Viscosity dependent empirical model

In order to determine the distributions of pressure, velocity, and temperature the principles of conservation of mass, conservation of momentum (Newton s Law) and conservation of energy (first law of Thermodynamics) are applied. These conservation principles represent empirical models of the behavior of the physical world. They do not, of course, always apply, e.g., there can be a conversion of mass into energy in some circumstances, but they are adequate for the analysis of the vast majority of engineering problems. These conservation principles lead to the so-called Continuity, Navier-Stokes and Energy equations respectively. These equations involve, beside the basic variables mentioned above, certain fluid properties, e.g., density, p viscosity, p conductivity, k and specific heat, cp. Therefore, to obtain the solution to the equations, the relations between these properties and the pressure and temperature have to be known. (Non-Newtonian fluids in which p depends on the velocity field are not considered here.) As discussed in the previous chapter, there are, however, many practical problems in which the variation of these properties across the flow field can be ignored, i.e., in which the fluid properties can be assumed to be constant in obtaining fire solution. Such solutions are termed constant... 

Attempts to use the analytical result of Equation 3 to correlate experimental data have consistently failed (17). Consequently, empirical and semi-empirical models which include various factors to account for evaporation and non-Newtonian behavior have been proposed (17) but these too have not been able to satisfactorily fit the available data. We have considered the coating flow problem with simultaneous solvent evaporation (11). In the regime of interface mass transfer controlled evaporation, i.e. at high solvent concentration, the fluid mechanics problem can be decoupled from the mass transfer problem via an experimental parameter a which measures the changing time-dependent kinematic viscosity due to solvent evaporation. An analytical expression for the film thickness has been obtained (11) ... 

To keep the particles in suspension, the flow should be at least 0.15m/sec faster than either 1) the critical deposition velocity of the coarsest particles, or 2) the laminar/turbulent flow transition velocity. The flow rate should also be kept below approximately 3 m/sec to minimize pipe wear. The critical deposition velocity is the fluid flow rate that will just keep the coarsest particles suspended, and is dependent on the particle diameter, the effective slurry density, and the slurry viscosity. It is best determined experimentally by slurry loop testing, and for typical slurries it will lie in the range from 1 m/s to 4.5 m/sec. Many empirical models exist for estimating the value of the deposition velocity, such as the following relations, which are valid over the ranges of slurry characteristics typical for coal slurries ... 

It is worth noting that neither Eq. [52] nor Eq. [53] takes into account the dependency of viscosity with shear rate, which may be encountered for values from about 0.2 (in the absence of other type of interactions). In addition, the aforementioned equations do not take into account the influence of particle size and shape, or particle size distribution (see following sections). To date, the complexity of the problem has made it very difficult to obtain a good, ali-encom-passing theoretical or empirical model. 

In fact the viscosity influences both the heat balance and the mass balance. It has been shown how the heat transfer coefficient is affected by the viscosity. But the energy dissipation by the stirrer is also strongly dependent on viscosity (see Section 11.4.4). Furthermore, viscosity affects the molecular diffusion, the mass transport, the mixing time, or the residence time distribution, and therefore the reaction rate. Since the reaction rates influence the chain length and particle sizes, they have a direct effect on the polymer properties. In turn they affect the viscosity and the shear forces - there is a feedback effect. Such complex interactions cannot be described by analytical equations, so empirical models must be used. Often... 

Several relations have been proposed in literature by giving the volume fraction at which co-continuity can be formed as a fimction of the viscosity ratio. These include the relations proposed by Paul and Barlow [65], Jordhamo et al. [66], Metelkin and Blekht [67], and Utracki [68]. All these relations describe the phase inversion as a function of the viscosity ratio. It has been shown by Willemse et al. that the viscosity ratio alone is not sufficient to predict the phase inversion point in all cases [69]. Parameters such as the interfacial tension, the absolute values of the viscosities rather than their ratio, the phase dimensions, and the mixing conditions can have an important effect on the formation of continuous phase structures. Therefore, Willemse et al. proposed a new empirical model by introducing the dependence of the formation of the continuous morphology on material properties (matrix viscosity, interfacial tension) and processing conditions via the consideration of the shape of the dispersed phase required for achieving phase cocontinuity [69]. 

