1. Overview
Volumetric efficiency (eta_v) is the ratio of actual gas volume delivered per stroke to the cylinder swept volume. It determines how much of the cylinder's displacement is productively used for gas compression and delivery. Understanding eta_v is essential for cylinder sizing, capacity prediction, and performance monitoring.
Definition
eta_v = V_actual / V_swept
Fraction of swept volume delivering gas
Capacity
Q = PD * eta_v
Actual flow = displacement x efficiency
Ideal (Cl=0)
eta_v = 100%
No clearance = all gas delivered (theoretical)
Zero Capacity
eta_v = 0%
Re-expansion fills entire stroke
Why Volumetric Efficiency Matters
| Impact Area | How eta_v Affects It | Consequence of Error |
|---|---|---|
| Cylinder sizing | PD = Q_required / eta_v | Undersized cylinder = insufficient capacity |
| Capacity prediction | Q_actual = PD * eta_v at conditions | Incorrect production forecasts |
| Power calculation | BHP proportional to mass flow | Wrong driver sizing |
| Capacity control range | eta_v decreases with added clearance | Turndown range miscalculation |
| Performance monitoring | Declining eta_v indicates wear | Missed maintenance opportunity |
Common confusion: Volumetric efficiency is NOT the same as isentropic (thermodynamic) efficiency. Volumetric efficiency measures capacity loss from the cylinder. Isentropic efficiency measures power loss from irreversibilities. A compressor can have 85% volumetric efficiency and 82% isentropic efficiency simultaneously.
2. Clearance Re-expansion
Clearance volume is the gas space remaining when the piston is at top dead center. This trapped gas must re-expand to suction pressure before the suction valve opens, consuming part of the stroke and reducing the volume available for new gas intake.
GPSA Volumetric Efficiency Equation
Theoretical volumetric efficiency (clearance effect only):
eta_v_theoretical = 1 - Cl * [r^(1/k) - 1]
Where:
Cl = Fractional clearance = V_clearance / V_swept
r = Compression ratio = P_discharge / P_suction (absolute)
k = Specific heat ratio (Cp/Cv)
Physical interpretation:
1 = Full swept volume available (100%)
Cl * [r^(1/k) - 1] = Fraction lost to re-expansion
The term r^(1/k) represents how much the clearance volume
expands when pressure drops from P_d to P_s:
V_4 = V_cl * (P_d / P_s)^(1/k) = V_cl * r^(1/k)
V_4 / V_cl = r^(1/k)
The lost volume is V_4 - V_cl = V_cl * [r^(1/k) - 1]
As a fraction of swept: Cl * [r^(1/k) - 1]
Clearance Effect on eta_v
| Clearance (Cl) | r = 1.5 | r = 2.0 | r = 3.0 | r = 4.0 | r = 5.0 |
|---|---|---|---|---|---|
| 5% | 98.3% | 96.2% | 92.7% | 89.4% | 86.5% |
| 10% | 96.6% | 92.4% | 85.4% | 78.9% | 73.1% |
| 15% | 94.9% | 88.6% | 78.1% | 68.3% | 59.6% |
| 20% | 93.2% | 84.9% | 70.7% | 57.8% | 46.1% |
| 30% | 89.7% | 77.3% | 56.1% | 36.7% | 19.2% |
| 40% | 86.3% | 69.7% | 41.5% | 15.6% | 0%* |
*Values shown for k = 1.27 (natural gas). Values below 40% are impractical.
Maximum Ratio for Given Clearance
At eta_v = 0 (zero capacity):
1 - Cl * [r_max^(1/k) - 1] = 0
r_max^(1/k) = 1 + 1/Cl
r_max = (1 + 1/Cl)^k
Example (k=1.27):
Cl = 10%: r_max = (1 + 10)^1.27 = 11^1.27 = 18.5
Cl = 20%: r_max = (1 + 5)^1.27 = 6^1.27 = 8.8
Cl = 30%: r_max = (1 + 3.33)^1.27 = 4.33^1.27 = 5.9
Cl = 40%: r_max = (1 + 2.5)^1.27 = 3.5^1.27 = 4.5
Note: These are theoretical maximums. Practical operation
requires eta_v > 40%, so usable ratios are much lower.
