Cross equation

The Cross equation assumes that a shear-thinning fluid has high and low shear-limiting viscosity (16) (eq. 4), where a and n are constants. 

Lai et al. [100] proposed the use of the Dow Rheology Index (DRI) as an indicator for comparing branching level in industrial polymers. For a linear polymer molecule, like unbranched polyethylene, the viscosity of the polymer as a function of the applied shear rate is given by the Cross equation [84,100],... 

The estimates do not even come close to satisfying the cross equation restrictions. The parameters in the cost function are extremely large, owing primarily to rather severe multicollinearity among the price tenns. 

Crop surveys Crospovidones Crossbar architectures Crossed polars Cross equation Cross-flow filtration... 

Carrean-Yasnda Model (full curve) Cross Model (full curve) For portions of the Cross equation that predict portions of the complete flow curve h-ih, 1 ho-1 . [1 + (k1y)2P 2 110-11 =(KiT Tl-11 1 oo... 

In Figure 6.14 we show a good match between a plot of Equation (6.28) (solid line) and the tabulated values (crosses). Equations (6.27) and (6.28) have then been included in our Fortran code in order to investigate the influence of radiation on the temperature distributions. 

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

This equation is similar to the Cross equation [3] for polymers to be discussed later. The Cross equation is also one of the key constitutive equations for ceramic suspensions at hig solids loading. 

At the percolation limit, the rheological behavior of the suspension changes from Newtonian to either the Cross equation with a low shear limit viscosity or the Bingham plastic equation with an apparent yield... 

FIGURE 12S Schematic of the low shear viscosity of TiOj as a function of pH. Near the zero point of charge (ZPC) the rheology is non-Newtonian for dilute suspensions, conforming to the Cross equation, which suggests that aggregation is responsible for this increase in viscosity. Away from the ZPC, the rheology is Newtonian for dilute suspensions. 

The Peclet number gauges the magnitude of the departure from equilibrium configuration of the particles. (Note that the rotational Peclet number, Pe, for a sphere has nearly the same definition only 6 is replaced by 8.) As such the Peclet number can be used in the Cross equation to determine the value of > .[= 8], giving... 

If we are to use a direct analogy of suspension rheology to the Cross equation derived for polymer solutions, we should consider that the... 

For the Cross equation, which is valid for monodisperse suspensions of hard spheres, the shear stress, t,, is given by... 

The apparent viscosity (tja) of the solution can be correlated with shear rate (y) using the Cross (Equation 2.14) or the Carreau (Equation 2.15) equations, respectively. 

With a lower power-law index, say, n = 0.2, the apparent viscosity would still be close to the limiting shear viscosity at A of 1.6 s and the shear rate of 0.1 s. For these conditions. Cross equation (17.7) gives 77 = T7o/1.23, and Carreau equation (17.8) gives 77 = 770/I.OI. However, at A of 16 s and the shear rate of 10 s (see above), the apparent viscosity would be much lower, that is, the shear thinning effect would be much more pronounced. Both Cross and Carreau equations give 77 = 77o/14,500, that is, 36 times lower apparent viscosity compared to that at n = 0.5. 

Tensile modulus, effect on, 174 Tensile strength, effect on, 174 Water absorption, effect on, 174 Covalent bonds, 161, 162 With wood fiber, 161 Cox-Merz rule, 640 Crane Plastics Co., 79, 80, 84, 87-91 Creep, 56, 293, 225, 260, 290-302 Creep-Recovery Test, 260 Creosote, 416, 417, 436 Critical moisture content, 413 Critical shear rate, 644, 664, 669 Critical shear stress, 658 Cross equation, 622, 623 Cross Timbers, 56, 58, 597 Sales, 58... 

Jones and co-workers (72, 88) found that the suspension viscosity variation with shear rate can be fitted fairly well by the Cross equation, equation 12, with m = 0.5 — 0.84. Both the low and high shear limit relative viscosities, Mro, Mroo> can be expressed by the Quemada s equation with 0max = 0.63 and 0.71, respectively. 

Doublier and Launay (1981) have calculated the four parameters of the Cross equation for guar gum by a computerized nonlinear regression... 

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