Continuous variable density reactions

In the reactors studied so far, we have shown the effects of variable holdups, variable densities, and higher-order kinetics on the total and component continuity equations. Energy equations were not needed because we assumed isothermal operations. Let us now consider a system in which temperature can change with time. An irreversible, exothermic reaction is carried out in a single perfectly mixed CSTR as shown in Fig. 3.3. 

Prompted by these considerations, Gillespie [388] introduced the reaction probability density function p (x, l), which is a joint probability distribution on the space of the continuous variable x (0 < x < oc) and the discrete variable l (1 = 1,..., to0). This function is used as p (x, l) Ax to define the probability that given the state n(t) at time t, the next event will occur in the infinitesimal time interval (t + x,t + x + Ax), AND will be an Ri event. Our first step toward finding a legitimate method for assigning numerical values to x and l is to derive, from the elementary conditional probability hi At, an analytical expression for p (x, l). To this end, we now calculate the probability p (x, l) Ax as the product po (x), the probability at time t that no event will occur in the time interval (t, t + x) TIMES a/ Ax, the subsequent probability that an R.i... 

We will begin the discussion of continuous reactors with the ideal CSTR, first with a constant-density example and then with a variable-density example. The ideal PFR then will be treated, for the constant-density and then the variable-density case. To point out the differences between constant- and variable-density systems, and the differences between how the different reactors are treated, all of our analysis will be based on Reaction (4-B) and the rate equation given by Eqn. (4-13). 

The primary process variables affecting the economics of sulfuric acid alkylation are the reaction temperature, isobutane recycle rate, reactor space velocity, and spent acid strength. To control fresh acid makeup, spent acid could be monitored by continuously measuring its density, the flow rate, and its temperature. This can reduce the acid usage in alkyla-tion units. 

The concepts in Chapters 2 and 3 are used in Chapter 4 to discuss the corrosion of so-called active metals. Chapter 5 continues with application to active/passive type alloys. Initial emphasis in Chapter 4 is placed on how the coupling of cathodic and anodic reactions establishes a mixed electrode or surface of corrosion cells. Emphasis is placed on how the corrosion rate is established by the kinetic parameters associated with both the anodic and cathodic reactions and by the physical variables such as anode/cathode area ratios, surface films, and fluid velocity. Polarization curves are used extensively to show how these variables determine the corrosion current density and corrosion potential and, conversely, to show how electrochemical measurements can provide information on the nature of a given corroding system. Polarization curves are also used to illustrate how corrosion rates are influenced by inhibitors, galvanic coupling, and external currents. 

Consider the control of a jacketed, continuous, stirred-tank reactor (CSTR) in which the exothermic reaction A — B is carried out. This system can be described by 10 variables, as shown in Figure 20.7 h, T, Ca, Cai, T Fi, F Fc, T and T oy diree of which are considered to be externally defined C I and Tco- Its model involves four equations, assuming constant fluid density. 

According to Berthellot, when substances of definite composition are formed, they are either (i) the result of a mutual saturation of an acid by a base, when the powers of each are just neutralised in a certain proportion (the change of reaction proceeding continuously) or (2) they are, exceptionally, due to the interference of other manifestations of the physical forces, such as (a) cohesion, when a precipitate of definite composition is formed because it happens to have the maximum density, or (b) elasticity, when a gaseous product is formed which escapes from the system, and as the most volatile product is favoured in the reaction. Solutions were regarded as true chemical compounds of variable composition. Salts crystallise out in a neutral state because in that neutral state the insolubility is greatest. ... 

In addition to phase behaviour, it is also important to evaluate the density of the reaction mixture. This variable plays a major role in both reaction equilibrium and kinetics. The control of density is more complex in supercritical reactors, where it can change dramatically with small perturbations in temperature, pressure or composition. The more direct application of density, in the case of continuous reactors, is to calculate the residence time of the reaction mixture for a given operating pressure and temperature. It is important to keep in mind that the volumetric flow measured downstream of the reactor can be very different from the flow inside the reactor due to the high variability of the density at supercritical conditions. In the case of batch reactors there are two... 

Appendix 3.Ill contains the equivalents of these equations for different variables (e.g., f and t), for heterogeneous catalytic reactions, and for the constant-density case. The numbering of the equations in Appendix 3.IIIA continues from Eqn. (3-33) above. 

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