Phase-Change Heat Transfer and Evaporation

The engineering problem

Phase change permits large energy transfer at nearly fixed saturation temperature, but the interfacial mechanics determine the heat flux. Boiling and condensation are not merely high values of a fixed heat-transfer coefficient. Vapor pressure, surface wetting, noncondensable gases and condensate drainage control process performance.

Scope and learning outcomes

Model latent heat, condensate films, evaporator duties and thermal limits of boiling without extrapolating educational estimates to safety-critical equipment. Students should be able to select a governing equation, construct a measurable boundary condition, calculate a plausible output with its units, and explain a failure case before applying a correlation.

Concept map

7.1 Sensible and latent heat accounting

An evaporator duty includes incoming sensible heating plus the enthalpy associated with vaporizing solvent; the latent component can dominate. An aqueous solution enters below boiling and leaves with specified vapor generation. Distinguish feed mass flow from evaporated mass flow; close the remaining-solids balance. Representative relationship: Qdot=mdotcp(Tsat−Tin)+mdot_vap*lambda.

7.2 Condensation films

Nusselt film-condensation theory relates the average h on a vertical plate to liquid density, viscosity, conductivity, gravity and length. Saturated steam condenses on the outside of a cooled vertical wall. Identify why shear, condensate turbulence and noncondensable gas invalidate the laminar-film estimate. Representative relationship: hbar=[0.943rho_l(rho_l−rho_v)glambdak_l^3/(mu_lL*dT)]^0.25.

7.3 Boiling regimes and heat-flux limits

Nucleate boiling, transition boiling and film boiling differ qualitatively; exceeding critical heat flux can cause wall-temperature excursions. A reactor jacket is supplied with a boiling utility at a specified vapor rate. Explain why an energy balance alone does not certify safe wall heat flux or prevent dryout. Representative relationship: Qdot≈mdot_vap*lambda (latent energy only).

7.4 Evaporator economy and multiple effects

Multiple effects reuse latent heat at successively lower pressures, but boiling point elevation, vapor carryover and heat-transfer area matter. A concentration plant considers a two-effect evaporator retrofit. List the extra thermophysical and mechanical data needed beyond a simple kg/kg economy estimate. Representative relationship: steam_economy=kg_evaporated/kg_live_steam.

A disciplined workflow for this unit

Start with a diagram of a process boundary, including fluid inlets, outlets, phase boundaries, hot and cold external surfaces and any energy storage. Record which values are measurements, which are assumed, and which are calculated. Write the conservation statement before inserting a transfer correlation. After obtaining a numerical result, test the limiting case: the heat flow should vanish with zero temperature difference in a passive linear resistance model, while doubling the wall area should increase conduction duty if all other conditions remain fixed. For nonlinear radiation a twofold temperature increase cannot be treated as a twofold radiation increase.

Report results with their uncertainty drivers. For a heated process tube, a thermocouple may measure external wall temperature but not internal wall temperature, and a flow meter may give volumetric rather than mass rate. An apparent discrepancy between measured and predicted duty may be the result of imperfect knowledge of heat capacities, uncontrolled bypass or fouling. If a correlation is used beyond the printed range, label the result an extrapolation instead of assigning spurious precision.

Unit investigation method

Use a blank process sheet with five sections: known properties, governing physics, calculation, validation, and operating decision. At minimum compare two physically plausible operating states and identify what extra measurement would most efficiently distinguish them. Do not rely on the number of digits printed by a simulation to argue that a real process is controlled to that precision.

Unit 7, Lesson 1: Sensible and latent heat accounting

Unit: Phase-change heat transfer and evaporation. Core formula: Qdot=mdotcp(Tsat−Tin)+mdot_vap*lambda. Suggested model: [Laboratory 13 — Evaporator thermal duty](../labs/lab-13.html).

Why this problem appears in process plants

An aqueous solution enters below boiling and leaves with specified vapor generation. An evaporator duty includes incoming sensible heating plus the enthalpy associated with vaporizing solvent; the latent component can dominate. This topic is a decision about where energy travels, which mechanism controls its rate, and what evidence is necessary to justify a calculation. The same reported heater duty can correspond to distinct wall temperatures depending on geometry, flow and material properties. A credible engineer must therefore distinguish the energy rate demanded by a process from the thermal resistance that allows the equipment to deliver that rate.

A process operator experiences this distinction as an operating constraint rather than as an equation. When product throughput increases, an unchanged heating utility can be insufficient even when the external wall stays hot. A measurement campaign should record mass flow, inlet and outlet conditions, utility conditions, local wall temperatures when available, and the operating history that may affect fouling. Without these signals, increasing utility temperature may mask the actual cause of poor performance and worsen degradation.

Physical mechanism and assumptions

An evaporator duty includes incoming sensible heating plus the enthalpy associated with vaporizing solvent; the latent component can dominate. The equation selected here, Qdot=mdotcp(Tsat−Tin)+mdot_vap*lambda, summarizes a limited physical model. Every term must refer to a declared boundary, an appropriate unit system, and a physically relevant value of temperature or material property. Before using the equation, identify whether the system is steady or transient, whether the fluid undergoes phase change, and whether the heat-transfer coefficient is known from a defensible correlation or simply a fitted number.

The engineer's first drawing should show an inward or outward normal for the relevant surface and a clear sign convention. Heat transport is directed down a temperature driving force in a passive system, although a controlled refrigeration cycle can move heat from a colder source to a hotter sink by supplying work. A temperature sensor located in a mixing chamber is not necessarily the right measurement of the wall temperature immediately adjacent to a process fluid. A fixed temperature boundary also differs from an imposed heat-flux boundary and gives a different mathematical problem.

Derivation and dimensional reasoning

Start from the first-law energy balance, accumulation = net incoming enthalpy + supplied heat + work, with the appropriate sign conventions. Introduce the constitutive expression Qdot=mdotcp(Tsat−Tin)+mdot_vap*lambda only after identifying the relevant energy-transfer mechanism. If a rate equation gives watts, confirm that conductivity is in W/(m K), surface area in m² and temperature difference in kelvin for a planar slab. The analogous exercise for a convection coefficient uses W/(m² K). For radiation, temperatures must be absolute because emission depends on the fourth power of thermodynamic temperature.

