Editing Compressor (section)

===Minimizing work required by a compressor===

====Comparing reversible to irreversible compressors====

Comparison of the differential form of the energy balance for each device.

Let <math> q </math> be heat, <math> w </math> be work, <math> ke </math> be kinetic energy, and <math> pe </math> be potential energy.

Actual Compressor:<br />
: <math>  \delta q_{act} - \delta w_{actelta q_{act}}{T} \geq 0</math>
<br />Furthermore, <math> ds \geq \frac{\delta q_{act}}{T} </math> and T is [absolute temperature] (<math> T \geq 0 </math>) which produces:<br /><math>   \delta w_{rev} \geq \delta w_{act} </math>

or<br /><math>   w_{rev} \geq w_{act} </math>

Therefore, work-consuming devices such as pumps and compressors (work is negative) require less work when they operate reversibly.<ref name="Cengel, Yunus A. 2012"/>

====Effect of cooling during the compression process====
[[File: Insetropic,polytropic,isothermal.jpg|thumb|P-v (Specific volume vs. Pressure) diagram comparing isentropic, polytropic, and isothermal processes between the same pressure limits.]]

[[isentropic]] process: involves no cooling,<br />
[[polytropic]] process: involves some cooling<br />
[[isothermal]] process: involves maximum cooling

By making the following assumptions the required work for the compressor to compress a gas from <math>P_1 </math> to <math> P_2</math> is the following for each process:
<br />













: <math>P_1 </math> and <math> P_2</math>

: Flow processes VdP
: All processes are internally reversible
: The gas behaves like an [[ideal gas]] with constant [[specific heats]]

Isentropic (<math> Pv^k = constant </math>, where <math>k = C_p/C_v</math>):
<br />
: <math> W_{comp,in}= \frac{kR(T_2-T_1)}{k-1}=\frac{kRT_1}{k-1} \left [  \left ( \frac{P_2}{P_1} \right ) ^{(k-1)/k} -1 \right ] </math>

Polytropic (<math> Pv^n = constant </math>):
<br />
: <math> W_{comp,in}= \frac{nR(T_2-T_1)}{n-1}=\frac{nRT_1}{n-1} \left [  \left ( \frac{P_2}{P_1} \right ) ^{(n-1)/n} -1 \right ] </math>

Isothermal (<math>T = constant</math> or <math>Pv = constant</math>):
<br />
:<math> W_{comp,in}= RT ln \left (  \frac{P_2}{P_1}\right ) </math>

By comparing the three internally reversible processes compressing an ideal gas from <math>P_1 </math> to <math> P_2</math>, the results show that isentropic compression (<math> Pv^k = constant </math>) requires the most work in and the isothermal compression(<math>T = constant</math> or <math>Pv = constant</math>) requires the least amount of work in. For the polytropic process (<math> Pv^n = constant </math>) work decreases as the exponent, n, decreases, by increasing the heat rejection during the compression process. One common way of cooling the gas during compression is to use cooling jackets around the casing of the compressor.<ref name="Cengel, Yunus A. 2012"/>