An assessment on modeling drying processes: Equilibrium multiphase model and the spatial reaction engineering approach (S-REA)

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An assessment on modeling drying processes: Equilibrium multiphase model and the spatial reaction engineering approach (S-REA) Aditya Putranto a,b , Xiao Dong Chen c,∗ a

Department of Chemical Engineering, Monash University, Clayton, Victoria, Australia Department of Chemical Engineering, Parahyangan Catholic University, Jalan Ciumbuleuit 94, Bandung, Indonesia c School of Chemical and Environmental Engineering, College of Chemistry, Chemical Engineering and Material Science, Soochow University, Jiangsu Province, People’s Republic of China b

a b s t r a c t An accurate drying model is useful in assisting process design and product design when drying is involved. Generally, drying models can be classified into equilibrium and non-equilibrium multiphase models. Both models employ a similar set of equations of conservation of heat and mass transfer. For the equilibrium ones, the moisture sorption isotherm relationship is used to link the concentration of water vapor inside the pore space and moisture content in and on solid matrix. On the other hand, the non-equilibrium ones have an expression of local evaporation/condensation rate to relate these relationships. Here, both models are compared and assessed to model several cases of convective and intermittent drying. The reaction engineering approach (REA) is used to describe the local evaporation/condensation rate in non-equlibrium model (i.e. the spatial reaction engineering approach (S-REA)). The results of modeling indicate that for the cases investigated, the equilibrium model does not work well. The S-REA can accurately model both convective and intermittent drying cases. For the interest of probing the dynamics of drying process, it is suggested that a non-equilibrium model is more appropriate. © 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Keywords: Drying; Equilibrium; Non-equilibrium; Multiphase; Model; Spatial reaction engineering approach (S-REA)

1.

Introduction

Drying is a process of water removal involving simultaneous heat and mass transfer process. Study of drying is important since drying is an energy-intensive process and affects the product quality. Several drying schemes, controlled drying operations and process intensification of drying need to be implemented to minimize energy consumption during drying and maintain the product quality of the materials being dried (Chou et al., 2000; Chua et al., 2003; Allanic et al., 2009). A reliable drying model is useful to assist in process design and maintaining product quality of drying. For process design, a reliable drying model can be used to design innovative dryers, evaluate the performance of existing dryer facilities,

explore various drying schemes and conditions in order to minimize the energy consumption during drying. For product quality, the effective drying model can be applied to predict the moisture content and temperature of the product so that the quality changes can be monitored. The drying models can be classified into empirical and mechanistic models. The empirical ones offer advantages of simplicity in mathematical formulation and solution but they cannot capture the physics of drying. Unlike the first ones, the mechanistic models, commonly implementing diffusion-based model, can reasonably well represent the physics of drying process. The models have been used widely to simulate the drying process with various degrees of success (Pakowski and Adamski, 2007; Mariani

∗

Corresponding author. Tel.: +86 18906053300. E-mail address: [email protected] (X.D. Chen). Received 21 November 2013; Received in revised form 30 August 2014; Accepted 8 October 2014 http://dx.doi.org/10.1016/j.cherd.2014.10.007 0263-8762/© 2014 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.

Please cite this article in press as: Putranto, A., Chen, X.D., An assessment on modeling drying processes: Equilibrium multiphase model and the spatial reaction engineering approach (S-REA). Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.10.007

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et al., 2008; Roberts et al., 2008). The diffusion-based models lump different transport processes including capillary flow, liquid diffusion, vapor diffusion and local evaporation/condensation into a single parameter termed as effective diffusivity (Nguyen et al., 2006; Batista et al., 2007; Corzo et al., 2008). The variability of the effective diffusivity is dependent on the drying condition, material structure and composition, and sometimes sample sizes (Chen, 2007). The diffusion-based models which implemented the effective diffusivity are not sufficient to describe the concentration of water vapor and local evaporation/condensation rate during drying. In order to obtain more comprehensive understanding of transport phenomena of drying process, the equilibrium and non-equilibrium multiphase drying approach can be applied (Zhang and Datta, 2004; Datta, 2007; Chen, 2007). In the equilibrium approach, the moisture content inside the samples is assumed to be in equilibrium with the concentration of water vapor at any time so that the water isotherm can be used to relate these relationships. The equilibrium approach does not require the expression of local evaporation rate. On the other hand, the non-equilibrium multiphase model requires a local evaporation rate to link the moisture content and concentration of water vapor inside the pore (Zhang and Datta, 2004; Datta, 2007). The local evaporation rate needs to be expressed as a function of difference between surface water vapor concentration and water vapor concentration inside the pore (Chen, 2007; Bazer-Bachi et al., 2011). The reaction engineering approach (REA) was initialy developed by Professor X.D. Chen in 1996 and has been used to model accurately various challenging drying cases including thin layer drying, drying of materials with thickness of several centimeters, infrared-heating drying, drying under time-varying humidity and drying air temperature, baking, roasting and heat treatment of wood (Chen and Xie, 1997; Chen and Lin, 2005; Putranto et al., 2010a,b, 2011a,b,c). The REA model in which the REA is used to model the global drying rate is then labeled as the lumped reaction engineering approach (L-REA) (Putranto and Chen, 2013). Because of the accuracy of the REA to model the global drying rate, the REA is applied to model the local evaporation rate. Coupling between the REA and a system of equations of mass and energy conservation yields a non-equilibrium multiphase model, called the spatial reaction engineering approach (S-REA) (Putranto and Chen, 2013). Although the use of equilibrium and non-equilibrium multiphase model have been suggested, so far there has been no direct comparison and assessment of the applicability of both models to model drying processes. The comparison is important to show the model which can represent drying processes more accurately. This study is aimed to compare and assess the applicability and effectiveness of the equilibrium multiphase model and the spatial reaction engineering approach (S-REA), a non-equilibrium multiphase drying model, to describe drying processes. The accuracy of both models to describe the convective and intermittent drying is assessed here. The outline of the paper is as follows: firstly, the reaction engineering approach (REA) is briefly reviewed followed by review of material and methods. The equilibrium multiphase model and the S-REA are explained subsequently. A relevant discussion of the applicability of both models to represent the convective and intermittent drying is provided.

