Identification of the potato moisture diffusion coefficient considering shrinkage: a nonlinear estimator approach to an inverse problem

Abstract

This study aimed to determine the moisture diffusion coefficients of potatoes by taking into account the shrinkage that occurs during the drying process. In addition, the study also considered the scenario where shrinkage was neglected for comparison purposes. The main objective of this study is to establish the importance of considering shrinking phenomena for proper modeling of the drying process and to highlight the advantages of a recently published methodology for estimating parameters. The potato samples, which were dried in a moisture balance, were prepared such that the moisture diffusion was unidimensional. The determination of the moisture diffusivity when shrinkage is neglected is based on the analytical solution of the diffusion equation when constant diffusivity is assumed. When shrinkage is considered, the diffusion coefficient is identified via a nonlinear estimator constructed based on a numerical solution to the diffusion equation. The shrinking rate and the drying kinetics are the information fed to the algorithm. Arrhenius-type expressions for the dependency of the diffusivity on temperature were obtained, and from them, activation energies were determined for both cases: considering and neglecting shrinkage. From the comparison of the effective diffusion coefficients obtained, we conclude that when shrinkage is neglected, the diffusion coefficients are underestimated. A model for the diffusion coefficient as a function of temperature and concentration is provided.

Introduction

In the design and optimization of drying operations, knowledge of the transport parameters of food products and their variation with temperature and moisture concentration during drying is necessary. Of particular importance is the availability of precise values of, or reliable expressions for, the moisture diffusion coefficients in the solids under study for designing, simulation, optimization, and control of the drying process. The shrinking of foodstuffs is an observable phenomenon associated with their drying that should be considered when determining moisture diffusion coefficients. Several authors have demonstrated in their studies (Rahman and Kumar (2007); Curcio and Aversa (2014); Adrover et al. (2019); Shahari and Hibberd, (2012)) the importance of considering shrinkage in the modeling of drying processes. Potatoes are the 4th most important crop for human nutrition worldwide. They have a high initial moisture content, which increases their susceptibility to deterioration. Therefore, potatoes must be dried for their preservation, which is a high-energy-consuming process. They also present a considerable reduction in size during drying. The objective of this work is to determine the moisture diffusion coefficient in potatoes considering the shrinking phenomenon associated with the drying process by applying a new methodology presented by Martínez et al. (2013) and Martínez and Vizcarra (2022a).

Among the works that calculate moisture diffusion coefficients of foodstuff without considering shrinkage and assume that they are dependent only on temperature, Boutelba et al. (2018) determined the effective diffusion coefficient of potatoes, and Radosław et al. (2020) estimated the moisture diffusion coefficient and the Biot number in the drying of parsley roots from the analytical solution to the unidimensional diffusion equation considering convective resistance at the solid–fluid interface through a Pareto optimization method. Additionally, Wang et al. (2020) studied the effect of the drying method and calculated potato diffusion coefficients by applying linear regression, corresponding to three different drying methods.

To highlight the importance of considering sample shrinkage in the determination of moisture diffusion coefficients, we also determine these coefficients while neglecting the shrinking phenomena and compare the values obtained in both cases. There have been previous works that pursue a similar objective, for example, Uddin and Hawlader (1990), Simal et al. (1996), Rahman and Kumar (2007), Hosain et al. (2013), Onu et al. (2017), Ortiz-García-Carrasco et al., 2015, and Boutelba et al. (2018). Neither of these works addresses the drying-shrinking process as a moving boundary problem. These authors departed from an analytical solution to the moisture diffusion equation by assuming constant diffusivity and neglecting shrinkage; subsequently, they modified the solution to introduce the shrinking effect.

The methodology proposed by Uddin and Hawlader (1990) is based on a modification of the one-term truncated infinity series solution to the diffusion equation assuming constant diffusivity and neglecting shrinkage. Considering only the first term of the infinite series solution and after taking the logarithms of both sides of the equation, the diffusion coefficients are obtained from the slope of the plot of the logarithm of the dimensionless concentration versus time. This plot should be linear if the assumed conditions are valid. When the plot is nonlinear due to the shrinking effect, the authors propose replacing the length with a modified length, related to the moisture content, such that the plot becomes a line and, from its slope, determines the diffusion coefficient. Examples of its application can be found in Hawlader et al. (1991), and Giovanelli et al. (2002).

In the approach followed by Rahman and Kumar (2006), the experimental moisture content-time curves were divided into small time intervals. The values of D and the particle radius were assumed to be constant over these short intervals. The value of the radius for each interval is calculated using an empirical relation for the shrinkage parameter as a function of the sample moisture content given by Ratti and Mujumdar (1993). Assuming constant radius and D values for each time interval, iterations were performed over D in the analytical solution until the difference between the predicted moisture content and the experimental moisture content was less than a preset tolerance for each interval. Studying potato cylinders and circular slices, they found that when no sample shrinkage was considered, the effective diffusivity was overestimated in the range of 75.9–128.1%. Da Silva et al. (2014) assumed an exponential dependency of the diffusion coefficient on the moisture concentration and determined the corresponding parameters via optimization. The shrinkage is included by correcting the banana dimensions in each integration time step according to a previously obtained expression for V/V_0. Ortiz-García-Carrasco et al., 2015 considered shrinkage and deformation in the estimation of diffusion coefficients. Adrober et al. (2019) model the drying-shrinking phenomenon as a moving boundary problem departing from the advection–diffusion equation and assume that the moisture diffusivity is only a function of temperature. The diffusivities are determined by a least square best-fit procedure.

As mentioned by Lentzou et al. (2019), inverse methods generally combine forward models with optimization algorithms to iteratively find the best parameter set that minimizes an objective function (the distance between computed and experimental values). The method applied here avoids the iterations inherent to the optimization algorithms but minimizes the distance between the computed and experimental values.

