Where does the 2.31 coefficient come from in the simplified compressor power formula?

The coefficient 2.31 is derived from combining the universal gas constant R (8.314 J/K/mol) with unit conversion factors from kg/s to t/h (dividing by 3600 and multiplying by 1000). Specifically, 8.314 / 3.6 = 2.3094, which rounds to 2.31.

What is the difference between general adiabatic head calculation and the simplified temperature-based formula?

The general formula relies on pressure ratios, compressibility factor Z, and gas isentropic coefficient k. The simplified formula substitutes pressure ratios with inlet and discharge temperature changes under perfect gas assumptions (Z = 1), making it extremely practical when suction and discharge temperatures are known.

Compressor power formula : step by step explanations

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Section summary
1. Compressor power simplified formula
2. Explanation of the formula : step by step

1. Compressor power simplified formula

In MyEngineeringTools page dedicated to compressor power calculation, there are 2 formula given, a general formula, and a simplified one. The simplified one is the following, for 1 compressor stage of a perfect gas, the isentropic compression is :

P_is = 2.31*(k/(k-1))*(T_dis-T_suct)/M*Q_m

\[ P_{is} = 2.31 \frac{k}{k-1} \frac{T_{dis} - T_{suct}}{M} Q_m \]

Calculation of power required for a compressor

Equation 1 : simplified compression power calculation formula

With : P_is=Power (kW)
T_suct=Temperature inlet compressor (K)
T_discharge=Temperature outlet compressor (K)
M=Molar weight of gas (g/mol)
Q_m=Compressor throughput (t/h)
k=Gas isentropic coefficient

⚠️ ENGINEERING NOTICE & EDUCATIONAL DISCLAIMER: This interactive calculator is provided exclusively for preliminary estimation and educational purposes. It is not intended for detailed design or equipment procurement without certified vendor rating. No warranty, expressed or implied, is provided, and no liability is assumed.

Interactive Compressor Power Calculator (Simplified Method)

Calculate isentropic compression power instantly. Toggle units between SI Metric and US Customary.

Unit System:

Inlet Temperature ($T_{suct}$) [°C]:

Discharge Temperature ($T_{dis}$) [°C]:

Molar Weight ($M$) [g/mol]:

Mass Throughput ($Q_m$) [t/h]:

Isentropic Coefficient ($k = C_p/C_v$):

Calculation Results

Inlet Temperature ($T_{suct}$): 298.15 K

Discharge Temperature ($T_{dis}$): 393.15 K

Temperature Elevation ($\Delta T$): 95.00 K

Mass Throughput ($Q_m$): 5.00 t/h (1.389 kg/s)

Isentropic Compression Power ($P_{is}$): -

🌿 Process Engineering Rules of Thumb & Safety Limits

Stage Temperature Limit: For reciprocating and centrifugal compressors, discharge temperature should generally not exceed 135°C to 150°C (275°F to 302°F) to prevent lubricating oil degradation, polymer formation, or valve coking.
Pressure Ratio per Stage: To maintain high adiabatic efficiency and avoid excessive discharge temperatures, single-stage pressure ratios ($P_2/P_1$) are typically limited to 3.0 - 4.0 for air/hydrocarbons.
Polytropic vs. Isentropic: The simplified formula assumes isentropic ($s = \text{const}$) compression with an ideal gas ($Z = 1$). Real gases require compressibility factor corrections ($Z \neq 1$) and polytropic efficiency adjustments.

Now where this formula is coming from, where the 2.31 coefficient is coming from ? As we received a lot of questions on this matter, we decided to propose a dedicated page explaining how to reach this simplified expression, from the general one, as usual in a step by step approach which is the trade mark of MyEngineeringTools.com.

