4.
Process Modeling
4.6. Case Studies in Process Modeling 4.6.4. Thermal Expansion of Copper Case Study


C/C Rational Function Model 
Since the Q/Q model did not describe the data well, we
next fit a cubic/cubic (C/C) rational function
model.
Based on the procedure described in 4.6.4.2, we fit the model: $$ y = A_0 + A_1 x + A_2 x^2 + A_3 x^3  B_1 x  B_2 x^2  B_3 x^3 + \varepsilon \, , $$ using the following seven representative points to generate the starting values for the C/C rational function. TEMP THERMEXP   10 0 30 2 40 3 50 5 120 12 200 15 800 20The coefficients from the preliminary linear fit of the seven points are: A0 = 2.323648e+00 A1 = 3.530298e01 A2 = 1.383334e02 A3 = 1.766845e04 B1 = 3.395949e02 B2 = 1.100686e04 B3 = 7.910518e06 

Nonlinear Fit Output 
The results of fitting the C/C model are shown below.
Parameter Estimate Stan. Dev t Value A0 1.07913 0.1710 6.3 A1 0.122801 0.1203E01 10.2 A2 0.408837E02 0.2252E03 18.2 A3 0.142848E05 0.2610E06 5.5 B1 0.576111E02 0.2468E03 23.3 B2 0.240629E03 0.1060E04 23.0 B3 0.123254E06 0.1217E07 10.1 Residual standard deviation = 0.0818 Residual degrees of freedom = 229The regression analysis yields the following estimated model. $$ \hat{y} = \frac{1.079  0.122x + 0.004097x^{2}  0.00000143x^{3}} {1  0.00576x + 0.000241x^{2}  0.000000123x^{3}} $$ 

Plot of C/C Rational Function Fit 
We generate a plot of the fitted rational
function model with the raw data.
The fitted function with the raw data appears to show a reasonable fit. 

6Plot for Model Validation 
Although the plot of the fitted function with the raw
data appears to show a reasonable fit, we need to
validate the
model assumptions.
The 6plot
is an effective tool for this purpose.
The 6plot indicates no significant violation of the model assumptions. That is, the errors appear to have constant location and scale (from the residual plot in row 1, column 2), seem to be random (from the lag plot in row 2, column 1), and approximated well by a normal distribution (from the histogram and normal probability plots in row 2, columns 2 and 3). 

Residual Plot 
We generate a fullsized residual plot in order to
show more detail.
The fullsized residual plot suggests that the assumptions of constant location and scale for the errors are valid. No distinguishing pattern is evident in the residuals. 

Conclusion  We conclude that the cubic/cubic rational function model does in fact provide a satisfactory model for this data set. 