In the present work, we determined the moisture diffusion coefficient in potatoes by employing two approaches: 1) neglecting the shrinkage of the sample and 2) considering the sample´s volume reduction and comparing the values obtained with both approaches. The first case constitutes a fixed boundary problem, and the second is a moving boundary problem. In the case in which shrinkage is neglected, the estimation of the diffusivity departs from the solution to the diffusion equation by separating variables while assuming that the sample size and geometry remain constant. In the second case, the approach followed here is based on a numerical solution to the moving boundary problem posed by the diffusion equation when the solid´s volume changes with time. A state observer is constructed based on a numerical solution to estimate diffusion coefficients. The estimator is driven by an error term, the difference between the experimental moisture content and the estimated moisture content, which minimizes the error through the adjustment of the diffusion coefficient at each integration time step. As a result, the estimator produces a set of time-diffusivity pairs of data. This approach for identifying the moisture content and diffusion coefficient of solids is based on a Luenberger observer (Bastin and Dochain 1990; Martínez et al. 2013) and has been explained in full detail by Martínez and Vizcarra (2022a), who applied this approach to a problem in spherical geometry. Here, the same approach is applied to a rectangular geometry. An Arrhenius-type expression for the dependency of the diffusivity on temperature was obtained, and the activation energies were determined.

In the majority of the studies cited above, the drying mechanism is convective or microwave drying, whereas, in this work, the sample drying was carried out in a thermobalance in which the heat drying mechanism is thermal radiation. There are other works in which the drying method involves infrared radiation. For example, Arranz et al. (2017) determined moisture diffusion coefficients in carrot slices dried in a thermobalance, and Wang et al. (2020) determined diffusion coefficients in potato cubes dried by infrared-assisted hot air drying.

Methodology

Experimental procedure

Fresh white round potatoes of the alfa variety were peeled and cut into rectangles of 1 cm × 1 cm × 2 cm. The surface area was 2 cm². The rectangles were immersed in a solution of 0.005 M Na₂S₂O₅ for 10 s to avoid oxidation during drying and the darkening of their surfaces. The samples were prepared such that moisture diffusion occurred in only one dimension. The shrinkage in the other dimensions is considered neglectable. The four faces of each rectangle were sealed with Teflon tape. The potato rectangles were placed over an aluminum dish in a thermobalance (accuracy of ± 0.01 g), and only one of its faces was in contact with air. The experiments were carried out at four temperature levels: 323, 333, 343, and 353 K. The setpoint temperature was established by the moisture balance for each of these temperatures. Weight loss, as well as size reduction, was registered during the drying time. The experiments for measuring the thickness as a function of time were performed independently of those for measuring the weight loss in the same thermobalance and were performed in quintuplicate. The thickness was measured with a digital Vernier caliper (accuracy: ± 0.1 mm). The samples were removed from the thermobalance at specified times, measured, and then returned to the thermobalance. The drying curves, as well as the shrinking curves, were obtained at each temperature level (Martínez et al., 2022b). The drying experiments in the thermobalance are considered isothermal.

Diffusion coefficient estimation neglecting the volume reduction

Under the following assumptions: 1. The moisture transfer is internally controlled and unidirectional; 2. The effective diffusivity accounts for all the moisture transfer mechanisms involved; 3. The effective diffusivity is temperature dependent; 4. The initial moisture content and temperature are uniform; 5. The external heat and mass transfer are negligible; 6. The shrinkage of the solids is negligible. The diffusion problem to be solved when solid shrinkage is neglected is:

$$\begin{array}{cc}\frac{\partial c}{\partial t}=D\frac\partial{\partial x}\left(\frac{\partial c}{\partial x}\right)&0\leq x<L_0,t>0\end{array}$$

(1a)

where c is the volumetric moisture concentration of the solid ((kg water/kg dry mass)/m³), D is the moisture diffusion coefficient (m²/s) and ${L}_{0}$ is the initial length of the sample (m). The boundary and initial conditions are:

$$\begin{array}{ccc}\frac{\partial c}{\partial x}=0&at&x=0,\;\forall\;t\end{array}$$

(1b)

$$\begin{array}{ccc}c=c_{eq}&at&x=L_0,\;\forall\;t\end{array}$$

(1c)

$$\begin{array}{cc}c=c_0 0\leq x<L_0,&t=0\end{array}$$

(1d)

where c₀ and c_eq are the initial and equilibrium moisture contents, respectively. An initial uniform moisture content is assumed (Eq. (1d)). The equilibrium moisture concentrations are calculated based on the GAB model given by McMinn and Magee (1999).

From the solution to the unidimensional diffusion problem (1a-1d) by separating variables assuming constant diffusivity, the following equation is obtained for the dimensionless average moisture content as a function of time:

$$\frac{\overline c\left(t\right)-c_{eq}}{{}_{C_0-c_{eq}}}\;=\;\frac8{\pi^2}\;{\textstyle\sum_{n=0}^\infty}\frac1{\left(2n+1\right)^2}\;{exp}^{\left(-\frac{\left(2n+1\right)^2\pi^2D\;t}{4\;L_0^2}\right)}$$

(2)

Taking into account only the first term of the series solution and after taking logarithms of both sides of the equation, the diffusion coefficients are obtained from the slope of the plot of the logarithm of the dimensionless concentration versus time (-π²D/4 ${L}_{0}^{2}$). The diffusion-temperature data are fitted to an Arrhenius model, Eq. (3), and the activation energy is obtained from the slope of the plot of Ln [D] versus 1/T, (Eq. (4)):

$$D={D}_{0}\cdot \text{Exp}\left[-\frac{\text{Ea}}{\text{R T}}\right]$$

(3)

$$\text{Ln}\left[D\right]=\text{Ln}[{D}_{0}]-\frac{\text{Ea}}{\text{R T}}$$

(4)

Diffusion coefficient estimation considering the volume reduction

To determine the effective diffusion coefficient considering the shrinkage effect, we applied a methodology proposed by Martínez and Vizcarra (2022). First, the mathematical model that describes the drying problem posed here is presented, followed by the construction of the estimator. Finally, the estimator tuning procedure is discussed.

