Boundary layer, laminar, natural

I. Turbulent, local flat plate, natural convection, vertical plate Turbulent, average, flat plate, natural convection, vertical plate Nsk. = — = 0.0299Wg=Ws = D x(l + 0.494W ) )- = 0.0249Wg=W2f X (1 + 0.494WE )- [S] Low solute concentration and low transfer rates. Use arithmetic concentration difference. Ncr > 10 " Assumes laminar boundary layer is small fraction of total. D [151] p. 225... 

In natural waters, unattached microorganisms move with the bulk fluid [55], so that no flux enhancement will occur due to fluid motion for the uptake of typical (small) solutes by small, freely suspended microorganisms [25,27,35,41,56,57], On the other hand, swimming and sedimentation have been postulated to alleviate diffusive transport limitation for larger organisms. Indeed, in the planar case (large r0), the diffusion boundary layer, 8, has been shown to depend on advection and will vary with D according to a power function of Da (the value of a is between 0.3 and 0.7 [43,46,58]). For example, in Chapter 3, it was demonstrated that in the presence of a laminar flow parallel to a planar surface, the thickness of the diffusion boundary layer could be estimated by ... 

Apart from the nature of the bulk flow, the hydrodynamic scenario close to the surfaces of drug particles has to be considered. The nature of the hydrodynamic boundary layer generated at a particle s surface may be laminar or turbulent regardless of the bulk flow characteristics. The turbulent boundary layer is considered to be thicker than the laminar layer. Nevertheless, mass transfer rates are usually increased with turbulence due to the presence of the viscous turbulent sub-layer. This is the part of the (total) turbulent boundary layer that constitutes the main resistance to the overall mass transfer in the case of turbulence. The development of a viscous turbulent sub-layer reduces the overall resistance to mass transfer since this viscous sub-layer is much narrower than the (total) laminar boundary layer. Thus, mass transfer from turbulent boundary layers is greater than would be calculated according to the total boundary layer thickness. 

In electrochemical reactors, the externally imposed velocity is often low. Therefore, natural convection can exert a substantial influence. As an example, let us consider a vertical parallel plate reactor in which the electrodes are separated by a distance d and let us assume that the electrodes are sufficiently distant from the reactor inlet for the forced laminar flow to be fully developed. Since the reaction occurs only at the electrodes, the concentration profile begins to develop at the leading edges of the electrodes. The thickness of the concentration boundary layer along the length of the electrode is assumed to be much smaller than the distance d between the plates, a condition that is usually satisfied in practice. 

Now return to a view of the nature of flow in the boundary layer. It has been called laminar, and so it is for values of the Reynolds number below a critical value. But for years, beginning about the time of Osborne Reynolds experiments and revelations in the field of fluid flow, it has been known that the laminar property disappears, and the flow suddenly becomes turbulent, when the critical VUv is reached. Usually flow starts over a surface as laminar but after passing over a suitable length the boundary layer becomes turbulent, with a thin laminar sublayer thought to exist because of damping of normal turbulent components at the surface. See Fig. 6. 

A numerical solution to the laminar boundary layer equations for natural convection can be obtained using basically the same method as applied to forced convection in Chapter 3. Because the details are similar to those given in Chapter 3, they will not be repeated here. 

Some of the more commonly used methods of obtaining solutions to problems involving natural convective flow have been discussed in this chapter. Attention has been given to laminar natural convective flows over the outside of bodies, to laminar natural convection through vertical open-ended channels, to laminar natural convection in a rectangular enclosure, and to turbulent natural convective boundary layer flow. Solutions to the boundary layer forms of the governing equations and to the full governing equations have been discussed. 

Metzner and Friend [Ind. Eng. Chem., 51, 879 (1959)] present relationships for turbulent heat transfer with nonnewtonian fluids. Relationships for heat transfer by natural convection and through laminar boundary layers are available in Skellands book (op. cit.). 

In contrast, stabilization of boundary layer by contouring the airfoil surface to achieve favourable pressure gradient as a passive way is found to be practical and attractive. The resultant section is known as the Natural Laminar Flow (NLF) airfoil and this is an area which has been under renewed investigation over the last three decades. Early efforts of designing... 

