Flowsheet Recycle Loops and Numerical Convergence

Scope: Resolve process loops without treating an unconverged iteration as a valid plant prediction.

End-to-end teaching case: A solvent/reagent recycle system with a purge, a splitter and a unit requiring repeated solution passes.

Engineering question driving this unit

Imagine that the simulated plant must answer a design question about recycles, numerical solvers and convergence. The answer should not be a screenshot of a flowsheet. It should be a transparent chain: defined physical boundary → fixed and calculated quantities → constitutive laws → numerical method → independent check → decision and evidence limitations. A solvent/reagent recycle system with a purge, a splitter and a unit requiring repeated solution passes. Your output should clearly distinguish an imposed operating condition from a calculated stream result.

Four-lesson learning sequence

1. [Recycle and purge as component balances](../lessons/unit-06-lesson-01.html) — Governing model: `I_loop = I_fresh/[1−r(1−p)]`. Lab: [Inert accumulation versus recycle purge](../labs/lab-16.html).

2. [Tear streams and fixed-point convergence](../lessons/unit-06-lesson-02.html) — Governing model: `x_(n+1)=b+a x_n; x*=b/(1−a)`. Lab: [Fixed-point recycle iteration and divergence](../labs/lab-17.html).

3. [Numerical error: Euler versus Runge–Kutta](../lessons/unit-06-lesson-03.html) — Governing model: `dy/dt=−k y; y_exact=exp(−kt)`. Lab: [Forward Euler versus fourth-order Runge–Kutta](../labs/lab-18.html).

4. [Diagnosing oscillation, divergence and false closure](../lessons/unit-06-lesson-04.html) — Governing model: `relative change = |x_(n+1)−x_n| / max(1,|x_n|)`. Lab: [Fixed-point recycle iteration and divergence](../labs/lab-17.html).

Working with the central flowsheet

The main scenario is not a plant drawing with arbitrary numbers copied from an operating facility. It is the explicitly synthetic instructional case: A solvent/reagent recycle system with a purge, a splitter and a unit requiring repeated solution passes. Begin by constructing a stream register in which each row contains a stream identifier, the material phase, independent composition variables, total-flow units, temperature, pressure, source block and destination block. For every unit operation attach a balance that can be evaluated independently of the flowsheet iteration. Never assume that graphical arrows prove mass or energy conservation.

A reproducible simulation study also requires a model-boundary register. Label one assumption as a conservation law, one as a constitutive correlation and one as a numerical choice. Conservation laws remain valid as mathematical accounting, subject to correct species and energy definitions. Constitutive correlations need a range of physical applicability. A numerical choice, such as a time step or root tolerance, can change the error without changing the model physics. If a prediction is sensitive to a correlation or solver choice, that sensitivity belongs in the reported result.

Quantities that must be explicit

Use an absolute reference state when an energy balance depends on enthalpy. Identify whether flow is molar, mass-based, or volumetric before using a rate expression. Distinguish local stream variables from plant-wide targets. A measured product purity and a computed purity are not interchangeable evidence. In a unit involving two phases, write the overall composition and each phase composition separately; an overall component balance remains the same even when the phase states change.

Record property assumptions, numerical solver settings and the sign convention for heat and work. For a negative heater duty, the arithmetic may be correct while the configured utility is wrong. For an infeasible flash, the absence of a two-phase root is physically informative; the application must not draw a plausible curve by extrapolating outside a vetted coefficient interval.

Suggested independent audit

Complete an algebraic limit test, a numerical benchmark and a deliberately adverse case. For the limit test, select a zero-flow, zero-duty, zero-conversion or unity-equilibrium case appropriate to the model. For the benchmark, reproduce one displayed result by hand or a separate computation and write down the governing equation and units. For the adverse case, use a nonconvergent recycle, an out-of-range property condition, a negative inventory attempt or a conflicting specification to verify that the model warns rather than silently changes assumptions.

Connection to the interactive laboratory

Every interactive model is a simplified teaching kernel and is accompanied by its own equation, parameter ranges, limitation statement and analysis prompt. For this unit, begin with [Laboratory 16: Inert accumulation versus recycle purge](../labs/lab-16.html) and check the labels before moving a slider. A plotted trend is a hypothesis to explain from the model, not measured evidence from an operating plant.

Unit mastery criteria

You can articulate why the selected model is mathematically closed, reproduce one calculated output, expose a parameter whose value matters to the result, describe the main omitted physical effect, and recommend what data or alternative model would be needed to use the estimate for a real design. Show the residual and a unit-bearing numerical result rather than only a completion mark.

[Unit assessment](../assessments/unit-06.html) · [Applied project](../projects/unit-06.html) · [Book contents](../index.html)

Unit 6, Lesson 1: Recycle and purge as component balances

Part of: [Recycles, Numerical Solvers and Convergence](../units/unit-06.html). Related interactive laboratory: [Lab 16, Inert accumulation versus recycle purge](../labs/lab-16.html). Lesson classification: synthetic teaching example, not a calibrated plant model.

What you will be able to demonstrate

After studying this lesson you should be able to define the quantities needed for recycle and purge as component balances, justify the selected model, work the reference arithmetic with consistent units, distinguish a numerical convergence problem from a physical-model error, and communicate a specific limitation. Your final calculation should identify both the output and the evidence required before it could inform a plant decision. Use the linked laboratory to demonstrate a nontrivial sensitivity, not merely to reproduce its default plot.

Physical interpretation and model boundary

A purge prevents a nonreacting component from accumulating indefinitely in a closed recycle. If a fraction r of a mixed process stream is returned and a fraction p of that return is purged, the effective inert return fraction is r(1−p). The steady-state inert balance is a geometric series. The result is sensitive when the return fraction approaches one: tiny parameter changes can greatly alter the inert flow.