Terms that look superficially similar can require fundamentally different models. A cylindrical pipe wall does not have a constant radial area; a nonlinear radiation exchange cannot be replaced by a temperature-independent resistance without specifying a linearization point. Correlations expressed with Reynolds and Prandtl numbers need fluid properties at the proper reference temperature and geometry. In a thermally developing flow, the local Nusselt number may differ from its fully developed average. Write these distinctions directly beside the calculation rather than after the final answer.

Worked investigation: evaporator thermal duty

We use a numerical teaching model with the following documented inputs: evaporation rate = 85; feed = 420; cp = 4.05; dT = 40; latent = 2250. The primary control evaporation rate is measured in kg/h. Other parameters are feed [kg/h]; cp [kJ/(kg K)]; dT [K]; latent [kJ/kg]. Using the model equation, the predicted total heating duty is 72.025 kW. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 85 to 143.4 kg/h while holding the other inputs fixed. The same model predicts 108.525 kW. Compare the absolute difference (36.5 kW) with the percentage difference, and interpret the sign using the physical mechanism. Does this change chiefly alter the driving force, the thermal resistance, the material storage, or the rate at which the process moves energy? Explain why interpreting the curve without checking the other controlled inputs could be misleading.

The control range is deliberately finite. At very large or small values a simplified correlation can violate its validity conditions even if the numerical program continues to return finite numbers. For this lab, the important limitation is: Assumes the specified feed stays as a liquid until vaporization and no heat losses or boiling point elevation. Evaporated rate must not exceed feed rate. Decide whether the default state is inside that limitation and state what kind of additional property data, geometry measurement, or experimental coefficient would be needed to increase confidence in the estimate.

Measurement strategy and uncertainty

Distinguish feed mass flow from evaporated mass flow; close the remaining-solids balance. Select one variable from the worked model that could realistically be measured on a plant floor, one that would normally be inferred, and one that is liable to drift over months of operation. Suggest how each could be independently checked. Thermal couples need installation and dynamic-response consideration, while flow meters require correct fluid properties and calibration. A sensor can be accurate at its measurement point yet irrelevant to the heat-transfer boundary being modeled.

Develop a first-order sensitivity calculation. If the output Y depends on parameter p, estimate dY/dp from two nearby runs of the simulator. If the measured parameter has uncertainty u(p), its approximate output contribution is abs(dY/dp)*u(p). This is a local linear approximation; it does not characterize poorly known geometry, phase-change instability or systematic installation bias. Keep uncertainty in the output's physical units until presenting any final percentage. Where a temperature difference is small, uncertainty can become the dominant proportion of the inferred transfer rate.

Process decision and limitation

Distinguish feed mass flow from evaporated mass flow; close the remaining-solids balance. An excellent model is still a screening tool if it ignores details that control real equipment performance. A preliminary calculation may support an experiment or identify the direction of a sensitivity, but it cannot substitute for a vendor curve, verified thermophysical properties, design margins, pressure-equipment requirements or an installed sensor audit. Present the conclusion as an evidence-backed statement with a named range of validity. Avoid claiming that a temperature or duty is safe solely because one steady-state expression has a positive numerical solution.

Independent worked case

[Read a fully contextualized numerical case: Evaporation duty and condensate economy](../worked-cases/unit-07-case-01.html).

Exercises

1. Starting with Qdot=mdotcp(Tsat−Tin)+mdot_vap*lambda, identify the independent variables and produce a dimensionally consistent derivation or explanation of each term. Give an example where using the wrong unit would change the answer by at least a factor of 100.
2. Repeat the worked experiment with the control variable reduced by 10%. Compute the new output, the percent change, and compare the result to a first-order sensitivity estimate. Explain whether a linear approximation is justified.
3. Distinguish feed mass flow from evaporated mass flow; close the remaining-solids balance. Describe the minimum sensor set and a method for recording uncertainty in the field.
4. Construct an alternative physical model for the same plant scenario by changing one boundary condition. State which observation could distinguish the two models.
5. Design a sensitivity chart comparing two material-property assumptions. What input dominates the uncertainty at the baseline and why?

Reproducible laboratory

Run [Laboratory 13: Evaporator thermal duty](../labs/lab-13.html), export 151 modeled points to CSV, and annotate where the correlation is appropriate. Cite the chosen property source and write an explicit statement about the limits of extrapolation. The laboratory provides a schematic response curve and numerical readout but cannot validate a design without measured plant data.

Unit 7, Lesson 2: Condensation films

Unit: Phase-change heat transfer and evaporation. Core formula: hbar=[0.943rho_l(rho_l−rho_v)glambdak_l^3/(mu_lL*dT)]^0.25. Suggested model: [Laboratory 14 — Laminar-film condensation](../labs/lab-14.html).

Why this problem appears in process plants

Saturated steam condenses on the outside of a cooled vertical wall. Nusselt film-condensation theory relates the average h on a vertical plate to liquid density, viscosity, conductivity, gravity and length. This topic is a decision about where energy travels, which mechanism controls its rate, and what evidence is necessary to justify a calculation. The same reported heater duty can correspond to distinct wall temperatures depending on geometry, flow and material properties. A credible engineer must therefore distinguish the energy rate demanded by a process from the thermal resistance that allows the equipment to deliver that rate.

A process operator experiences this distinction as an operating constraint rather than as an equation. When product throughput increases, an unchanged heating utility can be insufficient even when the external wall stays hot. A measurement campaign should record mass flow, inlet and outlet conditions, utility conditions, local wall temperatures when available, and the operating history that may affect fouling. Without these signals, increasing utility temperature may mask the actual cause of poor performance and worsen degradation.

Physical mechanism and assumptions

Nusselt film-condensation theory relates the average h on a vertical plate to liquid density, viscosity, conductivity, gravity and length. The equation selected here, hbar=[0.943rho_l(rho_l−rho_v)glambdak_l^3/(mu_lL*dT)]^0.25, summarizes a limited physical model. Every term must refer to a declared boundary, an appropriate unit system, and a physically relevant value of temperature or material property. Before using the equation, identify whether the system is steady or transient, whether the fluid undergoes phase change, and whether the heat-transfer coefficient is known from a defensible correlation or simply a fitted number.

The engineer's first drawing should show an inward or outward normal for the relevant surface and a clear sign convention. Heat transport is directed down a temperature driving force in a passive system, although a controlled refrigeration cycle can move heat from a colder source to a hotter sink by supplying work. A temperature sensor located in a mixing chamber is not necessarily the right measurement of the wall temperature immediately adjacent to a process fluid. A fixed temperature boundary also differs from an imposed heat-flux boundary and gives a different mathematical problem.

