# Modelling spatial distributions of moisture and quality during drying of granular baker's yeast.

INTRODUCTION

Drying is an important unit operation widely used in chemical, food processing, biotechnology and pharmaceutical industries. The process of drying commonly means moisture evaporation due to simultaneous heat and mass transfer processes (Nonhebel and Moss, 1971; Mujumdar, 1995). The main purpose is the removal of water to guarantee safe storage of the material (Strumillo and Kudra, 1986; Quirijns et al., 2000). Many products are sensitive to drying conditions (i.e., as temperature and moisture content) examples include: agricultural products, foods, pharmaceuticals, enzyme preparations (Adamiec et al., 1995), and bacterial (Lievense and van't Riet, 1993, 1994) and yeast cultures (Beker and Rapoport, 1987). One of major side effects of drying heat labile materials is the degradation of product quality. Provided that suitable mechanistic models are available for drying of granular materials, dynamic optimization and optimal control theory can be used to enhance the operations of drying processes (Quirijns et al., 1998, 2000).

A variety of dryers are used in industry for the drying of sensitive products (Mujumdar, 1995). The fluidized bed drying technique holds an important position among modern drying methods (Hovmand, 1995). In fluidized bed dryers, the material to be dried is loaded into the dryer through a perforated strain plate to form cylindrical or spherical granules. The granulated material is then simultaneously fluidized and its moisture is removed using a hot drying agent, typically conditioned air (Bayrock and Ingledew, 1997). Water transport from the granules is mainly through diffusion, depending on their size. Examples of granular materials from which water is removed are foods like grain and rice. An important limitation on the drying process is the moisture diffusion inside the granules (Parry, 1985). Several mathematical models that describe the performance of drying process have been presented in the literature. Kanarya (2002) and Turker et al. (2006) have modelled the drying performance of granular baker's yeast where the moisture diffusion limitation in the granules was neglected. In this approach, the resistance to mass transfer of moisture from a granule to the air was assumed to occur in the thin film surrounding the granule. This approach may be valid for small granules and for an approximately constant-rate drying period. Temple and van Boxtel (1999a,b) used the thin-layer drying model to simulate the fluidized bed drying of black tea. However, they had to reduce the drying rate using an efficiency factor of 0.6 to obtain good correspondence between model predictions and actual measurements.

[FIGURE 1 OMITTED]

Therefore, distributed parameter models are necessary to take into account all possible mechanisms of heat and mass transfer at the micro- (particle level) and macro-scales (plant level) in drying (van Ballegooijen et al., 1997). These models are normally comprised of non-linear partial differential equations and their solution requires numerical methods due to the complexity of analytical solutions (Gerald and Wheatly, 1984). Several authors have studied temperature and moisture profiles resulting from heat and mass transfer limitations inside the granules during the drying process, including Kerkhof and Schoeber (1974), Luyben et al. (1982), Quirijns et al. (1998), Zimmermann and Bauer (1986). A theory by Coumans (1987) was developed to describe the drying process and used to solve the resulting non-linear differential equations in drying process (Liou and Bruin, 1982a,b). Specifically, the diffusion coefficient used in the model was defined as a function of temperature and moisture content. In the mathematical drying model reported by Yuzgec et al. (2004), the granules in the dryer were assumed to have identical dimensions and to be non-shrinking. In this study, the particle based drying model for granular products is extended by integrating both granule shrinkage and the concept of product. By solving simultaneous mass and heat transfer equations at the particle level, local values of moisture, temperature and quality were calculated and used to estimate overall drying performance of a particle and as a result, the overall performance of the bed.

THE PARTICLE LEVEL MODEL

Modelling the drying of biological materials consists of four main parts: moisture diffusion equation, heat balance equation, granule shrinking model and the product activity (Figure 1). The model also includes the dependence of granular moisture content and temperature on several parameters, including moisture diffusion coefficient, heat and mass transfer coefficients and water activity. The batch fluidized bed is assumed to be an ideally mixed bed, with uniform air temperature and humidity, which equal the outgoing air conditions. The particles are assumed to be at the same stage of drying at any instant of the batch operation. Furthermore, there is no interaction between the particles, as far as drying is concerned.

Moisture Diffusion

A generalized formulation of the moisture diffusion equation for a granule is given by the following relation (Schoeber, 1976):

[partial derivative]([[rho].sub.s]X)/[partial derivative]t = 1/[r.sup.v] [partial derivative]/[partial derivative]r ([r.sup.v][[rho].sub.s]D(X, T) [partial derivative]X/[partial derivative]r) (1)

where X (kg water/kg dry solid) is the moisture content inside the granule, D ([m.sup.2]/s) is the moisture diffusion coefficient, which is a function of the material's moisture content (X) and temperature (T), and v represents geometry factor with v = 0 (slab), v = 1 (cylinder), v = 2 (sphere). The initial and boundary conditions are:

At t = 0, 0 [less than or equal to] r [less than or equal to] [R.sub.d] : X(0, r) = [X.sub.0] (2)

At t > 0 : [[partial derivative]X/[partial derivative]r|.sub.r=0] = 0 (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [j.sub.m,i] represents the moisture flux at the interface, k is the liquid film mass transfer coefficient around the granule, [[rho].sub.wv,i] is the water vapour concentration at the interface and [[rho].sub.wv,g] is the water vapour concentration in the bulk air.

