Equations of the Reactor

We now formulate six mass-balance DEs for the six reactions inside the reactor, as well as the energy-balance and pressure-drop equations. 

The mass-balance equations for the bulk phase of the reactor are given by 

Here the index j = 1,6 refers to the reactions (1) through (6) defined at the start of this section and the index i = 1,10 refers to the 10 components of the system. 

The solution to the reactor model differential equations (7.166) and (7.180) to (7.182) simulates the molar flow rates and the pressure drop and energy balance of the reactor. The solution of the catalyst pellet boundary value differential equations (7.172) and (7.173) provides the effectiveness factors r]j for each reaction labeled j = 1. 6 for use inside the differential equations (7.180) to (7.182). 

The main objective of the pellet model is to calculate the effectiveness factors for the six reactions that take place inside the reactor. These factors are defined as the ratios of the actual rates occurring inside the pellet and the rates occurring in the bulk phase, i.e., inside the pellet when diffusional resistances are negligible. 

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Substituting Eq.(7) into (4) the simplified equations of the reactor are obtained ... 

Considering the assumptions taken in section 2, the modelling equations of the reactor with two PI controllers are the following ... 

By substituting Eq.(33) into Eq.(27) the dimensionless equations of the reactor are simplified as follows ... 

This expression is also called the performance equation of the reactor and calculates the time required to achieve a certain conversion in a given reactor, or the reactor volume required to achieve a defined conversion in a given time. 

Hence, when designing a plug-flow reactor with a heating/cooling fluid whose temperature varies, we have to solve either Eq. 7.4.3 or 7.4.5 simultaneously with the design equations and the energy balance equation of the reactor. 

Equations (4.8a), (4.9a), and (4.10b) in Example 4.10 describe the dynamic behavior of a continuous stirred tank reactor with a simple, exothermic and irreversible reaction, A - B. Develop a numerical procedure that solves these equations and can be implemented on a digital computer. Also, describe a numerical procedure for solving the algebraic steady-state equations of the reactor above. (Note For this problem you need to be familiar with numerical techniques for the solution of differential and algebraic equations on a computer.)... 

Microkinetics concerns the chemical and physical kinetics at the scale of the particles (catalyst particles, liquid films and droplets, gas bubbles...) and their coupling in order to get the apparent reaction rate and selectivity equations. Macrokinetics refers to the transport of momentiam, mass and heat at the scale of the reactor, and its goal is the establishment of a model of the reactor. Combination of this model with the apparent reaction rate and selectivity equations allows to write the equations of the reactor. 

Using flow reactors imder steady-state conditions, we can easily collect data for process optimization, record activity, and selectivity and study catalyst life and deactivation processes. If we know the contacting pattern in the reactor, then we can explore the kinetics from the reactor performance equation. All of the flow reactors described previously present data for the average reactor concentration versus time. Generally the activity, selectivity, and stability are presented as a function of different process variables such as temperature, pressure, and space velocity. From conversion we can calculate the rate from the performance equation of the reactor, for example, for a CSTR ... 

Given the estimate of the reactor effluent in Example 4.2 for fraction of methane in the purge of 0.4, calculate the.actual separation in the phase split assuming a temperature in the phase separator of 40°C. Phase equilibrium for this mixture can be represented by the Soave-Redlich-Kwong equation of state. Many computer programs are available commercially to carry out such calculations. 

Computer simulation of the reactor kinetic hydrodynamic and transport characteristics reduces dependence on phenomenological representations and idealized models and provides visual representations of reactor performance. Modem quantitative representations of laminar and turbulent flows are combined with finite difference algorithms and other advanced mathematical methods to solve coupled nonlinear differential equations. The speed and reduced cost of computation, and the increased cost of laboratory experimentation, make the former increasingly usehil. 

Addition Chlorination. Chlorination of olefins such as ethylene, by the addition of chlorine, is a commercially important process and can be carried out either as a catalytic vapor- or Hquid-phase process (16). The reaction is influenced by light, the walls of the reactor vessel, and inhibitors such as oxygen, and proceeds by a radical-chain mechanism. Ionic addition mechanisms can be maximized and accelerated by the use of a Lewis acid such as ferric chloride, aluminum chloride, antimony pentachloride, or cupric chloride. A typical commercial process for the preparation of 1,2-dichloroethane is the chlorination of ethylene at 40—50°C in the presence of ferric chloride (17). The introduction of 5% air to the chlorine feed prevents unwanted substitution chlorination of the 1,2-dichloroethane to generate by-product l,l,2-trichloroethane. The addition of chlorine to tetrachloroethylene using photochemical conditions has been investigated (18). This chlorination, which is strongly inhibited by oxygen, probably proceeds by a radical-chain mechanism as shown in equations 9—13. 

Two variables of primary importance, which are interdependent, are reaction temperature and ch1orine propy1ene ratio. Propylene is typically used ia excess to act as a diluent and heat sink, thus minimising by-products (eqs.2 and 3). Since higher temperatures favor the desired reaction, standard practice generally involves preheat of the reactor feeds to at least 200°C prior to combination. The heat of reaction is then responsible for further increases in the reaction temperature toward 510°C. The chlorine propylene ratio is adjusted so that, for given preheat temperatures, the desired ultimate reaction temperature is maintained. For example, at a chlorine propylene molar ratio of 0.315, feed temperatures of 200°C (propylene) and 50°C (chlorine) produce an ultimate reaction temperature of approximately 500°C (10). Increases in preheat temperature toward the ultimate reactor temperature, eg, in attempts to decrease yield of equation 1, must be compensated for in reduced chlorine propylene ratio, which reduces the fraction of propylene converted and, thus aHyl chloride quantity produced. A suitable economic optimum combination of preheat temperature and chlorine propylene ratio can be readily deterrnined for individual cases. 

