Ind. Eng. Chem. Res. 2000, 39, 775-782
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Theoretical Study of Fluidized-Bed Drying with Microwave Heating Zhao Hui Wang and Guohua Chen* Department of Chemical Engineering, Biotechnology Research Institute, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong
A mathematical model of heat and mass transfer is developed for fluidized-bed drying with microwave heating. The numerical results show that the temperature is uniformly distributed within a particle and that the pressure distribution in the particle has a significant effect on microwave fluidized-bed drying. The electric field strength E is an important parameter affecting the magnitude and distribution of the moisture, temperature, and pressure within a particle. At E ) 10 000 V/m, a pressure difference of 1.3 MPa and a temperature of over 300 °C are obtained. Microwave fluidized-bed drying is capable of a much higher drying rate than conventional fluidized-bed drying, while it also maintains a low particle temperature. At a constant E, the drying time increases initially with the working load but gradually reaches a constant because of the limited moisture-carrying capacity of gas. The microwave power absorbed by the particles decreases during the drying process, and some of the microwave energy can be lost to the fluidized gas for certain operating conditions. Although the advantage of a high heatand mass-transfer coefficient associated with the fluidized bed does not contribute to water evaporation at a high load, fluidization is still important for achieving better product quality and a slightly higher drying rate when compared with microwave fixed-bed drying. Introduction Since the invention of the fluidized-bed dryer with microwave heating by Smith,1 there have been several reports concerning its drying results. Salek-Mery2 conducted a series of experimental studies with food materials. Kudra3 conducted experiments with wheat and alumina, two types of porous material. Doelling and co-workers4,5 discussed the construction and performance of a microwave fluidized-bed dryer in detail. Kaensup and co-workers6 compared its drying characteristics with conventional fluidized-bed drying based on results from drying pepper seeds. Feng and Tang7 reported experimental results for the microwave spoutedbed drying of diced apples. Although a combination of a fluidized-bed and microwave heating has been successfully used in biomass pyrolysis, polysilicon production, and catalyst calcination,8,9 the primary application of microwave fluidized-bed drying is in the dehydration of food or biomaterial. For biomaterial drying, the temperature of the material is a very important parameter that affects not only the drying time but more importantly the product quality. It is extremely difficult to measure the temperature of a particle during fluidized-bed drying, not to mention the temperature, moisture, and vapor pressure within a particle during drying with additional microwave heating. Therefore, theoretical analyses have to be carried out for this purpose. Such analyses are available for fluidized-bed drying10-12 or microwave drying13-16 but not for fluidized-bed drying with microwave heating. This lack of literature is probably due to the complex transport phenomena in the particles and in the fluidized bed. Recently, a comprehensive model for heat and mass transfer through porous materials was developed by the authors and successfully applied to convection drying,17 fixed-bed drying,18 and fluidized-bed drying.19 The * Corresponding author. Tel: (852)23587138. Fax: (852)23580054. E-mail:
[email protected].
model is extended in this paper to consider the effect of microwave heating on fluidized-bed drying. Interesting results are obtained, as will be seen subsequently. Mathematical Modeling Heat and Mass Transfer in a Particle. When the convective heat transfer in the material is neglected, the mathematical model for heat and mass transfer in a one-dimensional porous sphere can be obtained as19
∂S 1 ∂ ∂S ∂T 1 ∂ ) a r2 + 2 a3r2 ∂τ r2 ∂r 2 ∂r ∂r r ∂r
(1)
∂T 1 ∂ ∂T ∂S 1 ∂ ) b r2 + 2 b3r2 + q˘ ∂τ r2 ∂r 2 ∂r ∂r ∂r r
(2)
∂Pg ∂S ∂T ) c2 + c3 ∂r ∂r ∂r
(3)
a1 b1
(
(
c1
)
)
(
(
)
)
The mass transfer in the porous material includes the liquid capillary flow, gas flow, and vapor diffusion, which are quantified by Darcy’s law and Fick’s law, respectively. The coefficients in eqs 1-3 can be found in the Appendix, and the internal heat generation intensity in eq 2 can be expressed as
q˘ ) kE2
(4)
Because of the fact that the size of a particle is far smaller than the penetration depth of the microwave, the electric field strength is considered to be uniformly distributed within the particle. Heat and Mass Transfer in a Fluidized Bed. Here we consider the situation of a one-dimensional distribution fluidized bed along the bed height, as illustrated in Figure 1. The particles are considered well mixed, and the gas is assumed in plug flow. The heat conduction and vapor diffusion in the gas phase are neglected. In addition, the amount of the vapor dehydrated from the particles is considered negligible as compared with
10.1021/ie990428a CCC: $19.00 © 2000 American Chemical Society Published on Web 01/29/2000
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Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000
∂T ∂r
(15)
Kl ∂(Pg - Pc) F µl l ∂r
(16)
Kg ∂Pg ∂Fv F - Dv(1 - S) µg v ∂r ∂r
(17)
q ) -λ jl ) jv ) -
The results of heat- and mass-transfer coefficients obtained in a fixed bed by Ranz20 are widely applied to fluidized-bed drying in the literature21 and will also be applied in the present work:
Nu ) 2 + 1.8Re1/2Pr1/3 1/2
Sh ) 2 + 1.8Re Sc
Figure 1. Schematic of microwave fluidized-bed drying.
the total gas flow in the bed. Then the temperature and vapor density of the outlet gas from the fluidized bed are19
Tfout ) T|r)RP + (Tfin - T|r)RP)e-fh
(5)
Ffout ) Fv|r)RP + (Fvfin - Fv|r)RP)e-fm
(6)
where
fh ) SthfA
(7)
fm ) StmfA
(8)
fA ) NAP/AB
(9)
For a given fluidized bed and a given particle size, fA indicates the loading of wet materials in the fluidizedbed dryer. fA is an important parameter in the operation and design of a fluidized-bed dryer, as established in Wang and Chen.19 With T and Fvf known, the relative humidity of the gas can be determined. Because the particles stay mixed in the fluidized bed during the drying process, the boundary conditions vary constantly for each particle. To simplify the treatment of heat and mass transfer, an averaging method is adopted. Specifically, considering the total heat and mass balances between the particles and gas from the inlet to the outlet, the average temperature and vapor density of gas flow are calculated as19
1 - e-fh fh
T h f ) T|r)RP + (Tfin - T|r)RP)
-fm
1-e fm
Fjvf ) Fv|r)RP + (Fvfin - Fv|r)RP)
(10) (11)
The gas pressure in the fluidized bed is considered uniform and assumes the value of atmospheric pressure. Heat and Mass Transfer at the Interface between Particles and Gas.
where
h f) q|r)RP - jl|r)RP∆H ) Rh(T|r)RP - T
(12)
h vf) (jl + jv)|r)RP ) Rm(Fv|r)RP - P
(13)
Pg|r)RP ) Patm
(14)
1/3
(18) (19)
where the superficial velocity of the gas is used in the Reynolds number calculation. Boundary Conditions at the Particle Center. Because of symmetry, the boundary conditions at the particle center can be simply written as
| |
∂S ∂r
r)0
∂T ∂r
r)0
)0
(20)
)0
(21)
Boundary Conditions of Gas.
Tf|z)0 ) Tfin
(22)
RH|z)0 ) RHin
(23)
Initial Conditions.
