the thermophysical properties (of all materials entering the system) on the temperature, and • The dependence of the heat-transfer coefficients (on all boundaries of the system) on the temperature of the surface of the casting and mold. ANSYS has an integrated mesh generator (i.e. pre- processing) as well as graphical output (i.e. post-pro- cessing) of the results and the user can change the pouring parameters, the dimensions of the casting-mold system and the dimensions of the elementary mesh volume before the actual calculation. The density of the mesh makes it possible to approximate the linear distribution of temperatures between individual points of the 3D mesh, and even within time intervals. The accuracy of the numerical solution depends not only on the spatial and temporal discretization, but also on the precision with which the thermophysical proper- ties of all materials entering the system are determined, and also on how precisely the boundary conditions are derived. 3 THE PROBLEM The application of the 3D numerical model on a transient temperature field requires systematic experimentation, including the relevant measurement of the operational parameters directly in the foundry. The results of the measurement, which is focused on measuring temperatures, serve not only to verify the exactness of the model, but also to maintain continuity of the procedure: real process (risering, melting, pouring, solidification, etc.) → input data → numerical analysis → optimization → correction of real process (risering, melting, pouring, solidification, etc.). A real (500 × 1000 × 500) mm ductile cast-iron block had been used for the numerical calculation and the experiment (Figures 2 and 3). Temperature measurement (using thermocouples) and its successive confrontation with the calculation proved that it is possible to apply the numerical model on basic calculations of solidification and cooling of the casting. It is also possible to determine the temperature gradients, the rate of solidification and the local solidification times (i.e. the time for which the given point of the casting finds itself between the liquidus and solidus temperatures). The local solidification time θ (according to the analogy from steels) significantly influences the forming of the as solidified structure of the given material. F. KAVICKA ET AL.: NUMERICAL OPTIMIZATION OF THE METHOD OF COOLING ... Materiali in tehnologije / Materials and technology 43 (2009) 2, 73–78 75 Figure 3: A picture showing the forming of casting no. 2 with chills on three sides Slika 3: Formanje ulitka {t. 2 s hlajenjem na treh straneh Figure 2: A picture showing the forming of casting no. 1 with chills on one side Slika 2: Formiranje ulitka {t.1 s hlajenjem na eni strani Figure 4: The mesh inside the chills and casting no. 1 Slika 4: @i~na mre`ica znotraj hladil in ulitka {t. 1 The investigated experimental castings weigh approximately 2 t. They were cast into sand molds with various arrangements of steel chills of cylindrical shape. The dimensions of the castings, the mold, the chills and their arrangements are illustrated in Figures 2 and 3. Figures 4 and 5 show an example of the mesh generated by ANSYS. 4 TEMPERATURE MEASUREMENT The temperature was measured using K- and B-type thermocouples and special thermocouple probes of type PtRh6 – PtRh30. The recording was carried out by the data-acquisition device GRANT 1250. The measuring ends of the thermocouples were placed in holes of 2 mm in diameter. The initial temperature of the mold and chills was approximately 20 °C, the mold was filled through the top inlet gate with a melt at 1300 °C. The courses of the temperatures were measured on casting no. 1 for 19 h 11 min and 19 h 20 min on casting no. 2 after casting. The positions of the probes and a view of the actual installation in mold no. 2 are illustrated in Figures 6 and 7. 5 CALCULATION AND EXPERIMENTAL RESULTS The points in casting no. 2 and the chill were selected for comparison (Figures 3 and 6 respectively). The computation and experimental curves are in Figures 8 and 9. It is obvious, that in the compared points of the mold and chills, the results from the numerical model correspond to those measured, which applies to most other places where both the mold and chills were measured. In the centers of the casting it was not possible to carry out the comparison due to a failure in the thermocouple probes. The iso-zones, calculated in castings 1 and 2 and in the chills in various parts after casting, are illustrated in F. KAVICKA ET AL.: NUMERICAL OPTIMIZATION OF THE METHOD OF COOLING ... 76 Materiali in tehnologije / Materials and technology 43 (2009) 2, 73–78 Figure 8: The measured and calculated temperature history in casting no. 2, 5 mm beneath its surface Slika 8: Merjene in izra~unane temperature 5 mm pod povr{ino ulitka {t. 2 Figure 6: The measured positions for casting no. 2 Slika 6: Lega hladil pri ulitku {t. 2 Figure 7: The installations of the thermocouples while molding casting no. 2 Slika 7: Namestitev termo~lenov med formanjem ulitka {t. 2 Figure 5: A detailed view of the mesh inside the chills and mold for casting no. 1 Slika 5: Podroben pogled na `i~ne mre`ice znotraj hladil in forme za ulitek {t. 1 Figures 10 and 11. The comparison of the iso-zones, including the mushy zone in castings 1 and 2 shows that this time is, relatively, not very much influenced by the increase in the number of chills or by the increase in the number of walls on which the chills are mounted. The total solidification time of casting no. 1 is 5:08:10 hours and 4:33:35 hours of casting no. 2. Even the various arrangements of chills did not significantly influence the difference in the total solidification time. 6 CONCLUSION The numerical models of the temperature field of the solidifying castings of different authors have observed two main goals: directed solidification as the basic condition for the healthiness of a casting and the optimi- zation of the technology of pouring, while, optimizing the utility properties of the product. Achieving these goals is conditioned by the ability to analyze and to successively control the influence of the deciding factors that either characterize the solidification process or accompany it. It is advantageous to focus the analysis especially on a breakdown of the causes behind the formation of the non-homogeneities within the casting, considering the latent and structural changes, on the thermokinetic formation of the contractions and cavities, on the prediction of their forming, thus, managing to optimize the shape and size of the risers, the method of insulation, the treatment of the level, etc. The main economic goal observed is the saving of liquid material, molding and insulation materials, the saving of energy and the already mentioned optimization of pouring and the properties of the cast product. This paper has been discussing an original application of ANSYS for the investigation into a temperature field of a massive casting of ductile cast-iron with spheroidal graphite, which makes it possible to evaluate the local solidification times and the local rates of solidification and cooling. Using these parameters, together with the model of microstructural and chemical heterogeneity, it is possible to design the technology of pouring a casting of massive cast-iron with spheroidal graphite (i.e. for example a system of chills) in such a way so as to optimize the quality of the casting even from the viewpoint of its pouring structure. Acknowledgments. This analysis was conducted using a program devised within the framework of the GA CR projects No. 106/06/1210, 106/06/1225, 106/06/0393, 106/08/0606 and 106/08/1243, 106/09/0940. NOMENCLATURE c hv k x, y, z T  specific heat capacity specific volume enthalpy hv = h heat conductivity time axes in given directions temperature density J kg−1 K−1 J m−3 W m−1 K−1 s K kg m−3 F. KAVICKA ET AL.: NUMERICAL OPTIMIZATION OF THE METHOD OF COOLING ... Materiali in tehnologije / Materials and technology 43 (2009) 2, 73–78 77 Figure 11: The calculated iso-zones in casting no. 2 and in its chills a) τ = 1 h, b) τ = 4,5 h Slika 11: Izra~unane izo-temperaturne cone v ulitku {t. 2 s hladili a) τ = 1 h, b) τ = 4,5 h Figure 9: The measured and calculated temperature history in the chill for casting no. 2 Slika 9: Merjene in izra~unane temperature v hladilih ulitka {t. 2 Figure 10: The calculated iso-zones in casting no. 