This article was downloaded by: [University of Guelph] On: 06 October 2012, At: 09:42 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Drying Technology: An International Journal Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/ldrt20 Mathematical Modeling of a Continuous Vibrating Fluidized Bed Dryer for Grain Apolinar Picado a b & Joaquín Martínez a a Department of Chemical Engineering and Technology, KTH Royal Institute of Technology, Stockholm, Sweden b Faculty of Chemical Engineering, National University of Engineering (UNI), Managua, Nicaragua Version of record first published: 17 Sep 2012. To cite this article: Apolinar Picado & Joaquín Martínez (2012): Mathematical Modeling of a Continuous Vibrating Fluidized Bed Dryer for Grain, Drying Technology: An International Journal, 30:13, 1469-1481 To link to this article: http://dx.doi.org/10.1080/07373937.2012.690123 PLEASE SCROLL DOWN FOR ARTICLE Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
Transcript
This article was downloaded by: [University of Guelph]On: 06 October 2012, At: 09:42Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK
Drying Technology: An International JournalPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/ldrt20
Mathematical Modeling of a Continuous VibratingFluidized Bed Dryer for GrainApolinar Picado a b & Joaquín Martínez aa Department of Chemical Engineering and Technology, KTH Royal Institute of Technology,Stockholm, Swedenb Faculty of Chemical Engineering, National University of Engineering (UNI), Managua,Nicaragua
Version of record first published: 17 Sep 2012.
To cite this article: Apolinar Picado & Joaquín Martínez (2012): Mathematical Modeling of a Continuous Vibrating FluidizedBed Dryer for Grain, Drying Technology: An International Journal, 30:13, 1469-1481
To link to this article: http://dx.doi.org/10.1080/07373937.2012.690123
PLEASE SCROLL DOWN FOR ARTICLE
Full terms and conditions of use: http://www.tandfonline.com/page/terms-and-conditions
This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form toanyone is expressly forbidden.
The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses shouldbe independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims,proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly inconnection with or arising out of the use of this material.
Mathematical Modeling of a Continuous Vibrating FluidizedBed Dryer for Grain
Apolinar Picado1,2 and Joaquın Martınez11Department of Chemical Engineering and Technology, KTH Royal Institute of Technology,Stockholm, Sweden2Faculty of Chemical Engineering, National University of Engineering (UNI), Managua, Nicaragua
A mathematical model for the drying of grain in a continuousvibrating fluidized bed dryer was developed. Simple equipment andmaterial models were applied to describe the process. In the plug-flowequipment model, a thin layer of particles moving forward and wellmixed in the direction of the gas flow was examined. Mass and heattransfer within a single wet particle was described by effective trans-port coefficients. Assuming constant effective mass transport coef-ficient and thermal conductivity, analytical solutions of the massand energy balances were obtained. The variation in both transportcoefficients along the dryer was taken into account by a stepwiseapplication of the analytical solution in space intervals with averagedcoefficients from previous locations in the dryer. Calculation resultswere in fairly good agreement with experimental data from theliterature. However, the results depend strongly on relationships usedto determine the heat and mass transfer coefficients; because theresults from correlations found in the literature vary considerably,the correlations should be adapted to the specific equipment in orderto obtain reliable results.
Keywords Continuously worked dryer; Dryer simulation; Dry-ing modeling; Drying of particulate materials; Heatand mass transfer
INTRODUCTION
Continuous vibrating fluidized bed dryers have been suc-cessfully employed to dry a variety of particulate solids suchas inorganic salts, fertilizers, foodstuffs, pharmaceuticals,plastics, and coated materials. In general, vibrating flui-dized beds are used for drying cohesive, sticky, and agglom-erated materials that cannot be well fluidized due to verylarge or small diameter particles, wet particles, or a verythin bed layer. A vibrating fluidized bed dryer has severaladvantages compared to a conventional fluidized bed dryer:it allows reduction of gas velocity and pressure drop andprovides better uniformity of the agglomerate, gentle hand-ling, better control of residence time, and intensification of
heat and mass transfer, which can significantly reduce theenergy consumption and drying time.[1–4]
A number of studies have been conducted on variousaspects of this type of dryer. The drying of wheat particu-lates and instant pharmaceutical Bi Yan Ning (BYN)granules in a continuous pilot-plant vibrating fluidized beddryer[5] showed that the flow of particles could be consideredas plug-flow and that vibration intensity was the most sig-nificant factor affecting the particle mean residence timeand drying rate. In the optimization of a commercial-scalevibrofluidized bed dryer for paddy rice[6] it was found thatthe combined electrical power of the blower and vibrationmotors was approximately 55% of that used by the blowermotor alone in a conventional fluidized bed dryer. Inert sup-porting particles have been used to dry soybean milk[7] andliquid pastes.[8] Teflon pellets were found to be superior toglass beads because they resulted in higher heat transfercoefficients, higher liquid feed rates, and lower materialholdup. It was also concluded that vibration intensity shouldnot be adopted as a universal parameter to characterize thedynamics of vibrofluidized beds. In experiments usingground eggshell and an image analysis methodology[9] itwas found that vibration amplitude and the air flow hadthe most significant effect on the mean residence time andparticle dispersion. Other studies have also been done onminimum fluidization velocity,[10] heat transfer between thegas and solid,[11] and particle motion through the dryer.[12]
