Fluidization Davidson-Harrison model

This model represents one of the first modeling approaches used to describe large industrial beds that has been fully documented and discussed in the open literature. The model was adopted by the Shell company designing a fluidized bed reactor for their solid catalyzed hydrogen chloride oxygen process. However, the model has many of the same limitations as the Davidson-Harrison model because is was developed before the importance of bubble clouds and wakes were realized. 

This model bears a familial resemblance to some that were discussed earlier in this chapter. When the dispersion terms are discarded and appropriate changes in the names and significance of some of the parameters are recognized, then we end up basically at the Davidson-Harrison model for fluidized beds. 

Number of FEM elements that constitute a part of the domain Parameter in bubble size model for bubbling beds (—) Permeability constant in the Davidson-Harrison model characteristic of the particles and the fluidizing fluid... 

Bubble Dynamics. To adequately describe the jet, the bubble size generated by the jet needs to be studied. A substantial amount of gas leaks from the bubble, to the emulsion phase during bubble formation stage, particularly when the bed is less than minimally fluidized. A model developed on the basis of this mechanism predicted the experimental bubble diameter well when the experimental bubble frequency was used as an input. The experimentally observed bubble frequency is smaller by a factor of 3 to 5 than that calculated from the Davidson and Harrison model (1963), which assumed no net gas interchange between the bubble and the emulsion phase. This discrepancy is due primarily to the extensive bubble coalescence above the jet nozzle and the assumption that no gas leaks from the bubble phase. 

Derive the cloud thickness and volume ratio of cloud to bubble for a single bubble in a fluidized bed in terms of the Davidson-Harrison (1963) model. Since the cloud is the region established by the gas circulating in a closed loop between the bubble and its surrounding, the following assumptions can be employed (1) zero radial gas velocity outside the surface of the cloud sphere (r = Rc) (2) uniform gas velocity at far distance from the bubble, i.e., Uboo... 

Chapter 8 will reveal that this is one version of a well-known fluidized-bed reactor model due to Davidson and Harrison). Using this, make an analysis of the uniqueness of steady-state operation. 

The analysis of fluidized-bed reactors is based largely on the fluid mechanical model first described fully by Davidson and Harrison (1963) and modified later by a number of investigators (e.g., Jackson, 1963 Murray, 1965 Pyle and Rose, 1965 Kunii and Levenspiel, 1968a,b Rowe, 1971 Orcutt and Carpenter, 1971 Davidson and Harrison, 1971 Davidson et al., 1978 Van Swaaij, 1985). Our description of fluidized-bed reactor modeling will be based on the Kunii-Levenspiel adaptation (see Levenspiel, 1993). 

The modeling of fluidized beds begins with the analysis of the two most important hydro-dynamic flow models presented by Davidson (Davidson and Harrison, 1963) and Kunii and Levenspiel (1968). 

To describe the particle and gas flows around the bubble in fluidized beds, the pioneering model of Davidson and Harrison (1963) is particularly noteworthy because of its fundamental importance and relative simplicity. On the basis of some salient features of this model, a number of other models were developed [e.g., Collins, 1965 Stewart, 1968 Jackson, 1971]. The material introduced later follows Davidson and Harrison s approach. 

In the Davidson and Harrison s (1963) maximum stable bubble size model, the bubble disintegration takes place when the relative velocity between the bubble and the particles exceeds the particle terminal velocity. Considering that, for a vertical gas-solid flow system, choking occurs when the maximum stable bubble size is equal to the column size, Yang (1976) obtained the following choking criterion for fine particles fluidization ... 

