C-THERM TCi, Operator'S Manual, Page: 113 / 138

TCi Operator Manual
107
m
m m
   
)
1
( )
1
(
2
(35)
And therefore:
m
m
m
m



/ 1
) / 1 (
(36)
The relative variance in 1/m is the same as the relative variance in m .
From (17) we can calculate the variance in
2
e
as ( C is assumed constant for the set of
measurements):
M
m
e
)
1
(
2

 
(37)
And the relative variance as:
C
m
m
M
C
m
M
m
e
e



 


1
)
1
(
)
1
(
)
1
(
2
2
(38)
If C =0, i.e. sensor effusivity is 0, then:
I
I
m
m
m
m
e
e







3
1
)
1
(
2
2
(39)
In an ideal case where the sensor effusivity is 0, the relative variance in measured effusivity is
triple the relative variance of the supplied current. However, in reality
C is always a positive
number as can be seen from equation (19), and therefore the relative variance in measured
effusivity is always higher than the relative variance in
m or 1/m .
Conclusions:
1. Lower sensor effusivity will reduce C , and this in turn will reduce the variance in
measured effusivity. This is evident from equation (38).
2. Low effusivity materials will produce higher m and lower 1/m than high effusivity
materials. Therefore, the relative variance of measured effusivity is expected to be higher
for low effusivity materials.
The first conclusion proposes that it is desired to design the sensor such that it has minimum
effusivity. This principle is intuitively understood (in terms of sensitivity and consumed power) and
in fact has been an important factor in the sensor design.
9.7.6 Additional Notes – Heat Equation
A few words on the heat equation: For nomenclature see page 4.
'
2
2
G
x
T
t
T
c
p






 
(6)