C-THERM TCi, Operator'S Manual, Page: 114 / 138

TCi Operator Manual
108
This equation describes the dynamics of temperature change and heat flow in a medium in one
dimension (i.e. assume semi-infinite plane), with a source of heat
G’ per unit volume supplied at
the surface. The left side of the equation describes change of temperature in the medium versus
time. The right side is related mainly to change of temperature in space.
Because the

c
p
multiplies the temperature change in time, the larger the density and heat
capacity of the medium, the longer it would take the medium to change its temperature.
Conversely, materials with low density and heat capacity will experience a rapid change in
temperature. This can be intuitively understood.
The right side is not so intuitive because of the second order derivative. The larger the thermal
conductivity, the less steep the temperature profile will look inside the medium. Very small
thermal conductivity means that the temperature stays high at the surface and drops fast as
x
increases.
A simple case to look at is a medium at steady state (e.g. a wall with high temperature T
1
on one
side and low temperature T
2
on the other side). Also assume that G’ is 0. In steady state there is
no change in time, and therefore the derivative of T relative to time is 0. This means that the
second derivative of T versus x is also 0. This may happen only if the first derivative of T versus x
is constant,
2
2
0
x
T


 
(I-1)
) ( 0
x
T
x
T




 
(I-2)
t Cons
x
T
tan




(I-3)
This describes a constant heat flow and constant slope of T relative to space x. In other words the
temperature profile inside the wall is a straight line between the extreme temperatures.
Note: equation I-3 is not accurate theoretically if heat flow is not one-dimensional.
Equation (I-3) also describes the constant amount of energy flux (W/m
2
) that flows through the
wall. Assume the temperatures on the side of the wall are T
1
and T
2
, and its thickness is d, then
the energy flux passing through the wall is
d
T T
q
2 1




(I-4)