The process example is deliberately constrained to a solvent/reagent recycle system with a purge, a splitter and a unit requiring repeated solution passes. In a modeling report, the relevant boundary must include any place where mass, energy or momentum can enter or leave, and the time basis must match every rate. Write separate entries for imposed specifications and unknown results; otherwise a flowsheet editor may overconstrain the model and resolve conflicting values in an unexpected way. This distinction is central to recycle and purge as component balances because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: I_loop = I_fresh/[1−r(1−p)]. This compact relation is useful only with a defined variable list. Describe which quantities are known, which are solved, and how the physical units cancel. If a value supplied to the relationship is a concentration, specify whether it is molar, mass-based or dimensionless and do not use a rate parameter calibrated on a different basis. The governing relation is a simplifying model; the lesson checks whether the simplification is visible and auditable.

Worked reference exercise

Given and claimed result (synthetic inputs): An inert fresh input of 1 kmol/h, recycle fraction r=0.8 and purge fraction p=0.1 yield I_loop=1/0.28≈3.571 kmol/h.

Start the calculation by writing down the input values, their dimensions, and the full equation. Establish a consistent unit system before computing a result: use seconds for flows quoted per second, hours for rate coefficients quoted per hour, Kelvin for any absolute-temperature dependence, and concentration fractions only when the selected equation expects fractions. If a temperature rise appears in a sensible-heat expression, the increment is numerically the same in K and °C, but an absolute-temperature exponent requires Kelvin.

Substitute the values into the symbolic expression before rounding intermediate results. Then perform a deliberately independent calculation: for a linear balance, reconstruct the corresponding inflow and outflow totals; for a phase equilibrium, reconstruct the sum of liquid and vapor component fractions; for a reactor, evaluate the zero-time or zero-rate limit; for a numerical solver, compare with an analytic case. Use more digits in intermediate values than in the reported engineering result. A residual smaller than machine rounding is reassuring, but it does not validate an incorrect underlying assumption.

Next change one independent input and predict the direction of the response before looking at the updated model output. This step matters because a numerical graph may look smooth even when a sign convention has been reversed. Record the parameter change, the new output, the difference relative to baseline, and whether a conservation limit or an operating constraint becomes active. The result should be evaluated as a comparison within the stated model, not as a prediction of unseen equipment.

A reproducible simulation workflow

1 — Define the task. For recycle and purge as component balances, describe the required output and the acceptable numerical or physical error. Decide whether the result is a state, a rate, a duty, a composition or an economic metric. An error in this classification propagates into the entire flowsheet.

2 — Enumerate inputs. Use the lesson equation I_loop = I_fresh/[1−r(1−p)] to identify independent conditions. Place the measured or stipulated inputs in one column and solver-derived values in another. Inspect whether any input is absent, contradictory, extrapolated or implicitly supplied by an inaccessible property library.

3 — Build the minimum physical model. Start with the simplest conservation or equilibrium statement that describes the scenario. Before adding extra unit operations, establish a hand benchmark from the declared reference exercise. If the benchmark cannot be repeated, reduce the model instead of adding more parameters.

4 — Solve with diagnostics. Select direct substitution for explicit formulas, a bracketed root finder for a single well-behaved nonlinear root, and an integration method appropriate to an initial-value problem. Record the termination condition. Reject nonfinite values, negative physical inventories or phase classifications that violate the declared domains. A converged calculation can still be physically invalid.

5 — Challenge the answer. Calculate the loop inventory for purge fractions 0, 0.05, 0.1 and 0.2 and explain the steepness near a closed loop. Change the model coefficients within the stated range and repeat the conservation test. Distinguish sensitivity caused by engineering physics from sensitivity caused by a numerical tolerance, plotting interval or initial guess.

Numerical and physical verification

Model-specific benchmark: At purge 100%, loop inert equals fresh; for r<1 and p=0, result remains finite.

Most important restriction: This single-inert equation is not a model of reaction conversion, phase separation or pressure-driven recycle feasibility.

A useful validation hierarchy has several levels. First, dimensionally balance the equation and confirm that signed terms can be interpreted physically. Second, test one limiting value such as no flow, unity separation selectivity, zero temperature difference or negligible reaction time. Third, run the model at two different numerical resolutions and check convergence of the answer rather than the smoothness of the drawn line. Fourth, compare against an independent measured data set—but only if such data have actually been provided and their measurement uncertainty is understood.

For nonlinear or highly coupled problems, a small residual may coexist with a wrong root. Check phase validity, conservation and operating boundaries. For a strongly temperature-sensitive equilibrium, bracketing and property-correlation validity are separate tests. For a recycle loop, compare the new tear stream with the previous guess and compute physical flow residuals across the complete loop. For an optimization result, verify both the objective and the physical feasibility of the candidate solution.

Where this model stops being defensible

This single-inert equation is not a model of reaction conversion, phase separation or pressure-driven recycle feasibility. Record this exclusion prominently if the model is exported, reused or embedded in another page. If the omitted effect is expected to be large, replace the current model with a more detailed formulation and obtain appropriate parameter data. Adding decimal places cannot repair missing physics. Likewise, a chart generated using synthetic parameters must never be described as a measurement or a fitted empirical trend.

The operational interpretation is: Track where the purge is located, define the split basis, and verify the inert fresh input equals the final inert loss from the entire process boundary. Before making a final decision, identify a practical observation capable of distinguishing this explanation from an alternative. Such observations might include simultaneous inlet and outlet compositions, a pressure profile, a verified utility duty, a time-dependent level record or temperature-dependent pure-component data. Choose observations appropriate to the specific equipment, and document the measurement range and data quality.