Derivation and dimensional reasoning

Start from the first-law energy balance, accumulation = net incoming enthalpy + supplied heat + work, with the appropriate sign conventions. Introduce the constitutive expression hbar=[0.943rho_l(rho_l−rho_v)glambdak_l^3/(mu_lL*dT)]^0.25 only after identifying the relevant energy-transfer mechanism. If a rate equation gives watts, confirm that conductivity is in W/(m K), surface area in m² and temperature difference in kelvin for a planar slab. The analogous exercise for a convection coefficient uses W/(m² K). For radiation, temperatures must be absolute because emission depends on the fourth power of thermodynamic temperature.

Terms that look superficially similar can require fundamentally different models. A cylindrical pipe wall does not have a constant radial area; a nonlinear radiation exchange cannot be replaced by a temperature-independent resistance without specifying a linearization point. Correlations expressed with Reynolds and Prandtl numbers need fluid properties at the proper reference temperature and geometry. In a thermally developing flow, the local Nusselt number may differ from its fully developed average. Write these distinctions directly beside the calculation rather than after the final answer.

Worked investigation: laminar-film condensation

We use a numerical teaching model with the following documented inputs: plate length = 0.65; dT = 12; rhol = 960; rhov = 0.6; mul = 0.00028; kl = 0.68; latent = 2.25e+06. The primary control plate length is measured in m. Other parameters are dT [K]; rhol [kg/m³]; rhov [kg/m³]; mul [Pa s]; kl [W/(m K)]; latent [J/kg]. Using the model equation, the predicted mean film coefficient is 6935.44 W/(m² K). This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 0.65 to 1.026 m while holding the other inputs fixed. The same model predicts 6187.5 W/(m² K). Compare the absolute difference (-747.935 W/(m² K)) with the percentage difference, and interpret the sign using the physical mechanism. Does this change chiefly alter the driving force, the thermal resistance, the material storage, or the rate at which the process moves energy? Explain why interpreting the curve without checking the other controlled inputs could be misleading.

The control range is deliberately finite. At very large or small values a simplified correlation can violate its validity conditions even if the numerical program continues to return finite numbers. For this lab, the important limitation is: Nusselt laminar gravity film, vertical plate, saturated vapor, negligible gas shear. Decide whether the default state is inside that limitation and state what kind of additional property data, geometry measurement, or experimental coefficient would be needed to increase confidence in the estimate.

Measurement strategy and uncertainty

Identify why shear, condensate turbulence and noncondensable gas invalidate the laminar-film estimate. Select one variable from the worked model that could realistically be measured on a plant floor, one that would normally be inferred, and one that is liable to drift over months of operation. Suggest how each could be independently checked. Thermal couples need installation and dynamic-response consideration, while flow meters require correct fluid properties and calibration. A sensor can be accurate at its measurement point yet irrelevant to the heat-transfer boundary being modeled.

Develop a first-order sensitivity calculation. If the output Y depends on parameter p, estimate dY/dp from two nearby runs of the simulator. If the measured parameter has uncertainty u(p), its approximate output contribution is abs(dY/dp)*u(p). This is a local linear approximation; it does not characterize poorly known geometry, phase-change instability or systematic installation bias. Keep uncertainty in the output's physical units until presenting any final percentage. Where a temperature difference is small, uncertainty can become the dominant proportion of the inferred transfer rate.

Process decision and limitation

Identify why shear, condensate turbulence and noncondensable gas invalidate the laminar-film estimate. An excellent model is still a screening tool if it ignores details that control real equipment performance. A preliminary calculation may support an experiment or identify the direction of a sensitivity, but it cannot substitute for a vendor curve, verified thermophysical properties, design margins, pressure-equipment requirements or an installed sensor audit. Present the conclusion as an evidence-backed statement with a named range of validity. Avoid claiming that a temperature or duty is safe solely because one steady-state expression has a positive numerical solution.

Independent worked case

[Read a fully contextualized numerical case: Film condensation length effect](../worked-cases/unit-07-case-02.html).

Exercises

1. Starting with hbar=[0.943rho_l(rho_l−rho_v)glambdak_l^3/(mu_lL*dT)]^0.25, identify the independent variables and produce a dimensionally consistent derivation or explanation of each term. Give an example where using the wrong unit would change the answer by at least a factor of 100.
2. Repeat the worked experiment with the control variable reduced by 10%. Compute the new output, the percent change, and compare the result to a first-order sensitivity estimate. Explain whether a linear approximation is justified.
3. Identify why shear, condensate turbulence and noncondensable gas invalidate the laminar-film estimate. Describe the minimum sensor set and a method for recording uncertainty in the field.
4. Construct an alternative physical model for the same plant scenario by changing one boundary condition. State which observation could distinguish the two models.
5. Design a sensitivity chart comparing two material-property assumptions. What input dominates the uncertainty at the baseline and why?

Reproducible laboratory

Run [Laboratory 14: Laminar-film condensation](../labs/lab-14.html), export 151 modeled points to CSV, and annotate where the correlation is appropriate. Cite the chosen property source and write an explicit statement about the limits of extrapolation. The laboratory provides a schematic response curve and numerical readout but cannot validate a design without measured plant data.

Unit 7, Lesson 3: Boiling regimes and heat-flux limits

Unit: Phase-change heat transfer and evaporation. Core formula: Qdot≈mdot_vap*lambda (latent energy only). Suggested model: [Laboratory 13 — Evaporator thermal duty](../labs/lab-13.html).

Why this problem appears in process plants

A reactor jacket is supplied with a boiling utility at a specified vapor rate. Nucleate boiling, transition boiling and film boiling differ qualitatively; exceeding critical heat flux can cause wall-temperature excursions. This topic is a decision about where energy travels, which mechanism controls its rate, and what evidence is necessary to justify a calculation. The same reported heater duty can correspond to distinct wall temperatures depending on geometry, flow and material properties. A credible engineer must therefore distinguish the energy rate demanded by a process from the thermal resistance that allows the equipment to deliver that rate.

A process operator experiences this distinction as an operating constraint rather than as an equation. When product throughput increases, an unchanged heating utility can be insufficient even when the external wall stays hot. A measurement campaign should record mass flow, inlet and outlet conditions, utility conditions, local wall temperatures when available, and the operating history that may affect fouling. Without these signals, increasing utility temperature may mask the actual cause of poor performance and worsen degradation.