Recently, a model based on the concept of a drying coef-ficient to describe the resistance against internal moisturetransport was developed for a batch vibrating fluidizedbed dryer.[13] This type of dryer has also been modeled bya neural network approach[14] that takes into account theback-mixing effect by establishing interconnected dryingzones. The model included the mass and energy balancesfor each zone in the solid phase, and complete mixing wasassumed in the gas phase. Continuous vibrating fluidizedbed dryers have been simulated by incremental models. Inthis regard, most of the equipment models assume plug-flowof the solids, but nonideal solids flow has also been stud-ied.[15,16] Gas cross-flow has been studied to some extent.[17]
Correspondence: Apolinar Picado, Department of ChemicalEngineering and Technology, KTH Royal Institute of Tech-nology, Teknikringen 42, SE-100 44 Stockholm, Sweden; E-mail:[email protected]
Drying Technology, 30: 1469–1481, 2012
Copyright # 2012 Taylor & Francis Group, LLC
ISSN: 0737-3937 print=1532-2300 online
DOI: 10.1080/07373937.2012.690123
1469
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
An incremental model for drying of solids containingmulticomponent mixtures has also been developed.[18] Themodel qualitatively described well the main features ofmulticomponent drying, particularly the effects of solidresistance against mass transfer on the retention of volatilecomponents. Although significant experimental and theor-etical efforts have been made to understand the transportprocesses in vibrating fluidized bed dryers, the currentknowledge is still far from complete.[17,19]
The purpose of this study was to develop a model tosimulate the drying of wet grain in a continuous vibratingfluidized bed dryer. The model was developed by incorpor-ating a material model for a single wet grain, amenable tobe solved analytically, in an incremental equipment modelassuming plug-flow of the solids with gas cross-flow. Thevalidity of the model was tested by comparison of modelpredictions with experimental data from the literature.
MATHEMATICAL MODEL
A schematic description of the vibrating fluidized beddryer is shown in Fig. 1. Wet solid particles enter the dryerby a screw conveyor and are fluidized smoothly by thevibration combined with an upward flow of gas, normallyair coming from the bottom of the bed through a gasdistributor. In this way, the requirements of fluid dynamicsand heat and mass transfer are effectively decoupled,allowing a pseudo-fluidization of the bed with rather lowgas flow rates. In addition, an effective mixing of the par-ticles takes place and a homogeneous material at a verticalcross section of the dryer is usually obtained. As indicatedin the literature,[5,20] the residence time distribution of theparticles measured at the outlet does not differ much fromthat calculated for a plug-flow model. Due to mechanicalvibration and the slope of the dryer, solid particles moveforward along the dryer and a partially dry product exitsfrom the end.
The following assumptions are introduced to simplifythe complex characteristics of the process in the dryer:
� Solids are a spherical grain, isotropic, uniform insize, and homogenous.
� The grain is perfectly well mixed in the verticaldirection.
� Shrinkage of the grain during drying is negligible.� Physical properties of the dry grain remainconstant with time.
� The inlet distribution of the moisture content andtemperature is uniform.
� Heat and mass transfer inside the grain takes placeonly in the radial direction.
� Moisture at the solid surface is in equilibrium withthe gas humidity.
� The dryer is perfectly insulated.
Mass and Energy Balances
In the analysis of the dryer, it was assumed that the bedof particles was moving forward at a uniform velocity andthat the dryer had been operating for long enough to ensurethat steady-state conditions were reached. A moisturebalance applied to the differential volume element shownin Fig. 2 yields:
FsdXb
dz¼ � as M Gg ð1Þ
whereX is the solid moisture content on a dry basis,M is themolecular weight of the moisture, as is the specific evapor-ation area per unit bed volume, Gg is the molar evaporationflux, F is a mass flow per cross section in the direction of theflow, and z is the distance along the bed from the solids inlet.The subscripts s and b denote solid and bed, respectively.
Because all of the evaporated liquid goes to the gas, thefollowing equation gives the change in gas humidity:
FgdY ¼ �Fs HbdXb
dzð2Þ
FIG. 1. A plug-flow vibrating fluidized bed dryer. FIG. 2. Scheme of a differential dryer element.
1470 PICADO AND MARTINEZ
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
where Y is the gas humidity on a dry basis and Hb is thebed height. The subscript g denotes gas. The bed heightis calculated as
Hb ¼S
vqpð1� epÞð1� ebÞBð3Þ
where S is the flow of dry solids, v is the forward bedvelocity, q is the density, E is the porosity, and B is the dryerwidth. The subscript p denotes particle. Because heat lossesin the dryer are neglected, the energy balance over the samedifferential volume element becomes
dIg ¼ �HbFs
Fg
dIsdz
ð4Þ
where I is the enthalpy of the phases per unit mass on a drybasis. To integrate Eq. (1) to determine the changes inmean liquid content of the grain along the dryer, inaddition inlet conditions, the evaporation flux must be pro-vided. This flux depends on the temperature and liquidcontent at the surface of the particles. This informationcan be obtained by analyzing what happens with a singleparticle moving along the dryer.
