Estimating a pressure dependent thermal conductivity coefficient with applications in food technology

Figures & data

Table 1. Parameter setup. We have used parameters and dimensions as described in [9].

Name	Symbol	Value and dimension
Thermal expansion coefficient	α	$4.217\times {10}^{-4}{\mathrm{K}}^{-1}$
Density	ϱ	$1000.6\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$
Specific heat	$C_p$	$3780\mathrm{J}{\mathrm{k}\mathrm{g}}^{-1}{\mathrm{K}}^{-1}$
Heat exchange coefficient	h	$28\mathrm{W}{\mathrm{m}}^{-2}{\mathrm{K}}^{-1}$
Inner radius	$r_0$	$0\mathrm{m}$
Outer radius	R	$0.045\mathrm{m}$
Initial temperature	$T_0$	$295\mathrm{K}$
Pressure increase rate	β	$120/61\times {10}^6\mathrm{P}\mathrm{a}{\mathrm{s}}^{-1}$
Final time	$t_{\mathrm{f}}$	$1000\mathrm{s}$

Figure 1. Uncertainty propagation. Subplots in the top row depict 100 perturbed samples of the thermal conductivity coefficient, given by Equation (17(17) $k\left(t\right)=\arctan \frac{t}{30}+\mathrm{0.45.}$(17) ), for $\mathrm{S}\mathrm{N}\mathrm{R}=10$ and $\mathrm{S}\mathrm{N}\mathrm{R}={10}^3$ respectively. Subplots in the middle and the bottom rows depict the variance of the numerical simulation of forward mapping (2(2) $F\left(k\right)=T,$(2) ) acting on the above conductivity coefficients at r=0 and r=R respectively. The smoothing nature of the forward mapping makes it necessary to acquire temperature data at large integration times, e.g. 1000 s.

Figure 2. Example 1. (a) Trace plot. (b) Posterior distribution of ${\sigma }_2$. (c) True and estimators Middle row depicts 2000 samples of the posterior distribution of the thermal conductivity coefficient. At the bottom row are shown the corresponding temperatures evaluated at r=0 and r=R respectively. Hierarchical modelling.

Figure 3. Example 2. (a) Trace plot. (b) Posterior distribution of ${\sigma }_2$. (c) True and estimators Middle row depicts 2000 samples of the posterior distribution of the thermal conductivity coefficient. At the bottom row are shown the corresponding temperatures evaluated at r=0 and r=R respectively. Hierarchical modelling.

Figure 4. Example 3. (a) Trace plot. (b) Posterior distribution of ${\sigma }_2$. (c) True and estimators Middle row depicts 2000 samples of the posterior distribution of the thermal conductivity coefficient. At the bottom row are shown the corresponding temperatures evaluated at r=0 and r=R respectively. Hierarchical modelling.

Table 2. Absolute error. We have used ${10}^6$ samples of the MCMC to compute the absolute error $||k-k_{MAP}|{|}_{L^{\mathrm{\infty }}}$ for Examples 3.1, 3.2 and 3.3 at signal to noise ratio (SNR) ${10}^2$, ${10}^3$ and ${10}^4$. Although the MCMC is convergent in every case, a threshold effect is apparent at $\mathrm{S}\mathrm{N}\mathrm{R}={10}^3$.

SNR vs Example	Example 3.1	Example 3.2	Example 3.3
${10}^2$	$9.322234\times {10}^{-1}$	$3.319563\times {10}^{-1}$	$1.061651\times {10}^{-1}$
${10}^3$	$1.599418\times {10}^{-1}$	$1.191929\times {10}^{-1}$	$1.139235\times {10}^{-2}$
${10}^4$	$5.642748\times {10}^{-2}$	$2.620082\times {10}^{-2}$	$2.128011\times {10}^{-3}$

Table 3. Efficiency. We have used ${10}^6$ samples of the MCMC to compute the measure of effficiency IAT$/n$ for Examples 3.1, 3.2 and 3.3 at signal to noise ratio (SNR) ${10}^2$, ${10}^3$ and ${10}^4$.

SNR	Example 3.1	Example3.2	Example 3.3
${10}^2$	81.5	74.2	95.9
${10}^3$	55.1	39.6	25.8
${10}^4$	29.7	32.9	24.6

Smith NAS, Mitchell SL, Ramos AM. Analysis and simplification of a mathematical model for high-pressure food processes. Appl Math Comput. 2014;226:20–37.

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