7.2 Heat Capacity, Transfer Modes & Volumetric Expansion

Heat Capacity, Transfer Modes & Volumetric Expansion

Knowing that heat is energy in transit is not enough for calculations or system understanding. You must know how much heat changes temperature (heat capacity), how heat travels (conduction, convection, radiation), and how temperature change alters size (thermal expansion). These topics sit at the centre of engine cooling, cabin environmental control, fuel volume accounting, and structural clearances.

Heat Capacity and Specific Heat Capacity

Heat capacity of a body is the quantity of heat required to raise that body’s temperature by one kelvin (or one degree Celsius — the intervals are equal). A large engine block has a greater heat capacity than a small bolt of the same metal because more mass must be warmed.

Specific heat capacity (symbol c, often just “specific heat”) is the heat required to raise unit mass by one kelvin:

Q = m c ΔT

• Q = heat transferred (J)
• m = mass (kg)
• c = specific heat capacity (J/(kg·K) or J/(kg·°C))
• ΔT = temperature change (K or °C — same numerical change)

Rearrangements:

• c = Q / (m ΔT)
• ΔT = Q / (m c)
• m = Q / (c ΔT)

A high specific heat means the material absorbs a lot of energy for a modest temperature rise (water is notable: c ≈ 4200 J/(kg·K)). Metals typically have lower c than water, so the same heat input raises metal temperature more quickly for the same mass.

Worked example — oil warm-up

A mass of 25 kg of oil with c = 2000 J/(kg·K) is heated from 15 °C to 75 °C. Find Q.

ΔT = 75 − 15 = 60 K
Q = m c ΔT = 25 × 2000 × 60 = 3 000 000 J = 3.0 MJ

That energy came from combustion heat rejected through the oil cooler path, friction, and related losses — the formula does not care about the source, only the energy that entered the oil and the temperature rise.

Worked example — same heat, different ΔT

Apply Q = 3.0 MJ to 25 kg of water (c ≈ 4200 J/(kg·K)):

ΔT = Q / (m c) = 3 000 000 / (25 × 4200) ≈ 28.6 K

Same energy, smaller temperature rise — water’s higher specific heat “buffers” temperature change. Coolant choice and oil vs water behaviour in heat exchangers rest on this physics.

Sensible heat

Heat that changes temperature without changing state is sometimes called sensible heat (you can “sense” the temperature change). Heat that changes state at constant temperature (melting, boiling) is latent heat — section 7.4.

Modes of Heat Transfer

Heat always moves from higher temperature to lower temperature spontaneously. The three modes are conduction, convection, and radiation. Real aircraft systems usually combine all three.

Conduction

Conduction is heat transfer through matter by molecular or electronic interaction without bulk movement of the material as a whole. A hot cylinder head conducts heat into cooler fins and into bolted joints. Fourier’s law (awareness): heat flow rate increases with thermal conductivity, area, and temperature gradient, and decreases with thickness.

• Good conductors: most metals (aluminium, copper) — used for heat sinks, cylinder heads, heat exchanger cores.
• Poor conductors (insulators): air, many plastics, ceramics, lagging materials — used to protect structure and people from hot pipes and to retain heat where wanted.

Touch a metal tool and a wooden handle at the same room temperature: metal feels colder because it conducts heat away from your skin faster, not because it is actually colder.

Convection

Convection is heat transfer by bulk motion of a fluid (liquid or gas). Warm fluid moves, carrying internal energy with it.

• Natural (free) convection: buoyancy-driven — hot air rises, cooler air sinks (cooling fins in still air, cabin air stratification).
• Forced convection: fans, blowers, ram air, pumps — oil coolers, radiators, avionics bay fans, cabin packs with airflow.

Engine oil coolers and liquid coolant radiators rely on forced flow over large surface areas. Increasing airflow or oil flow increases convective heat removal for a given temperature difference (within design limits).

Radiation

Thermal radiation is emission of electromagnetic energy from a surface because of its temperature. It does not require a material medium — the Sun heats the Earth through space by radiation. Net radiation heat transfer between bodies depends strongly on temperature (roughly on T⁴ for idealised black bodies — Module 2 needs the idea, not full Stefan–Boltzmann algebra in every question), surface area, and surface properties (emissivity, absorptivity). Dark, rough surfaces generally radiate and absorb better than polished shiny surfaces.

In engines and brakes, hot components lose heat by radiation to cooler surroundings as well as by convection to air. Heat shields manage radiant load on adjacent structure and systems.

Combined cooling path (typical engine)

1. Combustion gas → metal (convection/conduction at the gas–wall interface).
2. Through the metal wall and fins (conduction).
3. From fins to cooling air (convection), plus radiation from hot surfaces.
4. Oil absorbs heat from bearings and pistons (convection in galleries, conduction at films) then rejects heat in an oil cooler to ram or fan air (forced convection).

Thermal Expansion

Most materials expand when heated and contract when cooled (water near 4 °C is a special case; Module 2 focuses on the general rule for solids and engineering liquids).

Linear expansion

For a solid rod or span of length L₀:

ΔL = α L₀ ΔT

• ΔL = change in length
• α = coefficient of linear expansion (1/K)
• L₀ = original length
• ΔT = temperature change

Longer members and larger ΔT produce larger growth. Different metals have different α — aluminium expands more per degree than steel generally — so mixed-material assemblies need design allowance.

Worked example. A steel spar length L₀ = 5.0 m, α = 12 × 10⁻⁶ /K, heated by 40 K:

ΔL = 12 × 10⁻⁶ × 5.0 × 40 = 0.0024 m = 2.4 mm

A few millimetres matters for close-fitting structure, control cable tension (with temperature), and precision assemblies.

Area and volumetric expansion

For isotropic solids, area expansion coefficient is approximately 2α, and volumetric coefficient β ≈ 3α:

ΔV = β V₀ ΔT

Liquids have their own volumetric expansion coefficients (often larger than solids). Fuel in a tank expands when temperature rises: a tank filled cold to the brim can overflow or pressurise vents when the aircraft sits in the sun. Conversely, fuel contracted in cold soak occupies less volume for the same mass — linking back to density from fluid dynamics.

Gases expand strongly with temperature at constant pressure (Charles’s law — next section); the β for an ideal gas at constant pressure is 1/T (absolute), much larger effect than solid expansion.

Aircraft structure and systems implications

• Expansion joints, gaps, and tolerances accommodate growth of skins, ducts, and pipes.
• Control cables may need temperature-aware rigging practices depending on design (differential expansion of cable vs structure).
• Tyres and oleo pressures change with temperature (gas laws + some volume change).
• Hot brakes and wheels: large ΔT, careful cooling, and respect for heat transfer to nearby components.
• Composite vs metal skin: different expansion behaviour must be designed for at joints.

Practical Summary Table

Formula Recap

• Q = m c ΔT
• Heat modes: conduction, convection, radiation
• ΔL = α L₀ ΔT; ΔV = β V₀ ΔT (β ≈ 3α for isotropic solids)

Master these and you can estimate temperature rise from heat input, explain how an oil cooler works, and predict why a cold-soaked full tank behaves differently after solar heating — all before gas laws formalise the behaviour of the working fluid itself.