Wall shear

Averaging the velocity using equation 50 yields the weU-known Hagen-Poiseuille equation (see eq. 32) for laminar flow of Newtonian fluids in tubes. The momentum balance can also be used to describe the pressure changes at a sudden expansion in turbulent flow (Fig. 21b). The control surface 2 is taken to be sufficiently far downstream that the flow is uniform but sufficiently close to surface 3 that wall shear is negligible. The additional important assumption is made that the pressure is uniform on surface 3. The conservation equations are then applied as follows ... 

Frictional loss. Frictional loss is due to wall shear forces. This loss varies from about 1-2% as the flow varies from a low-flow to a high-flow setting. 

In fact with a Newtonian liquid y = 4Q/ nR. This latter expression, viz. 4Q/ uR, is obviously much easier to calculate than the true wall shear rate and, since they are uniquely related and the simple expression is just as useful, in design practice it is very common when plotting flow curves to plot against... 

The latter expression is known as the apparent wall shear rate and usually given the symbol 7vi,a-... 

A slit die is designed on the assumption that the material is Newtonian, using apparent viscous properties derived from capillary rheometer measurements, at a particular wall shear stress, to calculate the volumetric flow rate through the slit for the same wall shear stress. Using the correction factors already derived, obtain an expression for the error involved in this procedure due to the melt being non-Newtonian. Also obtain an expression for the error in pressure drop calculated on the same basis. What is the magnitude of the error in each case for a typical power law index n = 0.377... 

As demonstrated, Eq. (7) gives complete information on how the weight fraction influences the blend viscosity by taking into account the critical stress ratio A, the viscosity ratio 8, and a parameter K, which involves the influences of the phenomenological interface slip factor a or ao, the interlayer number m, and the d/Ro ratio. It was also assumed in introducing this function that (1) the TLCP phase is well dispersed, fibrillated, aligned, and just forms one interlayer (2) there is no elastic effect (3) there is no phase inversion of any kind (4) A < 1.0 and (5) a steady-state capillary flow under a constant pressure or a constant wall shear stress. 

V = flow velocity T = average wall shear stress... 

Z will be zero at the pipe wall (ux = 0) and at the axis dux/l y = 0), and it will reach a maximum at some intermediate position in the cross-section. From equation 3.6, the wall shear stress R for laminar flow may be expressed in terms of the pressure gradient along the pipe (—AP/l) ... 

For a Newtonian fluid, equation 3.137 gives a wall shear rate of 8u/d (corresponding to equation 3.39) and a shear stress of 8fju/d (corresponding to equation 3.40). 

Working in terms of the apparent viscosity /rw, at the wall shear rate, by definition ... 

Values of n and k for the suspensions used are given in Table 5.2. Experimental results are shown in Figure 5.8 as wall shear stress R as a function of wall shear rate (dn /dyfr o using logarithmic coordinates. 

The shear stress Ri at the pipe wall in the upper portion of the pipe may be calculated on the assumption that the liquid above the bed is flowing through a non-circular duct, bounded at the top by the wall of the pipe and at the bottom by the upper surface of the bed. The hydraulic mean diameter may then be used in the calculation of wall shear stress. However, this does not take account of the fact that the bottom boundary, the top surface of the bed, is not stationary, and will have a greater effective roughness than the pipe... 

Onset of Nucleate Boiling in Conventional Size Channels where / = 2Tw/(pt/ ), tw is the wall shear stress. 

Equation (15) indicates that the wall shear stress changes with position on the cell surface and at each position the stress changes with time. Figures 9 and 10 show the distributions of the fluid shear rate in the eddy and the cell surface shear stress as a function of the dimensionless radius, R, and time, yt, respectively. 

Since the present study aims at carrying out the investigation of the break-down phenomenon and searching for the possible mechanism of the phenomenon, we have chosen the similar condition as in [1] for the wall shear stress to induce break-down The reference temperature in the degradation studies was 60 °C. This value may be lower than the value used in a typical DHS. In a low-pressure system, however, it was necessary to use lower the temperature to avoid the formation of bubbles. For parametric studies, one of the variables was varied while the other variables were fixed at the reference condition (Tanperature 60 °C Re 8,000 Surfactant concentration 200ppm Volume of solution charged 0.010 m ). 

The power of this technique is two-fold. Firstly, the viscosity can be measured over a wide range of shear rates. At the tube center, symmetry considerations require that the velocity gradient be zero and hence the shear rate. The shear rate increases as r increases until a maximum is reached at the tube wall. On a theoretical basis alone, the viscosity variation with shear rate can be determined from very low shear rates, theoretically zero, to a maximum shear rate at the wall, yw. The corresponding variation in the viscosity was described above for the power-law model, where it was shown that over the tube radius, the viscosity can vary by several orders of magnitude. The wall shear rate can be found using the Weissen-berg-Rabinowitsch equation ... 

The wall shear stress can be calculated on the basis of the fully developed pipe flow correlation (the hydraulic diameter concept). 

For turbulent flow, we shall use the Chilton-Colburn analogy [12] to derive an expression for mass transfer to the spherical surface. This analogy is based on an investigation of heat and mass transfer to a flat plate situated in a uniform flow stream. At high Schmidt numbers, the local mass transfer rate is related to the local wall shear stress by... 

For turbulent flow, the local wall shear stress, xr, is given by Eq. (25). Substituting Eqs. (48H50) into Eq. (47) and making use of Eq. (25), one arrives at an expression for the Sherwood number based upon the radius of the rotating hemisphere ... 

The Chilton-Colburn analogy can be also used to estimate the local mass transfer rate in laminar flow where the wall shear stress is related to the azimuthal velocity gradient by... 

That is, the removal of spherical particles from a flat surface is determined by the magnitude of the wall shear stress, x0. Visser (1) also claims that since the removal mechanism is unknown, it is not possible to relate the Fh (tangential force) to the Fa (adhesive force) on theoretical grounds. Therefore, he assumes that the tangential force required for particle release is proportional to the adhesive force. 

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. 

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