Advanced Heat Transfer/ I Conduction and Handout Lectures for MSc. / Power Chapter Four/ Steady-State TwoDimensional Conduction Heat Transfer

University of Anbar
College of Engineering
Mechanical Engineering Dept.




Advanced Heat Transfer/ I
Conduction and Radiation
Handout Lectures for MSc. / Power
Chapter Four/ Steady-State Two-
Dimensional Conduction Heat Transfer

Course Tutor
Assist. Prof. Dr. Waleed M. Abed

 J. P. Holman, “Heat Transfer”, McGraw-Hill Book Company, 6th Edition,
2006.
 T. L. Bergman, A. Lavine, F. Incropera, D. Dewitt, “Fundamentals of Heat
and Mass Transfer”, John Wiley & Sons, Inc., 7th Edition, 2007.
 Vedat S. Arpaci, “Conduction Heat Transfer”, Addison-Wesley, 1st Edition,
1966.
 P. J. Schneider, “Conduction Teat Transfer”, Addison-Wesley, 1955.
 D. Q. Kern, A. D. Kraus, “Extended surface heat transfer”, McGraw-Hill
Book Company, 1972.
 G. E. Myers, “Analytical Methods in Conduction Heat Transfer”, McGraw-
Hill Book Company, 1971.
 J. H. Lienhard IV, J. H. Lienhard V, “A Heat Transfer Textbook”, 4th
Edition, Cambridge, MA : J.H. Lienhard V, 2000.


Advanced Heat Transfer/ I
Conduction and Radiation

, Steady-State Two-Dimensional Conduction Heat Transfer Chapter: Four


Chapter Four
Steady-State Two-Dimensional Conduction Heat Transfer


4.1 Introduction
In many cases such problems are grossly oversimplified if a one-dimensional
treatment is used, and it is necessary to account for multidimensional effects. In
this chapter, we will focus on "analytical method" for treating two-dimensional
systems under steady-state conditions.


4.2 Boundary-value problems and characteristic-value problems
Consider an ordinary differential equation of second order which may result from
the differential formulation of a steady one-dimensional conduction problem. The
solution of this equation involves two arbitrary constants which are determined by
two conditions, each specified at one boundary of the problem. Problems of this
type are called boundary-value problems to distinguish them from initial-value
problems, in which all conditions are specified at one location. Reconsider the
differential equation

(4-1)
Assume that this homogeneous equation involves a parameter "λ" as

(4-2)

And is subject to homogeneous boundary conditions
( ) , and ( ) then the general solution of Equation (4-2) is
(4-3)
The use of ( ( ) ) results in and ,

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Advanced Heat Transfer/ I
Conduction and Radiation

, Steady-State Two-Dimensional Conduction Heat Transfer Chapter: Four
From ( ( ) ), combined with , gives . The
problem has nontrivial solutions only if λ satisfies the ( ). Therefore,
, where n= 1, 2, 3, ……, (4-4)

And the corresponding solutions of ( ) are,
( ), ( ) ( ) (4-5)
Note that no new solutions are obtained when "n" assumes negative integer values.
Thus, the foregoing boundary-value problem has no solution other than the trivial
solution y= 0, unless λ assumes one of the characteristic values given by Equation
4-4. Corresponding to each characteristic value of λn there exists a characteristic
function ( ) given by Equation 4-5, such that any constant multiple of this
function is a solution of the problem. It is important to note that the boundary-
value problem given by

; ( ) , ( )

has no solution other than the trivial solution y= 0 corresponding to λ= 0. Hence
there does not exist any set of characteristic values and characteristic functions for
this problem. This illustrates the fact that a boundary-value problem may or may
not be a characteristic-value problem. A boundary-value problem is a
characteristic-value problem when it has particular solutions that are periodic in
nature; the period and amplitude of these solutions may or may not be constant.
Therefore, in the next three sections the general properties of characteristic
functions are investigated.


4.3 Orthogonality of Characteristic Functions
By definition, two functions ( ) and ( ) are said to be orthogonal with
respect to a weighting function ( ), over a finite interval (a, b), if the integral of
the product ( ) ( ) ( ) over that interval vanishes as
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Advanced Heat Transfer/ I
Conduction and Radiation

, Steady-State Two-Dimensional Conduction Heat Transfer Chapter: Four

∫ ( ) ( ) ( ) , where m ≠ n (4-6)
Furthermore, a set of functions is said to be orthogonal in (a, b) if all pairs of
distinct functions in the set are orthogonal in (a, b). The word orthogonality comes
from vector analysis. Let ( ) denote a vector in 3D space whose rectangular
components are ( ), ( ), and ( ). Two vectors, ( ) and ( ),
are said to be orthogonal, or perpendicular to each other, if
( ) ( ) ∑ ( ) ( ) (4-7)
When the units of length on the coordinate axes vary from one axis to another, the
foregoing scalar product assumes the form
( ) ( ) ∑ ( ) ( ) ( ) (4-8)
Where the weighting numbers ( ), ( ), and ( ) depend upon the units of
length used along the three axes. The vectors in an N-Dimensional space having
components ( ), ( ), i= 1, 2, 3, ……, N are said to be orthogonal with
respect to the weighting numbers ( ).
It will now be shown that the characteristic functions of a characteristic-value
problem are orthogonal over a finite interval with respect to a weighting function.
To establish this fact, consider the characteristic-value problem composed of the
linear homogenous second-order differential equation of the general form

( ) ( ) ( ) (4-9)
∫ ( )
This equation, multiplied through by the factor ( ( )) and with the
functions defined as ( ) ( ) ( ) and ( ) ( ) ( ), may be rearranged
in the form

* ( ) + ( ) ( ) (4-10)

Which is more convenient for the following discussion.



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Advanced Heat Transfer/ I
Conduction and Radiation