Heat Conduction
30
15. Schneider, P. J., Conduction Heat Transfer. Addison-Wesley, 1955.
16. Shapiro, A. H., The Dynamics and Thermodynamics of Compressible Fluid Flow, vol. 1, The Ronald
Press, 1953.
17. Siegel, R., and Howell, J. R., Thermal Radiation Heat Transfer, 4th ed., Taylor and Francis, 2002.
18. Van Wylen, G. J., and Sonntag, R. E., Fundamentals of Classical Thermodynamics. 3rd ed., John
Wiley and Sons, 1986.
19. Zhang, Z. M., Nano/Microscale Heat Transfer, McGraw-Hill, 2007.
Problems
1.1 In most commonly encountered engineering problems involving steady flows of
fluids through various devices, the inlet and outlet flows are usually regarded as
one dimensional. Develop the first law of thermodynamics (i.e., the steady-state
steady-flow energy equation) for the device illustrated in Fig. 1.11.
1.2 Apply the first law of thermodynamics, Eq. (1.30), to a finite volume element in a
stationary solid. Let T(r, t) be the temperature distribution in the solid. Simplify
the relation you obtain to reduce it to an expression which relates the temperature
distribution to heat fluxes in the solid.
1.3 Prove that, under steady-state conditions and in the absence of internal energy
sources or sinks, the rate of heat transfer by conduction through a solid flat plate
whose surfaces are maintained at constant temperatures T1 and T2 is constant.
1.4 Obtain an expression for the steady-state temperature distribution in the flat plate
of Problem 1.3 if the thickness of the plate is L and its thermal conductivity k is
constant.
1.5 The plane wall shown in Fig. 1.12 has one surface maintained at T1 and the other
at T2. The temperature at the center plane is measured to be T3, and the rate of heat
flow through the wall is q. Assuming that the thermal conductivity of the wall
varies linearly with temperature, find an expression for the thermal conductivity
as a function of temperature and the rate of heat flow through the wall.
.
W
c.v
V1, A1, ρ1
V2, A2, ρ2
(1)
z1
(2)
q
FIGURE 1.11
Figure for Problem 1.1.
z2
Foundations of Heat Transfer
31
A
T1
k(T )
T3
T2
q
L
L
FIGURE 1.12
Figure for Problem 1.5.
1.6 Show that the gradient of the temperature distribution, ∇T at point P shown in Fig.
1.5, is a vector normal to the isothermal surface passing through P and pointing in
the direction of increasing temperature.
1.7 Obtain the relation given by Eq. (1.52).
1.8 The steady-state temperature distribution in a flat plate is given by
T (°C) = 150 − 400 x 2
where x is the distance in meters (m) along the width of the plate and is measured
from the surface at 150°C. Determine the heat fluxes at the two surfaces of the
plate. The thermal conductivity of the wall material is 40 W/(m·K), and the thickness of the wall is 0.25 m.
1.9 The instantaneous temperature distribution in a plane wall at a specific time t1
during a transient heat transfer process is shown in Fig. 1.13. Is the wall being
heated or cooled at t1? Explain.
T
k = const.
T(x, t1)
0
FIGURE 1.13
Figure for Problem 1.9.
x
Heat Conduction
32
1.10 The temperature profile at a location in water flowing over a flat surface is experimentally measured to be
T (°C) = 20 + 80e −800 y
where y is the distance in meters (m) measured normal to the surface with y = 0
corresponding to the surface. What is the value of heat transfer coefficient at this
location? Assume that the thermal conductivity of water is k = 0.62 W/(m·K).
1.11 Estimate the equilibrium temperature of a long rotating cylinder of diameter D,
and oriented in space with its axis normal to the sun’s rays. The cylinder is at a
location in space where the irradiation from the sun (i.e., energy incident on a
surface perpendicular to the sun’s rays per unit time per unit area) is 1500W/m2.
Assume that the absorptivity of the surface of the cylinder to solar radiation, αs, is
equal to its emissivity ε. and the outer space is a blackbody at 0 K.
1.12 A freighter vessel is adapted for the transportation of liquefied gas using spherical tanks. The tanks have a radius of 5 m and receive liquefied natural gas at a
temperature of −160°C. Assuming a heat transfer coefficient h = 25 W/m2C on the
external surface of the tank (without insulation) and the outside air at 20°C, what
is the heat transfer rate through the tank external surface, assuming the internal
temperature in the tank wall remains at −160°C and neglecting the thermal resistance of the tank wall?