Mathematical model

The following assumptions are made: 1. The moisture transfer is internally controlled and unidirectional (external mass transfer is negligible); 2. The product is homogeneous; 3. The initial moisture content is uniformly distributed inside the sample; 4. The effective diffusivity accounts for all the moisture transfer mechanisms involved; 5. The effective diffusivity is concentration and temperature dependent; 6. There are no temperature gradients inside the sample, and the temperature is constant and uniform; 7. Shrinkage has a significant effect.

When the concentration is a function of volume and time and when the volume is a function of time, the drying process constitutes a moving boundary problem. As shrinkage is assumed to occur in only one dimension, the x coordinate, the volume reduction corresponds to a reduction in the length L of the sample, and the dependency of the concentration on volume becomes dependent on length. In this case, it is necessary to evaluate the total derivative of the volumetric moisture concentration, $c(x\left(t\right),t)$:

$$\frac{dc}{dt}=\frac{\partial c}{\partial t}+\frac{\partial c}{\partial x}\frac{dx}{dt}$$

(5)

and the first term on the right side, $\frac{\partial c}{\partial t}$, is given by Fick’s law:

$$\frac{\partial c}{\partial t}=\frac{\partial }{\partial x}\left(D\frac{\partial c}{\partial x}\right)$$

(6)

Then, from Eqs. (5) and (6), the following equation is obtained:

$$\begin{array}{cc}\frac{dc(x\left(t\right),t)}{dt}=\frac\partial{\partial x}\left(D\frac{\partial c(x\left(t\right),t)}{\partial x}\right)+\frac{\partial c(x\left(t\right),t)}{\partial x}\frac{dx}{dt}&0\leq x<L,t>0\end{array}$$

(7)

which is transformed into a fixed boundary problem by the change of variable z = x/L (Crank 1984). z = 0 corresponds to the bottom of the potato cube, and z = l corresponds to the potato surface, which is in contact with air. In terms of the dimensionless concentration y = c/c(t = 0), Eq. (7) becomes:

$$\begin{array}{cc}\frac{dy}{dt}=\frac1{L^2}\frac\partial{\partial z}\left(D\frac{\partial y}{\partial z}\right)+\frac1L\frac{\partial y}{\partial z}z\frac{dL}{dt}&0\leq z<1,t>0\end{array}$$

(8a)

Because there is no diffusion at the bottom of the rectangle and moisture equilibrium at the solid‒gas interphase is assumed, the boundary conditions are:

$$\begin{array}{ccc}\frac{\partial y}{\partial z}=0&at&z=0,\;\forall\;t\end{array}$$

(8b)

$$\begin{array}{ccc}y=y_{eq}&at&z=1,\;\forall\;t\end{array}$$

(8c)

The assumption of uniform initial moisture content is expressed as:

$$\begin{array}{cc}y=y_0 0\leq z<1,&t=0\end{array}$$

(8d)

The concentrations at each time step averaged over the sample volume ($V={L}_{0}^{2}L$) are given by the following equation:

$$\overline{c }\left(t\right)=\frac{1}{{V}_{p}}{\int }_{0}^{{V}_{p}}c\left(x, t\right)dv$$

(9)

or, in terms of dimensionless concentration:

$$\overline{y }\left(t\right)={\int }_{z=0}^{z=1}y\left(z, t\right)dz$$

(10)

Numerical solution

A one-point collocation solution to Eq. (8a), which satisfies the boundary conditions, is constructed as the product of two functions: one of the spatial variable and another of time (y stands for the dimensionless concentration at the collocation point z):

$$y\left(t\right) =\left(1-{z}^{2}\right)\cdot \tau \left(t\right)+{y}_{eq}$$

(11)

$\uptau \left(\text{t}\right)$ is a function of the time to be determined.

Substituting Eq. (11) in Eq. (8a), assuming that D is constant (which can be sustained if the system dynamics are slower than the estimator dynamics) and neglecting the dependency of ${y}_{eq}$ on time, the following equation for the time function τ at the collocation point z is obtained:

$$\frac{d\tau }{dt}=-\left(\frac{\frac{2 D}{{L}^{2}} + \left(\frac{2 {z}^{2} }{L}\right)\cdot \frac{dL}{dt}}{\left(1-{z}^{2}\right)}\right)\cdot \tau$$

(12)

The time-dependent function τ(t) is, from Eq. (11), given by the following equation:

$$\tau \left(t\right) =(y\left(t\right) -{y}_{eq})/\left(1-{z}^{2}\right)$$

(13)

Averaged over the dimensionless volume, the average concentration is obtained:

$$\overline{y }\left(t\right)=\frac{2 }{3}\tau +{y}_{eq}$$

(14)

Estimator design

To estimate D from a set of dynamic moisture content and shrinking rate data, the coupled differential equations in Eqs. (12) and (15) are used:

$$\dot{D}=0$$

(15)

