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Theoretical and experimental study of heat and mass transfer mechanism during convective drying of multi-layered porous packed bed
International Communications in Heat and Mass Transfer 38 (2011) 900–905
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International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Theoretical and experimental study of heat and mass transfer mechanism during convective drying of multi-layered porous packed bed☆ Ratthasak Prommas Department of Mechanical Engineering, Rajamangala University of Technology Rattanakosin, 96 Mu 6 Putthamonthon Sai 5 Salaya, Putthamonthon, Nakhon Pathom, 73120, Thailand
a r t i c l e
i n f o
Available online 9 April 2011 Keywords: Convective drying Porous packed bed Multi-layered Capillary pressure
a b s t r a c t In this paper, the experimental validation of a combined mass and thermal model for a convective drying of multi-layered packed beds composed of glass beads, water and air is presented. The effects of the drying time, particle size and the layered structure on the overall drying kinetics are clarified in detail. Based on a completed model combining the temperature, total pressure and moisture equations, the results show that the convective drying kinetics strongly depend on the particle size as well as hydrodynamic properties and layered structure, considering the interference between capillary flow and vapor diffusion in the multi-layered porous packed bed. The predicted results are in a good agreement with the experimental results. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction From a theoretical standpoint, the drying process of porous media is a complicated process involving simultaneous, coupled heat and mass transfer phenomena. Modeling simultaneous heat and mass transport in porous media is of growing interest in a wide range of new technology. In order to improve process performance and energy utilization for new technologies related to tertiary oil recovery processes, medical application, geothermal analysis, freeze drying processes, forest product, building materials, food stuffs and microwave drying process have been applied. Knowledge of heat and mass transfer that occurs during convective drying of porous materials is crucial to reduce energy cost of drying, equipment and process design, and preserve the quality of products. The analysis of heat and mass transfer in porous materials has been the subject of theoretical and experimental work for several decades. Most theories have been proposed to explain the physical phenomena of drying process in porous materials: the diffusion theory, the capillary flow theory and the evaporation–condensation theory. A convenient starting point of drying theory is found, in the work of Whitaker [1] who has derived appropriate local volume averaged conservation equations for two-phase capillary flow in porous media. The mathematical models for simultaneous heat and mass transfer during convective drying of porous media have been studied by many authors [2–9]. These previous models are based on Whitaker's theory,
☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.03.031
taking into account the thermal and mass transfer as well as the total gaseous pressure. Recently, the evolution of phase distributions within the network during drying in capillary-porous media are visualized and compared to numerical discrete simulations [10]. Also many researchers such as Bae [11], Lamnatou [12], Bubnovich [13,14], Kaya [15,16] and Kim [17] have used mathematical principles to analyze heat and mass transfer in porous materials. This does not explain the mechanism of heat and mass transfer clearly. However, most of the previous works deal with the drying of uniform materials. Indeed, little effort has been reported on the study of drying process of non-uniform material i.e., multi-layered material especially a complete comparison between mathematical model and experimental data. Typical applications of non-uniform material include the tertiary oil recovery process, geothermal analysis, asphaltic concrete pavement process and preservation process of food stuffs. Therefore, knowledge of heat and mass transfer that occurs during convective drying of multi-layered porous materials is necessary to provide a basis for fundamental understanding of convective drying of non-uniform materials. There is one published study of heat and mass transport in non-uniform material, that by Plumb [18]. This paper contains a useful review, an experimental study of heat and mass transfer in non-uniform material, and supporting numerical work. From a macroscopic point of view, the effects of the particle sizes, hydrodynamic properties, and the layered location on the overall drying kinetics must be clarified in detail. Therefore, the specific objectives of this work are to: (1) solve the mathematical model numerically, (2) compare the numerical results with experimental measurements and (3) discuss the effect of particle size and the layered location on the overall drying kinetics.
R. Prommas / International Communications in Heat and Mass Transfer 38 (2011) 900–905
901
Nomenclature cp d D Dm p p0 S T U∞ w z
specific heat capacity, (J/kgK) diameter, (m) binary mass diffusion in plain media effective molecular mass diffusion pressure, (Pa) ambient pressure, (Pa) water saturation temperature, (°C) hot air velocity, (m/s) velocity, (m/s) coordinate axis, (m)
Greek symbols ϕ porosity, (m3/m3) ρ density, (kg/m3) ξ surface tension, (Pa m) λeff effective thermal conductivity, (W/mK) α, β coordinate transformation δ packed bed depth, (m)
Subscripts 0 initial a air c capillary C coarse bed e effective F fine bed g gas ir irreducible j layer number l liquid water p particle v water vapor ∞ ambient
The result presented here provides a basis for fundamental understanding of convective drying of multi-layered packed beds.
2. The characteristic of moisture transport in multi-layered porous packed beds Fig. 1 shows the typical multi-layered packed bed. The multilayered packed beds are arranged in different configurations as follows: (a) F–C bed, the fine particles (average diameter of 0.15 mm) are over the coarse particles (average diameter of 0.4 mm). (b) C–F bed, the coarse particles (average diameter of 0.4 mm) are over the fine particles (average diameter of 0.15 mm). It is observed that the moisture content profiles are not uniform in multi-layered packed beds. During convective drying higher moisture content occurs in the fine bed while the moisture content inside the coarse bed remains lower compared with the initial state. This is a result of capillary action. The physical and transport properties of the two sizes of particles composing the bed are shown in Table 1.
Fig. 1. The typical profile of moisture content in multi-layered packed bed during convective drying in the cases of: (a) C–F bed; (b) F–C bed.
