3.3 Effective Thermal Diffusivity and Conductivity
153
finally, the combined standard uncertainty of ¯
p j is given by
u ¯
p j =
u A, ¯
p j
2 +
u B, ¯
p j
2
(3.55)
The degree of freedom of type A standard uncertainty is
v A, ¯
p j = v f + v n = 8.
(3.56)
3.3.5.3 Standard and Expanded Uncertainties of Effective Thermal
Diffusivity
In this experiment, the final effective thermal diffusivity is given by the splinepiecewise cubic-polynomials with the interpolation points of vector P.
α(T ) = f s ( P, T ), T ∈ [T min , T max ],
(3.57)
where f s is the spline function mentioned in Sect. (3.3.4)
The Monte Carlo method is used to calculate the standard uncertainty of diffusivity as well as further conductivity to avoid using the lengthy uncertainty-propagation
formula. The fundamental method is to generate 10
6 random samples for each ¯
p j by
the normal distribution whose mean and standard deviation are ¯
p j and u ¯
p j , respectively. Then, 10
6 sets of diffusivities are calculated with the 10
6 random P at each
temperature. The standard uncertainty of α at each temperature is given by counting
the standard deviation of these 10
6 sets of diffusivities.
To give the expanded uncertainty, the effective degree of freedom of diffusivity
uncertainty should be calculated by the Welch-Satterthwaite formula as
v eff (T ) =
u
4
α (T )
10
j = 1
∂α(T )
∂ p j
4
u 4
A, ¯
p j
v A, ¯
p j
+
v 4
B, ¯
p j
v B, ¯
p j
(3.58)
where j indicates the index of the component of vector P, and u α (T ) is the standard
uncertainty of diffusivity obtained by the Monte Carlo method at temperature T . The
partial derivative can be calculated by a numerical method.
The expanded uncertainty with a 95% level of confidence can be given as
u α,95 (T ) = t 95 (v eff (T ))u α (T ).
(3.59)
The t 95 (v eff ) is the t-factor from t distribution of the parameter v eff , which means
the 95% fraction of the t distribution is encompassed by the interval −t 95 (v eff ) to
t 95 (v eff ).
153
finally, the combined standard uncertainty of ¯
p j is given by
u ¯
p j =
u A, ¯
p j
2 +
u B, ¯
p j
2
(3.55)
The degree of freedom of type A standard uncertainty is
v A, ¯
p j = v f + v n = 8.
(3.56)
3.3.5.3 Standard and Expanded Uncertainties of Effective Thermal
Diffusivity
In this experiment, the final effective thermal diffusivity is given by the splinepiecewise cubic-polynomials with the interpolation points of vector P.
α(T ) = f s ( P, T ), T ∈ [T min , T max ],
(3.57)
where f s is the spline function mentioned in Sect. (3.3.4)
The Monte Carlo method is used to calculate the standard uncertainty of diffusivity as well as further conductivity to avoid using the lengthy uncertainty-propagation
formula. The fundamental method is to generate 10
6 random samples for each ¯
p j by
the normal distribution whose mean and standard deviation are ¯
p j and u ¯
p j , respectively. Then, 10
6 sets of diffusivities are calculated with the 10
6 random P at each
temperature. The standard uncertainty of α at each temperature is given by counting
the standard deviation of these 10
6 sets of diffusivities.
To give the expanded uncertainty, the effective degree of freedom of diffusivity
uncertainty should be calculated by the Welch-Satterthwaite formula as
v eff (T ) =
u
4
α (T )
10
j = 1
∂α(T )
∂ p j
4
u 4
A, ¯
p j
v A, ¯
p j
+
v 4
B, ¯
p j
v B, ¯
p j
(3.58)
where j indicates the index of the component of vector P, and u α (T ) is the standard
uncertainty of diffusivity obtained by the Monte Carlo method at temperature T . The
partial derivative can be calculated by a numerical method.
The expanded uncertainty with a 95% level of confidence can be given as
u α,95 (T ) = t 95 (v eff (T ))u α (T ).
(3.59)
The t 95 (v eff ) is the t-factor from t distribution of the parameter v eff , which means
the 95% fraction of the t distribution is encompassed by the interval −t 95 (v eff ) to
t 95 (v eff ).