Non-isothermal systems

Adiabatic operation implies that there is no heat interaction between the reactor contents and their surroundings. Isothermal operation implies that the feed stream, the reactor contents, and the effluent stream are equal in temperature and have a uniform temperature throughout. The present chapter is devoted to the analysis of such systems. Adiabatic and other forms of non-isothermal systems are treated in Chapter 10. 

For the more general case of non-isothermal systems, the S VD of Y can still be used to partition the chemical species into reacting and conserved sub-spaces. Thus, in addition to the dependence on c, the transformed chemical source term for the chemical species, S, will also depend on /. The non-zero chemical source term for the temperature,, SY, must also be rewritten in terms of c in the transport equation for temperature. 

In non-isothermal systems, the Jacobian is usually most sensitive to the value of temperature T. Indeed, for low temperatures, the components of the Jacobian are often nearly zero, while at high temperatures they can be extremely large. 

The CSTR is, in many ways, the easier to set up and operate, and to analyse theoretically. Figure 6.1 shows a typical CSTR, appropriate for solution-phase reactions. In the next three chapters we will look at the wide range of behaviour which chemical systems can show when operated in this type of reactor. In this chapter we concentrate on stationary-state aspects of isothermal autocatalytic reactions similar to those introduced in chapter 2. In chapter 7, we turn to non-isothermal systems similar to the model of chapter 4. There we also draw on a mathematical technique known as singularity theory to explain the many similarities (and some differences) between chemical autocatalysis and thermal feedback. Non-stationary aspects such as oscillations appear in chapter 8. 

In chapters 2-5 two models of oscillatory reaction in closed vessels were considered one based on chemical feedback (autocatalysis), the other on thermal coupling under non-isothermal reaction conditions. To begin this chapter, we again return to non-isothermal systems, now in a well-stirred flow reactor (CSTR) such as that considered in chapter 6. 

Comparisons between properties of the reaction rate curve R for cubic autocatalysis without decay and the first-order non-isothermal system in an... 

In Chapter 4, we introduced transport equations that apply when there are fluxes other than those of matter that contribute to the entropy production. Assuming that both matter and electrons take part in the transport, Eqns. (4.16)-(4.17) have been derived. In non-isothermal systems, we can use the same set of equations but replace... 

Y, indicates that the summation is over (n-1) independent fluxes in the n-compo-nent crystal (see Eqn. (4.29)). Qf jj is the (isothermal) energy flux due to the flux of species j. In an isobaric but non-isothermal system, Xj =... 

The LDF model is a realistic representation of the system with a surface barrier. Otherwise, k can be treated as an apparent mass transfer coefficient irrespective of the true transport mechanism which can be directly used in the design and optimization of adsorbers. This concept has been successfully used to analyze column breakthrough data for practical non-isothermal systems [18-20]. It substantially... 

Kinetic polynomial can be applied to reactor design and reactor control. Modified kinetic polynomial can be used to describe more complicated [12] and realistic cases, especially non-ideal and non-isothermal systems. 

If a non-isothermal system is considered, the energy balance has to be satisfied along with the above-mentioned conditions. The term pseudoequilibrium is used here to indicate the limit conversion and to compare it with the equilibrium conversion of a traditional system. Since the membrane reactor is an open system, it is indeed not fully correct to use the term equilibrium . 

The work is organized in two parts in the first part kinetics is presented focusing on the reaction rates, the influence of different variables and the determination of specific rate parameters for different reactions both homogeneous and heterogeneous. This section is complemented with the classical kinetic theory and in particular with many examples and exercises. The second part introduces students to the distinction between ideal and non-ideal reactors and presents the basic equations of batch and continuous ideal reactors, as well as specih c isothermal and non-isothermal systems. The main emphasis however is on both qualitative and quantitative interpretation by comparing and combining reactors with and without diffusion and mass transfer effects, complemented with several examples and exercises. Finally, non-ideal and multiphase systems are presented, as well as specific topics of biomass thermal processes and heterogeneous reactor analyses. The work closes with a unique section on the application of theory in laboratory practice with kinetic and reactor experiments. 

Because the primary circuit is an almost closed and non-isothermal system, the above processes compete with reverse processes for example, particles and deposits may dissolve. 

The analogies between our autocatalytic sequence and the non-isothermal system may be taken further. Autocatalysis with a stable catalyst resembles non-isothermal reaction imder adiabatic conditions. In these cases, the amount of reactant A consumed is uniquely related to and completely fixed by the amount of catalyst B or heat produced, at all times. These are one-variable systems, the concentration of A cannot be varied independently of the concentration of B or the temperature-excess AT, respectively. 

When the second fastest mode relaxes, the trajectory will reach a stuface with dimension Ns—Nc—2. In a closed system, this process continues tmtil the trajectory in the space of concentrations reaches a 3D stuface, a 2D stuface (a curved plane) and a ID surface (a curved line) and finally ends up near the OD equilibrium point. Therefore, following the ideas of Roussel and Fraser, we can imagine the system collapsing onto a cascade of manifolds of decreasing dimension with the fastest modes collapsing first and the slowest last. For a non-isothermal system, temperature may also be a variable increasing the dimension of the phase space by 1, but the same principles apply. In our discussions, we denote Ns as the dimension of the full system which may include temperature as a variable. 

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