Crossflow coefficient

The experimental results were fitted by adjusting the two model parameters, namely the crossflow coefficient, k, and the core radius, r, considered invariant with height. Best fit values for k ranged from 0.08 to 0.11 m/s, and tJR was equal to 0.78. 

Patience and Chaouki [98] adopted the two-phase model of Brereton et al. [96] to interpret their gas RTD data obtained with a radioactive tracer gas. The two model parameters, crossflow coefficient, k, and (ratio between core and riser cross-sections), were evaluated by fitting the model to the experimental data. They found that the crossflow coefficients varied between 0.03 to 0.1 m/s, and varied from 0.98, at high gas velocities, to 0.5, at low velocities. They attributed gas crossflow between core and annulus by supposing that solids drag gas to the annulus as they condense along the wall and then carry it downward for a certain distance. Solids are reintroduced into the core as they are stripped off the wall and re-entrained into the core gas flow. They developed a correlation for describing this gas mass transfer based on the analogy with wetted wall towers, as ... 

The design calculations highlighted the shortcomings of the Kern method of exchanger design. The Kern method fails to account for shell-side inefficiencies such as bypassing, leakage, crossflow losses, and window losses. This leads to a marked overestimate of the shell-side heat-transfer coefficient and shell-side pressure drop. The Bell method is recommended to correct these deficiencies. 

On the contrary, no general expression is available for calculating the mass-transfer coefficient at the shell side. In the literature, in fact, different equations are proposed, depending on the type of module and on the type of flow (parallel or crossflow). Probably, this is due to the fact that the fluidodynamics of the stream sent outside the fibers is strongly affected by the phenomena of channeling or bypassing and it is not well defined as for the stream, which is sent into the fibers. Hereinafter some of the different expressions proposed are reported. 

In the fourth subtechnique, flow FFF (F/FFF), an external field, as such, is not used. Its place is taken by a slow transverse flow of the carrier liquid. In the usual case carrier permeates into the channel through the top wall (a layer of porous frit), moves slowly across the thin channel space, and seeps out of a membrane-frit bilayer constituting the bottom (accumulation) wall. This slow transverse flow is superimposed on the much faster down-channel flow. We emphasized in Section 7.4 that flow provides a transport mechanism much like that of an external field hence the substitution of transverse flow for a transverse (perpendicular) field is feasible. However this transverse flow—crossflow as we call it—is not by itself selective (see Section 7.4) different particle types are all transported toward the accumulation wall at the same rate. Nonetheless the thickness of the steady-state layer of particles formed at the accumulation wall is variable, determined by a combination of the crossflow transport which forms the layer and by diffusion which breaks it down. Since diffusion coefficients vary from species to species, exponential distributions of different thicknesses are formed, leading to normal FFF separation. 

Soc. Mech. Engrs. 74 (1952), 343-347 Distribution of heat transfer coefficients around circular cylindres in crossflow at Reynolds numbers from 20 to 500. 

Three-parameter PDE model (Van Swaaij et aL106) This model is largely used to correlate the RTD curves from a trickle-bed reactor. The model is based on the same concept as the crossflow or modified mixing-cell model, except that axial dispersion in the mobile phase is also considered. The model, therefore, contains three arbitrary parameters, two of which are the same as those used in the cross-flow model and the third one is the axial dispersion coefficient (or the Peclet number in dimensionless form) in the mobile phase (see Fig. 3-11). 

Shown in Figure 6.4 are the effects of the process stream temperature (35 to 50X) on the permeate flux at a crossflow velocity of 3 m/s and a TMP of 5 bars for a feed pH of 4.40 [Attia et al., 1991b]. The steady state flux increases from 20 to 130 L7hr-m as the temperature increases from 35 to 50 C. Correspondingly, the protein retention rate decreases from 98.6 to 95.6 %. This increase in the flux and decrease in the protein retention may be explained by the reduced viscosity, increased diffusion coefficients and increased solubility of the constituents in the membrane and solution as a result of raising the temperature. 

For the UF of proteins, the concentration polarization model has been found to predict the filtration performance reasonably well [56]. However, this model is inherently weak in describing the two-dimensional mass transport mechanism during crossflow filtration and does not take into account the solute-solute interactions on mass transport that occur extensively in colloids, especially during MF [21,44,158,159]. The diffusion coefficient, which is inversely proportional to the particle radius, is low and underestimates the movement of particles away from the membrane [56]. This results to the well-known flux paradox problem where the predicted permeate flux is as much as two orders of magnitude lower than the observed flux during MF of colloidal suspensions [56,58,158]. This problem has then been underlined by the experimental finding of a critical flux for colloids, which demonstrates the specificity of colloidal suspension filtration wherein just a small variation in physicochemical or hydrodynamic conditions induces important changes in the way the process has to be operated [21]. 

The flow around and therefore the heat transfer around an individual tube within the bundle is influenced by the detachment of the boundary layer and the vortices from the previous tubes. The heat transfer on a tube in the first row is roughly the same as that on a single cylinder with a fluid in crossflow, provided the transverse pitch between the tubes is not too narrow. Further downstream the heat transfer coefficient increases because the previous tubes act as turbulence generators for those which follow. From the fourth or fifth row onwards the flow pattern hardly changes and the mean heat transfer coefficient of the tubes approach a constant end value. As a result of this the mean heat transfer coefficient over all the tubes reaches for an end value independent of the row number. It is roughly constant from about the tenth row onwards. This is illustrated in Fig. 3.26, in which the ratio F of the mean heat transfer coefficient Oim(zR) up to row zR with the end value am (zR —> oo) = amoo is plotted against the row number zR. 

The value of the exponential dependence of the product flow rate on the crossflow velocity is indicative of the hydrodynamic conditions which prevail. The plateau values, or near plateau values of the product flow are shown in Figure 10, plotted versus the crossflow velocity, and the slope is found to have a value of 0.85. For a turbulent flow situation in a tube, the mass transfer coefficient is given by lD. 

MULTIPASS EX, a] ERS. In multipass shell-and-tube exchangers the flow pattern is complex itlrparallel, countercurrent, and crossflow all present. Under these conditions, even when the overall coefficient t/is constant, the LMTD cannot be used, Calculation procedures for multipass exchangers are given in Chap. 15. 

After the individual coefficients are known, the total area required is found in the usual way from the overall coefficient using an equation similar to Eq. (11.14). As discussed in the next section, the LMTD must often be corrected for the effects of crossflow. 

COEFFICIENTS FOR CROSSFLOW. In some exchangers such as air heaters the shell is rectangular and the number of tubes in each row is the same. Flow is directly across the tubes, and baffles are not needed. For such crossflow conditions the following equation is recommended ... 

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