EQUATION SHEET FOR EXAM #1
dT
dx
FOURIER’S LAW:
q   kA
STEFAN-BOLTZMANN LAW:
4
q net   A  Ts4  Tsur
 where   5.67  108
NEWTON’S LAW OF COOLING: q
W
m K
2
 hATs  T 
HEAT DIFFUSION EQUATION
(CARTESIAN COORDINATES)
  T   T   T
T
 k
 
 k
 
 k
  q  c p
 x  x  y  y  z  z 
t
HEAT DIFFUSION EQUATION
(CYLINDRICAL COORDINATES)
1   T 1   T   T
T
 kr
  2
 k
 
 k
  q  c p
r  r   r  r      z   z 
t
HEAT DIFFUSION EQUATION
(SPHERICAL COORDINATES)
T
T
1   2T
1
  T
1
 
 kr
  2
 k
  2
 k sin 
  q  c p
2
2
 r  r sin      r sin    
 
t
r  r
AREAS
Surface of a Sphere
A  4 r 2
Curved surface of a Circular Cylinder
A  2 r L
VOLUMES
Volume of a Sphere
Volume of a Circular Cylinder
4
   r3
3
   r2L
ME 309 HEAT TRANSFER
EXAM # 1
CLOSED BOOK AND NOTES
MONDAY 27 SEPTEMBER 1999
Radioactive wastes are packed into a long, thin-walled cylindrical container. The wastes generate thermal
energy nonuniformly according to the relation
generation per volume,
b.

q o is a constant, and ro is the radius of the container. Steady state is maintained
by submerging the container in a liquid that is at
a.

2
q  q o 1  r ro  , where q is the local rate of energy
T and provides a uniform convection coefficient, h.
Obtain an expression for the total rate at which energy is generated in a unit length of the container
and determine an expression for the temperature of the container wall.
Determine the temperature distribution T r of the waste in the container in terms of the
conductivity of the waste, k, and the variables named above.

ro
Note: The volumetric (per unit length) integral of a quantity f( r ) in a cylinder is given as:
 2 f r  r dr
0