Page 69 - Introduction to Computational Fluid Dynamics

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1D HEAT CONDUCTION
temperature to reach 140 C. What is the minimum value of n required to obtain
◦
an accurate estimate of t?
3. Repeat Problem 1 from the text using both explicit and implicit methods by
choosing N = 7, 12, and 22. Determine the largest allowable time step in the
explicit case. Compare your solution for the time required for adhesion with
the exact solution determined in the previous problem.
4. Evaluate Su N−1 and Sp N−1 for an unsteady problem when T N is specified as a
function of time. Assume an arbitrary value of ψ.
5. Consider a time-varying heat-flux-specified condition at i = 1. Hence, derive
Su 2 and Sp 2 for arbitrary ψ. Confirm the validity of the three-step procedure
following Equation 2.63 for ψ = 1.
6. Repeat Exercise 5 for a time-varying heat transfer coefficient boundary condi-
tion. Hence, confirm the validity of the procedure following Equation 2.65 for
ψ = 1.
7. Confirm the correctness of Equations 2.67 and 2.68.
8. Verify the entries in Tables 2.5 and 2.7 by carrying out the necessary
calculations.
9. Develop a TDMA routine in which the postulated equation is

T i = A i T i−1 + B i .
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10. Consider a slab of width b = 20 cm. At x = 0, T = 100 C and at x = b, q = 1
2
3
kW/m . The heat generation rate is q = 1,000 − 5 T W/m . Calculate the

steady-state temperature distribution with and without source-term linearisa-
tion. Compare the number of iterations required in the two cases for N = 22
and 42. Also calculate the heat flux at x = 0 and T b and check the overall heat
balance. Take k = 1 W/m-K. Use TDMA.
11. Consider a nuclear fuel rod of length L and diameter D. The two ends of the rod
are maintained at T 0 . The internal heat generation rate is q = a sin(πx/L),

where x is measured from one end of the rod and a is an arbitrary constant. The
rod loses heat by convection (coefficient h) to a coolant fluid at T ∞ .
(a) Nondimensionalise the steady-state heat conduction equation and identify
the dimensionless parameters. [Hint: Define θ = (T − T ∞ )/(T 0 − T ∞ ),
2
2
x = x/L, P 1 = aL /k (T 0 − T ∞ ), and P 2 = 4hL /(kD).]
∗
(b) Compute the temperature distribution in the rod and compare with the exact
solution for 0 < P 1 , P 2 < 10. Use source-term linearisation and TDMA.
Carry out an overall heat balance from the computed results
(c) Solve the problem for P 1 = P 2 = 10 using different underrelaxation pa-
rameters 0 <α < 2 for N = 22 and N = 42. Determine α opt in each case.
Use uniform grid spacing and the GS procedure.