Nodal Analysis for Energy Stored

[chriskay301]

Homework Statement

The temperature response of a fin is to be approximated using the finite difference method. The fin is 3 cm long and has a square cross-section that is 5 mm on a side. The fin is aluminum, with thermal properties ρ=2700 kg/m3, k=200 W/(m-K), and c=900 J/(kg-K). The fin starts at a uniform initial temperature of 100°C, and is suddenly exposed to convection with h=30 W/(m2-K) and a freestream temperature of 15°C. During this process the base of the fin (node 1) remains at the initial temperature. (a) Discretize the fin into 4 nodes, and write the appropriate equations (using control volumes and energy balances) for the node temperatures using the explicit formulation for the spatial derivatives. (b) Determine the maximum timestep (dt) that would be allowable if the solution is to remain stable. (c) Using a computer program with graphing capability (Excel, Matlab, ??), plot the temperatures at each of the nodes versus x using the timestep from part (b) for t=0*dt (initial time), t=1*dt, t=2*dt, t=10*dt, and t=40*dt. Your graph should look something like the below figure.

Homework Equations

Node 1 = T1 = 100C
I have node 2, 3 and 4 equations which I know are correct.
Example. T$^{n+1}_{2}$=(1-2Fo)T$^{n}_{2}$+Fo(T$^{n}_{1}$+T$^{n}_{3}$)

The Attempt at a Solution

I know that my equations for each node are right. But in order to solve the temp for T2 at n+1, I need the temp for T1, T2 and T3 at n. And I only know T1 at n. How would I get the other ones?

 

[rude man]

What is Fo?

Initially, n=0 and all four T are at 100C.
Increment n one unit at a time to see the finite-difference equation progress.

For stability I would recommend the z transform and use the theorem pertaining to the roots of the transformed equation to ensure stability.

 

[chriskay301]

Thanks for the reply.

Fo is just something my teacher uses to represent a bunch of constants pretty much. like rho, h, k

 

[rude man]

OK.
I would consider using the z transform on all three equations (node 1 temp. is not a variable). That would give you 3 algebraic equations in 3 unknowns: T2, T3, T4.

Nice thing about the z transform is it automatically includes initial conditions, just as the Laplace does for continuous systems.

Question to yourself: if any of nodes 2, 3 or 4 is stable, what does that tell you about the other two nodes' stability?

 

[rezeew]

To solve for the temperatures at each node, you can use the explicit formulation for the spatial derivatives as given in the homework equations. This will allow you to calculate the temperatures at each node at time step n+1 using the temperatures at time step n. You can start with the initial conditions where only T1 is known, and then use the equations to solve for T2, T3, and T4 at time step n+1. Then, you can use these new values to solve for T2, T3, and T4 at time step n+2, and so on until you reach the desired time steps. This iterative process will allow you to solve for the temperatures at each node at every time step.