Drying of a Single Particle
The drying of a single particle into an inert gas is sche-matically described in Fig. 3.
During the drying process, the moisture migrates fromthe center of the solid particle toward the surface, whereit evaporates. Migration occurs due to the moisture gradi-ent in the solid particle by several mechanisms: moleculardiffusion (liquid and vapor), capillary flow, Knudsen dif-fusion, surface diffusion, or combinations of the foregoingmechanisms. Usually, all of these mechanisms are lumpedinto an effective liquid transport mechanism using Fick’slaw for mass flux, which seems to describe the experimentaldata fairly well.[16,21]
Governing Equations
Based on the assumptions outlined above, a set ofspace-averaged governing equations can be written forthe particle. The longitudinal location of a particle in thedryer is related to the residence time (t) by the linear velo-city of the bed (v). This allows one to track the changes inmoisture content and temperature of particles as functionsof distance along the bed (z). By introducing this changein variables, the diffusion equation for the sphericalparticle is[22]
v@X
@z¼ Deff
@2X
@r2þ 2
r
@X
@r
� �ð5Þ
where r is the radial coordinate and Deff is the effectivemass transport coefficient that is a function of the liquidcontent, temperature, and structure of the solid and canbe expressed in the following general form:
Deff ¼ f T ; X ; structure of thematerialð Þ ð6Þ
A collection of such equations for various food materialscan be found elsewhere.[21] In the present study, a relation-ship reported by Maroulis et al.[23] was used.
If conduction is the only mechanism for heat transferwithin the particle, the corresponding equation to describetemperature changes is the conduction equation[24]:
v@T
@z¼ Dh
@2 T
@r2þ 2
r
@T
@r
� �ð7Þ
where Dh is the thermal diffusivity of the wet grain, and Tis the temperature.
Equations (5) and (7) represent a system of partial dif-ferential equations with the following inlet and boundaryconditions:
X ¼ X0 rf g;T ¼ T0 rf gAt z ¼ 0 and 0 � r � d; ð8Þ
@X
@r¼ 0;
@T
@r¼ 0 At z > 0 and r ¼ 0; ð9Þ
� qp Deff@X
@r¼ MGg; �keff
@T
@r
¼ h T � Tg;1� �
þ kGg At z > 0 and r ¼ d;ð10Þ
where d is the particle radius, k is the thermal conductivityof the wet solid, and k is the latent heat of vaporization.The subscripts 0, eff and 1 denote inlet, effective value,and gas bulk, respectively. The effective thermal conduc-tivity of the wet solid was computed using correlations thatcan be found elsewhere.[16]
FIG. 3. Schematic of drying of a single particle into an inert gas (color
figure available online).
MODELING OF A VIBRATING FLUIDIZED BED DRYER 1471
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
Mass and Heat Transfer Rates
At the interphase, the evaporation flux is simplyexpressed as
Gg ¼ km yd � y1ð Þ ð11Þ
where km is the mass transfer coefficient, and yd and y1 arethe vapor molar fractions at the gas–wet solid interface andin the gas bulk, respectively. In a hygroscopic material, thevapor pressure at the surface will be lower than the satu-rated value. This is accounted for by introducing a sorptionisotherm that relates the equilibrium vapor pressure at thesolid surface to the saturated value. Experimental equilib-rium data for different wet materials are conveniently cor-related by the following equation[25]:
w ¼ Pw
P0w
¼ exp � c1T þ c2
exp �c3Xð Þ� �
ð12Þ
where Pw is the partial pressure of water vapor. Thesuperscript 0 denotes a saturated value, which dependson the temperature only. The constants c1, c2, and c3can be found in the literature for several materials.[26] Thisinformation is inserted into Eq. (11), where yd is given byratio between the partial pressure of water vapor and thetotal pressure. When the grain is saturated with liquid andwater covers the particle surface, w becomes 1; otherwise,w becomes smaller than unity.
The convective heat flux can be calculated as:
q ¼ h Tg;1 � Td� �
ð13Þ
where h is a heat transfer coefficient between the dryinggas and particle. There are several empirical correlationsin the literature to calculate the gas-to-particle heattransfer coefficient for a vibrating fluidized bed dryer.[4]
Solution of the Model
Equation (1) is a nonlinear ordinary differential equa-tion that can be solved numerically using Euler’s or anyother standard method. The simultaneous solution ofEqs. (5) and (7), subjected to inlet and boundary con-ditions (8) through (10), provides the temperature andmoisture content gradients in the particle as well as massand heat transfer rates at the particle surface. Mass andheat fluxes are required in the mass and energy balancesalong the dryer. By assuming constant transport coeffi-cients as well as constant heat and mass transfer ratesin each integration step of Eq. (1), Eqs. (5) and (7) canbe solved analytically. In dimensionless form, the solutionof the mass balance is
u ¼ 2X1m¼ 1
e�Dd n2m s n2m þ ð/�1Þ2
n2m þ /ð/�1Þ
!
sinðnm fÞZ10
u0ðfÞ sinðnm fÞ df
8<:
9=;
ð14Þ
where the roots in Eq. (14) are defined implicitly by:
tanðnmÞ ¼ ð1� /Þ�1nm ð15Þ
Finally, the dimensionless moisture content, u, is trans-formed back to obtain the moisture content in the particle:
X ¼ /�1 u
f� yb
� �ð16Þ
Analogously, the solution for the heat transfer equation is
H ¼ 2X1m¼ 1
e�j n2h;m
s n2h;m þ ða�1Þ2
n2h;m þ aða�1Þ
!
sinðnh;m fÞZ10
H0ðfÞ sinðnh;m fÞ df
8<:
9=;
ð17Þ
with roots defined implicitly by:
tanðnh;mÞ ¼ ð1� aÞ�1nh;m ð18Þ
Substitution back to temperature:
T ¼ Tg �ðTg � T0Þ
a
Hf� b
� �ð19Þ
For details about the analytical solutions and dimen-sionless variables as well as other parameters of the solu-tions, see the Appendix.
The liquid content and the temperature at the center ofthe particle are undefined in Eqs. (14) and (17) and have tobe calculated as limit values. The integral terms in the solu-tions contain the inlet conditions and are obtained numeri-cally from the inlet liquid content or temperature gradients.To assess the convergence of the series, the solutions can beanalyzed by using short space steps and without includingthe sinus term that is constrained to defined boundaries.
Because these conditions change along the dryer, theanalytical solution is applied to an interval dz, with inletconditions and averaged transport coefficients correspond-ing to the outlet conditions of the previous step. As theintegration of Eq. (1) proceeds, the procedure is repeatedstep by step. The outlet composition of the gas at each stepdz is calculated using Eq. (2). Then the energy balance,Eq. (4), allows for the calculation of the exhaust gas
1472 PICADO AND MARTINEZ
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
enthalpy using the mean particle temperature to calculatethe outlet enthalpy of the wet solids. Because the gasenthalpy is a function of gas composition and temperature,the outlet gas temperature can be calculated from a non-linear equation that relates gas temperature with the gasenthalpy. Integration proceeds in this way until the dryerexit is reached.
Calculation of the roots of Eqs. (15) and (18) turned outto be very sensitive to the initial guess. An initial guessclose to the true value was obtained by linear interpolationof values tabulated in Carslaw and Jaeger.[24]
Interface heat and mass transfer coefficients togetherwith the physical properties of the particles, effective inter-nal transport coefficients, and equilibrium isotherms have agreat influence on the results. There are several correlationsavailable in the literature to calculate heat transfer coeffi-cients in vibrating fluidized bed dryers. However, they seemto be specific for the equipment and the conditions in whichthey were determined because they predict results that varyconsiderably. Consequently, model validation woulddemand the test of available correlations.
Mass transfer coefficients were calculated from heattransfer coefficients by applying the Lewis analogy becausethe Prandtl and Schmidt numbers are almost equal formixtures of water vapor and air.[16]
To calculate heat and mass transfer rates according toEqs. (11) and (13), the conditions at the bulk of the gas wereconsidered constant and equal to inlet gas conditions. If theevaporation flow is large compared to the gas flow, the gastemperature and humidity will change and the assumptionwill lead to an overestimation of the driving forces. Becausethe outlet gas temperature and composition were calculatedafter the solution of the equations for the particle in a givenstep, possible cooling and humidification of the gas afterpassing the bed had no restrictive effect on the drying ofthe particles during integration. However, the outlet tem-perature of the gas can be used to determine the reasonable-ness of the calculated results. In fact, the consistency of thetransfer coefficients calculated with correlations from theliterature was determined by checking the outlet gas tem-peratures and disregarding correlations that producedvalues lower than grain outlet temperature.
RESULTS AND DISCUSSION
Convergence of the Analytical Solution
The convergence of the solution was assessed by calculat-ing the number of series terms that have to be included to inorder to obtain an acceptable accuracy. The rate of conver-gence depends mainly on drying intensity; that is, on massand heat transfer rates, as revealed in Fig. 4, which showsthe calculations for the experimental conditions in Fig. 6and a case with 10 times higher mass and heat transfer rates.In the first case, the relative increment tends to one in less
than 10 terms. Here, the increment is relative to the totalsum after 100 terms.
Because calculation of the terms required to assessconvergence is part of the calculation of the solution, thenumber of series terms required to obtain satisfactory accu-racy may be automatically controlled at each integrationstep, thus introducing computational economies.
Heat and Mass Transfer Coefficients
Because the conditions in which the correlations tocalculate heat transfer coefficients were determined arenot always well defined, and there are uncertainties con-cerning the equipment in which the experimental data usedto verify simulations were obtained, several correlationsfrom the literature were tested (see Table 1). The resultsshowed that there were differences up to four orders ofmagnitude between the results of different correlations.
