Plantwide Integrated and Dynamic Simulation

Scope: Carry steady-state specifications into a credible dynamic study, while retaining equipment and operating constraints.

End-to-end teaching case: A storage tank upstream of a compressor and a packed-bed contactor with a designed operating envelope.

Engineering question driving this unit

Imagine that the simulated plant must answer a design question about integrated and dynamic flowsheets. 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 storage tank upstream of a compressor and a packed-bed contactor with a designed operating envelope. Your output should clearly distinguish an imposed operating condition from a calculated stream result.

Four-lesson learning sequence

1. [Inventories, initial conditions and dynamic responses](../lessons/unit-07-lesson-01.html) — Governing model: `A dh/dt = q_in−C sqrt(h)`. Lab: [Gravity-drained tank dynamic inventory](../labs/lab-19.html).

2. [Ideal-gas compressor performance estimates](../lessons/unit-07-lesson-02.html) — Governing model: `W = m_dot Cp T_in (r^((gamma−1)/gamma)−1)/eta_s`. Lab: [Ideal-gas compression energy versus pressure ratio](../labs/lab-20.html).

3. [Packed-bed resistance and pressure limitations](../lessons/unit-07-lesson-03.html) — Governing model: `ΔP/L = 150 μ(1−ε)^2 v/(ε^3 d_p^2) + 1.75 ρ(1−ε) v²/(ε^3 d_p)`. Lab: [Ergun equation pressure loss through packed bed](../labs/lab-21.html).

4. [Plant-wide integration, interlocks and scenario testing](../lessons/unit-07-lesson-04.html) — Governing model: `global residual = sum external input − sum external output − accumulation`. Lab: [Gravity-drained tank dynamic inventory](../labs/lab-19.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 storage tank upstream of a compressor and a packed-bed contactor with a designed operating envelope. 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 19: Gravity-drained tank dynamic inventory](../labs/lab-19.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-07.html) · [Applied project](../projects/unit-07.html) · [Book contents](../index.html)

Unit 7, Lesson 1: Inventories, initial conditions and dynamic responses

Part of: [Integrated and Dynamic Flowsheets](../units/unit-07.html). Related interactive laboratory: [Lab 19, Gravity-drained tank dynamic inventory](../labs/lab-19.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 inventories, initial conditions and dynamic responses, 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 steady-state flowsheet does not determine dynamic behavior unless holdup, inertia, thermal capacitance and control states are supplied. Tank level is a state because it has an initial value and a balance governing its rate of change. A dynamic liquid-storage model must explicitly handle the nonnegative boundary. Use it to test how an outlet restriction or changed feed rate alters transient inventory rather than assuming instantaneous steady state.

The process example is deliberately constrained to a storage tank upstream of a compressor and a packed-bed contactor with a designed operating envelope. 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 inventories, initial conditions and dynamic responses because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: A dh/dt = q_in−C sqrt(h). 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=2 m², inflow 0.020 m³/s and C=0.010 m^(5/2)/s, the steady head is 4 m.

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 inventories, initial conditions and dynamic responses, 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 A dh/dt = q_in−C sqrt(h) 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. Run a level trajectory from below and above equilibrium and explain which direction each should move. 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: Level remains nonnegative and trends toward h*=(q/C)².

Most important restriction: The square-root outlet law is an empirical hydraulic approximation and constant cross-sectional area is not valid for every vessel.

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 square-root outlet law is an empirical hydraulic approximation and constant cross-sectional area is not valid for every vessel. 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 the computed trajectory with the steady-state root and ensure the level remains nonnegative. If the numerical solver crosses zero under an extreme setting, its algorithm needs a boundary treatment. 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 Gravity-drained tank dynamic inventory model](../labs/lab-19.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: Run a level trajectory from below and above equilibrium and explain which direction each should move.

The laboratory also states: Constant area and empirical gravity drain; no tank overflow or pump cutoff logic. 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 inventories, initial conditions and dynamic responses, 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 A dh/dt = q_in−C sqrt(h) an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Run a level trajectory from below and above equilibrium and explain which direction each should move.