Guided laboratory investigation

Open [the interactive Inert accumulation versus recycle purge model](../labs/lab-16.html). Its sliders expose the primary assumptions; its output is computed locally in the browser, with no network dependency. Read the displayed equation and variable units before changing parameters. Export CSV to preserve numerical values for independent checking. The primary task is: Calculate the loop inventory for purge fractions 0, 0.05, 0.1 and 0.2 and explain the steepness near a closed loop.

The laboratory also states: Single inert, homogeneous recycle splitter, constant r and no in-loop inert reaction. Investigate a plausible operating setting and a limiting setting. Identify the predicted trend, the mechanism producing it, and one reason why the same trend may fail in a real plant. Do not call a laboratory “validated” merely because the curve responds to input changes; validation needs independent comparison and a defined intended use.

What to include in your solution notebook

Your notebook entry should contain a one-line physical question, a labeled sketch or stream inventory, the model equation, a variable-and-unit table, the reference case calculation, one sensitivity experiment, a balance or limiting-case residual and a transparent limitation. Finish with a claim that is no stronger than the evidence. For recycle and purge as component balances, distinguish a computed value from a specified value and explain whether extra physical observations would change confidence in the result.

Knowledge-check questions

1. Why is I_loop = I_fresh/[1−r(1−p)] an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Calculate the loop inventory for purge fractions 0, 0.05, 0.1 and 0.2 and explain the steepness near a closed loop.

1. What physical observation would reveal that this model is inadequate? State one measurable residual or comparative test.

1. If the simulation does not converge or produces an impossible value, what checks would you perform before changing a physical parameter?

[Return to Unit 6](../units/unit-06.html) · [Associated assessment](../assessments/unit-06.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 6, Lesson 2: Tear streams and fixed-point convergence

Part of: [Recycles, Numerical Solvers and Convergence](../units/unit-06.html). Related interactive laboratory: [Lab 17, Fixed-point recycle iteration and divergence](../labs/lab-17.html). Lesson classification: synthetic teaching example, not a calibrated plant model.

What you will be able to demonstrate

After studying this lesson you should be able to define the quantities needed for tear streams and fixed-point convergence, justify the selected model, work the reference arithmetic with consistent units, distinguish a numerical convergence problem from a physical-model error, and communicate a specific limitation. Your final calculation should identify both the output and the evidence required before it could inform a plant decision. Use the linked laboratory to demonstrate a nontrivial sensitivity, not merely to reproduce its default plot.

Physical interpretation and model boundary

Sequential modular simulation requires guessing a recycle stream, evaluating downstream units, comparing the returned estimate and repeating. The selected tear variable must carry all necessary stream properties, not merely a total flow. A linear scalar recurrence demonstrates the core stability condition for fixed-point iteration: the magnitude of the local derivative must be smaller than one. More advanced acceleration or equation-oriented solvers can change convergence behavior but do not remove the physical consistency tests.

The process example is deliberately constrained to a solvent/reagent recycle system with a purge, a splitter and a unit requiring repeated solution passes. In a modeling report, the relevant boundary must include any place where mass, energy or momentum can enter or leave, and the time basis must match every rate. Write separate entries for imposed specifications and unknown results; otherwise a flowsheet editor may overconstrain the model and resolve conflicting values in an unexpected way. This distinction is central to tear streams and fixed-point convergence because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: x_(n+1)=b+a x_n; x*=b/(1−a). This compact relation is useful only with a defined variable list. Describe which quantities are known, which are solved, and how the physical units cancel. If a value supplied to the relationship is a concentration, specify whether it is molar, mass-based or dimensionless and do not use a rate parameter calibrated on a different basis. The governing relation is a simplifying model; the lesson checks whether the simplification is visible and auditable.

Worked reference exercise

Given and claimed result (synthetic inputs): With a=0.6 and b=4, the fixed point is 10; with a=1.05 the naive iteration diverges despite a formal fixed point of −80.

Start the calculation by writing down the input values, their dimensions, and the full equation. Establish a consistent unit system before computing a result: use seconds for flows quoted per second, hours for rate coefficients quoted per hour, Kelvin for any absolute-temperature dependence, and concentration fractions only when the selected equation expects fractions. If a temperature rise appears in a sensible-heat expression, the increment is numerically the same in K and °C, but an absolute-temperature exponent requires Kelvin.

Substitute the values into the symbolic expression before rounding intermediate results. Then perform a deliberately independent calculation: for a linear balance, reconstruct the corresponding inflow and outflow totals; for a phase equilibrium, reconstruct the sum of liquid and vapor component fractions; for a reactor, evaluate the zero-time or zero-rate limit; for a numerical solver, compare with an analytic case. Use more digits in intermediate values than in the reported engineering result. A residual smaller than machine rounding is reassuring, but it does not validate an incorrect underlying assumption.

Next change one independent input and predict the direction of the response before looking at the updated model output. This step matters because a numerical graph may look smooth even when a sign convention has been reversed. Record the parameter change, the new output, the difference relative to baseline, and whether a conservation limit or an operating constraint becomes active. The result should be evaluated as a comparison within the stated model, not as a prediction of unseen equipment.

A reproducible simulation workflow

1 — Define the task. For tear streams and fixed-point convergence, describe the required output and the acceptable numerical or physical error. Decide whether the result is a state, a rate, a duty, a composition or an economic metric. An error in this classification propagates into the entire flowsheet.

2 — Enumerate inputs. Use the lesson equation x_(n+1)=b+a x_n; x*=b/(1−a) to identify independent conditions. Place the measured or stipulated inputs in one column and solver-derived values in another. Inspect whether any input is absent, contradictory, extrapolated or implicitly supplied by an inaccessible property library.

3 — Build the minimum physical model. Start with the simplest conservation or equilibrium statement that describes the scenario. Before adding extra unit operations, establish a hand benchmark from the declared reference exercise. If the benchmark cannot be repeated, reduce the model instead of adding more parameters.

