Gas Mixtures, Humidity and Phase-Change Balances

Gas pressure, humidity, phase changes and evaporation frequently introduce coupled mass and energy accounting. State thermodynamic-property assumptions.

Core learning goals

• Explain and apply Ideal-gas and mixture balances, including assumptions and independent verification.
• Explain and apply Humidity and air-water mixtures, including assumptions and independent verification.
• Explain and apply Evaporation and solids conservation, including assumptions and independent verification.
• Explain and apply Phase-change energy coupling, including assumptions and independent verification.

Lessons and reading order

• [Lesson 5.1: Ideal-gas and mixture balances](../lessons/unit-05-lesson-01.html)
• [Lesson 5.2: Humidity and air-water mixtures](../lessons/unit-05-lesson-02.html)
• [Lesson 5.3: Evaporation and solids conservation](../lessons/unit-05-lesson-03.html)
• [Lesson 5.4: Phase-change energy coupling](../lessons/unit-05-lesson-04.html)

Unit synthesis: identify what is known

Gas pressure, humidity, phase changes and evaporation frequently introduce coupled mass and energy accounting. State thermodynamic-property assumptions. A strong solution begins with a process sketch and a stream table. Write values to a stated basis, mark all missing data, and calculate degrees of freedom before attempting numerical iteration. A flow diagram is useful only if the stream symbols and assumptions can be traced into actual balance equations. If an unknown variable is introduced to close a balance, explain what physical instrument or property measurement would establish it.

Connecting the four lessons

The first lesson establishes a local quantity or concept, while the later lessons expose how its interpretation changes with additional process complexity. For example, mass fractions remain valid for a blend but reacting compositions need generation terms; heat duties may be linear in temperature under constant heat capacity but nonlinear near a phase transition. Develop a single reference stream table and compare how each analytical step changes when more units are added.

Diagnostic exercise

Assume an otherwise plausible design produces an impossible negative outlet or a reported overall recovery of 105%. Instead of clipping the number, trace which equation or boundary was omitted or whose units were inconsistent. Check whether the solver was overdetermined by a redundant balance or underdetermined by a missing composition. Document the correction and rerun your limiting-case tests.

A second diagnostic asks you to separate measured and assumed quantities. Label uncertainties if known; do not attach precision to idealized coefficients. A balance audit should preserve the original measurements and record any reconciliation separately. This habit will become important in the integrated case studies and final project.

Unit completion exercise

Draw one example flowsheet for this unit and formulate an overall balance plus two meaningful independent additional equations, if the process contains enough components. State why each equation is independent. Determine what unknowns remain and choose the cheapest credible measurement to close the problem. Explain whether the modified process is still steady, and assess one alternative operating condition without assuming unchanged performance.

Unit 5.1: Ideal-gas and mixture balances

Book 2: Material and Energy Balances — Gas mixtures and phase-change balances

What you will be able to do

Partial pressure is proportional to mole fraction for an ideal mixture. Interpret PV=nRT; p_i=y_iP and demonstrate how it follows from an appropriate boundary. Identify whether the equation describes an overall, species, elemental, or energy balance; discuss what is actually measured and the conditions under which the balance is valid.

Essential physical insight

Gas-bag measurements and nitrogen blanketing. The critical objective is not to memorize a plug-in formula but to recognize conserved quantities and define the physical process correctly. Break the problem into stream variables, known specifications, unknowns, and constitutive or property relationships. A numerical estimate is an engineering claim only if its basis and limitations are reproducible.

Deriving the balance from the process boundary

Start from the observations implied by ideal-gas and mixture balances. Partial pressure is proportional to mole fraction for an ideal mixture. Represent each material crossing the boundary with a signed rate and state its measurement basis. If the equipment retains material, include an accumulation term. A balance written per second cannot be directly combined with a plant log written per shift without converting the interval; similarly, mass fractions and molar compositions cannot be used in the same species account without a molecular-weight transformation. Before any algebra, list feed states, outlet states, any generation or consumption, and the measured inventory change.

