Question

Solar radiation is incident on the outer surface of a spaceship at a rate of \(400 \mathrm{Btu} / \mathrm{h} \cdot \mathrm{ft}^{2}\). The surface has an absorptivity of \(\alpha_{s}=0.10\) for solar radiation and an emissivity of \(\varepsilon=0.6\) at room temperature. The outer surface radiates heat into space at \(0 \mathrm{R}\). If there is no net heat transfer into the spaceship, determine the equilibrium temperature of the surface

Short answer

Answer: The equilibrium temperature of the surface is 334.2 Kelvin.

Step-by-step solution

Write down the given values

We are given the following information: - Solar radiation incident (\(I\)) on surface: \(400 \, \mathrm{Btu/h\cdot ft^2}\) - Absorptivity of surface (\(\alpha_s\)): \(0.10\) - Emissivity of surface (\(\varepsilon\)): \(0.6\) Note that we need to make sure that all values are in consistent units. The solar radiation needs to be converted from Btu/h\(\cdot\)ft² to \(W/m^2\).

Convert solar radiation to consistent units

Since we want to work in SI units, let's convert the given solar radiation from Btu/h\(\cdot\)ft² to \(W/m^2\). The conversion factors are: - 1 Btu/h = 0.293071 W - 1 ft² = 0.092903 m² So, the solar radiation in \(W/m^2\) is: \(I = 400\, \mathrm{Btu/h\cdot ft^2} \times \dfrac{0.293071\, \mathrm{W}}{1\, \mathrm{Btu/h}} \times \dfrac{1\, \mathrm{ft^2}}{0.092903\,\mathrm{m^2}}\) \(I = 1257.87\, \mathrm{W/m^2}\)

Calculate absorbed solar radiation

Since not all solar radiation is absorbed by the surface, we multiply incident solar radiation by the absorptivity: \(Q_{absorbed} = \alpha_s \times I\) \(Q_{absorbed} = 0.10 \times 1257.87\,\mathrm{W/m^2}\) \(Q_{absorbed} = 125.787\,\mathrm{W/m^2}\)

Apply the Stefan-Boltzmann Law

The Stefan-Boltzmann Law relates the radiated heat from a surface to its temperature by the equation: \(Q_{radiated} = \varepsilon \times \sigma \times T^4\) where \(\sigma\) is the Stefan-Boltzmann constant (\(5.67 \times 10^{-8}\) W/m²K⁴), \(\varepsilon\) is the emissivity, and \(T\) is the temperature in Kelvin. Since there is no net heat transfer into the spaceship, \(Q_{absorbed} = Q_{radiated}\), thus we have \(\varepsilon \times \sigma \times T^4 = Q_{absorbed}\)

Solve for the temperature

Now, we have everything we need to solve for the equilibrium temperature of the surface: \(T^4 = \dfrac{Q_{absorbed}}{\varepsilon \times \sigma}\) \(T^4 = \dfrac{125.787\,\mathrm{W/m^2}}{0.6 \times 5.67 \times 10^{-8}\,\mathrm{W/m^2 K^{4}}}\) \(T^4 = 3.680 \times 10^8\,\mathrm{K^4}\) Taking the fourth root of both sides to solve for the temperature: \(T = (3.680 \times 10^8\,\mathrm{K^4})^{1/4}\) \(T = 334.2\,\mathrm{K}\) The equilibrium temperature of the surface is 334.2 Kelvin.

Key concepts

Solar Radiation

Solar radiation is the stream of photons, or particles of light, emitted by the sun, which carry energy through space and onto various bodies like the Earth and satellites. This energy is instrumental in determining the climate and weather patterns on Earth and affects the thermal conditions of man-made objects in space, such as satellites and spacecraft. Understanding solar radiation is crucial when calculating the equilibrium temperature of a surface affected by it, as it dictates the amount of energy that a surface absorbs. To calculate this, we first measure the incident solar radiation power per unit area, which is expressed in units of watts per square meter (\(W/m^2\)). For objects in space, the solar radiation received needs to account for the material's specific properties, which determine how much of this radiation is absorbed and turned into heat.

Stefan-Boltzmann Law

The Stefan-Boltzmann Law is a fundamental principle in thermodynamics that describes how the radiated energy from a black body per unit area is directly proportional to the fourth power of its absolute temperature. It is mathematically expressed as \(Q_{radiated} = \varepsilon \times \sigma \times T^4\), where \(Q_{radiated}\) is the radiant heat energy emitted by the surface in watts per square meter (\(W/m^2\)), \(\varepsilon\) is the emissivity of the surface, \(\sigma\) is the Stefan-Boltzmann constant \(5.67 \times 10^{-8} W/m^2K^4\), and \(T\) is the absolute temperature in Kelvin. This law allows us to calculate the equilibrium temperature of an object, such as the surface of a spacecraft, when we know the energy it absorbs and emits. When an object is in thermal equilibrium, the energy it absorbs from solar radiation must equal the energy it radiates away into space.

Heat Transfer

Heat transfer is the process by which thermal energy moves from one place to another. In the context of our problem, this transfer happens through radiation, which does not require a medium and can occur through the vacuum of space. This is different from other heat transfer modes, such as conduction and convection, which do require a medium (a solid object or fluid, respectively) and are relevant in different contexts. In the vacuum of space, where the spacecraft resides, radiation is the sole method for heat transfer, restraining the spacecraft from gaining or losing heat through conduction or convection. Therefore, the equilibrium temperature of an object in space is achieved when the rate of energy absorbed via solar radiation equals the rate of energy emitted via radiative heat transfer.

Radiative Properties

Radiative properties of a material, including absorptivity and emissivity, define how a surface exchanges thermal radiation with its surroundings. Absorptivity (\(\alpha_s\)) determines the fraction of incoming radiation that a material absorbs and converts into heat, whereas emissivity (\(\varepsilon\)) describes the efficiency of a surface at emitting thermal radiation compared to an ideal black body. In our spaceship example, these properties are critical. With an absorptivity of 0.10, the spaceship surface absorbs 10% of the solar radiation it encounters. Meanwhile, its emissivity of 0.6 means it is 60% as effective at radiating heat as a perfect black body would be. By considering these properties, along with the Stefan-Boltzmann Law and the heat transfer concept, we can accurately calculate the equilibrium temperature of the spacecraft’s surface.