4 — Solve with diagnostics. Select direct substitution for explicit formulas, a bracketed root finder for a single well-behaved nonlinear root, and an integration method appropriate to an initial-value problem. Record the termination condition. Reject nonfinite values, negative physical inventories or phase classifications that violate the declared domains. A converged calculation can still be physically invalid.

5 — Challenge the answer. Compare a=0.4, 0.9 and 1.1; determine how the number of iterations required for a specified residual changes. Change the model coefficients within the stated range and repeat the conservation test. Distinguish sensitivity caused by engineering physics from sensitivity caused by a numerical tolerance, plotting interval or initial guess.

Numerical and physical verification

Model-specific benchmark: If |a|<1, x_n approaches b/(1−a); otherwise do not present terminal value as converged.

Most important restriction: Convergence of a scalar toy iteration does not establish convergence of a coupled nonlinear, multiphase recycle flowsheet.

A useful validation hierarchy has several levels. First, dimensionally balance the equation and confirm that signed terms can be interpreted physically. Second, test one limiting value such as no flow, unity separation selectivity, zero temperature difference or negligible reaction time. Third, run the model at two different numerical resolutions and check convergence of the answer rather than the smoothness of the drawn line. Fourth, compare against an independent measured data set—but only if such data have actually been provided and their measurement uncertainty is understood.

For nonlinear or highly coupled problems, a small residual may coexist with a wrong root. Check phase validity, conservation and operating boundaries. For a strongly temperature-sensitive equilibrium, bracketing and property-correlation validity are separate tests. For a recycle loop, compare the new tear stream with the previous guess and compute physical flow residuals across the complete loop. For an optimization result, verify both the objective and the physical feasibility of the candidate solution.

Where this model stops being defensible

Convergence of a scalar toy iteration does not establish convergence of a coupled nonlinear, multiphase recycle flowsheet. Record this exclusion prominently if the model is exported, reused or embedded in another page. If the omitted effect is expected to be large, replace the current model with a more detailed formulation and obtain appropriate parameter data. Adding decimal places cannot repair missing physics. Likewise, a chart generated using synthetic parameters must never be described as a measurement or a fitted empirical trend.

The operational interpretation is: Report both the update norm and actual material-balance residual. If the iteration diverges, change the solution method or model basis rather than inventing an endpoint. Before making a final decision, identify a practical observation capable of distinguishing this explanation from an alternative. Such observations might include simultaneous inlet and outlet compositions, a pressure profile, a verified utility duty, a time-dependent level record or temperature-dependent pure-component data. Choose observations appropriate to the specific equipment, and document the measurement range and data quality.

Guided laboratory investigation

Open [the interactive Fixed-point recycle iteration and divergence model](../labs/lab-17.html). Its sliders expose the primary assumptions; its output is computed locally in the browser, with no network dependency. Read the displayed equation and variable units before changing parameters. Export CSV to preserve numerical values for independent checking. The primary task is: Compare a=0.4, 0.9 and 1.1; determine how the number of iterations required for a specified residual changes.

The laboratory also states: Scalar linear model, pedagogical surrogate for multidimensional recycle tearing. Investigate a plausible operating setting and a limiting setting. Identify the predicted trend, the mechanism producing it, and one reason why the same trend may fail in a real plant. Do not call a laboratory “validated” merely because the curve responds to input changes; validation needs independent comparison and a defined intended use.

What to include in your solution notebook

Your notebook entry should contain a one-line physical question, a labeled sketch or stream inventory, the model equation, a variable-and-unit table, the reference case calculation, one sensitivity experiment, a balance or limiting-case residual and a transparent limitation. Finish with a claim that is no stronger than the evidence. For tear streams and fixed-point convergence, distinguish a computed value from a specified value and explain whether extra physical observations would change confidence in the result.

Knowledge-check questions

1. Why is x_(n+1)=b+a x_n; x*=b/(1−a) an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Compare a=0.4, 0.9 and 1.1; determine how the number of iterations required for a specified residual changes.

1. What physical observation would reveal that this model is inadequate? State one measurable residual or comparative test.

1. If the simulation does not converge or produces an impossible value, what checks would you perform before changing a physical parameter?

[Return to Unit 6](../units/unit-06.html) · [Associated assessment](../assessments/unit-06.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 6, Lesson 3: Numerical error: Euler versus Runge–Kutta

Part of: [Recycles, Numerical Solvers and Convergence](../units/unit-06.html). Related interactive laboratory: [Lab 18, Forward Euler versus fourth-order Runge–Kutta](../labs/lab-18.html). Lesson classification: synthetic teaching example, not a calibrated plant model.

What you will be able to demonstrate

After studying this lesson you should be able to define the quantities needed for numerical error: euler versus runge–kutta, justify the selected model, work the reference arithmetic with consistent units, distinguish a numerical convergence problem from a physical-model error, and communicate a specific limitation. Your final calculation should identify both the output and the evidence required before it could inform a plant decision. Use the linked laboratory to demonstrate a nontrivial sensitivity, not merely to reproduce its default plot.

Physical interpretation and model boundary

Time-step error can be measured against an analytic solution even when a graph looks smooth. Forward Euler advances using the derivative at the beginning of a step; classic fourth-order Runge–Kutta evaluates additional intermediate slopes. The order is not an unconditional guarantee of physical accuracy: discontinuities, positivity requirements and stiffness can defeat a convenient explicit step size. Use a refinement study and compare end-point errors.

The process example is deliberately constrained to a solvent/reagent recycle system with a purge, a splitter and a unit requiring repeated solution passes. In a modeling report, the relevant boundary must include any place where mass, energy or momentum can enter or leave, and the time basis must match every rate. Write separate entries for imposed specifications and unknown results; otherwise a flowsheet editor may overconstrain the model and resolve conflicting values in an unexpected way. This distinction is central to numerical error: euler versus runge–kutta because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: dy/dt=−k y; y_exact=exp(−kt). This compact relation is useful only with a defined variable list. Describe which quantities are known, which are solved, and how the physical units cancel. If a value supplied to the relationship is a concentration, specify whether it is molar, mass-based or dimensionless and do not use a rate parameter calibrated on a different basis. The governing relation is a simplifying model; the lesson checks whether the simplification is visible and auditable.