This equation represents the assumption that D is constant ($\dot{D}$ denotes the time derivative of D) and is justified if the observer dynamics are faster than the system dynamics. The structure of the Luenberger observer ($\widehat{\tau }$ denotes the estimated value of the function τ(t) and $\widehat{D}$ the estimated D value) is:

$$\frac{\text{d}\widehat{\uptau }}{\text{dt}}=-\left(\frac{\frac{2 \widehat{D}}{{L}^{2}} + \left(\frac{2 {z}^{2} }{L}\right)\cdot \frac{dL}{dt}}{\left(1-{z}^{2}\right)}\right)\cdot \widehat{\tau }+ {\omega }_{1}\left(y- \widehat{y}\right)$$

(16)

$$\frac{\text{d}\widehat{D}}{\text{dt}}= {\omega }_{2} \left(y- \widehat{y}\right)$$

(17)

The terms on the right-hand side, ${\omega }_{i}\left(y- \widehat{y}\right)$, have the function of driving the estimation error to zero. ω₁ and ω₂ are the filter gains that should be large enough, in magnitude, to make the observer dynamics faster than the system dynamics because, as mentioned, the constancy of D is sustained under the assumption that the observer dynamics, determined by the filter gains, is faster than the system dynamics. ω₁ and ω₂ have dimensions of t⁻¹ and L² t⁻², respectively.

Defining:

$${a}_{1}=-\frac{2 }{{L}^{2}\left(1-{z}_{c}^{2}\right)}$$

(18)

$${a}_{2}=-\frac{2 {z}_{c}^{2}\cdot \dot{L}}{L\left(1-{z}_{c}^{2}\right)}$$

(19)

Equations (16) and (17) are written as:

$$\dot{\widehat{\tau }}={a}_{1}\cdot \widehat{D}\cdot \widehat{\tau }+{a}_{2}\cdot \widehat{\tau }+ {\omega }_{1}\left({y}_{exp}- \widehat{\overline{y} }\right)$$

(20)

and

$$\dot{\widehat{D}}={\omega }_{2}\left({y}_{exp}- \widehat{\overline{y} }\right)$$

(21)

a₁ and a₂ change as the length of the solid changes, that is, as L changes and $\dot{L}\ne 0$.

Estimator tuning

The design of the adaptive gains is based on a linearized version of the process dynamics observation error presented in Appendix 1 (Dochain, 2003). To determine ω₁ and ω₂, the eigenvalues ($\lambda$ ₁ and $\lambda$ ₂) of matrix M (Appendix 1), which defines the estimation error dynamics, were calculated. Then, the gains were expressed in terms of the eigenvalues (the tuning parameters). The resulting equations for ω₁ and ω₂ are:

$${\omega }_{1}={a}_{2}+{a}_{1}\cdot \widehat{D}-{\lambda }_{1}-{\lambda }_{2}$$

(22)

$${\omega }_{2}=\frac{{\lambda }_{1}{\lambda }_{2}}{{a}_{1}\cdot \widehat{\tau }}$$

(23)

Large enough negative values must be assigned to $\lambda$ ₁ and $\lambda$ ₂ so that the error dynamics converge to zero fast enough to sustain the assumption of the constancy of D.

Diffusion model assessment

The estimation algorithm is supplied with functions that interpolate experimental moisture concentration–time data and length–time data, that is, ${y}_{exp}(t)$ and $L(t)$. $\dot{L}(t)$ is the time derivative of $L(t)$. The functions ${y}_{exp}(t)$, $L(t)$ and $\dot{L}(t)$ provide the required values at each integration time step for integrating the coupled Eqs. (20) and (21). The observer produces the estimated diffusion coefficient, $\widehat{D}$, and time function, $\widehat{\tau }$, at each integration time step. From $\widehat{\tau }$, the values of $\widehat{\overline{y} }$ are obtained (Eq. (14)). The difference ${(y}_{exp}- \widehat{\overline{y} })$ in Eqs. (20) and (21), weighted by ${\omega }_{i}$, drives the estimation error to zero. From the $\widehat{\overline{y} }$ and $\widehat{D}$ values produced by the estimator, a diffusion-concentration relationship is generated via nonlinear fitting.

Results and discussion

The experimental drying curves are presented in Figs. 1 and 2 (red continuous curves), with the volumetric moisture concentration in the former and the dimensionless form in the latter.

Fig. 1

Experimental and estimated average volumetric moisture content profiles at the four temperature levels

Fig. 2

Experimental and estimated dimensionless moisture content profiles at the four temperature levels

As explained above, experiments were carried out to measure the thickness variation at each temperature level during the drying time. The results of the experimental runs for each temperature were averaged and are shown as points in Fig. 3. The functions fitting the average of the experimental values, $\overline{L }$, at each temperature are shown as continuous lines in the same figure. The mathematical functions fitting the average length–time data are given in Appendix 2 (the L–t experimental data obtained in each of the experimental runs at each temperature and the $\overline{L }$-moisture content (db) dependency at each temperature are given as Supplementary material). The moisture content-time curves for T = 343 K and T = 353 K are nearly identical; however, their length–time curves are quite different.

Fig. 3

Average experimental thickness measurements and their fitting functions at four temperature levels

For some of the cases analyzed here, it seems that a stream of vapor breaks out through one of the solid faces, making a hole in it. From this moment onward, the drying process is no longer controlled by unidimensional diffusion, and the sample length remains nearly constant. The shrinking and crusting formation, as noted by Wang et al. (2020), could hinder the migration of internal moisture in the later stage of drying, in addition to the penetration of infrared radiation beyond the surface, resulting in the generation of heat energy to heat the potato interior through conduction, which may lead to an increase in the internal pressure that could cause the breakdown of the sample surface. This effect is notorious at higher temperatures. When holes are formed (see Figs. 4 and 5), mainly at higher temperatures, the diffusional model, described by Eqs. (7) and (8a), is no longer valid. From the moment when holes are formed, due to the sudden breaking of the crust, the thickness remains nearly constant. We consider the time intervals in which the diffusion model is considered valid.

