Larson differential equation

The population balanee eoneept enables the ealeulation of CSD to be made from basie kinetie data of erystal growth and nueleation and the development of this has been expounded by Randolph and Larson (1988), as summarized in Chapters 2 and 3. Bateh operation is, of eourse, inherently in the unsteady-state so the dynamie form of the equations must be used. For a well-mixed bateh erystallizer in whieh erystal breakage and agglomeration may be negleeted, applieation of the population balanee leads to the partial differential equation (Bransom and Dunning, 1949)... 

Unseeded Cooling Crystallization. In a similar fashion, a cooling profile can be derived for the unseeded case in which spontaneous nucleation and growth are allowed to occur at constant rates. The actual solutions to the resultant third-order differential equations found in the literature differ due to the different sets of the four initial conditions used by various authors (Karpinski et al. 1980b Nyvlt 1991 Randolph and Larson 1988). Understandably, all of them result in a cooling profile of the general form... 

As our second major topic, we present the simplest equations from each of the three important classes of constitutive equations, namely the differential equations from the retarded-motion expansion, the Maxwell-type differential equations, and the integral equations. Third and finally, we summarize the more accurate constitutive equations that we feel are the most promising for simply and realistically describing viscoelastic fluids and for modeling viscoelastic flows. More complete treatments of nonlinear constitutive equations are available elsewhere (Tanner, 1985 Bird et al., 1987 Larson, 1988 Joseph, 1990). Throughout this chapter, our examples are drawn from the literature on polymeric... 

Many improvements or modifications to the UCM model can be found in the literature. These csm lead to various classes of constitutive equations keeping the differential nature of the equation [2, 3, 35]. As pointed out by Larson [43], a systematic classification of these can be performed by rewritting the UCM model as ... 

R.G.Larson, Convected derivatives for differential constitutive equations, J. of Non-Newt. Fluid Mech. 2i (1987), 331-342. 

This latter approximation shows that the strain dependence of the Doi-Edwards equation is softer than that of the temporary network model roughly by the factor 1 -p (7i — 3)/5, There is also a differential approximation to the Doi-Edwards equation (Marmcci 1984 Larson 1984b) ... 

Differential models obtained by replacing the ordinary time derivative in Eq. 10.21 by one that can describe large, rapid deformations are able to describe some nonlinear viscoelastic phenomena, but only qualitatively. To improve on such models, it is necessary to introduce additional nonlinearity into the equation. In the popular Phan-Thien/Tanner model, the Gordon-Schowalter convected derivative is used, and nonlinearity is introduced by multiplying the stress term by a function of the trace of the stress tensor. The Giesekus and Leonov models are other examples of nonlinear differential models. All of the models mentioned above are described in the monograph by Larson [7j. 

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