Page 71 - Introduction to Computational Fluid Dynamics

P1: IWV/ICD
0 521 85326 5
10:49
May 25, 2005
CB908/Date
0521853265c02
50
2
to surroundings at 25 C. If h = 3(T rod − T ∞ ) 0.25 1D HEAT CONDUCTION
◦
W/m -K, perform the
following:
(a) Compute the variation of h with time at x = 5cmand x = 9cmovera
period of 1 min. Take t = 1 s and ψ = 1 and use TDMA.
(b) Compute the percentage reduction in the energy content of the rod at the
end of 1 min.
(c) Extend the calculation beyond 1 min and estimate the time required to reach
near steady state. (Hint: You will need to specify a criterion for steady state.)
16. Consider an unsteady conduction problem in which T 1 is given. However, at
x = L, the heat transfer coefficient is specified. By examining the discretised
equationforageneralnodei,fornodei = 2,andfornodei = N − 1,determine
the stability constraint on t. Assume uniform control volumes, constant area,

and conductivity with q = 0 and ψ = 0.
17. A semi-infinite solid is initially at 25 C. At t = 0, the solid surface (x = 0) is
◦
2
suddenly exposed to q w = 10 kW/m . A thermocouple is placed at x = 1mm
to apparently measure the surface temperature. Compute the temperature distri-
bution in the solid as a function of x and t and estimate the error in the thermo-
couple reading as a function of time. Carry out computations up to 1 s. Given
3
are the following: k = 80 W/m-K, ρ = 7, 870 kg/m , and C = 450 J/kg-K.
[Hint: The boundary condition at x =∞ is T L = 25 C at all times. Choose
◦
sufficiently large L (say 1 cm) and execute with t = 0.01 s.]
18. A laboratory built in the Antarctic has a composite wall made up of plaster
board (10 mm), fibreglass insulation (100 mm), and plywood (20 mm). The
inside room temperature is maintained at T i = 293 K throughout. The plywood
is exposed to an outside temperature T o that varies with time t (in hours) as
π
⎧
⎨ 273 + 5sin t for 0 ≤ t ≤ 12 h,
⎪
12
T o =
π
273 + 30sin t for 12 ≤ t ≤ 24 h.
⎪
⎩
12
(a) Compute the heat loss to the outside over a typical 24-h period (i.e., under
2
periodic steady state) in J/m .
(b) Plot the variation of interface temperatures between the plasterboard and
the fibreglass and between the fibreglass and the plywood as a function of
2
2
time. Assume: h i = 15 W/m -K and h o = 60 W/m -K. Material properties
are given in Table 2.9.
∗
19. Solve for fully developed laminar flow in a concentric annular (r = R i /R o =
0.6) duct. Compare the predicted velocity profile with the exact solution [33]

2

u 2 r r
= 1 − + B ln ,
u A R o R o