3. Experimental apparatus Fig. 2(a) shows the experimental convective drying system. The hot air, generated electrically travels through a duct toward the upper surface of two samples situated inside the test section. The outside walls of test section are covered with insulation to reduce heat loss to the ambient. The flow outlet and temperature can be adjusted at a control panel. As shown in Fig. 2(b), the samples are unsaturated packed beds composed of glass beads, water and air. The samples are prepared in two configurations in the: a single-layered packed bed (uniform packed bed) with bed depth δ = 40 mm (d = 0.15 mm (F bed) and d = 0.4 mm (C bed)) and a two-layered packed bed, respectively. The two-layered packed bed is arranged in different configurations in the: F–C bed (fine particles (d = 0.15 mm, δ = 20 mm) overlay the coarse particles (d = 0.4 mm, δ = 20 mm)), and C–F bed (coarse particles (d = 0.4 mm, δ = 20 mm) overlay the fine particles (d = 0.15 mm, δ = 20 mm)), respectively. The width and total length of all samples used in the experiments are 25 mm and 40 mm, respectively. In analysis, the main assumptions involved in the formulation of the transport model are: 1. The capillary porous material is rigid and no chemical reactions occur in the sample. 2. Local thermodynamic equilibrium among each phase is assumed. 3. The gas phase is ideal in the thermodynamic sense. 4. The contribution of convection to energy transport is included. 5. Darcy's law holds for the liquid and gas phases. 6. Gravity is included for the liquid and gas phases. 7. Permeabilities of liquid and gas can be expressed in terms of relative permeability. 8. In a macroscopic sense, the packed bed is assumed to be homogeneous and isotropic, and liquid water is not bound to the solid matrix. Therefore, the volume average model for a homogeneous and isotropic material can be used in the theoretical model and analysis. 9. A dry layer (evaporation front) is formed immediately after water saturation approaches the irreducible value. By conservation of mass and energy in the multi-layered packed bed, the governing equation of mass and energy for all phases can be derived
Table 1 The characteristics of water transport in porous materials. Diameter, d (mm)
Porosity, ϕ
Permeability, k (m2)
0.15 0.4
0.387 0.371
8.41 × 10− 12 3.52 × 10− 11
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R. Prommas / International Communications in Heat and Mass Transfer 38 (2011) 900–905
Fig. 2. Schematic of experimental facility: (a) equipment setup; (b) multi-layered porous packed bed (sample).
using the volume-averaging technique. The main basic equation for two-phase layer and dry layer are given by following equations: 4. Basic equation for two-phase layer 4.1. Mass conservation The microscopic mass conservation equations for liquid, water vapor, air, and gas phase, are written, respectively, as
Liquid phase ρl ϕj
Vapor phase
∂sj ∂t
+ ρl
∂wl; j ∂z
:
= −n
o i ∂n ∂h : + ρv ϕj 1−sj ρv wv; j = n ∂t ∂z
ð1Þ
The kinetic energy and pressure terms, which are usually unimportant are ignored. Local thermodynamics equilibrium among all phases is assumed. The temperature of the sample exposed to hot air stream with fixed properties is obtained by solving the conventional heat transport equation. Considering the enthalpy transport based on the water and gas flows, the conduction heat and latent heat transfer due to evaporation. The energy conservation equation is represented by: o i ∂qj : ∂ ∂ hn T + ρcp ρl cpl wl; j + ρa cpa + ρv cpv wg; j T + hlv n = − T;j ∂z ∂t ∂z
ð5Þ ð2Þ
Air phase
o i ∂n ∂h + ρa ϕj 1−sj ρa wa; j = 0 ∂t ∂z
ð3Þ
Gas phase
o i ∂n ∂h : + ρg ϕj 1−sj ρg wg; j = n ∂t ∂z
ð4Þ
:
4.2. Energy conservation
where j denotes layer number, n is the condensation rate or evaporation rate during phase change and ϕ is the porosity. The water vapor and air mass flux is the sum of the convective term with the gas superficial velocity and diffusive term.
where hlv is the latent heat of vaporization of water. 3.2.1. Phenomenological relations In order to complete the system of equations, the expressions for the superficial average velocity of the liquid and gas phases derived from the generalized Darcy's law in the following form are used: wl; j
" # Kj Krl; j ∂pg ∂pc; j =− − −ρl g μl ∂z ∂z
ð6Þ
" # Kj Krg; j ∂pg −ρg g μg ∂z
ð7Þ
wg;j = −
R. Prommas / International Communications in Heat and Mass Transfer 38 (2011) 900–905
where the capillary pressure pc related to the gas and liquid phases can be written as: ð8Þ
For the velocities of water vapor and air phases, the generalized Fick's law for a two-component gas mixture can be expressed as: ρv wv;j = ρv wg; j −ρg Dm
∂ ρv ∂z ρg
!
