Blasius

T] Low M.T. rates. Low mass-flux, constant property systems. Ns, % L local k. Use with arithmetic difference in concentration. Coefficient 0.323 is Blasius approximate solution. 

For turbulent flow in smooth tubes, the Blasius equation gives the friction facdor accurately for a wide range of Reynolds numbers. 

A good deal of work has been done on polymeric crown ethers during the last decade. Hogen Esch and Smid have been major contributors from the point of view of cation binding properties, and Blasius and coworkers have been especially interested in the cation selectivity of such species. Montanari and coworkers have developed a number of polymer-anchored crowns for use as phase transfer catalysts. Manecke and Storck have recently published a review titled Polymeric Catalysts , which may be useful to the reader in gaining additional perspective. 

Blasius and coworkers have offered a somewhat different approach to systems of this general type. In the first of these, shown in Eq. (6.20), he utilizes a hydroxymethyl-substituted 15-crown-5 residue as the nucleophile. This essentially similar to the Mon-tanari method. The second approach is a variant also, but more different in the sense that covalent bond formation is effected by a Friedel-Crafts alkylation. In the reaction... 

An alternative copolymerization is illustrated by the method of Blasius. In this preparation, a phenol-formaldehyde (novolac) type system is formed. Monobenzo-18-crown-6, for example, is treated with a phenol (or alkylated aromatic like xylene) and formaldehyde in the presence of acid. As expected for this type of reaction, a highly crosslinked resin results. The method is illustrated in Eq. (6.23). It should also be noted that the additional aromatic can be left out and a crown-formaldehyde copolymer can be prepared in analogy to (6.22). ... 

E Blasius and B Brozio, Chelating ion-exchange resins. In Chelates in Analytical Chemistry, H A Flaschka and A J Barnard (Eds), Vol. 1, Marcel Dekker, New York, 1967, p 49... 

Chelating ion-exchange resins. E. Blasius and B. Brozio, Chelates Anal. Chem., 1967, 1, 49-79 (149). 

Equation 3.11 is due to Blasius(6) and the others are derived from considerations of velocity profile. In addition to the Moody friction factor / = 8R/pu2, the Fanning or Darcy friction factor / = 2R/pu2 is often used. It is extremely important therefore to be clear about the exact definition of the friction factor when using this term in calculating head losses due to friction. 

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. 

Equation 3.175 reduces to the simple Blasius relation (equation 3.10) for a Newtonian fluid (n = 1). 

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,... 

The right-hand side of equation 10.224 gives numerical values which are very close to those obtained from the Blasius equation for the friction factor (j> for the turbulent flow of a fluid through a smooth pipe at Reynolds numbers up to about 106. 

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. 

If at a distance a from the leading edge the laminar sub-layer is of thickness 5 and the total thickness of the boundary layer is 8, the properties of the laminar sub-layer can be found by equating the shear stress at the surface as given by the Blasius equation (11.23) to that obtained from the velocity gradient near the surface. 

The shear stress at the walls is given by the Blasius equation (11.23) as ... 

Using the Blasius equation (equation 11.46) to give an approximate value for R/pu2 for a smooth pipe ... 

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. 

The Blasius relation between friction factor and Reynolds number for turbulent flow is ... 

For hydrodynamically smooth pipes, through which fluid is flowing under turbulent conditions, the shear stress is given approximately by the Blasius equation ... 

For flow in a smooth pipe, the friction factor for turbulent flow is given approximately by the Blasius equation and is proportional to the Reynolds number (and hence the velocity) raised to a power of -2. From equations 12.102 and 12.103, therefore, the heat and mass transfer coefficients are both proportional to w 75. 

Oil of density 950 kg/m3 and viscosity 10-2 Ns/m2 is to be pumped 10 km through a pipeline and the pressure drop must not exceed 2 x lt N/m2. What is the minimum diameter of pipe which will be suitable, if a flowrate of 50 tonne/h is to be maintained Assume the pipe wall to be smooth. Use either the pipe friction chart or the Blasius equation (R/pu1 = 0.0396/ -1/4). 

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. 

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