Question

An absorption air-conditioning system is to remove heat from the conditioned space at \(20^{\circ} \mathrm{C}\) at a rate of \(150 \mathrm{kJ} / \mathrm{s}\) while operating in an environment at \(35^{\circ} \mathrm{C}\). Heat is to be supplied from a geothermal source at \(140^{\circ} \mathrm{C}\). The minimum rate of heat supply is \((a) 86 \mathrm{kJ} / \mathrm{s}\) (b) \(21 \mathrm{kJ} / \mathrm{s}\) \((c) 30 \mathrm{kJ} / \mathrm{s}\) \((d) 61 \mathrm{kJ} / \mathrm{s}\) \((e) 150 \mathrm{kJ} / \mathrm{s}\)

Short answer

Answer: The minimum rate of heat supply from the geothermal source is (e) 150 kJ/s.

Step-by-step solution

Convert temperatures to Kelvin

To convert temperatures from Celsius to Kelvin, we add 273.15. \(T_{conditioned} = 20^{\circ}\mathrm{C} + 273.15 = 293.15 \mathrm{K}\) \(T_{environment} = 35^{\circ}\mathrm{C} + 273.15 = 308.15 \mathrm{K}\) \(T_{geothermal} = 140^{\circ}\mathrm{C} + 273.15 = 413.15 \mathrm{K}\)

Calculate Carnot Efficiency

Use the Carnot efficiency formula (\(\eta_{Carnot} = 1 - \frac{T_{low}}{T_{high}}\)) to determine the efficiency of the system: \(\eta_{Carnot} = 1 - \frac{T_{conditioned}}{T_{geothermal}} = 1 - \frac{293.15 \mathrm{K}}{413.15 \mathrm{K}} = 0.290\)

Calculate Minimum Rate of Heat Supply

We know that the air-conditioning system removes heat at a rate of 150 kJ/s. To determine the minimum rate of heat supply, we must find out how much heat needs to be added to the system in order to maintain this heat removal rate while considering the Carnot efficiency. The formula required to determine the minimum heat supply rate is: \(Q_{supply} = \frac{Q_{removed}}{\eta_{Carnot}}\) Substitute the known values: \(Q_{supply} = \frac{150 \mathrm{kJ/s}}{0.290} = 517.03 \mathrm{kJ/s}\) However, the system also rejects heat to the environment, which should be subtracted from the geothermal heat supply: \(Q_{rejected} = \frac{Q_{removed}}{\eta_{Carnot}} - Q_{removed}\) Substitute the known values: \(Q_{rejected} = 517.03 \mathrm{kJ/s} - 150 \mathrm{kJ/s} = 367.03 \mathrm{kJ/s}\) Finally, the minimum rate of heat supply is the difference between the geothermal heat supply and the rejected heat: \(Q_{min} = Q_{supply} - Q_{rejected} = 517.03 \mathrm{kJ/s} - 367.03 \mathrm{kJ/s} = 150 \mathrm{kJ/s}\) The minimum rate of heat supply is 150 kJ/s, which corresponds to the answer (e) 150 kJ/s.

Key concepts

Understanding Absorption Air-Conditioning Systems

Absorption air-conditioning systems are a type of cooling technology that differs from the standard vapor-compression cycle used in most air conditioners. Instead of using a mechanical compressor to drive the refrigeration cycle, absorption systems use a heat source to operate. This heat, often supplied by natural gas, solar power, or in this exercise, geothermal energy, is used to separate a refrigerant and an absorbent fluid. This separation allows the system to utilize heat to create a cooling effect inside the conditioned space.

One of the main advantages of absorption air-conditioning systems is that they can use waste heat or sustainable heat sources, which can reduce the environmental impact. In our exercise, the heat is drawn from a geothermal source. Understanding the concept of heat removal in this manner is crucial since it directly influences the determination of the efficiency and the subsequent calculations needed to find the rate of heat supply.

Harnessing Geothermal Energy

Geothermal energy is considered a renewable and sustainable resource, which involves harnessing the Earth's internal heat. This energy source can be used for heating and cooling purposes, among other applications. Geothermal power is reliable and has low emissions, making it a popular choice for environmentally conscious operations.

In the context of the exercise, geothermal energy is being utilized as the heat source for the absorption air-conditioning system. The geothermal source's temperature is specified at 140°C, which is quite substantial. By tapping into geothermal heat, the system can achieve the necessary driving force to evaporate the refrigerant without relying on electricity-based compressors. It's critical to consider that the efficiency and practicality of using geothermal energy depend heavily on the temperature of the source and the location.

Calculating the Rate of Heat Supply

The rate of heat supply is a measure of the energy provided to a system over a period of time, often expressed in kilojoules per second (kJ/s). It is vital in determining the amount of energy required to maintain a certain thermal output. In the context of our exercise, the heat is supplied from a geothermal source to the absorption air-conditioning system.

Calculating the minimum rate of heat supply involves utilizing the Carnot efficiency principle, which expresses the theoretical maximum efficiency of a heat engine based on the temperatures of the heat source and the heat sink. Once the efficiency is known, we can determine the amount of heat needed to maintain the system's operation. As we work through the formula outlined in the solution steps, we find that the minimum rate of heat supply corresponds to the heat removed from the conditioned space. In this exercise, the answer reflects the system's equilibrium where the supplied heat equals the heat removal rate to maintain a consistent indoor temperature.