C APTURA DE  R EQUISITOS

In order to capture detailed information about the distributions of moisture content and temperature throughout the material being dried, the S-REA has been developed. S-REA is a non-equilibrium multiphase drying approach in which the REA is implemented to represent the local phase change term. It is envisaged that, for a better understanding of the transport phenomena, application of effective liquid diffusion alone without source terms in both energy and mass conservation equations may not be sufficient as it cannot represent the water vapour concentration during drying. This could be affected by gas in the pore structure (Chen,2007). Also, vapour generation and transfer may affect other volatile transport in the same material. Traditionally, effective liquid diffusion has been used to simulate the detailed profiles of temperature and moisture content. This will be discussed further inChapter 4.

Equilibrium and non-equilibrium approaches can be implemented in the multiphase drying model mentioned previously (Zhang and Datta,2004; Datta,2007). By applying the equilibrium approach, it is assumed that vapour pressure inside the pores of the samples equilibrates with the liquid moisture content inside the same pores and the relationship can be described by the relevant equilibrium isotherm (Zhang and Datta, 2004). The equilibrium model has been applied to the baking process and a good agreement with experimental data has been shown (Ni et al.,1999; Zhang et al.,2005;

Zhang and Datta,2006). The equilibrium model has been implemented by combining the mass conservation of water in both liquid and vapour phases, which effectively resulted in the elimination of the source term. The moisture content and water vapour concentration are related by the available isotherm data (through a correlation equation) for the materials. Reasonable agreement with experimental data was shown in the cases investigated (Ni et al.,1999; Zhang et al.,2005; Zhang and Datta,2006). Similarly, the equilibrium model was applied by Aversa et al. (2010) to model the convective drying of food materials. The model has been shown to represent the experimental data reasonably well. Nevertheless, it has not been proven that use of equilibrium approach is valid in the case of heating hygroscopic materials (Zhang and Datta,2004).

For a more generic application of the multiphase drying model, it is suggested that the non-equilibrium approach is more appropriate. The good non-equilibrium multi-phase drying model is also useful in determining the appropriateness of the equilibrium approach for the situation of concern (Zhang and Datta,2004). In order to implement

z

y

x

Figure 3.1 Schematic diagram of a cube dried in a uniform convective environment.

the model, it is necessary to represent the internal evaporation rate explicitly and appro-priately. The internal evaporation/condensation rate is implemented in the multiphase drying model as a depletion term for the liquid phase and as a source term for the vapour phase. It has been proposed that the internal evaporation/wetting rate can be related to the difference of equilibrium vapour pressure and the vapour pressure at a particular time inside the pore spaces (Chong and Chen,1999; Scarpa and Milano,2002; Zhang and Datta,2004). In other words, gas must be present in the pores to permit this process to occur. The REA, in its lumped format, has been proven to model the global drying rate of various challenging drying cases accurately (Chen and Lin,2005; Chen, Pirini and Ozilgen,2001; Chen and Xie, 1997; Lin and Chen,2005;2006; 2007; Putranto et al.,2010a,b,2011a–e). It was expected that formulation of the L-REA may also be applicable in representing the source term in the S-REA approach; for instance, the same activation energy profile can be reserved for the same material.

The S-REA consists of a mass balance of liquid water, mass balance of water vapour and heat balance. For uniform convective drying of cubic object in a heated environment, three-dimensional modelling can be established. The mass balance of water in the liquid phase (liquid water) is written as (refer toFigure 3.1and Chong and Chen,1999; Chen, 2007; Kar and Chen,2011; Putranto and Chen,2013; Zhang and Datta,2004):

∂(CsX ) where X is the concentration of liquid water (kg H2O kg dry solids⁻¹), Csis the solid’s concentration (kg dry solids m⁻³), which can change if the structure is shrinking, Dwis liquid diffusivity (m²s⁻¹), ˙I is the evaporation or condensation rate (kg H2O m⁻³.s⁻¹) and ˙I is usually defined as positive when evaporation occurs locally. The liquid diffusivity represents the movement of liquid water inside the pore structure of the materials due to capillary action as a result of the water concentration gradient. In practice, the liquid diffusivity needs to be extracted from the available effective diffusivity data (Datta, 2007).

Reaction engineering approach II: S-REA 123

The mass balance of water vapour is expressed as (Chen, 2007; Chong and Chen, 1999; Kar and Chen,2011; Putranto and Chen,2013):

∂Cv

where Cvis the concentration of water vapour (kg m⁻³) and Dvis the effective water vapour diffusivity in pore channels (m²s⁻¹).

The heat balance is represented by the following equation (Chen,2007; Chong and Chen,1999; Kar and Chen,2011; Putranto and Chen,2013):

ρCp

where T is the sample temperature (K),HVis the vaporisation heat of water (J kg⁻¹), k is the sample thermal conductivity (W m⁻²K⁻¹),ρ is the sample density (kg m⁻³) and k andρ may be functions of temperature and moisture content.

