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Exercise set

Thermal Management and Heat Transfer Exercises

Worked mechanical engineering exercises for thermal management covering heat duty, thermal resistance, junction temperature, convection, coolant flow, heat exchangers, heat flux, and validation.

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Mechanical Engineering
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Exercise set
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v1.1.0 ยท reviewed

These exercises practise thermal management and heat-transfer calculations for mechanical systems, electronics, cooling loops, heat exchangers, and energy equipment. The goal is not only to calculate a temperature or heat rate. The goal is to decide whether the heat path is credible, where margin is lost, and what must be measured during validation.

Assume steady operation and constant properties unless an exercise states otherwise. Real designs should also check transients, radiation, contact resistance, fouling, fluid-property variation, control response, manufacturing tolerance, sensor uncertainty, and degraded operation.

How to Use These Exercises

For each calculation, define:

  1. the heat source and operating duty cycle;
  2. the heat path from source to sink;
  3. the temperature limit that controls the design;
  4. the boundary for heat duty, flow rate, and measured validation data;
  5. the margin between calculated operation and allowable operation.

The most common mistake is treating cooling capacity as a single plant number. Thermal failures often occur because the local heat path is weak: a poor interface, blocked channel, unbalanced flow branch, fouled surface, wrong sensor location, or underestimated heat flux.

Use the exercises as thermal release gates: accept or reject a heat-duty balance, change a coolant flow requirement, reduce a heat-flux concentration, challenge a thermal-resistance assumption, require transient testing, or block release when measured temperature, flow, pressure drop, and sensor uncertainty do not support the margin.

Exercise 1: Heat Duty from Temperature Rise

A coolant loop removes heat from a machine housing. The coolant mass flow rate is 0.85\ \text{kg/s}, the specific heat capacity is 3.9\ \text{kJ/(kg K)}, and the measured temperature rise across the housing is 7^\circ\text{C}.

Estimate the heat removed by the coolant.

Solution

For sensible heat transfer:

\dot Q=\dot m c_p \Delta T
\dot Q=(0.85)(3.9)(7)=23.2\ \text{kW}

The coolant removes approximately 23.2\ \text{kW}.

Engineering Comment

This result is only as good as the flow and temperature measurements. If sensors are placed where mixing is incomplete, or if part of the heat leaves through the structure, the coolant heat duty may not equal total heat generation.

Exercise 2: Heat Flux on a Cooled Surface

A power module dissipates 2.4\ \text{kW} through a cold plate contact area of 0.018\ \text{m}^2.

Find the average heat flux.

Solution

Heat flux is:

\displaystyle q''=\frac{\dot Q}{A}
\displaystyle q''=\frac{2400}{0.018}=133{,}000\ \text{W/m}^2

The average heat flux is:

133\ \text{kW/m}^2

Engineering Comment

Average heat flux can hide local hot spots. Semiconductor packages, battery cells, brake surfaces, and compact heat exchangers may have nonuniform heat generation. A thermal design should check peak local heat flux and interface quality, not only area-average loading.

Exercise 3: Junction Temperature from Thermal Resistance

A power device dissipates 75\ \text{W}. The measured ambient temperature is 42^\circ\text{C}. The effective junction-to-ambient thermal resistance under the installed airflow condition is 0.82^\circ\text{C/W}.

Estimate junction temperature and margin to a 125^\circ\text{C} limit.

Solution

Temperature rise:

\Delta T=P_D R_{\theta JA}
\Delta T=75(0.82)=61.5^\circ\text{C}

Junction temperature:

T_j=T_a+\Delta T
T_j=42+61.5=103.5^\circ\text{C}

Temperature margin:

M_T=125-103.5=21.5^\circ\text{C}

Engineering Comment

The calculation passes the steady-state limit, but it does not prove reliability. The engineer should also check transient overload, blocked airflow, board layout, contact pressure, thermal cycling, sensor uncertainty, and whether the thermal resistance value matches the installed configuration.

Exercise 4: Convection Coefficient from Measured Heat Transfer

A test plate rejects 900\ \text{W} to air. The exposed area is 0.60\ \text{m}^2. The average surface temperature is 68^\circ\text{C} and the bulk air temperature is 32^\circ\text{C}.

Estimate the average convection coefficient.

