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The validity of the S-REA in modelling convective drying is benchmarked against the experimental data of mango tissues (Vaquiro et al.,2009) and potato tissues (Srikiatden and Roberts,2008). The experimental data from drying of mango tissues are derived from the previous study (Vaquiro et al.,2009). For better understanding of the predictions, the necessary experimental details are summarised and reviewed in this section. The samples of mango tissues were formed as cubes with initial side lengths of 2.5 cm while the initial moisture content and temperature were 9.3 kg kg⁻¹and 10.8ºC, respectively.

The laboratory drier was described in Sanjuan et al. (2004). During drying the weight change of the sample and the centre temperature history were recorded. The drying air temperature and air velocity were controlled at preset values by PID control algorithms while air humidity was maintained constant during drying. The experimental setting for convective drying is shown inTable 3.1.

The density, thermal conductivity, heat capacity, equilibrium moisture content and shrinkage of the samples are presented in previous publication (Putranto et al.,2011a).

For convective drying of potato tissues, the experimental data were taken from the pre-vious work (Srikiatden and Roberts,2008). Their experimental details are also reviewed here for better understanding of the modelling approach (Roberts et al.,2002; Srikiatden and Roberts,2006; 2008). The cylindrical samples of Russet potatoes with diameters

Table 3.1 Experimental conditions of convective drying of mango tissues (Vaquiro et al.,2009).

Number

Air velocity (m s⁻¹)

Air temperature (°C)

Air humidity (kg H2O kg dry air⁻¹)

1 4 45 0.0134

2 4 55 0.0134

3 4 65 0.0134

of 1.4 and 2.8 cm were obtained using cylindrical cutters. The samples were sealed at their top and bottom ends with epoxy to establish approximately a one-dimensional (radial direction) moisture transfer. The experiments were conducted in a laboratory convective dryer with a drying air temperature of 70°C and axial velocity of 1.5 m s⁻¹. The experimental setup can be found in Srikiatden and Roberts (2006). The fan at the bottom of the sample draws air downward and this reduces the turbulence effect near the sample as the air moves downwards (Roberts et al.,2002).

The samples with the diameter of 1.4 cm were cut into two concentric parts for measurement of moisture content distribution, i.e. core and cortex (the core is a cylinder with the radius of 0.35 cm derived from the inner part of the potato tissues, while the cortex is a concentric shell derived from the outer part of the potato tissues). For the samples with the diameter of 2.8 cm, the samples were cut into four concentric parts;

i.e. core, cortex 1, cortex 2 and cortex 3. Similarly to the samples with the diameter of 1.4 cm, the core is a cylinder with a radius of 0.35 cm derived from the innermost part of the potato tissues. The procedures were repeated for a number of intervals (Srikiatden and Roberts,2006).

3.3.1 Mathematical modelling of convective drying of mango tissues using the S-REA Based on the experiments reported by Vaquiro et al. (2009) which have been used to help establish the REA, the samples were dried from three directions (x, y and z directions) so three-dimensional modelling of the S-REA for convective and intermittent drying of mango tissues needs to be set up, which is presented next.

The mass balance of water in the liquid phase (liquid water), the mass balance of water in the vapour phase (water vapour) and the heat balance are shown in Equations (3.1.1), (3.1.2) and (3.1.3), respectively, while the initial and boundary conditions for equations are shown in Equations (3.1.4)–(3.1.16). Since the sample dried was a cube shape dried uniformly from all directions (x, y and z directions) (Sanjuan et al.,2004; Vaquiro et al., 2009), the mass balance of water in liquid phase can be simplified into (Incropera and DeWitt,2002; Van der Sman,2003):

∂(CsX )

∂t = 3 ∂

∂x



D_w∂(CsX )

∂x

− ˙I, (3.3.1)

Reaction engineering approach II: S-REA 129

while the mass balance of water in vapour phase can be expressed as:

∂C_v

In addition, the heat balance can be represented as:

ρCp

The internal-surface water vapour concentration (Cv,s) and internal evaporation rate ( ˙I ) are evaluated using Equations (3.1.18) and (3.1.19).

The relative activation energy of convective drying of mango tissues is generated from one accurate drying run of convective drying mango tissues under constant environment conditions with a drying air temperature of 55°C (Vaquiro et al.,2009). The activation energy during drying is evaluated using Equation (2.1.5) and divided with the equilibrium activation energy represented in Equation (2.1.7) to yield the relative activation energy as mentioned in Equation (2.1.6). The relationship between the relative activation energy and average moisture content can be represented by a simple mathematical equation obtained by use of the least-squares method using Microsoft Excel (Microsoft Inc., 2012). The relative activation energy can be represented as:

Ev

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

−0.101(X − Xb)+ 1.053. (3.3.4)

The good agreement between the fitted and experimental relative activation energy is shown by R² of 0.999. Although Equation (3.3.4) involves Xbas mentioned earlier, all successful applications of REA so far suggest that the experiments carried out should dry the materials to Xbs of very small values in order to allow the correlations such as Equation (3.3–15) to cover the widest range of water content of practical interest. If Xb

is close to initial water content, the activation energy calculated from the laboratory data can be misleading.

