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Process Analysis & Optimization

A converged flowsheet answers one question: what happens at these conditions? Real engineering asks better ones — how sensitive is the answer, what input hits a target, which design is cheapest. This chapter turns the model from a calculator into a decision tool.

7.1 Sensitivity analysis

The first question is always "what if?" A sensitivity sweep varies one input

across a range and re-solves the whole flowsheet at each point, tracing an output :

$x_k$

The k-th value of the swept input (e.g. reflux ratio) — chosen at N evenly spaced points across the range.

$x_{\min},\; x_{\max}$

The lower and upper bound of the sweep range you set.

$N$

Number of sweep points — each one a full, independent re-solve of the flowsheet.

$y_k$

The tracked output (e.g. reboiler duty) at that point — the solver's real answer, not an interpolation.

For the ethanol-water column, sweeping the reflux ratio against the reboiler duty shows the central trade-off of distillation: more reflux sharpens the separation but costs energy, and the curve tells you where the knee is. MaximaLabs runs each point as a real re-solve (an enqueued job) and plots the response:

7.2 Design specs: state the result, solve for the input

Sensitivity answers "what does this input do?" A design spec answers the inverse — "what input gives me this result?" You name a manipulated variable and a target, and the solver root-finds the input that makes the target hit its setpoint:

$x$

The manipulated variable — a single flowsheet parameter you designate (e.g. reflux ratio).

$g(x)$

The target quantity as a function of that input (e.g. distillate ethanol mole fraction) — computed by a full flowsheet re-solve at each trial x.

$s^{*}$

The setpoint you want the target to hit (e.g. 0.8).

Instead of guessing reflux ratios until the purity comes out right, you state the purity and MaximaLabs finds the reflux — the same "Design Spec" idea as Aspen, or a SimCentral adjust. Each spec reads as a sentence and reports whether it was met on the next Run:

7.3 Case studies

A sensitivity sweep moves one variable continuously; a case study instead evaluates a small table of named, discrete scenarios — winter vs. summer feed, three catalyst activities, a set of turndown rates — each one a full set of

overrides, re-solved from scratch and laid side by side against whichever outputs you care about:

$x_c$

The full set of {node, param, value} overrides that define named case c (e.g. Winter feed's colder, wetter composition).

$C$

Number of named cases (here 3: Design, Winter feed, Turndown) — discrete scenarios you actually need answers for, not a swept continuum.

$f_o$

The flowsheet re-solve, read at tracked output o (e.g. distillate purity, reboiler duty).

$O$

Number of tracked outputs (here 2: distillate purity and reboiler duty).

$y_{c,o}$

The solved value of output o under case c — one cell of the results grid below.

Take the ethanol-water column and ask "what does the winter feed do to us?" — define Design (the current feed/reflux), Winter feed (colder, wetter feed composition), and Turndown (70% of design feed rate) as three cases, track distillate purity and reboiler duty as outputs, and MaximaLabs re-solves all three and returns one grid — no sweep range to define, because these are three specific, named operating points you actually need answers for, not a continuum:

The pattern each row shares: colder/wetter feed costs both purity and energy (more water to reboil off), while turndown trades duty for purity in the other direction — reading that off a grid, not three separate manual runs, is the entire point of a case study. It lives on the same Plots panel as the sweep (§7.1), and reuses the identical re-solve mechanics — a case study is a sensitivity sweep over discrete named points instead of a continuous range. For four full worked case studies — reaction, separation, and economics chained across whole flowsheets, not just one column's named scenarios — see Chapter 11.

7.4 Optimization

Sensitivity and case studies explore; optimization decides. Before you can even pose the problem, you need to know whether there's a decision to make at all — the same degrees-of-freedom count from Chapter 1, applied to the whole optimization problem rather than one flowsheet solve:

$N_f$

Degrees of freedom of the optimization problem — the count of variables genuinely free to be decided.

$\text{independent process variables}$

Every unknown quantity in the problem (flows, temperatures, pressures, compositions, equipment parameters) not already fixed by a spec.

$\text{independent equations}$

Every material/energy balance and equilibrium relation that must hold — each one removes one degree of freedom.

Three cases fall out of the sign of

: at the problem is exactly specified — there's nothing left to optimize, only a system to solve (this is every ordinary flowsheet Run in this Handbook so far); at it's over-specified — too many constraints for the variables you have, no feasible solution exists until you relax one; only at is the problem under-specified in the useful sense — genuinely free variables exist, and picking their best values is what optimization means. With real freedom established, you give an objective and let a solver search those free decision variables for the best feasible point:

$x$

The decision variables — the genuinely free variables N_f identified above (e.g. reflux ratio, feed preheat).

$f(x)$

The objective function being minimized (installed cost, energy, carbon — whichever you choose).

$g(x) = 0$

Equality constraints — the flowsheet's own material and energy balances, which must always hold exactly.

$h(x) \le 0$

Inequality constraints — product specs and equipment limits the solution must not violate (e.g. purity ≥ 95%).

$x^L,\; x^U$

Lower and upper bounds on each decision variable — the physically/operationally sane range the search stays inside.

The objective

might be installed cost, energy, or carbon; the equality constraints are the flowsheet's own balances; the inequalities are your product specs and equipment limits. MaximaLabs drives this with a real optimizer (Pyomo/SciPy) rather than an LLM guessing — the AI copilot can propose what to optimize and set it up, but the deterministic solver finds every number. The built-in objectives (lowest installed cost, lowest carbon footprint) are one click from the Simulation ▸ Optimizer menu; a custom objective and decision-variable set is a short spec away.

7.5 Try it

Sweep to understand, spec to hit a target, optimize to decide — all on the same converged model from the earlier chapters. Last stop: adding time, with dynamic simulation.

7.6 Exercises