Worked reference exercise

Given and claimed result (synthetic inputs): For k=1 s^-1 and step h=0.5 s, Euler gives y(1)=0.25, whereas exact is 0.3679; RK4 is much closer.

Start the calculation by writing down the input values, their dimensions, and the full equation. Establish a consistent unit system before computing a result: use seconds for flows quoted per second, hours for rate coefficients quoted per hour, Kelvin for any absolute-temperature dependence, and concentration fractions only when the selected equation expects fractions. If a temperature rise appears in a sensible-heat expression, the increment is numerically the same in K and °C, but an absolute-temperature exponent requires Kelvin.

Substitute the values into the symbolic expression before rounding intermediate results. Then perform a deliberately independent calculation: for a linear balance, reconstruct the corresponding inflow and outflow totals; for a phase equilibrium, reconstruct the sum of liquid and vapor component fractions; for a reactor, evaluate the zero-time or zero-rate limit; for a numerical solver, compare with an analytic case. Use more digits in intermediate values than in the reported engineering result. A residual smaller than machine rounding is reassuring, but it does not validate an incorrect underlying assumption.

Next change one independent input and predict the direction of the response before looking at the updated model output. This step matters because a numerical graph may look smooth even when a sign convention has been reversed. Record the parameter change, the new output, the difference relative to baseline, and whether a conservation limit or an operating constraint becomes active. The result should be evaluated as a comparison within the stated model, not as a prediction of unseen equipment.

A reproducible simulation workflow

1 — Define the task. For numerical error: euler versus runge–kutta, describe the required output and the acceptable numerical or physical error. Decide whether the result is a state, a rate, a duty, a composition or an economic metric. An error in this classification propagates into the entire flowsheet.

2 — Enumerate inputs. Use the lesson equation dy/dt=−k y; y_exact=exp(−kt) to identify independent conditions. Place the measured or stipulated inputs in one column and solver-derived values in another. Inspect whether any input is absent, contradictory, extrapolated or implicitly supplied by an inaccessible property library.

3 — Build the minimum physical model. Start with the simplest conservation or equilibrium statement that describes the scenario. Before adding extra unit operations, establish a hand benchmark from the declared reference exercise. If the benchmark cannot be repeated, reduce the model instead of adding more parameters.

4 — Solve with diagnostics. Select direct substitution for explicit formulas, a bracketed root finder for a single well-behaved nonlinear root, and an integration method appropriate to an initial-value problem. Record the termination condition. Reject nonfinite values, negative physical inventories or phase classifications that violate the declared domains. A converged calculation can still be physically invalid.

5 — Challenge the answer. Halve the Euler and RK4 step and compare final-time error ratios; explain deviations caused by step-count rounding. Change the model coefficients within the stated range and repeat the conservation test. Distinguish sensitivity caused by engineering physics from sensitivity caused by a numerical tolerance, plotting interval or initial guess.

Numerical and physical verification

Model-specific benchmark: RK4 should be more accurate for the selected nonstiff test; y(0)=1.

Most important restriction: Stiff kinetic systems often need implicit algorithms and event handling; the small offline demonstration is not a stiff solver.

A useful validation hierarchy has several levels. First, dimensionally balance the equation and confirm that signed terms can be interpreted physically. Second, test one limiting value such as no flow, unity separation selectivity, zero temperature difference or negligible reaction time. Third, run the model at two different numerical resolutions and check convergence of the answer rather than the smoothness of the drawn line. Fourth, compare against an independent measured data set—but only if such data have actually been provided and their measurement uncertainty is understood.

For nonlinear or highly coupled problems, a small residual may coexist with a wrong root. Check phase validity, conservation and operating boundaries. For a strongly temperature-sensitive equilibrium, bracketing and property-correlation validity are separate tests. For a recycle loop, compare the new tear stream with the previous guess and compute physical flow residuals across the complete loop. For an optimization result, verify both the objective and the physical feasibility of the candidate solution.

Where this model stops being defensible

Stiff kinetic systems often need implicit algorithms and event handling; the small offline demonstration is not a stiff solver. Record this exclusion prominently if the model is exported, reused or embedded in another page. If the omitted effect is expected to be large, replace the current model with a more detailed formulation and obtain appropriate parameter data. Adding decimal places cannot repair missing physics. Likewise, a chart generated using synthetic parameters must never be described as a measurement or a fitted empirical trend.

The operational interpretation is: A useful verification record states the equation, initial condition, final time, numerical step and error norm. Never equate increasing plot density with decreasing integration step. Before making a final decision, identify a practical observation capable of distinguishing this explanation from an alternative. Such observations might include simultaneous inlet and outlet compositions, a pressure profile, a verified utility duty, a time-dependent level record or temperature-dependent pure-component data. Choose observations appropriate to the specific equipment, and document the measurement range and data quality.

Guided laboratory investigation

Open [the interactive Forward Euler versus fourth-order Runge–Kutta model](../labs/lab-18.html). Its sliders expose the primary assumptions; its output is computed locally in the browser, with no network dependency. Read the displayed equation and variable units before changing parameters. Export CSV to preserve numerical values for independent checking. The primary task is: Halve the Euler and RK4 step and compare final-time error ratios; explain deviations caused by step-count rounding.

The laboratory also states: Fixed 4 s horizon, positive single-state nonstiff test; no adaptive or implicit solver. Investigate a plausible operating setting and a limiting setting. Identify the predicted trend, the mechanism producing it, and one reason why the same trend may fail in a real plant. Do not call a laboratory “validated” merely because the curve responds to input changes; validation needs independent comparison and a defined intended use.

