5-1
Solutions Manual
for
Heat and Mass Transfer: Fundamentals & Applications
Fourth Edition
Yunus A. Cengel & Afshin J. Ghajar
McGraw-Hill, 2011
Chapter 5
NUMERICAL METHODS IN HEAT
CONDUCTION
PROPRIETARY AND CONFIDENTIAL
This Manual is the proprietary property of The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and
protected by copyright and other state and federal laws. By opening and using this Manual the user
agrees to the following restrictions, and if the recipient does not agree to these restrictions, the Manual
should be promptly returned unopened to McGraw-Hill: This Manual is being provided only to
authorized professors and instructors for use in preparing for the classes using the affiliated
textbook. No other use or distribution of this Manual is permitted. This Manual may not be sold
and may not be distributed to or used by any student or other third party. No part of this Manual
may be reproduced, displayed or distributed in any form or by any means, electronic or
otherwise, without the prior written permission of McGraw-Hill.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.
, 5-2
Why Numerical Methods?
5-1C Analytical solutions provide insight to the problems, and allows us to observe the degree of dependence of solutions on
certain parameters. They also enable us to obtain quick solution, and to verify numerical codes. Therefore, analytical
solutions are not likely to disappear from engineering curricula.
5-2C Analytical solution methods are limited to highly simplified problems in simple geometries. The geometry must be such
that its entire surface can be described mathematically in a coordinate system by setting the variables equal to constants.
Also, heat transfer problems can not be solved analytically if the thermal conditions are not sufficiently simple. For example,
the consideration of the variation of thermal conductivity with temperature, the variation of the heat transfer coefficient over
the surface, or the radiation heat transfer on the surfaces can make it impossible to obtain an analytical solution. Therefore,
analytical solutions are limited to problems that are simple or can be simplified with reasonable approximations.
5-3C In practice, we are most likely to use a software package to solve heat transfer problems even when analytical
solutions are available since we can do parametric studies very easily and present the results graphically by the press of a
button. Besides, once a person is used to solving problems numerically, it is very difficult to go back to solving differential
equations by hand.
5-4C The energy balance method is based on subdividing the medium into a sufficient number of volume elements, and then
applying an energy balance on each element. The formal finite difference method is based on replacing derivatives by their
finite difference approximations. For a specified nodal network, these two methods will result in the same set of equations.
5-5C The analytical solutions are based on (1) driving the governing differential equation by performing an energy balance
on a differential volume element, (2) expressing the boundary conditions in the proper mathematical form, and (3) solving
the differential equation and applying the boundary conditions to determine the integration constants. The numerical solution
methods are based on replacing the differential equations by algebraic equations. In the case of the popular finite difference
method, this is done by replacing the derivatives by differences. The analytical methods are simple and they provide solution
functions applicable to the entire medium, but they are limited to simple problems in simple geometries. The numerical
methods are usually more involved and the solutions are obtained at a number of points, but they are applicable to any
geometry subjected to any kind of thermal conditions.
5-6C The experiments will most likely prove engineer B right since an approximate solution of a more realistic model is
more accurate than the exact solution of a crude model of an actual problem.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.
, 5-3
Finite Difference Formulation of Differential Equations
5-7C A point at which the finite difference formulation of a problem is obtained is called a node, and all the nodes for a
problem constitute the nodal network. The region about a node whose properties are represented by the property values at
the nodal point is called the volume element. The distance between two consecutive nodes is called the nodal spacing, and a
differential equation whose derivatives are replaced by differences is called a difference equation.
5-8 The finite difference formulation of steady two-dimensional heat conduction in a medium with heat generation and
constant thermal conductivity is given by
Tm −1, n − 2Tm, n + Tm +1, n Tm,n −1 − 2Tm, n + Tm, n +1 e&m, n
2
+ 2
+ =0
∆x ∆y k
in rectangular coordinates. This relation can be modified for the three-dimensional case by simply adding another index j to
the temperature in the z direction, and another difference term for the z direction as
Tm −1, n, j − 2Tm,n, j + Tm +1, n, j Tm,n −1, j − 2Tm,n, j + Tm,n +1, j Tm, n, j −1 − 2Tm, n, j + Tm, n, j +1 e& m, n, j
2
+ 2
+ 2
+ =0
∆x ∆y ∆z k
5-9 A plane wall with variable heat generation and constant thermal conductivity is subjected to uniform heat flux q& 0 at the
left (node 0) and convection at the right boundary (node 4). Using the finite difference form of the 1st derivative, the finite
difference formulation of the boundary nodes is to be determined.
Assumptions 1 Heat transfer through the wall is steady since there is no indication of change with time. 2 Heat transfer is
one-dimensional since the plate is large relative to its thickness. 3 Thermal conductivity is constant and there is nonuniform
heat generation in the medium. 4 Radiation heat transfer is negligible.
