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S-REA (spatial reaction engineering approach): An effective approach to model drying, baking and water vapor sorption processes
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Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
S-REA (spatial reaction engineering approach): An effective approach to model drying, baking and water vapor sorption processes Aditya Putranto a,b , Xiao Dong Chen c,∗ a
Department of Chemical Engineering, Monash University, Clayton Campus, Melbourne 3800, VIC, Australia Department of Chemical Engineering, Parahyangan Catholic University, Bandung, Indonesia c Suzhou Key Lab of Green Chemical Engineering, School of Chemical and Environmental Engineering, College of Chemistry, Chemical Engineering and Material Science, Soochow University, Suzhou, Jiangsu Province, China b
a r t i c l e
i n f o
a b s t r a c t
Article history:
An effective model of simultaneous heat and mass transfer processes is useful to assist
Received 29 January 2015
in process design, optimization of existing processes and monitoring product quality. Pre-
Received in revised form 29 April
viously, the lumped reaction engineering approach (L-REA) has been shown to model the
2015
global drying rate of several heat and mass transfer processes very well. Here, the REA
Accepted 2 May 2015
(reaction engineering approach) framework is implemented to model the local evapora-
Available online xxx
tion/condensation rate in drying of non-food materials, baking and water vapor sorption. The REA is combined with a set of equations of conservation of heat and mass transfer to
Keywords:
yield the S-REA (spatial reaction engineering approach) to model these processes. For mod-
S-REA (spatial reaction engineering
eling each process, the activation energy is generated from one accurate run and evaluated
approach)
according to environmental temperature and humidity. The relative activation energy imple-
Model
mented in the L-REA is implemented in the S-REA by applying local variables. The results
Heat and mass transfer
indicate that the S-REA is accurate to describe baking, drying and water vapor sorption pro-
Drying
cess which shows the applicability of the REA to model the local evaporation/condensation
Baking
rate of these processes. The S-REA is readily implemented to assist in process design, evalu-
Water vapor sorption
ation of existing processes and maintenance of product quality. In near future, by coupling with solid mechanics, the REA may be employed to predict material deformation and shape change during heat and mass transfer processes by coupling with solid mechanics. The development of the REA to describe migration of volatiles inside materials undergoing heat and mass transfer processes is also underway. © 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
1.
Introduction
Several heat and mass transfer processes including drying, baking and water vapor sorption are commonly applied in industry. For process design of new equipment and evaluation of current processes, a reliable mathematical model is useful. The model may also be used to predict product quality during the processes. The models may be classified into
empirical and mechanistic models. The empirical ones cannot capture physics during drying and cannot be extrapolated to other conditions (Page, 1949; Westerman et al., 1973; Overhults et al., 1973; Henderson, 1974; Wang and Singh, 1978). These models are not appropriate predictive tools. The mechanistic models usually employ diffusion-based models (Azzouz et al., 2002; Pakowski and Adamski, 2007; Mariani et al., 2008; Vaquiro et al., 2009). These may also include pore network and
∗
Corresponding author. Tel.: +86 18906053300. E-mail address: [email protected] (X.D. Chen). http://dx.doi.org/10.1016/j.cherd.2015.05.004 0263-8762/© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
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capillary system models (Younsi et al., 2006a,b, 2007; Kharaghani et al., 2012). These models capture the physics of the process in detail but they tend to require a lot of constants which need to be determined from several sets of experiments. They may also not be favorable for quick-decision-making in industry. The reaction engineering approach (REA), a semi-empirical approach, was proposed in 1996 and initially used to model thin layer drying of food materials (Chen and Lin, 2005; Lin and Chen, 2006, 2007). The model accurately captures the major physics of drying. The relative activation energy, a material characteristic, basically expresses the difficulty to remove the moisture from the materials. The removal of both free and bound moisture is captured by the relative activation energy. The REA has advantages of simplicity, accuracy and effectiveness with respect to generating the drying parameters. Only one accurate drying run is required to generate the parameters (Chen, 2008). The REA in its lumped format, later called as the lumped reaction engineering approach (L-REA), has accurately modeled several challenging heat and mass transfer processes. The L-REA describes to a high degree of accuracy of the following processes; time-varying drying, heat treatment of wood, baking and roasting (Putranto et al., 2011a,b,c; Putranto and Chen, 2012). For drying of non-food materials, predictions of spatial variables are essential to determine stress and cracking inside materials (Takhar et al., 2011). Extreme drying conditions may lead to cracking during drying of kaolin although it can be avoided by implementing drying under time-varying external conditions (Kowalski and Pawlowski, 2010a,b). Similarly, drying can alter the surface color, texture and solute distribution of food materials (Timoumi et al., 2007; Di Scala and Crapiste, 2008; Mrad et al., 2012; Ramallo and Mascheroni, 2012). The spatial models are important to predicting the quality changes so that the drying trajectories can be designed to minimize the quality loss during drying (Jin et al., 2013). For baking, understanding of the local variables will allow optimization of crispness, softness, color and volume change as a result of starch gelatinization, expansion and browning reactions (Hadiyanto et al., 2008). The temperature gradient inside the samples also needs to be studied since it induces the increase of local water vapor pressure which determines the evaporation/condensation inside the samples (De Vries et al., 1989). The spatial information regarding water content during vapor sorption is also useful to predicting surface color and material durability if exposed to fluctuating environmental conditions. The water vapor may also affect scission of macromolecular components inside wood which play an important role in changing the color of timber surface (Sharatt et al., 2011). The distribution of water vapor may also affect the ‘catalytic’ role of adsorbents since it may influence the mechanisms of component sorption (Popescu et al., 2014). As mentioned before, the diffusion-based models are spatial models that are commonly implemented to describe heat and mass transfer processes. However, for better understanding of transport phenomena, multiphase heat and transfer models need to be used (Zhang and Datta, 2004; Chen, 2007). There are two types of models that couple the vapor and liquid water in the solid matrix domain i.e. equilibrium and non-equilibrium models. The equilibrium model assumes that the moisture content inside the pore is in equilibrium with the concentration of water vapor and these can be linked by available isotherm data for the related materials. The model was applied to several cases and result
in a reasonably good agreement towards the experimental data (Zhang et al., 2005; Zhang and Datta, 2006; Curcio, 2010). The use of non-equilibrium model is encouraged as it is more general and can be used to assess the validity of the equilibrium model (Zhang and Datta, 2004; Chen, 2007). In non-equilibrium model, explicit formulation of local evaporation/condensation rate is needed. It was shown that the application of both liquid and vapor diffusion without use of source and depletion terms yielded unreasonable profiles of moisture content during drying (Chen, 2007; Kar, 2008). Due to the effectiveness of the REA to describe the global drying rate, it is appropriate to employ it to model the local evaporation/condensation rate. Here, basically, the REA is extrapolated to describe the drying rate in micro-scale of materials as affected by local structure and composition. The REA has been shown to describe well the local evaporation rate during convective drying of food and biological materials by coupling with a system of microscopic heat and mass balance to yield the spatial reaction engineering approach (S-REA) (Putranto and Chen, 2013a,b). Although the S-REA has been shown to describe accurately the convective and intermittent drying (Putranto and Chen, 2013a,b), each transport process may have unique mechanism as affected by local structure, porosity, variables and associated reactions. For baking, the transport process may be affected by starch gelatinization, protein denaturation and crust formation. Similarly, for drying of non-food materials, the local structure may not be significantly altered by the moisture as the initial moisture content is relatively low. It may be challenges by the REA to describe water vapor sorption since condensation should occur inside the materials. Therefore, the REA needs to be developed in order to describe the local rates of these processes. In this study, the REA is combined with a set of equations of conservation of heat and mass transfer to yield the S-REA to model drying of non-food materials, baking and water vapor sorption. This study aims to develop S-REA and implement it in the study of several heat and mass transfer processes. Its accuracy is examined by benchmarking model results against high quality experimental data reported in the literature. Firstly, experimental details are reviewed briefly followed up by the mathematical modeling using the S-REA for drying of non-food materials, baking and water vapor sorption. The accuracy of the S-REA approach for these processes is then discussed.
2.
Review of experimental details
Experimental data is used to assess the accuracy of the S-REA to model drying, baking and water vapor sorption. In order to better understand to experimental settings for developing the model, the experimental settings are reviewed briefly here. For drying, the applicability of the S-REA to describe the drying processes was benchmarked towards experimental data of Dhall et al. (2012). In this experiment, cylindrical honeycombs with diameter of 2.54 cm (refer to Fig. 1) were dried in commercial drier at relative humidity of 5% and temperature of 103 and 137 ◦ C. During drying, weight of the sample was recorded periodically so that the moisture content was determined. Similarly, the surface temperature was measured using FISO Fiber Optic Temperature Probes (Quebec, Canada).
Please cite this article in press as: Putranto, A., Chen, X.D., S-REA (spatial reaction engineering approach): An effective approach to model drying, baking and water vapor sorption processes. Chem. Eng. Res. Des. (2015), http://dx.doi.org/10.1016/j.cherd.2015.05.004
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1999; Zhang and Datta, 2004; Chen, 2007; Putranto and Chen, 2013a,b): 1 ∂ ∂(Cs X) = ∂t r ∂r
Fig. 1 – Schematic of honeycombs sample (Dhall et al., 2012).
