Chapter 2: Heat Conduction Equation

2-1 General Relation for Fourier’s Law of

Heat Conduction
2-2 Heat Conduction Equation
2-3 Boundary Conditions and Initial Conditions

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-1 General Relation for Fourier’s Law
of Heat Conduction (1)

The rate of heat conduction through a medium in a

specified direction (say, in the x-direction) is
expressed by Fourier’s law of heat conduction for
one-dimensional heat conduction as:
dT
Qcond  kA (W)
dx
Heat is conducted in the direction
of decreasing temperature, and thus
the temperature gradient is negative
when heat is conducted in the positive x-direction.
2-1

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-1 General Relation for Fourier’s Law
of Heat Conduction (2)

The heat flux vector at a point P on the surface of the

figure must be perpendicular to the surface, and it must
point in the direction of decreasing temperature
If n is the normal of the
isothermal surface at point P,
the rate of heat conduction at
that point can be expressed by
Fourier’s law as
dT
Qn  kA (W)
dn 2-2

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-1 General Relation for Fourier’s Law
of Heat Conduction (3)

In rectangular coordinates, the heat conduction vector

can be expressed in terms of its components as

Qn  Qx i  Qy j  Qz k
which can be determined from Fourier’s law as
 T
 x
Q   kA
x
x

 T
Qy  kAy
 y
 T
 z
Q   kA
 z
z
2-3

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-2 Heat Conduction Equation (1)

Rate of heat Rate of heat Rate of heat Rate of change of

conduction - conduction + generation inside the energy content
at x at x+ x the element
= of the element

Eelement
Qx Qx x  Egen,element 
t
Q 1  2 
Qx
Q x  x  Q x  x
x  ( x ) 2
 ...
x 2! x 2

  Q x  T  T
Qx  Qx  x     (kA )  (kA )
x x x x x
Neglect higher orders 2-4

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-2 Heat Conduction Equation (2)
 Eelement  Et t  Et  mc Tt t  Tt    cAx Tt t  Tt 


 Egen,element  egenVelement  egen Ax

Tt t  Tt
Qx  Qx x egen Ax   cAx
t
Dividing by Ax, taking the limit as x 0 and t 0,
and from Fourier’s law:
1   T  T
 kA 
 gen
e   c
A x  x  t
The area A is constant for a plane wall 
the one dimensional transient heat conduction equation in a plane wall
is   T  T
Variable conductivity:  k 
 gen
e   c
x  x  t 2-5

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-2 Heat Conduction Equation (3)

Constant conductivity:  2T egen 1 T k

  ; 
x 2
k  t c

The one-dimensional conduction equation may be reduces to the

following forms under special conditions

d 2T egen
1) Steady-state: 2
 0
dx k
 2T 1 T
2) Transient, no heat generation: 
x 2
 t

d 2T
3) Steady-state, no heat generation: 2
0
dx 2-6

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-2 Heat Conduction Equation (4)

General Heat Conduction Equation

Rate of heat Rate of heat Rate of heat Rate of change
conduction - conduction
+ generation
= of the energy
at x, y, and z at x+ x, y+ y, inside the content of the
and z+ z element element

Qx  Qy  Qz Qxx  Qy y  Qz z  Egen,element  Eelement

t
T
(kT )  egen  c
t
 2T  2T  2T egen
1) Steady-state:  
x 2 y 2 z 2

k
0
 2T  2T  2T 1 T
2) Transient, no heat generation:   
x 2 y 2 z 2  t
 2T  2T  2T
  0
3) Steady-state, no heat generation: x 2 y 2 z 2
2-7

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-2 Heat Conduction Equation (5)

Cylindrical Coordinates

1   T  1 T  T    T  T
 rk  2 k  k   egen   c
r r  r  r     z  z  t
2-8

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-2 Heat Conduction Equation (6)

Spherical Coordinates

1   2 T  1   T  1   T  T
 kr  2 2 k  2  k sin    egen   c
r r 
2
r  r sin      r sin      t

2-9

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-3 Boundary and Initial Conditions (1)

Specified Temperature Boundary Condition

Specified Heat Flux Boundary Condition
Convection Boundary Condition
Radiation Boundary Condition
Interface Boundary Conditions
Generalized Boundary Conditions

2-10

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-3 Boundary and Initial Conditions (2)

Specified Temperature Boundary Condition

For one-dimensional heat transfer
through a plane wall of thickness L,
for example, the specified
temperature boundary conditions
can be expressed as
T(0, t) = T1
T(L, t) = T2

The specified temperatures can be constant, which is the

case for steady heat conduction, or may vary with time. 2-11

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-3 Boundary and Initial Conditions (3)

Specified Heat Flux Boundary Condition

The heat flux in the positive x-direction
anywhere in the medium, including the
boundaries, can be expressed by Fourier’s
law of heat conduction as

dT Heat flux in the

q  k  positive x-
dx direction

The sign of the specified heat flux is determined by inspection: positive

if the heat flux is in the positive direction of the coordinate axis, and
negative if it is in the opposite direction.
2-12

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-3 Boundary and Initial Conditions (4)

Two Special Cases

Insulated boundary Thermal symmetry

k
T (0, t )
0 or
T (0, t )
0 
T L , t
2 0 
x x x
2-13

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-3 Boundary and Initial Conditions (5)

Convection Boundary Condition

Heat conduction Heat convection

at the surface in a
selected direction = at the surface in
the same direction

T (0, t )
k  h1 T1  T (0, t )
x
and
T ( L, t )
k  h2 T ( L, t )  T 2 
x 2-14

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-3 Boundary and Initial Conditions (6)

Radiation Boundary Condition

Heat conduction Radiation exchange

at the surface in a
selected direction
= at the surface in
the same direction

T (0, t )
k  1 Tsurr
4
 T (0, t ) 4

x
,1

and
T ( L, t )
k   2 T ( L, t )4  Tsurr
4

,2 
x 2-15

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013
 2-3 Boundary and Initial Conditions (7)

Interface Boundary Conditions

At the interface the requirements are:
(1) two bodies in contact must have the same temperature at the
area of contact,
(2) an interface (which is a
surface) cannot store any
energy, and thus the heat flux
on the two sides of an
interface must be the same.
TA(x0, t) = TB(x0, t)
and
TA ( x0 , t ) T ( x , t )
k A  k B B 0
x x 2-16

Heat Transfer Chapter 2: Heat Conduction Equation

Y.C. Shih September 2013