# The Effect Of The Fluidized Bed Model: Model Equations

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This chapter presents the model equations and assumptions. The physical properties of charcoal and air, the initial conditions, as well as the important system parameters are then outlined. Solution algorithm of the model equations is then explained.
The effect of the intra-particle diffusion is examined by means of effective mass transfer coefficient. It can be obtained by similarity with heat transfer by conduction. Finally, the criterion for choosing the optimum switching time is set.
4.1. Fluidized Bed Model
4.1.1. Model Assumptions
1- The bed is perfectly mixed hence; all particles have the same temperature and moisture content at any given time. Air temperature and humidity changes are functions of time.
2- Intra-particle diffusion resistance
4- Temperature gradient within the solid particles is considered.
5- The solid particles are assumed to be spheres with constant radii.
6- All particles are identical and have the same moisture content.
7- The problem is one-dimensional.
4.1.2. Model Equations
4.1.2.1. Mass and Energy Balances on Solids
The moisture content of solids is a function of time only due to perfect mixing and negligible diffusion resistance as stated in assumptions 1 and 2. Following assumption 3, air at the solid-air interface is saturated. The driving force of mass transfer is the difference between the vapor pressure of water vapor at the interface temperature and the partial pressure of water vapor in the air bulk at the bulk temperature. And the variation of moisture content of a single particle can be expressed by equation (4.1): (4.1)
Applying shell balance on a spherical particle, the energy balance equation of solid particles can be written as in equation (4.2): (4.2)
This matches assumptions 4 to 6 that there is a temperature profile inside the particles and the temperature varies with time and particle radius. In order to solve equation (2), two boundary conditions are needed and they are as
At :
B.C.2: At the solid-air interface, the heat transferred from air to solid by convection is consumed in changing the solid temperature and evaporating the moisture at the solid surface.
At :
Equation (2) with its boundary conditions can be solved using finite difference method. Another method of solution is using the collocation method. Simply; this method assumes that the temperature profile within the solid particle can be expressed as a binomial of the second degree, function in two collocation points; the surface collocation point at and an interior collocation point at where their temperatures are nominated (Ti) and (T1) respectively. The collocation method is explained in details in appendix A.
As a result of using this method, equation (4.2) and its boundary conditions can be replaced by equations (4.3) and (4.4). And T1 can be used to express the average temperature of the solid particle at time (t). (4.3)

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