1 and in its chills a) τ = 1 h, b) τ = 5 h Slika 10: Izra~unane izotemperaturne cone v ulitku {t. 1 s hladili a) τ = 1 h, b) τ = 5 h 7 REFERENCES 1 Popela P., Stochastic programming models and methods for tech- nical applications. Folia Fac. Sci.Nat. Univ. Masarykianae Brunen- sis, Mathematica, 11 (2002), 181–206 2 Svantner M., Honner M., The model of cooling of a casting, Research report of the Research Centre of New Technologies, West Bohemian University in Plzen, 2001 3 Kovarik J., Vavroch O., Determining thermophysical properties of mould mixtures, Research report VZVÚ 07687 Skoda Research Ltd., Plzen 1993 4 Kavi~ka F., Stetina J.: A numerical model of heat transfer in a system a plate casting-mold-suroundings for optimization. Pro- ceedings of the Conference of the ASME, Seattle, USA, July 2000, 161–168 5 Kavicka F., Stetina J.. Anumerical model of heat transfer in a system plate-mold-surroundings. Proceedings of the 6th International Con- ference on Advanced Computational Methods in Heat Transfer, Spain, Madrid, June 2000, 95–104 6 Kavicka F. et al.: Optimisation of properties and foundry technology of heavy weight ductile cast-iron castings. Final report of GACR project No.106/01/1164, Brno, 2003 F. KAVICKA ET AL.: NUMERICAL OPTIMIZATION OF THE METHOD OF COOLING ... 78 Materiali in tehnologije / Materials and technology 43 (2009) 2, 73–78 M. MARIN[EK: THE ELECTRICAL CONDUCTIVITY OF SINTERED LSM CERAMICS ELECTRICAL CONDUCTIVITY OF SINTERED LSM CERAMICS ELEKTRI^NA PREVODNOST SINTRANE LSM-KERAMIKE Marjan Marin{ek University of Ljubljana, Faculty of Chemistry and Chemical Technology, A{ker~eva 5, Ljubljana, Slovenia marjan.marinsenfkkt.uni-lj.si Prejem rokopisa – received: 2008-10-13; sprejem za objavo – accepted for publication: 2008-11-26 The carbonate co-precipitation route was applied for batch La0.85Sr0.15MnO3 (LSM) preparation as an alternative synthesis method to the solid-state reaction. Because co-precipitation is a wet-chemistry solution process, the maximum LSM homogeneity was achieved. The microstructural characteristics, such as the porosity and grain size of the prepared LSM elements, were controlled by subjecting the green bodies to various sintering conditions. The LSM sintered bodies with relative sintered densities as high as 95 % were prepared at sintering temperatures not higher than 1100 °C. The microstructure of the prepared LSM was characterized by digital online image analysis and the microstructural parameters were determined for the ceramic phase as well as for the porosity. The electrical characteristics of the sintered LSM elements were, for the first time in the literature, described with respect to a model for the sine-wave approximation of the conductivity change for porous materials. The observed results of the relative conductivity σ/σ0 vs. the relative density ρ/ρ0 dependence were essentially consistent with the sine-wave approximation. As an absolute value, the highest σ = 65 S/cm at 800 °C was measured for a sample with ρ/ρ0 = 99.58 %. Key words: LSM co-precipitation, microstructure, electrical conductivity, sine-wave approximation of conductivity La0,85Sr0,15MnO3 (LSM) je bil pripravljen po metodi karbonatne koprecipitacije kot alternativa metodi reakcije v trdnem. Z uporabo koprecipitacijske metode kot ene izmed tehnik mokre kemije nam je uspelo pripraviti zelo homogene LSM-prahove. Mikrostrukturne lastnosti kon~nih, pripravljenih LSM-elementov, kot sta poroznost in velikost zrn, smo spreminjali s sintranjem LSM-surovcev pri razli~nih pogojih. Uspelo nam je pripraviti sintrane LSM-elemente z relativno sintrano gostoto ≈95 % pri temperaturtah sintranja, ki niso bile vi{je kot 1100 °C. Bolj podrobna karakterizacija mikrostrukture sintranih elementov je bila opravljena z analizo digitalnih slik. Elektri~ne lastnosti sintranih LSM-elementov smo pojasnili s teoreti~nim modelom vrtenine za opis elektri~ne prevodnosti porozne keramike, kar je tudi prvi tovrsten opis LSM-keramike. Rezultati dolo~itve relativne prevodnosti σ/σ0 LSM-keramike kot funkcije njene relativne gostote ρ/ρ0 relativno dobro sledijo predlaganemu teoreti~nemu modelu. Najvi{jo absolutno izmerjeno vrednost elektri~ne prevodnosti σ = 65 S/cm pri 800 °C je imel vzorec z relativno gostoto ρ/ρ0 = 99,58 %. Klju~ne besede: LSM-koprecipitacija, mikrostruktura, elektri~na prevodnost, "sine-vawe" model prevodnosti 1 INTRODUCTION Lanthanum strontium manganite (La1–xSrxMnO3, LSM) has been extensively used as a cathode material for solid-oxide fuel cells (SOFCs). It offers a high elec- tronic conductivity, a high catalytic activity for oxygen reduction as well as chemical and thermal compatibility with the yttria-stabilized zirconia (YSZ) electrolyte at the operating temperature1,2. In the present generation of SOFCs, the nominal composition of La1-xSrxMnO3 (x < 0.2) is normally used3,4. The use of LSM-based cathode materials depends not only on their chemical, structural and thermodynamic characteristics, but also on their final microstructure, grain size, pore size and pore-size distribution5–7. Various preparation techniques have been reported for LSM synthesis. In general, perovskite manganites are synthesized at high temperatures using a standard cera- mic technique. However, when utilizing a solid-state reaction for LSM preparation, the homogeneity and final microstructure of the material are more difficult to con- trol, due to the fact that the conventional ceramic syn- thesis process is based on the diffusion of components in the solid state at high temperatures. In this respect, several preparation techniques based on solution chemistry methods, such as the citrate-gel process8-12, the sol-gel process13-15, combustion syntheses16,17 and the co-precipitation technique18,19 have also been tested for the preparation of highly homogenous and fine LSM powders. The characterization of LSM and some other perov- skite powders such as SOFC cathodes or oxygen mem- branes prepared by different chemical routes was made by Sfeir et al.20, while a comprehensive study of the effect of the synthesis route on the catalytic activity of LSM was performed by Bell et al.21. The results of these studies implied that the carbonate co-precipitation syn- thesis route is especially interesting since it delivers a finer powder with a more homogeneous composition and surface structure and is, thus, more suited to mixed-con- ductor applications in SOFC systems. The groundwork of the co-precipitation method for the LSM preparation was done by Tanaka et al., where Na2CO3 was used as the precipitating agent22. However, one of the serious drawbacks of the method is the incorporation of Na+ ions into the precipitate due to the use of Na2CO3. Another co-precipitation-based LSM preparation was developed Materiali in tehnologije / Materials and technology 43 (2009) 2, 79–84 79 UDK 66:546.654:54.022:620.1 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 43(2)79(2009) by Ghosh et al.,19 using ammonium carbonate as the precipitating agent. LSM is a perovskite oxide ABO3 where the La3+ ions at the A-sites are partially substituted by Sr2+ ions. LaMnO3 is a p-type semiconductor due to the small polaron hopping of holes23 between the Mn3+ and Mn4+ ions. The doping of Sr into LaMnO3 increases the electrical conductivity considerably because of the increased number of holes24. However, in order to serve as a cathode material in an operating fuel cell, besides the appropriate electro-catalytic properties, the LSM also has to exhibit suitable microstructure characteristics, including ≈30 % open porosity. Changing the porosity, going from a dense to a more porous structure, will again alter some of the electrical characteristics of the material, since the apparent conductivity of sintered materials is sensitive to their relative density. To describe the rela- tionship between the conductivity of the porous material and its relative density, a novel mathematical approach was proposed25. In this new method of interpretation, where as a first approximation, porous material is represented by a uniaxial string of spheres along the direction of the potential gradient and then remodeled into a rotating body, the conductivity of highly porous ceramic materials can be expressed with a model of a rotating sine-wave function. By changing the ratio of the contribution of two sine-waves, one representing the shape of each grain and the other the shape of the bottleneck between the particles, the change of the electrical conductivity σ vs. the change of the material’s relative density ρ/ρ0 can be represented. In the present work, the carbonate co-precipitation route for batch LSM preparation was applied. Being a solution process, the maximum product homogeneity with minimum secondary-phase addition can be achieved. The aim of the study is a description of the electrical conductivity behavior of the prepared LSM bodies with respect to the material’s microstructure characteristics. For the fist time, the specific electrical conductivity of porous LSM is explained with the model of a rotating sine-wave function. 