In order to assess the applicability of the S-REA to model bread baking, the results of modeling are validated towards experimental data of Chinese bread baking of Zhang and Chen (2014). Chinese bread was made from dough using the following ingredients: bread flour (150 g), sugar (15 g), fine salt (2.5 g), butter (4.5 g), instant yeast (1.5 g), and UHT skim milk (100 g). First, dough was made in a mixer (Twinbird, Japan) then was proved at ambient temperature for 45 min. Subsequently, individual pieces (miniature breads) of 0.50 and 1.00 g (spherical shape) of dough were weighted accurately and placed in an aluminum alloy dish. Experiments were carried out by using at temperature of 175 ◦ C. Each experiment was carried out for 8 min. Weight loss measurements were conducted during baking at interval of 1 min. The sample core temperature during baking was measured at interval of 1 min using K-type thermocouples (Omega, USA) connected to a data logger (Pico, UK) which was incorporated to a PC. The experiments were done by triplicate with the oven under steady state conditions. In addition, the validity of the S-REA on modeling of water vapor sorption was assessed towards the experimental data of Li et al. (2007). Prior to experiments of water vapor sorption, silica gels with diameter of 3 mm are dried at 413 K for 4 h. For water vapor sorption, 2 g of silica gel was fed inside a covered airtight flask, placed on the microbalance located inside the sorption chamber. During the water vapor sorption, cycle of gas was used to adjust the temperature and relative humidity inside the chamber. The temperature was controlled with accuracy of ±5% and the relative humidity was temperature and relative humidity of the chamber is adjusted by using cycle of gas. The relative humidity is controlled with accuracy of ±3% by application of humidifier and dehumidifier. As soon as both temperature and relative humidity are set, the flask cover was opened to allow the water vapor sorption to start. The weight of the sample was recorded during the process until constant weight was achieved indicating the saturation of the samples. The similar procedures were repeated for experiments at various levels of humidity.
3. Mathematical modeling of several heat and mass transfer processes using the spatial reaction engineering approach (S-REA) 3.1.
Dw r
∂(Cs X) ∂r
.
−I
(1)
where Dw is the local surface diffusivity (m2 s−1 ), X is the concentration of liquid water (kg H2 O kg dry solids−1 ), Cs is the solids concentration (kg dry solids m−3 ) which can change if the structure changes, I˙ is the evaporation or condensation rate (kg H2 O m−3 s−1 ) and I˙ is positive when evaporation occurs locally. The mass balance of water in the vapor phase (water vapor) in spherical coordinate system is expressed as (Chong and Chen, 1999; Zhang and Datta, 2004; Chen, 2007; Putranto and Chen, 2013a,b): 1 ∂ ∂Cv = ∂t r ∂r
Dv r
∂Cv ∂r
+ I˙
(2)
where Dv is the local effective vapor diffusivity (m2 s−1 ) and Cv is the concentration of water vapor (kg H2 O m−3 ). The heat balance is represented by the following equation (Chong and Chen, 1999; Zhang and Datta, 2004; Chen, 2007; Putranto and Chen, 2013a,b): Cp
1 ∂ ∂T = ∂t r ∂r
∂T kr − I˙Hv
(3)
∂r
where T is the sample temperature (K), k is the local thermal conductivity of sample (W m−1 K−1 ), Hv is the water vaporization heat (J kg−1 ), Cp is the local specific heat of sample (W m−1 K−1 ) and is the local sample density (kg m−3 ). The initial and boundary conditions for Eqs. (1)–(3) are: t = 0,
X = Xo ,
r = 0,
dX = 0, dr
Cv = Cvo ,
T = To
dCv = 0, dr
dT =0 dr
C
(4) (5)
r = R,
−Cs Dw
−Dv
k
dX = hm εw dr
dCv = hm εv dr
v,s
ε
C
v,s
ε
dT = h (Tb − T) dr
− v,b
− v,b
(6)
(7)
(8)
where R is the sample radius (m), h is the heat transfer coefficient (W m−2 K−1 ), εw and εv are fraction of surface area influenced by liquid water and water vapor respectively. I˙ is the local evaporation/condensation rate within the solid structure described as (Kar and Chen, 2010, 2011; Putranto and Chen, 2013a,b):
S-REA for modeling of drying I˙ = hmin Ain (Cv,s − Cv )
Based on the experimental details summarized in Section 2, the S-REA is set up. It consists of a set of equations of microscopic heat and mass balance (in spherical coordinate system) in which the REA is used to describe the local evaporation/condensation rate. The mass balance of liquid water in Lagrangian coordinate, which allows expansion and contraction of the matrix, is written as (Chong and Chen,
(9)
where hmin is the internal mass transfer coefficient (m s−1 ), Ain is the total internal surface area available for phase change (m2 m−3 ), Cv,s is the internal-solid-surface water vapor concentration (kg m−3 ). Although the void space inside the samples may be large enough, the incorporation of the advection term in the S-REA is not necessary since its term has
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been captured by the local evaporation/condensation term as represented by Eq. (9). By implementing the REA, the internal-surface water vapor concentration can be written as (Kar and Chen, 2010, 2011; Putranto and Chen, 2013a,b):
Cv,s = exp
−E v
RT
Cv,sat
(10)
where Cv,sat is the internal-saturated water vapor concentration (kg m−3 ). Therefore, the local evaporation rate can be expressed as (Kar and Chen, 2010, 2011; Putranto and Chen, 2013a,b):
I˙ = hmin Ain exp
−E v
RT
Cv,sat − Cv
2
Based on the experimental settings mentioned in Section 2, the S-REA in spherical coordinates, similar to the one shown in Eqs. (1)–(3), is setup. In addition, the initial and boundary conditions shown in Eqs. (4)–(8) are implemented. The relative activation energy was generated from baking run at temperature of 205 ◦ C and can be written as:
5 4 3 Ev = −55.754 X − Xb + 113.95 X − Xb − 96.768 X − Xb Ev,b
+40.609 X − Xb
2
(13)
− 8.385 X − Xb + 1.006
(11)
5 4 3 Ev = −138, 685 X − Xb + 50, 405 X − Xb − 6953.5 X − Xb Ev,b
S-REA for modeling bread baking
The relative activation energy of honeycombs is generated from one accurate drying run at temperature of 103 ◦ C. The activation energy during drying is evaluated using Eq. (A2) and divided with the equilibrium activation energy represented in Eq. (A4) to yield the relative activation energy as mentioned in Eq. (A3). The relationship between the relative activation energy and average moisture content is assumed to be represented by simple mathematical equation obtained by least square method using Microsoft Excel® (Microsoft Corp, 2014). The relative activation energy was found to be suitably fitted as:
+466.12 X − Xb
3.2.
(12)
− 17.616 X − Xb + 1.0053
The good agreement between the fitted and experimental relative activation energy is shown by R2 of 0.995. For modeling using the S-REA here, the relative activation energy shown in ¯ in Eq. (12) is Eq. (12) is used but the average moisture content X substituted by the local moisture content (X) as the REA is used to represent the local evaporation rate instead of the global drying rate of the whole sample. The effective vapor diffusivity (Dv ), tortuosity (), solid concentration (Cs ) and porosity (ε) of the samples are evaluated using the procedures explained previously (Putranto and Chen, 2013a,b) and these transport parameters are summarized in Appendix B. According to Dhall et al. (2012), no shrinkage was observed during drying. In order to yield the spatial profiles, Eqs. (1)–(3) are solved in conjunction with the initial and boundary conditions shown in Eqs. (4)–(9) as well as the transport parameters, the relative and equilibrium activation energy mentioned above. Method of lines is used to solve the partial differential equations by first transforming the equations into a set of ordinary differential equations which is then solved using ‘ode23s’ solver, available in Matlab® , implements Rosenbrock formula of order 2. The equations are solved from t = 0 until the end of the processes. In the modeling, the radius change is incorporated by moving 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. Moving mesh was found to give better agreement towards experimental data than fixed coordinate (immobilizing boundary) (Thuwapanichayanan et al., 2008; Putranto and Chen, 2013a,b).
The effective liquid diffusivity (Dw ) of baking of bread is expressed as (Ni et al., 1999): Dw = 1 × 10−6 exp(−2.8 + 2X)ε
(14)
where ε is the fraction volume taken by the vapor. The radius change during baking is incorporated in the modeling and described as:
Rs = −0.008X2 + 0.003X + 0.0073
(15)
where Rs is the sample radius (m) and X is the moisture content (kg H2 O kg dry solids−1 ). The numerical method, similar to the one mentioned in Section 3.1, is implemented to solve the equations to yield the spatial profiles of moisture content, concentration of water vapor and moisture content. In order to incorporate the radius change shown in Eq. (15) in the modeling, moving mesh is used.
3.3.
