# Section 1.2.1: Basic Steady State Material Balances

In discussing material balances we will refer to streams of material in a continuous chemical process containing components or chemical species. Streams are characterised most importantly by their flowrates and/or the flowrates of the species they contain. When we refer to a flow or quantity of material in the following, a flowrate is always implied unless otherwise stated.

Flows will usually be molar, i.e. kmol/unit time. the units will generally be omitted in non-numerical examples. In modelling systems without reaction it is sometimes convenient to work in mass flow units. There is no standard notation for flowrates. We will use a variety of notations suited to the problem in hand.

We will often denote a stream in a process by a letter symbol, e.g. the `the feed stream F to the process'. It is then convenient to refer to the flowrate of each species i in F by:

fi

If the species are A, B, C etc we will typically identify their flowrates as:

fA, fB ...
or:
fa, fb ...

It will sometimes, however, be convenient or necessary to use different notation, which will be introduces as required.

When developing mathematical models material balances should normally be written in the form of component balances in terms of component flows.

## Simple balances

For a simple `do nothing' process from which all materials entering in a single feed stream F emerge without reaction or reparation in a single product stream P, we can write a balance equation for each component i as:

fi - pi = 0 ; i = 1, 2, ...

Note:

• By always writing component balances, and not writing any total material balances, we ensure that we always write only the allowed number of independent material balances, namely the same as the number of components.
• By writing balances in terms of flowrates, and avoiding terms in composition, we maintain as many linear equations as possible. This is normally a good idea as linear equations are easier to solve than nonlinear.
• In this very simple situation there are no other equations available to describe the process. If the equations were to be solved as written there must be as many equations as unknowns, so that half the component flows must be known to allow the rest to be determined.

## Stream Mixing

Suppose a feed stream F and a recycle stream R with component flows ri mix to give a single product stream P, then there is again a balance equation for each component:

fi + ri - pi = 0 ; i = 1, 2, ...

If there are e.g. 3 components then there will be 3 equations containing 9 variables. If the equations are to be solved, then 6 quantities must be known, e.g. all the flows in both input streams F and R, to leave as many equations as unknowns.

This is an appropriate point to consider how stream compositions should be introduced into a mass balance model. If rather than specifying both F and P it is required that sufficient F, containing say pure component 1, should be added to maintain a specified constant mol fraction composition of 1, x in P. By the definition of mol fraction:

x = f1 / (f1 + f2 + f3)

As written this equation looks nonlinear in the flows because some of them appear in the denominator. However because x is constant, the equation becomes linear if written:

f1 = x (f1 + f2 + f3)

Or:

(1-1/x) f1 + f2 + f3 = 0

## Stream Splitting and Separations

Consider a stream F which is split into a main product stream P and a purge stream V.

There is still a balance equation for each species:

fi - pi - vi = 0

Additionally another equation for each component defines the fraction split. If this were a total splitter with fraction S of the total feed (and thus of each component) going to P then:

S fi - pi = 0

These equations are linear provided S is a constant. Note that if feed stream component flows and the split fraction are known then the equations can be solved for both sets of outlet component flows.

If the the device were a separator rather than a total splitter then its operation can be defined in terms of s separate split or recovery of each component, si

si fi - pi = 0

Again these are linear for known si and may be solved if feed and recoveries are known. Separator equations are linear if specifications are given in terms of:

• Specified recoveries or component splits
• Specified component or total flows, or specified ratios of flows
• Specified fractional compositions
They will be non linear if e.g. ratios of mol fractions, such as `k-values' are are specified. (Except in the special case of a binary system.)

## Reactions

In a reacting system, moles are not conserved, and so the component balance must contain a term to account for the generation or removal of material by reaction. There are several ways of expressing this, the easiest to understand being to use an extent of reaction, here given the symbol E.

Assuming the reactor has an inlet stream F and an outlet P, the balance equations are:

fi - pi + E ni = 0

Here ni is the signed stoichiometric coefficient for the species, i.e. the stoichiometric coefficient as it appears in the reaction, but with a positive sign for a product (which is created in the reaction) and a negative sign for a reactant (which disappears).

Again the equations are are linear. They remain linear for a fixed conversion reaction, since the extent can be written as a linear expression in the feed flow of the reference reactant. Nonlinear equations are required however in the case of an equilibrium reaction.

## Whole Processes

The material balance equations for a complete process are simply all of those for each of its individual units, mixers, separators reactors etc. It can be seen that with many quite sensible forms of specification these equations are all linear. Advantage will be taken of this by using the special solution methods available for sets of linear equations.

Energy balances are also linear if written in terms of enthalpies. However, they in general become nonlinear if temperatures are included. This is less of a problem than might be supposed, since in many cases temperatures may be thought of as specifications rather than unknowns to be calculated.