Analysis The boundary conditions at the left and right boundaries can be expressed analytically as
dT (0)
at x = 0: −k = q0
dx
dT ( L)
at x = L : −k = h[T ( L) − T∞ ] e(x)
dx q0
h, T∞
Replacing derivatives by differences using values at the closest nodes, the ∆x
finite difference form of the 1st derivative of temperature at the
boundaries (nodes 0 and 4) can be expressed as 0• • • • •
1 2 3 4
dT T1 − T0 dT T4 − T3
≅ and ≅
dx left, m = 0 ∆x dx right, m =4 ∆x
Substituting, the finite difference formulation of the boundary nodes become
T1 − T0
at x = 0: −k = q0
∆x
T4 − T3
at x = L : −k = h[T4 − T∞ ]
∆x
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.
, 5-4
5-10 A plane wall with variable heat generation and constant thermal conductivity is subjected to insulation at the left (node
0) and radiation at the right boundary (node 5). Using the finite difference form of the 1st derivative, the finite difference
formulation of the boundary nodes is to be determined.
Assumptions 1 Heat transfer through the wall is steady since there is no indication of change with time. 2 Heat transfer is
one-dimensional since the plate is large relative to its thickness. 3 Thermal conductivity is constant and there is nonuniform
heat generation in the medium. 4 Convection heat transfer is negligible.
Analysis The boundary conditions at the left and right boundaries can be expressed analytically as
dT (0) dT (0)
At x = 0: −k = 0 or =0
dx dx
dT ( L)
At x = L : −k = εσ [T 4 ( L) − T surr
4
] e(x) Radiation
dx Insulated
Tsurr
Replacing derivatives by differences using values at the closest nodes, ∆x ε
the finite difference form of the 1st derivative of temperature at the
boundaries (nodes 0 and 5) can be expressed as 0• • • • • •
1 2 3 4 5
dT T1 − T0 dT T5 − T4
≅ and ≅
dx left, m = 0 ∆x dx right, m =5 ∆x
Substituting, the finite difference formulation of the boundary nodes become
T1 − T0
At x = 0: −k =0 or T1 = T0
∆x
T5 − T4
At x = L : −k = εσ [T54 − Tsurr
4
]
∆x
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.
Solutions Manual
for
Heat and Mass Transfer: Fundamentals & Applications
Fourth Edition
Yunus A. Cengel & Afshin J. Ghajar
McGraw-Hill, 2011
Chapter 5
NUMERICAL METHODS IN HEAT
CONDUCTION
PROPRIETARY AND CONFIDENTIAL
This Manual is the proprietary property of The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and
protected by copyright and other state and federal laws. By opening and using this Manual the user
agrees to the following restrictions, and if the recipient does not agree to these restrictions, the Manual
should be promptly returned unopened to McGraw-Hill: This Manual is being provided only to
authorized professors and instructors for use in preparing for the classes using the affiliated
textbook. No other use or distribution of this Manual is permitted. This Manual may not be sold
and may not be distributed to or used by any student or other third party. No part of this Manual
may be reproduced, displayed or distributed in any form or by any means, electronic or
otherwise, without the prior written permission of McGraw-Hill.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.
, 5-2
Why Numerical Methods?
5-1C Analytical solutions provide insight to the problems, and allows us to observe the degree of dependence of solutions on
certain parameters. They also enable us to obtain quick solution, and to verify numerical codes. Therefore, analytical
solutions are not likely to disappear from engineering curricula.
5-2C Analytical solution methods are limited to highly simplified problems in simple geometries. The geometry must be such
that its entire surface can be described mathematically in a coordinate system by setting the variables equal to constants.
Also, heat transfer problems can not be solved analytically if the thermal conditions are not sufficiently simple. For example,
the consideration of the variation of thermal conductivity with temperature, the variation of the heat transfer coefficient over
the surface, or the radiation heat transfer on the surfaces can make it impossible to obtain an analytical solution. Therefore,
analytical solutions are limited to problems that are simple or can be simplified with reasonable approximations.
5-3C In practice, we are most likely to use a software package to solve heat transfer problems even when analytical
solutions are available since we can do parametric studies very easily and present the results graphically by the press of a
button. Besides, once a person is used to solving problems numerically, it is very difficult to go back to solving differential
equations by hand.
5-4C The energy balance method is based on subdividing the medium into a sufficient number of volume elements, and then
applying an energy balance on each element. The formal finite difference method is based on replacing derivatives by their
finite difference approximations. For a specified nodal network, these two methods will result in the same set of equations.
