Biot Number
The Biot number (Bi) is a dimensionless quantity in heat transfer analysis that helps in determining whether the heat conduction within a body is faster than the heat transfer across the boundary. It is calculated using the formula:
\[\begin{equation}Bi = \frac{hD}{k}\end{equation}\]
where \(h\) is the heat transfer coefficient, \(D\) is the characteristic length (typically the diameter for spheres), and \(k\) is the thermal conductivity. If Bi < 0.1, the temperature within the object can be assumed uniform due to rapid conduction compared to convection (lumped system analysis). However, if Bi is significantly larger than 0.1, temperature gradients within the object need to be considered for accurate analysis.
Fourier Number
The Fourier number (Fo) is another dimensionless number that characterizes transient heat conduction. It is a ratio of conductive heat transfer to the energy stored and is found using the formula:
\[\begin{equation}Fo = \frac{\alpha t}{r^{2}}\end{equation}\]
where \(\alpha\) is the thermal diffusivity, \(t\) is the time, and \(r\) is the radius (or half the characteristic length) of the body. The Fourier number provides insight into how much time is required for a material to reach a certain level of thermal equilibrium, with higher values indicating a closer approach to uniform temperature distribution.
Analytical One-Term Approximation Method
The analytical one-term approximation method simplifies the solution of the complex transient heat conduction equation by using a single term from the infinite series. This method is particularly useful when dealing with spherical, cylindrical, or planar geometries and gives a reasonable approximation when the first term dominates the series, which is often the case for short times or objects with high thermal diffusivity. By applying this method, estimates for temperature distribution, as shown in the exercise above, can be determined effectively without detailed numerical analysis.
Heat Transfer Coefficient
The heat transfer coefficient (\(h\)) quantifies the convective heat transfer between a solid surface and the surrounding fluid. It is measured in watts per square meter-kelvin \(\left(\frac{W}{m^2 \cdot K}\right)\). This value depends on the properties of the fluid, the flow conditions, and the surface geometry. A higher \(h\) means that heat will be transferred more efficiently from the surface into the fluid or vice versa.
Thermal Diffusivity
Thermal diffusivity (\(\alpha\)) is a measure of how quickly a material can conduct heat relative to how much heat it can store. It is given by:
\[\begin{equation}\alpha = \frac{k}{\rho c_{p}}\end{equation}\]
where \(k\) is thermal conductivity, \(\rho\) is density, and \(c_{p}\) is the specific heat capacity. The higher the thermal diffusivity, the faster the heat spreads through the material.
Specific Heat Capacity
Specific heat capacity (\(c_p\)) represents how much energy is needed to raise the temperature of a unit mass of a substance by one degree Celsius. It is a crucial property in heat transfer calculations because it indicates the ability of a material to store thermal energy. Specific heat capacity impacts how rapidly a material can change its temperature when exposed to heat or how well it can maintain temperature in the face of heat loss or gain.
Thermal Conductivity
Thermal conductivity (\(k\)) is a fundamental material property that measures the ability of a material to conduct heat. It is commonly found in the units \(\frac{W}{m \cdot K}\). Materials with high thermal conductivity can transfer heat quickly, such as metals, while materials with low thermal conductivity, such as plastics or wood, act as insulators. Thermal conductivity plays a vital role in determining the heat transfer rate through an object and is an essential parameter in calculating the Biot number and thermal diffusivity.
Lumped System Analysis
Lumped system analysis is an approach to simplify the otherwise complicated differential equations in transient heat conduction problems. By assuming that the temperature variation inside a body is negligible relative to the variation at the surface, we can treat the entire object at a single, uniform temperature. This is suitable when the Biot number is less than 0.1 and leads to an ordinary differential equation that's straightforward to solve. It helps in approximately calculating the total heat transfer over time as demonstrated in the given solution.
Spherical Heat Conduction
Spherical heat conduction refers to the transfer of heat within spherical objects. This process is governed by the spherical version of the heat equation, which is generally more complex than its Cartesian counterpart due to geometric considerations. In practice, formulas are often derived for specific boundary conditions and simplified scenarios, such as in this exercise, where the analytical one-term approximation method is used to approximate the temperature distribution within the apples, treated as spherical objects.
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