UDK 621.771:621.892 Original scientific article/Izvimi znanstveni članek ISSN 1580-2949 MTAEC9, 45(1)75(2011) SIMILARITY CRITERIA AND EFFECT OF LUBRICANT INERTIA AT COLD ROLLING MERILA PODOBNOSTI IN VPLIV VZTRAJNOSTI MAZIVA PRI HLADNEM VALJANJU Dušan Curcija1, Ulja Mamuzic1, Franc Vodopivec2 iCroatian Metallurgical Society, Berislaviceva 6, 10000 Zagreb, Croatia 2Institute of Metals and technology, Lepi pot 11, 1000 Ljubljana, Slovenia plutonijanac21 @net.hr Prejem rokopisa - received: 2010-09-01; sprejem za objavo - accepted for publication: 2011-01-17 Modern rolling mills operate at speed requiring the consideration of inertial forces for the explanation of the behaviour of lubricant layer in the enter gap of the metal deformation zone. For this calculation a similarity criterium and the improved Mizun-Grudev equation were applied. Good results were obtained in comparison with the numerical Monte Carlo method. A significant point of the calculation is the definition of the initial state in the point of singularity. Key words: cold rolling, lubrification, lubricant inertia, gripping angle, Reynolds equation, Monte Carlo method Moderne valjarne obratujejo pri hitrostih valjanja, ki zahtevajo upoštevanje sile vztrajnosti za razlago vedenja plasti maziva na vhodni reži zone deformacije metala. Za izračun sta bili uporabljeni merilo podobnosti in Mizum-Gridevova enačba. Dobljeni so dobri rezultati v primerjavi z numerično metodo Monte Carlo. Pomembna točka izračuna je definicija začetnega stanja v točki singularnosti. Ključne besede: hladno valjanje, mazanje, vztrajnost maziva, kot oprijema, Reynoldsova enačba, metoda Monte Carlo 1 INTRODUCTION The lubricant achieves in the enter gap of the metal deformation1-5 zone a wedge shape determined with the geometry of the rolls and the rolled sheet, as shown in Figure 1. The flow of lubricant in the rolling gap can be described with the simplified Reynol's differential 6-9 equations: dp d2 v dp d;=0 dv dx dv - = 0 Z =- dx -dv+C (x) (1a) (1b) (1c) (1d) According to the differential equation (1b), the pressure in the lubricant layer is constant over the gap height and that it changes along the layer length, only, and the approximate analytical solution of equation (1a) is: 1 dpy u dx 2 (v) Assuming boundary conditions from Figure 1 v^ = ^ v = 0 v = v v = e( x) (1e) (1f) the integration constants are: C = vRx -v 0 -1 dp e(x) 1 e( x) u dx 2 C 2 = v 0 Including (1g) and (1h) in (1d) we obtain 1 dp 2 , ^ r vRx - v 0 1 v --v -£(x)v + ----v +v (1g) (1h) 7 Lv -e(x)v J+ . . iv +v 0 (1i) 2u d^ L £(x) J 0 The lubricant consumption along the strip perimeter is: 1 dp r ^ + 1 ß(x) =J vxdy =-J2U Yx ^ 0.03 rad. Further, equation (8) preserves the same law of iso values than the Mizunov-Grudev equation. It follows, that by applying the criteria of iso values it is possible by cold rolling to pass over from a greater to a smaller gripping angle and to calculate the lubricant layer thickness according to equation (8) using the relation at = (m)±(1'3)a. The values in Table 2 cannot be calculated using equation (1) developed to a polynome of the third or higher order but only to the square polynome according to equation (5b). Figure 3: Modelled solutions for Figure 2 at a = 0.03 rad. Same denotations as in Figure 2. The tool REVSURF allows the rotation around the linearisatioin method (curve 3). Slika 3: Modelirane re{itve za sliko 2 pri a = 0.03 rad. Iste ozna~be kot na sliki 2. Orodje REVSURF omogo~a rotacijo okoli metode linearizacije (krivulja 3). M 0 In Figure 2 the comparison of results of the calculation of the lubricant layer thickness according to different methods is shown. The figures show that the linearisation method can be applied only for the dressing rolling. For cold rolling the Mizunov-Grudev solution differs from the numerical solution for aproximately a = 