Question

An ordinary egg can be approximated as a \(5.5-\mathrm{cm}-\) diameter sphere whose properties are roughly \(k=0.6 \mathrm{~W} / \mathrm{m} \cdot \mathrm{K}\) and \(\alpha=0.14 \times 10^{-6} \mathrm{~m}^{2} / \mathrm{s}\). The egg is initially at a uniform temperature of \(8^{\circ} \mathrm{C}\) and is dropped into boiling water at \(97^{\circ} \mathrm{C}\). Taking the convection heat transfer coefficient to be \(h=\) \(1400 \mathrm{~W} / \mathrm{m}^{2} \cdot \mathrm{K}\), determine how long it will take for the center of the egg to reach \(70^{\circ} \mathrm{C}\). Solve this problem using analytical one-term approximation method (not the Heisler charts).

Short answer

Question: Determine the time it takes for the center of an egg to reach 70°C when dropped into boiling water. Consider the following properties for the egg: diameter - 5.5 cm; thermal conductivity - 0.6 W/m⋅K; convection heat transfer coefficient - 1400 W/m²⋅K; initial temperature - 8°C; water temperature - 97°C. Answer: It takes approximately 1568.13 seconds (or about 26 minutes) for the center of the egg to reach 70°C.

Step-by-step solution

Calculate the Biot number (Bi)

To find the Biot number, we need to use the following formula: \(Bi = \frac{hL_c}{k}\), where: \(h = 1400 \mathrm{~W/m^2 \cdot K}\) - Convection heat transfer coefficient \(L_c = \frac{D}{6}\) - Characteristic length (the diameter of the sphere divided by 6) \(D = 5.5 \mathrm{~cm} = 0.055 \mathrm{~m}\) - Diameter of the egg \(k = 0.6 \mathrm{~W/m \cdot K}\) - Thermal conductivity of the egg Substitute values and calculate the Biot number: \(Bi = \frac{1400 \times 0.055/6}{0.6} = 21.1667\)

Find the time factor, \(\tau\)

To find the time factor, using the analytical one-term approximation, look for a value of the Fourier number that satisfies the following equation: \(T_\infty - T_{1} = (T_\infty - T_i) e^{- Bi \tau}\) Where: \(T_\infty = 97^{\circ} \mathrm{C}\) \(T_1 = 70^{\circ} \mathrm{C}\) \(T_i = 8^{\circ} \mathrm{C}\) (You can find a value for \(\tau\) by examining a standard chart that plots values of \(e^{-x\tau}\) against \(\tau\) for various Biot numbers). Rearrange the equation to find \(\tau\) and substitute the known values: \(\tau = -\frac{\ln[(T_\infty - T_{1})/(T_\infty - T_i)]}{Bi} = -\frac{\ln[(97-70)/(97-8)]}{21.1667} = 0.2976\)

Calculate the time required to reach the desired temperature, \(t\)

To find the time required to reach the desired temperature, use the Fourier number: \(Fo = \frac{\alpha \cdot t}{L_c^2}\) Where \(Fo\) is the Fourier number. Now, plug in the given values and solve for t: \(t = \frac{Fo \cdot L_c^2}{\alpha} = \frac{0.2976 \cdot (0.055/6)^2}{0.14 \times 10^{-6}} = 1568.13 \mathrm{~s}\) So, it will take approximately \(1568.13\) seconds (or about \(26\) minutes) for the center of the egg to reach \(70^{\circ} \mathrm{C}\).

Key concepts

Biot number calculation

The Biot number (Bi) is a dimensionless quantity used in heat transfer calculations to quantify the ratio of resistance to heat conduction within an object to the resistance to convection from its surface to the surrounding fluid. In practical terms, it indicates whether an object is thermally thick or thin. Calculating the Biot number involves determining the characteristic length (\r\(L_c\)), the thermal conductivity (\r\(k\)) of the material, and the convection heat transfer coefficient (\r\(h\)).

For spherical objects, like our example with the egg, the characteristic length is often taken as the radius of the sphere divided by three (\r\(D/6\)), which considers the volume to surface area ratio. Using this definition, the Biot number for the egg is calculated by the provided formula \r\(Bi = \frac{hL_c}{k}\). With the given values, we get a high Biot number, indicating that conduction resistance inside the egg dominates over the convective resistance at the surface; thus, the temperature within the egg will not change evenly.

Fourier number in heat conduction

The Fourier number (Fo) is another critical dimensionless number in heat transfer, particularly in transient heat conduction problems. It is defined as the ratio of heat conduction rate to the heat storage rate within a body and is used to determine the temperature distribution in an object over time. The higher the Fourier number, the more uniform the temperature within the object is likely to be.

The Fourier number is given by \r\(Fo = \frac{\alpha \cdot t}{L_c^2}\) where \r\(\alpha\) is the thermal diffusivity of the material, \r\(t\) is the time, and \r\(L_c\) is the characteristic length. It's used in conjunction with the Biot number to find solutions to transient conduction problems, like how long it takes for the center of the egg to heat up in the example provided.

Analytical one-term approximation method

The analytical one-term approximation method simplifies complex heat transfer problems. It uses a single exponential term to predict temperature changes within an object over time, which is particularly helpful when dealing with simple shapes like spheres, cylinders, or slabs.

In the case of the egg, this method calculates the time needed for the center to reach a certain temperature. The equation for the temperature in a semi-infinite solid exposed to a step change in surface temperature applies here as an approximation. By manipulating the equation \r\(T_\infty - T_{1} = (T_\infty - T_i) e^{- Bi \tau}\) where \r\(T_\infty\) is the surrounding temperature, \r\(T_1\) is the desired center temperature, and \r\(T_i\) is the initial temperature, we can solve for the dimensionless time factor \r\(\tau\) and subsequently for the actual time (\r\(t\)). This method provides a reliable estimate of the heating time needed without relying on complex numerical methods or extensive charts.

Convection heat transfer coefficient

The convection heat transfer coefficient (\r\(h\)) is a measure of the convective heat transfer between a surface and a fluid in motion. It is an empirical value that depends on various factors, including fluid velocity, viscosity, and temperature difference between the surface and the fluid. In our context, it is essential for calculating the Biot number and, by extension, determining heat transfer behavior in transient conduction problems.

For the egg dropped into boiling water, a high \r\(h\) value of \r\(1400 \mathrm{~W/m^2 \cdot K}\) implies that heat is efficiently transferred from the water to the egg, enhancing the overall heat transfer rate. Such information is crucial for predicting how quickly an object will reach thermal equilibrium with its surroundings and must be accurately determined to ensure precise thermal analyses.