An astronaut in orbit has initially heated a gray body copper ball of 5cm diameter to a
0 uniform temperature of 250 C and then suddenly moved the ball to the outside of the
satellite. The gray body copper ball has an emissivity of 0.8 and a reflectivity of 0.2. Using
the lumped-heat-capacity method of analysis, calculate the ball temperature after 30 minutes,
if the astronaut is (a) facing away from the sun (no solar radiation) and (b)
2 facing toward the sun (with solar radiation intensity of 1395 W/m ). Here we neglect the
radiation due to the astronaut. For a boundary layer flow with U =constant over a porous plate as shown in Figure 1, a
suction velocity Vo (< 0) is introduced at the wall to delay flow separation. (a) By integrating
the boundary layer equations from porous wall across the boundary layer, show that the
integral momentum equation is given by and (b) obtain the integral energy equation. (c) Perform the dimensionless analysis on the
integral equations and discuss the effect of Vo on boundary layer thickness. o A vertical cylinder 1.8m high and 7.5cm in diameter is maintained at a temperature of 93 C
o in an atmospheric environment of 30 C. Calculate the heat lost by free convection
from this cylinder. Hint: For boundary layer thinner than cylinder diameter the cylinder
may be treated as a vertical flat plate. In the manufacture of steel wires, the melted steel becomes solidified and cooled as the wire
is drawn from the die at a constant speed U as shown in Figure 2. Because the metal is so hot,
the thermal radiation cannot be neglected, particularly at the initial cooling. From the
governing energy equation for steady convective heat transfer, (a) show that the equation
governing the averaged temperature over the cross sectional area A of the wire is given by The above equation is nonlinear in and cannot be solved analytically. However, to have a
quick look of the cooling behavior, we can approximate the radiation heat transfer coefficient hr (T 2 T2 )(T T ) by a constant to render equation (1) into a linear
equation. With hc and hr being assumed constants, (b) solve equation (1) subjected to the boundary conditions of T Tw at x =0
and T T
at x = and (c) discuss the
characteristics of the solution. An infinite plane wall of 20cm thick with non-uniform linear internal heat generation of q 4 ax with a = 2000kW/m is exposed to different environments on the two sides as shown in Figure 3. Calculate the maximum temperature in the plane wall if the thermal
0 conductivity of the plane wall is k = 20 W/m- C. Show that the equations of similarity solutions for two-dimensional laminar free convection
over a vertical flat plat subject to a specified surface temperature Tw can be expressed as where the similarity variables , and the dimensionless stream function f and temperature are given as (Hint: following the procedure of the Falkner-Skan transformation given in Section 6.3)
—————————————————————————————————————Figure 1
y u = U , T = T Figure 2
0 u = 0,
v = Vo (< 0),
T=T
x
w Figure 3
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