Analytical solution of freeze-drying mathematical model based in Darcy’s law: application to an orange juice-based cake

Figures & data

Figure 1. Freeze drying chamber for orange juice based cake experimental dynamics.

Figura 1. Cámara de secado por congelación utilizada para las dinámicas experimentales de la torta con base de jugo naranja

Table 1. Variables and properties for heat transfer equations at experimental set 1.

Tabla 1. Variables y propiedades para las ecuaciones de transferencia de calor en el conjunto experimental 1

Variable/Property	Value	Reference
$N$	15	This study
$T_0$, $T_f$	−27, 30 ${\hspace{0.17em}}^{\mathrm{o}}\mathrm{C}$	Experimental
$T_h$, $T_{env}$	30, 30 ${\hspace{0.17em}}^{\mathrm{o}}\mathrm{C}$	Experimental
$A$, $l$	2.6x10^−3 ${\mathrm{m}}^2$, 0.01 $\mathrm{m}$	Experimental
$p_{\mathrm{\infty }}$	50 $\mathrm{P}\mathrm{a}$	Experimental
$m_{s0}$	5.4x10^−3 $\mathrm{k}\mathrm{g}$	Experimental
${\left(\mathrm{\Delta }X/\mathrm{\Delta }t\right)}_{exp}$	9.2x10^−5 $\mathrm{k}\mathrm{g}\mathrm{w}\cdot \mathrm{k}\mathrm{g}\mathrm{d}{\mathrm{m}}^{-1}\cdot {\mathrm{s}}^{-1}$	Experimental, averaged form three replicates.
$\mathrm{\Delta }{H}_s$	2.78X10^6 $\mathrm{J}\cdot \mathrm{k}{\mathrm{g}}^{-1}$	Geankoplis (1998)
$k_{\beta }$, $k_{\gamma }$	2.5, 0.15 $\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot \mathrm{k}{\mathrm{g}}^{-1}$	Lopez-Quiroga et al. (2012)
${\alpha }_{\beta }$, ${\alpha }_{\gamma }$	(5.2, 1.0)x10^−7 ${\mathrm{m}}^2\cdot {\mathrm{s}}^{-1}$	Lopez-Quiroga et al. (2012)
$h_{\beta }$, $h_{\gamma }$	8, 12 $\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot \mathrm{k}{\mathrm{g}}^{-1}$	Empirical correlations natural convection.

Table 2. Variables and fitted properties obtained by non-linear least squares of Eq. (15)(15) $X=\frac{l-{\left(\left(n+2\right)a_2{a}_1{K}^{\ast }t\right)}^{1/\left(n+2\right)}}{a_2}$(15) on experimental freeze drying kinetic at set 1.

Tabla 2. Variables y propiedades ajustadas obtenidas por mínimos cuadrados no lineales de la ec. (15) sobre la cinética de liofilización experimental del conjunto 1

Variable/Property	Value	Reference
$X_0$	4 $\mathrm{k}\mathrm{g}\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\cdot {\left(\mathrm{k}\mathrm{g}\mathrm{d}\mathrm{r}\mathrm{y}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\right)}^{-1}$	Experimental, averaged of three independent replicates
$T_{i\beta \gamma }$	−12.1 ${\hspace{0.17em}}^{\mathrm{o}}\mathrm{C}$	Result for pseudo-stationary heat transfer
${\rho }_{wi}$	900 $\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^{-3}$	Geankoplis (1998)
$K$, $n=0$	2.50x10^−11 ${\mathrm{m}}^2$	Fitted
$K^{\ast }$, $n$	2.0x10^−12 ${\mathrm{m}}^{2.5}$, 0.5	Fitted

Figure 2. Temperature evolution in the 15 nodes of frozen zone obtained by solution of Eqs. (5c)(5c) $\frac{{\mathrm{\Psi }}_{\beta 2}-{\mathrm{\Psi }}_{\beta 0}}{2\mathrm{\Delta }{\xi }_{\beta }}=B{i}_{\beta }\left({\mathrm{\Psi }}_{\beta 1}-{\mathrm{\Psi }}_h\right)\mathrm{a}\mathrm{t}j=1$(5c) -(10c(10c) $-\frac{{\mathrm{\Psi }}_{\gamma N+2}-{\mathrm{\Psi }}_{\gamma N}}{2\mathrm{\Delta }{\xi }_{\gamma }}=B{i}_{\gamma }\left({\mathrm{\Psi }}_{\gamma N+1}-{\mathrm{\Psi }}_{env}\right)\mathrm{a}\mathrm{t}i=N+1$(10c) ) with $l_{\beta }=l/2$ at variables and properties listed in Table 1.

