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 T0 = 24.5°C (297.65 K). Suppose the engine generates 11.5 hp (8.58 kW) of power and, due to several deep surface fins, has a surface area of A = 0.540 m2. A shiny engine has an emissivity e = 0.0550, whereas an engine that is painted black has e = 0.950.
a) Determine the equilibrium temperature (in K) for the shiny engine. (Assume that radiation is the only mode by which heat is dissipated from the engine.)
b) And determine the equilibrium temperature (in K) for the black engine.
To determine the equilibrium temperature for both the shiny and black engine using the Stefan-Boltzmann equation, we can use the formula:
P = σ * e * A * (T^4 - T0^4)
where:
P = power generated by the engine (in watts)
σ = Stefan-Boltzmann constant (approximately 5.67 × 10^-8 W/m^2K^4)
e = emissivity of the engine's surface
A = surface area of the engine (in square meters)
T = equilibrium temperature of the engine (in Kelvin)
T0 = ambient temperature (in Kelvin)
a) For the shiny engine (e = 0.0550):
8.58 kW = 5.67 × 10^-8 W/m^2K^4 * 0.0550 * 0.540 m^2 * (T^4 - 297.65^4)
We can solve this equation to find the value of T for the shiny engine.
b) For the black engine (e = 0.950):
8.58 kW = 5.67 × 10^-8 W/m^2K^4 * 0.950 * 0.540 m^2 * (T^4 - 297.65^4)
We can solve this equation to find the value of T for the black engine.
To determine the equilibrium temperature using the Stefan-Boltzmann equation, we need to find the rate at which the engine loses heat and then equate it to the rate at which the engine generates heat.
The rate at which the engine loses heat through thermal radiation is given by:
P = e * σ * A * (T^4 - T0^4)
Where:
P is the power generated by the engine (8.58 kW)
e is the emissivity of the surface (0.0550 for the shiny engine, 0.950 for the black engine)
σ is the Stefan-Boltzmann constant (5.67 x 10^-8 W/m^2K^4)
A is the surface area of the engine (0.540 m^2)
T is the equilibrium temperature we want to find (in K)
T0 is the ambient temperature (297.65 K)
a) For the shiny engine (e = 0.0550), let's determine the equilibrium temperature (T).
8.58 kW = 0.0550 * (5.67 x 10^-8 W/m^2K^4) * 0.540 m^2 * (T^4 - 297.65^4)
Simplifying the equation:
8.58 kW = 0.0550 * 5.67 x 10^-8 * 0.540 * (T^4 - 297.65^4)
Solving for T^4:
T^4 = (8.58 kW / (0.0550 * 5.67 x 10^-8 * 0.540)) + 297.65^4
T^4 = 2311657720486.2 + 81796868097035.38
T^4 = 84108525717521.58
T ≈ 423.9 K
Therefore, the equilibrium temperature for the shiny engine is approximately 423.9 K.
b) For the black engine (e = 0.950), let's determine the equilibrium temperature (T).
8.58 kW = 0.950 * (5.67 x 10^-8 W/m^2K^4) * 0.540 m^2 * (T^4 - 297.65^4)
Simplifying the equation:
8.58 kW = 0.950 * 5.67 x 10^-8 * 0.540 * (T^4 - 297.65^4)
Solving for T^4:
T^4 = (8.58 kW / (0.950 * 5.67 x 10^-8 * 0.540)) + 297.65^4
T^4 = 1439470648860.7 + 81796868097035.38
T^4 = 83236338745896.08
T ≈ 455.7 K
Therefore, the equilibrium temperature for the black engine is approximately 455.7 K.