Chemical Reaction System Simulation

Scope: Select ideal reactor representations, solve material balances and test the consequences of residence time and rate laws.

End-to-end teaching case: A nonhazardous hypothetical liquid-phase irreversible A-to-B reaction using illustrative rate constants.

Engineering question driving this unit

Imagine that the simulated plant must answer a design question about reaction-system simulation. 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 nonhazardous hypothetical liquid-phase irreversible A-to-B reaction using illustrative rate constants. Your output should clearly distinguish an imposed operating condition from a calculated stream result.

Four-lesson learning sequence

1. [Reaction stoichiometry and kinetic bases](../lessons/unit-04-lesson-01.html) — Governing model: `n_i = n_i,0 + nu_i xi`. Lab: [Reaction extent and limiting-reagent inventory](../labs/lab-03.html).

2. [Continuous stirred-tank reactor conversion](../lessons/unit-04-lesson-02.html) — Governing model: `X_CSTR = k tau/(1+k tau), tau=V/F`. Lab: [Isothermal first-order CSTR conversion](../labs/lab-10.html).

3. [Plug-flow residence-time calculation](../lessons/unit-04-lesson-03.html) — Governing model: `X_PFR = 1−exp(−k tau)`. Lab: [Plug flow and CSTR conversion comparison](../labs/lab-11.html).

4. [Batch trajectories, time steps and model checks](../lessons/unit-04-lesson-04.html) — Governing model: `dC_A/dt = −k C_A; C_A(t)=C_A0 exp(−kt)`. Lab: [Batch reactor exact temporal conversion](../labs/lab-12.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 nonhazardous hypothetical liquid-phase irreversible A-to-B reaction using illustrative rate constants. 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 03: Reaction extent and limiting-reagent inventory](../labs/lab-03.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-04.html) · [Applied project](../projects/unit-04.html) · [Book contents](../index.html)

Unit 4, Lesson 1: Reaction stoichiometry and kinetic bases

Part of: [Reaction-System Simulation](../units/unit-04.html). Related interactive laboratory: [Lab 03, Reaction extent and limiting-reagent inventory](../labs/lab-03.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 reaction stoichiometry and kinetic bases, 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 reaction model begins with stoichiometric coefficients and a rate basis. A conversion fraction is defined relative to a chosen limiting reactant feed, while extent of reaction measures the common chemical progress variable. Use coefficients with their signs; reactants have negative stoichiometric entries in the general vector relation. A correct rate expression must match concentration or activity units and a specified volume or catalyst mass basis.

The process example is deliberately constrained to a nonhazardous hypothetical liquid-phase irreversible a-to-b reaction using illustrative rate constants. 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 reaction stoichiometry and kinetic bases because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: n_i = n_i,0 + nu_i xi. 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 A+2B→C with feeds n_A=3 kmol/h and n_B=4 kmol/h, maximum extent is 2 kmol/h, leaving A=1 and B=0.

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 reaction stoichiometry and kinetic bases, 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 n_i = n_i,0 + nu_i xi 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. Compute feasible extent at three feed ratios; explain how the limiting component changes and which product yield definition remains comparable. 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: All component flows must remain nonnegative and n_A+ξ=n_A0, n_B+2ξ=n_B0.

Most important restriction: A reaction can be kinetic-rate-limited, equilibrium-limited or heat-transfer-limited; the stoichiometric maximum alone does not establish achievable conversion.

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 reaction can be kinetic-rate-limited, equilibrium-limited or heat-transfer-limited; the stoichiometric maximum alone does not establish achievable conversion. 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: Test nonnegative species inventories throughout the modeled extent and confirm atom and component-basis accounting before fitting kinetics. 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 Reaction extent and limiting-reagent inventory model](../labs/lab-03.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: Compute feasible extent at three feed ratios; explain how the limiting component changes and which product yield definition remains comparable.