Write the central relationship as PV=nRT; p_i=y_iP. Read every variable aloud with its units, explaining which terms are independent measurements and which are derived. For a process with reaction, the component generation term is not optional. For a nonreactive mixer, a nonzero computed generation indicates a modeling or measurement mistake. Use unit consistency as the first diagnostic check, and then compare the predicted result against simple physical limits.

A worked calculation and its interpretation

Illustrative case. A 2 m^3 vessel at 300 K and 100 kPa holds n=1002/(8.314300)=0.0802 kmol of ideal gas. This numerical calculation is a worked illustration and not a measurement from an operating facility. Write the stream table with at least one named component, a physical total rate, and a units column. Substitute numbers into the governing equation and solve the unknown before rounding. Keep at least one more significant figure in intermediate calculations than will be reported to the operator.

The numerical result has value only within the stated assumptions. Use absolute temperature and coherent energy units; R=8.314 kPa m^3/(kmol K). Confirm that any calculated component fraction is between zero and one, that recoveries never exceed their admissible range and that energy duties obey the stated sign convention. A unit conversion error can shift a prediction by factors of 1,000 or 3,600 even when the arithmetic appears internally consistent.

Why the answer changes in a real plant

Consider the industrial case: Gas-bag measurements and nitrogen blanketing. Before trusting a software result, identify temperature and pressure effects, instrument calibration, inventory fluctuations and unrecorded process streams. A perfectly balanced flowsheet can still be a poor representation of reality if one instrument reads systematically low. Conversely, a balance residual does not by itself prove a physical leak or chemical loss.

To decide whether further detail is necessary, first quantify the decision at issue. A rough balance may be sufficient to choose between two pilot runs but not to specify relief equipment, guarantee product quality, or assert environmental compliance. Explain which property or stream measurement would most reduce uncertainty, and how it could be obtained without introducing a new uncontrolled hazard.

Guided redesign exercise

Imagine a proposed 20% increase in the process feed or load. Identify which terms in PV=nRT; p_i=y_iP scale linearly and which could change because of capacity limits, residence time, equilibrium, or heat transfer. Recompute the balance on a consistent time basis and explain why unchanged efficiency or composition may be an unreasonable assumption. Do not silently extrapolate a laboratory coefficient outside the range where it was measured.

Prepare a second scenario in which the least certain input changes by 10% while the other stated conditions remain fixed. Estimate whether the output changes proportionally. If the result violates a conservation limit, retrace variable definitions. If it remains feasible, identify an observation that could distinguish between the original and changed scenarios.

Verification and critical-thinking questions

1. State the system boundary and draw the streams for ideal-gas and mixture balances; label every unknown rather than assigning guessed values.
2. Explain the difference between an identity enforced by conservation and an empirical correlation that may fail at changed conditions.
3. What measurement would most directly test the assumption: partial pressure is proportional to mole fraction for an ideal mixture?
4. Recompute the illustrative case using a different but consistent calculation basis. Which outputs must remain invariant?
5. Construct a limiting-case test and explain precisely what would constitute a physically impossible result.

Practical check: Use absolute temperature and coherent energy units; R=8.314 kPa m^3/(kmol K). A satisfactory answer should separate the equation, numerical result, process interpretation, and limitations rather than ending with a number alone.

Practical laboratory connection

The laboratory models in this volume can illustrate how a calculated response changes as conditions vary. Locate a related simulation in the book index, write down a prediction before operating its controls, and test whether the result is monotone, bounded or approaching an asymptotic limit. Compare at least one displayed numerical result with an independent calculation. A graphical match is not sufficient: test numerical values and units separately.

Further study

Consult the freely accessible LearnChemE Material and Energy Balances resources and the NIST Chemistry WebBook for property-data methodology. Use a suitable authoritative property model for real design rather than applying constant heat capacities or ideal-mixture assumptions indiscriminately.

