1. The problem statement, all variables and given/known data
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.
2. Relevant equations
Node 1 = T1 = 100C
I have node 2, 3 and 4 equations which I know are correct.
Example. T[itex]^{n+1}_{2}[/itex]=(1-2Fo)T[itex]^{n}_{2}[/itex]+Fo(T[itex]^{n}_{1}[/itex]+T[itex]^{n}_{3}[/itex])
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?
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.
2. Relevant equations
Node 1 = T1 = 100C
I have node 2, 3 and 4 equations which I know are correct.
Example. T[itex]^{n+1}_{2}[/itex]=(1-2Fo)T[itex]^{n}_{2}[/itex]+Fo(T[itex]^{n}_{1}[/itex]+T[itex]^{n}_{3}[/itex])
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?
0 commentaires:
Enregistrer un commentaire