Nonisothermal system

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

Equation (15.46) is applicable to nonisothermal systems since there is no chemical reaction. 

Three main flow patterns exist at various points within the tube bubble, annular, and dispersed flow. In Section I, the importance of knowing the flow pattern and the difficulties involved in predicting the proper flow pattern for a given system were described for isothermal processes. Nonisother-mal systems may have the added complication that the same flow pattern does not exist over the entire tube length. The point of transition from one flow pattern to another must be known if the pressure drop, the holdups, and the interfacial area are to be predicted. In nonisothermal systems, the heat-transfer mechanism is dependent on the flow pattern. Further research on predicting flow patterns in isothermal systems needs to be undertaken... 

In general, when designing a batch reactor, it will be necessary to solve simultaneously one form of the material balance equation and one form of the energy balance equation (equations 10.2.1 and 10.2.5 or equations derived therefrom). Since the reaction rate depends both on temperature and extent of reaction, closed form solutions can be obtained only when the system is isothermal. One must normally employ numerical methods of solution when dealing with nonisothermal systems. 

For isothermal systems this equation, together with an appropriate expression for rv, is sufficient to predict the concentration profiles through the reactor. For nonisothermal systems, this equation is coupled to an energy balance equation (e.g., the steady-state form of equation 12.7.16) by the dependence of the reaction rate on temperature. 

For a nonisothermal system the unsteady-heat balance equation (dynamic model) is given... 

The same very simple principles apply to the heat-balance equations for nonisothermal system and also to distributed systems as will be shown in the following subsections. The same principles also apply to heterogeneous systems. 

Nonisothermal systems are accounted for by the introduction of temperature-control units into the generic reactor unit representation. These units consist of elements associated with the manipulation of temperature changes and constitute temperature profiles (profile-based approach) and heaters/coolers (unit-based approach). The assumption of thermal equilibrium between the contacting phases reduces the need for a single temperature per shadow reactor compartment. The profile-based system (PBS) finds the optimum profiles without considering the details of heat transfer mechanisms. Because the profiles are imposed rather than... 

It should be understood that this rate expression may in fact represent a set of diffusion and mass transfer equations with their associated boundary conditions, rather than a simple explicit expression. In addition one may write a differential heat balance for a column element, which has the same general form as Eq. (17), and a heat balance for heat transfer between particle and fluid. In a nonisothermal system the heat and mass balance equations are therefore coupled through the temperature dependence of the rate of adsorption and the adsorption equilibrium, as expressed in Eq. (18). 

For a system with n components (including nonad-sorbable inert species) there are n — 1 differential mass balance equations of type (17) and n — 1 rate equations [Eq. (18)]. The solution to this set of equations is a set of n — 1 concentration fronts or mass transfer zones separated by plateau regions and with each mass transfer zone propagating through the column at its characteristic velocity as determined by the equilibrium relationship. In addition, if the system is nonisothermal, there will be the differential column heat balance and the particle heat balance equations, which are coupled to the adsorption rate equation through the temperature dependence of the rate and equilibrium constants. The solution for a nonisothermal system will therefore contain an additional mass transfer zone traveling with the characteristic velocity of the temperature front, which is determined by the heat capacities of adsorbent and fluid and the heat of adsorption. A nonisothermal or adiabatic system with n components will therefore have n transitions or mass transfer zones and as such can be considered formally similar to an (n + 1)-component isothermal system. 

The rational design of a reaction system to produce a desired polymer is more feasible today by virtue of mathematical tools which permit one to predict product distribution as affected by reactor type and conditions. New analytical tools such as gel permeation chromatography are beginning to be used to check technical predictions and to aid in defining molecular parameters as they affect product properties. The vast majority of work concerns bulk or solution polymerization in isothermal batch or continuous stirred tank reactors. There is a clear need to develop techniques to permit fuller application of reaction engineering to realistic nonisothermal systems, emulsion systems, and systems at high conversion found industrially. A mathematical framework is also needed which will start with carefully planned experimental data and efficiently indicate a polymerization mechanism and statistical estimates of kinetic constants rather than vice-versa. 

The mathematical complexity involved with temperature variations has limited most of the studies cited in this paper to the isothermal case. Since few commercial polymerization reactor systems can or should operate isothermally, there is a clear need to develop techniques to permit fuller application of reaction engineering to nonisothermal systems. In polymerizations as in simpler reactions, changes in temperature or temperature profile can have larger effects on rate and distribution than even reactor type. 

In a nonisothermal system, an electric current (flow) may be coupled with a heat flow this effect is known as the thermoelectric effect. There are two reciprocal phenomena of thermoelectricity arising from the interference of heat and electric conductions the first is called the Peltier effect. This effect is known as the evolution or the absorption of heat at junctions of metals resulting from the flow of an electric current. The other is the thermoelectric force resulting from the maintenance of the junctions made of two different metals at different temperatures. This is called the Seebeck effect. Temperature measurements by thermocouples are based on the Seebeck effect. 

When the steady-state approximation of Section B.2.5.2 is applied to nonisothermal systems involving molecular transport, it is sometimes referred to as the extended steady-state approximation. The simplifications that have just been indicated to follow from the application of this approximation underscore the importance of having methods for ascertaining the validity of the approximation. Criteria for the applicability of the extended steady-state approximation have been developed by Giddings and Hirsch-felder [64] and improved by Millan and Da Riva [89]. The discussion... 

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... 

The solution of the nonlinear optimization problem (PIO) gives us a lower bound on the objective function for the flowsheet. However, the cross-flow model may not be sufficient for the network, and we need to check for reactor extensions that improve our objective function beyond those available from the cross-flow reactor. We have already considered nonisothermal systems in the previous section. However, for simultaneous reactor energy synthesis, the dimensionality of the problem increases with each iteration of the algorithm in Fig. 8 because the heat effects in the reactor affect the heat integration of the process streams. Here, we check for CSTR extensions from the convex hull of the cross-flow reactor model, in much the same spirit as the illustration in Fig. 5, except that all the flowsheet constraints are included in each iteration. A CSTR extension to the convex hull of the cross-flow reactor constitutes the addition of the following terms to (PIO) in order to maximize (2) instead of [Pg.279]

A considerable amount of experimental information has been obtained for difiusion-reaction-deactivation in pellets by the use of so-called "single pellet difiiision reactors". These come in two forms the first, origiiMy developed by Balder and Petersen [11] ("Petersen s Pellet Poisoner"), relies upon the analysis of concentrations of reactants and their variation with time the second, fi om Kehoe and Butt [12] ("Kehoe s Katalyst Killer") involves measurement of temperature profiles within the pellet and their variation with time. The first is usefiil for deactivation in isothermal systems the second, at the expense of more complexity, can be used for both isothermal and nonisothermal systems. It is not possible to go into a long discussion of these systems here, but some discussion of the more simple reactor, for isothermal systems, will be useful as an example. 

For a nonisothermal system, the differential heat balance for the mobile phase can be written as [52]... 

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