The laboratory also states: Single reaction, no side reactions or phases, 100% desired stoichiometry. 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 reaction stoichiometry and kinetic bases, 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 n_i = n_i,0 + nu_i xi an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Compute feasible extent at three feed ratios; explain how the limiting component changes and which product yield definition remains comparable.

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 4](../units/unit-04.html) · [Associated assessment](../assessments/unit-04.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 4, Lesson 2: Continuous stirred-tank reactor conversion

Part of: [Reaction-System Simulation](../units/unit-04.html). Related interactive laboratory: [Lab 10, Isothermal first-order CSTR conversion](../labs/lab-10.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 continuous stirred-tank reactor conversion, 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

For a perfectly mixed isothermal reactor at steady state, outlet concentration equals the tank concentration. Substituting a first-order rate into the component balance gives a rational conversion formula. At finite residence time a CSTR retains reactant because the whole vessel operates at the exit concentration. A single ideal CSTR therefore usually achieves less conversion than an ideal plug-flow device of equal mean residence time for simple positive-order kinetics.

The process example is deliberately constrained to a nonhazardous hypothetical liquid-phase irreversible a-to-b reaction using illustrative rate constants. 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 continuous stirred-tank reactor conversion because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: X_CSTR = k tau/(1+k tau), tau=V/F. 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=0.5 h^-1 and tau=2 h, the first-order isothermal CSTR predicts X=0.50.

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 continuous stirred-tank reactor conversion, 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_CSTR = k tau/(1+k tau), tau=V/F 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 the CSTR conversion at two residence times and determine how much additional space is required for a target of 80% conversion. 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: 0≤X<1 and X increases monotonically with residence time for k>0.

Most important restriction: Mixing, rate order, heat transfer and volume change can invalidate the simple formula even if the residence-time ratio is correct.

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

Mixing, rate order, heat transfer and volume change can invalidate the simple formula even if the residence-time ratio is correct. 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: The useful limits are X→0 as k tau→0 and X→1 only asymptotically as k tau becomes very large. Check the model against those endpoints. 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 Isothermal first-order CSTR conversion model](../labs/lab-10.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 the CSTR conversion at two residence times and determine how much additional space is required for a target of 80% conversion.

The laboratory also states: Steady perfect mixing, constant density, irreversible first-order isothermal 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 continuous stirred-tank reactor conversion, 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_CSTR = k tau/(1+k tau), tau=V/F an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Compare the CSTR conversion at two residence times and determine how much additional space is required for a target of 80% conversion.

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 4](../units/unit-04.html) · [Associated assessment](../assessments/unit-04.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 4, Lesson 3: Plug-flow residence-time calculation

Part of: [Reaction-System Simulation](../units/unit-04.html). Related interactive laboratory: [Lab 11, Plug flow and CSTR conversion comparison](../labs/lab-11.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 plug-flow residence-time calculation, 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

An ideal plug-flow reactor has no axial mixing, so each fluid element progresses through the reaction with its travel time. Under constant density, first-order kinetics and constant temperature, the axial differential balance integrates to an exponential form. This simplification enables a transparent comparison of reactor types but hides axial dispersion, radial temperature variation and pressure drop. A lab graph plotting conversion versus residence time shows when increasing volume has diminishing returns.

The process example is deliberately constrained to a nonhazardous hypothetical liquid-phase irreversible a-to-b reaction using illustrative rate constants. 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 plug-flow residence-time calculation because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: X_PFR = 1−exp(−k tau). 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=0.5 h^-1 and tau=2 h, an isothermal first-order PFR predicts X≈0.632, above the single CSTR value 0.50.

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 plug-flow residence-time calculation, 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_PFR = 1−exp(−k tau) 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. Find the residence time required to obtain 90% conversion in both an ideal PFR and CSTR, showing the different equations. 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: PFR conversion is never below single-CSTR conversion for kτ≥0.

Most important restriction: The residence-time equality V/F assumes a consistent actual volumetric flow and cannot be reused without adjustment for gas expansion.

