Heat Exchangers and Thermal Networks

The engineering problem

A heat exchanger has at least two coupled energy balances and a local heat-flow law. Hot-side duty and cold-side gain must agree within heat loss and measurement uncertainty. Specifying one temperature approach does not determine exchanger area until capacity rates and an overall thermal resistance model are available.

Scope and learning outcomes

Design and interpret exchanger duties using energy balances, LMTD, epsilon-NTU, overall coefficients, fouling and approaches to temperature crossing. 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

6.1 Countercurrent LMTD

Log-mean temperature difference represents an integrated driving force; countercurrent terminal differences are not those of a parallel-flow unit. A cold feed is preheated by a hot product stream before a reactor. Check terminal temperatures, signs, pinch and whether a correction factor is needed for multiple passes. Representative relationship: Qdot=UALMTD.

6.2 Effectiveness-NTU interpretation

Effectiveness compares realized duty to the reversible sensible-temperature-span limit and depends on capacity-rate ratio and configuration. A pilot heat exchanger has area fixed but changing stream flow rates. Use the Cr→1 limiting form to avoid numerical cancellation and verify hot and cold outlet temperatures. Representative relationship: NTU=UA/Cmin; eps=(1−exp[−NTU(1−Cr)])/(1−Crexp[−NTU(1−Cr)]).

6.3 Overall heat coefficient and fouling

Convection, wall conduction and deposit resistance combine in series on a chosen area basis; fouling decreases U when other conditions remain fixed. A fouled exchanger requires hotter utility to reach the same process outlet. Explain how area basis and wall geometry change resistance accounting. Representative relationship: 1/U=1/hh+Rw+Rf+1/hc (same-area approximation).

6.4 Thermal design checks and bypass

Bypass, maldistribution, pressure drop and property drift can all cause an exchanger to miss duty even when clean calculations predict adequate UA. A countercurrent cooler loses duty after a maintenance shutdown. Plan inlet/outlet temperature, mass flow and pressure-drop measurements to distinguish fouling from bypass. Representative relationship: Q_hot≈Q_cold; dE≈Q_hot−Q_cold.

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 6, Lesson 1: Countercurrent LMTD

Unit: Heat exchangers and thermal networks. Core formula: Qdot=UALMTD. Suggested model: [Laboratory 16 — LMTD driving force](../labs/lab-16.html).

Why this problem appears in process plants

A cold feed is preheated by a hot product stream before a reactor. Log-mean temperature difference represents an integrated driving force; countercurrent terminal differences are not those of a parallel-flow unit. 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

Log-mean temperature difference represents an integrated driving force; countercurrent terminal differences are not those of a parallel-flow unit. The equation selected here, Qdot=UALMTD, 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=UALMTD 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: lmtd driving force

We use a numerical teaching model with the following documented inputs: hot inlet = 150; Thout = 95; Tcin = 35; Tcout = 72. The primary control hot inlet is measured in °C. Other parameters are Thout [°C]; Tcin [°C]; Tcout [°C]. Using the model equation, the predicted countercurrent lmtd is 68.6069 K. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 150 to 162.8 °C while holding the other inputs fixed. The same model predicts 74.3396 K. Compare the absolute difference (5.73272 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: Terminal differences must be positive; no multipass correction. 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

Check terminal temperatures, signs, pinch and whether a correction factor is needed for multiple passes. 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

Check terminal temperatures, signs, pinch and whether a correction factor is needed for multiple passes. 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: Countercurrent terminal differences](../worked-cases/unit-06-case-01.html).

Exercises

1. Starting with Qdot=UALMTD, 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. Check terminal temperatures, signs, pinch and whether a correction factor is needed for multiple passes. 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 16: LMTD driving force](../labs/lab-16.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 6, Lesson 2: Effectiveness-NTU interpretation

Unit: Heat exchangers and thermal networks. Core formula: NTU=UA/Cmin; eps=(1−exp[−NTU(1−Cr)])/(1−Crexp[−NTU(1−Cr)]). Suggested model: [Laboratory 15 — Counterflow exchanger effectiveness](../labs/lab-15.html).

