# 8: Transient Mass Balances

- Page ID
- 44504

Our analysis of transient systems begins with the *molar form* of Axiom I for species \(A\) given by

Axiom I: \[\frac{d}{dt} \int_{\mathscr{V}_{a} (t)}c_{A} dV + \int_{\mathscr{A}_{a} (t)}c_{A} (\mathbf{v}_{A} -\mathbf{w})\cdot \mathbf{n} dA =\int_{\mathscr{V}_{a} (t)}R_{A} dV \label{1}\]

Here one must remember that \(\mathscr{V}_{a} (t)\) represents an *arbitrary*, moving control volume having a surface \(\mathscr{A}_{a} (t)\) that moves with a speed of displacement given by \(\mathbf{w}\cdot \mathbf{n}\). The second axiom requires that atomic species be conserved and is stated as

Axiom II: \[\sum_{A = 1}^{A = N} N_{JA} R_{A} =0 , \quad J = 1,2,...,T \label{2}\]

One can use this form to develop (see Section 6.1) a useful constraint on the molar rates of reaction given by

\[\sum_{A = 1}^{A = N} MW_{A} R_{A} =0 \label{3}\]

The *mass form* of Axiom I will be useful in our analysis of biomass production in Sec. 8.4 and this form is obtained from Equation \ref{1} by multiplying by the molecular mass. The result is given by

\[\frac{d}{dt} \int_{\mathscr{V}_{a} (t)}\rho_{A} dV + \int_{\mathscr{A}_{a} (t)}\rho_{A} (\mathbf{v}_{A} -\mathbf{w})\cdot \mathbf{n} dA =\int_{\mathscr{V}_{a} (t)} r_{A} dV \label{4}\]

and it is often used with a constraint on the species mass rates of production that takes the form

\[\sum_{A = 1}^{A = N} r_{A} =0 \label{5}\]

We will use all of these forms in our study of transient systems.

- 8.1: Perfectly Mixed Stirred Tank
- 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.

- 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.

- 8.4: Biomass Production
- Biological compounds are produced by living cells, and the design and analysis of biological reactors requires both macroscopic balance analysis and kinetic studies of the complex reactions that occur within the cells. Given essential nutrients and a suitable temperature and pH, living cells will grow and divide to increase the cell mass. Cell mass production can be achieved in a chemostat where nutrients and oxygen are externally supplied.

- 8.5: Batch Distillation
- Distillation is a common method of separating the components of a solution. The degree of separation that can be achieved depends on the vapor-liquid equilibrium relation and the manner in which the distillation takes place. Salt and water are easily separated in solar ponds in a process that is analogous to batch distillation. In that case the separation is essentially perfect since a negligible amount of salt is present in the vapor phase leaving the pond.