Unit 5.2: Humidity and air-water mixtures

Book 2: Material and Energy Balances — Gas mixtures and phase-change balances

What you will be able to do

Humidity ratio is water-vapor mass per dry-air mass, not a mass fraction of humid air. Interpret w=0.62198p_v/(P-p_v); p_v=phip_sat(T) and demonstrate how it follows from an appropriate boundary. Identify whether the equation describes an overall, species, elemental, or energy balance; discuss what is actually measured and the conditions under which the balance is valid.

Essential physical insight

Dryer inlet-air measurements and ventilation moisture loads. The critical objective is not to memorize a plug-in formula but to recognize conserved quantities and define the physical process correctly. Break the problem into stream variables, known specifications, unknowns, and constitutive or property relationships. A numerical estimate is an engineering claim only if its basis and limitations are reproducible.

Deriving the balance from the process boundary

Start from the observations implied by humidity and air-water mixtures. Humidity ratio is water-vapor mass per dry-air mass, not a mass fraction of humid air. Represent each material crossing the boundary with a signed rate and state its measurement basis. If the equipment retains material, include an accumulation term. A balance written per second cannot be directly combined with a plant log written per shift without converting the interval; similarly, mass fractions and molar compositions cannot be used in the same species account without a molecular-weight transformation. Before any algebra, list feed states, outlet states, any generation or consumption, and the measured inventory change.

Write the central relationship as w=0.62198p_v/(P-p_v); p_v=phip_sat(T). Read every variable aloud with its units, explaining which terms are independent measurements and which are derived. For a process with reaction, the component generation term is not optional. For a nonreactive mixer, a nonzero computed generation indicates a modeling or measurement mistake. Use measurable stream composition as the first diagnostic check, and then compare the predicted result against simple physical limits.

A worked calculation and its interpretation

Illustrative case. At 25°C and 50% relative humidity, p_sat≈3.17 kPa, p_v≈1.585 kPa, and w≈0.00988 kg/kg dry air. This numerical calculation is a worked illustration and not a measurement from an operating facility. Write the stream table with at least one named component, a physical total rate, and a units column. Substitute numbers into the governing equation and solve the unknown before rounding. Keep at least one more significant figure in intermediate calculations than will be reported to the operator.

The numerical result has value only within the stated assumptions. Psychrometric formulas are sensitive to saturation pressure and barometric pressure. Confirm that any calculated component fraction is between zero and one, that recoveries never exceed their admissible range and that energy duties obey the stated sign convention. A unit conversion error can shift a prediction by factors of 1,000 or 3,600 even when the arithmetic appears internally consistent.

Why the answer changes in a real plant

Consider the industrial case: Dryer inlet-air measurements and ventilation moisture loads. Before trusting a software result, identify temperature and pressure effects, instrument calibration, inventory fluctuations and unrecorded process streams. A perfectly balanced flowsheet can still be a poor representation of reality if one instrument reads systematically low. Conversely, a balance residual does not by itself prove a physical leak or chemical loss.

To decide whether further detail is necessary, first quantify the decision at issue. A rough balance may be sufficient to choose between two pilot runs but not to specify relief equipment, guarantee product quality, or assert environmental compliance. Explain which property or stream measurement would most reduce uncertainty, and how it could be obtained without introducing a new uncontrolled hazard.

Guided redesign exercise

Imagine a proposed 20% increase in the process feed or load. Identify which terms in w=0.62198p_v/(P-p_v); p_v=phip_sat(T) scale linearly and which could change because of capacity limits, residence time, equilibrium, or heat transfer. Recompute the balance on a consistent time basis and explain why unchanged efficiency or composition may be an unreasonable assumption. Do not silently extrapolate a laboratory coefficient outside the range where it was measured.

Prepare a second scenario in which the least certain input changes by 10% while the other stated conditions remain fixed. Estimate whether the output changes proportionally. If the result violates a conservation limit, retrace variable definitions. If it remains feasible, identify an observation that could distinguish between the original and changed scenarios.

