Minimum fluidising velocity

At the point of incipient fluidisation, the bed voidage, s f, depends on the shape and size range of the particles, but is approximately equal to 0.4 for isometric particles. The minimmn fluidising velocity V f for a power-law fluid in streamline flow is then obtained by substituting s = Smf in equation (5.67). Although this equation applies oifly at low values of the bed Reynolds numbers ( 1), this is not usually a limitation at the high apparent viscosities of most non-Newtonian materials. 

At high velocities, the flow may no longer be streamline, and a more general equation must be used for the pressme gradient in the bed, such as equation (5.56). Substituting e = s f and Vo = Vm/, this equation becomes  

Replacement of fmf by the Galileo number eliminates the unknown velocity which appears in Rej y. Multiplying both sides of equation (5.69) by 

For a given liquid (known m, n, p) and particle (ps, d, Smf) combination, equation (5.70) can be solved for Re y which in turn enables the value of the minimmn fluidising velocity to be calculated, as illustrated in example 5.7. 

A bed consists of uniform glass spheres of size 3.57mm diameter (density = 2500 kg/m ). What will be the minimum fluidising velocity in a polymer solution of density, 1000kg/m, with power-law constants n = 0.6 and m = 0.25 Pa-s Assume the bed voidage to be 0.4 at the point of incipient fluidisation. 

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The minimum fluidisation velocity of the particles is achieved when the adsorbent becomes suspended in the liquid. This occurs when the drag forces exerted by the upward flow of the liquid phase ate equal to the weight of particles in the liquid. Therefore, at minimum fluidising conditions, it can be described by the following expression ... 

However, on die basis of the relation between pressure drop and die minimum fluidisation velocity of particles, the point of transition between a packed bed and a fluidised bed has been correlated by Ergun41 using (17.7.2.3). This is obtained by summing the pressure drop terms for laminar and turbulent flow regions. 

If fihll is unknown, the following equation suggested by Wen and Yu41 can be used to determine the minimum fluidisation velocity for the whole range of Reynolds numbers by assuming ... 

Equation (17.7.2.9) was originally used to correlate the minimum fluidisation velocity for gas-solid fluidisation beds but has been successfully employed by Lan and his co-workers42 for adsorbents in the field of direct recovery using liquid-solid systems (Figure 17.4). 

The operational window of a fluidised bed process is defined by the minimum fluidisation velocity, Uml, at which a settled bed of adsorbent beads starts to fluidise and the terminal velocity ( /t) at which the bed stabilises and adsorbent beads are entrained from the bed. 

Detailed consideration of the interaction between particles and fluids is given in Volume 2 to which reference should be made. Briefly, however, if a particle is introduced into a fluid stream flowing vertically upwards it will be transported by the fluid provided that the fluid velocity exceeds the terminal falling velocity m0 of the particle the relative or slip velocity will be approximately o- As the concentration of particles increases this slip velocity will become progressively less and, for a slug of fairly close packed particles, will approximate to the minimum fluidising velocity of the particles. (See Volume 2, Chapter 6.)... 

Obtain a relationship for the ratio of the terminal falling velocity of a particle to the minimum fluidising velocity for a bed of similar particles. It may be assumed that Stokes Law and the Carman-Kozeny equation are applicable. What is the value of the ratio if the bed voidage at the minimum fluidising velocity is 0.4 ... 

Writing u = umf, the minimum fluidising velocity in Ergun s equation and substituting... 

Empirical relationships for the minimum fluidising velocity are presented as a function of Reynolds number and this problem illustrates the importance of using the equations applicable to the particle Reynolds number in question. 

What will be the minimum fluidising velocity of the system Stokes law states that the force on a spherical particle = 3jtpduo. 

The theoretical value of the minimum fluidising velocity may be calculated from the equations given in Chapter 4 for the relation between pressure drop and velocity in a fixed packed bed, with the pressure drop through the bed put equal to the apparent weight of particles per unit area, and the porosity set at the maximum value that can be attained in the fixed bed. 

As the upward velocity of flow of fluid through a packed bed of uniform spheres is increased, the point of incipient fluidisation is reached when the particles are just supported in the fluid. The corresponding value of the minimum fluidising velocity (umf) is then obtained by substituting emf into equation 6.3 to give ... 

Substituting e = emf at the incipient fluidisation point and for —AP from equation 6.1, equation 6.6 is then applicable at the minimum fluidisation velocity umf, and gives ... 

Wen and Yu(6) have examined the relationship between voidage at the minimum fluidising velocity, emf, and particle shape, (ps, which is defined as the ratio of the diameter of the sphere of the same specific as the particle d, as used in the Ergun equation to the diameter of the sphere with the same volume as the particle dp. 

The minimum fluidising velocity is a function of both emf and 4>s, neither of which is easily measured or estimated, and Wen and Yu have shown that these two quantities are, in practice, inter-related. These authors have published experimental data of emf and characterised particles, and it has been shown that the relation between these two quantities is essentially independent of particle size over a wide range. It has also been established that the following two expressions give reasonably good correlations between emf and [Pg.297]

Minimum fluidising velocity in terms of terminal failing velocity... 

The minimum fluidising velocity, umf, may be expressed in terms of the free-falling velocity o of the particles in the fluid. The Ergun equation (equation 6.11) relates the Galileo number Ga to the Reynolds number Re mj in terms of the voidage < , / at the incipient fluidisation point. 

Figure 6.4. Ratio of terminal falling velocity to minimum fluidising velocity, as a function of Galileo number... 

An alternative method of calculating the value of Re mf (and hence //,/) is to substitute for Re 0 from equation 6.21 into equation 6.35, and to put the voidage e equal to its value emf at the minimum fluidising velocity. 

For inelastic fluids exhibiting power-law behaviour, the bed expansion which occurs as the velocity is increased above the minimum fluidising velocity follows a similar pattern to that obtained with a Newtonian liquid, with the exponent in equation 6.31 differing by no more than about 10 per cent. There is some evidence, however, that with viscoelastic polymer solutions the exponent may be considerably higher. Reference may be made to work by Srimvas and Chhabra(15) for further details. 

Thus, the bubbling region, which is an important feature of beds operating at gas velocities in excess of the minimum fluidising velocity, is usually characterised by two phases — a continuous emulsion phase with a voidage approximately equal to that of a bed at its minimum fluidising velocity, and a discontinous or bubble phase that accounts for most of the excess flow of gas. This is sometimes referred to as the two-phase theory of fluidisation. 

Because minimum fluidising velocity is not very sensitive to the pressure in the bed, much greater mass flowrates of gas may be obtained by increasing the operating pressure. 

The influence of pressure, over the range 100-1600 kN/m2, on the fluidisation of three grades of sand in the particle size range 0.3 to 1 mm has been studied by Olowson and Almstedt(50) and it was showed that the minimum fluidising velocity became less as the pressure was increased. The effect, most marked with the coarse solids, was in agreement with that predicted by standard relations such as equation 6.14. For fine particles, the minimum fluidising velocity is independent of gas density (equation 6.5 with Ps >> P), and hence of pressure. 

The onset of turbulent fluidisation appears to be almost independent of bed height, or height at the minimum fluidisation velocity, if this condition is sufficiently well defined. It is, however, strongly influenced by the bed diameter which clearly imposes a maximum on the size of the bubble which can form. The critical fluidising velocity tends to become smaller as the column diameter and gas density, and hence pressure, increase. Particle size distribution appears to assert a strong influence on the transition velocity. With particles of wide size distributions, pressure fluctuations in the bed are smaller and the transition velocity tends to be lower. 

In a deep bed, the situation is more complex and the value of E may be considerably greater at the bed inlet than at its inner surface. In addition, the gas velocity increases towards the centre because of the progressively reducing area of flow. These factors combine so that the value of the minimum fluidising velocity increases with radius in the bed. Thus, as flowrate is increased, fluidisation will first occur at the inner surface of the bed and it will then take place progressively further outwards, until eventually the whole bed will become fluidised. In effect, there are therefore two minimum fluidising velocities of interest, that at which fluidisation first occurs at the inner surface, and that at which the whole bed becomes fluidised. 

If a gas is introduced at the bottom of a bed of solids fluidised by a liquid, the expansion of the bed may either decrease or increase, depending on the nature of the solids and particularly their inertia. It is generally found that the minimum fluidising velocity of the liquid is usually reduced by the presence of the gas stream. Measurements are difficult to make accurately, however, because of the fluctuating flow pattern which develops. 

Figure 6.25. Heat transfer coefficient from a surface to 78 xm glass spheres fluidised by air. AB = fixed bed B = minimum fluidising velocity C = maximum coefficient CD = falling coefficient D = minimum coefficient DE = final region of increasing coefficient...

Fluidised beds may be divided into two classes. In the first, there is a uniform dispersion of the particles within the fluid and the bed expands in a regular manner as the fluid velocity is increased. This behaviour, termed particulate fluidisation, is exhibited by most liquid-solids systems, the only important exceptions being those composed of fine particles of high density. This behaviour is also exhibited by certain gas-solids systems over a very small range of velocities just in excess of the minimum fluidising velocity—particularly where the particles are approximately spherical and have very low free-falling velocities. In particulate fluidisation the rate of movement of the particles is comparatively low, and the fluid is predominantly in piston-type flow with some back-mixing, particularly at low flowrates. Overall turbulence normally exists in the system. 

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