Velocity power law

Laminar or power law velocity distribution in which the linear velocity varies with radial position in a cylindrical vessel. Plug flow exists along any streamline and the mean concentration is found by integration over the cross section. 

Since U is a function of z, some mean value must be used. The appropriate value is the mean through the plume. However, the time-averaged wind speed at the stack height is commonly used. Often, even this value may not be known, in which case an estimate must be made. This estimate could be based on an assumed power law velocity profile such as ... 

Evaluate the kinetic energy correction factor a in Bernoulli s equation for turbulent flow assuming that the 1/7 power law velocity profile [Eq. (6-36)] is valid. Repeat this for laminar flow of a Newtonian fluid in a tube, for which the velocity profile is parabolic. 

If a power law velocity distribution is assumed in the boundary layer, i.e., if it is assumed that ... 

To assess the physical deviation between the average of products and the product of averages a momentum velocity correction factor can be defined by Cm = vz) / v1)a- By use of the Hagen-Poiseuille law (1.353) and the power law velocity profile (1.354) it follows that at steady state Cm has a value of about 0.95 for turbulent flow and 0.75 for laminar flow [55]. In practice a value of 1 is used in turbulent flow so that v1)a is simply replaced by the averaged bulk velocity vz) - On the other hand, for laminar flows a correction factor is needed. For more precise calculations a simplified (not averaged ) 2D model is often considered for ideal axisymmetric pipe flows [52, 69]. 

Unfortunately, it is not possible to derive an analogue velocity profile for turbulent flow in an anal dical manner based on the generalized momentum equations. However, a number of entirely empirical relations of similar simplicity exist for the velocity profile in turbulent pipe flow. One such relation often found in introductory textbooks on engineering fluid flow is the power law velocity profile. ... 

The Power Law Velocity Distribution. The solution for the power law velocity distribution is introduced in Prandtl [42] in the following form ... 

Friction Factor. From the power law velocity distribution of Eq. 5.65, the friction factor can be expressed as ... 

The analogous function for a flat plate is given by Eq. 6.66. These results were improved upon in Ref. 49, where a power-law velocity profile was assumed, ulue = (ylye)d with d found to best fit Hartree s calculations [45] as listed in Table 6.4. The form of the kernel function (Eq. 6.124) is the same, but the exponents are changed to... 

Painter, S., Cvetkovic, V., and Selroos, J., 2002. Power-law velocity distributions in fracture networks Numerical evidence and... 

Afzal, N., Seena, A., Bushra, A., 2007. Power law velocity profile in fully developed turbulent pipe and channel flows. J. Hydraul. Eng. 133, 1080-1086. 

A typical range for n in the power law model is 0.40-0.70, which gives a range of values from 1.75 to 1.11 for the n-related factor in Equation 3.73. The wall shear rate is therefore higher for a power law fluid than for a Newtonian fluid, as can be seen from Figure 3.17, in which the slope of the power law velocity profile is clearly steeper at the capillary wall. 

The erosion rate, E, is conunonly given in terms of mass or volume of material removed per unit mass of erodent impacted, volume being preferred because it permits thickness loss comparisons between materials of different density. Implicit is the assumption that the dimensions of the eroded area and the particle concentration are unimportant, which is a good approximation for dilute flows. Metals and ceramics differ in the dependence of E on a, as mentioned above, and also in their response to velocity and particle size and shape. E generally shows a power-law velocity dependence ... 

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