Power law, rheology

A typical graph of drag ratio as a function of superficial air velocity is shown in Figure 5.5 in which each curve refers to a constant superficial liquid velocity. The liquids in question exhibited power law rheology and the corresponding values of the Metzner and Reed Reynolds numbers ReMR based on the superficial liquid velocity uL (see Chapter 3) are given. The following characteristics of the curves may be noted ... 

The material behaviors considered will include linear elasticity plus linear or nonlinear creep behavior. The nonlinear case will be restricted to power-law rheologies. In some cases the elasticity will be idealized as rigid. In ceramics, it is commonly the case that creep occurs by mass transport on the grain boundaries.1 This usually leads to a linear rheology. In the models considered,... 

There are many pseudoplastic food products that display more complex rheological behavior and with a yield stress that can be characterized in two ways, either by an extension of the power law rheological equation of Herschel-Bulkley ... 

Parameters of the power-law rheological equation for pseudoplastic substances... 

It should be noted that for a sufficiently wide range of shear rates in real fluids, k and n are not constant. This does not prevent one from widely using the power-law rheological equation, since in practice one usually deals with a rather narrow range of shear rates. 

By plotting these data on linear and logarithmic scales, ascertain the type of fluid behaviour, e.g. Newtonian, or shear-thinning, or shear-thickening, etc. Also, if the liquid is taken to have power-law rheology, calculate the consistency and flow-behaviour indices respectively for this liquid. 

A viscous material (power-law rheology) is to be processed in a mixing vessel under laminar conditions. A sample of the material is tested using a laboratory scale mixer. If the mixing time is to be the same in the small and large-scale equipments, estimate the torque ratio for a scale-up factor of 5 for the range 1 > n > 0.2. 

For non-Newtonian flow (based on a power law rheological model) ... 

Pressure Drop Prediction for Slurries Exhibiting Power-law Rheology... 

Outline the steps in the procedure for predicting pipeline pressure drop for slurries exhibiting power-law rheology. 

Moat lubricants which contain polymers will show a constant viscosity Newtonian behavior up to a critical shear rate. Above that shear rate, the viscosity of these polymer solutions decreases with increasing shear rate and as the first approximation this Shear-thinning region can be represented by a power-law rheological model. At very high shear rates, polymer solutions will also characteristically approach an upper Newtonian limit. 

Incorporation of viscosity variations in non-elastic generalized Newtonian flow models is based on using empirical rheological relationships such as the power law or Carreau equation, described in Chapter 1. In these relationships fluid viscosity is given as a function of shear rate and material parameters. Therefore in the application of finite element schemes to non-Newtonian flow, shear rate at the elemental level should be calculated and used to update the fluid viscosity. The shear rale is defined as the second invariant of the rate of deformation tensor as (Bird et at.., 1977)... 

Rheology. Flow properties of latices are important during processing and in many latex appHcations such as dipped goods, paint, inks (qv), and fabric coatings. For dilute, nonionic latices, the relative latex viscosity is a power—law expansion of the particle volume fraction. The terms in the expansion account for flow around the particles and particle—particle interactions. For ionic latices, electrostatic contributions to the flow around the diffuse double layer and enhanced particle—particle interactions must be considered (92). A relative viscosity relationship for concentrated latices was first presented in 1972 (93). A review of empirical relative viscosity models is available (92). In practice, latex viscosity measurements are carried out with rotational viscometers (see Rpleologicalmeasurement). 

The rheological properties of a particular suspension may be approximated reasonably well by either a power-law or a Bingham-plastic model over the shear rate range of 10 to 50 s. If the consistency coefficient k is 10 N s, /m-2 and the flow behaviour index n is 0.2 in the power law model, what will be the approximate values of the yield stress and of the plastic viscosity in the Bingham-plastic model ... 

A Newtonian liquid of viscosity 0.1 N s/m2 is flowing through a pipe of 25 mm diameter and 20 m in lenglh, and the pressure drop is 105 N/m2. As a result of a process change a small quantity of polymer is added to the liquid and this causes the liquid to exhibit non-Newtonian characteristics its rheology is described adequately by the power-law model and the flow index is 0.33. The apparent viscosity of the modified fluid is equal to ihc viscosity of the original liquid at a shear rate of 1000 s L... 

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. 

In a series of experiments on the flow of flocculated kaolin suspensions in laboratory and industrial scale pipelines(26-27-2Sl, measurements of pressure drop were made as a function of flowrate. Results were obtained using a laboratory capillary-tube viscometer, and pipelines of 42 mm and 205 mm diameter arranged in a recirculating loop. The rheology of all of the suspensions was described by the power-law model with a power law index less than unity, that is they were all shear-thinning. The behaviour in the laminar region can be described by the equation ... 

A liquid w hose rheology can be represented by the power law model is flowing under streamline conditions through a pipe of 5 mm diameter. If the mean velocity of flow in I nt/s and the velocity at the pipe axis is 1.2 m/s, what is the value of the power law index n ... 

The rheological behavior of the compound can be described by the well-known power law h(x) describes the distance between the surface given by y = 0 and the roll surface. 

The polymer rheology is modeled by extending the usual power-law equation to include second-order shear-rate effects and temperature dependence assuming Arrhenius type relationship. 

Many materials are conveyed within a process facility by means of pumping and flow in a circular pipe. From a conceptual standpoint, such a flow offers an excellent opportunity for rheological measurement. In pipe flow, the velocity profile for a fluid that shows shear thinning behavior deviates dramatically from that found for a Newtonian fluid, which is characterized by a single shear viscosity. This is easily illustrated for a power-law fluid, which is a simple model for shear thinning [1]. The relationship between the shear stress, a, and the shear rate, y, of such a fluid is characterized by two parameters, a power-law exponent, n, and a constant, m, through... 

The theoretical basis for spatially resolved rheological measurements rests with the traditional theory of viscometric flows [2, 5, 6]. Such flows are kinematically equivalent to unidirectional steady simple shearing flow between two parallel plates. For a general complex liquid, three functions are necessary to describe the properties of the material fully two normal stress functions, Nj and N2 and one shear stress function, a. All three of these depend upon the shear rate. In general, the functional form of this dependency is not known a priori. However, there are many accepted models that can be used to approximate the behavior, one of which is the power-law model described above. 

Other schemes have been proposed in which data are fit to a lower, even order polynomial [19] or to specific rheological models and the parameters in those models calculated [29]. This second approach can be justified in those cases when the range of behavior expected for the shear viscosity is limited. For example, if it is clear that power-law fluid behavior is expected over the shear rate range of interest, then it would be possible to calculate the power-law parameters directly from the velocity profile and pressure drop measurement using the theoretical velocity profile... 

Cloud, J.E. and Clark, P.E. "Stimulation Fluid Rheology III. Alternatives to the Power Law Fluid Model for Crosslinked Fluids," SPE paper 9332, 1980 SPE Annual Technical Conference and Exhibition, Dallas, September 21 24. 

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