2. Brief review of the reaction engineering approach (REA) The general REA is an application of chemical reaction engineering principles to model drying kinetics in which evaporation is modeled as zero-order kinetics with activation energy while condensation is described as first order wetting reaction with respect to drying air solvent vapor concentration without activation energy (Chen and Xie, 1997; Chen, 2008). This approach has been firstly employed to express the overall drying rate for the entire object being dried – a lumped approach. A summary of the developments of the lumped approach of REA for various challenging drying cases was presented in previous publications (Chen and Xie, 1997; Chen, 2008; Putranto et al., 2010a,b, 2011a,b,c). By using the reaction engineering approach (REA), the drying rate of a material is expressed as (Chen and Xie, 2007; Chen, 2008; Putranto et al., 2010a,b, 2011a,b,c):



ms

dX = −hm A exp dt

 −E  v

RT

Cv,sat (T) − Cv,b

 (1)

where ms is the dried mass of thin layer material (kg), X is the average moisture content on dry basis (kg kg−1 ), t is time (s), Cv,sat is the saturated water vapor concentration (kg m−3 ), Cv,b is the water vapor concentration in the drying medium (kg m−3 ), hm is the mass transfer coefficient (m s−1 ), A is surface area of the material (m2 ), T is the sample temperature (K) and Ev is the activation energy (J mol−1 ). Eq. (1) is the core of the lumped reaction engineering approach, i.e. the L-REA. It must be noted that the L-REA does not assume uniform moisture content. When material is ‘thermally’ thin, the surface temperature is considered to be the same as the sample temperature (Chen and Peng, 2005; Patel and Chen, 2008), i.e. Ts ≈ T. The activation energy is calculated by re-arranging Eq. (1) as follows:

 Ev = −RTs ln

−ms (dX/dt)(1/hm A) + Cv,b Cv,sat

 (2)

The dependence of activation energy on average moisture content on a dry basis (X) can be normalized as: Ev = f (X − Xb ) Ev,b

(3)

where f is a function of water content difference. Ev,b is the ‘equilibrium’ activation energy representing the maximum Ev determined by the relative humidity and temperature of the drying air: Ev,b = −RTb ln (RHb )

(4)

RHb is the relative humidity of drying air, Xb is the equilibrium moisture content under the condition of the drying air (kg kg−1 ) and Tb is the drying air temperature (K). The relative activation energy generated from one accurate drying run can be used to project other drying runs provided the same material with the similar initial moisture content since it would collapse the similar profiles of relative activation energy (Chen and Lin, 2005; Chen, 2008).

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Table 1 – Experimental conditions of convective drying of mango tissues (Vaquiro et al., 2009). Number

Air velocity (m s−1 )

1 2 3

Air temperature (◦ C)

4 4 4

Air humidity (kg H2 O kg dry air−1 )

45 55 65

0.0134 0.0134 0.0134

Table 2 – Scheme of intermittent drying of mango tissues (Vaquiro et al., 2009). Drying air temperature (◦ C) 65

3.

Period of first heating (s)

Period of resting (at 27 ◦ C ± 1.6) (s)

7800

Materials and methods

The experimental data of drying of mango tissues are derived from the previous study (Vaquiro et al., 2009). The data sets are used since these provide comprehensive data of both convective and intermttent drying. For better understanding of the predictions, the necessary experimental details are summarized and reviewed in this section. The experimental setup is shown in Fig. 1. The samples of mango tissues were made cubes with initial side-length of 2.5 ± 0.1 cm and weight of 14.9 ± 1 g. The initial moisture content was 9.3 ± 2.2 kg kg−1 and the initial temperature was 10.8 ± 1.8 ◦ C. The laboratory drier was described in Sanjuan et al. (2004). The drying air with the velocity of 4 m s−1 was introduced to the drying chamber shown in Fig. 1. The air was heated to the drying air temperature i.e. 45, 55 and 65 ◦ C using heating elements. The drying air velocity and temperatures were controlled during experiments PID control algorithms (Sanjuan et al., 2004). The experimental setting of convective drying is summarized in Table 1. During drying, the cube sample was contained in a sample holder and suspended in a weight balance. The weight and center sample temperatures were automatically measured. AOAC method 934.06 was used to determine the moisture content (Sanjuan et al., 2004). For intermittent drying, the experimental setup was similar to the convective drying. The intermittency was created by schemes of heating and resting period listed in Table 2. During resting period, the samples stay at environment with ambient temperature of 27 ± 1.6 ◦ C and relative humidity of 60%. The first heating, resting and second heating period was allowed for 7800, 10,800 and 16,200 s, respectively (Vaquiro et al., 2009).

10,800

16,200

4. Mathematical modeling using equilibrium multiphase model 4.1. Modeling of convective drying using equilibrium multiphase model In this section, the mathematical model of convective drying of mango tissues using equilibrium model is presented. The mass balance of water in the liquid phase (liquid water) is written as (Chen, 2007; Kar and Chen, 2010, 2011): ∂(Cs X) ∂ = ∂t ∂x +

 Dw

∂ ∂z

∂(Cs X) ∂x

 Dw



∂ ∂y

+

∂(Cs X) ∂z

 Dw

∂(Cs X) ∂y



− I˙

(5)

where X is the concentration of liquid water (kg H2 O kg dry solids−1 ), Dw is the capillary water diffusivity (m2 s−1 ), Cs is the solids concentration (kg dry solids m−3 ) which can change if the structure is shrinking, I˙ is the evaporation or condensation rate (kg H2 O m−3 s−1 ) and I˙ is >0 when evaporation occurs locally. The mass balance of water vapor is expressed as (Chen, 2007; Kar and Chen, 2010, 2011): ∂ ∂Cv = ∂t ∂x

 Dv

∂Cv ∂x



+

∂ ∂y

 Dv

∂Cv ∂y



+

∂ ∂z

 Dv

∂Cv ∂z



+ I˙

(6)

Dv is the effective water vapor diffusivity (m2 s−1 ) and Cv is the concentration of water vapor inside the pore (kg m−3 ). The energy balance can be expressed as (Chen, 2007; Kar and Chen, 2010, 2011): Cp

Fig. 1 – Drying equipment of intermittent drying of mango tissues (Sanjuan et al., 2004). 1, frame; 2, fan; 3, anemometer; 4, heating element; 5, pneumatic valve; 6, temperature control and measurement; 7, sample holder; 8, balance; 9, filter; 10, air compressor.

Periodofsecondheating(s)

∂ ∂T = ∂t ∂x

 ∂T  k

∂x

+

∂ ∂y

 ∂T  k

∂y

+

∂ ∂z

 ∂T  k − I˙Hv ∂z

(7)

where T is the sample temperature (K), k is the thermal conductivity (W m−1 K−1 ), Hv is the vaporization heat of water (J kg−1 ),  is the sample density (kg m−3 ) and Cp is the sample heat capacity (J kg−1 K−1 ). The sample dried was cube shape. It is dried uniformly from all directions (x, y and z directions) since it was suspended on a weight balance (contained inside a sample holder) and exposed to uniform drying air in all directions (Sanjuan et al., 2004; Vaquiro et al., 2009) so the mass balance of water in liquid phase can be simplified into (Incropera and de Witt, 2002; Van der Sman, 2003): ∂(Cs X) ∂ =3 ∂t ∂x

 Dw

∂(Cs X) ∂x

− I˙

(8)

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while the mass balance of water in vapor phase can be expressed as (Incropera and deWitt, 2002; Van der Sman, 2003): ∂Cv ∂ =3 ∂t ∂x



∂Cv ∂x

Dv



+ I˙

Table 3 – Parameters used in both equilibrium and S-REA models. Parameters

(9)

Density () Specific heat (Cp ) Thermal conductivity (k) Heat transfer coefficient (h) Mass transfer coefficient (hm ) Porosity (ε) Solid concentration (Cs ) Tortuosity () Liquid diffusivity (Dw ) Vapor diffusivity (Dv ) Shrinkage coefficient (Sb )

In addition, the energy balance can be represented as (Incropera and deWitt, 2002; Van der Sman, 2003): Cp

∂T ∂ =3 ∂t ∂x

 ∂T  k − I˙Hv

(10)

∂x

Eqs. (8) and (9) can be combined to yield: ∂(Cs X) ∂ ∂Cv + =3 ∂t ∂t ∂x

 Dw

∂(Cs X) ∂x

+3

∂ ∂x

 Dv

∂Cv ∂x

 (11)

Value Eq. (B5) this paper Eq. (B9) this paper Eq. (B6) this paper Eq. (24) Vaquiro et al. (2009) Eq. (25) Vaquiro et al. (2009) Eq. (A5) this paper Eq. (A4) this paper Eq. (A3) this paper Eq. (A6) this paper Eq. (A1) this paper Eq. (B13) this paper

In addition, Eqs. (8) and (10) can be combined to result in: ∂ ∂T =3 Cp ∂t ∂x

 ∂T   ∂  k

∂x

− 3

∂x

.

∂(Cs X) Dw ∂x





−

k

∂(Cs X) Hv ∂t

dT = h(Tb − T) dx (convective boundary for heat transfer)

(12) For the equilibrium multiphase approach, the concentration of water vapor is expressed through equilibrium relationships as a function of local moisture content and temperature (Zhang and Datta, 2004; Curcio, 2010). The moisture sorption isotherm of the mango tissues can be represented as (Basu et al., 2006; Vaquiro et al., 2009): X = 0.112

CG KG RHs (1 − KG RHs )(1 + (CG − 1)KG RHs )

(14)

For yielding the spatial profiles of moisture content and temperature, the mass and energy balances shown in Eqs. (11) and (12) are solved simultaneously in conjunction with the available moisture sorption isotherm indicated in Eqs. (13)–(15) as well as the initial and boundary conditions shown in Eqs. (17)–(20). The set of partial differential equations are solved using method of lines to result in a set of simultaneous ordinary differential equations by firstly discretizing the spatial derivatives into increments (Constantinides, 1999; Chapra, 2006). Solver ode23s available in Matlab® is used to solve the set of ordinary differential equations simultaneously. For the modeling, the transport coefficients and parameters shown and summarized in Appendix A and Table 3 are used. The physical properties of mango tissues are shown in Appendix B.

