Laminar Sublayer Region

In this laminar sublayer region, the velocity of the fluid, v, is assumed to be linear with respect to x. At the interface of the two regions, x = 8, the velocities are equal, that is Eb = z x = s- The concentration of the transferring species, Ca, is expected to depend on both x and z. At the interface, Cb = Cp x = 5. 

Consider now, a mass balance in a shell of volume Ax Azw and length L over the laminar sublayer  

Division by AxAzw and allowing Ax and Az to go to zero in the limiting process results in 

FIGURE 1.2 Laminar sublayer. (From Huang, CAe/w.Eng.Senei. Vol 59,pp 1191-1198. With permission.) 

assuming that the mass flux of A in the z-directi(Mi is controlled by bulk flow 

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Equation (6-31) applies to the laminar sublayer region in a Newtonian fluid, which has been found to correspond to 0 < y+ < 5. The intermediate region, or buffer zone, between the laminar sublayer and the turbulent boundary layer can be represented by the empirical equation... 

Consider an alternative approach that does not rely on the knowledge of the eddy diffusivity as a function of the distance y from the wall. Here, we examine the mass transfer for a turbulent flowing fluid in a smooth tube. In the tube, a turbulent core region and a laminar sublayer region are considered separately as contributing to the total mass transfer of the transferring species from the fluid toward the wall as well as away from the wall. 

When the flow in the boundary layer is turbulent, streamline flow persists in a thin region close to the surface called the laminar sub-layer. This region is of particular importance because, in heat or mass transfer, it is where the greater part of the resistance to transfer lies. High heat and mass transfer rates therefore depend on the laminar sublayer being thin. Separating the laminar sub-layer from the turbulent part of the boundary... 

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. 

When a fluid is in turbulent flow past a rigid surface, fluctuations of velocity in the direction normal to the surface are inhibited, and very close to the surface they may he negligible. Then the Reynolds shear stress is small compared with the viscous stresses, and it has been common to describe the region as a laminar sublayer. In fact, turbulent fluctuations of velocity in planes parallel to the wall are considerable in comparison with the mean velocity. 

The presence of the solid wall has a considerable influence on the turbulence structure near the wall. Because there can be no flow normal to the wall near the wall, v decreases as the wall is approached and as a result the turbulent stress and turbulent heat transfer rate are negligible in the region very near the wall. This region in which the effects of the turbulent stress and turbulent heat transfer rate can be neglected is termed the sublayer or, sometimes, the laminar sublayer [1],[2], [26],[27],[28],[29]. In this sublayer ... 

In terms of our previous qualitative discussion, the laminar sublayer is the region where 0, the buffer layer has eM r, and the turbulent layer has cm v. Therefore, taking e,v = 0 in Eq. (5-69) and integrating yields... 

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). 

A boundary layer is a region of a fluid next to a solid that is dominated by the shearing stresses originating at the surface of the solid such layers arise for any solid in a fluid, such as a leaf in air. Adjacent to the leaf is a laminar sublayer of air (Fig. 7-6), where air movement is predominantly parallel to the leaf surface. Air movement is arrested at the leaf surface and has increasing speed at increasing distances from the surface. Diffusion... 

Figure 7-6. Schematic illustration of originally nonturbulent air (straight anrows in upwind side on left) flowing over the top of a flat leaf, indicating the laminar sublayer (shorter straight anrows), the turbulent region (curved arrows), and the effective boundary layer thickness, 5bl. The length of an arrow indicates the relative speed, and the curvature indicates the local direction of air movement. A similar airflow pattern occurs on the lower leaf surface.

Turbulent flow over a flat plate is characterized by three re-gions f l (a) a viscous sublayer often called the laminar sublayer, which exists right next to the plate, (b) an adjacent turbulent boundary layer, and (c) the turbulent core. Viscous forces dominate inertial forces in the viscous sublayer, which is relatively quiescent compared to the other regions and is therefore also called the laminar sublayer. This is a bit of a misnomer, since it is not really laminar. It is in this viscous sublayer that the velocity changes are the greatest, so that the shear is largest. Viscous forces become less dominant in the turbulent boundary layer. These forces are not controlling factors in the turbulent core. 

The velocity profile is replaced by a straight line in the laminar sublayer, b) In the completely turbulent region, a long distance away from the wall, y — 00 and therefore y+ — 00, the second term outweighs the first, and it holds that... 

In reality the laminar sublayer is continuously transformed into the fully turbulent region. A transition region exists between the two, known as the buffer layer, so that the wall law of velocity can be split into three areas, whose boundaries are set by experimentation. The laminar sublayer extends over the region... 

The near-wall region is composed of three layers as shown in Fig. 6.31. The layer immediately adjacent to the surface (y+ 5) is called the laminar sublayer where, because of the presence of the surface, the turbulence has been damped into a fluctuating laminar flow. In this layer, viscosity predominates over the eddy viscosity, and the velocity distribution may be approximated by... 

Early theories for transpiration of air into air [114, 115] were based on the Couette flow approximation. Reference 114 extended the Reynolds analogy to include mass transfer by defining a two-part boundary layer consisting of a laminar sublayer and a fully turbulent core. Here, t = 0 in the sublayer (y < y ), and t = OAy and (i = 0 in the fully turbulent region. The density was permitted to vary with temperature. The effect of foreign gas injection in a low-speed boundary layer was studied in Ref. 116, and all these theories were improved upon in Ref. 117. 

The large values indicate a fully turbulent region. We expect that du fdx2 should be independent of V in this region, since the thickness of the laminar sublayer, proportional to v/m, is on the order of 0.01 cm. Thus we can set F2(x3U. /v) = constant = a, and... 

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