Verification and critical-thinking questions

1. State the system boundary and draw the streams for humidity and air-water mixtures; label every unknown rather than assigning guessed values.
2. Explain the difference between an identity enforced by conservation and an empirical correlation that may fail at changed conditions.
3. What measurement would most directly test the assumption: humidity ratio is water-vapor mass per dry-air mass, not a mass fraction of humid air?
4. Recompute the illustrative case using a different but consistent calculation basis. Which outputs must remain invariant?
5. Construct a limiting-case test and explain precisely what would constitute a physically impossible result.

Practical check: Psychrometric formulas are sensitive to saturation pressure and barometric pressure. A satisfactory answer should separate the equation, numerical result, process interpretation, and limitations rather than ending with a number alone.

Practical laboratory connection

The laboratory models in this volume can illustrate how a calculated response changes as conditions vary. Locate a related simulation in the book index, write down a prediction before operating its controls, and test whether the result is monotone, bounded or approaching an asymptotic limit. Compare at least one displayed numerical result with an independent calculation. A graphical match is not sufficient: test numerical values and units separately.

Further study

Consult the freely accessible LearnChemE Material and Energy Balances resources and the NIST Chemistry WebBook for property-data methodology. Use a suitable authoritative property model for real design rather than applying constant heat capacities or ideal-mixture assumptions indiscriminately.

Unit 5.3: Evaporation and solids conservation

Book 2: Material and Energy Balances — Gas mixtures and phase-change balances

What you will be able to do

Nonvolatile solute balances set concentrate flow independently of energy demand. Interpret Fx_F=Px_P; V=F-P and demonstrate how it follows from an appropriate boundary. Identify whether the equation describes an overall, species, elemental, or energy balance; discuss what is actually measured and the conditions under which the balance is valid.

Essential physical insight

Fruit concentrate and industrial brine evaporation. The critical objective is not to memorize a plug-in formula but to recognize conserved quantities and define the physical process correctly. Break the problem into stream variables, known specifications, unknowns, and constitutive or property relationships. A numerical estimate is an engineering claim only if its basis and limitations are reproducible.

Deriving the balance from the process boundary

Start from the observations implied by evaporation and solids conservation. Nonvolatile solute balances set concentrate flow independently of energy demand. Represent each material crossing the boundary with a signed rate and state its measurement basis. If the equipment retains material, include an accumulation term. A balance written per second cannot be directly combined with a plant log written per shift without converting the interval; similarly, mass fractions and molar compositions cannot be used in the same species account without a molecular-weight transformation. Before any algebra, list feed states, outlet states, any generation or consumption, and the measured inventory change.

Write the central relationship as Fx_F=Px_P; V=F-P. Read every variable aloud with its units, explaining which terms are independent measurements and which are derived. For a process with reaction, the component generation term is not optional. For a nonreactive mixer, a nonzero computed generation indicates a modeling or measurement mistake. Use physical limit cases as the first diagnostic check, and then compare the predicted result against simple physical limits.

A worked calculation and its interpretation

Illustrative case. Feed 1000 kg/h at 10 wt% nonvolatile solids concentrated to 40 wt% gives P=250 kg/h and V=750 kg/h vapor. This numerical calculation is a worked illustration and not a measurement from an operating facility. Write the stream table with at least one named component, a physical total rate, and a units column. Substitute numbers into the governing equation and solve the unknown before rounding. Keep at least one more significant figure in intermediate calculations than will be reported to the operator.

The numerical result has value only within the stated assumptions. Do not assume all solvent vaporizes when the outlet still contains liquid solvent. Confirm that any calculated component fraction is between zero and one, that recoveries never exceed their admissible range and that energy duties obey the stated sign convention. A unit conversion error can shift a prediction by factors of 1,000 or 3,600 even when the arithmetic appears internally consistent.

