Transient analysis in heat transfer

[svishal03]

Hi All,

I'd be grateful if helped in this understanding;

I'm following the attached pages to solve a numerical problem by Forward difference, backward difference and central difference- in fact to be precise, deriving the transient solution approach to a heat transfer problem---

In the attached, pages 279 and 280 (don't worry I've not attached a 280 page book but just extracyed pages) derive a solution by forward difference

Pages 283-285 derive by backward and central difference,

He (the author) says forward and backward difference are different because the way the capacitance matrix i.e. [C] matrix and conductivity matrix i.e.[K] matrix are treated.I'm not able to apprciate the difference,Can anyone throw some light on this and explain the difference between forward and backwartd difference?

Vishal

 

[AlephZero]

Compare equations 7.119 and 7.122. Those are the two approximations (forward and backward difference) for dT/dt at time t. (The book leaves out the "at time t" in 7.119)

7.119 says an approximate value of dT/dt at time t is estimated from the values of T at times t and t PLUS delta t.

7.122 says an approximate value of dT/dt at time t is estimated from the values of T at times t and t MINUS delta t.

 

[svishal03]

Hi AlephZero,

Thanks for the reply, yes I do appreciate what you said, but my qustion was with regard to the statement;



If we compare Equation 7.125 with Equation 7.121, we find that the major difference
lies in the treatment of the conductance matrix. In the latter case, the
effects of conductance are, in effect, updated during the time step. In the case of
the forward difference method, Equation 7.121, the conductance effects are held
constant at the previous time step.

I'm not able to appreciate the difference concerning the treatment of the conductance matrix as stated above.

Will be grateful if helped.

Vishal

 

[AlephZero]

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.

 

[svishal03]

Thanks a lot, but as you say that in Forward difference the temperature obtained is the one at the beginning of the time step, that is, in every following time step the temperature is that corresponding to the end of the preceeding time step, right?

Similarly in backward difference its the converse, the the temperature obtained is the one at the end of the time step, that is, in every following time step the temperature is that corresponding to the beginning of that time step, right?

So, how do you say one is superior to the other?

 

[AlephZero]

There are standard techniques to study the accuracy and stability of the variious difference methods. I don't know your textbook, but it will probably deal with them later on.

The usual choice in thermal modelling is between backward difference or central difference, or a method which is "in between" them and has some advantages of both.

Forward difference doesn't have any "good" properties as a general purpose method, but it could be useful in some very specialized situations.

 

[svishal03]

My textbook is Fundamentals of finite element methood by Dsavid hutton. I'm referring the chapter for thermal as I need it for some projects coming up.Please can you suggest some textbook for heat transfer which explains fundamentals as well as FE approach for different ele ments and boundary conditions?