These results also revealed that heat transfer coefficients ina vibrating fluidized bed were less than those in a conventional
FIG. 4. Rate of convergence of the analytical solution as a function of the
number of eigenvalues. Tg¼ 380.15K, Y0¼ 0.0024 kg=kg, S¼ 30.7 kg=h,
one at a given particle Reynolds number. This was presum-ably due to the differences in gas–particle relative motionsbetween the equipment. Heat transfer coefficients used inthe following simulations were calculated using Vyletok andMushtayev’s correlation.[4]
Model Validation
In order to investigate the validity of the model predic-tions, the theoretical results were compared with pilot-scaleexperimental data from the literature. The material underconsideration was corn grits. The experimental vibratingfluidized bed dryer consisted of a chamber with a rectangu-lar cross section for the gas flow of 1m� 0.35m and aheight of 0.1m. Details of the experimental procedurecan be found elsewhere.[27] The experimental data sets wereeither cited directly from tables or read (digitized) fromexperimental points on figures in Alvarez et al.[27] The basicphysical properties of corn grits used in the simulation aresummarized in Table 2.
TABLE 1Correlations for the heat transfer coefficient in a vibrating fluidized bed dryer
Author Correlation h0 (kW=m2K) he (kW=m2K)
Chlenov and Mikhailov[4] Nu ¼ 1:16Re Hb
A
� ��0:8 f ugg
� �0:960.4855 0.3564
Sbrodov[4] Nu ¼ 0:142ReC0:04v 0.0932 0.0833
Vyletok and Mushtayev[4] h ¼ 10:4ug
1�ebAxg
� �0:10.0098 0.0098
Pakowski et al.[4] Nu ¼ 0:827Re1:04C0:483v
dpHb
� �1:170.1369 0.1217
Vega et al.[29] Nu ¼ 0:2345Re0:0076C�0:613v
Adp
� ��0:702
0.0718� 10�3 0.0710� 10�3
Wang et al.[30] Nu ¼ 0:15Re0:21C0:90v
dpHb
� �0:540.0359 0.0354
Yu et al.[31] Nu ¼ 0:0138ReC0:392v
dpHb
� �0:3740.0121 0.0112
Brod et al.[32] Nu ¼ 70:48Re0:47C�1:19v
dp /s
Hb
� �2:210.2038� 10�3 0.2034� 10�3
Rezende and Finzer[33] Nu ¼ 0:1926Re0:156C0:889v 0.1574 0.1564
0.35;0.30 FIG. 5. Sorption isotherms and effective mass transport coefficient for
corn.
1474 PICADO AND MARTINEZ
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
Sorption isotherms for corn and the effective masstransfer coefficient for mass transport inside the corn usedin the simulations are shown in Fig. 5.
Figures 6 and 7 show a comparison between the meanmoisture content and mean solid temperature predictedby the mathematical model and experimental data for thesame operating conditions. Both cases exhibited fairly goodagreement with the experimental data. As can be seen in thefigures, the predicted and experimental moisture contentsexhibited a smooth exponential decay curve over the wholelength of the dryer that characterizes the drying in thefalling rate period for hygroscopic materials. Air dryingof many food and agricultural products[16] such as corndisplays no constant rate period and the internal resistanceto moisture transfer controls the drying process.
The temperature of the solids progressively increases asthey move from the feed point to the dryer exit. In the endstages of drying, the mathematical model underestimatesthe solid temperatures of corn grits. However, it is difficult
to discern whether it is due to limitations of the model oruncertainties in the experimental data. To some extent,deviations can be attributed to the estimated propertiesthat depend on the internal structure and the nature ofthe solid, such as effective transport coefficients and sorp-tion isotherms, but also to uncertainties of the experi-mental measurements. Due to the complex flow patternin the dryer it is not always possible to obtain representa-tive samples. In addition, measuring the true particle tem-perature is a difficult task.
The liquid content and temperature profiles for particles atdifferent locations along the bed length revealed that theinternal resistance for mass transfer was greater than theinternal resistance against heat transfer (see Fig. 8). Tempera-ture profiles in the particles were flat, whereas liquid contentprofiles in the solid exhibited pronounced gradients. Thebasic data for the calculations correspond to the experimentshown in Fig. 6. The low resistance against heat transferwithin the solid may be explained by the relatively high value
FIG. 6. Comparison between experimental and predicted moisture
content and solids temperature along the bed length. ug¼ 0.36m=s, Tg¼380.15K, Y0¼ 0.0024kg=kg, S¼ 30.7kg=h, Hb¼ 0.025m, v¼ 4.2� 10�3
m=s, A¼ 0.025m, f¼ 20Hz (color figure available online).
FIG. 7. Comparison between experimental and predicted moisture
content and solids temperature along the bed length. ug¼ 0.95m=s, Tg¼368.15K, Y0¼ 0.0024 kg=kg, S¼ 40.7 kg=h, Hb¼ 0.05m, v¼ 2.8� 10�3
m=s, A¼ 0.03m, f¼ 40Hz (color figure available online).
MODELING OF A VIBRATING FLUIDIZED BED DRYER 1475
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
of the thermal conductivity of water and its contribution tothe effective thermal conductivity of the moist solid.
Effects of Operational Conditions on Drying Kinetics
Because knowledge of the parameters that significantlyimpact drying behavior is useful in designing dryers, simu-lations were extended to predict the effects of operatingparameters. The basic data used in the simulations thatfollow were the same as those in Fig. 6. In all cases, a�15% variation in the operating parameters was applied.
