Boundary layers for flow

The development of the boundary layer for flow over a flat plate, and the different flow regimes. 

Hydrodynamic boundary layer for flow over a flat plate. 

Hydrodynamic and thermal boundary layers for flow over a solid surface. 

Boundary-layer development. The boundary layer that develops within an impeller passage causes the flowing fluid to experience a smaller exit area as shown in Figure 6-23. This smaller exit is due to small flow (if any) within the boundary layer. For the fluid to exit this smaller area, its velocity must increase. This increase gives a higher relative exit velocity. 

Figures 4.34 and 4.35 represent two extreme cases. Drying processes represent the case shown in Fig. 4.34 and distillation processes represent Fig. 4.35. Neither case represents a convective mass transfer case while the gas flow is in the boundary layer, other flows are Stefan flow and turbulence. Thus Eqs. (4.243) and (4.244) can seldom be used in practice, but their forms are used in determining the mass transfer factor for different cases.

This expression is applicable only to the region of fully developed flow. The heat transfer coefficient for the inlet length can be calculated approximately, using the expressions given in Chapter 11 for the development of the boundary layers for the flow over a plane surface. It should be borne in mind that it has been assumed throughout that the physical properties of the fluid are not appreciably dependent on temperature and therefore the expressions will not be expected to hold accurately if the temperature differences are large and if the properties vary widely with temperature. 

Derive the momentum equation for the flow of a fluid over a plane surface for conditions where the pressure gradient along the surface is negligible. By assuming a sine function for the variation of velocity with distance from the surface (within the boundary layer) for streamline flow, obtain an expression for the boundary layer thickness as a function of distance from the leading edge of the surface. 

The boundary layer thickness gradually increases until a critical point is reached at which there is a sudden thickening of the boundary layer this reflects the transition from a laminar boundary layer to a turbulent boundary layer. For both types, the flow outside the boundary layer is completely turbulent. In that part of the boundary layer near the leading edge of the plate the flow is laminar and consequently this is known as a... 

The radial velocity profile is linear and the circumferential velocity is zero outside the viscous boundary layer, which indicates that the vorticity is constant in that region. Thus, for substantial ranges of the flow and rotation Reynolds numbers, the flow is inviscid, but rotational, outside the viscous boundary layer. For sufficiently low flow, the boundary-layer can grow to fill the gap, eliminating any region of inviscid flow. 

Consider convection with incompressible, laminar flow of a constant-temperature fluid over a flat plate maintained at a constant temperature. With the velocity distributions found in either Prob. 10.1 or Prob. 10.2, compute the dimensionless temperature distribution within the thermal boundary layer for the Peclet number equal to 0.1,1.0,10.0,100.0. Use the ADI method. 

The way in which the momentum integral equation is applied will be discussed in detail in the next chapter. Basically, it involves assuming the form of the velocity profile, i.e., of the variation of u with y in the boundary layer. For example, in laminar flow a polynomial variation is often assumed. The unknown coefficients in this assumed form are obtained by applying the known condition on velocity at the inner and outer edges of the boundary layer. For example, the velocity must be zero at the wall while at the outer edge of the boundary layer it must become equal to the freestream velocity, u. Thus, two conditions that the assumed velocity profile must satisfy are ... 

Numerically determine the local Nusselt number variation with two-dimensional turbulent boundary layer air flow over an isothermal flat plate for a maximum Reynolds number of 107. Assume that transition occurs at a Reynolds number of 5 X 105. Compare the numerical results with those given by the Reynolds analogy. 

With this solution for the velocity components, the energy equation gives if a boundary layer-type flow is assumed ... 

The zones where these gradients occur are often called boundary layers. For example, the aerodynamic boundary layer is the region near a surface where viscous forces predominate. Boundary layers exist with both laminar and turbulent flow and may be either solely laminar or turbulent with a laminar sublayer themselves (Landau and Lifshitz, 1959). 

Above inviscid mechanism of instability is often encountered in free shear layers and jets. A fundamental difference between flows having an inflection point (such as in free shear layer, jets and wakes and the cross flow component of some three-dimensional boundary layers) and flows without inflection points (as in wall bounded flows in channel or in boundary layers) exists. Flows with inflection points are susceptible to temporal instabilities for very low Reynolds numbers. One can find detailed accounts of invis-... 

During laminar flow in a tube, the magnitude of the dimensionless Prandtl number Pr is a measure of the relative growth of the velocity and thermal boundary layers. For fluids with Pr = I, such as gases, the two boundary layers essentially coincide with each other. For fluids with Pr > I, such as oils, the velocity boundary layer outgrows the thermal boundary layer. As a result, the hydrodynamic entry length is smaller than the thermal entry length. The opposite is tnie for fluids with Pr < 1 such as liquid metals. 

The development of a concentration boundary layer for species A during external flow on a flat surface. 

Heat and mass transfer apparatus normally consist of channels, frequently tubes, in which a fluid is heated, cooled or changes its composition. While the boundary layers in flow over bodies, for example over a flat plate, can develop freely without influence from neighbouring restrictions, in channels it is completely enclosed and so the boundary layer cannot develop freely. In the following the flow, and then the heat and mass transfer in tubes will be discussed. After this we will study flow through packed and fluidised beds. 

Figure 11-5. A schematic representation showing the two-layer structure of the thermal-momentum boundary layer for large Pe, with Re 1 and Pr < C 1. In this limit, the distance required for 9 to approach its ffee-stream value is much larger than the distance required for the velocity to approach the potential-flow form. Thus convection within the thermal boundary layer is dominated by the outer limit of the momentum boundary-layer velocity distribution (or equivalently, the inner limit as we approach the body surface of the potential-flow distribution).

As Re increases, the adjacent vortices are elongated and shedding of vortices occurs (Karman s vortices are formed). Finally, for Re > 1000, the remote wake becomes completely turbulent [117]. At the same time, the separation point moves toward the midsection and even a bit farther upstream. For such values of Re, we can speak about a pronounced boundary layer. In a large part of the boundary layer, the flow remains laminar [486], Strong turbulence within the boundary layer occurs for considerably higher Reynolds numbers (Re 2x 105), at which the cylinder drag drops rapidly [117], This phenomenon is called the drag crisis. 

Figures 6.2 and 6.3 indicate the order of magnitude of concentration polarization for laminar and turbulent flows through tubular membranes. The diagrams illustrate the dependence of the concentration boundary layer on flow conditions along the membrane (Re) and on the permeation flux (Pew).

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