Question

Somebody proposes the following system to cool a house in the summer: Compress the regular outdoor air, let it cool back to the outdoor temperature, pass it through a turbine, and discharge the cold air leaving the turbine into the house. From a thermodynamic point of view, is the proposed system sound?

Short answer

Answer: The proposed system appears to be thermodynamically sound in principle, as it involves adiabatic processes and heat exchange with the surroundings. However, its practicality and efficiency depend on the energy required for the compression process and the actual cooling effect achieved. It may be more efficient to consider conventional cooling methods, such as air conditioning or evaporative cooling.

Step-by-step solution

Understanding the proposed system

First, let's understand the process in the proposed system: 1. The regular outdoor air is compressed, which increases its pressure and temperature. 2. This compressed air cools back to the outdoor temperature by rejecting heat to the surroundings. 3. The cooled compressed air passes through a turbine, where it expands and cools even further. 4. The cold air leaving the turbine is discharged into the house.

Analyzing the compression process

During the compression process, the work is done on the air, increasing the internal energy, and therefore raising the temperature and pressure. This can be described as an adiabatic process (assuming no heat exchange during compression). The relationship between the temperature and pressure for an adiabatic compression can be expressed as: \(T_2 = T_1 * (P_2/P_1)^{(\gamma-1)/\gamma}\) where \(T_1\) is the initial temperature, \(P_1\) is the initial pressure, \(T_2\) is the final temperature, \(P_2\) is the final pressure, and \(\gamma\) is the adiabatic index (which is around 1.4 for air).

Analyzing the cooling process

After compression, the compressed air is allowed to cool back to the outdoor temperature by rejecting heat to the surroundings. This cooling process can be described as an isobaric process (constant pressure). As the temperature decreases, the internal energy of the air also decreases.

Analyzing the expansion process

The cooled compressed air then passes through a turbine, where it expands. This expansion can also be assumed as an adiabatic process. The equation for the relationship between the temperature and pressure for an adiabatic expansion is the same as for the adiabatic compression (Step 2). Since the air expands in the turbine, its pressure and temperature decrease, and work is extracted in the form of mechanical energy. This energy can be used to run the compressor in the first step and other components of the system.

Evaluating the system's efficiency

From the analysis in the previous steps, the air loses energy in the form of heat to the surroundings and mechanical work is extracted from it in the turbine. Thus, the air coming out of the turbine is colder than the regular outdoor air. This seems like a good way to cool the house from a thermodynamic point of view. However, one should consider the energy spent in the compression process. If the energy required to compress the air is greater than the cooling effect provided by the system, then the system would be inefficient and not thermodynamically sound. In conclusion, the proposed system appears to be thermodynamically sound in principle, but its practicality and efficiency would be determined by the energy required for the compression process and the actual cooling effect achieved. It is recommended to consider other more conventional and efficient cooling methods, such as air conditioning or evaporative cooling, for practical applications.