Non-isothermal problems

Problems involving non-isothermal conditions are difficult to solve even in Elasticity Theory. Our interest lies mainly in problems where a material inhomogeneity derives from spacial temperature variations. In such cases, the instantaneous elastic portion of the viscoelastic functions remains independent of spacial co-ordinates, so that the limiting instantaneous problem is homogeneous. The inhomogeneity is only in the specifically viscoelastic portion of the equations. 

We will outline below a method due to Fichera (1965), Edelstein (1969a, b) and used extensively by F. Williams (1975) which is applicable to such problems. In fact, it is applicable in principle to any problem for which the elastic Green s Functions are known, though it will not yield analytic results in any except the simplest cases. 

The method involves transforming the integro-differential equations determining the behaviour of the material into an integral equation which can be solved by standard methods. Our object will be to sketch briefly the principle of the approach, which in fact is very simple. Later, in Chap. 6, it will be applied to the solution of a specific problem. 

The principle of the method may be written down very simply in formal operator terms, if one remembers that the Green s function of a differential operator is simply the inverse of the operator. Let the equation to be solved have the form 

General methods of solving viscoelastic boundary value problems are described in this chapter. 

---

To applying Eq. (2.47) to non-isothermal problems, it is necessary to generalize it by introducing temperature-dependent constants. The basic approach was proposed by Ziabicki94,95 who developed a quasi-static model of non-isothermal crystallization in the form of a kinetic rate equation ... 

Hence, when solving a non-isothermal problem the question arises -is this a problem where the equations of motion and energy are coupled To address this question we can go back to Example 6.1, a simple shear flow system was analyzed to decide whether it can be addressed as an isothermal problem or not. In a simple shear flow, the maximum temperature will occur at the center of the melt. By substituting y = h/2 into eqn. (6.5), we get an equation that will help us estimate the temperature rise... 

Mass conservation equations apply to water and air. When the porous medium is deformable, the momentum balance equation (mechanical equilibrium) is also taken into account. In non-isothermal problems, the internal energy balance for the total porous medium must be considered. The basic equations solved by the finite element code CODE BRIGHT are ... 

It is now further assumed that the functions Tl for all beads of all species are identical functions of position of time, namely T (r, t). This assumption, along with Eq. (12.11), will be referred to as the assumption of equilibration in momentum space. That is, we assume that the distnbution function in momentum space for a particular bead is the same as that for a bead in a system at equilibrium at the fluid temperature surrounding the bead in question. By allowing for the variation of the temperature over the full extent of a molecule, we are then in a position to study non-isothermal problems--that is, situations in which a = V In r IS nonzero. 

IV. Aging or Non-isothermal Problems. Most of the techniques described above can be generalized to handle aging materials or non-isothermal problems, though in the latter case with some restriction on the temperature field. Sect. 2.12 outlines a method whereby problems not solvable by any of the above techniques may sometimes be solved. The method relies upon an iterative solution of the integral equation (2.12.5). 

In chapter 1, the properties of the viscoelastic functions are explored in some detail. Also the boundary value problems of interest are stated. In chapter 2, the Classical Correspondence Principle and its generalizations are discussed. Then, general techniques, based on these, are developed for solving non-inertial isothermal problems. A method for handling non-isothermal problems is also discussed and in chapter 6 an illustrative example of its application is given. Chapter 3 and 4 are devoted to plane isothermal contact and crack problems, respectively. They utilize the general techniques of chapter 2. The viscoelastic Hertz problem and its application to impact problems are discussed in chapter 5. Finally in chapter 7, inertial problems are considered. 

The theoretical description of a non-isothermal viscoelastic flow presents a conceptual difficulty. To give a brief explanation of this problem we note that in a non-isothennal flow field the evolution of stresses will be affected by the... 

When equilibrium between the fluid and the solid cannot be assumed, it may still be possible to obtain analytical solutions for beds operating non-isothermally. In general, however, it will be necessary to look for numerical solutions. This problem has been summarised by Ruth vex 16. ... 

A comprehensive review of electrothermal atomization devices has been published (94). The review includes a discussion of commonly encountered problems such as atom loss through non-pyrolytic graphite, non-isothermal conditions, differences in peak height and peak area measurement, etc. 

However, the situation is different if one considers the total transformation, including the solidus and peritectic type reactions where substantial solid state difflision is needed to obtain complete equilibrium. Unless very slow cooling rates are used, or some further control mechanism utilised in the experiment, it is quite common to observe significant undercooling below the equilibrium temperature of transformation. The following sections will briefly describe determinations of phase diagrams where non-isothermal techniques have been successfully used, and possible problems associated with non-equilibrium effects will be discussed. 

In practice, of course, it is rare that the catalytic reactor employed for a particular process operates isothermally. More often than not, heat is generated by exothermic reactions (or absorbed by endothermic reactions) within the reactor. Consequently, it is necessary to consider what effect non-isothermal conditions have on catalytic selectivity. The influence which the simultaneous transfer of heat and mass has on the selectivity of catalytic reactions can be assessed from a mathematical model in which diffusion and chemical reactions of each component within the porous catalyst are represented by differential equations and in which heat released or absorbed by reaction is described by a heat balance equation. The boundary conditions ascribed to the problem depend on whether interparticle heat and mass transfer are considered important. To illustrate how the model is constructed, the case of two concurrent first-order reactions is considered. As pointed out in the last section, if conditions were isothermal, selectivity would not be affected by any change in diffusivity within the catalyst pellet. However, non-isothermal conditions do affect selectivity even when both competing reactions are of the same kinetic order. The conservation equations for each component are described by... 

The affect of diffusion on catalyst selectivity in porous catalysts operating under non-isothermal conditions has been examined by a number of workers. The mathematical problem has been comprehensively stated in a paper [21] which also takes into account the affect of surface diffusion on selectivity. For consecutive first-order exothermic reactions, the selectivity increases with an increase in Thiele modulus when the parameter A (the difference between the activation energy for reaction... 

A classic chemical engineering problem of the form under consideration here is that of a non-isothermal reaction occurring in a catalytic particle or packed bed into which a single gaseous participant diffuses from a surrounding reservoir (Hatfield and Aris 1969 Luss and Lee 1970 Aris 1975 Burnell et al. 1983). This scenario is also appropriate to the technologically important problem of spontaneous combustion of stockpiled, often cellulosic, material in air (Bowes 1984). If we represent the concentration of the gaseous species as c, the mass- and heat-balance equations for reaction in an infinite slab are... 

The two Proceedings of the Royal Society Papers (Reprints K and L) are a matched pair, exploring the model reaction that Schmidt and Takoudis had devised [177] A + S <-> AS, B + S BS, AS + BS + 2S -> C + 45. Here, the autocatalytic element is the vacant site, just as B is in the Gray-Scott reaction and heat is in the non-isothermal exothermic case. The two reprints, although not an absolutely comprehensive treatment of this model, have a satisfying completeness. The tale of students who worked on this class of problem includes Alhumaizi, Cordonier, Farr, Jorgenson, Kevrekidis, McKar-nin, and Takoudis their papers are listed in the Index of Co-Authors. 

We shall consider, in turn, the various problems which have to be faced when designing isothermal, adiabatic and other non-isothermal tubular reactors, and we shall also briefly discuss fluidised bed reactors. Problems of instability arise when inappropriate operating conditions are chosen and when reactors are started up. A detailed discussion of this latter topic is outside the scope of this chapter but, since reactor instability is undesirable, we shall briefly inspect the problems involved. 

There are several aspects of thermal sensitivity and instability which are important to consider in relation to reactor design. When an exothermic catalytic reaction occurs in a non-isothermal reactor, for example, a small change in coolant temperature may, under certain circumstances, produce undesirable hotspots or regions of high temperature within the reactor. Similarly, it is of central importance to determine whether or not there is likely to be any set of operating conditions which may cause thermal instability in the sense that the reaction may either become extinguished or continue at a higher temperature level as a result of fluctuations in the feed condition. We will briefly examine these problems. 

The non-isothermal nature of the process is one of the most important problems in the gelation and curing of ester-based compounds, since the increase in temperature in the reaction can reach... 

The key to modelling the crystallization process is the derivation a kinetic equation for a(t,T). It is possible to find different versions of this equation, including the classical Avrami equation, which allows adequate fitting of the experimental data. However, this equation is not convenient for solving processing problems. This is explained by the need to use a kinetic equation for non-isothermal conditions, which leads to a cumbersome system of interrelated differential and integral equations. The problem with the Avrami equation is that it was derived for isothermal conditions and... 

Polymer synthesis is usually accompanied by a strong exothermal effect and, as a general rule, real process proceeds in non-isothermal conditions. This problem was considered in Section 2.6 for polyester synthesis, and the results discussed above are of generally applicable to all non-isothermal processes. 

The results of the calculations shown in Fig. 2.32 represent a complete quantitative solution of the problem, because they show the decrease in the induction period in non-isothermal curing when there is a temperature increase due to heat dissipation in the flow of the reactive mass. The case where = 0 is of particular interest. It is related to the experimental observation that shear stress is almost constant in the range t < t. In this situation the temperature dependence of the viscosity of the reactive mass can be neglected because of low values of the apparent activation energy of viscous flow E, and Eq. (2.73) leads to a linear time dependence of temperature ... 

Injection molding is a rather complex process during which non-Newtonian as well as non-isothermal effects play significant roles. Here, we present a couple of problems that are relatively simple to allow an analytical solution. Injection molding is discussed further in the next chapters. 

Although we analyze most polymer processes as isothermal problems, many are non-isothermal even at steady state conditions. The non-isothermal effects during flow are often difficult to analyze, and make analytical solutions cumbersome or, in many cases impossible. The non-isothermal behavior is complicated further when the energy equation and the momentum balance are fully coupled. This occurs when viscous dissipation is sufficiently high to raise the temperature enough to affect the viscosity of the melt. 

When analyzing non-isothermal flow problems, we often assume that the viscosity decays exponentially with temperature following the relation... 

---