Lumped-REA (L-REA)
2.5 Convective drying of particulates or thin layer products modelled using the L-REA
The L-REA (lumped reaction engineering approach) has been used to describe the convective drying of droplets of whey protein concentrate and thin layer of a mixture of polymer solution (Lin and Chen, 2007; Allanic et al.,2009). For the convective drying of droplets of whey protein concentrate, the experimental setup is similar to that explained in Section 2.3. The droplets are suspended in a glass-filament convective dryer, and the mass and temperature are recorded during drying. The deflection of the glass filament is captured and converted to droplet weight. The measurement of weight also takes into account the drag force. A video camera system is used to monitor the droplet diameter change during drying. A calibrated thermocouple is used to record the sample temperature during drying. The thermocouple is connected to a picometer and the data is obtained from the data logger. The repeatability of the weight loss and temperature measurement is±0.01 mg and 0.1 °C, respectively. Drying air with the velocity of 0.45 m s−1 and a temperature of 70°–110 °C is used. Initial droplet diameter and solids concentration are 1.45 mm and 30% wt., respectively (Lin and Chen, 2007).
For convective drying a mixture of polymer solutions, the experimental data used in the current work are derived from the study reported by Allanic et al. (2009) and the experimental conditions are shown inTable 2.1.
In order to better understand the modelling presented here, the details of experiment are briefly described here (Allanic et al.,2006; 2009). Materials used in this exper-iment were a mixture of equal proportions of partially hydrolysed polyvinyl alcohol (80%wt.) and glycerol with 88%wt. of water, then 8 ml of the mixture was poured into a 90-mm-diameter Petri dish so the initial thickness of the sample was 1.3 mm. During
Reaction engineering approach I: L-REA 51
Table 2.1 Experimental conditions of convective drying of a mixture of polymer solutions (Allanic et al.,2009).
Number
Air velocity (m s−1)
Air temperature (°C)
Air relative humidity (%)
1 2.8 55 12
2 1 35 30
3 1 55 12
drying, shrinkage occurred and the relationship between thickness (m) and moisture content on a dry basis could be correlated in the linear form:
e= ed(1+ λX), (2.5.1)
where e is the thickness of product (m), edis the thickness of the dried product (m), X is the moisture content on a dry basis (kg kg−1), andλ is the linear shrinkage coefficient (=1.3).
The weight measurement was accurate to about 0.2 g. During drying, regulated drying air temperature with particular velocity and temperature was fed into the rectangular casing so that it flowed gently above the sample. The drying air temperatures and velocities used for each experiment are listed inTable 2.1. The stable humidity of drying air was maintained and measured using a capacitive transmitter sensor. The temperature of the sample surface was measured using an optical pyrometer and the temperature of the upper and lower side of the Petri dish was measured with thermocouples with an uncertainty of about 2.5°C. The results of this previous study showed that temperature gradient inside the Petri dish and product can be ignored (Allanic et al.,2006).
2.5.1 Mathematical modelling of convective drying of droplets of whey protein concentrate (WPC) using the L-REA
The L-REA explained in Section 2.1 is implemented here to model the moisture content and temperature profiles during the convective drying of WPC. The relative activation energy is generated from one accurate drying run. The activation energy and equilibrium activation energy are evaluated using Equations (2.1.5) and (2.1.7), respectively. The mass balance implementing the L-REA shown in Equation (2.1.4) is used with the convective mass transfer coefficient (hm), determined based on the work of Lin and Chen (2002):
Sh= 1.54 + 0.54 Re0.5Sc0.333, (2.5.2) where Sh is the Sherwood number, Re is the Reynolds number and Sc is the Schmidt number.
The heat balance of the convective drying of WPC can be represented as:
d(mCpT )
dt ≈ h A(Tb− T ) + ms
d X
dt Hv, (2.5.3)
ΔE/ΔEb
1.0
0.8
0.6
0.4
0.2
0.0
0.0 0.5 1.0 1.5 2.0 2.5
X–Xb (kg/kg)
67.5°C 87.1°C 106.6°C Curve fitted
Figure 2.7 The relative activation energy of convective drying of WPC at different drying air temperatures.[Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Lin and X.D.
