2-1
Solutions Manual for
Heat and Mass Transfer: Fundamentals & Applications
6th Edition
Yunus A. Çengel, Afshin J. Ghajar
McGraw-Hill Education, 2020
Chapter 2
HEAT CONDUCTION EQUATION
PROPRIETARY AND CONFIDENTIAL
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2-2
Introduction
2-1C The term steady implies no change with time at any point within the medium while transient implies variation with
time or time dependence. Therefore, the temperature or heat flux remains unchanged with time during steady heat transfer
2-6C The phrase “thermal energy generation” is equivalent to “heat generation,” and they are used interchangeably. They
2-7C The heat transfer process from the kitchen air to the refrigerated space is
transient in nature since the thermal conditions in the kitchen and the
refrigerator, in general, change with time. However, we would analyze this
problem as a steady heat transfer problem under the worst anticipated conditions
2-3
2-8C Heat transfer through the walls, door, and the top and bottom sections of an oven is transient in nature since the thermal
conditions in the kitchen and the oven, in general, change with time. However, we would analyze this problem as a steady
heat transfer problem under the worst anticipated conditions such as the highest temperature setting for the oven, and the
2-9C Heat transfer to a potato in an oven can be modeled as one-dimensional since temperature differences (and thus heat
transfer) will exist in the radial direction only because of symmetry about the center point. This would be a transient heat
2-10C Assuming the egg to be round, heat transfer to an egg in boiling water can be modeled as one-dimensional since
temperature differences (and thus heat transfer) will primarily exist in the radial direction only because of symmetry about
2-11C Heat transfer to a hot dog can be modeled as two-dimensional since temperature differences (and thus heat transfer)
will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in the
2-12C Heat transfer to a roast beef in an oven would be transient since the temperature at any point within the roast will
2-13C Heat loss from a hot water tank in a house to the surrounding medium can be considered to be a steady heat transfer
2-4
2-14C Heat transfer to a canned drink can be modeled as two-dimensional since temperature differences (and thus heat
transfer) will exist in the radial and axial directions (but there will be symmetry about the center line and no heat transfer in
2-15 A certain thermopile used for heat flux meters is considered. The minimum heat flux this meter can detect is to be
determined.
2-5
2-6
2-20E The power consumed by the resistance wire of an iron is given. The heat generation and the heat flux are to be
determined.
Assumptions Heat is generated uniformly in the resistance wire.
q = 1000 W
L = 15 in
D = 0.08 in
2-7
Heat Conduction Equation
2-21C The one-dimensional transient heat conduction equation for a plane wall with constant thermal conductivity and heat
T
e
T
1
gen
2
2-8
2-24 We consider a thin cylindrical shell element of thickness r in a long cylinder (see Fig. 2-14 in the text). The density of
the cylinder is
, the specific heat is c, and the length is L. The area of the cylinder normal to the direction of heat transfer at
any location is
rLA
2=
where r is the value of the radius at that location. Note that the heat transfer area A depends on r in
this case, and thus it varies with location. An energy balance on this thin cylindrical shell element of thickness r during a
small time interval t can be expressed as
t
E
EQQ rrr
=+− + element
element
where
2-9
2-10
2-28 For a medium in which the heat conduction equation is given in its simplest by
0
1
gen =+
e
dr
dT
rk
dr
d
r
:
2-29 For a medium in which the heat conduction equation is given by
0
1
gen =+
+
e
z
T
k
zr
T
kr
rr
:
2-30 For a medium in which the heat conduction equation is given in its simplest by
02
2
2
=+ dr
dT
dr
Td
r
:
2-31 For a medium in which the heat conduction equation is given by
t
T
αr
T
r
r
r
=
11 2
2
2-32 For a medium in which the heat conduction equation is given by
t
TT
r
r
T
r
r
r
=
+
1
sin
11
2
2
22
2
2
2-11
2-33 We consider a small rectangular element of length x, width y, and height z = 1 (similar to the one in Fig. 2-20). The
density of the body is and the specific heat is c. Noting that heat conduction is two-dimensional and assuming no heat
generation, an energy balance on this element during a small time interval t can be expressed as
=
+
−
element theof
content energy the
of change of Rate
and +
at surfaces at the
conductionheat of Rate
and at surfaces
at the conduction
heat of Rate
yyxxyx
or
t
E
QQQQ yyxxyx
=−−+ ++ element
Noting that the volume of the element is
1
element == yxzyx
V
, the change in the energy content of the element can
be expressed as
2-12
2-34 We consider a thin ring shaped volume element of width z and thickness r in a cylinder. The density of the cylinder is
and the specific heat is c. In general, an energy balance on this ring element during a small time interval t can be
expressed as
t
E
QQQQ zzzrrr
=−+− ++ element
)()(
But the change in the energy content of the element can be expressed as
2
z
r
rr
z
2-13
2-35 Consider a thin disk element of thickness z and diameter D in a long cylinder. The density of the cylinder is , the
specific heat is c, and the area of the cylinder normal to the direction of heat transfer is
4/
2
DA
=
, which is constant. An
energy balance on this thin element of thickness z during a small time interval t can be expressed as
=
+
−
element theof
content energy the
of change of Rate
element the
inside generation
heat of Rate
+at surface
at the conduction
heat of Rate
at surface the
at conduction
heat of Rate
zzz
or,
E
2-14
Boundary and Initial Conditions; Formulation of Heat Conduction Problems
2-36C The mathematical expressions of the thermal conditions at the boundaries are called the boundary conditions. To
2-37C The mathematical expression for the temperature distribution of the medium initially is called the initial condition.
