Bingham plastic fluids laminar flow

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. 

Pigford (P5) has stated that the heat transfer coefficients of Bingham-plastic fluids in laminar flow will be greater than those of Newtonian fluids by a factor of approximately 1 + ( ). Furthermore, the heat... 

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian liquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic fluid described by Eq. (6-3), in a pipe of diameter D and a pressure drop per unit length AP/L, the flow rate is given by... 

For the Bingham plastic fluid, there is no abrupt transition from laminar to turbulent flow, as is observed for Newtonian fluids. Instead, a gradual deviation from purely laminar flow to fully turbulent flow occurs. For turbulent flow, the friction factor can be represented by the empirical expression of Darby and Melson (1982) (as modified by Darby et al., 1992) ... 

Figure 3.4 Schematic velocity distribution for laminar flow of a Bingham plastic fluid in a pipe...

A similar procedure can, in principle, be used for other rheological models by inserting an appropriate expression for shear stress in equation (3.62). The analogous result for the laminar flow of Bingham plastic fluids in this geometry is given here ... 

Figure 3.20 Dimensionless flowrate for Bingham plastic fluids in laminar flow through an annulus...

Figure 3.21 Chart for the estimation ofpressure gradient for laminar flow of Bingham plastic fluids in an annulus...

Laminar boundary layer flow of Bingham plastic fluids over a plate... 

To calculate the pressure drop for a Bingham plastic fluid with a yield stress, methods are available for laminar flow and are discussed in detail elsewhere (Cl, PI, S2). 

The transition to turbulent flow begins at Re R in the range of 2,000 to 2,500 (Metzuer and Reed, AIChE J., 1, 434 [1955]). For Bingham plastic materials, K and n must be evaluated for the condition in question in order to determine Re R and establish whether the flow is laminar. An alternative method for Bingham plastics is by Hanks (Hanks, AIChE J., 9, 306 [1963] 14, 691 [1968] Hanks and Pratt, Soc. Petrol. Engrs. J., 7, 342 [1967] and Govier and Aziz, pp. 213-215). The transition from laminar to turbulent flow is influenced by viscoelastic properties (Metzuer and Park, J. Fluid Mech., 20, 291 [1964]) with the critical value of Re R increased to beyond 10,000 for some materials. 

Corresponding expressions for the friction loss in laminar and turbulent flow for non-Newtonian fluids in pipes, for the two simplest (two-parameter) models—the power law and Bingham plastic—can be evaluated in a similar manner. 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, extreme care should be exercised in its application, because any application involving extrapolation beyond the range of shear stress (or shear rate) represented by the data used to determine the model parameters can lead to misleading or erroneous results. 

In laminar flow of Bingham-plastic types of materials the kinetic energy of the stream would be expected to vary from V2/2gc at very low flow rates (when the fluid over the entire cross section of the pipe moves as a solid plug) to V2/gc at high flow rates when the plug-flow zone is of negligible breadth and the velocity profile parabolic as for the flow of Newtonian fluids. McMillen (M5) has solved the problem for intermediate flow rates, and for practical purposes one may conclude... 

The shearing characteristics of non-Newtonian fluids are shown in Fig. 3.24 of Volume 1. This type of fluid remains rigid when the shear stress is less than the yield stress Ry and flows like a Newtonian fluid when the shear stress exceeds Ry. Examples of Bingham plastics are many fine suspensions and pastes including sewage sludge and toothpaste. The velocity profile in laminar flow is shown in Fig. 3c. 

Laminar flow conditions cease to exist at Rcmod = 2100. The calculation of the critical velocity corresponding to Rcmod = 2100 requires an iterative procedure. For known rheology (p, m, n, Xq) and pipe diameter (D), a value of the wall shear stress is assumed which, in turn, allows the calculation of Rp, from equation (3.9), and Q and Qp from equations (3.14b) and (3.14a) respectively. Thus, all quanties are then known and the value of Rcmod can be calculated. The procedure is terminated when the value of x has been found which makes RCjnod = 2100, as illustrated in example 3.4 for the special case of n = 1, i.e., for the Bingham plastic model, and in example 3.5 for a Herschel-Bulkley fluid. Detailed comparisons between the predictions of equation (3.34) and experimental data reveal an improvement in the predictions, though the values of the critical velocity obtained using the criterion Rqmr = 2100 are only 20-25% lower than those predicted by equation (3.34). Furthermore, the two... 

The heating of a viscous fluid in laminar flow in a tube of radius R (diameter, D) will now be considered. Prior to the entry plane z < 0), the fluid temperature is uniform at Tf for z > 0, the temperature of the fluid will vary in both radial and axial directions as a result of heat transfer at the tube wall. A thermal energy balance will first be made on a differential fluid element to derive the basic governing equation for heat transfer. The solution of this equation for the power-law and the Bingham plastic models will then be presented. 

Laminar flow will now be considered for both power-law and Bingham plastic model fluids, followed by a short discussion for turbiflent flow. 

Although there has been little systematic study of deposition in tm-bulent flow of non-Newtonian slurries, the existing evidence for aqueous slurries suggests that the correlations in Equation 3 may be used for estimation purposes if the viscosity p is replaced by the high shear rate asymptotic quotient of shear stress and shear rate. Thus for example, a Bingham carrier fluid would be represented by its plastic viscosity. It must be emphasized that this recommendation is a tentative one and is restricted to turbulent slurry flows. The situation for laminar slurry flows will be quite different. 

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