T|τ)0 ) T0
(24)
S|τ)0 ) S0
(25)
Physical Properties. In the present work, the physical properties of apple are used in the simulation. Apple is chosen as the biomaterial not only because its physical properties are available but also because experimental results have been reported for drying apple with a microwave fluidized-bed dryer.7 Detailed data of the physical properties are listed in Table 1. By using the average method, the heat capacity and thermal conductivity can be obtained as
Fc ) FlclS + Fgcg(1 - S) + Fscs(1 - )
(26)
λ ) λlS + λg(1 - S) + λs(1 - )
(27)
and the microwave energy dissipation coefficient of the material is
k ) klS + ks(1 - )
(28)
On the basis of Kelvin’s equation, the vapor density in the material is
Fv )
P(T) -Pc/FlRvT e RvT
(29)
where P(T) is the saturated vapor pressure. The capillary pressure can be written as22
Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 777 Table 1. Physical Properties for the Drying Model
Table 2. Typical Operating Conditions
symbol
value
unit
ref
symbol
value
unit
cg cl cs D ∆H KD kl ks Scr λg λl λs µg µl
1006 4180 1810 2.56 × 10-5 2.443 09 × 106 1 × 10-19 1.3 0.5 0.3 0.7 0.025 0.65 0.78 1.83 × 10-5 (0.458 509-5.304 74 × 10-3T + 2.312 31 × 10-5T2 - 4.491 61 × 10-8T3 + 3.276 81 × 10-11T4 1.29 1000 1600 0.121 978 - 0.000 168 3T
J/(kg‚K) J/(kg‚K) J/(kg‚K) m2/s J/kg m2 W/m3‚(V/m)-2 W/m3‚(V/m)-2
E Tfin RHin dP fA u Patm T0 S0
2000 60 10 5 100 2 1.01325 × 105 20 0.95
V/m °C % mm
J/(s‚m‚K) J/(s‚m‚K) J/(s‚m‚K) kg/(m‚s) kg/(m‚s)
25 25 26 25 25 assumeda 27 27 assumeda 28 25 25 26 25 25
kg/m3 kg/m3 kg/m3 kg/s2
25 25 28 25
Fg Fl Fs σ
m/s Pa °C
a Assumption made based on the experimental values of similar materials.
Pc ) J(S) σx/KD
(30)
where the Leverett function23 J(S) is
J(S) )
0.020023 0.009547 + - 0.12S + 0.4415 (31) S S - 1.028
The permeabilities of the porous particle for liquid flow and gas flow are respectively
Kl ) KDKrl
(32)
Kg ) KDKrg
(33)
where the relative permeability of liquid flow24 is
{(
)
S - Scr Krl ) 1 - Scr 0
3
S > Scr
(34)
S e Scr
The relative permeability of gas in the material is
Krg ) 1 - Krl
(35)
Numerical Results Heat and Mass Transfer during Microwave Fluidized-Bed Drying. Equations 1-25 were solved numerically with the finite difference method. The iteration is stopped when
| | |
Sj+1 i
-
Sji
| | |
Sji
Tj+1 - Tji i Tji
Pj+1 - Pji i Pji
< 10-6
< 10-6 < 10-6
It should be noted that a stable electromagnetic field in the fluidized bed is assumed and a uniform and stable electric field strength in the particle is used in the
Figure 2. Profiles of saturation, temperature, and pressure in the particle during drying (E ) 2000 V/m): (a) saturation, (b) temperature, (c) pressure. τ/s: (1) 0; (2) 30; (3) 100; (4) 400; (5) 700; (6) 1000; (7) 1100.
present calculation. Table 2 lists the typical operating conditions in the simulation. In the following discussion, only those values different from the respective ones listed in Table 2 are noted. Numerical calculations show that the results are more sensitive to the permeability of the porous material than other parameters. Detailed analyses are available elsewhere.17 Figure 2 shows the profiles of saturation, temperature, and pressure in a particle under typical operating conditions. When the moisture at the surface of the particle is larger than its critical value, capillary flow is the main mechanism for moisture transport (curves 2 and 3 in Figure 2a). When the moisture at the surface
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falls below its critical value, there are three typical regions: a capillary flow region near the center, an evaporation-condensation region near the surface, and a transition region in between (curves 4 and 5 in Figure 2a). After the moisture near the center of the particle falls below its critical value, the moisture transport is by evaporation-condensation only. The criteria for distinguishing the three regions have been discussed in detail in a previous publication.17 Although moisture is distributed nonuniformly during almost the full course of the drying process (Figure 2a), the temperature distribution in the material is almost flat (Figure 2b). During the initial period of drying, the moisture remains relatively constant (curve 2 in Figure 2a) while the particle temperature rises by over 30 °C (curve 2 in Figure 2b). Almost all of the heat absorbed by the material is used to increase the temperature of the material. When the moisture decreases significantly (curves 3-5 in Figure 2a), the temperature varies within a rather small range around 60 °C (curves 3-5 in Figure 2b). Obviously, the microwave energy absorbed by the particles is mainly consumed in the evaporation of the moisture during this drying period. As the moisture in the particle becomes negligible at τ ) 1000 s, the temperature of the particle rises rapidly again. The temperature approaches a steady distribution (curve 7 in Figure 2b) after the moisture content of each point in the material reaches its local equilibrium value (curve 7 in Figure 2a). Careful examination of Figure 2b shows that there is a minimum temperature observable within the material. This minimum temperature appears near the particle surface at the beginning of drying (curves 2 and 3) and moves into the transition region when the surface saturation decreases below the critical saturation (curves 4 and 5). According to the findings of Wang and Chen,17 the transition region has the most intensive evaporation, which explains why a minimum temperature is observed there. At the end of drying, the temperature decreases from the center of the material to the surface, as expected. Now all of the microwave energy absorbed by the