S-REA for modeling water vapor sorption process
The spatial reaction engineering approach (S-REA), similar to the ones shown in Eqs. (1)–(3), is setup. Similarly, the initial and boundary conditions shown in Eqs. (4)–(8) are used. The relative activation energy of silica gel can be expressed as (Putranto and Chen, 2014):
4 3 2 Ev = −4.19 X − Xb − 15.66 X − Xb + 3.192 X − Xb Ev,b
− 10.063 X − Xb + 1
(16)
Eq. (16) has been used to model the global wetting rate of silica gel successfully (Putranto and Chen, 2014). Here, by ¯ with local moisreplacing the average moisture content (X) ture content (X), Eq. (16) is used to describe the local wetting rate of silica gel. The numerical method mentioned above is implemented here to yield the spatial profiles during water vapor sorption. During experiments, no volume change was observed (Li et al., 2007). Similarly, the modeling by Putranto and Chen (2014) did not incorporate the volume change during water sorption process.
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Fig. 2 – Sensitivity of effective liquid diffusivity towards profiles of moisture content during drying of honeycombs at temperature of 137 ◦ C.
5
Fig. 4 – Profiles of moisture content during drying of honeycombs at temperature of 103 and 137 ◦ C.
4. Results of mathematical modeling of several heat and mass transfer processes using the lumped reaction engineering approach (S-REA) 4.1. Modeling of drying using the spatial reaction engineering approach (S-REA) Figs. 1–6 indicate the results of modeling of drying of honeycombs at drying air temperature of 103 and 137 ◦ C. As suggested by Dhall et al. (2012), the liquid diffusivity of 10−10 m2 s−1 is used. Interestingly, as shown in Figs. 2 and 3, the liquid diffusivity of lower than 1 × 10−10 m2 s−1 does not result in any noticeable differences in the moisture content and temperature profiles. This may indicate that the modeling is independent on surface diffusivity and hence the surface diffusion term may be neglected in the modeling. This is similar to modeling heat treatment of wood under linearlyincreased gas temperature in which the profiles of moisture content and temperature are independent on the liquid diffusivity so that the surface diffusion term is neglected (Putranto and Chen, 2013b).
Fig. 3 – Sensitivity of effective liquid diffusivity towards profiles of temperature during drying of honeycombs at temperature of 137 ◦ C.
Fig. 5 – Profiles of surface temperature during drying of honeycombs at temperature of 103 and 137 ◦ C. Without using surface diffusion term on the mass balance (refer to Eq. (1)), the results of modeling of drying at temperature of 103 and 137 ◦ C are shown in Figs. 4 and 5. A good agreement towards the experimental data is shown (R2 of higher than 0.998 for moisture content and R2 of higher than 0.993 for temperature). The S-REA is accurate to model
Fig. 6 – Spatial profiles of moisture content during drying of honeycombs at temperature of 137 ◦ C.
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Fig. 7 – Spatial profiles of temperature during drying of honeycombs at temperature of 137 ◦ C.
both moisture content and temperature profiles at drying air temperature of 103 and 137 ◦ C. Benchmarks towards modeling implemented by Dhall et al. (2012), shown in Fig. 5, indicate that the S-REA results in closer agreement towards the experimental data. This could be because of the sufficiency of the combination of the REA and a set of equations of microscopic heat and mass balances to describe the drying processes. Here, both moisture content and temperature profiles are independent on the surface diffusivity which may be because of the low initial moisture content and high porosity of the samples. The movement of moisture by capillary flow and moisture diffusion may be much faster than vapor diffusion and evaporation so that the transport phenomena inside the samples are governed by the vapor diffusion and evaporation/condensation. Similar phenomena also occurs during heat treatment of wood reported previously (Putranto and Chen, 2013b). Fig. 6 indicates the spatial profiles of moisture content during drying of honeycombs at temperature of 137 ◦ C. The moisture content at the outer part of the samples is lower than that at the inner part of the samples which may indicate that the moisture migrates outwards during drying. Initially, the gradient of moisture content inside the samples is relatively large. Nevertheless, the gradient decreases as drying progresses and at the end of drying, no noticeable gradient are observed which indicates that the equilibrium conditions may have been attained. Fig. 7 shows the spatial profiles of temperature during drying of honeycombs at temperature of 137 ◦ C. The temperature of the samples at the outer part is higher than that of the inner part which indicates that the heat is penetrated inwards during drying for evaporating water and increasing sample temperature. During drying, the temperature increases to approach drying air temperature and the gradient of temperature decreases. At the end of drying, no noticeable differences of temperature inside the samples are observed, in agreement with the spatial profiles of moisture content shown in Fig. 6. Figs. 2–7 show that the S-REA is accurate to model drying of non-food materials. Previously, the S-REA was implemented to describe drying of food and biological materials and the results of modeling match very well with the experimental data (Putranto and Chen, 2013a). The accuracy of the S-REA may be because of the applicability of the REA to describe the local evaporation/condensation rate. The combination of
Fig. 8 – Profiles of average moisture content during baking of bread at baking temperature of 175 ◦ C. relative and equilibrium activation energy yield unique activation energy which may allow the representation of local changes as affected by environmental conditions. For modeling of drying of samples with low moisture content, such as honeycombs, the surface diffusion term is not necessary to be included but the evaporation/condensation term needs to be incorporated. This also highlights the importance of local evaporation/condensation rate in the modeling of heat and mass transfer processes. It has been shown here that the REA models the local evaporation rate of non-food materials accurately. Because of the accuracy of the S-REA, it may be adopted for process design and product quality of several materials during drying. By using the S-REA, transport phenomena inside porous materials can be better understood. For process design, the S-REA may yield as reliable model to explore various drying schemes favorable for energy minimization. The S-REA may also be used to effectively fine-tune the operating conditions of drying of food materials to minimize nutrient degradation, protein denaturation, color change and texture loss. Similarly, for drying of non-food materials, the S-REA may be applied to design cycle time and intermittency to minimize cracking or material deformation.
4.2. Modeling of bread baking using the spatial reaction engineering approach (S-REA) Figs. 8 and 9 show the results of modeling at baking temperature of 175 ◦ C. The moisture content profiles during baking of 0.5 and 1 g bread are shown in Fig. 8. A good agreement towards the experimental data is shown (R2 of 0.992 and 0.988 for 0.5 and 1 g bread, respectively). For the temperature profiles, as shown in Fig. 9, the S-REA predictions match well with the experimental data. The good agreement is shown by R2 of 0.963 and 0.985 for 0.5 and 1 g bread, respectively. The combination between the REA and a system of equations of conservation of heat and mass transfer seems to be sufficient to describe the transport phenomena during bread baking. Fig. 10 shows the spatial profiles of moisture content during baking of 1 g of bread at baking temperature of 235 ◦ C. The profiles at the outer part of the samples are lower than those at the inner part which indicate that the moisture migrate outwards during baking. The profiles decrease during baking time since the moisture in the samples depletes as baking progresses.
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Fig. 9 – Profiles of core temperature during baking of bread at baking temperature of 175 ◦ C.
Fig. 10 – Spatial profiles of moisture content during bread baking of 1 g of bread at baking temperature of 235 ◦ C.
Fig. 12 – Sensitivity of the effective liquid diffusivity towards the moisture content profiles during water vapor sorption at relative humidity of 10%.
heat by convection from the environment and used for water vaporization. The excessive heat may be penetrated inwards by conduction to increase the centre temperature. The profiles of temperature here corresponds well the concentrations of water vapor. As explained above, the higher temperature at the outer part of the samples seems to result in higher local evaporation rate and thus concentration of water vapor near the surface of the samples. Figs. 8–11 indicate that the S-REA models well the bread baking process. It is obvious that the success of this approach is based on the accurate representation of the evaporation and condensation behavior by the REA. Similar to drying of honeycombs and biological materials presented previously (Putranto and Chen, 2013a), the REA seems to be able to capture the environmental conditions and reflect them to the changes in local structure during baking. Several chemical changes during baking covering starch gelatinization, starch retrogradation and protein denaturation seem to be captured well in the relative energy mentioned in Eq. (13). The S-REA is not only accurate to describe drying processes but also bread baking. The spatial profiles generated by the S-REA are useful for predictions of quality changes during baking especially crust formation and surface browning. The profiles of moisture content and temperature can be combined with established equations to project the color kinetics and probiotics survival during breadbaking. The S-REA may also be adopted in process design to explore various baking schemes favorable for energy minimization. Alternatively, it can also be implemented computational fluid dynamics (CFD)-based simulations to predict the flowfield inside baking chamber—a multiscale modeling exercise.
4.3. Modeling of water vapor sorption using the spatial reaction engineering approach (S-REA) Fig. 11 – Spatial profiles of temperature during bread baking of 1 g of bread at baking temperature of 235 ◦ C. Fig. 11 shows the spatial profiles of temperature during baking of 1 g bread at baking temperature of 235 ◦ C. The temperature at the outer part is higher than that of the inner part of the samples. This may indicate that the samples receive
Figs. 12–15 indicate the results of modeling of water vapor sorption using the S-REA. As suggested by Pesaran and Mills, 1987), the effective liquid diffusivity of silica gel is:
Dw = 1.6 × 10−6 exp −0.974 × 10−3
Hads T
(17)
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Fig. 13 – Profiles of moisture content during water vapor sorption at various relative humidity.
Fig. 14 – Spatial profiles of moisture content inside silica gel during water vapor sorption at relative humidity of 10%.
Fig. 15 – Spatial profiles of concentration of water vapor inside silica gel during water vapor sorption at relative humidity of 10%.
where Hads is the heat of adsorption (J kg−1 ) dan Dw is the surface diffusivity (m2 s−1 ). In this study, the surface diffusivity is varied from 1 × 10−10 to 1 × 10−30 m2 s−1 and the results are shown in Fig. 12. Similar to drying of honeycombs, the varied surface diffusivity does not yield any noticeable difference in the profiles of moisture content. This may indicate that the modeling is independent on surface diffusivity and the surface diffusion term can be neglected. Without using surface diffusion term on the mass balance, the results of modeling of sorption of water vapor into silica gel are shown in Figs. 13–15. A good agreement towards the experimental data is shown in Fig. 13. For sorption of water vapor into silica gel at relative humidity of 10%, the S-REA models well the moisture content during water vapor sorption (R2 of 0.993). Similarly, a good agreement towards experimental data is shown (R2 of 0.992) for water vapor sorption at relative humidity of 45%. The results of modeling using the S-REA at higher relative humidity i.e. 65 also match well with the experimental data. The S-REA models well the average moisture content during water vapor sorption at various relative humidity. Although the surface diffusion term is not implemented in the modeling, the results match very well with the experimental data. This could be because the porosity of the samples is relatively high and the samples are relatively small. Similar to drying mentioned in Section 4.1 and heat treatment of wood reported previously (Putranto and Chen, 2013b), condensation of water vapor and vapor diffusion may be the governing transport processes during water vapor sorption. This also emphasizes the importance of local condensation rate in modeling of water vapor sorption. This expression seems to be necessary to accurately model condensation in heat and mass transfer processes. Here, the REA can describe the local condensation rate well. Fig. 14 shows the spatial profiles of moisture content during water vapor sorption at relative humidity of 10%. There is no noticeable difference of gradient of moisture content during water vapor sorption. Similarly, as shown in Fig. 15, the concentration of water vapor inside the samples is basically uniform which could be because of the small sample used in the experiments. Nevertheless, for larger samples, the S-REA is useful to better understand the transport phenomena during water vapor sorption because of the generation of spatial profiles as affected by local structure and variables. Figs. 13–15 indicate that the S-REA models accurately the water vapor sorption which may be because the REA describes well the local condensation rate. The applicability of the REA indicates that the relative activation energy used to model the global wetting rate (refer to Eq. (16)) (Putranto and Chen, 2014) is also applicable to model the local wetting rate of water vapor sorption. Similar to ‘local’ relative activation energy for drying, the relative activation energy mentioned in Eq. (16) is essentially an extrapolation of relative activation energy to describe the wetting rate at micro scale. Fig. 16 shows that both the L-REA (lumped reaction engineering approach) (Putranto and Chen, 2014) and the S-REA models accurately the water vapor sorption. While the L-REA provides simplicity in mathematical modeling, the S-REA yields advantages of generating the spatial profiles. It is proven here that the REA is not only accurate to describe the local evaporation rate but also the local condensation rate. The applicability of the REA here is facilitated by the two terms of the REA (i.e. evaporation and condensation) as shown in Eq. (11). Because of the accuracy of the S-REA
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Appendix A. Brief review of the reaction engineering approach (REA) (Chen and Putranto, 2013) By using the REA, the mass balance of water during heat and mass transfer processes can be expressed as (Chen and Xie, 1997; Chen, 2008): ms
Fig. 16 – Profiles of average moisture content during water vapor sorption at various relative humidity modeled by the L-REA and S-REA.
here, the S-REA is readily implemented to design new water vapor sorption process and evaluate the existing facilities. The development of the REA to describe drying of solutes dissolved in organic substances is now underway. It is anticipated that the REA is also able to describe the migration of volatiles inside the materials so that the composition of each component involved in heat and mass transfer processes can better represented. Similarly, for the prediction of material deformation, the REA may be combined with solid mechanics to yield the profiles of stress-strain during heat and mass transfer processes. This development may also enable the prediction of material shape changes during transport processes.
5.
Conclusions
In this paper, the REA is used to describe local evaporation/condensation rate in several heat and mass transfer processes including drying, baking and water vapor sorption. The REA framework is combined with a set of equations of conservation of heat and mass transfer processes to yield the spatial reaction engineering approach (S-REA). The relative activation energy, generated in one accurate experiment and implemented to represent the global drying rate, is applied here to model the local evaporation/condensation rate. When validated towards experimental data, it is indicated that the S-REA describes accurately drying, baking and water vapor sorption. In drying and water vapor sorption, the surface diffusion term may not be used in the modeling because of the low moisture content and high porosity of the samples. The accuracy of the S-REA could be because of the accuracy and flexibility of the relative activation energy to capture the environmental change and reflect to the change of internal behavior as affected by the local variables and structures. The REA framework is readily adopted to assist in process design and maintaining product quality during several heat and mass transfer processes in industries. The development of the REA to predict drying of materials dissolved in non-water components as well as stress-strain is now underway.
¯ dX = −hm A exp dt
−E v
RTs
v,sat (Ts ) − v,b
(A1)
¯ is the average moisture content on dry basis, hm is the where X mass transfer coefficient (m s−1 ), A is the surface area (m2 ), Ev is the activation energy (J mol−1 ), Ts is the sample temperature (K), ms is the dried mass of sample (kg), v,sat is the saturated water vapor concentration (kg m−3 ) and v,b is the concentration of water vapor in drying medium (kg m−3 ). Eq. (A1) indicates that the REA is expressed in first order ordinary differential equation with respect to time. Eq. (A1) is the core of the reaction engineering approach, further called as the lumped reaction engineering approach (L-REA). The L-REA does not assume uniform moisture content but it evaluates average moisture content of the samples during drying. The activation energy (Ev ) is determined experimentally by placing the parameters required for Eq. (A1) in its rearranged form:
Ev = −RTs ln
¯ −ms dX/dt (1/hm A) + v,b v,sat
(A2)
¯ where dX/dt is experimentally determined. Besides the average moisture content, the surface area, temperature and mass transfer coefficient need to be measured or known. The dependence of activation energy on average moisture ¯ can be normalized as: content on a dry basis (X) Ev ¯ − Xb ) = f (X Ev,b
(A3)
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 )
(A4)
RHb is the relative humidity of drying air and Tb is the drying air temperature (K). In order to generate the relative activation energy (Ev /Ev,b ) shown by Eq. (A4), the activation energy (Ev ) can be evaluated by Eq. (A2) from one accurate drying experiment. So far, the experiments conducted to generate the relationship (Eq. (A3)) generally employed fairly dry air so the relationship ¯ − Xb ) covers a complete range of water content difference (X while Xb in the experiments for generating REA parameters is set to be very small value. The activation energy is divided by the equilibrium activation energy (Ev,b ) indicated by Eq. (A4) to yield the relative activation energy during drying. This is a normalization process. For similar drying condition and initial water content, it is possible to obtain the necessary REA parameters (apart from the equilibrium isotherm), expressed in the relative activation energy (Ev /Ev,b ) as indicated in Eq. (A3) in one accurate drying experiment. The relative activation energy (Ev /Ev,b )
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generated can then be used to project to other drying conditions provided the material has the same initial moisture content (Chen and Lin, 2005; Chen, 2008).
Appendix B. Procedures to determine effective vapor diffusivity, tortuosity, solid concentration and porosity of the samples The effective vapor diffusivity (Dv ) is deduced from (Bird et al., 2002): Dv = Dvo
ε
(B1)
where ε is the porosity, is the tortuosity, Dvo is the water vapor diffusivity (m2 s−1 ) which is dependent on temperature, which can be expressed as (Slattery and Bird, 1958): Dvo = 2.09 × 10−5 + 2.137 × 10−7 (T − 273.15)
(B2)
The tortuosity () of the samples is generally related to the porosity (ε). The relationship can be represented as (Audu and Jeffreys, 1975; Gimmi et al., 1993): = ε−n
(B3)
where n is the value between 0 and 0.5 (Audu and Jeffreys, 1975; Gimmi et al., 1993). There is no noticeable effect of various values of n on the profiles of liquid water concentration, water vapor concentration and temperature as witnessed in many numerical tests performed in this study and n = 0.5 is chosen in this study. Cs is the solid concentration which can be expressed by (Kar and Chen, 2010, 2011; Putranto and Chen, 2013a,b): Cs =
1−ε (1/s ) + (X/w )
(B4)
where s is the dry solid density (kg m−3 ) and w is the density of water (kg m−3 ), ε is the porosity, dependent on shrinkage and local moisture content. This can be determined according to Madiouli et al. (2007): ε=1−
V0 (1 − ε0 ) V
¯ +1 (s /w ) X 1 + (s /w ) X0
(B5)
where V0 is the initial volume of the sample (m3 ) and ε0 is the initial porosity of the sample.
