Pressure Drop-Velocity Relationship

The form of the curve of pressure drop against velocity for the fixed and fluidized beds should provide considerable amount of information on the structure of the bed. Deviations from the idealized behavior are attributable to intraparticle forces and maldistribution of the fluid in the bed. For practical 

Pressvire drop over fixed and fluidized beds as a function of fluid velocity. 

Due to the complexity of interaction of fluid-solid systems is not possible to predict precisely the behavior of fluidized-bed operations. With solid-gas 

If the bed expands the product (1 - e) will remain constant. Using the values of each term proper for the conditions of incipient fluidization  

For fine particles the pressure drop-velocity relationship will be given by the Carman-Kozeny equation, which will take the following form for incipient fluidization  

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The Richardson-Zaki equation has been found to agree with experimental data over a wide range of condifions. Equally, if is possible fo use a pressure drop-velocity relationship such as Ergun to determine minimum fluidization velocity, just as for gas-solid fluidizafion. An alternative expression, which has the merit of simplicify, is fhaf of Riba ef al. (1978)... 

Ratio of wake volume to bubble volume Constant in pressure drop-velocity relationship, Eq. (101) Constant in pressure drop-velocity relationship, Eq. (102) Constant in pressure drop-velocity relationship, Eq. (101) Constant in pressure drop-velocity relationship, Eq. (102) Proportionality constant in Eq. (31)... 

Note that for n = 1 and k = /i, Equations (4.7) and (4.8) reduce to the familiar Hagen-Poiseuille equation which describes the pressure drop-velocity relationship for the laminar flow of a Newtonian fluid. 

The gas flowing upward relative to the solids generates a frictional pressure drop. The relationship between the pressure drop per unit length (AP/Lg) and the relative velocity for a particular material is determined by the fluidization curve for that material. Normally, this fluidization curve is generated in a fluidization column where the solids are not flowing. However, the relationship also applies for solids flowing in a standpipe. 

The shear stress is hnear with radius. This result is quite general, applying to any axisymmetric fuUy developed flow, laminar or turbulent. If the relationship between the shear stress and the velocity gradient is known, equation 50 can be used to obtain the relationship between velocity and pressure drop. Thus, for laminar flow of a Newtonian fluid, one obtains ... 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

Typical velocities in plate heat exchangers for waterlike fluids in turbulent flow are 0.3-0.9 meters per second (m/s) but true velocities in certain regions will be higher by a factor of up to 4 due to the effect of the corrugations. All heat transfer and pressure drop relationships are, however, based on either a velocity calculated from the average plate gap or on the flow rate per passage. 

The emphasis in this chapter is on the generalization of piston flow to situations other than constant velocity down the tube. Real reactors can closely approximate piston flow reactors, yet they show many complications compared with the constant-density and constant-cross-section case considered in Chapter 1. Gas-phase tubular reactors may have appreciable density differences between the inlet and outlet. The mass density and thus the velocity down the tube can vary at constant pressure if there is a change in the number of moles upon reaction, but the pressure drop due to skin friction usually causes a larger change in the density and velocity of the gas. Reactors are sometimes designed to have variable cross sections, and this too will change the density and velocity. Despite these complications, piston flow reactors remain closely akin to batch reactors. There is a one-to-one correspondence between time in a batch and position in a tube, but the relationship is no longer as simple as z = ut. 

So far, some researchers have analyzed particle fluidization behaviors in a RFB, however, they have not well studied yet, since particle fluidization behaviors are very complicated. In this study, fundamental particle fluidization behaviors of Geldart s group B particle in a RFB were numerically analyzed by using a Discrete Element Method (DEM)- Computational Fluid Dynamics (CFD) coupling model [3]. First of all, visualization of particle fluidization behaviors in a RFB was conducted. Relationship between bed pressure drop and gas velocity was also investigated by the numerical simulation. In addition, fluctuations of bed pressure drop and particle mixing behaviors of radial direction were numerically analyzed. 

Now, the relationship between velocity (vT) and the heat transfer coefficient (hr) needs to be determined to relate pressure drop to hT. 

The relationship between flow rate, pressure drop, and pipe diameter for water flowing at 60°F in Schedule 40 horizontal pipe is tabulated in Appendix G over a range of pipe velocities that cover the most likely conditions. For this special case, no iteration or other calculation procedures are required for any of the unknown driving force, unknown flow rate, or unknown diameter problems (although interpolation in the table is usually necessary). Note that the friction loss is tabulated in this table as pressure drop (in psi) per 100 ft of pipe, which is equivalent to 100pef/144L in Bernoulli s equation, where p is in lbm/ft3, ef is in ft lbf/lbm, and L is in ft. 

Ishii and Murakami (1991) evaluated the CFB scaling relationships of Horio et al. (1989) using two cold CFB models. Solids flux, pressure drop, and optical probe measurements were used to measure a large number of hydrodynamic parameters to serve as the basis for the comparison. Fair to good similarity was obtained between the beds. Dependent hydrodynamic parameters such as the pressure drop and pressure fluctuation characteristics, cluster length and voidage, and the core diameter were compared between the two beds. The gas-to-solid density ratio was not varied between the beds. As seen in Table 7, the dimensionless solids flux decreased as the superficial velocity was increased because the solids flux was held constant. 

The anticipated pressure drop is calculable from the relationships in Fig. 21. Assuming the cyclone is located within, or attached externally to, the shell of an 8 diameter fluidized bed reactor operating at a superficial gas velocity of 2 fi/sec, then ... 

Figure 1.3 Relationship between bed pressure drop and superficial fluidizing velocity.

Expressions for minimum fluidizing velocify can be derived by examining fhe relationship between the velocity of a fluid passing through a packed bed of particles and the resultant pressure drop across the bed. Consequently it is necessary to define a number of bulk particle properties which influence fluidized bed behaviour. 

From the above relationships it can also be shown that the pressure drop AP (Pa) in the laminar flow of a Newtonian fluid ofviscosity // (Pa s) through a straight round tube of diameter d (m) and length L (m) at an average velocity of v (ms ) is given by Equation 2.9, which expresses the Hagen-Poiseuille law ... 

Thus, the pressure drop AP for laminar flow through a tube varies in proportion to the viscosity n, the average flow velocity v, and the tube length L, and in inverse proportion to the square of the tube diameter d. Since v is proportional to the total flow rate Q (m s ) and to d, C P should vary in proportion to ft, Q, L, and The principle of the capillary tube viscometer is based on this relationship. 

Schweitzer 22(f), using ratios of orifice length to diameter less than 4 to 1, finds the relationship V = c /(2qAF)/p for the velocity of the liquid jet, where equation c is a velocity coefficient, AP is the pressure drop, and p is the specific gravity of the oil being tested. The viscosity plays no role in the liquid jet velocity for low L/D ratios. Poi-seuille s formula applied for long orifices. 

In experimental work at the Texas Electric laboratory this relationship has been established (1). The work reported applied to fluids in turbulent flow and hence to relatively high velocities. Since that time the studies have been extended to lower velocities. To support thin membranes, a pierced, corrugated material was placed in the streams with the corrugations in the direction of flow, in a manner to cause the smallest pressure drop this obstruction induced strong mixing and simulated turbu-... 

Incipient or minimum fluidization can be characterized by the relationship of the dynamic pressure drop, Apd, to the gas velocity. The dynamic pressure gradient, dpd/dH, can be related to the total pressure gradient, dp/dH, by... 

For a gas-solid system, pg is negligibly small compared to (—dp/dH). Consequently, dpd/dH can be approximated by dp/dH. The relationship of pressure drop through the bed, Apb, and superficial gas velocity U for fluidization with uniform particles is illustrated in Fig. 9.5. In the figure, as U increases in the packed bed, Apb increases, reaches a peak, and then drops to a constant. As U decreases from the constant Apb, Apb follows a different path without passing through the peak. The peak under which the bed is operated is denoted the minimum fluidization condition, and its corresponding superficial gas velocity is defined as the minimum fluidization velocity, Umf. 

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