# 8.2: Batch Reactor

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In many chemical process industries, the *continuous* reactor is the most common type of chemical reactor. Petroleum refineries, for example, run day and night and units are shut down on rare occasions. However, small-scale operations are a different matter and economic considerations often favor batch reactors for small-scale systems. The fermentation process that occurs during winemaking is an example of a batch reactor, and experimental studies of chemical reaction rates are often carried out in batch systems.

The analysis of a **batch reactor**, such as the liquid phase reactor shown in Figure \(\PageIndex{1}\), begins with the general form of the species mole balance

\[\frac{d}{dt} \int_{\mathscr{V}_{a} (t)}c_{A} dV + \int_{\mathscr{A}_{a} (t)}c_{A} (\mathbf{v}_{A} -\mathbf{w})\cdot \mathbf{n} dA =\int_{\mathscr{V}_{a} (t)}R_{A} dV \label{22}\]

The batch reactor, by definition, has no entrances on exits, thus this result reduces to

\[\frac{d}{dt} \int_{\mathscr{V}(t)}c_{A} dV =\int_{\mathscr{V}(t)}R_{A} dV \label{23}\]

Here we have replaced \(\mathscr{V}_{a} (t)\) with \(\mathscr{V}(t)\) since the control volume is no longer *arbitrary* but is specified by the process under consideration. In terms of the volume averaged values of \(c_{A}\) and \(R_{A}\), we can express our macroscopic balance as

\[\frac{d}{dt} \left[\langle c_{A} \rangle \mathscr{V}(t)\right]=\langle R_{A} \rangle \mathscr{V}(t) \label{24}\]

In some batch reactors the control volume is a function of time; however, in this development we assume that variations of the control volume are negligible so that Equation \ref{24} reduces to

\[\frac{d\langle c_{A} \rangle }{dt} =\langle R_{A} \rangle \label{25}\]

The simplicity of this form of the *macroscopic species mole balance* makes the constant volume batch reactor an especially useful tool for studying the rate of production of species \(A\). Often it is important that the batch reactor be *perfectly mixed*; however, we will avoid imposing that condition for the time being.

As an example, we consider the thermal decomposition of dimethyl ether in the constant volume batch reactor illustrated in Figure \(\PageIndex{2}\). The chemical species involved are \(\ce{C2H6O}\) which decomposes to produce \(\ce{ CH4}\), \(\ce{H2}\) and \(\ce{CO}\).

The visual representation of the atomic matrix is given by

\[ \text{ Molecular Species}\to \ce{CH4} \quad \ce{H2} \quad \ce{CO} \quad \ce{C2H6O} \\ \begin{matrix} {carbon} \\ {hydrogen} \\ oxygen \end{matrix}\begin{bmatrix} { 1} & { 0} & {1} & { 2 } \\ { 4} & { 2} & {0} & { 6 } \\ {0} & { 0} & {1} & { 1 } \end{bmatrix} \label{26}\]

and Axiom II takes the form

Axiom II: \[\begin{bmatrix} { 1} & { 0} & {1} & { 2 } \\ { 4} & { 2} & {0} & { 6 } \\ {0} & { 0} & {1} & { 1 } \end{bmatrix} \begin{bmatrix} {R_{ \ce{CH4} } } \\ {R_{ \ce{H2} } } \\ {R_{ \ce{CO}} } \\ R_{\ce{C2H6O}} \end{bmatrix} = \begin{bmatrix} {0} \\ {0} \\ {0} \\ {0} \end{bmatrix} \label{27}\]

Making use of the *row reduced echelon form* of the atomic matrix and applying the pivot theorem given in Sec. 6.4 leads to

\[\begin{bmatrix} {R_{ \ce{CH4} } } \\ {R_{ \ce{H2} } } \\ {R_{ \ce{CO}} } \end{bmatrix} = \begin{bmatrix} -1 \\ -1 \\ -1 \end{bmatrix} \begin{bmatrix} R_{\ce{C2H6O}} \end{bmatrix} \label{28}\]

in which \(\ce{C2H6O}\) has been chosen as the *pivot species*. Hinshelwood and Asky^{3} found that the reaction could be expressed as a first order decomposition providing a rate equation of the form