Heat Balance

The heat balance can be described as heat transfer both to and from the surface and within the material. The heat balance for a granule, with general geometry, is expressed by the following non-linear partial differential equation (Bird et al., 1960; Quirijns et al., 1998, 2000):

[partial derivative](T([[rho].sub.s][c.sub.p,s] + [[rho].sub.m][c.sub.p,m]))/[partial derivative]t = 1/[r.sup.v] [partial derivative]/[partial derivative]r ([r.sup.v] [lambda] [partial derivative]T/[partial derivative]r) (5)

where T is the temperature, [[rho].sub.m] is the moisture concentration, [[rho].sub.s] is the dry solid concentration inside the granule, [c.sub.p,s] and [c.sub.p,m] are the heat capacities of the solid and moisture, and [lambda] is the thermal conductivity of the granule. The initial and boundary conditions are given by:

t = 0, 0 [less than or equal to] r [less than or equal to] [R.sub.d] : T(0, r) = [T.sub.0] (6)

t > 0 : [[partial derivative]T/[partial derivative]|.sup.r=0] = 0 (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [j.sub.T,i] is the heat flux at the interface, [alpha] is the heat transfer coefficient, [T.sub.a] is the inlet air temperature and [DELTA][H.sub.v] is the evaporation enthalpy of water.

Product Quality

The product quality can be described with first-order kinetics as (Lievense, 1991):

dQ/dt = -[k.sub.e]Q (10)

where Q is the concentration of the active product and [k.sub.e] the specific rate of product activity. According to the Arrhenius equation, the rate of the product activity can be written as a function of the temperature (Liou and Bruin, 1982a,b; Liou et al., 1984; Lievense, 1991):

[k.sub.e] = [k.sub.[infinity]]exp(-[E.sub.ai]/RT) (11)

where [k.sub.[infinity]] is the frequency factor and [E.sub.a] is the activation energy. Lievense (1991) has presented following equations describing the dependency of [k.sub.e] on temperature and moisture:

ln([k.sub.e]) = [([a.sub.1] - [a.sub.2]/RT) X + ([b.sub.1] - [b.sub.2]/RT)] + [1 - exp(p[X.sup.q])][([a.sub.3] - [a.sub.4]/RT) X + ([b.sub.3] - [b.sub.4]/RT)] (12)

where p,q,[a.sub.i],[b.sub.i] are the parameter values in the equation. If p < 0 and q [greater than or equal to] 1, at high moisture content, exp(p[X.sup.q]) [approximately equal to] 0 and ln([k.sub.e]) consists of the linear sum of the two parts; at low moisture content, exp(p[X.sup.q]) [approximately equal to] 1 and ln([k.sub.e]) is described with the first linear part of the equation. The magnitudes of first and second terms of the right-hand side of Equation (12) along with ln([k.sub.e]) as a function of both temperature and moisture content are shown in Figure 2. It appears that the product activity is less sensitive to temperature at lower moisture contents.

Granule Shrinking

The volume of the granule consists of volumes of both moisture and solid:

V = [V.sub.m] + [V.sub.s] (13)

According to Coumans (1987), the volume of the granule is a linear function of the average moisture content [bar.X] during shrinkage, expressed as:

V = [V.sub.s] (1 + [tau] [d.sub.s]/[d.sub.m] [bar.X])

where [tau] is the shrinkage coefficient where 0 [less than or equal to] [tau] [less than or equal to] 1. The solid mass balance is given by:

[[bar.[rho]].sub.s]V = [d.sub.s][V.sub.s] (15)

where [[bar.[rho]].sub.S] is the average of the solid mass concentration. If Equations (14) and (15) are combined:

[[bar.[rho]].sub.S] = 1/(1/[d.sub.s]) + [tau]([bar.X]/[d.sub.m]) (16)

The average moisture concentration [[bar.[rho]].sub.m] is given by:

[[bar.[rho]].sub.m] = [bar.X](1/[d.sub.s]) + [tau]([bar.X]/ [d.sub.m]) (17)

The granule diameter can be considered to be a function of the shrinkage coefficient and average moisture content and is described by the following equation:

[R.sub.d] = [R.sub.o] [([d.sub.m] + [tau][d.sub.s] [bar.X]/[d.sub.m] + [tau] [d.sub.s] [[bar.X].sub.o]).sup.0.5] (18)

The dependence of granule diameter on shrinking coefficient is shown in Figure 3. The average values for moisture content and temperature in the granule are determined by an integral over the geometry defined respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Note that L in Equation (19) is the cylinder length.

Parameters Used in the Model

The diffusion coefficient in Equation (1) depends on the moisture content and temperature of the granules as given by following equation (Liou, 1982; Luyben et al., 1982; Liou et al., 1984):

D = [D.sub.ref]exp([E.sub.a,D]/R (1/[T.sub.ref] - 1/T)) [[X - [X.sub.e]/[X.sub.0] - [X.sub.e]].sup.z] (21)

[E.sub.a,D] = 80 000 (1/1 + 10X + 0.147) (22)

where [E.sub.a,D] represents the activation energy for diffusion, z is the power in concentration dependence of diffusion coefficient, R = 8.314 J/mol K (gas constant), [T.sub.ref] = 323 K. The values of [D.sub.ref](7.89 x [10.sup.-10] [m.sup.2]/s) and z(0.154) in Equation (21) for baker's yeast were obtained using the experimental data. The dependency of moisture diffusion coefficient on moisture content and temperature is shown in Figure 4 for baker's yeast.

