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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.

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Degrees of Freedom

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  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 • 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)

  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 don’t interact strongly (single recycle w/o reactor)

  5. Degrees of Freedom CONV-II Now what? 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. 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

  13. 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!

  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 After Gaussian Elimination, get 3 independent reactions! But…Which three?

  17. DOF Reactive Systems Example Table of Stoichiometric Coefficients

  18. DOF Reactive Systems Example After Gaussian Elimination, get the following three independent reactions:

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