(15)

4.2. Modeling of intermittent drying using equilibrium multiphase model

(13)

where X is the moisture content (kg kg−1 ), RHs is the internal solid-surface relative humidity, CG and KG are coefficients which can be represented as (Basu et al., 2006; Marinos-Kouris and Maroulis, 2007; Garcia-Perez et al., 2008): CG = CG0 exp KG = KG0 exp

 7005  RT

 H M − 56, 596  v w RT

where R = 8.314 J mol−1 K−1 , RH is the relative humidity of air, T is the temperature (K), Mw is the molecular weight of water (kg kmol−1 ) and Hv is the vaporization enthalpy of water (kJ kg−1 ). The water vapor concentration inside the pores can be expressed as: Cv = Cv,sat RHs

(16)

The initial and boundary conditions for mass and heat balances expressed in Eqs. (11) and (12), respectively are: t = 0,

X = X0 ,

T = T0

(initial condition, uniform initial concentrations and temperature) (17)

x = 0,

dX = 0, dx

dT = 0 (symmetry boundary) dx

x = L,

dX − Cs Dw = hm dx

C

v,s

ε

− v,b

(18)

For modeling intermittent drying using equilibrium multiphase model, the mass and energy balances shown in Eqs. (11) and (12) in conjunction with the moisture sorption isotherm shown in Eqs. (13)–(15), used for modeling convective drying, are still implemented. The initial and boundary conditions shown in Eqs. (17)–(20) are used but they implement the corresponding drying settings during intermittent drying. Here, the intermittency of temperature is used so that Eq. (20) needs to implement the corresponding drying air temperature in each drying period. The method of solution is similar to that for modeling of convective drying using the equilibrium multiphase model mentioned above. The transport coefficients and parameters used for the convective drying are also applied here.

5. Mathematical modeling using spatial reaction engineering approach (S-REA) 5.1.



(convective boundary for water transfer)

(20)

(19)

Modeling of convective drying using S-REA

For modeling using the S-REA, the mass balances of liquid water and water vapor are shown in Eqs. (8) and (9),

Please cite this article in press as: Putranto, A., Chen, X.D., An assessment on modeling drying processes: Equilibrium multiphase model and the spatial reaction engineering approach (S-REA). Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.10.007

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respectively while the energy balance is indicated in Eq. (10). The initial and boundary conditions for Eqs. (8)–(10) are: t = 0,

X = X0 ,

x = 0,

dX = 0, dx

x = L,

− Cs Dw

−Dv k

dCv = hm εv dx

Cv = Cv0 ,

T = T0

dCv = 0, dx dX = hm εw dx

C

v,s

ε

− v,b

(initial condition, uniform initial concentrations and temperature)

dT = 0 (symmetry boundary) dx

C

v,s

ε

− v,b

(21)

(22)

 (convective boundary for liquid water transfer)

(23)

 (convective boundary for water vapor transfer)

(24)

dT = h(Tb − T) (convective boundary for heat transfer) dx (25)

The local evaporation rate within the solid structure (I˙), shown in Eqs. (8)–(10), is described as: I˙ = hm,in Ain (Cv,s − Cv )

(26)

where hm,in is the internal mass transfer coefficient (m s−1 ) and Ain is the total internal surface area per unit volume available for phase change (m2 m−3 ). By implementing the REA, internal-surface water vapor concentration can be written as (Kar and Chen, 2010, 2011): Cv,s = exp

 −E  v

RT

(27)

Cv,sat

where Cv,s is the internal-solid-surface water vapor concentration (kg m−3 ) and Cv,sat is the internal-saturated water vapor concentration (kg m−3 ). Therefore, Eq. (29) can be expressed as (Kar and Chen, 2010, 2011):



I˙ = hm,in Ain exp

 −E  v

RT

Cv,sat − Cv

 (28)

The relative activation energy (Ev /Ev,b ) is generated from one accurate drying run i.e. convective drying of mango tissues at drying air temperature of 55 ◦ C. The activation energy is calculated using Eq. (2) and divided by the equilibrium activation energy shown in Eq. (4). The relative activation energy, indicated in Eq. (3), is then correlated against the difference of moisture content (X − Xb ). Solver available in Microsoft Excel® is used to give the relationship of relative activation energy and the moisture content difference. The relative activation energy of convective drying of mango tissues at drying air temperature of 55 ◦ C is written as:

replaced by the local moisture content (X) since the ‘local’ relative activation energy needs to be implemented to describe the local evaporation rate.hm,in (shown in Eq. (28)) represents the inverse of resistance of transfer of liquid water on the pore surface to the vapor on the void space inside the samples. The value of hm,in grow from small value to the value of Dv /rp (rp = cell radius) (when constriction factor is 1) (Kar, 2008; Kar and Chen, 2010, 2011). In this study, hm,in of 0.01 m s−1 is used. The value of hm,in of 0.01 m s−1 is also in the order of Dv /rp thus it is a fundamental value (Kar and Chen, 2010, 2011). The internal surface area is evaluated according to the procedures outlined in Kar and Chen (2010, 2011) based on the area of single cells inside the samples or particles and number of cells per unit volume inside the samples explained in Appendix C. In order to yield the spatial profiles of moisture content, water vapor concentration and temperature, the mass balances of liquid water and water vapor as well as the energy balance shown in Eqs. (11), (12) and (13), respectively are solved in conjunction with the local evaporation rate using the REA indicated in Eq. (28). Similar to modeling of the convective drying using equilibrium multiphase model, method of lines (Constantinides, 1999; Chapra, 2006) is used here to solve the set of partial differential equations. Both equilibrium and SREA models implemented the same transport coefficients and parameters as summarized in Table 3.

Ev ¯ − Xb )3 + 9.74 × 10−3 (X ¯ − Xb )2 = −9.92 × 10−4 (X Ev,b ¯ − Xb ) + 1.053 − 0.101(X

(29)

The good agreement between the fitted and experimental activation energy is shown in Fig. 2 and confirmed by R2 of 0.999. The relative activation energy shown in Eq. (29) has been used to model the convective and intermittent drying of mango tissues using the lumped reaction engineering approach (L-REA) well (Putranto et al., 2011a,b). In S-REA, the REA is basically extrapolated to model the local evaporation rate. For this purpose, Eq. (29) can still be used to describe the ¯ is local evaporation rate but the average moisture content (X)

Fig. 2 – The relative activation energy (Ev /Ev,b ) of convective drying of mango tissues at air velocity of 4 m s−1 , drying air temperature of 55 ◦ C, and air humidity of 0.0134 kg H2 O/kg dry air (Putranto et al., 2011a,b).

Please cite this article in press as: Putranto, A., Chen, X.D., An assessment on modeling drying processes: Equilibrium multiphase model and the spatial reaction engineering approach (S-REA). Chem. Eng. Res. Des. (2014), http://dx.doi.org/10.1016/j.cherd.2014.10.007

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Fig. 4 – Profiles of center temperature during convective drying at drying air temperature of 45, 55 and 65 ◦ C using equilibrium multiphase drying model. Fig. 3 – Profiles of average moisture content during convective drying at drying air temperature of 45, 55 and 65 ◦ C using equilibrium multiphase drying model.

5.2.

Modeling of intermittent drying using S-REA

For modeling intermittent drying using the S-REA, the mass and heat balances represented in Eqs. (8)–(10) in conjunction with the local evaporation rate using the REA shown in Eq. (28) is still implemented. In order to incorporate the effect of intermittency, the equilibrium activation energy shown in Eq. (4) needs to be calculated according the corresponding drying air humidity and temperature in each drying period. The equilibrium activation energy is then combined with the relative activation energy shown in Eq. (29). Similarly, the boundary conditions of the mass and energy balances need to incorporate the time-varying drying air settings. Since the intermittent drying under time-varying drying air temperature is employed here, the corresponding drying air temperature in each drying period is used in Eq. (25). The method of solution of lines (Constantinides, 1999; Chapra, 2006), as mentioned above, is used for solving the set of partial differential equations.

6.

Results and discussion

The equilibrium multiphase drying model and spatial reaction engineering approach (S-REA) are used to model the convective and intermittent drying. Figs. 3 and 4 indicate the results of modeling of convective drying using the equilibrium multiphase approach. For convective drying at all observed drying air temperatures, as shown in Fig. 3, the equilibrium one overestimates the drying rate. The time scales associated with achieving equilibrium are much larger. Consequently, the moisture loss predicted by the equilibrium model is highly overestimated. Fig. 4 shows the profiles of center temperature at drying air temperature of 45, 55 and 65 ◦ C, modeled using the equilibrium model. For convective drying of mango tissues at 45, 55 and 65 ◦ C, the equilibrium multiphase approach results in underestimations of the temperature profiles. This is reasonable since the equilibrium model predicts higher drying rate than the actual one which requires higher amount of heat for evaporation.

Figs. 5 and 6 show the results of modeling using the S-REA. The profiles of average moisture content of convective drying at drying air temperature of 45, 55 and 65 ◦ C are shown in Fig. 5. The results of modeling using the S-REA match well with the experimental data. The good agreement is also indicated by R2 of 0.998, 0.998 and 0.996 for convective drying at drying air temperature of 45, 55 and 65 ◦ C, respectively. For the temperature profiles, the S-REA describes the center temperature profiles accurately as shown in Fig. 6. The good agreement is also shown by R2 of 0.998, 0.985 and 0.994, respectively. This may indicate that the moisture content inside solid matrix is not under instant equilibrium condition with the concentration of water vapor in the accessible pore. The transfer of moisture to water vapor i.e. the evaporation process seems to be better described by rate-based approach. Here, the REA describes well the local evaporation process. The spatial profiles of moisture content during convective drying at drying air temperature of 55 ◦ C modeled using the equilibrium model and S-REA are shown in Fig. 7. During drying, the moisture content profiles predicted using the equilibrium one is lower than those modeled by the S-REA which is because of overestimation of the drying rate by the equilibrium model as mentioned above. Both models predict that the moisture content at the outer part of samples is lower than that at the core of samples which indicate that the moisture migrates outwards during drying. The gradient of moisture content inside the samples is relatively high at the beginning of drying and it decreases during drying. However, the equilibrium model results in slightly higher gradient of moisture content inside the samples than the S-REA. This could be because the equilibrium model requires higher supply of moisture to the vapor phase which is then transferred outwards to the drying medium. Both models show that the gradient of moisture content inside the samples decreases during drying and at the end of drying process, no noticeable gradient is observed since final equilibrium condition is approached. Fig. 8 represents the spatial profiles of concentration of water vapor during convective drying predicted by the equilibrium model and S-REA at drying air temperature of 55 ◦ C. Both models estimate that the profiles increase during drying. The gradient of concentration of water vapor inside the samples is relatively low and this decreases toward the end of drying.

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Fig. 5 – Profiles of average moisture content during convective drying at drying air temperature of 45, 55 and 65 ◦ C using spatial reaction engineering approach (S-REA).

Fig. 6 – Profiles of center temperature during convective drying at drying air temperature of 45, 55 and 65 ◦ C using spatial reaction engineering approach (S-REA).

The equilibrium model yields higher profiles of concentration of water vapor than the S-REA. This is in agreement with the lower profiles of moisture content during drying as shown in Fig. 7. The equilibrium one predicts the profiles at the outer part of the samples are higher than that at the inner part which could be because of higher temperature at the outer part. On the other hand, the S-REA predicts the maximum concentration of water vapor occurs at particular position inside the samples. The local evaporation rate at the core of the samples is relatively low since the porosity is low. At the outer part of the samples, the local evaporation rate seems to be enhanced due to higher temperature and higher porosity. However, this is balanced by the higher diffusive flux at the outer part of the samples. The combination of this seems to result in maximum concentration of water vapor at particular position inside the samples. Fig. 9 indicates the spatial profiles of temperature at drying air temperature of 55 ◦ C modeled using the equilibrium model and S-REA. Both models predict that the temperature increases during drying to approach the drying air

Fig. 7 – Spatial profiles of moisture content during convective drying at drying air temperature of 55 ◦ C predicted using equilibrium model and S-REA. (Red *) equilibrium multiphase model t = 1000s; (blue *) S-REA t = 1000 s; (red -) equilibrium multiphase model t = 3000 s; (blue -) S-REA t = 3000 s; (red –), equilibrium multiphase model t = 5000 s; (blue –) S-REA t = 5000 s; (red ) equilibrium multiphase model t = 10,000 s; (blue ) S-REA t = 10,000 s; (red ) equilibrium multiphase model t = 20,000 s; (blue ) S-REA t = 20,000 s; (red ) equilibrium multiphase model t = 30,000 s; (blue ) S-REA t = 30,000 s). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

temperature. However, the profiles of temperature predicted by the equilibrium model are lower than those modeled by the S-REA. This may be due to overestimation of drying rate by the equilibrium one as mentioned above. Both models show that the temperature at the outer part of the samples is higher than that of the inner part. This could be because the samples receive heat from the drying air, which is used for water evaporation and any left is penetrated inwards by conduction to raise the samples.

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Fig. 8 – Spatial profiles of concentration of water vapor during convective drying at drying air temperature of 55 ◦ C using equilibrium model and S-REA. (Red *) equilibrium multiphase model t = 1000 s; (blue *) S-REA t = 1000 s; (red -) equilibrium multiphase model t = 3000 s; (blue -) S-REA t = 3000 s; (red –) equilibrium multiphase model t = 5000 s; (blue –) S-REA t = 5000 s; (red ) equilibrium multiphase model t = 10,000 s; (blue ) S-REA t = 10,000 s; (red ) equilibrium multiphase model t = 20,000 s; (blue ) S-REA t = 20,000 s; (red ) equilibrium multiphase model t = 30,000 s; (blue ) S-REA t = 30,000 s). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 10 – Profiles of average moisture content during intermittent drying at drying air temperature of 65 ◦ C using equilibrium multiphase drying model.

Fig. 11 – Profiles of center temperature during intermittent drying at drying air temperature 65 ◦ C using equilibrium multiphase drying model.

Fig. 9 – Spatial profiles of temperature during convective drying at drying air temperature of 55 ◦ C using equilibrium model and S-REA. (Red *) equilibrium multiphase model t = 1000 s; (blue *) S-REA t = 1000 s; (red -) equilibrium multiphase model t = 3000 s; (blue -) S-REA t = 3000 s; (red –) equilibrium multiphase model t = 5000 s; (blue –) S-REA t = 5000 s; (red ) equilibrium multiphase model t = 10,000 s; (blue ) S-REA t = 10,000 s; (red ) equilibrium multiphase model t = 20,000 s; (blue ) S-REA t = 20,000 s; (red ) equilibrium multiphase model t = 30,000 s; (blue ) S-REA t = 30,000 s). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Turn to intermittent drying, the results of modeling using the equilibrium model and S-REA are shown in Figs. 10–16. Figs. 10 and 11 show the average moisture content and center temperature profiles during drying predicted using the equilibrium model. As shown in Fig. 10, the model can only match with the experimental data at the beginning of drying. This is probably because the samples are still saturated with liquid water near the start. The equilibrium model can predict the trend of intermittency but the model results in lower moisture content than the experimental data, similar to the predictions of convective drying as shown in Fig. 3. The overestimation of the drying rate seems to be because under instant equilibrium condition, higher amount of moisture needs to be extracted from the solid matrix. Fig. 11 indicates the center temperature profiles during intermittent drying predicted using the equilibrium one. The model underestimates the temperature profiles during first heating and resting period which could be because higher amount of heat required for vaporization of water due to overestimation of drying rate. Similar profiles are observed for convective drying as shown in Fig. 4.

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Fig. 12 – Profiles of average moisture content during intermittent drying at drying air temperature of 65 ◦ C using spatial reaction engineering approach (S-REA). The results of modeling of intermittent drying using the S-REA are represented in Figs. 12 and 13. The S-REA models the average moisture content accurately during intermittent drying as shown in Fig. 12 (R2 of 0.998). The good agreement between the predicted and experimental data of temperature is indicated in Fig. 13 (R2 of 0.997). It can be said that the S-REA models the profiles of moisture content and temperature of the intermittent drying better than the equilibrium model. This could be because during intermittent drying, the moisture content inside solid matrix is not under instant equilibrium condition with the concentration of water vapor inside the accessible pore. Therefore, similar to the convective drying, the evaporation process during intermittent drying is better described using rate-based approach i.e. the REA here. Fig. 14 shows the spatial profiles of moisture content during intermittent drying modeled using the equilibrium model and the S-REA. Similar to convective drying, the equilibrium one yields lower profiles than the S-REA which is due to overestimation of the drying rate as mentioned above. Both models show that the moisture content at the outer part of the

Fig. 13 – Profiles of center temperature during intermittent drying at drying air temperature of 65 ◦ C using spatial reaction engineering approach (S-REA).