Why this problem appears in process plants

A pilot heat exchanger has area fixed but changing stream flow rates. Effectiveness compares realized duty to the reversible sensible-temperature-span limit and depends on capacity-rate ratio and configuration. 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

Effectiveness compares realized duty to the reversible sensible-temperature-span limit and depends on capacity-rate ratio and configuration. The equation selected here, NTU=UA/Cmin; eps=(1−exp[−NTU(1−Cr)])/(1−Crexp[−NTU(1−Cr)]), 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 NTU=UA/Cmin; eps=(1−exp[−NTU(1−Cr)])/(1−Crexp[−NTU(1−Cr)]) 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: counterflow exchanger effectiveness

We use a numerical teaching model with the following documented inputs: UA = 1800; Ch = 2800; Cc = 1900; Thin = 160; Tcin = 25. The primary control UA is measured in W/K. Other parameters are Ch [W/K]; Cc [W/K]; Thin [°C]; Tcin [°C]. Using the model equation, the predicted heat-transfer duty is 134.788 kW. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 1800 to 4712 W/K while holding the other inputs fixed. The same model predicts 202.985 kW. Compare the absolute difference (68.1967 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: Single-phase countercurrent; constant capacity rates, no external loss; not multiple pass. 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

Use the Cr→1 limiting form to avoid numerical cancellation and verify hot and cold outlet temperatures. 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

Use the Cr→1 limiting form to avoid numerical cancellation and verify hot and cold outlet temperatures. 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: Capacity-limited heat exchanger](../worked-cases/unit-06-case-02.html).

Exercises

1. Starting with NTU=UA/Cmin; eps=(1−exp[−NTU(1−Cr)])/(1−Crexp[−NTU(1−Cr)]), 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. Use the Cr→1 limiting form to avoid numerical cancellation and verify hot and cold outlet temperatures. 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 15: Counterflow exchanger effectiveness](../labs/lab-15.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 6, Lesson 3: Overall heat coefficient and fouling

Unit: Heat exchangers and thermal networks. Core formula: 1/U=1/hh+Rw+Rf+1/hc (same-area approximation). Suggested model: [Laboratory 17 — Heat-exchanger fouling penalty](../labs/lab-17.html).

Why this problem appears in process plants

A fouled exchanger requires hotter utility to reach the same process outlet. Convection, wall conduction and deposit resistance combine in series on a chosen area basis; fouling decreases U when other conditions remain fixed. 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

Convection, wall conduction and deposit resistance combine in series on a chosen area basis; fouling decreases U when other conditions remain fixed. The equation selected here, 1/U=1/hh+Rw+Rf+1/hc (same-area approximation), 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 1/U=1/hh+Rw+Rf+1/hc (same-area approximation) 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: heat-exchanger fouling penalty

We use a numerical teaching model with the following documented inputs: deposit resistance = 0.001; hh = 850; hc = 1100; Rw = 0.0002; A = 28; LMTD = 30. The primary control deposit resistance is measured in m² K/W. Other parameters are hh [W/(m² K)]; hc [W/(m² K)]; Rw [m² K/W]; A [m²]; LMTD [K]. Using the model equation, the predicted available duty is 255.664 kW. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 0.001 to 0.00148 m² K/W while holding the other inputs fixed. The same model predicts 223.074 kW. Compare the absolute difference (-32.5898 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: Same-area resistance approximation, fixed LMTD and unchanged film coefficients. 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 how area basis and wall geometry change resistance accounting. 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 how area basis and wall geometry change resistance accounting. 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: Fouling at fixed thermal driving force](../worked-cases/unit-06-case-03.html).

Exercises

1. Starting with 1/U=1/hh+Rw+Rf+1/hc (same-area approximation), 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 how area basis and wall geometry change resistance accounting. 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 17: Heat-exchanger fouling penalty](../labs/lab-17.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 6, Lesson 4: Thermal design checks and bypass

Unit: Heat exchangers and thermal networks. Core formula: Q_hot≈Q_cold; dE≈Q_hot−Q_cold. Suggested model: [Laboratory 15 — Counterflow exchanger effectiveness](../labs/lab-15.html).

