Thermodynamic Property Packages and Phase Equilibria

Scope: Use a property basis consistent with phase, pressure and temperature, and distinguish physical-property error from numerical error.

End-to-end teaching case: A volatile binary mixture entering a vapor-liquid equilibrium vessel operated around atmospheric pressure.

Engineering question driving this unit

Imagine that the simulated plant must answer a design question about property models and thermodynamic equilibrium. 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 volatile binary mixture entering a vapor-liquid equilibrium vessel operated around atmospheric pressure. Your output should clearly distinguish an imposed operating condition from a calculated stream result.

Four-lesson learning sequence

1. [Property packages and reference states](../lessons/unit-02-lesson-01.html) — Governing model: `h(T) − h(T_ref) = integral(Cp(T), T_ref to T)`. Lab: [Ideal ethanol–water bubble-point solver](../labs/lab-04.html).

2. [Antoine vapor pressure and Raoult-law bubble point](../lessons/unit-02-lesson-02.html) — Governing model: `P = x_ethanol P_sat,ethanol(T) + (1−x_ethanol) P_sat,water(T)`. Lab: [Ideal ethanol–water bubble-point solver](../labs/lab-04.html).

3. [Isothermal flash and Rachford–Rice logic](../lessons/unit-02-lesson-03.html) — Governing model: `sum_i z_i (K_i−1)/(1+beta (K_i−1)) = 0`. Lab: [Binary isothermal flash phase fraction](../labs/lab-05.html).

4. [Chemical equilibrium versus phase equilibrium](../lessons/unit-02-lesson-04.html) — Governing model: `K_eq(T) = K_ref exp[−ΔH/R (1/T − 1/T_ref)]`. Lab: [Van ’t Hoff reaction-equilibrium shift](../labs/lab-06.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 volatile binary mixture entering a vapor-liquid equilibrium vessel operated around atmospheric pressure. 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 04: Ideal ethanol–water bubble-point solver](../labs/lab-04.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-02.html) · [Applied project](../projects/unit-02.html) · [Book contents](../index.html)

Unit 2, Lesson 1: Property packages and reference states

Part of: [Property Models and Thermodynamic Equilibrium](../units/unit-02.html). Related interactive laboratory: [Lab 04, Ideal ethanol–water bubble-point solver](../labs/lab-04.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 property packages and reference states, 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 property method is a set of relationships defining phase equilibrium, density, enthalpy, viscosity and other quantities in a usable range. An equation of state and a liquid activity-coefficient model make different assumptions about interactions. Identify the species and phases actually present before selecting one. Even a simple fixed heat capacity requires an enthalpy reference, so it should never be combined blindly with latent-heat terms from another basis.

The process example is deliberately constrained to a volatile binary mixture entering a vapor-liquid equilibrium vessel operated around atmospheric pressure. 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 property packages and reference states because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: h(T) − h(T_ref) = integral(Cp(T), T_ref to T). 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 Cp=4 kJ/(kg K), a 5 kg/s single-phase stream heated by 20 K requires 400 kW of sensible heat.

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 property packages and reference states, 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 h(T) − h(T_ref) = integral(Cp(T), T_ref to T) 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 the effect of a ±10% Cp uncertainty on a heater duty and explain why the outlet temperature alone does not reveal reference-state consistency. 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 the bubble root total partial pressure equals applied pressure; vapor mole fractions sum to one.

Most important restriction: Constant heat capacity does not describe vaporization, strong temperature dependence or a reacting mixture across a wide range.

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

Constant heat capacity does not describe vaporization, strong temperature dependence or a reacting mixture across a wide range. 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: Property consistency must be checked through limiting behavior, phase flags and units; when trusted data are unavailable, label every numerical constant as synthetic and avoid claiming predictive accuracy. 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 ethanol–water bubble-point solver model](../labs/lab-04.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 the effect of a ±10% Cp uncertainty on a heater duty and explain why the outlet temperature alone does not reveal reference-state consistency.

The laboratory also states: Ideal Raoult-law mixture; DOES NOT represent ethanol–water azeotropy. Correlation overlap strictly 344–363 K: outside this range report no validated root. 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 property packages and reference states, 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 h(T) − h(T_ref) = integral(Cp(T), T_ref to T) an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Compute the effect of a ±10% Cp uncertainty on a heater duty and explain why the outlet temperature alone does not reveal reference-state consistency.

