Model of Davidson and Harrison

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. 

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 first term in each case arises from bulk flow of gas into the floor of an isolated bubble and out the roof, as required by the hydrodynamic model of Davidson and Harrison (27). The weight of experimental evidence, from studies of cloud size (28,29), from chemical reaction studies (e.g. 30), and from interphase transfer studies (e.g. 31,32), is that this term is better described by the theory proposed by Murray (33). The latter leads to a reduction in the first term by a factor of 3. Some enhancement of the bulk flow component occurs for interacting bubbles (34,35), but this enhancement for a freely bubbling bed is only of the order of 20-30% (35), not the 300% that would be required for the bulk flow term Equations (1) and (2) to be valid. 

Fryer and Potter (1972), using the model of Davidson and Harrison, reported that a bubble size found at about 0.4h could be used as the single bubble size in that model. Earlier in this paper, the bubble size found at 0.5h was used arbitrarily in calculating the conversion in an ammonia oxida tion system using the K-L model. 

Thus, the modeling of the gas-solid noncatalytic reactor is far more complex than that of its catalytic counterpart. Even so, simplified models have been used to get a qualitative (and to some extent, quantitative) feel for the performance of the reactor. Thus, the two-phase model of Davidson and Harrison (1963) has been used by Campbell and Davidson (1975) to analyze the data on the combustion of carbon particles for short periods of combustion in a batch reactor. The model has also been used and considerably extended by Amundson (see Bukur et al., 1977). Tigrel and Pyle (1971) have used this model for the not-too-different problem of catalyst deactivation. Kunii and Levenspiel (1969) and Kato and Wen (1969) have extended their models to gas-solid noncatalytic systems. A particularly useful model that takes account of some of the complexities in practical systems has been suggested by Chen and Saxena (1978). 

As a first approximation, the analysis in reference [68] uses the well-known model of Davidson and Harrison [65] in which the bubble is assumed to be a spherical cavity without particles and in which the dispersed phase is characterized by uniform concentration < )o everywhere outside the bubble. Relative interstitial fluid velocity, u, and mean particle velocity, w, can then be found on the basis of 1) a simple filtration flow model for a homogeneous porous body containing a spherical cavity, and 2) an ideal fluid model for flow around a sphere. In particular, the vertical components of these velocities along vertical axis z of the coordinate system having its origin at the bubble center are ... 

Another representative two-phase model is the one proposed by Partridge and Rowe (1966). In this model, the two-phase theory of Toomey and Johnstone (1952) is still used to estimate the visible gas flow, as in the model of Davidson and Harrison (1963). However, this model considers the gas interchange to occur at the cloud-emulsion interphase, i.e., the bubble and the cloud phase are considered to be well-mixed, the result being called bubble-cloud phase. The model thus interprets the flow distribution in terms of the bubble-cloud phase and the emulsion phase. With the inclusion of the clouds, the model also allows reactions to take place in the bubble-cloud phase. The rate of interphase mass transfer proposed in the model, however, considers the diffusive mechanism only (i.e., without throughflow) and is much lower than that used in the model of Davidson and Harrison (1963). 

The model of Davidson and Harrison (1963), which assumes perfect mixing in the dense phase (DPPM), underestimates seriously the overall conversion for the reaction studied. While the counterpart model that assumes piston flow in the dense phase (DPPF) gives much better predictions of overall conversion, still the predicted concentration profiles in the individual phases are in poor agreement with the observed profiles. 

Krishna [1981] tried to confirm values and trends for A/starting from the model of Davidson and Harrison [1971] derived from hydrodynamic observations and theory. Davidson and Harrison expressed the interaction between the bubble phase and the emulsion in terms of two contributions cross flow and diffusion. The total rate of gas transfer, Q (in m%), is derived to be... 

Figure 11.6 Two>phase model of Davidson and Harrison. (Davidson, Harrison, Darton, LaNauze, in Chemical Reactor Theory A Review, ed. Lapidus and Amundson, 1979. Reprinted by permission of Prentice-Hall, Inc., Englewood Cliffs, NJ.)...

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. 

We wish to see what the overall conversion of a continuous mixture will be, but first we have to ask which parameters will depend on x, the index variable of the continuous mixture. Clearly k, the rate constant in the Damkdhler number, will be a function of x, and if monotonic, it can be put equal to Da.x. The parameter P is clearly hydrodynamic, and so, for the most part, are the terms in the Davidson number. The only term in Equation 6.21 of Davidson and Harrison that might depend on x is the gas phase diffusivity, and this appears under a square root sign in the second of two terms. Tr was found to be virtually constant with a value close to three in a series of experiments by Orcutt which Davidson and Harrison analyze. We will, therefore, assume that only the Damkdhler number varies with x, and that this variation is linear. Then for the well-mixed model... 

Here, Pc is the mixture density of the dense phase. U up i is defined by J Uf-U/), where Uf and U are mean velocities of the dilute and dense phases, respectively. This definition of mesoscale slip velocity differs a little bit from that in the cluster-based EMMS model, because the continuous phase transforms from the dilute phase to the dense phase. And their quantitative difference is l-f)PgUgc/Pc, which is normally negligible for gas-solid systems. Similarly, the closure of Fdi switches to the determination of bubble diameter. And it is well documented in literature ever since the classic work of Davidson and Harrison (1963). Compared to cluster diameter, bubble diameter arouses less disputes and hence is easier to characterize. The visual bubbles are normally irregular and in constantly dynamic transformation, which may deviate much from spherical assumption. Thus, the diameter of bubble here can also be viewed as drag-equivalent definition. 

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. 

The Davidson and Harrison (1963) model assumed there was no net exchange of gas between the bubble and the emulsion phase. The validity of this assumption was later questioned by Botterill et al. (1966), Rowe and Matsuno (1971), Nguyen and Leung (1972), and Barreto et al. (1983). The predicted bubble volume, if assumed no net gas exchange, was considerably larger than the actual bubble volume experimentally observed. 

The authors have collected data for flow rates up to 15 cm/3sec using water as the liquid, and have found the model to be applicable. Figure 9 shows a comparison made between the theoretical values of Kumar and Kuloor (K18), the data obtained at higher flow rates by various investigators, and the Davidson and Harrison (D6) equation. 

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). 

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. 

In reactor modeling we need to deal with the behavior of the bubbling bed as a whole rather than single rising bubbles. In extending the simple two-phase theory, Davidson and Harrison [29] proposed that the average velocity of bubbles in a freely bubbling bed can be approximated by... 

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... 

The Davidson and Harrison [29, 30] bubbling bed reactor model represents one of the first modeling attempts that was based on bubble dynamics. The model rest on the following assumptions ... 

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... 

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. 

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 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). 

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