Equations Modified Cross

When using simple rotating cup or spindle viscometers, the modified Cross equation can be simplified and rearranged (12) to the form ... 

Shen (1990) reviews the flow modelling of injection moulding of thermosets. The kinetic model was the Kamal model (as shown above in Equation (6.31)) and the chemoviscosity model is given by a modified WLF model for low shear mtes (as shown by Equation (6.35)) and a modified Cross model for high shear (as shown by Equation (6.36)), namely... 

A possibly better description of the behavior comes from the modified Cross equation (75), particularly if there is a second Newtonian plateau ... 

As will be illustrated later, certain important calculations (e.g., the determination of velocity profiles and flow rates in mbes) are facilitated by flow equations that are explicit in T, that is, have the form y = -/(t) or tj = tj(t). The power law can be written in either form. Some of the equations not given here are in or can be put in one of these forms. Unfortunately, they tend not to fit data as well as the Carreau and modified Cross equations, neither of which can be made explicit in t. This has led to the practice of fitting flow data with polynomials such as... 

FIGURE 14.7 Temperature dependence of zero-shear viscosity. Zero-shear viscosities from modified Cross fits to the L-80 polyisohutylene data from Figure 14.6a [9] and plotted according to Equation 14.10. 

The modified Cross equation gives the temperature dependence of viscosity at finite shear rates. With other equations, the temperature dependence of all the constants would have to be known. Another approach has used an equation that is independent of any model for the flow curve. Since the viscosity is a function of temperature and shear stress or shear rate ... 

Note that the above calculations are facilitated if the flow curve can be written as an explicit function of t, that is, in the form y = y(T)or rj = r (t). Although the power law meets that criterion, neither the Carreau Equation 14.7 nor the modified Cross Equation 14.8 meets the same. For this reason, it is sometimes viewed as being easier to fit data with a polynomial such as Equations 14.9a or A.9b, rather than to use an established constitutive equation to make these calculations (but note the cautions ). 

Compare the truncated (rj o = 0) form of the Carreau Equation 14.7 and the modified Cross Equation 14.8 for fitting these data. 

R.A. Stratton (/. Colloid Interface Sci. 22, 517 (1966)) provides r versus y data for five essentially monodisperse polystyrenes of different molecular weight. Do these data better fit the Carreau or modified Cross equations Stratton obtains rj values by extrapolating plots of rj versus t to t = 0. Do these equations give substantially different values Stratton s values at 183 °C are as follows ... 

Figure 15.7 illustrates that the modified Cross equation (15.9) reduces to the power law when (Ctjoyy 1. Assume that the temperature dependence of ijo is given by (15.11). What is the relation between and j in the power-law region according to the modified Cross equation ... 

Add one constant to the modified Cross equation (15.9) so that it will reproduce an upper-Newtonian region. 

In the capillary method, the time required for a liquid to flow through a capillary tube is determined. The melt under investigation flows with a constant rate through a tube with a small, definite cross-sectional area, such as a cylindrical capillary. The viscosity can be measured in an absolute way from the pressure drop. This method can yield the most reliable absolute data, the viscosity being given by a modified Hagen-Poiseuille equation ... 

At this point, only one type of H% is considered. To account for possible interference effects or cross correlations among two different relaxation interactions X and V with ranks l and /, respectively, the above equation is modified to give39... 

This relationship becomes very useful because dipole oscillator strengths are available for a wide selection of atoms and molecules. The authors also provide a means of modifying this equation for use when oscillator strengths are not available the interested reader is directed to their work [39] for additional details. The strength of their work is also evident in that it provides a functional relationship for the determination of total cross section as well. By integration of the single dilferential cross sections, one obtains... 

Optimizing solvents and solvent mixtures can be done empirically or through modeling. An example of the latter involves a single Sanchez-Lacombe lattice fluid equation of state, used to model both phases for a polymer-supercritical fluid-cosolvent system. This method works well over a wide pressure range both volumetric and phase equilibrium properties for a cross-linked poly(dimethyl siloxane) phase in contact with CO2 modified by a number of cosolvents (West et al., 1998). 

When Eqn. 1 is used to calculate the quantity of gas 1 (Qi, in moles) transported by molecular diffusion through a tube in a given time, one must also consider the cross-sectional area of the tube (A, in cm ) and the time (t, in sec.) for which the flux persists. The modified equation becomes ... 

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