Physical mechanism and assumptions

Nucleate boiling, transition boiling and film boiling differ qualitatively; exceeding critical heat flux can cause wall-temperature excursions. The equation selected here, Qdot≈mdot_vap*lambda (latent energy only), summarizes a limited physical model. Every term must refer to a declared boundary, an appropriate unit system, and a physically relevant value of temperature or material property. Before using the equation, identify whether the system is steady or transient, whether the fluid undergoes phase change, and whether the heat-transfer coefficient is known from a defensible correlation or simply a fitted number.

The engineer's first drawing should show an inward or outward normal for the relevant surface and a clear sign convention. Heat transport is directed down a temperature driving force in a passive system, although a controlled refrigeration cycle can move heat from a colder source to a hotter sink by supplying work. A temperature sensor located in a mixing chamber is not necessarily the right measurement of the wall temperature immediately adjacent to a process fluid. A fixed temperature boundary also differs from an imposed heat-flux boundary and gives a different mathematical problem.

Derivation and dimensional reasoning

Start from the first-law energy balance, accumulation = net incoming enthalpy + supplied heat + work, with the appropriate sign conventions. Introduce the constitutive expression Qdot≈mdot_vap*lambda (latent energy only) only after identifying the relevant energy-transfer mechanism. If a rate equation gives watts, confirm that conductivity is in W/(m K), surface area in m² and temperature difference in kelvin for a planar slab. The analogous exercise for a convection coefficient uses W/(m² K). For radiation, temperatures must be absolute because emission depends on the fourth power of thermodynamic temperature.

Terms that look superficially similar can require fundamentally different models. A cylindrical pipe wall does not have a constant radial area; a nonlinear radiation exchange cannot be replaced by a temperature-independent resistance without specifying a linearization point. Correlations expressed with Reynolds and Prandtl numbers need fluid properties at the proper reference temperature and geometry. In a thermally developing flow, the local Nusselt number may differ from its fully developed average. Write these distinctions directly beside the calculation rather than after the final answer.

Worked investigation: evaporator thermal duty

We use a numerical teaching model with the following documented inputs: evaporation rate = 85; feed = 420; cp = 4.05; dT = 40; latent = 2250. The primary control evaporation rate is measured in kg/h. Other parameters are feed [kg/h]; cp [kJ/(kg K)]; dT [K]; latent [kJ/kg]. Using the model equation, the predicted total heating duty is 72.025 kW. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 85 to 143.4 kg/h while holding the other inputs fixed. The same model predicts 108.525 kW. Compare the absolute difference (36.5 kW) with the percentage difference, and interpret the sign using the physical mechanism. Does this change chiefly alter the driving force, the thermal resistance, the material storage, or the rate at which the process moves energy? Explain why interpreting the curve without checking the other controlled inputs could be misleading.

The control range is deliberately finite. At very large or small values a simplified correlation can violate its validity conditions even if the numerical program continues to return finite numbers. For this lab, the important limitation is: Assumes the specified feed stays as a liquid until vaporization and no heat losses or boiling point elevation. Evaporated rate must not exceed feed rate. Decide whether the default state is inside that limitation and state what kind of additional property data, geometry measurement, or experimental coefficient would be needed to increase confidence in the estimate.

Measurement strategy and uncertainty

Explain why an energy balance alone does not certify safe wall heat flux or prevent dryout. Select one variable from the worked model that could realistically be measured on a plant floor, one that would normally be inferred, and one that is liable to drift over months of operation. Suggest how each could be independently checked. Thermal couples need installation and dynamic-response consideration, while flow meters require correct fluid properties and calibration. A sensor can be accurate at its measurement point yet irrelevant to the heat-transfer boundary being modeled.

Develop a first-order sensitivity calculation. If the output Y depends on parameter p, estimate dY/dp from two nearby runs of the simulator. If the measured parameter has uncertainty u(p), its approximate output contribution is abs(dY/dp)*u(p). This is a local linear approximation; it does not characterize poorly known geometry, phase-change instability or systematic installation bias. Keep uncertainty in the output's physical units until presenting any final percentage. Where a temperature difference is small, uncertainty can become the dominant proportion of the inferred transfer rate.

Process decision and limitation

Explain why an energy balance alone does not certify safe wall heat flux or prevent dryout. An excellent model is still a screening tool if it ignores details that control real equipment performance. A preliminary calculation may support an experiment or identify the direction of a sensitivity, but it cannot substitute for a vendor curve, verified thermophysical properties, design margins, pressure-equipment requirements or an installed sensor audit. Present the conclusion as an evidence-backed statement with a named range of validity. Avoid claiming that a temperature or duty is safe solely because one steady-state expression has a positive numerical solution.

Independent worked case

[Read a fully contextualized numerical case: Critical heat-flux caution](../worked-cases/unit-07-case-03.html).

Exercises

1. Starting with Qdot≈mdot_vap*lambda (latent energy only), identify the independent variables and produce a dimensionally consistent derivation or explanation of each term. Give an example where using the wrong unit would change the answer by at least a factor of 100.
2. Repeat the worked experiment with the control variable reduced by 10%. Compute the new output, the percent change, and compare the result to a first-order sensitivity estimate. Explain whether a linear approximation is justified.
3. Explain why an energy balance alone does not certify safe wall heat flux or prevent dryout. Describe the minimum sensor set and a method for recording uncertainty in the field.
4. Construct an alternative physical model for the same plant scenario by changing one boundary condition. State which observation could distinguish the two models.
5. Design a sensitivity chart comparing two material-property assumptions. What input dominates the uncertainty at the baseline and why?

Reproducible laboratory

Run [Laboratory 13: Evaporator thermal duty](../labs/lab-13.html), export 151 modeled points to CSV, and annotate where the correlation is appropriate. Cite the chosen property source and write an explicit statement about the limits of extrapolation. The laboratory provides a schematic response curve and numerical readout but cannot validate a design without measured plant data.

Unit 7, Lesson 4: Evaporator economy and multiple effects

Unit: Phase-change heat transfer and evaporation. Core formula: steam_economy=kg_evaporated/kg_live_steam. Suggested model: [Laboratory 13 — Evaporator thermal duty](../labs/lab-13.html).

