Plastic system, Bingham

The system shows a (dynamic) yield stress cr that can be obtained by extrapolation to zero shear rate [8]. Clearly, at and below cr the viscosity ri-roo. The slope of the hnear curve gives the plastic viscosity ri i- Some systems, such as clay suspensions, may show a yield stress above a certain clay concentration. 

The Bingham equation describes the shear stress/shear rate behaviour of many shear thiiming materials at low shear rates. Unfortunately, the value of obtained depends on the shear rate ranges used for the extrapolation procedure. 

---

Both Eqs. (20) and (21) are applicable for small deviations from Newtonian behavior only. They suggest an enhancement in mass transfer due to shear-thinning behavior. Figure 10 shows the extent of enhancement in mass transfer attributable to the power law index. Bhavaraju et al. (1978) also reported a similar increase in the value of the mass transfer coefficient for bubbles rising (Re 1) in Bingham plastic systems ... 

The apparent viscosity that is to be used in the Reynolds number has to be measured in the viscometer at the pipeline shear rate. This can be obtained by fitting the flow curve to the power law relationship given by equation (4.25). In turbulent non-Newtonian flow the friction factor is a unique function of the Reynold s number. For Bingham plastic systems, the Reynold number is calculated by using the plastic viscosity since it remains constant with increasing shear rates. For pseudoplastic flow, the Reynold s number is calculated using an estimated apparent viscosity that is obtained by extrapolation to infinite shear rate. 

You want to determine how fast a rock will settle in mud, which behaves like a Bingham plastic. The first step is to perform a dimensional analysis of the system. 

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... 

So far we have restricted our discussion to Newtonian liquids, but the analysis will change somewhat if the systems are non-Newtonian. A useful illustration of the problems that arise is the case of a Bingham plastic. This gives us a linear response, as does a Newtonian liquid, but in this case there is an intercept or yield stress. The constitutive equation for a Bingham plastic is... 

A viscometer can be used to study the yield stress and viscosity of cement pastes (Section 1.3.1). This is carried out by plotting the shear rate against shear stress as shown in Fig. 2.4 for cement pastes of various water cement ratios. These cement pastes are generally considered to exhibit Bingham plastic behavior where the yield value is the intercept on the shear stress axis and is related to cohesion, and the slope of the line is the apparent viscosity which is related to the consistency or workability of the system. The following general observations can be made ... 

Using the rheology of the ceramic system, we can write in terms of the velocity gradient dvjdx. Then, by integration, the velocity profile can be obtained. These velocity profiles for Newtonian, Bingham plastic, and Crossian rheology follow. 

Vehicles that exhibit the unusual property of Bingham-type plastic rheological flow are characterized by the need to overcome a finite yield stress before flow is initiated. Permanent suspension of most pharmaceutical systems requires yield-stress values of at least 2-5 Pa (20-50 dyn/cm ). Bingham plastic flow is rarely produced by pharmaceutical gums and hydrophilic colloids. National Formulary (NF) carbomers exhibit a sufficiently high yield value at low solution concentration and low viscosity to produce permanent suspensions. The carbomers, however, require a pH value between 6 and 8 for maximum suspension performance. The polymer is essentially incompatible... 

If the decrease in viscosity is very large at small shear rates, the system is sometimes called pseudoplastic (curves 4 and 5). Commonly, concentrated suspensions show a plastic behavior, that is, there is no response until a limiting yield stress oy has been exceeded. If the flow is linear above oy, the system is called Bingham plastic (curve 4) and can be expressed by the Bingham model (5) ... 

One further feature must be mentioned about pharmaceutical suspensions, namely, their desirable rheolt ical properties (7). In practice, a Bingham plastic" behavior is most used a minimum shear stress yield stress) is needed for the suspension to begin to flow. For tower stresses—and, of course, when the system is left undisturbed—the viscosity is so high that the particles will likely remain homogeneously dispersed. According to Falkiewicz (7). thixotropy is another flow characteristic that can be useful, since in thixotropic fluids a finite lime is needed to rebuild the structure after, for instance, shaking it for administration. For this reason, most formulations contain thixotropic flow regulators intended to confer optima viscous flow propertie.s to the suspensions. The reader is referred to Chapter 5 of this book for details. 

The plastic and pseudo-plastic systems are described by the Bingham equation. 

An examination of equation (2.14) shows that for any fluid with a finite yield point, the versus Xb curve approaches the Xb axis at zero slope, due to the requirement for such a system that the shear rate must become zero at finite Xb. This may lead to apparent shear-thinning characteristics being ascribed to systems, irrespective of the actual form of their flow curves above the yield point, i.e., whether Bingham plastic, shear-thickening (with a yield stress), or shear-thinning (with a yield stress). 

Viscometric measurements suggest that an aqueous carbopol solution behaves as a Bingham plastic fluid with yield stress of 1.96 Pa and plastic viscosity 3.80 Pa s. The liquid flows down a plate inclined at an angle 6 to the horizontal. Derive an expression for the volumetric flow rate per unit width of the plate as a function of the system variables. Then, show that the following experimental results for 0 = 5° are consistent with the theoretical predictions. 

Plug flow n. Movement of a material as a unit without shearing within the mass. This is an extreme seldom realized in practice, but can occur over the center of a Bingham-plastic stream or in a system where the fluid does not wet the bounding walls. As compared with Newtonian flow, the more pseudoplastic the plastic melt, the more nearly sluggish is its flow. 

Slow sedimentation of particles will occur, for example, in an activated sludge or in fine particle catalyst suspensions. For those lands of systems, a homt eneous distribution of solids is characteristic. Here, the liftoff from the vessel bottom as well as the state of a homogeneous suspension can be achieved with a comparably low power input or only slight movement of the liquid. On the other hand, at higher solids concentrations a pseudoplastic flow characteristic of the suspension can occur. As an example, concentrations of only 6% of fibrous material - typically known from paper industry - can lead to this non-Newtonian behavior Frequently observed in suspensions with high solids concentrations is a Bingham plastic behavior. In this case, if a certain amount of shear is not introduced by agitation, the system behaves like an elastic solid body or a gel. 

Plastic systems initially behave as sohd (Bingham) bodies (for T < Tq) and resist deformation until a yield stress (tq) is reached. 

To date many paint manufacturers use one-point measurement for measurement of paint consistency. This can be carried out, for example, with a simple Brookfield viscometer using one spindle at a given rpm. This one-point measurement can be misleading [49]. To illustrate this point let us consider three systems, namely Newtonian, Bingham plastic and pseudoplastic with thixotropy as illustrated in Fig. 4.27. At a specific shear rate, all the three systems show the same apparent viscosity although their flow behavior (using the full shear-stress curves) is significantly different [49]. These systems will show entirely different behavior on application at the shear rate at which the apparent viscosity is the same. A study of the flow curves indicates that the Newtonian system will flow at extremely low shear rates, whereas the plastic and thixotropic systems will show reluctance to do so because of their yield values. This is clearly reflected in the final film properties. Once the yield value is overcome, the... 

---