Numerical simulation and experimental investigation of plug-flow fluidized bed drying under dynamic conditions

Journal of Food Engineering 137 (2014) 64–75

Contents lists available at ScienceDirect

Journal of Food Engineering journal homepage: www.elsevier.com/locate/jfoodeng

Numerical simulation and experimental investigation of plug-flow fluidized bed drying under dynamic conditions Majid Khanali ⇑, Shahin Rafiee, Ali Jafari Department of Agricultural Machinery Engineering, Faculty of Agricultural Engineering and Technology, College of Agriculture and Natural Resources, University of Tehran, Karaj, Iran

a r t i c l e

i n f o

Article history: Received 9 December 2013 Received in revised form 19 March 2014 Accepted 21 March 2014 Available online 1 April 2014 Keywords: Drying Dynamic Fluidized bed Modeling Plug-flow

a b s t r a c t In this study, a mathematical model was presented for prediction of the plug-flow fluidized bed drying process under dynamic conditions resulted from the transient of inlet dry solids mass flow rate. The model previously developed and successfully validated for plug-flow fluidized bed drying process under steady-state condition was the starting point of this study. This model was extended in order to account for the mass and energy transfers between solids and gas phases at dynamic conditions. Additionally, a mass balance equation of dry-based solid holdup and a mass flow rate relationship for outlet solids were developed to predict the transient response of outlet dry solids mass flow rate. The model equations were solved numerically using the finite difference method. To validate the model, drying of rough rice in a laboratory-scale plug-flow fluidized bed dryer was investigated under dynamic conditions. A very satisfactory agreement between simulated and measured results was achieved. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction The goal of any industrial drying process is to produce a product of desired quality at maximum throughput and minimum cost and to maintain these consistently. Good quality implies that the product corresponds to a number of technical, chemical, and biological parameters, each within specified limits. Drying process is one of the most energy-intensive unit operations due to the high latent heat of vaporization and the inherent inefficiency of using hot air as the drying medium. Application of an automatic model-based control system to industrial dryer offers an opportunity to improve the dryer operation and its efficiency. The basic idea of this control system is to use a dynamic model of the drying process that predicts the drying behavior under dynamic conditions caused by the transient of operating parameters (Mujumdar, 2006). A continuous fluidized bed dryer is operated with a continuous stream of solid particles flowing within the bed. The inlet dry solids mass flow rate is an important factor affecting the extent of drying of the outlet product by changing the residence time of solids in the dryer. In continuous dryers, the inlet solids mass flow rate is a common manipulated variable set using the control system (Mujumdar, 2006). Although a large number of studies in modeling and simulation of continuous fluidized bed dryers have been devoted for steady-state conditions (Nilsson and Wimmerstedt, 1987; Fyhr ⇑ Corresponding author. Tel.: +98 2632801011; fax: +98 2632808138. E-mail address: [email protected] (M. Khanali). http://dx.doi.org/10.1016/j.jfoodeng.2014.03.020 0260-8774/Ó 2014 Elsevier Ltd. All rights reserved.

et al., 1999; Izadifar and Mowla, 2003; Wanjari et al., 2006; Baker et al., 2006; Ramli and Daud, 2007; Bizmark et al., 2010; Apolinar and Martínez, 2012; Khanali et al., 2013), studying the fluidized bed drying process under dynamic conditions rigorously has received little attention. Burgschweiger and Tsotsas (2002) investigated the continuous well-mixed fluidized bed drying process under both steady-state and dynamic conditions. The mixing behavior and residence time distribution of particles in the dryer were considered as a continuous stirred tank reactor. The mass flow rates of solids and gas, air heater capacity, inlet solids moisture content, and inlet gas temperature were varied systematically and a very good agreement between measured and calculated results was obtained. Abdel-Jabbar et al. (2002) presented a model to simulate the dynamic behavior of a continuous well-mixed fluidized bed dryer by combining the drying kinetics for diffusioncontrolled system and residence time density function. Only one study on the mathematical modeling of an industrial plug-flow fluidized bed drying process under dynamic conditions has been published by Tacidelli et al. (2012). The proposed model was based on the two-fluid model consists of the gas and particulate phases. The flow of the gas through the bed was considered as a plug and the particulate phase was treated as a perfect mixture. To validate the model, the dynamic responses of the drying process at the dryer outlet to the step changes of +10% and 10% in the inlet solids moisture content were simulated. The objective of the present work was to study the mathematical modeling and experimental investigation of the plug-flow fluidized bed drying process under dynamic conditions. This

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

65

Nomenclature a specific surface area of expanded bed (m2/m3) a1, a2, a3 constant parameter b1, b2, b3 constant parameter c1, c2, c3, c4 constant parameter C specific heat (J/kg K) D dispersion coefficient (m2/s) E mean relative error (%) g acceleration of gravity (m/s2) h length of the discrete segment along the dryer length (m) hbed bed height (m) hv heat of vaporization (J/kg) hweir weir height (m) H enthalpy (J/kg) k1 constant parameter k2 constant parameter L bed length (m) M solids moisture content (dry basis) (kg water/kg dry solid) n number of time steps N number of discrete segments along the bed length _p mass flow rate of solids (kg dry solid/s) m P number of discrete segments along the bed height q constant parameter r constant parameter R drying rate (kg moisture/kg dry solid s) S dry-based solid holdup (kg dry solid) t time (s) T temperature (K or °C)

modeling was an extension of the steady-state plug-flow fluidized bed drying model developed by Khanali et al. (2013). In this way, a mathematical model was developed to predict the solids moisture content and temperature and outlet gas humidity and temperature along the dryer length as well as the outlet solids moisture content and temperature under dynamic conditions resulted from the transients of inlet solids mass flow rate. The accuracy of the model was also investigated by comparison with the experimental data of plug-flow fluidized bed drying process under dynamic conditions.

u U w x X X Y z

longitudinal solid flow velocity (m/s) gas velocity (m/s) bed width (m) bed length coordinate (m) arbitrary drying attribute Mean value of arbitrary drying attribute gas humidity (kg water/kg dry gas) bed height coordinate (m)

Greek letters a heat transfer coefficient (W/m2 K) q density (kg/m3) n solids concentration in the bed (kg dry solid/ m3 fluidized bed) Subscripts 0 superficial bed bed exp experimental g gas in input out output p particle pre predicted ref reference v water vapor w water

model. Important assumptions and features of this model are as follow:  Dispersed plug-flow of solids within the bed from inlet to outlet of the dryer, i.e. the combination of ideal plug-flow at constant longitudinal solid flow velocity (bulk flow) and longitudinal dispersion.  Ideal plug-flow of the gas from bottom to top of the bed.  Perfect mixing of particles in vertical direction, which results the uniform moisture content and temperature in the major control volume.  Consideration of uniform physical and fluidization properties of the particles along the dryer length.

2. Development of the model The study of drying process under dynamic conditions can be conducted by considering the start-up behavior or the stepresponses resulted from the transient of various operating parameters. Here, it is refrained from discussing different types of startup behavior and is concentrated only on the transients initiated by step change in inlet solids mass flow rate. The schematic differential model of the proposed plug-flow fluidized bed drying process at dynamic condition is presented in Fig. 1. The bed of the dryer is divided horizontally into major control volumes of dx length (hbedwdx) as Fig. 1a and each of these control volumes is divided vertically into minor control volumes of dz height (wdxdz) as Fig. 1b. This differential scheme of the dryer model at dynamic condition is similar to the scheme given by Khanali et al. (2013) for the dryer model under steady-state condition except for the accumulation terms that shows the change of moisture or energy in the major and minor control volumes with respect to time. On the other hand, if the accumulation terms in the dynamic model are set to zero, it becomes a steady-state

In the case of the first above assumptions, a dimensionless group (D/uL), known as the dispersion number can be used to characterize the dispersion phenomenon or the deviation from ideal plug-flow. When the dispersion number is zero, the flow is ideal plug-flow; for values of dispersion number less than or equal to 0.01, the particles flow can be considered approximately as plugflow with small dispersion, and the values of dispersion number more than 0.01 show a large deviation from ideal plug-flow (Khanali et al., 2013). In order not to disturb the main text, a complete definition about the model parameters such as bed porosity, solids concentration in the bed, specific surface area of expanded bed, dispersion coefficient, heat transfer coefficient, and thermodynamic equations of air-vapor system can be found in the previous work by Khanali et al. (2013). The balance equations corresponding to the scheme of Fig. 1 and expressing the model are given in the following subsections.

66

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

Fig. 1. Scheme of the plug-flow fluidized bed dryer model at dynamic condition: (a) major control volume and (b) minor control volume.

2.1. Mass balance on moisture in the particles The mass balance on moisture in the particles over the major control volume under dynamic conditions considering the input and output rates of moisture by both bulk flow and longitudinal dispersion of the particles, the moisture evaporation rate, and the accumulation rate of moisture inside the major control volume is as follows:

    @M @M dx þ Dhbed wn hbed uwnM  hbed uwn M þ @x @x      @M @ @M dx  Rhbed wndx þ  Dhbed wn @x @x @x @M dx ¼ hbed wn @t

ð1Þ

2.3. Energy balance in the particles The energy balance in the particles over the major control volume under dynamic conditions is developed by considering the energy transfer rate by both bulk flow and longitudinal dispersion of the particles, the heat transfer rate between the particles and the gas, and the accumulation rate of energy inside the major control volume:

    @Hp @Hp hbed uwnHp  hbed uwn Hp þ dx þ Dhbed wn dx @x      @Hp @ @Hp dx þ ahbed waðT g;in  T p Þdx þ  Dhbed wn @x @x @x  Rhbed wnðC w ðT p  T ref Þ þ hv þ C v ðT g;in  T p ÞÞdx @Hp ¼ hbed wn dx @t

Simplifying the above equation and dividing by hbedwn dx gives:

@ 2 M u @M R 1 @M   ¼ @x2 D @x D D @t

ð2Þ

where R the drying rate of rough rice determined experimentally can be expressed as follows (Khanali et al., 2013):

R ¼ k1 expðk2 MÞ;

0:1 6 M 6 0:58ðd:b:Þ

ð3Þ

in which

k1 ¼ 106 ð0:497T g  159:40Þ and k2

ð6Þ

Simplifying the above equation and dividing by hbedwn dx gives: 2

@ Hp u @Hp aaðT g;in  T p Þ  þ  RðC w ðT p  T ref Þ þ hv þ C v ðT g;in  T p ÞÞ Dn @x2 D @x 1 @Hp ¼ ð7Þ D @t 2.4. Energy balance in the gas

¼ 0:236T g þ 95:84 ðT g in KÞ: 2.2. Mass balance on moisture in the gas The mass balance on moisture in the gas over the minor control volume at any specific location along the bed length (x) under dynamic conditions considering the input and output rates of moisture by bulk flow of the gas, the rate of moisture transfer between the particles and the gas, and the accumulation rate of humidity within the minor control volume of drying gas is as follows:

    @Y dz dx þ Rwndxdz U 0 Ywqg dx  U 0 wqg Y þ @z @Y dxdz ¼ wqg @t

  @Hg U 0 wqg Hg dx  U 0 wqg Hg þ dz dx  awaðT g  T p Þdxdz @z @Hg þRwnðC w ðT p  T ref Þ þ hv þ C v ðT g  T p ÞÞdxdz ¼ wqg dxdz ð8Þ @t Simplifying the above equation and dividing by wdxdz gives:

ð4Þ

Simplifying the above equation and dividing by wdxdz gives:

@Y Rn 1 @Y  ¼ @z qgU0 U 0 @t

The energy balance in the gas over the minor control volume at any specific location along the bed length (x) under dynamic conditions considering the input and output rates of energy by bulk flow of the gas, the rate of energy transfer between the particles and the gas, and the accumulation rate of energy inside the minor control volume can be written as follows:

ð5Þ

@Hg  aaðT g  T p Þ þ RnðC w ðT p  T ref Þ þ hv þ C v ðT g  T p ÞÞ @z @Hg ð9Þ ¼ qg @t

 U 0 qg

2.5. The outflow equation of the particles The operation of the particles outflowing from the bed is depicted schematically in Fig. 2. At steady-state operation of the dryer, the dry-based solid holdup within the bed is not changed

67

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

Fig. 2. Schematic representation of solids outflow from the bed.

with time and the dry solids mass flow rate at the outlet is equal to that at the inlet of dryer assuming that the elutriation from the bed is negligible. During dynamic conditions resulted from the transient of inlet dry solids mass flow rate, the accumulation rate of solids inside the bed is nonzero and the balance equation of the mass of solids in the apparatus (dry-based solid holdup) is an ordinary differential equation with respect to time as follows:

dS _ p;in  m _ p;out ¼m dt

ð10Þ

As shown in Fig. 2, the liquid like behavior of a fluidized bed allows the solids to be handled like a fluid, and continuous feeding and withdrawal therefore become possible. With the similarity between the behavior of fluidized bed with that of a liquid, a Torricelli relationship can be developed to predict the mass flow rate of outlet solids based on the dry solid holdup and the bed height as follows:

