Pipe flow wall shear rate

With turbulent channel flow the shear rate near the wall is even higher than with laminar flow. Thus, for example, (du/dy) ju = 0.0395 Re u/D is vaHd for turbulent pipe flow with a hydraulically smooth wall. The conditions in this case are even less favourable for uniform stress on particles, as the layer flowing near the wall (boundary layer thickness 6), in which a substantial change in velocity occurs, decreases with increasing Reynolds number according to 6/D = 25 Re", and is very small. Considering that the channel has to be large in comparison with the particles D >dp,so that there is no interference with flow, e.g. at Re = 2300 and D = 10 dp the related boundary layer thickness becomes only approx. 29% of the particle diameter. It shows that even at Re = 2300 no defined stress can be exerted and therefore channels are not suitable model reactors. 

The second subscript N is a reminder that this is the wall shear rate for a Newtonian fluid. The quantity (8u/d,), or the equivalent form in equation 3.13, is known as the flow characteristic. It is a quantity that can be calculated for the flow of any fluid in a pipe or tube but it is only in the case of a Newtonian fluid in laminar flow that it is equal to the magnitude of the shear rate at the wall. 

The 17, 170, A, and n are all parameters that are used to fit data, taken here as 17 = 0.05, 170 = 0.492, A = 0.1, and n = 0.4. A plot of the viscosity vs shear rate is given in Figure 10.8. For small shear rates, the viscosity is essentially constant, as it is for a Newtonian fluid. For extremely large shear rates, the same is true. For moderate shear rates, though, the viscosity changes with shear rate. In pipe flow, or channel flow, the shear rate is zero at the centerline and reaches a maximum at the wall. Thus, the viscosity varies greatly from the centerline to the wall. This complication is easily handled in FEMLAB. 

Equation (3.41) can now be used directly to give the pressine drop for any pipe diameter if the flow is turbulent. Alternatively, this approach can be used to construct a wall shear stress (T )-apparent wall shear rate (SV/D) turbulent flow line for any pipe diameter. This method is particularly suitable when either the basic rheological measurements for laminar flow are not available or it is not possible to obtain a satisfactory lit of such data. The application of this method is illustrated in example 3.8. 

Tube or pipe viscometers take many forms, but they should all be able to give the pressure-drop P as a function of flow rate Q for situations where the tube is long enough to be able to neglect entrance and end effects, say L/a > 50. In this case we can calculate the viscosity as a function of the wall shear rate, y, which is given by... 

The shear rate at the wall of the pipe is related to the volumetric flow rate Q, which can be measured experimentally. The volumetric flow rate also can be obtained by integrating the velocity profile over the cross-sectional area of the pipe. Figure 8.13 shows the velocity profile of a polymer fiuid in a pipe with radius R. The velocity of the fluid at distance r from the pipe center is u. The velocity at the wall i.e., r = R)is zero since there is no slip at the wall. Shear rate is the same as the velocity gradient, and hence the shear rate at any point in the pipe is defined as -du Jdr. 

For steady-state laminar flow of any time-independent viscous fluid, at average velocity V in a pipe of diameter D, the Rabinowitsch-Mooney relations give a general relationship for the shear rate at the pipe wall. 

The production of turbulence is maximum close to walls, where both shear rate and turbulent viscosity, ut, are high. In pipe flow, the maximum is close to y+ = 12. A proper design of a chemical reactor for efficient mixing at low Re should allow the generated turbulence to be transported with the mean flow from the region where it is produced to the bulk of the fluid where it should dissipate. 

Like the von Karman equation, this equation is implicit in/. Equation (6-46) can be applied to any non-Newtonian fluid if the parameter n is interpreted to be the point slope of the shear stress versus shear rate plot from (laminar) viscosity measurements, at the wall shear stress (or shear rate) corresponding to the conditions of interest in turbulent flow. However, it is not a simple matter to acquire the needed data over the appropriate range or to solve the equation for / for a given flow rate and pipe diameter, in turbulent flow. 

Another definition is based, not on the true shear rate at the wall, but on the flow characteristic. This quantity, which may be called the apparent viscosity for pipe flow, is given by... 

Mooney clearly showed that the relationship between the shear stress at the wall of a pipe or tube, DAP/4L, and the term 8V/D is independent of the diameter of the tube in laminar flow. This statement is rigorously true for any kind of flow behavior in which the shearing rate is only a function of the applied shearing stress.1 This relationship between DAP/4L and 8VJD may be conveniently determined in a capillary-tube viscometer, for example. Once this has been done over the range of... 

A somewhat more general approach which has been shown to be at least partially useful for the latter two problems as well as for correlation of laminar-flow data is that of Metzner and Reed (Mil). Their work is based upon an equation developed by Rabinowitsch (Rl) and Mooney (M15) for the calculation of shear rates at the wall of a tube or pipe ... 

V/D Shear rate of a Newtonian fluid at the wall of a pipe (laminar flow), sec.-1... 

If one considers fluid flowing in a pipe, the situation is highly illustrative of the distinction between shear rate and flow rate. The flow rate is the volume of liquid discharged from the pipe over a period of time. The velocity of a Newtonian fluid in a pipe is a parabolic function of position. At the centerline the velocity is a maximum, while at the wall it is a minimum. The shear rate is effectively the slope of the parabolic function line, so it is a minimum at the centerline and a maximum at the wall. Because the shear rate in a pipe or capillary is a function of position, viscometers based around capillary flow are less useful for non-Newtonian materials. For this reason, rotational devices are often used in preference to capillary or tube viscometers. 

Q = Volume flow rate D = pipe diameter t w = wall shear stress Y = Apparent shear rate... 

Consider, first, the simple Reynolds analogy for pipe flows. The pipe wall temperature will be assumed to be uniform. If y is the distance from the wall to the point in the flow being considered as shown in Fig. 7.2, the total shearing stress and heat transfer rate are again written as ... 

Some simple methods of determining heat transfer rates to turbulent flows in a duct have been considered in this chapter. Fully developed flow in a pipe was first considered. Analogy solutions for this situation were discussed. In such solutions, the heat transfer rate is predicted from a knowledge of the wall shear stress. In fully developed pipe flow, the wall shear stress is conventionally expressed in terms of the friction factor and methods of finding the friction factor were discussed. The Reynolds analogy was first discussed. This solution really only applies to fluids with a Prandtl number of 1. A three-layer analogy solution which applies for all Prandtl numbers was then discussed. 

An outline of the steps involved to derive the equations for shear rate and shear stress for fully developed flow in a tube is given in Appendix 3-B. The shear stress (ctw) is given by Equation (3.34) and the shear rate by Equation 3.35, where the subscript w is used to emphasize that the values obtained are those at the pipe wall. 

The shear rate at the wall (7 ) for laminar flow in a pipe can be calculated as follows ... 

Silverman [21] derived velocity correlations between a rotating cylinder (mO, pipe flow (m2), annulus flow (1/3), and an impinging jet (wall jet region only, 1/4), as listed in Table 2. These equations assume that the appropriate transformations are to be made on the basis of equal mass-transfer rates for the different geometries. Silverman [21] also explored the case where the equality of surface shear stress is the appropriate criterion, on the basis that the equality of the shear stress will ensure the same corrosion processes for the various geometries. We stress that the equations listed in Table 2 must be used with great caution because they are based on the... 

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