Lumped-REA (L-REA)
2.6 Convective drying of thick samples modelled using the L-REA
For studying simulations of convective drying of thick samples using the L-REA, the experimental data are derived from the work of Vaquiro et al. (2009) on convective drying of mango tissues. Mango tissues used for drying experiments were formed into cubes of side lengths of 2.5 cm, with initial moisture content of 9.3 kg kg−1 and an initial temperature of 10.8°C. Drying was conducted in a laboratory dryer described in detail by Sanjuan et al. (2004). The drying air temperature and air velocity were controlled at preset values by PID control algorithms while air humidity was main-tained at a constant during drying. Details of the experimental conditions are listed in Table 2.3. The weight of the sample was measured periodically to record weight loss as well as centre temperatures every 2 min.
2.6.1 Formulation of the L-REA for convective drying of thick samples
In order to model the convective drying of thick samples, the original formulation of the L-REA can still be implemented. However, the temperature of concern is the surface
temperature (Ts) (Putranto et al.,2011a,b). Therefore, the drying rate of the material can
where ms is the dried mass of thin layer material (kg), t is time (s), X is moisture content on a dry basis (kg kg−1),ρv,s is the vapour concentration at the material-air interface (kg m−3),ρv,bis the vapour concentration in the drying medium (kg m−3), hm
is the mass transfer coefficient (m s−1) and A is the surface area of the material (m2).
The mass transfer coefficient (hm) is determined based on the established Sherwood number correlations for the geometry and flow condition of concern or established experimentally for the specific drying conditions involved (Lin and Chen,2002; Kar and Chen,2009). The surface vapour concentration (ρv,s) can be scaled against saturated vapour concentration (ρv,sat) using the following equation (Chen and Xie,1997; Chen, 2008):
whereEvrepresents the additional difficulty in removing moisture from the material beyond the free water effect. ThisEv is moisture-content (X) dependent. Ts is the surface temperature of the material being dried, andρv,satfor water can be estimated at the surface material being dried by the following equation:
ρv,sat = 4.844 × 10−9(Ts− 273)4− 1.4807 × 10−7(Ts− 273)3+ 2.6572
×10−5(Ts− 273)2− 4.8613 × 10−5(Ts− 273) + 8.342 × 10−3, (2.6.3)
based on the data summarised by Keey (1992).
The mass balance (Equation 2.6.1) is then expressed as:
ms
The activation energy (Ev) is determined experimentally by placing the parameters required for Equation (2.6.4) in its rearranged form:
Ev = −RTsln
The equilibrium activation energy (Ev,b) is still evaluated by Equation (2.1.7). It can be shown that the general formulation of the L-REA shown in Equation (2.1.4) can still be implemented but the temperature of concern is the surface temperature (Ts).
Reaction engineering approach I: L-REA 63
2.6.2 Prediction of surface sample temperature
For large sample slabs, prediction of sample temperature may be necessary since the temperature may be not uniform inside the sample. The sample temperature may be approximated using a simple parabolic equation (Chen,2008):
T = a + bx2. (2.6.6)
If Tois the centre sample temperature (K) and L is the half-thickness of the sample as a characteristic slab length, Equation (2.6.6) is rewritten as:
T = To+
Ts− To
L2
x2, (2.6.7)
Tavgis determined by:
Tavg=
L 0
T (x)d x
L . (2.6.8)
By combining Equations (2.6.7) and (2.6.8), Tavgis expressed as:
Tavg= 1 3Ts+2
3To. (2.6.9)
For a sample heated in a convective environment, the boundary condition at sample surface (x= L) can be written as: Also note that Equation (2.6.7) satisfies the boundary condition at centre (x= 0), which can be expressed as:
d T d x
x=0= 0. (2.6.11)
By combining Equation (2.6.7) to (2.6.10), Tsand Toare expressed as:
Ts = Tavg+h L
Equation (2.6.12) and (2.6.13) clearly show that Tsand Toare represented as functions of Tavgand Tb.
The temperature profile prediction described previously seems to be valid for drying conditions suitable for the boundary conditions mentioned (i.e. there is symmetry at centre and at the surface; heat gained by convection from drying air is balanced by conduction heat inside the sample and heat for water evaporation). The prediction is in agreement with Pang (1994), who conducted convective drying of softwood and
heartwood with a half-thickness of 2.5 cm. It was observed that the boundary conditions indicated in Equations (2.6.10) and (2.6.11) fulfil the drying conditions of Pang (1994).