What to include in your solution notebook

Your notebook entry should contain a one-line physical question, a labeled sketch or stream inventory, the model equation, a variable-and-unit table, the reference case calculation, one sensitivity experiment, a balance or limiting-case residual and a transparent limitation. Finish with a claim that is no stronger than the evidence. For numerical error: euler versus runge–kutta, distinguish a computed value from a specified value and explain whether extra physical observations would change confidence in the result.

Knowledge-check questions

1. Why is dy/dt=−k y; y_exact=exp(−kt) an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Halve the Euler and RK4 step and compare final-time error ratios; explain deviations caused by step-count rounding.

1. What physical observation would reveal that this model is inadequate? State one measurable residual or comparative test.

1. If the simulation does not converge or produces an impossible value, what checks would you perform before changing a physical parameter?

[Return to Unit 6](../units/unit-06.html) · [Associated assessment](../assessments/unit-06.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 6, Lesson 4: Diagnosing oscillation, divergence and false closure

Part of: [Recycles, Numerical Solvers and Convergence](../units/unit-06.html). Related interactive laboratory: [Lab 17, Fixed-point recycle iteration and divergence](../labs/lab-17.html). Lesson classification: synthetic teaching example, not a calibrated plant model.

What you will be able to demonstrate

After studying this lesson you should be able to define the quantities needed for diagnosing oscillation, divergence and false closure, justify the selected model, work the reference arithmetic with consistent units, distinguish a numerical convergence problem from a physical-model error, and communicate a specific limitation. Your final calculation should identify both the output and the evidence required before it could inform a plant decision. Use the linked laboratory to demonstrate a nontrivial sensitivity, not merely to reproduce its default plot.

Physical interpretation and model boundary

A solver can stop because an iteration limit is reached, a limit cycle forms, derivatives are undefined or a secondary criterion appears satisfied. Distinguish solver status from physical closure. Detect repeated states, monotonic divergence and values moving outside acceptable domains. Separate machine precision issues from nonlinear conditioning: shrinking a tolerance cannot repair an inconsistent flowsheet.

The process example is deliberately constrained to a solvent/reagent recycle system with a purge, a splitter and a unit requiring repeated solution passes. In a modeling report, the relevant boundary must include any place where mass, energy or momentum can enter or leave, and the time basis must match every rate. Write separate entries for imposed specifications and unknown results; otherwise a flowsheet editor may overconstrain the model and resolve conflicting values in an unexpected way. This distinction is central to diagnosing oscillation, divergence and false closure because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: relative change = |x_(n+1)−x_n| / max(1,|x_n|). This compact relation is useful only with a defined variable list. Describe which quantities are known, which are solved, and how the physical units cancel. If a value supplied to the relationship is a concentration, specify whether it is molar, mass-based or dimensionless and do not use a rate parameter calibrated on a different basis. The governing relation is a simplifying model; the lesson checks whether the simplification is visible and auditable.

Worked reference exercise

Given and claimed result (synthetic inputs): An iteration at x_n=1000 and x_(n+1)=1000.01 has a small relative update yet may violate a 10 kmol/h component balance.

Start the calculation by writing down the input values, their dimensions, and the full equation. Establish a consistent unit system before computing a result: use seconds for flows quoted per second, hours for rate coefficients quoted per hour, Kelvin for any absolute-temperature dependence, and concentration fractions only when the selected equation expects fractions. If a temperature rise appears in a sensible-heat expression, the increment is numerically the same in K and °C, but an absolute-temperature exponent requires Kelvin.

Substitute the values into the symbolic expression before rounding intermediate results. Then perform a deliberately independent calculation: for a linear balance, reconstruct the corresponding inflow and outflow totals; for a phase equilibrium, reconstruct the sum of liquid and vapor component fractions; for a reactor, evaluate the zero-time or zero-rate limit; for a numerical solver, compare with an analytic case. Use more digits in intermediate values than in the reported engineering result. A residual smaller than machine rounding is reassuring, but it does not validate an incorrect underlying assumption.

Next change one independent input and predict the direction of the response before looking at the updated model output. This step matters because a numerical graph may look smooth even when a sign convention has been reversed. Record the parameter change, the new output, the difference relative to baseline, and whether a conservation limit or an operating constraint becomes active. The result should be evaluated as a comparison within the stated model, not as a prediction of unseen equipment.

A reproducible simulation workflow

1 — Define the task. For diagnosing oscillation, divergence and false closure, describe the required output and the acceptable numerical or physical error. Decide whether the result is a state, a rate, a duty, a composition or an economic metric. An error in this classification propagates into the entire flowsheet.

2 — Enumerate inputs. Use the lesson equation relative change = |x_(n+1)−x_n| / max(1,|x_n|) to identify independent conditions. Place the measured or stipulated inputs in one column and solver-derived values in another. Inspect whether any input is absent, contradictory, extrapolated or implicitly supplied by an inaccessible property library.

3 — Build the minimum physical model. Start with the simplest conservation or equilibrium statement that describes the scenario. Before adding extra unit operations, establish a hand benchmark from the declared reference exercise. If the benchmark cannot be repeated, reduce the model instead of adding more parameters.

4 — Solve with diagnostics. Select direct substitution for explicit formulas, a bracketed root finder for a single well-behaved nonlinear root, and an integration method appropriate to an initial-value problem. Record the termination condition. Reject nonfinite values, negative physical inventories or phase classifications that violate the declared domains. A converged calculation can still be physically invalid.

5 — Challenge the answer. Describe a three-part diagnostic using iteration trend, balance residual and sensitivity to initial guess. Change the model coefficients within the stated range and repeat the conservation test. Distinguish sensitivity caused by engineering physics from sensitivity caused by a numerical tolerance, plotting interval or initial guess.