(

_ p;out ¼ m

0 S Lhbed

hbed 6 hweir pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðhbed  hweir Þ 2gðhbed  hweir Þ hbed > hweir

ð11Þ

It must be noted that, in a fluidized bed, the bed surface is not stable and horizontal like a liquid, thus the considered bed height is actually an average term. The bed height (hbed) based on the solids concentration in the bed, the dry-based solid holdup, and the dimension of the dryer vessel can be calculated as:

hbed ¼

S nwL

ð12Þ

3. Solution of the model equations The present plug-flow fluidized bed drying model under dynamic condition consists of a set of four partial differential equations [Eqs. (2), (5), (7), and (9)], the ordinary differential equation of mass balance dry-based solid holdup [Eq. (10)], and the outflow equation of outlet solids [Eq. (11)]. The solution procedures of the differential equations of the model are given in the following subsections. Based on the solution methods, a program was developed and implemented in MATLAB (version 7.8) environment to simulate the drying process under dynamic conditions. Drying process under dynamic conditions resulted from the disturbance of operating conditions can be considered as an extension of the drying process under steady-state conditions. Thus, once the drying behavior under steady-state condition is identified, the solution of the drying model under dynamic condition can proceed. On the other hand, when a disturbance happens, the originally balanced steady-state condition cannot be maintained and a transient response develops.

In this study, all the differential equations of the model are solved based on the finite difference method. In this way, time and geometry derivatives are discretized using the finite difference approximations and iterations are carried out for both time and geometry coordinates. In each time step of calculation, the drybased solid holdup is calculated using Eqs. (10) and (11) at first and then the solid holdup dependent model parameters such as longitudinal solid flow velocity and solids concentration in the bed are calculated to prepare all terms in the balance equations of the model. The steady-state drying attributes (i.e., the longitudinal profiles of solids moisture content and temperature, the humidity and temperature of outlet gas, and the dry-based solid holdup) before starting the drying process under dynamic condition is determined based on the analysis of the plug-flow fluidized bed drying process under steady-state condition developed by Khanali et al. (2013). The boundary conditions applied in this study are similar to those presented in Khanali et al. (2013). In this modeling, the independent variables are location along the dryer length (x), location along the dryer height (z), and time (t). To solve the partial differential equations by numerical method, the system is discretized along the length and height of the bed. The length and height of the bed are divided into N and P segments, respectively. The size of each segment along the length and height of the bed are h = L/N and Dz = hbed/P, respectively. The number of time steps is dependent on the total drying time and the length of each time step. The grid points along the length and height of the bed and the time grid points have the coordinates xi = ih, i = {0, 1, . . ., N}; z1 = 1Dz, 1 = {0, 1, . . ., P}; and tj = jDt, j = {1, 2, . . ., n}, respectively. 3.1. Solution of Eq. (2) By substituting Eq. (3) into Eq. (2), the result is a second-order nonlinear partial differential equation as follows:

@ 2 Mðx; tÞ u @Mðx; tÞ k1 1 @Mðx; tÞ   expðk2 Mðx; tÞÞ ¼ @x2 D @x D @t D

ð13Þ

It is worth mentioning that the step change in inlet solids mass flow rate at dynamic conditions leads to the variation of solid holdup and residence time of particles within the bed, thus the longitudinal solid flow velocity is changed until the new steady state is reached which causes the longitudinal solid flow velocity is considered as a time-dependent parameter uj in the discrete form. The approximation of Eq. (13) for any interior grid point along the bed length {xi = ih, i = 1, 2, . . ., N  1} at each time step (j = j) can be done by replacing the first and second derivatives with respect to x and the first derivative with respect to t by central and backward difference approximations, respectively, as follows:

Miþ1;j  2Mi;j þ M i1;j h ¼

2



uj M iþ1;j  M i1;j k1  expðk2 M i;j Þ D 2h D

1 Mi;j  M i;j1 D Dt

ð14Þ

Simplifying the above equation gives:

!     2 h h h 1  aj M i1;j þ 2  M i;j þ 1 þ aj M iþ1;j 2 2 D Dt 2

¼ h r i;j

ð15Þ

in which

aj ¼ 

uj k1 Mi;j1 and r i;j ¼ expðk2 M i;j Þ  : D D D Dt

By employing the forward difference approximation, the boundary condition for point x0 at time step j is written as:

68

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

tj 2 D=ðhuj Þ 6 1 þ D=ðhuj Þ 6 6 2 6 1  h=2aj 2  h =DDt 6 6 6 6 0 1  h=2aj 6 6 . .. 6 .. . 6 6 6 0 0 6 6 6 6 0 0 6 4 0 0

0 1 þ h=2aj 2

2  h =DDt .. .

  D D M 0;j  1þ M1;j ¼ Min huj huj

0 0 0

.. . 0 .. . 0 .. . 0 .. .. . . .. 2 . 2  h =DDt .. . 1  h=2aj .. . 0

0 0 0 .. . 1 þ h=2aj 2

2  h =DDt 1

ð16Þ

For the boundary condition in point xN at time step j, the backward difference approximation is used which leads to:

M N1;j  MN;j ¼ 0

ð17Þ

In order to numerical determination of unknown variables (solids moisture content) for all the grid points along the bed length at time step j, the system of equations can be expressed in matrix notation (AX = B) as follows: The elements of matrix (A) are time-dependent variables which are dependent on longitudinal solid flow velocity at time steps j. The algorithm used to calculate the longitudinal solid flow velocity at each time step is given later. The interior elements of vector (B) are nonlinear functions (ri,j) of unknown variables at time steps j and j  1. For the first time step (j = 1), the initial conditions, i.e. the values of solids moisture content for all the grid points along the bed length at steady-state condition must be used. For the time step j(j P 2), the system of nonlinear equations (AX = B) is solved using the iterative method. The solution starts with an initial guess of solution vector (X), then vector (B) is determined and the derived system of linear equations is solved by the Gauss elimination method. This procedure is repeated until the change in solution vector (X) becomes negligible. In this regard, the iteration was continued until the stopping criterion, i.e., the absolute difference between the successive values of the sum of the components of vector (kD1 expðk2 XÞ) become smaller than 0.001 or the iteration number exceeded 1000. It should be noted that the calculated values of moisture content for any grid point along the bed length at each time step are used for the next time step. Determination of longitudinal solid flow velocity at any time step consists of the execution of the following algorithm: 1. At first time step, the solid holdup and the outlet solids mass flow rate assumed to be equal to those at steady-state condition. 2. The bed height at each time step (j = j) can be determined as follows:

hbed;j ¼

Sj nLw

¼

3

ð19Þ

3. The average residence time of particles within the bed based on the dry-based solid holdup and the average mass flow rate of _ P;in;j þ m _ P;out;j Þ=2], can be calculated dry solids at time step j [ðm as follows:

0

7 2 3 3 2 M 7 in M 0;j 7 7 7 6 2 0 h r 1;j 7 7 6 M 1;j 7 7 6 7 6 7 6 7 6 7 6 M 7 6 h2 r 7 7 6 6 2;j 7 2;j 7 0 7 6 7 6 7 6 .. 7 7 6 .. .. 76 . 7 7 . 7¼6 . 7 6 7 6 7 7 6 7 6 2 7 7 6 M N2;j 7 6 h rN2;j 7 0 7 6 7 6 2 7 4 M N1;j 7 6 5 4 h rN1;j 7 7 5 1 þ h=2aj 7 7 M N;j 0 5 1

ð20Þ

ð18Þ

2 _ p;in;j þ m

69

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

4. The longitudinal solid flow velocity at time step j is calculated as follows:

L uj ¼  tj

ð21Þ

5. The variation of solid holdup at time step j based on Eq. (10) can be calculated as follows:

_ p;in;j  m _ p;out;j ÞDt DSj ¼ ð m

ð22Þ

6. The solid holdup at time step j + 1 can be calculated as follows:

Sjþ1 ¼ DSj þ Sj

ð23Þ

7. The outlet solids mass flow rate at time step j + 1 can be calculated as follows:

( _ p;out;jþ1 ¼ m

0

hbed;j 6 hweir

Sj ðhbed;j Lhbed;j

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  hweir Þ 2gðhbed;j  hweir Þ hbed;j > hweir

ð24Þ

Simplifying the above equation gives:

1 1 1 Y l1;j þ Y l;j þ Y lþ1;j 2Dz U 0 Dt 2Dz nj k1 1 ¼ expðk2 M i;j Þ þ Y l;j1 U 0 Dt qg U 0

ð28Þ

Above equation is only valid for interior grid points along the bed height. For the first grid point corresponding to l = 0, an auxiliary point is introduced at the gas inlet that in this point, the gas humidity is equal to the boundary condition. For this grid point, the same procedure applied in Eq. (28) is employed which results in:

nj k1 1 1 1 1 expðk2 Mi;j Þ þ Y 0;j þ Y 1;j ¼ Y 0;j1 þ Y in U 0 Dt 2Dz U 0 Dt 2 Dz qg U 0 ð29Þ For the last grid point corresponding to l = P, backward difference approximation is used instead of central difference approximation which  leads to:

nj k1 1 1 1 1 Y P;j ¼ þ expðk2 Mi;j Þ þ Y P1;j þ Y P;j1 U 0 Dt Dz Dz U 0 Dt qg U 0

ð30Þ 3.2. Solution of Eq. (5) By substituting the drying rate of solid particles located at each grid point (xi) along the dryer length based on Eqs. (3), (5), the result is:

@Yðz; tÞ nk1 1 @Yðz; tÞ  expðk2 M i Þ ¼ @z U0 @t qg U 0

ð25Þ

where Mi, the solids moisture content at grid point (xi) along the dryer length is a known parameter calculated by solution of Eq. (2) at each time step. Thus Eq. (25) is a first-order nonlinear partial differential equation with dependent variable of gas humidity (Y) and independent variables of time (t) and location along the bed height (z). For solving the above equation, the boundary condition at the gas inlet (the distributor) is as follows:

Yð0; tÞ ¼ Y in

ð26Þ

To approximate Eq. (24) for any interior grid point along the bed height {zl = lDz, l = 1, 2, . . ., P  1} at each time step (j = j) and grid point (xi) along the dryer length, the derivatives with respect to x and t were replaced by central and backward difference approximations, respectively, as follows:

To determine the unknown variables (gas humidity) for all the grid points along the bed height at time step j and grid point (xi), the system of equations are written and the resulting system of linear equations is solved using the Gauss elimination method. For the first time step (j = 1), the initial conditions, i.e. the solids moisture content for grid point (xi) and the values of gas humidity along the bed height at grid point (xi) during steady-state condition must be used. 3.3. Solution of Eq. (7) By substituting Eqs. (3), (5), (7), the resulting differential equation can be written as follows:

@ 2 Hp ðx; tÞ u @Hp ðx; tÞ aðT g;in  T p ðx; tÞÞa k1  þ  expðk2 MÞ @x2 D @x Dn D  ðC w ðT p ðx; tÞ  T ref Þ þ hv þ C v ðT g;in  T p ðx; tÞÞÞ ¼

1 @Hp ðx; tÞ D @t

ð31Þ

Considering the solids enthalpy as a function of solids moisture content and temperature, Eq. (31) can be rewritten as follows:

Y lþ1;j  Y l1;j 2 Dz nj k1  qg U 0   @ 2ÞMðx;tÞ @Mðx;tÞ @Mðx;tÞ  expðk aa 2 M i;j DC  uC  C   k C expðk Mðx; tÞÞ þ k C expðk MÞ w w w 1 w 2 1 v 2 2 qbed @x @t @x a21 ¼ ; DðC p þ C w Mðx; tÞÞ ¼ U0 Y l;j  Y l;j1  Dt ð27Þ

a3 ¼

 DC w T ref

@ 2 Mðx;tÞ @x2

 uC w T ref

@Mðx;tÞ @x

 C w T ref

@Mðx;tÞ @t

 k1 C w T ref expðk2 Mðx; tÞÞ  DðC p þ C w Mðx; tÞÞ

aaT g;in n

 þ k1 hv expðk2 Mðx; tÞÞ þ k1 C v T g;in expðk2 Mðx; tÞÞ

:

