12.7 Air & Gas Compressors: Reciprocating, Rotary & Multi-Stage

Why Compressors Matter in a Mine

Compressed air is the workhorse utility of underground and surface coal operations: rock drills, pneumatic loaders, hoist controls, instrument air and dust-suppression sprays all depend on it. Compressed air is also notoriously inefficient — a large fraction of the input electrical energy is rejected as heat — so understanding where that loss arises is a genuine operational skill, not just an exam topic.

Classification

Work of Compression: Reciprocating, No Clearance

For a polytropic process $pv^{n} = C$, the work per cycle for an ideal compressor without clearance is

$W = \frac{n}{n-1}\,p_1V_1\left[\left(\frac{p_2}{p_1}\right)^{(n-1)/n} - 1\right]$

The index $n$ decides how much work is required:

For isothermal compression the expression degenerates to

$W_{\text{iso}} = p_1V_1\ln\frac{p_2}{p_1}$

Isothermal work is the minimum, and this single fact drives the entire design of compressor cooling. On a $p$-$V$ diagram the isothermal curve lies below the adiabatic, and the area between them is the work saved by cooling.

Isothermal efficiency

$\eta_{\text{iso}} = \frac{\text{Isothermal work}}{\text{Actual indicated work}}$

This is the standard yardstick for a reciprocating compressor, and improving it means improving cooling — water jackets, finned cylinders, and above all intercooling between stages.

Clearance Volume and Volumetric Efficiency

A real cylinder cannot have zero clearance: valves, gaskets and thermal expansion require a small volume $V_c$ at top dead centre. That trapped high-pressure gas re-expands on the suction stroke before the inlet valve can open, so less fresh air is drawn in than the swept volume $V_s$.

Defining the clearance ratio $C = V_c/V_s$, the volumetric efficiency is

$\boxed{\eta_v = 1 + C - C\left(\frac{p_2}{p_1}\right)^{1/n}}$

Two consequences follow directly:

1. $\eta_v$ falls as the pressure ratio rises. At a sufficiently high pressure ratio, the re-expanding clearance gas fills the entire swept volume and $\eta_v$ reaches zero — the compressor delivers nothing at all.
2. $\eta_v$ falls as clearance increases. Hence the practical limit on single-stage pressure ratio, typically about 5 to 8.

Importantly, clearance does not affect the work per unit mass delivered in the ideal analysis, because the work recovered during re-expansion equals that spent compressing the clearance gas. It reduces capacity, not efficiency — a distinction frequently tested.

Worked example. A compressor has 5% clearance and operates at a pressure ratio of 6 with $n = 1.3$.

$\eta_v = 1 + 0.05 - 0.05\times6^{1/1.3} = 1.05 - 0.05\times4.03 = 1.05 - 0.202 = 0.848$

So 84.8%. At a pressure ratio of 20 the same machine would give $1.05 - 0.05\times10.6 = 0.52$, and the case for staging becomes obvious.

Multi-Stage Compression with Intercooling

Splitting the compression into stages with a heat exchanger between them achieves three things at once:

1. Reduced work, by approaching isothermal compression in steps.
2. Improved volumetric efficiency, since each stage has a modest pressure ratio.
3. Lower delivery temperature, protecting lubricant and valves.

Optimum intermediate pressure

For a two-stage machine with perfect intercooling — that is, the gas returns to its original suction temperature between stages — the total work is

$W = \frac{n}{n-1}\,RT_1\left[\left(\frac{p_i}{p_1}\right)^{(n-1)/n} + \left(\frac{p_2}{p_i}\right)^{(n-1)/n} - 2\right]$

Differentiating with respect to $p_i$ and setting the result to zero gives

$\boxed{p_i = \sqrt{p_1 p_2}}$

The optimum intermediate pressure is the geometric mean of suction and delivery pressures, which means both stages operate at the same pressure ratio. For $z$ stages the generalisation is

$\text{Pressure ratio per stage} = \left(\frac{p_{z+1}}{p_1}\right)^{1/z}$

Under these conditions each stage does equal work, and the total is

$W = z\,\frac{n}{n-1}RT_1\left[\left(\frac{p_{z+1}}{p_1}\right)^{(n-1)/nz} - 1\right]$

Worked example. Air is compressed from 1 bar to 25 bar in two stages with perfect intercooling. The optimum intermediate pressure is

$p_i = \sqrt{1\times25} = 5 \text{ bar}$

so each stage works at a pressure ratio of 5.

Perfect versus imperfect intercooling

Perfect intercooling returns the gas exactly to the initial temperature; on a $p$-$V$ diagram the second-stage suction point falls on the original isothermal through the suction state. Imperfect intercooling leaves the gas warmer, so the second stage draws in a larger volume and the work saving is reduced. Aftercooling, downstream of the last stage, removes moisture but saves no compression work.

Rotary Compressors

Screw compressor

Two helical rotors mesh so that trapped pockets of gas are progressively reduced in volume as they travel axially. Screw compressors dominate industrial and mining plant air because they deliver a continuous, pulsation-free flow, tolerate dusty environments, and have few wearing parts. Oil-flooded designs inject oil for sealing, cooling and lubrication, achieving compression close to isothermal in a single stage.

Roots blower

Two lobed rotors transfer gas without reducing its volume internally — compression occurs entirely by back-flow when the discharge port opens. There is no internal compression, so the process is far from isothermal and efficiency is poor at any significant pressure ratio. Roots blowers are used where large volumes at low pressure are needed, such as ventilation boosting.

Centrifugal and axial compressors

These are dynamic machines: energy is added as kinetic energy by an impeller or rotor and converted to pressure in a diffuser or stator. Their analysis follows Euler's turbomachine equation. Two characteristic instabilities must be recognised:

• Surge: a system-wide flow oscillation and reversal that occurs when flow is throttled below a critical value. It is violent and can destroy the machine.
• Choking: the limiting maximum flow, reached when the velocity at some section becomes sonic, so no further increase in flow is possible.

Efficiencies to Distinguish

A point worth remembering: isothermal efficiency is the standard for reciprocating machines with cooling, whereas isentropic efficiency is the standard for uncooled dynamic machines, because the appropriate ideal differs between the two.