1. Treating simulation data with mathematical functions using chi-square technique **formulas achieved by using Chi - square based Java applets onhttp://csbsju.edu *relationship btw outputs ( filling,spread ) and inputs ( thickness red.,friction,)

2. Thickness reduction ( r ) variation Rolling model Filling factor = H 1 t (H 1= central height) 3. Fixed condition with: friction 0.15; sample width 40mm, thickness 5mm, groove 10mm and 20 degrees. 0.00 40.00 5.00 0 1.22 42.36 4.22 40 1.07 41.52 4.32 35 0.95 41.27 4.45 30 0.76 41.06 4.51 25 0.59 41.03 4.59 20 0.24 40.33 4.74 10 Filling factor (as drawn) S (mm) spread H r(mm)(central) Thickness reduction,% 4. Other fixed conditions,relationship btw Hr and r (thickness red.) Drawn with Origin 7.0 from simulation results The central (max) height 5. Linear model : y = a + bx( http://www.physics.csbsju.edu/stats/WAPP.html ) Hr = 4.992 0.019r Test with r = 40 Hr = 4.992 0.019 x 40 = 4.232 Measured H r (40%) = 4.22 mmerror (relative) = (4.232 4.22)/4.22 =0.28% Test with r = 10 Hr = 4.992 0.019 x 10 = 4.80. error = (4.8 4.74)/4.74 =1.25% 6. Graph showing the regression line and the data points Regression line Data point 7. Filling factor and thickness reduction 8. Rolled width vs thickness reduction Plotted with Origin 9. Quadratic model : y = a + bx + cx 2( http://www.physics.csbsju.edu/cgi-bin/stats/WAPP ) Sr = 40.03 + 0.01862r + 0.0009461r 2 Test with r = 35Sr = 40.03 + 0.01862x35 + 0.0009461x35 2= 41.84 Measured S r (35%) = 4.1.52 mmerror (relative) = (41.84 41.52)/41.52 =0.77% 10. Graph showing the regression curve and the data points Regression curve Data point 11. 12. Friction factor ( m ) variation 13. Friction factor variation data from DEFORM 4.22/4.76 4.47/40.97 4.57/40.68 0.60 4.27/41.73 4.44/41.02 4.50/40.65 0.40 4.27/41.70 4.48/41.05 4.54/40.76 0.35 4.28/41.75 4.51/41.04 4.53/40.79 0.30 4.25/41.90 4.48/41.17 4.53/40.82 0.25 4.18/42.28 4.47/41.24 4.57/40.89 0.20 4.22/42.36 4.45/41.27 4.59/40.96 0.15 40% 30% 20% 14. Not significant effect on central height 15. Try parabolic model for (1) (1) On width Half quadratic curve 16. Sr = 43.61 10.48m + 14.36m 2

Test with m = 0.20

Sr = 43.61 10.48x0.2 + 14.36x0.04)

= 42.09

Error = (42.28 42.09)/42.09

=0.45%

17. Quadratic graph 18.

Central height & filling vs thickness reductionrelationship can be treated withlinear function

Spreading vs thickness reductioncan be well treated withquadratic relationship

Friction does not affect central height

Spread vs frictioncan be described by aquadratic function

Upcoming:thickness, width, vacancy, inclined angle ,

Conclusions 19. Appendix 20. Appendix