material is carried away by the gas through the fluidized bed. An insignificant pressure difference is noticed in the particle for the drying conditions of Figure 2, as shown in Figure 2c. The pressure decreases from the center to the surface of the particle throughout the entire drying process. The maximum pressure difference is about 7.5 kPa between the values at the center and at the surface. Thus, the gas within the particle flows from the particle center to the surface, carrying the vapor out of the material via convection. Figure 2c also shows that the pressure gradient is very small in the capillary region. The gradient becomes greater in the transition region and the evaporation-condensation region. As a result, gas convection plays a more important role in vapor transfer in the transition and evaporation-condensation regions than in the capillary flow region. This is different from low-intensity convection drying in which the gas pressure is considered uniform and no gas convection appears in the material.17 It is expected that the moisture, temperature, and pressure distributions will be affected by the electric field strength, and this is indeed shown to be the case in Figure 3. Compared with Figure 2a, the moisture distribution at a high electric field strength (Figure 3a) shows a gradual increase from the center to the surface
Figure 3. Profiles of saturation, temperature, and pressure in the particle during drying (E ) 10 000 V/m): (a) saturation, (b) temperature, (c) pressure. τ/s: (1) 0; (2) 1.5; (3) 5; (4) 10; (5) 20; (6) 35; (7) 50.
of the particle during a certain period of time, curves 5 and 6. This has been explained by Turner and Jolly14 as a pump effect for the situation with internal microwave heating. The present numerical calculation also shows that this effect is more significant when the wet material is more permeable. The general trend for the temperature variation at this high electric field is similar to that observed in Figure 2b but with a different magnitude. The material temperature rises rapidly to over 100 °C after only 5 s of drying (curve 3 in Figure 3b). Then the moisture in the material evaporates at a temperature of around 180 °C (curves 3-6 in Figure 3b). When the drying process ends, the particle temperature reaches about 250 °C (curve 7 in Figure 3b). Such findings show two effects: first the temperature distribution is no longer uniform, and second the highest temperature can be much higher than the threshold temperature for heat-sensitive materials, which is 60 °C for the apple used here. It should be noted that the temperature at the particle surface can also be much higher than the inlet gas temperature of 60 °C. The air flow in this situation plays a cooling role rather than supplying heat to the wet particles.
Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 779
Figure 4. Microwave power variation absorbed by the particles during drying.
Although in real drying an apple particle cannot maintain its physical properties as listed in Table 1 under such severe conditions, the hypothetical condition does generate some interesting results. These results not only help to understand the transport phenomena involved but also may be useful for processing rigid and heatinsensitive materials. A pressure difference of about 1.3 MPa is found for the drying conditions of Figure 3 (curve 4 in Figure 3c). Such a large pressure difference can easily make the surface of material crack. The high internal pressure is also harmful to the quality of the material although the mass transport in the material is significantly increased by the pressure gradient. In this case, the convective vapor transfer in the particle can no longer be neglected. According to eqs 4 and 28, the microwave heating power applied to the particles varies in the drying process because of the decreasing moisture content in the material. Figure 4 shows the microwave power variation during drying for the conditions shown in Table 1. Assuming a stable and uniform electric field strength in the particle, the microwave power absorbed per unit of material (dry basis) markedly decreases with time. As a result, the microwave power absorbed by the fluidized-bed dryer is reduced and reaches the minimum value by the end of drying. This situation will be encountered for batch fluidized-bed drying with microwave heating but not for continuous operations. The microwave power that is supplied after drying is used solely to heat the gas through the fluidized bed via the particles. Because the microwave power supplied per unit of material (dry basis) for the operation of E ) 10 000 V/m can be about 25 times that at E ) 2000 V/m, the drying time of the former operation is reduced to about 1/20 of the latter. The selection of a large or small electric field strength requires a consideration of the properties of the wet material and the dryer. Figure 5 shows the outlet gas temperature variations during microwave fluidized-bed drying at two different electric field strengths. For both cases, the temperature changes insignificantly during a relatively long period of time with regard to the total drying process, about 130 and 57 °C for E ) 10 000 V/m and for E ) 2000 V/cm, respectively. There is a temperature drop for drying at a higher electrical field strength because of the significant decrease of absorbed microwave energy with moisture content and the increase in the drying rate. At the end of drying, the temperatures reach their own equilibrium values. Because of microwave heating, these values at the end of drying are higher than 60
Figure 5. Temperature variations of outlet gas during drying: (1) E ) 2000 V/m; (2) 10 000 V/m.
