134
3 Experiments in Pebble Bed Heat Transfer
The statistical error of measured temperature in the inverse method was studied
by [16–18]. Here is a brief description without derivation:
V ( P) = σ
2
( J
T J)
−1
(3.33)
V ( P) is a variance-covariance matrix, and σ is the aforementioned standard deviation
of the measured temperature. The diagonal terms of V ( P) are the variance of each
parameter p i and its standard deviation is the square root of variance. Here the
quadratic polynomial function is selected to illustrate this issue for convenience.
The maximum error of temperature appears at a maximum temperature of 1,028
◦ C.
Therefore, a conservative σ at 4.7
◦ C in Eq. (3.33), is chosen. Consequently, the
standard uncertainties of p 1 , p 2 , and p 3 , caused by the statistical error of measured
temperatures, are given by 1.7779 × 10
−9 , 6.1511 × 10
−12 , and 7.5659 × 10
−15 ,
respectively. According to the error propagation formula and Eq. (3.7), the standard
uncertainty of α is given by
u α,s (T ) =
∂α
∂ p 1
u p 1 ,s
2 +
∂α
∂ p 2
u p 2 ,s
2 +
∂α
∂ p 3
u p 3 ,s
2
(3.34)
=
u 2
p 1 ,s +
T u p 2 ,s
2 +
T 2 u p 3 ,s
2
(3.35)
In Fig. (3.8a), the statistical uncertainty of measured temperatures is shown, which
is negligible due to heat dissipation characteristic in dynamic heat conduction.
The installation position accuracy of thermocouples is set as 10mm in this facility.
However, this position error can be regarded as the statistical uncertainty of the
five circumferential temperature series. In other words, five independent-position
data are used to estimate this uncertainty. Therefore, five sets of data in different
circumferential coordinates are used to solve the inverse method. Subsequently, the
standard deviations of α at different temperatures and the total uncertainty at different
temperatures are calculated by
u α, p (T ) =
1
5
5
i = 1
(α i (T ) − ¯
α(T ))
2
(3.36)
u α (T ) =
u α,s (T ) 2 + u α, p (T ) 2
(3.37)
The standard uncertainty and value of α are shown in Fig. (3.8b). The main uncertainty is caused by the installation position.
The combined uncertainty of effective thermal conductivity is given by the error
propagation formula based on Eq. (3.10).
u α = λ
u α
α
2 +
u c p
c p
2
(3.38)
3 Experiments in Pebble Bed Heat Transfer
The statistical error of measured temperature in the inverse method was studied
by [16–18]. Here is a brief description without derivation:
V ( P) = σ
2
( J
T J)
−1
(3.33)
V ( P) is a variance-covariance matrix, and σ is the aforementioned standard deviation
of the measured temperature. The diagonal terms of V ( P) are the variance of each
parameter p i and its standard deviation is the square root of variance. Here the
quadratic polynomial function is selected to illustrate this issue for convenience.
The maximum error of temperature appears at a maximum temperature of 1,028
◦ C.
Therefore, a conservative σ at 4.7
◦ C in Eq. (3.33), is chosen. Consequently, the
standard uncertainties of p 1 , p 2 , and p 3 , caused by the statistical error of measured
temperatures, are given by 1.7779 × 10
−9 , 6.1511 × 10
−12 , and 7.5659 × 10
−15 ,
respectively. According to the error propagation formula and Eq. (3.7), the standard
uncertainty of α is given by
u α,s (T ) =
∂α
∂ p 1
u p 1 ,s
2 +
∂α
∂ p 2
u p 2 ,s
2 +
∂α
∂ p 3
u p 3 ,s
2
(3.34)
=
u 2
p 1 ,s +
T u p 2 ,s
2 +
T 2 u p 3 ,s
2
(3.35)
In Fig. (3.8a), the statistical uncertainty of measured temperatures is shown, which
is negligible due to heat dissipation characteristic in dynamic heat conduction.
The installation position accuracy of thermocouples is set as 10mm in this facility.
However, this position error can be regarded as the statistical uncertainty of the
five circumferential temperature series. In other words, five independent-position
data are used to estimate this uncertainty. Therefore, five sets of data in different
circumferential coordinates are used to solve the inverse method. Subsequently, the
standard deviations of α at different temperatures and the total uncertainty at different
temperatures are calculated by
u α, p (T ) =
1
5
5
i = 1
(α i (T ) − ¯
α(T ))
2
(3.36)
u α (T ) =
u α,s (T ) 2 + u α, p (T ) 2
(3.37)
The standard uncertainty and value of α are shown in Fig. (3.8b). The main uncertainty is caused by the installation position.
The combined uncertainty of effective thermal conductivity is given by the error
propagation formula based on Eq. (3.10).
u α = λ
u α
α
2 +
u c p
c p
2
(3.38)