Heat Transfer: Finite Difference method using MATLAB

[aznkid310]

Homework Statement

I have uploaded the problem statement and corresponding programs.

Reader is the problem statement
Reader1 is another file that is suppose to help

Homework Equations

I'm just having trouble getting started. I can do the analytical part on paper, but I don't know how to implement this in MATLAB.

In the program for problem 1:

What is the 'A' and 'C' matrix?

I can get some values on paper, but, for example, what is A(i, i+1)? I realize this this the temperature to the left of the node I am interested in, but do I type in some sort of equation?

Similarly for A(i,i), A(i, i-1), C(n)...

Is C(1) = -200 from my solution?

It seems I have to do what is says in the reader1 file, but I don't know how to use it

The Attempt at a Solution

For #1a:

T(x) = C1x + C2
Boundary Conditions: T(0) = 500
T(L = 1m) = 300

Thus T(x) = 500 - 200x

q = -kA(dT/dx) = 37200 Watts, where k = 186 W/m.k

energy balance for internal nodes: q1 + q2 +qV = 0

q1 = kA[ (T_i-1) - T_i]
q2 = kA[(T_i+1) - T_i]
qV = dx = internal heat generation assuming dy=dz = infinite

Finite Diff. eqn: (T_i-1) - (2T_i) + (T_i+1) = 0

For the left boundary node:

q_left side = 0 b/c T = 500k is constant?

therefore q_right side + qV = 0

q_right side = kA[(T_i+1) - T_i] and qV is the same from before

Right Boundary Node:

q_left side = kA[(T_i-1) - T_i] and qV stays the same

How is this implemented in MATLAB?

For #1b:

Boundary Conditions: T(0) = 500
-kA(dT/dx) @ x=1m = hA(Ts -T_inf) [Convective BC]

Assuming C1 is the same from part a (is this correct?): T_inf = 462.8 K

 

[Valkarie]

h = h_inf = 20 W/m^2.KFinite Diff. eqn: (T_i-1) - (2T_i) + (T_i+1) = 0For the left boundary node:q_left side = 0 b/c T = 500k is constant?therefore q_right side + qV = 0q_right side = kA[(T_i+1) - T_i] and qV is the same from beforeRight Boundary Node: q_left side = kA[(T_i-1) - T_i] and qV stays the sameq_right side = hA[T_s - T_inf]How is this implemented in MATLAB?

 

[Mene]

(from previous calculations)

Finite Diff. eqn: (T_i-1) - (2T_i) + (T_i+1) = -hA(T_i - T_inf)

For the left boundary node:

q_left side = 0 b/c T = 500k is constant?

therefore q_right side + qV = 0

q_right side = -hA(T_i - T_inf) and qV is the same from before

Right Boundary Node:

q_left side = kA[(T_i-1) - T_i] and qV stays the same

To implement this in MATLAB, you would first need to define your variables and parameters, such as the length of the domain, the number of nodes, the thermal conductivity, and the boundary conditions. Then, you would need to create a loop that goes through each node and calculates the temperature values using the finite difference equations. This can be done by creating a matrix for the temperature values and using the A and C matrices to calculate the temperature values at each node. You can also use the built-in functions in MATLAB for matrix operations and solving systems of equations. It may also be helpful to plot the temperature distribution at each iteration to see how it changes over time. You can refer to the reader1 file for more guidance on how to set up the code.