Single Particle Model

In the single particle model, each electrode is represented by a single spherical particle, the surface area of which is equivalent to that of the active area of the solid phase in the porous electrode [22, 54, 55). This model assumes that the transport limitation due to the electrolyte phase of the cell is negligible. Therefore, the equations for electrolyte phase potential and lithium ion concentration in the electrolyte phase are not solved. 

Finally, it is important to briefly review the simplified treatments of the solid phase lithium diffusion reported in the literature in an attempt to reduce the computational complexity and overheads involved in solving the lithium diffusion equation in the active material concurrent to the electrolyte phase transport. 

Doyle et al. [17], used Duhamel s superposition integral to numerically solve the solid phase diffusion, as described by Equation 25.36 The exact solution obtained for spherical particles provides a considerable improvement in computational speed however, it is limited to restrictive assumptions (e.g., perfectly spherical particles and constant diffusion coefficient in the solid phase, where the exact integral solution is possible). This approach still requires the numerical solution of the solid phase diffusion at each control volume. 

The simplest model of this kind can be represented by one in which an isolated particle surrounded by gas is in contact with or in the vicinity of the heating surface for a certain time, during which the heat transfer between the particle and the heating surface takes place by transient conduction, as shown in Fig. 12.4. In terms of the model, the Fourier equation of thermal conduction can be expressed as 

The boundary and initial conditions forEqs. (12.3) and (12.4) are given by the following  

For the case without particle-surface contact, only T =TS should be imposed. 

12 / Heat and Mass Transfer Phenomena in Fluidization Systems 

Sample isotherm simulation results for a glass sphere of 200 /tm in contact with a heat transfer surface surrounded by static air are shown in Fig. 12.5 for two contact times, i.e., 1.2 ms and 52.4 ms [Botterill and Williams, 1963]. The initial temperature difference between the sphere and the surface is 10°C. It is seen that at the instant of contact heat begins to flow around the upper surface of the sphere and significant heat transfer takes place at 

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To be able to mathematically simulate and really understand the thermochemical conversion process of a packed bed both a single particle model and a bed model must be included in the overall bed model (CFSD code ... 

Figure 12.4. Transient heat conduction in the single-particle model (from Botterill, 1975) ...

The interpretation of the interband transition is based on a single particle model, although in the final state two particles, an electron and a hole, exist. In some semiconductors, however, a quasi one-particle state, an exciton, is formed upon excitation [23,24]. Such an exciton represents a bound state, formed by an electron and a hole, as a result of their Coulomb attraction, i.e. it is a neutral quasi-particle, which can move through the crystal. Its energy state is close to the conduction band (transition 3 in Fig. 2), and it can be split into an independent electron and a hole by thermal excitation. Therefore, usually... 

Depending on the boundary conditions, the regime in which the char conversion takes place, might vary between shrinking core and reacting core behaviour, as illustrated in Fig.4 for conversion of a char particle with oxygen and carbondioxide based on the single particle model. Kinetic data for char reactions is taken from [14]. 

McKenna, T.F. Soares, J.B.P. Single particle modelling for olefin polymerization on supported catalysts a review and proposals for future developments. Chem. Eng. Sci. 2001, 56, 3931-3949. 

This formulation is particularly convenient when Euler-Lagrange simulations are used to approximate the disperse multiphase flow in terms of a fimte sample of particles. As discussed in Sections 5.2 and 5.3, although some of the mesoscale variables are intensive (i.e. independent of the particle mass), it is usually best to start with a conserved extensive variable (e.g. particle mass or particle momentum) when deriving the single-particle models. For example, in Chapter 4 we found that must have at least one component, corresponding to the fluid mass seen by a particle, in order to describe cases in which the disperse-phase volume fraction is not constant. 

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