Isothermal viscosity behavior equation

Although we analyze most polymer processes as isothermal problems, many are non-isothermal even at steady state conditions. The non-isothermal effects during flow are often difficult to analyze, and make analytical solutions cumbersome or, in many cases impossible. The non-isothermal behavior is complicated further when the energy equation and the momentum balance are fully coupled. This occurs when viscous dissipation is sufficiently high to raise the temperature enough to affect the viscosity of the melt. 

It is evident from the discussions in Chapters 2 and 3 that the various aspects of hydrodynamic lubrication problems can range from the classical isoviscous, isothermal solutions of the simple Reynolds equation to short-time squeeze film behavior on impact. In dealing with elastohydrodynamic and impact problems, viscosity can no longer be taken as constant but instead must be introduced in a manner which correctly accounts for its response to temperature and pressure. As background for the appreciation of the intricacies of such problems we shall examine the effects of temperature and pressure on the viscosity of liquids, particularly those which can be used as lubricants. 

Equation 20 shows that a porous medium is permeative, that is, a shear factor exists to account for the microscopic momentum loss. Our preliminary study recently reveals that, however, a porous medium is not only permeative but dispersive as well. The dispersivity of a porous medium has been traditionally characterized through heat transfer (in a single- or multifluid flow) and mass transfer (in a multifluid flow) studies. For an isothermal single-fluid flow, the dispersivity of a porous medium is characterized by a flow strength and a porous medium property-de-pendent apparent viscosity. For simplicity, we discuss the single-fluid flow behavior in this chapter without considering the dispersivity of the porous medium. 

Under isothermal conditions, these four equations are sufficient to describe the flow of water (or air and any other gas or liquid with so-called Newtonian behavior of the viscosity). However, in most cases of industrial interest (i.e., at large scale), these equations cannot be solved using analytical techniques. The momentum balance is nonlinear in velocity, which makes analytical solution virtually always impossible. This is reflected in the properties of the flow of water it is in many cases turbulent. This means that the flow is inherently transient in time a steady state solution only exists for the time-averaged flow. The real flow shows a wide variety of structures, both in time and in space the flow field is built up of eddies of all kinds of sizes that have a finite life time. They come and disappear. These eddies make the solution very difficult. However, they are also vital to the processes we are running they make flow so effective in transport and mixing. Without them, we would have to rely on diffusion, which is a very slow process, and life on a larger scale as we know it would not have been possible. 

Many of the materials described as viscoplastic also exhibit time-dependent effects associated with a change in structure. This behavior is characterized by a reversible decrease in shear viscosity with time under isothermal conditions. Materials that fit this description are called thixotropic and one can describe them using the same constitutive equation suggested for incompressible viscoplastic materials. However now the modulus, Eq. 155, evolves with time through the gel strength F and the characteristic time k. Evolution equations are supplied in the... 

The excess viscosity has been determined for each datum by the use of equation (14.46) and subtracting the dilute-gas value, and the critical enhancement, A c, from the experimental value, T). For this purpose, reported by the experimentalists, rather than the value obtained from equation (14.47), has been preferred. This choice minimizes the influence of systematic errors in the individual measurements and forces the data of each author to a proper asymptotic behavior for the dilute-gas state. Furthermore, the experimental excess viscosity obtained in this fashion is independent of the choice for a dilute-gas viscosity correlation. Unfortunately, the majority of measurements on viscosity of ethane have been performed at pressures above 0.7 MPa and hence only a few authors reported a >7 value. Therefore, in order to estimate the experimental zero-density viscosity of each isotherm, again an iterative procedure had to be used (Hendl et al. 1994). The correction introduced by the extrapolation to zero density is small and in general does not exceed 0.5%. 

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