Why this problem appears in process plants

A concentration plant considers a two-effect evaporator retrofit. Multiple effects reuse latent heat at successively lower pressures, but boiling point elevation, vapor carryover and heat-transfer area matter. This topic is a decision about where energy travels, which mechanism controls its rate, and what evidence is necessary to justify a calculation. The same reported heater duty can correspond to distinct wall temperatures depending on geometry, flow and material properties. A credible engineer must therefore distinguish the energy rate demanded by a process from the thermal resistance that allows the equipment to deliver that rate.

A process operator experiences this distinction as an operating constraint rather than as an equation. When product throughput increases, an unchanged heating utility can be insufficient even when the external wall stays hot. A measurement campaign should record mass flow, inlet and outlet conditions, utility conditions, local wall temperatures when available, and the operating history that may affect fouling. Without these signals, increasing utility temperature may mask the actual cause of poor performance and worsen degradation.

Physical mechanism and assumptions

Multiple effects reuse latent heat at successively lower pressures, but boiling point elevation, vapor carryover and heat-transfer area matter. The equation selected here, steam_economy=kg_evaporated/kg_live_steam, summarizes a limited physical model. Every term must refer to a declared boundary, an appropriate unit system, and a physically relevant value of temperature or material property. Before using the equation, identify whether the system is steady or transient, whether the fluid undergoes phase change, and whether the heat-transfer coefficient is known from a defensible correlation or simply a fitted number.

The engineer's first drawing should show an inward or outward normal for the relevant surface and a clear sign convention. Heat transport is directed down a temperature driving force in a passive system, although a controlled refrigeration cycle can move heat from a colder source to a hotter sink by supplying work. A temperature sensor located in a mixing chamber is not necessarily the right measurement of the wall temperature immediately adjacent to a process fluid. A fixed temperature boundary also differs from an imposed heat-flux boundary and gives a different mathematical problem.

Derivation and dimensional reasoning

Start from the first-law energy balance, accumulation = net incoming enthalpy + supplied heat + work, with the appropriate sign conventions. Introduce the constitutive expression steam_economy=kg_evaporated/kg_live_steam only after identifying the relevant energy-transfer mechanism. If a rate equation gives watts, confirm that conductivity is in W/(m K), surface area in m² and temperature difference in kelvin for a planar slab. The analogous exercise for a convection coefficient uses W/(m² K). For radiation, temperatures must be absolute because emission depends on the fourth power of thermodynamic temperature.

Terms that look superficially similar can require fundamentally different models. A cylindrical pipe wall does not have a constant radial area; a nonlinear radiation exchange cannot be replaced by a temperature-independent resistance without specifying a linearization point. Correlations expressed with Reynolds and Prandtl numbers need fluid properties at the proper reference temperature and geometry. In a thermally developing flow, the local Nusselt number may differ from its fully developed average. Write these distinctions directly beside the calculation rather than after the final answer.

Worked investigation: evaporator thermal duty

We use a numerical teaching model with the following documented inputs: evaporation rate = 85; feed = 420; cp = 4.05; dT = 40; latent = 2250. The primary control evaporation rate is measured in kg/h. Other parameters are feed [kg/h]; cp [kJ/(kg K)]; dT [K]; latent [kJ/kg]. Using the model equation, the predicted total heating duty is 72.025 kW. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 85 to 143.4 kg/h while holding the other inputs fixed. The same model predicts 108.525 kW. Compare the absolute difference (36.5 kW) with the percentage difference, and interpret the sign using the physical mechanism. Does this change chiefly alter the driving force, the thermal resistance, the material storage, or the rate at which the process moves energy? Explain why interpreting the curve without checking the other controlled inputs could be misleading.

The control range is deliberately finite. At very large or small values a simplified correlation can violate its validity conditions even if the numerical program continues to return finite numbers. For this lab, the important limitation is: Assumes the specified feed stays as a liquid until vaporization and no heat losses or boiling point elevation. Evaporated rate must not exceed feed rate. Decide whether the default state is inside that limitation and state what kind of additional property data, geometry measurement, or experimental coefficient would be needed to increase confidence in the estimate.

Measurement strategy and uncertainty

List the extra thermophysical and mechanical data needed beyond a simple kg/kg economy estimate. Select one variable from the worked model that could realistically be measured on a plant floor, one that would normally be inferred, and one that is liable to drift over months of operation. Suggest how each could be independently checked. Thermal couples need installation and dynamic-response consideration, while flow meters require correct fluid properties and calibration. A sensor can be accurate at its measurement point yet irrelevant to the heat-transfer boundary being modeled.

Develop a first-order sensitivity calculation. If the output Y depends on parameter p, estimate dY/dp from two nearby runs of the simulator. If the measured parameter has uncertainty u(p), its approximate output contribution is abs(dY/dp)*u(p). This is a local linear approximation; it does not characterize poorly known geometry, phase-change instability or systematic installation bias. Keep uncertainty in the output's physical units until presenting any final percentage. Where a temperature difference is small, uncertainty can become the dominant proportion of the inferred transfer rate.

Process decision and limitation

List the extra thermophysical and mechanical data needed beyond a simple kg/kg economy estimate. An excellent model is still a screening tool if it ignores details that control real equipment performance. A preliminary calculation may support an experiment or identify the direction of a sensitivity, but it cannot substitute for a vendor curve, verified thermophysical properties, design margins, pressure-equipment requirements or an installed sensor audit. Present the conclusion as an evidence-backed statement with a named range of validity. Avoid claiming that a temperature or duty is safe solely because one steady-state expression has a positive numerical solution.

Independent worked case

[Read a fully contextualized numerical case: Multiple-effect evaporator estimate](../worked-cases/unit-07-case-04.html).

Exercises

1. Starting with steam_economy=kg_evaporated/kg_live_steam, identify the independent variables and produce a dimensionally consistent derivation or explanation of each term. Give an example where using the wrong unit would change the answer by at least a factor of 100.
2. Repeat the worked experiment with the control variable reduced by 10%. Compute the new output, the percent change, and compare the result to a first-order sensitivity estimate. Explain whether a linear approximation is justified.
3. List the extra thermophysical and mechanical data needed beyond a simple kg/kg economy estimate. Describe the minimum sensor set and a method for recording uncertainty in the field.
4. Construct an alternative physical model for the same plant scenario by changing one boundary condition. State which observation could distinguish the two models.
5. Design a sensitivity chart comparing two material-property assumptions. What input dominates the uncertainty at the baseline and why?

