Question

An air-cooled motorcycle engine loses a significant amount of heat through thermal radiation according to the Stefan-Boltzmann equation. Assume that the ambient temperature is \(T_{0}=27^{\circ} \mathrm{C}(300 \mathrm{~K})\). Suppose the engine generates 15 hp \((11 \mathrm{~kW})\) of power and, due to several deep surface fins, has a surface area of \(A=0.50 \mathrm{~m}^{2}\). A shiny engine has an emissivity \(e=0.050\), whereas an engine that is painted black has \(e=0.95 .\) Determine the equilibrium temperatures for the black engine and the shiny engine. (Assume that radiation is the only mode by which heat is dissipated from the engine.)

Short answer

Answer: The approximate equilibrium temperatures for the black engine and shiny engine are 223.2 °C (496.2 K) and 566.6 °C (839.6 K), respectively.

Step-by-step solution

Write the Stefan-Boltzmann equation

The equation governing the power loss per unit area due to thermal radiation is: P/A = eσ(T^4 - T0^4)

Determine the power lost as a result of heat dissipation (P)

The problem states that the engine generates 15 hp (11 kW) of power. The power dissipated as heat is also equal to 15 hp (11 kW), as radiation is the only mode that heat is dissipated from the engine. P = 11 kW

Substitute the known values into the equation for both black and shiny engines

The equation becomes: 11 = e * 0.50 * 5.67 * 10^{-8} * (T^4 - 300^4) For the black engine, e = 0.95 and for the shiny engine, e = 0.050.

Solve the equation for the temperature of the black engine (T1) and the shiny engine (T2)

For the black engine: 11 = 0.95 * 0.50 * 5.67 * 10^{-8} * (T1^4 - 300^4) And for the shiny engine: 11 = 0.050 * 0.50 * 5.67 * 10^{-8} * (T2^4 - 300^4) Solving these equations for T1 and T2, we get: T1 ≈ 496.2 K (223.2 °C) for the black engine T2 ≈ 839.6 K (566.6 °C) for the shiny engine Hence, the equilibrium temperatures for the black engine and shiny engine are approximately 223.2 °C (496.2 K) and 566.6 °C (839.6 K), respectively.

Key concepts

Stefan-Boltzmann Equation

The Stefan-Boltzmann equation is a fundamental principle in thermal physics that describes how objects emit thermal radiation. This equation can be written as \( P/A = e \sigma (T^4 - T_0^4) \), where:

• \( P \) is the power emitted per unit area.
• \( A \) is the surface area of the object.
• \( e \) is the emissivity of the surface.
• \( \sigma \) is the Stefan-Boltzmann constant \( (5.67 \times 10^{-8} \, Wm^{-2}K^{-4}) \).
• \( T \) is the absolute temperature of the object.
• \( T_0 \) is the ambient, or surrounding, temperature.

The equation tells us that the thermal power emitted depends on both the temperature difference between the object and its environment and the properties of the object's surface. So, when we want to find how much heat an engine radiates, we can apply this equation, considering not only its temperature but also its emissivity and the surrounding temperature.

Emissivity

Emissivity is a measure of how effectively a surface emits thermal radiation. It is a dimensionless value that ranges from 0 to 1.

• An emissivity of 1 means the surface is a perfect black body emitter, radiating the maximum possible energy at a given temperature.
• An emissivity of 0 means the surface does not emit thermal radiation.
• A shiny, polished surface, like the shiny engine, typically has a low emissivity.
• An engine painted in matte black usually has a high emissivity, absorbing and emitting more radiation.

In our problem, the shiny engine has an emissivity \( e \) of 0.050, while the black paint increases the emissivity to 0.95. This significantly affects the equilibrium temperature since a higher emissivity promotes efficient heat dissipation, keeping the engine cooler.

Equilibrium Temperature

The equilibrium temperature of an object in thermal radiation is the temperature at which the energy emitted by the object equals the energy it absorbs from its environment.
For the motorcycle engine, the equilibrium temperature is where the power it radiates matches the power it's generating and dissipating as heat, due to thermal radiation being the only mode of heat dissipation.
Using the emissivities of the engines, we found:

• A black engine has a lower equilibrium temperature of approximately 496.2 K (223.2 °C).
• A shiny engine reaches a much higher equilibrium temperature of about 839.6 K (566.6 °C).

This shows that engines with higher emissivity materials stabilize at cooler temperatures, improving performance and avoiding overheating.

Heat Dissipation

Heat dissipation is the process by which an object loses heat to its surroundings. In the context of the motorcycle engine, this was through thermal radiation. Since it's the sole mode of heat loss, it plays a crucial role in determining the engine's temperature.
Several factors influence how well heat dissipation occurs:

• Surface Area: Larger surface areas, aided by fins in engines, increase the area available for heat loss.
• Emissivity: High emissivity surfaces lose heat more effectively, as seen with the black engine.
• Temperature Difference: Greater differences between engine and ambient temperatures drive higher heat dissipation rates.

Effective heat dissipation ensures that the engine runs efficiently without reaching excessively high temperatures that might damage its components.