The temperature dependence of viscosity can often be as important as its shear rate dependence for nonisothermal processing problems (e.g.. Tanner, 1985). For all liquids, viscosity decreases with increasing temperature and decreasing pressure. A useful empirical model for both effects on the limiting low shear rate viscosity is... 

Adhesive pastes are non-Newtonian fluids whose viscosity depends upon temperature, time, and shear rate. They are applied to the substrates by means of either stamping (pin transfer), screen printing, or dispensing. The performance of the last technique depends on a key factor the paste rheology. There exists at least six semi-empirical models to describe the rheological response of non-Newtonian fluids [4]. For shear-thinning fluids (thixotropic materials), it has been reported... 

Bersted, B. H. An empirical model relating the molecular weight distribution of high-density to the shear dependence of the steady shear viscosity. /. AppL Polym. Sd. (1975) 19, pp. 2167-2177... 

The techniques that have been described so far to measure the molar masses of polymers in solution depend upon the equilibrium properties of the polymer solution. It is possible to relate the molar mass of the polymer to the solution properties through theoretical (e.g. thermodynamic) equations and the measurements are normally extrapolated to zero concentration where the solutions exhibit ideal behaviour. It is also possible to determine molar masses by studying the transport properties of polymer solutions which are usually analysed in terms of hydrodynamic models. These properties can be divided into two categories one of which involves the motion of the molecules through a solvent which is itself stationary (e.g. ultracentifuge) and the other deals with the effect of polymer molecules upon the motion of the whole solution (e.g. solution viscosity). The theoretical models which have been devised to explain the transport properties are by no means as well developed as those used to explain, for example, the thermodynamic properties of polymer solutions and so transport properties are normally analysed using semi-empirical... 

A basic theme throughout this book is that the long-chain character of polymers is what makes them different from their low molecular weight counterparts. Although this notion was implied in several aspects of the discussion of the shear dependence of viscosity, it never emerged explicitly as a variable to be investi-tated. It makes sense to us intuitively that longer chains should experience higher resistance to flow. Our next task is to examine this expectation quantitatively, first from an empirical viewpoint and then in terms of a model for molecular motion. 

There are basically three types of approaches to define the solid stress tensor, or more specifically the solid viscosity. In the early hydrodynamic models— developed by Jackson and his co-workers (Anderson and Jackson, 1967 Anderson et al., 1995), Kuipers et al., (1992), and Tsuo and Gidaspow (1990)—the viscosity is defined as an empirical constant, and also the dependence of the solid phase pressure on the solid volume fraction is determined from experiments. The advantage of this model is its simplicity, the drawback is that it does not take into account the underlying characteristics of the solid phase rheology. 

Temperature Dependence of Pure Metal Viscosity. Practically speaking, empirical and semiempirical relationships do a much better job of correlating viscosity with nsefnl parameters such as temperature than do equations like (4.7). There are nnmerons models and their resnlting equations that can be used for this purpose, and the interested student is referred to the many excellent references listed at the end of this chapter. A useful empirical relationship that we have already studied, and that is applicable to viscosity, is an Arrhenius-type relationship. For viscosity, this is... 

It must, of course, be clearly understood that e and e are not, like v and a, properties of the fluid involved alone but depend primarily on the turbulence structure at the point under consideration and hence on the mean velocity and temperature at this point and the derivatives of these quantities as well as on the type of flow being considered. The use of e and e does not, in itself, constitute the use of an empirical turbulence model. It is only when attempts are made to describe the variation of 6 and eh through the flow field on the basis of certain usually rather limited experimental measurements that the term eddy viscosity turbulence model is applicable. In fact, even when advanced turbulence models are used, it is often convenient to express the end results in terms of the eddy viscosity and eddy diffusivity. 

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