Fig. 4

Photograph of the top of the potato surface after drying at T = 353 K

Fig. 5

Photograph of the potato sample sealed by the sides cut by halves after drying

Diffusion coefficient calculation neglecting volume reduction

As explained above, diffusion coefficients are obtained from the slope of the plot of the logarithm of the dimensionless concentration versus time (-π² D/4L²). The values obtained when shrinkage is neglected are reported in Table 1. Onu et al. (2017) studied the drying of potato slices in a hot-air convection drier and the effects of slice thickness, air velocity, and temperature. These authors calculated diffusion coefficients neglecting shrinkage and reported that they depend on the air velocity and slice thickness. The higher values reported by them are shown in Table 1 for comparison purposes.

Table 1 Diffusion coefficients × 10⁹ [m²/s]

The dependency of the moisture diffusivity on temperature is modeled through the Arrhenius expression (Eq. (3)). The activation energies, Ea, obtained from the plot of Ln[D] versus 1/T (Fig. 6) are reported in Table 2.

Fig. 6

Plot of Ln [D] versus 1/T, from which the activation energies are obtained. The red points and the fitted red line correspond to the values of D obtained without considering shrinkage. The black points and the fitted black line correspond to the case in which shrinkage is considered

Table 2 Activation energies

The activation energy, Ea, obtained with the diffusivity values at the four temperature levels reported in Table 1 is 25.80 kJ/mol. The temperature dependence of the average effective diffusivity fitted to an Arrhenius model is given by the following equation:

$$D(T)=1.5100\times {10}^{-5}\cdot \text{Exp}\left[-\frac{25.80}{8.314\times {10}^{-3}\text{ T}}\right]$$

(24)

Onu et al. (2017) reported activation energies in the convective drying of potato slices at different air velocities, neglecting shrinkage ranging from 28 to 37 kJ/mol. The value reported by Srikiatden and Roberts (2006) is 25.77 kJ/mol, similar to the value obtained in this work.

Diffusion coefficient estimation considering volume reduction

Moisture content (db)–time data and radius–time experimental data were fitted by continuous functions. The moisture content and radius as functions of time are the information fed to the estimator for obtaining the diffusion–time set of values at each temperature level. By integrating the coupled Eqs. (20) and (21) and calculating the adaptive gains given by Eqs. (22) and (23), at each integration time step, the average dimensionless concentration (calculated with Eq. (14)) and estimated moisture diffusivity are obtained at each integration step. The average volumetric moisture concentration is obtained from the dimensionless concentration and the initial concentration. The estimated concentrations and the experimental concentrations fed to the estimator at each temperature are shown graphically in Figs. 1 and 2, with the volumetric concentration in the former and the dimensionless concentration in the latter. In these figures, it can be seen that the estimated moisture content curves match the experimental curves, except for a very short time at the beginning of the integration process, which was due to the time required for the observer to adjust the required D value, the learning time.

The graphs of $\widehat{D}$ vs. time are shown in Fig. 7. To highlight how fast the observer adjusts the diffusivity value even though the initial diffusivity guess is far from the required value, Fig. 8 presents a case for T = 323 K, in which the output time interval in the integration algorithm is 100 s, and the initial guess of D is far from the required value for matching the estimated concentration to the experimental one. The initial estimate for $\widehat{D}$ (1.2 × 10^–9) is far from the required for matching the estimated concentration to the experimental concentration, and after 100 s, it has already been adjusted to the value required for matching the concentrations (2.93 × 10^–9). At 100 s, the observer adjusted the value of $\widehat{D}$ to the required value to minimize the difference between the experimental moisture content and the estimated moisture content. Figure 9 is a kind of zoom for the same run of the integration algorithm shown in Fig. 8, now with an output time interval of 1 s. The estimated concentration matches the experimental concentration at 0.1 s, and the estimated diffusivity oscillates from the initial guess, $\widehat{D}$ = 1.2 × 10^–9, to $\widehat{D}$ = 1.32 × 10^–7, and then to $\widehat{D}$ = 2.88 × 10^–9. This oscillation is hidden when the output time in the algorithm is 100 s.

Fig. 7

Diffusion coefficients estimated by the observer along the experimental drying time starting from the initial guess provided to the algorithm at each temperature

Fig. 8

Estimated moisture diffusivity vs time (left) and concentrations, estimated and experimental, (right) for the case at T = 323 K when the output time interval in the integration algorithm is 100 s

Fig. 9

Estimated moisture diffusivity vs time (left) and concentrations, estimated and experimental, (right) for the case at T = 323 K with an output time interval of 1 s

The diffusivity values vary with time, which can indicate the concentration dependency of the diffusivity on the temperature. Commonly, diffusivity decreases as drying proceeds (Giovanelli et al. (2002)), but in particular, for T=343 K, the diffusivity starts decreasing and later increases. This could be due to a case-hardening effect that can be more severe at higher temperatures. As noted by Hassini et al. (2007), contradictory results have been reported about the effect of moisture content on diffusivity. Some authors have reported that diffusivity decreases with decreasing water content, but other authors have reported the opposite trend for some food products, particularly for microwave-convective drying of potatoes and infrared radiation drying of onion slices. Hassini et al. (2007) found that during the drying of potato slices, long drying times and low moisture contents increase the drying speed, as if the diffusivity greatly increases, and proposed as a possible explanation the transition of the water transfer mechanism from liquid to vapor diffusion. Hosain et al. (2013) found that in the microwave drying of potato slices, the effective moisture diffusivity increased with decreasing moisture content of the samples.

