Wall shear rate

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-... 

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 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 ... 

Equation (6-43) describes the laminar flow of a power law fluid in a tube. Since a power law fluid is defined by the relation r = myn, rearrange Eq. (6-43) to show that the shear rate at the tube wall for a power law fluid is given by yw = (8V/D)(3n + l)/4w where 8 V/D is the wall shear rate for a Newtronian fluid. 

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 solution to the problem of determining the wall shear rate for a non-Newtonian fluid in laminar flow in a tube relies on equation 2.6. 

Rearranging equation 3.18 gives the wall shear rate yw as... 

As the wall shear rate ywN for a Newtonian fluid in laminar flow is equal to (- 8u/di), equation 3.20 can be expressed as... 

Calculate the true wall shear rate from equation 3.20 with the derivative determined in 3. In general, the plot of ln(8u/d,) against lnr. will not be a straight line and the gradient must be e iluated at the appropriate points on the curve. 

Recall that the wall shear rate for a Newtonian fluid in laminar flow in a tube is equal to —8w/d,. In the case of a non-Newtonian fluid in laminar flow, the flow characteristic is no longer equal to the magnitude of the wall shear rate. However, the flow characteristic is still related uniquely to tw because the value of the integral, and hence the right hand side of equation 3.17, is determined by the value of tw. 

Equation 3.29 is helpful in showing how the value of the correction factor in the Rabinowitsch-Mooney equation corresponds to different types of flow behaviour. For a Newtonian fluid, n = 1 and therefore the correction factor has the value unity. Shear thinning behaviour corresponds to < 1 and consequently the correction factor has values greater than unity, showing that the wall shear rate yw is of greater magnitude than the value for Newtonian flow. Similarly, for shear thickening behaviour, yw is of a... 

Equations 4 and 5 have been used to predict flux values for a variety of macromolecular solutions and channel geometries ( ). The theoretical values were in good agreement with the experimental values. Figure 13 illustrates the 0.33 power dependence on wall shear rate per unit channel length (U/dj L). 

Figure 13. Flux dependence on wall shear rate in laminar flow...

Van Reis et al. [92] reported the scale-up of a HF system for the recovery of human tissue plasminogen activator (t-PA) produced by recombinant CHO cells from the 2.5-m to the 180-m scale. A robust and reproducible process was achieved by combining hnear scale-up principles, control of fluid dynamic parameters and experimentally defined limits of product retention, which meant maintaining channel length, wall shear rate and flux constant. 

Kempken et al. [113] employed a rotating disc filter to harvest CHO cells, and observed that the filter could be operated at low transmembrane-pressure with high wall shear rates, leading to high filtrate flow rates, high product yields and minimum fouling. They concluded that their system offered a powerful alternative to conventional tangential flow filtration. 

Tu, as a function of Newtonian wall shear rate, F, for a number of common polymers. Consider the melt spinning of PS at a volumetric flow rate of 4.06 x 10 cm /s through a spinneret that contains 100 identical holes of radius 1.73 X 10 cm and length 3.46 x 10 cm. Assume that the molecular weight distribution is broad. 

The ratio of entrance pressure drop to shear stress at the capillary wall versus Newtonian wall shear rate, r.i, PP n, PS o, LDPE +, HDPE , 2.5% PIBin mineral oil x, 10% PIB in decalin a, NBS-OB oil. Reprinted, by permission, from Z. Tadmorand C. G. Gogos, Principles of Polymer Processing, p. 537. Copyright 1979 by John Wiley Sons, Inc. 

Figure 3 shows a curve of the 13 vol. % 4 graphite slurry in water as determined with a capillary viscometer. This is the same material examined on the cone and plate unit. The apparent viscosity is 1.6 poise at a shear rate of 103 sec. 1 and decreases to a value of 0.39 poise at a shear rate of 104 sec."1, the viscosity data being corrected for the true wall shear rate. The flow curves obtained from both instruments agree quite closely. 

In a 1991 study by van Reis et al. (5), a filtration operation as applied to harvest of animal cells was optimized by the use of dimensional analysis. The fluid dynamic variables used in the scale-up work were the length of the fibers (L, per stage), the fiber diameter (D), the number of fibers per cartridge (k), the density of the culture (p), and the viscosity of the culture (p). From these variables, scale-up parameters such as wall shear rate (y ) and its effect on flux (L/m /h) were derived. Based on these calculations, an optimum wall shear rate for membrane utilization, operating time, and flux was found. However, because there is no single mathematical expression relating all of these parameters simultaneously, the optimal solution required additional experimental research. 

In the case where a liquid suspension of fine particles of radius r (cm) flows along a solid surface at a wall shear rate (s ), the effective diffusivity Dp (cm s ) of particles in the direction perpendicular to the surface can be correlated by the following empirical equation [7] ... 

Fig. 4. SEM images of a SMY deposited over a polysulfone ultrafiltration membrane at (A) low magnification (<- 200) and (B) high magnification (<->700). The image is captured during the protein filtration portion of cycle 10. The primary feed contained 5 g/L of cellulase, and the secondary feed contained 5.36 g/L of yeast. The cycle conditions were tf=300 s, tsf= 30 s, and tb = 2, with an average TMP of 15 psi maintained during forward as well as reverse filtration and a wall shear rate of 100 s...

Fig. 9. Permeate flux vs time during ultrafiltration of 5.0 g/L of cellulase. The solid line represents SMY and backflushing with Pf= 30 psi, Ph = 15 psi, f( = 300 s, tsf= 5 s, and tb = 2s the line with short and long dashes represents SMY and backflushing under the same conditions but with t = 10 s. The yeast concentration in the secondary feed was 4.0 g/L. The dashed line is the permeate flux obtained without deposition of a secondary membrane or backflushing. A wall shear rate of 1300 s 1 was used. LMH = L(m2 h).

Fig. 10. Average permeate flux after 6000 s of ultrafiltration of 5.0 g/L of cellulase for different concentrations of yeast in secondary feed for wall shear rates of 400 s 1 (— —) and 100 s x with Pf = Ph = 7.5 psi, tf=300 s, f(, = 3s,and tsf= 30 s. Also...

Fig. 12.11 Uncorrected shear stress versus Newtonian wall shear rate for ABS Cycolac T measured and calculated using various thermal boundary conditions. Dq — 0.319 cm L/Dq = 30 T0 — 505 K. [Reprinted by permission from H. W. Cox and C. W. Macosko, Viscous Dissipation...

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