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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.

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.

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.

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.

A wall is constructed of an outer layer of 0.5 inch thick
plywood (k = 0:80 Btu/h-ft-F), an inner core of 5 inch thick fiberglass
insulation (k = 0:020 Btu/h-ft-F), and an inner layer of 0.5 inch thick
sheetrock (k = 0:10 Btu/h-ft-F) (Figure P13–13). The inside temperature is 65 F
with h = 1:5 Btu/h-ft
2
– F, while the outside temperature is 0 F with
h = 4 Btu/h-ft
2
– F. Determine the temperature at the interfaces of
the materials and the rate of heat flow in Btu/h through the wall.

A large plate of stainless steel with thickness of 5 cm and
thermal conductivity of k = 15 W/ m-C is subjected to an internal uniform heat
generation throughout the
plate at constant rate of Q = 10 10
6
W/ m
3
.
One side of the plate is maintained at 0 C by ice water, and the other side is
subjected to convection to an environment at T
x
= 35 C, with heat
transfer coefficient h = 40 W/ m
2
– C, as shown in Figure P13–14. Use
two elements in a finite element model to estimate the temperatures at each
surface and in the middle of the plate’s thickness. Assume a one-dimensional
heat transfer through the plate.

The base plate of an iron is 0.6 cm thick. The plate is
subjected to 100 W of power (provided by resistance heaters inside the iron, as
shown in Figure P13–15), over a base plate cross-sectional area of 250 cm
2
,
resulting in a uniform flux generated on the inside surface. The thermal
conductivity of the metal base plate is k = 20 W/ m-C. The outside ambient air
temperature is 20 C with a heat transfer coefficient of 20W/ m
2
– C
at steady-state conditions. Assume one-dimensional heat transfer through the
plate thickness. Using three elements, model the plate to determine the
temperatures at the inner surface and interior one-third points.

A hot surface is cooled by attaching fins (called pin fins)
to it, as shown in Figure P13–16. The surface of the plate (left end of the
pin) is 90 C. The fins are 6 cm long and have a cross-sectional area of 5
10
6
m
2
with a perimeter of 0.006 m. The fins are made of copper (k
= 400 W/ m-C). The temperature of the surrounding air is T
=
20 C with heat transfer coefficient on the surface (including the end surface)
of h = 10W/ m
2
– C. A model of the typical fin is also shown in
Figure P13–16. Use three elements in your finite element model to determine the
temperatures along the fin length.

Use the direct method to derive the element equations for
the one-dimensional steady-state conduction heat-transfer problem shown in
Figure P13–17. The bar is insulated all around and has cross-sectional area A,
length L, and thermal conductivity K
xx
. Determine the relationship
between nodal temperatures t
1
and t
2
( F) and the thermal
inputs F
1
and F
2
(in Btu). Use Fourier’s law of heat
conduction for this case.

1. Express the stiffness matrix and the force matrix for
convection from the left end of a bar, as shown in Figure P13–18. Let the
cross-sectional area of the bar be A, the convection coefficient be h, and the
free stream temperature be T∞.
2. For the element shown in Figure P13–19, determine the [k]
and { f } matrices. The conductivities are K
xx
= K
yy
= 15
Btu/(h-ft-F) and the convection coefficient is h = 20 Btu/(h-ft
2
–
F). Convection occurs across the i-j surface. The free-stream temperature is Ty
= 70 F. The coordinates are expressed in units of feet. Let the line source be
Q = 150 Btu/(h-ft) as located in the figure. Take the thickness of the element
to be 1 ft.