Reproducible laboratory

Run [Laboratory 13: Evaporator thermal duty](../labs/lab-13.html), export 151 modeled points to CSV, and annotate where the correlation is appropriate. Cite the chosen property source and write an explicit statement about the limits of extrapolation. The laboratory provides a schematic response curve and numerical readout but cannot validate a design without measured plant data.

Applied project 07: Phase-change heat transfer and evaporation

Project brief

Phase change permits large energy transfer at nearly fixed saturation temperature, but the interfacial mechanics determine the heat flux. Boiling and condensation are not merely high values of a fixed heat-transfer coefficient. Vapor pressure, surface wetting, noncondensable gases and condensate drainage control process performance. Build a small but defensible engineering evaluation for a hypothetical process facility. The target is not an equipment purchase specification: it is a traceable analysis of what would need to be known before design or procurement. Model latent heat, condensate films, evaporator duties and thermal limits of boiling without extrapolating educational estimates to safety-critical equipment.

Process case and data plan

The scenario begins with incomplete information. A process technician has measured bulk inlet and outlet temperatures, approximate flow and a surface temperature, while operations has an estimate of annual hours. Choose one of the four lesson topics below and define a physically plausible dataset with at least eight rows. Use SI units. You may use representative approximate fluid properties if you label them as assumptions; do not describe fabricated samples as field measurements.

• Option 1: Sensible and latent heat accounting. An aqueous solution enters below boiling and leaves with specified vapor generation. Distinguish feed mass flow from evaporated mass flow; close the remaining-solids balance. Begin with Qdot=mdotcp(Tsat−Tin)+mdot_vap*lambda and then examine which neglected effect might dominate.
• Option 2: Condensation films. Saturated steam condenses on the outside of a cooled vertical wall. Identify why shear, condensate turbulence and noncondensable gas invalidate the laminar-film estimate. Begin with hbar=[0.943rho_l(rho_l−rho_v)glambdak_l^3/(mu_lL*dT)]^0.25 and then examine which neglected effect might dominate.
• Option 3: Boiling regimes and heat-flux limits. A reactor jacket is supplied with a boiling utility at a specified vapor rate. Explain why an energy balance alone does not certify safe wall heat flux or prevent dryout. Begin with Qdot≈mdot_vap*lambda (latent energy only) and then examine which neglected effect might dominate.
• Option 4: Evaporator economy and multiple effects. A concentration plant considers a two-effect evaporator retrofit. List the extra thermophysical and mechanical data needed beyond a simple kg/kg economy estimate. Begin with steam_economy=kg_evaporated/kg_live_steam and then examine which neglected effect might dominate.

Work packages

A — Model the thermal circuit. Produce a diagram of thermal boundaries and flow directions. Record each resistance, energy-storage term, or imposed heat flux. Separate first-law calculations from empirical closure equations. Verify one result using an independent algebraic or numerical method.

B — Establish a baseline. Choose reasonable conditions, calculate heat duty in watts or kilowatts and identify one required material property that is not supplied. Define how a student would obtain that property from an authoritative source and which temperature or pressure governs the lookup.

C — Experiment and plot. Run two relevant interactive laboratories and export the curves. Compare at least three operating points, keeping all but one input fixed at a time. Include a table with the model assumption, the changed parameter and the resulting percentage change. Explain why some apparent trends would fail in a full system because stream capacity rates, pressure drop or properties co-vary.

D — Measurement and risk. Write a list of five instruments or process observations necessary to validate the simplified model. Include likely systematic and random errors, a rough range of plausible input uncertainty, and a note about unavailable information such as surface condition or contact resistance. Do not infer any pressure-vessel or worker-safety adequacy from the simulator.

E — Decision memorandum. Provide a two-page recommendation that ranks measurement, retrofit and operating-control options by estimated benefit and uncertainty. For energy economics, use explicit unit price, operating hours and equipment efficiency assumptions. Highlight which conclusions would change under a realistic alternative case.

Acceptance criteria

Deliver a readable process schematic, a table of inputs and sources, equation derivations with units, one independent verification, at least two experimental response plots, a measurement strategy and a decision memo. A technically unsupported precise number is not acceptable. The model is a teaching artifact, not a certified plant design.

Worked case 7.1: Evaporation duty and condensate economy

Process situation and calculation

An evaporator heats a 500 kg/h feed through 30 K with cp=4.0 kJ/(kg K), then removes 100 kg/h of water with assumed latent heat 2250 kJ/kg. Sensible heat is 500×4×30=60,000 kJ/h; latent heat is 225,000 kJ/h. Total input is 285,000 kJ/h or 79.17 kW, excluding losses.

Physical interpretation and possible error

Because latent duty dominates, overlooking vapor production gives a large error. Check that evaporation never exceeds liquid feed, and verify boiling-point elevation and latent heat at operating pressure if the solution contains solutes.

Verification tasks

1. Annotate a sketch with heat-flow directions and temperature measurement positions. Explain whether the quoted numerical value is a rate, a transport coefficient, a stored-energy scale or a transient timescale.
2. Repeat the calculation after changing one physical parameter by 10%. Identify whether the result responds linearly, logarithmically or with a power law, and derive that dependence rather than relying only on a graph.
3. State which property or correlation requires independent external verification. What uncertainty would a surface thermocouple, flow meter or geometry measurement introduce?
4. Describe the limiting condition under which the numerical relationship is not meaningful. Do not generalize beyond its stated geometry or thermal regime.

Scope: This is an illustrative, parameter-specific calculation for education, not an equipment rating or validation of actual process data.

Worked case 7.2: Film condensation length effect

Process situation and calculation

For laminar gravity condensation on a vertical plate, the average Nusselt film coefficient varies approximately with plate length to the power −1/4 when the other properties and temperature difference remain fixed. Doubling length therefore reduces average coefficient by about 2^(-1/4)=0.841, or 16%.

Physical interpretation and possible error

The total condensate production may still increase with larger area. This distinction between a lower area-average coefficient and a larger total duty matters when comparing short and long condenser plates.