Why the answer changes in a real plant

Consider the industrial case: Fruit concentrate and industrial brine evaporation. Before trusting a software result, identify temperature and pressure effects, instrument calibration, inventory fluctuations and unrecorded process streams. A perfectly balanced flowsheet can still be a poor representation of reality if one instrument reads systematically low. Conversely, a balance residual does not by itself prove a physical leak or chemical loss.

To decide whether further detail is necessary, first quantify the decision at issue. A rough balance may be sufficient to choose between two pilot runs but not to specify relief equipment, guarantee product quality, or assert environmental compliance. Explain which property or stream measurement would most reduce uncertainty, and how it could be obtained without introducing a new uncontrolled hazard.

Guided redesign exercise

Imagine a proposed 20% increase in the process feed or load. Identify which terms in Fx_F=Px_P; V=F-P scale linearly and which could change because of capacity limits, residence time, equilibrium, or heat transfer. Recompute the balance on a consistent time basis and explain why unchanged efficiency or composition may be an unreasonable assumption. Do not silently extrapolate a laboratory coefficient outside the range where it was measured.

Prepare a second scenario in which the least certain input changes by 10% while the other stated conditions remain fixed. Estimate whether the output changes proportionally. If the result violates a conservation limit, retrace variable definitions. If it remains feasible, identify an observation that could distinguish between the original and changed scenarios.

Verification and critical-thinking questions

1. State the system boundary and draw the streams for evaporation and solids conservation; label every unknown rather than assigning guessed values.
2. Explain the difference between an identity enforced by conservation and an empirical correlation that may fail at changed conditions.
3. What measurement would most directly test the assumption: nonvolatile solute balances set concentrate flow independently of energy demand?
4. Recompute the illustrative case using a different but consistent calculation basis. Which outputs must remain invariant?
5. Construct a limiting-case test and explain precisely what would constitute a physically impossible result.

Practical check: Do not assume all solvent vaporizes when the outlet still contains liquid solvent. A satisfactory answer should separate the equation, numerical result, process interpretation, and limitations rather than ending with a number alone.

Practical laboratory connection

The laboratory models in this volume can illustrate how a calculated response changes as conditions vary. Locate a related simulation in the book index, write down a prediction before operating its controls, and test whether the result is monotone, bounded or approaching an asymptotic limit. Compare at least one displayed numerical result with an independent calculation. A graphical match is not sufficient: test numerical values and units separately.

Further study

Consult the freely accessible LearnChemE Material and Energy Balances resources and the NIST Chemistry WebBook for property-data methodology. Use a suitable authoritative property model for real design rather than applying constant heat capacities or ideal-mixture assumptions indiscriminately.

Unit 5.4: Phase-change energy coupling

Book 2: Material and Energy Balances — Gas mixtures and phase-change balances

What you will be able to do

Latent heat can dominate sensible heating even when temperature does not change. Interpret Q=mcp(T_b-T_in)+m_vapor*lambda and demonstrate how it follows from an appropriate boundary. Identify whether the equation describes an overall, species, elemental, or energy balance; discuss what is actually measured and the conditions under which the balance is valid.

Essential physical insight

Steam-jacket sizing and evaporator energy audits. The critical objective is not to memorize a plug-in formula but to recognize conserved quantities and define the physical process correctly. Break the problem into stream variables, known specifications, unknowns, and constitutive or property relationships. A numerical estimate is an engineering claim only if its basis and limitations are reproducible.

Deriving the balance from the process boundary

Start from the observations implied by phase-change energy coupling. Latent heat can dominate sensible heating even when temperature does not change. Represent each material crossing the boundary with a signed rate and state its measurement basis. If the equipment retains material, include an accumulation term. A balance written per second cannot be directly combined with a plant log written per shift without converting the interval; similarly, mass fractions and molar compositions cannot be used in the same species account without a molecular-weight transformation. Before any algebra, list feed states, outlet states, any generation or consumption, and the measured inventory change.

Write the central relationship as Q=mcp(T_b-T_in)+m_vapor*lambda. Read every variable aloud with its units, explaining which terms are independent measurements and which are derived. For a process with reaction, the component generation term is not optional. For a nonreactive mixer, a nonzero computed generation indicates a modeling or measurement mistake. Use independent balance closure as the first diagnostic check, and then compare the predicted result against simple physical limits.

A worked calculation and its interpretation

Illustrative case. Heating and vaporizing 100 kg/h water from 20°C to 100°C with cp=4.18 kJ/(kg K) and λ=2257 kJ/kg needs 33,440+225,700=259,140 kJ/h. This numerical calculation is a worked illustration and not a measurement from an operating facility. Write the stream table with at least one named component, a physical total rate, and a units column. Substitute numbers into the governing equation and solve the unknown before rounding. Keep at least one more significant figure in intermediate calculations than will be reported to the operator.

The numerical result has value only within the stated assumptions. Using one constant heat capacity across a two-phase region without latent heat severely underpredicts duty. Confirm that any calculated component fraction is between zero and one, that recoveries never exceed their admissible range and that energy duties obey the stated sign convention. A unit conversion error can shift a prediction by factors of 1,000 or 3,600 even when the arithmetic appears internally consistent.

Why the answer changes in a real plant

Consider the industrial case: Steam-jacket sizing and evaporator energy audits. Before trusting a software result, identify temperature and pressure effects, instrument calibration, inventory fluctuations and unrecorded process streams. A perfectly balanced flowsheet can still be a poor representation of reality if one instrument reads systematically low. Conversely, a balance residual does not by itself prove a physical leak or chemical loss.

To decide whether further detail is necessary, first quantify the decision at issue. A rough balance may be sufficient to choose between two pilot runs but not to specify relief equipment, guarantee product quality, or assert environmental compliance. Explain which property or stream measurement would most reduce uncertainty, and how it could be obtained without introducing a new uncontrolled hazard.

Guided redesign exercise

Imagine a proposed 20% increase in the process feed or load. Identify which terms in Q=mcp(T_b-T_in)+m_vapor*lambda scale linearly and which could change because of capacity limits, residence time, equilibrium, or heat transfer. Recompute the balance on a consistent time basis and explain why unchanged efficiency or composition may be an unreasonable assumption. Do not silently extrapolate a laboratory coefficient outside the range where it was measured.

Prepare a second scenario in which the least certain input changes by 10% while the other stated conditions remain fixed. Estimate whether the output changes proportionally. If the result violates a conservation limit, retrace variable definitions. If it remains feasible, identify an observation that could distinguish between the original and changed scenarios.

Verification and critical-thinking questions

1. State the system boundary and draw the streams for phase-change energy coupling; label every unknown rather than assigning guessed values.
2. Explain the difference between an identity enforced by conservation and an empirical correlation that may fail at changed conditions.
3. What measurement would most directly test the assumption: latent heat can dominate sensible heating even when temperature does not change?
4. Recompute the illustrative case using a different but consistent calculation basis. Which outputs must remain invariant?
5. Construct a limiting-case test and explain precisely what would constitute a physically impossible result.

Practical check: Using one constant heat capacity across a two-phase region without latent heat severely underpredicts duty. A satisfactory answer should separate the equation, numerical result, process interpretation, and limitations rather than ending with a number alone.

Practical laboratory connection

The laboratory models in this volume can illustrate how a calculated response changes as conditions vary. Locate a related simulation in the book index, write down a prediction before operating its controls, and test whether the result is monotone, bounded or approaching an asymptotic limit. Compare at least one displayed numerical result with an independent calculation. A graphical match is not sufficient: test numerical values and units separately.

Further study

Consult the freely accessible LearnChemE Material and Energy Balances resources and the NIST Chemistry WebBook for property-data methodology. Use a suitable authoritative property model for real design rather than applying constant heat capacities or ideal-mixture assumptions indiscriminately.