70

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

@ 2 T p ðx; tÞ @T p ðx; tÞ 1 @T p ðx; tÞ þ a1 þ a2 T p ðx; tÞ ¼ a3 þ @x2 @x D @t

ð32Þ

T p0;j 

  DðC p þ C w M0;j Þ T p1;j  T p0;j ¼ T p;in uj ðC p þ C w M in Þ h

ð37Þ

Simplifying the above equation gives:

in which

b1j T p0;j þ b2j T p1;j ¼ T p;in

2DC w @Mðx;tÞ  uðC p þ C w Mðx; tÞÞ @x a1 ¼ ; DðC p þ C w Mðx; tÞÞ

ð38Þ

in which

b1j ¼ 1 þ

and As shown, the coefficients a1, a2, and a3 are dependent on solids moisture content [M(x, t)], its h i first and second derivatives with @Mðx;tÞ @ 2 Mðx;tÞ , and its first derivative with respect respect and 2 h to ix @x @x to t @Mðx;tÞ . @t The values of M(x, t) for all grid points along the bed length at each time step (j = j) are found by solving Eq. (2). The values of parameter @Mðx;tÞ at point x0, interior grid points along the bed @x length, and point xN are discretized using forward, central, and backward difference approximations, respectively. The values of 2

parameter @ Mðx;tÞ for each grid point along the bed length can be @x2 calculated using Eq. (13) as follows:

@ 2 Mðx; tÞ 1 @Mðx; tÞ u @Mðx; tÞ k1 ¼ þ þ expðk2 Mðx; tÞÞ @x2 D @t D @x D

The boundary condition for point xN at time step j using the backward difference approximation can be written as follows:

T pN 1;j  T pN;j ¼ 0

ð39Þ

To determine the unknown variables (solids temperature) for all the grid points along the bed length at time step j, the system of equations are written and the resulting system of linear equations is solved using the Gauss elimination method. For the first time step (j = 1), the initial conditions, i.e. the values of solids temperature for all the grid points along the bed length at steady-state condition must be used. 3.4. Solution of Eq. (9)

ð33Þ

The values of parameter @Mðx;tÞ for each grid point xi at each time @t step (j = j) can be discretized using backward difference approximation as follows:

@Mðx; tÞ M i;j  Mi;j1 ¼ @t Dt

DðC p þ C w M 0;j Þ DðC p þ C w M0;j Þ and b2j ¼ : huj ðC p þ C w Min Þ huj ðC p þ C w M in Þ

ð34Þ

Substituting for solids drying rate from Eqs. (3), (5), (7), (9) leads to the following differential equation:

@Hg ðz; tÞ  aaðT g ðz; tÞ  T pi Þ þ nk1 expðk2 Mi ÞðC w ðT pi @z  T ref Þ þ hv þ C v ðT g ðz; tÞ  T pi ÞÞ

U 0 qg

¼ qg

@Hg ðz; tÞ @t

ð40Þ

To approximate Eq. (32) for each interior grid point along the bed length {xi = ih, i = 1, 2, . . ., N  1} at time step j, the first and second derivatives with respect to x and first derivative with respect to t can be replaced with central and backward difference approximations, respectively, as follows:

The boundary condition for the gas temperature at the gas inlet (the distributor) is as follows:

T piþ1;j  2T pi;j þ T pi1;j

Considering the gas enthalpy as a function of gas humidity and temperature, Eq. (40) can be rewritten as follows:

2

h

¼ a3i;j þ

þ a1i;j

T piþ1;j  T pi1;j þ a2i;j T pi;j 2h

1 T pi;j  T pi;j1 D Dt

ð35Þ

T g0;t ¼ T g;in

@T g ðz; tÞ @T g ðz; tÞ þ b1 T g ðz; tÞ ¼ b2 þ b3 @z @t

ð41Þ

ð42Þ

in which Simplifying the above equation gives: 2

c1i;j T pi1;j þ c2i;j T pi;j þ c3i;j T piþ1;j ¼ h c4i;j

ð36Þ

b1 ¼

U 0 qg C v @Yðz;tÞ þ qg C v @Yðz;tÞ þ aa  nk1 C v expðk2 M i Þ @z @t ; U 0 qgðC g þCv Yðz;tÞÞ b2 ¼





qg ðT ref C v  hv 0 Þ U 0 @Yðz;tÞ þ @Yðz;tÞ þ nk1 expðk2 M i ÞðT pi ðC w  C v Þ þ hv  T ref C w Þ @z @t b3 ¼ : U 0 qg ðC g þ C v Yðz; tÞÞ

in which 2

c1i;j c3i;j

h h 2 ¼ 1  a1i;j ; c2i;j ¼ 2 þ a2i;j h  ; 2 DDt Tp h ¼ 1 þ a1i;j ; and c4i;j ¼ a3i;j  i;j1 : 2 DDt

The boundary condition for point x0 at time step j based on the forward difference approximation can be written as follows:

and As shown, the coefficients b1, b2, and b3 are dependent on solids moisture content and temperature at grid point xi [Mi and T pi ], gas humidity [Y(z, t)], and its first derivatives with respect to z and t [@YðZ;tÞ and @YðZ;tÞ ]. @z @t The values of solids moisture content and temperature for grid point xi [Mi and T pi ] at each time step (j = j) are found by solving Eqs. (2) and (7), respectively. The values of Y(z, t) for all grid points

1 ; U0

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

71

Fig. 3. Schematic diagram of the laboratory scale plug flow fluidized bed dryer: (1) centrifugal fan, (2) pitot tube, (3) electrical heater, (4) bin, (5) tracer inlet port, (6) screw conveyor, (7) drying vessel, (8) solid outlet port, (9) plenum, (10) baffle plate, (11) control unit.

Fig. 4. Dynamic response of outlet dry solids mass flow rate to step changes in inlet dry solid mass flow rate.

Fig. 6. Dynamic response of solids moisture content along the bed length.

points along the bed height are discretized by the backward difference approximation based on the values of gas humidity at time steps j and j  1. The discretization of Eq. (42) for any interior grid point along the bed height {zl = lDz, l = 1, 2, . . ., P  1} at time step j can be done using the first derivatives with respect to x and t based on central and backward difference approximations, respectively, as follows:

  1 b2 1 b2 T gl;j þ ¼ T gl1;j þ b1  Tg T g þ b3 2Dz 2Dz lþ1;j Dt Dt l;j1

Fig. 5. Dynamic response of outlet solids moisture content.

Eq. (43) is only valid for interior grid points along the bed height. For the first grid point corresponding to l = 0, an auxiliary point is introduced at the gas inlet that in this point, the gas temperature is equal to the boundary condition. By employing the same procedure applied in Eq. (43), the result for the first grid point (l = 0) is as follows:

  b2 1 b2 1 T g0;j þ b1  þ Tg ¼ Tg T g;in þ b3 2Dz 1;j Dt Dt 0;j1 2Dz along the bed height at time step j can be found by solving Eq. (5). The values of parameter @YðZ;tÞ at point z0, interior grid points along @z the bed height, and point zP are discretized using forward, central, and backward difference approximations, respectively. At grid point xi and time step j, the values of parameter @YðZ;tÞ for all the grid @t

ð43Þ

ð44Þ

For the last grid point corresponding to l = P, the backward difference approximation is used instead of central difference approximation which leads to:

72

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

Fig. 7. The dynamic response of outlet solids temperature: (a) simulation results and (b) experimental results.