2 EXPERIMENTAL PROCEDURE In order to prepare stock solutions of 0.5-M lantha- num nitrate and strontium nitrate, lanthanum oxide (La2O3) and strontium carbonate (SrCO3) were separately, carefully dissolved in concentrated nitric acid, while manganous nitrate (Mn(NO3)2 · 4H2O) was dissolved in distilled water. For the La0.85Sr0.15MnO3 preparation, predetermined amounts of each solution were then mixed. The mixed solution was added drop-wise to a precipitating bath containing an aqueous solution of ammonium carbonate in which the amount of ammo- nium carbonate was in 50 % excess for the complete precipitation of the mixed La-Mn-Sr-precursor. The pH of the precipitating bath was kept constant at 8.0 by small additions of aqueous ammonia. The temperature of the reaction mixture in the precipitating bath was adjusted to 65 °C and kept constant. The precipitation reaction took place under a CO2 atmosphere to prevent any undesired oxidation of the manganese to MnO2. The reaction time for the complete co-precipitation process was approximately 3 h. Typical quantities of the initial precursors consumed in one cycle of the bath precipitation were 100 g of (NH4)2CO3 dissolved in 2.5 L of H2O, 854 mL of 0.5-M Mn-precursor solution, 363 mL of 0.5-M La-precursor solution, and 128 mL of 0.5-M Sr-precursor solution. Such a reaction mixture yielded approximately 100 g of the final, calcined LSM powder. The filtered precipitate was washed several times with a 0.125-M aqueous ammonium-carbonate solution and then dried first at room temperature in a CO2 atmosphere for several hours and afterwards for six hours at 110 °C in air. Prior to calcination in a muffle furnace at 1000 °C for one hour, the dried powder was ground in an agate mortar. Calcined LSM powders were then wet milled in a ball mill in isopropanol. To achieve a higher morphological homogeneity the wet-milled powders were further subjected to atritor milling. Thus prepared, the LSM powders were pressed into tablets (Φ = 6 mm, h 4 mm, P = 200 MPa) and sintered at various temperatures from 1000 °C to 1330 °C for one hour. The samples were characterized with the X-ray powder-diffraction technique using a D4 ENDEAVOR diffractometer. The shrinkage during the sintering was measured with a LEITZ WETZLAR heating microscope. After sintering and polishing the samples were thermally etched and analyzed by SEM (Zeiss FE SUPRA 35 VP). The quantitative analysis of the microstructures was performed using a Zeiss KS300 3.0 image analyzer. In order to get accurate data on the electrical resistivity of the prepared and sintered pellets as a function of temperature, the four-point electrical resistivity method was used. 3 RESULTS AND DISCUSSION The precipitated and calcined powders were sub- mitted for an XRD examination (Figure 1). For XRD study purposes the calcination of the precursor carbo- nate-hydroxide powder was carried out in the tempera- ture range 1000–1400 °C. According to Figure 1, calcination at 1000 °C is quite sufficient for a complete perovskite LSM phase formation. Another apparent characteristic from the XRD results is that only traces of the residual secondary phases La2O3 or La(OH)3 are still present after the calcination (La(OH)3 resulting from the reaction of the La2O3 with moisture). Additionally, no Mn-oxide secondary phases were detected with the XRD analysis. During the calcination at higher temperatures (1100–1400 °C) the secondary phases are completely M. MARIN[EK: THE ELECTRICAL CONDUCTIVITY OF SINTERED LSM CERAMICS 80 Materiali in tehnologije / Materials and technology 43 (2009) 2, 79–84 dissolved in the perovskite structure. The relatively small amount of secondary phases in the sample calcined up to 1000 °C makes the co-precipitation method favorable when compared to synthesis processes that are based on the diffusion of components in the solid state. Namely, if the solid-state reaction is employed for the LSM prepa- ration, the amount of secondary phases is normally greater. At this point, it is necessary to mention that secon- dary phases may nevertheless appear in the LSM struc- ture, even in the case when the calcination temperature exceeds 1100 °C. Specifically, a precise SEM analysis revealed that calcination or sintering of the prepared LSM at temperatures above 1100 °C caused the repre- cipitation of MnO2 at the LSM grain-boundary region (Figure 2). The amount of reprecipitated MnO2 at 1100 °C is relatively low (the volume fraction below 1 % of the material) but is increased to 3.9 % when the material is thermally treated at 1330 °C for 1 h. One of the principal characteristics LSM has to exhibit is sintering at relatively low temperatures (less than 1200 °C). This is very important from the applica- tion point of view. If the material is used as a cathode in SOFC systems and co-sintered with other cell layers, then sintering at temperatures above 1200 °C may cause some highly undesirable reactions with neighboring materials26,27. For this reason, the morphological homo- geneity of LSM powders is essential for achieving material densification at relatively low temperatures. During LSM sinterability tests, the best results were achieved when a combination of milling methods was used (grinding in an agate mortar, wet milling in a ball mill and atritor milling). After wet milling in a ball mill in isopropanol, the average particle size dav. was deter- mined to be 2.45 µm (standard deviation σ 1.98 µm). After 3 hours of additional atritor milling dav. was lowered to 0.57 µm (σ 0.52 µm). Such a combination of homogenization operations substantially lowered the sintering temperature of the LSM tablets (Figure 3). After successfully reducing the sintering temperature, a series of tablets in the green state was prepared and sintered at various temperatures in order to alter the densities of the sintered elements. Prior to establishing the LSM’s electrical characteristics with respect to its microstructure, a complete, quantitative microstructure analysis was performed on sintered and polished elements (Table 1, Figure 4). According to the results summarized in Table 1, rather dense elements with relative densities of more than 92 % can be prepared at sintering temperatures Ts as low as 1090 °C. This is of prime importance when multilayered structures of SOFCs are prepared; it is very important that the material is capable of densification at relatively low temperatures, resulting in good particle-to-particle connections, especially in cases where a high electrical conductivity is required. In contrast to the sintered density, the porosity of the sintered elements decreased with an increased sintering temperature. The elements’ porosity was determined from the material density as the geometrical porosity ε (ε M. MARIN[EK: THE ELECTRICAL CONDUCTIVITY OF SINTERED LSM CERAMICS Materiali in tehnologije / Materials and technology 43 (2009) 2, 79–84 81 Figure 2: Microstructure of LSM sample thermally treated at 1330 °C for 1 hour Slika 2: Mikrostruktura LSM-vzorca, obdelanega pri 1330 °C 1 h Figure 3: Shrinkage curves of LSM tablets after powder wet milling or after additional atritor milling Slika 3: Krivulje sintranja LSM-surovca po mokrem mletju prahu oziroma po dodatnem atritorskem mletju prahu Figure 1: XRD patterns of the synthesized powders calcined at diffe- rent temperatures Slika 1: Pra{kovni posnetki sintetiziranih vzorcev, kalciniranih pri razli~nih temperaturah = 1 – ρ/ρ0) and from the quantitative microstructure analysis ε’. Both values, ε and ε’, are in relatively good agreement. The sintering temperature also substantially influences the size of the LSM particles. The mean particle size in each case was determined as a diameter of the area analogue circle d as well as the intercept length in the x or y directions dx and dy. With increasing