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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). Convergence.

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degrees of freedom
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)
convergence
Convergence
  • What’s 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 we’re dealing with?
    • Not always well-defined or behaved
    • Therefore, perturb system with small changes (D)
convergence methods
Convergence Methods
  • Newton-Raphson
    • Numeric approximation to derivative
    • Given initial/current value of x, determine next value
convergence methods4
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 don’t interact strongly (single recycle w/o reactor)
degrees of freedom5
Degrees of Freedom

CONV-II

Now what?

1

2

3

CONV-I

  • Guess CONV-I
  • Iterate to converge CONV-II
  • Iterate CONV-I
degrees of freedom6
Degrees of Freedom
  • Two approaches
    • Sequential modular strategy
    • Simultaneous strategy (equation-oriented approach)
  • ASPEN can solve with either approach
complex systems
Complex Systems
  • Partitioning
    • How will I break the process up?
  • Precedence Ordering
    • What order will I solve blocks?
    • Which block solutions precede others?
  • Tearing
tearing and converging of streams
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)
tearing and converging of streams9
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)
degrees of freedom10
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

-

-

-

degrees of freedom11
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

dof example flash separation

VAPOR

FEED

Equimolar Propane and n-Butane

40˚C

10 bar

1 kmol/hr

T, P

LIQUID

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

dof example flash separation13

VAPOR

FEED

Equimolar Propane and n-Butane

40˚C

10 bar

1 kmol/hr

T, P

LIQUID

DOF Example – Flash Separation

DOF = 11 – 4 – 3 – 3 = 1

Choose FLASH Operating P

Problem Well-Specified!

dof reactive systems
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

=

dof reactive systems example
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

dof reactive systems example16
DOF Reactive Systems Example

After Gaussian Elimination, get 3 independent reactions!

But…Which three?

dof reactive systems example17
DOF Reactive Systems Example

Table of Stoichiometric Coefficients

dof reactive systems example18
DOF Reactive Systems Example

After Gaussian Elimination, get the following three independent reactions: