Friction factor Blasius

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. 

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. 

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. 

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

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. 

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. 

Equation (6-37) represents the friction factor for Newtonian fluids in smooth tubes quite well over a range of Reynolds numbers from about 5000 to 105. The Prandtl mixing length theory and the von Karman and Blasius equations are referred to as semiempirical models. That is, even though these models result from a process of logical reasoning, the results cannot be deduced solely from first principles, because they require the introduction of certain parameters that can be evaluated only experimentally. 

For turbulent flow in smooth tubes, the semiempirical Prandtl-von Karman/Nikuradse or Blasius models represent the friction factor quite well. Whether a tube is hydraulically smooth or rough depends upon... 

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) ... 

In developing their correlation, Lockhart and Martinelli assumed that the friction factors could be determined from equations of the same form as 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... 

The Bernoulli equation can now be written for the liquid in channel flow in the bottom part of the tube, and for the liquid in slug flow in the upper part. The acceleration terms are then neglected, and the friction factors for each type of liquid flow found from the Blasius equation and from true Reynolds numbers. The resulting equations cannot be readily evaluated because of the two hydraulic-radius terms involved in the two types of flow, and an unknown fraction defining the relative mass of liquid in each part of the tube. 

With the Blasius equation (6.96), the friction factor and the pressure gradient become, with this model,... 

F or turbulent pipe flow, the friction velocity u = Vx ,/p used earlier in describing the universal turbulent velocity profile may be used as an estimate for V Together with the Blasius equation for the friction factor from which e may be obtained (Eq. 6-214), this provides an estimate for the energy-containing eddy size in turbulent pipe flow ... 

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. 

In addition, there are several correlations for the friction factor. For smooth pipes, one of the simplest correlations is the Blasius equation ... 

For turbulent flow (2100 < / e < 100,000) in long, smooth pipes, the Blasius formula may be used to estimate the friction factor ... 

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... 

Many useful correlations have been published to determine the friction factor for fully developed turbulent flow of Newtonian fluids in smooth pipes. One of the earliest correlations was given by Blasius (8) as follows ... 

The friction factor is calculated in the same way as for single-phase flow, for example, according to Blasius... 

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... 

For any given Reynolds number, one can read the value of friction factor, and then use Eq. (1) to determine the pressure drop. In the turbulent region, for Re > 2200, one correlation is the Blasius formula [Eq. (3)] ... 

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