2-38C A heat transfer problem that is symmetric about a plane, line, or point is said to have thermal symmetry about that
2-39C The boundary condition at a perfectly insulated surface (at x = 0, for example) can be expressed as
0
),0(
or 0
),0( =
=
−x
tT
x
tT
k
which indicates zero heat flux.
2-42 Heat conduction through the bottom section of an aluminum pan that is used to cook stew on top of an electric range is
considered. Assuming variable thermal conductivity and one-dimensional heat transfer, the mathematical formulation (the
differential equation and the boundary conditions) of this heat conduction problem is to be obtained for steady operation.
Assumptions 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity is given to be variable. 3
C108)(
==
L
TLT
2-15
2-43 Heat conduction through the bottom section of a steel pan that is used to boil water on top of an electric range is
considered. Assuming constant thermal conductivity and one-dimensional heat transfer, the mathematical formulation (the
differential equation and the boundary conditions) of this heat conduction problem is to be obtained for steady operation.
Assumptions 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity is given to be constant. 3
2-16
2-45 Heat is generated in a long wire of radius ro covered with a plastic insulation layer at a constant rate of
gen
e
. The heat
flux boundary condition at the interface (radius ro) in terms of the heat generated is to be expressed. The total heat generated
in the wire and the heat flux at the interface are
)(
2
genwiregengen
o
LreeE
==
V
2-46 A long pipe of inner radius r1, outer radius r2, and thermal conductivity
k is considered. The outer surface of the pipe is subjected to convection to a
egen
h, T
2-17
2-48 Water flows through a pipe whose outer surface is wrapped with a thin electric heater that consumes 300 W per m
length of the pipe. The exposed surface of the heater is heavily insulated so that the entire heat generated in the heater is
transferred to the pipe. Heat is transferred from the inner surface of the pipe to the water by convection. Assuming constant
thermal conductivity and one-dimensional heat transfer, the mathematical formulation (the differential equation and the
boundary conditions) of the heat conduction in the pipe is to be obtained for steady operation.
Assumptions 1 Heat transfer is given to be steady and one-dimensional. 2 Thermal conductivity is given to be constant. 3
There is no heat generation in the medium. 4 The outer surface at r = r2 is subjected to uniform heat flux and the inner surface
at r = r1 is subjected to convection.
Analysis The heat flux at the outer surface of the pipe is
2
W300
Q
Q
2-49 A spherical container of inner radius
r
1
, outer radius
r2
, and thermal conductivity k is
given. The boundary condition on the inner surface of the container for steady one-dimensional
conduction is to be expressed for the following cases:
2-50 A spherical shell of inner radius r1, outer radius r2, and thermal
conductivity k is considered. The outer surface of the shell is subjected to
k
2-18
2-51 A spherical container consists of two spherical layers A and B that are at
perfect contact. The radius of the interface is ro. Assuming transient one-
dimensional conduction in the radial direction, the boundary conditions at the
interface can be expressed as
2-52 A spherical metal ball that is heated in an oven to a temperature of Ti throughout is dropped into a large body of water
at T where it is cooled by convection. Assuming constant thermal conductivity and transient one-dimensional heat transfer,
the mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction
problem is to be obtained.
Assumptions 1 Heat transfer is given to be transient and one-dimensional. 2 Thermal conductivity is given to be constant. 3
There is no heat generation in the medium. 4 The outer surface at r = r0 is subjected to convection.
Analysis Noting that there is thermal symmetry about the midpoint and convection at the outer surface, the differential
i
TrT
r
=
)0,(
2-53 A spherical metal ball that is heated in an oven to a temperature of Ti throughout is allowed to cool in ambient air at T
by convection and radiation. Assuming constant thermal conductivity and transient one-dimensional heat transfer, the
mathematical formulation (the differential equation and the boundary and initial conditions) of this heat conduction problem
is to be obtained.
Assumptions 1 Heat transfer is given to be transient and one-dimensional. 2 Thermal conductivity is given to be variable. 3
There is no heat generation in the medium. 4 The outer surface at r = ro is subjected to convection and radiation.
ro
2-19
Solution of Steady One–Dimensional Heat Conduction Problems
2-54C Yes, the temperature in a plane wall with constant thermal conductivity and no heat generation will vary linearly
during steady one-dimensional heat conduction even when the wall loses heat by radiation from its surfaces. This is because
22 /dxTd
2-55C Yes, this claim is reasonable since in the absence of any heat generation the rate of heat transfer through a plain wall
2-56C Yes, this claim is reasonable since no heat is entering the cylinder and thus there can be no heat transfer from the
2-57C Yes, in the case of constant thermal conductivity and no heat generation, the temperature in a solid cylindrical rod
2-20
2-58 A large plane wall is subjected to specified heat flux and temperature on the left surface and no conditions on the right
surface. The mathematical formulation, the variation of temperature in the plate, and the right surface temperature are to be
determined for steady one-dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional since the wall is large relative to its thickness, and the thermal
conditions on both sides of the wall are uniform. 2 Thermal conductivity is constant. 3 There is no heat generation in the
wall.
Properties The thermal conductivity is given to be k =2.5 W/m°C.
Analysis (a) Taking the direction normal to the surface of the wall to