Degrees of Freedom

1
Degrees of Freedom

• Suppose we have the following process

1
2
3
f(y) returning/calculated value
Guess y
Tear stream

• Why tear the stream?
• So we can insert solver/convergence block
• Iterate to convergence criteria
• y f(y)-y 0 (desired)

2
Convergence

• Whats the problem (mathematically)?
• Find roots for function y(x) 0
• Given a starting value x p
• Therefore, best approximation for y(p) is
• Where Jy is the Jacobian
• So what is y, the function were dealing with?
• Not always well-defined or behaved
• Therefore, perturb system with small changes (D)

3
Convergence Methods

• Newton-Raphson
• Numeric approximation to derivative
• Given initial/current value of x, determine next
  value

4
Convergence Methods

• Broyden
• Quasi-Newton Method
• Computes whole Jacobian only at first iteration
• Uses finite differences for derivatives and
  Jacobian
• Good for processes with O(100) equations
• Secant
• Linearizes the system
• Use succession of secant lines to approximate a
  roof for function f
• Wegstein
• Bounded, relaxed method
• Works well for processes where components/units
  dont interact strongly (single recycle w/o
  reactor)

5
Degrees of Freedom

• Now what?

CONV-II
1
2
3
CONV-I

• Guess CONV-I
• Iterate to converge CONV-II
• Iterate CONV-I

6
Degrees of Freedom

• Two approaches
• Sequential modular strategy
• Simultaneous strategy (equation-oriented
  approach)
• ASPEN can solve with either approach

7
Complex Systems

• Partitioning
• How will I break the process up?
• Precedence Ordering
• What order will I solve blocks?
• Which block solutions precede others?
• Tearing

8
Tearing and Converging of Streams

• How many streams will require iterations?
• Which stream(s) selected for iteration?
• What order should tear streams be updated/solved?
• What numerical scheme used to update the
  successive values of the iterated streams?
• Note
• ASPEN always defaults to recycle streams as
  convergence blocks (that is, it tears the recycle
  stream)
• You can define/put in your own convergence block
  (could make a more informed choice)

9
Tearing and Converging of Streams

• The maximum number of streams that have to be
  torn is given by the number of mixers in the
  flowsheet
• Essential mixers
• Non-essential mixers (must eliminate to solve)

10
Degrees of Freedom

• Degrees of Freedom

Total Number of Independent Stream Variables
Total Number of Independent Balance Equations
(Mass, Energy, etc.)
Total Number of Specified Independent Stream
Variables
Total Number of Subsidiary Relations
-
-
-
11
Degrees of Freedom

• Total Number of Subsidiary Relations
• Mathematical relationships/constraints
• Equilibrium constraints (phase/chemical
  equilibrium, PVT relationships, etc.)
• Sum of mole fractions
• Split ratios
• Splitter restrictions
• (N - 1)(S - 1)
• Where

N Number of Exiting Streams S Number of
Species
12
DOF Example Flash Separation

• Number of Independent Stream Variables 11 (F,
  V, L, zA, zB, yA, yB, xA, xB, T, P)
• Number of Independent Equations 4
• Number of Known/Specified Stream Variables 3
  (zA, T, F)
• Number of Subsidiary Relations 3

13
DOF Example Flash Separation

• DOF 11 4 3 3 1
• Choose FLASH Operating P
• Problem Well-Specified!

14
DOF Reactive Systems

• Species balance
• Element balance

Total Number of Independent Stream Variables
Number of Species in Each Stream
Number of Independent Reactions


Total Number of Independent Stream Variables
Number of Species in Each Stream
Number of Independent Reactions


Total Number of Independent Balance Equations
Number of Elements in System

15
DOF Reactive Systems Example
Q Are these linearly independent? A Probably
Not! Q What is the maximum number of
independent reactions we can write? A Depends on
element balance
16
DOF Reactive Systems Example
Species Elements NH3 O2 NO H2O N2 NO2
N 1 0 1 0 2 1
H 3 0 0 2 0 0
O 0 2 1 1 0 2
After Gaussian Elimination, get 3 independent
reactions! ButWhich three?
17
DOF Reactive Systems Example
Table of Stoichiometric Coefficients
RXN NH3 O2 NO H2O NO2 N2
1 -4 -5 4 6 0 0
2 -4 -3 0 6 0 2
3 -4 0 -6 6 0 5
4 0 -1 -2 0 2 0
5 0 1 -2 0 0 1
6 0 -2 0 0 2 -1
18
DOF Reactive Systems Example
After Gaussian Elimination, get the following
three independent reactions