Particle bed model

The net force (F) was considered to be the difference between the drag force (Fd) and the effective weight (We). Fd was derived on the basis of particle bed model. For instance, for the case of a solid-hquid fluidized bed. 

Foscolo PU, Gibilaro LG. Fluid d5Uiamic stability of fluidised suspensions the particle bed model. Chem Eng Sci 42 1489-1500, 1987. 

We now turn to the key concept of the particle bed model, the introduction of which gives rise to a quite different conclusion for system stability... 

We are now in a position to write down the closed formulation of the conservation equations for the particle and fluid phases that defines the one-dimensional particle bed model. The particle phase momentum equation, derived in Chapter 7, adopts the net primary force term F of eqn (8.4) and is augmented by the elasticity term Taz, which, in terms of the dynamic-wave velocity, becomes ppU dejdz. 

In Chapters 9 and 12, the predietions of eqn (8.36) will be compared with the copious body of observations reported for the stability condition of a wide variety of experimentally investigated fluidized beds. By way of introduction to these comparisons, an example of the particle bed model predictions of the effect of particle diameter on the stability of gas fluidized beds reflecting the observations reported at the start of this chapter, which confirmed the existence of the stable, homogeneously-fluidized state is illustrated in Figure 8.3. 

Foscolo, P.U. and Gibilaro, L.G. (1987). Fluid-dynamic stability of fluidized suspensions the particle bed model. Chem. Eng. ScL, 42, 1489. 

It has long been known that such disturbance sources can give rise to the phenomenon of premature bubbling that is to say a measured Smb value lower than that obtained in a disruption-free system. For this reason the homogeneous expansion behaviour of fine-powder, gas-fluidized beds has sometimes been referred to as metastable - because the stability can be destroyed by simply disrupting the regular flow operation. In Chapter 14, an analysis that employs the full, non-linear, particle bed model formulation provides a simple explanation of this phenomenon. 

The particle bed model readily accommodates variations in g. When the comparisons with model predictions that we now reproduce (Gibilaro et al., 1986) and the experiments to which they refer were first performed the whole exercise appeared purely academic. It was subsequently gratifying to learn that a space-exploration study had given serious consideration to the feasibility of operating fluidized bed reactors under conditions of greatly reduced gravitational field strength. Without an effective model, it is quite impossible to predict what effect such an environment would have on the overall performance. 

The particle bed model formulation so far employed omits consideration of inertial coupling phenomena. These arise when a body submerged in a fluid is subjected to a net force causing it to accelerate as it does so, some fluid is carried with the body, effectively increasing its inertial mass. Clearly, this added mass will be negligible for gas-fluidized systems, but for liquid fluidization of relatively low-density particles it could well be important. A remarkably simple method of approximating this effect for the problem in hand has been derived by Wallis (1990). His analysis leads to the conclusion that some correction can be obtained by simply increasing the value of both the particle and fluid densities in the model formulation (and hence in any derived result) by one-half the fluid density ... 

The direct verification of the kinematic- and dynamic-wave speed expressions, which in some respects effectively define the particle bed model, provides further support for the formulation. Still required, however, is an experimental method for measuring dynamic-wave speeds in the particle phase of expanded beds, at void fractions greater than e f. 

In the preceding chapters we first considered the primary forces acting on a fluidized particle in a bed in equilibrium, and then the elastic forces between particles that come into play under non-equilibrium conditions. These two effects provide closure for the particle bed model, formulated in terms of the particle-and fluid-phase conservation equations for mass and momentum. Up to now, applications have focused on the stability of the state of homogeneous particle suspension, in particular for gas-fluidized systems for which the condition that particle density is much greater than fluid density enables the particle-phase equations to be decoupled and treated independently. The analysis has involved solely the linearized forms of these equations, and has led to a stability criterion that broadly characterizes fluidized systems according to three manifestations of the fluidized state always stable - the usual case for liquids always unstable - the usual case for gases and transitional behaviour - involving a switch, at a critical fluid flux, from the stable to the unstable condition. This characterization has... 

Figure 10.1 Powder classification map for fluidization by ambient air. Heavy lines, empirically determined boundaries of Geldart light broken lines, boundary predictions of the particle bed model.

Also shown in Figure 10.1, by means of light, broken lines, are predictions of the group boundaries obtained from the particle bed model. Only the... 

The above speculations, linking fluidization quality to perturbation-amplitude growth and decay rates, are now examined. Although bubble-related phenomena clearly imply conditions outside the linear response limit of the system, initial growth rates, obtainable from the linearized relations, can be so large in these cases that they could be expected to play a major role in subsequent developments. The linearized particle bed model delivers explicit relations for perturbation-wave velocity and amplitude growth rate, thereby enabling the above considerations to be... 

The water-copper system falls well between the other two. This also corresponds to qualitative observations of fluidization quality in such systems. Although transition points are relatively sharp, the bubbles that result remain very small and there is no evidence of metastable behaviour, referred to in Chapter 9 with reference to the phenomenon of premature bubbling in gas-fluidized systems, and analysed in Chapter 14 on the basis of the unlinearized particle bed model equations the manifestations of instability remain significantly less pronounced than is the case for the air-alumina system. 

Consider a homogeneously fluidized bed in equilibrium. If the particles are now subjected to a small force, they will move to restore the equilibrium condition. How fast they do this will depend on the specific system properties the greater the velocity of the particles, the more uniformly held together will be the suspension, and vice versa. A parameter that could provide a measure of this effect, the bulk mobility Bp of the particles , has been proposed by Batchelor (1988) in the development of a model for fluidization that is structurally similar to the particle bed model. He defines Bp as the ratio of the (small additional) mean velocity, relative to zero-volume-flux axes, to the (small additional) steady force applied to each particle of a homogeneous dispersion . For a bed initially in a state of equilibrium, this becomes ... 

Table 12.1 Comparison of the one- and two-phase particle bed models fluidization by air and water (at temperatures of 20°C and ambient pressure, unless otherwise indicated)... 

Predictions of the single-phase particle bed model were confronted with experimental observations of gas-fluidized beds in Chapter 9. The assumption of pp pf, which enabled the fluid-phase equations to be effectively removed from consideration in this case, would appear to render this approximation inappropriate for most cases of liquid fluidization. The above results, however, show that the single-phase approximation leads to stability predictions in reasonable harmony with the full two-phase model for liquid fluidization over a substantial range of particle density, down to perhaps three times that of the fluid. In this section we confront reported experimental observations relating to the stability of... 

Although there is far less experimental data available on the stability of liquid-fluidized beds than there is for gas beds, the results presented above provide further reassuring evidence for the basic integrity of the particle bed model formulation. In some respects the liquid system comparisons go further than those for gas systems, in that they allow for the interpretation of instabilities that fall well short of bubbling behaviour, relating this phenomenon to model predictions of growth- and decay-rates of void fraction perturbations. 

Table 13.1 shows three pairs of fluidized systems, each representing a water- and a gas-fluidized bed, which are approximately matched with regard to the one-dimensional scaling rules. The results for fluidization with synthesis gas at 124 bar (Jacob and Weimer, 1987) and with water at 10 °C (Gibilaro et al., 1986) have been referred to in Chapters 9 and 12 respectively, where they are shown to support the predictions of the particle bed model for the minimum bubbling void fraction 6 the high pressure carbon tetrafluoride results (at 21 bar and 69 bar for systems I and III respectively) were reported by Crowther and Whitehead (1978). 

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