[FIGURE 4 OMITTED]

According to Josic (1982), the heat capacities of granular baker yeast, [c.sub.p,s] can be estimated as a polynomial function of the dry solid content Y (percent kg dry solid/kg total) (Kanarya, 2002):

[c.sub.p,s] = -[6.10.sup.-6][Y.sup.3] + [8.10.sup.-4][Y.sup.2] - [534.10.sup.-4]Y + 4.26 (23)

Y = 100 1/1 + [bar.X] (24)

Water activity is an alternative method of describing equilibrium relative humidity. Guggenheim-Anderson-de Boer (GAB) equation, given as Equation (25), was considered for this model (Van den Berg et al., 1978; Josic, 1982; Temple and van Boxtel, 1999a):

X(t, [R.sub.d]) = WCK [a.sub.w]/(1 - K [a.sub.w]) (1 - K [a.sub.w] + CK [a.sub.w]) (25)

where W, C, K are the fitting parameters in the GAB equation and [a.sub.w] is the dimensionless water activity. For other temperatures, water activity can be calculated using the following relation (Lievense, 1991):

ln [a.sub.w]([T.sub.ref]) [a.sub.w](T) = [E.sub.a,s]/R [1/[T.sub.ref] - 1/T] (26)

where [E.sub.a,s] represents the activation energy for sorption. The parameters in the GAB equation for baker's yeast were obtained by fitting to the experimental data. The water activity as function of moisture content and temperature is shown Figure 5.

The heat transfer coefficient [alpha] used in Equation (9) can be described by Equation (27) according to Bird et al. (1960) and Liou et al. (1984):

[alpha] = [[alpha].sub.g] ([gamma]/[e.sup.[gamma]] - 1) (27)

[gamma] = [j.sub.m,i][c.sub.p,wv]/[[alpha].sub.g] (28)

where [[alpha].sub.g] is the heat transfer coefficient in the gas phase and [c.sub.p,wv] is the heat capacity of the water vapour. The heat transfer coefficient in the gas phase was found by the relation between the heat and mass transfer coefficients in the gas phase:

[FIGURE 5 OMITTED]

[[alpha].sub.g] = [k.sub.g][([d.sub.a][c.sub.p,a]).sup.1/3][([[lambda].sub.a]/ [D.sub.w,a]).sup.2/3] (29)

where [d.sub.a] is the air density, [c.sub.p,a] the heat capacity of air, [[lambda].sub.a] the thermal conductivity of air, [k.sub.g] the mass transfer coefficient in the gas phase and [D.sub.w,a] the diffusion coefficient of water in air. The mass transfer coefficient in gas phase [k.sub.g] can be calculated by (Liou et al., 1984):

[k.sub.g] = k([X'.sub.wv,i] - [X'.sub.wv,g])/ln[(1 - [X'.sub.wv,g])/ (1 - [X'.sub.wv,i])] (30)

where the liquid film mass transfer coefficient k is calculated from Equation (4) (Temple and van Boxtel, 1999c; T" urker et al., 2006) and X' is the moisture content on total basis (kg/kg total) as given by the following equation:

[X'.sub.wv] = [[rho].sub.wv] / [[rho].sub.wv] + [[rho].sub.a] (31)

where [[rho].sub.wv] represents the water vapour concentration in air. [[rho].sub.wv] is calculated by multiplying [[rho].sub.wv,sat] obtained from Table 1 and the water activity [a.sub.w] obtained from the GAB equation (Equations (25) and (26)).

Materials and Methods

The micro-organism Saccharomyces cerevisiae, known as baker's yeast, was used to test the model in this study. The experiments conducted with a fluid bed dryer. Yeast cake was extruded into the dryer through a perforated plate, with the perforations' diameter selected to obtain the desired granule size. Two granule shapes were used throughout this work: cylindrical (obtained at shorter drying periods) and spherical (obtained at longer residence times in the bed). The experimental data set used for comparison with model predictions was obtained from different loadings to the dryer using different perforated plates to get different granule sizes: one with 0.006 m holes and the other 0.012 m holes. The fluid bed equipment included a centrifugal fan to supply air drawn from ambient air. Air inlet temperature was maintained at 100[degrees]C. The temperature and humidity of air at inlet and outlet and its flow rate were measured on-line and registered on a computer in order to establish continuous material and energy balances for the prediction of the moisture, temperature and quality of the product. The quality of the product was measured as the amount of carbon dioxide produced upon introduction of the yeast into dough per unit time. This method is commonly used in yeast industry to assess the performance of the baker's yeast (Reed and Nagodawithana, 1991). Relative activity is expressed as the ratio of the activity of the product at time t to the activity of the yeast cake introduced into the dryer. The parameters used for drying model are shown in Table 1.

[FIGURE 6 OMITTED]

NUMERICAL METHOD

There are two non-linear partial differential equations in this model. Because of complex nature of the analytical solutions, numerical methods (i.e., the Crank-Nicholson method)were used for the solution of the non-linear temperature and moisture diffusion equations (Gerald and Wheatly, 1984). The equations were subject to a Robin boundary condition, in which the mass flux at the interface is variable. The program algorithm is shown in Figure 6. The sample time was chosen as 1 s and the particle was divided into ten grids, which strikes a compromise between convergence and computational time.

RESULTS AND DISCUSSION

Simulation of Non-Shrinking Granules

The moisture content profiles, temperature profiles and the product activity inside the non-shrinking cylindrical and spherical granules are shown in Figures 7 and 8. The average values of these profiles are given in Figures 7d and 8d.

As can be seen from these figures, the moisture moves from the centre towards the surface of the granule and the gradients become steeper towards the surface of the granule due to rapid evaporation. However, temperature gradients within the granule are not significant. The quality decreases from the surface towards the centre due to the higher moisture content inside the granule. In fact, this is the result of two computing mechanisms between the effects of moisture and temperature as described by Equation (12). The results for a non-shrinking spherical granule show trends similar to those observed for a non-shrinking cylindrical granule. The temperature gradients are insignificant within the spherical granule. As a result, the quality is mainly a function of the moisture content within the granule. Therefore, the energy equation can be simplified and reduced to an ordinary differential equation.We have tested both partial and ordinary differential equations and obtained the similar results. The simulation results for non-shrinking cylindrical and spherical granules are shown in Figure 9 together with experimental data. The mathematical model provides moisture concentration and temperature of the granular product. The model shows relatively good agreement with the experimental data.

