## Material Balances

Law of Conservation of Mass: Mass can neither be created nor destroyed (except in nuclear reactions).

Because of this, we can write equations called "mass balances" or "material balances". Any process being studied must satisfy balances on the total amount of material, on each chemical component, and on individual atomic species. Later in the course, we'll use the Law of Conservation of Energy (1st Law of Thermodynamics) to write similar balance equations for energy.

#### General Balance Equation

(Applies to mass, components, energy, etc.)

The system is any process or portion of a process chosen by the engineer for analysis. A system is said to be "open" if material flows across the system boundary during the interval of time being studied; "closed" if there are no flows in or out.

Accumulation is usually the rate of change of holdup within the system -- the change of material within the system. It may be positive (material is increasing), negative (material decreasing), or zero (steady state).

If the system does not change with time, it is said to be at steady state, and the net accumulation will be zero.

For our purposes, the generation and consumption terms are the consequence of chemical reaction. Note that while the total mass of a system and elements (or "atoms") are conserved, individual species are not.

If there is no chemical reaction, the production and consumption terms are typically zero.

Your bank statement can be thought of as a "dollar balance". Specifically, an "integral form balance" on dollars.

### Forms of Balance Equations

#### Differential Form

All the terms are rates, so the balance describes an instant in time. Usually the best choice for a continuous process. When formulated for an instant in time, the result is an ordinary differential equation. This is what we used on steady flow systems in Thermo I.

#### Integral Form

(Also called cumulative form) Written using total amounts as terms, so it describes the overall effect. Often a good choice for batch processes. In Thermo I, we used these on both closed and uniform state problems.

In this class, most problems will be differential balances on steady state systems. Consequently, accumulation will usually be zero.

The flow terms can usually be easily identified from the problem statement. If the process is batch, these may be zero.

Production and consumption are almost never present when balancing total mass, and are only present in component balances when reaction occurs.

EXAMPLE:

The figure describes a mixing tank problem. Find all flows and compositions.

Since we are doing a total material balance, the production and consumption terms are zero. We'll make a "differential" balance and use flow rates (lb/hr) for all terms.
where the flowrates are in units of pounds per hour.

If we assume the system is at steady state (and there is no indication not to), accumulation is zero, and:

Now we need to figure out the composition of the product stream. Start with the general equation and write a component balance on compound A:

There is no reaction, so no production or consumption. We're assuming steady state, so accumulation is zero. Consequently:

Similar balances are done on compounds B and C:

It is always smart to check answers for consistency. Here, we do this by summing the mole fractions:

The solution checks.

In the example, we wrote and solved one total material balance and three component balances, for a total of four balance equations describing a three component system. Is this the right number?

• We could not have written more.
• We could have written one less and gotten the final value by difference -- because only three of the balance equations are linearly independent.
In general, need NC balances where NC is the number of components.

Try, whenever possible, to write the easy balance equations! In the example, this might have lead to the sequence:

• Write the overall material balance -- only addition is required
• Write the component B balance -- fewer non-zero terms
• Write either A or C
• Get solution by difference

Balances can also be written on atoms -- which like total mass, are always conserved. These "atom balances" are useful in certain classes of balance problems where reaction is present.

EXAMPLE:

Consider the complete combustion of methane:

You could write a component balance on methane (assuming steady state):

or a total material balance:
or an atom balance on O:
Which would be most useful?

To review, solving a problem goes like this:

2. Eliminate negligible terms (based on the problem description)
3. Define the remaining terms (use variables first to get a more general solution before you plug in numbers)
4. Solve for unknowns.

WARNING: Mass and atoms are conserved. Moles are conserved only when there is no reaction. Volume is NOT conserved.

You may write balances on total mass, total moles, mass of a compound, moles of an atomic species, moles of a compound, mass of a species, etc.

References:

1. Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2nd Edition, John Wiley, 1986, pp. 83, 85-90.
2. Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2005 3rd Edition, 2005, p. 83, 85-89.
3. Himmelblau, D.M., Basic Principles and Calculations in Chemical Engineering (6th Ed.), Prentice Hall, 1996, pp. 141-149.

R.M. Price
Original: 6/6/94
Modified: 9/12/94, 9/4/96, 9/8/98; 12/25/2004