Stagnation flow mass transfer

Any consideration of mass transfer to or from drops must eventually refer to conditions in the layers (usually thin) of each phase adjacent to the interface. These boundary layers are envisioned as extending away from the interface to a location such that the velocity gradient normal to the general flow direction is substantially zero. In the model shown in Fig. 8, the continuous-phase equatorial boundary layer extends to infinity, but the drop-phase layer stops at the stagnation ring. At drop velocities well above the creeping flow region there is a thin laminar sublayer adjacent to the interface and a thicker turbulent boundary layer between this and the main body of the continuous phase. 

The mechanism of mass transfer to the external flow is essentially the same as for spheres in Chapter 5. Figure 6.8 shows numerically computed streamlines and concentration contours with Sc = 0.7 for axisymmetric flow past an oblate spheroid (E = 0.2) and a prolate spheroid (E = 5) at Re = 100. Local Sherwood numbers are shown for these conditions in Figs. 6.9 and 6.10. Figure 6.9 shows that the minimum transfer rate occurs aft of separation as for a sphere. Transfer rates are highest at the edge of the oblate ellipsoid and at the front stagnation point of the prolate ellipsoid. 

In an ideal stagnation flow, a certain amount of the flow that enters through the inlet manifold can leave without entering the thermal or mass-transfer boundary layers above the surface. For an axisymmetric, finite-gap, flow, determine how the bypass fraction depends on the separation distance and the inlet velocity. 

While our primary interest in this text is internal flow, there are certain similarities with the classic aerodynamics-motivated external flows. Broadly speaking, the stagnation flows discussed in Chapter 6 are classified as boundary layers where the outer flow that establishes the stagnation flow has a principal flow direction that is normal to the solid surface. Outside the boundary layer, there is typically an outer region in which viscous effects are negligible. Even in confined flows (e.g., a stagnation-flow chemical-vapor-deposition reactor), it is the existence of an inviscid outer region that is responsible for some of the relatively simple correlations of diffusive behavior in the boundary layer, like heat and mass transfer to the deposition surface. 

Fig. 4. Mass transfer from an impinging jet electrode. (From ref. 48 by permission of the publisher, The Electrochemical Society, Inc.). I, Potential core region II, established flow region III, stagnation region IV, wall-jet region.

The electrode is uniformly accessible to the diffusing ions within dimensionless electrode radius, 0.1 < R/d < 1.0, for turbulent nozzle flow and, 0.1 < R/d < 0.5, for laminar nozzle flow. Within the region of uniform accessibility, the mass transport rate is relatively independent of the electrode size in both laminar and turbulent flow for 0.2 < Hjd < 6, where H is the nozzle-to-plate distance. Beyond the region of uniform accessibility, the mass transfer rate decreases with the radial distance. In the intermediate range, 1 < R/d < 4, the turbulent impinging jet changes from the stagnation flow to the wall-jet flow and for R/d > 4 the wall-jet flow predominates (- wall-jet electrode). 

M. T. Scholtz and O. Trass, "Mass Transfer in a Nonuniform Impinging Jet I. Stagnation Flow-Velocity and Pressure Distribution," AIChE Journal, 16 (1970) 82-90. 

Freely rotating cylinder. Now let us consider convective mass transfer to the surface of a circular cylinder freely suspended in an arbitrary linear shear Stokes flow (Re -> 0). In view of the no-slip condition, the cylinder rotates at a constant angular velocity equal to the angular velocity of the flow at infinity. The fluid velocity distribution is described by formulas (2.7.11). The streamline pattern qualitatively differs from that for the case of a fixed cylinder. For 0 0, there are no stagnation points on the surface of the cylinder and there exist two qualitatively different types of flow. For 0 < Ifigl < 1, there are both closed and open streamlines in the flow, the region filled with closed streamlines is adjacent to the surface of the cylinder, and streamlines far from the cylinder are open (Figure 2.11). For Ifl l > 1, all streamlines are open. 

Given the availability of hollow fiber membranes adequately permeable to substrates and products, and the control of fluid flow all around the fibers in the bundle in order to assure uniform flow distribution and to avoid stagnation (in order to reduce mass transfer diffusional resistances), the technique offers several advantages. Enzyme proteins can be easily retained within the core of the fibers with no deactivation due to coupling agents or to shear stresses, and the enzyme solution can be easily recovered and/or recycled. 

J. Stevens and B. W. Webb, Measurements of Flow Structure in the Stagnation Zone of Impinging Free-Surface Jets, Int. J. Heat Mass Transfer, 36, pp. 4283-4286,1993. 

When the fluid approaches the sphere from above, the fluid initially contacts the sphere at 0 = 0 (i.e., the stagnation point) because polar angle 6 is defined relative to the positive z axis. This is convenient because the mass transfer boundary layer thickness Sc is a function of 6, and 5c = 0 at 0 = 0. In the laminar and creeping flow regimes, the two-dimensional fluid dynamics problem is axisymmetric (i.e., about the z axis) with... 

Figure 11-1 Thickness of the mass transfer boundary layer around a solid sphere, primarily in the creeping flow regime. This graph in polar coordinates illustrates 8c 9) divided by the sphere diameter vs. polar angle 9, and the fluid approaches the solid sphere horizontally from the right. No data are plotted at the stagnation point, where 9=0.

These factors create prerequisites for impurities accumulation in the zones with limited mass transfer to the main sodium flow, namely surface of cover gas plenum of the reactor and possible stagnation zones. At the same time, corrosion products would deposit on the non-isothermal surfaces of the intermediate heat exchanger until the certain moment determined by the critical thickness of deposits layer. 

The Stanton tube is a rectangular-shaped pitot tube located very close to the boundary wall, and the mean velocity measured from this pitot tube pressure difference is directly related to the shear stress. The Preston tube is similar to the concept of the Stanton tube using a pitot static tube close to the surface, and the difference between the stagnation pressure at the center of the tube from the static pressure is related to the shear stress. The electrochemical or mass transfer probe is flush mounted with the wall, and the concentration at the wall element is maintained constant. The measurement of mass transfer rate between the fluid and the wall element is used for determination of the wall shear stress. One of the limitations of the mass transfer probe is that at very high flow rates, the mass transfer rate becomes large and it may not be possible to maintain the wall concentration constant. A detailed discussion on the above three techniques can be found in Hanratty and Campbell [1]. 

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