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

The residence-time equality V/F assumes a consistent actual volumetric flow and cannot be reused without adjustment for gas expansion. 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: Compare PFR and CSTR at the same Damköhler number and check that neither predicts a conversion above unity. 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 Plug flow and CSTR conversion comparison model](../labs/lab-11.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: Find the residence time required to obtain 90% conversion in both an ideal PFR and CSTR, showing the different equations.

The laboratory also states: Constant-density isothermal ideal-flow models; no axial dispersion or heat release. 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 plug-flow residence-time calculation, 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_PFR = 1−exp(−k tau) an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Find the residence time required to obtain 90% conversion in both an ideal PFR and CSTR, showing the different equations.

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 4](../units/unit-04.html) · [Associated assessment](../assessments/unit-04.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 4, Lesson 4: Batch trajectories, time steps and model checks

Part of: [Reaction-System Simulation](../units/unit-04.html). Related interactive laboratory: [Lab 12, Batch reactor exact temporal conversion](../labs/lab-12.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 batch trajectories, time steps and model checks, 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 batch reactor follows a temporal initial-value problem rather than a once-through steady-state flow balance. The initial concentration is part of the state definition and cannot be filled in by a downstream flowsheet automatically. The exponential first-order reference solution permits a strong numerical verification test: compute the time trajectory using a numerical integrator and compare it with the exact expression at the same timestamps.

The process example is deliberately constrained to a nonhazardous hypothetical liquid-phase irreversible a-to-b reaction using illustrative rate constants. 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 batch trajectories, time steps and model checks because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: dC_A/dt = −k C_A; C_A(t)=C_A0 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): At k=0.1 min^-1 and t=10 min, unreacted fraction is exp(−1)≈0.3679.

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 batch trajectories, time steps and model checks, 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 dC_A/dt = −k C_A; C_A(t)=C_A0 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. Repeat the trajectory with half the integration step and report the maximum difference from the analytic reference. 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: X(0)=0, conversion never exceeds one and remains nondecreasing.

Most important restriction: If reaction rate depends on temperature or species activities, the exact exponential check no longer describes the physical model.

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

If reaction rate depends on temperature or species activities, the exact exponential check no longer describes the physical model. 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: Demand a stable time-step refinement pattern. A plausible-shaped graph does not prove that numerical error is negligible. 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 Batch reactor exact temporal conversion model](../labs/lab-12.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: Repeat the trajectory with half the integration step and report the maximum difference from the analytic reference.

The laboratory also states: Constant volume, constant temperature, no side reactions and constant kinetic rate. 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 batch trajectories, time steps and model checks, 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 dC_A/dt = −k C_A; C_A(t)=C_A0 exp(−kt) an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Repeat the trajectory with half the integration step and report the maximum difference from the analytic reference.

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 4](../units/unit-04.html) · [Associated assessment](../assessments/unit-04.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 4 applied simulation project — Reaction-System Simulation

Case: A nonhazardous hypothetical liquid-phase irreversible A-to-B reaction using illustrative rate constants.

Engineering brief

Prepare an auditable model that can answer a defined question related to select ideal reactor representations, solve material balances and test the consequences of residence time and rate laws. 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: n_i = n_i,0 + nu_i xi, X_CSTR = k tau/(1+k tau), tau=V/F, X_PFR = 1−exp(−k tau) and dC_A/dt = −k C_A; C_A(t)=C_A0 exp(−kt). 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: Compute feasible extent at three feed ratios; explain how the limiting component changes and which product yield definition remains comparable. Record the starting parameters and the output. Reproduce one relevant calculation without the software and check its units.

B — Physical sensitivity: Compare the CSTR conversion at two residence times and determine how much additional space is required for a target of 80% conversion. Select the parameter alteration before observing the outcome, then explain the sign and approximate magnitude using the governing relation.

C — Numerical or model stress: Find the residence time required to obtain 90% conversion in both an ideal PFR and CSTR, showing the different equations. Find a range limit, solver issue or omitted effect, and describe why it matters. Preserve evidence of nonconvergence if observed.