The average values of the estimated diffusivities, $\widehat{D}(t)$, produced by the observer are calculated for comparison purposes, with those values obtained when shrinkage is neglected and those obtained considering shrinkage but not considering concentration dependence. To obtain the average $\overline{\widehat{D} }$ values, the simulations for obtaining the curves $\widehat{D}$ vs. t were run again, taking the initial guess for $\widehat{D}$ as the value with which the observer matches the experimental moisture content obtained in the previous simulation. These values, reported in Table 1, are obtained with Eq. (25) (the arithmetic averages are the same as those obtained with Eq. (25)).

$$\overline{\widehat{D} }=\frac{1}{tf-ti}{\int }_{ti}^{tf}\widehat{D}(t) dt$$

(25)

The initial time, ti, is zero because, as explained above, in the simulations for obtaining the average $\overline{\widehat{D} }$ values, the initial guess for $\widehat{D}$ was the value with which the observer matches the experimental moisture content obtained in a previous simulation. The final time, tf, for obtaining $\overline{\widehat{D} }$ at T = 323 K is 16,000 s, and that at T = 333 K is 8200 s, taking into account only the time range in which $\overline{\widehat{D} }$ decreases. For T = 343 K and T = 353 K is 10,800 s. tf can be seen in Fig. 7 for each temperature. As mentioned above, the moisture content-time curves for T = 343 K and T = 353 K are nearly identical, but although their dimensionless length values at the end of the drying experiments for these two temperatures are close (see Fig. 3), their length–time curves are quite different. This could explain the different forms in which the estimated moisture diffusivity varies with time at these two temperatures, as shown in Fig. 7, and the fact that their $\overline{\widehat{D} }$ values are similar.

From the results shown in Table 1, it can be appreciated that, in our study, the values obtained neglecting the shrinking effect are underestimated; their values are approximately two or three times less than the values obtained when the size reduction was taken into account. With respect to the values calculated by Onu et al. (2017), reported in Table 1, the rest of the values in that table overestimate the diffusion coefficients. Hassini et al. (2007) reported that “the differences between the data corrected and noncorrected for shrinkage are very significant, nearly one order of magnitude”. Rahman and Kumar (2006) concluded that when the effect of sample shrinkage is not considered, the effective diffusion coefficients are overestimated. The fact that in this study, the diffusivities when shrinkage is neglected are underestimated is not inherent to the observer methodology. When this methodology was applied to the identification of moisture diffusivities in peas (Martínez and Vizcarra (2022)), the results indicated that when shrinkage was not considered, the effective diffusion coefficients were overestimated. For the case analyzed here, for potatoes dried in a thermobalance, the effect of the reduction in the length traveled by water molecules could be more than counterbalanced by the effect of the crust formed near the surface, resulting in lower diffusion coefficients when shrinkage is neglected than when shrinkage is taken into consideration. Compared with the values obtained by other authors, our values are generally greater. Wang et al. (2020) obtained diffusion coefficient values between 0.90 × 10^–9 and 1.35 × 10^–9 m²/s for potatoes at 333 K without considering shrinkage, depending on the drying method. Hassini et al. (2007) obtained moisture diffusivity values in the convective drying of potato slabs that corrected for shrinkage in the range of 3.55 × 10^–10 – 1.92 × 10^–9 m²/s. Hosain et al. (2013) reported effective moisture diffusivities in the microwave drying of potato slices, calculated with the first term of Eq. (2), which ranges from approximately 1.0 × 10^–8 to 3.8 × 10^–8 m²/s depending on the microwave power. Boutelba et al. (2018) reported effective moisture diffusivities in the convective drying of potato parallelepipeds, calculated with Eq. (2), which ranged from approximately 0.2 × 10^–9 to 2.0 × 10^–9 m²/s. These latter values are similar to the values reported in this work when shrinkage is not considered.

The diffusion coefficient values at the same temperatures considered in this study considering shrinkage reported by other authors are shown, for comparison purposes, in Table 1. Those reported by Adrover et al. (2019) are lesser than ours but of the same order of magnitude. Those reported by Ortiz-García-Carrasco et al., 2015 are one order of magnitude lesser than the $\overline{\widehat{D} }$ obtained in this work.

It should be noted that, as mentioned by Lentzou et al. (2019), effective diffusion coefficients should be smaller than the self-diffusion coefficient of water. The diffusion coefficients determined in this work satisfy this condition.

The form in which $\widehat{D}$ depends on y is obtained from the data produced by the estimator. The $\widehat{D}$-c dependencies are shown graphically in Fig. 10. In Appendix 3, the $\widehat{D}$-m dependencies are shown graphically, as well as a fifth-degree polynomials fitting each curve.

Fig. 10

Plots showing the diffusion coefficient-volumetric concentration relationship produced by the estimator for the 4 temperatures analyzed

In Fig. 11 are shown the graphs of the diffusion coefficient vs db moisture concentration estimated by the observer in the db moisture content range 1.5–4 kg/kg for the 4 temperatures analyzed. It can be appreciated that the graphs at the temperatures 323 K, 333 K, and 343 K follow a similar pattern, however, the graph at 353 K shows a different pattern. The diffusion coefficient-db moisture concentration data were fitted by an exponential function, Eq. (26), at each temperature:

Fig. 11

Plots showing the diffusion coefficient-db moisture concentration estimated by the observer in the db moisture content range 1.5–4 kg/kg for the 4 temperatures analyzed

$$\widehat{D}=A\cdot \text{Exp}[B\cdot m]$$

(26)

In Table 3 are reported the coefficients A and B corresponding to each temperature. As a measure of the goodness of the fit the R² coefficients are reported. The fitted curves (in red) as well as the estimated diffusion coefficients at 323 K, 333 K, and 343 K are shown in Fig. 12.

