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

Authors: Alessandra Adrover; Antonio Brasiello
Journal: International Journal of Chemical Engineering
 (2019)

---

## 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 velocityv, 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 X/X0 vs. rescaled volume V/V0: a fully analytical approach and a shortcut numerical one.


---

## Body

## 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].LetLr be a characteristic length of the food sample, ϕ the point-wise water volume fraction, ϕ0 the uniform initial water volume fraction, and D the water diffusivity at the operating temperature.By introducing the dimensionless space and time variablesτ=tD/Lr2, x⟶x/Lr, V⟶V/Lr3, and S⟶S/Lr2 and the dimensionless differential operators ∇⟶∇/Lr and ∇⋅⟶∇/Lr⋅, the moving boundary model equations for the normalized water volume fraction ψ=ϕ/ϕ0 attain the form:(1)∂ψ∂τ=∇⋅∇ψ−vψ=∇⋅∇ψ1−αψϕ0ψ,x∈Vτ,(2)−∇ψ⋅nxb=Bimψxb−ψeq,xb∈Sτ,(3)dxbdτ=vxb=αψϕ0∇ψxb,xb∈Sτ,(4)v=αψϕ0∇ψ,Bim=hmLrDKeqρairρs,(5)V0=V0,S0=S0,ψx,0=1,where Bim is the mass transfer Biot number, hm is a mass transfer coefficient, ρs is the solid (pulp) density, ρair is the air density at the operating temperature, Keq is the water partition ratio between the gas and the solid phases H=Keqρw/ρsϕ, H being the absolute air humidity (kg water/kg dry air).The core of the model is the shrinkage velocityv, proportional to the water concentration gradient times a proportionality function αψ tuning, at each point of the system, the relationship between water flux and volume reduction. The shrinkage velocity v evaluated at the boundary Sτ 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 αψ.In dealing with a discoid sample (radiusR0, thickness L0) with R0>L0, we choose as a reference length Lr=L0 so that the dimensionless initial domain is ρ,ζ∈0,R0/L0×−1/2,1/2, R0/L0 being the discoid aspect ratio.For high values of the aspect ratioR0/L0, a one-dimensional model can be readily adopted, describing the time evolution of the rescaled water volume fraction ψζ,τ along the axial coordinate (associated with the smallest initial dimension L0) and the time evolution of the dimensionless sample thickness L˜τ=Lτ/L0 (uniform along the radial direction):(6)∂ψ∂τ=∂∂ζ∂ψ∂ζ1−αψϕ0ψ,ζ∈−L˜τ2,L˜τ2,(7)−∂ψ∂ζζ=L˜/2=BimψL˜/2−ψeq,(8)∂ψ∂ζζ=−L˜/2=Bimψ−L˜/2−ψeq,or equivalently∂ψ∂ζζ=0=0,(9)dL˜/2dτ=αψϕ0∂ψ∂ζL˜/2,L˜0=1.In this 1-d approach, both radial shrinkage and water flux from the discoid lateral surface are neglected.


## 3. Estimation of the Shrinkage Factorαψ

If we adopt a 1-d model for describing the drying process of a discoid sample, the shrinkage factorαψ 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 L/L0 vs. the moisture ratio as follows:(10)dX/X0dτ=1ϕ0αψPdL/L0dτ⟶αψP≃1ϕ0dL/L0dX/X0,where ψP=ψxP is the rescaled water volume fraction evaluated, at each time instant t, in a suitable point xP, 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 X/X0=Mw/Mw0=W−Wd/W0−Wd, where Mw is the amount of water at time t, Mw0=ρwV0ϕ0 is the initial amount of water in the sample, W is the sample weight (water + pulp) at time t, and Wd is the dry sample weight. In terms of dimensionless variables the moisture ratio is X/X0=∫VψxdV=∫−L˜/2L˜/2ψζdζ) X/X0.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 ζP=L˜τ/2 and evolving in time together with sample thickness L˜τ.From equation (10), it is evident that the thickness calibration curve G and, more specifically, its derivative G′ actually furnish an experimentally derived shrinkage factor αXX/X0:(11)αX=G′X/X0ϕ0,LL0=GXX0,G′XX0=dGdX/X0,that depends on the integral quantity X/X0 and not on the required probe point concentration ψP.In order to recoverαψP from αXX/X0, it is necessary to identify a function gψP,Bim relating X/X0 to ψP, thus obtaining(12)X/X0=gψP,Bim⟶αψP=αXgψP,Bim=G′gψP,Bimϕ0.By considering thatψP explores the entire range of values of ψ (from ψP=1 at t=0 to ψP=ψeq for t⟶∞), once αψP has been derived, we can adopt the same expression for αψ to evaluate the shrinkage velocity at each point in the domain.The simplest model that can be adopted for thegψP,Bim function is a linear model, i.e., X/X0=ψP that implies αψP=αXψP=G′ψP/ϕ0 on the boundary and consistently(13)αψ=G′ψϕ0,at each point in the sample domain.By observing that(14)XX0=ψavLL0,ψav=1L˜∫−L˜/2L˜/2ψζdζ≥ψP∀t,∀Bim,it is easy to see that the linear approximation X/X0=ψP underestimates X/X0 at short time scales, when L/L0≃1, while it overestimates X/X0 at large time scales, when ψav≃ψP and L/L0≪1 (Figure 8 in Appendix). However, the linear approximation gψP,Bim=ψP, independent of Bim, 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 αψ.In order to derive a more accurate explicit expression forgψP,Bim, 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 GX/X0 and (ii) a shortcut numerical approach [22] that can handle any nonlinear function GX/X0.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 ratioR0/L0>5 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 R0=35 mm and initial thickness L0=6 mm were prepared. The discoid aspect ratio is R0/L0≃5.83. Chayote drying (at T=40,50,60,70°C) was carried out in a convective dryer with air velocity 2 (m/s). The initial moisture content was 93.38±1.03 g water/100 g product. The initial water volume fraction was ϕ0=0.525 [31]. Available experimental data are the moisture ratio X/X0 vs. time (min) and and the thickness calibration curves L/L0 vs. X/X0 at the four temperatures analyzed.In [21, 22], Adrover et al. have shown that low values of Bim 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 Bim, namely, Bim∈1,2.Figures1(a) and 1(b) show experimental data for L/L0 vs. X/X0 and two different approximating functions, both valid in the entire range of temperature analyzed: a second-order polynomial function G2ndX/X0 (Figure 1(a)) and a more accurate fourth-order polynomial function G4thX/X0 (Figure 1(b)):(15)G2ndXX0=1−∑j=12γj1−XX0j,γ1=0.460,γ2=0.378,(16)G4thXX0=1−∑j=14βj1−XX0j,β1=0.411,β2=1.155,β3=−2.131,β4=1.445.Figure 1
Experimental dataL/L0 vs. X/X0 for chayote discoid samples at four different temperatures T=40,50,60,and70°C. Least-square best fit with (a) a second-order polynomial function G2nd (equation (15)) and (b) a fourth-order polynomial function G4th (equation (16)).
(a)
(b)The fourth-order polynomial functionG4thX/X0 better captures the initial linear behaviour (for large values of X/X0) as well as the significant increase of the volume reduction rate for larger time scales (small X/X0 values). Although both functions G4th and G2nd approximate well the global behaviour (R2=0.9811 for G2nd and R2=0.9887 for G4th), the two different approximations lead to very different experimental shrinkage functions αXX/X0=G′X/X0/ϕ0 as shown in Figure 2.Figure 2
Experimental shrinkage functionsαXX/X0=G′X/X0/ϕ0 obtained from G2ndX/X0 (equation (15)) and from G4thX/X0 (equation (16)).We adopt theG2nd quadratic function for applying the fully analytical approach and the more accurate fourth-order function G4th for the shortcut numerical approach. The final goal is to estimate with both approaches the effective water diffusion coefficient D and its dependence on temperature T and to compare the results.It should be observed that there is no physical difference between the shrinkage functionαψ obtained with the fully analytical approach and that obtained from the shortcut approach. The two approaches furnish different approximations of the shrinkage function αψ that is an intrinsic property, a sort of fingerprint, of the material under investigation.In point of fact, during dehydration, the material exhibits structural changes. As a consequence, sample volume can be reduced by an amount that can be greater, less, or equal to the volume of released water. The introduction of the shrinkage functionαψ allows us to take this feature into account because α represents the proportionality factor between the pointwise shrinkage velocity and the local water flux. This proportionality factor changes with the water content in a nonlinear fashion for many materials as can be deduced, in a straightforward way, from the nonlinear behaviour of GX/X0 or, better to say, from the nonconstant behaviour of G′X/X0.For a material exhibiting a linear calibration curveGX/X0=1−ϕ0δ1−X/X0, the corresponding shrinkage function is constant αψ=G′X/X0/ϕ0=δ. If δ=1, the material exhibits ideal shrinkage because volume reduction equals, at each time instant, the volume of released water.However, in most cases,GX/X0 is a nonlinear function of X/X0. In the specific case of the chayote, GX/X0 is an increasing nonlinear function of X/X0, better approximated by a fourth-order polynomial if we want to accurately describe its behaviour for small values of X/X0, that is, on long time scales of the dehydration process. This implies that while sample volume reduction and volume of released water are comparable with each other at short time scales, on the contrary, on longer time scales, when the rescaled moisture content X/X0 is below 40%, a structural collapse of the sample occurs and the volume reduction is two to three times larger than the volume of release water (Figure 2).


