# 8.2: Batch Reactor

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 Asky3 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.