Steady Conduction and Thermal Resistance Networks

The engineering problem

Conduction moves thermal energy through solids and stationary fluids because microscopic fluctuations and molecular interactions redistribute energy. For constant conductivity the temperature profile is linear in a one-dimensional slab with no generation, but cylindrical geometry changes the area available for flow with radius.

Scope and learning outcomes

Analyze planar and cylindrical conduction, composite media, contact resistance, and insulation economics while defending boundary conditions. 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

2.1 Planar multilayer conduction

A slab, a refractory lining, and a steel wall experience the same steady heat rate only when side losses and heat generation are negligible. A dryer wall combines steel, refractory and fibrous insulation. Estimate layer-interface temperatures and flag conditions under which the stated k changes strongly with temperature. Representative relationship: Qdot=(T_hot−T_cold)/(Σ L/(k*A)).

2.2 Cylindrical walls and critical insulation radius

Radial area increases with radius, making cylindrical resistance logarithmic; added insulation can increase loss around a very fine wire before it reduces it. An instrument trace heater is surrounded by an insulating jacket. Explain why rcrit is not a universal rule for a large pipe and why radiation changes the optimum. Representative relationship: R_cyl=ln(r2/r1)/(2pik*L); rcrit=k/h.

2.3 Contact resistance and thermal bridges

Finite roughness, interface pressure, oxides and gaps introduce interfacial resistance not captured by bulk material conductivity. A bolted heated plate shows a high temperature drop across a gasket. Identify independent tests that can separate contact resistance from an incorrect substrate k. Representative relationship: DeltaT_interface=Qdot*R_contact.

2.4 Temperature-dependent conduction

Real conductivity can vary with temperature, and heat-generating solids can have an interior temperature maximum. An electrically heated polymer block is thickened for mechanical strength. Check whether heat-generation density or surface loss dominates peak temperature. Representative relationship: q_x=−k(T)dT/dx; d2T/dx2=−qvol/k.

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 2, Lesson 1: Planar multilayer conduction

Unit: Steady conduction and insulation. Core formula: Qdot=(T_hot−T_cold)/(Σ L/(k*A)). Suggested model: [Laboratory 01 — Single-layer wall conduction](../labs/lab-01.html).

Why this problem appears in process plants

A dryer wall combines steel, refractory and fibrous insulation. A slab, a refractory lining, and a steel wall experience the same steady heat rate only when side losses and heat generation are negligible. 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

A slab, a refractory lining, and a steel wall experience the same steady heat rate only when side losses and heat generation are negligible. The equation selected here, Qdot=(T_hot−T_cold)/(Σ L/(k*A)), 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=(T_hot−T_cold)/(Σ L/(k*A)) 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: single-layer wall conduction

We use a numerical teaching model with the following documented inputs: thickness = 0.08; k = 1.2; A = 2; dT = 55. The primary control thickness is measured in m. Other parameters are k [W/(m K)]; A [m²]; dT [K]. Using the model equation, the predicted conductive heat rate is 1650 W. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 0.08 to 0.1072 m while holding the other inputs fixed. The same model predicts 1231.34 W. Compare the absolute difference (-418.657 W) 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: Constant conductivity, 1D steady slab, fixed surface temperatures. 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

Estimate layer-interface temperatures and flag conditions under which the stated k changes strongly with temperature. 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

Estimate layer-interface temperatures and flag conditions under which the stated k changes strongly with temperature. 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: Refractory wall thermal drop](../worked-cases/unit-02-case-01.html).

Exercises

1. Starting with Qdot=(T_hot−T_cold)/(Σ L/(k*A)), 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. Estimate layer-interface temperatures and flag conditions under which the stated k changes strongly with temperature. 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 01: Single-layer wall conduction](../labs/lab-01.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 2, Lesson 2: Cylindrical walls and critical insulation radius

Unit: Steady conduction and insulation. Core formula: R_cyl=ln(r2/r1)/(2pik*L); rcrit=k/h. Suggested model: [Laboratory 03 — Radial pipe-insulation resistance](../labs/lab-03.html).