[FIGURE 7 OMITTED]

Simulation of Shrinking Granules

For the shrinking spherical and cylindrical granules, the moisture profiles and quality distribution inside a granule are shown in Figure 10. The product activity inside a granule is shown to be a function of time and radial distance in Figure 11.

The moisture profiles inside a shrinking granule reach the desired value in less time than a non-shrinking granule under identical conditions. The model that considers granule shrinkage gives a more realistic representation of experimental moisture and temperature measurements. The product activity is a function of both temperature and moisture content and thus is reduced towards the centre from the surface of a granule because of the higher moisture content at the centre. The product activity for a shrinking sphere is similar to that for a cylinder. The difference between the activity inside a spherical granule and a cylindrical one is clearly apparent during the last drying period. At the end of this period, the product activity for a spherical granule is lower than that inside a cylindrical one.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

The effect of inlet air temperature is shown for cylindrical and spherical granules in Figures 12 and 13, respectively. At higher inlet air temperatures, the progress of drying is faster than at lower air inlet temperatures. Similarly, higher inlet air temperatures negatively affect the quality and the rate of quality reduction.

The predictions of the mathematical model for batch drying are shown in Figure 14 together with the appropriate data sets. In the spherical granule data set, there were only experimental values of temperature and dry solid content. As can be seen from this figure, the model based on granule shrinking gives a more realistic representation of experimental moisture and temperature measurements that the model that does not consider shrinkage (Figure 9).

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

Desired trajectories of average moisture content, dry solid content and product temperature have been obtained during the simulation for both types of products. The predicted values of moisture content and product temperature show no remarkable differences from the experimental measurements. The product activity was predicted during drying and compared to experimentally obtained measurements for both granule shapes (see Figure 15). The quality predictions are reasonable, considering the uncertainties involved in determination of product quality.

Predicted changes in granule diameter with time are shown in Figure 16 together with experimental measurements. A good fit with experimental measurements was obtained when shrinkage coefficient [tau] =0.6 was used for both cylindrical and spherical granules.

CONCLUSIONS

A distributed parameter model was developed for the fluid bed drying cylindrical and spherical granular baker's yeast, by considering the drying process as a simultaneous heat and mass transfer problem. In one variation of the model, granular shrinkage was considered. The quality was also integrated into the model. The model predictions were better than those obtained with previous models, especially during the falling rate period of drying. Compared to the lumped parameter model, this model provides significantly improved predictions during the drying of granular product by providing spatial distributions of moisture and quality (Turker et al., 2006). The model accurately predicted the change of granule size during drying, which is normally neglected in drying studies. The model also provides estimates of the product quality, which is an important drying parameter. The mechanistic nature of this model also allows one to develop an optimal drying strategy in terms of energy efficiency and quality. We have used this model to develop optimal trajectories using genetic algorithms and found consistent observations with industrial results (Yuzgec, et al., 2006).

ACKNOWLEDGEMENTS

The authors would like to thank the editors and reviewers for their constructive, valuable and informative comments.

Manuscript received October 10, 2005; revised manuscript received May 15, 2006; accepted for publication May 15, 2006.

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Ugur Yuzgec, (1) * Mustafa Turker (2) and Yasar Becerikli (3)

(1.) Department of Electronic & Telecommunication Engineering, Kocaeli University, 41040 Kocaeli, Turkey

(2.) Pakmaya, P.O. Box 149, 41001 Kocaeli, Turkey

(3.) Wireless Communications and Information Systems Research Unit (WINS) and Computer Engineering Department, Kocaeli University, 41040 Kocaeli, Turkey

* Author to whom correspondence may be addressed. E-mail address: uyuzgec@kocaeli.edu.tr

Drying is an important unit operation widely used in chemical, food processing, biotechnology and pharmaceutical industries. The process of drying commonly means moisture evaporation due to simultaneous heat and mass transfer processes (Nonhebel and Moss, 1971; Mujumdar, 1995). The main purpose is the removal of water to guarantee safe storage of the material (Strumillo and Kudra, 1986; Quirijns et al., 2000). Many products are sensitive to drying conditions (i.e., as temperature and moisture content) examples include: agricultural products, foods, pharmaceuticals, enzyme preparations (Adamiec et al., 1995), and bacterial (Lievense and van't Riet, 1993, 1994) and yeast cultures (Beker and Rapoport, 1987). One of major side effects of drying heat labile materials is the degradation of product quality. Provided that suitable mechanistic models are available for drying of granular materials, dynamic optimization and optimal control theory can be used to enhance the operations of drying processes (Quirijns et al., 1998, 2000).

A variety of dryers are used in industry for the drying of sensitive products (Mujumdar, 1995). The fluidized bed drying technique holds an important position among modern drying methods (Hovmand, 1995). In fluidized bed dryers, the material to be dried is loaded into the dryer through a perforated strain plate to form cylindrical or spherical granules. The granulated material is then simultaneously fluidized and its moisture is removed using a hot drying agent, typically conditioned air (Bayrock and Ingledew, 1997). Water transport from the granules is mainly through diffusion, depending on their size. Examples of granular materials from which water is removed are foods like grain and rice. An important limitation on the drying process is the moisture diffusion inside the granules (Parry, 1985). Several mathematical models that describe the performance of drying process have been presented in the literature. Kanarya (2002) and Turker et al. (2006) have modelled the drying performance of granular baker's yeast where the moisture diffusion limitation in the granules was neglected. In this approach, the resistance to mass transfer of moisture from a granule to the air was assumed to occur in the thin film surrounding the granule. This approach may be valid for small granules and for an approximately constant-rate drying period. Temple and van Boxtel (1999a,b) used the thin-layer drying model to simulate the fluidized bed drying of black tea. However, they had to reduce the drying rate using an efficiency factor of 0.6 to obtain good correspondence between model predictions and actual measurements.

