Prandtl boundary layer

Typically, the effects of the viscous forces originate at the solid boundary of the body of fluid. The fluid contained in the region of substantial velocity change is called the hydrodynamic boundary layer, Prandtl, 1904 [2]. Similarly, if the fluid and the solid are at different temperatures, the region of substantial temperature change in the fluid is... 

Further, empirical and semi-empirical relations or calculations of boundary layers (Prandtl s equations)... 

Initially it was assumed that no solution movement occurs within the diffusion layer. Actually, a velocity gradient exists in a layer, termed the hydrodynamic boundary layer (or the Prandtl layer), where the fluid velocity increases from zero at the interface to the constant bulk value (U). The thickness of the hydrodynamic layer, dH, is related to that of the diffusion layer ... 

The velocity of the fluid may be assumed to obey the Prandtl one seventh power law, given by equation 11.26. If the boundary layer thickness S is replaced by the pipe radius r, this is then given by ... 

It will be shown that the momentum and thermal boundary layers coincide only if the Prandtl number is unity, implying equal values for the kinematic viscosity (p./p) and the thermal diffusivity (DH = k/Cpp). 

In the Taylor-Prandtl modification of the theory of heat transfer to a turbulent fluid, it was assumed that the heat passed directly from the turbulent fluid to the laminar sublayer and the existence of the buffer layer was neglected. It was therefore possible to apply the simple theory for the boundary layer in order to calculate the heat transfer. In most cases, the results so obtained are sufficiently accurate, but errors become significant when the relations are used to calculate heat transfer to liquids of high viscosities. A more accurate expression can be obtained if the temperature difference across the buffer layer is taken into account. The exact conditions in the buffer layer are difficult to define and any mathematical treatment of the problem involves a number of assumptions. However, the conditions close to the surface over which fluid is flowing can be calculated approximately using the universal velocity profile,(10)... 

For a Prandtl number, Pr. less than unity, the ratio of the temperature to the velocity boundary layer thickness is equal to Pr 1Work out the thermal thickness in terms of the thickness of the velocity boundary layer... 

Fluid flow and reaction engineering problems represent a rich spectrum of examples of multiple and disparate scales. In chemical kinetics such problems involve high values of Thiele modulus (diffusion-reaction problems), Damkohler and Peclet numbers (diffusion-convection-reaction problems). For fluid flow problems a large value of the Mach number, which represents the ratio of flow velocity to the speed of sound, indicates the possibility of shock waves a large value of the Reynolds number causes boundary layers to be formed near solid walls and a large value of the Prandtl number gives rise to thermal boundary layers. Evidently, the inherently disparate scales for fluid flow, heat transfer and chemical reaction are responsible for the presence of thin regions or "fronts in the solution. 

In the flow, the thin layer of liquid that is directly adjacent to the solid is retained by molecular forces and does not move. The liquid s velocity relative to the sohd increases from zero at the very surface to the bulk value v which is attained some distance away from the surface. The zone within which the velocity changes is called the Prandtl or hydrodynamic boundary layer. 

The second approach assigns thermal resistance to a gaseous boundary layer at the heat transfer surface. The enhancement of heat transfer found in fluidized beds is then attributed to the scouring action of solid particles on the gas film, decreasing the effective film thickness. The early works of Leva et al. (1949), Dow and Jacob (1951), and Levenspiel and Walton (1954) utilized this approach. Models following this approach generally attempt to correlate a heat transfer Nusselt number in terms of the fluid Prandtl number and a modified Reynolds number with either the particle diameter or the tube diameter as the characteristic length scale. Examples are ... 

An important concept in fluid mechanics is the hydrodynamic boundary layer (also known as Prandtl layer) or region where the effective disturbance... 

Figure 4 Hydrodynamic boundary layer development on the semi-infinite plate of Prandtl. <5D = laminar boundary layer, <5t = turbulent boundary layer, /vs = viscous turbulent sub-layer, <5ds = diffusive sub-layer (no eddies are present solute diffusion and mass transfer are controlled by molecular diffusion—the thickness is about 1/10 of <5vs)> B = point of laminar—turbulent transition. Source From Ref. 10.

