DegreeofFreedom Analysis

1
Degree-of-Freedom Analysis

• A degree-of-freedom analysis (DFA) is simply an
  accounting of the number of unknowns in a problem
  and the number of independent equations that can
  be written. The difference between the number of
  unknowns and the number of independent equations
  is the number of degrees-of-freedom, DF or ndf,
  of the process.
• Possible outcomes of a DFA
• ndf 0, there are n independent equations and n
  unknowns. The problem can be solved.
• ndf gt 0, there are more unknowns that independent
  equations. The problem is underspecified. ndf
  more independent equations or specifications are
  needed to solve the problem.
• ndf lt 0, there are more independent equations
  than unknowns. The problem is overspecified with
  redundant and possibly inconsistent relations.

2
Sources of Equations

• Sources of equations that relate unknown process
  variables include
• Material balances for a nonreactive process,
  usually but not always, the maximum number of
  independent equations that can be written equals
  the number of chemical species in the process
• Energy balances 2nd half of course
• Process specifications given in the problem
  statement
• Physical properties and laws e.g., density
  relation, gas law
• Physical constraints e.g., mass or mole
  fractions must add to 1
• Stoichiometric relations systems with reaction

3
Independent Equations

• A set of equations are independent if you cannot
  derive one by adding and subtracting combinations
  of the others.
• Is this set of equations independent?

Matrices An n x n matrix A has a rank r lt n
only if A0 If A0 then A is called a
singular matrix. Otherwise, it is non
singular. An n x n matrix has rank r n only if
A ? 0 The rank of matrix A is equal to the
maximum number of linearly independent rows (or
columns) of A.
row reduce
Rank 3. No non-zero rows in reduced form
4
Independent Equations
Are these sets of equations independent?
5
Example

• One thousand kg/h of an ethanol/methanol stream
  is to be separated in a distillation column. The
  feed has 40 wt ethanol and the distillate has
  90 methanol by weight. Eighty weight percent of
  the methanol is to be recovered as distillate.
  Determine the wt methanol in the bottoms
  product.

distillate
feed
bottoms
Remember, for a non-reactive system with N
species, there are N independent equations. It
is possible to formulate N1 material balance
equations but only N of them will be independent!!
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Notes on DFA
There are two common situations where you will
find fewer independent equations than species,
and they are

• Balance around a divider (splitter)
• Single input two or more outputs with same
  composition
• Only 1 independent equation
• Splitters are used for
• Purge streams (reactor systems with recycle)
• Total condensers at the top of distillation
  columns

7
Notes on DFA

• If two species are in the same ratio to each
  other wherever they appear in a process and this
  ratio is incorporated in the flowchart labeling,
  balances on those species will not be independent
  equations.
• Situation occurs frequently when air is present
  in a nonreactive process (21 mol O2 79 mol N2)
• E.g., vapourization of liquid carbon
  tetrachloride into an air stream

n3 mol O2/s 3.76n3 mol N2/s n4 mol CCl4(v)/s
n1 mol O2/s 3.76n1 mol N2/s
Best to treat air as a single species in this
situation
n5 mol CCl4(v)/s
n2 mol CCl4(v)/s