Chen, The reaction engineering approach to modelling the cream and whey protein concentrate droplet drying, 437–443, Copyright (2012), with permission from Elsevier.]
where m is the mass of droplets during drying (kg), Cpis the specific heat of the samples (J kg−1 K−1), T is the sample temperature (K), Tb is the drying medium temperature (K),Hvis the vaporisation heat of water (J kg−1) and h is the heat transfer coefficient (W m−2K−1) which can be evaluated by (Lin and Chen,2002):
N u= 2.04 + 0.62 Re0.5Pr0.333, (2.5.4) where Nu is the Nusselt number and Pr is the Prandtl number.
The relative activation energy of the WPC is generated from one accurate drying run.
It is shown inFigure 2.7and can be expressed as (Lin and Chen,2007):
Ev
Ev,b = 1.335 − 0.3669 exp
exp(X− Xb)0.3011
, (2.5.5)
while the droplet diameter changes during drying can be expressed as (Lin and Chen, 2007):
d
d0 = 0.873 + 0.127 X− Xb
X0− Xb. (2.5.6)
The good agreement between Equation (2.5.6) and experimental diameter changes during drying is shown inFigure 2.8.
The profiles of moisture content and temperature during drying are generated by solving the mass balance implementing the L-REA and the heat balance shown in Equations (2.1.5) and (2.5.3), respectively, in conjunction with the equilibrium activation energy, relative activation energy and droplet diameter changes during drying shown in Equations (2.1.5), (2.1.7) and (2.5.6), respectively.
Reaction engineering approach I: L-REA 53
d/d0
1.0
0.9
0.9 1.0
0.0 0.5 1.0 1.5 2.0 2.5
X–Xb (kg kg–1)
67.5°C 87.1°C 106.6°C Curve fitted
Figure 2.8 The droplet diameter changes during convective drying of WPC.[Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineering
approach to modelling the cream and whey protein concentrate droplet drying, 437–443, Copyright (2012), with permission from Elsevier.]
2.5.2 Mathematical modelling of convective drying of a mixture of polymer solutions using the L-REA
The L-REA shown in Equation (2.1.4) used here is similar to formulation of the L-REA used in the convective drying of WPC. The activation energy and equilibrium activation energy is also calculated using Equations (2.1.5) and (2.1.7), respectively.
During convective drying, it can be seen that the sample is heated from the upper side due to forced convective heat transfer from the drying air, while the lower side is heated through the Petri dish which is heated by drying air from below (refer toFigure 2.9).
The heat balance can be written as:
d(ρ CpeT )
dt = htop(Tb− T ) + Ubottom(Tb− T ) +ms
A d X
dt Hv(T ), (2.5.7) where ρ is sample density (kg m−3), Cp is sample heat capacity (J kg−1 K−1), e is sample thickness (m), T is sample temperature (K), Tb is drying air tempera-ture (K), ms is mass of dried product (kg), A is surface area of product (m2), X is moisture content of product (kg kg−1), Hv is enthalpy of vaporisation (J kg−1), htop is heat transfer coefficients at the upper surface of the dish and Ubottom repre-sents the overall heat transfer coefficient from the lower side including convection (natural) along and conduction through the Petri dish. Rearranging Equation (2.5.7) results in:
d(ρ CpeT )
dt = Utotal(Tb− T ) +ms
A d X
dt Hv(T ), (2.5.8)
Petri dish 0
e(t)
z
Product
Thermocoupl Convection Conduction
Evaporation Diffusion
Figure 2.9 Heat transfer mechanisms of the convective drying of a mixture of polymer solutions.
[Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright
(2012), with permission from Elsevier.]
where Utotalrepresents the sum of the convective heat transfer coefficient (W m−2K−1) at the top and that at bottom. This value can be deduced from the constant rate period of drying. The heat balance for this period can be expressed as:
Utotal(Tb− T ) = ms
A d X
dt Hv(T ). (2.5.9)
For this experiment, the activation energy is determined based on previously published experimental data (Allanic et al.,2009) using Equation (2.1.5). The vapour concentration in the environment is determined from the corresponding relative humidity and drying air temperature reported previously (shown inTable 2.1). The mass transfer coefficient was deduced from the established Sherwood number correlation. Based on drying kinetics data, the relative activation energy (Ev/Ev,b) for convective drying calculated through this exercise is expressed as:
Ev
Ev,b = exp
− 1.0794(X − Xb)1.28
. (2.5.10)
Only one set of drying data was necessary and this was taken from experiment at a drying air temperature of 35°C, drying air velocity of 1 m s−1 and relative humidity of 30% (Allanic et al.,2009). This is of a similar format to that proposed previously (Chen and Xie,1997; Chen and Lin, 2005). As Figure 2.10shows, there is excellent agreement between correlated and experimental activation energy (R2 = 0.9892). At high water content, moisture removal is easy, as shown by the low activation energy, and this increases during drying as moisture content decreases indicating greater difficulty
Reaction engineering approach I: L-REA 55
0 1 2 3 4 5 6 7 8
ΔEv/ΔEv,b
X–Xb (kg water/kg dry solid) –0.2
0 0.2 0.4 0.6 0.8 1.0
Data Fitted curve
Figure 2.10 Normalised activation energy and fitted curve of polyvinyl alcohol/glycerol/water under convective drying at an air temperature of 35°C and relative humidity of 30%. [Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen
and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction engineering approach (REA), 348–357, Copyright
(2012), with permission from Elsevier.]
in removing moisture. Also, this correlation ensures that Ev/Ev,b is1 when the moisture content approaches equilibrium (i.e. X= Xb).
In order to evaluate the moisture content and the temperature profile as a function of drying time, the mass balance and heat balance expressed in Equations (2.1.4) and (2.5.8), respectively were solved simultaneously in conjunction with the equilibrium and relative activation energy shown in Equations (2.1.7) and (2.5.10), respectively.
2.5.3 Results of modelling convective drying of droplets of WPC using the L-REA
The results of modelling of the convective drying of the droplets of WPC using the L-REA are shown inFigure 2.11. Generally, the predictions using the L-REA match well with the experimental data. For the convective drying at a drying air temperature of 67.5°C, the results of modelling of moisture content and temperature profiles match the experimental data well as shown inFigure 2.11(a). In addition, the L-REA describes well the moisture content and temperature profiles of the convective drying of WPC at drying air temperatures of 87.1° and 106.6 °C, as depicted in Figures 2.11(b) and (c), respectively. For all experiments, the average absolute differences between the
Droplet weight (kg) Droplet temperature (°C)
Droplet weight (kg) Droplet temperature (°C)
4.0E-07
Droplet weight (kg) Droplet temperature (°C)
4.0E-07
Figure 2.11 The comparison between experimental and model prediction using the L-REA of convective drying of WPC at drying air temperatures of (a) 67.5°C (b) 87.1 °C (c) 106.6 °C.
[Reprinted from Chemical Engineering and Processing, 46, S.X.Q. Lin and X.D. Chen, The reaction engineering approach to modelling the cream and whey protein concentrate droplet
drying, 437–443, Copyright (2012), with permission from Elsevier].
Reaction engineering approach I: L-REA 57
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
X(kg water/kg dry solid)
t(s) 0
1 2 3 4 5 6 7 8
Model Data
Figure 2.12 Moisture content profile of convective drying at air temperature of 55°C, air velocity of 2.8 m s−1and air relative humidity of 12%.[Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction
engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]
experiments and predictions are about 2.1% of initial droplet weight for the droplet weight prediction and about 1.9°C for the temperature prediction (Lin and Chen,2007).
It has been shown that the L-REA can model the convective drying of WPC accurately.
This could be due to the accuracy of the relative activation energy in capturing the physics of the convective drying of WPC. The combination between the equilibrium relative energy and relative activation energy shown by Equations (2.1.7) and (2.5.10), respectively seems to be sufficient to describe the change of internal behaviour of the WPC droplets during drying. Therefore, it can be said that the L-REA can describe the drying kinetics of the particulates well.