Effect of Gas Velocity and Temperature
Figure 9 depicts the changes in moisture content andsolid temperature at different inlet air velocities. Three mainmechanisms determine the drying rate in a convective dry-ing process: heat and mass transfer in the gas side (externaltransfer), thermodynamic equilibrium at the interphase, andheat and mass transfer within the wet solid (internal trans-fer). In drying processes controlled by external drying
conditions, a higher gas velocity induces faster heat andmass transfer rates because the mass and heat transfer coef-ficients increase with the gas velocity. The drying rate alsoincreases due to increased gas temperature and reducedgas humidity. Both variables enhance mass and heat trans-fer because they increase the driving forces. In solid-sidecontrolled processes, an indirect and less discernible influ-ence remains. For instance, external conditions determinewhen the transition to a solid-side controlled process occurs;highly intensive initial drying results in an earlier transition.External conditions also contribute to shape the tempera-ture and gas composition at the interphase. For these rea-sons, higher gas velocity and temperature as well as lowergas humidity improve the drying rate even in a processunder control of internal mass and heat transfer, althoughindirectly and to a lesser extent than in a process governedby the external conditions. As shown in Figs. 9 and 10, bothmoisture content and temperature of the solid behave as
FIG. 8. Moisture content and temperature profiles for particles along the
FIG. 9. Effect of gas velocity on drying behavior. Tg¼ 380.15K, Y0¼0.0024 kg=kg, S¼ 30.7 kg=h, Hb¼ 0.025m, v¼ 4.2� 10�3m=s, dp¼ 1.7�10�3m, A¼ 0.025m, f¼ 20Hz.
1476 PICADO AND MARTINEZ
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
expected according to drying theory with the changes in gasvelocity and temperature.
Under the conditions of the present simulations, theresistance against heat transfer inside the solid is muchlower than the resistance against mass transfer inside thesolid. This means that convection heat not used to evapor-ate water is used to increase the solids temperature (seeFig. 10), which in turn leads to higher coefficients ofmoisture transfer inside the solid and higher partial vaporpressure at the solid–gas interphase and hence to higherdrying rates.
Effect of Vibration Intensity
It was found that the vibration intensity, Cv, had only aslight influence on the drying rate (results not shown). Thiscould be due to the operational range of the vibrationintensity, Cv¼ 28–52, used in the calculations. Accordingto some authors,[4] the optimum range of vibration
intensity to obtain a suitable bed structure and high dryingrates is Cv¼ 1–6. To be consistent with the other para-meters employed in the experiments, we used the range ofthe experimental conditions reported by Alvarez et al.[27]
for calculations.
Effect of Particle Size
Figure 11 illustrates the effect of particle diameter on thedrying process. It was observed that particle diameterexerted a significant influence on the drying process; largerparticles required much longer dryer length to reach thesame final liquid content due to the increase in internalresistance against heat and mass transfer.
The results of the simulations depend strongly on heatand mass transfer coefficients. As shown in Table 1, theresults of the empirical correlations from the literature varygreatly. It is evident that mass and heat transfer coefficientsare dependent on the experimental conditions and the
FIG. 10. Effect of gas temperature on drying behavior. ug¼ 0.36m=s,
specific equipment in which the experiments are performed.Therefore, to obtain reliable results from simulations, thetransfer coefficients or at least the constants included inthe correlations should be determined from experimentalresults in the same equipment.
As mentioned previously, for the sake of simplicity, theconditions of the gas were assumed constant and equal tothe inlet conditions. With this assumption, the drivingforces are overestimated because unless the bed is very thinand evaporation is low compared to the gas flow, the airshould be colder and more saturated with vapors at thedryer exit. Even when inlet gas conditions are assumedconstant, driving forces change along the dryer due tochanges in the liquid content and particle temperature. Ifthe changes in gas composition due to the evaporationrates obtained in the simulations are considered, drivingforces could be reduced by as much as 50%. In fact, thechanges in gas conditions can be considered in the drivingforces at each step by solving iteratively until reachingconvergence of the exit conditions. However, consideringthat the variation in transfer coefficients calculated by cor-relations from the literature is of four orders of magnitudeand that the objective of this work is not to optimize cor-relation parameters, such a refining of the driving forceswas not considered necessary at this stage.
CONCLUSIONS
A mathematical model for grain drying in a vibratingfluidized bed dryer was developed and validated. Thepredicted moisture content and temperature using theproposed model showed fairly good agreement with theexperimental data. However, the results depended stronglyon correlations used to determine heat and mass transfercoefficients. Because the results from correlations found inthe literature vary considerably, the transfer coefficientsor at least the constants included in the correlations shouldbe determined from experimental results in the same equip-ment to obtain more reliable results. Consequently, anequipment model for this type of dryer will never be com-pletely predictive. Simulations based on this model wereconducted to study the effects of operating parameters suchas gas velocity, gas temperature, vibration intensity, andparticle size on the moisture content and solids temperature.An increase in gas velocity and temperature induced fasterdrying. As the particle diameter was increased, the dryingprocess slowed down. In these calculations, vibration para-meters had no appreciable effect on the drying process. Theanalytical solution allows for rapid and reliable predictionsof drying processes with proper parameters. Althoughthe comparison between experiments and calculations ispromising, further model validation with other materialswould be valuable to make the mathematical model a usefultool for process exploration and optimization of this typeof dryer.