sintering temperatures, all the values d , dx and dy increased and the LSM particles become increasingly spherical (shape factor Ψ). As mentioned previously, temperatures above 1100 °C may cause re-precipitation of the secondary phase MnO2. The values d , dx, dy and Ψ describing the formed MnO2 particles are summarized in Table 1 in brackets. Since the LSM in SOFC applications is prepared as a relatively thin porous layer, it is necessary to establish the relationship between the material’s microstructure and its electrical characteristics. The appropriate electrical conductivity of the material prepared as a dense element is one of the principal requirements. However, considering only the electrical conductivity data of the dense element without relating this data to the real material microstructure is insufficient. The electrical conductivity of the porous material is described not only by its specific electrical conductivity σ0 but also by the element’s relative density and the degree of connection between the particles along the potential gradient. In this respect, the sine-wave approximation considers a highly porous ceramic element as the repeating pattern of a uniaxial string of spheres. In this simplified approach to describing the microstructure, the volume of the rotating string is controlled by the contribution of two sine- waves. The progress of the sintering is described by changing the ratio of the contribution of the two sine-waves to the string volume (changing the parameter c as described by Mizusaki et al23). Finally, the change of the relative specific electrical conductivity σ/σ0 vs. the change in the material’s relative density ρ/ρ0 can be presented. Results describing the apparent conductivity of the sintered materials vs. the temperature with respect to their relative density or porosity are shown in Figure 5. The apparent conductivity increases with temperature, indicating the semi-conductive nature of the LSM cera- mics. Since the LSM’s density was controlled through a sintering process (higher densities were obtained if Ts was higher) it was to be expected that the specific con- ductivities σ should also reach higher values with progress in the particle-to-particle contact. As an M. MARIN[EK: THE ELECTRICAL CONDUCTIVITY OF SINTERED LSM CERAMICS 82 Materiali in tehnologije / Materials and technology 43 (2009) 2, 79–84 Table 1: Microstructural parameters of the sintered LSM bodies Tabela 1: Mikrostrukturne zna~ilnosti sintranih LSM-tabletk sample A B C D E F G sintering temperature Ts / °C 1000 1015 1024 1060 1090 1100 1330 relative sintered density ρs / % 57.3 65.8 70.1 83.0 92.7 94.1 98.2 porosity (geometrical) ε / % 42.7 34.2 29.9 17.0 7.3 5.9 1.8 porosity (microstructural) ε' / % 43.2 28.6 25.4 15.9 6.0 4.4 0.9 mean particle diameter d ·102 / µm 5.7 8.9 20.0 28.0 29.0 34.0 (16.0) 506.0 (33.0) standard deviation ζ·102 / µm 4.9 5.9 7.9 12.6 18.8 25.4 163.2 shape factor Ψ 0.53 0.69 0.73 0.79 0.94 0.93 (0.63) 0.90 (0.51) intercept length in x dx . 102 / µm 8.7 11.1 24.0 31.0 38.0 40.0 (16.0) 520.0 (39.0) intercept length in y dy . 102 / µm 8.2 8.2 22.0 33.0 30.0 43.0 (15.0) 749.0 (24.0) No. of analyzed particles 782 855 848 419 136 110 90 vol. fraction of MnO2 phase / % / / / / / 1.16 3.94 Figure 4: Microstructure of LSM elements sintered at various tempe- ratures Slika 4: Mikrostruktura LSM-tabletk, sintranih pri razli~nih tempera- turah absolute value, the highest σ = 65 S/cm at 800 °C was measured for a sample with ρ/ρ0 = 99.58 %. The relationship relative conductivity (σ/σ0) vs. relative density (ρ/ρ0) or porosity (ε) is demonstrated in Figure 6. For the σ0 value, the highest measured σ value was adopted (σ0 = 65 S/cm), while ρ0 was the theoretical density of La0.85Sr0.15MnO3 (ρ0 = 6.595 g/cm3). The observed results of the relative conductivity σ/σ0 vs. relative density ρ/ρ0 dependence are essentially consi- stent with the sine-wave approximation of the conduc- tivity change for porous materials. The scattering of the observed results around the theoretically predicted curves may be attributed to the inaccuracy of the con- ductivity measurements or some microstructure defects in the measured bodies. Because of the simplification of the sine-wave modeling regarding the material’s microstructure, it is impossible to discuss precisely the relationship between the theoretical predictions and the experimental results. That is to say, contrary to the assumption used in the mathematical model, the grain size and shapes are not homogeneous and the packing of the grains is disordered. To incorporate the inhomo- geneity and disorder, such as micro-cracks, the low-scale material inhomogeneity, the inhomogeneous grain size and shape or the local packing disorder into the sine-wave model, a more advanced approach to a mathematical description of the microstructure will be needed. However, a comparison of the results presented in Figure 6 revealed that the LSM’s relative conductivity convergences on 0 at the relative density 0.45–0.55, a value close to the relative density of as-pressed powders (green density). Such a relationship is not surprising; ceramics are generally prepared from green parts that consist of isolated particles of material. In as-pressed powders, the particles are isolated, except for the point-contacts between them, i.e., it is not the material but the void space that is continuous. The electrical conduction in green wares is strongly hindered by the limited contact between the particles, although the apparent density is normally in the range 40–60 %. With increasing sintering temperatures, the relative sintered densities increase, ensuring a better contact between the particles, which is demonstrated through higher relative conductivities. 4 CONCLUSIONS LSM was prepared using the carbonate co-precipi- tation route. The relatively small amount of secondary phases in the synthesized and subsequently calcined sample up to 1000 °C makes the co-precipitation method favorable when compared to synthesis processes that are based on the diffusion of components in the solid state. During LSM sinterability tests, the best results were achieved when a combination of milling methods was used (grinding in an agate mortar, wet milling in a ball mill and atritor milling). Such combinations of homoge- nization operations may substantially lower the sintering temperature of the LSM tablets. Rather dense elements, with relative densities greater than 92 %, can be prepared at sintering temperatures Ts as low as 1090 °C. The electrical conductivity behavior of the prepared LSM bodies was determined with respect to the material’s microstructure characteristics. The apparent conductivity of the prepared LSM bodies increased with temperature, indicating their semi-conductive nature. The relationship between the relative conductivity (σ/σ0) vs. the relative density (ρ/ρ0) or the porosity (ε) was essentially con- M. MARIN[EK: THE ELECTRICAL CONDUCTIVITY OF SINTERED LSM CERAMICS Materiali in tehnologije / Materials and technology 43 (2009) 2, 79–84 83 Figure 6: Calculated relationships between the relative conductivity σ/σ0 and the relative density ρ/ρ0 or porosity ε for different parameters c; (solid, dashed and dotted lines as described by Mizusaki et al25) and the observed trend of the measured data ( LSM tablets). The relative shape parameter c determines the contribution of the two sine-waves in describing the shape of the cross-section of the particle during sintering. Slika 6: Odnos med relativno prevodnostjo σ/σ0 in relativno gostoto ρ/ρ0 oziroma poroznostjo ε LSM-tablet z ozirom na mikrostrukturni parameter c (polna in ~rtkane ~rte). Parameter c dolo~a prispevek obeh sinusnih funkcij, ki opisujeta obliko delca med sintranjem. 