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Calculate the [k] and { f } matrices for the element shown
in Figure P13–20. The conductivities are K
xx
= K
yy
= 15
W/(m C) and the convection coefficient is h = 20 W/(m
2
C).
Convection occurs across the i-m surface. The free-stream temperature is Ty =
15 C. The coordinates are shown expressed in units of meters. Let the line
source be Q = 100 W/m as located in the figure. Take the thickness of the
element to be 1 m.

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.

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.

For the square two-dimensional body shown in Figure P13–25,
determine the temperature distribution. Let K
xx
= K
yy
=
10 Btu/(h-ft-F) and h = 10 Btu/(h-ft
2
– F). The top face is
maintained at 500 F, the left face is maintained at 100 F, and the
other two faces are exposed to an environmental
(free-stream) temperature of 100 F. Also, plot the temperature contours on the
body.

1. The cross section of a tall chimney shown in Figure
P13–27 has an inside surface temperature of 330 F and an exterior temperature
of 130 F. The thermal conductivity is K = 0:5 Btu/(h-ft-F). Determine the
temperature distribution within the chimney per unit length.
2. The square duct shown in Figure P13–28 carries hot gases
such that its surface temperature is 570 F. The duct is insulated by a layer of
circular fiberglass that has a thermal conductivity of K = 0:020 Btu/(h-ft-F).
The outside surface temperature of
the fiberglass is maintained at 110 F. Determine the
temperature distribution within the fiberglass.

1. The buried pipeline in Figure P13–29 transports oil with
an average temperature of 60 F. The pipe is located 15 ft below the surface of
the earth. The thermal conductivity of the earth is 0.6 Btu/(h-ft-F). The
surface of the earth is 50 F. Determine the temperature distribution in the
earth.
2. For the circular body with holes shown in Figure P13–31,
determine the temperature distribution. The inside surfaces of the holes have
temperatures of 150 C. The outside of the circular body has a temperature of 30
C. Let K
xx
= K
yy
= 10 W/(m C).

A 10-in.-thick concrete bridge deck is embedded with heating
cables, as shown in Figure P13–30. If the lower surface is at 0 F, the rate of
heat generation (assumed to be the same in each cable) is 100 Btu/(h-in.) and
the top surface of the concrete is at 35 F. The thermal conductivity of the
concrete is 0.500 Btu/(h-ft-F). What is the temperature distribution in the
slab? Use symmetry in your model.

For the square two-dimensional body shown in Figure P13–32,
determine the temperature distribution. Let K
xx
= K
yy
=
10 W/(m C) and h = 10 W/(m
2
C). The top face is maintained at 100 C,
the left face is maintained at 0 C, and the other two faces are exposed to a
free-stream temperature of 0 C. Also, plot the temperature contours on the
body.

A 200-mm-thick concrete bridge deck is embedded with heating
cables as shown in Figure P13–33. If the ambient temperature under the deck is
10
0
C with h = 10 W/(m
2
– C) and the ambient air temperature above the
deck is 10 C with h = 10 W/(m
2
– C), what is the temperature
distribution in the slab? The heating cables are line sources generating heat
of Q = 50 W/m. The thermal conductivity of the concrete is 1:2 W/m-C. Use
symmetry in your model.

1. For the two-dimensional body shown in Figure P13–34, determine the temperature distribution. Let the left and right ends have constant temperatures of 200 C and 100 C, respectively. Let K
xx
= K
yy
= 5 W/(m C). The body is insulated along the top and bottom.
2. For the two-dimensional body shown in Figure P13–35, determine the temperature distribution. The top and bottom sides are insulated. The right side is subjected to heat transfer by convection. Let K
xx
= K
yy
= 10 W/(m C).