The temperatures in several positions (x= 0, 7, 13, 19 and 25 cm from centre) were measured during the drying time and a plot of the temperature profiles against positions during drying time revealed parabolic profiles.
For drying of mango tissue, as mentioned before, the sample was heated uniformly from all directions (Sanjuan et al.,2004). It is reasonable to assume that the temperature profiles would be similar in the x, y and z directions. Because of this, the approximation of the temperature profiles can be simplified into one dimension.
It is also observed that for drying of mango tissues, there is symmetry at the centre and at the surface; heat received by convection from drying air is balanced by conduction heat inside the sample and heat for water evaporation is represented by the boundary conditions shown in Equations (2.6.10) and (2.6.11) (Sanjuan et al., 2004; Incropera and DeWitt, 2002). Therefore, similarly to Equations (2.6.7), (2.6.12) and (2.6.13), showing the temperature distribution inside mango and apple tissues, surface and centre temperature can be represented as:
T = To+
The equivalent radius for cubes is the side length (Incropera and DeWitt,2002; Radziem-ska and Lewandowski,2008). Because of the symmetry principle used for Equations (2.6.14) to (2.6.16), the equivalent radius (r) used for cubes in this study are half the side length.
2.6.3 Modelling convective drying thick samples of mango tissues using the L-REA For drying thick samples of mango tissues convectively, the relative activation energy (Ev/Ev,b) is generated from continuous convective drying runs at 55°C (Vaquiro et al.,2009). Based on drying kinetics data, the relative activation energy (Ev/Ev,b) of convective drying of mango tissues is expressed as:
Ev
Ev,b = −9.92 × 10−4(X− Xb)3+ 9.74 × 10−3(X− Xb)2
− 0.101(X − Xb)+ 1.053. (2.6.17)
A good agreement between the fitted (Equation 2.6.17) and experimental activation energy is shown inFigure 2.18(R2(0.997)). This format of correlation is similar to that proposed by Kar (2008) to describe the activation energy of drying porcine skin. The
Reaction engineering approach I: L-REA 65
0 1 2 3 4 5 6 7 8 9 10
ΔEv/ΔEv,b
X–Xb 0.1
0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Model Data
Figure 2.18 The relative activation energy (Ev/Ev,b) of convective drying of mango tissues at an air velocity of 4 m s−1, drying air temperature of 55°C and air humidity of 0.0134 kg H2O kg
dry air−1.[Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor &
Francis Ltd.]
format of the equation could be varied but for this study the Equation (2.6.17) seems to represent the activation energy well. It may be observed that the decrease of moisture content results in the increase of activation energy, which indicates greater difficulty in removing water. This equation also yieldsEv/Ev,bapproaching1 as the material is dried.
The heat balance for convective drying of mango tissues can be written as:
d(mCpTavg)
dt ≈ h A (Tb− Ts)+ ms
d X
dt HV, (2.6.18)
where m is the sample mass (kg), Cpis the heat capacity of the sample (J kg−1 K−1), h is the heat transfer coefficient (W m−2K−1) andHVis the latent heat of vaporisation of water (J kg−1). The drying rate dX/dt is negative when drying occurs. In order to yield both profiles of moisture content and temperature of mango tissues during drying, the mass implementing the L-REA and heat balance shown in Equations (2.6.4) and (2.6.18) are solved simultaneously in conjunction with the equilibrium and relative activation energy shown in Equations (2.1.7) and (2.6.17), respectively. The surface temperature predicted by Equation (2.6.15) is used in the mass balance implementing the L-REA and heat balance.
Table 2.4 R2and RMSE of modelling of convective drying of mango tissues using the L-REA.
Number
Velocity (m s−1)
Air temperature (°C)
Air humidity
(kg H2O kg dry air−1) R2X RMSE X R2T RMSE T
1 4 45 0.0134 0.998 0.08 0.993 0.61
2 4 55 0.0134 0.998 0.1 0.982 1.12
3 4 65 0.0134 0.996 0.14 0.984 1.41
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
X(kg water/kg dry solid)
× 104 0
1 2 3 4 5 6 7 8 9 10
Model 65°C Model 55°C Model 45°C Data 65°C Data 55°C Data 45°C
t(s)
Figure 2.19 Moisture content profile of convective mango tissues at air temperatures of 45°, 55°
and 65°C (modelled using the L-REA which incorporates the temperature distribution inside the sample).[Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction
engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor & Francis Ltd.]