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Contents lists available at ScienceDirect
Chemical Engineering Research and Design journal homepage: www.elsevier.com/locate/cherd
S-REA (spatial reaction engineering approach): An effective approach to model drying, baking and water vapor sorption processes Aditya Putranto a,b , Xiao Dong Chen c,∗ a
Department of Chemical Engineering, Monash University, Clayton Campus, Melbourne 3800, VIC, Australia Department of Chemical Engineering, Parahyangan Catholic University, Bandung, Indonesia c Suzhou Key Lab of Green Chemical Engineering, School of Chemical and Environmental Engineering, College of Chemistry, Chemical Engineering and Material Science, Soochow University, Suzhou, Jiangsu Province, China b
a r t i c l e
i n f o
a b s t r a c t
Article history:
An effective model of simultaneous heat and mass transfer processes is useful to assist
Received 29 January 2015
in process design, optimization of existing processes and monitoring product quality. Pre-
Received in revised form 29 April
viously, the lumped reaction engineering approach (L-REA) has been shown to model the
2015
global drying rate of several heat and mass transfer processes very well. Here, the REA
Accepted 2 May 2015
(reaction engineering approach) framework is implemented to model the local evapora-
Available online xxx
tion/condensation rate in drying of non-food materials, baking and water vapor sorption. The REA is combined with a set of equations of conservation of heat and mass transfer to
Keywords:
yield the S-REA (spatial reaction engineering approach) to model these processes. For mod-
S-REA (spatial reaction engineering
eling each process, the activation energy is generated from one accurate run and evaluated
approach)
according to environmental temperature and humidity. The relative activation energy imple-
Model
mented in the L-REA is implemented in the S-REA by applying local variables. The results
Heat and mass transfer
indicate that the S-REA is accurate to describe baking, drying and water vapor sorption pro-
Drying
cess which shows the applicability of the REA to model the local evaporation/condensation
Baking
rate of these processes. The S-REA is readily implemented to assist in process design, evalu-
Water vapor sorption
ation of existing processes and maintenance of product quality. In near future, by coupling with solid mechanics, the REA may be employed to predict material deformation and shape change during heat and mass transfer processes by coupling with solid mechanics. The development of the REA to describe migration of volatiles inside materials undergoing heat and mass transfer processes is also underway. © 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
1.
Introduction
Several heat and mass transfer processes including drying, baking and water vapor sorption are commonly applied in industry. For process design of new equipment and evaluation of current processes, a reliable mathematical model is useful. The model may also be used to predict product quality during the processes. The models may be classified into
empirical and mechanistic models. The empirical ones cannot capture physics during drying and cannot be extrapolated to other conditions (Page, 1949; Westerman et al., 1973; Overhults et al., 1973; Henderson, 1974; Wang and Singh, 1978). These models are not appropriate predictive tools. The mechanistic models usually employ diffusion-based models (Azzouz et al., 2002; Pakowski and Adamski, 2007; Mariani et al., 2008; Vaquiro et al., 2009). These may also include pore network and
∗
Corresponding author. Tel.: +86 18906053300. E-mail address: [email protected] (X.D. Chen). http://dx.doi.org/10.1016/j.cherd.2015.05.004 0263-8762/© 2015 The Institution of Chemical Engineers. Published by Elsevier B.V. All rights reserved.
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capillary system models (Younsi et al., 2006a,b, 2007; Kharaghani et al., 2012). These models capture the physics of the process in detail but they tend to require a lot of constants which need to be determined from several sets of experiments. They may also not be favorable for quick-decision-making in industry. The reaction engineering approach (REA), a semi-empirical approach, was proposed in 1996 and initially used to model thin layer drying of food materials (Chen and Lin, 2005; Lin and Chen, 2006, 2007). The model accurately captures the major physics of drying. The relative activation energy, a material characteristic, basically expresses the difficulty to remove the moisture from the materials. The removal of both free and bound moisture is captured by the relative activation energy. The REA has advantages of simplicity, accuracy and effectiveness with respect to generating the drying parameters. Only one accurate drying run is required to generate the parameters (Chen, 2008). The REA in its lumped format, later called as the lumped reaction engineering approach (L-REA), has accurately modeled several challenging heat and mass transfer processes. The L-REA describes to a high degree of accuracy of the following processes; time-varying drying, heat treatment of wood, baking and roasting (Putranto et al., 2011a,b,c; Putranto and Chen, 2012). For drying of non-food materials, predictions of spatial variables are essential to determine stress and cracking inside materials (Takhar et al., 2011). Extreme drying conditions may lead to cracking during drying of kaolin although it can be avoided by implementing drying under time-varying external conditions (Kowalski and Pawlowski, 2010a,b). Similarly, drying can alter the surface color, texture and solute distribution of food materials (Timoumi et al., 2007; Di Scala and Crapiste, 2008; Mrad et al., 2012; Ramallo and Mascheroni, 2012). The spatial models are important to predicting the quality changes so that the drying trajectories can be designed to minimize the quality loss during drying (Jin et al., 2013). For baking, understanding of the local variables will allow optimization of crispness, softness, color and volume change as a result of starch gelatinization, expansion and browning reactions (Hadiyanto et al., 2008). The temperature gradient inside the samples also needs to be studied since it induces the increase of local water vapor pressure which determines the evaporation/condensation inside the samples (De Vries et al., 1989). The spatial information regarding water content during vapor sorption is also useful to predicting surface color and material durability if exposed to fluctuating environmental conditions. The water vapor may also affect scission of macromolecular components inside wood which play an important role in changing the color of timber surface (Sharatt et al., 2011). The distribution of water vapor may also affect the ‘catalytic’ role of adsorbents since it may influence the mechanisms of component sorption (Popescu et al., 2014). As mentioned before, the diffusion-based models are spatial models that are commonly implemented to describe heat and mass transfer processes. However, for better understanding of transport phenomena, multiphase heat and transfer models need to be used (Zhang and Datta, 2004; Chen, 2007). There are two types of models that couple the vapor and liquid water in the solid matrix domain i.e. equilibrium and non-equilibrium models. The equilibrium model assumes that the moisture content inside the pore is in equilibrium with the concentration of water vapor and these can be linked by available isotherm data for the related materials. The model was applied to several cases and result
in a reasonably good agreement towards the experimental data (Zhang et al., 2005; Zhang and Datta, 2006; Curcio, 2010). The use of non-equilibrium model is encouraged as it is more general and can be used to assess the validity of the equilibrium model (Zhang and Datta, 2004; Chen, 2007). In non-equilibrium model, explicit formulation of local evaporation/condensation rate is needed. It was shown that the application of both liquid and vapor diffusion without use of source and depletion terms yielded unreasonable profiles of moisture content during drying (Chen, 2007; Kar, 2008). Due to the effectiveness of the REA to describe the global drying rate, it is appropriate to employ it to model the local evaporation/condensation rate. Here, basically, the REA is extrapolated to describe the drying rate in micro-scale of materials as affected by local structure and composition. The REA has been shown to describe well the local evaporation rate during convective drying of food and biological materials by coupling with a system of microscopic heat and mass balance to yield the spatial reaction engineering approach (S-REA) (Putranto and Chen, 2013a,b). Although the S-REA has been shown to describe accurately the convective and intermittent drying (Putranto and Chen, 2013a,b), each transport process may have unique mechanism as affected by local structure, porosity, variables and associated reactions. For baking, the transport process may be affected by starch gelatinization, protein denaturation and crust formation. Similarly, for drying of non-food materials, the local structure may not be significantly altered by the moisture as the initial moisture content is relatively low. It may be challenges by the REA to describe water vapor sorption since condensation should occur inside the materials. Therefore, the REA needs to be developed in order to describe the local rates of these processes. In this study, the REA is combined with a set of equations of conservation of heat and mass transfer to yield the S-REA to model drying of non-food materials, baking and water vapor sorption. This study aims to develop S-REA and implement it in the study of several heat and mass transfer processes. Its accuracy is examined by benchmarking model results against high quality experimental data reported in the literature. Firstly, experimental details are reviewed briefly followed up by the mathematical modeling using the S-REA for drying of non-food materials, baking and water vapor sorption. The accuracy of the S-REA approach for these processes is then discussed.
2.
Review of experimental details
Experimental data is used to assess the accuracy of the S-REA to model drying, baking and water vapor sorption. In order to better understand to experimental settings for developing the model, the experimental settings are reviewed briefly here. For drying, the applicability of the S-REA to describe the drying processes was benchmarked towards experimental data of Dhall et al. (2012). In this experiment, cylindrical honeycombs with diameter of 2.54 cm (refer to Fig. 1) were dried in commercial drier at relative humidity of 5% and temperature of 103 and 137 ◦ C. During drying, weight of the sample was recorded periodically so that the moisture content was determined. Similarly, the surface temperature was measured using FISO Fiber Optic Temperature Probes (Quebec, Canada).
