An efficient optimization methodology of respiration rate parameters coupled with transport properties in mass balances to describe modified atmosphere packaging systems

Figures & data

Figure 1. Two-dimensional cross-section of the polypropylene container. The container is open at the top for allowing the mass transfer of $O_2$ and $C{O}_2$ through the film, whereas the other three sides are considered impervious, i.e. no mass transfer through them.

Figure 2. Experimental schematic for pMAP of each lot stored in triplicate for the two films: PSF530 (25 µm thickness) and PPCX (35 µm thickness).

Table 1. Physical parameters for general mass balance principle determined experimentally, except for the values of $P_{O_2}^{\mathrm{ext}}$, $P_{C{O}_2}^{\mathrm{ext}}$, and R that were obtained from the reference [14].

Parameter	Symbol	Value	Units
Area of package surface	S	4.0e−2	m${}^2$
External $O_2$ partial pressure	$P_{O_2}^{\mathrm{ext}}$	21	kPa
External $C{O}_2$ partial pressure	$P_{C{O}_2}^{\mathrm{ext}}$	3.5e−2	kPa
Package head space volume	V	5e−4	m${}^3$
Universal gas constant	R	8.3145e−3	kPa m${}^3$ mol${}^{-1}$ K${}^{-1}$
Storage temperature	T	283	K
Product mass	m	1.25e−1	Kg

Table 2. Parameters specific to each film type.

Parameter	Symbol	Unit	Film PSF530	Film PPCX
Thickness	ℓ	m	2.5e−5	3.5e−5
$O_2$ permeability	$P_{O_2}$	$O_2$ mol m${}^{-1}$ h${}^{-1}$ Kpa${}^{-1}$	6.9e−11	1.44e−9
$C{O}_2$ permeability	$P_{C{O}_2}$	$C{O}_2$ mol m${}^{-1}$ h${}^{-1}$ Kpa${}^{-1}$	7.67e−10	2.88e−9

Table 3. Input arguments for lsqnonlin solver.

Description	Input argument
Residuals vector	${\mathbf{U}}_i^j-\mathbf{U}(i,{\mathit{\theta }}^{\mathit{j}})i=1,\dots ,I.$
Initial parameters vector	$({\mathit{\theta }}^{\mathit{j}}{)}^0$
Lower bound for parameters vector	${\mathit{\theta }}_L=(0,0,0)$
Upper bound for parameters vector	${\mathit{\theta }}_U=+\mathrm{\infty }\cdot (1,1,1)$

Figure 3. Schematic diagram of the numerical methodology.

Figure 4. Fitted curves for the exponential model. (a) Film PSF530 (b) Film PPCX.

Figure 5. Fitted curves for the MMU model. (a) Film PSF530. (b) Film PPCX.

Figure 6. Fitted curves for the MMC model. (a) Film PSF530. (b) Film PPCX.

Table 4. Values for initial $({\mathit{\theta }}^j{)}^{(0)}$ and optimal ${\hat{\mathit{\theta }}}^j$ nonlinear LSE, and normalized standard errors ${\mathit{n}\mathit{s}\mathit{e}}^j$ for exponential model.

	Film PSF530			Film PPCX
Parameter	$({\mathit{\theta }}^1{)}^{(0)}$	${\hat{\mathit{\theta }}}^1$	${\mathit{n}\mathit{s}\mathit{e}}^1$	$({\mathit{\theta }}^2{)}^{(0)}$	${\hat{\mathit{\theta }}}^2$	${\mathit{n}\mathit{s}\mathit{e}}^2$
$\begin{array}{c}{A}_1\\ A_2\\ K\end{array}$	$\begin{array}{c}3.41\\ 5.5\\ 10.7\end{array}$	$\begin{array}{c}3.3301e-2\\ 1.0844e+3\\ 3.1839\end{array}$	$\begin{array}{c}1.7507\\ 1.3354\\ 4.3112e-1\end{array}$	$\begin{array}{c}6.82\\ 5.5\\ 10.7\end{array}$	$\begin{array}{c}2.3114e-2\\ 2.6974e+2\\ 3.2514\end{array}$	$\begin{array}{c}1.3579\\ 3.8445\\ 4.9309e-1\end{array}$

Table 5. Convergence metrics, initial $S(({\mathit{\theta }}^j{)}^{(0)})$ and optimal $S({\hat{\mathit{\theta }}}^j)$ values and fit performance for the exponential model.

