2-81
Special Topic: Review of Differential equations
2-126C We utilize appropriate simplifying assumptions when deriving differential equations to obtain an equation that we
can deal with and solve.
2-127C A variable is a quantity which may assume various values during a study. A variable whose value can be changed
2-128C A differential equation may involve more than one dependent or independent variable. For example, the equation
t
txT
k
e
x
txT
),(
1
),( gen
2
2
t
txW
t
txT
x
txW
x
txT
+
=
+
),(
1
),(
1
),(),(
2
2
has 2 dependent (T and W) and 2 independent variables (x and t).
2-129C Geometrically, the derivative of a function y(x) at a point represents the slope of the tangent line to the graph of the
2-130C The order of a derivative represents the number of times a function is differentiated, whereas the degree of a
2-82
2-83
Review Problems
2-142 A plane wall is subjected to uniform heat flux on the left surface, while the right surface is subjected to convection and
radiation heat transfer. The boundary conditions and the differential equation of this heat conduction problem are to be
obtained.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivity is constant. 3 There is no heat
generation in the wall. 4 The left surface at x = 0 is subjected to uniform heat flux while the right surface at x = L is subjected
to convection and radiation. 5 The surrounding temperature is T∞ = Tsurr.
Analysis Taking the direction normal to the surface of the wall to be the x
direction with x = 0 at the left surface, the differential equation for heat
conduction can be expressed as
2=
Td
2-84
2-144E A large plane wall is subjected to a specified temperature on the left (inner) surface and solar radiation and heat loss
by radiation to space on the right (outer) surface. The temperature of the right surface of the wall and the rate of heat transfer
are to be determined when steady operating conditions are reached.
Assumptions 1 Steady operating conditions are reached. 2 Heat transfer is
one-dimensional since the wall is large relative to its thickness, and the
thermal conditions on both sides of the wall are uniform. 3 Thermal
properties are constant. 4 There is no heat generation in the wall.
Properties The properties of the plate are given to be k = 1.2 Btu/hft°F and
= 0.80, and
60.0=
s
.
Analysis In steady operation, heat conduction through the wall must be equal
to net heat transfer from the outer surface. Therefore, taking the outer surface
2-145 A spherical vessel is subjected to uniform heat flux on the inner surface, while the outer surface is subjected to
convection and radiation heat transfer. The boundary conditions and the differential equation of this heat conduction problem
are to be obtained.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivity is constant. 3 There is no heat
generation in the wall. 4 The inner surface at r = r1 is subjected to uniform heat flux while the outer surface at r = r2 is
subjected to convection and radiation. 5 The surrounding temperature is T∞ = Tsurr.
Analysis For one-dimensional heat transfer in the radial r direction,
the differential equation for heat conduction in spherical coordinate
can be expressed as
2=
dT
d
qsolar
520 R
T2
2-85
2-146 Heat is generated at a constant rate in a short cylinder. Heat is lost from the cylindrical surface at r = ro by convection
to the surrounding medium at temperature
T
with a heat transfer coefficient of h. The bottom surface of the cylinder at r = 0
is insulated, the top surface at z = H is subjected to uniform heat flux
h
q
, and the cylindrical surface at r = ro is subjected to
convection. The mathematical formulation of this problem is to be expressed for steady two-dimensional heat transfer.
Assumptions 1 Heat transfer is given to be steady and two-dimensional. 2 Thermal conductivity is constant. 3 Heat is
generated uniformly.
Analysis The differential equation and the boundary conditions for this heat conduction problem can be expressed as
0
1gen
2
2
=+
+
k
e
z
T
r
T
r
rr
H
q
z
HrT
k
z
rT
=
=
),(
0
)0,(
]),([
),(
0
),0(
−=
−
=
TzrTh
r
zrT
k
r
zT
o
o
2-147 A small hot metal object is allowed to cool in an environment by convection. The differential equation that describes
the variation of temperature of the ball with time is to be derived.
Assumptions 1 The temperature of the metal object changes uniformly with time during cooling so that T = T(t). 2 The
density, specific heat, and thermal conductivity of the body are constant. 3 There is no heat generation.
Analysis Consider a body of arbitrary shape of mass m, volume
V
, surface area A, density
, and specific heat cp initially at a
uniform temperature Ti. At time t = 0, the body is placed into a medium at temperature
T
, and heat transfer takes place
between the body and its environment with a heat transfer coefficient h.
During a differential time interval dt, the temperature of the body rises by a
differential amount dT. Noting that the temperature changes with time only, an energy
balance of the solid for the time interval dt can be expressed as
energy in the decrease The
body thefromfer Heat trans
h
T
egen
qH
z
ro
h
T
m, c, Ti
A
2-86
2-148 A large plane wall is subjected to convection on the inner and outer surfaces. The mathematical formulation, the
variation of temperature, and the temperatures at the inner and outer surfaces to be determined for steady one-dimensional
heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivity is constant. 3 There is no heat
generation.
Properties The thermal conductivity is given to be k = 0.77 W/m°C.
Analysis (a) Taking the direction normal to the surface of the wall to be the x direction with x = 0 at the inner surface, the
mathematical formulation of this problem can be expressed as
0
2
2
=
dx
Td
and
dx
dT
kTTh )0(
)]0([ 11 −=−
])([
)(
22
−=− TLTh
dx
LdT
k
(b) Integrating the differential equation twice with respect to x yields
1
C
dx
dT =
21
)( CxCxT +=
where C1 and C2 are arbitrary constants. Applying the boundary conditions give
x = 0:
12111 )]0([ kCCCTh −=+−
x = L:
])[( 22121
−+=− TCLChkC
Substituting the given values, these equations can be written as
12 77.0)22(8 CC −=−
)82.0)(12(77.0 211 −+=− CCC
Solving these equations simultaneously give
26.18 84.38 21 =−= CC
Substituting
21 and CC
into the general solution, the variation of temperature is determined to be
xxT 84.3826.18)( −=
(c) The temperatures at the inner and outer surfaces are
C10.5
C18.3
=−=
=−=
2.084.3826.18)(
084.3826.18)0(
LT
T
k
h1
T1
L
h2
T2
2-87
2-149 The base plate of an iron is subjected to specified heat flux on the left surface and convection and radiation on the right
surface. The mathematical formulation, and an expression for the outer surface temperature and its value are to be determined
for steady one-dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional. 2 Thermal conductivity is constant. 3 There is no heat
generation. 4 Heat loss through the upper part of the iron is negligible.
Properties The thermal conductivity and emissivity are given to be k = 18 W/m°C and = 0.7.
Analysis (a) Noting that the upper part of the iron is well insulated and thus the entire
heat generated in the resistance wires is transferred to the base plate, the heat flux
through the inner surface is determined to be
2
24
base
0
0 W/m667,66
m 10150
W1000 =
== −
A
Q
q
Taking the direction normal to the surface of the wall to be the x direction with x = 0 at
the left surface, the mathematical formulation of this problem can be expressed as
2
Td
h
T
Tsurr
q
2-88
2-150 A 30 m2 concrete slab with embedded heating cable melts snow at a rate of 0.1 kg/s. Formulate the temperature
profile in the concrete slab in terms of the snow melt rate. The power density for the embedded heater is to be determined
whether it is in compliance with the NFPA 70 code.
Assumptions1 Heat transfer is steady. 2 One dimensional heat conduction through the concrete slab. 3 The bottom surface at
x = 0 is subjected to uniform heat flux from the heating cable. 4 The upper surface at x = L is at a constant temperature of 0°C
from the snow melt. 5 There is no heat generation in the concrete slab. 6 Thermal properties are constant.
Properties The latent heat of fusion for water is 333.7 kJ/kg (Table A-2).
