Boundary layer thickness dimensionless

Because D is independently determined, and p is obtainable from initial conditions and thermod5mamic equilibrium, the problem of determining the convective dissolution rate now becomes the problem of estimating the boundary layer thickness. In fluid dynamics, the boundary layer thickness appears in a dimensionless number, the Sherwood number Sh ... 

The boundary layer equations may be obtained from the equations provided in Tables 6.1-6.3, with simplification and by an order-of-magnitude study of each term in the equations. It is assumed that the main flow is in the x direction. The terms that are too small are neglected. Consider the momentum and energy equations for the two-dimensional, steady flow of an incompressible fluid with constant properties. The dimensionless equations are given by Eqs. (6.46) to (6.48). The principal assumption made in the boundary layer is that the hydrodynamic boundary layer thickness 8 and the thermal boundaiy layer thickness 8t are small compared to a characteristic dimension L of the body. In mathematical terms,... 

It is also necessary to determine the local value of the dimensionless boundary layer thickness in order to find the mixing length distribution. It is usually adequate to take this as the value of Y at which U reaches 0.99 of its free-stream value. 

Here, L is a reference length that characterizes the size of the surface, e.g., its length. Twr is again some reference wall temperature. If some measure of the boundary layer thickness, 8, is also introduced, then the governing equations can be written in terms of the following dimensionless variables ... 

Dimensionless numbers have proved useful for analyzing relationships between heat transfer and boundary layer thickness for leaves. In particular, the Nusselt number increases as the Reynolds number increases for example, Nu experimentally equals 0.97 Re0-5 for flat leaves (Fig. 7-9). By Equations 7.18 and 7.19, d/8bl is then equal to 0.97 (vd/v)V2y so for air temperatures in the boundary layer of 20 to 25°C, we have... 

The bottom line is essentially Equation 7.10, indicating that dimensionless numbers can be used to estimate average boundary layer thicknesses, as indicated above. 

In order to estimate the order of magnitude of the individual terms in these equations, dimensionless quantities are introduced. It is useful to measure the distance y from the wall relative to a mean boundary layer thickness ... 

As according to (3.170), S (vxjwoo) /2 holds for the velocity boundary layer, the distance y from the wall can usefully be related to the boundary layer thickness <5, so introducing a dimensionless variable... 

Intuitively one could imagine that the boundary layer as a whole can be characterized in terms of the boundary layer thickness and related dimensionless groups. However, experimental data reveals that the laminar shear is dominant near the wall (i.e., in the inner wall layer), and turbulent shear dominates in the outer wall layer. There is also an intermediate region, called the overlap wall region, where both laminar and turbulent shear are important. 

The dimensionless group H = / (Pe/4) is proportional to the ratio of the particle diameter to the concentration boundary layer thickness given by (3.27)... 

T Tobias number (ratio of mass transport to ohmic resistance), dimensionless Wa Wagner number, (ratio of activation to ohmic resistance), dimensionless aa,ac, transfer coefficients, anodic and cathodic, respectively, dimensionless 8C equivalent mass transfer boundary layer thickness (Nemst-type), cm r overpotential, V... 

Thus, Eqs. 6.49 and 6.50 indicate that the skin friction coefficient and Stanton number remain equal to their constant-property values. In terms of these dimensionless transfer coefficients, the effects of the linear dependence of viscosity on temperature just cancel those of the perfect gas variation of the density. It should be noted, however, that the density variation itself still affects the boundary layer thickness. 

Creeping flow of an incompressible Newtonian fluid around a solid sphere corresponds to g (0) = I sin0. For any flow regime that does not include turbulent transport mechanisms, the dimensionless boundary layer thickness is... 

In dimensionless notation, the generalized expression for the simplified mass transfer boundary layer thickness is... 

Dimensionless Molar Density. The final form of the mass transfer equation for Cp, y, t), which will be used to calculate the concentration profile and boundary layer thickness of species A in the liquid phase, is... 

This result can be written in terms of the important dimensionless numbers for mass and heat transfer. A completely dimensionless expression is obtained via division of the boundary layer thickness by the cylindrical radius / . If the Reynolds number is defined using R as the characteristic length, instead of the cylindrical diameter, then... 

The Nernst boundary layer thickness is a simple characteristic of the mass transfer but its definition is formal since no boundary layer is in fact stagnant and least of all boundary layers on gas-evolving electrodes furthermore, the Schmidt number, known to influence mass transfer, is not incorporated in the usual dimensionless form. For this reason, lines representing data from gas evolution in two different solutions can be displaced from one another because of viscosity differences. Nevertheless, the exponent in the equation = aib (32)... 

Rewritting (6) under the dimensionless formulation, the small quantity 1/Re appears on the right hand-side, and the contribution of viscous stresses may be considered as a small perturbation to an ideal fluid motion (in ideal flows right hand-side of (6) is zero). The boundary layer equations are obtained as velocity asymptotic expansions which must satisfy the perturbed equation (Table 3). Physically, this is equivalent to say that the velocity gradient in the flow direction is very small compared with the normal one, and that the normal velocity component is much smaller than the axial one. It can be shown that the ratio of the transverse to longitudinal velocities is about Re and that the boundary layer thickness varies as Re" (Fig. 17). Such considerations may be applied to temperature and concentration profiles and lead to the so called Thermal boundary layer and Diffusion boundary layer . According to similitude laws, Pr, Le, Sc numbers allows a comparison to be made of these different layer thicknesses ... 

For the disc system, an earlier analysis of V. Karman [6], allowed us to derive the rate of strain tensor, and thus to calculate an elongational gradient equal to 2 F.Q [2], where F is the dimensionless radial velocity component which is only dependent on the reduced axial coordinate n = y / This elongational gradient reaches a maximum value of 0.36 Q for an approximate distance roughly equal to the boundary layer thickness. 

Lev que s problem was extracted from the rescaled mass balance in Equation 8.28. As can be seen, this equation is the basis of a perturbation problem and can be decomposed into several subproblems of order 0(5 ). The concentration profile, the flux at the wall, and consequently the mixing-cup concentration (or conversion) can all be written as perturbation series on powers of the dimensionless boundary layer thickness. This series is often called as the extended Leveque solution or Lev jue s series. Worsoe-Schmidt [71] and Newman [72] presented several terms of these series for Dirichlet and Neumann boundary conditions. Gottifredi and Flores [73] and Shih and Tsou [84] considered the same problem for heat transfer in non-Newtonian fluid flow with constant wall temperature boundary condition. Lopes et al. [40] presented approximations to the leading-order problem for all values of Da and calculated higher-order corrections for large and small values of this parameter. 

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. 

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