Hello again folks(adsbygoogle = window.adsbygoogle || []).push({});

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

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Heat TransferAssigning direction to the temperature gradient

**Physics Forums | Science Articles, Homework Help, Discussion**