Figure 6. Drying rate curves with different heating methods: (1) E ) 2000 V/m; Tf ) 60 °C; fA ) 100; (2) E ) 2000 V/m; Tf ) 60 °C; fA ) 1000; (3) E ) 0 V/m; Tf ) 300 °C; fA ) 100; (4) E ) 0 V/m; Tf ) 60 °C; fA ) 100; (5) E ) 0 V/m; Tf ) 60 °C; fA ) 1000.
°C, which is the inlet gas temperature and also the equilibrium value for conventional fluidized-bed drying. Comparison with the Conventional FluidizedBed Drying. To make a comparison with conventional fluidized-bed drying, a simulation is also carried out at E ) 0 V/m. Figure 6 shows the drying rate curves under five different operating conditions, some with and some without microwave heating. The following observations can be made. First, the effects of the limited moisturecarrying capacity of the gas through the fluidized bed become significant with the increase in the particles charged in the conventional fluidized-bed dryer, indicated by the dramatic decrease in the drying rate when fA rises from 100 (curve 4) to 1000 (curve 5). Second, the conventional fluidized-bed dryer has a limitation in terms of its drying rate, shown by the limited increase of the constant drying rate when the inlet gas temperature is increased to 300 °C (curve 3). Third, microwave heating can greatly increase the drying rate, demonstrated by curves 1 and 2. The maximum drying rate, 0.0028/s, is 50 times higher than those without microwave heating at Tf ) 60 °C and fA ) 1000. Consequently, the drying time required to reach 5% of the initial saturation is reduced from 25 600 to 870 s for the case at fA ) 1000 and from 4930 to 785 s for the case at fA ) 100 when microwave heating is applied at only E ) 2000 V/m. It is useful to note that the drying time obtained with microwave heating compares favorably with that reported by Feng and Tang7 obtained from drying apple dice. The fourth observation is that the loading effect is much smaller for microwave fluidized-bed drying than for conventional fluidized-bed drying. Such a phenomenon will be analyzed subsequently.
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Figure 7. Variations of drying times with drying load for different heating methods: (1) E ) 0 V/m; Tf ) 60 °C; (2) E ) 2000 V/m; Tf ) 60 °C.
Figure 7 shows the drying time at different working loads for conventional and microwave fluidized-bed drying. Here the drying time is obtained when the moisture left in a particle is 1.5% of its initial value. The figure shows that the drying time increases with the working load for both cases. The limited moisturecarrying capacity of the gas through the fluidized bed has an effect on both drying methods; otherwise, the drying time would be unrelated to the working load and would simply stay constant with fA. However, a difference exists between the two curves in the figure. The drying time increases obviously with the value of fA for conventional fluidized-bed drying (curve 1). For microwave fluidized-bed drying (curve 2), the drying time rises only for small fA values and stays relatively constant when fA is larger than 100. With the increase of particle load in the fluidized bed, the microwave power supplied to the dryer increases because of the assumption of a steady electric field strength. The energy from volumetric heating increases proportionally with the increase of the working load, whereas the flowing gas has a limit in terms of the heat and mass it can carry. When the loading of wet material is small, i.e., fA is small, the effects from gas as well as microwave heating are important. Because the capacity of the gas effect is limited, its effect per unit of wet material would decrease with the increase of load, resulting in an increase in drying time. Because a further increase of load makes the effect of gas per unit of wet material approach zero, the only significant effect is then from microwave heating, which is a constant. Thus, the drying time does not change with fA for microwave fluidized-bed drying. The consumption of total energy will, of course, increase proportionally with fA. Thus, there exists a critical fA, above which the contribution of fluidizing gas to dehydration is insignificant. The value of this critical fA depends on E, u, and Tfin as well as on the inlet gas RH. For the conditions shown in Table 1, this critical value is about 100. The direct implementation of the phenomenon analyzed above is that the advantage of high heat-transfer coefficient for fluidized-bed drying does not show for the region where fA is greater than its critical value. Along with the increase of fA, the calculation shows that the outlet gas temperature becomes higher than the inlet temperature even at earlier stages of the drying process and is much higher than the inlet temperature by the end of drying for a large load operation. In this case, the result of the higher heat-transfer coefficient is that a larger amount of the microwave energy that is absorbed by the particle is lost to the fluidizing gas. This
Figure 8. Particle temperature variations in drying (fA ) 100): (a) fixed-bed drying; (b) fluidized-bed drying. s: r/RP ) 0. - -: r/RP ) 0.5. - -: r/RP ) 1.