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BELI^: MODELLING THE CHARACTERISTICS OF AN INVERTED MAGNETRON USING NEURAL NETWORKS MODELLING THE CHARACTERISTICS OF AN INVERTED MAGNETRON USING NEURAL NETWORKS MODELIRANJE KARAKTERISTIKE INVERTNEGA MAGNETRONA Z NEVRONSKIMI SISTEMI Igor Beli~ Institute of Metals and Technology, Lepi pot 11, 1000 Ljubljana, Slovenia igor.belicimt.si Prejem rokopisa – received: 2009-02-10; sprejem za objavo – accepted for publication: 2009-03-10 The inverted magnetron or cold cathode gauge (CCG) is a device used as a vacuum gauge. It is a very robust device, with mostly very positive properties. The problem with its use lies in its nonlinear, temporary, variable characteristic and the fact that the theory of its operation is not thoroughly understood. Neural networks are, therefore, an ideal solution for building a nonlinear characteristics model, based on a set of measured points. Such a model is valid for some certain period of time. When the characteristic of the CCG is altered significantly (due to aging and contamination), the process of recalibration needs to be done, where again neural networks provide a very easy-to-use and robust tool. In the article the simulation of the CCG characteristics is presented. It is meant to provide sufficiently large sets of data to enable a study of the modelling properties of the used neural networks. The CCG characteristic was split into several segments, each of which was modelled by a separate neural network. The results of the study are presented. The study ended in a practically usable methodology for employing neural networks to calibrate (or recalibrate) the CCGs. Keywords: inverted magnetron, CCG, modelling, approximation, neural networks, calibration Invertni magnetron ali merilnik s hladno katodo (CCG) je naprava, ki se uporablja kot grobi merilnik tlaka v vakuumskih sistemih. To so robustne naprave s celo vrsto dobrih lastnosti. Problem prakti~ne uporabe je, da je karakteristika CCG zelo nelinearna, ~asovno spremenljiva in da teorija delovanja ni povsem znana. Zato so nevronski sistemi idealno orodje za gradnjo nelinearnega modela, ki je zgrajen na mno`ici izmerjenih to~k. Tak model je uporaben v nekem ~asovnem obdobju. Ko se karakteristika CCG preve~ spremeni (zaradi staranja in kontaminacije naprave), je treba narediti rekalibracijo. Tudi pri rekalibraciji so nevronski sistemi uporabljeni kot orodje, ki je robustno in enostavno za uporabo. V prispevku je opisana simulacija karakteristike CCG. Namenjena je generiranju zadostnega {tevila to~k, ki so omogo~ile {tudijo lastnosti modeliranja z nevronskimi sistemi. Celotna karakteristika CCG je bila razdeljena na nekaj segmentov, pri ~emer je bil vsak segment posebej modeliran s svojim nevronskim sistemom. Predstavljeni so rezultati {tudije. Rezultat {tudije je prakti~no uporabna metodologija modeliranja karakteristike CCG z nevronskimi sistemi, ki jih uporabimo za kalibracijo (rekalibracijo) merilnika. Klju~ne besede: invertni magnetron, CCG, modeliranje, aproksimacija, nevronski sistemi, kalibracija 1 INTRODUCTION The inverted magnetron or cold cathode gauge (CCG) is normally used as a coarse pressure gauge in the range from 1·10–12 to 1·10–2 mbar. During our work the range from 1·10–9 to 1·10–5 mbar was used. (In the field of vacuum phisics the mbar is commonly used. The SI unit is Pa. 1 bar = 105 N/m2 = 105 Pa; 1 mbar = 1 hPa) On the principles of CCG operation, our research group has already published several articles 1,2,3,4,5. In the scope of this article only a very brief overview of the CCG’s operating principles is given. In the inverted magnetron the electrons are trapped in perpendicular magnetic and electric fields 5. The electrons are moving on cycloid trajectories around the anode, which is placed inside the discharge cell. The kinetic energy of electrons is high enough to ionize the atoms and molecules of the vacuum chamber’s atmo- sphere inside the magnetron cell. After the collision of the electron with an atom/molecule, the kinetic energy of the electron decreases, therefore it is drawn into a cycloid trajectory closer to the anode. After a series of collisions, the electron reaches the anode and therefore contributes to the anode current. Due to the higher mass/ charge ratio, the ions take wider cycloid trajectories than electrons and they hit the cathode. By doing so, new electrons emerge from the cathode surface and they add to the electron cloud within the magnetron cell 6,7. Some ions are trapped on the cathode surface and therefore they no longer contribute to the chamber’s atmosphere. This causes an unwanted pumping effect of the CCG gauge. The gauge itself lowers the pressure inside the vacuum chamber. Inverted magnetrons are very robust devices. They use very little power for their operation, they have a very high sensitivity, they operate without a hot cathode and they are relatively cheap 8. Usually, they act as relative pressure gauges used for large vacuum systems, such as accelerators, as well as in vacuum systems where the additional RF pollution caused by the gauge (for exam- ple, the hot filament cathode) cannot be tolerated 9,10,11,12. The cold cathode gauges compared to the hot cathode gauges also show a very low level of thermic outgassing, they do not emit the unwanted x-rays, nor do they cause Materiali in tehnologije / Materials and technology 43 (2009) 2, 85–95 85 UDK 533.5:681.2.08 ISSN 1580-2949 Original scientific article/Izvirni znanstveni ~lanek MTAEC9, 43(2)85(2009) electron stimulated desorption. The electron cloud is provided solely by the self-sustaining mechanism of the vacuum chamber atmosphere’s atoms/molecules ionisa- tion. Although there are always enough electrons in the rotating field of the CCG the anode current rises with the pressure. The property that makes the use of the inverted magnetron problematic is its non-linear characteristic between the registered ion current and the actual pressure in the vacuum chamber. At very low pressures the device does not start easily and it can take some time to form the spatial charge in the CCG cell. Devices without the starter that provides the initial electron cloud might not start in UHV (ultra-high-vacuum) conditions. In addition to the many advantages of CCGs, these instruments are restricted in their use by a decreasing measuring accuracy over the operating time 13,14 as the internal electrodes become contaminated. Thus, to guarantee a consistently high measuring accuracy, this gauge type needs to be calibrated regularly after a fixed operating time period. The calibration process can be improved by the use of neural network modelling. The process of modelling the characteristics of the inverted magnetron (CCG – cold cathode gauge) using the neural networks is presented. The characteristics were obtained on a calibration ultra-high-vacuum system which consists of the test chamber, the extractor gauge, the spinning rotor gauge, and the gas manifold with the precision valve. The magnetron ion current was measured simultaneously with the high-voltage measure- ments between the cathode and the anode, all at different pressures, that vary form 10–9 do 10–5 mbar. The working voltage (cathode-anode) was varied in the range from 1.2kV to 9kV. For all measurements, the magnetic field density remained at 1.3T. A very positive attribute of the CCG is its extremely low thermal outgassing values, and it can be used for measurements of low-pressure values suitable for ultra-high-vacuum systems. An unwanted property of the inverted magnetron is its highly nonlinear dependence between the ion current and the pressure in the vacuum system (Figure 1). In some areas the CCG characteristic can also show discon- tinuities. The mechanisms of operation of such a complicated device as an inverted magnetron are not understood in detail. Consequently, classical mathematical modelling is not appropriate to cover the analytical needs for devices that serve as measurement equipment. For the inverted magnetron in the role of a vacuum-system pressure gauge the dependence between the ion current, the operating voltage and the pressure in the vacuum system must be known. In addition, in the process of magnetron calibration, its characteristics must be measured. Usually, the number of measured points is, from the practical point of view, limited. The role of the neural network is to model the characteristic in the whole usable space between the measured points. The complete set of measured points is used as the training set for the multi-layer neural network with the classical error-backpropagation training scheme. The formed model must be able to reconstruct the input–output relationship, where the input consists of the ion current and the working voltage, while the vacuum system pressure represents the output value. The built model makes it possible to use the inverted magnetron as a pressure gauge. In the CCG’s lifetime, its characteristic changes, and therefore it needs to be recalibrated several times. The use of a neural network to model the CCG’s characte- ristic is proposed. Nonlinear CCG characteristic (I/p) is normally approximated-modelled piecewise using Equation (1). I = k pn (1) with the sensitivity I/p = k p(n–1) (2) where I represents the ion current, n and k are the constants that are different for the observed part of the CCG characteristic. In the literature 15 the values of n are listed from 1.05 to 2. The constant k depends on the magnetic flux density, the geometry of the discharge chamber and the gases present in the chamber. The constant n depends primarily on the magnetic flux density, the operating voltage and, again, on the geometry of the device 7. The theory of magnetron operation is not known in such a detail as to enable the theoretical mathematical model to cover the device’s operation for measurement purposes 16. The relationship between the ion current and the pressure above the so-called "magnetron knee" is usually obtained in the logarithm tables of measured values 17,18. The tables are formed in a time-consuming calibration process. Furthermore, the use of such tables makes operating the magnetron clumsy. The values between those covered in the table are usually calculated I. BELI^: MODELLING THE CHARACTERISTICS OF AN INVERTED MAGNETRON USING NEURAL NETWORKS 86 Materiali in tehnologije / Materials and technology 43 (2009) 2, 85–95 Figure 1: The nonlinear characteristic I/U of the magnetron (Measu- rements were conducted by dr. Alenka Vesel and dr. Miran Mozeti~, both from the Josef Stefan Institute, Ljubljana, Slovenia.) Slika 1: Nelinearna karakteristika I/U magnetrona (Meritve sta izvedla dr. Alenka Vesel in dr. Miran Mozeti~, oba IJS, Ljubljana, Slovenija) by linear interpolation, which introduces additional errors into the measurements. The introduction of neural networks reduces the number of measured points needed for the calibration process. Since the neural network builds the nonlinear CCG characteristic, the linear interpolation is no longer needed and, consequently, any error produced by the linear interpolation is avoided. The important properties of the inverted magnetron can be summarized in several points: • The principle of the magnetron’s operation is not known in such a detail as to enable a concise mathe- matical model; • The CCG characteristic is nonlinear and in some places even discontinued; • Usually, a very coarse piecewise mathematical model is applied; • The operation of the CCG as a pressure gauge is stable and repeatable, although due to contamination and aging process it needs recalibrations. • It is usually used as a coarse relative pressure gauge. 1.1 A testing ground for the neural network modelling In the process of modelling it is of vital importance to have a reasonably large amount of data to first build the model and second to validate its operation. It is rare to have a situation where there is a large amount of data readily to hand. Therefore, it is very good practice to form some kind of generator that is able to provide the amount of data needed to asses all the necessary aspects of the formed model. The simulated data is intended only to enable a thorough analysis of the modelling process alone, before it can be used on "live" data (Figure 2). By no means is the simulation intended to clarify the physical phenomena that take place in the CCG. The simulation of the inverted magnetron characte- ristic uses the basic Equation (1), which combines the pressure in the vacuum chamber and the ion current of the inverted magnetron. Equation (1) also includes two parameters that depend on the magnetic flux density, the operating voltage, the geometry, and the materials used to fabricate the device. Figure 2 depicts the measured characteristic (U-p-I) measured at four different pressure values, with different operating voltages between 2.5 kV and 7.5 kV. There- fore, for each operating voltage we have four different ion-current values for different chamber pressures. At first we have to assess the values for parameters k and n. From four different characteristic points, the least-squares method was used to calculate k and n at all voltages. The upper part of Table 1 contains the assessed values that use the measured values of the CCG. These values represent the initial assessment of where the simulated values should be. In the simulated characte- ristic, a slightly narrower range was used (Table 1 – I. BELI^: MODELLING THE CHARACTERISTICS OF AN INVERTED MAGNETRON USING NEURAL NETWORKS Materiali in tehnologije / Materials and technology 43 (2009) 2, 85–95 87 Figure 3: Simulation of the ideal characteristic of the inverted magnetron. Different curves are due to the different operating voltage U, with the appropriate parameters k and n. The right-hand figure is the log 10 of the figure on the left-hand side. Slika 3: Simulirana idealna karakteristika invertnega magnetrona. Parameter pri razli~nih krivuljah je delovna napetost U s pripadajo~ima k in n. Desna slika je deseti{ki logaritem leve slike. Figure 2: The measured characteristic of the inverted magnetron (log values for p and I) (Measurements were obtained by dr. Bojan Erjavec, IMT, Ljubljana, Slovenia.). Please note that the operating range of the device spans several decades, which complicates the modelling process. Slika 2: Izmerjena karakteristika invertnega magnetrona (logaritem- ske vrednosti) (Meritve je izvedel dr. Bojan Erjavec, IMT). Zaradi merilnega podro~ja, ki obsega podro~je ve~ dekad, je logaritmiranje nujno, sicer grafi~en prikaz ne bi bil smiseln. lower part). The values used to create the simulation are printed in the same table. Tabela 1: Measured and simulated values for the parameters k and n Tabela 1: Izra~unane in simulirane vrednosti za parametra k in n MEASURED VALUES Operating voltage U/kV  2.5 O.9130 1.0026 3 1.3866 0.9973 3.5 1.8302 1.0371 4 2.2576 1.0657 4.5 2.7023 1.1448 5 3.0021 1.1127 5.5 3.2945 1.0753 6 3.5140 1.0412 6.5 3.9903 1.0319 7 4.3364 1.0218 7.5 4.9055 1.0398 SIMULATED VALUES Operating voltage U/kV  1 1.0000 1.0000 1.3 1.0500 1.0200 1.6 1.1000 1.0400 1.9 1.1500 1.0600 2.2 1.2000 1.0800 2.5 1.2500 1.1000 2.8 1.3000 1.1200 3.1 1.3500 1.1400 3.4 1.4000 1.1600 3.7 1.4500 1.1800 4 1.5000 1.2000 For the selected values of k and n (lower part of Table 1), the ideal characteristic U-p-I is generated. This follows the logarithm (base 10) of the pressure p and the ion current I, while keeping the operating voltage con- stant (Figure 3). It is not our goal to simulate the ideal characteristic, in fact we need the characteristic that includes the departures from such idealizations. The ideal characte- ristic is therefore modified in a few steps. All the modi- fications are made on data in log space. The first modification changes the value of the ion current versus pressure (Figure 4). The modification follows Equation (3). lg (I2) = lg (I1) + sin ((lg(p) + 7)/4) (3) Here, I1 represents the ion current prior to the modifi- cation, while the current I2 represents the value after it. The second modification bends the CCG characte- ristic with the regard to the operating voltage – Equa- tion (4). lg (I3) = lg (I2) + sin ((U – 1)/3) (4) The current I2 is the value prior to, and I3 is after, the second modification (Figure 5). The third modification introduces random fluctuat- ions to the so-far modified characteristic. The modifica- tion follows equation (5). lg (I4) = lg (I3) + m Rand() (5) As in previous modifications, the current I3 holds the value prior to, and the I4 after, the modification. The generator of random numbers is denoted by Rand(). It generates pseudo random numbers with values from –1 to +1, while the parameter m sets the magnitude of the influence of the randomization process. The result of the third modification is shown in Figure 6. The three modifications form the simulated CCG characteristic, which is presented in 3D in Figure 7. The data of the simulated CCG characteristic is gathered in Table 2. The same data can also be presented in the para- meterized graph shown in Figure 8. The similarity between the characteristic from Figure 1 and Figure 8 is obvious. The similarity between the actual CCG characteristic and its simulated counterpart is close enough to enable a study of the modelling properties of the neural network. I. BELI^: MODELLING THE CHARACTERISTICS OF AN INVERTED MAGNETRON USING NEURAL NETWORKS 88 Materiali in tehnologije / Materials and technology 43 (2009) 2, 85–95 Figure 4: The first modification of the ideal CCG characteristic – it bends the characteristic with regard to the pressure. Different curves have a different operating voltage U, and the parameters k and n. The right-hand figure is the log 10 of the figure on the left-hand side. Slika 4: Prva korekcija idealne karakteristike – ukrivljenost glede na tlak. Parameter pri razli~nih krivuljah je delovna napetost U s pripadajo~ima k in n. Desna slika je deseti{ki logaritem leve slike. 