Stefan-Boltzmann law
The Stefan-Boltzmann law is crucial in understanding radiative heat transfer, especially when dealing with objects like our spherical tank in the exercise. This law states that the energy radiated per unit surface area of a black body is directly proportional to the fourth power of the black body's thermodynamic temperature. Formally, it is expressed as
\( q = \epsilon \sigma A (T1^4 - T2^4) \)
where \( q \) is the radiative heat transfer rate, \( \epsilon \) is the emissivity of the material, \( \sigma \) is the Stefan-Boltzmann constant, \( A \) is the surface area, and \( T1 \) and \( T2 \) are the absolute temperatures of the bodies involved. The law applies perfectly for idealistic black bodies, but for real materials, emissivity comes into play, which represents how closely a material's thermal radiation characteristic follows a perfect black body. In our case, the polished surfaces have an emissivity of 0.15, which indicates that they radiate much less than a perfect black body would.
Fourier's law of heat conduction
Fourier's law is the cornerstone of conduction heat transfer — the type of heat transfer that occurs from molecule to molecule within a material. It mathematically defines the rate at which heat energy flows through a material as a result of temperature differences. The law's formula is open to interpretation in terms of its variables:
\( q = k A \frac{(T2-T1)}{t} \)
Here, \( q \) represents the heat transfer rate by conduction, \( k \) is the thermal conductivity of the material, \( A \) is the cross-sectional area through which heat is conducted, \( T1 \) and \( T2 \) are the temperatures at the two sides of the material, and \( t \) is the thickness of the material through which heat is being conducted. The given exercise uses Fourier's law to calculate the conduction through the steel walls of the spherical tank. With the assumed values for the thermal conductivity, temperature difference, and wall thickness, you can see how the heat transfer is significant when compared to radiation.
Thermal conductivity
Thermal conductivity, denoted as \( k \), is a property that quantifies a material's ability to conduct heat. It's an intrinsic characteristic of a material, implying that some materials are naturally better conductors (such as metals) than others (like wood or rubber). In the context of the exercise, the steel of the tank walls has a thermal conductivity of \( 16 \frac{W}{mK} \), suggesting that it’s a good conductor of heat. This thermal conductivity value is what we plug into Fourier’s law of heat conduction to find the rate of heat transfer through the walls of the tank. The higher the thermal conductivity, the more efficient the material is in transferring heat.
Radiative heat transfer
When it comes to radiative heat transfer, it's important to know that this is energy transferred in the form of electromagnetic waves, primarily in the infrared spectrum. Unlike conductive and convective heat transfer, radiation doesn't require any medium to occur and can happen in a vacuum. In our exercise, radiative heat transfer between the inner and outer walls of the tank is considered, and the heat transfer rate is determined using the Stefan-Boltzmann law. The concept is essential for understanding how heat energy escapes or is absorbed by objects without direct contact.
Conductive heat transfer
Conductive heat transfer is the process of heat being transferred through materials, atom by atom, molecule by molecule. It’s the underlying principle when you touch a hot object and feel the heat, or why a spoon in a pot of hot soup also gets hot. In our exercise scenario, conductive heat transfer is reviewed and calculated for both the inner and outer walls of the tank. The conductivity of the material and the temperature gradient are fundamental, as they directly affect the rate at which heat is transferred.
Heat of fusion
The heat of fusion is thermodynamics that represents the amount of energy needed to change a substance's state from solid to liquid at its melting point, without changing its temperature. This concept is typically measured in Joules per kilogram (J/kg). For instance, the heat of fusion for ice is \( 334000 \frac{J}{kg} \), which is applied in our exercise to determine the mass of the ice that melts over a 24-hour period due to the heat transfer to the iced water.
Spherical coordinates
Spherical coordinates are a system for representing the points in a three-dimensional space through three coordinates: the radius \( r \), the polar angle \( \theta \), and the azimuthal angle \( \phi \). These coordinates are particularly useful when dealing with problems involving spheres, as in our example of a spherical tank. To calculate properties like the surface area of a sphere, we often use the radius in the formula \( A = 4\pi r^2 \), which helps us to determine important variables in our heat transfer calculations. The use of spherical coordinates can simplify complex problems involving spheres by aligning with the symmetry of the sphere itself.