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 7](../units/unit-07.html) · [Associated assessment](../assessments/unit-07.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 7, Lesson 2: Ideal-gas compressor performance estimates

Part of: [Integrated and Dynamic Flowsheets](../units/unit-07.html). Related interactive laboratory: [Lab 20, Ideal-gas compression energy versus pressure ratio](../labs/lab-20.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 ideal-gas compressor performance estimates, 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 an ideal gas with constant heat capacity and a specified isentropic efficiency, compressing to a higher pressure raises gas temperature and requires shaft work. The isentropic temperature ratio depends on gamma and pressure ratio. Real efficiency relates the ideal enthalpy rise to actual compression work. A pressure-ratio sweep visualizes the rising cost of compression but cannot itself evaluate compressor maps, surge, choking or intercooling.

The process example is deliberately constrained to a storage tank upstream of a compressor and a packed-bed contactor with a designed operating envelope. 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 ideal-gas compressor performance estimates because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: W = m_dot Cp T_in (r^((gamma−1)/gamma)−1)/eta_s. 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 m=1 kg/s, Cp=1 kJ/(kg K), T_in=300 K, gamma=1.4, eta_s=0.75 and pressure ratio 4, idealized duty is about 194 kW.

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 ideal-gas compressor performance estimates, 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 W = m_dot Cp T_in (r^((gamma−1)/gamma)−1)/eta_s 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. Investigate the energy penalty of increasing pressure ratio from 2 to 5 and distinguish isentropic efficiency from motor efficiency. 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 r=1, power=0; lower efficiency raises power.

Most important restriction: Real gases, variable Cp and mechanical losses can change the requirement; no vendor operating envelope is encoded in this 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

Real gases, variable Cp and mechanical losses can change the requirement; no vendor operating envelope is encoded in this 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: At pressure ratio one, compression duty should be zero. Reduce efficiency and confirm actual work increases at fixed pressure ratio. 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 Ideal-gas compression energy versus pressure ratio model](../labs/lab-20.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: Investigate the energy penalty of increasing pressure ratio from 2 to 5 and distinguish isentropic efficiency from motor efficiency.

The laboratory also states: Ideal gas and constant heat capacity, no compressor map, no surge, no drive losses. 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 ideal-gas compressor performance estimates, 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 W = m_dot Cp T_in (r^((gamma−1)/gamma)−1)/eta_s an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Investigate the energy penalty of increasing pressure ratio from 2 to 5 and distinguish isentropic efficiency from motor efficiency.

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 7](../units/unit-07.html) · [Associated assessment](../assessments/unit-07.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 7, Lesson 3: Packed-bed resistance and pressure limitations

Part of: [Integrated and Dynamic Flowsheets](../units/unit-07.html). Related interactive laboratory: [Lab 21, Ergun equation pressure loss through packed bed](../labs/lab-21.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 packed-bed resistance and pressure limitations, 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

The Ergun correlation estimates pressure loss through a packed bed by combining a viscous term proportional to superficial velocity and an inertial term proportional to its square. It provides a useful scaling law connecting particle size, porosity, fluid properties and throughput. A packed bed also has heat transfer, mass transfer, maldistribution and potential reaction constraints, none of which can be derived from the pressure-drop curve alone.

The process example is deliberately constrained to a storage tank upstream of a compressor and a packed-bed contactor with a designed operating envelope. 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 packed-bed resistance and pressure limitations because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: ΔP/L = 150 μ(1−ε)^2 v/(ε^3 d_p^2) + 1.75 ρ(1−ε) v²/(ε^3 d_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): For fixed particle diameter, reducing bed void fraction strengthens both the viscous and inertial pressure-drop contributions.

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 packed-bed resistance and pressure limitations, 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 ΔP/L = 150 μ(1−ε)^2 v/(ε^3 d_p^2) + 1.75 ρ(1−ε) v²/(ε^3 d_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. Vary particle diameter from 1 mm to 4 mm and estimate the corresponding pressure-drop sensitivity. 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: Pressure loss is nonnegative and increases with superficial velocity.

Most important restriction: The expression is an empirical model for particular packed-bed conditions; two-phase flow, bed compression and extreme particle shapes require further evidence.

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 expression is an empirical model for particular packed-bed conditions; two-phase flow, bed compression and extreme particle shapes require further evidence. 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: Verify that pressure loss vanishes with superficial velocity and that smaller particles produce greater predicted loss at the same conditions. 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 Ergun equation pressure loss through packed bed model](../labs/lab-21.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: Vary particle diameter from 1 mm to 4 mm and estimate the corresponding pressure-drop sensitivity.