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 2](../units/unit-02.html) · [Associated assessment](../assessments/unit-02.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 2, Lesson 2: Antoine vapor pressure and Raoult-law bubble point

Part of: [Property Models and Thermodynamic Equilibrium](../units/unit-02.html). Related interactive laboratory: [Lab 04, Ideal ethanol–water bubble-point solver](../labs/lab-04.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 antoine vapor pressure and raoult-law bubble point, 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

Antoine parameters encode pure-component saturated vapor pressures. Raoult’s law then uses liquid mole fractions as weights in an ideal-solution vapor-pressure sum. A bubble point is not a weighted average of pure boiling temperatures. Use Kelvin if the selected coefficient table defines Kelvin; do not substitute Celsius into a denominator calibrated for another scale. Constrain numerical bracketing to a temperature interval where both pure-component parameter correlations are documented.

The process example is deliberately constrained to a volatile binary mixture entering a vapor-liquid equilibrium vessel operated around atmospheric pressure. 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 antoine vapor pressure and raoult-law bubble point because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: P = x_ethanol P_sat,ethanol(T) + (1−x_ethanol) P_sat,water(T). 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 a specified liquid ethanol fraction, the bubble temperature is the root of summed ideal partial pressures minus total pressure.

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 antoine vapor pressure and raoult-law bubble point, 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 P = x_ethanol P_sat,ethanol(T) + (1−x_ethanol) P_sat,water(T) 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 bubble point at two ethanol fractions and verify that the calculated vapor fraction of ethanol differs from the liquid fraction. 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 the bubble root total partial pressure equals applied pressure; vapor mole fractions sum to one.

Most important restriction: Ethanol/water is nonideal in important regimes; the ideal calculation is a pedagogical benchmark, not a rigorous azeotrope 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

Ethanol/water is nonideal in important regimes; the ideal calculation is a pedagogical benchmark, not a rigorous azeotrope 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: For the interactive example, NIST Antoine correlations are applied within their stated overlapping temperature ranges only. A failed root bracket is a useful status, not an invitation to return a fabricated temperature. 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 ethanol–water bubble-point solver model](../labs/lab-04.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 bubble point at two ethanol fractions and verify that the calculated vapor fraction of ethanol differs from the liquid fraction.

The laboratory also states: Ideal Raoult-law mixture; DOES NOT represent ethanol–water azeotropy. Correlation overlap strictly 344–363 K: outside this range report no validated root. 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 antoine vapor pressure and raoult-law bubble point, 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 P = x_ethanol P_sat,ethanol(T) + (1−x_ethanol) P_sat,water(T) an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Find the bubble point at two ethanol fractions and verify that the calculated vapor fraction of ethanol differs from the liquid fraction.

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 2](../units/unit-02.html) · [Associated assessment](../assessments/unit-02.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 2, Lesson 3: Isothermal flash and Rachford–Rice logic

Part of: [Property Models and Thermodynamic Equilibrium](../units/unit-02.html). Related interactive laboratory: [Lab 05, Binary isothermal flash phase fraction](../labs/lab-05.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 isothermal flash and rachford–rice logic, 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 isothermal flash divides a feed into vapor and liquid while conserving each component. K values relate equilibrium vapor and liquid compositions; they are not separation efficiencies. The Rachford–Rice relation follows by eliminating phase compositions from component balances. Test the function at beta=0 and beta=1. Opposite signs indicate a root in the physical vapor-fraction interval; otherwise the state is single phase under the specified constants.

The process example is deliberately constrained to a volatile binary mixture entering a vapor-liquid equilibrium vessel operated around atmospheric pressure. 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 isothermal flash and rachford–rice logic because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: sum_i z_i (K_i−1)/(1+beta (K_i−1)) = 0. 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 z_A=0.5, K_A=2 and K_B=0.5, the two-phase solution is beta=0.5, x_A=1/3, y_A=2/3.

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 isothermal flash and rachford–rice logic, 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 sum_i z_i (K_i−1)/(1+beta (K_i−1)) = 0 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. At a selected feed composition, recompute both phase fractions and the A component balance from the chart data. 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: Two-phase solutions satisfy component closure within tolerance and 0≤β≤1; no false interior root.

Most important restriction: Constant arbitrary K values neglect temperature, pressure and mixture dependence; two-phase roots are not evidence that the chosen property package is appropriate.