Table 3 Coefficients of Eq. (26) at each temperature

Fig. 12

Diffusion coefficients estimated by the observer in the db moisture content range 1.5–4 kg/kg at each temperature and their exponential fitting by Eq. (26) (red curves)

For obtaining an expression of the diffusion coefficient as a function of temperature and moisture content the coefficients A and B in Eq. (26) were fitted as functions of the temperature for 323 K ≤ T ≤ 343 K and 1.5 kg/kg ≤ m ≤ 4.0:

$$A={a}_{1}+{a}_{2}\cdot \text{T}+{a}_{3}\cdot {T}^{2}$$

(27)

$$B={b}_{1}+{b}_{2}\cdot \text{T}$$

(28)

Then, the diffusion coefficient as a function of T and m is given by:

$$\widehat{D}[T,m]=({a}_{1}+{a}_{2}\cdot \text{T}+{a}_{3}\cdot {T}^{2})\cdot \text{Exp}[({b}_{1}+{b}_{2}\cdot \text{T})\cdot m]$$

(29)

With

$${a}_{1}=-2.1291\times {10}^{-7}$$

$${a}_{2}=1.1861\times {10}^{-9}$$

$${a}_{3}=-1.6176\times {10}^{-12}$$

$${b}_{1}=0.0672$$

$${b}_{2}=0.00017$$

In Fig. 13 are shown the fitted functions with Eq. (29) and the diffusion coefficients estimated by the observer. It can be appreciated that the fitted curves obtained with Eq. (29) are practically indistinguishable from those obtained with Eq. (26) and shown in Fig. 12.

Fig. 13

Diffusion coefficients estimated by the observer in the db moisture content range 1.5–4 kg/kg at each temperature and their exponential fitting (red curves) with Eq. (29)

Figure 14 shows the dependency of D with T and m as given by Eq. (29) for 323 K ≤ T ≤ 343 K and 1.5 kg/kg ≤ m ≤ 4.0. As expected, in those ranges D increases with T and m.

Fig. 14

Graphic of the exponential fitting function, Eq. (29), of the diffusion coefficients estimated by the observer as a function of the moisture content and temperature in the db moisture content range 1.5–4 kg/kg

The dependency of the moisture diffusivity on temperature is modeled through the Arrhenius expression (Eq. (3)). The activation energy, Ea, obtained from the plot of Ln[$\overline{\widehat{D} }$] versus 1/T (Fig. 6) is reported in Table 2. The Ea values obtained considering and neglecting shrinkage are similar. The temperature dependence of the average effective diffusivity considering shrinkage fitted to an Arrhenius model is given by the following equation:

$$D(T)=2.7024\times {10}^{-5}\cdot \text{Exp}\left[-\frac{25.14}{8.314\times {10}^{-3}\text{ T}}\right]$$

(30)

Conclusions

In this work, an observer-based methodology was applied to the identification of moisture diffusion coefficients in potato slabs dried in a thermobalance, taking into account the shrinking associated with the drying process. To emphasize the importance of the shrinking phenomenon in the estimation of moisture diffusion coefficients, the coefficients were also determined by neglecting shrinkage. From the comparison of the effective diffusion coefficients obtained with both methods, it is concluded that when shrinkage is neglected, the diffusion coefficients obtained are underestimated (the diffusion coefficients considering shrinkage are approximately 2–3 times greater than those obtained neglecting shrinkage). The results of the present work allow us to conclude that shrinkage cannot be neglected if the objective is to obtain reliable values of effective diffusion coefficients. The activation energy obtained when the volume reduction associated with the drying process is neglected is similar to that estimated when shrinkage is considered. An exponential model for the diffusion coefficient as a function of temperature and concentration is provided. The observer-based methodology here applied is easily implemented, does not require good guesses of the diffusion coefficient, and converges very quickly. The solution to the diffusion equation neglecting shrinkage could be used as a preliminary estimation of the moisture diffusivity.

Appendices

Appendix 1 Error dynamics

We define the errors in the estimation of the water concentration, the $\tau$ function and in the diffusion coefficient as:

$${\epsilon }_{y}={y}_{exp}-\widehat{\overline{y} }$$

(31)

$${\epsilon }_{\tau }={\tau }_{exp}-\widehat{\tau }$$

(32)

$${\epsilon }_{D}= {D}_{s}-\widehat{D}$$

(33)

where ${\tau }_{exp}$ denotes the value of the function $\tau (t)$ that satisfies Eq. (14) with ${y}_{exp}(t)$ instead of $\overline{y }\left(t\right)$. The respective dynamics of the errors are given by the next equations:

$${\dot{\varepsilon }}_{y}={\dot{y}}_{exp}-\dot{\widehat{\overline{y}} }$$

(34)

$${\dot{\varepsilon }}_{\tau }={\dot{\tau }}_{exp}-\dot{\widehat{\tau }}$$

(35)

$${\dot{\varepsilon }}_{D}={\dot{D}}_{s}-\dot{\widehat{D}}$$

(36)

The next equations are obtained from Eqs. (12) and (16):

$${\dot{\tau }}_{exp}={a}_{1}\cdot {D}_{s}\cdot {\tau }_{exp}+{a}_{2}\cdot {\tau }_{exp}$$

(37)

$$\dot{\widehat{\tau }}={a}_{1}\cdot \widehat{D}\cdot \widehat{\tau }+{a}_{2}\cdot \widehat{\tau }+ {\omega }_{1}\left({y}_{exp}- \widehat{\overline{y} }\right)$$

(38)

