D,15 • VARIABLE THERMAL PROPERTIES 



It will now be observed from Fig. D,15a that in this range the curves 

 can be approximated rather closely with straight hnes. If this is done, 

 then the differential equation and boundary conditions become linear and 

 the solution can be written immediately on the basis of the simple slab 

 solution with constant thermal properties. The essential feature of this 

 method of linearization is the fact that now there exists at least a quali- 

 tative criterion whereby one may judge the accuracy of the solution; in 

 general, the closer the fit between the straight hnes and the curves in the 

 range — 1 ^ 6 ^ Gm the more accurate the solution. However, it is not 

 necessary to carry out the solution by this Hnearization method because 

 of the following result. 



0.8 



0.6 



9 0.4 



0.2 



0.2 0.4 0.6 



Dimensionless time 



0.8 



1.0 



Fig. D,15b. Comparison between exact results and 

 results based on average c for small N. 



Suppose that two straight lines are drawn so as to minimize the mean 

 square error between these lines and the curves representing /i(G) and 

 /2(0) in a range — 1 <6^0^O. Suppose further that a third straight 

 Une, line A in the figure, is drawn, passing through the origin (0 = 0) 

 and intersecting the curve /3(0) at = — i(l — 0) (i.e. at correspond- 

 ing to the midpoint of the range) . Then it is found that the results based 

 upon this linearization are exactly the same as the results which would be 

 obtained if average thermal properties were assumed at a temperature 

 corresponding precisely to the mid-interval. This then is the geometric 

 meaning of the use of average thermal properties. 



Observing then that the generally used method of average thermal 

 properties corresponds to the fitting of certain straight lines to the graphs 



< 283 ) 