Figura 2. Evolución de la temperatura en los 15 nodos de la zona congelada obtenida por la solución de las ecuaciones (5 c)-(10 c) con$l_{\beta }=l/2$ en las variables y propiedades enumeradas en la Tabla 1

Figure 3. Temperature evolution in the 15 nodes of porous zone obtained by solution of Eqs. (5c)(5c) $\frac{{\mathrm{\Psi }}_{\beta 2}-{\mathrm{\Psi }}_{\beta 0}}{2\mathrm{\Delta }{\xi }_{\beta }}=B{i}_{\beta }\left({\mathrm{\Psi }}_{\beta 1}-{\mathrm{\Psi }}_h\right)\mathrm{a}\mathrm{t}j=1$(5c) -(10c(10c) $-\frac{{\mathrm{\Psi }}_{\gamma N+2}-{\mathrm{\Psi }}_{\gamma N}}{2\mathrm{\Delta }{\xi }_{\gamma }}=B{i}_{\gamma }\left({\mathrm{\Psi }}_{\gamma N+1}-{\mathrm{\Psi }}_{env}\right)\mathrm{a}\mathrm{t}i=N+1$(10c) ) with $l_{\beta }=l/2$ at variables and properties listed in Table 1.

Figura 3. Evolución de la temperatura en los 15 nodos de la zona porosa obtenida por la solución de las ecuaciones (5 c)-(10 c) con$l_{\beta }=l/2$ en las variables y propiedades enumeradas en la Tabla 1

Figure 4. Experimental (o) and fitted (lines) freeze drying dynamics at condition listed in Table 2. Discontinuous line represents the ideal Darcy’s Law (Eq. 15(15) $X=\frac{l-{\left(\left(n+2\right)a_2{a}_1{K}^{\ast }t\right)}^{1/\left(n+2\right)}}{a_2}$(15) with $n=0$) and continuous line the non-ideal one. (Horizontal line at zero moisture represents the physical limit for Eq. 15(15) $X=\frac{l-{\left(\left(n+2\right)a_2{a}_1{K}^{\ast }t\right)}^{1/\left(n+2\right)}}{a_2}$(15) domain).

Figura 4. Dinámica de liofilización experimental (o) y ajustada (líneas) en las condiciones enumeradas en la Tabla 2. La línea discontinua representa la ley de Darcy ideal (Ec. 15 con) y la línea continua la no-ideal. (La línea horizontal a humedad cero representa el límite físico para el dominio de la Ec. 15)

Table 3. Variables and properties for heat transfer equations at experimental set 2.

Tabla 3. Variables y propiedades para las ecuaciones de transferencia de calor en el conjunto experimental 2

Variable/Property	Value	Reference
$N$	15	This study
$T_0$, $T_f$	−42, 40 ${\hspace{0.17em}}^{\mathrm{o}}\mathrm{C}$	Experimental
$T_h$, $T_{env}$	40, 30 ${\hspace{0.17em}}^{\mathrm{o}}\mathrm{C}$	Experimental
$A$, $l$	2.6x10^−3 ${\mathrm{m}}^2$, 0.01 $\mathrm{m}$	Experimental
$p_{\mathrm{\infty }}$	50 $\mathrm{P}\mathrm{a}$	Experimental
$m_{s0}$	4.6x10^−3 $\mathrm{k}\mathrm{g}$	Experimental
${\left(\mathrm{\Delta }X/\mathrm{\Delta }t\right)}_{exp}$	1.5x10^−4 $\mathrm{k}\mathrm{g}\mathrm{w}\cdot \mathrm{k}\mathrm{g}\mathrm{d}{\mathrm{m}}^{-1}\cdot {\mathrm{s}}^{-1}$	Initial guess at −10.5°C
$\mathrm{\Delta }{H}_s$	2.78X10^6 $\mathrm{J}\cdot \mathrm{k}{\mathrm{g}}^{-1}$	Geankoplis (1998)
$k_{\beta }$, $k_{\gamma }$	2.5, 0.15 $\mathrm{W}\cdot {\mathrm{m}}^{-1}\cdot \mathrm{k}{\mathrm{g}}^{-1}$	Lopez-Quiroga et al. (2012)
${\alpha }_{\beta }$, ${\alpha }_{\gamma }$	(5.2, 1.0)x10^−7 ${\mathrm{m}}^2\cdot {\mathrm{s}}^{-1}$	Lopez-Quiroga et al. (2012)
$h_{\beta }$, $h_{\gamma }$	14, 12 $\mathrm{W}\cdot {\mathrm{m}}^{-2}\cdot \mathrm{k}{\mathrm{g}}^{-1}$	Empirical correlations natural convection with conduction

Table 4. Variables and properties used for simulation of experimental freeze drying kinetic at set 2 with Eq. (15)(15) $X=\frac{l-{\left(\left(n+2\right)a_2{a}_1{K}^{\ast }t\right)}^{1/\left(n+2\right)}}{a_2}$(15) .