1 T g ination method. For the first time step (j = 1), the initial conditions, Dz P1;j i.e. the solidsmoisture content and temperature for grid point (xi) b2 1 þ and b1 the þ valuesT gof P;j gas humidity along the bed height at grid point Dt Dz (xi) during steady-state condition must be used.

b2 Tg Dt P;j1 þ b34. Drying experiment ¼

ð45Þ

Fig. 8. Dynamic response of solids temperature along the bed length.

To determine the unknown variables (the values of gas temperature) for all the grid points along the bed height at and time step j and grid point (xi), the system of equations are written and the resulting system of linear equations is solved using the Gauss elim-

The experiment was performed in a laboratory scale plug-flow fluidized bed dryer shown in Fig. 3. The drying gas, hot air, was supplied by a centrifugal fan and heated by a controllable electrical heater before entering the plenum. The drying vessel was 100 cm in length and 8 cm in width. The distributor plate was a 2 mm thick perforated steel plate with 2 mm diameter holes on a 5 mm triangular pitch (open area = 0.15). Due to the fluidization, particles move axially along the dryer and exit from the outlet. For the output, a manually controllable sliding weir was used. The inlet solids mass flow rate was adjusted by changing the rotational speed of the screw conveyor with a three phase electric motor equipped with an inverter (LS, SV004iG5-4, Korea).

Fig. 9. Dynamic response of outlet gas temperature: (a) simulation results and (b) experimental results.

Fig. 10. Dynamic response of outlet gas humidity: (a) simulation results and (b) experimental results.

73

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

Drying experiment was started when the bed was empty by setting the superficial fluidization velocity, the inlet gas temperature, and the inlet dry solid mass flow rate at 2.5 m/s, 70 °C, and 135 g/min, respectively. The inlet gas humidity, the inlet rough rice moisture content, and the weir height were kept constant during experiment at 0.0118 kg/kg, 0.3 d.b., and 0.1 m, respectively. After 30 min from the start of the experiment that ensured the achievement of steady-state conditions, the inlet dry solids mass flow rate was abruptly reduced from 135 to 46 g/min and after 60 min the inlet dry solids mass flow rate was abruptly increased to 96 g/min and the experiment continued for 40 min. Thus, the experiment was last for 130 min. It is important to note that after each of the two step changes of solids mass flow rate, the experiment was continued until the steady-state condition was achieved. The criteria for achieving the steady-state conditions imply the stable experimental drying attributes such as outlet solids mass flow rate and moisture content. These criteria were checked experimentally through measurement of desirable drying attributes (see Figs. 4 and 5). It must be noted that all the drying operating conditions, except the mass flow rate of dry solids, were constant during experiment. To measure the outlet solids mass flow rate, the outlet solids from the dryer were collected every minute during experiment and their mass was measured. To measure the outlet solids moisture content and temperature, about 10 g of the outlet solids was sampled for determination of solid moisture content and temperature. The solid temperature was taken immediately after sampling using an infrared contactless hand held temperature measuring instrument (Testo 860-T3, Germany, accuracy ±0.75 °C). The measurement was performed by using the appropriate mode incorporated on the instrument for rough rice. This mode adjusts the emissivity to provide the accurate temperature measurement. The solid moisture content was determined by drying the particles at temperature of 105 °C for 24 h in a hot air oven (ASAE, 1984). The relative humidity and temperature of the drying gas coming out of the bed at 0.1, 0.5, and 0.9 m from the solid inlet port were measured on-line by the air humidity-temperature measurement sensors (SHT15, USA, relative humidity accuracy ±0.02 RH, temperature accuracy ±0.4 °C). Iranian long grain rough rice was used in the experiment. The particles were cleaned manually and foreign matter such as stones and straw were removed, then they were wetted with demineralized water, mixed and kept in a cold storage at 3–5 °C for four days prior to the experiment in order to ensure uniform water concentration throughout the grain. Based on the methods proposed by Mohsenin (1986), the measured values of rough rice equivalent diameter, solids true density, and porosity at stagnant sate were 3.43  103 m, 857 kg/m3, and 0.48, respectively. The measured value of rough rice minimum fluidization velocity was 1.7 m/s (Khanali et al., 2013). The specific heat of the rough rice and its specific surface area at stagnant state were considered equal to 1110 J/kg K and 1132 m2/m3, respectively (Brooker et al., 1992).

5. Results and discussion Based on Khanali et al. (2013), at steady-state fluidized bed dying experiments for inlet dry solids mass flow rate of 46, 96, and 135 g/min, superficial fluidization velocity of 2.5 m/s, and weir height of 0.1 m, the values of dispersion number were in the range of 0.3–0.68, which are so much higher than 0.01. Considering the similarity between the experimental conditions of this study with the steady-state one, the flow of solids within the bed in this study can be considered as a dispersed plug-flow with large deviation from ideal plug-flow. On the other hand, the assumption of the sol-

ids dispersed plug-flow through the bed was confirmed and verified. The effects of the transients initiated by step changes of inlet dry solids mass flow rate on outlet dry solids mass flow rate, solids moisture content, solids temperature, and outlet gas humidity and temperature are presented in Figs. 4–10. As shown in these figures, the time variable presented on the abscissa is started from 1200 s after the start of the experiment described in previous section. On the other hand, it is refrained from discussing or presenting the drying behavior in time interval 0–1200 s which deals with the start-up behavior of the process. Instead, drying process was studied in time interval 1200–7800 s from the start of the mentioned experiment. Considering that in this study the dynamic condition is only resulted from the step changes of inlet dry solids mass flow rate, in the remainder of this section, it is refrained from repeating that dynamic condition resulted from the step changes in the inlet dry solids mass flow rate. For each drying attribute, the simulation model performance is determined by calculation of the mean relative error (E) and correlation coefficient (R2) as follows:

E¼

n 1X jX pre;j  X exp;j j  100 n j¼1 X pre;j

Pn

j¼1 ðX pre;j

R2 ¼ 1  P n

j¼1 ðX

 X exp;j Þ2 2

ð46Þ

ð47Þ

 X pre;j Þ

where X is an arbitrary drying attribute such as outlet solids mass flow rate, solids moisture content, solids temperature, outlet gas humidity or temperature and subscripts ‘‘pre’’ and ‘‘exp’’ denote the predicted and the experimental data, respectively. The dynamic response of the outlet dry solids mass flow rate is shown as solid curve in Fig. 4. For a new steady state (the constant mass flow rate of outlet dry solids) to be reached at the outlet of the dryer after each of the two successive step changes, about 500 s are necessary. Duration and course of the dynamic response of the outlet dry solids mass flow rate is properly predicted by the model. The values of mean relative error and correlation coefficient of the model in predicting the outlet dry solids mass flow rate during dynamic experiment were 12.05% and 0.96, respectively. At steady-state condition, the dry solids mass flow rate at both inlet and outlet ports of the dryer are identical, thus the solid holdup within the bed is constant. At dynamic condition, the difference _ P;in and m _ P;out leads to the change of solid holdup. After between m the first step change and before attaining the constant outlet dry _ P;out > m _ P;in , solids mass flow rate (new steady-state condition), m thus the solid holdup is being decreased. After the second step _ P;out < m _ P;in , thus the dryer is filled up until a new steady change, m state with a higher solid holdup has been attained. The change of solid holdup due to each of the two step changes corresponds to the area between the broken and the solid curves in Fig. 4. The dynamic response of the outlet solids moisture content is shown in Fig. 5. As can be seen, the duration and course of the transients are properly predicted by the model. The values of the mean relative error and correlation coefficient of the model in predicting the outlet solids moisture content was 2.1% and 0.99, respectively. The solids moisture content at the outlet reacts to each of the transients by, first, a rather quick response covering about 80% of the interval between the new and the old steady-state conditions in a time interval equal to 1000 s, as shown in Fig. 5. However, the remaining part of the dynamic response curve is much slower, which correlate with the relatively long time necessary in order to reach the new operating condition of constant outlet dry solids mass flow rate. The lower solids moisture content at lower inlet dry solids mass flow rate is owing to the increased residence time of material in the dryer

74

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

which in turn leads to the higher degree of material drying and vice versa. Similar dynamic response of average moisture content of outlet solids to the step changes in mass flow rate of inlet solids in the continuous well-mixed fluidized bed dryer has been reported by Burgschweiger and Tsotsas (2002). A comparison of Figs. 4 and 5 indicates that the outlet dry solids mass flow rate reacts faster than the outlet solids moisture content. The dynamic behavior of outlet dry solids mass flow rate is directly influenced by the bed height. Varying the inlet dry solids mass flow rate resulted in the obvious change of solid holdup, thus the bed height and consequently the outlet dry solids mass flow rate will be changed fast. The outlet solids moisture content is obviously influenced by the relatively large moisture content inertia contained in the large amount of solid holdup within the bed, thus the outlet solids moisture content reacts slowly to the step change in inlet solids mass flow rate. The simulated dynamic responses of solids moisture content along the bed length at 0.1, 0.5, and 0.9 m from solid inlet port of the dryer are demonstrated in Fig. 6. As shown, the response times of solids moisture content along the bed length decrease by increase in the distance from the solid inlet port because of the increased contained material along the bed length that results in the increased moisture content inertia. In this study, due to the off-line measurement procedure of solids moisture content that was very time-consuming, it was refrained from measuring this parameter along the bed length. However these results show the ability of the model to predict the outlet solids moisture content in the dryers with different length. The dynamic response of the outlet solids temperature is shown in Fig. 7. As shown, the simulated values of outlet solids temperature are limited in the range of 69.83–69.93 °C. The trend of the dynamic response of outlet solids temperature is similar to that of outlet solids moisture content, i.e., the quick response at the initial stage of transient followed by the slow response until reaching the new steady-state condition. Also, similar to the dynamic response of outlet solids moisture content, the outlet dry solids mass flow rate reacts faster than the outlet solids temperature because of the influence of moisture content inertia of solids inside the bed. The higher outlet solids temperature at lower inlet dry solids mass flow rate is owing to the increased residence time of material in the dryer which in turn leads to the higher solids temperature and vice versa. Similar result has been presented by Burgschweiger and Tsotsas (2002). The measured results of outlet solids temperature at dynamic experiment are varied from 67.5 to 68 °C which are about 2 °C less than the simulated ones. The difference between measured and simulated results can be due to the off-line solids temperature measurement procedure described by Khanali et al. (2013) which results in reduction of measured outlet solids temperature. Thus, due to this measurement limit, the correlation coefficient of the model in this case was 0.83. However, mean relative error of the model in predicting the outlet solids temperature at dynamic condition was 3.15% which is an indication of the ability of the model in predicting the trend of outlet solids temperature. It is important to note that the small change in outlet solids temperature as a result of step change in inlet solids mass flow rate was due to the major effect of drying gas temperature on solids temperature which is held constant during experiment. However, mean relative error of the model in predicting the outlet solids temperature at dynamic condition was 3.15% which is an indication of the satisfactory accuracy of the model. The correlation coefficient of the model between the measured and simulated results was 0.13. Fig. 8 shows the simulation results of dynamic responses of solids temperature along the bed length at 0.1, 0.5, and 0.9 m from solid inlet port. As can be seen, similar to the dynamic behavior of solids moisture content along the bed length, the response times of solids temperature along the bed length