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1999; Zhang and Datta, 2004; Chen, 2007; Putranto and Chen, 2013a,b): 1 ∂ ∂(Cs X) = ∂t r ∂r
Fig. 1 – Schematic of honeycombs sample (Dhall et al., 2012).
In order to assess the applicability of the S-REA to model bread baking, the results of modeling are validated towards experimental data of Chinese bread baking of Zhang and Chen (2014). Chinese bread was made from dough using the following ingredients: bread flour (150 g), sugar (15 g), fine salt (2.5 g), butter (4.5 g), instant yeast (1.5 g), and UHT skim milk (100 g). First, dough was made in a mixer (Twinbird, Japan) then was proved at ambient temperature for 45 min. Subsequently, individual pieces (miniature breads) of 0.50 and 1.00 g (spherical shape) of dough were weighted accurately and placed in an aluminum alloy dish. Experiments were carried out by using at temperature of 175 ◦ C. Each experiment was carried out for 8 min. Weight loss measurements were conducted during baking at interval of 1 min. The sample core temperature during baking was measured at interval of 1 min using K-type thermocouples (Omega, USA) connected to a data logger (Pico, UK) which was incorporated to a PC. The experiments were done by triplicate with the oven under steady state conditions. In addition, the validity of the S-REA on modeling of water vapor sorption was assessed towards the experimental data of Li et al. (2007). Prior to experiments of water vapor sorption, silica gels with diameter of 3 mm are dried at 413 K for 4 h. For water vapor sorption, 2 g of silica gel was fed inside a covered airtight flask, placed on the microbalance located inside the sorption chamber. During the water vapor sorption, cycle of gas was used to adjust the temperature and relative humidity inside the chamber. The temperature was controlled with accuracy of ±5% and the relative humidity was temperature and relative humidity of the chamber is adjusted by using cycle of gas. The relative humidity is controlled with accuracy of ±3% by application of humidifier and dehumidifier. As soon as both temperature and relative humidity are set, the flask cover was opened to allow the water vapor sorption to start. The weight of the sample was recorded during the process until constant weight was achieved indicating the saturation of the samples. The similar procedures were repeated for experiments at various levels of humidity.
3. Mathematical modeling of several heat and mass transfer processes using the spatial reaction engineering approach (S-REA) 3.1.
Dw r
∂(Cs X) ∂r
.
−I
(1)
where Dw is the local surface diffusivity (m2 s−1 ), X is the concentration of liquid water (kg H2 O kg dry solids−1 ), Cs is the solids concentration (kg dry solids m−3 ) which can change if the structure changes, I˙ is the evaporation or condensation rate (kg H2 O m−3 s−1 ) and I˙ is positive when evaporation occurs locally. The mass balance of water in the vapor phase (water vapor) in spherical coordinate system is expressed as (Chong and Chen, 1999; Zhang and Datta, 2004; Chen, 2007; Putranto and Chen, 2013a,b): 1 ∂ ∂Cv = ∂t r ∂r
Dv r
∂Cv ∂r
+ I˙
(2)
where Dv is the local effective vapor diffusivity (m2 s−1 ) and Cv is the concentration of water vapor (kg H2 O m−3 ). The heat balance is represented by the following equation (Chong and Chen, 1999; Zhang and Datta, 2004; Chen, 2007; Putranto and Chen, 2013a,b): Cp
1 ∂ ∂T = ∂t r ∂r
∂T kr − I˙Hv
(3)
∂r
where T is the sample temperature (K), k is the local thermal conductivity of sample (W m−1 K−1 ), Hv is the water vaporization heat (J kg−1 ), Cp is the local specific heat of sample (W m−1 K−1 ) and is the local sample density (kg m−3 ). The initial and boundary conditions for Eqs. (1)–(3) are: t = 0,
X = Xo ,
r = 0,
dX = 0, dr
Cv = Cvo ,
T = To
dCv = 0, dr
dT =0 dr
C
(4) (5)
r = R,
−Cs Dw
−Dv
k
dX = hm εw dr
dCv = hm εv dr
v,s
ε
C
v,s
ε
dT = h (Tb − T) dr
− v,b
− v,b
(6)
(7)
(8)
where R is the sample radius (m), h is the heat transfer coefficient (W m−2 K−1 ), εw and εv are fraction of surface area influenced by liquid water and water vapor respectively. I˙ is the local evaporation/condensation rate within the solid structure described as (Kar and Chen, 2010, 2011; Putranto and Chen, 2013a,b):
S-REA for modeling of drying I˙ = hmin Ain (Cv,s − Cv )
Based on the experimental details summarized in Section 2, the S-REA is set up. It consists of a set of equations of microscopic heat and mass balance (in spherical coordinate system) in which the REA is used to describe the local evaporation/condensation rate. The mass balance of liquid water in Lagrangian coordinate, which allows expansion and contraction of the matrix, is written as (Chong and Chen,
(9)
where hmin is the internal mass transfer coefficient (m s−1 ), Ain is the total internal surface area available for phase change (m2 m−3 ), Cv,s is the internal-solid-surface water vapor concentration (kg m−3 ). Although the void space inside the samples may be large enough, the incorporation of the advection term in the S-REA is not necessary since its term has
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been captured by the local evaporation/condensation term as represented by Eq. (9). By implementing the REA, the internal-surface water vapor concentration can be written as (Kar and Chen, 2010, 2011; Putranto and Chen, 2013a,b):
Cv,s = exp
−E v
RT
Cv,sat
(10)
where Cv,sat is the internal-saturated water vapor concentration (kg m−3 ). Therefore, the local evaporation rate can be expressed as (Kar and Chen, 2010, 2011; Putranto and Chen, 2013a,b):
I˙ = hmin Ain exp
−E v
RT
Cv,sat − Cv
2
Based on the experimental settings mentioned in Section 2, the S-REA in spherical coordinates, similar to the one shown in Eqs. (1)–(3), is setup. In addition, the initial and boundary conditions shown in Eqs. (4)–(8) are implemented. The relative activation energy was generated from baking run at temperature of 205 ◦ C and can be written as:
5 4 3 Ev = −55.754 X − Xb + 113.95 X − Xb − 96.768 X − Xb Ev,b
+40.609 X − Xb
2
(13)
− 8.385 X − Xb + 1.006
(11)
5 4 3 Ev = −138, 685 X − Xb + 50, 405 X − Xb − 6953.5 X − Xb Ev,b
S-REA for modeling bread baking
The relative activation energy of honeycombs is generated from one accurate drying run at temperature of 103 ◦ C. The activation energy during drying is evaluated using Eq. (A2) and divided with the equilibrium activation energy represented in Eq. (A4) to yield the relative activation energy as mentioned in Eq. (A3). The relationship between the relative activation energy and average moisture content is assumed to be represented by simple mathematical equation obtained by least square method using Microsoft Excel® (Microsoft Corp, 2014). The relative activation energy was found to be suitably fitted as:
+466.12 X − Xb
3.2.
(12)
− 17.616 X − Xb + 1.0053
The good agreement between the fitted and experimental relative activation energy is shown by R2 of 0.995. For modeling using the S-REA here, the relative activation energy shown in ¯ in Eq. (12) is Eq. (12) is used but the average moisture content X substituted by the local moisture content (X) as the REA is used to represent the local evaporation rate instead of the global drying rate of the whole sample. The effective vapor diffusivity (Dv ), tortuosity (), solid concentration (Cs ) and porosity (ε) of the samples are evaluated using the procedures explained previously (Putranto and Chen, 2013a,b) and these transport parameters are summarized in Appendix B. According to Dhall et al. (2012), no shrinkage was observed during drying. In order to yield the spatial profiles, Eqs. (1)–(3) are solved in conjunction with the initial and boundary conditions shown in Eqs. (4)–(9) as well as the transport parameters, the relative and equilibrium activation energy mentioned above. Method of lines is used to solve the partial differential equations by first transforming the equations into a set of ordinary differential equations which is then solved using ‘ode23s’ solver, available in Matlab® , implements Rosenbrock formula of order 2. The equations are solved from t = 0 until the end of the processes. In the modeling, the radius change is incorporated by moving 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. Moving mesh was found to give better agreement towards experimental data than fixed coordinate (immobilizing boundary) (Thuwapanichayanan et al., 2008; Putranto and Chen, 2013a,b).
The effective liquid diffusivity (Dw ) of baking of bread is expressed as (Ni et al., 1999): Dw = 1 × 10−6 exp(−2.8 + 2X)ε
(14)
where ε is the fraction volume taken by the vapor. The radius change during baking is incorporated in the modeling and described as:
Rs = −0.008X2 + 0.003X + 0.0073
(15)
where Rs is the sample radius (m) and X is the moisture content (kg H2 O kg dry solids−1 ). The numerical method, similar to the one mentioned in Section 3.1, is implemented to solve the equations to yield the spatial profiles of moisture content, concentration of water vapor and moisture content. In order to incorporate the radius change shown in Eq. (15) in the modeling, moving mesh is used.
3.3.