Why this problem appears in process plants

A countercurrent cooler loses duty after a maintenance shutdown. Bypass, maldistribution, pressure drop and property drift can all cause an exchanger to miss duty even when clean calculations predict adequate UA. 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

Bypass, maldistribution, pressure drop and property drift can all cause an exchanger to miss duty even when clean calculations predict adequate UA. The equation selected here, Q_hot≈Q_cold; dE≈Q_hot−Q_cold, 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 Q_hot≈Q_cold; dE≈Q_hot−Q_cold 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: counterflow exchanger effectiveness

We use a numerical teaching model with the following documented inputs: UA = 1800; Ch = 2800; Cc = 1900; Thin = 160; Tcin = 25. The primary control UA is measured in W/K. Other parameters are Ch [W/K]; Cc [W/K]; Thin [°C]; Tcin [°C]. Using the model equation, the predicted heat-transfer duty is 134.788 kW. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 1800 to 4712 W/K while holding the other inputs fixed. The same model predicts 202.985 kW. Compare the absolute difference (68.1967 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: Single-phase countercurrent; constant capacity rates, no external loss; not multiple pass. 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

Plan inlet/outlet temperature, mass flow and pressure-drop measurements to distinguish fouling from bypass. 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

Plan inlet/outlet temperature, mass flow and pressure-drop measurements to distinguish fouling from bypass. 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: Heat-balance mismatch test](../worked-cases/unit-06-case-04.html).

Exercises

1. Starting with Q_hot≈Q_cold; dE≈Q_hot−Q_cold, 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. Plan inlet/outlet temperature, mass flow and pressure-drop measurements to distinguish fouling from bypass. 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 15: Counterflow exchanger effectiveness](../labs/lab-15.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 06: Heat exchangers and thermal networks

Project brief

A heat exchanger has at least two coupled energy balances and a local heat-flow law. Hot-side duty and cold-side gain must agree within heat loss and measurement uncertainty. Specifying one temperature approach does not determine exchanger area until capacity rates and an overall thermal resistance model are available. 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. Design and interpret exchanger duties using energy balances, LMTD, epsilon-NTU, overall coefficients, fouling and approaches to temperature crossing.

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: Countercurrent LMTD. A cold feed is preheated by a hot product stream before a reactor. Check terminal temperatures, signs, pinch and whether a correction factor is needed for multiple passes. Begin with Qdot=UALMTD and then examine which neglected effect might dominate.
• Option 2: Effectiveness-NTU interpretation. A pilot heat exchanger has area fixed but changing stream flow rates. Use the Cr→1 limiting form to avoid numerical cancellation and verify hot and cold outlet temperatures. Begin with NTU=UA/Cmin; eps=(1−exp[−NTU(1−Cr)])/(1−Crexp[−NTU(1−Cr)]) and then examine which neglected effect might dominate.
• Option 3: Overall heat coefficient and fouling. A fouled exchanger requires hotter utility to reach the same process outlet. Explain how area basis and wall geometry change resistance accounting. Begin with 1/U=1/hh+Rw+Rf+1/hc (same-area approximation) and then examine which neglected effect might dominate.
• Option 4: Thermal design checks and bypass. A countercurrent cooler loses duty after a maintenance shutdown. Plan inlet/outlet temperature, mass flow and pressure-drop measurements to distinguish fouling from bypass. Begin with Q_hot≈Q_cold; dE≈Q_hot−Q_cold 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 6.1: Countercurrent terminal differences

Process situation and calculation

A countercurrent exchanger receives hot fluid at 150°C and releases it at 95°C, while cold fluid enters at 35°C and exits at 72°C. The terminal differences are 150−72=78 K and 95−35=60 K. LMTD=(78−60)/ln(78/60)≈68.6 K.

Physical interpretation and possible error

Substituting 150−35 and 95−72 instead would mix the wrong ends for countercurrent flow. LMTD is not automatically applicable without correction to a complicated shell-and-tube exchanger with multiple passes.

