Sources and References

Primary texts — Biegler, L.T., Grossmann, I.E., and Westerberg, A.W., Systematic Methods of Chemical Process Design, Prentice Hall, 1997; Seader, J.D., Seider, W.D., and Lewin, D.R., Process Design Principles: Synthesis, Analysis and Evaluation, 4th ed., Wiley, 2017.

Supplementary texts — Finlayson, B.A., Introduction to Chemical Engineering Computing, 2nd ed., Wiley, 2014; Luyben, W.L., Distillation Design and Control Using Aspen Simulation, 2nd ed., Wiley, 2013.

Online resources — MIT OCW 10.34 “Numerical Methods Applied to Chemical Engineering”; NIST WebBook and REFPROP public summaries; DWSIM open-source process simulator documentation; Aspen technical publications and academic tutorials.

Chapter 1: The Role of Process Models

1.1 Why Model?

Process models support design (sizing equipment, predicting performance), control (designing and tuning loops), optimization (steady-state and dynamic), safety analysis (hazard propagation), operator training (simulators), and regulatory validation. A model is neither truth nor fiction but a purposeful abstraction; fit for purpose matters more than completeness.

1.2 Taxonomy of Models

• First-principles (white-box): mass, energy, momentum balances and constitutive laws.
• Empirical (black-box): data-driven, from regression to neural nets.
• Gray-box: first-principles structure with empirical submodels.

Steady-state vs. dynamic; deterministic vs. stochastic; lumped vs. distributed; linear vs. nonlinear. The choice depends on timescales, data availability, and the decisions the model supports.

1.3 Model Building Workflow

Specify purpose → sketch structure → write equations → identify unknowns, parameters, and data → select solution method → solve → validate against data or limiting cases → refine. Iteration is the norm; the first model is rarely the last.

Chapter 2: Equation-Oriented vs. Sequential Modular

2.1 Sequential Modular (SM) Approach

Each unit is a black box with subroutines computing outputs from inputs. The flowsheet is solved one unit at a time in calculation order. Simple, modular, and easy to extend, but recycle streams require tear streams whose values are iterated until convergence.

Convergence strategies: direct substitution (slow but robust), Wegstein (accelerated), dominant eigenvalue methods, Broyden. For each recycle loop, choice of tear streams—independent subset that, when specified, allows sequential solution—dictates performance.

2.2 Equation-Oriented (EO) Approach

All equations are assembled into one large sparse system \( F(x) = 0 \) and solved simultaneously, typically by Newton’s method:

Sparse LU decomposition handles the Jacobian. EO solvers handle recycle and design specifications uniformly, enable rigorous derivatives for optimization, and are dominant in modern simulators for large problems. Initialization and scaling are more demanding than for SM.

2.3 Partitioning and Precedence Ordering

For a process flowsheet, the directed graph of unit dependencies can be partitioned into strongly connected components (SCCs) by Tarjan’s algorithm. Each SCC contains a minimal set of mutually dependent units (a recycle loop); acyclic subgraphs between SCCs can be solved sequentially. This reduces an \( N \)-unit problem to a series of smaller subproblems.

2.4 Tearing Heuristic

Finding a minimum set of tear streams is NP-hard in general; heuristics (Barkley-Motard, Upadhye-Grens, Motard-Westerberg) give near-optimal solutions in polynomial time. Good tearing dramatically speeds SM convergence.

Chapter 3: Numerical Methods for Engineering Systems

3.1 Linear Systems

Dense LU, Cholesky (symmetric positive definite), QR (least-squares), SVD (rank-deficient). For sparse systems, Markowitz pivoting preserves sparsity; iterative methods (GMRES, BiCGStab) work well with good preconditioners (ILU, multigrid).

3.2 Nonlinear Systems

Newton-Raphson requires initial guess and Jacobian. Variants:

• Damped Newton: line search or trust region stabilizes convergence far from the solution.
• Quasi-Newton (Broyden): Jacobian approximated and updated.
• Homotopy continuation: embed problem in a family \( F(x, t) = 0 \) with known solution at \( t = 0 \); track solutions as \( t \to 1 \).

Convergence checking uses \( \|F\| \), \( \|\Delta x\| \), and change in both; a single criterion is insufficient.

3.3 Differential-Algebraic Equations (DAEs)

Many process models take the form \( f(\dot x, x, y, t) = 0 \), mixing differential and algebraic variables. The index of a DAE measures the difficulty: index 1 (most problems) solvable by BDF methods; higher index requires reformulation via index reduction (Pantelides algorithm).

