Degree of Freedom Analysis

Independent Equations

Now for a quick review of algebra. Take the pair of equations below.

5x + 3y = 75
10x + 6y = 150

These equations are not independent. The bottom equation is merely the top equation multiplied by two. Any attempt to solve these equations is useless, because the equations are dependent- they do not convey unique information. For a system of algebraic equations to be solvable, there must be exactly as many independent equations as there are unknowns.

In chemical engineering terms, the number of independent equations that can be obtained from a non-reactive system is equal to the number of unique molecular species in that system. You can write a balance on atomic species, molecular species, or total mass, but you will never get more independent equations than individual chemical species.

Degrees of Freedom

Not all problems have solutions, and not all problems have unique solutions. Problems in which there is insufficient information are underspecified. Problems with too much information are overspecified. A problem with just the right amount of information is uniquely specified. A degree of freedom analysis is used to determine whether a process is uniquely specified.

To perform a degree of freedom analysis for a given system, you must:

Nunknown find the number of unknowns in the problem
+Nreaction add the number of reactions in the system
-Nmaterial subtract the number of independent material balances
-Nother and subtract the number of other relationships given in the problem statement
= Ndof
If Ndof equals zero, the problem is uniquely specified. If is Ndof greater than zero, the problem is underspecified. Finally, as you may have guessed, if Ndof is less than zero, the problem is overspecified.

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