Question

Consider a spherical shell of inner radius \(r_{1}\), outer radius \(r_{2}\), thermal conductivity \(k\), and emissivity \(\varepsilon\). The outer surface of the shell is subjected to radiation to surrounding surfaces at \(T_{\text {surr }}\), but the direction of heat transfer is not known. Express the radiation boundary condition on the outer surface of the shell.

Short answer

Answer: The radiation boundary condition on the outer surface of a spherical shell is given by: \(-k \int_{r_1}^{r_2} \frac{dT}{dr} \frac{4\pi r^2}{r_2 - r_1} dr = \varepsilon \sigma(T_2^4 - T_{\text{surr}}^4) 4\pi r_2^2\) where \(k\) is the thermal conductivity of the shell, \(r_1\) and \(r_2\) are the inner and outer radii, \(\varepsilon\) is the emissivity, \(\sigma\) is the Stefan-Boltzmann constant, \(T_2\) is the temperature at the outer surface of the shell, and \(T_{\text{surr}}\) is the surrounding temperature. This equation takes into account both heat transfer due to conduction and radiation and considers the emissivity of the spherical shell.

Step-by-step solution

Determine heat transfer due to conduction in the shell

The heat transfer due to conduction in the shell can be calculated using the Fourier's law of heat conduction. For a spherical shell, the heat conduction equation is given by: \(Q_{\text{cond}} = -k \int_{r_1}^{r_2} \frac{dT}{dr} \frac{4\pi r^2}{r_2 - r_1} dr\) where \(Q_{\text{cond}}\) is the heat transfer due to conduction, \(k\) is the thermal conductivity, \(r_1\) and \(r_2\) are the inner and outer radii of the shell, and \(T\) is the temperature within the shell.

Determine heat transfer due to radiation from the outer surface of the shell

The heat transfer due to radiation from the outer surface of the shell can be calculated using the Stefan-Boltzmann law for a spherical surface, which is given by: \(Q_{\text{rad}} = \varepsilon \sigma(T_2^4 - T_{\text{surr}}^4) 4\pi r_2^2\) where \(Q_{\text{rad}}\) is the heat transfer due to radiation, \(\varepsilon\) is the emissivity of the shell, \(\sigma\) is the Stefan-Boltzmann constant, \(T_2\) is the temperature at the outer surface of the shell, and \(T_{\text{surr}}\) is the surrounding temperature.

Establish the radiation boundary condition using conservation of energy

Conservation of energy requires that the heat transfer due to conduction within the shell must equal the heat transfer due to radiation from its outer surface. Therefore, we can formulate the radiation boundary condition as: \(Q_{\text{cond}} = Q_{\text{rad}}\) Substituting the expressions for \(Q_{\text{cond}}\) and \(Q_{\text{rad}}\), we get: \(-k \int_{r_1}^{r_2} \frac{dT}{dr} \frac{4\pi r^2}{r_2 - r_1} dr = \varepsilon \sigma(T_2^4 - T_{\text{surr}}^4) 4\pi r_2^2\) This equation expresses the radiation boundary condition on the outer surface of the shell and takes both the heat transfer due to conduction and radiation into account, as well as the emissivity of the spherical shell.

Key concepts

Fourier's Law of Heat Conduction

Heat conduction is the process by which heat energy is transmitted through materials from higher temperature regions to lower temperature regions. Fourier's law of heat conduction is the governing principle behind this process, and it states that the rate at which heat is transferred through a material is proportional to the negative gradient of temperatures and the area through which the heat is being transferred.

Mathematically, it is expressed as: \[\begin{equation}Q_{\text{cond}} = -kA\frac{dT}{dx}\end{equation}\]where:

• Q_{\text{cond}} is the heat transfer rate due to conduction (in watts, W),
• k represents the material's thermal conductivity (in watts per meter kelvin, W/mK),
• A is the cross-sectional area through which the heat flows (in square meters, m²), and
• dT/dx is the temperature gradient in the direction of heat flow (in kelvins per meter, K/m).

In the context of our spherical shell problem, we apply Fourier's law to the specific geometry, which accounts for the radial distribution of heat.

Stefan-Boltzmann Law

The Stefan-Boltzmann law is an essential concept when discussing thermal radiation, which is the emission of electromagnetic waves due to an object's temperature. This law states that the total energy radiated per unit surface area of a black body in unit time is directly proportional to the fourth power of the black body's absolute temperature. For non-black bodies, the law includes the emissivity factor. The law can be mathematically represented as: \[\begin{equation}Q_{\text{rad}} = \varepsilon \sigma A T^4\end{equation}\]where:

• Q_{\text{rad}} is the radiative heat transfer (in watts, W),
• \varepsilon is the emissivity of the material (dimensionless),
• \sigma is the Stefan-Boltzmann constant (5.67 \times 10^{-8} W/m²K⁴),
• A is the radiating surface area (in square meters, m²), and
• T is the absolute temperature of the surface (in kelvins, K).

In our exercise, we use this law to calculate the heat transfer due to radiation from the outer surface of the spherical shell.

Conservation of Energy

The principle of conservation of energy states that energy cannot be created or destroyed in an isolated system; it can only be transformed from one form to another. In the realm of thermodynamics, this principle implies that the total rate of energy transfer into a system must equal the total rate of energy transfer out of the system, assuming the system is in a steady state.

This fundamental principle allows us to set up equations that balance the heat entering and leaving a system. In our problem involving the spherical shell, the heat conducted through the shell must be equal to the heat radiated from its surface, assuming there are no other forms of heat transfer or energy storage occurring. This provides us with the necessary condition to write out the radiation boundary condition, ensuring that the shell's energy remains constant over time.

Thermal Conductivity

Thermal conductivity, symbolized as 'k', is a material property that measures a substance's ability to conduct heat. It fundamentally describes how well heat is transferred through the material due to temperature differences. Materials with high thermal conductivity, such as metals, are good heat conductors, whereas materials with low thermal conductivity, like wood or foam insulation, are considered good thermal insulators.

The thermal conductivity plays a critical role in Fourier's law for heat conduction; it is what we use to calculate the amount of heat that will flow through a section of our spherical shell. The higher the thermal conductivity of the shell's material, the more heat will be conducted for a given temperature gradient.

Emissivity

Emissivity (denoted as \(\varepsilon\)) is a measure of a material's ability to emit energy as thermal radiation. It is expressed as a ratio from 0 to 1, where a value of 1 indicates a perfect black body that emits radiation most efficiently, and a value of 0 represents a perfect reflector that emits no radiation. Real objects usually have emissivities between these two extremes.

This property is essential in the Stefan-Boltzmann law for calculating the radiative heat loss from the surface of an object. In our exercise, the emissivity of the spherical shell's material will determine how much energy it emits as thermal radiation to its surroundings. Understanding emissivity is crucial for accurate thermal analysis in applications ranging from building insulation to spacecraft design.