Laminar and power law flows

Plug flow is an idealization. Deviations arise with viscous or non-Newtonian fluids. A mathematically simple deviation from the plug flow pattern is that of power law fluids whose velocity in a tube depends on the radial position, (3 = r/R, according to the equation, 

Along any particular radial position, all molecules have the same residence time, that is, plug flow conversion is achieved on that streamline. The average over the cross section will be different. 

The equation for conversion in laminar flow is developed in problem P4.08.01 which also compares the performance with that in plug flow when the rate equation is first or second order. The residual concentration is higher (the conversion is lower) with laminar flow. A similar result is found in problem P4.08.09 which is for straight line radial variation or n = eo. Numerical magnitudes of the variation of concentration over a cross section are found in problem P4.08.07. Other problems point out the errors in calculated specific rates when the data are assumed to be plug flow instead of laminar. 

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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. 

The power law model is very popular for representing the viscosity of a wide variety of non-Newtonian fluids because of its simplicity and versatility. However, care should be exercised in its application for reliable results, the range of shear stress (or shear rate) expected in the application should not extend beyond the range of the rheological data used to evaluate the model parameters. Both laminar and turbulent pipe flow of highly loaded slurries of line particles, for example, can often be adequately represented by either of these two models, as shown by Darby et al. (1992). 

As with slurries following a power-law flow model, it is necessary to reliably predict the pressure drop in a horizontal pipe of diameter D under laminar, fully developed flow conditions. A fundamental analysis of the Bingham plastic model yields the following expression for the mean velocity in terms of the yield stress Ty and the wall shear stress tq. 

Turbulent versus laminar flow. The subject of turbulent versus laminar flow is important operationally, but flow turbulence, despite the advanees made during the past decades, is still empirieally grounded and represents a researeh discipline in its own right. We will not extend our annular flow work to turbulent regimes, except note that the ideas discussed so far (and next for power law flows) can be applied to appropriate eurve-fitted velocity profiles. 

For laminar flow of power law fluids in channels of noncircular cross section, see Schecter AIChE J., 7, 445 48 [1961]), Wheeler and Wissler (AJChE J., 11, 207-212 [1965]), Bird, Armstrong, and Hassager Dynamics of Polymeric Liquids, vol. 1 Fluid Mechanics, Wiley, New York, 1977), and Skelland Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967). 

Experimental evidence regarding the power law is somewhat contradictory. A constant value of 3 = 0..5 is considered to give a good fit to experimental data by many authors. According to Awbi, p depends on the flow regime and has a value of 0..5 for fully turbulent flow and 1,0 for laminar flow.- fn practice the value of P tends to be between 0.6 and 0.7. 

As in the case of Newtonian fluids, one of the most important practical problems involving non-Newtonian fluids is the calculation of the pressure drop for flow in pipelines. The flow is much more likely to be streamline, or laminar, because non-Newtonian fluids usually have very much higher apparent viscosities than most simple Newtonian fluids. Furthermore, the difference in behaviour is much greater for laminar flow where viscosity plays such an important role than for turbulent flow. Attention will initially be focused on laminar-flow, with particular reference to the flow of power-law and Bingham-plastic fluids. 

What will be the pressure drop, when the suspension is flowing under laminar conditions in a pipe 200 m long and 40 mm diameter, when the centre line velocity is 1 m/s, according to the power-law model Calculate the centre-line velocity for this pressure drop for the Bingham-plastic model. 

The critical value of the Reynolds number (Remit) for the transition from laminar to turbulent flow may be calculated from the Ryan and Johnson001 stability parameter, defined earlier by equation 3.56. For a power-law fluid, this becomes ... 

Equation 5.2 is found to hold well for non-Newtonian shear-thinning suspensions as well, provided that the liquid flow is turbulent. However, for laminar flow of the liquid, equation 5.2 considerably overpredicts the liquid hold-up e/,. The extent of overprediction increases as the degree of shear-thinning increases and as the liquid Reynolds number becomes progressively less. A modified parameter X has therefore been defined 16 171 for a power-law fluid (Chapter 3) in such a way that it reduces to X both at the superficial velocity uL equal to the transitional velocity (m )f from streamline to turbulent flow and when the liquid exhibits Newtonian properties. The parameter X is defined by the relation... 

In a series of experiments on the flow of flocculated kaolin suspensions in laboratory and industrial scale pipelines(26-27-2Sl, measurements of pressure drop were made as a function of flowrate. Results were obtained using a laboratory capillary-tube viscometer, and pipelines of 42 mm and 205 mm diameter arranged in a recirculating loop. The rheology of all of the suspensions was described by the power-law model with a power law index less than unity, that is they were all shear-thinning. The behaviour in the laminar region can be described by the equation ... 

Equation 11.12 does not fit velocity profiles measured in a turbulent boundary layer and an alternative approach must be used. In the simplified treatment of the flow conditions within the turbulent boundary layer the existence of the buffer layer, shown in Figure 11.1, is neglected and it is assumed that the boundary layer consists of a laminar sub-layer, in which momentum transfer is by molecular motion alone, outside which there is a turbulent region in which transfer is effected entirely by eddy motion (Figure 11.7). The approach is based on the assumption that the shear stress at a plane surface can be calculated from the simple power law developed by Blasius, already referred to in Chapter 3. 

A fluid which exhibits non-Newtonian behaviour is flowing in a pipe of diameter 70 mm and the pressure drop over a 2 m length of pipe is 4 x 104 N/m2. A pitot lube is used to measure the velocity profile over the cross-section. Confirm that the information given below is consistent with the laminar flow of a power-law fluid. Calculate the power-law index n and consistency coefficient K. 

A highly concentrated suspension of flocculated kaolin in water behaves as a pseudo-homogeneous fluid with shear-thinning characteristics which can be represented approximately by the Ostwald-de Waele power law, with an index of 0.15. It is found that, if air is injected into the suspension when in laminar flow, the pressure gradient may be reduced even though the flowrate of suspension is kept constant, Explain how this is possible in slug flow and estimate the possible reduction in pressure gradient for equal volumetric flowrates of suspension and air. 

Because the shear stress and shear rate are negative in pipe flow, the appropriate form of the power law model for laminar pipe flow is... 

The flow distribution in a cylindrical vessel has the shape of an isosceles triangle with apex on the axis, thus u = u0(j 3j, /3 = r/R. Find the mean velocity and the mean conversion of a reaction with a power law rate equation. Compare with laminar and uniform flows. 

For laminar flow of a power-law fluid through a cylindrical tube, the relation between mean velocity u and pressure drop —AP is given by ... 

Transitional Flow. Reynolds numbers and friction factors at which the flow changes from laminar to turbulent are indicated by the breaks in the plots of Figures 6.4(a) and (b). For Bingham models, data are shown directly on Figure 6.6. For power-law liquids an equation for the critical Reynolds number is due to Mishra and Triparthi [Trans. IChE 51, T141 (1973)],... 

Figure 6.5. Friction factors in laminar and turbulent flows of power-law and Bingham liquids, (a) For pseudoplastic liquids represented by tw = K [WID) , with K and n constant or dependent on r l/V/ = [4.0/(n )0 75] log10[Re /( "2)] — 0.40/(k )1 2j, [Dodge and Metzner, AIChE J. 5, 159 (7959)]. (b) For Bingham plastics, ReB = DVp/pB, He = 10D2plp% [Hanks and Dadia, AIChE J. 17, 554 (J971)].

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