Chemical kinetic rate equation: \[R_{\ce{C2H6O}} =-k c_{\ce{C2H6O}} \label{29}\]

If we let dimethyl ether be species \(A\), we can express the reaction rate equation as

\[R_{A} =- k c_{A} \label{30}\]

Use of this result in Equation \ref{25} leads to

\[\frac{d\langle c_{A} \rangle }{dt} =- k \langle c_{A} \rangle \label{31}\]

and we require only an initial condition to complete our description of this process. Given the following initial condition

I.C. \[\langle c_{A} \rangle =c_{A}^{ o} , \quad t=0 \label{32}\]

we find the solution for \(\langle c_{A} \rangle\) to be

\[\langle c_{A} \rangle =c_{A}^{ o} e^{ - kt} \label{33}\]

This simple exponential decay is a classic feature of first order, irreversible processes. One can use this result along with experimental data from a batch reactor to determine the first order rate coefficient, \(k\). This is often done by plotting the logarithm of \(\langle c_{A} \rangle /c_{A}^{ o}\) versus \(t\), as illustrated in Figure \(\PageIndex{3}\), and noting that the slope is equal to \(- k\).

If the rate coefficient in Equation \ref{33} is known, one can think of that result as a *design equation*. The idea here is that one of the key features of the design of a batch reactor is the specification of the *process time*. Under these circumstances, one is inclined to plot \(\langle c_{A} \rangle /c_{A}^{ o}\) as a function of time and this is done in Figure \(\PageIndex{4}\). The situation here is very similar to the mixing process described in the previous section. In that case the *characteristic time* was the average residence time, \(\mathscr{V}/Q\), while in this case the *characteristic time* is the inverse of the rate coefficient, \(k^{-1}\). When the rate coefficient is known one can quickly deduce that the *process time* is on the order of \(3 k^{-1}\) to \(4 k^{-1}\).

Very few reactions are as simple as the first order irreversible reaction; however, it is a useful model for certain decomposition reactions.

Example \(\PageIndex{1}\): First Order, Reversible Reaction in a Batch Reactor

A variation of the first order *irreversible* reaction is the first order *reversible* reaction described by the following chemical kinetic schemata:

\[A \xleftarrow[k_2]{\xrightarrow{k_1}} B \label{1}\tag{1}\]

Here \(k_{1}\) is the forward reaction rate coefficient and \(k_{2}\) is the reverse reaction rate coefficient. The *net rate of production* of species \(A\) can be modeled on the basis of the *pictures* represented by Eqs. \ref{1} and this leads to a *chemical reaction rate equation* of the form

Chemical reaction rate equation: \[ R_{A} =- k_{1} c_{A} + k_{2} c_{B} \label{2}\tag{2}\]

Here we remind the reader that in this text we use arrows to represent *pictures* and equal signs to represent *equations*. Given an initial condition of the form

I.C. \[c_{A} =c_{A}^{ o} , \quad c_{B} =0 , \quad t=0 \label{3}\tag{3}\]

we want to derive an expression for the concentration of species \(A\) as a function of time for the batch reactor illustrated in Figure \(\PageIndex{4}\).

We begin the analysis with the species mole balance for a constant control volume

\[ \frac{d}{dt} \int_{\mathscr{V}}c_{A} dV =\int_{\mathscr{V}}R_{A} dV \label{4}\tag{4}\]

and express this result in terms of volume averaged quantities to obtain

\[ \frac{d\langle c_{A} \rangle }{dt} =\langle R_{A} \rangle \label{5}\tag{5}\]

The chemical kinetic rate equation given by Equation \ref{2} can now be used to write Equation \ref{5} in the form

\[ \frac{d\langle c_{A} \rangle }{dt} =- k_{1} \langle c_{A} \rangle + k_{2} \langle c_{B} \rangle \label{6}\tag{6}\]