2.6.4 Results of convective drying thick samples of mango tissues using the L-REA FromFigures 2.19and2.20, a good agreement between the experimental and predicted data is observed for convective drying of mango tissues at drying air temperatures of 45°, 55° and 65 °C. The good predictions made by using the REA are further revealed by R2and RMSE presented inTable 2.4, which shows all modelling of these cases yield R2 of moisture content and temperature profiles higher than 0.996 and 0.982, respectively, as well as RMSE of moisture content and temperature profiles lower than 0.14 and 1.41, respectively. On the other hand,Figures 2.21and2.22show discrepancies between the predicted and experimental data. It is clear that the L-REA with the approximation of
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Figure 2.20 Temperature profile of convective mango tissues at air temperatures of 45°, 55° and 65°C (modelled using the L-REA which incorporates the temperature distribution inside the
sample).[Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor &
Francis Ltd.]
Moisture content (kg water/kg dry solid)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Figure 2.21 Moisture content profile of convective mango tissues at air temperatures of 45°, 55°
and 65°C (modelled using the L-REA without approximation of temperature distribution inside the sample).[Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley,
Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor &
Francis Ltd.]
280 300
290 310 320 330 340
Centre temperature (K)
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
t(s) × 104
Model 65°C Model 55°C Model 45°C Data 65°C Data 55°C Data 45°C
Figure 2.22 Temperature profile of convective mango tissues at air temperatures of 45°, 55° and 65°C (modelled using the L-REA without approximation of temperature distribution inside the sample).[Reprinted from Drying Technology, 29, A. Putranto, X.D. Chen and P.A. Webley, Modelling of drying of food materials with thickness of several centimeters by the reaction engineering approach (REA), 961–973, Copyright (2012), with permission from Taylor &
Francis Ltd.]
temperature distribution inside the sample is necessary, and this model describes both moisture content and centre sample temperature profile well during drying.
Vaquiro et al. (2009) used diffusion-based modelling to represent the data and our REA compares well with the modelling by Vaquiro et al. (2009). Modelling by Vaquiro et al. (2009) showed a kink in the beginning of the temperature profile that was not observed by modelling using the REA. For drying at 65°C, both the REA and modelling proposed by Vaquiro et al. (2009) showed a slight overestimation of the temperature profile during drying times of 5000–20 000 s.
It can be said that the L-REA, with the prediction of sample temperature as explained in Section 2.6.2, is accurate enough to describe continuous convective drying of mango tissues well. It also compares favourably with the model proposed by Vaquiro et al.
(2009) in spite of the simplicity of the L-REA. While the results are accurate, the modelling itself is still simple and only requires a short computational time to predict the drying kinetics accurately. This shows that the L-REA is effective for modelling
‘thick’ samples of mango tissues.
A new and innovative application of the L-REA has been implemented in this study to describe both the moisture content and sample temperature profile of convective drying large samples of mango tissues. For this purpose, the activation energy and the saturation vapour concentration are evaluated at the surface temperature. The remaining principles
Reaction engineering approach I: L-REA 69
Table 2.5 Schemes of intermittent drying of mango tissues (Vaquiro et al.,2009).
Drying air temperature (°C)
Period of first heating (s)
Period of resting (at 27°C ± 1.6) (s)
Period of second heating (s)
45 16 200 10 800 36 360
55 9 480 10 800 33 720
65 7 800 10 800 16 200
are similar to those of the L-REA used to describe the drying kinetics of thin layers or small objects published previously. Results indicate that the REA models both moisture content and temperature of convective drying of large samples of mango tissues very well. When compared to the experimental data published by Vaquiro et al. (2009), a similar if not better agreement is observed against diffusion-based models. While the results are accurate, the effectiveness of the L-REA is also revealed as the modelling itself is still simple and only requires a short amount of computational time. Therefore, this work has extended the application of the L-REA to handle drying of thick samples substantially. The L-REA can model not only the drying of thinlayer or small objects, but also drying of thick samples.