How Do Feasible Path And Infeasible Path Optimization Approaches Differ? Choose The Best Method For Batch Distillation

Feasible path and infeasible path optimization differ primarily in when process feasibility is enforced. In a feasible path method, the optimizer changes the control profile, such as reflux ratio, and integrates the complete batch distillation DAE/ODE model at every iteration, so each evaluated trajectory satisfies the process equations. In an infeasible path method, the optimizer discretizes both the control profile and the dynamic process equations, solving the resulting algebraic optimization problem even when intermediate iterates do not yet satisfy the model equations.

Feasible path optimization provides physically consistent trajectories throughout the search but can be computationally expensive because it repeatedly integrates the dynamic model and evaluates gradients. Infeasible path optimization usually offers more direct algebraic gradient calculations and potentially better computational performance, but feasibility is restored only as the optimization converges.

Why Batch Distillation Requires Dynamic Optimization

The Process State Changes Continuously

Batch distillation is inherently unsteady-state. As the batch proceeds, volatile components are depleted from the reboiler and column, causing liquid holdups, temperatures, compositions, and separation behavior to change over time.

A steady-state optimization model cannot adequately represent these transient effects. The pilot plant therefore requires a dynamic model expressed through ordinary differential equations (ODEs) or differential-algebraic equations (DAEs).

The Control Profile Determines the Separation Path

The main decision variable is often a time-varying reflux ratio, although reboiler heat duty and termination thresholds may also be optimized. The selected control trajectory affects product purity, recovery, distillate volume, batch time, and energy use.

Thermodynamic distillation maps also constrain the attainable residue and distillate compositions. This means product specifications cannot generally be selected independently of recovery and other intensive operating quantities.

The Optimization Objective Is Operational

Typical objectives include:

Minimizing batch time while meeting purity and recovery targets.
Maximizing purified distillate collected within a fixed time.
Maximizing profit by balancing product value, utility costs, raw material costs, and cycle overhead.

The optimization must also respect constraints such as maximum reboiler temperature, minimum product purity, and minimum reboiler liquid level.

How Feasible Path Optimization Works

Only the Control Is Directly Discretized

In the feasible path approach, the control profile is parameterized using time intervals, control vector parameterization, or another finite-dimensional representation. For example, the reflux ratio may be defined by a sequence of piecewise-constant or piecewise-linear values.

For every trial control profile, the full dynamic model is integrated forward in time. The resulting state trajectory provides the compositions, temperatures, holdups, and other variables needed to evaluate the objective and constraints.

Every Evaluated Trajectory Satisfies the Model

Because the DAE or ODE system is integrated at each iteration, the calculated trajectory is consistent with the process equations, subject to numerical integration accuracy. The optimizer therefore searches over controls while the process states are generated by the model.

This is the defining feature of a feasible path method: dynamic feasibility is maintained during the optimization search.

Gradients Are the Main Computational Cost

The objective and constraint gradients must describe how changes in reflux, heat duty, or switching times affect the entire state trajectory. In batch distillation, this sensitivity calculation can account for a substantial share of total computation time.

Common approaches include:

Finite differences, which are simple but require repeated model evaluations and are sensitive to perturbation size.
Adjoint equations, which integrate the process model forward and an adjoint system backward, making them efficient when there are many control variables relative to constraints.
State trajectory sensitivities, which integrate the states and their sensitivities simultaneously and can provide strong numerical stability by avoiding backward integration.

How Infeasible Path Optimization Works

States and Controls Are Both Optimization Variables

The infeasible path approach discretizes the dynamic equations and the control profile. The values of the process states at selected time points become optimization variables alongside the reflux ratio, heat duty, or other controls.

The differential and algebraic equations are then converted into a set of algebraic equality constraints, commonly through a direct transcription or collocation procedure.

Feasibility Is Enforced Through Constraints

Intermediate optimization iterates may violate the discretized process equations. These iterates are therefore called infeasible paths: the proposed state and control values do not necessarily form a physically valid batch trajectory during the early stages of the search.

The nonlinear programming solver progressively reduces these equation residuals until the final solution satisfies the dynamic model within specified tolerances.

Gradients Come From the Algebraic Formulation

Once the dynamic model has been discretized, gradients can be calculated directly from the resulting algebraic equations and their derivatives. The optimizer does not necessarily need to integrate the complete dynamic model separately at every iteration in the same way as a feasible path method.

This formulation can be attractive for large, constrained problems because the dynamic equations, path constraints, control bounds, and endpoint specifications are handled together in one nonlinear programming problem.

The Practical Difference for Pilot Plant Optimization

Feasible Path Emphasizes Valid Trajectories

Feasible path optimization is useful when it is important to inspect or evaluate physically consistent trajectories throughout the search. Each control trial produces a process trajectory that obeys the integrated model, which can simplify interpretation and support simulation-based validation.

It is also a natural choice when an existing dynamic simulator already provides reliable forward integration and when the number of control parameters is moderate.

Infeasible Path Emphasizes Unified Optimization

Infeasible path optimization treats the control profile and discretized state trajectory as a single optimization problem. This makes it easier to impose state constraints at specific time points or throughout the batch, including temperature limits, purity bounds, holdup restrictions, and reboiler operating limits.

Its algebraic structure can also provide efficient derivative calculations, especially when paired with analytic or automatic differentiation.

Both Methods Can Represent the Same Control Problem

The distinction is not between dynamic and nondynamic optimization. Both approaches can solve a dynamic batch distillation problem governed by DAEs or ODEs.

The difference is the numerical treatment of the process states: feasible path methods generate states by integration, while infeasible path methods optimize discretized states subject to dynamic-equation constraints.

