Bingham-Newtonian Liquid

For non-Newtonian liquids and suspensions, an apparent viscosity is determined using correlations which include power input and the Reynolds number. Scale-up comparisons based on heat generation data only were determined by comparison of results from RC1 experiments and from a 675-liter reactor [208]. In the experiments, a Bingham plastic fluid was used to determine the film heat transfer coefficient. This presents a worst case because of the low thermal conductivity of the Bingham plastic. Calculated inside film heat transfer coefficients determined in the RC1 tests were about 60% lower than the values determined in the pilot plant reactor, even though substantial effort was made to obtain both geometric and kinematic similarity in the pilot reactor. 

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... 

The melt flow under isothermal conditions, when it is described by the rheological equation for the Newtonian or power law liquid, has been studied in detail63 66). The flow of the non-Newtonian liquid in the channels of non-round cross section for the liquid obeying the Sutterby equation have also been studied 67). In particular, the flow in the channels of rectangular and trigonal cross section was studied. In the analysis of the non-isothermal flow, attention should be paid to the analysis 68) of pseudo-plastic Bingham media. 

The Bingham Fluid. The Bingham fluid is an empirical model that represents the rheological behavior of materials that exhibit a no flow region below certain yield stresses, tv, such as polymer emulsions and slurries. Since the material flows like a Newtonian liquid above the yield stress, the Bingham model can be represented by... 

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... 

Both polymeric and some biological reactors often contain non-Newtonian liquids in which viscosity is a function of shear rate. Basically, three types of non-Newtonian liquids are encountered power-law fluids, which consist of pseudoplastic and dilatant fluids viscoplastic (Bingham plastic) fluids and viscoelastic fluids with time-dependent viscosity. Viscoelastic fluids are encountered in bread dough and fluids containing long-chain polymers such as polyamide and polyacrylonitrite that exhibit coelastic flow behavior. These... 

In concentrated suspensions, the particles touch each other. If there is also an attraction between the particles, the suspension may not flow when the shear stress is small it is a solid (Figure C4-14). The stress at which the liquid starts moving is known as the yield stress. Once the liquid yields, it often behaves like a Newtonian liquid with a constant differential viscosity. The behaviour of such Bingham fluids is similar to that of shear thinning fluids ... 

FIG. 155. Types of rheological behaviour (a) Newtonian liquid (b) anomalous (pseudoplastic) liquid (c) Bingham body (d) real plastic body (e) thixotropic body (f) dilatant body. The viscosity is given by the tangent of the indicated angle. 

Figure 4-2. Flow curves for various ideal rheological bodies. A Newtonian liquid. B Pseudoplastic fluid. C Dilatant fluid. D Bingham plastic iii is the yield value). E Pseudoplastic material with a yield value. F Dilatant material with a yield value.

LAMINAR FLOW OF NON-NEWTONIAN LIQUIDS. Because of the difference in the relation between shear stress and velocity gradient, the shape of the velocity profile for non-newtonian liquids differs from that of a newtonian liquid. In the more complicated situations of non-newtonian flow the shape of the profile is determined experimentally. For the simpler cases such as the power-law model [Eq. (3.7)] or the Bingham model [Eq. (3.6)] the same methods used for determining the flow parameters of a newtonian fluid can be used for non-newtonian fluids in these categories. 

The term viscosity, as used above, is not accurate. Dense media at higher solid concentrations exhibit characteristics of non-Newtonian liquids, namely, Bingham plastic or pseudo-plastic behavior, as shown in Fig. 14. ... 

Liquids that follow Newton s law are called Newtonian liquids. In non-Newtonian liquids, the quantity rj, which can be calculated from the quotient, aij/D, also changes with the velocity gradient, or with the shear stress. Newtonian behavior is usually observed for the limiting case D - 0 or Gij - 0. Melts and macromolecular solutions often exhibit non-Newtonian behavior. Non-Newtonian liquids are classified as dilatant, Bingham body, pseudoplastic, thixotropic, or rheopectic liquids. 

A plastic material is one that shows little or no deformation up to a certain level of stress. Above this yield stress the material flows readily. Plasticity is conunon to widely different materials. Many metals yield at strains less than 1%. Concentrated suspensions of solid particles in Newtonian liquids often show a yield stress followed by nearly Newtonian flow. These materials are called viscoplastic or Bingham plastics after E. C. Bingham, who first described paint in this way in 1916. House paint and food substances like margarine, mayonnaise, and ketchup are good examples of viscoplastic materials. 

Perhaps the best picture of a viscoplastic fluid is that of a very viscous, even solidlike, material at low stresses. Over a narrow stress range, which can often be modeled as a single yield stress, its viscosity drops dramatically. This is shown clearly in Figure 2.5.5b, where viscosity drops over five decades as shear stress increases from 1 to 3 Pa. (The drop is even more dramatic in Figure 10.7.2.) Above this yield stress the fluid flows like a relatively low viscosity, even Newtonian, liquid. Because of the different behaviors exhibited by these fluids, the model (Bingham, Casson, etc.) and the range of shear rates used to calculate the parameters must be chosen carefully. In Section 10.7 we will discuss microstructural bases for r. It is also important to note that experimental problems like wall slip are particularly prevelant with viscoplastic materials. Aspects of slip are discussed in Section 5.3. 

Figure 8.9 Shear stress-streun rate plots of various fluids. For shear-shinning liquids (e.g. paint), viscosity is reduced with increasing shear rate and vice versa for shear thickening liquids (e.g. quicksemd). Water and petrol are Newtonian liquids. Bingham plastics, where a yield stress is required for flow, can be, for example, toothpaste and butter...

A theoretical idealisation of pseudo-plastic with yield value behaviour is the Bingham plastic (see curve 3 of Fig. 6.3), which deforms elastically until the yield stress Ty is reached and then flows like a Newtonian liquid with 7 linearly related to t. When r is greater than Ty ... 

Plastic fluids are Newtonian or pseudoplastic liquids that exhibit a yield value (Fig. 3a and b, curves C). At rest they behave like a solid due to their interparticle association. The external force has to overcome these attractive forces between the particles and disrupt the structure. Beyond this point, the material changes its behavior from that of a solid to that of a liquid. The viscosity can then either be a constant (ideal Bingham liquid) or a function of the shear rate. In the latter case, the viscosity can initially decrease and then become a constant (real Bingham liquid) or continuously decrease, as in the case of a pseudoplastic liquid (Casson liquid). Plastic flow is often observed in flocculated suspensions. 

A vertical belt is moving upward continuously through a liquid bath, at a velocity V. A film of the liquid adheres to the belt, which tends to drain downward due to gravity. The equilibrium thickness of the film is determined by the steady-state condition at which the downward drainage velocity of the surface of the film is exactly equal to the upward velocity of the belt. Derive an equation for the film thickness if the fluid is (a) Newtonian (b) a Bingham plastic. 

Feed rheology (liquid) Newtonian/pseudoplastic/dilatant/Bingham plastic/thixotropic/rheopectic/viscoelastic. 

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