Brake Thermal Capacity⚠ unverified
Mechanical / Clutches Brakes · Compute the temperature rise in a brake from dissipated energy
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| E | E | J | 1.0 | Energy dissipated |
| t | t | s | 1.0 | Time over which the energy is dissipated |
| A | A | m**2 | 1.0 | Heat-transfer surface area |
| h | h | — | 1.0 | Convective heat-transfer coefficient, in watts per square metre per kelvin (W/(m**2*K)) |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| result | ΔT | K | Temperature rise, in kelvin (K). Returns 0.0 when ``A * h * t`` is not positive |
The science & history
Understanding the Parameters
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Energy $E$ over time $t$ — their ratio $E/t$ is the average heat rate (power) the brake must reject. A brake dumping the same energy over a longer time has a lower average power and a lower steady temperature — the reason gentle, spread-out braking (or engine braking on a descent) keeps brakes cooler than repeated hard stops.
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Surface area $A$ and convection coefficient $h$ — together, $hA$ is the brake's thermal conductance to the air (W/K). A larger cooling area (fins, vanes) or a higher $h$ (airflow, ventilation) rejects more heat per degree of temperature rise, lowering $\Delta T$. This is why brakes are finned, vented, and placed in the airstream.
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The steady-state picture — this is a convective-balance estimate: it gives the temperature at which heat-in equals heat-out. The transient spike during a single stop can be much higher (governed by heat flux and the material's heat capacity, Disk Brake Heat Flux); this formula addresses the sustained/repeated-duty temperature.
Derivation (Approaching a Proof)
At thermal steady state, the heat the brake rejects by convection equals the heat it must dissipate. Newton's law of cooling (Convection Heat Transfer) gives the convective heat rate for a surface at temperature rise $\Delta T$ above ambient:
$$q_{\text{out}} = h A\,\Delta T.$$
The heat rate the brake must reject is the dissipated energy $E$ divided by the time $t$ over which it occurs:
$$q_{\text{in}} = \frac{E}{t}.$$
Setting $q_{\text{out}} = q_{\text{in}}$ (steady balance) and solving for the temperature rise:
$$h A\,\Delta T = \frac{E}{t} \;\Longrightarrow\; \Delta T = \frac{E}{A h\, t}.$$
So the temperature rise is the average power ($E/t$) divided by the convective conductance ($hA$) — exactly the thermal-resistance relation $\Delta T = q\,R_{\text{conv}}$ with $R_{\text{conv}} = 1/(hA)$ (Thermal Resistance Convection). It assumes all rejection is convective (ignoring radiation and conduction into the hub) and that the brake reaches a steady temperature; it therefore estimates the sustained operating temperature, not the peak of a single stop.
Dimensional check. $\dfrac{E}{A h t} = \dfrac{\text{J}}{\text{m}^2\cdot(\text{W/m}^2\text{K})\cdot\text{s}} = \dfrac{\text{J}}{\text{W}\cdot\text{s}/\text{K}} = \dfrac{\text{J}}{\text{J/K}} = \text{K}$. ✓
History and Development
Convective thermal balance is the basis of brake and clutch thermal rating (Shigley, SAE thermal brake tests). The distinction between the transient flash/surface temperature and the steady convective temperature is central to brake design — fade on a long descent is a steady-heat problem, while thermal cracking on a single hard stop is a transient-flux problem.
Related Concepts: Convection Heat Transfer, Thermal Resistance Convection, Disk Brake Heat Flux, Disk Clutch Heat Generation, Brake Fade Factor
Notes: Steady convective-balance estimate ($\Delta T = q/(hA)$); the transient single-stop spike is higher (see Disk Brake Heat Flux). $h$ in W/(m²·K) (see note). Ignores radiation and conduction into the hub.