Solution

Use:

\dot Q=hA(T_s-T_\infty)

Rearrange:

\displaystyle h=\frac{\dot Q}{A(T_s-T_\infty)}
\displaystyle h=\frac{900}{0.60(68-32)}
h=41.7\ \text{W/(m}^2\text{K)}

Engineering Comment

The result is an average coefficient for the tested condition. It should not be reused blindly for another orientation, flow rate, surface roughness, enclosure geometry, or temperature range. Convection correlations and tests must match the physical situation.

Exercise 5: Coolant Flow for a Battery Module

A battery module must reject 18\ \text{kW} to a liquid loop. The coolant specific heat is 3.6\ \text{kJ/(kg K)}. The design limits coolant temperature rise across the module to 5^\circ\text{C}.

Find the required mass flow rate.

Solution

Use:

\dot Q=\dot m c_p \Delta T

Rearrange:

\displaystyle \dot m=\frac{\dot Q}{c_p\Delta T}
\displaystyle \dot m=\frac{18}{(3.6)(5)}=1.0\ \text{kg/s}

Engineering Comment

The flow rate is only the thermal balance. A battery cooling design must also check cell-to-cell temperature spread, pressure drop, pump power, leak risk, coolant compatibility, isolation, cold-start viscosity, and behavior during partial blockage or pump fault.

Exercise 6: Heat Exchanger Duty Reconciliation

A water-to-water heat exchanger is tested. The hot stream flow is 2.2\ \text{kg/s} and cools from 72^\circ\text{C} to 54^\circ\text{C}. The cold stream flow is 2.6\ \text{kg/s} and warms from 28^\circ\text{C} to 42^\circ\text{C}. Use c_p=4.18\ \text{kJ/(kg K)} for both streams.

Compare the hot-side and cold-side heat duties.

Solution

Hot-side heat duty:

\dot Q_h=\dot m_h c_p(T_{h,in}-T_{h,out})
\dot Q_h=(2.2)(4.18)(72-54)=165.5\ \text{kW}

Cold-side heat duty:

\dot Q_c=\dot m_c c_p(T_{c,out}-T_{c,in})
\dot Q_c=(2.6)(4.18)(42-28)=152.2\ \text{kW}

Relative mismatch using hot-side duty as reference:

\displaystyle \epsilon=\frac{165.5-152.2}{165.5}=0.080=8.0\%

Engineering Comment

An 8 percent mismatch may be acceptable for a rough field test or unacceptable for a formal acceptance test, depending on the measurement uncertainty and heat-loss assumptions. The engineer should review flowmeter calibration, temperature sensor placement, external heat loss, incomplete mixing, and property values.

Exercise 7: Thermal Resistance Network

A component dissipates 120\ \text{W}. Heat flows through three effective thermal resistances in series:

R_1=0.12^\circ\text{C/W}
R_2=0.18^\circ\text{C/W}
R_3=0.25^\circ\text{C/W}

The heat sink base is held at 58^\circ\text{C}. Estimate the component temperature.

Solution

Total thermal resistance:

R_{total}=R_1+R_2+R_3
R_{total}=0.12+0.18+0.25=0.55^\circ\text{C/W}

Temperature rise:

\Delta T=PR_{total}
\Delta T=120(0.55)=66^\circ\text{C}

Component temperature:

T=58+66=124^\circ\text{C}

Engineering Comment

The result is close to many electronics and material limits. Before accepting it, check whether each resistance is measured or assumed, whether contact pressure is stable, whether interface material pumps out over time, and whether the heat sink base temperature remains 58^\circ\text{C} during degraded operation.

Exercise 8: Thermal Expansion Strain

An aluminum component is constrained by surrounding structure. Its temperature rises from 20^\circ\text{C} to 85^\circ\text{C}. Use a coefficient of thermal expansion:

\alpha=23\times10^{-6}\ \text{K}^{-1}

Find the free thermal strain. Explain why the constrained case is more severe.

Solution

Temperature change:

\Delta T=85-20=65\ \text{K}

Free thermal strain:

\epsilon_T=\alpha\Delta T
\epsilon_T=(23\times10^{-6})(65)=0.00150

This is:

1500\ \mu\varepsilon

Engineering Comment

If the part were free to expand, this strain would mainly produce displacement. If expansion is constrained, thermal strain can become thermal stress. The actual stress depends on stiffness, constraints, geometry, creep, yielding, contact, and load path. Thermal management should therefore consider mechanical accommodation, not only temperature.