Numerical and physical verification

Model-specific benchmark: If |a|<1, x_n approaches b/(1−a); otherwise do not present terminal value as converged.

Most important restriction: A falsely converged recycle may produce believable energy consumption numbers while violating conservation upstream of the tear.

A useful validation hierarchy has several levels. First, dimensionally balance the equation and confirm that signed terms can be interpreted physically. Second, test one limiting value such as no flow, unity separation selectivity, zero temperature difference or negligible reaction time. Third, run the model at two different numerical resolutions and check convergence of the answer rather than the smoothness of the drawn line. Fourth, compare against an independent measured data set—but only if such data have actually been provided and their measurement uncertainty is understood.

For nonlinear or highly coupled problems, a small residual may coexist with a wrong root. Check phase validity, conservation and operating boundaries. For a strongly temperature-sensitive equilibrium, bracketing and property-correlation validity are separate tests. For a recycle loop, compare the new tear stream with the previous guess and compute physical flow residuals across the complete loop. For an optimization result, verify both the objective and the physical feasibility of the candidate solution.

Where this model stops being defensible

A falsely converged recycle may produce believable energy consumption numbers while violating conservation upstream of the tear. Record this exclusion prominently if the model is exported, reused or embedded in another page. If the omitted effect is expected to be large, replace the current model with a more detailed formulation and obtain appropriate parameter data. Adding decimal places cannot repair missing physics. Likewise, a chart generated using synthetic parameters must never be described as a measurement or a fitted empirical trend.

The operational interpretation is: Retain numeric diagnostics and an explicit nonconverged status instead of showing an attractive final dashboard. Escalate to a better solver only after checking the engineering specification. Before making a final decision, identify a practical observation capable of distinguishing this explanation from an alternative. Such observations might include simultaneous inlet and outlet compositions, a pressure profile, a verified utility duty, a time-dependent level record or temperature-dependent pure-component data. Choose observations appropriate to the specific equipment, and document the measurement range and data quality.

Guided laboratory investigation

Open [the interactive Fixed-point recycle iteration and divergence model](../labs/lab-17.html). Its sliders expose the primary assumptions; its output is computed locally in the browser, with no network dependency. Read the displayed equation and variable units before changing parameters. Export CSV to preserve numerical values for independent checking. The primary task is: Describe a three-part diagnostic using iteration trend, balance residual and sensitivity to initial guess.

The laboratory also states: Scalar linear model, pedagogical surrogate for multidimensional recycle tearing. Investigate a plausible operating setting and a limiting setting. Identify the predicted trend, the mechanism producing it, and one reason why the same trend may fail in a real plant. Do not call a laboratory “validated” merely because the curve responds to input changes; validation needs independent comparison and a defined intended use.

What to include in your solution notebook

Your notebook entry should contain a one-line physical question, a labeled sketch or stream inventory, the model equation, a variable-and-unit table, the reference case calculation, one sensitivity experiment, a balance or limiting-case residual and a transparent limitation. Finish with a claim that is no stronger than the evidence. For diagnosing oscillation, divergence and false closure, distinguish a computed value from a specified value and explain whether extra physical observations would change confidence in the result.

Knowledge-check questions

1. Why is relative change = |x_(n+1)−x_n| / max(1,|x_n|) an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Describe a three-part diagnostic using iteration trend, balance residual and sensitivity to initial guess.

1. What physical observation would reveal that this model is inadequate? State one measurable residual or comparative test.

1. If the simulation does not converge or produces an impossible value, what checks would you perform before changing a physical parameter?

[Return to Unit 6](../units/unit-06.html) · [Associated assessment](../assessments/unit-06.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 6 applied simulation project — Recycles, Numerical Solvers and Convergence

Case: A solvent/reagent recycle system with a purge, a splitter and a unit requiring repeated solution passes.

Engineering brief

Prepare an auditable model that can answer a defined question related to resolve process loops without treating an unconverged iteration as a valid plant prediction. The main deliverable is not a large software project; it is a compact chain of engineering reasoning supported by one or more simulation results and at least one hand-checkable benchmark. Adopt an explicitly synthetic set of operating inputs unless verified experimental data are supplied.

Model specification

Use a coherent subset of these relationships: I_loop = I_fresh/[1−r(1−p)], x_(n+1)=b+a x_n; x*=b/(1−a), dy/dt=−k y; y_exact=exp(−kt) and relative change = |x_(n+1)−x_n| / max(1,|x_n|). Select only relations required by your declared boundary. Describe the balance basis and state or flow variables, and include a table distinguishing imposed specifications from computed stream properties. Confirm that the system is neither underspecified nor contradictory.

Proposed investigation sequence

A — Baseline: Calculate the loop inventory for purge fractions 0, 0.05, 0.1 and 0.2 and explain the steepness near a closed loop. Record the starting parameters and the output. Reproduce one relevant calculation without the software and check its units.

B — Physical sensitivity: Compare a=0.4, 0.9 and 1.1; determine how the number of iterations required for a specified residual changes. Select the parameter alteration before observing the outcome, then explain the sign and approximate magnitude using the governing relation.

C — Numerical or model stress: Halve the Euler and RK4 step and compare final-time error ratios; explain deviations caused by step-count rounding. Find a range limit, solver issue or omitted effect, and describe why it matters. Preserve evidence of nonconvergence if observed.

D — Decision test: Describe a three-part diagnostic using iteration trend, balance residual and sensitivity to initial guess. Write a recommendation conditional on the mathematical assumptions and any unresolved physical constraints.

Acceptance criteria

A reviewer must be able to identify where every numeric input came from, rerun the model offline, understand at least two output curves, see a component or energy balance residual, and read a limitation statement relevant to the chosen process. Provide both ordinary operating and stress cases. Compare your conclusions with what would change if a rate coefficient, property choice, equipment geometry or feed state were uncertain.