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

Constant arbitrary K values neglect temperature, pressure and mixture dependence; two-phase roots are not evidence that the chosen property package is appropriate. 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: Check beta bounds, both composition summations and an overall component balance. The browser lab includes phase-state notices where a two-phase solution does not exist. 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 Binary isothermal flash phase fraction model](../labs/lab-05.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: At a selected feed composition, recompute both phase fractions and the A component balance from the chart data.

The laboratory also states: Constant user-supplied equilibrium ratios. They are NOT real component property predictions. 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 isothermal flash and rachford–rice logic, 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 sum_i z_i (K_i−1)/(1+beta (K_i−1)) = 0 an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. At a selected feed composition, recompute both phase fractions and the A component balance from the chart data.

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 2](../units/unit-02.html) · [Associated assessment](../assessments/unit-02.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 2, Lesson 4: Chemical equilibrium versus phase equilibrium

Part of: [Property Models and Thermodynamic Equilibrium](../units/unit-02.html). Related interactive laboratory: [Lab 06, Van ’t Hoff reaction-equilibrium shift](../labs/lab-06.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 chemical equilibrium versus phase equilibrium, 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 chemical-equilibrium constant relates reaction activities after a stoichiometric reaction has occurred. A phase-equilibrium K value instead links vapor and liquid composition of one component; using the same letter does not make the models interchangeable. Van ’t Hoff’s relation predicts how the equilibrium constant changes with temperature when the standard reaction enthalpy is approximated as constant. It captures a useful direction-of-change test for exothermic and endothermic examples.

The process example is deliberately constrained to a volatile binary mixture entering a vapor-liquid equilibrium vessel operated around atmospheric pressure. 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 chemical equilibrium versus phase equilibrium because several different assumptions can yield similar plotted curves while representing different physical devices.

Governing relation: K_eq(T) = K_ref exp[−ΔH/R (1/T − 1/T_ref)]. 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 ⇌ B with ideal activities and K_eq=4, x_B at equilibrium is K/(1+K)=0.80.

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

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

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

A reproducible simulation workflow

1 — Define the task. For chemical equilibrium versus phase equilibrium, 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 K_eq(T) = K_ref exp[−ΔH/R (1/T − 1/T_ref)] 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. Switch ΔH from negative to positive and interpret the reversal in the conversion–temperature slope. 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: Product fraction remains between zero and one; negative ΔH causes conversion to decline as T increases.

Most important restriction: Constant reaction enthalpy and ideal activities are modeling choices, not universal descriptions of real reaction chemistry.

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

Constant reaction enthalpy and ideal activities are modeling choices, not universal descriptions of real reaction chemistry. 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: Confirm that an exothermic reaction favors reactants as temperature rises and that a product fraction stays within the admissible interval. Report the assumption before optimizing temperature. 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 Van ’t Hoff reaction-equilibrium shift model](../labs/lab-06.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: Switch ΔH from negative to positive and interpret the reversal in the conversion–temperature slope.

The laboratory also states: Ideal activities, constant ΔH over temperature range, single reversible 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 chemical equilibrium versus phase equilibrium, 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 K_eq(T) = K_ref exp[−ΔH/R (1/T − 1/T_ref)] an appropriate representation for the specified instructional boundary, and which input must be independently known?

1. Switch ΔH from negative to positive and interpret the reversal in the conversion–temperature slope.

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 2](../units/unit-02.html) · [Associated assessment](../assessments/unit-02.html) · [All laboratories](../LABORATORY-MANUAL.html)

Unit 2 applied simulation project — Property Models and Thermodynamic Equilibrium

Case: A volatile binary mixture entering a vapor-liquid equilibrium vessel operated around atmospheric pressure.

Engineering brief

Prepare an auditable model that can answer a defined question related to use a property basis consistent with phase, pressure and temperature, and distinguish physical-property error from numerical error. 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: h(T) − h(T_ref) = integral(Cp(T), T_ref to T), P = x_ethanol P_sat,ethanol(T) + (1−x_ethanol) P_sat,water(T), sum_i z_i (K_i−1)/(1+beta (K_i−1)) = 0 and K_eq(T) = K_ref exp[−ΔH/R (1/T − 1/T_ref)]. 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 the effect of a ±10% Cp uncertainty on a heater duty and explain why the outlet temperature alone does not reveal reference-state consistency. Record the starting parameters and the output. Reproduce one relevant calculation without the software and check its units.

B — Physical sensitivity: Find the bubble point at two ethanol fractions and verify that the calculated vapor fraction of ethanol differs from the liquid fraction. Select the parameter alteration before observing the outcome, then explain the sign and approximate magnitude using the governing relation.

