Reference no: EM132388508 , Length: 7 pages

The University of Queensland
School of Engineering
MECH2700 Engineering Analysis I (2019)
Assignment: Computational Heat Transfer

Assignment Tasks
Preliminary analysis

Task 1.

Show that the second derivative of T at node i may be approximated by the following finite- difference formula:

d²T/dx²|x=x_i ≈ (T_i+1 - 2T_i + T_i-1) /?x²

where T_i is shorthand for T (x = x_i) and ?x is the node spacing.

The finite-difference form of the governing equation may now be written for all interior nodes, after substituion and rearrangement, as:

T_i-1 - (2 + ?x²hP)kA_c) T_i + Ti+1 = -?x²hPT_a/kA_c , i = 1...n - 1

Note that the equations are linear in the unknown temperatures, T_i, and so may be assembled into matrix form. These equations are valid for all interior nodes, ie., up to i = n 1. The final node, n, must be treated specially - the treatment of the end node will form Task 2.

Task 2.

Show that the first derivative of T may be approximated, using a one-sided difference, as:

dT/dx|x=x_i ≈ (T_i - T_i-1) /?x (7)

Use this approximation in the boundary condition, Equation 3, to show that the finite-difference equation for the boundary node n is:

-k/?x T_n-1 + (h + k/?x)T_n = hT_a     (8)

Task 3.

For n = 5, write out the five finite-difference equations (i = 1...5) and then assemble the matrix of coefficients, A, and the vector ?b in the system of equations, AT? = ?b. Include a sixth line in your matrix system at the top: the trivial equation, T0 = Tw, for the node n = 0. This will be useful later when you try to build the matrix system in your computer code. What is special about the structure of matrix A?

Computer implementation

Task 4.

Write a Python program to solve for the temperatures in the fin using one of the direct solvers for systems of linear equations. Take advantage of the special structure of A to make your code more computationally efficient than the general method presented in lectures.

Task 5.
Plot the temperatures for n = 5, 10, 20, 200 against the analytic solution. (Hint: The analytic solution may be found in the text by Incropera and De Witt, "Fundamentals of Heat and Mass Transfer". Look for the section on "Fins of uniform cross-sectional area". The fin in this problem is of finite length with a convective end boundary condition.)

Task 6.
As separate figures, plot the temperature at the fin tip and the relative error at the fin tip (||T_n - T_exact /T_exact) for a range of values of n to show convergence of the numerical solution as ?x = L/n approaches zero. What value of n is required to give a numerical solution that is within 0.1% of the analytic solution, in terms of the relative error in temperature at the convective boundary node?

Task 7.
Based on a heat flux balance, the heat transfer rate from the entire fin is equal to the heat transfer rate through the base of the fin (ie. the conduction from the cylinder wall into the fin). Compute the heat transfer rate for the fin using the finite-difference approximation:

q = kA_c (T_w - T₁)/?x (9)

Note that T₁ is the value at the first node computed as part of the solution procedure. Plot the value for heat transfer rate, q, as ?x approaches zero. What value of n is required to give a numerical estimate for heat transfer rate that is within 0.1% of the analytic solution?

For the fin to achieve its purpose and maintain the combustor wall temperature at 95% of the material limit, the heat transfer rate from the fin must balance the heat transfer rate to the section of combustor wall it is responsible for cooling, which here is q_comb = 14 kW. According to your analysis, does the fin provide sufficient cooling? If not, how might it's geometry be varied to improve its performance?

Attachment:- Engineering Analysis.rar