Originally, the concept of the Prandtl boundary layer was developed for hydraulically even bodies. It is assumed that any characteristic length L on the particle surface is much greater than the thickness (<5hl) of the boundary layer itself (L > Ojil) Provided this assumption is fulfilled, the concept can be adapted to curved bodies and spheres, including real drug particles. Furthermore, the classical ( macroscopic ) concept of the hydrodynamic boundary layer is valid solely for high Reynolds numbers of Re>104 (14,15). This constraint was overcome for the microscopic hydrodynamics of dissolving particles by the convective diffusion theory (9). 

The Schmidt number is the ratio of kinematic viscosity to molecular diffusivity. Considering liquids in general and dissolution media in particular, the values for the kinematic viscosity usually exceed those for diffusion coefficients by a factor of 103 to 104. Thus, Prandtl or Schmidt numbers of about 103 are usually obtained. Subsequently, and in contrast to the classical concept of the boundary layer, Re numbers of magnitude of about Re > 0.01 are sufficient to generate Peclet numbers greater than 1 and to justify the hydrodynamic boundary layer concept for particle-liquid dissolution systems (Re Pr = Pe). It can be shown that [(9), term 10.15, nomenclature adapted]... 

K° consists of a combination of Prandtl s original proportionality constant used for the hydrodynamic boundary layer at a semi-infinitive plate, Ke, and a constant, K, characterizing a particular hydrodynamic system that is under consideration. The latter constant has to be determined experimentally. 

Ludwig Prandtl introduced the concept of boundary layers in 1904. Since they play a large role in determining the parameters of fluid flow, they have been extensively studied. A boundary layer is that region near a surface where the fluid flow is dominated by the presence of the surface. The fluid cannot flow through the surface but there is always some attraction between the molecules of the fluid and those of the surface, the surface tension effect. In addition, at low velocities, the viscous forces in the fluid dominate the kinetic forces. Therefore, the fluid immediately adjacent to the surface is restrained in its normal tendency to move with the rest of the fluid. The result of this restraint is a velocity gradient. The velocity increases from effectively zero at the surface to the nominal fluid velocity at some distance away. 

The gas film coefficient is dependent on turbulence in the boundary layer over the water body. Table 4.1 provides Schmidt and Prandtl numbers for air and water. In water, Schmidt and Prandtl numbers on the order of 1,000 and 10, respectively, results in the entire concentration boundary layer being inside of the laminar sublayer of the momentum boundary layer. In air, both the Schmidt and Prandtl numbers are on the order of 1. This means that the analogy between momentum, heat, and mass transport is more precise for air than for water, and the techniques apphed to determine momentum transport away from an interface may be more applicable to heat and mass transport in air than they are to the liquid side of the interface. 

In developing the boundary layer concept, Prandtl suggested an order-of-magnitude evaluation of the terms in the Navier-Stokes equations, which provides an expression (with an unknown proportionality constant) for the... 

As suggested by Prandtl, the entire zone of motion can be subdivided into two regions a boundary layer region near the plate of thickness 6h = <5,(x), in... 

Regarding the heat transfer coefficient, let us write that the thickness 8 of the thermal boundary layer is related, for sufficiently large Prandtl numbers, to that of the hydrodynamic boundary layer by... 

It is difficult to solve the system of Eqs. (39)—(41) for these boundary conditions. However, certain simplifying assumptions can be made, if the Prandtl number approaches large values. In this case, the thermal boundary layer becomes very thin and, therefore, only the fluid layer near the plate contributes significantly to the heat transfer resistance. The velocity components in Eq. (41) can then be approximated by the first term of their Taylor series expansions in terms of y. In addition, because the nonlinear inertial terms are negligible near the wall, one can further assume that the combined forced and free convection velocity is approximately equal to the sum of the velocities that would exist when these effects act independently. Therefore, for assisting flows at large Prandtl numbers (theoretically for Pr -> oo), Eq. (41) can be rewritten in the form ... 

As with other matters concerned with transport to electrodes, detailed treatments were set up very early. Various boundary layers at interfaces under flow were suggested by Prandtl as early as 1904. Three are shown in Fig. (7.93). The 8y is die well-known diffusion layer due to Nernst (Section 7.9.9). The 8 is the thermal boundary layer and the 8V signifies the thickness of the layer (Prandtl s layer) in a flowing liquid in which the velocity slows an approach peipendicular to the surface. 