[FIGURE 1 OMITTED]

Therefore, distributed parameter models are necessary to take into account all possible mechanisms of heat and mass transfer at the micro- (particle level) and macro-scales (plant level) in drying (van Ballegooijen et al., 1997). These models are normally comprised of non-linear partial differential equations and their solution requires numerical methods due to the complexity of analytical solutions (Gerald and Wheatly, 1984). Several authors have studied temperature and moisture profiles resulting from heat and mass transfer limitations inside the granules during the drying process, including Kerkhof and Schoeber (1974), Luyben et al. (1982), Quirijns et al. (1998), Zimmermann and Bauer (1986). A theory by Coumans (1987) was developed to describe the drying process and used to solve the resulting non-linear differential equations in drying process (Liou and Bruin, 1982a,b). Specifically, the diffusion coefficient used in the model was defined as a function of temperature and moisture content. In the mathematical drying model reported by Yuzgec et al. (2004), the granules in the dryer were assumed to have identical dimensions and to be non-shrinking. In this study, the particle based drying model for granular products is extended by integrating both granule shrinkage and the concept of product. By solving simultaneous mass and heat transfer equations at the particle level, local values of moisture, temperature and quality were calculated and used to estimate overall drying performance of a particle and as a result, the overall performance of the bed.

THE PARTICLE LEVEL MODEL

Modelling the drying of biological materials consists of four main parts: moisture diffusion equation, heat balance equation, granule shrinking model and the product activity (Figure 1). The model also includes the dependence of granular moisture content and temperature on several parameters, including moisture diffusion coefficient, heat and mass transfer coefficients and water activity. The batch fluidized bed is assumed to be an ideally mixed bed, with uniform air temperature and humidity, which equal the outgoing air conditions. The particles are assumed to be at the same stage of drying at any instant of the batch operation. Furthermore, there is no interaction between the particles, as far as drying is concerned.

Moisture Diffusion

A generalized formulation of the moisture diffusion equation for a granule is given by the following relation (Schoeber, 1976):

[partial derivative]([[rho].sub.s]X)/[partial derivative]t = 1/[r.sup.v] [partial derivative]/[partial derivative]r ([r.sup.v][[rho].sub.s]D(X, T) [partial derivative]X/[partial derivative]r) (1)

where X (kg water/kg dry solid) is the moisture content inside the granule, D ([m.sup.2]/s) is the moisture diffusion coefficient, which is a function of the material's moisture content (X) and temperature (T), and v represents geometry factor with v = 0 (slab), v = 1 (cylinder), v = 2 (sphere). The initial and boundary conditions are:

At t = 0, 0 [less than or equal to] r [less than or equal to] [R.sub.d] : X(0, r) = [X.sub.0] (2)

At t > 0 : [[partial derivative]X/[partial derivative]r|.sub.r=0] = 0 (3)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

where [j.sub.m,i] represents the moisture flux at the interface, k is the liquid film mass transfer coefficient around the granule, [[rho].sub.wv,i] is the water vapour concentration at the interface and [[rho].sub.wv,g] is the water vapour concentration in the bulk air.

Heat Balance

The heat balance can be described as heat transfer both to and from the surface and within the material. The heat balance for a granule, with general geometry, is expressed by the following non-linear partial differential equation (Bird et al., 1960; Quirijns et al., 1998, 2000):

[partial derivative](T([[rho].sub.s][c.sub.p,s] + [[rho].sub.m][c.sub.p,m]))/[partial derivative]t = 1/[r.sup.v] [partial derivative]/[partial derivative]r ([r.sup.v] [lambda] [partial derivative]T/[partial derivative]r) (5)

where T is the temperature, [[rho].sub.m] is the moisture concentration, [[rho].sub.s] is the dry solid concentration inside the granule, [c.sub.p,s] and [c.sub.p,m] are the heat capacities of the solid and moisture, and [lambda] is the thermal conductivity of the granule. The initial and boundary conditions are given by:

t = 0, 0 [less than or equal to] r [less than or equal to] [R.sub.d] : T(0, r) = [T.sub.0] (6)

t > 0 : [[partial derivative]T/[partial derivative]|.sup.r=0] = 0 (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

where [j.sub.T,i] is the heat flux at the interface, [alpha] is the heat transfer coefficient, [T.sub.a] is the inlet air temperature and [DELTA][H.sub.v] is the evaporation enthalpy of water.

Product Quality

The product quality can be described with first-order kinetics as (Lievense, 1991):

dQ/dt = -[k.sub.e]Q (10)

where Q is the concentration of the active product and [k.sub.e] the specific rate of product activity. According to the Arrhenius equation, the rate of the product activity can be written as a function of the temperature (Liou and Bruin, 1982a,b; Liou et al., 1984; Lievense, 1991):

[k.sub.e] = [k.sub.[infinity]]exp(-[E.sub.ai]/RT) (11)

where [k.sub.[infinity]] is the frequency factor and [E.sub.a] is the activation energy. Lievense (1991) has presented following equations describing the dependency of [k.sub.e] on temperature and moisture:

ln([k.sub.e]) = [([a.sub.1] - [a.sub.2]/RT) X + ([b.sub.1] - [b.sub.2]/RT)] + [1 - exp(p[X.sup.q])][([a.sub.3] - [a.sub.4]/RT) X + ([b.sub.3] - [b.sub.4]/RT)] (12)

where p,q,[a.sub.i],[b.sub.i] are the parameter values in the equation. If p < 0 and q [greater than or equal to] 1, at high moisture content, exp(p[X.sup.q]) [approximately equal to] 0 and ln([k.sub.e]) consists of the linear sum of the two parts; at low moisture content, exp(p[X.sup.q]) [approximately equal to] 1 and ln([k.sub.e]) is described with the first linear part of the equation. The magnitudes of first and second terms of the right-hand side of Equation (12) along with ln([k.sub.e]) as a function of both temperature and moisture content are shown in Figure 2. It appears that the product activity is less sensitive to temperature at lower moisture contents.

