# Material Balance Equation

DEFINITION

A balance is a method of accounting for something (here mass or material).

As we discussed when we defined balances earlier, the general material balance simply accounts for where things come/go and how their total number (or amount) changes. This led us to the following generic expression:

IN - OUT + GENERATION - CONSUMPTION = ACCUMULATION

where IN and OUT are the inputs and outputs to the system, respectively. In material and energy balances these will correspond to process streams and their contents. The GENERATION and CONSUMPTION accounted for ways in we could change the "stuff" in out system without flows in and out. In material and energy balances, this will correspond to chemical reactions.

In general, the types of systems that we have defined thus far (batch, continuous, etc.) are most easily handled by defining two alternate forms of this balance equation, in practice.

DEFINITION

A differential balance is a balance at one particular instant in time -- deals with rates (for mass balances: mass per time [kg/s]).

This type of balance is best suited for continuous processes and may be written more precisely in mathematical form as:

$\frac{dM}{dt} = \dot M_I - \dot M_O + G - C$

Where dM/dt denotes the rate of change of the material M (i.e., ACCUMULATION), G and C denote the rate of generation and consumption (respectively), and the overdots denote flowrates. (NOTE: all terms have units of mass/time.)

There are some special cases:

IN + GENERATION = OUT + CONSUMPTION
$0 = \dot M_I - \dot M_O + G - C$

For steady state, with no reaction

IN = OUT
$0 = \dot M_I - \dot M_O$

EXAMPLE

Let's look at an example.

A process running at steady-state involves a 100 kg/min stream with a mixture of Water (80 kg/min) and Sodium Hydroxide (20 kg/mm) being fed to a separator.

The mass flow of one of the two outlet streams (40 kg/min) is analyzed and found to contain 5 kg/min NaOH. Using differential balance equations, determine the mass flow rates of the remaining streams/components.

#### Flowchart

Write and simplify a balance equation on Water.

DEFINITION

An integral balance deals with the entire time of the process at once (so it uses amounts rather than rates: e.g., mass NOT mass/time).

This form of the equation is best suited for batch or semi-batch operation. A mathematical form for this equation can be derived by simply integrating the differential balance over the length of time the system is operating (i.e., from tinitial to tfinal. First we simply multiply both sides of the differential balance equation by dt to give:

$dt \frac{dM}{dt} = dt \dot M_I - dt \dot M_O + dt G - dt C$

which we can rearrange and then integrate:

$\int_{t_{initial}}^{t_{final}} \frac{dM}{dt} dt = \int_{t_{initial}}^{t_{final}} \dot M_I dt - \int_{t_{initial}}^{t_{final}} \dot M_O dt + \int_{t_{initial}}^{t_{final}} G dt - \int_{t_{initial}}^{t_{final}} C dt$

which ultimately leads to our final form of:

$M_{t_{final}} - M_{t_{initial}} = \int_{t_{initial}}^{t_{final}} \dot M_I dt - \int_{t_{initial}}^{t_{final}} \dot M_O dt + \int_{t_{initial}}^{t_{final}} G dt - \int_{t_{initial}}^{t_{final}} C dt$

There are some special cases:

For a closed system there is no input or output:

$M_{t_{final}} - M_{t_{initial}} = \int_{t_{initial}}^{t_{final}} G dt - \int_{t_{initial}}^{t_{final}} C dt$

For a system with no chemical reactions, the generation and consumption terms cancel:

$M_{t_{final}} - M_{t_{initial}} = \int_{t_{initial}}^{t_{final}} \dot M_I dt - \int_{t_{initial}}^{t_{final}} \dot M_O dt$

For a closed system with no reaction (we get a really boring problem!):

$M_{t_{final}} - M_{t_{initial}} = 0$

EXAMPLE

Let's look at an example.

The safety division of your company tells you that you are not allowed to dump acid that contains concentrations greater than a certain threshold (5% by mass) into the sink.

You have two solutions, one (1L) solution contains a mass fraction of 0.01 HCI in water and another (500 g) which has a mass fraction of 0.1 HCI. You wonder if you should combine the two and then dump them in the sink, or just cut your losses, dump the "legal" one, and properly dispose of the concentrated one. How would you determine which path to choose?

#### Flowchart

Can you write the proper integral balance?

OUTCOME:

Explain the origin and physical meaning of each of the terms in the General Mass Balance Equation