Solutions Manual for Heat and Mass Transfer: Fundamentals & Applications 5th Edition Yunus A. Cengel & Afshin J. Ghajar McGraw-Hill, 2015

5-1


Solutions Manual
for
Heat and Mass Transfer: Fundamentals & Applications
5th Edition
Yunus A. Cengel & Afshin J. Ghajar
McGraw-Hill, 2015




Chapter 5
NUMERICAL METHODS IN HEAT
CONDUCTION

(Last updated August 1, 2015)




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, 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.




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, 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
  0
x 2
y 2
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
   0
x 2
y 2
z 2 k




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, 5-4

5-9 Finite difference formulation for an interior node, boundary node subject to convection and constant heat flux in case of
variable thermal conductivity is to be determined.
Analysis The one dimensional steady state heat conduction equation with variable thermal conductivity is expressed as
d  dT 
k   e  0
dx  dx 
Using Eq. (5-6), the first derivative of the temperature at the midpoints surrounding the node with variable thermal
conductivity can be expressed for x as

k T 
dT  T  T  T  T 
 k o 1   m1 m  m1 m and k T 
dT  T  T  T  T 
 k o 1   m1 m  m1 m
dx m
1  2  x dx m
1  2  x
2 2

Using the definition of second derivative as the derivative of the first derivative we get

 T  T  T  T   T  T  T  T 
k o 1   m1 m  m1 m  k o 1   m1 m  m1 m
d  dT   2  x  2  x
k   e   em  0
dx  dx  x
Simplifying above equation yields,

Tm1  Tm 1   Tm1  Tm   Tm1  Tm 1   Tm1  Tm   em x
2
0
 2   2  ko


 Tm1  2Tm  Tm1  

2
T2
m1 
 2Tm2  Tm21  em
x 2
ko
0

For left boundary node exposed to constant heat flux apply energy balance to the half volume around the boundary node with
all the heat transfer entering the volume element.
Replacing k by k(T) in Eq. 5-22 we get,

 T  T  T  T 
k o 1   m1 m  m1 m  q  em
x 
 0  Tm1  Tm   Tm21  Tm2 
qx em x 2
 0  
 2  x 2 2 ko 2k o

For right boundary node exposed to convection environment, apply energy balance to the half volume around the boundary
node with all the heat transfer entering the volume element.

 T  T  T  T  e x
k o 1   m1 m  m1 m  hT  Tm   m  0
 2  x 2


T  hx
T  Tm   em x  0
2

 Tm1  Tm   2
m1  Tm2 
2 ko 2k o




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