The relative activation energy correlated with Equation (3.3.4) has been implemented to model the convective and intermittent drying of mango tissues using the L-REA and the results of modelling already matched well with experimental data (Putranto et al., 2011a,b). For modelling using the S-REA here, the relative activation energy shown in Equation (3.3.4) is used but the average moisture content X in Equation (3.3.4) is substituted by the local moisture content (X) as the REA is used to represent the local evaporation rate instead of the overall drying rate of the whole sample. In addition, it is emphasised that, for the S-REA, the equilibrium relative activation energy (Ev,b) is evaluated at corresponding humidity and temperature inside the pores of the samples under equilibrium condition.

The effective vapour diffusivity (Dv), tortuosity (τ), solid concentration (Cs) and porosity (ε) are deduced using Equations (3.2.1)–(3.2.5). Similarly, the internal mass transfer coefficient (hm,in) is evaluated using the procedures described in Section 3.2.

The effective diffusivity of mango tissues presented by Vaquiro et al. (2009) is

Equation (3.3.5) can be used as an approximation to determine the liquid water diffusivity of mango tissues but a little adjustment of the constant is needed in order to match the prediction with the experimental data of moisture content and temperature. The liquid water diffusivity used in this study can be expressed as:

D_w= 2.933 × 10⁻³exp

In order to yield the spatial profiles of moisture content, water vapour concentration and temperature of the convective of mango tissues, the mass and heat balances shown in Equations (3.3.1)–(3.3.3) in conjunction with the initial and boundary conditions represented in Equations (3.1.4)–(3.1.16) and the relative activation energy shown by Equation (3.3.4) are solved by the method of lines (Chapra,2006; Constatinides,1999).

By this method, the partial differential equations are transformed into a set of ordinary differential equations with respect to time by firstly discretising the spatial derivatives.

The ordinary differential equations are then solved simultaneously by ode23s in Matlab (Mathworks Inc., 2012). The spatial derivative here is discretised into 10 increments;

application of 200 increments has been conducted and there is no real difference in the profiles observed as shown inFigure 3.2.

The shrinkage (Putranto et al., 2011a) is incorporated in the modelling by a mov-ing mesh in which the number of intervals is kept constant but the intervals of each increment are allowed to change according to the shrinkage relationship. The moving mesh was found to give better agreement with experimental data than a fixed coordinate (immobilising boundary) (Thuwapanichayanan et al.,2008).

The average moisture content of mango tissues during convective drying is evaluated by:

The profiles of average moisture content and centre temperature are then validated against the experimental data of Vaquiro et al. (2009).

3.3.2 Mathematical modelling of convective drying of potato tissues using the S-REA

In the experiments reported by Srikiatden and Roberts (2008) which are of interest, the samples were covered at both the top and bottom end to promote the one-dimensional

Reaction engineering approach II: S-REA 131

Moisture content (kg water/kg dry solids)

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Figure 3.2 Moisture content profiles of the convective drying of mango tissues at a drying air temperature of 45°C solved by the method of lines with 10 and 200 spatial increments.

[Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for

drying of food and biological materials, 55–67, Copyright (2012), with permission from John Wiley & Sons Inc.]

drying condition with respect to radial direction (Srikiatden and Roberts,2008) so one-dimensional modelling (at radial directions) of the S-REA of the convective drying of cylindrical potato tissues is possible and can be represented by a set of equations of conservation next.

The mass balance of liquid water can be represented as (Chen,2007; Chong and Chen, 1999; Kar and Chen,2011; Zhang and Datta,2004):

∂(CsX ) solids concentration (kg dry solids m⁻³), ˙I is the evaporation or wetting rate (kg H2O m⁻³s⁻¹) and ˙I is>0 when evaporation occurs locally.

The mass balance of water vapour can be expressed as (Chen,2007; Chong and Chen, 1999; Kar and Chen,2011; Zhang and Datta,2004):

∂C_v

In addition, the heat balance can be written as (Chen,2007; Chong and Chen,1999;

Kar and Chen,2011; Zhang and Datta,2004):

ρCp

where T is the sample temperature (K).