The first terms on the right-hand side of the last two Eqs. are non-linear. Next, a linear approximation of these two terms is made through a Taylor series approximation, in the following way:

Defining

$$f\left(\tau ,{D}_{s}\right)={a}_{1}\cdot {D}_{s}\cdot {\tau }_{exp}$$

(39)

and

$$f\left(\widehat{\tau },\widehat{D}\right)={a}_{1}\cdot \widehat{D}\cdot \widehat{\tau }$$

(40)

The Taylor series approximation is given by the next equation:

$$f\left(\tau ,{D}_{s}\right)=f\left(\widehat{\tau }+{\epsilon }_{\tau },\widehat{D}+{\epsilon }_{D}\right)=f\left(\widehat{D},\widehat{\tau }\right)+{\left[\frac{\partial f}{\partial \tau }\right]}_{\widehat{D},\widehat{\tau }}\cdot {\epsilon }_{\tau }+{\left[\frac{\partial f}{\partial D}\right]}_{\widehat{D},\widehat{\tau }}\cdot {\epsilon }_{D}={a}_{1}\cdot \widehat{D}\cdot \widehat{\tau }+{a}_{1}\cdot \widehat{D}\cdot {\epsilon }_{\tau }+{a}_{1}\cdot \widehat{\tau }\cdot {\epsilon }_{D}$$

(41)

therefore

$$f\left(\tau ,{D}_{s}\right)-f\left(\widehat{\tau },\widehat{D}\right)={a}_{1}\cdot \widehat{D}\cdot {\epsilon }_{\tau }+{a}_{1}\cdot \widehat{\tau }\cdot {\epsilon }_{D}$$

(42)

Then

$${\dot{\epsilon }}_{\tau }={\dot{\tau }}_{exp}-\dot{\widehat{\tau }}={a}_{1}\cdot \widehat{D}\cdot {\epsilon }_{\tau }+{a}_{1}\cdot \widehat{\tau }\cdot {\epsilon }_{D}+{a}_{2}\cdot {\epsilon }_{\tau }-{\omega }_{1}\left({y}_{exp}- \widehat{\overline{y} }\right)$$

(43)

$${\dot{\epsilon }}_{\tau }=\dot{ \tau }-\dot{\widehat{\tau }}={a}_{1}\cdot \widehat{D}\cdot {\epsilon }_{\tau }+{a}_{1}\cdot \widehat{\tau }\cdot {\epsilon }_{D}+{a}_{2}\cdot {\varepsilon }_{\tau }-{\omega }_{1}\cdot {\epsilon }_{y}$$

(44)

The dynamics of ${\epsilon }_{D}$ is:

$${\dot{\varepsilon }}_{D}=\dot{{D}_{s}}-\dot{\widehat{D}}=- \dot{\widehat{D}}$$

(45)

$${\dot{\epsilon }}_{D}=-{\omega }_{2} \left({y}_{exp}- \widehat{\overline{y} }\right)=-{\omega }_{2}\cdot {\epsilon }_{y}$$

(46)

Then, the error dynamics is:

$$\dot{\epsilon }=\left[\begin{array}{c}{\dot{\epsilon }}_{\tau }\left(\widehat{\tau }, \widehat{D}\right)\\ {\dot{\epsilon }}_{D}(\widehat{\tau }, \widehat{D})\end{array}\right]=\mathbf{M}\left[\begin{array}{c}{\epsilon }_{\tau }\\ {\epsilon }_{D}\end{array}\right]=\mathbf{M}\cdot \epsilon$$

(47)

where the matrix M is:

$$\mathbf{M}= \left[\begin{array}{cc}{a}_{1}\cdot \widehat{D}+{a}_{2}-{\omega }_{1}& {a}_{1}\cdot \widehat{\tau }\\ -{\omega }_{2}& 0\end{array}\right]$$

(48)

It is clear that ${\epsilon }_{\tau }=0$ and ${\epsilon }_{D}=0$ are an equilibrium point for the error dynamics, this is,

ϵ = 0 is a solution of this algebraic equation which indicates that the constructed observer may have ϵ = 0 at steady state.

Appendix 2 $\overline{L }$- t function fitting.

Tables 4 and 5

The $\overline{L }$-t dependencies for T = 323 K and T = 333 K were fitted by the following function (bidose response model):

$$\overline{L }\left(t\right)={a}_{1}+ {(a}_{2}-{a}_{1})(\frac{p}{1+ {10}^{\left(\left({a}_{3}-t\right)\right){h}_{1}))}}+ \frac{\left(1-p\right)}{1+ {10}^{\left(\left({a}_{4}-t\right)\right){h}_{2}))}} )$$

(49)

Table 4 Coefficients for eq. (49)

For T = 343 K and T = 353 K, the data were fitted by the following function (dose‒response model):

$$\overline{L }\left(t\right)={b}_{1}+ \frac{{(b}_{2}-{b}_{1})}{1+ {10}^{\left(\left({b}_{3}-t\right)p\right)}}$$

(50)

Table 5 Coefficients for Eq. (50)

Appendix 3 $\widehat{D}$- c polynomial fitting

Figure 15

Table 6

The $\widehat{D}$-m dependencies were fitted by a fifth-degree polynomial:

$$\widehat{D}(m)={a}_{0}+{a}_{1}m+{a}_{2}{m}^{2}+{a}_{3}{m}^{3}+{a}_{4}{m}^{4}+{a}_{5}{m}^{5}$$

(51)

m (kg water/kg dry solid); $\widehat{D}$ (m² s^-1)

Fig. 15

Plots showing the diffusion coefficient-moisture concentration (db) relationship and the polynomial fittings to the relationships produced by the estimator for the 4 temperatures analyzed

Table 6 Coefficients for eq. (51) for each temperature