Method for Bingham plastics

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. 

Weltmann (W4) presented this relationship on a friction factor-Reynolds number diagram similar to Fig. 4 for Bingham-plastic fluids. Excellent agreement between predicted and measured results was found by Salt for two carboxymethylcellulose solutions Weltmann shows no data to support her somewhat more useful rearrangement but cites three literature references for this purpose. Review of these shows that none dealt explicitly with this method of approach, as claimed. 

In the same way as there are many equations for predicting friction factor for turbulent Newtonian flow, there are munerous equations for time-independent non-Newtonian fluids most of these are based on dimensional considerations combined with experimental observations [Govier and Aziz, 1982 Heywood and Cheng, 1984]. There is a preponderance of correlations based on the power-law fluid behaviour and additionally some expressions are available for Bingham plastic fluids [Tomita, 1959 Wilson and Thomas, 1985], Here only a selection of widely used and proven methods is presented. 

Newtonian fluids can be correlated by this method that is, the same correlation applies to both Newtonian and non-Newtonian fluids when the Newtonian Reynolds number is replaced by either Eq. (7-40) for the power law fluid model or Eq. (7-41) for the Bingham plastic fluid model. As a first approximation, therefore, we may assume that the same result would apply to friction loss in valves and fittings as described by the 2-K or 3-K models [Eq. 7-38)]. 

For greater concentrations of fine particles the suspension is more likely to be non-Newtonian, in which case the viscous properties can probably be adequately described by the power law or Bingham plastic models. The pressure drop-flow relationship for pipe flow under these conditions can be determined by the methods presented in Chapters 6 and 7. 

Despite the fact that equation (3.37) is applicable to all kinds of time-independent fluids, numerous workers have presented expressions for turbulent flow friction factors for specific fluid models. For instance, Tomita [1959] applied the concept of the Prandtl mixing length and put forward modified definitions of the friction factor and Reynolds number for the turbulent flow of Bingham Plastic fluids in smooth pipes so that the Nikuradse equation, i.e. equation (3.37) with n = 1, could be used. Though he tested the applicability of his method using his own data in the range 2000 < Reg(l — 4>f 3 — )< 10, the validity of this approach has not been established using independent experimental data. 

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). 

Most adsorbed surfactant and polymer coils at the oil-water (0/W) interface show non-Newtonian rheological behavior. The surface shear viscosity Pg depends on the applied shear rate, showing shear thinning at high shear rates. Some films also show Bingham plastic behavior with a measurable yield stress. Many adsorbed polymers and proteins show viscoelastic behavior and one can measure viscous and elastic components using sinusoidally oscillating surface dilation. For example the complex dilational modulus c obtained can be split into an in-phase (the elastic component e ) and an out-of-phase (the viscous component e") components. Creep and stress relaxation methods can be applied to study viscoelasticity. 

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