∂ ρa ∂z ρg
ρa wa; j = ρa wg; j −ρg Dm
ð9Þ
2ϕ ð1−sÞD 3−ϕ
ð10Þ
ð11Þ
where D is the binary mass diffusion in continuous media. Fourier's law is used to define the heat flux through the porous medium qj = −λeff
∂T : ∂z
ð12Þ
4.3. Equilibrium relations The system of conservation equations obtained for multiphase transport mode requires constitutive equations for relative permeabilities Kr, capillary pressure pc, capillary pressure functions (Leverett functions) J, and the effective thermal conductivity λeff. A typical set of constitutive relationships for liquid and gas system is given by [6]: 3
ð13Þ
Krl = se
3
Krg = ð1−se Þ
ð14Þ
where se is the effective water saturation considered the irreducible water saturation sir and defined by se =
s−sir : 1−sir
ð15Þ
The capillary pressure pc is further assumed to be adequately represented by Leverett's well known J(se) functions. The relationship between the capillary pressure and the water saturation is represented by using Leverett functions J(se): ξ pc = pg −pl = pffiffiffiffiffiffiffiffiffiffiffi J ðse Þ K=ϕ
ð16Þ
where ξ is the gas–liquid interfacial tension. As indicated at the same water saturation, a smaller particle size corresponds to a higher capillary pressure. Furthermore, [19] and [20] correlated capillary pressure data obtained by Leverett as follows to give J(se): 0:217
J ðse Þ = 0:325ð1 =se −1Þ
:
ð17Þ
The effective thermal conductivity of the capillary porous medium is represented as a function of water saturation as [21]: λeff =
0:8 : 1 + 3:78e−5:95s
The experimental results for convective drying layered packed beds are compared with model simulations. In order to test the validity of the mathematical model, the results are divided into two parts covering moisture profiles and temperature profiles. One aspect of model verification was to compare drying data from experiments run under different conditions with mathematical simulations using parameter values obtained from Table 1. 5.1. Moisture profiles within multi-layered packed bed
!
and Dm is the effective molecular mass diffusion [7] Dm =
5. Result and discussion
In this study, we assume that the volumes of the upper layer and lower layer are equal. The distributions of water saturation in the layered packed beds differ greatly depending on the structure of layered packed beds even if the total volume of water that exists among the pores in the packed beds is identical. Fig. 4(a) and (b) shows the predicted moisture content profiles at various times and locations within the single-layered packed bed (F bed and C bed) and two-layered packed bed (F–C bed and C–F bed), which correspond to that of s0 = 0.5, Ta = 80 (°C) and U∞ = 1.2 (m/s), the other physical properties are given in Table 1. In the case of a single-layered packed bed, the moisture content continuously decreases toward the surface. The decrease in surface saturation, sets up a saturation gradient, which draws liquid water toward the surface through capillary action while water vapor moves towards the surface due to a gradient in the vapor partial pressure. However, the internal moisture transport is mainly attributable to capillary flow of liquid water through the voids during the initial stage of drying. The following discussion is concerned with the effect of particle size on moisture migration mechanism under the same conditions for the single-layered packed bed [22]. In Fig. 4(a), the observed moisture content profiles at the leading edge of the sample in the case of fine bed (F bed) are higher than those in the case of coarse bed (C bed). This is because the fine bed or small particle size (corresponding to a higher capillary pressure) can cause moisture to reach the surface at a higher rate than in the case of the coarse bed, i.e., large particle size. On the other hand, in the case of coarse bed, the moisture profiles at the leading edge of the sample are always lower as compared with the small particle size due to the lower capillary pressure inside the sample and the gravitational effect. Continued drying eventually causes the average moisture level inside the packed bed to decrease, especially at the leading edge of the upper layer (Fig. 4(b)). When the moisture content at the upper surface (surface saturation) approaches the irreducible where the liquid water becomes discontinuous (pendular state), the liquid water supply to the surface by capillary action becomes insufficient to replace the liquid being evaporated.
Surface Saturation Coefficient ϕ[-]
pc = pg −pl :
903
1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
Average Saturation s[-] ð18Þ
Fig. 3. The surface saturation coefficient as a function of average water saturation.
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R. Prommas / International Communications in Heat and Mass Transfer 38 (2011) 900–905
The latter arises from the fact that the dry layer takes place over a small effective surface or on a front retreating from the surface into the interior of the sample dividing it into two layers, a dry layer and a two-phase layer. The discontinuity of the temperature gradient close to the drying front is a result of heat flux necessary for evaporation. In addition the effective thermal conductivity falls considerably resulting in a resistance of the heat flow rate. A further consequence of premature drying of the outer dry layer is that the local temperature of the sample will reach that of convection drying. In the case of two-layered packed beds (F–C bed and C–F bed), the moisture content profiles become discontinuous at the interface between the two layers [23], namely, the moisture content in the upper layer is higher than that for the lower layer in F–C bed, while the moisture content in the upper layer is lower than that for the C–F bed. This is because the equilibrium water saturation under the same capillary pressure differs according to the particle sizes with the smallest particle size corresponding to higher water saturation. Therefore, the drying kinetics is strongly influenced by the difference in water saturation. Continuing the drying process (Fig. 4(b)) causes the average moisture level inside the F–C bed to decrease quickly in comparison with the other cases. In the case of the C–F bed, in contrast to the F–C bed, the moisture content in the upper layer (C bed) is very low and the dry layer is formed rapidly while the moisture content in the lower layer (F bed) remains high. This is due to the C bed (which corresponds to a lower capillary pressure) located above the F bed
(a) 1.0 0.9 0.8 0.7 0.6 0.5
5.2. Temperature profiles within multi-layered packed bed
t=15 [min]
0.4
F bed C bed F-C bed C-F bed
0.3 0.2 0.1 0.0
0
5
10
15
20
25
30
35
retarding the upward migration of liquid water through the interface between two layers and also due to the effect of gravity. Figs. 5 and 6 show the variations of drying mass and temperature with respect to time for the four packed beds (F, C, F–C and C–F beds) under conditions. It is observed that the variations of drying mass and temperature are interrelated. The drying speed is greatly different although the initial water saturation is the same. The F–C bed dries the fastest, and the C–F bed dries the slowest. This is due to the difference in the moisture content in the neighborhood of the drying surface as shown by Fig. 3. In the microscopic sense, the drying rate rises quickly in the early stages of drying (which corresponds to higher moisture content at the upper surface) and then decreases and reaches a plateau before decreasing again when the formation of a drying front is established. It is evident from the figure that the increase in drying rate can become significant when the F–C bed is utilized. This is a result of the strong effect of capillary pressure in F–C bed. For this case the capillary pressure easily overcomes the resistance caused by the lower permeability and maintains a supply of liquid water near the surface. However, in the case of C–F bed the formation of a dry layer is established in the early stage of drying while the moisture content within the lower layer is still high. This results in a lower drying rate when compare with other cases. The efficiency of the drying process is the lowest in this case. Further, the observed drying rates of the uniform beds (F bed and C bed) are similar. However, the drying rate for the F bed is slightly higher than that for the C bed. This is because the F bed composed of small particles leads to a higher capillary pressure resulting in a faster drying time in comparison to that for the large particles (C bed). Late in the drying process, the drying rate suddenly decreases with time due to the decrease in surface moisture content and the thermal resistance increases because of the formation of a dry layer. Overall the simulated results are in good agreement with the experimental results for convective drying.