does not mean that gas fluidization makes no contribution to microwave fluidized-bed drying. In fact, fluidization is still important for controlling the properties of dried products for the uniform temperature distribution noticed before. This advantage will be further illustrated in the next section, where microwave fluidized-bed drying is compared with microwave fixed-bed drying. Comparison with Microwave Fixed-Bed Drying. Based on the numerical method developed by the authors,18 microwave fixed-bed drying at u ) 0.5 m/s is calculated in the present work. The differential bed area factor ∆fA ) 5 is used in the numerical calculation and the microwave energy is assumed to be uniformly supplied to the wet particles. Figure 8a shows the temperature variations of the particles at the inlet and the outlet of the fixed-bed dryer at fA ) 100. For the particles at the inlet, the temperature is about 55 °C during most of the drying process and the largest value is about 70 °C by the end of drying. For the particles at the outlet, the temperature can reach 90 °C in the early stage of drying and more than 150 °C by the end of the process. It should be noted that the magnitude of the temperature increase is related to the microwave heating power. When this heating power is small or the fixed-bed height is not very high, the outlet temperature should be lower than the inlet temperature.29 This temperature nonuniformity results from the nonuniform distribution of gas parameters in the fixed bed. Such a temperature nonuniformity among wet particles would result in nonuniform quality of dried products. For the same operating conditions, if the particles are fluidized, the temperature varies only slightly, staying around 60 °C from the beginning to nearly the completion of the drying process, as shown in Figure 8b. The maximum temperature after the completion of drying is below 90 °C. Because of fluidization, all of the particles in the fluidized-bed dryer have the same status.
Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000 781
Figure 9. Variations of drying times with drying load for different drying methods: (1) fixed bed (u ) 0.5 m/s); (2) fluidized bed (u ) 2 m/s).
Because of fluidization, the contribution of the gas to the moisture removal at smaller fA is much more significant for the fluidized bed than for the fixed bed. In addition, even in the microwave heating control region, fA > 60, the drying time for fixed-bed drying is over 10% longer than the time for fluidized-bed drying because of the nonuniformity of the particles in the fixed bed, as shown in Figure 9. Conclusion The moisture distribution in a wet particle during microwave fluidized-bed drying can be in the capillary flow region, evaporation-condensation region, and transition region. The temperature in the particle is distributed relatively uniformly during most of the drying process, with an insignificant minimum temperature observed in the transition region. The pressure distribution in the particle may have a significant effect on the microwave fluidized-bed drying process, while the microwave heating power plays an important role in affecting the magnitude as well as the distribution of moisture, temperature, and pressure in the wet particle. The microwave power decreases in the drying process, and some of the microwave power absorbed by the particles can be lost to the fluidizing gas. The drying rate for microwave fluidized-bed drying can be much higher than the conventional fluidized-bed drying, while it also keeps the particle temperature low. When fixed