Applied project 5: Liquid buffer tank during a production changeover

Scenario and process objective

A well-mixed buffer vessel remains at 2.0 m³ with equal inlet and outlet flow of 0.25 m³/h. A liquid containing 1.0 g/L marker is suddenly replaced by feed containing 6.0 g/L marker. The operator mistakenly assumes a new product-quality target is met as soon as 2.0 m³ of fluid has passed through the tank.

The project must be documented as a sequence of claims: identify what conservation enforces, which property data are assumed, which quantities are operator observations, and which decisions remain uncertain. Prepare a separate stream table for every external feed or outlet. Do not insert convenient values for missing streams and then present the case as uniquely solved. A correctly identified underdetermined model is more valuable than an unjustified numerical answer.

Calculation and reference result

The residence time is τ=V/F=8 h. The ideal outlet is C(t)=6−5 exp(−t/8) g/L. At one residence time, C≈4.161 g/L, not 6 g/L. To reach 95% of the total concentration change requires about −τ ln(0.05)=23.97 h. This is a model prediction under perfect mixing, not an observed plant guarantee.

The figures above are idealized training values; they are not operational measurements. Repeat the calculation independently using base SI units and show at least one back-substitution into the original overall or species balance. Keep dimensional conversions visible, especially if a heat rate passes between kJ/h and kW or a composition passes between percent and fraction.

Model criticism and operating context

Sampling delays, dead zones, nonconstant flow and multiple connected vessels change the step response. If the vessel is partially drained, the volume balance must be solved together with tracer mass. A physical test should verify mixing and the instrument response before relying on the predicted release time.

A design report needs a statement of operating range and expected effects that the simplified equations leave out. For example, thermodynamic equilibrium may be inappropriate during a rapid transient; a nominal heat-transfer coefficient may depend on fouling or regime; a flow meter may give an average that is misaligned with an inventory sampling time. Identify which omitted effect could reverse the preferred operating decision.

Required investigation

Estimate the amount of off-specification outflow until the 95% target is met. Investigate whether the objective is concentration, color, conductivity or another measured attribute and explain why different proxies may be misleading.

Provide a graphical sensitivity study with clearly labeled axes and units. Use at least two independent checks on the reported values. One check must be a conservation identity and the other a dimensional, physical-bounds, or alternative-method check. If the case is not fully determined, specify the least expensive measurement that would make the next calculation defensible, and quantify how measurement uncertainty could influence the decision.

Extension: implementation and quality assurance

Design a two-stage implementation plan: first, a low-cost experiment or audit, and second, a revised model calibrated with those observations. Decide how you would document instrumentation, calibration, sampling and data provenance. Provide a concise statement of operational risks that would prevent deployment of the model without expert supervision.

Decision memo template

• What is known: state the measured feed and product values, date range, units and calculation basis.
• What is calculated: identify conserved totals, composition conversions and any heat duty.
• What is assumed: list ideal-gas behavior, perfect mixing, constant heat capacity, reaction completion or transfer efficiency only if used.
• What remains uncertain: name unmeasured streams, material properties or error correlations.
• What to do next: propose a specific measurement and the change in decision it could permit.

Questions for viva or peer discussion

1. Which equation would become invalid if inventory changed over the measurement window?
2. Can a numerically closed overall mass balance conceal a large component imbalance? Construct an example.
3. What is the most dangerous units mistake for this case, and what magnitude would it introduce?
4. Explain the effect of a 10% feed increase on both production and utility rates, including capacity limits.
5. What evidence would allow a simplified teaching model to support a plant engineering decision?

Assessment — Unit 5: Gas mixtures and phase-change balances

These questions demand a calculation or a defended explanation. State units and assumptions for every numerical answer.

Topic 5.1: Ideal-gas and mixture balances

1. Derivation. Build a balance for: Gas-bag measurements and nitrogen blanketing. Derive the expression `P*V=n*R*T; p_i=y_i*P` or explain why an additional relationship is necessary.

2. Quantitative check. Recalculate this illustrative scenario without quoting the reported result: A 2 m^3 vessel at 300 K and 100 kPa holds n=100*2/(8.314*300)=0.0802 kmol of ideal gas. Record a self-check based on conservation.

