Question

1). Consider 1D heat conduction in a solid plate as shown. The temperatures at two boundaries are 20 K and 10 K, respectively. lm- 2 1 1 3 4 5 T20 K T = 10 K 0.25m 0.25m 0.25% 0.25 (a) Write down the governing equation for the temperature distribution inside the plate. Assume no heat source inside the entire plate. (6) The domain has been discretized using 5 equally spaced grids. Discretize the governing equation in (a) using finite volume method. Write down the discrete algebraic equations for node 2, 3, and 4. (c) Use Gauss-Seidel iteration to solve the system of equations to obtain the temperatures at node 2. 3 and 4. Carry out 4 iterations. (d) Bonus Question. If there is a 10 w/m heat generation in the 0.25m length of the plate middle as shown below. What are the temperatures at node 2, 3 and 4? (The thermal conductivity of the plate k = 1w/m.K)) 0375 m2 225 1m Q=10 w/m3 MİLL141 1 2 3 4 5 T2-20 K Ts=10 K 0.25m 0.25m 0.25m 0.25m

Answer 1

Given, one dimensional heat conduction in a solid plate. Temperature $T_{1}$ = 20 K and $T_{5}$ = 10 K.

(a) Governing equation for temperature distribution inside the plate :

3.2 += 0

where, k = thermal conductivity of the plate

T = temperature

x = coordinate

Q= Heat source.

Since, no heat source is assumed. So Q =0.

Therefore, governing equation is  

T 172 -0

The solution for the discretized equation is ,

4,1, = AT + AT

where $A_{p} = \frac{k_{p}A}{\Delta x}$ ,
kwA Au
and
ke A A =
, p is the point of interest, e is east side and w is the west side

.

(b) Given, , i.e; spacing between nodal points.

Ar= 0.25m

Using finite control volume method,

Node 2, $T_{2}$	20k1-2 + 2k2-3T3 = 3k2T
Node 3, $T_{3}$	k2-31, + k3-414 = 2k3Tz
Node 4, $T_{4}$	2k3-4T3 + 10k4-5 = 3k4T4