Why this problem appears in process plants

An instrument trace heater is surrounded by an insulating jacket. Radial area increases with radius, making cylindrical resistance logarithmic; added insulation can increase loss around a very fine wire before it reduces it. 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

Radial area increases with radius, making cylindrical resistance logarithmic; added insulation can increase loss around a very fine wire before it reduces it. The equation selected here, R_cyl=ln(r2/r1)/(2pik*L); rcrit=k/h, 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 R_cyl=ln(r2/r1)/(2pik*L); rcrit=k/h 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: radial pipe-insulation resistance

We use a numerical teaching model with the following documented inputs: outer radius = 0.09; r1 = 0.025; k = 0.045; Len = 4; dT = 80. The primary control outer radius is measured in m. Other parameters are r1 [m]; k [W/(m K)]; Len [m]; dT [K]. Using the model equation, the predicted heat rate is 70.6343 W. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 0.09 to 0.114 m while holding the other inputs fixed. The same model predicts 59.6299 W. Compare the absolute difference (-11.0044 W) 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: Steady radial conduction, no external surface film; r2 must exceed r1. 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 rcrit is not a universal rule for a large pipe and why radiation changes the optimum. 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 rcrit is not a universal rule for a large pipe and why radiation changes the optimum. 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 cylinder radius example](../worked-cases/unit-02-case-02.html).

Exercises

1. Starting with R_cyl=ln(r2/r1)/(2pik*L); rcrit=k/h, 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 rcrit is not a universal rule for a large pipe and why radiation changes the optimum. 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 03: Radial pipe-insulation resistance](../labs/lab-03.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 2, Lesson 3: Contact resistance and thermal bridges

Unit: Steady conduction and insulation. Core formula: DeltaT_interface=Qdot*R_contact. Suggested model: [Laboratory 02 — Composite wall heat flow](../labs/lab-02.html).

Why this problem appears in process plants

A bolted heated plate shows a high temperature drop across a gasket. Finite roughness, interface pressure, oxides and gaps introduce interfacial resistance not captured by bulk material conductivity. 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

Finite roughness, interface pressure, oxides and gaps introduce interfacial resistance not captured by bulk material conductivity. The equation selected here, DeltaT_interface=Qdot*R_contact, 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 DeltaT_interface=Qdot*R_contact 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: composite wall heat flow

We use a numerical teaching model with the following documented inputs: insulation thickness = 0.07; k1 = 0.08; k2 = 35; L2 = 0.01; A = 2; dT = 95. The primary control insulation thickness is measured in m. Other parameters are k1 [W/(m K)]; k2 [W/(m K)]; L2 [m]; A [m²]; dT [K]. Using the model equation, the predicted net heat rate is 217.072 W. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 0.07 to 0.0988 m while holding the other inputs fixed. The same model predicts 153.811 W. Compare the absolute difference (-63.2614 W) 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: Pure steady conduction; interfaces perfectly bonded; no external films. 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 independent tests that can separate contact resistance from an incorrect substrate k. 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 independent tests that can separate contact resistance from an incorrect substrate k. 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: Gasket contact-temperature discontinuity](../worked-cases/unit-02-case-03.html).

Exercises

1. Starting with DeltaT_interface=Qdot*R_contact, 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 independent tests that can separate contact resistance from an incorrect substrate k. 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 02: Composite wall heat flow](../labs/lab-02.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 2, Lesson 4: Temperature-dependent conduction

Unit: Steady conduction and insulation. Core formula: q_x=−k(T)dT/dx; d2T/dx2=−qvol/k. Suggested model: [Laboratory 03 — Radial pipe-insulation resistance](../labs/lab-03.html).