0.06 rad, while equation (8) differs for only a = 0.025 rad. For the range a = 0.02 to a = 0.03 rad on Figure 2 the criterium of similarity can be used and an approximate solution can be obtained with interpolation from known values. The criterium of similarity is based on equation (6) and the selected transfer function was a hyperbola. The thickness of the lubricant layer is calculated using the linearisation method and applying the standard data in Table 2 over the guider to the case on Figure 2. The hyperbola is used to transmit the lubricant layer thickness defined by the equation (8), as in this case, or using the boundary conditions for a = 0.02 and a = 0.03, as shown in Table 3. In this table the comparison is given of the lubricant layer thickness calculated using the similarity criteria and the Monte Carlo method for the gripping angles indicated. The guider is defined for R = 0.35 and A = 8 ■ 105 m1. It is clear from Table 3 that the methods of linearisation according to equation (6), of similarity criteria and of the improved Mizunov-Grudev equation are complementar and represent an algorithm for the thickness of the lubricant layer suitable for fast practical application. In Figure 3 the AutoCAD modelling of Figure 2 using different modelling methods is shown. In the initial part of abscissa a good approximation is obtained between the numerical and the linearisation method and that in the border part of abscissa with a = 0.03 the agreement between the corrected equation (9) and the numerical Monte-Carlo calculation is acceptable. Straight lines obtained with linear programming and derivation at the point of singularity are shown in Figure 4. The linear representation was selected to obtain the optimal transfer of similarity criteria to the cylinder-clone for which the solution of differential equation is obtained using the known solutions for the guider and the standard on the base of only one solution for the singular point (a*, £*o). This approach is valid for gripping angles in the range 0.0124 rad to 0.0404 rad with allowed angle change for Aa ~ 0.00362 rad. In the second part of Table 3 the transfer of similarity criteria from the standard over the guider to the cylinder clone using the straight lines 4, 5 and 6 in Figure 5, is shown. If the strainght lines transferring the accuracy from the first to the fourth quadrant become unreliable for the transfer of similarity criteria, it is possible to use, as help, the singular point derivation with direction of transfer of similarity criteria to the first quadrant for which the calculation of the lubricant layer thickness is performed. The singular point transfers well the calculation of the lubricant film, although the gripping angle is relatively distant. It is usefull to remind that the method of derivation in Table 3 offers the possibility of increasing the mathematical accuracy in comparison to the Monte-Carlo method. In this case, the simple analytical definition would approach a form approaching equation (8). Further, data in Table 3 show that the improved equation (8) can, for smooth surfaces of rolls and of rolled metal, substitute the Monte Carlo method. In this case, the transfer of similarity criteria for the standard and the guider can be calculated using equation (8), while the clone cylinder is defined mathematically with differential equations connecting the longitudinal and transverse roughness and with possible analytical solutions for the point of singularity. The solution of differential equations related to the equation (1a), as for instance equation (2), can be obtained through the solution for one point, as boundary condition, in this case the solution for the singular point. If the accuracy as in Table 1 is expected, the calculation of the lubricant layer for smooth surfaces can not be simplified further. Using