NOMENCLATURE
A Amplitude of vibration (m)as Specific area per unit bed volume (m2 m�3)B Dryer width (m)cp Specific heat (J kg�1 K�1)c1, c2, c3 Constant with dimensions as defined by Eq. (12)D Mass transport coefficient (m2 s�1)Dh Thermal diffusivity (m2 s�1)dp Particle diameter (m)F Mass flow per cross section (kg m�2 s�1)f Frequency of vibration (Hz)Gg Molar evaporation flux (k molm�2 s�1)g Acceleration due to gravity (m s�2)Hb Bed height (m)h Heat transfer coefficient (W m�2 K�1)I Enthalpy per unit mass, dry basis (J kg�1)k Thermal conductivity (W m�1 K�1)km Mass transfer coefficient (kmolm�2 s�1)L Dryer length (m)M Molecular weight of the moisture (kg kmol�1)Nu Nusselt numberP Pressure (Pa)Pw Partial pressure of water vapor (Pa)q Heat flux (J m�2 s�1)Re Reynolds numberr Radial coordinate (m)S Flow of dry solids (kg s�1)T Temperature (K)t Time (s)u Transformed and dimensionless moisture
contentug Gas velocity (m s�1)V Flow of dry air (kg s�1)v Forward bed velocity (m s�1)X Solids moisture content, dry basis (kg kg�1)Y Gas humidity, dry basis (kg kg�1)y Molar fraction in the gas phase (kmol kmol�1)z Distance along the length of the bed (m)
Greek Letters
Cv Vibration intensity [� Ax2=g]d Particle radius (m)e PorosityH Transformed and dimensionless temperaturek Latent heat of vaporization (J kmol�1)nh,m Roots related to heat transfernm Roots related to mass transferq Density (kg m�3)us Sphericityw Relative humidityx Angular frequency [� 2pf] (rad s�1)
Subscripts and Superscripts
b Bed
1478 PICADO AND MARTINEZ
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
e Exiteff Effective valueg Gasp Particles Solidt Totald Interface1 Gas bulk0 Inlet0 Saturated value
ACKNOWLEDGMENTS
The authors gratefully acknowledge the financial sup-port provided by the Swedish International DevelopmentCooperation Agency (Sida). A. Picado also acknowledgescomplementary funding support from the Linder Foun-dation and AForsk.
REFERENCES
1. Gupta, R.; Mujumdar, A.S. Aerodynamic and thermal characteristics
of vibrated fluid beds—A review. In Drying’80: Vol. 1. Developments
in Drying; Mujumdar, A.S., Ed.; Hemisphere Publishing: New York,
1980; 141–150.
2. Danielsen, S.; Hovmand, S. Drying of granulated product in a vibrated
fluid bed. In Drying’80: Vol. 1. Developments in Drying; Mujumdar,
A.S., Ed.; Hemisphere Publishing: New York, 1980; 194–199.
3. Strumillo, C.; Pakowski, Z. Drying of granular products in vibroflui-
dized beds. In Drying’80: Vol. 1. Developments in Drying; Mujumdar,
A.S., Ed.; Hemisphere Publishing: New York, 1980; 211–226.
4. Pakowski, Z.; Mujumdar, A.S.; Strumillo, C. Theory and application
of vibrated beds and vibrated fluid beds for drying processes.
In Advances in Drying, Vol. 3; Mujumdar, A.S., Ed.; Hemisphere
Publishing: New York, 1984; 245–306.
5. Han, W.; Mai, B.; Gu, T. Residence time distribution and drying char-
acteristics of a continuous vibro-fluidized bed. Drying Technology
fer in a vibrated fluidized bed. In Proceedings of the 8th International
Drying Symposium (IDS ‘92), Montreal, Canada, August 2–5,
1992.
32. Brod, F.P.R.; Pecora, A.A.B.; Park, K.J. Heat transfer in vibro-
fluidized beds: A comparative study. In Proceedings of the 16th Brazi-
lian Congress of Mechanical Engineering (COBEM 2001), Uberlandia,
Brazil, November 26–30, 2001 (in Portuguese).
33. Rezende, D.R.; Finzer, J.R.D. Dimensionless on coffee fruit drying in
the vibrated tray dryer. FAZU emRevista 2008, 5, 73–78 (in Portuguese).
