Question

An ordinary egg can be approximated as a \(5.5-\mathrm{cm}-\) diameter sphere. 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 properties of egg to be \(\rho=1020 \mathrm{kg} / \mathrm{m}^{3}\) and \(c_{p}=3.32 \mathrm{kJ} / \mathrm{kg} \cdot^{\circ} \mathrm{C}\) determine how much heat is transferred to the egg by the time the average temperature of the egg rises to \(70^{\circ} \mathrm{C}\) and the amount of exergy destruction associated with this heat transfer process. Take \(T_{0}=25^{\circ} \mathrm{C}\).

Short answer

Answer: The amount of heat transferred to the egg is 19.26 kJ, and the amount of exergy destruction associated with this heat transfer process is 42.25 kJ.

Step-by-step solution

Calculate the volume and mass of the egg

To calculate the volume of the egg, we can use the volume formula for a sphere: \(V = \dfrac{4}{3}\pi r^3\) The diameter of the egg is given as 5.5 cm, so the radius is half of that: \(r = \dfrac{5.5}{2} = 2.75\, \text{cm} = 0.0275\, \text{m}\). Now, we can calculate the volume: \(V = \dfrac{4}{3}\pi (0.0275)^3 = 9.176 \times 10^{-5}\, \text{m}^3\) Using the density, we can determine the mass of the egg: \(m = \rho V = (1020\, \text{kg} / \text{m}^3)(9.176 \times 10^{-5}\, \text{m}^3) = 0.0936\, \text{kg}\)

Determine the amount of heat transfer

We use the mass, specific heat capacity, and temperature change to calculate the heat transfer: \(q = m c_p \Delta T\) The temperature change is given by: \(\Delta T = T_{\text{final}} - T_{\text{initial}} = 70^{\circ} \mathrm{C} - 8^{\circ} \mathrm{C} = 62^{\circ} \mathrm{C}\) We substitute these values into the formula with \(c_p = 3.32\, \text{kJ} / \text{kg} \cdot^{\circ} \mathrm{C}\): \(q = (0.0936\, \text{kg})(3.32\, \text{kJ} / \text{kg} \cdot^{\circ} \mathrm{C})(62^{\circ} \mathrm{C}) = 19.26\, \text{kJ}\)

Calculate the initial and final entropy of the egg

We use the formula for the entropy change of a constant specific heat process: \(\Delta S = m c_p \ln\left(\dfrac{T_f}{T_i}\right)\) Calculate the initial and final entropy of the egg: \(S_i = m c_p \ln\left(\dfrac{T_i}{T_0}\right)\) \(S_i = (0.0936\, \text{kg})(3.32\, \text{kJ} / \text{kg} \cdot^{\circ} \mathrm{C})\ln\left(\dfrac{8^{\circ} \mathrm{C}}{25^{\circ} \mathrm{C}}\right) = -0.3985\, \text{kJ}/^{\circ} \mathrm{C}\) \(S_f = m c_p \ln\left(\dfrac{T_f}{T_0}\right)\) \(S_f = (0.0936\, \text{kg})(3.32\, \text{kJ} / \text{kg} \cdot^{\circ} \mathrm{C})\ln\left(\dfrac{70^{\circ} \mathrm{C}}{25^{\circ} \mathrm{C}}\right) = 1.490\, \text{kJ}/^{\circ} \mathrm{C}\)

Calculate the entropy change of the egg and the surrounding medium (boiling water)

The entropy change of the egg is simply the difference between the final and initial entropy: \(\Delta S_{\text{egg}} = S_f - S_i = 1.490 - (-0.3985) = 1.8885\, \text{kJ}/^{\circ} \mathrm{C}\) The entropy change of the surrounding medium (boiling water) can be calculated using the heat transfer and its temperature: \(\Delta S_{\text{water}} = -\dfrac{q}{T_{\text{water}}} = -\dfrac{19.26\, \text{kJ}}{97^{\circ} \mathrm{C}} = -0.1986\, \text{kJ}/^{\circ} \mathrm{C}\)

Determine the exergy destruction associated with the heat transfer process

Exergy destruction is given by the total entropy change multiplied by the ambient temperature: \(X_{\text{destruction}} = T_0 (\Delta S_{\text{egg}} + \Delta S_{\text{water}})\) \(X_{\text{destruction}} = (25^{\circ} \mathrm{C})(1.8885\, \text{kJ}/^{\circ} \mathrm{C} - 0.1986\, \text{kJ}/^{\circ} \mathrm{C}) = 42.25\, \text{kJ}\) The amount of heat transferred to the egg is 19.26 kJ, and the amount of exergy destruction associated with this heat transfer process is 42.25 kJ.

Key concepts

Entropy Change

Entropy is a fundamental concept in thermodynamics often associated with the level of disorder in a system, and it's measured in units of joules per Kelvin (\text{J/K}). It's a core element in predicting how systems will evolve over time and under what conditions processes will occur.

When heat is transferred to or from an object, like the ordinary egg in our problem, there's a change in the system’s entropy. For a constant specific heat process, the entropy change (\text{𝚫S}) can be mathematically expressed as:
\[\text{𝚫S} = m c_p \text{ln}\bigg(\frac{T_f}{T_i}\bigg)\]
where:

• \text{m} represents mass,
• \text{c_p} is the specific heat capacity,
• \text{T_f} and \text{T_i} are the final and initial temperatures, respectively.

This formula shows that entropy change depends on the amount of heat transferred and the temperature at which the transfer occurs.

It's integral to understand that the total entropy of the universe increases in a spontaneous process and this principle underpins the second law of thermodynamics.

Specific Heat Capacity

Specific heat capacity (\text{c_p}) is a property of a material that indicates how much energy is needed to raise the temperature of a unit mass of the substance by one degree Celsius (or Kelvin). It's typically expressed in units like joules per kilogram per degree Celsius (\text{J/kg·°C}).
The higher the specific heat capacity of a material, the more energy it can store per unit mass for a given temperature change. This is why materials with high specific heat capacities, like water, are effective at retaining heat and regulating temperature.

In our exercise, we use the specific heat capacity of an egg to calculate the heat transfer:\[q = m c_p \text{𝚫T}\]
where \text{𝚫T} is the temperature change. Since we know the initial and final temperatures of the egg, along with its mass and specific heat capacity, we can determine the total heat absorbed by the egg as it reaches the desired temperature.

Exergy Destruction

Exergy destruction is a measure of the energy that is no longer available for doing useful work due to inefficiencies or irreversible processes in a thermodynamic system. It's linked to the concept of entropy creation and can be seen as 'wasted' energy.

Exergy destruction is not a physical quantity that can be measured directly but it can be calculated if we know the entropy change of the system and its surroundings. The second law of thermodynamics posits that in all real processes, some exergy is destroyed because of irreversibilities, such as friction, mixing, or heat transfer through a finite temperature difference.
Using the ambient temperature (\text{T_0}), and the entropy changes of the system (\text{𝚫S_egg}) and the surroundings (\text{𝚫S_water}), we can find the exergy destruction with the formula:\[X_{\text{destruction}} = T_0 (\text{𝚫S}_{\text{egg}} + \text{𝚫S}_{\text{water}})\]
In the exercise, this value represents the potential useful work lost in the process of heating the egg, illuminating the importance of considering energy efficiency in thermodynamic processes.