a complex method of statistical analysis of data in Table 2 and for the rolls of diameter of R = 0.3 m the following relation was developed for the lubricant layer thickness: Figure 4: The method of linearisation (1, 2, 3) through the points (0 : e10) and (a* : e*0) and of derivation in the point (a* : e*0) (4, 5, 6) Standard (3, 4), guider (1, 5), Clone cylinder (2, 6) Slika 4: Metoda linearizacije (1, 2, 3) skozi točke (0 : e10) in (a* : e*0) in derivacije v točki (a* : e*0) (4, 5, 6). Standard (3, 4), vodilo (1, 5), valj klon (2, 6) Figure 5: Relationship lubricant layer thickness - rolling gap geometrical shape Slika 5: Odvisnost debeline plasti maziva od geometrične oblike va-ljalniske reže S 0. 8519 B -0. 2633 C 0.0346 L -0.0015 Rsq 0. 99999 F 66829 . 2 d . f. 5 Q, W Abscissa and ordinate W = (e M/ e T)«2 (9) The results of the calculation are shown in Table 4. Table 4: Some statistical data on the regression analysis Tabela 4: Nekateri statisti~ni podatki o regresijski analizi 0.8519 -0.2633 0.0346 -0.0015 0.99999 66829.2 a/rad 0 0.011335 0.02 0 03 0 04 0 05 e0/m 15. 863 • 10-6 12. 255 • 10-6 9 .416 • 10-6 7.244 • 10-6 5 .797 • 10-6 4. 695 • 10-6 eIN0/m 15. 863 • 10-6 12. 335 • 10-6 9 . 542 • 10-6 7.447 • 10-6 5 . 959 • 10-6 4.906 • 10-6 3 EFFECTS OF LUBRICANT INERTIA The effect of lubricant inertia depends on the speed of the processing, the gap geometry and the rheological characteristics of the lubricant. In this work only investigations of the effect in the gap geometry is discussed. It was deduced with a numerical solution of the equation (1) applying the Monte-Carlo method with A = 1 965 512 m-1, R = 0.2 m, p = 854 kg/m3, dressing angle 0.02 rad and other data as in Table 2 and it is shown in Figure 4. The reliable analysis range lies both sides near of the point 2 in Figure 5. On the arc 2^3 of the curve, by increasing of the strip lubricant layer thickness, the thickness of the lubricant layer in the initial section of the metal deformation zone is increased and the effect of inertia forces is decreased, while, on the arc 2^1 of the Figure 6: Change of pressure gradient for the gripping angler « = 0,02 rad and rolls diameter R = 0.2 m in dependence of the ratio (-a) / (£-a) in the range of - 0.001 m to - 0.000015 m and other parameters as for Table 2 Slika 6: Sprememba gradienta pritiska za kot oprijema a = 0,02 rad in premer valjev R = 0,2 m v odvisnosti od razmerja (-a)/(£-a) v obmo~ju od -0.001 m do - 0.000015 m in drugih pamaterov kot za tabelo 2 curve the effect is opposite. For a selected processing, the vertex at the point 2 can be deduced with a regression analysis for determined processing parameters. In Table 5 data on the effect of inertia forces on ec calculated from equations (1a) and (2) using the Monte-Carlo method are given. The calculation was similar as for the curves in Figure 2 and the data in Table 2. For the arc 2^3 and dressing processing the correction for the lubricant layer thickness would be unnecessary, while, considering the inertia forces it is significant for the arc 2^1. If the working velocity is increased from 16 m/s, as in Table 5, to 50 m/s for the angle 0,05 rad ec = 13.854 • 10-6 m is deduced, while the consideration of inertia forces gives eINc= 14,951 • 10-6m. Data in Table 5 show that the shape of the gap in the area (-a, 0) prevails over the lubricant rheological cha- Table 5: Effect of inertia forces(£'N0) for dressing rolling Tabela 5: Vpliv sil vztrajnosti (£'N0) za oblikovalno valjanje a/rad 0 0.011335 0.02 0.03 0.04 0.05 ec/m 15.863 • 10- 12.255 • 10- 9.416 • ^ 7 .244 • 1C-6 5.797 • 1C-6 4.695 • 1C-6 eINc/m 15.863 • 1C-6 12.335 • 1C-6 9.542 • 1C-6 7.447 • 1C-6 5.959 • 1C-6 4.906 • 1C-6 Table 6: Dependence of the lubricant layer thickness on the gripping angle and the lubricant layer thickness on the strip ahead the deformation