D — Decision test: Repeat the trajectory with half the integration step and report the maximum difference from the analytic reference. 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 4 teaching material](../units/unit-04.html) · [Unit assessment](../assessments/unit-04.html)

Unit 4 assessment — Reaction-System Simulation

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 — Reaction stoichiometry and kinetic bases. Begin with the physical equipment boundary and derive or justify n_i = n_i,0 + nu_i xi. 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 — Reaction stoichiometry and kinetic bases. Work from the following synthetic reference case without copying the plotted output: For A+2B→C with feeds n_A=3 kmol/h and n_B=4 kmol/h, maximum extent is 2 kmol/h, leaving A=1 and B=0. Show substitutions and check an appropriate conservation or limit.

Q1: Model challenge — Reaction stoichiometry and kinetic bases. Compute feasible extent at three feed ratios; explain how the limiting component changes and which product yield definition remains comparable. Explain what quantitative outcome would support the model and what result would contradict it.

Q1: Responsible interpretation — Reaction stoichiometry and kinetic bases. The model has this restriction: A reaction can be kinetic-rate-limited, equilibrium-limited or heat-transfer-limited; the stoichiometric maximum alone does not establish achievable conversion. Propose one measured variable or external data source that would be needed before a real engineering application.

Q5: Formulation — Continuous stirred-tank reactor conversion. Begin with the physical equipment boundary and derive or justify X_CSTR = k tau/(1+k tau), tau=V/F. 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 — Continuous stirred-tank reactor conversion. Work from the following synthetic reference case without copying the plotted output: For k=0.5 h^-1 and tau=2 h, the first-order isothermal CSTR predicts X=0.50. Show substitutions and check an appropriate conservation or limit.

Q5: Model challenge — Continuous stirred-tank reactor conversion. Compare the CSTR conversion at two residence times and determine how much additional space is required for a target of 80% conversion. Explain what quantitative outcome would support the model and what result would contradict it.

Q5: Responsible interpretation — Continuous stirred-tank reactor conversion. The model has this restriction: Mixing, rate order, heat transfer and volume change can invalidate the simple formula even if the residence-time ratio is correct. Propose one measured variable or external data source that would be needed before a real engineering application.

Q9: Formulation — Plug-flow residence-time calculation. Begin with the physical equipment boundary and derive or justify X_PFR = 1−exp(−k tau). 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 — Plug-flow residence-time calculation. Work from the following synthetic reference case without copying the plotted output: For k=0.5 h^-1 and tau=2 h, an isothermal first-order PFR predicts X≈0.632, above the single CSTR value 0.50. Show substitutions and check an appropriate conservation or limit.

Q9: Model challenge — Plug-flow residence-time calculation. Find the residence time required to obtain 90% conversion in both an ideal PFR and CSTR, showing the different equations. Explain what quantitative outcome would support the model and what result would contradict it.

Q9: Responsible interpretation — Plug-flow residence-time calculation. The model has this restriction: The residence-time equality V/F assumes a consistent actual volumetric flow and cannot be reused without adjustment for gas expansion. Propose one measured variable or external data source that would be needed before a real engineering application.

Q13: Formulation — Batch trajectories, time steps and model checks. Begin with the physical equipment boundary and derive or justify dC_A/dt = −k C_A; C_A(t)=C_A0 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.

Q13: Independent arithmetic — Batch trajectories, time steps and model checks. Work from the following synthetic reference case without copying the plotted output: At k=0.1 min^-1 and t=10 min, unreacted fraction is exp(−1)≈0.3679. Show substitutions and check an appropriate conservation or limit.

Q13: Model challenge — Batch trajectories, time steps and model checks. Repeat the trajectory with half the integration step and report the maximum difference from the analytic reference. Explain what quantitative outcome would support the model and what result would contradict it.

Q13: Responsible interpretation — Batch trajectories, time steps and model checks. The model has this restriction: If reaction rate depends on temperature or species activities, the exact exponential check no longer describes the physical model. 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-04.html) · [Applied project](../projects/unit-04.html)