MODELING OF A VIBRATING FLUIDIZED BED DRYER 1479
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
APPENDIX: ANALYTICAL SOLUTION OF THEEQUATIONS FOR THE PARTICLE
Equations (5) and (7) can be rewritten by introducingthe following dimensionless variables:
s ¼ z
L; f ¼ r
d; h ¼ Tg � T
Tg � T0
� �ðA1Þ
The system of partial differential Eqs. (5) and (7)becomes
@X
@s¼ Dd
@2X
@f2þ 2
f@X
@f
� �;@h@s
¼ j@2h
@f2þ 2
f@h@f
� �ðA2Þ
with
Dd ¼ LDeff
vd2; j ¼ LDh
vd2ðA3Þ
The inlet and boundary conditions are
At s ¼ 0 and 0 � f � 1;X ¼ X0 ff g; h ¼ h0 ff g ðA4Þ
Ats > 0 and f ¼ 0;@X
@f¼ 0;
@h@f
¼ 0 ðA5Þ
At s > 0 and f ¼ 1;� @X
@f¼ /X þ yb;�
@h@f
¼ ahþ b
ðA6Þ
with
/ ¼ ML
vdqpDdkm Kw� �
; yb
¼ � ML
vdqpDdkm y1ð Þ;Kw ¼ wP0
w
X Pt
ðA7Þ
and
a ¼ h
keffd ; b ¼ �
a k Gg
� �h ðTg � T0Þ
ðA8Þ
Equations (A2) may be now transformed into equationsthat describe linear flow in one direction. By introducingthe following new dependent variables, the boundary con-ditions are also transformed to take advantage of existinganalytical solutions:
u ¼ f ð/X þ ybÞ;H ¼ f ðahþ bÞ ðA9Þ
The set of Eqs. (A2) becomes
@u
@s¼ Dd
@2u
@f2;@H@s
¼ j@2H
@f2ðA10Þ
The new inlet and boundary conditions are
At s ¼ 0 and 0 � f � 1; u ¼ u0 ff g; H ¼ H0 ff g ðA11Þ
At s > 0 and f ¼ 0; u ¼ 0;H ¼ 0 ðA12Þ
At s > 0 and f ¼ 1;@u
@fþ /� 1ð Þu ¼ 0;
@H@f
þ a� 1ð ÞH ¼ 0 ðA13Þ
Because the transformed moisture equation is notexplicitly dependent on temperature, it can be solved separ-ately. The solution of Eqs. (A10) for those inlet and bound-ary conditions is contained in Carslaw and Jaeger.[24]
For the mass transfer equation:
u ¼ 2X1m¼ 1
e�Dd n2m s n2m þ ð/�1Þ2
n2m þ /ð/�1Þ
!
sinðnm fÞZ10
u0ðfÞ sinðnm fÞ df
8<:
9=;
ðA14Þ
The roots in Eq. (A14) are defined implicitly by:
tanðnmÞ ¼ ð1� /Þ�1nm ðA15Þ
Finally, by using Eq. (A9), u is transformed back toobtain the moisture content:
X ¼ /�1 u
f� yb
� �ðA16Þ
At the center of the particle, when f¼ 0, the moisturecontent is undetermined and therefore the expression mustbe evaluated as a limit. The limit of the expression is relatedto the derivative of the transformed moisture with respectto the dimensionless radius:
limf!0
u
f
� �¼ lim
f!0
du
df
� �ðA17Þ
1480 PICADO AND MARTINEZ
Dow
nloa
ded
by [
Uni
vers
ity o
f G
uelp
h] a
t 09:
42 0
6 O
ctob
er 2
012
By evaluating the derivative of Eq. (A14) at f¼ 0:
limf!0
du
df
� �¼ 2
X1m¼ 1
e�Dd n2m s n2m þ ð/� 1Þ2
n2m þ /ð/� 1Þ
!
nm
Z10
u0ðfÞ sinðnm fÞ df
8<:
9=;
ðA18Þ
The solution for the heat transfer equation is
H ¼ 2X1m¼ 1
e�j n2h;m
s n2h;m þ ða�1Þ2
n2h;m þ aða�1Þ
!
sinðnh;m fÞZ10
H0ðfÞ sinðnh;m fÞ df
8<:
9=;
ðA19Þ
with roots defined implicitly by:
tanðnh;mÞ ¼ ð1� aÞ�1nh;m ðA20Þ
Substitution back to temperature:
T ¼ Tg �ðTg � T0Þ
a
Hf� b
� �ðA21Þ
The values of the center are calculated using a similarrelation between the limits. In this case,
limf!0
Hf
� �¼ lim
f!0
dHdf
� �ðA22Þ
Applied to Eq. (A19):
limf!0
dHdf
� �¼ 2
X1m¼ 1
e�j n2h;m
s n2h;m þ ða� 1Þ2
n2h;m þ aða� 1Þ
!
nh;m
Z10
H0ðfÞ sinðnh;m fÞ df
8<:
9=;
ðA23Þ
To determine the optimal number of series terms that isnecessary to obtain a satisfactory solution, the rate of con-vergence of the series can be examined by evaluating
SN ¼ 2XNm¼ 1
e�Dd n2m s n2m þ ð/�1Þ2
n2m þ /ð/� 1Þ
!
Z10
u0ðfÞ sinðnm fÞ df
8<:
9=;
ðA24Þ
SNh¼ 2
XNm¼ 1
e�j n2h;m
s n2h;m þ ða�1Þ2
n2h;m þ aða�1Þ
!
Z10
H0ðfÞ sinðnh;m fÞ df
8<:
9=;
ðA25Þ
with a chosen short space step without including the sinusterm from the original solution because it is constrainedto defined boundaries. The integral terms in the solutionscontain the inlet conditions and are obtained numericallyfrom the inlet liquid content or temperature gradients. Bychoosing an appropriate value of space step and solvingEqs. (A24) and (A25) for an increasing number of eigenva-lues until the series converges, the optimal number of seriesterms can be determined with a given truncation error. Lunaet al.[28] reported this approach for drying calculations.