40
Depth z[mm]
(b) 0.7
Fig. 6 gives the measured and predicted temperature profiles with respect to time within the single-layered and two-layered packed beds. Results are for s0 = 0.5,Ta = 80(°C), U∞ = 1.2 (m/s) and bed depth (z) of 5 mm in F and F–C beds. The other physical properties are given in Table 1. It is observed that at the early stage of the drying process the temperature profiles in both cases are nearly the same. As drying progresses, the temperature increases. This is because the latent heat transfer due to evaporation is retained due to the decline of the mass transfer rate together with the decreases of average moisture content. Nevertheless, the temperature profile in F–C bed rises at a higher rate
0.6 18 0.5 0.4 t=810 [min] 0.3
F bed C bed F-C bed C-F bed
0.2 0.1 0.0
0
5
10
15
20
25
30
35
40
Depth z[mm] Fig. 4. The predicted result of distribution of water saturation in multi-layered porous packed bed as a function of locations (a) initial water saturation and (b) long stage of drying.
Mass of Drying M[g]
16
T∞ =80.0oC ρ∞ =0.55 U∞ =1.2 m/s si=0.5
Formation of dry layer
14 12 10 8
Cal.
6 4 2 0 0
Exp.
d=0.15 mm d=0.40 mm F-C Beds C-F Beds 2
4
6
8
10
12
14
16
18
Elapsed Time t[h] Fig. 5. The variation of drying mass with respect to time.
20
R. Prommas / International Communications in Heat and Mass Transfer 38 (2011) 900–905
80
z=5mm Temperature T[oC]
70
differentiates it from the others. It has a shorter drying time due to the strong effect of capillary action. 3. It is also found that the drying rate depends strongly on the moisture content at the heating surface. 4. The predicted results for the temperature distribution within the multi-layered packed beds and the drying rate are in good agreement with experimental results using packed beds composed of glass beads.
Formation of dry layer
60
50 Cal.
40
Acknowledgment
Exp.
d=0.15mm F-C Beds
30 0
2
4
6
8
905
10
Elapsed Time t[h]
The authors gratefully acknowledge Institute of Research and Development (IRD) of Rajamangala University of Technology Rattanakosin and Commission on Higher Education, Thailand for providing financial support for this study.
Fig. 6. The temperature profiles with respect to time.
References than that in the F bed. This is because the dry layer formed earlier in the F–C bed and an abrupt temperature rise occurs as the dry bulb temperature is approached. On the other hand, in the case of the F bed the temperature increases more slowly in comparison with the F–C bed due to the late formation of dry layer. The prediction of the formation of a dry layer estimated from the drop in the surface mass of drying and rise in the drying temperature is marked in Figs. 5 and 6, respectively. At the longer drying time the temperature profile at any instant tends to assume constant shape throughout the region. In this period the vapor diffusion effect plays an important role in the moisture migration mechanism because of the sustained vaporization that is generated within the sample. The simulated results are in agreement with the experimental results for convective drying. Finally, the observation of drying rate and temperature profiles depicted in Figs. 5 and 6 for the convective drying of multi-layered packed beds verifies that the predicted and experimental results are qualitatively consistent. Discrepancy may be attributed to uncertainties in the thermal and physical property data. Moreover, the discrepancy may be attributed to the impact of variations in humidity and room temperature. Based on model simulations, because the total pressure is not extreme and the packed bed is dried at atmospheric condition there is never a large pressure gradient within packed bed. In fact constant pressure-drying model might be a reasonable modeling assumption when drying temperature is lower than the boiling point. From this paper, the capability of the mathematical model to correctly handle the field variations at the interfaces between materials of different hydrodynamic properties is illustrated. With further quantitative validation of the mathematical model, it is clear that the model can be used as a real tool for investigating in detail convective drying of porous materials at a fundamental level.
