Department of polymer And Petrochemical Engineering
Heat And Mass Transfer
Assistant lecture:Qusai A.Mahdi
Chapter three
Steady-State Conduction two dimensions
For steady state, two dimensional heat transfers by conduction with no
heat generation the general heat conduction equation reduced to
assuming constant thermal conductivity. The solution to this equation
may be obtained by analytical, numerical, or graphical techniques.The
objective of any heat-transfer analysis is usually to predict heat flow or
the temperature that results from a certain heat flow. The solution to
Equation (3-1) will give the temperature in a two-dimensional body as a
function of the two independent space coordinates x and y. Then the heat
flow in the x and y directions may be calculated from the Fourier
equations .....(3 ? 2)
? ?
? ?
T x
qx kAx
.........(3 ? 3)
? ?
? ?
T y
q kA
y y
So if the temperature distribution inthe material is known, we may easily
establish the heat flow.
MATHEMATICAL ANALYSIS OF TWO-DIMENSIONAL HEAT
CONDUCTION
To solve Equation (3-1), the separation-of-variables method is used. The
essential point of this method is that the solution to the differential
equation is assumed to take a product form
T = XY where X = f(x) Y = f(y)
.
d Y. ....(3 5)
. d Y. ..(3 4)
2
2
2
2
2
2
2
2
by subsituting in controlling differntia l eqn
dy
d Y
dy
T
and
dx
d X
dx
T
dx
dX
Y
dx
dT
? ?
? ? ?
? 1 2 2 ? 1 2 2 .....(3? 6)
dy
d Y
dx Y
d X
X
2 0......(3-1)
2
2
2
?
? ?
?
? ?
T y
T x
Department of polymer And Petrochemical Engineering
Heat And Mass Transfer
Assistant lecture:Qusai A.Mahdi
Each side of Equation (3-6) is independent of the other because x and y
are independent variables. This requires that each side be equal to some
constant. We may thus obtain two ordinary differential equations in terms
of this constant .
2
2
2
2
2
1 1
? ? ? ?
dy
d Y
dx Y
d X
X
This equation gives two differential equations
2 0
2
2
? X ?
dx
d X
? …(3-7)
2 0
2
2
? Y ?
dy
d Y
? ….(3-8)
Where ?2is called the separation constant. Its value must be determined
from the boundary conditions. Consider a rectangular plate shown in
figure ,three sides of plate maintained at constant temperature (T1). The
boundary conditions with a sine-wave temperature distribution impressed
on the upper edge of the plate. Thus
(1)T=T1 at y = 0
(2)T = T1 at x = 0
(3)T=T1 at x = W
(4)
x W
Tm T
m
?
? sin at y = H
Where (Tm) is the amplitude of the sine function.
The first step in solving the problem is examine the value of (?2),there are
three possibilities (?2=0, ?2<0,and ?2 >0) one of this value is acceptable
,and the other will be neglected.
For ?2=0 0 and 0
2
2
2
2
? ?
dy
d Y
dx
d X
The solutions are X=(C1+C2x)
Y=(C3+C4y) and
T= (C1+C2x) *(C3+C4y)…(3-9) This function cannot fit the sine-function
boundary condition, so the?2=0 solution may be excluded
Department of polymer And Petrochemical Engineering
Heat And Mass Transfer
Assistant lecture:Qusai A.Mahdi
For ?2 < 0
( )( cos sin ) .....(3-10)
( cos sin )
( )
5 6 7 8
7 8
5 6
T C e C e C y C y
Y C y C y
X C e C e
x x
x x
? ?
? ?
? ?
? ?
? ? ?
? ?
? ?
? ?
Again, the sine-function boundary condition cannot be satisfied, so this
solution is excluded also.
For ?2 > 0
( cos sin )( )....(3 11)
( )
( cos sin )
9 10 11 12
11 12
9 10
? ? ? ?
? ?
? ?
?
?
y y
y y
T C x C x C e C e
Y C e C e
X C x C x
? ?
? ?
? ?
? ?
Now, it is possible to satisfy the sine-function boundary condition; so we shall attempt
to satisfy the other conditions. The algebra is somewhat easier to handle when the
substitution is made as
? ? T ? T1
The transform of the boundary conditions are
(1)? = 0 at y = 0 (2)? = 0 at x = 0
(3)? = 0 at x = W (4)? = W
x
Tm
?
sin
at y = H ..(3-12)
Applying these conditions, we have
sin ( cos sin )( )...[ ]
0 ( cos sin )( )....[ ]
0 ( ).....[ ]
0 ( cos sin )( )...[ ]
9 10 11 12
9 10 11 12
9 11 12
9 10 11 12
C x C x C e C e d
x W
T
C W C W C e C e c
C C e C e b
C x C x C C a
H H
m
y y
y y
? ?
? ?
? ?
? ? ?
? ?
? ?
? ? ?
? ? ?
? ?
? ? ?
?
?
?
Accordingly C11=-C12 and C9=0
From [c]
0 ? C10C12 sin ?W(e?y ? e??y )
This requires tha
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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