2. Explanation of the formula : step by step

STEP 1 : general equation

We need to start from a more general equation for isentropic compressor power calculation. According to [Perry], the adiabatic head of a compressor is given as :

H_ad = (k*Z*R*T₁)/(k-1)*[(P₂/P₁)^(k-1)/k-1]

\[ H_{ad} = \frac{k \cdot Z \cdot R \cdot T_1}{k-1} \left[ \left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} - 1 \right] \]

With : H_ad= Adiabatic head (N.m/kg)
Z = gas compressibility factor (can be defined on an Amagat diagram by calculating the reduced pressure and reduced temperature of the gas)
P₁ = Pressure inlet compressor (kPa)
P₂ = Pressure outlet compressor (kPa)
k=Gas isentropic coefficient
R = 8314/molecular weight (J/(kg.K))

The work required during the compression is equal to the adiabatic head multiplied by the mass flow rate of gas, and divided by 1000 in order to express it in kW :

P_is (kW) = H_ad * Q_m / 1000

\[ P_{is} (\text{kW}) = \frac{H_{ad} \cdot Q_m}{1000} \]

With : P_is=Power (kW)
H_ad= Adiabatic head (N.m/kg)
Q_m=Compressor throughput (kg/s)
1000 W/kW

This formula is also given on the other page [Perry] :

P_is = (k*Z*R*T₁)/(k-1)*[(P₂/P₁)^(k-1)/k-1]*Q_m

\[ P_{is} = \frac{k \cdot Z \cdot R \cdot T_1}{k-1} \times \left[ \left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} - 1 \right] \times Q_m \]

Equation 2 : general compression power calculation formula [Perry]

With : P_is=Power (W)
Z = gas compressibility factor (can be defined on an Amagat diagram by calculating the reduced pressure and reduced temperature of the gas)
P₁ = Pressure inlet compressor (kPa)
P₂ = Pressure outlet compressor (kPa)
Q_m=Compressor throughput (kg/s)
k=Gas isentropic coefficient
R = [8314/molecular weight (J/(kg.K))]/1000 = 8.314/molecular weight

STEP 2 : Assumptions and simplifications

The general expression is using the gas compressibility factor Z, however in the simplified version we assumed that the gas is a perfect gas, as a consequence the factor is assumed to be 1 : Z = 1

R is actually equal to 8.314 (J/K/kmol) / M (kg/kmol) (see above), it can thus be replaced in the formula :

\[ P_{is} = \frac{8.314}{M} \times \frac{k \cdot T_1}{k-1} \times \left[ \left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} - 1 \right] \times Q_m \]

STEP 3 : Introducing isentropic compression

It is possible, via isentropic compression, to relate the pressure change to the temperature change :

\[ \frac{T_2}{T_1} = \left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}}, \quad k = \frac{C_p}{C_v} \]

Calculation Temperarure Elevation in Compressor

It can then be rearranged :

\[ P_{is} = \frac{8.314}{M} \times \frac{k \cdot T_1}{k-1} \times \left[ \frac{T_2}{T_1} - 1 \right] \times Q_m \]

\[ P_{is} = 8.314 \times \frac{k}{k-1} \times \left[ \frac{T_2 - T_1}{M} \right] \times Q_m \]

STEP 4 : Convert mass flow to t/h

The simplified formula is based on t/h while the general formula is considering kg/s.

We can then convert :

Qm_kgs / 1000 * 3600 = Qm_th

Qm_kgs = Qm_th * 1000 / 3600

\[ P_{is} = 8.314 \times \frac{k}{k-1} \times \left[ \frac{T_2 - T_1}{M} \right] \times \left( \frac{Q_{m,th} \cdot 1000}{3600} \right) \]

Then 8314 / 3600 = 2.31, gives :

\[ P_{is} = 2.31 \times \frac{k}{k-1} \times \left[ \frac{T_2 - T_1}{M} \right] \times Q_{m,th} \]

Which is the simplified expression given above.

Source

[Perry] Perry's Chemical Engineer's Handbook, Section 10 Transport and storage of fluids, page 10-45, McGraw-Hill, 2008