Page 178 - Fundamentals of The Finite Element Method for Heat and Fluid Flow

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a procedure that has a certain number of future time steps (R) from the starting point, and
for a one-sensor problem, this is given as follows:
R TRANSIENT HEAT CONDUCTION ANALYSIS
r
(U k+r−1 − T ∗ )Z
k+r−1 r
∗ r=1
q k = q + R (6.77)
k
r 2
(Z )
r
r=1
The calculation of the sensitivity coefficient is very important in the above equation. It
is normally calculated by solving the following equation:
2
∂Z ∂ Z
ρc p = k 2 (6.78)
∂t ∂x
with
∂Z
−k = 1at x = 0 (6.79)
∂x
∂Z
k = 0at x = l (6.80)
∂x
and with an initial condition of Z = 0at t = 0. Using the above procedure, the inverse
heat conduction problem may be solved via the following steps.
∗
(i) Assume q = 0 in the first time interval.
k
(ii) Calculate T k+r−1 for r = 1, 2,... ,R (for all sensors) employing the finite element
method and assumed heat flux at the left-hand side q k = q using Equations 6.66
∗
k
to 6.69.
(iii) Calculate q k from Equation 6.77.

∗
(iv) Set q = q k−1 and go to step (ii) and continue until convergence is achieved.
k
6.9 Summary

In this chapter, we have introduced the transient heat conduction problem and demonstrated
solutions of such a problem via many numerical examples. However, the problems discussed
in this chapter are only the ‘tip of the iceberg’. We recommend that the readers formulate
their own transient heat conduction problems and solve them using the transient computer
programs available from the authors (see Chapter 10). For transient convection problems,
the readers should refer to Chapters 7 and 9.

6.10 Exercise

◦
Exercise 6.10.1 A large block of steel with a thermal conductivity of 40 W/m C and a ther-
2
◦
mal diffusivity of 1.5 × 10 −5 m /s is initially at a uniform temperature of 25 C. The surface