Granule Shrinking

The volume of the granule consists of volumes of both moisture and solid:

V = [V.sub.m] + [V.sub.s] (13)

According to Coumans (1987), the volume of the granule is a linear function of the average moisture content [bar.X] during shrinkage, expressed as:

V = [V.sub.s] (1 + [tau] [d.sub.s]/[d.sub.m] [bar.X])

where [tau] is the shrinkage coefficient where 0 [less than or equal to] [tau] [less than or equal to] 1. The solid mass balance is given by:

[[bar.[rho]].sub.s]V = [d.sub.s][V.sub.s] (15)

where [[bar.[rho]].sub.S] is the average of the solid mass concentration. If Equations (14) and (15) are combined:

[[bar.[rho]].sub.S] = 1/(1/[d.sub.s]) + [tau]([bar.X]/[d.sub.m]) (16)

The average moisture concentration [[bar.[rho]].sub.m] is given by:

[[bar.[rho]].sub.m] = [bar.X](1/[d.sub.s]) + [tau]([bar.X]/ [d.sub.m]) (17)

The granule diameter can be considered to be a function of the shrinkage coefficient and average moisture content and is described by the following equation:

[R.sub.d] = [R.sub.o] [([d.sub.m] + [tau][d.sub.s] [bar.X]/[d.sub.m] + [tau] [d.sub.s] [[bar.X].sub.o]).sup.0.5] (18)

The dependence of granule diameter on shrinking coefficient is shown in Figure 3. The average values for moisture content and temperature in the granule are determined by an integral over the geometry defined respectively:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] 19)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (20)

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

Note that L in Equation (19) is the cylinder length.

Parameters Used in the Model

The diffusion coefficient in Equation (1) depends on the moisture content and temperature of the granules as given by following equation (Liou, 1982; Luyben et al., 1982; Liou et al., 1984):

D = [D.sub.ref]exp([E.sub.a,D]/R (1/[T.sub.ref] - 1/T)) [[X - [X.sub.e]/[X.sub.0] - [X.sub.e]].sup.z] (21)

[E.sub.a,D] = 80 000 (1/1 + 10X + 0.147) (22)

where [E.sub.a,D] represents the activation energy for diffusion, z is the power in concentration dependence of diffusion coefficient, R = 8.314 J/mol K (gas constant), [T.sub.ref] = 323 K. The values of [D.sub.ref](7.89 x [10.sup.-10] [m.sup.2]/s) and z(0.154) in Equation (21) for baker's yeast were obtained using the experimental data. The dependency of moisture diffusion coefficient on moisture content and temperature is shown in Figure 4 for baker's yeast.

[FIGURE 4 OMITTED]

According to Josic (1982), the heat capacities of granular baker yeast, [c.sub.p,s] can be estimated as a polynomial function of the dry solid content Y (percent kg dry solid/kg total) (Kanarya, 2002):

[c.sub.p,s] = -[6.10.sup.-6][Y.sup.3] + [8.10.sup.-4][Y.sup.2] - [534.10.sup.-4]Y + 4.26 (23)

Y = 100 1/1 + [bar.X] (24)

Water activity is an alternative method of describing equilibrium relative humidity. Guggenheim-Anderson-de Boer (GAB) equation, given as Equation (25), was considered for this model (Van den Berg et al., 1978; Josic, 1982; Temple and van Boxtel, 1999a):

X(t, [R.sub.d]) = WCK [a.sub.w]/(1 - K [a.sub.w]) (1 - K [a.sub.w] + CK [a.sub.w]) (25)

where W, C, K are the fitting parameters in the GAB equation and [a.sub.w] is the dimensionless water activity. For other temperatures, water activity can be calculated using the following relation (Lievense, 1991):

ln [a.sub.w]([T.sub.ref]) [a.sub.w](T) = [E.sub.a,s]/R [1/[T.sub.ref] - 1/T] (26)

where [E.sub.a,s] represents the activation energy for sorption. The parameters in the GAB equation for baker's yeast were obtained by fitting to the experimental data. The water activity as function of moisture content and temperature is shown Figure 5.

The heat transfer coefficient [alpha] used in Equation (9) can be described by Equation (27) according to Bird et al. (1960) and Liou et al. (1984):

[alpha] = [[alpha].sub.g] ([gamma]/[e.sup.[gamma]] - 1) (27)

[gamma] = [j.sub.m,i][c.sub.p,wv]/[[alpha].sub.g] (28)

where [[alpha].sub.g] is the heat transfer coefficient in the gas phase and [c.sub.p,wv] is the heat capacity of the water vapour. The heat transfer coefficient in the gas phase was found by the relation between the heat and mass transfer coefficients in the gas phase:

[FIGURE 5 OMITTED]

[[alpha].sub.g] = [k.sub.g][([d.sub.a][c.sub.p,a]).sup.1/3][([[lambda].sub.a]/ [D.sub.w,a]).sup.2/3] (29)

where [d.sub.a] is the air density, [c.sub.p,a] the heat capacity of air, [[lambda].sub.a] the thermal conductivity of air, [k.sub.g] the mass transfer coefficient in the gas phase and [D.sub.w,a] the diffusion coefficient of water in air. The mass transfer coefficient in gas phase [k.sub.g] can be calculated by (Liou et al., 1984):