1.2 The neural-network modelling of the CCG characte- ristic The calibration process for the inverted magnetron is a time-consuming task. The neural-network modelling of the characteristic must provide a reduction of the required number of calibration points and it should model the characteristic in the whole usable space. The central problem of modelling the CCG is the fact that its operation spans a large range, which is true for the current (10–11 A to 10–4 A) as well as for the pressure (10–9 to 10–6 mbar). I. BELI^: MODELLING THE CHARACTERISTICS OF AN INVERTED MAGNETRON USING NEURAL NETWORKS Materiali in tehnologije / Materials and technology 43 (2009) 2, 85–95 89 Figure 5 : The second modification – the characteristic is bent with regard to the operating voltage. Different curves have a different operating voltage U, and the parameters k and n. The right-hand figure is the log 10 of the figure on the left-hand side. Slika 5: Druga korekcija karakteristike – ukrivljenost glede na napetost. Parameter pri razli~nih krivuljah je delovna napetost U s pripadajo~ima k in n. Desna slika je deseti{ki logaritem leve slike. Figure 6 : The third modification – the randomization process. Different curves have a different operating voltage U, and the parameters k and n. The right-hand figure is the log 10 of the figure on the left-hand side. Slika 6: Tretja korekcija karakteristike – naklju~na sprememba. Parameter pri razli~nih krivuljah je delovna napetost U s pripadajo~ima k in n. Desna slika je deseti{ki logaritem leve slike. Figure 8: The parameterized view of the simulated characteristic of the CCG. The pressure p is the parameter for the presented curves. The higher curve is obtained at higher pressure. Slika 8: Primer parametriziranega prikaza simulirane karakteristike invertnega magnetrona. Parameter je tlak v vakuumski komori – vi{ja krivulja je dobljena pri vi{jem tlaku. Figure 7: An example of the simulated characteristic of the mag- netron Slika 7: Primer simulirane karakteristike magnetrona The neural-network approximation requires that both the input and the output values are mapped in the range from 0 to 1 (or in some versions from –1 to +1). The main problem of mapping is that the small values are modelled with a very low precision. The problem is addressed in detail in 19. Basically, we have two strate- gies to deal with the problem, one is to transform the data in the log space, and the other is to split the charac- teristic into the appropriate number of segments 5,20. The solution to the problem of modelling the large data range with the neural networks can not be found in the literature. In such cases it is the usual approach to use the log transformation of the whole data space and then execute the modelling in log space. Nowadays, com- puters are very fast, they provide very large memory capacities, and so there is no difficulty in addressing the problem from another perspective. Instead of performing the log transformation, the data space can be segmented I. BELI^: MODELLING THE CHARACTERISTICS OF AN INVERTED MAGNETRON USING NEURAL NETWORKS 90 Materiali in tehnologije / Materials and technology 43 (2009) 2, 85–95 Table 2: The simulated CCG characteristic. The ion current of the inverted magnetron I/A in relation to the pressure p/mbar, and the operating voltage U/kV. The CCG characteristic is divided into 10 segments for further processing. The central part of the table represents the ion current I/A. The segments overlapping areas are shaded in gray. Tabela 2: Simulirana karakteristika; katodni tok invertnega magnetrona I/A v odvisnosti od tlaka p/mbar in delovne napetosti U/kV. Celotna karakteristika je zaradi potreb v nadaljevanju razdeljena na 10 segmentov. Vse vrednosti v osrednjem delu tabele so katodni tok I/A. Sivo obarvana polja vsebujejo podatke, kjer se segmenti glede na vrednost tlaka p prekrivajo.  4 3.7 3.4 3.1 2.8 2.5 2.2 1.9 1.6 1.3 1  I/A 1.00E-09 1.00E-10 7.37E-11 5.07E-11 3.05E-11 1.53E-11 6.19E-12 2.00E-12 5.20E-13 1.13E-13 2.14E-14 3.79E-15 2.00E-09 2.13E-10 1.63E-10 1.16E-10 7.22E-11 3.75E-11 1.57E-11 5.24E-12 1.41E-12 3.17E-13 6.24E-14 1.14E-14 3.00E-09 3.52E-10 2.74E-10 1.99E-10 1.27E-10 6.71E-11 2.87E-11 9.77E-12 2.69E-12 6.16E-13 1.24E-13 2.31E-14 4.00E-09 5.15E-10 4.07E-10 3.00E-10 1.93E-10 1.04E-10 4.51E-11 1.56E-11 4.35E-12 1.01E-12 2.06E-13 3.91E-14 5.00E-09 7.01E-10 5.60E-10 4.17E-10 2.72E-10 1.48E-10 6.49E-11 2.27E-11 6.40E-12 1.51E-12 3.10E-13 5.95E-14 6.00E-09 9.10E-10 7.34E-10 5.52E-10 3.63E-10 1.99E-10 8.81E-11 3.11E-11 8.86E-12 2.10E-12 4.37E-13 8.46E-14 7.00E-09 1.14E-09 9.27E-10 7.03E-10 4.66E-10 2.58E-10 1.15E-10 4.09E-11 1.17E-11 2.80E-12 5.87E-13 1.15E-13 8.00E-09 1.39E-09 1.14E-09 8.70E-10 5.81E-10 3.23E-10 1.45E-10 5.19E-11 1.50E-11 3.61E-12 7.61E-13 1.50E-13 9.00E-09 1.67E-09 1.37E-09 1.05E-09 7.07E-10 3.96E-10 1.79E-10 6.44E-11 1.87E-11 4.53E-12 9.60E-13 1.90E-13 1.00E-08 1.96E-09 1.62E-09 1.25E-09 8.46E-10 4.76E-10 2.16E-10 7.82E-11 2.28E-11 5.56E-12 1.19E-12 2.36E-13 2.00E-08 6.01E-09 5.15E-09 4.11E-09 2.88E-09 1.67E-09 7.87E-10 2.95E-10 8.92E-11 2.25E-11 4.96E-12 1.02E-12 3.00E-08 1.20E-08 1.05E-08 8.52E-09 6.08E-09 3.61E-09 1.73E-09 6.63E-10 2.05E-10 5.26E-11 1.18E-11 2.49E-12 4.00E-08 1.97E-08 1.75E-08 1.44E-08 1.05E-08 6.30E-09 3.07E-09 1.19E-09 3.73E-10 9.72E-11 2.22E-11 4.73E-12 5.00E-08 2.92E-08 2.61E-08 2.19E-08 1.60E-08 9.75E-09 4.80E-09 1.88E-09 5.96E-10 1.57E-10 3.63E-11 7.82E-12 6.00E-08 4.03E-08 3.64E-08 3.07E-08 2.27E-08 1.40E-08 6.93E-09 2.74E-09 8.77E-10 2.34E-10 5.44E-11 1.18E-11 7.00E-08 5.29E-08 4.83E-08 4.10E-08 3.05E-08 1.89E-08 9.47E-09 3.78E-09 1.22E-09 3.27E-10 7.67E-11 1.68E-11 8.00E-08 6.72E-08 6.16E-08 5.27E-08 3.95E-08 2.47E-08 1.24E-08 4.99E-09 1.62E-09 4.37E-10 1.03E-10 2.28E-11 9.00E-08 8.29E-08 7.65E-08 6.58E-08 4.96E-08 3.12E-08 1.58E-08 6.38E-09 2.08E-09 5.65E-10 1.34E-10 2.98E-11 1.00E-07 1.00E-07 9.28E-08 8.03E-08 6.09E-08 3.84E-08 1.96E-08 7.95E-09 2.61E-09 7.12E-10 1.70E-10 3.79E-11 2.00E-07 3.43E-07 3.30E-07 2.95E-07 2.32E-07 1.51E-07 7.98E-08 3.36E-08 1.14E-08 3.22E-09 7.97E-10 1.84E-10 3.00E-07 6.97E-07 6.83E-07 6.25E-07 5.00E-07 3.33E-07 1.79E-07 7.70E-08 2.67E-08 7.70E-09 1.94E-09 4.58E-10 4.00E-07 1.14E-06 1.14E-06 1.05E-06 8.55E-07 5.79E-07 3.16E-07 1.38E-07 4.83E-08 1.41E-08 3.63E-09 8.66E-10 5.00E-07 1.66E-06 1.67E-06 1.57E-06 1.29E-06 8.81E-07 4.86E-07 2.14E-07 7.61E-08 2.25E-08 5.84E-09 1.41E-09 6.00E-07 2.25E-06 2.28E-06 2.16E-06 1.79E-06 1.24E-06 6.88E-07 3.06E-07 1.10E-07 3.28E-08 8.57E-09 2.09E-09 7.00E-07 2.89E-06 2.96E-06 2.82E-06 2.36E-06 1.64E-06 9.20E-07 4.12E-07 1.49E-07 4.48E-08 1.18E-08 2.90E-09 8.00E-07 3.58E-06 3.69E-06 3.54E-06 2.98E-06 2.09E-06 1.18E-06 5.32E-07 1.93E-07 5.86E-08 1.55E-08 3.85E-09 9.00E-07 4.32E-06 4.48E-06 4.32E-06 3.66E-06 2.58E-06 1.46E-06 6.64E-07 2.43E-07 7.40E-08 1.98E-08 4.92E-09 1.00E-06 5.09E-06 5.31E-06 5.15E-06 4.38E-06 3.10E-06 1.77E-06 8.08E-07 2.97E-07 9.11E-08 2.44E-08 6.11E-09 2.00E-06 1.43E-05 1.54E-05 1.55E-05 1.36E-05 9.97E-06 5.90E-06 2.78E-06 1.06E-06 3.36E-07 9.34E-08 2.42E-08 3.00E-06 2.48E-05 2.73E-05 2.80E-05 2.51E-05 1.88E-05 1.13E-05 5.46E-06 2.12E-06 6.87E-07 1.95E-07 5.15E-08 4.00E-06 3.58E-05 3.99E-05 4.16E-05 3.79E-05 2.87E-05 1.76E-05 8.60E-06 3.39E-06 1.11E-06 3.20E-07 8.59E-08 5.00E-06 4.69E-05 5.29E-05 5.57E-05 5.13E-05 3.94E-05 2.44E-05 1.21E-05 4.81E-06 1.60E-06 4.64E-07 1.26E-07 6.00E-06 5.79E-05 6.60E-05 7.01E-05 6.52E-05 5.05E-05 3.15E-05 1.57E-05 6.33E-06 2.12E-06 6.22E-07 1.70E-07 7.00E-06 6.88E-05 7.90E-05 8.45E-05 7.92E-05 6.18E-05 3.89E-05 1.96E-05 7.93E-06 2.68E-06 7.92E-07 2.18E-07 8.00E-06 7.95E-05 9.18E-05 9.89E-05 9.33E-05 7.33E-05 4.65E-05 2.35E-05 9.60E-06 3.26E-06 9.72E-07 2.70E-07 9.00E-06 8.99E-05 1.04E-04 1.13E-04 1.07E-04 8.49E-05 5.41E-05 2.76E-05 1.13E-05 3.87E-06 1.16E-06 3.24E-07 1.00E-05 1.00E-04 1.17E-04 1.27E-04 1.21E-04 9.65E-05 6.19E-05 3.16E-05 1.31E-05 4.49E-06 1.35E-06 3.79E-07 2.00E-05 1.88E-04 2.27E-04 2.56E-04 2.53E-04 2.08E-04 1.38E-04 7.31E-05 3.12E-05 1.11E-05 3.46E-06 1.01E-06 3.00E-05 2.56E-04 3.16E-04 3.63E-04 3.66E-04 3.07E-04 