Why this problem appears in process plants

An electrically heated polymer block is thickened for mechanical strength. Real conductivity can vary with temperature, and heat-generating solids can have an interior temperature maximum. 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

Real conductivity can vary with temperature, and heat-generating solids can have an interior temperature maximum. The equation selected here, q_x=−k(T)dT/dx; d2T/dx2=−qvol/k, 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_x=−k(T)dT/dx; d2T/dx2=−qvol/k 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: radial pipe-insulation resistance

We use a numerical teaching model with the following documented inputs: outer radius = 0.09; r1 = 0.025; k = 0.045; Len = 4; dT = 80. The primary control outer radius is measured in m. Other parameters are r1 [m]; k [W/(m K)]; Len [m]; dT [K]. Using the model equation, the predicted heat rate is 70.6343 W. This is an arithmetic result for assumed properties, not a measured industrial duty or a certified equipment rating.

Increase the primary control from 0.09 to 0.114 m while holding the other inputs fixed. The same model predicts 59.6299 W. Compare the absolute difference (-11.0044 W) 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: Steady radial conduction, no external surface film; r2 must exceed r1. 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 whether heat-generation density or surface loss dominates peak temperature. 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 whether heat-generation density or surface loss dominates peak temperature. 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: Volumetric heating in a slab](../worked-cases/unit-02-case-04.html).

Exercises

1. Starting with q_x=−k(T)dT/dx; d2T/dx2=−qvol/k, 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 whether heat-generation density or surface loss dominates peak temperature. 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 03: Radial pipe-insulation resistance](../labs/lab-03.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 02: Steady conduction and insulation

Project brief

Conduction moves thermal energy through solids and stationary fluids because microscopic fluctuations and molecular interactions redistribute energy. For constant conductivity the temperature profile is linear in a one-dimensional slab with no generation, but cylindrical geometry changes the area available for flow with radius. 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. Analyze planar and cylindrical conduction, composite media, contact resistance, and insulation economics while defending boundary conditions.

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: Planar multilayer conduction. A dryer wall combines steel, refractory and fibrous insulation. Estimate layer-interface temperatures and flag conditions under which the stated k changes strongly with temperature. Begin with Qdot=(T_hot−T_cold)/(Σ L/(k*A)) and then examine which neglected effect might dominate.
• Option 2: Cylindrical walls and critical insulation radius. An instrument trace heater is surrounded by an insulating jacket. Explain why rcrit is not a universal rule for a large pipe and why radiation changes the optimum. Begin with R_cyl=ln(r2/r1)/(2pik*L); rcrit=k/h and then examine which neglected effect might dominate.
• Option 3: Contact resistance and thermal bridges. A bolted heated plate shows a high temperature drop across a gasket. Identify independent tests that can separate contact resistance from an incorrect substrate k. Begin with DeltaT_interface=Qdot*R_contact and then examine which neglected effect might dominate.
• Option 4: Temperature-dependent conduction. An electrically heated polymer block is thickened for mechanical strength. Check whether heat-generation density or surface loss dominates peak temperature. Begin with q_x=−k(T)dT/dx; d2T/dx2=−qvol/k 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 2.1: Refractory wall thermal drop

Process situation and calculation

A 100 mm refractory layer has k=0.75 W/(m K), wall area 4 m² and surface temperatures 750°C and 110°C. The predicted steady one-dimensional heat flow is 0.75×4×640/0.1=19.2 kW. At k=0.65, holding the two surface temperatures fixed, the calculated duty falls to 16.64 kW.

Physical interpretation and possible error

These surface conditions are highly demanding and require checking refractory service ratings and temperature-dependent conductivity. The second result is not necessarily an operating savings: changed conductivity may cause hot-face or cold-face temperatures to shift when external films are included.

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 2.2: Critical cylinder radius example

Process situation and calculation

A heated sensor cable has a conducting core radius of 2 mm, outer ambient film coefficient h=10 W/(m² K), and insulation k=0.04 W/(m K). The ideal critical radius is k/h=0.004 m. Added insulation from 2 to 3 mm can increase heat loss in the simplified model because added convection area dominates the initial rise in radial resistance.

Physical interpretation and possible error

Large process pipes with outer radii far above 4 mm are already beyond this critical radius. On a pipe of 50 mm radius, more insulation generally adds resistance, but complete losses also require radiation, supports, moisture and insulation condition.

