For greater efficiency air compression should be isothermal as this requires the minimum work input.
In practice Isothermal compression is not possible, an ideal Isothermal cycle requires sufficient time to allow all the required heat to be transferred out of the cylinder, practicality dictates that the piston must have a relatively high speed to give a reasonable output, Cylinder cooling on a single stage compressor gives better efficiency but there is a limitation in the surface area to cylinder volume that can be used for cooling effect, but multistage compressors with an efficient extended surface inter stage cooler gives cycle improved compression efficiency better approaching that of the isothermal.
In theory the greater the number of stages the closer the curve will approach the ideal isothermal compression curve, however there is an increase in cost, complexity, and the law of diminishing returns limit the number.
In practice Isothermal compression is not possible, an ideal Isothermal cycle requires sufficient time to allow all the required heat to be transferred out of the cylinder, practicality dictates that the piston must have a relatively high speed to give a reasonable output, Cylinder cooling on a single stage compressor gives better efficiency but there is a limitation in the surface area to cylinder volume that can be used for cooling effect, but multistage compressors with an efficient extended surface inter stage cooler gives cycle improved compression efficiency better approaching that of the isothermal.
In theory the greater the number of stages the closer the curve will approach the ideal isothermal compression curve, however there is an increase in cost, complexity, and the law of diminishing returns limit the number.