Understanding the Trade-offs

Computational Expense

Feasible path methods may require many forward integrations and repeated sensitivity calculations. Finite-difference gradients can be particularly expensive because each perturbed control profile may require another complete batch simulation.

Infeasible path methods avoid relying solely on repeated forward simulations, but they can create large nonlinear programs with many state variables and equality constraints. The resulting problem may demand substantial memory and a capable sparse NLP solver.

Numerical Robustness

Feasible path optimization benefits from the fact that every simulated trajectory respects the model dynamics. However, difficult regions such as sharp composition transitions, changing phase behavior, or near-singular DAEs can make integration and sensitivity calculations challenging.

Infeasible path methods can move through dynamically inconsistent intermediate states. This flexibility may improve the optimizer's ability to reach a solution, but poor scaling, weak initialization, or ill-conditioned discretized equations can make convergence difficult.

Constraint Handling

Feasible path methods can evaluate state constraints along the integrated trajectory, but accurate enforcement requires suitable time grids and event handling. Important violations between evaluation points can otherwise be missed.

Infeasible path methods naturally include state values and path constraints in the NLP. Their accuracy still depends on mesh resolution and discretization quality, so a converged NLP solution must be checked against the original continuous-time model.

Initialization and Model Quality

A feasible path method generally needs an initial control profile, but the corresponding state trajectory is generated by the simulator. This can make initialization relatively straightforward when the model integrates reliably.

An infeasible path method may require initial guesses for both controls and states. A dynamically consistent initial trajectory is helpful but is not always required; the solver can reduce the initial model residuals during optimization.

Validation Requirements

Neither approach eliminates the need for independent verification. The final control policy should be simulated with the original high-fidelity DAE/ODE model, checked against all safety and product constraints, and tested under parameter variations.

For a pilot plant, nominal feasibility is not sufficient if uncertainty in feed composition, thermodynamic parameters, heat transfer, or measurement conditions can cause purity or temperature limits to be violated.

Choosing Between the Two Approaches

When Feasible Path Is Appropriate

Feasible path optimization is often suitable when the pilot plant already has a trusted forward simulator, the number of control variables is limited, and physically valid intermediate trajectories are valuable for analysis.

It is also useful when robust trajectory sensitivities or online sensitivity information are more important than solving a very large optimization problem in one unified formulation.

When Infeasible Path Is Appropriate

Infeasible path optimization is often preferable when the problem contains many state and path constraints, multiple control intervals, switching decisions, or tightly coupled operating limits.

It is particularly attractive when direct transcription, collocation, sparse derivatives, and nonlinear programming infrastructure are available.

A Hybrid Workflow Can Be Effective

The methods can be used together. A feasible simulation can generate a high-quality initial state trajectory, after which an infeasible path formulation can refine the control and state variables while enforcing detailed constraints.

Conversely, an infeasible path solution can provide an initial control policy for a high-fidelity feasible-path simulation and subsequent sensitivity-based refinement.

Making the Right Choice for Your Goal

The best formulation depends on the plant model, optimization scale, constraint structure, and available derivative technology.

If your primary focus is reliable forward simulation and physically interpretable trajectories: Use a feasible path method with efficient sensitivity equations or adjoint gradients, then validate the final reflux and heat-duty profiles against the full pilot-plant model.
If your primary focus is large-scale constrained optimization: Use an infeasible path formulation with direct transcription or collocation, sparse nonlinear programming, and careful scaling of dynamic and operating constraints.
If your primary focus is real-time or rapidly repeated optimization: Consider a reduced polynomial or surrogate model for screening, followed by high-fidelity dynamic validation before applying the operating policy.
If your primary focus is robust pilot-plant operation: Either formulation should be tested against worst-case parameter variations, temperature limits, purity requirements, reboiler level constraints, and uncertainty in feed conditions.

The right choice is the one that balances computational cost, constraint reliability, derivative quality, and confidence that the optimized policy will remain safe and effective on the actual batch distillation pilot plant.

Summary Table:

Aspect	Feasible Path	Infeasible Path
State treatment	States are generated by integrating the ODE/DAE model.	States and controls are optimized together.
Feasibility	Every evaluated trajectory satisfies the process model.	Feasibility is restored through dynamic-equation constraints.
Main strength	Physically consistent, interpretable trajectories.	Unified handling of large-scale constraints.
Main challenge	Repeated simulations and sensitivity calculations can be expensive.	Large nonlinear programs may require substantial memory and careful scaling.
Suitable applications	Trusted simulators, moderate control dimensions, trajectory analysis.	Many state constraints, control intervals, switching decisions, or tight limits.
Validation	Verify the final policy with the original high-fidelity model.	Check the discretized solution against the continuous-time model.
Recommended practice	Use sensitivity or adjoint gradients when possible.	Use direct transcription or collocation with sparse derivatives.

Optimize Your Pilot Plant with Confidence

Need to select the right optimization approach for your process? Contact LABPARK to discuss an Educational or Vocational Unit Operations Pilot Plant tailored to your needs. LABPARK provides pilot plants for chemical engineering, bioprocess and biotechnology, and environmental and water treatment applications.

Our systems help universities, research institutes, and enterprises evaluate dynamic process behavior, develop control strategies, train operators, and validate optimized operating policies under realistic conditions. We can support batch distillation studies involving reflux control, heat-duty optimization, purity and recovery targets, temperature limits, reboiler levels, and model-based validation. Contact LABPARK to identify the appropriate configuration, integrate your process objectives, and build a reliable platform for research, teaching, scale-up, and industrial process improvement.