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Fig. 14 – Spatial profiles of moisture content during intermittent drying at drying air temperature of 65 ◦ C using equilibrium model and S-REA. (Red *) equilibrium multiphase model t = 1000 s; (blue *) S-REA t = 1000 s; (red -) equilibrium multiphase model t = 3000 s; (blue -) S-REA t = 3000 s; (red –) equilibrium multiphase model t = 5000 s; (blue –) S-REA t = 5000 s; (red ) equilibrium multiphase model t = 10,000 s; (blue ) S-REA t = 10,000 s; (red ) equilibrium multiphase model t = 15,000 s; (blue ) S-REA t = 15,000 s; (red ) equilibrium multiphase model t = 20,000 s; (blue ) S-REA t = 20,000 s; (red -·) equilibrium multiphase model t = 30,000 s; (blue -·) S-REA t = 30,000 s; (red ) equilibrium multiphase model t = 35,000 s; (blue ) S-REA t = 35,000 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

samples is lower than that of the inner part which indicates that the moisture migrates outwards during intermittent drying. The gradient of moisture content inside the samples decreases during drying. During drying time of 7800–18,600 s, the gradient is relatively low because of the samples are under resting period which results in low temperature of the samples. During second heating period, the gradient is also relatively low as the moisture content approaches equilibrium. However, the gradient of moisture content inside the samples during intermittent drying predicted by the equilibrium model is higher than that modeled by the S-REA which may be because under equilibrium condition, higher amount of moisture needs to be supplied to the vapor phase. The difference of gradient of moisture content predicted by the two models is not significant during resting period which may be because the temperature of the samples is relatively low. During second heating period, the difference of gradient is also not significant which could be because the final equilibrium condition is approached. The spatial profiles of water vapor concentration inside the samples of intermittent drying described by both models are shown in Fig. 15. Both models predict that the profiles increase during drying but the profiles predicted by the equilibrium model are higher than those estimated by the S-REA. This is in agreement with higher profiles of spatial moisture content during intermittent drying as shown in Fig. 14. In addition, the spatial distribution of concentration of water vapor during intermittent drying predicted by both models is similar to that of convective drying. During resting period, the gradient

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Fig. 15 – Spatial profiles of concentration of water vapor during intermittent drying at drying air temperature of 65 ◦ C using equilibrium model and S-REA. (Red *) equilibrium multiphase model t = 1000 s; (blue *) S-REA t = 1000 s; (red -) equilibrium multiphase model t = 3000 s; (blue -) S-REA t = 3000 s; (red –) equilibrium multiphase model t = 5000 s; (blue –) S-REA t = 5000 s; (red ) equilibrium multiphase model t = 10,000 s; (blue ) S-REA t = 10,000 s; (red ) equilibrium multiphase model t = 15,000 s; (blue ) S-REA t = 15,000 s; (red ) equilibrium multiphase model t = 20,000 s; (blue ) S-REA t = 20,000 s; (red -·) equilibrium multiphase model t = 30,000 s; (blue -·) S-REA t = 30,000 s; (red ) equilibrium multiphase model t = 35,000 s; (blue ) S-REA t = 35,000 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) of concentration of water vapor inside the samples predicted by the two models is relatively small which could be because of relatively low temperature of the samples. The gradient during second heating period predicted by both models is also relatively small which may be because of the final equilibrium condition is approached. Fig. 16 represents the spatial profiles of temperature of intermittent drying predicted using the equilibrium model and the S-REA. Similar to the convective drying, the profiles predicted by the equilibrium model are lower than those modeled by the S-REA. This may be because under equilibrium condition, larger amount of heat is required for water evaporation since more moisture needs to be supplied to the vapor phase as mentioned above. Both models predict that the temperature increases during first heating, decreases during resting and increases again during second heating. It is emphasized here that for the equilibrium model no adjustment of diffusivity is conducted. This is unlike the common diffusion models implementing effective diffusivity (Pakowski and Adamski, 2007; Mariani et al., 2008; Vaquiro et al., 2009). Vaquiro et al. (2009) conducted modeling of convective and intermittent drying of mango tissues using the same sets of data cited here. It is explained that ten experiments were required to establish the effective diffusivity function followed up by validating the function with five experiments. The effective diffusivity was extracted from these experimental data and multi-objective optimization was conducted subsequently to fit the model with the data. Fig. 17 shows the results of modeling of convective drying of mango tissues using equilibrium model, S-REA and effective diffusion

Fig. 16 – Spatial profiles of temperature during intermittent drying at drying air temperature of 65 ◦ C using equilibrium model and S-REA. (Red *) equilibrium multiphase model t = 1000 s; (blue *) S-REA t = 1000 s; (red -) equilibrium multiphase model t = 3000 s; (blue -) S-REA t = 3000 s; (red –) equilibrium multiphase model t = 5000 s; (blue –) S-REA t = 5000 s; (red ) equilibrium multiphase model t = 10,000 s; (blue ) S-REA t = 10,000 s; (red ) equilibrium multiphase model t = 15,000 s; (blue ) S-REA t = 15,000 s; (red ) equilibrium multiphase model t = 20,000 s; (blue ) S-REA t = 20,000 s; (red -·) equilibrium multiphase model t = 30,000 s; (blue -·) S-REA t = 30,000 s; (red ) equilibrium multiphase model t = 35,000 s; (blue ) S-REA t = 35,000 s. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 17 – Comparison of average moisture content profiles of convective drying at drying air temperature of 55 ◦ C using equilibrium model, S-REA and effective diffusion model. Implemented by Vaquiro et al. (2009). model (implemented by Vaquiro et al. (2009)). As mentioned before, the equilibrium model can only match with the experimental data at the beginning of drying period while the S-REA models the profiles well along the convective drying. Similarly, the effective diffusion model shows a good agreement toward the experimental data. Nevertheless, this could be because of the fitting of the effective diffusivity function.

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It has been shown here that the convective and intermittent drying cannot be well represented by the equilibrium model. On the other hand, the S-REA describes accurately both drying schemes. The S-REA models are more appropriate to be implemented here since the time scale for achieving equilibrium far exceeds the drying period. This seems to make the moisture content on the solid matrix not under equilibrium condition with the concentration of water vapor inside the pore space of the samples. The evaporation rate from is better represented using rate-based approach in which the REA is used here. The REA provides sound formulation of the local evaporation/condensation rate of water.

