Question

Eggs with a mass of \(0.15 \mathrm{~kg}\) per egg and a specific heat of $3.32 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}$ are cooled from \(32^{\circ} \mathrm{C}\) to \(10^{\circ} \mathrm{C}\) at a rate of 300 eggs per minute. The rate of heat removal from the eggs is (a) \(11 \mathrm{~kW}\) (b) \(80 \mathrm{~kW}\) (c) \(25 \mathrm{~kW}\) (d) \(657 \mathrm{~kW}\) (e) \(55 \mathrm{~kW}\)

Short answer

Answer: (e) 55 kW

Step-by-step solution

Calculate the heat loss of a single egg

First, we need to find the heat loss of a single egg while cooling from \(32^{\circ} \mathrm{C}\) to \(10^{\circ} \mathrm{C}\). We'll use the formula: \(Q = mc\Delta T\) Where, \(Q\) is the heat loss, \(m\) is the mass of the egg (\(0.15 \mathrm{~kg}\)), \(c\) is the specific heat of the egg (\(3.32 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}\)), \(\Delta T\) is the change in temperature (\(32^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C}\))

Calculate the heat loss of a single egg

Now calculate the heat loss by substituting the given values in the formula: \(Q= (0.15\mathrm{~kg}) (3.32 \mathrm{~kJ} / \mathrm{kg} \cdot{ }^{\circ} \mathrm{C}) (32^{\circ} \mathrm{C} - 10^{\circ} \mathrm{C}) = 10.392 \mathrm{~kJ}\) So, the heat loss of a single egg while cooling is \(10.392 \mathrm{~kJ}\).

Calculate the total heat loss per minute

As we know that the process of cooling happens at the rate of \(300\) eggs per minute. So, we have to find the total heat loss per minute. Multiply the heat loss of a single egg by the rate of cooling: \(Q_{\text{total}}= (10.392 \mathrm{~kJ})(300 \text{ eggs per minute}) = 3117.6 \mathrm{~kJ/minute}\)

Convert the total heat loss per minute to kW

We need to find the rate of heat removal from the eggs, which is given in the problem statement in kilowatts (kW). To convert the total heat loss per minute to kW, divide by the number of seconds in a minute and multiply by \(1000\) to convert kJ to J: \(P = \frac{Q_\text{total}}{60\mathrm{~s}} = \frac{3117.6 \mathrm{~kJ}}{60\mathrm{~s}} \times \frac{1000 \mathrm{~J}}{1 \mathrm{~kJ}} = 51.96 \mathrm{~kW}\)

Select the correct answer

Now we can choose the correct answer from the given options. The closest value to our calculated value is \(55 \mathrm{~kW}\). Therefore, the correct answer is (e) \(55 \mathrm{~kW}\).