With these kinetic data and a knowledge of the reactor configuration, the development of a computer simulation model of the esterification reaction is iavaluable for optimising esterification reaction operation (25—28). However, all esterification reactions do not necessarily permit straightforward mathematical treatment. In a study of the esterification of 2,3-butanediol and acetic acid usiag sulfuric acid catalyst, it was found that the reaction occurs through two pairs of consecutive reversible reactions of approximately equal speeds. These reactions do not conform to any simple first-, second-, or third-order equation, even ia the early stages (29). 

The balances are made over a differential volume dV,. of the reactor. Rate equation ... 

Packed Red Reactors The commonest vessels are cylindrical. They will have gradients of composition and temperature in the radial and axial directions. The partial differential equations of the material and energy balances are summarized in Table 7-10. Example 4 of Modeling of Chemical Reactions in Sec. 23 is an apphcation of such equations. 

The equations that have been developed for design using these pseudo constants are based on steady-state mass balances of the biomass and the waste components around both the reactor of the system and the device used to separate and recycle microorganisms. Thus, the equations that can be derived will be dependent upon the characteristics of the reactor and the separator. It is impossible here to... 

Initial conditions for the system of differential equations shown before are given by the values of state variables known at the inlet of the reactor ... 

The rate equation involves a mathematical expression describing the rate of progress of the reaction. To predict the size of the reactor required in achieving a given degree of conversion of reactants and a fixed output of the product, the following information is required ... 

Ca plug flow)- the case where the effluent composition is fixed instead of the reactor size. Equations 8-152 and 8-154 can be manipulated to show that for small Dg /uL,... 

Steady state models of the automobile catalytic converter have been reported in the literature 138), but only a dynamic model can do justice to the demands of an urban car. The central importance of the transient thermal behavior of the reactor was pointed out by Vardi and Biller, who made a model of the pellet bed without chemical reactions as a onedimensional continuum 139). The gas and the solid are assumed to have different temperatures, with heat transfer between the phases. The equations of heat balance are ... 

The kinetic models are the same until the final stage of the solution of the reactor balance equations, so the description of the mathematics is combined until that point of departure. The models provide for the continuous or intermittent addition of monomer to the reactor as a liquid at the reactor temperature. 

The most important characteristic of an ideal batch reactor is that the contents are perfectly mixed. Corresponding to this assumption, the component balances are ordinary differential equations. The reactor operates at constant mass between filling and discharge steps that are assumed to be fast compared with reaction half-lives and the batch reaction times. Chapter 1 made the further assumption of constant mass density, so that the working volume of the reactor was constant, but Chapter 2 relaxes this assumption. 

The difficulty disappears when the mixing and mass transfer steps are fast compared with the reaction steps. The contents of the reactor remain perfectly mixed even while new ingredients are being added. Compositions and reaction rates will be spatially uniform, and a flow term is simply added to the mass balance. Instead of Equation (2.30), we write... 

The dAc/dz term is usually zero since tubular reactors with constant diameter are by far the most important application of Equation (3.7). For the exceptional case, we suppose that Afz) is known, say from the design drawings of the reactor. It must be a smooth (meaning differentiable) and slowly varying function of z or else the assumption of piston flow will run into hydrodynamic as well as mathematical difficulties. Abrupt changes in A. will create secondary flows that invalidate the assumptions of piston flow. 

An integral form of Equation (3.15) was used to derive the pressure ratio for scaleup in series of a turbulent liquid-phase reactor, Equation (3.34). The integration apparently requires ji to be constant. Consider the case where ii varies down the length of the reactor. Define an average viscosity... 

Thermal effects can be the key concern in reactor scaleup. The generation of heat is proportional to the volume of the reactor. Note the factor of V in Equation (5.32). For a scaleup that maintains geometric similarity, the surface area increases only as Sooner or later, temperature can no longer be controlled,... 

The boundary conditions normally associated with Equation (9.14) are known as the Danckwerts or closed boundary conditions. They are obtained from mass balances across the inlet and outlet of the reactor. We suppose that the piping to and from the reactor is small and has a high Re. Thus, if we were to apply the axial dispersion model to the inlet and outlet streams, we would find = 0, which is the definition of a closed system. See... 

Free Enzymes in Flow Reactors. Substitute t = zju into the DDEs of Example 12.5. They then apply to a steady-state PFR that is fed with freely suspended, pristine enzyme. There is an initial distance down the reactor before the quasisteady equilibrium is achieved between S in solution and S that is adsorbed on the enzyme. Under normal operating conditions, this distance will be short. Except for the loss of catalyst at the end of the reactor, the PFR will behave identically to the confined-enzyme case of Example 12.4. Unusual behavior will occur if kfis small or if the substrate is very dilute so Sj Ej . Then, the full equations in Example 12.5 should be (numerically) integrated. 

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