The structure of turbulence in the transition zone from a fully turbulent fluid to a nonfluid medium (often called the Prandtl layer) has been studied intensively (see, for instance, Williams and Elder, 1989). Well-known examples are the structure of the turbulent wind field above the land surface (known as the planetary boundary layer) or the mixing regime above the sediments of lakes and oceans (benthic boundary layer). The vertical variation of D(x) is schematically shown in Fig. 19.8b. Yet, in most cases it is sufficient to treat the boundary as if D(x) had the shape shown in Fig. 19.8a. 

Fig. 6.10 Nondimensional axial-velocity gradients and scaled radial velocities at the viscous boundary-layer edge as a function of Reynolds number in a finite-gap stagnation flow. The Prandtl number is Pr = 0.7 and the flow is isothermal in all cases. The outer edge of the boundary layer is defined in two ways. One is the z position of maximum V velocity and the other is the z. at which T — 0.01. As Re - oo, du/dz - —2 and V — 1, which are the values in the inviscid semi-infinite stagnation flow regions.

There is a natural draw rate for a rotating disk that depends on the rotation rate. Both the radial velocity and the circumferential velocity vanish outside the viscous boundary layer. The only parameter in the equations is the Prandtl number in the energy equation. Clearly, there is a very large effect of Prandtl number on the temperature profile and heat transfer at the surface. For constant properties, however, the energy-equation solution does not affect the velocity distributions. For problems including chemistry and complex transport, there is still a natural draw rate for a given rotation rate. However, the actual inlet velocity depends on the particular flow circumstances—there is no universal correlation. 

Fig. 6.16 Nondimensional velocity and temperature profiles in a finite gap with a rotating surface. In all cases the Prandtl number is 0.7 and the forced-flow Reynolds number is Rey = 100. The profiles are illustrated for four values of the rotation Reynolds number Re = G1L2/v. The viscous boundary layers are close to the surface. With the exception of the axial velocity, the plots show the range 0 < z < 0.2, with the small insets illustrating the entire gap 0 < z < 1.

Determine the effect of Prandtl number on the thermal boundary-layer thickness. Consider the range 1 < Pr < 100. 

The expressions given by Eqns. (3.4-9) correspond to the theory of the boundary layer. Similar expressions are obtained with different theories. In practical work, expressions of the type given below are used for different arrangements. Generally, the exponent of the Reynolds number is less than unity, while the exponent of the Schmidt and Prandtl numbers has been kept as 1/3. The usual expressions for the Nusselt and Sherwood numbers are ... 

In many respects, similar to the diffusion layer concept, there is that of the hydrodynamic boundary layer, <5H. The concept was due originally to Prandtl [16] and is defined as the region within which all velocity gradients occur. In practice, there has to be a compromise since all flow functions tend to asymptotic limits at infinite distance this is, to some extent, subjective. Thus for the rotating disc electrode, Levich [3] defines 5H as the distance where the radial and tangential velocity components are within 5% of their bulk values, whereas Riddiford [7] takes a figure of 10% (see below). It has been shown that... 

Keulegan (K13) applied the semiempirical boundary-layer concepts of Prandtl and von K arm an to the case of turbulent flow in open channels, taking into account the effects of channel cross-sectional shape, roughness of the wetted walls, and the free surface. Most of the results are applicable mainly to deep rough channels and bear little relation to the flow of thin films. 

Many of the phenomena of the boundary layer are explainable on the basis of the theory advanced by Prandtl al the University of Gottingen laboratory nearly half a century ago. In the same flow-research group were others, like Blasius, who broadened and experimentally confirmed the original hypotheses. 

The flow sensitivity of the electrode has the same origin, as has been pointed out previously. A stagnant (Prandtl) boundary layer of thickness 5 forms around the spherical electrode (radius ro) placed in the liquid of kinematic viscosity v which is moving with linear velocity U. 

A comparison of Eqs. (3.52) and (3.53) and also of their boundary conditions as given in Eqs. (3.24) and (3.54) respectively, shows that these equations are identical in all respects. Therefore, for the particular case of Pr equal to one, the distribution of 9 through the boundary layer is identical to the distribution of uJu ). In this par-ticular case, therefore, Fig. 3.4 also gives the temperature distribution and the two boundary layer thicknesses are identical in this case. Now many gases have Prandtl numbers which are not very different from 1 and this relation between the velocity and temperature fields and the results deduced from it will be approximately correct for them. 

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