5-5C The analytical solutions are based on (1) driving the governing differential equation by performing an energy balance
on a differential volume element, (2) expressing the boundary conditions in the proper mathematical form, and (3) solving
the differential equation and applying the boundary conditions to determine the integration constants. The numerical solution
methods are based on replacing the differential equations by algebraic equations. In the case of the popular finite difference
method, this is done by replacing the derivatives by differences. The analytical methods are simple and they provide solution
functions applicable to the entire medium, but they are limited to simple problems in simple geometries. The numerical
methods are usually more involved and the solutions are obtained at a number of points, but they are applicable to any
geometry subjected to any kind of thermal conditions.
5-6C The experiments will most likely prove engineer B right since an approximate solution of a more realistic model is
more accurate than the exact solution of a crude model of an actual problem.
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.
, 5-3
Finite Difference Formulation of Differential Equations
5-7C A point at which the finite difference formulation of a problem is obtained is called a node, and all the nodes for a
problem constitute the nodal network. The region about a node whose properties are represented by the property values at
the nodal point is called the volume element. The distance between two consecutive nodes is called the nodal spacing, and a
differential equation whose derivatives are replaced by differences is called a difference equation.
5-8 The finite difference formulation of steady two-dimensional heat conduction in a medium with heat generation and
constant thermal conductivity is given by
Tm −1, n − 2Tm, n + Tm +1, n Tm,n −1 − 2Tm, n + Tm, n +1 e&m, n
2
+ 2
+ =0
∆x ∆y k
in rectangular coordinates. This relation can be modified for the three-dimensional case by simply adding another index j to
the temperature in the z direction, and another difference term for the z direction as
Tm −1, n, j − 2Tm,n, j + Tm +1, n, j Tm,n −1, j − 2Tm,n, j + Tm,n +1, j Tm, n, j −1 − 2Tm, n, j + Tm, n, j +1 e& m, n, j
2
+ 2
+ 2
+ =0
∆x ∆y ∆z k
5-9 A plane wall with variable heat generation and constant thermal conductivity is subjected to uniform heat flux q& 0 at the
left (node 0) and convection at the right boundary (node 4). Using the finite difference form of the 1st derivative, the finite
difference formulation of the boundary nodes is to be determined.
Assumptions 1 Heat transfer through the wall is steady since there is no indication of change with time. 2 Heat transfer is
one-dimensional since the plate is large relative to its thickness. 3 Thermal conductivity is constant and there is nonuniform
heat generation in the medium. 4 Radiation heat transfer is negligible.
Analysis The boundary conditions at the left and right boundaries can be expressed analytically as
dT (0)
at x = 0: −k = q0
dx
dT ( L)
at x = L : −k = h[T ( L) − T∞ ] e(x)
dx q0
h, T∞
Replacing derivatives by differences using values at the closest nodes, the ∆x
finite difference form of the 1st derivative of temperature at the
boundaries (nodes 0 and 4) can be expressed as 0• • • • •
1 2 3 4
dT T1 − T0 dT T4 − T3
≅ and ≅
dx left, m = 0 ∆x dx right, m =4 ∆x
Substituting, the finite difference formulation of the boundary nodes become
T1 − T0
at x = 0: −k = q0
∆x
T4 − T3
at x = L : −k = h[T4 − T∞ ]
∆x
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.
, 5-4
5-10 A plane wall with variable heat generation and constant thermal conductivity is subjected to insulation at the left (node
0) and radiation at the right boundary (node 5). Using the finite difference form of the 1st derivative, the finite difference
formulation of the boundary nodes is to be determined.
Assumptions 1 Heat transfer through the wall is steady since there is no indication of change with time. 2 Heat transfer is
one-dimensional since the plate is large relative to its thickness. 3 Thermal conductivity is constant and there is nonuniform
heat generation in the medium. 4 Convection heat transfer is negligible.
Analysis The boundary conditions at the left and right boundaries can be expressed analytically as
dT (0) dT (0)
At x = 0: −k = 0 or =0
dx dx
dT ( L)
At x = L : −k = εσ [T 4 ( L) − T surr
4
] e(x) Radiation
dx Insulated
Tsurr
Replacing derivatives by differences using values at the closest nodes, ∆x ε
the finite difference form of the 1st derivative of temperature at the
boundaries (nodes 0 and 5) can be expressed as 0• • • • • •
1 2 3 4 5
dT T1 − T0 dT T5 − T4
≅ and ≅
dx left, m = 0 ∆x dx right, m =5 ∆x
Substituting, the finite difference formulation of the boundary nodes become
T1 − T0
At x = 0: −k =0 or T1 = T0
∆x
T5 − T4
At x = L : −k = εσ [T54 − Tsurr
4
]
∆x
PROPRIETARY MATERIAL. © 2011 The McGraw-Hill Companies, Inc. Limited distribution permitted only to teachers and educators for course
preparation. If you are a student using this Manual, you are using it without permission.