3. Error diagnosis. A colleague proposes an answer that contradicts this principle: Use absolute temperature and coherent energy units; R=8.314 kPa m^3/(kmol K). Identify the failed assumption and describe a corrective measurement.

4. Sensitivity. Change the most influential given input by +10%, leaving other inputs fixed. Determine the new result where the data suffice; otherwise explicitly name the missing information.

5. Boundary test. Present a physically meaningful limit and show how the predicted result behaves. State whether a thermodynamic property correlation remains valid in that limit.

Topic 5.2: Humidity and air-water mixtures

1. Derivation. Build a balance for: Dryer inlet-air measurements and ventilation moisture loads. Derive the expression `w=0.62198*p_v/(P-p_v); p_v=phi*p_sat(T)` or explain why an additional relationship is necessary.

2. Quantitative check. Recalculate this illustrative scenario without quoting the reported result: At 25°C and 50% relative humidity, p_sat≈3.17 kPa, p_v≈1.585 kPa, and w≈0.00988 kg/kg dry air. Record a self-check based on conservation.

3. Error diagnosis. A colleague proposes an answer that contradicts this principle: Psychrometric formulas are sensitive to saturation pressure and barometric pressure. Identify the failed assumption and describe a corrective measurement.

4. Sensitivity. Change the most influential given input by +10%, leaving other inputs fixed. Determine the new result where the data suffice; otherwise explicitly name the missing information.

5. Boundary test. Present a physically meaningful limit and show how the predicted result behaves. State whether a thermodynamic property correlation remains valid in that limit.

Topic 5.3: Evaporation and solids conservation

1. Derivation. Build a balance for: Fruit concentrate and industrial brine evaporation. Derive the expression `F*x_F=P*x_P; V=F-P` or explain why an additional relationship is necessary.

2. Quantitative check. Recalculate this illustrative scenario without quoting the reported result: Feed 1000 kg/h at 10 wt% nonvolatile solids concentrated to 40 wt% gives P=250 kg/h and V=750 kg/h vapor. Record a self-check based on conservation.

3. Error diagnosis. A colleague proposes an answer that contradicts this principle: Do not assume all solvent vaporizes when the outlet still contains liquid solvent. Identify the failed assumption and describe a corrective measurement.

4. Sensitivity. Change the most influential given input by +10%, leaving other inputs fixed. Determine the new result where the data suffice; otherwise explicitly name the missing information.

5. Boundary test. Present a physically meaningful limit and show how the predicted result behaves. State whether a thermodynamic property correlation remains valid in that limit.

Topic 5.4: Phase-change energy coupling

1. Derivation. Build a balance for: Steam-jacket sizing and evaporator energy audits. Derive the expression `Q=m*cp*(T_b-T_in)+m_vapor*lambda` or explain why an additional relationship is necessary.

2. Quantitative check. Recalculate this illustrative scenario without quoting the reported result: Heating and vaporizing 100 kg/h water from 20°C to 100°C with cp=4.18 kJ/(kg K) and λ=2257 kJ/kg needs 33,440+225,700=259,140 kJ/h. Record a self-check based on conservation.

3. Error diagnosis. A colleague proposes an answer that contradicts this principle: Using one constant heat capacity across a two-phase region without latent heat severely underpredicts duty. Identify the failed assumption and describe a corrective measurement.

4. Sensitivity. Change the most influential given input by +10%, leaving other inputs fixed. Determine the new result where the data suffice; otherwise explicitly name the missing information.

5. Boundary test. Present a physically meaningful limit and show how the predicted result behaves. State whether a thermodynamic property correlation remains valid in that limit.

Grading rubric

Award credit for a correctly defined boundary and flow basis (25%), an equation with independent unknowns (25%), numerical units and magnitude checks (20%), critical assumptions and uncertainty (20%), and readable presentation (10%). More digits are not a substitute for property data.