microwave heating electric strength is maintained, there exists a critical loading to the fluidized bed, beyond which the drying time is constant. The particle temperature can be higher than the gas temperature in this case, and the advantage of a high heattransfer coefficient in conventional fluidized-bed drying becomes a disadvantage in microwave fluidized-bed drying. Fluidization is, however, still important in achieving better product quality and a slightly higher drying rate when compared with microwave fixed-bed drying. Acknowledgment The authors are grateful for the financial support from BRI of HKUST and ITDC of Hong Kong SAR Government. Nomenclature a1, a2, a3 ) coefficients of mass-transfer equations AP ) surface area of a single particle, m2 AB ) flow area of a fluidized bed, m2
b1, b2, b3 ) coefficients of heat-transfer equations c ) specific heat capacity, J/(kg‚K) c1, c2, c3 ) coefficients of pressure distribution equations D ) diffusivity, m2 dP ) particle diameter, m E ) electric field strength, V/m f ) fraction of moisture content left in the particle to the initial moisture content fA ) area factor of the fluidized bed fh ) modified Stanton number of heat transfer in the fluidized bed, SthfA fm ) modified Stanton number of mass transfer in the fluidized bed, StmfA H ) height of the fluidized bed, m ∆H ) evaporation heat, J/kg j ) mass flux, kg/(m2‚s) K ) permeability, m2 KD ) permeability, m2 Kr ) relative permeability k ) microwave energy dissipation coefficient, W/m3‚(V/m)-2 N ) total number of particles in the fluidized bed Nu ) Nusselt number, RhdP/λg P ) pressure, Pa Patm ) atmospheric pressure, Pa ∆P ) P - Patm Pc ) capillary pressure, Pa Pr ) Prandtl number, cgµg/λg p ) microwave power, kW/kg (dry basis) q ) heat flux, J/(m2‚s) q˘ ) internal heat generation intensity, W/m3 r ) space axis, m Rg ) gas constant of air, m2/(s2‚K) Rv ) gas constant of vapor, m2/(s2‚K) RP ) particle radius, m Re ) Reynolds number, udPFg/µg RH ) relative humidity, % S ) saturation Scr ) critical saturation Sc ) Schmidt number, µg/FgD Sh ) Sherwood number, RmdP/D Sth ) Stanton number of heat transfer, Nu/(Re × Pr) Stm ) Stanton number of mass transfer, Sh/(Re × Sc) T ) temperature, °C or K u ) superficial of gas, m/s z ) axis along bed height, m Greek Letters Rh ) heat-transfer coefficient, J/(s‚m2‚K) Rm ) mass-transfer coefficient, m/s ) porosity, 1 - s λ ) thermal conductivity, J/(s‚m‚K) µ ) dynamic viscosity, kg/(m‚s) F ) mass density, kg/m3 σ ) surface tension, kg/s2 τ ) time, s Superscript j ) iteration number in numerical calculation Subscripts f ) gas in the fluidized bed g ) gas in the material i ) node in numerical calculation in ) inlet l ) liquid out ) outlet s ) solid matrix S ) saturation T ) temperature v ) vapor 0 ) initial
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Ind. Eng. Chem. Res., Vol. 39, No. 3, 2000
overbar ) average value in the fluidized bed
Appendix The coefficients of eqs 1-3 are
a1 ) Fl
c1 )
(A1)
a2 )
c2d1 + d2 c1
(A2)
a3 )
c3d1 + d3 c1
(A3)
b1 ) Fc
(A4)
b2 ) λ + ∆HDvT
(A5)
b3 ) ∆HDvS
(A6)
(
)
Kg Pg Dg(1 - S) - Fv + µg RgT Rg T c2 ) Dv(1 - S)
c3 ) Dg(1 - S) d1 )
Pg Rg T
2
(A7)
∂Fv ∂T
+ Dv(1 - S)
(A8) ∂Fv ∂T
Kl Kg F + F µl l µg g
(A9)
(A10)
d2 ) -
Kl ∂Pc ∂Fv + Dv(1 - S) Fl µl ∂S ∂S
(A11)
d3 ) -
Kl ∂Pc ∂Fv Fl + Dv(1 - S) µl ∂T ∂T
(A12)
DvT )
c3 Kg ∂Fv Fv + Dv(1 - S) c1 µg ∂T
(A13)
DvS )
c2 K g ∂Fv F + Dv(1 - S) c 1 µg v ∂S
(A14)
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Received for review June 15, 1999 Revised manuscript received November 15, 1999 Accepted December 2, 1999 IE990428A
