Question

For minimum compression work in a 2 - stage reciprocating air compressor, which of the following expressions gives the ratio of low pressure cylinder to high pressure cylinder diameters?

Where P₁ P₂ and P₃ are suction, intermediate and delivery pressures respectively.

1. \({\left( {\frac{{{P_3}}}{{{P_1}}}} \right)^{\frac{1}{4}}}\)
2. \({\left( {\frac{{{P_3}}}{{{P_1}}}} \right)^{\frac{1}{3}}}\)
3. \({\left( {\frac{{{P_3}}}{{{P_1}}}} \right)^{\frac{1}{2}}}\)
4. \({\left( {\frac{{{P_3}}}{{{P_2}}}} \right)^{\frac{1}{4}}}\)

Answer 1

Correct Answer - Option 3 : \({\left( {\frac{{{P_3}}}{{{P_1}}}} \right)^{\frac{1}{2}}}\)

Concept:

In a two-stage compressor there is intercooling between the suction and delivery pressure at constant pressure by this intercooling some reduction in work supplied to the compressor another advantage of using multi-stage compression is that greater compression can be achieved compared to the single-stage compression.

For intercooling between the stages using one intercooler the perfect conditions are

\({P_i} = \sqrt {{P_H}{P_L}} \)

Where P_L and P_H are the suction and delivery pressures respectively.

Cylinder dimensions in multistage compression:

From mass conservation the mass flow rate of air across all the cylinders will be same, for perfect intercooling, if all the cylinders have same stroke length and volumetric efficiency then

\({P_1}D_1^2 = {P_2}D_2^2 = {P_3}D_3^2\)

Where P is the pressure and the D is the diameter

Calculation:

Let us consider D₃ as the diameter of high pressure cylinder and D_1­ as the diameter of low pressure cylinder then,

\({P_1}D_1^2 = {P_3}D_3^2\)

⇒ \(\frac{{{D_1}}}{{{D_3}}} = {\left( {\frac{{{P_3}}}{{{P_1}}}} \right)^{\frac{1}{2}}}\)