Growth coefficient

Eliminating c and introducing an overall crystal growth coefficient, KG gives the approximate relation ... 

Figure 14.5 Behavior of the perturbation-amplitude growth coefficients 1 /rv and l/rs...

Upon applying the interface condition, Eq. (4), an expression for the growth coefficient is obtained ... 

Zener gives numerical solutions for the growth coefficients for one- and three-dimensional growth, and also asymptotic expansions for large and small values of the growth coefficients ... 

The maximum strain rate (e < Is1) for either extensional rheometer is often very slow compared with those of fabrication. Fortunately, time-temperature superposition approaches work well for SAN copolymers, and permit the elevation of the reduced strain rates kaj to those comparable to fabrication. Typical extensional rheology data for a SAN copolymer (h>an = 0.264, Mw = 7 kg/mol,Mw/Mn = 2.8) are illustrated in Figure 13.5 after time-temperature superposition to a reference temperature of 170°C [63]. The tensile stress growth coefficient rj (k, t) was measured at discrete times t during the startup of uniaxial extensional flow. Data points are marked with individual symbols (o) and terminate at the tensile break point at longest time t. Isothermal data points are connected by solid curves. Data were collected at selected k between 0.0167 and 0.0840 s-1 and at temperatures between 130 and 180 °C. Also illustrated in Figure 13.5 (dashed line) is a shear flow curve from a dynamic experiment displayed in a special format (3 versus or1) as suggested by Trouton [64]. The superposition of the low-strain rate data from two types (shear and extensional flow) of rheometers is an important validation of the reliability of both data sets. 

Figure 13.5 Dependences of the reduced tensile stress growth coefficient ti (i,t)/ar at 170°C on reduced time fay and reduced strain rate iaj for a SAN resin (wAN = 0.264, Mw = 78 kg/mol, Mw/M = 2.8) during the startup of uniaxial extensions flow. Also illustrated (dashed curve) are dynamic shear viscosity data displayed as 3 r7 (c<, 170°C) versus or7 as suggested by Trouton [64]. Reproduced from L. Li, T. Masuda and M. Takahashi, J.Rheol., 34(1), 103(1990), with permission of the American Institute of Physics...

Figure 8. Tensile stress-growth coefficient of sample SI at 123 °C...

Further tests have been carried out on sample SI at two higher strain rates. The stress-growth coefficient corresponding to these experiments is represented in Fig. 8, where the... 

ICg being the overall crystal growth coefficient and n the order of the crystal growth process. For more details reference 2 should be consulted. 

From the plot, compute the cell growth coefficient, Y, and the cell decay rate, b. 

Equations which govern the higher cumulant growth coefficients K n > 3, can also be derived using this method [14] by evaluating the coefficients of higher order derivatives of V x,t). 

INDIVIDUAL AND OVERALL GROWTH COEFFICIENTS. In mass-transfer operations it is generally assumed that equilibrium exists at the interface between phases. If this were true in crystallization, the concentration of the solution at the face of the crystal would be the saturation value y, and the total driving force for mass transfer would be y — y, where y is the concentration at a distance from the crystal face. Because of the surface reaction, however, a driving force is needed for the interfacial step, and the concentration at the interface is therefore y, where y < y < y. Only y — / remains as the driving force for mass transfer. This is illustrated in Fig. 27.8. 

Surface-growth coefficients. Much research on the growth of crystals by the interfacial reaction has been done and reported in standard monographs on crystallization. Although a coherent theory of crystal growth has evolved, numerical data on of a kind that can be used in design are scarce. 

As Nj 0, corresponding to quasi-steady motion, the growth coefficient should be reduced by a factor of In 2, as in Eq. (102), in agreement with the numerical solution. 

Spectral frequency Dimensionless recombination progress variable Gas density Initial gas density Laboratory time Exponential growth coefficient Stoichiometric equations for chemical reaction... 

Out-of-phase component of complex viscosity Shear stress growth coefficient Shear stress decay coefficient Tensile stress growth coefficient Tensile stress decay coefficient... 

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