The initial and boundary conditions of Equations (3.3.8)–(3.3.10) are:

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

concentrations and temperature), (3.3.11)

r = 0,d X

dr = 0, dC_v

dr = 0, d T

dr = 0 (symmetrical condition), (3.3.12) r= R, −CsD_wd X liquid water transfer), (3.3.13)

−DvdC_v

dr = h(Tb− T ) (convective boundary for heat transfer). (3.3.15) Similar to the convective drying of mango tissues, the internal solid-surface water vapour concentration and the local evaporation rate (˙I ) are evaluated using Equations (3.1.17) and (3.1.18). The relative activation energy of convective drying of potato tissues is generated from one accurate drying run of the convective drying of potato tissues with a diameter of 1.4 cm at a drying air temperature of 70°C (Srikiatden and Roberts, 2008). The activation energy during drying was evaluated using Equation (2.1.5) based on the experimental data of moisture content and surface temperature during drying (Srikiatden and Roberts,2008). It is then divided with the equilibrium activation energy represented in Equation (2.1.7) to yield the relative activation energy as mentioned in Equation (2.1.6). The relationship between the relative activation energy and average moisture content can be represented by a simple mathematical equation obtained by the least-squares method using Microsoft Excel (Microsoft Inc., 2012). The relative activation energy can be represented as:

E_v

Ev,b = exp

−0.364(X − Xb)^0.876

. (3.3.16)

Similar to modelling of convective drying of mango tissues, for modelling using the S-REA, the average moisture content X in Equation (3.3.16) is substituted by the local moisture content (X) as the REA is then able represent the local evaporation or condensation rate here instead of the global drying rate.

The effective liquid diffusivity (Dw) is shown in Equation (3.2.7) while the effective vapour diffusivity (Dv), tortuosity (τ), solid concentration (Cs) and porosity (ε) are

Reaction engineering approach II: S-REA 133

deduced using Equations (3.2.1)–(3.2.5). Similarly, the internal mass transfer coefficient (hm,in) is evaluated using the procedures explained in Section 3.2.

The average moisture content in the core of potato tissues (Xcore) is evaluated by:

Xcore=

Rcore

0

X (r )r dr

Rcore

0

rdr

. (3.3.17)

The average moisture content in cortex (Xcortex) is evaluated by:

Xcortex3=

Rsample out

Rcortex in

X (r )r dr

R_{cortex out}

Rcortex in

r dr

. (3.3.18)

The results of modelling average moisture content in core and cortex (hence the spa-tial distribution) are validated against the experimental data of Srikiatden and Roberts (2008). Similarly to the convective drying of mango tissues, the mass and heat balances are shown in Equations (3.3.8)–(3.3.10) in conjunction with the initial and boundary conditions indicated in Equations (3.3.11)–(3.3.15). The application of 10 and 200 increments did not result in noticeable differences in the profiles.

3.3.3 Results of modelling of convective drying of mango tissues using the S-REA The S-REA is used to model the convective drying of mango tissues at drying air tem-peratures of 45°, 55° and 65 °C. The original formulation of the L-REA is implemented in the partial differential equation set for transport in porous media to represent the local drying or condensation rate. It is thus coupled with the system of equations of conserva-tion to describe the spatial profiles of moisture content, water vapour concentraconserva-tion and temperature. It is noted that, if locally there is no vacant pore space which is connected with other pores or channels, the internal mass transfer area should be considered to be zero; hence the REA term is zero if the pores or channels are fully hydrated. In this study, the internal mass transfer coefficient (hm,in: see Equation3.3.12) is chosen to 0.01 m s⁻¹ as the sensitivity analysis indicates that hm,in is likely to be higher than 0.01 m s⁻¹, but any higher than this and it does not give any noticeable difference in the profiles of moisture content and temperature predicted. More importantly, the value of 0.01 m s⁻¹ is also in the order of Dv/rp (Kar and Chen,2010; 2011); thus it is a fundamental value.

The good agreement between the predicted and experimental data of the average moisture content and the centre temperature is shown inFigures 3.3and3.4. It is also supported by R² of moisture content higher than 0.996 and R² of temperature higher than 0.985 as listed inTable 3.2. The results of the S-REA modelling match well with the experimental data. Benchmarks against the diffusion-based model (Vaquiro et al.,

0.5 0

Moisture content (kg water/kg dry solids)

°C Data 55°C S-REA 65°C Data 65°C

1 1.5 2 2.5

t(s)

3 3.5 4 4.5 5

× 10⁴ 0

1 2 3 4 5 6 7 8

Figure 3.3 Average moisture content profiles of mango tissues during convective drying at different drying air temperatures.[Reprinted from AIChE Journal, 59, Aditya Putranto, Xiao

Dong Chen, Spatial reaction engineering approach as an alternative for nonequilibrium multiphase mass-transfer model for drying of food and biological materials, 55–67, Copyright

(2012), with permission from John Wiley & Sons Inc.]

0

Temperature (K)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

S-REA 45°C