2.5.4 Results of modelling convective drying of a thin layer of a mixture of polymer solutions using the L-REA
Figures 2.12to 2.17 present results of the simulated drying profiles and temperature profiles using the L-REA. It can be seen that generally, all moisture content and tem-perature profiles agree well with the experimental data supported by R2 and RMSE of moisture content and temperature profile presented inTable 2.2.Figures 2.12and2.13 show the moisture content and temperature profile of convective drying conducted at 55°C and relative humidity of 12% at air velocity of 2.8 m s−1.Figure 2.12indicates that
Table 2.2 R2and RMSE of modelling of a mixture of polymer solutions using the L-REA.
Number R2X R2T RMSE X RMSE T
1 0.999 0.958 0.071 2.263
2 0.998 0.991 0.083 0.436
3 0.997 0.975 0.104 1.448
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
Temperature(K)
t(s) 295
300 305 310 315 320 325 330
Model Data
Figure 2.13 Product temperature profile of convective drying at an air temperature of 55°C, air velocity of 2.8 m s−1and air relative humidity of 12%.[Reprinted from Chemical Engineering and Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The
reaction engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]
moisture content profile can be predicted very well by the L-REA. Similarly, the temper-ature profile shown byFigure 2.13indicates very small differences between predicted and experimental data. This modelling is comparable with modelling of drying kinetics conducted by Allanic et al. (2009). Slight discrepancies with experimental temperature data were also shown although the model employed was based on a diffusion partial differential equation with fitted diffusivity (Allanic et al.,2009).
Figures 2.14and2.15provide results of modelling of drying conducted at 35°C and relative humidity of 30% at an air velocity of 1 m s−1 using the REA. A very good prediction of both moisture content and temperature data was observed. Compared with the simulation using the model proposed previously, it is apparent that the L-REA gives
0 2000 4000 6000 8000 10000 12000
X(kg water/kg dry solid)
t(s) 0
1 2 3 4 5 6 7 8
Model Data
Figure 2.14 Moisture content profile of convective drying at an air temperature of 35°C, air velocity of 1 m s−1and air relative humidity of 30%.[Reprinted from Chemical Engineering and
Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction
engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]
0 2000 4000 6000 8000 10000 12000
Temperature(K)
t(s) 294
296 298 300 302 304 306 308
Model Data
Figure 2.15 Product temperature profile of convective drying at air temperature of 35°C, air velocity of 1 m s−1and air relative humidity of 30%.[Reprinted from Chemical Engineering and
Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction
engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]
0 1000 2000 3000 4000 5000 6000 7000 8000
X(kg water/kg dry solid)
t(s) 0
1 2 3 4 5 6
Figure 2.16 Product temperature profile of convective drying at an air temperature of 55°C, air velocity of 1 m s−1and air relative humidity of 12%.[Reprinted from Chemical Engineering and
Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction
engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]
0 1000 2000 3000 4000 5000 6000 7000 8000
Temperature(K)
t(s) 300
305 310 315 320 325
Model Data
Figure 2.17 Product temperature profile of convective drying at an air temperature of 55°C, air velocity of 1 m s−1and air relative humidity of 12%.[Reprinted from Chemical Engineering and
Processing: Process Intensification, 49, A. Putranto, X.D. Chen and P.A. Webley, Infrared and convective drying of thin layer of polyvinyl alcohol (PVA)/glycerol/water mixture – The reaction
engineering approach (REA), 348–357, Copyright (2010), with permission from Elsevier.]
Reaction engineering approach I: L-REA 61
Table 2.3 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
better results because the diffusion model shows slight discrepancies in the moisture content profile during drying times around 4000–10 000 s (Allanic et al.,2009). The L-REA can be used to describe this well, as shown inFigure 2.14.
Similarly,Figures 2.16and2.17show a good agreement between estimated and exper-imental moisture and temperature data. Despite the simplicity of L-REA, it compares well with the model proposed before, which shows some discrepancies of moisture profile during drying times around 3000–6000 s (Allanic et al.,2009).
Overall the L-REA can be used successfully to model the thin layer drying of a mixture of polyvinyl alcohol, glycerol and water. The L-REA is shown to be able to model not only the convective drying of particulate or thin layer of food materials which has been proven before (Chen and Lin,2005; Lin and Chen,2005; 2006; 2007), but also that of thin layers of non-food materials. The accuracy of the L-REA could be due to the accuracy of the relative activation energy in describing the change of internal behaviour during drying.