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 6.2: Capacity-limited heat exchanger

Process situation and calculation

A hot stream has capacity rate 2.8 kW/K and a cold stream 1.9 kW/K. Entering at 160°C and 25°C, the thermodynamic sensible-exchange upper bound is Cmin×ΔTin=1.9×135=256.5 kW. Any finite single-pass exchanger with positive temperature difference has an actual effectiveness below unity.

Physical interpretation and possible error

Computing a heat duty above 256.5 kW would violate the simplified energy bound regardless of how impressive the specified heat-transfer area appears. In detailed process design, heat losses and phase changes alter the balance and must be treated explicitly.

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 6.3: Fouling at fixed thermal driving force

Process situation and calculation

For two same-area film coefficients hhot=850 and hcold=1100 W/(m² K), and wall resistance 0.0002 m² K/W, the clean U is 1/(1/850+1/1100+0.0002)≈437 W/(m² K). Adding 0.001 m² K/W of deposit resistance reduces U to about 304 W/(m² K).

Physical interpretation and possible error

At unchanged exchanger area and LMTD, capacity decreases by roughly 30%. Actual exchanger terminal temperatures are not held fixed when fouling develops, so duty and the LMTD should be solved together instead of multiplying by a fixed percentage.

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 6.4: Heat-balance mismatch test

Process situation and calculation

A cooling service reports hot-stream duty 125 kW and cold-stream gained duty 114 kW. The difference is 11 kW, or about 9% of hot duty. Before diagnosing heat loss, compare the uncertainties in flow and temperature measurements; for small temperature rises, a single thermocouple bias can explain a surprising share of the apparent discrepancy.

Physical interpretation and possible error

Cross-check conductivity estimates or fouling with the observed pressure drop. A hidden bypass can create the same delivered-temperature symptom as low overall heat-transfer coefficient without producing the same hydraulic signature.

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 11 — Radiation between two plates

Linked unit: 5. Measured output: Net plate radiation (kW). Equation: Q=σ A(Th⁴−Tc⁴)/(1/e1+1/e2−1).

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: Large diffuse gray parallel plates, view factor≈1, no participating gas. 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: hot temperature, unit K. Initial value: 800 K. Predicted output: 29.88434 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 524 K, the reference model gives 4.57328 kW. At the second test value 1106 K, it gives 112.1854 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 12 — Ideal radiation shield

Linked unit: 5. Measured output: Shielded radiant duty (kW). Equation: Q=σ A(Th⁴−Tc⁴)/(1/e1+2/es+1/e2−2).

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: One thin ideal adiabatic shield, equal areas, no gas radiation or convection. 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: shield emissivity, unit fraction. Initial value: 0.15 fraction. Predicted output: 5.283857 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 0.22 fraction, the reference model gives 7.639165 kW. At the second test value 0.79 fraction, it gives 24.57809 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?

Unit 06 — Assessment: Heat exchangers and thermal networks

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 6.1: Countercurrent LMTD

1. Derivation: Starting with `Qdot=U*A*LMTD`, 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 cold feed is preheated by a hot product stream before a reactor. 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: Check terminal temperatures, signs, pinch and whether a correction factor is needed for multiple passes. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Topic 6.2: Effectiveness-NTU interpretation

1. Derivation: Starting with `NTU=U*A/Cmin; eps=(1−exp[−NTU(1−Cr)])/(1−Cr*exp[−NTU(1−Cr)])`, 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 pilot heat exchanger has area fixed but changing stream flow rates. 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: Use the Cr→1 limiting form to avoid numerical cancellation and verify hot and cold outlet temperatures. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Topic 6.3: Overall heat coefficient and fouling

1. Derivation: Starting with `1/U=1/hh+Rw+Rf+1/hc (same-area approximation)`, 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 fouled exchanger requires hotter utility to reach the same process outlet. 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 how area basis and wall geometry change resistance accounting. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Topic 6.4: Thermal design checks and bypass

1. Derivation: Starting with `Q_hot≈Q_cold; dE≈Q_hot−Q_cold`, 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 countercurrent cooler loses duty after a maintenance shutdown. 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: Plan inlet/outlet temperature, mass flow and pressure-drop measurements to distinguish fouling from bypass. 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.