Uniform Free-Stream Conditions

These relationships may be used to obtain cone flow results from the flat-plate results of the section on uniform free-stream conditions. Real-gas solutions for air obtained in this manner are given in Ref. 17. 

Evaluation of the wall region eddy viscosity has been facilitated by the fact that the shear stress remains essentially constant across the wall layer. On a flat plate with uniform free-stream conditions, the shear stress drops less than 10 percent from its wall value across the wall layer. The equation governing low-speed flow on a flat plate in the absence of a pressure gradient is... 

The quantity cy in Eqs. 6.219 and 6.221 is found from the equations given in the section on uniform free-stream conditions. For example, at moderate Reynolds numbers on a flat plate, the skin friction coefficient is given by... 

The form of Eq. 6.241 applies for Re, < 4 x 106, where the Blasius skin friction equation (Eq. 6.16) is reasonably accurate and 2 St cf = PrM and Pr-04 for laminar and turbulent flow, respectively. It also uses the laminar reference enthalpy approach to define p p /pcpc (see the section on uniform free-stream conditions) and uses the turbulent boundary layer transfor-... 

The critical Reynolds number Reor is typically taken as 5 x 105, lie, < Re, < 3 x 107, and 0.7 < Pr < 400. The fluid properties are evaluated at the film temperature (7, + I )/2 where 7, is the free-stream temperature and 7 is the surface temperature. Equation (5-60) also apphes to the uniform heat flux boundary condition provided h is based on the average temperature difference between 7 and 7, ... 

In reviewing our analysis of (9-152) leading to (9-141), we may note that we have used the conditions (9-143) and (9-144) only on the velocity field. Thus, as stated earlier, the result (9-141) or (9-142) is valid in streaming flow for any heated body with a uniform surface temperature provided that these conditions are satisfied and that /V Higher-order terms in (9-142) will depend on the details of the flow, and thus on the Reynolds number Re, as well as the shape and orientation of the body relative to the free stream. However, in the creeping-flow limit, Brenner was able to extend (9-142) to one additional term for a particle of arbitrary shape,... 

Problem 9-13. Heat Transfer From a Circular Cylinder in an Inviscid Fluid. Determine the dimensionless rate of heat transfer from a circular cylinder of radius a immersed in a uniform flow of an inviscid fluid. Because there is no vorticity in the free stream, the velocity field can be written in terms of a potential, u = V, with V2 = 0. The boundary conditions are u U as r -> oo, and u n = 0 at r = a. The cylinder is maintained at temperature 7), while the fluid far from the cylinder is at 7o. 

Besides non-ideal operation, non-ideal module construction influences the module performance as well. Module performance refers to the recovery, which is the ratio of a product stream to the feed stream of the target compound, for a given product purity. The objective of manufacturing process is to produce uniform, defect free hollow fibers. Unfortunately, real membrane fibers are not uniform since small changes in production conditions as well as manufacturing tolerances result in different fiber properties. Variations in fiber dimensions (diameters, length, and membrane thickness) and fiber properties (permeability) as well as manufacturing defects (blocked fibers, pinholes) have an impact on the module performance. In particular, the variation of fiber diameters, which is one of the major drawbacks in module performance, was well... 

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