Blasius equation

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

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. 

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. 

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. 

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). 

In the above equations, a is a coefficient with the value of 1.0 for single phase flow and 2.0 for multi-phase flow [6], and pis an adjustable coefficient and has a value of 2.1 by fitting the experimental results for the two phase flow. The flow resistant coefficient is determined by the Blasius equation. 

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. 

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 ... 

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 ... 

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. 

At high Reynolds numbers, for example, Blasius equation is... 

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 ... 

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