Film	$||\mathbf{\nabla }S({\hat{\mathit{\theta }}}^j)|{|}_{\mathrm{\infty }}$	${\delta }_S^{(k)}$	k + 1	$S(({\mathit{\theta }}^j{)}^{(0)})$	$S({\hat{\mathit{\theta }}}^j)$	$({\tilde{\sigma }}^j{)}^2$	RMSE	$R^2$
PSF530	1.49e−13	1.53e−19	30	1.9435e−1	3.5920e−7	1.4967e−9	3.8931e−5	0.9914
PPCX	3.11e−13	4.33e−18	33	2.0853e−1	4.4406e−7	1.8502e−9	4.3286e−5	0.9892

Table 6. Values for initial $({\mathit{\theta }}^j{)}^{(0)}$ and optimal ${\hat{\mathit{\theta }}}^j$ nonlinear LSE, and normalized standard errors ${\mathit{n}\mathit{s}\mathit{e}}^j$ for the MMU model.

	Film PSF530			Film PPCX
Parameter	$({\mathit{\theta }}^1{)}^{(0)}$	${\hat{\mathit{\theta }}}^1$	${\mathit{n}\mathit{s}\mathit{e}}^1$	$({\mathit{\theta }}^2{)}^{(0)}$	${\hat{\mathit{\theta }}}^2$	${\mathbf{n}\mathbf{s}\mathbf{e}}^2$
$\begin{array}{c}{V}_{max}\\ K_m\\ K_u\end{array}$	$\begin{array}{c}1.3000e-3\\ 2.0000e-3\\ 1.3100e-1\end{array}$	$\begin{array}{c}8.2862e-5\\ 4.5195e-3\\ 1.5954e-5\end{array}$	$\begin{array}{c}9.7539e+1\\ 1.8543e+2\\ 9.7607e+1\end{array}$	$\begin{array}{c}1.3000e-3\\ 2.0000e-3\\ 1.3100e-1\end{array}$	$\begin{array}{c}7.5057e-5\\ 5.7782e-4\\ 3.5235e-5\end{array}$	$\begin{array}{c}3.4779e+1\\ 2.7269e+2\\ 3.4809e+1\end{array}$

Table 7. Convergence metrics, initial $S(({\mathit{\theta }}^j{)}^{(0)})$ and optimal $S({\hat{\mathit{\theta }}}^j)$ values and fit performance for the MMU model.

Film	$||\mathbf{\nabla }S({\hat{\mathit{\theta }}}^j)|{|}_{\mathrm{\infty }}$	${\delta }_S^{(k)}$	k + 1	$S(({\mathit{\theta }}^j{)}^{(0)})$	$S({\hat{\mathit{\theta }}}^j)$	$({\tilde{\sigma }}^j{)}^2$	RMSE	$R^2$
PSF530	4.8e−16	1.10e−20	75	7.7628	3.7158e−6	1.5483e−8	1.2521e−4	0.9107
PPCX	5.71e−7	4.66e−21	43	6.9119	2.5631e−6	1.0680e−8	1.0399e−4	0.9379

Table 8. Values for initial $({\mathit{\theta }}^j{)}^{(0)}$ and optimal ${\hat{\mathit{\theta }}}^j$ nonlinear LSE, and normalized standard errors ${\mathit{n}\mathit{s}\mathit{e}}^j$ for the MMC model.

