Fluid laminar layer thickness

For turbulent flow of a fluid past a solid, it has long been known that, in the immediate neighborhood of the surface, there exists a relatively quiet zone of fluid, commonly called the Him. As one approaches the wall from the body of the flowing fluid, the flow tends to become less turbulent and develops into laminar flow immediately adjacent to the wall. The film consists of that portion of the flow which is essentially in laminar motion (the laminar sublayer) and through which heat is transferred by molecular conduction. The resistance of the laminar layer to heat flow will vaiy according to its thickness and can range from 95 percent of the total resistance for some fluids to about I percent for other fluids (liquid metals). The turbulent core and the buffer layer between the laminar sublayer and turbulent core each offer a resistance to beat transfer which is a function of the turbulence and the thermal properties of the flowing fluid. The relative temperature difference across each of the layers is dependent upon their resistance to heat flow. 

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... 

In this approach, it is assumed that turbulence dies out at the interface and that a laminar layer exists in each of the two fluids. Outside the laminar layer, turbulent eddies supplement the action caused by the random movement of the molecules, and the resistance to transfer becomes progressively smaller. For equimolecular counterdiffusion the concentration gradient is therefore linear close to the interface, and gradually becomes less at greater distances as shown in Figure 10.5 by the full lines ABC and DEF. The basis of the theory is the assumption that the zones in which the resistance to transfer lies can be replaced by two hypothetical layers, one on each side of the interface, in which the transfer is entirely by molecular diffusion. The concentration gradient is therefore linear in each of these layers and zero outside. The broken lines AGC and DHF indicate the hypothetical concentration distributions, and the thicknesses of the two films arc L and L2. Equilibrium is assumed to exist at the interface and therefore the relative positions of the points C and D are determined by the equilibrium relation between the phases. In Figure 10.5, the scales are not necessarily the same on the two sides of the interface. 

HARRIOTT 25 suggested that, as a result of the effects of interfaeial tension, the layers of fluid in the immediate vicinity of the interface would frequently be unaffected by the mixing process postulated in the penetration theory. There would then be a thin laminar layer unaffected by the mixing process and offering a constant resistance to mass transfer. The overall resistance may be calculated in a manner similar to that used in the previous section where the total resistance to transfer was made up of two components—a Him resistance in one phase and a penetration model resistance in the other. It is necessary in equation 10.132 to put the Henry s law constant equal to unity and the diffusivity Df in the film equal to that in the remainder of the fluid D. The driving force is then CAi — CAo in place of C Ao — JPCAo, and the mass transfer rate at time t is given for a film thickness L by ... 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

Continuous Cylindrical Surface The continuous surface shown in Fig. 6-48fe is applicable, for example, for a wire drawn through a stagnant fluid (Sakiadis, AIChE J., 7, 26-28, 221-225, 467- 72 [1961]). The critical-length Reynolds number for transition is Re, = 200,000. The laminar boundary layer thickness, total drag, and entrainment flow rate may be obtained from Fig. 6-49 the drag and entrainment rate are obtained from the momentum area 0 and displacement area A evaluated at x = L. 

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 ... 

The thickness of the velocity boundary layer is normally defined as the distance from the solid body to the fluid layer at which the flow velocity reaches 99% of the free stream velocity, as illustrated in Figure 2.17. For a flat-plate body emerging in an incompressible and laminar fluid, the boundary layer thickness is given by... 

There are insnfficient data in the literatnre to provide a reliable estimate of the effect of roughness on friction loss for non-Newtonian flnids in tnrbnlent flow. However, the influence of roughness is normally neglected, since the laminar bonndary layer thickness for such fluids is typically much larger than for Newtonian fluids (i.e., the flow conditions most often fall in the hydraulically smooth range for common pipe materials). An expression by Darby et al. (1992) for / for the power law flnid, which applies to both laminar and turbulent flow, is... 

The concentration gradients in an asymmetric membrane are complex because the driving force for diffusion in the skin layer is the concentration gradient of gas dissolved in the dense polymer, and the driving force in the porous support layer is a concentration or pressure gradient in the gas-filled pore. When the porous layer is thick, diffusion does not contribute very much to the flux, and gas flows by laminar flow in the tortuous pores. For high-flux membranes, there may also be significant mass-transfer resistances in the fluid boundary layers on both sides. 

The principle of FFF can be explained best with the aid of Fig. 7.1. A lateral field acting across a narrow channel, composed usually of two planparallel walls, interacts with molecules or particles of a solute and compresses them to one of the channel walls in the direction of x-axis perpendicular to this wall. Hence a concentration gradient is established in the direction of the x-axis. This concentration gradient induces a diffusion flow in the reverse direction. After a certain time a steady state has been reached and the distribution of the solute across the channel can be characterized by a mean layer thickness /. At a laminar isothermal flow of a Newtonian fluid along a narrow channel, usually a parabolic velocity profile is formed inside the channel. It means that the molecules or the particles of the solute are transported in the direction of the longitudinal axis of the channel at varying... 

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... 

The thickness of the diffusion layer is strongly related to the fluid flow conditions and only in rather simple cases 6 can be calculated as function of characteristic hydrodynamic parameters. An order of magnitude is about one tenth of the hydrodynamic laminar boundary layer thickness. Because the diffusion layer is very small, one can totalize the phenomena occurring in it and suppose that the diffusion layer belongs to the electrode. This will lead to the concept of concentration overpotential (see section 1.ii.3). 

