# 8.1: Perfectly Mixed Stirred Tank

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In the chemical process industries, one encounters a system used for mixing which is referred to as a “completely mixed stirred tank” or a “perfect mixer.” When used as a continuous reactor, such a system is often identified as a *continuous stirred tank reactor* or CSTR as an abbreviation. The essential characteristic of the perfectly mixed stirred tank is that the concentration in the tank is assumed to be uniform and equal to the effluent concentration even when the inlet conditions to the tank are changing with time. While this is impossible to achieve in any real system, it does provide an attractive model that represents an important limiting case for real stirred tank reactors and mixers.

As an example of a mixing process, we consider the system illustrated in Figure \(\PageIndex{1}\). The volumetric flow rate entering and leaving the system is assumed to be constant, thus the control volume is fixed in space. However, the concentration of the inlet stream is subject to changes, and we would like to know how the concentration in the tank responds to these changes.

Since no chemical reaction is taking place, we can express Equation \((8.3)\) as

\[\frac{d}{dt} \int_{\mathscr{V}(t)}c_{A} dV + \int_{\mathscr{A}(t)}c_{A} (\mathbf{v}_{A} -\mathbf{w})\cdot \mathbf{n} dA =0 \label{6}\]

While the gas-liquid interface may be moving normal to itself, it is reasonable to assume that there is no mass transfer of species \(A\) at that interface, thus \((\mathbf{v}_{A} -\mathbf{w})\cdot \mathbf{n}=0\) everywhere except at Streams #1 and #2. In addition, since the volumetric flow rates entering and leaving the tank are equal, it is reasonable to treat the control volume as a constant so that Equation \ref{6} simplifies to

\[\frac{d}{dt} \int_{\mathscr{V}}c_{A} dV + \int_{\mathscr{A}_{ e} }c_{A} \mathbf{v}_{A} \cdot \mathbf{n} dA =0 \label{7}\]

Here \(A_{ e}\) represents the area of *both* the entrance and the exit. Use of the traditional assumption for entrances and exits, \(\mathbf{v}_{A} \cdot \mathbf{n}= \mathbf{v}\cdot \mathbf{n}\), along with the flat velocity profile restriction, allows us to write Equation \ref{7} as

\[\frac{d}{dt} \int_{\mathscr{V}}c_{A} dV + \langle c_{A} \rangle_{2} Q - \langle c_{A} \rangle_{ 1} Q=0 \label{8}\]

Here \(\langle c_{A} \rangle_{ 1}\) and \(\langle c_{A} \rangle_{2}\) represent the *area-averaged* concentrations^{1} at the entrance and exit respectively. The *volume-averaged* concentration can be defined by

\[\langle c_{A} \rangle =\frac{1}{\mathscr{V}} \int_{\mathscr{V}}c_{A} dV \label{9}\]

and use of this definition in Equation \ref{8} leads to

\[\underbrace{ \mathscr{V} \frac{d\langle c_{A} \rangle }{dt} }_{\begin{array}{c}\text{rate of accumulation} \\ \text{of species A} \end{array}}=\underbrace{ \langle c_{A} \rangle_{ 1} Q }_{\begin{array}{c} \text{rate at which species A} \\ \text{enters the control volume} \end{array}}-\underbrace{ \langle c_{A} \rangle_{2} Q }_{\begin{array}{c} \text{rate at which species A} \\ \text{leaves the control volume} \end{array}} \label{10}\]

Here we have *two unknowns*, \(\langle c_{A} \rangle\) and \(\langle c_{A} \rangle_{2}\), and only a *single equation*; thus we need *more information* if we are to solve this problem. Under certain circumstances the two concentrations are essentially equal and we express this limiting case as

\[\underbrace{ \langle c_{A} \rangle }_{\begin{array}{c} \text{volume average} \\ \text{concentration in} \\ \text{ the tank} \end{array}}=\underbrace{ \langle c_{A} \rangle_{2} }_{\begin{array}{c} \text{area average } \\ \text{concentration} \\ \text{in the effluent} \end{array}} \label{11}\]

This allows us to write Equation \ref{10} in terms of the *single unknown*, \(\langle c_{A} \rangle\), in order to obtain

\[\mathscr{V}\frac{d\langle c_{A} \rangle }{dt} + \langle c_{A} \rangle Q=\langle c_{A} \rangle_{1} Q \label{12}\]

One must be very careful to understand that Equation \ref{11} is an *approximation* based on the assumption that the difference between \(\langle c_{A} \rangle_{2}\) and \(\langle c_{A} \rangle\) is *small enough* so that it can be neglected^{2}.

