Blasius expression

Turbulent mass transfer near a wall can be represented by various physical models. In one such model the turbulent flow is assumed to be composed of a succession of short, steady, laminar motions along a plate. The length scale of the laminar path is denoted by x0 and the velocity of the liquid element just arrived at the wall by u0. Along each path of length x0, the motion is approximated by the quasi-steady laminar flow of a semiinfinite fluid along a plate. This implies that the hydrodynamic and diffusion boundary layers which develop in each of the paths are assumed to be smaller than the thickness of the fluid elements brought to the wall by turbulent fluctuations. Since the diffusion coefficient is small in liquids, the depth of penetration by diffusion in the liquid element is also small. Therefore one can use the first terms in the Taylor expansion of the Blasius expressions for the velocity components. The rate of mass transfer in the laminar microstructure can be obtained by solving the equation... 

The friction factors required in Equation (10) above and for the individual-phase pressure drops in the Martinelli parameter (Equation 11) were computed using / = 64/Re for Re < 2100 and the Blasius expression... 

The shear stresses within the fluid are responsible for the frictional force at the walls and the velocity distribution over the cross-section. A given assumption for the shear stress at the walls therefore implies some particular velocity distribution. It will be shown in Chapter 11 that the velocity at any point in the cross-section will be proportional to the one-seventh power of the distance from the walls if the shear stress is given by the Blasius equation (equation 3.11). This may be expressed as ... 

Yooi24) has proposed a simple modification to the Blasius equation for turbulent flow in a pipe, which gives values of the friction factor accurate to within about 10 per cent. The friction factor is expressed in terms of the Metzner and Reed(I8) generalised Reynolds number ReMR and the power-law index n. 

By substituting the well-known Blasius relation for the friction factor, Eq. (45) in Table VII results. Van Shaw et al. (V2) tested this relation by limiting-current measurements on short pipe sections, and found that the Re and (L/d) dependences were in accord with theory. The mass-transfer rates obtained averaged 7% lower than predicted, but in a later publication this was traced to incorrect flow rate calibration. Iribame et al. (110) showed that the Leveque relation is also valid for turbulent mass transfer in falling films, as long as the developing mass-transfer condition is fulfilled (generally expressed as L+ < 103) while Re > 103. The fundamental importance of the Leveque equation for the interpretation of microelectrode measurements is discussed at an earlier point. 

The friction factor plot is available in many handbooks, so that given a value of Re, one can find the corresponding value of /. In the context of numerical optimization, however, using a graph is a cumbersome procedure. Because all of the constraints should be expressed as mathematical relations, we select the Blasius correlation for a smooth pipe (Bird et al., 1964) ... 

Considerable effort has been expended in trying to And algebraic expressions to relate/to Re and eld,. For turbulent flow in smooth pipes, the simplest expression is the Blasius equation ... 

This is consistent with the Blasius type of expression used for the friction factors in deriving the Martinelli parameter. Using the value n = 0.20 and expressing the ratio of flow rates in terms of the quality... 

Expressions for the Nusselt number for zero suction are available for various values of m [4], For the Blasius flow (m = 0), the expression has the form... 

Assuming large Prandtl numbers, one can substitute only the first terms of Blasius series [Eqs. (68a)] for the velocity components u and v, and the algebraic evaluation leads to the expression ... 

A particularly simple expression is obtained for the multiplier in terms of the Blasius equation ... 

Blasius R, Reuter S, Henry E, Dicato M, Diederich M. 2006. Curcumin regulates signal transducer and activator of transcription (STAT) expression in K562 cells. Biochem Pharmacol 72 1547-1554. 

The relation between cost per unit length C of a pipeline installation and its diameter d is given by C = a + bd where a and b are independent of pipe size. Annual charges are a fraction of the capital cost. Obtain an expression for the optimum pipe diameter on a minimum cost basis for a fluid of density p and viscosity p flowing at a mass rate of G. Assume that the fluid is in turbulent flow and that the Blasius equation is applicable, that is the friction factor is proportional to the Reynolds number to the power of minus one quarter. Indicate clearly how the optimum diameter depends on flowrate and fluid properties. 

This result has the following consequence for the completeness of basis function constructed from the eigenvectors obtained by stability analysis of external flows. It has been clearly shown by Mack (1976) that internal flows, like the channel flow, has denumerable infinite number of eigen modes and any arbitrary applied disturbance can be expressed in terms of this complete basis set. However, for external flows, as we have seen for the Blasius flow in Table 2.1 that there are only a few discrete eigenmodes and it is not possible to express any arbitrary functions in terms of these only, in the absence of any other singularities for this flow. 

Exact solution. Friction coefficient. Following Blasius [43], we express the fluid velocity components via the stream function L according to (1.1.6) and substitute them into (1.7.2). Then we seek the stream function in the form... 

Duvoix, A., Schnekenburger, M., Delhalle, S., Blasius, R., Borde-Chiche, P., Morceau, F., Dicato, M., and Diederich, M. (2004) Expression of glutathione S-transferase PI -1 in leukemic cells is regulated by inducible AP-1 binding. Cancer Lett. 216, 207-219. 

Several studies have been reported to determine friction losses in turbulent flow of slurries. Hannah et al. (29) presented an approach in which they compared expressions for the friction pressure of the slurry and clean fluid. In their analysis, they assumed Blasius (30) turbulent Fanning friction factor versus Reynolds number equation for Newtonian fluids. The following expression for estimating slurry friction pressure knowing the clean fluid friction pressure is proposed. 

Of much greater interest is the case where Sc= v D>, since our principal concern is with dilute solutions. For this situation the diffusion boundary layer is imbedded in the viscous boundary layer, and the velocity it sees is that close to the wall. Solution to the steady, Blasius, flat plate, viscous boundary layer equation shows the velocity components close to the wall (y[Pg.108]

No exact mathematical analysis of the conditions within a turbulent fluid has yet been developed, though a number of semi-theoretical expressions for the shear stress at the walls of a pipe of circular cross-section have been suggested, including that proposed by Blasius. ... 

Blasius has given the following approximate expression for the shear stress at a plane smooth surface over which a fluid is flowing with a velocity Us, for conditions... 

Obviously, exact computation of the shear stresses is limited either to laminar flows or simple geometries, and yet is complicated. Thus, the more conventional way to evaluate the wall and interfacial shear stresses is to adopt the Blasius equation, whereby wall shear stresses x, x are expressed in terms of the phases velocity heads and appropriate friction factors, /, f ... 

Introduction and derivation of integral expression. In the solution for the laminar boundary layer on a fiat plate, the Blasius solution is quite restrictive, since it is for laminar flow over a flat plate. Other more complex systems cannot be solved by this method. An approximate method developed by von Karman can be used when the configuration is more complicated or the flow is turbulent. This is an approximate momentum integral analysis of the boundary layer using an empirical or assumed velocity distribution. 

In Section 3.IOC an exact solution was obtained for the hydrodynamic boundary layer for isothermal laminar flow past a plate and in Section 5.7A an extension of the Blasius solution was also used to derive an expression for convective heat transfer. In an analogous manner we use the Blasius solution for convective mass transfer for the same geometry and laminar flow. In Fig. 7.9-1 the concentration boundary layer is shown where the concentration of the fluid approaching the plate is and in the fluid adjacent to the surface. 

---