Verification tasks

1. Annotate a sketch with heat-flow directions and temperature measurement positions. Explain whether the quoted numerical value is a rate, a transport coefficient, a stored-energy scale or a transient timescale.
2. Repeat the calculation after changing one physical parameter by 10%. Identify whether the result responds linearly, logarithmically or with a power law, and derive that dependence rather than relying only on a graph.
3. State which property or correlation requires independent external verification. What uncertainty would a surface thermocouple, flow meter or geometry measurement introduce?
4. Describe the limiting condition under which the numerical relationship is not meaningful. Do not generalize beyond its stated geometry or thermal regime.

Scope: This is an illustrative, parameter-specific calculation for education, not an equipment rating or validation of actual process data.

Worked case 7.3: Critical heat-flux caution

Process situation and calculation

An energy balance indicates that a boiling jacket needs 300 kW to satisfy reactor demand. This single number does not determine the wall temperature or safe attainable local flux because heated area, wetting, nucleation sites, vapor removal and operating pressure all affect the boiling regime.

Physical interpretation and possible error

A local dry patch can experience film boiling even while the area-average duty meets the calculated target. Certified heater designs require applicable process safety analysis and appropriate boiling data; this textbook does not implement a critical-heat-flux correlation.

Verification tasks

1. Annotate a sketch with heat-flow directions and temperature measurement positions. Explain whether the quoted numerical value is a rate, a transport coefficient, a stored-energy scale or a transient timescale.
2. Repeat the calculation after changing one physical parameter by 10%. Identify whether the result responds linearly, logarithmically or with a power law, and derive that dependence rather than relying only on a graph.
3. State which property or correlation requires independent external verification. What uncertainty would a surface thermocouple, flow meter or geometry measurement introduce?
4. Describe the limiting condition under which the numerical relationship is not meaningful. Do not generalize beyond its stated geometry or thermal regime.

Scope: This is an illustrative, parameter-specific calculation for education, not an equipment rating or validation of actual process data.

Worked case 7.4: Multiple-effect evaporator estimate

Process situation and calculation

A plant evaporates 3000 kg/h of solvent using 1900 kg/h of live steam. Its observed steam economy is 3000/1900≈1.58 kg/kg. This number alone does not imply one exact number of effects because feed preheating, heat leaks and vapor recompression can affect the apparent balance.

Physical interpretation and possible error

Check condensate subcooling, boiling-point elevation and whether the feed sensible duty is included. Steam economy should be compared at similar product concentration and feed temperature before claiming process improvements.

Verification tasks

1. Annotate a sketch with heat-flow directions and temperature measurement positions. Explain whether the quoted numerical value is a rate, a transport coefficient, a stored-energy scale or a transient timescale.
2. Repeat the calculation after changing one physical parameter by 10%. Identify whether the result responds linearly, logarithmically or with a power law, and derive that dependence rather than relying only on a graph.
3. State which property or correlation requires independent external verification. What uncertainty would a surface thermocouple, flow meter or geometry measurement introduce?
4. Describe the limiting condition under which the numerical relationship is not meaningful. Do not generalize beyond its stated geometry or thermal regime.

Scope: This is an illustrative, parameter-specific calculation for education, not an equipment rating or validation of actual process data.

Laboratory 13 — Evaporator thermal duty

Linked unit: 7. Measured output: Total heating duty (kW). Equation: Q=(feed cp ΔT+vap latent)/3600.

Experiment objective and theory

This interactive experiment isolates one thermal relationship so that parameter sensitivity can be observed without the distraction of a full process model. It does not solve an entire plant; the objective is to learn which inputs the mathematical expression regards as controlling, which inputs the engineer would have to obtain from measurements, and which unmodeled effects could reverse an apparent benefit. Start by writing the direction of expected heat transfer. For an entirely passive system, check that the response is compatible with the stated hot and cold temperatures and the sign convention.

Validity boundary: Assumes the specified feed stays as a liquid until vaporization and no heat losses or boiling point elevation. Evaporated rate must not exceed feed rate. The simplified correlation treats the remaining conditions as fixed even if that assumption would not survive an actual change in flow, phase or material temperature. If the laboratory displays a finite number at an invalid combination, do not call that number a validated prediction. Temperature-dependent properties should be evaluated at a defensible reference state; this model deliberately does not call a property database.

Default test and sensitivity measurements

Primary variable: evaporation rate, unit kg/h. Initial value: 85 kg/h. Predicted output: 72.025 kW. Your result may round on the web page. Confirm the orders of magnitude independently with the printed equation.

At the first selected test value 94 kg/h, the reference model gives 77.65 kW. At the second test value 361 kg/h, it gives 244.525 kW. Do not draw a straight-line conclusion from these two numbers without checking the full response curve. Export the CSV, calculate a central finite difference near the default value, and compare it with the curvature of the graph. Keep the parameter-change interval small enough for a sensitivity calculation but large enough to avoid rounding effects.

Controlled-variable experiment

1. Record the baseline output and plot before touching any slider. Explain whether a temperature difference, resistance, heat capacity or dimensionless correlation establishes the result.
2. Move the primary slider to its low and high limits. Identify the graph region where an extrapolation of the physics would be most dangerous.
3. Restore the baseline. Change one secondary material or geometry parameter by approximately 10%. State the expected direction before observing it; then describe the difference between your prediction and the numerical response.
4. Use reset to reproduce the default experiment, and export the CSV. Check the exported data for units, sign and monotonicity. A curve with a sharp numerical discontinuity should be investigated rather than smoothed away automatically.
5. Prepare a four-column comparison of baseline, adjusted input, adjusted output and physical interpretation. Finish with a statement of the lab model's limitations and what one real measurement would validate it.

Interpretation and field extension

A computational curve is a relationship under controlled assumptions, not a set of experimental observations. In a plant, thermal conductivity, viscosity and heat capacity change with temperature; fouling deposits grow in time; and equipment area can be effectively reduced by flow maldistribution. Explain whether the chosen experiment can distinguish these effects. If not, propose a paired measurement, such as wall temperatures and stream flow, that creates additional evidence. Never claim improved process safety or energy economics solely from a single slider result.

Questions: (a) Which input appears with the highest exponent in the local law? (b) Could an unmeasured contact resistance mimic the same trend? (c) Does the graph represent a local coefficient, an area-average value or a whole-unit thermal duty? (d) What would an alternative geometry change? (e) What independent instrument or property reference would be required before industrial scale-up?

Laboratory 14 — Laminar-film condensation

Linked unit: 7. Measured output: Mean film coefficient (W/(m² K)). Equation: h=0.943[rho_l(rho_l−rho_v)g latent k³/(mu L ΔT)]^0.25.