It is convenient to divide Equation \ref{12} by the volumetric flow rate and express the result as

\[{\tau } \frac{d\langle c_{A} \rangle }{dt} + \langle c_{A} \rangle =\langle c_{A} \rangle_{1} \label{13}\]

Here \({\tau }\) represents the average residence time that is defined explicitly by

\[\left\{\begin{array}{c} \text{average} \\ \text{residence} \\ \text{time} \end{array}\right\} = \frac{\mathscr{V}}{Q} ={\tau } \label{14}\]

In general, we are interested in processes for which the inlet concentration, \(\langle c_{A} \rangle_{ 1}\), is a function of time, and a classic example is illustrated in Figure \(\PageIndex{2}\). There we have indicated that \(\langle c_{A} \rangle_{ 1}\) undergoes a sudden change from \(c_{A}^{ o}\) to \(c_{A}^{1}\), and we want to determine how the concentration in the tank, \(\langle c_{A} \rangle\), changes because of this change in the inlet concentration. The initial value problem associated with the sudden change in the inlet concentration is given by

\[{\tau } \frac{d\langle c_{A} \rangle }{dt} + \langle c_{A} \rangle =c_{A}^{1} , \quad t \geq 0 \label{15a}\]

IC. \[\langle c_{A} \rangle =c_{A}^{ o} , \quad t=0 \label{15b}\]

Equations \ref{15a} - \ref{15b} are consistent with a situation in which the inlet concentration is fixed at \(c_{A}^{ o}\) for some period of time and then *instantaneously* switched from \(c_{A}^{ o}\) to \(c_{A}^{1}\) at \(t=0\).

To solve Equation \ref{15a}, we separate variables to obtain

\[{\tau } \frac{d\langle c_{A} \rangle }{\langle c_{A} \rangle -c_{A}^{1} } =-dt \label{16}\]

The integrated form can be expressed as

\[\int_{\eta = c_{A}^{ o} }^{\eta = \langle c_{A} \rangle }\frac{d\eta }{\eta -c_{A}^{1} } =- {\tau }^{-1} \int_{\xi = 0}^{\xi = t}d\xi \label{17}\]

in which \(\eta\) and \(\xi\) are the dummy variables of integration. Carrying out the integration leads to

\[\ln \left[\frac{\langle c_{A} \rangle -c_{A}^{1} }{c_{A}^{ o} - c_{A}^{1} } \right]=- {t / {\tau }} \label{18}\]

which can be represented as

\[\langle c_{A} \rangle =c_{A}^{1} + \left(c_{A}^{ o} -c_{A}^{1} \right)e^{ - {t / {\tau }} } \label{19}\]

This result is illustrated in Figure \(\PageIndex{3}\), and there we see that a new steady-state condition is achieved for times on the order of *three to four residence times*.

Even though *perfect mixing* can never be achieved in practice and one can never change the inlet concentration *instantaneously* from one value to another, the results shown in Figure \(\PageIndex{3}\) can be used to provide an estimate of the *response time* of a mixer and this qualitative information is extremely useful. Experiments can be performed in systems similar to that shown in Figure \(\PageIndex{1}\) by suddenly changing the inlet concentration and continuously measuring the outlet concentration. If the results are in good agreement with the curve shown in Figure \(\PageIndex{3}\), one concludes that the system behaves as a perfectly mixed stirred tank *with respect to a passive mixing process*.

The solution to the mixing process described in the previous paragraphs was especially easy since the concentration of the inlet concentration was a constant for all times greater than or equal to zero. The more general case would replace Eqs. \ref{15a} - \ref{15b} with

\[{\tau }\frac{d\langle c_{A} \rangle }{dt} + \langle c_{A} \rangle =f(t) , \quad t \geq 0 \label{20}\]

I.C. \[\langle c_{A} \rangle =c_{A}^{ o} , \quad t=0 \label{21}\]

The solution of Equation \ref{20} can be obtained by means of a *transformation* known as the *integrating factor method* and this is left as an exercise for the student (see Section 8.6, Problems 4 and 5).