7.

Conclusion

In this study, we have implemented both equilibrium and nonequilibrium drying model to simulate the local and overall drying behavior of the same material under the same drying conditions. Although both models employ the set of equations of conservation of heat and mass transfer, the difference comes when describing the local rates of evaporation and condensation. However, the REA is suitably used to represent the local phase change rate. The non-equilibrium model has been shown to predict better results quantitatively and qualitatively.

(A1)

Dv is the effective water vapor diffusivity et al., 2002). Dv0 = 2.09 × 10−5 + 2.137 × 10−7 (T − 273.15)

(m2 s−1 )

(Bird

(A2)

Dv0 is the water vapor diffusivity (m2 s−1 ) (Slattery and Bird, 1958).  = ε−n

(A3)

 is the tortuosity, ε is the porosity and n = 0.5 (Audu and Jeffreys, 1975; Gimmi et al., 1993). Cs =

1−ε (1/s ) + (X/w )

(A4)

Cs is the solid concentration (Kar, 2008; Kar and Chen, 2010, 2011):

ε=1−

V0 (1 − ε0 ) V



¯ +1 (s /w )X 1 + (s /w )X0

(A5)

ε is the porosity (Madiouli et al., 2007), s and w are the density of solid and water, respectively (kg m−3 ), V0 and V are the initial volume and sample volume (m3 ).

 Dw = 2.933 × 10−3 exp −

31.924 × 103 8.314T

Dw is the liquid water diffusivity (m2 s−1 ) (Putranto and Chen, 2013).

Appendix B. Properties of mango tissues w = 997 + 3.14 × 10−3 T − 3.76 × 10−3 T 2

(B1)

where w is density of water (kg m−3 ), T is sample temperature (◦ C) (Toledo, 2006; Sahin and Sumnu, 2006; Lozano, 2006; Floury et al., 2008). s = 1599 − 0.31T

(B2)

where s is density of solids (kg m−3 ), T is sample temperature (◦ C) (Toledo, 2006; Sahin and Sumnu, 2006; Lozano, 2006; Floury et al., 2008). kw = 0.571 × 10−3 + 1.76 × 10−6 T − 6.7 × 10−9 T 2

(B3)

where kw is thermal conductivity of water, T is sample temperature (◦ C) (Toledo, 2006; Sahin and Sumnu, 2006; Lozano, 2006; Floury et al., 2008). ks = 0.201 × 10−3 + 1.39 × 10−6 T − 4.33 × 10−9 T 2

(B4)

where ks is thermal conductivity of solids, T is sample temperature (◦ C) (Toledo, 2006; Sahin and Sumnu, 2006; Lozano, 2006; Floury et al., 2008).

Appendix A. Transport properties used for the modeling of equilibrium and non-equilibrium multiphase model ε Dv = Dv0 

11

 X −1.885×10−2  X+1 (A6)

1−w w 1 + =  w s

(B5)

where  is sample density (kg m−3 ), w is mass fraction of water (by weight) (Toledo, 2006; Sahin and Sumnu, 2006; Lozano, 2006; Floury et al., 2008). k=

kw s X + ks w s X + w

(B6)

where k is sample thermal conductivity, X is moisture content on dry basis (kg kg−1 ) (Toledo, 2006; Sahin and Sumnu, 2006; Lozano, 2006; Floury et al., 2008). Cpwater = −2 × 10−9 T 5 + 4.39 × 10−6 T 4 − 3.56 × 10−3 T 3 + 1.4327T 2 − 285.6T + 26779.6

(B7)

where Cpwater is specific heat of water (J kg−1 K−1 ), T is temperature (K) (Toledo, 2006; Sahin and Sumnu, 2006; Lozano, 2006; Floury et al., 2008). Cps = −3.9 × 10−3 T 3 + 0.13T 2 + 30T + 650

(B8)

where Cps is specific heat of solid (J kg−1 K−1 ), T is temperature (◦ C) (Toledo, 2006; Sahin and Sumnu, 2006; Lozano, 2006; Floury et al., 2008). Cpmix = wCpwater + (1 − w)Cps

(B9)

where Cpmix is specific heat of sample (J kg−1 K−1 ), x is mass fraction of water (by weight) (Toledo, 2006; Sahin and Sumnu, 2006; Lozano, 2006; Floury et al., 2008). Xe = 0.112

CG KG RH (1 − KG RH)(1 + (CG − 1)KG RH)

(B10)

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CG = CG0 exp KG = KG0 exp

 7005 

(B11)

RT

 H M − 56596  v w RT

(B12)

where Xe is the equilibrium moisture content (kg kg−1 ), R = 8.314 J mol−1 K−1 , RH is relative humidity of air, T is temperature (K), Mw is molecular weight of water (kg kmol−1 ), and Hv is vaporization enthalpy of water (kJ kg−1 ) (Basu et al., 2006; Marinos-Kouris and Maroulis, 2007; Garcia-Perez et al., 2008). The shrinkage model is expressed as (Dissa et al., 2008): Sb =

w + s X w + s X0

S = S0 Sb (2/3)

(B13) (B14)

where Sb is shrinkage coefficient, S0 is initial surface area (m2 ), S is surface area (m2 ), X0 is initial water content on dry basis (kg kg−1 ).

Appendix C. Evaluation of internal surface area (Ain ) (Kar, 2008; Kar and Chen, 2010) Ap = 4rp2

(C1)

4 3 r 3 p

(C2)

Vp =

mp = p Vp (1 − vw )

(C3)

N=

ms mp

(C4)

np =

N Vs

(C5)

Ain = np Ap

(C6)

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