The values of the laminar boimdary layer thickness and of the frictional drag are not very sensitive to the form of approximations used for the velocity distribution, as illustrated by Skelland [1967] for various choices of velocity proflles. The resulting values of C(n) are compared in Table 7.1 with the more reflned values obtained by Acrivos et al. [1960] who solved the differential momentum and mass balance equations numerically the two values agree within 10% of each other. Schowalter [1978] has discussed the extension of the laminar boimdary layer analysis for power-law fluids to the more complex geometries of two- and three-dimensional flows. 

From this equation, the dependency of the mass transfer coefficient yff,- on the diffusion coefficient D, and the boundary layer thickness d of the fluid flow, may be seen. The laminar boundary layer and turbulent bulk cannot be distinguished exactly, due to the continous transition the boundary layer thickness 3 is, therefore, a formal complementary variable. 

Mass transfer coefficients are the basis for models where the dissolved species are transported by a combination of diffusive and advective processes. The diffusive mass transfer coefficient ko, m/sec) is based on boundary layer theory. The basic premise of boundary layer theory is that, for laminar ffow, the ffuid velocity adjacent to a solid surface is zero (the no slip condition ) and the velocity increases as a parabolic function of distance away from the surface until it matches the velocity of the bulk fluid (Figure 7.5). This means that there is a thin layer of fluid with a thickness of 5d (m) adjacent to the surface that is effectively static. The rate of mass transport through this layer is limited by the diffusion rate of the dissolved species. The diffusional boundary layer is much thinner than the velocity boundary layer. For laminar flow past a flat surface, the thickness of the diffusional boundary layer is related to the thickness of the velocity boundary layer (Sy) by the Schmidt number, which compares the fluid viscosity to the diffusivity (Probstein, 1989). 

Figure 7.5. The velocity boundary layer thickness (S)/) snd the diffusional boundary layer thickness So) for laminar flow past a flat surface.The diffusional boundary layer is approximated as a thin layer of static fluid at the solid surface where only diffusional mass transport occurs. See discussions in Probstein (1989), Denny (1993), and Vogel (1994) for more details.

It is further of interest to note that in the particular case of thin laminar liquid layers in channel flow, the liquid layer thickness (as appears in Fr, Equation 27), can be obtained explicitly by solving the steady-state two-fluid momentum equations. In this case, the transitional criterion for wind generated waves, > 1 reduces to a critical superficial gas phase Reynolds number, Re > 1.113 x 10 [103]. 

If the velocity profile at the entrance region of a tube is flat, a certain length of the tube is necessary for the velocity profile to be fully established. This length for the establishment of fully developed flow is called the transition length or entry length. This is shown in Fig. 2.10-6 for laminar flow. At the entrance the velocity profile is flat i.e., the velocity is the same at all positions. As the fluid progresses down the tube, the boundary-layer thickness increases until finally they meet at the center of the pipe and the parabolic velocity profile is fully established. 

Since the Stokes layer thickness S itself depends on the flow frequency, transition to turbulence depends on the Womersley number as well as the Reynolds number. Experimental measurements of the velocity made in rigid tubes by noninvasive optical techniques [21] have shown that over a range of values of a > 8, the flow was found to be fuUy laminar for Re < 500, where Re is the Reynolds number based on the Stokes layer thickness rather than tube diameter. That is, Re = US/v. For 500 < Rej < 1300, the core flow remained laminar while the Stokes layer became unstable during the deceleration phase of fluid motion. This turbulence was most intense in an annular region near the tube wall. These results are in accord with theoretical predictions of instabilities in Stokes layers [22,23]. For higher values of Rej, instability can be expected to spread across the tube core. 

The above equation indicates that c is a constant following a fluid particle. Suppose we have c = Cq in the free stream and the wall boundary condition for a reacting surface is = 0. The solution of equation (4.75) cannot satisfy the boundary conditions at the reaction surface. Evidently, near the surface, there must be a thin diffusion boundary layer of thickness djy within which the concentration changes rapidly (see Figure4.15). This reasoning parallels the Prandtl boundary layer argument for viscous flow past a solid boundary at high Reynolds number. If is the Prandtl viscous boundary layer thickness for steady unbounded laminar flow, we know that... 

At 700°C and 1 atm this leads to a diffusion constant of 0.81 cm /s The flow field around a superheater tube is very complex involving both laminar and turbulent boundary layers and the estimation of the local boundary layer thicknesses (velocity, diffusion and thermal boundary layers) around the tube requires computer simulations with computational fluid dynamic (CFD) software packages. However, for this rough analysis an average value of the thermal boundary layer thickness around the tube is enough and can be estimated if the average Nusselt number around the tube is known... 

External Fluid Film Resistance. A particle immersed ia a fluid is always surrounded by a laminar fluid film or boundary layer through which an adsorbiag or desorbiag molecule must diffuse. The thickness of this layer, and therefore the mass transfer resistance, depends on the hydrodynamic conditions. Mass transfer ia packed beds and other common contacting devices has been widely studied. The rate data are normally expressed ia terms of a simple linear rate expression of the form... 

As velocity continues to rise, the thicknesses of the laminar sublayer and buffer layers decrease, almost in inverse proportion to the velocity. The shear stress becomes almost proportional to the momentum flux (pk ) and is only a modest function of fluid viscosity. Heat and mass transfer (qv) to the wall, which formerly were limited by diffusion throughout the pipe, now are limited mostly by the thin layers at the wall. Both the heat- and mass-transfer rates are increased by the onset of turbulence and continue to rise almost in proportion to the velocity. 

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. 

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... 

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