# Compressor Head and Power: Basic Equation

** Theoretical Work**

Theoretical work or compressor head is the substance in the compressor design. General energy equation is used in deriving the head equation as follows:

As elevation differences are not significant with gas, the z term can be eliminated, and then the equation will be as follows:

The velocity can be considered as part of the enthalpy if the enthalpy is defined as the stagnation or total enthalpy. Then the equation will be as follows:

h2 - h1 = -W + Qh

If the process is adiabatic (no heat transfer) then Qh=0. Enthalpy Equation in the differential form as follows:

dH = du + Pdv + vdP

According to second law of thermodynamics, du = Tds – Pdv, so the equation:

dH = Tds + vdP

If the process is assumed reversible, so the entropy as constant and therefore ds = 0 and Tds = 0. The equation will be:

dH = vdP

For isentropic (reversible adiabatic and constant entropy) will be as follows:

Pvk = constant = C

P = Cvk

dP = C(-k)v-k-1dv

Substituting in to enthalpy equation:

dH = C(-k)v-kdv

The next step integrates from state point 1 to 2 and assuming k constant over the path yields,

Substitute equation C = P1v1k = Pv2k in the equation dH = C(-k)v-kdv, will be as follows:

According to perfect gas law equation Pv = RT, and substituting to the equation above will be as follows:

By regrouping the above equation and substituting in the energy equation with Qh = 0, the equation is developed as follows:

According to equation C = P1v1k = Pv2k and perfect gas equation below

We find that T2/T1 = rpk-1/k, Where rp = P2/P1

Substituting the equation above would give the equation as follows:

According to equation 4 and according to compressibility factor Pv = ZRT, then equation will be

Since the compressibility does not change the isentropic temperature rise, it can be out form ΔT portion of equation. Assume that Zavg = (Z2 + Z1)/2, the equation can be written as follows:

The same process to obtain equation 8, the head equation with compressibility will be as follows:

For polytropic (reversible) process, the polytropic exponent shall be used which related to isentropic process as follows:

So, by substituting n for k, the head equation for polytropic process will be as follows:

Please note that R in this equation is specific gas constant, so if using universal gas constant Ru shall be divided by molecular mass M as per equation below:

R = Ru/M (Equation 14)

**Power**

Input shaft power is the head of the compressor multiplied by weight flow and devided by the efficiency and then the result is added by mechanical losses. In general the equation will be as follows;

For isentropic process:

For polytropic process: