Adiabatic conditions, steady-state equations

Another type of stability problem arises in reactors containing reactive solid or catalyst particles. During chemical reaction the particles themselves pass through various states of thermal equilibrium, and regions of instability will exist along the reactor bed. Consider, for example, a first-order catalytic reaction in an adiabatic tubular reactor and further suppose that the reactor operates in a region where there is no diffusion limitation within the particles. The steady state condition for reaction in the particle may then be expressed by equating the rate of chemical reaction to the rate of mass transfer. The rate of chemical reaction per unit reactor volume will be (1 - e)kCAi since the effectiveness factor rj is considered to be unity. From equation 3.66 the rate of mass transfer per unit volume is (1 - e) (Sx/Vp)hD(CAG CAl) so the steady state condition is ... 

Write the steady-state mass and heat balance equations for this system, assuming constant physical properties and constant heat of reaction. (Note Concentrate your modeling effort on the adiabatic nonisothermal reactor, and for the rest of the units, carry through a simple mass and heat balance in order to define the feed conditions for the reactor.)... 

The ordinary differential equations describing a steady-state adiabatic PFR can be written with axial length z as the independent variable. Alternatively the weight of catalyst w can be used as the independent variable. There are three equations a component balance on the product C, an energy balance, and a pressure drop equation based on the Ergun equation. These equations describe how the molar flowrate of component C, temperature T, and the pressure P change down the length of the reactor. Under steady-state conditions, the temperature of the gas and the solid catalyst are equal. This may or may not be true dynamically ... 

Consider an exothermic irreversible reaction with first order kinetics in an adiabatic continuous flow stirred tank reactor. It is possible to determine the stable operating temperatures and conversions by combining both the mass and energy balance equations. For the mass balance equation at constant density and steady state condition,... 

This is the classical argument introduced by van Heerden in 1953 for the adiabatic stirred tank. It is a most important one to grasp firmly for it can be used in more complicated situations to get some insight into the stability of a system. However, its limitations must be also thoroughly understood. In particular, it can be used to establish instability, but it does not count conclusively for stability because of several reasons. First, we should be suspicious of a single condition for a system in which there are two variables. Second, the diagram for the heat generation was drawn in a rather special way, for the steady state-mass balance equation, f 7), was first... 

Moving on to compressible flow, it is first of all necessary to explain the physics of flow through an ideal, frictionless nozzle. Chapter S shows how the behaviour of such a nozzle may be derived from the differential form of the equation for energy conservation under a variety of constraint conditions constant specific volume, isothermal, isentropic and polytropic. The conditions for sonic flow are introduced, and the various flow formulae are compared. Chapter 6 uses the results of the previous chapter in deriving the equations for frictionally resisted, steady-state, compressible flow through a pipe under adiabatic conditions, physically the most likely case on... 

Analytical studies of the TRAM have profited greatly from topological studies of the structure of the reactor equations. Luss and Amundson obtained a sufBcient condition for uniqueness of the steady state in adiabatic reactors by the use of fixed-point methods in deriving a linearized eigenvalue problem. Gavalas and Luss produced equivalent criteria for adiabatic TRAM and McGowin and Perlmutter and Han and Agrawal extended these results to non-adiabatic reactors. 

Solution The entropy balance of adiabatic steady-state process is given bvea. r6.281 with Q 0 mass flow rate of the inlet and outlet streams are equal G y virtue of the steady-state condition on the mass balance equation) therefore, the entropy balance becomes... 

It is impossible to create an isothermal process in plug flow reactors as it requires the variation of thermal transfei along the reactor length, according to the kinetics of heat emission. Therefore, plug flow reactors run under adiabatic conditions or at least imder nonisothermal mode conditions with external heat removal. The heat balance equation for steady state conditions for the micro volume of a reactor can be written in the form [4] ... 

Indeed, for a certain range of operating conditions, three steady-state profiles are possible with the same feed conditions, as is shown in Fig. 11.6-2. The outer two of these steady state profiles are stable, at least to small perturbations, whereas the middle one is unstable. Which steady-state profile will be predicted by steady-state computations depends on the initial guesses of Ca and T involved in the integration of this two-point boundary value problem. Physically, this means that the steady state actually experienced depends on the initial profile in the reactor. For all situations where the initial values are different from the feed conditions, transient equations have to be considered to make sure that the correct steady-state profile is predicted. To avoid those transient computations when they are unnecessary, it is useful to know a priori if more than one steady-state profile is possible. From Fig. 11.6-2 it is seen that a necessary and sufficient condition for uniqueness of the steady-state profile in an adiabatic reactor is that the curve Tq = f[1 I)] has no hump. 

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