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MODELLING OF GAS AND SOLID PHASE

To describe the simultaneous agglomeration and drying process, we use a heterogeneous fluidized bed model which incorporates the active bypass. This model is based on the assumption, that a certain fraction of fluidization gas passes the solid phase as a bypass. Burgschweiger (2000) applied this model for drying of porous materials using adsorption isotherms. To model the heat and mass transfer processes, the following assumptions are made

The bypass fraction is free of solids and it is in plug flow.
All solids are in the suspension phase. The suspension is in plug flow. No back mixing occurs.
The particles are ideal mixed.
Vapor and heat transfer take place between suspension and bypass phase.
Vapor and heat transfer take place between surface of particles and gas in suspension phase. Water sprayed in is deposited on particles.
The wall may exchange heat with environment, particles, suspension gas and bypass gas.

In Figure 1 the balance scheme is shown. All heat, mass and enthalpy fluxes between solid and gas phase considered in the model are depicted in Figure 1.

Figure 1: Balance scheme of the fluid bed model

Modeling of solid phase

The agglomeration process for batch vessels is described by means of the one-dimensional PBE

(10)

The parameter influencing the shape of PPD is the agglomeration rate β defined by

(11)

Since the model describes heat and mass transfer processes, we need the corresponding kinetic expressions. According to Groenewold and Tsotsas (1997) the mass transfer between the solids and the suspension gas is determined by

(12)

where Yeq denotes the equilibrium moisture content. This property depends on the sorption equilibrium of the solids and can be expressed in terms of the sorption isotherm. The heat transfer between the granules and the gas phase is given by

(13)

The kinetic expressions in equations (12) and (13) contain two more properties of solid phase, the temperature of particles Jp and the moisture content X. It is clear that these properties have to be considered in the model as the combined description of agglomeration and drying is the objective of this study. Since the population balance for agglomeration is not capable to predict intensive properties of solid phase (temperature and moisture content) directly, we need to express those properties in terms of the corresponding extensive properties. In our case the temperature and moisture content will be represented by the enthalpy and the mass of liquid respectively. According to previous section, see equation (4), balances for the enthalpy of particles

(14)

and the mass of liquid

(15)

can be formulated. To incorporate the solid and gas phase in one model, additional terms have to be included in equations (14) and (15), which take the heat, mass and enthalpy flux between these two phases into account (see Figure 1). Thus the coupling of the solid and gas phase is established.

The Gas Phase

According to Figure 1, the following mass and heat balances for suspension and bypass gas phases can be derived. The moisture and enthalpy in suspension gas phase are given by

(16)

(17)

Analog to that, that balances for the bypass gas phase can be deduced

(18)

(19)

The parameter n is the ratio of gas flowing through the bypass to total gas flow rate. This parameter depends on Geldart classification of particles and bed height. A briefer discussion of this model is given in Peglow (2005).

SIMULATION RESULTS

In this section a simulation example of an agglomeration process is provided. Such a process is usually characterized by three stages. Firstly the primary particles are dried and heated. In a second stage the particles are wetted by spraying in a liquid binder. At this stage the formation of agglomerates occurs. Finally the wet agglomerates will be dried up to a moisture content defined by product specifications. In our simulation it will be assumed, that the particle size distribution will change only during spraying and remains constant in the first and last stage. The main process parameters are summarized in Table 1. The model has been implemented in the MATLAB suite. For the numerical integration we used the integration routine ode15s. The ode15s is an implicit procedure based on numerical differentiation equipped with an automatic step size control. It is suited for stiff systems. Figures 2, 3 and 4 reflect the transient behavior of properties of solid phase for all three stages. Additionally Figure 5 depicts the progression of mean and outlet moisture of gas phase. As mentioned above the first stage provides drying and heating of particles. During this process the outlet humidity of the gas phase decreases to the value of gas inlet moisture. At the end of this stage the moisture content of solids is equal to equilibrium moisture content determined by sorption isotherm of the granules. In the second stage, the agglomeration stage, a certain amount of liquid is sprayed onto the solids. Thus the moisture content of solids increases rapidly. The temperature of granules decreases due to evaporation. Figure 4 shows that small particles are drier than larger particles. This effect is caused by the size dependency of heat and mass transfer coefficients. Since a size independent kernel has been assumed for this agglomeration stage, a change of particle size distribution can be observed. Any type of agglomeration kernel can be chosen, but for simplicity size-independent kernel has been assumed. The actual value of β0 can be calculated by defining the degree of aggregation Iagg. Here we set this value to Iagg = 0.8. Finally, the particles are dried in the last stage. Since no liquid is sprayed onto the granules, the moisture content decreases and the particle temperature increases again. No change on particle size occurs, since the agglomeration rate is set to zero. Summarizing we can conclude, that the model is capable to predict the transient behavior of solid phase and gas phase properties as well. In the next section we want to turn to the experimental validation of this model.

Figure 2: Evolution of Particle Size Distribution

Figure 3: Progression of particle enthalpy temperature

Figure 4: Progression of amount of particle moisture content

Figure 5: Progression of outlet and mean gas moisture content

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