Laminar flow Bingham plastic

Non-Newtonian Flow For isothermal laminar flow of time-independent non-Newtonian hquids, integration of the Cauchy momentum equations yields the fully developed velocity profile and flow rate-pressure drop relations. For the Bingham plastic flmd 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... 

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. 

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. 

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. 

For the Bingham plastic, there is no abrupt transition from laminar to turbulent flow as is observed for Newtonian fluids. Instead, there is a gradual deviation from purely laminar flow to fully turbulent flow. For turbulent flow, the friction factor can be represented by the empirical expression of Darby and Melson (1981) [as modified by Darby et al. (1992)] ... 

Pressure drop for Bingham plastics in laminar flow... 

Friction factor chart for laminar flow of Bingham plastic materials. (See Friction Factor Charts on page 349.)... 

Friction factor chart for laminar flow of Bingham plastic materials... 

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... 

Mori and Ototake (M17) have presented a mathematical analysis of the laminar flow of Bingham-plastic materials in the annulus between two concentric pipes. The complex results have been shown in convenient graphical form which enables one to solve for the flow rate corresponding to a given pressure gradient. 

The problem of flow through fittings and annular spaces has dealt only with the laminar region the work on annuli was further restricted to the relatively unimportant case of Bingham-plastic behavior. The annular studies to date were quite well chosen in the sense that assumption of Bingham-plastic properties has led to a well-developed method of attack which may not have been possible if more complex non-Newtonian... 

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... 

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)].

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. 

Figure 6.5. Friction factors in laminar and turbulent flows = K 8VID)", with K and n constant or dependent AIChE J. 5, 189 (1959)]. (b) For Bingham plastics, Re ...

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