Question

Consider isentropic compressor of a vaporcompression refrigeration cycle. What are the isentropic efficiency and second-law efficiency of this compressor? Justify your answers. Is the second-law efficiency of a compressor necessarily equal to its isentropic efficiency? Explain.

Short answer

Explain. Answer: Yes, in the case of an isentropic compressor, the isentropic efficiency and the second-law efficiency are equal to each other, and both are 100%. In an isentropic compressor, the actual work input is equal to the isentropic work input as there are no losses or irreversibilities. Keep in mind, this equality is valid for the theoretical case of an isentropic compressor. However, in real-world compressors, the isentropic and second-law efficiencies are generally not equal, as they involve losses and irreversibilities.

Step-by-step solution

Define isentropic efficiency

The isentropic efficiency of a compressor is defined as the ratio of the actual work input required to compress the refrigerant to the ideal (isentropic) work input. This efficiency takes into account the losses due to irreversibilities in the compression process. Isentropic efficiency (η_isentropic) = (Isentropic work input) / (Actual work input)

Define second-law efficiency

The second-law efficiency of a compressor is defined as the ratio of the minimum theoretical work input required to achieve the desired effect (increase in pressure and temperature of the refrigerant) to the actual work input. Second-law efficiency (η_second-law) = (Minimum theoretical work input) / (Actual work input)

The isentropic efficiency

Since we are considering an isentropic compressor, which is a theoretical case without any friction or other losses, the actual work input will be equal to the isentropic work input. Consequently, the isentropic efficiency will be equal to 1 (or 100%). η_isentropic = (Isentropic work input) / (Actual work input) = (Isentropic work input) / (Isentropic work input) = 1

The second-law efficiency

In an isentropic compressor, the actual work input is equal to the isentropic work input. Therefore, the second-law efficiency of the compressor will also be equal to 1 (or 100%), because the minimum theoretical work input is equal to the actual work input. η_second-law = (Minimum theoretical work input) / (Actual work input) = (Isentropic work input) / (Isentropic work input) = 1

Comparing isentropic and second-law efficiencies

In the case of an isentropic compressor, the isentropic efficiency and the second-law efficiency are equal to each other, and both are 100%. This equality is valid for the isentropic compressor, where the compression process ideally doesn't involve any losses or irreversibilities. However, in real-world compressors, there are always some losses and irreversibilities, leading to a decrease in both isentropic and second-law efficiencies. Therefore, for a real compressor, the second-law efficiency is generally not equal to its isentropic efficiency. The second-law efficiency is often higher than the isentropic efficiency because the actual work input includes losses that are not considered in the minimum theoretical work input.