Review Checklist

Before accepting a thermal-management calculation, check:

  • whether the heat source and duty cycle are realistic;
  • whether the thermal boundary includes all relevant heat paths;
  • whether material and fluid properties match the operating temperature;
  • whether average heat flux hides local hot spots;
  • whether contact resistance, fouling, and flow imbalance are included;
  • whether steady-state margin is adequate for transient and degraded operation;
  • whether measurement uncertainty is small enough for the acceptance decision;
  • whether sensor locations capture the limiting junction, interface, surface, or coolant condition rather than a convenient average;
  • whether thermal expansion and thermal stress can change the mechanical design.
  • whether blocked flow, fouled surfaces, pump degradation, fan failure, or control lag would erase the stated thermal margin.

Good thermal engineering turns heat calculations into evidence. The design is credible when measured heat duty, temperature, flow, pressure drop, sensor location, and operating mode all support the same physical explanation.

REF

See also

  1. Thermal Management, Heat Transfer, and Cooling Systems Mechanical Engineering
  2. Thermal Resistance Network Formula Sheet Mechanical Engineering
  3. Heat Sink and Forced Convection Sizing Exercises Mechanical Engineering
  4. Electronics Thermal Management Project Mechanical Engineering
  5. Thermal Runaway and Cooling Failure Case Study Mechanical Engineering
  6. Thermal Energy Systems Formula Sheet Energy Engineering
  7. Thermal Energy Systems and Heat Exchangers Energy Engineering
  8. Energy Efficiency and Heat Recovery Systems Energy Engineering
  9. Data Center Power and Cooling Exercises Energy Engineering
  10. Data Center Power and Cooling Formula Sheet Energy Engineering
  11. How Liquid Cooling Works in Data Centers Energy Engineering
  12. Mechanical Fluid Flow and Piping Systems Mechanical Engineering
  13. Mechanical Fluid Flow and Piping Systems Exercises Mechanical Engineering
  14. Mechanical Stress Analysis Mechanical Engineering
  15. Mechanical Stress Analysis Formula Sheet Mechanical Engineering
  16. Mechanical Stress Analysis Exercises Mechanical Engineering
  17. Power and Sensor Interface Electronics Electronic Engineering
  18. Power and Sensor Interface Electronics Exercises Electronic Engineering
  19. Power Electronics and Converter Systems Electrical Engineering
  20. Materials Selection and Mechanical Properties Materials Engineering
  21. Liquid Cooling Energy Engineering method
  22. Heat Duty Chemical Engineering quantity
  23. Heat Exchanger Energy Engineering device
  24. Heat Flux Energy Engineering quantity
  25. Junction Temperature Electronic Engineering quantity
  26. Thermal Stress Mechanical Engineering phenomenon
  27. Thermal Efficiency Energy Engineering metric
  28. Efficiency Energy Engineering metric
  29. Exergy Energy Engineering quantity
  30. Entropy Energy Engineering quantity
  31. Power Electrical Engineering quantity
  32. Flow Rate Mechanical Engineering quantity
  33. Density Materials Engineering quantity
  34. Viscosity Mechanical Engineering quantity
  35. Reynolds Number Mechanical Engineering quantity
  36. Nusselt Number Energy Engineering quantity
  37. Grashof Number Energy Engineering quantity
  38. Mass Balance Chemical Engineering model
  39. Continuity Equation Mechanical Engineering model
  40. Bernoulli Equation Mechanical Engineering model
  41. Hydraulics Mechanical Engineering concept
  42. Laminar Flow Mechanical Engineering phenomenon
  43. Turbulence Mechanical Engineering phenomenon
  44. Cavitation Mechanical Engineering phenomenon
  45. Water Hammer Mechanical Engineering phenomenon
  46. Thermocouple Electronic Engineering device
  47. Transducer Electronic Engineering device
  48. Validation Industrial and Management Engineering method
  49. Uncertainty Analysis Mathematical Engineering method
  50. Error Budget Electronic Engineering method
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