Solutions Manual for Heat and Mass Transfer:
Fundamentals & Applications Fourth Edition
, 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- -
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
, 5-2
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.
, 5-3
Why Numerical Methods?
5-1 C 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.
5-2 C 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.
5-3 C 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.
5-4 C 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.
5-5 C 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.
5-6 C 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.
Fundamentals & Applications Fourth Edition
, 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- -
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
, 5-2
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.
, 5-3
Why Numerical Methods?
5-1 C 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.
5-2 C 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.
5-3 C 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.
5-4 C 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.
5-5 C 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.
5-6 C 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.