The laboratory also states: Empirical Ergun packed-bed formula; no two-phase flow, changing voidage or non-Darcy entrance effects. 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 packed-bed resistance and pressure limitations, 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 ΔP/L = 150 μ(1−ε)^2 v/(ε^3 d_p^2) + 1.75 ρ(1−ε) v²/(ε^3 d_p) an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Vary particle diameter from 1 mm to 4 mm and estimate the corresponding pressure-drop sensitivity.

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 7](../units/unit-07.html) · [Associated assessment](../assessments/unit-07.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 7, Lesson 4: Plant-wide integration, interlocks and scenario testing

Part of: [Integrated and Dynamic Flowsheets](../units/unit-07.html). Related interactive laboratory: [Lab 19, Gravity-drained tank dynamic inventory](../labs/lab-19.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 plant-wide integration, interlocks and scenario testing, 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

Integrated process simulation means that states and constraints pass through multiple units with correct sequencing. An event affecting inventory may later affect pressure, temperature or composition elsewhere. Test credible startup, shutdown, feed upset and utility-loss scenarios, and distinguish protective interlocks from ordinary control loops. An offline process teaching model must never be used to justify bypassing a safety instrumented function.

The process example is deliberately constrained to a storage tank upstream of a compressor and a packed-bed contactor with a designed operating envelope. 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 plant-wide integration, interlocks and scenario testing because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: global residual = sum external input − sum external output − accumulation. 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): A feed surge doubling tank inflow for one minute can change the downstream compressor inlet condition even when the nominal flowsheet is balanced.

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 plant-wide integration, interlocks and scenario testing, 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 global residual = sum external input − sum external output − accumulation 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. Define an upset scenario and list three measurements needed to decide whether a predicted high-level alarm is credible. 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: Level remains nonnegative and trends toward h*=(q/C)².

Most important restriction: Combining individually reasonable unit models can still create incorrect results if variable bases, time scales or initial states conflict at interfaces.

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

Combining individually reasonable unit models can still create incorrect results if variable bases, time scales or initial states conflict at interfaces. 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: Keep the operating envelope visible, including nonnegative inventories, equipment pressure limits, temperature targets and model validity ranges. 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 Gravity-drained tank dynamic inventory model](../labs/lab-19.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: Define an upset scenario and list three measurements needed to decide whether a predicted high-level alarm is credible.

The laboratory also states: Constant area and empirical gravity drain; no tank overflow or pump cutoff logic. 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 plant-wide integration, interlocks and scenario testing, 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 global residual = sum external input − sum external output − accumulation an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Define an upset scenario and list three measurements needed to decide whether a predicted high-level alarm is credible.

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 7](../units/unit-07.html) · [Associated assessment](../assessments/unit-07.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 7 applied simulation project — Integrated and Dynamic Flowsheets

Case: A storage tank upstream of a compressor and a packed-bed contactor with a designed operating envelope.

Engineering brief

Prepare an auditable model that can answer a defined question related to carry steady-state specifications into a credible dynamic study, while retaining equipment and operating constraints. 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: A dh/dt = q_in−C sqrt(h), W = m_dot Cp T_in (r^((gamma−1)/gamma)−1)/eta_s, ΔP/L = 150 μ(1−ε)^2 v/(ε^3 d_p^2) + 1.75 ρ(1−ε) v²/(ε^3 d_p) and global residual = sum external input − sum external output − accumulation. 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: Run a level trajectory from below and above equilibrium and explain which direction each should move. Record the starting parameters and the output. Reproduce one relevant calculation without the software and check its units.

B — Physical sensitivity: Investigate the energy penalty of increasing pressure ratio from 2 to 5 and distinguish isentropic efficiency from motor efficiency. Select the parameter alteration before observing the outcome, then explain the sign and approximate magnitude using the governing relation.

C — Numerical or model stress: Vary particle diameter from 1 mm to 4 mm and estimate the corresponding pressure-drop sensitivity. Find a range limit, solver issue or omitted effect, and describe why it matters. Preserve evidence of nonconvergence if observed.

