Question

It is often stated that the refrigerator door should be opened as few times as possible for the shortest duration of time to save energy. Consider a household refrigerator whose interior volume is \(0.9 \mathrm{m}^{3}\) and average internal temperature is \(4^{\circ} \mathrm{C} .\) At any given time, one-third of the refrigerated space is occupied by food items, and the remaining \(0.6 \mathrm{m}^{3}\) is filled with air. The average temperature and pressure in the kitchen are \(20^{\circ} \mathrm{C}\) and \(95 \mathrm{kPa}\), respectively. Also, the moisture contents of the air in the kitchen and the refrigerator are 0.010 and \(0.004 \mathrm{kg}\) per \(\mathrm{kg}\) of air, respectively, and thus \(0.006 \mathrm{kg}\) of water vapor is condensed and removed for each kg of air that enters. The refrigerator door is opened an average of 20 times a day, and each time half of the air volume in the refrigerator is replaced by the warmer kitchen air. If the refrigerator has a coefficient of performance of 1.4 and the cost of electricity is 11.5 cents per \(\mathrm{kWh}\), determine the cost of the energy wasted per year as a result of opening the refrigerator door. What would your answer be if the kitchen air were very dry and thus a negligible amount of water vapor condensed in the refrigerator?

Short answer

Question: Determine the annual cost of the energy wasted as a result of opening the refrigerator door, given the properties of the refrigerator, the cost of electricity, and details about the opening and closing of the door. Also, find the cost when the kitchen air is very dry. Answer: To determine the annual cost of the energy wasted, follow the steps mentioned in the solution, which include calculating the mass of the air replaced, finding the change in enthalpy due to air and moisture, and finding the total energy wasted per opening. Multiply the energy wasted per opening by the number of door openings per day and the cost of electricity, and then calculate the annual cost. When the kitchen air is very dry, repeat these steps without considering the enthalpy change due to moisture.

Step-by-step solution

1. Calculate the mass of the air replaced.

First, we need to find the mass of the air in the refrigerator that is replaced each time the door is opened. We are given that half of the air volume in the refrigerator (\(0.6 m^3\)) is replaced each time the door is opened. The mass of the replaced air can be found using the Ideal Gas Law: \(PV = mRT\), where P is the pressure, V is the volume, m is the mass, R is the gas constant, and T is the temperature. We need to convert the given values to SI units, and we know that the gas constant for air, \(R_{air} = 287 \frac{J}{kg * K}\).

2. Calculate the change in enthalpy due to the air replaced.

Now we have to calculate the change in enthalpy due to the air replaced in the refrigerator. We will need to use the temperature of the kitchen air and the refrigerator air. The enthalpy change for air can be expressed as \(\Delta H = c_p * (T_{kitchen} - T_{ref})\), where \(c_p = 1006 J/(kg \cdot K)\) is the specific heat of dry air at constant pressure, and \(T_{kitchen}\) and \(T_{ref}\) are the temperatures in the kitchen and refrigerator, respectively.

3. Calculate the mass of the moisture condensed.

Next, we need to determine the mass of the moisture condensed as the air moves from the kitchen into the refrigerator. We are told that there is \(0.006 kg\) per \(kg\) of air, and we found the mass of the air in step 1. Thus, we simply calculate the mass of the moisture condensed, \(m_{moisture} = 0.006 * m_{air}\).

4. Calculate the change in enthalpy due to the moisture condensed.

We need to determine the change in enthalpy of the water vapor due to moisture condensation as it moves from the kitchen air to the refrigerator. The enthalpy change can be expressed as \(\Delta H_{moisture} = c_{p, water} * (T_{kitchen} - T_{ref})\), where \(c_{p, water} = 4190 J/(kg \cdot K)\) is the specific heat of water at a constant pressure.

5. Calculate the total change in enthalpy for the air and moisture.

Now, we can find the total change in enthalpy by adding the enthalpy change due to the air replaced and the enthalpy change due to the moisture condensed: \(\Delta H_{total} = \Delta H_{air} + \Delta H_{moisture}\).

6. Find the energy wasted per door opening.

The energy wasted per door opening can be calculated by using the equation: \(Q = \frac{\Delta H_{total}}{COP}\), where \(Q\) is the energy wasted and \(COP\) is the coefficient of performance of the refrigerator.

7. Calculate the annual cost of the energy wasted.

Now that we have found the energy wasted per door opening, we can calculate the annual cost of the energy wasted. We are given that the refrigerator is opened 20 times per day and that the cost of electricity is 11.5 cents per kWh. We can then find the total energy wasted per year and multiply it by the cost of electricity to get the total cost per year.

8. Repeat the calculation for dry kitchen air.

To find the result when the kitchen air is very dry, we need to repeat steps 1-7 without considering the enthalpy change due to moisture since it is negligible. This will give us the cost of the energy wasted per year when the kitchen air is very dry.

Key concepts

Ideal Gas Law

Understanding the Ideal Gas Law is crucial when studying thermodynamics, especially when you want to determine the characteristics of gases under various conditions. In the context of a refrigerator, it's used to calculate the mass of air replaced each time the door is opened. The Ideal Gas Law is expressed as \(PV = mRT\), where P (pressure), V (volume), m (mass), R (specific gas constant), and T (temperature) are all in SI units. For air, the specific gas constant is \(287 J/(kg \times K)\).

When the refrigerator door is opened, the warm air from the kitchen enters and mixes with the cold air inside, resulting in a change of pressure and temperature which can be described using the Ideal Gas Law. This calculation is fundamental to understanding the subsequent energy exchange that takes place due to the door opening.

Enthalpy Change Calculation

The enthalpy change calculation is used to determine the amount of heat transfer during a process where pressure is constant, like when air enters a refrigerator. It's given by the formula \(\Delta H = c_p \times (T_{kitchen} - T_{ref})\), where \(\Delta H\) is the change in enthalpy, \(c_p\) is the specific heat capacity (in joules per kilogram per kelvin), and \(T_{kitchen}\) and \(T_{ref}\) are the temperatures of the kitchen and refrigerator. This is significant because it influences the energy required to restore the refrigerator to its initial temperature after warmer air enters.

Specific heat capacity, which varies slightly with temperature, is essentially the energy required to raise the temperature of a kilogram of a substance by one Kelvin. In this case, knowing that the specific heat of the air and the moisture are different allows us to calculate the total enthalpy change when both are considered.

Coefficient of Performance

The coefficient of performance (COP) is a measure of a refrigerator's efficiency. It is defined as the ratio of the heat removed from the refrigerated space to the work input, essentially measuring how well a refrigerator turns energy (from electricity) into cooling. This is expressed as:\[COP = \frac{\text{Cooling effect}}{\text{Work input}}\].

A higher COP means more efficient cooling for less electrical energy, making it an important consideration for cost and energy savings. When calculating the energy waste due to door openings, the COP helps to find out how much extra energy the refrigerator will need to consume to compensate for the heat introduced by the warmer air; the lower the COP, the higher the energy consumption for the same amount of cooling.

Electricity Cost Calculation

The electricity cost calculation is a practical application of thermodynamics, translating the energy waste to a monetary value. It involves multiplying the energy consumed in kilowatt-hours (kWh) by the cost of electricity per kWh. The formula is:\[\text{Cost} = \text{Energy wasted (kWh)} \times \text{Cost per kWh}\].

In the exercise, the energy wasted due to door openings over a year is converted into a cost by considering the number of times the door is opened daily and the cost of electricity. This helps in understanding the real-world implications of energy efficiency and wastage. Knowing how actions like frequently opening the refrigerator door can impact electricity costs incentivizes consumers to adopt more energy-efficient habits.