Editing Compressor (section)

==Temperature==
{{main|Gas laws}}
Compression of a gas increases its [[temperature]].

For a [[polytropic process|polytropic transformation]] of a gas:

: <math>\begin{cases}
pV^n=\text{constant}=p_1V_1^n=p_2V_2^n \Rightarrow \frac { p_2 }{ p_1 }\ = \left( \frac{ V_1 } { V_2 } \right) ^ n & \\
\frac{p^{\frac{n-1}{n}}}{T}=\text{constant}=\frac{p_1^{\frac{n-1}{n}}}{T_1}=\frac{p_2^{\frac{n-1}{n}}}{T_2} \Rightarrow \left( \frac{ p_2 } { p_1 } \right) ^ \frac{n-1}{n} = \frac{ T_2 } { T_1 } &
\end{cases}</math>

The work done for [[polytropic process|polytropic compression]] (or expansion) of a gas into a closed cylinder.
: <math>W = \int_{V_1}^{V_2} p dV = p_1 V_1^n \int_{V_1}^{V_2} V^{-n} dV = \frac{p_1 V_1^n}{1-n} (V_2^{1-n} - V_1^{1-n}) = \frac{p_1 V_1^n}{1-n} V_1^{1-n} \left( \frac{V_2^{1-n}}{V_1^{1-n}} -1 \right) = \frac{p_1 V_1}{1-n} \left( \frac{V_2^{1-n}}{V_1^{1-n}} -1 \right) = </math>

: <math> = \frac{p_1 V_1}{1-n} \left( \left( \frac{V_1}{V_2} \right)^{n-1} -1 \right) = \frac{p_1 V_1}{1-n} \left( \left( \frac{p_2}{p_1} \right)^{\frac{n-1}{n}} -1 \right) = \frac{p_1 V_1}{1-n} \left( \frac{T_2}{T_1} -1 \right)</math>

so
: <math>W = - \frac{p_1 V_1}{n-1} \left( \left( \frac{p_2}{p_1} \right)^{\frac{n-1}{n}} -1 \right) </math>

in which ''p'' is pressure, ''V'' is volume, ''n'' takes different values for different compression processes (see below), and 1 & 2 refer to initial and final states.

{{anchor|Adiabatic}}
* [[Adiabatic process|Adiabatic]] – This model assumes that no energy (heat) is transferred to or from the gas during the compression, and all supplied work is added to the internal energy of the gas, resulting in increases of temperature and pressure. Theoretical temperature rise is:<ref>''Perry's Chemical Engineer's Handbook'' 8th edition
Perry, Green, page 10-45 section 10-76</ref>
: <math> T_2 = T_1 \left(\frac {p_2}{p_1}\right)^{(\kappa-1)/\kappa} </math>
with ''T''<sub>1</sub> and ''T''<sub>2</sub> in degrees [[Rankine scale|Rankine]] or [[kelvin]]s, ''p''<sub>2</sub> and ''p''<sub>1</sub> being absolute pressures and <math>\kappa = </math> [[Heat capacity ratio|ratio of specific heat]]s (approximately 1.4 for air).  The rise in air and temperature ratio means compression does not follow a simple pressure to volume ratio. This is less efficient, but quick. Adiabatic compression or expansion more closely model real life when a compressor has good insulation, a large gas volume, or a short time scale (i.e., a high power level). In practice there will always be a certain amount of heat flow out of the compressed gas. Thus, making a perfect adiabatic compressor would require perfect heat insulation of all parts of the machine. For example, even a bicycle tire pump's metal tube becomes hot as you compress the air to fill a tire. The relation between temperature and compression ratio described above means that the value of <math>n</math> for an adiabatic process is <math>\kappa</math> (the ratio of specific heats).

{{anchor|Isothermal}}
* [[Isothermal process|Isothermal]] – This model assumes that the compressed gas remains at a constant temperature throughout the compression or expansion process. In this cycle, internal energy is removed from the system as heat at the same rate that it is added by the mechanical work of compression. Isothermal compression or expansion more closely models real life when the compressor has a large heat exchanging surface, a small gas volume, or a long time scale (i.e., a small power level). Compressors that utilize inter-stage cooling between compression stages come closest to achieving perfect isothermal compression. However, with practical devices perfect isothermal compression is not attainable. For example, unless you have an infinite number of compression stages with corresponding intercoolers, you will never achieve perfect isothermal compression.

For an isothermal process, <math>n</math> is 1, so the value of the work integral for an isothermal process is:
: <math> W = \int_{V_1}^{V_2} p dV = p_1 V_1 \int_{V_1}^{V_2} \frac{1}{V} dV =  p_1 V_1 \ln \frac{V_2}{V_1} = - p_1 V_1 \ln \left( \frac {p_2} {p_1} \right) </math>

When evaluated, the isothermal work is found to be lower than the adiabatic work.

{{anchor|Polytropic}}
* [[Polytropic process|Polytropic]] – This model takes into account both a rise in temperature in the gas as well as some loss of energy (heat) to the compressor's components. This assumes that heat may enter or leave the system, and that input shaft work can appear as both increased pressure (usually useful work) and increased temperature above adiabatic (usually losses due to cycle efficiency). Compression efficiency is then the ratio of temperature rise at theoretical 100 percent (adiabatic) vs. actual (polytropic). Polytropic compression will use a value of <math>n</math> between 0 (a constant-pressure process) and infinity (a constant volume process). For the typical case where an effort is made to cool the gas compressed by an approximately adiabatic process, the value of <math>n</math> will be between 1 and <math>\kappa</math>.