zone. Calculated using equations (1a) i (2). Tabela 6: Odvisnost debeline plasti maziva od kota oprijema in debeline plasti maziva na traku pred zono deformacije. Izra~unano z uporabo ena~b (1a) in (2). a/rad ea/ m ea/m ea/m ea/m ea/m 0.001 0.002 0.003 0,004 0.005 Without inertia 12 613- 1C- 12 642- 1C- 12 648- 10- 12 651- 10- 12 652- 10- a = 0 01 12 793 ■ 1C-6 12 853 ■ 1C-6 12 884 ■ 1C-6 12 903 ■ 1C-6 12 912 ■ 10-6 Without inertia 9 390 ■ 10- 9 407 ■ 10- 9 412 ■ 10- 9 414 ■ 10- 9 415 ■ 10- a = 0 02 9 491 ■ 10-6 9 539 ■ 10-6 9 562 ■ 10-6 9 578 ■ 10-6 9 589 ■ 10-6 Without inertia 7 225 ■ 10- 7 238 ■ 10- 7 241 ■ 10- 7 242 ■ 10- 7 243 ■ 10- a = 0 03 7 286 ■ 10-6 7 322 ■ 10-6 7 340 ■ 10-6 7 352 ■ 10-6 7 361 ■ 10-6 Without inertia 5 786 ■ 10- 5 793 ■ 10- 5 795 ■ 10- 5 796 ■ 10- 5 796 ■ 10- a = 0 04 5 820 ■ 10-6 5 848 ■ 10-6 5 862 ■ 10-6 5 871 ■ 10-6 5 878 ■ 10-6 Without inertia 4 788 ■ 10- 4 793 ■ 10- 4 794 ■ 10- 4 795 ■ 10- 4 795 ■ 10- a = 0 05 4 809 ■ 10-6 4 830 ■ 10-6 4 841 ■ 10-6 4 848 ■ 10-6 4 854 ■ 10-6 S B C L R sq F racteristics and the kinematics of the processing because the gripping angle approaches to zero. For this reason, the inertia forces do not affect significantly the lubricant behaviour by dressing processes, while these forces shold be considered by cold rolling. In Table 6 the effect of £a (lubricant layer thickness on the sheet) on £0 (lubricant layer thickness at entrance cross section) is shown, as deduced using equation (2) and the Monte-Carlo calculation with R = 0.2 m and other data, as for Table 5. The value of £0is increased for ~1-2 %, while the value of £a increases for the sheet for five times for the rolling velocity up to 16 m/s. In Table 6 the effect of lubricant height on the sheet on the lubricant sheet on the entering section of the deformation zone is shown as function of lubricant inertia and gripping angle. The inertia effect increases significantly with the gripping angle and even faster with the lubricant height on the sheet ahead the rolls. On Figure 6 the change of pressure gradient ahead the rolling gap is shown with the maximum for x = -0.00025 m. It is very difficult to obtain approximate analytical solutions of the differential equation (2). For this reason, it is analysed applying the numerical Monte-Carlo integration. The analysis should be complemented13 considering surface roughness and with correlations with dependences to dynamkcal viscosity and rolling pressure. 4 CONCLUSIONS On the base of the results of the calculations presented the following conclusions are proposed: - The criteria of similarity are transferred acceptably with the solution of the equation (1a) from the standard over the guider to the cylinder clone applying the method of linearisation up to a gripping angle determined by linear programming. For a greater gripping angle the transfer of similarity criteria can be achieved using straight lines obtained with derivation in the point of singularity. - The effect of inertia forces can be neglected for lower dressing speed, while this effect becomes significant for incresead speed of cold rolling. - The use of the improved Mizunov-Grudev equation (8) gives solutions in good agreement with the numerical solution of equation (1) applying the Monte-Carlo method. Also, it is more practical for use than the transcendent equation (7). - For a determined relation rolls diametere versus gripping angle very similar results are obtained using the Mizunov - Grudev relation and the transcendent equation (7). 5 REFERENCES 1 D. Čurčija, I. Mamuzic, Schemes of metal-working processes and the related tribological equations of fluid mechanics, Mater. Tehnol., 43 (2009) 1, 23-30 2 R. Boman, Jean-Philippe Ponthot, Numerical simulation of lubricated contact in rolling processes, J. Mater. Process. 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