decrease by increasing the distance from solid inlet port. The increased response time is due to the increased contained material along the bed length that results in the increased thermal inertia. It is worthwhile to note that the small difference between the simulated solids temperatures at various locations along the bed length is in accordance with the results given by Khanali et al., 2013, that the solids temperature profile along the bed increases within a short distance from the solid inlet port and approaches a temperature nearly the same as the inlet gas temperature, remaining nearly constant until the outlet of the dryer. Similar to the measurement of solids moisture content, it is refrained from measuring the solids temperature along the bed length. However, these results can be used to analyze the dryer behavior with different bed lengths. Fig. 9 shows both the simulation and measured results of outlet gas temperature along the bed length at 0.1, 0.5, and 0.9 m from solid inlet port as well as the simulated results of mean outlet gas temperature from the dryer (T gout;pre ) at dynamic conditions. Comparison of Figs. 8 and 9a indicates that the dynamic behavior of outlet gas temperature along the bed length is in accordance with the dynamic response of solids temperature within the bed. The higher outlet gas temperature at lower inlet dry solids mass flow rate is owing to the increased solids temperature as a result of increased residence time of material in the dryer which in turn leads to the higher outlet gas temperature and vice versa. Based on simulation results, the mean values of outlet gas temperature from the dryer are higher than the values of outlet gas temperature at 0.1 m and lower than the values of outlet gas temperature at 0.5 and 0.9 m from solid inlet port. The measured values of outlet gas temperature along the bed length are less than the simulated ones. This difference between measured and simulated values to some extent is due to the heat loss resulted from the higher position of outlet gas temperature measurement points than the position of bed surface where the outlet gas temperature values are calculated. Thus, due to this difference between measured and simulated values, the correlation coefficient of the model in this case was 0.81. However, the values of mean relative error of the model in predicting outlet gas temperature along the bed length at 0.1, 0.5, and 0.9 m from solid inlet port were 6.54%, 3.05%, and 2.71%, respectively, which shows that experimental data are in good agreement with predicted results by the model. Fig. 10 depicts both the predicted and the experimental results of outlet gas humidity along the bed length at 0.1, 0.5, and 0.9 m from solid inlet port as well as the simulated results of mean outlet gas humidity from the dryer (Yout,pre) at dynamic conditions. Because of the moisture transfer between solid particles and drying gas, the dynamic behavior of outlet gas humidity along the bed length is in accordance with the dynamic response of solids moisture content within the bed. Based on simulation results, the values of mean outlet gas humidity from the dryer are lower than the values of outlet gas humidity at 0.1 m and higher than the values of outlet gas humidity at 0.5 and 0.9 m from solid inlet port. The correlation coefficient of the model in predicting the outlet gas humidity at various points along the bed was 0.99. The values of mean relative error of the model in predicting the outlet gas humidity along the bed length at 0.1, 0.5, and 0.9 m from solid inlet port were 1.24%, 1.05%, and 0.87%, respectively. These findings demonstrate the suitability of the model for simulation purposes. It is important to note that in all of the dynamic behavior curves presented in Figs. 5–10, the final steady-state conditions (solids moisture content, solids temperature, and outlet gas temperature and humidity) agree with the results from the steady-state model, which have been presented by Khanali et al. (2013).

M. Khanali et al. / Journal of Food Engineering 137 (2014) 64–75

6. Conclusions A differential control volume method for modeling the plugflow fluidized bed drying process under dynamic conditions was presented in this work. The model was examined by the experimental data from laboratory-scale plug-flow fluidized bed rough rice dryer and satisfactory agreement between simulated and measured results was achieved. The duration and course of the dynamic response of the outlet solids mass flow rate was properly predicted by the model. Both the simulated responses of outlet solids moisture content and temperature to transients initiated by step changes of inlet dry solids mass flow rate agreed with experimental data with correct trends and magnitudes. The simulated response times of both solids moisture content and temperature inside the bed decreased by increasing distance from solid inlet port of the dryer. Both solids moisture content and temperature at the dryer outlet reacted slower than the outlet dry solids mass flow rate because the response behaviors of solids moisture content and temperature are obviously influenced by the relatively large moisture content inertia and thermal inertia, respectively. The dynamic behaviors of outlet gas humidity and temperature along the bed length were respectively in accordance with the dynamic responses of solids moisture content and temperature. With this dynamic model it is possible to simulate transients in the drying process. Analysis of these transients and their responses opens the way for efficient model-based automatic control scheme and an improved assessment of product quality. Design of a control structure for plug-flow fluidized bed drying process based on this study will be subject of future work. Acknowledgments The authors gratefully acknowledge the financial support under Grant Number 31303-114505 provided by University of Tehran.

75

They also gratefully acknowledge the valuable comments of the reviewers. References Abdel-Jabbar, N.M., Jumah, R.Y., Al-Haj Ali, M.Q., 2002. Dynamic analysis of continuous well-mixed fluidized/spouted bed dryers. Drying Technol. 20 (1), 37–54. Apolinar, P., Martínez, J., 2012. Mathematical modeling of a continuous vibrating fluidized bed dryer for grain. Drying Technol. 30, 1469–1481. ASAE, 1984. Standard S352.1: Moisture Measurements. Grain and Seeds, Agricultural Engineer Yearbook. American Society of Agriculture Engineers, St Joseph, MI. Baker, C.G.J., Khan, A.R., Ali, Y.I., Damyar, K., 2006. Simulation of plug flow fluidized bed dryers. Chem. Eng. Process. 45, 641–651. Bizmark, N., Mostoufi, N., Sotudeh-Gharebagh, R., Ehsani, H., 2010. Sequential modeling of fluidized bed paddy dryer. J. Food Eng. 101, 303–308. Brooker, D.B., Bakker-Arkema, F.W., Hall, C.W., 1992. Drying and Storage of Grains and Oilseeds. Van Nostrand Reinold, New York. Burgschweiger, J., Tsotsas, E., 2002. Experimental investigation and modelling of continuous fluidized bed drying under steady-state and dynamic conditions. Chem. Eng. Sci. 57, 5021–5038. Fyhr, C., Kemp, I.C., Wimmerstedt, R., 1999. Mathematical modelling of fluidized bed dryers with horizontal dispersion. Chem. Eng. Process. 38, 89–94. Izadifar, M., Mowla, D., 2003. Simulation of a cross-flow continuous fluidized bed dryer for paddy rice. J. Food Eng. 58, 325–329. Khanali, M., Rafiee, S., Jafari, A., Hashemabadi, S.H., 2013. Experimental investigation and modeling of plug-flow fluidized bed drying under steadystate conditions. Drying Technol. 31, 414–432. Mohsenin, N.N., 1986. Physical Properties of Plant and Animal Materials. Gordon and Breach Science Publishers, New York. Mujumdar, A.S., 2006. Handbook of Industrial Drying. Taylor and Francis Group, London. Nilsson, L., Wimmerstedt, R., 1987. Drying in longitudinal-flow vibrating fluid-beds. Drying Technol. 5 (3), 337–361. Ramli, W., Daud, W., 2007. A cross-flow model for continuous plug flow fluidizedbed cross-flow dryers. Drying Technol. 25, 1229–1235. Tacidelli, A.R., Neto, A.T.P., Brito, R.P., De Arau´jo, A.C.B., Vasconcelos, L.G.S., Alves, J.J.N., 2012. Modeling and simulation of industrial PVC drying in fluidized beds with internal heat source. Chem. Eng. Technol. 35 (12), 2107–2119. Wanjari, A.N., Thorat, B.N., Baker, C.G.J., Mujumdar, A.S., 2006. Design and modeling of plug flow fluid bed dryers. Drying Technol. 24, 147–157.