# Degrees of Freedom Analysis

When attempting to solve a material balance problem, two questions that one may ask are:

• How many equations do I need?
• Where do they come from?

DEFINITION

A Degrees of Freedom Analysis is used to answer these two questions. (For now we will only consider non-reactive systems.)

Counting unknowns is simple! Just look at your (carefully drawn) flowchart. As you (should) remember from algebra, the number of equations necessary is equal to the number of unknowns.

There are multiple places that the necessary equations might come from.

• Material Balance Equations = Number of independent species
• Energy Balance Equations = - These provide additional relationships, but will be covered later in the course
• Process Specifications or Constraints = - These are provided in the problem statement.
• Physical Properties/Laws = - These include relations about the properties of the materials in the process. They may include relationships between volumetric flowrates and mass flowrates (density!), or equilibrium constraints (for example, dissolving sugar in water).
• Physical Constraints = - These include simple physical requirements, like the total mass in a stream is equal to the sum of the masses of components in the stream or (equivalently) the sum of mole (or mass) fractions must be equal to 1.
• Stoichiometric relations = - If a chemical reaction occurs, the stoichiometry of the reaction describes a relationship between the amounts of reactants and the amounts of the products. (For example A + 2B -> C would tell us that you must react two (somethings) of B for every C that you make!)

OUTCOME:

Perform a degree-of-freedom analysis for a single-unit mass balance process