Degrees of Freedom

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# Degrees of Freedom - PowerPoint PPT Presentation

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

CONV-II

Now what?

1

2

3

CONV-I

• Guess CONV-I
• Iterate to converge CONV-II
• Iterate CONV-I
Degrees of Freedom
• Two approaches
• Sequential modular strategy
• Simultaneous strategy (equation-oriented approach)
• ASPEN can solve with either approach
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
• 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)
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 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 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

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

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
• 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

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 Example

After Gaussian Elimination, get 3 independent reactions!

But…Which three?

DOF Reactive Systems Example

Table of Stoichiometric Coefficients

DOF Reactive Systems Example

After Gaussian Elimination, get the following three independent reactions: