Portable compressor power calculation

[joe609]

Hello

I have been trying to calculate the theoretical power required for compression of a small portable 2 stage compressor with the following spec.

Volume flow rate Q = 2x10^-5 m³/s
Inlet pressure P₁ = 1 Bar (10⁵Pa)
Outlet pressure = 200 Bar
interstage pressure P_x = 14 Bar
k = 1.4
N = number of stages
Motor power = 400 w

Using the equation for power during adiabatic compression

$$\large \large p=\frac{k}{k-1}P_{1}QN \begin{bmatrix} \left ( \frac{P_{x}}{P_{1}} \right )^{\left ( \frac{k-1}{k} \right )} & -1 \end{bmatrix}$$

$$\large \large p=\frac{1.4}{1.4-1}10^{5} \times 2\times 10^{-5} \times 2 \begin{bmatrix} 14^{0.3}-1 & \end{bmatrix}$$


I get a theoretical power of 16 watts. Why the huge discrepancy between the theoretical and the actual power required? Is my calculation correct? I am aware there are mechanical losses and inefficiency's in the pump but are they so great as to require 25 x the power?
Thanks very much for any help.

 

[darkside00]

What is the source of that equation? you can't even run a typical light bulb on 16 watts

 

[Mech_Engineer]

The flow rate of the compressor is very low, keeping the theoretical power requirements low. Calculating the power required based on isentropic enthalpy change between the initial and final states results in about 30 watts, and a final temperature of 1363 K (assuming the input temperature is 300 K). The efficiency of such a compressor would be low because a lot of energy is lost to heat; cooling the output air from 1363 K back down to 300K is lost energy.