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 2.3: Gasket contact-temperature discontinuity

Process situation and calculation

An exchanger plate transfers 8 kW through a gasketed contact with overall interface resistance 0.0015 K/W. The predicted contact temperature discontinuity is 8000×0.0015=12 K. If measured surface thermocouples show a 22 K difference, at least one modeled term, measurement position or contact assumption needs revision.

Physical interpretation and possible error

Do not infer a material conductivity from the two readings without estimating heat flux, effective area and lateral leakage. Increasing bolt preload can change the real contact resistance, but requires mechanical constraints and may damage the gasket.

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 2.4: Volumetric heating in a slab

Process situation and calculation

A thin electrical heating element creates 20 kW/m³ uniformly in a polymer slab of half-thickness 10 mm and conductivity k=0.2 W/(m K). For a plane wall with both surfaces fixed to the same temperature and zero slope at the center, the peak center-to-surface rise is qvol L²/(2k)=20,000×0.01²/(0.4)=5 K.

Physical interpretation and possible error

If the half-thickness doubles to 20 mm, the rise becomes 20 K, four times larger. That quadratic geometry effect can be missed when a total electrical-power budget is substituted for a local heat-generation density.

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 03 — Radial pipe-insulation resistance

Linked unit: 2. Measured output: Heat rate (W). Equation: Q=2π k Len ΔT / ln(r2/r1).

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: Steady radial conduction, no external surface film; r2 must exceed r1. 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: outer radius, unit m. Initial value: 0.09 m. Predicted output: 70.6343 W. Your result may round on the web page. Confirm the orders of magnitude independently with the printed equation.

At the first selected test value 0.072 m, the reference model gives 85.53479 W. At the second test value 0.198 m, it gives 43.72197 W. 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 04 — Straight-fin efficiency

Linked unit: 4. Measured output: Fin efficiency (fraction). Equation: η=tanh(mL)/(mL); m=sqrt(hP/(kAc)).

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: Constant properties; adiabatic fin tip; uniform rectangular cross-section. 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: fin length, unit m. Initial value: 0.045 m. Predicted output: 0.9192584 fraction. Your result may round on the web page. Confirm the orders of magnitude independently with the printed equation.

At the first selected test value 0.028 m, the reference model gives 0.9667704 fraction. At the second test value 0.097 m, it gives 0.7222875 fraction. 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 18 — Heat recovery minimum approach

Linked unit: 8. Measured output: Optimistic recovery ceiling (kW). Equation: Qscreen=Cmin max(0,Thi−Tci−ΔTmin).

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: Screening ceiling only; not a network pinch analysis or exchanger rating. 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: minimum approach, unit K. Initial value: 12 K. Predicted output: 313.6 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 11.6 K, the reference model gives 314.88 kW. At the second test value 40.4 K, it gives 222.72 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 02 — Assessment: Steady conduction and insulation

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 2.1: Planar multilayer conduction

1. Derivation: Starting with `Qdot=(T_hot−T_cold)/(Σ L/(k*A))`, 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 dryer wall combines steel, refractory and fibrous insulation. 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: Estimate layer-interface temperatures and flag conditions under which the stated k changes strongly with temperature. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Topic 2.2: Cylindrical walls and critical insulation radius

1. Derivation: Starting with `R_cyl=ln(r2/r1)/(2*pi*k*L); rcrit=k/h`, 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 instrument trace heater is surrounded by an insulating jacket. 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 rcrit is not a universal rule for a large pipe and why radiation changes the optimum. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Topic 2.3: Contact resistance and thermal bridges

1. Derivation: Starting with `DeltaT_interface=Qdot*R_contact`, 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 bolted heated plate shows a high temperature drop across a gasket. 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 independent tests that can separate contact resistance from an incorrect substrate k. State how the result would change for a tenfold increase in one relevant physical length, and whether the original correlation remains valid.

Topic 2.4: Temperature-dependent conduction

1. Derivation: Starting with `q_x=−k(T)dT/dx; d2T/dx2=−qvol/k`, 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 electrically heated polymer block is thickened for mechanical strength. 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 whether heat-generation density or surface loss dominates peak temperature. 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.