Balance equations Batch reactor

A batch reactor is usually well mixed, so that we may neglect spatial variations in the temperature and species concentration. The energy balance on batch reactors is found by setting equal to zero in Equation (9-10) yielding... 

Unsteady material and energy balances are formulated with the conservation law, Eq. (7-68). The sink term of a material balance is and the accumulation term is the time derivative of the content of reactant in the vessel, or 3(V C )/3t, where both and depend on the time. An unsteady condition in the sense used in this section always has an accumulation term. This sense of unsteadiness excludes the batch reactor where conditions do change with time but are taken account of in the sink term. Startup and shutdown periods of batch reactors, however, are classified as unsteady their equations are developed in the Batch Reactors subsection. For a semibatch operation in which some of the reactants are preloaded and the others are fed in gradually, equations are developed in Example 11, following. 

From the mass balance equation for a batch reactor... 

Balance equations for batch reactors may all be viewed as special cases of the following general equation... 

A batch reactor has no input or output of mass after the initial charging. The amounts of individual components may change due to reaction but not due to flow into or out of the system. The component balance for component A, Equation (1.6), reduces to... 

Reactor Performance Measures. There are four common measures of reactor performance fraction unreacted, conversion, yield, and selectivity. The fraction unreacted is the simplest and is usually found directly when solving the component balance equations. It is a t)/oo for a batch reaction and aout/ciin for a flow reactor. The conversion is just 1 minus the fraction unreacted. The terms conversion and fraction unreacted refer to a specific reactant. It is usually the stoichiometrically limiting reactant. See Equation (1.26) for the first-order case. 

In a batch vessel, the question of good mixing will arise at the start of the batch and whenever an ingredient is added to the batch. The component balance, Equation (1.21), assumes that uniform mixing is achieved before any appreciable reaction occurs. This will be true if Equation (1.55) is satisfied. Consider the same vessel being used as a flow reactor. Now, the mixing time must be short compared with the mean residence time, else newly charged... 

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 component balance for a batch reactor. Equation (1.21), still holds when there are multiple reactions. However, the net rate of formation of the component may be due to several different reactions. Thus,... 

Suppose there are N components involved in a set of M reactions. Then Equation (1.21) can be written for each component using the rate expressions of Equations (2.7) or (2.8). The component balances for a batch reactor are... 

A relatively simple example of a confounded reactor is a nonisothermal batch reactor where the assumption of perfect mixing is reasonable but the temperature varies with time or axial position. The experimental data are fit to a model using Equation (7.8), but the model now requires a heat balance to be solved simultaneously with the component balances. For a batch reactor. 

In the above reactions, I signifies an initiator molecule, Rq the chain-initiating species, M a monomer molecule, R, a radical of chain length n, Pn a polymer molecule of chain length n, and f the initiator efficiency. The usual approximations for long chains and radical quasi-steady state (rate of initiation equals rate of termination) (2-6) are applied. Also applied is the assumption that the initiation step is much faster than initiator decomposition. ,1) With these assumptions, the monomer mass balance for a batch reactor is given by the following differential equation. 

Aqueous Phase Hass Balances. The usual material balances for the active species in the aqueous solution are considered. With respect to the case of homopolymerization (4) the conplexity of the resulting equations is increased because of the cross propagation and termination terms. For the batch reactor considered in this wortt, the following equations arise ... 

Generally, the temperature changes with time or, equivalently, with distance from the reactor inlet (for flow reactors). This change is usually controlled well in reaction calorimeters but can become uncontrolled in other conventional laboratory flow or (semi)batch reactors. The balance equations of a batch reactor for a single reaction of a-th order kinetics are given by ... 

It becomes necessary to incorporate a total mass balance equation into the reactor model, whenever the total quantity of material in the reactor varies, as in the cases of semi-continuous or semi-batch operation or where volume changes occur, owing to density changes in flow systems. Otherwise the total mass balance equation can generally be neglected. 

For batch reactors, there is no flow into or out of the system, and those terms in the component balance equation are therefore zero. 

For semi-batch reactors, there is inflow but no outflow from the reactor and the outflow term in the above balance equation is therefore zero. 

The component mass balance, when coupled with the heat balance equation and temperature dependence of the kinetic rate coefficient, via the Arrhenius relation, provide the dynamic model for the system. Batch reactor simulation examples are provided by BATCHD, COMPREAC, BATCOM, CASTOR, HYDROL and RELUY. 

For a batch reactor, under constant volume conditions, the component mass balance equation can be represented by... 

There are a variety of limiting forms of equation 8.0.3 that are appropriate for use with different types of reactors and different modes of operation. For stirred tanks the reactor contents are uniform in temperature and composition throughout, and it is possible to write the energy balance over the entire reactor. In the case of a batch reactor, only the first two terms need be retained. For continuous flow systems operating at steady state, the accumulation term disappears. For adiabatic operation in the absence of shaft work effects the energy transfer term is omitted. For the case of semibatch operation it may be necessary to retain all four terms. For tubular flow reactors neither the composition nor the temperature need be independent of position, and the energy balance must be written on a differential element of reactor volume. The resultant differential equation must then be solved in conjunction with the differential equation describing the material balance on the differential element. 

The starting point for the development of the basic design equation for a well-stirred batch reactor is a material balance involving one of the species participating in the chemical reaction. For convenience we will denote this species as A and we will let (— rA) represent the rate of disappearance of this species by reaction. For a well-stirred reactor the reaction mixture will be uniform throughout the effective reactor volume, and the material balance may thus be written over the entire contents of the reactor. For a batch reactor equation 8.0.1 becomes... 

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. 

Consider a reaction represented by A +. . . - products taking place in a batch reactor, and focus on reactant A. The general balance equation, 1.51, may then be written as a material balance for A with reference to a specified control volume (in Figure 2.1, this is the volume of the liquid). 

As in the case of a batch reactor, the balance equation 2.3-3 or 2.3-4 may appear in various forms with other measures of flow and amounts. For a flow system, the fractional conversion of A (/A), extent of reaction (f), and molarity of A (cA) are defined in terms of Fa rather than nA ... 

In this chapter, we first consider uses of batch reactors, and their advantages and disadvantages compared with continuous-flow reactors. After considering what the essential features of process design are, we then develop design or performance equations for both isothermal and nonisothermal operation. The latter requires the energy balance, in addition to the material balance. We continue with an example of optimal performance of a batch reactor, and conclude with a discussion of semibatch and semi-continuous operation. We restrict attention to simple systems, deferring treatment of complex systems to Chapter 18. 

If the batch reactor operation is both nonadiabatic and nonisothermal, the complete energy balance of equation 12.3-16 must be used together with the iiaterial balance of equation 2.2-4. These constitute a set of two simultaneous, nonlincmr, first-flijer ordinary differential equations with T and fA as dependent variables and I as Iidependent variable. The two boundary conditions are T = T0 and fA = fAo (usually 0) at I = 0. These two equations usually must be solved by a numerical procedure. (See problem 12-9, which may be solved using the E-Z Solve software.)... 

The interpretation of cA(t) comes from the realization that each cylindrical shell passes through the vessel as an independent batch. Thus, cA(/) is obtained by integration of the material balance for a batch reactor (BR). Accordingly, we may rewrite equation 16.2-11, in terms of either cA(x) or fA(x), as... 

The material balance for a batch reactor may be used to develop a differential equation which may be solved for the cA(t) profile (see equation 3.4-1) ... 

The heat balance for a batch reactor, where Qin is zero, shows that heat accumulation is the difference between heat production and heat removal, from Equation (3-1), leading to Equation (3-7) ... 

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