3.4 Integration for Dynamic Models

Explicit methods (RK4) are simple but conditionally stable, bad for stiff problems. Implicit methods (backward Euler, BDF, Rosenbrock, implicit RK) handle stiffness at the cost of solving a nonlinear system per step. Adaptive step size control monitors local error estimate and adjusts.

Chapter 4: Thermodynamic Models

4.1 Equation of State Selection

• Ideal gas: low pressure, far from critical, no association.
• Cubic equations (RK, SRK, Peng-Robinson): general-purpose for hydrocarbons and petrochemicals.
• Lee-Kesler, BWR: improved accuracy at a higher parameter cost.
• Multiparameter Helmholtz-energy models (GERG-2008): reference-quality for natural gas and pure fluids.
• SAFT and PC-SAFT: molecules with association, polymers, electrolytes.

4.2 Activity Coefficient Models

For liquid-phase nonidealities: Wilson, NRTL, UNIQUAC, UNIFAC. Binary parameters must be regressed from data; UNIFAC estimates from group contribution when no data exist. Predictive versions (mod-UNIFAC, PSRK) extend coverage.

4.3 Mixing Rules

Cubic EOS mixing rules: van der Waals (classical), Huron-Vidal/MHV1/MHV2 (derived from activity coefficient models at infinite pressure). Binary interaction parameters \( k_{ij} \) tune predictions; their values come from VLE data fits.

4.4 Flash Calculations

At specified \( T, P \), inside-out and successive substitution algorithms solve Rachford-Rice for vapor fraction and compositions. At specified \( P, H \) or \( P, S \) (energy-specified flash), outer iteration on temperature is added. Robust flash is the backbone of every process simulator.

Chapter 5: Unit Operation Models

5.1 Heat Exchanger Models

Shortcut: \( Q = UA \Delta T_{lm} F \). Rigorous stream-wise discretization handles phase change and multi-pass. Two-phase and zone models (subcooled liquid, two-phase, superheated vapor) accommodate condensers and reboilers accurately.

5.2 Distillation

Tray-by-tray models consist of MESH equations (mass, equilibrium, summation, energy) on each stage. Solution methods: bubble-point (Wang-Henke, Amundson-Pontinen), Naphtali-Sandholm (simultaneous with Newton). Azeotropic and extractive distillation require careful activity coefficient modeling; reactive distillation adds kinetic terms to mass balances.

5.3 Reactor Models

Ideal reactors (PFR, CSTR) translate directly to ODEs/algebraic equations with rate laws. Packed-bed reactors incorporate pressure drop (Ergun) and catalyst effectiveness. Non-isothermal operation couples energy balance with reaction. Compartment models approximate imperfectly mixed real reactors from residence-time distribution data.

5.4 Separation Trains

Separation sequences are assembled from columns, flashes, extractors, and recycle. Models for multi-component separation (Underwood equations, Fenske-Underwood-Gilliland) provide initial estimates; rigorous simulation refines.

Chapter 6: Optimization and Control Applications

6.1 Process Optimization

Nonlinear programming (NLP) formulation: minimize objective (cost, energy, environmental impact) subject to model equations and inequality constraints. Active-set (SQP) and interior-point methods dominate modern solvers (IPOPT, SNOPT, CONOPT). Mixed-integer nonlinear (MINLP) handles discrete decisions (unit selection, stream matching).

6.2 Parameter Estimation

Given data \( y_i = h(x_i; \theta) + \varepsilon_i \), least squares minimizes \( \sum_i \|y_i - h(x_i; \theta)\|^2 \). Weighting by inverse variance (\( \chi^2 \)) handles measurements of different precision. Sensitivity analysis identifies poorly determined parameters; confidence intervals from the Hessian at optimum.

6.3 Dynamic Simulation

Startup, shutdown, process upsets, batch operations all require time-resolved simulation. Model predictive control (MPC) uses repeated solution of an optimization problem over a receding horizon, with state estimation (Kalman filter or moving-horizon estimator) closing the loop on measurements.

6.4 Commercial Simulators

Aspen Plus / HYSYS, UniSim, ProMax, CHEMCAD, DWSIM (open-source), gPROMS, and others. Each balances ease of use, rigor, library breadth, dynamic capability, and price. Engineers should learn at least one deeply and understand the underlying numerical choices (thermodynamic method, flowsheet solution strategy) rather than trust default settings.