Turbulent-Boundary-Layer Thickness

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. 

Boundary layer flows are a special class of flows in which the flow far from the surface of an object is inviscid, and the effects of viscosity are manifest only in a thin region near the surface where steep velocity gradients occur to satisfy the no-slip condition at the solid surface. The thin layer where the velocity decreases from the inviscid, potential flow velocity to zero (relative velocity) at the sohd surface is called the boundary layer The thickness of the boundary layer is indefinite because the velocity asymptotically approaches the free-stream velocity at the outer edge. The boundaiy layer thickness is conventionally t en to be the distance for which the velocity equals 0.99 times the free-stream velocity. The boundary layer may be either laminar or turbulent. Particularly in the former case, the equations of motion may be simphfied by scaling arguments. Schhchting Boundary Layer Theory, 8th ed., McGraw-HiU, New York, 1987) is the most comprehensive source for information on boundary layer flows. 

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... 

The precise transition from laminar to turbulent flow occurs at different values of Re depending on geometry. Even in turbulent flow there exists a thin laminar hydrodynamic sublayer of thickness 8h near the metal surface. If mass transport is also occurring at the surface, there will be a diffusional boundary layer of thickness 8d. 8h is a function of v while 8d is a function of D. The Schmidt number quantifies a relationship between these two parameters ... 

The very shallow ( 1 txm) layer of air in contact with the surfaces, where stream lines for air-flow are approximately parallel to the surface and turbulence is suppressed by the viscosity of the air, is known as the laminar boundary layer. The thickness of this layer is determined by physical characteristics of the object (size, shape and surface structure) and by the velocity of air outside the laminar boundary layer. Although gases and small particles are transported by the same mechanism and at about the same rates in the free atmosphere, their transport through the laminar boundary layer differs considerably. Gases must diffuse through this layer by molecular diffusion, the concentration gradient and diffusion coefficient for the gas in question controlling the rate of transfer. 

At very low pH, the rate of dissolution is so fast that the rate is limited by the transport of the reacting species between the bulk of the solution and the surface of the mineral (Berner and Morse, 1974). The rate can then be described in terms of transport (molecular or turbulent diffusion) of the reactants and products through a stagnant boundary layer. The thickness of this layer depends on the stirring and the local turbulence. 

A later study by Hancock [142] revealed a significant dependence of the boundary layer momentum thickness on the free-stream turbulence level. Consequently, significant effects were observed when skin friction data were compared at the same momentum-thickness... 

Reynolds number Ree. The correlated data are shown in Fig. 6.50, where the increase in the skin friction is plotted versus a parameter that accounts for both the intensity and the scale of the free-stream turbulence. The effect of free-stream turbulence is primarily in the outer region of the boundary layer, where the law of the wake is modified, and, hence, the boundary layer integral thicknesses are modified. 

In convective heat transport heat transfer takes place via a fluid flowing on a wall. If the fluid flows solely as the result of buoyancy forces, then this is known as free convection, as opposed to forced convection, for example, by pumps or compressors. Since effective heat exchange is only possible in the case of turbulent flow in or around the tube, the heat transfer can be described by a two-film model, in which the wall is regarded as a hydrodynamic boundary layer of thickness 8 and a heat-transfer coefficient a is defined (Equation 2.3.1-10) ... 

However, the transition Reynolds number depends on free-stream turbulence and may range from 3 X 10 to 3 X lO ". The laminar boundary layer thickness 8 is a function of distance from the leading edge ... 

This form of attack, especially as affecting copper alloys in sea water, has been widely studied since the pioneer work of Bengough and May . Impingement attack of sea water pipe and heat exchanger systems is considered in Sections 1.6 and 4.2. In such engineering systems the water flow is invariably turbulent and the thickness of the laminar boundary layer is an important factor in controlling localised corrosion. 

When a fluid flowing at a uniform velocity enters a pipe, the layers of fluid adjacent to the walls are slowed down as they are on a plane surface and a boundary layer forms at the entrance. This builds up in thickness as the fluid passes into the pipe. At some distance downstream from the entrance, the boundary layer thickness equals the pipe radius, after which conditions remain constant and fully developed flow exists. If the flow in the boundary layers is streamline where they meet, laminar flow exists in the pipe. If the transition has already taken place before they meet, turbulent flow will persist in the... 

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