Reporting format

Submit a concise technical memo containing: process objective; stream and parameter tables; equations; one worked benchmark; labeled model plots; CSV appendix; a verification checklist; a short uncertainty or sensitivity analysis; and a recommendation marked explicitly as an educational screening result. Do not call it a validated industrial design.

[Unit 6 teaching material](../units/unit-06.html) · [Unit assessment](../assessments/unit-06.html)

Unit 6 assessment — Recycles, Numerical Solvers and Convergence

Assessment design: 16 substantive problems, four for each lesson. These are not auto-scored trivia questions. Show equations, units, actual calculation steps, error checks and model validity statements. Do not present the synthetic values as physical measurements.

Questions

Q1: Formulation — Recycle and purge as component balances. Begin with the physical equipment boundary and derive or justify I_loop = I_fresh/[1−r(1−p)]. Define each variable, its physical units and its status as an input, state or model output. Name one plausible extra process effect requiring an additional equation.

Q1: Independent arithmetic — Recycle and purge as component balances. Work from the following synthetic reference case without copying the plotted output: An inert fresh input of 1 kmol/h, recycle fraction r=0.8 and purge fraction p=0.1 yield I_loop=1/0.28≈3.571 kmol/h. Show substitutions and check an appropriate conservation or limit.

Q1: Model challenge — Recycle and purge as component balances. Calculate the loop inventory for purge fractions 0, 0.05, 0.1 and 0.2 and explain the steepness near a closed loop. Explain what quantitative outcome would support the model and what result would contradict it.

Q1: Responsible interpretation — Recycle and purge as component balances. The model has this restriction: This single-inert equation is not a model of reaction conversion, phase separation or pressure-driven recycle feasibility. Propose one measured variable or external data source that would be needed before a real engineering application.

Q5: Formulation — Tear streams and fixed-point convergence. Begin with the physical equipment boundary and derive or justify x_(n+1)=b+a x_n; x*=b/(1−a). Define each variable, its physical units and its status as an input, state or model output. Name one plausible extra process effect requiring an additional equation.

Q5: Independent arithmetic — Tear streams and fixed-point convergence. Work from the following synthetic reference case without copying the plotted output: With a=0.6 and b=4, the fixed point is 10; with a=1.05 the naive iteration diverges despite a formal fixed point of −80. Show substitutions and check an appropriate conservation or limit.

Q5: Model challenge — Tear streams and fixed-point convergence. Compare a=0.4, 0.9 and 1.1; determine how the number of iterations required for a specified residual changes. Explain what quantitative outcome would support the model and what result would contradict it.

Q5: Responsible interpretation — Tear streams and fixed-point convergence. The model has this restriction: Convergence of a scalar toy iteration does not establish convergence of a coupled nonlinear, multiphase recycle flowsheet. Propose one measured variable or external data source that would be needed before a real engineering application.

Q9: Formulation — Numerical error: Euler versus Runge–Kutta. Begin with the physical equipment boundary and derive or justify dy/dt=−k y; y_exact=exp(−kt). Define each variable, its physical units and its status as an input, state or model output. Name one plausible extra process effect requiring an additional equation.

Q9: Independent arithmetic — Numerical error: Euler versus Runge–Kutta. Work from the following synthetic reference case without copying the plotted output: For k=1 s^-1 and step h=0.5 s, Euler gives y(1)=0.25, whereas exact is 0.3679; RK4 is much closer. Show substitutions and check an appropriate conservation or limit.

Q9: Model challenge — Numerical error: Euler versus Runge–Kutta. Halve the Euler and RK4 step and compare final-time error ratios; explain deviations caused by step-count rounding. Explain what quantitative outcome would support the model and what result would contradict it.

Q9: Responsible interpretation — Numerical error: Euler versus Runge–Kutta. The model has this restriction: Stiff kinetic systems often need implicit algorithms and event handling; the small offline demonstration is not a stiff solver. Propose one measured variable or external data source that would be needed before a real engineering application.

Q13: Formulation — Diagnosing oscillation, divergence and false closure. Begin with the physical equipment boundary and derive or justify relative change = |x_(n+1)−x_n| / max(1,|x_n|). Define each variable, its physical units and its status as an input, state or model output. Name one plausible extra process effect requiring an additional equation.

Q13: Independent arithmetic — Diagnosing oscillation, divergence and false closure. Work from the following synthetic reference case without copying the plotted output: An iteration at x_n=1000 and x_(n+1)=1000.01 has a small relative update yet may violate a 10 kmol/h component balance. Show substitutions and check an appropriate conservation or limit.

Q13: Model challenge — Diagnosing oscillation, divergence and false closure. Describe a three-part diagnostic using iteration trend, balance residual and sensitivity to initial guess. Explain what quantitative outcome would support the model and what result would contradict it.

Q13: Responsible interpretation — Diagnosing oscillation, divergence and false closure. The model has this restriction: A falsely converged recycle may produce believable energy consumption numbers while violating conservation upstream of the tear. Propose one measured variable or external data source that would be needed before a real engineering application.

Required deliverables

Hand in a stream table or variable table, four independent calculations, one CSV export from a relevant laboratory, at least two numerical residuals and a short paragraph explaining which uncertainty would most influence the engineering choice. A technically correct calculation without clear assumptions and units is incomplete.

Assessment rubric (20 points)

• Conservation-law and constitutive-model formulation: 5 points.

• Correct numerical work, correct units and transparent assumptions: 5 points.

• Independent numerical or limiting-case verification: 5 points.

• Careful interpretation, validity range and communication: 5 points.

[Unit lessons](../units/unit-06.html) · [Applied project](../projects/unit-06.html)