6. Conclusions The experiments and theoretical analysis presented in this paper describe many important interactions within multi-layered porous materials during convective drying. The following paragraphs summarize the conclusions of this study: 1. A generalized mathematical model for convective drying of multilayered packed beds is proposed. It is used successfully to describe the drying phenomena under various conditions. 2. The effects of particle sizes and layered structure on the overall drying kinetics are clarified. The drying rate in the case of the F bed (fine particles) is slightly higher than that of the C bed (coarse particles). This is because the higher capillary pressure for the F bed results in the maintenance of a wetted drying surface for a longer period of time. The F–C bed displays the drying curve which
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Contents lists available at ScienceDirect
International Communications in Heat and Mass Transfer j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / i c h m t
Theoretical and experimental study of heat and mass transfer mechanism during convective drying of multi-layered porous packed bed☆ Ratthasak Prommas Department of Mechanical Engineering, Rajamangala University of Technology Rattanakosin, 96 Mu 6 Putthamonthon Sai 5 Salaya, Putthamonthon, Nakhon Pathom, 73120, Thailand
a r t i c l e
i n f o
Available online 9 April 2011 Keywords: Convective drying Porous packed bed Multi-layered Capillary pressure
a b s t r a c t In this paper, the experimental validation of a combined mass and thermal model for a convective drying of multi-layered packed beds composed of glass beads, water and air is presented. The effects of the drying time, particle size and the layered structure on the overall drying kinetics are clarified in detail. Based on a completed model combining the temperature, total pressure and moisture equations, the results show that the convective drying kinetics strongly depend on the particle size as well as hydrodynamic properties and layered structure, considering the interference between capillary flow and vapor diffusion in the multi-layered porous packed bed. The predicted results are in a good agreement with the experimental results. © 2011 Elsevier Ltd. All rights reserved.
1. Introduction From a theoretical standpoint, the drying process of porous media is a complicated process involving simultaneous, coupled heat and mass transfer phenomena. Modeling simultaneous heat and mass transport in porous media is of growing interest in a wide range of new technology. In order to improve process performance and energy utilization for new technologies related to tertiary oil recovery processes, medical application, geothermal analysis, freeze drying processes, forest product, building materials, food stuffs and microwave drying process have been applied. Knowledge of heat and mass transfer that occurs during convective drying of porous materials is crucial to reduce energy cost of drying, equipment and process design, and preserve the quality of products. The analysis of heat and mass transfer in porous materials has been the subject of theoretical and experimental work for several decades. Most theories have been proposed to explain the physical phenomena of drying process in porous materials: the diffusion theory, the capillary flow theory and the evaporation–condensation theory. A convenient starting point of drying theory is found, in the work of Whitaker [1] who has derived appropriate local volume averaged conservation equations for two-phase capillary flow in porous media. The mathematical models for simultaneous heat and mass transfer during convective drying of porous media have been studied by many authors [2–9]. These previous models are based on Whitaker's theory,
☆ Communicated by W.J. Minkowycz. E-mail address: [email protected]. 0735-1933/$ – see front matter © 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.icheatmasstransfer.2011.03.031
taking into account the thermal and mass transfer as well as the total gaseous pressure. Recently, the evolution of phase distributions within the network during drying in capillary-porous media are visualized and compared to numerical discrete simulations [10]. Also many researchers such as Bae [11], Lamnatou [12], Bubnovich [13,14], Kaya [15,16] and Kim [17] have used mathematical principles to analyze heat and mass transfer in porous materials. This does not explain the mechanism of heat and mass transfer clearly. However, most of the previous works deal with the drying of uniform materials. Indeed, little effort has been reported on the study of drying process of non-uniform material i.e., multi-layered material especially a complete comparison between mathematical model and experimental data. Typical applications of non-uniform material include the tertiary oil recovery process, geothermal analysis, asphaltic concrete pavement process and preservation process of food stuffs. Therefore, knowledge of heat and mass transfer that occurs during convective drying of multi-layered porous materials is necessary to provide a basis for fundamental understanding of convective drying of non-uniform materials. There is one published study of heat and mass transport in non-uniform material, that by Plumb [18]. This paper contains a useful review, an experimental study of heat and mass transfer in non-uniform material, and supporting numerical work. From a macroscopic point of view, the effects of the particle sizes, hydrodynamic properties, and the layered location on the overall drying kinetics must be clarified in detail. Therefore, the specific objectives of this work are to: (1) solve the mathematical model numerically, (2) compare the numerical results with experimental measurements and (3) discuss the effect of particle size and the layered location on the overall drying kinetics.
R. Prommas / International Communications in Heat and Mass Transfer 38 (2011) 900–905
901
Nomenclature cp d D Dm p p0 S T U∞ w z
specific heat capacity, (J/kgK) diameter, (m) binary mass diffusion in plain media effective molecular mass diffusion pressure, (Pa) ambient pressure, (Pa) water saturation temperature, (°C) hot air velocity, (m/s) velocity, (m/s) coordinate axis, (m)
Greek symbols ϕ porosity, (m3/m3) ρ density, (kg/m3) ξ surface tension, (Pa m) λeff effective thermal conductivity, (W/mK) α, β coordinate transformation δ packed bed depth, (m)
Subscripts 0 initial a air c capillary C coarse bed e effective F fine bed g gas ir irreducible j layer number l liquid water p particle v water vapor ∞ ambient
The result presented here provides a basis for fundamental understanding of convective drying of multi-layered packed beds.
2. The characteristic of moisture transport in multi-layered porous packed beds Fig. 1 shows the typical multi-layered packed bed. The multilayered packed beds are arranged in different configurations as follows: (a) F–C bed, the fine particles (average diameter of 0.15 mm) are over the coarse particles (average diameter of 0.4 mm). (b) C–F bed, the coarse particles (average diameter of 0.4 mm) are over the fine particles (average diameter of 0.15 mm). It is observed that the moisture content profiles are not uniform in multi-layered packed beds. During convective drying higher moisture content occurs in the fine bed while the moisture content inside the coarse bed remains lower compared with the initial state. This is a result of capillary action. The physical and transport properties of the two sizes of particles composing the bed are shown in Table 1.
Fig. 1. The typical profile of moisture content in multi-layered packed bed during convective drying in the cases of: (a) C–F bed; (b) F–C bed.
3. Experimental apparatus Fig. 2(a) shows the experimental convective drying system. The hot air, generated electrically travels through a duct toward the upper surface of two samples situated inside the test section. The outside walls of test section are covered with insulation to reduce heat loss to the ambient. The flow outlet and temperature can be adjusted at a control panel. As shown in Fig. 2(b), the samples are unsaturated packed beds composed of glass beads, water and air. The samples are prepared in two configurations in the: a single-layered packed bed (uniform packed bed) with bed depth δ = 40 mm (d = 0.15 mm (F bed) and d = 0.4 mm (C bed)) and a two-layered packed bed, respectively. The two-layered packed bed is arranged in different configurations in the: F–C bed (fine particles (d = 0.15 mm, δ = 20 mm) overlay the coarse particles (d = 0.4 mm, δ = 20 mm)), and C–F bed (coarse particles (d = 0.4 mm, δ = 20 mm) overlay the fine particles (d = 0.15 mm, δ = 20 mm)), respectively. The width and total length of all samples used in the experiments are 25 mm and 40 mm, respectively. In analysis, the main assumptions involved in the formulation of the transport model are: 1. The capillary porous material is rigid and no chemical reactions occur in the sample. 2. Local thermodynamic equilibrium among each phase is assumed. 3. The gas phase is ideal in the thermodynamic sense. 4. The contribution of convection to energy transport is included. 5. Darcy's law holds for the liquid and gas phases. 6. Gravity is included for the liquid and gas phases. 7. Permeabilities of liquid and gas can be expressed in terms of relative permeability. 8. In a macroscopic sense, the packed bed is assumed to be homogeneous and isotropic, and liquid water is not bound to the solid matrix. Therefore, the volume average model for a homogeneous and isotropic material can be used in the theoretical model and analysis. 9. A dry layer (evaporation front) is formed immediately after water saturation approaches the irreducible value. By conservation of mass and energy in the multi-layered packed bed, the governing equation of mass and energy for all phases can be derived
Table 1 The characteristics of water transport in porous materials. Diameter, d (mm)
Porosity, ϕ
Permeability, k (m2)
0.15 0.4
0.387 0.371
8.41 × 10− 12 3.52 × 10− 11
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Fig. 2. Schematic of experimental facility: (a) equipment setup; (b) multi-layered porous packed bed (sample).
using the volume-averaging technique. The main basic equation for two-phase layer and dry layer are given by following equations: 4. Basic equation for two-phase layer 4.1. Mass conservation The microscopic mass conservation equations for liquid, water vapor, air, and gas phase, are written, respectively, as
Liquid phase ρl ϕj
Vapor phase
∂sj ∂t
+ ρl
∂wl; j ∂z
:
= −n
o i ∂n ∂h : + ρv ϕj 1−sj ρv wv; j = n ∂t ∂z
ð1Þ
The kinetic energy and pressure terms, which are usually unimportant are ignored. Local thermodynamics equilibrium among all phases is assumed. The temperature of the sample exposed to hot air stream with fixed properties is obtained by solving the conventional heat transport equation. Considering the enthalpy transport based on the water and gas flows, the conduction heat and latent heat transfer due to evaporation. The energy conservation equation is represented by: o i ∂qj : ∂ ∂ hn T + ρcp ρl cpl wl; j + ρa cpa + ρv cpv wg; j T + hlv n = − T;j ∂z ∂t ∂z
ð5Þ ð2Þ
Air phase
o i ∂n ∂h + ρa ϕj 1−sj ρa wa; j = 0 ∂t ∂z
ð3Þ
Gas phase
o i ∂n ∂h : + ρg ϕj 1−sj ρg wg; j = n ∂t ∂z
ð4Þ
:
4.2. Energy conservation
where j denotes layer number, n is the condensation rate or evaporation rate during phase change and ϕ is the porosity. The water vapor and air mass flux is the sum of the convective term with the gas superficial velocity and diffusive term.
where hlv is the latent heat of vaporization of water. 3.2.1. Phenomenological relations In order to complete the system of equations, the expressions for the superficial average velocity of the liquid and gas phases derived from the generalized Darcy's law in the following form are used: wl; j
" # Kj Krl; j ∂pg ∂pc; j =− − −ρl g μl ∂z ∂z
ð6Þ
" # Kj Krg; j ∂pg −ρg g μg ∂z
ð7Þ
wg;j = −
R. Prommas / International Communications in Heat and Mass Transfer 38 (2011) 900–905
where the capillary pressure pc related to the gas and liquid phases can be written as: ð8Þ
For the velocities of water vapor and air phases, the generalized Fick's law for a two-component gas mixture can be expressed as: ρv wv;j = ρv wg; j −ρg Dm
∂ ρv ∂z ρg
!
∂ ρa ∂z ρg
ρa wa; j = ρa wg; j −ρg Dm
ð9Þ
2ϕ ð1−sÞD 3−ϕ
ð10Þ
ð11Þ
where D is the binary mass diffusion in continuous media. Fourier's law is used to define the heat flux through the porous medium qj = −λeff
∂T : ∂z
ð12Þ
4.3. Equilibrium relations The system of conservation equations obtained for multiphase transport mode requires constitutive equations for relative permeabilities Kr, capillary pressure pc, capillary pressure functions (Leverett functions) J, and the effective thermal conductivity λeff. A typical set of constitutive relationships for liquid and gas system is given by [6]: 3
ð13Þ
Krl = se
3
Krg = ð1−se Þ
ð14Þ
where se is the effective water saturation considered the irreducible water saturation sir and defined by se =
s−sir : 1−sir
ð15Þ
The capillary pressure pc is further assumed to be adequately represented by Leverett's well known J(se) functions. The relationship between the capillary pressure and the water saturation is represented by using Leverett functions J(se): ξ pc = pg −pl = pffiffiffiffiffiffiffiffiffiffiffi J ðse Þ K=ϕ
ð16Þ
where ξ is the gas–liquid interfacial tension. As indicated at the same water saturation, a smaller particle size corresponds to a higher capillary pressure. Furthermore, [19] and [20] correlated capillary pressure data obtained by Leverett as follows to give J(se): 0:217
J ðse Þ = 0:325ð1 =se −1Þ
:
ð17Þ
The effective thermal conductivity of the capillary porous medium is represented as a function of water saturation as [21]: λeff =
0:8 : 1 + 3:78e−5:95s
The experimental results for convective drying layered packed beds are compared with model simulations. In order to test the validity of the mathematical model, the results are divided into two parts covering moisture profiles and temperature profiles. One aspect of model verification was to compare drying data from experiments run under different conditions with mathematical simulations using parameter values obtained from Table 1. 5.1. Moisture profiles within multi-layered packed bed
!
and Dm is the effective molecular mass diffusion [7] Dm =
5. Result and discussion
In this study, we assume that the volumes of the upper layer and lower layer are equal. The distributions of water saturation in the layered packed beds differ greatly depending on the structure of layered packed beds even if the total volume of water that exists among the pores in the packed beds is identical. Fig. 4(a) and (b) shows the predicted moisture content profiles at various times and locations within the single-layered packed bed (F bed and C bed) and two-layered packed bed (F–C bed and C–F bed), which correspond to that of s0 = 0.5, Ta = 80 (°C) and U∞ = 1.2 (m/s), the other physical properties are given in Table 1. In the case of a single-layered packed bed, the moisture content continuously decreases toward the surface. The decrease in surface saturation, sets up a saturation gradient, which draws liquid water toward the surface through capillary action while water vapor moves towards the surface due to a gradient in the vapor partial pressure. However, the internal moisture transport is mainly attributable to capillary flow of liquid water through the voids during the initial stage of drying. The following discussion is concerned with the effect of particle size on moisture migration mechanism under the same conditions for the single-layered packed bed [22]. In Fig. 4(a), the observed moisture content profiles at the leading edge of the sample in the case of fine bed (F bed) are higher than those in the case of coarse bed (C bed). This is because the fine bed or small particle size (corresponding to a higher capillary pressure) can cause moisture to reach the surface at a higher rate than in the case of the coarse bed, i.e., large particle size. On the other hand, in the case of coarse bed, the moisture profiles at the leading edge of the sample are always lower as compared with the small particle size due to the lower capillary pressure inside the sample and the gravitational effect. Continued drying eventually causes the average moisture level inside the packed bed to decrease, especially at the leading edge of the upper layer (Fig. 4(b)). When the moisture content at the upper surface (surface saturation) approaches the irreducible where the liquid water becomes discontinuous (pendular state), the liquid water supply to the surface by capillary action becomes insufficient to replace the liquid being evaporated.
Surface Saturation Coefficient ϕ[-]
pc = pg −pl :
903
1.0 0.8 0.6 0.4 0.2 0.0 0.0
0.2
0.4
0.6
0.8
1.0
Average Saturation s[-] ð18Þ
Fig. 3. The surface saturation coefficient as a function of average water saturation.
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The latter arises from the fact that the dry layer takes place over a small effective surface or on a front retreating from the surface into the interior of the sample dividing it into two layers, a dry layer and a two-phase layer. The discontinuity of the temperature gradient close to the drying front is a result of heat flux necessary for evaporation. In addition the effective thermal conductivity falls considerably resulting in a resistance of the heat flow rate. A further consequence of premature drying of the outer dry layer is that the local temperature of the sample will reach that of convection drying. In the case of two-layered packed beds (F–C bed and C–F bed), the moisture content profiles become discontinuous at the interface between the two layers [23], namely, the moisture content in the upper layer is higher than that for the lower layer in F–C bed, while the moisture content in the upper layer is lower than that for the C–F bed. This is because the equilibrium water saturation under the same capillary pressure differs according to the particle sizes with the smallest particle size corresponding to higher water saturation. Therefore, the drying kinetics is strongly influenced by the difference in water saturation. Continuing the drying process (Fig. 4(b)) causes the average moisture level inside the F–C bed to decrease quickly in comparison with the other cases. In the case of the C–F bed, in contrast to the F–C bed, the moisture content in the upper layer (C bed) is very low and the dry layer is formed rapidly while the moisture content in the lower layer (F bed) remains high. This is due to the C bed (which corresponds to a lower capillary pressure) located above the F bed
(a) 1.0 0.9 0.8 0.7 0.6 0.5
5.2. Temperature profiles within multi-layered packed bed
t=15 [min]
0.4
F bed C bed F-C bed C-F bed
0.3 0.2 0.1 0.0
0
5
10
15
20
25
30
35
retarding the upward migration of liquid water through the interface between two layers and also due to the effect of gravity. Figs. 5 and 6 show the variations of drying mass and temperature with respect to time for the four packed beds (F, C, F–C and C–F beds) under conditions. It is observed that the variations of drying mass and temperature are interrelated. The drying speed is greatly different although the initial water saturation is the same. The F–C bed dries the fastest, and the C–F bed dries the slowest. This is due to the difference in the moisture content in the neighborhood of the drying surface as shown by Fig. 3. In the microscopic sense, the drying rate rises quickly in the early stages of drying (which corresponds to higher moisture content at the upper surface) and then decreases and reaches a plateau before decreasing again when the formation of a drying front is established. It is evident from the figure that the increase in drying rate can become significant when the F–C bed is utilized. This is a result of the strong effect of capillary pressure in F–C bed. For this case the capillary pressure easily overcomes the resistance caused by the lower permeability and maintains a supply of liquid water near the surface. However, in the case of C–F bed the formation of a dry layer is established in the early stage of drying while the moisture content within the lower layer is still high. This results in a lower drying rate when compare with other cases. The efficiency of the drying process is the lowest in this case. Further, the observed drying rates of the uniform beds (F bed and C bed) are similar. However, the drying rate for the F bed is slightly higher than that for the C bed. This is because the F bed composed of small particles leads to a higher capillary pressure resulting in a faster drying time in comparison to that for the large particles (C bed). Late in the drying process, the drying rate suddenly decreases with time due to the decrease in surface moisture content and the thermal resistance increases because of the formation of a dry layer. Overall the simulated results are in good agreement with the experimental results for convective drying.
40
Depth z[mm]
(b) 0.7
Fig. 6 gives the measured and predicted temperature profiles with respect to time within the single-layered and two-layered packed beds. Results are for s0 = 0.5,Ta = 80(°C), U∞ = 1.2 (m/s) and bed depth (z) of 5 mm in F and F–C beds. The other physical properties are given in Table 1. It is observed that at the early stage of the drying process the temperature profiles in both cases are nearly the same. As drying progresses, the temperature increases. This is because the latent heat transfer due to evaporation is retained due to the decline of the mass transfer rate together with the decreases of average moisture content. Nevertheless, the temperature profile in F–C bed rises at a higher rate
0.6 18 0.5 0.4 t=810 [min] 0.3
F bed C bed F-C bed C-F bed
0.2 0.1 0.0
0
5
10
15
20
25
30
35
40
Depth z[mm] Fig. 4. The predicted result of distribution of water saturation in multi-layered porous packed bed as a function of locations (a) initial water saturation and (b) long stage of drying.
Mass of Drying M[g]
16
T∞ =80.0oC ρ∞ =0.55 U∞ =1.2 m/s si=0.5
Formation of dry layer
14 12 10 8
Cal.
6 4 2 0 0
Exp.
d=0.15 mm d=0.40 mm F-C Beds C-F Beds 2
4
6
8
10
12
14
16
18
Elapsed Time t[h] Fig. 5. The variation of drying mass with respect to time.
20
R. Prommas / International Communications in Heat and Mass Transfer 38 (2011) 900–905
80
z=5mm Temperature T[oC]
70
differentiates it from the others. It has a shorter drying time due to the strong effect of capillary action. 3. It is also found that the drying rate depends strongly on the moisture content at the heating surface. 4. The predicted results for the temperature distribution within the multi-layered packed beds and the drying rate are in good agreement with experimental results using packed beds composed of glass beads.
Formation of dry layer
60
50 Cal.
40
Acknowledgment
Exp.
d=0.15mm F-C Beds
30 0
2
4
6
8
905
10
Elapsed Time t[h]
The authors gratefully acknowledge Institute of Research and Development (IRD) of Rajamangala University of Technology Rattanakosin and Commission on Higher Education, Thailand for providing financial support for this study.
Fig. 6. The temperature profiles with respect to time.
References than that in the F bed. This is because the dry layer formed earlier in the F–C bed and an abrupt temperature rise occurs as the dry bulb temperature is approached. On the other hand, in the case of the F bed the temperature increases more slowly in comparison with the F–C bed due to the late formation of dry layer. The prediction of the formation of a dry layer estimated from the drop in the surface mass of drying and rise in the drying temperature is marked in Figs. 5 and 6, respectively. At the longer drying time the temperature profile at any instant tends to assume constant shape throughout the region. In this period the vapor diffusion effect plays an important role in the moisture migration mechanism because of the sustained vaporization that is generated within the sample. The simulated results are in agreement with the experimental results for convective drying. Finally, the observation of drying rate and temperature profiles depicted in Figs. 5 and 6 for the convective drying of multi-layered packed beds verifies that the predicted and experimental results are qualitatively consistent. Discrepancy may be attributed to uncertainties in the thermal and physical property data. Moreover, the discrepancy may be attributed to the impact of variations in humidity and room temperature. Based on model simulations, because the total pressure is not extreme and the packed bed is dried at atmospheric condition there is never a large pressure gradient within packed bed. In fact constant pressure-drying model might be a reasonable modeling assumption when drying temperature is lower than the boiling point. From this paper, the capability of the mathematical model to correctly handle the field variations at the interfaces between materials of different hydrodynamic properties is illustrated. With further quantitative validation of the mathematical model, it is clear that the model can be used as a real tool for investigating in detail convective drying of porous materials at a fundamental level.
6. Conclusions The experiments and theoretical analysis presented in this paper describe many important interactions within multi-layered porous materials during convective drying. The following paragraphs summarize the conclusions of this study: 1. A generalized mathematical model for convective drying of multilayered packed beds is proposed. It is used successfully to describe the drying phenomena under various conditions. 2. The effects of particle sizes and layered structure on the overall drying kinetics are clarified. The drying rate in the case of the F bed (fine particles) is slightly higher than that of the C bed (coarse particles). This is because the higher capillary pressure for the F bed results in the maintenance of a wetted drying surface for a longer period of time. The F–C bed displays the drying curve which
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