In order to eliminate \(\langle c_{B} \rangle\) from this result, we note that the development leading to Equation \ref{5} can be

repeated for species \(B\), and the use of \(R_{B} =-R_{A}\) on the basis of Eqs. \ref{1} leads to

\[ \frac{d\langle c_{B} \rangle }{dt} =\langle R_{B} \rangle =- \langle R_{A} \rangle \label{7}\tag{7}\]

From Eqs. \ref{5} and \ref{6} it is clear that

\[ \frac{d\langle c_{B} \rangle }{dt} =- \frac{d\langle c_{A} \rangle }{dt} \label{8}\tag{8}\]

indicating that the rate of *increase* of the concentration of species \(B\) is equal in magnitude to the rate of *decrease* of the concentration of species \(A\). We can use Equation \ref{8} and the initial conditions to obtain

\[ \langle c_{B} \rangle =- \left(\langle c_{A} \rangle - c_{A}^{ o} \right) \label{9}\tag{9}\]

This result allows us to eliminate \(\langle c_{B} \rangle\) from Equation \ref{6} leading to

\[ \frac{d\langle c_{A} \rangle }{dt} =- (k_{1} +k_{2} )\langle c_{A} \rangle + k_{2} c_{A}^{ o} \label{10}\tag{10}\]

One can separate variables and form the indefinite integral to obtain

\[ \frac{1}{(k_{1} +k_{2} )} \ln \left[(k_{1} +k_{2} )\langle c_{A} \rangle - k_{2} c_{A}^{ o} \right]=- t + C_{1} \label{11}\tag{11}\]

where \(C_{1}\) is the constant of integration. This constant can be determined by application of the initial condition which leads to

\[ \ln \left[\left(\frac{k_{1} +k_{2} }{k_{1} } \right)\frac{\langle c_{A} \rangle }{c_{A}^{ o} } - \frac{k_{2} }{k_{1} } \right]=- (k_{1} +k_{2} ) t \label{12}\tag{12}\]

An explicit expression for \(\langle c_{A} \rangle\) can be extracted from Equation \ref{12} and the result is given by

\[ \langle c_{A} \rangle =c_{A}^{ o} \left[\frac{k_{2} }{k_{1} +k_{2} } + \frac{k_{1} }{k_{1} +k_{2} } e^{-(k_{1} +k_{2} ) t} \right] \label{13}\tag{13}\]

It is always useful to examine any special case that can be extracted from a general result, and from Equation \ref{13} we can obtain the result for a first order, irreversible reaction by setting \(k_{2}\) equal to zero. This leads to

\[ \langle c_{A} \rangle =c_{A}^{ o} e^{-k_{1} t} , \quad k_{2} = 0 \label{14}\tag{14}\]

which was given earlier by Equation \ref{33}. Under *equilibrium conditions*, Equation \ref{2} reduces to

\[ k_{1} c_{A} =k_{2} c_{B} , \quad \text{ for } R_{A} =0 \label{15}\tag{15}\]

and this can be expressed as

\[ c_{A} =K_{eq} c_{B} , \quad \text{ at equilibrium} \label{16}\tag{16}\]

Here \(K_{eq}\) is the *equilibrium coefficient* defined by

\[ K_{eq} ={k_{2} / k_{1} } \label{17}\tag{17}\]

The general result expressed by Equation \ref{13} can also be written in terms of \(k_{1}\) and \(K_{eq}\) to obtain

\[ \langle c_{A} \rangle =c_{A}^{ o} \left[\frac{K_{eq} }{1+K_{eq} } + \frac{1}{1+K_{eq} } e^{ - k_{1} { (1}+K_{eq} ) t} \right] \label{18}\tag{18}\]

When \(K_{eq} <<1\) we see that this result reduces to Equation \ref{14} as expected. In the design of a batch reactor for a reversible reaction, knowledge of the equilibrium coefficient (or equilibrium relation) is crucial since it immediately indicates the limiting concentration of the reactants and products.