For cubic objects being dried as an example, the initial and boundary conditions for Equations (3.1.1) to (3.1.3) may be written as:

t = 0, X = Xo, Cv = Cvo, T = To (initial condition, uniform initial

concentrations and temperature), (3.1.4)

x= 0, y = 0, z = 0,d X

d x = 0,d X

d y = 0,d X

d z = 0 (symmetrical boundary), (3.1.5) dC_v

d x = 0, dC_v

d y = 0dC_v

d z = 0 (symmetrical boundary), (3.1.6) d T

d x = 0, d T

d y = 0, d T

d z = 0 (symmetrical boundary), (3.1.7) x= L, −CsD_wd X

d x = h(Tb− T ) (convective boundary for heat transfer), (3.1.10) y= L, −CsD_wd X

(convective boundary for vapour transfer), (3.1.12)

kd T

d y = h(Tb− T ) (convective boundary for heat transfer), (3.1.13) z= L, −CsD_wd X

for liquid transfer), (3.1.14)

−D_vdC_v

d z = h(Tb− T ) (convective boundary for heat transfer), (3.1.16) where εw andεv are the fractions of surface area covered by liquid water and water vapour, respectively.

The internal evaporation rate ( ˙I ) can be described as:

I˙= hm_{i n}Ai n(C_v,s− C_v), (3.1.17) where Ain is the internal surface area per unit volume available for phase change (m²m⁻³) and hm,inis the internal surface mass transfer coefficient (m s⁻¹).

By implementing the REA, internal-surface water vapour concentration can be written as (Kar and Chen,2010;2011; Putranto and Chen,2013):

C_v,s= exp

−E_v RT



C_v,sat, (3.1.18)

where Cv,sis the internal-solid surface water vapour concentration (kg m⁻³), Cv,satis the internal saturated water vapour concentration (kg m⁻³) andEvis the activation energy (J mol⁻¹) similar to the one described in Equation (2.1.4).

Therefore, the internal evaporation rate can be expressed as (Kar and Chen,2010;

2011; Putranto and Chen,2013):

I˙= hm_{i n}Ai n

In Equation (3.1.19), the REA is used to describe the local evaporation rate as affected by pore structure (porosity, shrinkage, local moisture content and local temperature).

These microstructural effects can be ‘encapsulated’ in the term hm,inAin.

In particular, Ainis clearly influenced by structural or microstructural of the material of concern. It is interesting to note that when Ainis zero (i.e. for a ‘non-porous’ or ‘voidless’

material), ˙I becomes zero. Then, the liquid transfer (may be a kind of diffusion form) is predominant. In this case, for soft materials such as polymeric and biological entities, the free volume concept may be used to predict the effective liquid diffusivity (Vrentas and Duda,1977; Van der Sman, 2007a,b; Van der Sman et al., 2012). Free Volume Flory–Huggins (FVFH) theory, an extension of the classical Flory–Huggins theory, can be used to describe the thermodynamics of food materials. The chemical potential of moisture transfer can be assumed due to osmotic, elastic and ionic contributions.

Reaction engineering approach II: S-REA 125

The Flory–Huggins theory can also be implemented to predict the mutual diffusivity (Dm). By using Darken’s relation, the mutual diffusivity can be expressed as (Van der Sman et al.,2012):

Dm= Q

φDs,s+ (1 − φ)Ds,w

, (3.1.20)

whereφ is the volume fraction of polymer, χ is the Flory–Huggins interaction parameter, Ds,sand Ds,ware the self-diffusivity of polymer and water, respectively, and Q is described using Flory–Huggins theory as (Van der Sman et al.,2012):

Q= 1 − 2χφ(1 − φ). (3.1.21)

The Stokes–Einstein relation can be used to predict Ds,sgiven by (Van der Sman et al., 2012):

Ds,s= Ds,oηw

ηeff

, (3.1.22)

where Ds,ois the polymer self-diffusivity at infinite dilution,ηwis the viscosity of water andηeffis the viscosity of the polymer solution.

Ds,wcan be predicted by Vrentas and Duda’s free volume theory (Vrentas and Duda, 1977), which can be written as:

ln Ds,w

D_w,o = −E

RT − y_wV_w^∗+ ςysV_s^•

y_wK_ww(Ksw− Tg,w+ T ) + ysK_ws(Kss− Tg,s− T ), (3.1.23) whereE is the activation energy, Kij is free volume parameter(s), Vi* is parameters related to the volume of the molecule andζ is the shape factor. The mutual diffusivity (Dm) may then be used as effective liquid diffusivity (Van der Sman,2012).