Experiment objective and theory

This interactive experiment isolates one thermal relationship so that parameter sensitivity can be observed without the distraction of a full process model. It does not solve an entire plant; the objective is to learn which inputs the mathematical expression regards as controlling, which inputs the engineer would have to obtain from measurements, and which unmodeled effects could reverse an apparent benefit. Start by writing the direction of expected heat transfer. For an entirely passive system, check that the response is compatible with the stated hot and cold temperatures and the sign convention.

Validity boundary: Nusselt laminar gravity film, vertical plate, saturated vapor, negligible gas shear. The simplified correlation treats the remaining conditions as fixed even if that assumption would not survive an actual change in flow, phase or material temperature. If the laboratory displays a finite number at an invalid combination, do not call that number a validated prediction. Temperature-dependent properties should be evaluated at a defensible reference state; this model deliberately does not call a property database.

Default test and sensitivity measurements

Primary variable: plate length, unit m. Initial value: 0.65 m. Predicted output: 6935.436 W/(m² K). Your result may round on the web page. Confirm the orders of magnitude independently with the printed equation.

At the first selected test value 0.68 m, the reference model gives 6857.643 W/(m² K). At the second test value 2.42 m, it gives 4992.847 W/(m² K). Do not draw a straight-line conclusion from these two numbers without checking the full response curve. Export the CSV, calculate a central finite difference near the default value, and compare it with the curvature of the graph. Keep the parameter-change interval small enough for a sensitivity calculation but large enough to avoid rounding effects.

Controlled-variable experiment

1. Record the baseline output and plot before touching any slider. Explain whether a temperature difference, resistance, heat capacity or dimensionless correlation establishes the result.
2. Move the primary slider to its low and high limits. Identify the graph region where an extrapolation of the physics would be most dangerous.
3. Restore the baseline. Change one secondary material or geometry parameter by approximately 10%. State the expected direction before observing it; then describe the difference between your prediction and the numerical response.
4. Use reset to reproduce the default experiment, and export the CSV. Check the exported data for units, sign and monotonicity. A curve with a sharp numerical discontinuity should be investigated rather than smoothed away automatically.
5. Prepare a four-column comparison of baseline, adjusted input, adjusted output and physical interpretation. Finish with a statement of the lab model's limitations and what one real measurement would validate it.

Interpretation and field extension

A computational curve is a relationship under controlled assumptions, not a set of experimental observations. In a plant, thermal conductivity, viscosity and heat capacity change with temperature; fouling deposits grow in time; and equipment area can be effectively reduced by flow maldistribution. Explain whether the chosen experiment can distinguish these effects. If not, propose a paired measurement, such as wall temperatures and stream flow, that creates additional evidence. Never claim improved process safety or energy economics solely from a single slider result.

Questions: (a) Which input appears with the highest exponent in the local law? (b) Could an unmeasured contact resistance mimic the same trend? (c) Does the graph represent a local coefficient, an area-average value or a whole-unit thermal duty? (d) What would an alternative geometry change? (e) What independent instrument or property reference would be required before industrial scale-up?

Unit 07 — Assessment: Phase-change heat transfer and evaporation

Instructions: Work at least three problems with complete units, a diagram, assumptions and a sensitivity check. These questions emphasize independent explanation rather than recognizing a repeated numerical template.

Topic 7.1: Sensible and latent heat accounting

1. Derivation: Starting with `Qdot=mdot*cp*(Tsat−Tin)+mdot_vap*lambda`, derive the expression from stated physical assumptions or explain the empirical closure that prevents a first-principles derivation. Define the system boundary and identify which values are measurable.

2. Engineering investigation: An aqueous solution enters below boiling and leaves with specified vapor generation. Produce two competing hypotheses that could explain a 12% deviation between the calculated heat duty and the measured duty; for each give an instrument or a controlled test.

3. Critical evaluation: Distinguish feed mass flow from evaporated mass flow; close the remaining-solids balance. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Topic 7.2: Condensation films

1. Derivation: Starting with `hbar=[0.943*rho_l*(rho_l−rho_v)*g*lambda*k_l^3/(mu_l*L*dT)]^0.25`, derive the expression from stated physical assumptions or explain the empirical closure that prevents a first-principles derivation. Define the system boundary and identify which values are measurable.

2. Engineering investigation: Saturated steam condenses on the outside of a cooled vertical wall. Produce two competing hypotheses that could explain a 12% deviation between the calculated heat duty and the measured duty; for each give an instrument or a controlled test.

3. Critical evaluation: Identify why shear, condensate turbulence and noncondensable gas invalidate the laminar-film estimate. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Topic 7.3: Boiling regimes and heat-flux limits

1. Derivation: Starting with `Qdot≈mdot_vap*lambda (latent energy only)`, derive the expression from stated physical assumptions or explain the empirical closure that prevents a first-principles derivation. Define the system boundary and identify which values are measurable.

2. Engineering investigation: A reactor jacket is supplied with a boiling utility at a specified vapor rate. Produce two competing hypotheses that could explain a 12% deviation between the calculated heat duty and the measured duty; for each give an instrument or a controlled test.

3. Critical evaluation: Explain why an energy balance alone does not certify safe wall heat flux or prevent dryout. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Topic 7.4: Evaporator economy and multiple effects

1. Derivation: Starting with `steam_economy=kg_evaporated/kg_live_steam`, derive the expression from stated physical assumptions or explain the empirical closure that prevents a first-principles derivation. Define the system boundary and identify which values are measurable.

2. Engineering investigation: A concentration plant considers a two-effect evaporator retrofit. Produce two competing hypotheses that could explain a 12% deviation between the calculated heat duty and the measured duty; for each give an instrument or a controlled test.

3. Critical evaluation: List the extra thermophysical and mechanical data needed beyond a simple kg/kg economy estimate. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Assessment rubric

Conceptual accuracy (30%): state the correct physics, not just a numerical recipe. Traceable units and sign conventions (20%): make every conversion explicit. Sensitivity and validity (25%): quantify a change and identify non-applicable regimes. Engineering communication (25%): diagrams, model limits and measurements must support the recommendation.

The exercises are deliberately open; different property assumptions can produce different numerical outcomes. Clearly distinguish supplied values, literature property values and measurements. A plausible order of magnitude with transparent limitations is more useful than an unjustifiably precise number.