RMP Lecture Notes

Problem Specification and Degrees of Freedom

Have you ever set up a set of equations and discovered that despite doing all the algebra right you still couldn't get an answer? Some problems just can't be solved -- they are not specified correctly.

Other problems have multiple solutions; and this can also pose difficulties.

In order for there to be a unique solution for a system of linear equations (nonlinear equations have their own, impenetrable uniqueness rules) of a problem with N unknowns, you need N independent linear equations. The independence requirement is important. Equations are independent is no equation can be obtained by linear combination of others in the set.

EXAMPLE:
Solve the system of equations: Case I

There are three unknowns in the system (x,y,z) and three equations. These equations are independent, so you can find the unique solution x=6, y=-5, z=5.

Now consider a second set of equations:

These equations are not independent (why not?) so you can't get a unique solution. You can get some information relating the variables (x=3z-9), but that is all.

And a third set:

Once again, the equations are not independent (why?), so a unique solution is impossible.

In both the 2nd and 3rd sets, there are 3 unknowns, but only 2 independent equations. Since there aren't enough equations, the problem is said to be underspecified and cannot be solved unless we can come up with another equation or fix the value of one of the unknowns.

One more case:

This time there are more equations than unknowns. The problem is overspecified. It will not have an unique solution. It can be solved by an optimization approach (least squares, etc.) to get the "best" answer that most closely solves the most equations.

When we set up material and energy balance problems, we face similar constraints. For each problem, we have an assortment of equations that might be written. It is usually safe to plan that the number of independent equations will be the same as the number of components in the system; after all, we can write a component balance for each. Remember though, that if we write a full set of component balances, the total material balance is the sum of the component balances, and thus is not independent.

For each material balance problem, you should be considering:

Material balances -- if there are NC species, we can write NC independent component or material balances.
Energy balance -- one can be written for each system whenever energy is transferred
Constraints -- the sum of the mole fractions in a phase always totals one; reaction stoichiometry may impose constraints
Physical properties -- density can be related to mass and volume and may depend on pressure and temperature, etc.
Specifications -- you may be given fixed flow ratios, compositions, etc. Your basis, once chosen, is also a specification, so you need to be careful that you don't overspecify by picking a basis and using a "given" flow at the same time.

Degrees of Freedom

One of the technologist's most widely used, yet least understood and appreciated, tools is a technique known as degrees of freedom analysis.

The "degrees of freedom" of a problem are a way to measure whether a system is properly specified. It uses the equation:

where

f is the number of degrees of freedom. It will equal zero when a problem is properly specified
V is the number of independent variables
E is the number of independent equations
S is the number of specifications imposed on the variables

Often, the equation is presented without the specifications term. In that case, you are expected to recognize which variables are already specified and omit them from the variable total.

Let's apply the degrees of freedom relationship to the four math example cases done earlier.

f=3-3-0=0, fully specified
f=3-2-0=1, underspecified
f=3-2-0=1, underspecified
f=3-4-0=-1, overspecified

Notice that if you choose a basis, it adds a specification, and so could be used to make the second and third cases viable (if you chose x=1, say).

You can often work out the degrees of freedom before you write the equations, and it is usually wise to check that a problem is properly specified before you commit to the mathematics.

References:

Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2nd Edition, John Wiley, 1986, pp. 103-105.
Felder, R.M. and R.W. Rousseau, Elementary Principles of Chemical Processes, 2005 3rd Edition, 2005, p. 96-101.
Himmelblau, D.M., Basic Principles and Calculations in Chemical Engineering, 6th Edition, Prentice-Hall, 1996, pp. 153-56, 159-61.

R.M. Price
Original: 7/14/94
Modified: 9/16/97; 1/4/2005