Understanding Multi-Stage Air Compressor Pressure

This question involves calculating the ideal intermediate pressure in a multi-stage air compressor. A multi-stage compressor increases pressure in steps (stages) using multiple cylinders, typically with decreasing diameter and increasing rotational speed. The goal is to achieve the final delivery pressure more efficiently and manage the heat generated during compression.

Ideal Pressure Ratio Concept

For optimal efficiency in an ideal multi-stage compressor (ignoring losses), the pressure ratio across each stage should be equal. The pressure ratio ($r_p$) is the ratio of the delivery pressure to the suction pressure for a single stage.

Let:

• $P_1$ be the initial suction pressure.
• $P_{delivery}$ be the final delivery pressure.
• $n$ be the number of stages.
• $r_p$ be the ideal pressure ratio per stage.
• $P_{int}$ be the ideal intermediate pressure at the end of the first stage.

The overall pressure ratio is given by:

$$ \frac{P_{delivery}}{P_1} = (r_p)^n $$

Calculating the Pressure Ratio

Given:

• Suction Pressure ($P_1$) = 1 bar
• Delivery Pressure ($P_{delivery}$) = 125 bar
• Number of stages ($n$) = 3

First, we find the overall pressure ratio:

$$ \text{Overall Ratio} = \frac{125 \text{ bar}}{1 \text{ bar}} = 125 $$

Now, we relate the overall ratio to the per-stage ratio:

$$ 125 = (r_p)^3 $$

To find the ideal pressure ratio per stage ($r_p$), we take the cube root of 125:

$$ r_p = \sqrt[3]{125} $$

$$ r_p = 5 $$

So, the ideal pressure ratio for each stage is 5.

Determining Intermediate Pressure

The intermediate pressure at the end of the first stage ($P_{int}$) is calculated by multiplying the initial suction pressure ($P_1$) by the pressure ratio per stage ($r_p$):

$$ P_{int} = P_1 \times r_p $$

Substituting the values:

$$ P_{int} = 1 \text{ bar} \times 5 $$

$$ P_{int} = 5 \text{ bar} $$

Verification (Optional)

We can verify this by calculating the pressure at the end of the second stage:

$$ P_{int2} = P_{int} \times r_p = 5 \text{ bar} \times 5 = 25 \text{ bar} $$

And finally, the delivery pressure:

$$ P_{delivery} = P_{int2} \times r_p = 25 \text{ bar} \times 5 = 125 \text{ bar} $$

This matches the given delivery pressure, confirming our calculation.

Conclusion

The ideal intermediate pressure at the end of the first stage for a 3-stage air compressor with a suction pressure of 1 bar and a delivery pressure of 125 bar is 5 bar.