International Journal of Chemical Engineering

Volume 2019, Article ID 3926897, 13 pages

https://doi.org/10.1155/2019/3926897

## A Moving Boundary Model for Isothermal Drying and Shrinkage of Chayote Discoid Samples: Comparison between the Fully Analytical and the Shortcut Numerical Approaches

^{1}Dipartimento di Ingegneria Chimica Materiali e Ambiente, Università degli Studi di Roma “La Sapienza”, via Eudossiana 18, 00184 Roma, Italy^{2}Dipartimento di Ingegneria Industriale, Università degli Studi di Salerno, via Giovanni Paolo II, 132 84084 Fisciano, Salerno, Italy

Correspondence should be addressed to Alessandra Adrover; ti.1amorinu@revorda.ardnassela

Received 30 November 2018; Accepted 13 March 2019; Published 16 May 2019

Academic Editor: Xunli Zhang

Copyright © 2019 Alessandra Adrover and Antonio Brasiello. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A moving boundary model for food isothermal drying and shrinkage is applied to predict the time decay of water content and sample volume, as well as water diffusivity for chayote discoid slices in the temperature range 40–70°C. The core of the model is the shrinkage velocity , assumed equal to the water concentration gradient times a shrinkage function *α* representing the constitutive equation of the food material under investigation. The aim is to provide a case study to analyze and quantify differences and accuracies of two different approaches for determining the shrinkage function *α* from typical experimental data of moisture content vs. rescaled volume : a fully analytical approach and a shortcut numerical one.

#### 1. Introduction

Mathematical modeling in chemical engineering historically provided the necessary support for understanding physics and transport phenomena that underlie the chemical processes. Mathematical models are also useful tools for the optimization of experimental campaigns and for scale-up from laboratory to industrial scale. During decades, more and more mathematical modeling techniques have been developed [1, 2] bridging different spatial and temporal scales from the microscopic to the macroscopic ones [2, 3] through the mesoscale [4–6]. The translation of chemical processes into equations has always been the unavoidable step for efficient plant design, process analysis, or control.

Food process engineering represents one of the most promising research fields in chemical engineering that could benefit the enhancements of the theoretical research. It was quite intentionally forgotten, in the past, due to the complexity of food materials and of related transformation processes. Among them, drying is undoubtedly one of the most investigated, probably for its huge industrial impact [7–9]. Actually, the drying process is very complex as it implies material transformation at several spatial scales (e.g., porous structure variation and volume changes).

Far from accomplishing a multiscale mathematical formulation able to account for the complexity of the all the phenomena involved [10–12], mathematical models for drying try to capture those features that are more important from the technological point of view. Shrinkage is the most relevant phenomenon connected to drying since it influences consumer quality perception, costs for transportation, and storage [13]. Among the wide literature on this topic, it is worth mentioning some works: [14] in which shrinkage kinetic laws are discussed; [15–17] in which applications of mathematical modeling techniques to food drying can be found; [10] in which a general discussion on advanced computational modeling for drying can be found; [13, 18–20] in which different advanced approaches for linking water content evolution to shrinkage can be found.

In particular, Adrover et al. [21, 22] recently developed a mathematical model with moving boundary for drying of food materials by suitably modifying a classical moving boundary model originally developed in [23, 24] for mass transport, swelling, and dissolution in polymers [25, 26]. The novelty consists in the introduction of a new constitutive equation linking the boundary movement to the water concentration gradient through a proportionality factor *α* representing the fingerprint of the food material under investigation.

In [21, 22], the authors were able to predict both water content evolution and volume reduction following two different approaches for the determination of the proportionality factor *α* from experiments: a fully analytical approach and a shortcut one. Reasonably, from a computational point of view, the first one could be more difficult to apply in many cases of practical interest, e.g., for very complex geometries of food samples, while the second one could not be accurate.

In this paper, the comparison between the two approaches is carried out by using literature experimental data [27] on chayote discoid slices. In this case, due to both the regular shape and the aspect ratio of the samples, both approaches can be easily adopted. The aim is to provide a case study to test differences in terms of accuracy of prediction of water content evolution, volume reduction, and other significant physical quantities such as the water diffusivity.

#### 2. Moving Boundary Model for Discoid Samples

We briefly recall the model equations derived in [21, 22].

Let be a characteristic length of the food sample, *ϕ* the point-wise water volume fraction, the uniform initial water volume fraction, and *D* the water diffusivity at the operating temperature.

By introducing the dimensionless space and time variables , , , and and the dimensionless differential operators and , the moving boundary model equations for the normalized water volume fraction attain the form:where is the mass transfer Biot number, is a mass transfer coefficient, is the solid (pulp) density, is the air density at the operating temperature, is the water partition ratio between the gas and the solid phases , *H* being the absolute air humidity (kg water/kg dry air).

The core of the model is the shrinkage velocity , proportional to the water concentration gradient times a proportionality function tuning, at each point of the system, the relationship between water flux and volume reduction. The shrinkage velocity evaluated at the boundary controls the movement of the boundary itself. The time evolution of the sample boundary, as described by equation (3), is consistent with the classical description of boundary movement induced by the transfer of a diffusing substance across the interface [23, 24, 28]. In point of fact, equation (3) represents a generalization of a classical Stefan condition because it accounts for structural changes of the material during the drying process through the introduction of the shrinkage function .

In dealing with a discoid sample (radius , thickness ) with , we choose as a reference length so that the dimensionless initial domain is , being the discoid aspect ratio.

For high values of the aspect ratio , a one-dimensional model can be readily adopted, describing the time evolution of the rescaled water volume fraction along the axial coordinate (associated with the smallest initial dimension ) and the time evolution of the dimensionless sample thickness (uniform along the radial direction):

In this 1-d approach, both radial shrinkage and water flux from the discoid lateral surface are neglected.

#### 3. Estimation of the Shrinkage Factor

If we adopt a 1-d model for describing the drying process of a discoid sample, the shrinkage factor can be assumed *a priori* or it can be estimated from the *thickness calibration curve* [29, 30], i.e., from experimental data of rescaled thickness vs. the moisture ratio as follows:where is the rescaled water volume fraction evaluated, at each time instant *t*, in a suitable point , called probe point P, placed on the sample surface and evolving in time together with the surface itself. The moisture ratio (or rescaled moisture content) is defined as , where is the amount of water at time *t*, is the initial amount of water in the sample, *W* is the sample weight (water + pulp) at time *t*, and is the dry sample weight. In terms of dimensionless variables the moisture ratio is ) .

In Adrover et al. [21, 22], we have shown that a proper choice of the probe point is a point exhibiting the maximum displacement (shrinkage). In this 1-d problem, since *ψ* is exclusively a function of *ζ*, the probe point is necessarily the surface point located at and evolving in time together with sample thickness .

From equation (10), it is evident that the thickness calibration curve *G* and, more specifically, its derivative actually furnish an experimentally derived shrinkage factor :that depends on the integral quantity and not on the required probe point concentration .

In order to recover from , it is necessary to identify a function relating to , thus obtaining

By considering that explores the entire range of values of *ψ* (from at to for ), once has been derived, we can adopt the same expression for to evaluate the shrinkage velocity at each point in the domain.

The simplest model that can be adopted for the function is a linear model, i.e., that implies on the boundary and consistentlyat each point in the sample domain.

By observing thatit is easy to see that the linear approximation underestimates at short time scales, when , while it overestimates at large time scales, when and (Figure 8 in Appendix). However, the linear approximation , independent of , and the resulting shrinkage function equation (13) represents an acceptable compromise between simplicity and a reasonable physical description of the drying process. Moreover, equation (13) represents a good starting point for a more accurate estimate of the shrinkage factor .

In order to derive a more accurate explicit expression for , we can adopt two different strategies: (i) a fully analytical approach [21] that can be easily applied for foods characterized by linear of quadratic calibration curves and (ii) a shortcut numerical approach [22] that can handle any nonlinear function .

In order to compare the two different approaches, we focus on experimental data of convective hot-air drying of chayote discoid samples characterized by a high initial aspect ratio so that the 1-d model, described above, can be reasonably applied.

#### 4. Chayote Discoid Samples Air-Drying

We analyze experimental data of convective hot-air drying of chayote discoid samples (data from [27]). Fruits were washed and peeled. Cylindrical slices with initial radius mm and initial thickness mm were prepared. The discoid aspect ratio is . Chayote drying (at ) was carried out in a convective dryer with air velocity 2 (m/s). The initial moisture content was g water/100 g product. The initial water volume fraction was [31]. Available experimental data are the moisture ratio vs. time (min) and and the thickness calibration curves vs. at the four temperatures analyzed.

In [21, 22], Adrover et al. have shown that low values of lead to a smoother and flat boundary profile of the sample, while higher values lead to pronounced cusps. By considering the air velocity in the convective dryer and since no information are reported in [27] regarding significant nonuniformities of the sample thickness along the radial direction, we assume in the further analysis a low value for , namely, .

Figures 1(a) and 1(b) show experimental data for vs. and two different approximating functions, both valid in the entire range of temperature analyzed: a second-order polynomial function (Figure 1(a)) and a more accurate fourth-order polynomial function (Figure 1(b)):