Happel’s cell model

Figure 8.5.1 Happel s cell model for a spherical particle.

Further, in Happel s cell model, there is no interaction between the contiguous cells through the cell surface boundaries. The number of spherical collectors in a UBE, n is obtained as follows ... 

Consider now Happel s cell model (Happel, 1958) in the spherical collector model. The total rate at which the particles enter one cell in HappeTs model is given by the following expression ... 

Cell models constitute a second major class of empirical developments. Among these, only two will be mentioned here as constituting the most successful and widely used. The first, due to Happel (1957,1958), is useful for estimating the effective viscosity and settling velocity of suspensions. Here, the suspension is envisioned as being composed of fictitious identical cells, each containing a single spherical particle of radius a surrounded by a concentric spherical envelope of fluid. The radius b of the cell is chosen to reproduce the suspension s volume fraction

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In order to include the coupling between the rugged laminar flow in a porous medium and the molecular diffusion, Horvath and Lin [50] used a model in which each particle is supposed to be surrounded by a stagnant film of thickness 5. Axial dispersion occurs only in the fluid outside this stagnant film, whose thickness decreases with increasing velocity. In order to obtain an expression for S, they used the Pfeffer and Happel "free-surface" cell model [52] for the mass transfer in a bed of spherical particles. According to the Pfeffer equation, at high values of the reduced velocity the Sherwood number, and therefore the film mass transfer coefficient, is proportional to... 

In Happel s model the porous medium is taken to be a random assemblage that is assumed to consist of a number of cells, each of which contains a particle surrounded by a fluid envelope. The fluid envelope is assumed to contain the same volumetric proportion of fluid to solid as exists in the entire assemblage. This determines the envelope radius of each cell. For illustration we... 

In the development of an expression for Et, a more detailed model of the porous medium is needed. Of the three types of models of a granular filter as a porous medium, the capillaric model is not preferred. For the sake of simplicity, we will consider one of the other two, namely the spherical collector model. In this model, the filter grain is assumed to be a sphere. There are a number of alternative approaches based on a spherical collector. We will illustrate the approach by Happel (1958). In Happel s model, the granular porous medium is assumed to consist of a large collection of identical cells, where each cell consists of a spherical particle of radius ((dgr)/2) (i.e. half of the average grain diameter) surrounded by a liquid envelope of radius b, such that the void volume of this cell is identical to the void volume of the porous medium ... 

Often, the mass-transfer correlation of Pfeffer (1964) based on Happel s (1958) cell model is used ... 

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