2.08E-04 1.12E-04 4.91E-05 1.78E-05 5.67E-06 1.68E-06 4.00E-05 3.11E-04 3.89E-04 4.54E-04 4.64E-04 3.96E-04 2.72E-04 1.49E-04 6.59E-05 2.43E-05 7.84E-06 2.36E-06 5.00E-05 3.56E-04 4.51E-04 5.33E-04 5.51E-04 4.75E-04 3.30E-04 1.83E-04 8.18E-05 3.05E-05 9.94E-06 3.02E-06 6.00E-05 3.95E-04 5.05E-04 6.02E-04 6.28E-04 5.46E-04 3.83E-04 2.14E-04 9.67E-05 3.64E-05 1.20E-05 3.68E-06 7.00E-05 4.29E-04 5.53E-04 6.64E-04 6.98E-04 6.11E-04 4.32E-04 2.44E-04 1.11E-04 4.20E-05 1.39E-05 4.31E-06 8.00E-05 4.59E-04 5.95E-04 7.20E-04 7.62E-04 6.71E-04 4.78E-04 2.71E-04 1.24E-04 4.74E-05 1.58E-05 4.93E-06 9.00E-05 4.86E-04 6.33E-04 7.70E-04 8.20E-04 7.27E-04 5.20E-04 2.97E-04 1.37E-04 5.25E-05 1.76E-05 5.53E-06 1.00E-04 5.09E-04 6.68E-04 8.16E-04 8.74E-04 7.79E-04 5.60E-04 3.22E-04 1.49E-04 5.75E-05 1.94E-05 6.11E-06 into the convenient sub-spaces and the modelling process should be executed for each segment separately (Figure 9). Thus, separate models are created for each separate segment. It is of vital importance that the data is segmented in such a way that the segments are not too wide, and that we have enough data for each separate segment to do the modelling. The segmentation theory shows the following important details: • Both the input and output spaces are divided into several sub-spaces called segments. Each segment has its own multiplication constant to map the area close to the 0, 1 interval. • The neural-network training tolerance is valid for each segment only. • When all the models are formed, the process of merging them again into the single characteristic has to be accomplished. The modelling error is again valid for each segment separately. The actual output value of the model ya is calculated from the output value y of the neural network using the equation ya = kiy (6) where ki denotes the multiplication constant of the i-th segment, ya is the scaled value of the value y produced by the neural-network model. The error produced by the model of the i-th segment can be calculated as it stands in Equation (7). ∆ ∆y F y y ii n ia = ⇒ = ∑ ∂∂1 ∆ ∆y k yia = ; ki << 1 (7) where the function F represents the modelled function and the meaning of the other symbols is the same as in Equation (6). We have found that reasonably good modelling results can be acheived if at least five data points are available for each segment (it is true for our CCG example). The division of the modelled space depends on various parameters. The most important parameters are the shape of the modelled function and the admi- ssible relative error that the model should fulfil. Table 2 shows the segmentation of the CCG characteristic into 10 segments. The segments should overlap in order to allow the merging of the segments when the separate segments are modelled. The overlapping region is shown with the gray background (Table 2). Neural networks (due to the pre-set training criteria) perform well at higher values, so when two segments are to be merged, one segment has locally high values, while the other is to be joined with the locally low values. During the merging process it is more likely that the data from the segment that is to be merged with the locally high values I. BELI^: MODELLING THE CHARACTERISTICS OF AN INVERTED MAGNETRON USING NEURAL NETWORKS Materiali in tehnologije / Materials and technology 43 (2009) 2, 85–95 91 Figure 9: The segmentation strategy – the complete data space is segmented into several sub-spaces. Each segment is then modelled separately Slika 9: Strategija segmentacije – delitve podro~ja na ve~ podpodro- ~ij. Vsak segment je modeliran posebej Figure 10: The CCG characteristic has been split into 10 segments. Each segment is modelled on its own – a separate neural network. Slika 10: Celotna karakteristika invertnega magnetrona je razdeljena na 10 segmentov, vsak segment modelira svoj nevronski sistem. are more accurately modelled, and some kind of weighting (linear, nonlinear, etc.) should be used. 2 EXPERIMENTAL Table 2 holds the data of the CCG characteristic, which has been divided into 10 segments. Figure 10 graphically represents the segmenting process. The segments were formed in such a way as to ensure that for each segment the ion current I covers as little area as possible. The main idea is that each segment should cover such data space to ensure that the model will produce results with acceptable errors. Table 3: The data space covered by the separate segments Tabela 3: Segmenti karakteristike invertnega magnetrona in podro~ja, ki jih posamezni segmenti obsegajo Segment   1 1.00E-09 to 6.00E-09 3.79E-15 to 8.86E-12 2 6.00E-09 to 5.00E-08 8.46E-14 to 5.96E-10 3 5.00E-08 to 4.00E-07 7.82E-12 to 4.83E-08 4 4.00E-07 to 4.00E-06 8.66E-10 to 3.39E-06 5 4.00E-06 to 1.00E-04 8.59E-08 to 1.49E-04 6 1.00E-09 to 6.00E-09 5.20E-13 to 9.10E-10 7 6.00E-09 to 5.00E-08 8.86E-12 to 2.92E-08 8 5.00E-08 to 4.00E-07 5.96E-10 to 1.14E-06 9 4.00E-07 to 4.00E-06 4.83E-08 to 4.16E-05 10 4.00E-06 to 1.00E-04 3.39E-06 to 8.74E-04 Table 3 shows the chosen segments and the area coverage for the pressure p as well as for the CCG ion current I. A brief inspection of the segments reveals that some segments still cover an area that spans well over two decades. It is a necessary trade off since the introduction of even more segments would require more data points, which does not represent the problem in the simulated environment, but for the real CCG calibration it can pose a problem. For testing purposes a 5% training tolerance was selected. 2.1 The testing environment In the experimental work we studied the modelling capabilities of neural networks, while at the same time we were seeking the neural-network architecture that would show the best modelling properties for the given problem. The approximation theory for use with the neural networks was corrected and published 20. In the same publication, the concept of the neural-network training stability was introduced. The training stability deals with the variability of various possible neural- network models and sets the boundary where all possible models (obtained with different configurations) give their results. The testing of various neural-network architectures was organised in an orderly fashion (Table 4), where the set of numbers represents the number of artificial neural cells in the appropriate layer. For clarification please refer to 20. For example, the notation 2 10 20 1 means that the input layer consists of 2 neurons, the first hidden layer of 10 neurons, the second hidden layer of 20 neurons, and finally the output layer contains 1 neuron. Since the experiment took quite some time to complete, it was necessary to develop a system that controls the experiments and in the case of power failure resumes with work where it has been interrupted. The log file was I. BELI^: MODELLING THE CHARACTERISTICS OF AN INVERTED MAGNETRON USING NEURAL NETWORKS 92 Materiali in tehnologije / Materials and technology 43 (2009) 2, 85–95 Table 4: The organisation of the different neural-network configurations included in the experiment Tabela 4: Seznam preizku{enih konfiguracij nevronskih sistemov CONFIGURATION CONFIGURATION CONFIGURATION CONFIGURATION 1 2 5 5 1 21 2 5 10 5 1 41 2 10 15 5 1 61 2 15 20 5 1 2 2 5 10 1 22 2 5 10 10 1 42 2 10 15 10 1 62 2 15 20 10 1 3 2 5 15 1 23 2 5 10 15 1 43 2 10 15 15 1 63 2 15 20 15 1 4 2 5 20 1 24 2 5 10 20 1 44 2 10 15 20 1 64 2 15 20 20 1 5 2 10 5 1 25 2 5 15 5 1 45 2 10 20 5 1 65 2 20 5 5 1 6 2 10 10 1 26 2 5 15 10 1 46 2 10 20 10 1 66 2 20 5 10 1 7 2 10 15 1 27 2 5 15 15 1 47 2 10 20 15 1 67 2 20 5 15 1 8 2 10 20 1 28 2 5 15 20 1 48 2 10 20 20 1 68 2 20 5 20 1 9 2 15 5 1 29 2 5 20 5 1 49 2 15 5 5 1 69 2 20 10 5 1 10 2 15 10 1 30 2 5 20 10 1 50 2 15 5 10 1 70 2 20 10 10 1 11 2 15 15 1 31 2 5 20 15 1 51 2 15 5 15 1 71 2 20 10 15 1 12 2 15 20 1 32 2 5 20 20 1 52 2 15 5 20 1 72 2 20 10 20 1 13 2 20 5 1 33 2 10 5 5 1 53 2 15 10 5 1 73 2 20 15 5 1 14 2 20 10 1 34 2 10 5 10 1 54 2 15 10 10 1 74 2 20 15 10 1 15 2 20 15 1 35 2 10 5 15 1 55 2 15 10 15 1 75 2 20 15 15 1 16 2 20 20 1 36 2 10 5 20 1 56 2 15 10 20 1 76 2 20 15 20 1 17 2 5 5 5 1 37 2 10 10 5 1 57 2 15 15 5 1 77 2 20 20 5 1 18 2 5 5 10 1 38 2 10 10 10 1 58 2 15 15 10 1 78 2 20 20 10 1 19 2 5 5 15 1 39 2 10 10 15 1 59 2 15 15 15 1 79 2 20 20 15 1 20 2 5 5 20 1 40 2 10 10 20 1 60 2 15 15 20 1 80 2 20 20 20 1                                                            !   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