C — Numerical or model stress: At a selected feed composition, recompute both phase fractions and the A component balance from the chart data. Find a range limit, solver issue or omitted effect, and describe why it matters. Preserve evidence of nonconvergence if observed.

D — Decision test: Switch ΔH from negative to positive and interpret the reversal in the conversion–temperature slope. 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 2 teaching material](../units/unit-02.html) · [Unit assessment](../assessments/unit-02.html)

Unit 2 assessment — Property Models and Thermodynamic Equilibrium

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 — Property packages and reference states. Begin with the physical equipment boundary and derive or justify h(T) − h(T_ref) = integral(Cp(T), T_ref to T). 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 — Property packages and reference states. Work from the following synthetic reference case without copying the plotted output: For Cp=4 kJ/(kg K), a 5 kg/s single-phase stream heated by 20 K requires 400 kW of sensible heat. Show substitutions and check an appropriate conservation or limit.

Q1: Model challenge — Property packages and reference states. Compute the effect of a ±10% Cp uncertainty on a heater duty and explain why the outlet temperature alone does not reveal reference-state consistency. Explain what quantitative outcome would support the model and what result would contradict it.

Q1: Responsible interpretation — Property packages and reference states. The model has this restriction: Constant heat capacity does not describe vaporization, strong temperature dependence or a reacting mixture across a wide range. Propose one measured variable or external data source that would be needed before a real engineering application.

Q5: Formulation — Antoine vapor pressure and Raoult-law bubble point. Begin with the physical equipment boundary and derive or justify P = x_ethanol P_sat,ethanol(T) + (1−x_ethanol) P_sat,water(T). 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 — Antoine vapor pressure and Raoult-law bubble point. Work from the following synthetic reference case without copying the plotted output: At a specified liquid ethanol fraction, the bubble temperature is the root of summed ideal partial pressures minus total pressure. Show substitutions and check an appropriate conservation or limit.

Q5: Model challenge — Antoine vapor pressure and Raoult-law bubble point. Find the bubble point at two ethanol fractions and verify that the calculated vapor fraction of ethanol differs from the liquid fraction. Explain what quantitative outcome would support the model and what result would contradict it.

Q5: Responsible interpretation — Antoine vapor pressure and Raoult-law bubble point. The model has this restriction: Ethanol/water is nonideal in important regimes; the ideal calculation is a pedagogical benchmark, not a rigorous azeotrope model. Propose one measured variable or external data source that would be needed before a real engineering application.

Q9: Formulation — Isothermal flash and Rachford–Rice logic. Begin with the physical equipment boundary and derive or justify sum_i z_i (K_i−1)/(1+beta (K_i−1)) = 0. 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 — Isothermal flash and Rachford–Rice logic. Work from the following synthetic reference case without copying the plotted output: For z_A=0.5, K_A=2 and K_B=0.5, the two-phase solution is beta=0.5, x_A=1/3, y_A=2/3. Show substitutions and check an appropriate conservation or limit.

Q9: Model challenge — Isothermal flash and Rachford–Rice logic. At a selected feed composition, recompute both phase fractions and the A component balance from the chart data. Explain what quantitative outcome would support the model and what result would contradict it.

Q9: Responsible interpretation — Isothermal flash and Rachford–Rice logic. The model has this restriction: Constant arbitrary K values neglect temperature, pressure and mixture dependence; two-phase roots are not evidence that the chosen property package is appropriate. Propose one measured variable or external data source that would be needed before a real engineering application.

Q13: Formulation — Chemical equilibrium versus phase equilibrium. Begin with the physical equipment boundary and derive or justify K_eq(T) = K_ref exp[−ΔH/R (1/T − 1/T_ref)]. 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 — Chemical equilibrium versus phase equilibrium. Work from the following synthetic reference case without copying the plotted output: For A ⇌ B with ideal activities and K_eq=4, x_B at equilibrium is K/(1+K)=0.80. Show substitutions and check an appropriate conservation or limit.

Q13: Model challenge — Chemical equilibrium versus phase equilibrium. Switch ΔH from negative to positive and interpret the reversal in the conversion–temperature slope. Explain what quantitative outcome would support the model and what result would contradict it.

Q13: Responsible interpretation — Chemical equilibrium versus phase equilibrium. The model has this restriction: Constant reaction enthalpy and ideal activities are modeling choices, not universal descriptions of real reaction chemistry. 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-02.html) · [Applied project](../projects/unit-02.html)