In this section we derive the equation of motion that governs the natural convection flow in laminar boundary layer. The conservation of mass and energy equations derived in Chapter 6 for forced convection are also applicable for natural convection, but tlie momentum equation needs to be modified to incorporate buoyancy. 

Consider a vertical hot flat plate immersed in a quiescent fluid body. We assume the natural convection flow to be steady, laminar, and two-dimensional, and the fluid to be Newtonian with constant properties, including density, with one exception the density difference p — is to be considered since it is this density difference between the inside and the outside of the boundary layer that gives rise to buoyancy force and sustains flow. (This is known as the Boussines.q approximation.) We take the upward direction along the plate to be X, and the direction normal to surface to be y, as shown in Fig. 9-6. Therefore, gravelly acts in the —.t-direclion. Noting that the flow is steady and two-dimensional, the.t- andy-compoijents of velocity within boundary layer are II - u(x, y) and v — t/(.Y, y), respectively. 

The simulations were performed assuming that the flow is laminar. Additionally, the contact angle is assumed to be known. The initial velocity is assumed to be zero everywhere in the domain. The initial fluid temperature profile is taken to be linear in the natural convection thermal boundary layer and the thermal boundary layer thickness, 5j, is evaluated using the correlation for the turbulent natural convection on a horizontal plate as, Jj. =1. 4(vfiCil ... 

Gemmerich J, Hasse L (1992) Small-scale surface streaming under natural conditions as effective in air-sea gas exchange. Tellus 44B 150-159 Howarth L (1937) On the solution of the laminar boundary layer equations. Proc Royal Soc London A 164 547-579... 

In Chapter 5, we learned the foundations of convection. Integrating the governing equations for laminar boundary layers, we obtained expressions for the heat transfer associated with forced convection over a horizontal plate and natural convection about a vertical plate. We also found analytically, as well as by the analogy between heat and momentum, that the thermal and momentum characteristics of laminar flow over a flat plate are related by... 

It is understood that the economical success of any membrane process depends primarily on the quality of the membrane, specifically on flux, selectivity and service lifetime. Consideration of only the transport mechanisms in membranes, however, will in general, lead to an overestimation of the specific permeation rates in membrane processes. Formation of a concentration boundary layer in front of the membrane surface or within the porous support structure reduces the permeation rate and, in most cases, the product quality as well. For reverse osmosis. Figure 6.1 shows how a concentration boundary layer (concentration polarization) forms as a result of membrane selectivity. At steady state conditions, the retained components must be transported back into the bulk of the liquid. As laminar flow is present near the membrane surface, this backflow is of diffusive nature, i.e., is based on a concentration gradient. At steady state conditions, the concentration profile is calculated from a mass balance as... 

The equation for the laminar Nusselt number Nut is obtained in a two-step procedure. In the first step, not only is the flow idealized as everywhere laminar, but the boundary layer is treated as thin. There results from this idealization the equation for the laminar thin-layer Nusselt number Nur. As already explained, natural convection boundary layers are generally not thin, so the second step is to correct Nur to account for thick boundary layers. This correction uses the method of Langmuir [175]. The corrected Nusselt number is the laminar Nusselt number Nuc. 

The details of the flow in the mixed convection regime have been clarified by Gilpin et al. [113]. After an initial development of the laminar forced convection boundary layer, rolls with axes aligned with the flow appear at the location marked Onset in Fig. 46. These persist until the end of the transition regime, marked Breakup, after which the motion appears as fully detached turbulent natural convection flow. 

When a nondeformable object is implanted in the flow field and the streamlines and equipotentials are distorted, the nature of the interface does not affect the potential flow velocity profiles. However, the results should not be used with confidence near high-shear no-slip solid-liquid interfaces because the theory neglects viscous shear stress and predicts no hydrodynamic drag force. In the absence of accurate momentum boundary layer solutions adjacent to gas-liquid interfaces, potential flow results provide a reasonable estimate for liquid-phase velocity profiles in Ihe laminar flow regime. Hence, potential flow around gas bubbles has some validity, even though an exact treatment of gas-Uquid interfaces reveals that normal viscous stress is important (i.e., see equation 8-190). Unfortunately, there are no naturally occurring zero-shear perfect-slip interfaces with cylindrical symmetry. 

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