D — Decision test: Define an upset scenario and list three measurements needed to decide whether a predicted high-level alarm is credible. 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 7 teaching material](../units/unit-07.html) · [Unit assessment](../assessments/unit-07.html)

Unit 7 assessment — Integrated and Dynamic Flowsheets

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 — Inventories, initial conditions and dynamic responses. Begin with the physical equipment boundary and derive or justify A dh/dt = q_in−C sqrt(h). 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 — Inventories, initial conditions and dynamic responses. Work from the following synthetic reference case without copying the plotted output: With A=2 m², inflow 0.020 m³/s and C=0.010 m^(5/2)/s, the steady head is 4 m. Show substitutions and check an appropriate conservation or limit.

Q1: Model challenge — Inventories, initial conditions and dynamic responses. Run a level trajectory from below and above equilibrium and explain which direction each should move. Explain what quantitative outcome would support the model and what result would contradict it.

Q1: Responsible interpretation — Inventories, initial conditions and dynamic responses. The model has this restriction: The square-root outlet law is an empirical hydraulic approximation and constant cross-sectional area is not valid for every vessel. Propose one measured variable or external data source that would be needed before a real engineering application.

Q5: Formulation — Ideal-gas compressor performance estimates. Begin with the physical equipment boundary and derive or justify W = m_dot Cp T_in (r^((gamma−1)/gamma)−1)/eta_s. 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 — Ideal-gas compressor performance estimates. Work from the following synthetic reference case without copying the plotted output: For m=1 kg/s, Cp=1 kJ/(kg K), T_in=300 K, gamma=1.4, eta_s=0.75 and pressure ratio 4, idealized duty is about 194 kW. Show substitutions and check an appropriate conservation or limit.

Q5: Model challenge — Ideal-gas compressor performance estimates. Investigate the energy penalty of increasing pressure ratio from 2 to 5 and distinguish isentropic efficiency from motor efficiency. Explain what quantitative outcome would support the model and what result would contradict it.

Q5: Responsible interpretation — Ideal-gas compressor performance estimates. The model has this restriction: Real gases, variable Cp and mechanical losses can change the requirement; no vendor operating envelope is encoded in this model. Propose one measured variable or external data source that would be needed before a real engineering application.

Q9: Formulation — Packed-bed resistance and pressure limitations. Begin with the physical equipment boundary and derive or justify ΔP/L = 150 μ(1−ε)^2 v/(ε^3 d_p^2) + 1.75 ρ(1−ε) v²/(ε^3 d_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.

Q9: Independent arithmetic — Packed-bed resistance and pressure limitations. Work from the following synthetic reference case without copying the plotted output: For fixed particle diameter, reducing bed void fraction strengthens both the viscous and inertial pressure-drop contributions. Show substitutions and check an appropriate conservation or limit.

Q9: Model challenge — Packed-bed resistance and pressure limitations. Vary particle diameter from 1 mm to 4 mm and estimate the corresponding pressure-drop sensitivity. Explain what quantitative outcome would support the model and what result would contradict it.

Q9: Responsible interpretation — Packed-bed resistance and pressure limitations. The model has this restriction: The expression is an empirical model for particular packed-bed conditions; two-phase flow, bed compression and extreme particle shapes require further evidence. Propose one measured variable or external data source that would be needed before a real engineering application.

Q13: Formulation — Plant-wide integration, interlocks and scenario testing. Begin with the physical equipment boundary and derive or justify global residual = sum external input − sum external output − accumulation. 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 — Plant-wide integration, interlocks and scenario testing. Work from the following synthetic reference case without copying the plotted output: A feed surge doubling tank inflow for one minute can change the downstream compressor inlet condition even when the nominal flowsheet is balanced. Show substitutions and check an appropriate conservation or limit.

Q13: Model challenge — Plant-wide integration, interlocks and scenario testing. Define an upset scenario and list three measurements needed to decide whether a predicted high-level alarm is credible. Explain what quantitative outcome would support the model and what result would contradict it.

Q13: Responsible interpretation — Plant-wide integration, interlocks and scenario testing. The model has this restriction: Combining individually reasonable unit models can still create incorrect results if variable bases, time scales or initial states conflict at interfaces. 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-07.html) · [Applied project](../projects/unit-07.html)