Prandtl boundary layer/thickness

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 ... 

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... 

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). 

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

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. 

Variation of boundary layer thickness ratio A with Prandtl number. 

Let us first consider the simple flat plate with a liquid metal flowing across it. The Prandtl number for liquid metals is very low, of the order of 0.01. so that the thermal-boundary-layer thickness should be substantially larger than the hydrodynamic-boundary-layer-thickness. The situation results from the high values of thermal conductivity for liquid metals and is depicted in Fig. 6-15. Since the ratio of 8/8, is small, the velocity profile has a very blunt shape over most of the thermal boundary layer. As a first approximation, then, we might assume a slug-flow model for calculation of the heat transfer i.e., we take... 

Suppose the fluid is highly conducting, such as a liquid metal. In this case, the thermal-boundary-layer thickness will be much greater than the hydrodynamic thickness. This is evidenced by the fact that the Prandtl numbers for liquid metals are very low, of the order of 0.01. For such a fluid, then, we might approximate the actual fluid behavior with a slug-flow model for energy transport in the thermal boundary layer, as outlined in Sec. 6-5. We assume a constant velocity profile... 

The thickness of the Nernst layer increases with the square root of time until natural - convection sets in, after which it remains constant. In the presence of forced convection (stirring, electrode rotation) (see also Prandtl boundary layer), the Nernst-layer thickness depends on the degree of convection that can be controlled e.g., by controlling the rotation speed of a -> rotating disk electrode. See also - diffusion layer. See also Fick s law. 

Further increase in anode potential gets into limiting current plateau range (El in Fig. 10.6). In this region, the potential is so high that the electrochemical reaction is faster than mass transport that is, Cu ions produced on the anode surface in a unit time are more than those that mass transport can remove from the anode surface into the bulk solution. As a result, a Cu ion concentrated layer is developed inside the Prandtl boundary layer [4]. The concentrated Cu layer is called Nernst layer or diffusion layer, which has a thickness of [13]... 

Solving Eq. (S-58 numerically for the temperature profile for different Prandtl numbers, and using the definition of the thermal boundary layer, it is determined that 8/S, = Pr. Then the thermal boundary layer thickness becomes... 

Eq. (3.150) represents the wall law for turbulent flow, first formulated by Prandtl in 1925. The functions f(y+) and g(y+) are of a universal nature, because they are independent of external dimensions such as the height of a channel and are valid for all stratified flows independent of the boundary layer thickness. 

However, the Blasius function f(rj) is available only as the numerical solution of the Blasius equation, and it is thus inconvenient to evaluate this formula for H(r] ). A simpler alternative is to numerically integrate the Blasius equation and the thermal energy equation (11-19) simultaneously. The function H(r]), obtained in this manner, is plotted in Fig. 11-2 for several different values of the Prandtl number, 0.01 < Pr < 100. As suggested earlier, it can be seen that the thermal boundary-layer thickness depends strongly on Pr. For Pr 1, the thermal layer is increasingly thin relative to the Blasius layer (recall that / ->. 99 for rj 4). The opposite is true for Pr <[Pg.773]

It may be noted that the functional dependence of the local heat flux onx is the same as the shear stress. This is a consequence of the fact that the thickness of the thermal boundary layer varies with x in the same way as the momentum boundary-layer thickness. Furthermore, the form of the correlations for large and small Prandtl numbers are also quite similar. However, this latter observation may be somewhat misleading. In the case Pr 1, the heat... 

The temperature profiles for different Pr and (i /pe are indicated in Fig. 6.6. Note that the curves for Pr. = 10 apply equally well for greater Prandtl numbers because of the use of the thermal boundary layer thickness parameter as the abscissa (see Fig. 6.2). 

We recall that an estimate of the characteristic Prandtl viscous boundary layer thickness 8y for steady unbounded flow is given by... 

By assuming that the velocity was a function of y yj xv)X Blasius was able to solve Prandtl s equations for the steady-flow laminar boundary layer on a flat plate. He found that the laminar boundary-layer thickness is proportional to the square root of the length down the plate. 

J9A,mix in the expressions for 5c and Sc represents a diffusivity instead of a molecular transport property, one must replace a, mix by the thermal diffusivity 0 (= kidpCp, where p = density, Cp = specific heat, and kjc = thermal conductivity) to calculate the analogous heat transfer boundary layer thickness Sj and the Prandtl number [i.e., Pr = d/p)ja. In the creeping flow regime, where g 9) = I sine. 

Simplify the radial conduction term, (l/r)(9/9r)(r3r/3r), at large Prandtl numbers when the thermal boundary layer thickness is small relative to the radius of the cylinder. This is the locally flat approximation. 

This means that the transfer of momentum and heat are directly analogous and the boundary-layer thickness 3 for the velocity profile (hydrodynamic boundary layer) and the thermal boundary-layer thickness Sj- are equal. This is important for gases, where the Prandtl numbers are close to 1. 

FIGURE 2.1. Velocity profile for fully developed turbulent flow, u = Fluid velocity d = boundary layer thickness 3p = thickness of Prandtl boundary layer. 

Thickness of the diffusion layer Thickness of the Prandtl boundary layer Defined by Eq. (5.34)... 

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... 

The above arrangement could have been applied to heat and momentum transfer with C and D replaced by T and a, respectively. The resulting formula would be the same as that above, but the Schmidt number is replaced by Prandtl number and the dimensionless diffusion coefficient is replaced by dimensionless heat transfer coefficient. Let us derive the expression for diffusion boundary layer thickness. 

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 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 ... 

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.

As follows from the hydrodynamic properties of systems involving phase boundaries (see e.g. [86a], chapter 2), the hydrodynamic, Prandtl or stagnant layer is formed during liquid movement along a boundary with a solid phase, i.e. also at the surface of an ISE with a solid or plastic membrane. The liquid velocity rapidly decreases in this layer as a result of viscosity forces. Very close to the interface, the liquid velocity decreases to such an extent that the material is virtually transported by diffusion alone in the Nernst layer (see fig. 4.13). It follows from the theory of diffusion transport toward a plane with characteristic length /, along which a liquid flows at velocity Vo, that the Nernst layer thickness, 5, is given approximately by the expression,... 

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... 

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