Pipe 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... 

For a Bingham plastic fluid flow in a circular pipe and annular space, the effective viscosities are given as [61]. 

For pipe flow of Bingham plastic type drilling fluid, the following can be used ... 

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. 

Fluids whose behaviour can be approximated by the power-law or Bingham-plastic equation are essentially special cases, and frequently the rheology may be very much more complex so that it may not be possible to fit simple algebraic equations to the flow curves. It is therefore desirable to adopt a more general approach for time-independent fluids in fully-developed flow which is now introduced. For a more detailed treatment and for examples of its application, reference should be made to more specialist sources/14-17) If the shear stress is a function of the shear rate, it is possible to invert the relation to give the shear rate, y = —dux/ds, as a function of the shear stress, where the negative sign is included here because velocity decreases from the pipe centre outwards. 

A Bingham plastic material is flowing under streamline conditions in a pipe of circular cross-section. What are the conditions for one half of the total flow to be within the central core across which the velocity profile is fiat The shear stress acting within die fluid Ry varies with velocity gradient du,/dy according to the relation ... 

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. 

The inclusion of significant fitting friction loss in piping systems requires a somewhat different procedure for the solution of flow problems than that which was used in the absence of fitting losses in Chapter 6. We will consider the same classes of problems as before, i.e. unknown driving force, unknown flow rate, and unknown diameter for Newtonian, power law, and Bingham plastics. The governing equation, as before, is the Bernoulli equation, written in the form... 

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. 

The slurry behaves as a non-Newtonian fluid, which can be described as a Bingham plastic with a yield stress of 40 dyn/cm2 and a limiting viscosity of 100 cP. Calculate the pressure gradient (in psi/ft) for this slurry flowing at a velocity of 8 ft/s in a 10 in. ID pipe. 

As the shear stress for flow in a pipe varies from zero at the centre-line to a maximum at the wall, genuine flow, ie deformation, of a Bingham plastic occurs only in that part of the cross section where the shear stress is greater than the yield stress ry. In the part where r< rv the material remains as a solid plug and is transported by the genuinely flowing outer material. 

Buckingham (BIO) integrated the rheological equation [Eq. (2)] for the isothermal flow of Bingham plastics in round pipes. The familiar and important resulting equation ... 

Since the relationship between 8V/D and DAP/4L is independent of pipe diameter, the same is true of Eqs. (17) to (19) inclusive. They are applicable to all four types of common flow behavior, i.e., to pseudoplastic, Newtonian, Bingham-plastic, and dilatant fluids. 

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. 

A Bingham-plastic fluid (yield stress 14.35 N/m2 and plastic viscosity 0.150 Ns/m2) is flowing through a pipe of diameter 40 mm and length 200 m. Starting with the rheological equation, show that the relation between pressure gradient —AP/l and volumetric flowrate Q is ... 

Figure 28. Comparison of calculated frictional pressures for flow of a drilling fluid in a pipe using (a) power law and (b) Bingham plastic models. (Data from reference 106.)...

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