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Equation-oriented vs sequential-modular flowsheet convergence

Sequential-modular solves one unit at a time in topological order and tears recycles, accelerating them with Wegstein. Equation-oriented makes every stream in the plant an unknown and solves them simultaneously with a damped Newton on a sparse finite-difference Jacobian. The difference shows on tight recycles: measured on the same loop, 78 tear passes against 5 Newton iterations.

convergencerecycleequation-orientedhow-to
Concept

Sequential-modular solves the flowsheet the way you would by hand: sort the units topologically, solve each one from its inlets, move on. A recycle breaks that order, so the solver cuts the loop at a tear stream, guesses it, sweeps the loop, compares the guess with what came back, and repeats — accelerated with Wegstein.

Equation-oriented refuses the ordering. Every inter-unit stream becomes an unknown, every unit contributes residual equations, and one damped Newton solves the whole plant at once on a sparse finite-difference Jacobian. There is no tear stream because there is no sequence to break.

The math
Show the governing equations
xk+1=q xk+(1−q) G(xk)(sequential-modular, Wegstein on the tear)x_{k+1} = q\,x_k + (1-q)\,G(x_k) \quad\text{(sequential-modular, Wegstein on the tear)}
J(xk) Δx=−F(xk)(equation-oriented, one Newton over every stream)J(x_k)\,\Delta x = -F(x_k) \quad\text{(equation-oriented, one Newton over every stream)}
G is one full pass around the loop; F is the stacked residual of every unit at once. The first converges linearly in the number of loop sweeps, the second quadratically near the solution.
Execution

Measured on the same flowsheet

A tight ammonia synthesis loop — about 10% conversion per pass, a 5:1 recycle-to-fresh ratio, argon accumulating until a purge balances it — solved both ways:

Sequential-modularEquation-oriented
Iterations78 tear passes5 Newton steps
Tear streams1none
Final residual9.0e-76.9e-12
Wall clock2.2 s3.8 s

The last row is the one usually left out. Equation-oriented took fifteen times fewer iterations and converged five orders of magnitude tighter, and was still slower on the clock, because each Newton step costs a finite-difference Jacobian over every stream in the plant while a tear pass is just one cheap sweep. EO wins where sequential-modular struggles or stalls — very tight recycles, many interacting loops, specs that must hold simultaneously — not automatically everywhere.

Execution

Which to reach for

  • Once-through or loose recycle: sequential-modular. It is the default and it is cheaper.
  • Recycle ratio above roughly 3:1, or a loop that will not settle: equation-oriented.
  • Several design specs that must all hold at once: equation-oriented, since the specs become equations in the same system rather than nested outer loops.
  • A flowsheet that fails to converge in one mode is worth trying in the other before touching the model — they fail on different things.
Try it
The high recycle ammonia eo flowsheet in the MaximaLabs workbench

The example as it opens in the workbench. Click to enlarge.

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