Tutorial 2-1

MIN 305 Heat & Mass Transfer
Tutorial Sheet 2: 1-D steady state conduction without heat generation
1. The rear window of an automobile is defogged by attaching a thin, transparent, film-type

heating element to its inner surface. By electrically heating this element, a uniform heat flux
may be established at the inner surface. For 4-mm-thick window glass, determine the electrical
power required per unit window area to maintain an inner surface temperature of 15°C when
the interior air temperature and convection coefficient are Ti = 25°C and hi = 10 W/m2 K,
while the exterior (ambient) air temperature and convection coefficient are To = 10°C and
ho=65 W/m2 K.
2. Consider a 15 cm X 20 cm epoxy glass laminate (k = 0.17 W/m·°C)
whose thickness is 0.13 cm. In order to reduce the thermal resistance
across its thickness, cylindrical cop- per fillings (k = 386 W/m·°C) of
0.05 cm diameter are to be planted throughout the board, with a
center-to-center distance of 0.15 cm. Determine the new value of the
thermal resistance of the epoxy board for heat conduction across its
thickness as a result of this modification. Take 0.02 in=0.05 cm and
0.06 in =0.15 cm.
3. Consider a plane composite wall that is composed of two materials of thermal conductivities
kA 0.1 W/m K and kB 0.04 W/m K and thicknesses LA 10 mm and LB 20 mm. The
contact resistance at the interface between the two materials is known to be 0.30 m2 K/W.
Material A adjoins a fluid at 200°C for which h =10 W/m2 K, and material B adjoins a fluid
at 40°C for which=20W/m2 K.
(a) What is the rate of heat transfer through a wall that is 2 m high by 2.5 m wide? (b) Sketch
the temperature distribution.
4. Consider a 5-m-high, 8-m-long, and 0.22-m-thick wall

whose representative cross section is as given in the
figure. The thermal conductivities of various materials
used, in W/m·K, are kA 5kF 52,kB 58,kC 520,kD
515,andkE 535.The left and right surfaces of the wall
are maintained at uniform temperatures of 300°C and
100°C, respectively. Assuming heat transfer through
the wall to be one-dimensional, determine (a) the rate
of heat transfer through the wall; (b) the temperature
at the point where the sections B, D, and E meet; and
(c) the temperature drop across the section F. Disregard any contact resistances at the
interfaces
5. A firefighter’s protective clothing, referred to as a turnout coat, is typically constructed as an
ensemble of three layers separated by air gaps, as shown schematically. Representative
dimensions and thermal
conductivities for the
layers are noted in the
table. The air gaps
between the layers are 1
mm thick, and heat is
transferred by conduction
and radiation ex- change
through the stagnant air.
The linearized radiation coefficient for a gap may be
approximated as, hrad= 𝜎(𝑇1 + 𝑇2 )(𝑇12 + 𝑇22 ) ≈
3 where T
4𝜎𝑇𝑎𝑣𝑔 avg, represents the average
temperature of the surfaces comprising the gap, and
the radiation flux across the gap may be expressed as
qrad = hrad (T1 - T2).
(a) Represent the turnout coat by a thermal circuit, labeling all the thermal resistances.
Calculate the thermal resistances per unit area (m2 K/W) for each of the layers, as well as for
the conduction and radiation processes in the gaps. Assume that a value of Tavg = 470 K may
be used to approximate the radiation resistance of both gaps. Comment on the relative
magnitudes of the resistances.
(b) For a pre-flash-over fire environment in which fire- fighters often work, the typical radiant
heat flux on the fire-side of the turnout coat is 0.25 W/cm2. What is the outer surface
temperature of the turnout coat if the inner surface temperature is 66C, a condition that would
result in burn injury?
6. Measurements show that steady-state conduction through a plane

wall without heat generation produced a convex temperature
distribution such that the mid- point temperature was ∆To higher
than expected for a linear temperature distribution. Assuming that
the thermal conductivity has a linear dependence on temperature, k
=ko(1+𝛼T ), where is a constant, develop a relationship to evaluate
in terms of ∆To, T1, and T2.
7. A truncated solid cone is of circular cross section, and its diameter
is related to the axial coordinate by an expression of the form D=
ax3/2, where a =1.0 m-1/2. The sides are well insulated, while the
top surface of the cone at x1 is maintained at T1 and the bottom sur-
face at x2 is maintained at T2.
(a) Obtain an expression for the temperature distribution T(x).
(b) What is the rate of heat transfer across the cone if it is

constructed of pure aluminum with x1 0.075 m, T1 100°C, x2
0.225 m, and T2 20°C?
8. Electric current flows through a long rod generating thermal energy at a uniform volumetric
rate of q ̇ = 2X106 W/m3. The rod is concentric with a hollow ceramic cylinder, creating an
enclosure that is filled
with air. The thermal
resistance per unit
length due to radiation
between the enclosure
surfaces is Rrad =0.30
m.K/W, and the
coefficient associated
with free convection
in the enclosure is h
is 20 W/m2 K.
(a) Construct a thermal circuit that can be used to calculate the surface temperature of the
rod, Tr. Label all temperatures, heat rates, and thermal resistances, and evaluate each
thermal resistance.
(b) Calculate the surface temperature of the rod for the prescribed conditions
9. Steam flowing through a long,

thin-walled pipe maintains the
pipe wall at a uniform
temperature of 500 K. The pipe
is covered with an insulation
blanket comprised of two
different materials, A and B.
The interface between the two
materials may be assumed to
have an infinite contact
resistance, and the entire outer
surface is exposed to air for
which T∞ =300 K and h = 25 W/m2 K.
_
(a) Sketch the thermal circuit of the system. Label (using the above symbols) all pertinent
nodes and resistances.
(b) For the prescribed conditions, what is the total heat loss from the pipe? What are the outer
surface temperatures Ts,2(A) and Ts,2(B)?
10. A bakelite coating is to be used with a 10-mm-diameter conducting rod, whose surface is
maintained at 200°C by passage of an electrical current. The rod is in a fluid at 25°C, and the
convection coefficient is 140 W/m2 K. What is the critical radius associated with the
coating? What is the heat transfer rate per unit length for the bare rod and for the rod with a
coating of bakelite that corresponds to the critical radius? How much bakelite should be
added to reduce the heat transfer associated with the bare rod by 25%?
11. A hollow aluminum sphere, with an electrical heater in the center, is used in tests to
determine the thermal conductivity of insulating materials. The inner and outer radii of the
sphere are 0.15 and 0.18 m, respectively, and testing is done under steady-state conditions
with the inner surface of the aluminum maintained at 250°C. In a particular test, a spherical
shell of insulation is cast on the outer surface of the sphere to a thickness of 0.12 m. The
system is in a room for which the air temperature is 20°C and the convection coefficient at
the outer surface of the insulation is 30 W/m2 K. If 80 W are dissipated by the heater under
steady-state conditions, what is the thermal conductivity of the insulation?
12. A transistor, which
may be approximated
as a hemispherical heat
source of radius ro 0.1
mm, is embedded in a
large silicon substrate
(k = 125 W/m K) and
dissipates heat at a rate
q. All boundaries of the
silicon are maintained
at an ambient
temperature of T∞ 27°C, except for the top surface, which is well insulated. Obtain a general
_
expression for the substrate temperature distribution and evaluate the surface temperature of
the heat source for q= 4 W.
13. A storage tank consists of a cylindrical section that has a length and inner diameter of L2m
and Di 1m, respectively, and two hemispherical end sections. The tank is constructed from
20-mm-thick glass (Pyrex) and is exposed to ambient air for which the temperature is 300 K
and the convection coefficient is 10 W/m2 K. The tank is used to store heated oil, which
maintains the inner surface at a temperature of 400 K. Determine the electrical power that
must be supplied to a heater submerged in the oil if the prescribed conditions are to be
maintained. Radiation effects may be neglected, and the Pyrex may be assumed to have a
thermal conductivity of 1.4 W/m K.

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MIN 305 Heat & Mass Transfer

Tutorial Sheet 2: 1-D steady state conduction without heat generation

1. The rear window of an automobile is defogged by attaching a thin, transparent, film-type

4. Consider a 5-m-high, 8-m-long, and 0.22-m-thick wall

6. Measurements show that steady-state conduction through a plane

(a) Obtain an expression for the temperature distribution T(x).

(b) What is the rate of heat transfer across the cone if it is

9. Steam flowing through a long,

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