# Transient analysis in heat transfer

by svishal03
Tags: analysis, heat, transfer, transient
 Engineering Sci Advisor HW Helper Thanks P: 7,279 Transient analysis in heat transfer Sorry, but I'm not really sure what the book is trying to say here! Equation 7.125 in the book is wrong. If you compare it with 7.124, the $F_Q$ and $F_g$ terms should be at time $t_{i+1}$, not time [itex]t_i[itex]. After 7.125 he says "the coefficient matrix on the left hand side changes at each time step". That is not correct, unless he is talking about (a) changing the time step size during the solution or (b) What happens if the matrices C and K are functions of temperature - but he doesn't seem to have discussed that anywhere earlier. The main difference between the forward and backward differences is what happens if you take arge time steps. In the forward difference method, you calculate a temperature gradient that is independent of the time step, and then assume the temp gradient remains constant through the whole of the step. It should be clear that will give nonsense if the step is very large, because you are assuming the temperature will change at the same rate "for ever". On the other hand, in the backward difference method, effectively you calculate what the temperature gradient would be when you get to the END of the step, assuming it remained constant through the step. In a sense that is "self correcting", because if the step size changes, the constant (or averate) temperature gradient during the step also changes. If you look at the correct version of 7.125 and take delta t to be very large, it is approximately the same as the steady state heat conduction equation at the end of the step. The terms involving C are the only ones NOT multiplied by delta t, so if delta t is very large the only significant terms in 7.125 are K, F_Q and F_g. So however large the time step is, the numerical solution will always correspond to something "phyisically sensible" even if it is not an accurate solution of the transient thermal problem.