Isothermal reactors design equations

Most reactors used in industrial operations run isother-mally. For adiabatic operation, principles of thermodynamics are combined with reactor design equations to predict conversion with changing temperature. Rates of reaction normally increase with temperature, but chemical equilibrium must be checked to determine ultimate levels of conversion. The search for an optimum isothermal temperature is common for series or parallel reactions, since the rate constants change differently for each reaction. Special operating conditions must be considered for any highly endothermic or exothermic reaction. 

Isothermal Reactor Design Conversion Chapter 5 Combining Equations (5-22) and (5-23) gives... 

Since there is no change in moles on reaction, and since the reactor is isothermal, the design equations for A and R are given by Eqn. (3-31),... 

Example 2.11 Suppose initially pure A dimerizes, 2A —> B, isothermally in the gas phase at a constant pressure of 1 atm. Find a solution to the batch design equation and compare the results with a hypothetical batch reactor in which the reaction is 2A B - - C so that there is no volume change upon reaction. 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

Two-phase mass transfer and heat transfer without phase change are analogous, and the results of mass-transfer studies can be used to help clarify the heat-transfer problems. Cichy et al. (C5) have formulated basic design equations for isothermal gas-liquid tubular reactors. The authors arranged the common visually defined flow patterns into five basic flow regimes, each... 

Since enthalpy is a state variable, the integral on the right side of equation 10.2.6 is independent of the path of integration, and it is possible to rewrite this equation in a variety of forms that are more convenient for use in reactor design analyses. One may evaluate this integral by allowing the reaction to proceed isothermally at the initial temperature from extent 0 to extent and then heating the final product mixture at constant pressure and composition from the initial temperature to the final temperature. 

Although semi-analytical solutions are available in some cases [5], these are cumbersome and it is more usual to employ a numerical method. A simple example is presented below which illustrates the solution of the design equation for a batch reactor operated isothermally the adiabatic operation of the same system is then examined. 

Numerical integration of design equation for a batch reactor operated non-isothermally... 

The various types of reactors employed in the processing of fluids in the chemical process industries (CPI) were reviewed in Chapter 4. Design equations were also derived (Chapters 5 and 6) for ideal reactors, namely the continuous flow stirred tank reactor (CFSTR), batch, and plug flow under isothermal and non-isothermal conditions, which established equilibrium conversions for reversible reactions and optimum temperature progressions of industrial reactions. 

The solution of the type B starts in the attraction area of the stable part of the isotherm. It consists of a straight segment followed by segment of the isotherm lying on its stable part. The end point of the shock path will be called a drop point and denoted (u, gCu )). The length of the reactor is calculated by applying the design equation to the continuous part ... 

In the next sections, the design equations for the three ideal reactor t)q>es will be derived for isothermal conditions. In practice, the heat effects associated with chemical reactions usually result in non-isothermal conditions. The application of the law of conservation of energy leads to the so-called energy balance equation. This derivation is analogous to the derivation of the mass balance equations, and will not be treated here (see for instance, [4,5]). However, it should be noted that under non-isothermal conditions, the energy balance equation should always be solved simultaneously with the corresponding mass balance equation, since the reaction rate depends not only on composition but also on temperature. 

In this section we shall be concerned with more realistic models of tubular reactors. The isothermal reactor is obviously the simplest type, but it implies that either there are no large heat effects or that they can be completely dominated by temperature control. The reactor with an optimal temperature profile is clearly the most desirable, but this means that the rate of heat exchange can be regulated precisely at each point. Between these two extremes there is a range of designs about which something should be said. We shall not always solve the equations in detail but we shall try to show the important features of the behavior of the reactor by means of examples. 

Equations (13-16a) are of the same form as the equations for homogeneous plug-flow reactors [Eq. (4-5)] they are the constant-density version because the system is isothermal, and because for gaseous reactions no allowance has been made for a change in number of moles. Equation (12-2) is the analog of Eq. (3-13) for heterogeneous reactions and is applicable for variable-density conditions. The application of these equations to reactor design is the same as discussed, for example, in Example 4-6. 

The main difficulty in determining the reaction rate r is that the extent is not a measurable quantity. Therefore, we have to derive a relationship between the reaction rate and the appropriate measurable quantity. We do so by using the design equation and stoichiometric relations. Also, since the characteristic reaction time is not known a priori, we write the design equation in terms of operating time rather than dimensionless time. Assume that we measure the concentration of species j, Cj(t), as a function of time in an isothermal, constant-volume batch reactor. To derive a relation between the reaction rate, r, and Cj(t), we divide both sides of Eq. 6.2.4, by obtain... 

We start the analysis of plug-flow reactors by considering isothermal operations with single reactions. For isothermal operations, rf0/rfT = O, and we have to solve only the design equations. The energy balance equation provides the heating (or cooling) load necessary to maintain isothermal conditions. Furthermore, for isothermal operations, the reaction rate depends only on the species concentrations, and Eq. 7.1.5 reduces to... 

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