Solutions Manual for Heat and Mass Transfer: Fundamentals & Applications Fourth Edition Yunus A. Cengel & Afshin J. Ghajar

5-1


Solutions Manual t




for
Heat and Mass Transfer: Fundamentals & Applications
t t t t t t




Fourth Edition t




Yunus A. Cengel & Afshin J. Ghajar
t t t t t t




t McGraw-Hill, 2011 t




Chapter 5 t




t NUMERICAL METHODS IN HEAT t t t




CONDUCTION




PROPRIETARY AND CONFIDENTIAL t t




This Manual is the proprietary property of The McGraw-Hill Companies, Inc. (“McGraw-Hill”) and
t t t t t t t t t t t t


protected by copyright and other state and federal laws. By opening and using this Manual the user agrees
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to the following restrictions, and if the recipient does not agree to these restrictions, the Manual should be
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promptly returned unopened to McGraw-Hill: This Manual is being provided only to authorized
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professors and instructors for use in preparing for the classes using the affiliated textbook. No
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other use or distribution of this Manual is permitted. This Manual may not be sold and may not be
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distributed to or used by any student or other third party. No part of this Manual may be
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reproduced, displayed or distributed in any form or by any means, electronic or otherwise,
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without the prior written permission of McGraw-Hill.
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PROPRIETARY tMATERIAL. t© t2011 tThe tMcGraw-Hill tCompanies, tInc. t Limited tdistribution tpermitted tonly tto tteachers tand teducators tfor tcourse
t preparation. t If tyou tare ta tstudent tusing tthis tManual, tyou tare tusing tit twithout tpermission.

, 5-2
Why Numerical Methods?
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5-1 C tAnalytical tsolutions tprovide tinsight tto tthe tproblems, tand tallows tus tto tobserve tthe tdegree tof tdependence tof tsolutions ton
certain parameters. They also enable us to obtain quick solution, and to verify numerical codes. Therefore, analytical solutions
t t t t t t t t t t t t t t t t t t

are not likely to disappear from engineering curricula.
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5-2 C tAnalytical tsolution tmethods tare tlimited tto thighly tsimplified tproblems tin tsimple tgeometries. tThe tgeometry tmust tbe tsuch
that its entire surface can be described mathematically in a coordinate system by setting the variables equal to constants.
t t t t t t t t t t t t t t t t t t t

Also, heat transfer problems can not be solved analytically if the thermal conditions are not sufficiently simple. For example,
t t t t t t t t t t t t t t t t t t

the consideration of the variation of thermal conductivity with temperature, the variation of the heat transfer coefficient over
t t t t t t t t t t t t t t t t t t

the surface, or the radiation heat transfer on the surfaces can make it impossible to obtain an analytical solution. Therefore,
t t t t t t t t t t t t t t t t t t t t

analytical solutions are limited to problems that are simple or can be simplified with reasonable approximations.
t t t t t t t t t t t t t t t t




5-3 C t In tpractice, twe tare tmost tlikely tto tuse ta tsoftware tpackage tto tsolve theat ttransfer tproblems teven twhen tanalytical
solutions are available since we can do parametric studies very easily and present the results graphically by the press of a
t t t t t t t t t t t t t t t t t t t t t

button. Besides, once a person is used to solving problems numerically, it is very difficult to go back to solving differential
t t t t t t t t t t t t t t t t t t t t t

equations by hand.
t t t




5-4 C tThe tenergy tbalance tmethod tis tbased ton tsubdividing tthe tmedium tinto ta tsufficient tnumber tof tvolume telements, tand tthen
applying an energy balance on each element. The formal finite difference method is based on replacing derivatives by their
t t t t t t t t t t t t t t t t t t t

finite difference approximations. For a specified nodal network, these two methods will result in the same set of equations.
t t t t t t t t t t t t t t t t t t t




5-5 C tThe tanalytical tsolutions tare tbased ton t(1) tdriving tthe tgoverning tdifferential tequation tby tperforming tan tenergy tbalance ton
a differential volume element, (2) expressing the boundary conditions in the proper mathematical form, and (3) solving the
t t t t t t t t t t t t t t t t t t

differential equation and applying the boundary conditions to determine the integration constants. The numerical solution
t t t t t t t t t t t t t t t

methods are based on replacing the differential equations by algebraic equations. In the case of the popular finite difference
t t t t t t t t t t t t t t t t t t t

method, this is done by replacing the derivatives by differences. The analytical methods are simple and they provide solution
t t t t t t t t t t t t t t t t t t t

functions applicable to the entire medium, but they are limited to simple problems in simple geometries. The numerical
t t t t t t t t t t t t t t t t t t

methods are usually more involved and the solutions are obtained at a number of points, but they are applicable to any
t t t t t t t t t t t t t t t t t t t t t

geometry subjected to any kind of thermal conditions.
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5-6 C tThe texperiments twill tmost tlikely tprove tengineer tB tright tsince tan tapproximate tsolution tof ta tmore trealistic tmodel tis
t more accurate than the exact solution of a crude model of an actual problem.
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PROPRIETARY tMATERIAL. t© t2011 tThe tMcGraw-Hill tCompanies, tInc. t Limited tdistribution tpermitted tonly tto tteachers tand teducators tfor tcourse
t preparation. t If tyou tare ta tstudent tusing tthis tManual, tyou tare tusing tit twithout tpermission.

, 5-3
Finite Difference Formulation of Differential Equations
t t t t t




5-7 C tA tpoint tat twhich tthe tfinite tdifference tformulation tof ta tproblem tis tobtained tis tcalled ta tnode, tand tall tthe tnodes tfor ta
problem constitute the nodal network. The region about a node whose properties are represented by the property values at
t t t t t t t t t t t t t t t t t t t

the nodal point is called the volume element. The distance between two consecutive nodes is called the nodal spacing, and a
t t t t t t t t t t t t t t t t t t t t t

differential equation whose derivatives are replaced by differences is called a difference equation.
t t t t t t t t t t t t t




5-8 The tfinite tdifference tformulation tof tsteady ttwo-dimensional theat tconduction tin ta tmedium twith theat tgeneration tand
constant thermal conductivity is given by
t t t t t t



Tm1,n  2Tm,n Tm1,n Tm,n1  2Tm,n Tm,n1 e˙m,n
  0
t t t t t
t t t t t t t
t t t

x2 y2 k

in rectangular coordinates. This relation can be modified for the three-dimensional case by simply adding another index j to
t t t t t t t t t t t t t t t t t t

the temperature in the z direction, and another difference term for the z direction as
t t t t t t t t t t t t t t t



Tm1,n, j  2Tm,n, j Tm1,n, j t t Tm,n1, j  2Tm,n, j Tm,n1, j t t Tm,n, j1  2Tm,n, j Tm,n, j1 t t e˙m,n, j
   0
t t t t t t t t t t t t t t t t t t t t t t
t t t t

x 2
t y 2
t z 2
t k




5-9 A tplane twall twith tvariable theat tgeneration tand tconstant tthermal tconductivity tis tsubjected tto tuniform theat tflux t q˙0 t at tthe
left (node 0) and convection at the right boundary (node 4). Using the finite difference form of the 1st derivative, the finite
t t t t t t t t t t t t t t t t t t t t t

difference formulation of the boundary nodes is to be determined.
t t t t t t t t t t



Assumptions 1 Heat transfer through the wall is steady since there is no indication of change with time. 2 Heat transfer is
t t t t t t t t t t t t t t t t t t t t t

one-dimensional since the plate is large relative to its thickness. 3 Thermal conductivity is constant and there is nonuniform
t t t t t t t t t t t t t t t t t t t

heat generation in the medium. 4 Radiation heat transfer is negligible.
t t t t t t t t t t t



Analysis The boundary conditions at the left and right boundaries can be expressed analytically as
t t t t t t t t t t t t t t




at x = 0: dT(0)
k q0
t t
t t t
t t t

dx
dT(L)
k  h[T(L) T ]
t t

at x = L : t t t t t t t t t t
t e(x)
dx q0
h, T
x
t

Replacing derivatives by differences using values at the closest nodes, the
t t t t t t t t t t

finite difference form of the 1st derivative of temperature at the
0    
t t t t t t t t t t t

tboundaries (nodes 0 and 4) can be expressed as t t t t t t t t
t
1 2 3 4
dT T1 T0 dT T4 T3
 
t t
t t
t and t

dx left, tm t t0 x dx right, tm t4 x

Substituting, the finite difference formulation of the boundary nodes become
t t t t t t t t t


T1 T0
k q
t
t t t
at x = 0: t t t
t t t

x 0


T4 T3
k  h[T T ]
t
t t t
at x = L : t t t t
t t

t
t

x 4 t




PROPRIETARY tMATERIAL. t© t2011 tThe tMcGraw-Hill tCompanies, tInc. t Limited tdistribution tpermitted tonly tto tteachers tand teducators tfor tcourse
t preparation. t If tyou tare ta tstudent tusing tthis tManual, tyou tare tusing tit twithout tpermission.

, 5-4
5-10 A tplane twall twith tvariable theat tgeneration tand tconstant tthermal tconductivity tis tsubjected tto tinsulation tat tthe tleft t(node t0)
and radiation at the right boundary (node 5). Using the finite difference form of the 1st derivative, the finite difference
t t t t t t t t t t t t t t t t t t t t

formulation of the boundary nodes is to be determined.
t t t t t t t t t



Assumptions 1 Heat transfer through the wall is steady since there is no indication of change with time. 2 Heat transfer is
t t t t t t t t t t t t t t t t t t t t t

one-dimensional since the plate is large relative to its thickness. 3 Thermal conductivity is constant and there is nonuniform
t t t t t t t t t t t t t t t t t t t


heat generation in the medium. 4 Convection heat transfer is negligible.
t t t t t t t t t t t



Analysis The boundary conditions at the left and right boundaries can be expressed analytically as
t t t t t t t t t t t t t t



dT(0) dT(0)
k 0
t t

0
t t
At x = t t
t t t
t


0: dx
t
t or
dx
dT(L)
k  [T 4 (L) T 4
t t
At
: x=L
t
t t t t t t t t t t t t t t ] Radiation
dx
surr e(x)
Insulated
Tsurr
Replacing derivatives by differences using values at the closest nodes,
t t t t t t t t t x 
the finite difference form of the 1st derivative of temperature at the
t t t t t t t t t t t t      
boundaries (nodes 0 and 5) can be expressed as
t t t t t t t t 0
1 2 3 4 t t 5
dT T1 T0 dT T5 T4
 
t t
t t
t and t

dx left, tm t t0 x dx right, tm t5 x

Substituting, the finite difference formulation of the boundary nodes become
t t t t t t t t t


T1 T0
k 0
t
t t t
At x = t t
t t t
or T1  T0 t

0:
t
x
T5 T4
k  [T 4 T 4
t
t t t
At x = L t t t t t t t t t t t t t ]
:
t


x 5 surr




PROPRIETARY tMATERIAL. t© t2011 tThe tMcGraw-Hill tCompanies, tInc. t Limited tdistribution tpermitted tonly tto tteachers tand teducators tfor tcourse
t preparation. t If tyou tare ta tstudent tusing tthis tManual, tyou tare tusing tit twithout tpermission.