A Derivation of the Heat Conduction Equation
Ernesto GutierrezMiravete
September 19, 1994
1 The Thermal Energy Balance
Consider a small region inside a solid material which is being heated (or
cooled). The molecules inside the region vibrate due to thermal excitation.
A quantitative expression of molecular thermal excitation is provided by the
specific internal energy u (i.e. internal energy per unit
volume).
The specific internal energy of a heated (or cooled) solid body changes when
energy is added to or substracted from the body.
Consider the small region inside the body to be a tetrahedron with a vertex at
the origin of a rectangular cartesian system of coordinates and with three
of its sides along the three coordinate axes.
Let a_{x}, a_{y}, a_{z} denote the areas of the tetrahedron faces
perpendicular to the Ox, Oy, and Oz axes, respectively, and let
a_{o} be the area of the fourth face. Further, let [(n_{o})\vec] the
outward normal to the inclined
face. Assume that energy enters (or exits) the tetrahedron through all faces.
A simple energy balance (i.e. time rate of change of specific internal energy
= total energy flux through all boundaries) for the tetrahedral element gives



ó õ


ó õ


ó õ


¶u ¶t

dV = 
ó õ


ó õ

f(x,y,z, 
® i

,t) da_{x} + 
ó õ


ó õ

f(x,y,z, 
® j

,t) da_{y} + 
 


ó õ


ó õ

f(x,y,z, 
® k

,t) da_{z} + 
ó õ


ó õ

f(x,y,z, 
® n_{o}

,t) da_{o} 

 

Here, the f's are the energy flows through the different faces of the
element (i.e. energy per unit area per unit time). These flows are
regarded as positive if they are in the direction of, respectively,
[i\vec], [j\vec], [k\vec] or [(n_{o})\vec].
The total volume of the tetrahedron is [1/3] h a_{o},
where h is the perpendicular distance from the origin to the face of
area a_{o}, from the mean value theorem one can write

1 3

h a_{o} 

= a_{x} 
q_{x}

+ a_{y} 
q_{y}

+ a_{z} 
q_{z}

+ a_{o} 


 (1) 
where the bars denote mean values and
are used to simplify notation.
Now, taking the limit as h ® 0 while maintaining
[(n_{o})\vec] constant,
the term [1/3] h a_{o}[`([(¶u )/(¶t)])] ® 0 and since
f(x,y,z, 
® n_{o}

,t) =  f(x,y,z, 
® n_{o}

,t) 
 (5) 
one obtains,
f(x,y,z, 
® n_{o}

,t) = q_{x} cos( 
® n_{o}

,x) + q_{y} cos( 
® n_{o}

,y) + q_{z} cos( 
® n_{o}

,z) 
 (6) 
where cos[((n_{o})\vec],x_{i}) are direction cosines of the outward normal
to the face of area a_{o}.
Thus, the function f(x,y,z,[n\vec],t) is the projection of
certain vector [q\vec] = (q_{x},q_{y},q_{z}) onto the direction of
the normal vector [(n_{o})\vec].
The vector [q\vec], analogous to the mass flux (i.e. mass crossing an unit
area per unit time) in fluid flow
problems, is called the heat flux vector and represents energy
transferred per unit time across an unit area.
Finally, for a small region of arbitrary shape, the energy balance equation is

ó õ


ó õ


ó õ


¶u ¶t

dV =  
ó õ


ó õ


® q

· 
® n

da 
 (7) 
where [n\vec] is the normal to the region's boundary.
The divergence (Green's) theorem can be applied to the right hand side of
the above equation to give

ó õ


ó õ


ó õ


¶u ¶t

dV =  
ó õ


ó õ


ó õ

Ñ· 
® q

dV 
 (8) 
Since this equation is valid for any volume, the integrands are equal and

¶u ¶t

=  Ñ· 
® q

=  div ( 
® q

) 
 (9) 
This is the general form of the differential thermal energy balance
equation or simply the energy equation.
2 The Heat Flux Vector and the Heat Conduction Equation
The temperature change at any point in a body is the result of heat
propagation processes. The rate of temperature change with distance along
a particular direction [n\vec] is
where ÑT is the temperature gradient vector.
For an isotropic medium, if the temperature increases in a direction
normal to a surface S, then the heat flow across S must be negative and
proportional to the rate of temperature increase along that direction,
(Fourier's law) i.e.
where k is some positive scalar quantity which may depend on the medium and
the temperature and is called the thermal conductivity.
For small variations in temperature k may be assumed
to depend on position only. Substitution of Fourier's law
in the integral form of the energy equation gives

ó õ


ó õ


ó õ


¶u ¶t

dV = 
ó õ


ó õ


ó õ

Ñ·(k ÑT) dV 
 (12) 
On the other hand, from thermodynamics we know that the time rate of change of
specific internal energy is given by
where r is the density and c is the specific heat.
Combining the above with the fact that
if the value of the integral is zero the argument must be zero,
the integral form of the energy equation acquires the differential form

¶(rc T) ¶t

= Ñ·(k ÑT) = div( k ÑT) 
 (14) 
We will call this equation the heat conduction equation.
Adopting rectangular cartesian coordinates (x,y,z) one obtains

¶(rc T) ¶t

= 
¶ ¶x

k ( 
¶T ¶x

) + 
¶ ¶y

k ( 
¶T ¶y

) + 
¶ ¶z

k ( 
¶T ¶z

) 
 (15) 
In cylindrical polar coordinates (r,q,z)

¶(rc T) ¶t

= 
1 r


¶ ¶r

(r k ( 
¶T ¶r

)) + 
1 r^{2}


¶ ¶q

( k ( 
¶T ¶q

)) + 
¶ ¶z

( k ( 
¶T ¶z

)) 
 (16) 
and in Spherical Coordinates (r,j,q)

¶(rc T) ¶t

= 
1 r^{2}


¶ ¶r

( r^{2} k ( 
¶T ¶r

)) + 
1 r^{2} sinq


¶ ¶q

( sinq k ( 
¶T ¶q

)) + 
1 r^{2} sinq^{2}


¶ ¶j

( k ( 
¶T ¶j

)) 
 (17) 
Finally, if r, c and k are all assumed constant one obtains
what is often also called the heat conduction equation:
where
is the thermal diffusivity.
3 References
[1] S.L. Sobolev ``Partial Differential Equations of Mathematical Physics'',
Dover Publications, New York, 1989, Sections 1.1 and 1.5.
[2] F.B. Hildebrand, ``Advanced Calculus for Applications'', 2nd. ed.,
PrenticeHall, Inc., Englewood Cliffs, NJ, 1976. Chapter 6.
File translated from T_{E}X by T_{T}H, version 2.34.
On 10 Jan 2001, 11:53.
Updated: 20010110, 11:53