	Film PSF530			Film PPCX
Parameter	$({\mathit{\theta }}^1{)}^{(0)}$	${\hat{\mathit{\theta }}}^1$	${\mathit{n}\mathit{s}\mathit{e}}^1$	$({\mathit{\theta }}^2{)}^{(0)}$	${\hat{\mathit{\theta }}}^2$	${\mathit{n}\mathit{s}\mathit{e}}^2$
$\begin{array}{c}{V}_{max}\\ K_m\\ K_c\end{array}$	$\begin{array}{c}1.3000e-2\\ 1.0000e-3\\ 1.3100e-1\end{array}$	$\begin{array}{c}8.2931e-5\\ 4.5195e-3\\ 1.7791e-5\end{array}$	$\begin{array}{c}5.0071e+1\\ 2.6455e+2\\ 2.1470e+2\end{array}$	$\begin{array}{c}1.3000e-2\\ 1.0000e-3\\ 1.3100e-1\end{array}$	$\begin{array}{c}7.9734e-5\\ 9.2464e-4\\ 7.7281e-6\end{array}$	$\begin{array}{c}9.0115e+1\\ 1.7132e+2\\ 8.1337e+1\end{array}$

Table 9. Convergence metrics, initial $S(({\mathit{\theta }}^j{)}^{(0)})$ and optimal $S({\hat{\mathit{\theta }}}^j)$ values and fit performance for the MMC model.

Film	$||\mathbf{\nabla }S({\hat{\mathit{\theta }}}^j)|{|}_{\mathrm{\infty }}$	${\delta }_S^{(k)}$	k + 1	$S(({\mathit{\theta }}^j{)}^{(0)})$	$S({\hat{\mathit{\theta }}}^j)$	$({\tilde{\sigma }}^j{)}^2$	RMSE	$R^2$
PSF530	7.77e−16	2.46e−20	29	7.9580	3.1769e−6	1.3237e−8	1.1578e−4	0.9237
PPCX	1.13e−7	1.02e−20	76	6.1954	4.3293e−6	1.8039e−8	1.3516e−4	0.8951

Figure 7. Simulated data obtained from the exponential model. (a) Film PSF530. (b) Film PPCX.

Table 10. Relative errors and convergence metrics for perturbed data with Gaussian noise.

Film	$E_{{\mathit{\theta }}^{j,p}}^R$	$E_{n_{O_2}^j}^R$	$E_{n_{C{O}_2}^j}^R$	$||\mathbf{\nabla }S({\hat{\mathit{\theta }}}^{j,p})|{|}_{\mathrm{\infty }}$	${\delta }_{S,p}^{(k)}$	k + 1	$S({\hat{\mathit{\theta }}}^{j,p})$	$S({\hat{\mathit{\theta }}}^j)$
PSF530	1.2366e−1	1.0279e−1	9.1993e−2	1.32e−13	1.86e−20	6	4.7668e−6	6.1732e−6
PPCX	9.7632e−2	8.0255e−2	9.6583e−2	9.93e−14	3.05e−20	6	4.0784e−6	4.1311e−6

Table 11. Relative errors and convergence metrics for perturbed initial parameters with Gaussian noise.

Film	$E_{({\mathit{\theta }}^{j,p}{)}^{(0)}}^R$	$E_{{\hat{\mathit{\theta }}}^{j,p}}^R$	$||\mathbf{\nabla }S({\hat{\mathit{\theta }}}^{j,p})|{|}_{\mathrm{\infty }}$	${\delta }_{S,p}^{(k)}$	k + 1	$S({\hat{\mathit{\theta }}}^{j,p})$	$S(({\mathit{\theta }}^{j,p}{)}^{(0)})$
PSF530	2.7304	5.1672e−8	2.89e−13	2.27e−18	19	3.5920e−7	2.8542
PPCX	1.2413e + 2	1.5921e−8	3.07e−14	3.69e−19	66	4.4406e−7	1.4472e−2

Yunus CA, Afshin GJ. Heat and mass transfer: fundamentals and applications. Chapter 14. New York: McGraw-Hill Education; 2015. p. 835–906.

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