The two-phase theory of fluidization has been extensively used to describe fluidization (e.g., see Kunii and Levenspiel, Fluidization Engineering, 2d ed., Wiley, 1990). The fluidized bed is assumed to contain a bubble and an emulsion phase. The bubble phase may be modeled by a plug flow (or dispersion) model, and the emulsion phase is assumed to be well mixed and may be modeled as a CSTR. Correlations for the size of the bubbles and the heat and mass transport from the bubbles to the emulsion phase are available in Sec. 17 of this Handbook and in textbooks on the subject. Davidson and Harrison (Fluidization, 2d ed., Academic Press, 1985), Geldart (Gas Fluidization Technology, Wiley, 1986), Kunii and Levenspiel (Fluidization Engineering, Wiley, 1969), and Zenz (Fluidization and Fluid-Particle Systems, Pemm-Corp Publications, 1989) are good reference books. 

The ultimate cause of bubble formation is the universal tendency of gas-solid flows to segregate. Many studies on the theory of stability [3, 4] have shown that disturbances induced in an initially homogeneous gas-solid suspension do not decay but always lead to the formation of voids. The bubbles formed in this way exhibit a characteristic flow pattern whose basic properties can be calculated with the model of Davidson and Harrison [30], Figure 5 shows the streamlines of the gas flow relative to a bubble rising in a fluidized bed at minimum fluidization conditions (e = rmf). The characteristic parameter is the ratio a of the bubble s upward velocity u, to the interstitial velocity of the gas in the suspension surrounding the bubble ... 

The direct contact model has some difiiculties, however. In fluidized beds, gas bubbles of very low solid content are usually considered to exist in the dense phase (H14, K13, T19). Also, the cloud layer is negligibly thin, due to small (/ r for the usual fluid catalyst beds, according to equa-ticMis of Davidson and Harrison (D3) and Murray (M47). The streamlines of gas phase through a bubble have been observed to pass through the cloud, but not through the bubble wake (R17). Thus there seems little possibility of believing that the bubble gas is in direct contact with a substantial amount of catalyst in the bubble phase (see also Secticxi VI,A). Furthermore, the direct contact model is applied to the data by Gilliland and Knudsen, and v in Eq. (7-9) is calculated to fit the data. Calculation (M26) shows that the volume of catalyst, with an apparent density the same as for the emulsion, which contacts the bubble gas freely exceeds the volume of bubble gas itself (v/ib = 3.3, 2.0, and 1.5, respectively, for Uc. = 10, 20, and 30 cm/sec). This seems to be unsound physically. 

The first hydrodynamic model proposed for fluid-bed reactor design (see Davidson and Harrison, 1963) is simple but is the basis of most models developed since. A sketch of the model appears in Figure CS5.1a. Three main groups are involved U for fluidization, for reaction, and Y for mass transfer. Equations can be derived both for plug flow and mixed flow of emulsion gas. The simpler mixed-flow model is usually adequate (with predictions close to those of the plug-flow model) and is given by... 

Figure 8.2 Schematic of the two-phase model of a fluidized bed according to Davidson and Harrison. [After J.J. Carberry, Chemical and Catalytic Reaction Engineering, with permission of McGraw-Hill Book Company, New York, NY, (1976).]...

The first model for the movement of both gas and solids and the pressure distribution around single rising bubbles was given by Davidson and Harrison [17] (Chap. 4). Two versions of the model were developed, considering two- and three-dimensional fluidized beds. The theory is based on the following assumptions ... 

Almost all of the models proposed to date are based on the two phase theory of fluidization originally proposed by Toomey and Johnstone (97) and later modified by Davidson and Harrison (98). According to the theory, the fliiidized bed is assumed to consist of two phases, viz., l) a continuous, dense particulate phase (emulsion phase) and 2) a discontinuous, lean gas phase (bubble phase) with exchange of gas between the bubble phase and emiilsion phase. The gas flow rate through the emulsion phase is assumed to be at minimum fluidization and that in excess of the minimum fluidization velocity passes throu the bubble phase. This formulation of the two phase theory is based on the ass mq)tion that the voidage of the emulsion phase remains constant. However, as pointed out by Rowe (22) and Horio and Wen (lOO) this assumption may be an over-simplification. In particular, experiments with fine powders (dp < 60 ym) conducted by Rowe show that the dense phase voidage changes with gas velocity, and as much as 30 percent of the gas flow occurs interstitially. This effect can be... 

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