Blasius law

The friction factor L of the liquid stream, according to Blasius law, is... 

The friction factor, still using Blasius law (4.138), and (4.144), can be eliminated, giving... 

As an example we will calculate Xtt for the case where both phases are turbulent, a state that occurs frequently. Presuming the validity of Blasius law for the friction factor... 

Formula (1.7.8) is known as the Blasius law for the drag in longitudinal flat-plate flow. This formula can be used in laminar flow, that is, for Ret < 3.5 x 105. 

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. 

Thus, the pipe friction chart for a Newtonian fluid (Figure 3.3) may be used for shearthinning power-law fluids if Remit is used in place of Re. In the turbulent region, the ordinate is equal to (R/pu2)n 0 fn5. For the streamline region the ordinate remains simply R/pu2, because Reme has been defined so that it shall be so (see equation 3.140). More recently, Irvine(25j has proposed an improved form of the modified Blasius equation which predicts the friction factor for inelastic shear-thinning polymer-solutions to within 7 per cent. 

Irvine, T. F. Chern. Eng. Comm. 65 (1988) 39. A generalized Blasius equation for power law fluids,... 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

A simple approximate form of the relation between u+ and y+ for the turbulent flow of a fluid in a pipe of circular cross-section may be obtained using the Prandtl one-seventh power law and the Blasius equation. These two equations have been shown (Section 11.4) to be mutually consistent. 

Show that, if the Blasius relation is used for the shear stress R at the surface, the thickness of the laminar sub-layer <5, is approximately 1.07 times that calculated on the assumption that the velocity profile in the turbulent fluid is given by PrandtFs one seventh power law. 

Since the forcing terms in equation (34) all vanish in this problem, we obtain equation (39), in which for simplicity we shall introduce the further assumption that C = 1—that is, p/t = [see equation (30)]. This assumption (that pp does not vary across the boundary layer) often is reasonable for gases if changes in the average molecular weight are negligible, then— because of the constancy of the pressure—the ideal-gas law implies that p 1/T, in which case constancy of pp corresponds to p T, a dependence close to the kinetic-theory predictions discussed in Appendix E. With C = 1, equation (39) is the Blasius equation [4], F " -F FF" = 0, and in view of equation (28), the boundary conditions implied by equations (48) and (49) are F co) = 1 and F (0) = 0. Use may be made of the present formula for p, C = 1, F (0) = 0, and equations (27) and (29) to ascertain the boundary condition implied by equation (50) the calculation results in... 

The shear stresses over the flow boundaries can be rigorously derived as an integral part of the solution of the flow field only in laminar flows. The need for closure laws arise already in single-phase, steady turbulent flows. The closure problem is resolved by resorting to semi-empirical models, which relate the characteristics of the turbulent flow field to the local mean velocity profile. These models are confronted with experiments, and the model parameters are determined from best fit procedure. For instance, the parameters of the well-known Blasius relations for the wall shear stresses in turbulent flows through conduits are obtained from correlating experimental data of pressure drop. Once established, these closure laws permit formal solution to the problem to be found without any additional information. 

The power-law equation does not hold, as y goes to zero at the wall. Another useful relation is the Blasius correlation for shear stress for pipe flow, which is consistent at the wall for the wall shear stress Tq. For boundary-layer flow over a flat plate, it becomes... 

In a similar fashion, the integral momentum analysis method used for the turbulent hydrodynamic boundary layer in Section 3.10 can be used for the thermal boundary layer in turbulent flow. Again, the Blasius 7-power law is used for the temperature distribution. These give results that are quite similar to the experimental equations as given in Section 4.6. 

The coefficient f depends on the Reynolds number for flow within the tube. In laminar flow, the Hagen-Poiseuille law can be applied. In turbulent flow the Blasius equation is used. The main difficulty is the evaluation of water pressure drop during transition boiling. The pressure drop consists of three components friction (APf), acceleration (APJ and static pressure (APg). In once-through horizontal tubes boiler APg=0. The Lockard-Martinelli formulation is used to estimate the friction term. 

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