S-REA for modeling water vapor sorption process
The spatial reaction engineering approach (S-REA), similar to the ones shown in Eqs. (1)–(3), is setup. Similarly, the initial and boundary conditions shown in Eqs. (4)–(8) are used. The relative activation energy of silica gel can be expressed as (Putranto and Chen, 2014):
4 3 2 Ev = −4.19 X − Xb − 15.66 X − Xb + 3.192 X − Xb Ev,b
− 10.063 X − Xb + 1
(16)
Eq. (16) has been used to model the global wetting rate of silica gel successfully (Putranto and Chen, 2014). Here, by ¯ with local moisreplacing the average moisture content (X) ture content (X), Eq. (16) is used to describe the local wetting rate of silica gel. The numerical method mentioned above is implemented here to yield the spatial profiles during water vapor sorption. During experiments, no volume change was observed (Li et al., 2007). Similarly, the modeling by Putranto and Chen (2014) did not incorporate the volume change during water sorption process.
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Fig. 2 – Sensitivity of effective liquid diffusivity towards profiles of moisture content during drying of honeycombs at temperature of 137 ◦ C.
5
Fig. 4 – Profiles of moisture content during drying of honeycombs at temperature of 103 and 137 ◦ C.
4. Results of mathematical modeling of several heat and mass transfer processes using the lumped reaction engineering approach (S-REA) 4.1. Modeling of drying using the spatial reaction engineering approach (S-REA) Figs. 1–6 indicate the results of modeling of drying of honeycombs at drying air temperature of 103 and 137 ◦ C. As suggested by Dhall et al. (2012), the liquid diffusivity of 10−10 m2 s−1 is used. Interestingly, as shown in Figs. 2 and 3, the liquid diffusivity of lower than 1 × 10−10 m2 s−1 does not result in any noticeable differences in the moisture content and temperature profiles. This may indicate that the modeling is independent on surface diffusivity and hence the surface diffusion term may be neglected in the modeling. This is similar to modeling heat treatment of wood under linearlyincreased gas temperature in which the profiles of moisture content and temperature are independent on the liquid diffusivity so that the surface diffusion term is neglected (Putranto and Chen, 2013b).
Fig. 3 – Sensitivity of effective liquid diffusivity towards profiles of temperature during drying of honeycombs at temperature of 137 ◦ C.
Fig. 5 – Profiles of surface temperature during drying of honeycombs at temperature of 103 and 137 ◦ C. Without using surface diffusion term on the mass balance (refer to Eq. (1)), the results of modeling of drying at temperature of 103 and 137 ◦ C are shown in Figs. 4 and 5. A good agreement towards the experimental data is shown (R2 of higher than 0.998 for moisture content and R2 of higher than 0.993 for temperature). The S-REA is accurate to model
Fig. 6 – Spatial profiles of moisture content during drying of honeycombs at temperature of 137 ◦ C.
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Fig. 7 – Spatial profiles of temperature during drying of honeycombs at temperature of 137 ◦ C.
both moisture content and temperature profiles at drying air temperature of 103 and 137 ◦ C. Benchmarks towards modeling implemented by Dhall et al. (2012), shown in Fig. 5, indicate that the S-REA results in closer agreement towards the experimental data. This could be because of the sufficiency of the combination of the REA and a set of equations of microscopic heat and mass balances to describe the drying processes. Here, both moisture content and temperature profiles are independent on the surface diffusivity which may be because of the low initial moisture content and high porosity of the samples. The movement of moisture by capillary flow and moisture diffusion may be much faster than vapor diffusion and evaporation so that the transport phenomena inside the samples are governed by the vapor diffusion and evaporation/condensation. Similar phenomena also occurs during heat treatment of wood reported previously (Putranto and Chen, 2013b). Fig. 6 indicates the spatial profiles of moisture content during drying of honeycombs at temperature of 137 ◦ C. The moisture content at the outer part of the samples is lower than that at the inner part of the samples which may indicate that the moisture migrates outwards during drying. Initially, the gradient of moisture content inside the samples is relatively large. Nevertheless, the gradient decreases as drying progresses and at the end of drying, no noticeable gradient are observed which indicates that the equilibrium conditions may have been attained. Fig. 7 shows the spatial profiles of temperature during drying of honeycombs at temperature of 137 ◦ C. The temperature of the samples at the outer part is higher than that of the inner part which indicates that the heat is penetrated inwards during drying for evaporating water and increasing sample temperature. During drying, the temperature increases to approach drying air temperature and the gradient of temperature decreases. At the end of drying, no noticeable differences of temperature inside the samples are observed, in agreement with the spatial profiles of moisture content shown in Fig. 6. Figs. 2–7 show that the S-REA is accurate to model drying of non-food materials. Previously, the S-REA was implemented to describe drying of food and biological materials and the results of modeling match very well with the experimental data (Putranto and Chen, 2013a). The accuracy of the S-REA may be because of the applicability of the REA to describe the local evaporation/condensation rate. The combination of
Fig. 8 – Profiles of average moisture content during baking of bread at baking temperature of 175 ◦ C. relative and equilibrium activation energy yield unique activation energy which may allow the representation of local changes as affected by environmental conditions. For modeling of drying of samples with low moisture content, such as honeycombs, the surface diffusion term is not necessary to be included but the evaporation/condensation term needs to be incorporated. This also highlights the importance of local evaporation/condensation rate in the modeling of heat and mass transfer processes. It has been shown here that the REA models the local evaporation rate of non-food materials accurately. Because of the accuracy of the S-REA, it may be adopted for process design and product quality of several materials during drying. By using the S-REA, transport phenomena inside porous materials can be better understood. For process design, the S-REA may yield as reliable model to explore various drying schemes favorable for energy minimization. The S-REA may also be used to effectively fine-tune the operating conditions of drying of food materials to minimize nutrient degradation, protein denaturation, color change and texture loss. Similarly, for drying of non-food materials, the S-REA may be applied to design cycle time and intermittency to minimize cracking or material deformation.
4.2. Modeling of bread baking using the spatial reaction engineering approach (S-REA) Figs. 8 and 9 show the results of modeling at baking temperature of 175 ◦ C. The moisture content profiles during baking of 0.5 and 1 g bread are shown in Fig. 8. A good agreement towards the experimental data is shown (R2 of 0.992 and 0.988 for 0.5 and 1 g bread, respectively). For the temperature profiles, as shown in Fig. 9, the S-REA predictions match well with the experimental data. The good agreement is shown by R2 of 0.963 and 0.985 for 0.5 and 1 g bread, respectively. The combination between the REA and a system of equations of conservation of heat and mass transfer seems to be sufficient to describe the transport phenomena during bread baking. Fig. 10 shows the spatial profiles of moisture content during baking of 1 g of bread at baking temperature of 235 ◦ C. The profiles at the outer part of the samples are lower than those at the inner part which indicate that the moisture migrate outwards during baking. The profiles decrease during baking time since the moisture in the samples depletes as baking progresses.
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Fig. 9 – Profiles of core temperature during baking of bread at baking temperature of 175 ◦ C.
Fig. 10 – Spatial profiles of moisture content during bread baking of 1 g of bread at baking temperature of 235 ◦ C.
Fig. 12 – Sensitivity of the effective liquid diffusivity towards the moisture content profiles during water vapor sorption at relative humidity of 10%.
heat by convection from the environment and used for water vaporization. The excessive heat may be penetrated inwards by conduction to increase the centre temperature. The profiles of temperature here corresponds well the concentrations of water vapor. As explained above, the higher temperature at the outer part of the samples seems to result in higher local evaporation rate and thus concentration of water vapor near the surface of the samples. Figs. 8–11 indicate that the S-REA models well the bread baking process. It is obvious that the success of this approach is based on the accurate representation of the evaporation and condensation behavior by the REA. Similar to drying of honeycombs and biological materials presented previously (Putranto and Chen, 2013a), the REA seems to be able to capture the environmental conditions and reflect them to the changes in local structure during baking. Several chemical changes during baking covering starch gelatinization, starch retrogradation and protein denaturation seem to be captured well in the relative energy mentioned in Eq. (13). The S-REA is not only accurate to describe drying processes but also bread baking. The spatial profiles generated by the S-REA are useful for predictions of quality changes during baking especially crust formation and surface browning. The profiles of moisture content and temperature can be combined with established equations to project the color kinetics and probiotics survival during breadbaking. The S-REA may also be adopted in process design to explore various baking schemes favorable for energy minimization. Alternatively, it can also be implemented computational fluid dynamics (CFD)-based simulations to predict the flowfield inside baking chamber—a multiscale modeling exercise.
4.3. Modeling of water vapor sorption using the spatial reaction engineering approach (S-REA) Fig. 11 – Spatial profiles of temperature during bread baking of 1 g of bread at baking temperature of 235 ◦ C. Fig. 11 shows the spatial profiles of temperature during baking of 1 g bread at baking temperature of 235 ◦ C. The temperature at the outer part is higher than that of the inner part of the samples. This may indicate that the samples receive
Figs. 12–15 indicate the results of modeling of water vapor sorption using the S-REA. As suggested by Pesaran and Mills, 1987), the effective liquid diffusivity of silica gel is:
Dw = 1.6 × 10−6 exp −0.974 × 10−3
Hads T
(17)
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Fig. 13 – Profiles of moisture content during water vapor sorption at various relative humidity.