[k.sub.g] = k([X'.sub.wv,i] - [X'.sub.wv,g])/ln[(1 - [X'.sub.wv,g])/ (1 - [X'.sub.wv,i])] (30)

where the liquid film mass transfer coefficient k is calculated from Equation (4) (Temple and van Boxtel, 1999c; T" urker et al., 2006) and X' is the moisture content on total basis (kg/kg total) as given by the following equation:

[X'.sub.wv] = [[rho].sub.wv] / [[rho].sub.wv] + [[rho].sub.a] (31)

where [[rho].sub.wv] represents the water vapour concentration in air. [[rho].sub.wv] is calculated by multiplying [[rho].sub.wv,sat] obtained from Table 1 and the water activity [a.sub.w] obtained from the GAB equation (Equations (25) and (26)).

Materials and Methods

The micro-organism Saccharomyces cerevisiae, known as baker's yeast, was used to test the model in this study. The experiments conducted with a fluid bed dryer. Yeast cake was extruded into the dryer through a perforated plate, with the perforations' diameter selected to obtain the desired granule size. Two granule shapes were used throughout this work: cylindrical (obtained at shorter drying periods) and spherical (obtained at longer residence times in the bed). The experimental data set used for comparison with model predictions was obtained from different loadings to the dryer using different perforated plates to get different granule sizes: one with 0.006 m holes and the other 0.012 m holes. The fluid bed equipment included a centrifugal fan to supply air drawn from ambient air. Air inlet temperature was maintained at 100[degrees]C. The temperature and humidity of air at inlet and outlet and its flow rate were measured on-line and registered on a computer in order to establish continuous material and energy balances for the prediction of the moisture, temperature and quality of the product. The quality of the product was measured as the amount of carbon dioxide produced upon introduction of the yeast into dough per unit time. This method is commonly used in yeast industry to assess the performance of the baker's yeast (Reed and Nagodawithana, 1991). Relative activity is expressed as the ratio of the activity of the product at time t to the activity of the yeast cake introduced into the dryer. The parameters used for drying model are shown in Table 1.

[FIGURE 6 OMITTED]

NUMERICAL METHOD

There are two non-linear partial differential equations in this model. Because of complex nature of the analytical solutions, numerical methods (i.e., the Crank-Nicholson method)were used for the solution of the non-linear temperature and moisture diffusion equations (Gerald and Wheatly, 1984). The equations were subject to a Robin boundary condition, in which the mass flux at the interface is variable. The program algorithm is shown in Figure 6. The sample time was chosen as 1 s and the particle was divided into ten grids, which strikes a compromise between convergence and computational time.

RESULTS AND DISCUSSION

Simulation of Non-Shrinking Granules

The moisture content profiles, temperature profiles and the product activity inside the non-shrinking cylindrical and spherical granules are shown in Figures 7 and 8. The average values of these profiles are given in Figures 7d and 8d.

As can be seen from these figures, the moisture moves from the centre towards the surface of the granule and the gradients become steeper towards the surface of the granule due to rapid evaporation. However, temperature gradients within the granule are not significant. The quality decreases from the surface towards the centre due to the higher moisture content inside the granule. In fact, this is the result of two computing mechanisms between the effects of moisture and temperature as described by Equation (12). The results for a non-shrinking spherical granule show trends similar to those observed for a non-shrinking cylindrical granule. The temperature gradients are insignificant within the spherical granule. As a result, the quality is mainly a function of the moisture content within the granule. Therefore, the energy equation can be simplified and reduced to an ordinary differential equation.We have tested both partial and ordinary differential equations and obtained the similar results. The simulation results for non-shrinking cylindrical and spherical granules are shown in Figure 9 together with experimental data. The mathematical model provides moisture concentration and temperature of the granular product. The model shows relatively good agreement with the experimental data.

[FIGURE 7 OMITTED]

Simulation of Shrinking Granules

For the shrinking spherical and cylindrical granules, the moisture profiles and quality distribution inside a granule are shown in Figure 10. The product activity inside a granule is shown to be a function of time and radial distance in Figure 11.

The moisture profiles inside a shrinking granule reach the desired value in less time than a non-shrinking granule under identical conditions. The model that considers granule shrinkage gives a more realistic representation of experimental moisture and temperature measurements. The product activity is a function of both temperature and moisture content and thus is reduced towards the centre from the surface of a granule because of the higher moisture content at the centre. The product activity for a shrinking sphere is similar to that for a cylinder. The difference between the activity inside a spherical granule and a cylindrical one is clearly apparent during the last drying period. At the end of this period, the product activity for a spherical granule is lower than that inside a cylindrical one.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

The effect of inlet air temperature is shown for cylindrical and spherical granules in Figures 12 and 13, respectively. At higher inlet air temperatures, the progress of drying is faster than at lower air inlet temperatures. Similarly, higher inlet air temperatures negatively affect the quality and the rate of quality reduction.

The predictions of the mathematical model for batch drying are shown in Figure 14 together with the appropriate data sets. In the spherical granule data set, there were only experimental values of temperature and dry solid content. As can be seen from this figure, the model based on granule shrinking gives a more realistic representation of experimental moisture and temperature measurements that the model that does not consider shrinkage (Figure 9).

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

[FIGURE 14 OMITTED]

[FIGURE 15 OMITTED]

[FIGURE 16 OMITTED]

Desired trajectories of average moisture content, dry solid content and product temperature have been obtained during the simulation for both types of products. The predicted values of moisture content and product temperature show no remarkable differences from the experimental measurements. The product activity was predicted during drying and compared to experimentally obtained measurements for both granule shapes (see Figure 15). The quality predictions are reasonable, considering the uncertainties involved in determination of product quality.

Predicted changes in granule diameter with time are shown in Figure 16 together with experimental measurements. A good fit with experimental measurements was obtained when shrinkage coefficient [tau] =0.6 was used for both cylindrical and spherical granules.

CONCLUSIONS

A distributed parameter model was developed for the fluid bed drying cylindrical and spherical granular baker's yeast, by considering the drying process as a simultaneous heat and mass transfer problem. In one variation of the model, granular shrinkage was considered. The quality was also integrated into the model. The model predictions were better than those obtained with previous models, especially during the falling rate period of drying. Compared to the lumped parameter model, this model provides significantly improved predictions during the drying of granular product by providing spatial distributions of moisture and quality (Turker et al., 2006). The model accurately predicted the change of granule size during drying, which is normally neglected in drying studies. The model also provides estimates of the product quality, which is an important drying parameter. The mechanistic nature of this model also allows one to develop an optimal drying strategy in terms of energy efficiency and quality. We have used this model to develop optimal trajectories using genetic algorithms and found consistent observations with industrial results (Yuzgec, et al., 2006).

ACKNOWLEDGEMENTS

The authors would like to thank the editors and reviewers for their constructive, valuable and informative comments.

NOMENCLATURE a water activity [a.sub.j] parameter values in Equation (12) (j=1-4) [b.sub.j] parameter values in Equation (12) (j=1-4) [C.sub.p] heat capacity (J/kg K) C parameter in the GAB equation d density (kg/[m.sup.3]) D diffusion coefficient ([m.sup.2]/s) [E.sub.a] activation energy (J/mol) [E.sub.a,D] activation energy for diffusion (J/mol) [E.sub.a,s] activation energy for sorption (J/mol) [j.sub.m,i] moisture flux at the interface (kg/[m.sup.2] s) [j.sub.T,i] heat flux at the interface (J/[m.sub.2] s) k mass transfer coefficient (1/s) [k.sub.e] specific rate of product activity (1/s) [k.sub.[infinity]] frequency factor (1/s) K parameter in the GAB equation L length of the cylindrical granule (m) p parameter value in Equation (22) q parameter value in Equation (22) Q concentration of the active product (kg/kg dry solid) r radial coordinate (m) R gas constant=8.314 J [mol.sup.-1] [K.sup.-1] [R.sub.d] radius (m) t time (min) T temperature (K) v geometry factor V total volume of the granule ([m.sup.3]) W parameter in the GAB equation X moisture content (kg water/kg dry solid) [X.sub.e] equilibrium moisture content (kg water/kg dry solid) X' moisture content (kg/kg total) Y dry solid content percent (kg dry solid/kg total) z power in diffusion coefficient expression, Equation (21) Greek Symbols [alpha] heat transfer coefficient (J/[m.sup.2] s K) [lambda] thermal conductivity of granule (J/m s K) [gamma] parameter in Equation (27) [rho] concentration (kg/[m.sup.3]) [DELTA][H.sub.v] evaporation enthalpy of water (J/kg) [tau] shrinkage coefficient Subscripts a air g gas i interface m moisture ref reference sat saturation w water wv water vapour 0 initial

Manuscript received October 10, 2005; revised manuscript received May 15, 2006; accepted for publication May 15, 2006.

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Zimmermann, K. and W. Bauer, "The Influence of Drying Conditions upon Reactivation of Baker's Yeast," Proc. 4th International Congress of Engineering and Food (ICEF 4), Edmonton, AB, Canada (1986).

Ugur Yuzgec, (1) * Mustafa Turker (2) and Yasar Becerikli (3)

(1.) Department of Electronic & Telecommunication Engineering, Kocaeli University, 41040 Kocaeli, Turkey

(2.) Pakmaya, P.O. Box 149, 41001 Kocaeli, Turkey

(3.) Wireless Communications and Information Systems Research Unit (WINS) and Computer Engineering Department, Kocaeli University, 41040 Kocaeli, Turkey

* Author to whom correspondence may be addressed. E-mail address: uyuzgec@kocaeli.edu.tr

Table 1. Physical data used in the model Thermal conductivity Luyben et al. [[lambda].sub.a] = of air (J/m s K) (1982) 4.5 x [10.sup.-3] + 7.26 x [10.sup.-5]T Diffusion coefficient Luyben et al. [D.sub.w,a] = 5.28 x of water vapour in air (1982) [10.sup.-9][T.sup.3/2] ([m.sup.2]/s) Evaporation enthalpy of Perry (1984) [DELTA][H.sub.v] (T) = water (J/kg) 2501(1.73 (1-[(T/647.3))).sup.0.38] Density of air ASHRAE (1981) [d.sub.a] =353.128/T (kg/[m.sup.3]) Density of water vapour ASHRAE (1981) [d.sub.wv] =220.705/T (kg/[m.sup.3]) Saturated water vapour Liou (1982) [[rho].sub.wv,sat] = concentration in air ((2.1936 x [10.sup.-3]) (kg/[m.sup.3]) /T) exp((-7246.5822)/T) + 77.641232 + 5.7447142 x [10.sup.-3]T - 8.2470402 ln T Heat capacity of air [C.sub.p,a] = 1010 J/kg/K Heat capacity of water [C.sub.p,w] = 4184 J/kg/K Heat capacity of vapour [C.sub.p,wv] = 2000 J/kg/K

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Author: | Yuzgec, Ugur; Turker, Mustafa; Becerikli, Yasar |
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Publication: | Canadian Journal of Chemical Engineering |

Date: | Aug 1, 2008 |

Words: | 5178 |

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