AnalysisTaking the direction normal to the surface of the concrete slab to be the x direction with x = 0 at the bottom surface
(the surface that is in contact with the heater surface), the differential equation for heat conduction can be expressed as
𝑑2𝑇
𝐴𝑠
30 m2=𝟏𝟏𝟏𝟐 W/m2<1300 W/m2
2-89
2-151 A series of ASME SA-193 carbon steel bolts are bolted on the upper surface of a metal plate. The upper surface is
exposed to convection with the ambient air. The bottom surface is subjected to a uniform heat flux. Formulate the variation of
temperature in the metal plate, and determine the temperatures at x = 0, 1.5, and 3.0 cm. The compliance of the SA-193 bolts
with the ASME Boiler and Pressure Vessel Code (ASME BPVC.IV-2015, HF-300) is to be determined.
Assumptions1 Heat transfer is steady. 2 One dimensional heat conduction through the metal plate. 3 The bottom surface at x
= 0 is subjected to uniform heat flux while the upper surface at x = L is at uniform temperature. 4 There is no heat generation
in the plate. 5 Thermal properties are constant.
Properties The thermal conductivity of the metal plate is given as 15 W/m·K.
Analysis Taking the direction normal to the surface of the plate to be the x direction with x = 0 at the bottom surface, the
differential equation for heat conduction can be expressed as
𝑑2𝑇
𝑑𝑥2= 0
Integrating the differential equation twice with respect to x yields
𝑑𝑇
2-90
2-91
2-153 A 10-m tall exhaust stack discharging exhaust gases at a rate of 1.2 kg/s is subjected to solar radiation and convection
at the outer surface. The variation of temperature in the exhaust stack and the inner surface temperature of the exhaust stack
are to be determined.
Assumptions 1 Heat conduction is steady and one-dimensional and there is thermal symmetry about the centerline. 2 Thermal
properties are constant. 3 There is no heat generation in the pipe.
Properties The constant pressure specific heat of exhaust gases is given to be 1600 J/kg ∙ °C and the pipe thermal
conductivity is 40 W/m ∙ K. Both the emissivity and solar absorptivity of the exhaust stack outer surface are 0.9.
Analysis The outer and inner radii of the pipe are
dr
dr
and
Lr
Q
A
Q
dr
rdT
k
s1
loss
1,
loss
1
2
)(
==−
(heat flux at the inner exhaust stack surface)
r
dr
2-92
Integrating with respect to r again gives
21 ln)( CrCrT +=
where
1
C
and
2
C
are arbitrary constants. Applying the boundary conditions gives
:
1
rr =
1
1
1
loss
1
2
1
)(
r
C
Lr
Q
kdr
rdT =−=
→
kL
Q
Closs
12
1
−=
→
C
C
loss
1
Q
1
loss
Q
1
2
2-93
2-154E A steam pipe is subjected to convection on the inner surface and to specified temperature on the outer surface. The
mathematical formulation, the variation of temperature in the pipe, and the rate of heat loss are to be determined for steady
one-dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional since the pipe is long relative to its thickness, and there is
thermal symmetry about the center line. 2 Thermal conductivity is constant. 3 There is no heat generation in the pipe.
Properties The thermal conductivity is given to be k = 8 Btu/hft°F.
Analysis (a) Noting that heat transfer is one-dimensional in the radial r direction, the mathematical formulation of this
problem can be expressed as
0=
dr
dT
r
dr
d
The boundary conditions for this problem are:
)(
1rTTh
rdT
Steam
250F
T =160F
2-94
2-155 A compressed air pipe is subjected to uniform heat flux on the outer surface and convection on the inner surface. The
mathematical formulation, the variation of temperature in the pipe, and the surface temperatures are to be determined for
steady one-dimensional heat transfer.
Assumptions 1 Heat conduction is steady and one-dimensional since the pipe is long relative to its thickness, and there is
thermal symmetry about the center line. 2 Thermal conductivity is constant. 3 There is no heat generation in the pipe.
Properties The thermal conductivity is given to be k = 14 W/mK.
Analysis (a) Noting that the 85% of the 300 W generated by the strip heater is transferred to the pipe, the heat flux through
the outer surface is determined to be
2
W 300850
.
Q
Q