Fig. 14 – Spatial profiles of moisture content inside silica gel during water vapor sorption at relative humidity of 10%.
Fig. 15 – Spatial profiles of concentration of water vapor inside silica gel during water vapor sorption at relative humidity of 10%.
where Hads is the heat of adsorption (J kg−1 ) dan Dw is the surface diffusivity (m2 s−1 ). In this study, the surface diffusivity is varied from 1 × 10−10 to 1 × 10−30 m2 s−1 and the results are shown in Fig. 12. Similar to drying of honeycombs, the varied surface diffusivity does not yield any noticeable difference in the profiles of moisture content. This may indicate that the modeling is independent on surface diffusivity and the surface diffusion term can be neglected. Without using surface diffusion term on the mass balance, the results of modeling of sorption of water vapor into silica gel are shown in Figs. 13–15. A good agreement towards the experimental data is shown in Fig. 13. For sorption of water vapor into silica gel at relative humidity of 10%, the S-REA models well the moisture content during water vapor sorption (R2 of 0.993). Similarly, a good agreement towards experimental data is shown (R2 of 0.992) for water vapor sorption at relative humidity of 45%. The results of modeling using the S-REA at higher relative humidity i.e. 65 also match well with the experimental data. The S-REA models well the average moisture content during water vapor sorption at various relative humidity. Although the surface diffusion term is not implemented in the modeling, the results match very well with the experimental data. This could be because the porosity of the samples is relatively high and the samples are relatively small. Similar to drying mentioned in Section 4.1 and heat treatment of wood reported previously (Putranto and Chen, 2013b), condensation of water vapor and vapor diffusion may be the governing transport processes during water vapor sorption. This also emphasizes the importance of local condensation rate in modeling of water vapor sorption. This expression seems to be necessary to accurately model condensation in heat and mass transfer processes. Here, the REA can describe the local condensation rate well. Fig. 14 shows the spatial profiles of moisture content during water vapor sorption at relative humidity of 10%. There is no noticeable difference of gradient of moisture content during water vapor sorption. Similarly, as shown in Fig. 15, the concentration of water vapor inside the samples is basically uniform which could be because of the small sample used in the experiments. Nevertheless, for larger samples, the S-REA is useful to better understand the transport phenomena during water vapor sorption because of the generation of spatial profiles as affected by local structure and variables. Figs. 13–15 indicate that the S-REA models accurately the water vapor sorption which may be because the REA describes well the local condensation rate. The applicability of the REA indicates that the relative activation energy used to model the global wetting rate (refer to Eq. (16)) (Putranto and Chen, 2014) is also applicable to model the local wetting rate of water vapor sorption. Similar to ‘local’ relative activation energy for drying, the relative activation energy mentioned in Eq. (16) is essentially an extrapolation of relative activation energy to describe the wetting rate at micro scale. Fig. 16 shows that both the L-REA (lumped reaction engineering approach) (Putranto and Chen, 2014) and the S-REA models accurately the water vapor sorption. While the L-REA provides simplicity in mathematical modeling, the S-REA yields advantages of generating the spatial profiles. It is proven here that the REA is not only accurate to describe the local evaporation rate but also the local condensation rate. The applicability of the REA here is facilitated by the two terms of the REA (i.e. evaporation and condensation) as shown in Eq. (11). Because of the accuracy of the S-REA
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Appendix A. Brief review of the reaction engineering approach (REA) (Chen and Putranto, 2013) By using the REA, the mass balance of water during heat and mass transfer processes can be expressed as (Chen and Xie, 1997; Chen, 2008): ms
Fig. 16 – Profiles of average moisture content during water vapor sorption at various relative humidity modeled by the L-REA and S-REA.
here, the S-REA is readily implemented to design new water vapor sorption process and evaluate the existing facilities. The development of the REA to describe drying of solutes dissolved in organic substances is now underway. It is anticipated that the REA is also able to describe the migration of volatiles inside the materials so that the composition of each component involved in heat and mass transfer processes can better represented. Similarly, for the prediction of material deformation, the REA may be combined with solid mechanics to yield the profiles of stress-strain during heat and mass transfer processes. This development may also enable the prediction of material shape changes during transport processes.
5.
Conclusions
In this paper, the REA is used to describe local evaporation/condensation rate in several heat and mass transfer processes including drying, baking and water vapor sorption. The REA framework is combined with a set of equations of conservation of heat and mass transfer processes to yield the spatial reaction engineering approach (S-REA). The relative activation energy, generated in one accurate experiment and implemented to represent the global drying rate, is applied here to model the local evaporation/condensation rate. When validated towards experimental data, it is indicated that the S-REA describes accurately drying, baking and water vapor sorption. In drying and water vapor sorption, the surface diffusion term may not be used in the modeling because of the low moisture content and high porosity of the samples. The accuracy of the S-REA could be because of the accuracy and flexibility of the relative activation energy to capture the environmental change and reflect to the change of internal behavior as affected by the local variables and structures. The REA framework is readily adopted to assist in process design and maintaining product quality during several heat and mass transfer processes in industries. The development of the REA to predict drying of materials dissolved in non-water components as well as stress-strain is now underway.
¯ dX = −hm A exp dt
−E v
RTs
v,sat (Ts ) − v,b
(A1)
¯ is the average moisture content on dry basis, hm is the where X mass transfer coefficient (m s−1 ), A is the surface area (m2 ), Ev is the activation energy (J mol−1 ), Ts is the sample temperature (K), ms is the dried mass of sample (kg), v,sat is the saturated water vapor concentration (kg m−3 ) and v,b is the concentration of water vapor in drying medium (kg m−3 ). Eq. (A1) indicates that the REA is expressed in first order ordinary differential equation with respect to time. Eq. (A1) is the core of the reaction engineering approach, further called as the lumped reaction engineering approach (L-REA). The L-REA does not assume uniform moisture content but it evaluates average moisture content of the samples during drying. The activation energy (Ev ) is determined experimentally by placing the parameters required for Eq. (A1) in its rearranged form:
Ev = −RTs ln
¯ −ms dX/dt (1/hm A) + v,b v,sat
(A2)
¯ where dX/dt is experimentally determined. Besides the average moisture content, the surface area, temperature and mass transfer coefficient need to be measured or known. The dependence of activation energy on average moisture ¯ can be normalized as: content on a dry basis (X) Ev ¯ − Xb ) = f (X Ev,b
(A3)
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 )
(A4)
RHb is the relative humidity of drying air and Tb is the drying air temperature (K). In order to generate the relative activation energy (Ev /Ev,b ) shown by Eq. (A4), the activation energy (Ev ) can be evaluated by Eq. (A2) from one accurate drying experiment. So far, the experiments conducted to generate the relationship (Eq. (A3)) generally employed fairly dry air so the relationship ¯ − Xb ) covers a complete range of water content difference (X while Xb in the experiments for generating REA parameters is set to be very small value. The activation energy is divided by the equilibrium activation energy (Ev,b ) indicated by Eq. (A4) to yield the relative activation energy during drying. This is a normalization process. For similar drying condition and initial water content, it is possible to obtain the necessary REA parameters (apart from the equilibrium isotherm), expressed in the relative activation energy (Ev /Ev,b ) as indicated in Eq. (A3) in one accurate drying experiment. The relative activation energy (Ev /Ev,b )
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generated can then be used to project to other drying conditions provided the material has the same initial moisture content (Chen and Lin, 2005; Chen, 2008).
Appendix B. Procedures to determine effective vapor diffusivity, tortuosity, solid concentration and porosity of the samples The effective vapor diffusivity (Dv ) is deduced from (Bird et al., 2002): Dv = Dvo
ε
(B1)
where ε is the porosity, is the tortuosity, Dvo is the water vapor diffusivity (m2 s−1 ) which is dependent on temperature, which can be expressed as (Slattery and Bird, 1958): Dvo = 2.09 × 10−5 + 2.137 × 10−7 (T − 273.15)
(B2)
The tortuosity () of the samples is generally related to the porosity (ε). The relationship can be represented as (Audu and Jeffreys, 1975; Gimmi et al., 1993): = ε−n
(B3)
where n is the value between 0 and 0.5 (Audu and Jeffreys, 1975; Gimmi et al., 1993). There is no noticeable effect of various values of n on the profiles of liquid water concentration, water vapor concentration and temperature as witnessed in many numerical tests performed in this study and n = 0.5 is chosen in this study. Cs is the solid concentration which can be expressed by (Kar and Chen, 2010, 2011; Putranto and Chen, 2013a,b): Cs =
1−ε (1/s ) + (X/w )
(B4)
where s is the dry solid density (kg m−3 ) and w is the density of water (kg m−3 ), ε is the porosity, dependent on shrinkage and local moisture content. This can be determined according to Madiouli et al. (2007): ε=1−
V0 (1 − ε0 ) V
¯ +1 (s /w ) X 1 + (s /w ) X0
(B5)
where V0 is the initial volume of the sample (m3 ) and ε0 is the initial porosity of the sample.
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