Fluid stagnant laminar

The phenomenon of concentration polarization, which is observed frequently in membrane separation processes, can be described in mathematical terms, as shown in Figure 30 (71). The usual model, which is weU founded in fluid hydrodynamics, assumes the bulk solution to be turbulent, but adjacent to the membrane surface there exists a stagnant laminar boundary layer of thickness (5) typically 50—200 p.m, in which there is no turbulent mixing. The concentration of the macromolecules in the bulk solution concentration is c,. and the concentration of macromolecules at the membrane surface is c. 

The existence of a stagnant laminar fluid film adjacent to the interface is not difficult to visualize, in the case where the interface is stationary, as when the fluid flows along a solid surface. However, this situation seems rather unrealistic with the fluid-fluid interface, as when the surface of the liquid in an agitated vessel is in contact with a gas phase above, or if gas bubbles move upward through a liquid, or when one liquid phase is in contact with another liquid phase in an extractor. 

This sequence is illustrated in Fig. 1. Steps 1 and 7 are typically complicated by the presence of a thin stagnant (laminar flow) fluid layer between the pellet exterior surface and the bulk fluid. 

In the previous sections of this chapter and Chapter 6 we have emphasized molecular diffusion in stagnant fluids or fluids in laminar flow. In many cases the rate of diffusion is slow, and more rapid transfer is desired. To do this, the fluid velocity is increased until turbulent mass transfer occurs. 

Continuous Flat Surface Boundaiy layers on continuous surfaces drawn through a stagnant fluid are shown in Fig. 6-48. Figure 6-48 7 shows the continuous flat surface (Saldadis, AIChE J., 7, 26—28, 221-225, 467-472 [1961]). The critical Reynolds number for transition to turbulent flow may be greater than the 500,000 value for the finite flat-plate case discussed previously (Tsou, Sparrow, and Kurtz, J. FluidMech., 26,145—161 [1966]). For a laminar boundary layer, the thickness is given by... 

Continuous Cylindrical Surface The continuous surface shown in Fig. 6-48b is apphcable, for example, for a wire drawn through a stagnant fluid (Sakiadis, AIChE ]., 7, 26-28, 221-225, 467-472 [1961]). The critical-length Reynolds number for transition is Re = 200,000. The laminar boundary laver thickness, total drag, and entrainment flow rate may be obtained from Fig. 6-49 the drag and entrainment rate are obtained from the momentum area 0 and displacement area A evaluated at x = L. 

When fluid is pumped through a cell such as that shown in Fig. 12, transport of dissolved molecules from the cell inlet to the IRE by convection and diffusion is an important issue. The ATR method probes only the volume just above the IRE, which is well within the stagnant boundary layer where diffusion prevails. Figure 13 shows this situation schematically for a diffusion model and a convection-diffusion model (65). The former model assumes that a stagnant boundary layer exists above the IRE, within which mass transport occurs solely by diffusion and that there are no concentration gradients in the convection flow. A more realistic model of the flow-through cell accounts for both convection and diffusion. As a consequence of the relatively narrow gap between the cell walls, the convection leads to a laminar flow profile and consequently to concentration gradients between the cell walls. 

Stagnant Flow situation considered in the discussion Fluid at 0f numerical solution of laminar natural Temp. T convection. 

Film theory predicts that the mass transfer coefficient for a phase (or the overall mass transfer coefficient) is proportional to the diffusion coefficient and inversely proportional to the thickness of the stagnant zone. The diffusion coefficient can be calculated from either the Wilke-Chang or the FSG equations. However, 6 is difficult (if not impossible) to determine. Hence, mass transfer coefficients are often determined from empirical correlations. Also, Film theory is based on the assumption that the bulk fluid phases are perfectly mixed. While this might approach reality for well-mixed turbulent systems, this is certainly not the case for laminar systems. 

D. Laminar, stationary disk [T] Stagnant fluid. Use arithmetic concentration difference. [138] p. 240... 

In order to include the coupling between the rugged laminar flow in a porous medium and the molecular diffusion, Horvath and Lin [50] used a model in which each particle is supposed to be surrounded by a stagnant film of thickness 5. Axial dispersion occurs only in the fluid outside this stagnant film, whose thickness decreases with increasing velocity. In order to obtain an expression for S, they used the Pfeffer and Happel "free-surface" cell model [52] for the mass transfer in a bed of spherical particles. According to the Pfeffer equation, at high values of the reduced velocity the Sherwood number, and therefore the film mass transfer coefficient, is proportional to... 

Convection. Stirring or hydrodynamic transport. Generally fluid flow occurs because of natural convection (convection caused by density gradients) md forced convection, and may be characterized by stagnant regions, laminar flow, and turbulent flow. 

In Chapter 2, the design of the so-called ideal reactors was discussed. The reactor ideahty was based on defined hydrodynamic behavior. We had assumedtwo flow patterns plug flow (piston type) where axial dispersion is excluded and completely mixed flow achieved in ideal stirred tank reactors. These flow patterns are often used for reactor design because the mass and heat balances are relatively simple to treat. But real equipment often deviates from that of the ideal flow pattern. In tubular reactors radial velocity and concentration profiles may develop in laminar flow. In turbulent flow, velocity fluctuations can lead to an axial dispersion. In catalytic packed bed reactors, irregular flow with the formation of channels may occur while stagnant fluid zones (dead zones) may develop in other parts of the reactor. Incompletely mixed zones and thus inhomogeneity can also be observed in CSTR, especially in the cases of viscous media. 

Diffusion in a laminar falling film. In Section 2.9C we derived the equation for the velocity profile in a falling film shown in Fig. 7.3-la. We will consider mass transfer of solute A into a laminar falling film, which is important in wetted-wall columns, in developing theories to explain mass transfer in stagnant pockets of fluids, and in turbulent mass transfer. The solute A in the gas is absorbed at the interface and then diffuses a distance into the liquid so that it has not penetrated the whole distance x = <5 at the wall. At steady state the inlet concentration = 0. At a point z distance from the inlet the concentration profile of is shown in Fig. 7.3- la. 

First let us examine mass transport through this film under isothermal conditions by employing the continuity equations for mass (a mass balance) and for momentum (an energy balance). In this stagnant film, which can correspond to the laminar boundary layer that develops when a fluid passes over a flat surface, there is no motion of the fluid, hence the latter equation is irrelevant. The continuity equation for mass describes the spacial dependence of concentration in terms of the velocities parallel, u, and perpendicular, V, to the surface ... 

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