Category: Writing

A tapered aluminum fin (k = 200W/ m-C), shown in Figure
P13–12, has a circular cross section with base diameter of 1 cm and tip
diameter of 0.5 cm. The base is maintained at 200 C and looses heat by
convection to the surroundings at T
x
= 10 C, h = 150W/ m
2
–
C. The tip of the fin is insulated. Assume one-dimensional heat flow and
determine the temperatures at the quarter points along the fin. What is the
rate of heat loss in Watts through each element? Use four elements with an
average cross-sectional area for each element.

Condensing steam is used to maintain a room at 20 C. The
steam flows through pipes that keep the pipe surface at 100 C. To increase heat
transfer from the pipes, stainless steel fins (k = 15W/ m-C), 20 cm long and
0.5 cm in diameter, are welded to the pipe surface as shown in Figure P13–11. A
fan forces the room air over the pipe and fins,
resulting in a heat transfer coefficient of 50W/ m
2
–
C at the base surface of the fin where it is welded to the pipe. However, the
air flow distribution increases the heat transfer coefficient to 80W/ m
2
–
C at the fin tip. Assume the variation in heat transfer coefficient to then
vary linearly from left end to right end of the fin surface. Determine the
temperature distribution at L/4 locations along the fin. Also determine the
rate of heat loss from each fin.

For the composite wall of a house, shown in Figure P13–10,
determine the temperatures at the inner and outer surfaces and at the
interfaces. The wall is composed of 2.5 cm thick plaster wall (k = 0:20W/ m-C)
on the inside, a 9 cm thick layer of fiberglass insulation (k = 0:038W/ m-C),
and a 1.25 cm plywood layer (k = 0:12 W/ m-C) on the outside. Assume the inside
room air is 20 C with convection coefficient of 10W/ m
2
– C and the
outside air at =10 C with convection coefficient of 20 W/ m
2
– C.
Also, determine the rate of heat transfer through the wall in Watts. Assume
one-dimensional heat flow through the wall thickness.

A double-pane glass window shown in Figure P13–9, consists of two 4-mm thick layers of glass with k = 0.80 W/m-C separated by a 10 mm thick stagnant air space with k = 0.025 W/m-C. Determine (a) the temperature at both surfaces of the inside layer of glass and the temperature at the outside surfaces of glass, and (b) the steady rate of heat transfer in Watts through the double pane. Assume the inside room temperature T
i1
= 20 C with hi = 10 W/m
2
-C and the outside temperature T
01
= 0 C with h
0
= 30 W/m
2
-C. Assume one-dimensional heat flow through the glass.

A composite wall is shown in Figure P13–8. For element 1,
let K
xx
= 5 W/(m-C), for element 2 let K
xx
= 10 W/(m-C),
for element 3 let K
xx
= 15 W/(m-C). The left end has a heat source
of 600 W applied to it. The right end is held at 10C. Determine the left end
temperature and the interface temperatures and the heat flux through element 3.

1. For the composite wall shown in Figure P13–6, determine
the interface temperatures. What is the heat flux through the 8-cm portion? Use
the finite element method. Use three elements with the nodes shown. 1 cm = 0:01
m.
2. For the composite wall idealized by the one-dimensional
model shown in Figure P13–7, determine the interface temperatures. For element
1, let K
xx
= 5 W/(m C); for element 2, K
xx
= 10 W/(m C);
and for element 3, K
xx
= 15 W/(m C). The left end has a constant
temperature of 200 C and the right end has a constant temperature of 600 C.

The fin shown in Figure P13–5 is insulated on the perimeter.
The left end has a constant temperature of 100 C. A positive heat flux of q
x
= 1000 W/m
2
acts on the right end. Let K
xx
= 6 W/(m C)
and cross-sectional area A = 0:1 m
2
. Determine the temperatures at
L/4, L/2, 3L/4, and L, where L = 0:4 m.

1. For the body shown in Figure P13–23, determine the
temperature distribution. Surface temperatures are shown in the figure. The
body is insulated along the top and bottom edges, and K
xx
= K
yy
= 1:0 Btu/(h-in.-F). No internal heat generation is present.
2. For the square two-dimensional body shown in Figure
P13–24, determine the temperature distribution. Let K
xx
= K
yy
= 10 Btu/(h-ft-F). The top surface is maintained at 500 F and the other three
sides are maintained at 100 F. Also, plot the temperature contours on the body.

The rod of 1-in. radius shown in Figure P13–4 generates heat
internally at the rate of uniform Q = 10;000 Btu/(h-ft
3
) throughout
the rod. The left edge and perimeter of the rod are insulated, and the right
edge is exposed to an environment of Ty = 100 F. The convection heat-transfer
coefficient between the wall and the environment is h = 100 Btu/(h-ft
2
–
F). The thermal conductivity of the rod is K
xx
= 12 Btu/(h-ft-F). The
length of the rod is 3 in. Calculate the temperature distribution in the rod.
Use at least three elements in your finite element model.

For the square two-dimensional body shown in Figure P13–21,
determine the temperature distribution. Let K
xx
= K
yy
=
25 Btu/(h-ft-F) and h = 10 Btu/(h-ft
2
– F). Convection occurs across
side 4–5. The free-stream temperature is Ty = 50 F. The temperatures at nodes 1
and 2 are 100 F. The dimensions of the body are shown in the figure. Take the
thickness of the body to be 1 ft.