Nonisothermal processing

Finally, in the case of nonisothermal processes, the overall heat transfer in the process must be analyzed, preferably in terms of elementary process steps similar to those discussed for mass transfer. 

Since thermal diffusion is a nonisothermal process and thus cannot be considered as driven by chemical potential gradients, we must go directly to the solute flux equations to understand the capacity of thermal diffusion for separation. The basic law expressing the flux density caused by thermal diffusion [46-48] is... 

The specific models will be further subdivided into isothermal and non-isothermal models. This distinction is justified because mathematical modeling of a nonisothermal system involves a heat balance in addition to coverage equations (or reactor mass balances), and therefore introduces strong Arrhenius-type nonlinearities into the coverage equations. Nonisothermal processes are much more dependent on the reactor type and the form of the catalyst (supported, wire, foil, or single crystal). Thus these heat balance equations that describe them must take into account the type of catalyst and... 

Perfectly isothermal systems are rare in chemical engineering practice and many processes, such as distillation, gas absorption, stripping, condensation, and evaporation, involve the simultaneous transfer of mass and energy across fluid-fluid interfaces. Representative temperature profiles in some nonisothermal processes are shown in Figure 11.1. The temperature profile also has a large influence in chemically reacting systems. For nonisothermal systems it is important to consider simultaneous heat transfer even though we are primarily interested in the mass transfer process. 

Figure 11.1. Typical temperature and composition profiles in some nonisothermal processes.

Several recently published reviews74-7sum up the results of investigations in this field, but the individual works differ in value. Three problem areas concerning modeling are examined below. These include methods for determining the kinetic constants, analysis of nonisothermal processes, and applied calculations. 

The application of Bodenstein s principle (a quasi-steady state with respect to radicals) to nonisothermal processes is shown not always to be correct. 

For nonisothermal processes, the change of energy functions contains a change of standard state. The standard-state values need to be known. The ideal-gas standard-state values are given in standard tables. These have to be looked up or, alternatively, heat capacity data are employed instead of the standard-state values. Consider the change of enthalpy the difference is found by integrating the heat capcity. 

Ziabicki [1967 1976] based his analysis on the assumption that any nonisothermal process can be treated as a combination of several isothermal crystallization steps ... 

Denq, B.L., Chin, W.Y., Lin, K.F. Kinetic model of thermal degradation of polymers from nonisothermal process. J. Appl. Polym. Sd. 66, 1855-1867 (1997)... 

The analysis of the above equations is often applied for obtaining the nucleation data from isothermal and nonisothermal crystallization experiments. Several simplifications of the equations are developed and used for isothermal crystallization (with instantaneous or spontaneous nucleations only) and nonisothermal processes with a constant cooling rate. It was found that the crystallization of iPP follows the dependence log [1 — a(f)] t where n is around three for relatively low supercoolings which indicates instantaneous character of primary nucleation. 

It is well known that the extent to which isothermal crystallization in polymers occurs can be profoundly influenced by molecular orientation. Recently Nakamura et al. (275) have performed a detailed theoretical analysis of the relationships between crystallization temperature, crystallinity, orientation and cooling conditions for nonisothermal processes. These investigators have also evaluated the crystallization rates under molecular orientation during melt spinning experiments and... 

Quench. Attempts have been made to model this nonisothermal process (32-35), but the complexity of the actual system makes quench design an art. 

Ziabicki proposed to analyze nonisothermal processes as a sequence of isothermal steps [170-172]. The proposed equation is a series expansion of the Avrami equation. In quasi-static conditions, provided that nucleation and growth of the crystals are governed by thermal mechanisms only, that their time dependence comes from a change in external conditions, and that the Avrami exponent is constant throughout the whole process, the nonisothermal crystallization kinetics can be expressed in terms of an observable half-time of crystallization, T]/2, a function of time, and of the external conditions applied. The following equation was derived for the dependence of the total volume of the growing crystal, E(t), with time ... 

In nonisothermal processes, the fluid properties depend on temperature and an energy conservation equation is to be added. 

The temperature dependence of viscosity can often be as important as its shear rate dependence for nonisothermal processing problems (e.g.. Tanner, 1985). For all liquids, viscosity decreases with increasing temperature and decreasing pressure. A useful empirical model for both effects on the limiting low shear rate viscosity is... 

A change in internal energy for a nonisothermal process can be calculated by carrying out a line integral of dU. 

The isothermal Newtonian model is a useful model, because it reveals most of the characteristics of the tubular film blowing process. Nevertheless, it suffers from two disadvantages the actual film blowing process is basically a nonisothermal process, and the polymer melt is non-Newtonian in character. In this section we address the nonisothermal case, and in the next section the matter of the non-Newtonian character of the polymer melt. 

Compression molding is a highly nonisothermal process. In the processing of blanks consisting of a thermoplastic reinforced with long fibers, the blanks are preheated and then placed in the mold, which is at a temperature lower than that of the blanks. In the case of thermoplastic prepregs, the... 

For nonisothermal processes, exponential nonlinearity, expressed through dependence of rate and equihbrium constants on temperature, results in appearance of thermo-kinetic oscillations in temperature and concentrations. 

Despite of the faet that the approaches do represent a signifieant step toward the description of morphology, it has been pointed out in the hterature that Nakamura s equation (Eq.(4) (6)) should include the induced time.[5] For simplicity, instantaneous nucleation is assumed, and this makes the use of a detailed nucleation model imnecessary. The nonisothermal processes can be considered as a combination of finite steps of the isothermal process. The following expression is generally used to determine the onset of crystallization imder nonisothermal conditions [6]... 

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