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This thread is regarding the Finite difference scheme for a 1-dimensional Heat transfer problem withnon-uniform cross-sectional area.As seen in https://www.physicsforums.com/showthread.php?t=397891", when the element has constant cross-sectional area, things cancel nicely. But when this is not the case, some parts get tricky and I hope to have these areas (no pun intended ) addressed.

Shown below is a (super-awesome MS word) drawing of an object subjected to a fixed temperature at point 1 and convection at point 3. Now the dotted lines are simply a construct that we used in class to help deal with the area problem; I will explain how we used it shortly but for now just note that it is there and that I will commonly refer to it as the "pseudo-element" (PE2) that bounds point 2 (pseudo because this is not really an element since this os not FEA it is finite difference).

Now I would like to derive the equations necessary for a finite difference scheme for this problem. Since the conditions at point 1 (p1) are known, I will move to p2 and p3 and write the corresponding energy balance for each point (pseudo-element).

Point 2:

Point 2 is lies at the center of dotted lines that bound the PE. Point 2 also lies at [itex]\Delta x = 0.5 m[/itex] with respect to the entire structure. The cross-sectional areas along the dotted linesaandbare given byAand_{a}A._{b}

Writing the energy balance

There is a conduction term due to the heat fluxqexiting (an assumption) at_{a}Aand a conduction term due to the heat flux_{a}qentering (an assumption) at_{b}A. There is also an energy storage term [itex]\Delta U = \rho*V_{element}\,dT[/itex]._{b}

The sign-convention dictates that heat flux into the element and out of the element are negative.

Here is where I get confused:

The conduction heat flux atAis due to the temperature gradient between p1 and p2. My professor wrote that the conduction term at_{a}ais given by

[tex]-KA_a\frac{(T_2 - T_1)}{\Delta x}\qquad (1)[/tex]

and the conduction atb:

[tex]KA_b\frac{(T_3 - T_2)}{\Delta x}\qquad (2)[/tex]

I am just not sure how we define the temperature gradient? That is why in the first term is it T2 - T1 instead of T1 - T2 ?

Maybe a silly question, but it seems like if I assumed that the flux at 'a' was entering and at 'b' was exiting then some sort of switch would need to be made.

~Casey

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# Heat TransferAssigning direction to the temperature gradient

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