Tabla 4. Variables y propiedades utilizadas para la simulación de la cinética de liofilización experimental en el conjunto 2 con la ecuación (15)

Variable/Property	Value	Reference
$X_0$	4.9 $\mathrm{k}\mathrm{g}\mathrm{w}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{r}\cdot {\left(\mathrm{k}\mathrm{g}\mathrm{d}\mathrm{r}\mathrm{y}\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{t}\mathrm{e}\mathrm{r}\right)}^{-1}$	Experimental, averaged of three independent replicates
$T_{i\beta \gamma }$	−9.7 ${\hspace{0.17em}}^{\mathrm{o}}\mathrm{C}$	Result for pseudo-stationary heat transfer
${\rho }_{wi}$	900 $\mathrm{k}\mathrm{g}\cdot {\mathrm{m}}^{-3}$	Geankoplis (1998)
$K$, $n=0$	2.50x10^−11 ${\mathrm{m}}^2$	Fitted
$K^{\ast }$, $n$	2.0x10^−12 ${\mathrm{m}}^{2.5}$, 0.5	Fitted

Figure 5. Temperature evolution in the 15 nodes of frozen zone obtained by solution of Eqs. (5c)(5c) $\frac{{\mathrm{\Psi }}_{\beta 2}-{\mathrm{\Psi }}_{\beta 0}}{2\mathrm{\Delta }{\xi }_{\beta }}=B{i}_{\beta }\left({\mathrm{\Psi }}_{\beta 1}-{\mathrm{\Psi }}_h\right)\mathrm{a}\mathrm{t}j=1$(5c) -(10c(10c) $-\frac{{\mathrm{\Psi }}_{\gamma N+2}-{\mathrm{\Psi }}_{\gamma N}}{2\mathrm{\Delta }{\xi }_{\gamma }}=B{i}_{\gamma }\left({\mathrm{\Psi }}_{\gamma N+1}-{\mathrm{\Psi }}_{env}\right)\mathrm{a}\mathrm{t}i=N+1$(10c) ) with $l_{\beta }=l/2$ at variables and properties listed in Table 3.

Figura 5. Evolución de la temperatura en los 15 nodos de la zona congelada obtenida por la solución de las ecuaciones (5 c)-(10 c) con$l_{\beta }=l/2$ en las variables y propiedades enumeradas en la Tabla 3

Figure 6. Temperature evolution in the 15 nodes of porous zone obtained by solution of Eqs (5c)(5c) $\frac{{\mathrm{\Psi }}_{\beta 2}-{\mathrm{\Psi }}_{\beta 0}}{2\mathrm{\Delta }{\xi }_{\beta }}=B{i}_{\beta }\left({\mathrm{\Psi }}_{\beta 1}-{\mathrm{\Psi }}_h\right)\mathrm{a}\mathrm{t}j=1$(5c) -(10c(10c) $-\frac{{\mathrm{\Psi }}_{\gamma N+2}-{\mathrm{\Psi }}_{\gamma N}}{2\mathrm{\Delta }{\xi }_{\gamma }}=B{i}_{\gamma }\left({\mathrm{\Psi }}_{\gamma N+1}-{\mathrm{\Psi }}_{env}\right)\mathrm{a}\mathrm{t}i=N+1$(10c) ) with $l_{\beta }=l/2$ at variables and properties listed in Table 3.

Figura 6. Evolución de la temperatura en los 15 nodos de la zona porosa obtenida por la solución de las ecuaciones (5 c)-(10 c) con$l_{\beta }=l/2$ en las variables y propiedades enumeradas en la Tabla 3

Figure 7. Experimental (o) and predicted (lines) freeze drying dynamics at condition listed in Table 4. Discontinuous line represents the ideal Darcy’s law (Eq. 15(15) $X=\frac{l-{\left(\left(n+2\right)a_2{a}_1{K}^{\ast }t\right)}^{1/\left(n+2\right)}}{a_2}$(15) with $n=0$) and continuous line the non-ideal one. (Horizontal line at zero moisture represents the physical limit for Eq. 15(15) $X=\frac{l-{\left(\left(n+2\right)a_2{a}_1{K}^{\ast }t\right)}^{1/\left(n+2\right)}}{a_2}$(15) domain).

Figura 7. Dinámicas de liofilización experimental (o) y predicha (líneas) en las condiciones enumeradas en la Tabla 4. La línea discontinua representa la ley de Darcy ideal (Ec. 15 con n = 0) y la línea continua la no-ideal. (La línea horizontal a humedad cero representa el límite físico para el dominio de la Ec. 15)

Geankoplis, C. J. (1998). Procesos de transporte y operaciones unitarias (3th ed.). México, D.F: CECSA

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Lopez-Quiroga, E., Antelo, L. T., & Alonso, A. A. (2012). Time-scale modeling and optimal control of freeze–drying. Journal of Food Engineering, 111(4), 655–666. https://doi.org/https://doi.org/10.1016/j.jfoodeng.2012.03.001

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