## 5. Analytical Approach

The analytical approach can be easily applied when the calibration curveL/L0=GX/X0 is well approximated by a linear or a quadratic function because the first step is to enforce equation (14) that can be rewritten as(17)XX0=ψavLL0=ψavGXX0.By solving equation (17) with respect to X/X0, we obtain an analytic expression for X/X0 as a function of ψav.For a linearG function GX/X0=1−γ1−X/X0, equation (17) reads as(18)XX0=ψav1−γ1−XX0,that can be solved with respect to X/X0, thus obtaining(19)XX0=qψav=ψav1−γ1−ψavγ.Analogously, for a quadraticG function, e.g., equations (15) and (17) read as(20)XX0=ψav1−γ11−XX0−γ21−XX02,that, solved with respect to X/X0, gives the following expression for qψav:(21)XX0=qψav=−b+b2−4ac2a,aψav=γ2ψav,bψav=1−γ1+2γ2ψav,cψav=γ1+γ2−1ψav.In the case of the calibration curve being an n-th order polynomial, then the identification of theqψav function would require the solution of an n-th order equation in X/X0 and this would be extremely difficult if not analytically impossible.Thesecond step in the analytical approach is to relate ψav to ψP by adopting the following model that has been specifically derived for a one-dimensional drying and shrinkage process (Appendix) and that explicitly takes into account the influence of Bim as follows:(22)ψavψP;Bim=1−θBim1−ψP/1−ψeq21+θBim/1−ψeq−1−δBim1−ψP/1−ψeq1/2+δBim1−ψP/1−ψeq2,(23)θBim=1−ψeqBim,(24)δBim=1−ψeq+θBim3−2σBim31−ψeq,where(25)σBim=BimLeq/L02λ02,(26)2λ0tanλ0−BimLeq/L0=0.The symbolλ0 represents the smallest positive root of equation (26), and Leq/L0 is the asymptotic rescaled thickness. The analytic derivation of the equations (22)–(26) relating ψav to ψP and Bim is reported in Appendix and is independent of the shrinkage function αψ.By making use of the analytic expressionX/X0=qψav, equations (19) or (21) (the latter in the case of chayote discoid samples), and the analytical expression for ψavψP;Bim, equations (22)–(26), the more accurate expression for the shrinkage factor is obtained:(27)αψ=1ϕ0G′qψavψ;Bim.This can be used to solve the moving boundary model equations (6)–(9).Figure3(a) shows the behaviour of αψ obtained with the linear approximation X/X0=ψP, i.e., αψ=G′ψ/ϕ0 (equation (13), red dashed line) and with the nonlinear model equation (27) (blue curve) for chayote discoid samples with thickness calibration curve G2ndX/X0 (equation (15)).Figure 3
(a)αψ obtained with the linear approximation X/X0=ψP, i.e., αψ=G′ψ/ϕ0 (equation (13)) and with the nonlinear model equation (27) for chayote discoid samples. (b) Comparison between experimental data for L/L0 vs. X/X0 for chayote discoid samples and the 1-d moving boundary model predictions obtained with the two different models for αψ shown in (a). For both figures, the thickness calibration curve adopted is G2ndX/X0 (equation (15)) and Bim=2.
(a)
(b)Figure3(b) shows the corresponding 1-d moving boundary model predictions for L/L0 vs. X/X0 and the comparison with experimental data. We observe that the model prediction for L/L0 vs. X/X0 with αψ obtained with the linear approximation (red dotted curve) is already satisfactory (R2=0.935), but the improvement in model predictions obtained by adopting the nonlinear model equation (27) is significant (blue curve, R2=0.980) and leads to a more accurate estimate of water diffusivity.


## 6. Shortcut Numerical Approach

In Section5, we have shown that the analytical approach requires two steps in order to evaluate an accurate expression for the gψP,Bim function, directly relating X/X0 to ψP and Bim. The first step is to evaluate the qψav function relating X/X0 to ψav. The second step is to evaluate the function ψavψP,Bim relating ψav to ψP and Bim.The basic idea behind the shortcut numerical approach is to make a direct use of numerical data obtained with the moving boundary model in order to obtain an accurate expression directly forX/X0=gψP,Bim.The shortcut numerical approach stems from the following observation. In Section5, we have shown that the shrinkage function αψ=G′ψ/ϕ0 (equation (13)) obtained with the linear approximation X/X0=ψP gives reasonably good results in terms of model predictions of experimental data for L/L0 vs. X/X0. Figure 4 supports this observation as it shows the good agreement between the experimental thickness calibration data for chayote discoid samples and 1-d moving boundary model predictions (red dotted curve) with αψ given by equation (13) adopting, in this case, the fourth-order more accurate approximating function G4thX/X0 (equation (16)).Figure 4
Comparison between experimental data forL/L0 vs. X/X0 for chayote discoid samples and the 1-d moving boundary model predictions obtained with the two different models for αψ shown in Figure 5(b). The red dotted curve shows model prediction with the α function adopting the linear approximation X/X0=ψP (equation (13)). The blue curve shows model predictions with the α function adopting the nonlinear approximation equations (28) and (29). The thickness calibration curve adopted is G4thX/X0 (equation (16)) and Bim=2.Therefore, thefirst step in the shortcut approach is the numerical integration of the moving boundary model with the shrinkage function equation (13) obtained with the linear approximation X/X0=ψP (and with the more accurate G4thX/X0 function) in order to obtain a numerical accurate estimate of the gψP,Bim function, i.e., X/X0=gψP,Bim (Figure 5(a), dots).Figure 5
(a)X/X0 vs. ψP obtained from the numerical integration of the 1-d moving boundary model with αψ=G′ψ/ϕ0 and Bim=2. Continuous lines represent the best fit curves SψP,Bim (equation (28)) with κ1=0.247, κ2=0.243, and n=0.044. (b) Shrinkage factor αψ obtained with the linear approximation X/X0=ψP (equation (13)) and with the nonlinear approximation (equations (28) and (29)). For both figures, GX/X0=G4thX/X0 (equation (16)).
(a)
(b)Thesecond step is to obtain an analytic expression for the gψP,Bim function by a least-square best fit of numerical data X/X0 vs. ψP by means of the versatile sigmoid function:(28)gψP,Bim≃SψP,Bim=1−κ11−ψP21+κ1−1−κ21−ψPn+κ21−ψP2,where κ1, κ2, and n<2 are best fit values (Figure 5(a), continuous line).The more accurate expression for the shrinkage functionαψ reads as(29)αψ=αXSψ,Bim=G′Sψ,Bimϕ0.Figure5(b) shows the significant difference between the shrinkage function αψ obtained with the linear approximation X/X0=ψP, i.e., αψ=G′ψ/ϕ0 (equation (13), red dashed line) and with the nonlinear model equations (28) and (29) (blue curve) for chayote discoid samples with thickness calibration curve G4thX/X0 (equation (16)).Figure4 shows the corresponding 1-d moving boundary model predictions for L/L0 vs. X/X0 and the comparison with experimental data. Like for the fully analytical approach, we observe that the model prediction for L/L0 vs. X/X0 with αψ obtained with the linear approximation (red dotted curve) is quite satisfactory but the adoption of the nonlinear more refined model for αψ, equations (28) and (29), permits the model to perfectly reproduce the experimental thickness calibration data in the entire range of X/X0 values.


## 7. Estimation of Water Diffusivity

All the above analysis is independent of the water diffusivityD as the thickness calibration curve L/L0 vs. X/X0 is time independent.The moving boundary model is a time-dependent model usually written in terms of dimensionless space and time variables. Specifically, the dimensionless time adopted isτ=tD/L02, L0 being the initial sample thickness, i.e., L0=6 mm.Once the shrinkage factorαψ has been determined, resulting the same for the four temperatures under investigation, since no significant differences have been observed in the thickness calibration curves at the different temperatures, the moving boundary model furnishes a unique time-dependent curve X/X0=fτ as well as a unique time-dependent curve L/L0 vs. τ for the four temperatures.The value of water diffusivityDT at each operating temperature is determined by means of a least-square best fit of the model curve X/X0=fτL02/DT onto the experimental data for time decay of the moisture ratio X/X0 vs. t at the corresponding temperature T, for the same value of Bim, namely, Bim=2, adopted for the estimate of the shrinkage function αψ.We followed two different approaches for estimating the shrinkage functionαψ: the analytical and the shortcut numerical approaches leading to two different expressions for αψ, namely, α2ndψ (equation (27)) for the fully analytical approach and α4thψ (equation (29)) for the shortcut numerical approach, and therefore to two different time-dependent curves X/X0=fτ, f2ndτ and f4thτ, respectively.Figures6(a) and 6(b) show the comparison between experimental data for X/X0 vs. (t) (min) for chayote discoid samples and the two model curves.Figure 6
Experimental dataX/X0 vs. (t) (min) for chayote discoid samples at four different temperatures T=40,50,60,and70°C. Least-square best fit with (a) f2ndτL02/D2ndT from the analytical approach and (b) f4thτL02/D4thT from the shortcut numerical approach. L0=6 mm. Bim=2.
(a)
(b)f
2
nd
τ
L
0
2
/
D
2
nd
T and f4thτL02/D4thT with the best fit diffusivity values are shown in Figure 7. Specifically, from the analytical approach, one obtains D2nd40°C=6.92×10−10 (m2/s), D2nd50°C=9.27×10−10 (m2/s), D2nd60°C=1.09×10−9 (m2/s), and D2nd70°C=1.35×10−9 (m2/s), with an Arrhenius correlation function DT=D0exp−E/RT with D02nd=1.318×10−6 (m2/s) and E/R=2358.38 (K). From the shortcut numerical approach, one obtains D4th40°C=6.71×10−10 (m2/s), D4th50°C=9×10−10 (m2/s), D4th60°C=1.06×10−9 (m2/s), and D4th70°C=1.31×10−9 (m2/s) with an Arrhenius correlation function with D04th=1.28×10−6 (m2/s) and the same activation energy E/R=2358.38 (K).Figure 7
Estimated diffusivitiesD2ndT from the analytical approach and D4thT from the shortcut numerical approach. Continuous line represents the Arrhenius correlation function DT=D0exp−E/RT with D04th=1.28×10−6 (m2/s) and E/R=2358.38 (K).The agreement between model curves and experimental data for the time decay of the moisture ratio is extremely good for both approaches, and differences between estimated diffusivities are below 3% for all temperatures analyzed.Diffusivities estimated from both approaches are comparable with literature data for other vegetable products such as carrots (2.58×10−10/1.72×10−9 (m2/s), 60–90°C), mango (2.61×10−10/1.30×10−9 (m2/s), 40–70°C), and potatoes (3.55×10−10/1.92×10−9 (m2/s), 40–85°C) in which shrinkage has also been considered [32–34].This result confirms that both approaches are robust and reliable and can be indifferently adopted for predicting the time decay of the moisture ratio as well as sample shrinkage during the entire drying process.


## 8. Conclusions

In this paper, the comparison between the two approaches developed in [21, 22] for mathematical modeling of food drying with shrinkage is carried out.The developed mathematical model consists of a mass balance equation for water volume fraction evolution, coupled to an equation for the movement of the boundary. The relationship between water concentration and boundary movement is introduced into the system through the dependence of a shrinkage velocity on the concentration gradient times a shrinkage factorα. This represents the constitutive equation of the material.The two developed approaches (fully analytical and shortcut), precisely dealing with the determination of the shrinkage factorα, were analyzed here and applied to the literature data of chayote discoid sample drying.We showed that, for the case discussed here, both approaches are able to provide accurate predictions of the experimental data for moisture fraction and thickness reduction, as well as a good estimation of the physical quantities involved (first of all, the water diffusion coefficient) with almost the same computational efforts.However, this last consideration is true only in this case. The adoption of a one-dimensional simplified model is, in fact, due to conditions of the samples (the symmetrical shape and the value of the aspect ratio) which, as an instance, are in general not found in industrial applications.For such complex cases, although it is reasonable to assume no differences in terms of prediction capabilities, the computational efforts connected to the fully analytical approach could be very huge compared to the shortcut one or, worst, it could not be pursued at all. Hence, the adoption of the shortcut approach, having demonstrated its efficiency here, can be an attractive opportunity (the only one in the worst cases) to have an accurate description of the evolution of the system through a mathematical model based on balance equations.



---

*Source: 3926897-2019-05-16.xml*