Design guideline: For initial sizing, use a clearance of 10-15% for moderate ratios (r < 3) and 8-12% for higher ratios. Additional clearance can always be added via pockets for capacity control, but reducing fixed clearance requires cylinder modification.
3. Loss Factors
In addition to clearance re-expansion, several other factors reduce volumetric efficiency. The overall equation accounts for all these losses.
Complete Volumetric Efficiency Equation
Overall volumetric efficiency (GPSA):
eta_v = 1 - Cl * [r^(1/k) - 1] - L_v - L_r - L_g
Where:
Cl * [r^(1/k) - 1] = Clearance re-expansion loss
L_v = Valve loss factor (0.02-0.06)
L_r = Piston ring leakage factor (0.01-0.05)
L_g = Gas heating/compressibility factor (0.00-0.03)
Combined loss factor (simplified):
eta_v = 1 - Cl * [r^(1/k) - 1] - L_total
Where L_total = L_v + L_r + L_g = 0.03-0.12
Typical L_total values by machine condition:
New, well-maintained: L_total = 0.03-0.05
Normal operation: L_total = 0.05-0.08
Aged, worn: L_total = 0.08-0.12
Individual Loss Factors
| Loss Factor | Symbol | Range | Cause | How to Reduce |
|---|---|---|---|---|
| Valve losses | L_v | 0.02-0.06 | Pressure drop through suction valves reduces effective suction pressure | Larger valves, poppet type, lower speed |
| Ring leakage | L_r | 0.01-0.05 | Gas leaks past piston rings from high to low pressure | Replace worn rings, proper ring material |
| Gas heating | L_g | 0.00-0.03 | Gas heats up in cylinder before suction valve closes, reducing density | Cooling, lower clearance, higher speed |
| Moisture content | - | 0.00-0.02 | Condensation reduces effective gas volume | Adequate suction scrubbing |
Valve Loss Detail
Suction valve pressure drop effect:
The effective suction pressure in the cylinder is:
P_cyl_suction = P_line - delta_P_sv
This increases the effective compression ratio:
r_eff = P_d / (P_s - delta_P_sv) > r_nominal
And the effective clearance expansion factor:
r_eff^(1/k) > r^(1/k)
Approximate valve loss on eta_v:
L_v = Cl * [(r_eff^(1/k) - 1) - (r^(1/k) - 1)]
+ delta_P_sv / P_s
For typical suction valve drop of 2% of P_s:
L_v ~ 0.02 + Cl * [small correction] ~ 0.02-0.04
Discharge valve drop does NOT directly affect eta_v
(but increases power and discharge temperature).
Compressibility Factor (Z) Effect
When Z changes between suction and discharge:
The re-expansion follows actual gas behavior, not ideal PV^k.
The corrected volumetric efficiency:
eta_v = 1 - Cl * [(Z_s / Z_d) * r^(1/k) - 1] - L_total
Where:
Z_s = Compressibility at suction
Z_d = Compressibility at discharge
If Z_s > Z_d (common at moderate pressures):
(Z_s/Z_d) > 1 and eta_v is LOWER than calculated with Z=1
If Z_s < Z_d (high pressures near critical):
(Z_s/Z_d) < 1 and eta_v is HIGHER than expected
This can occur at very high suction pressures
4. Capacity Control Effects
Capacity control methods work by modifying volumetric efficiency. Understanding how each method affects eta_v is crucial for predicting actual capacity at part-load conditions.
Clearance Pocket Control
Adding clearance reduces eta_v proportionally:
eta_v = 1 - (Cl_fixed + Cl_pocket) * [r^(1/k) - 1] - L_total
Required pocket volume for target capacity:
To reduce capacity from Q_design to Q_target:
eta_v_target = (Q_target / Q_design) * eta_v_design
Cl_total = (1 - eta_v_target - L_total) / [r^(1/k) - 1]
Cl_pocket = Cl_total - Cl_fixed
Variable Volume Pocket (VVP):
Continuously adjustable clearance
Provides smooth capacity control from ~50% to 100%
Fixed Volume Pocket (FVP):
Discrete clearance steps (on/off)
Each pocket provides a specific capacity step
Multiple pockets give multiple steps
Head-end unloader (HEU):
Holds suction valve open on HE
Reduces capacity by ~50% (for DA cylinder)
CE continues to compress normally
Capacity Control Steps Example
| Configuration | Cl_total | eta_v | Capacity (%) |
|---|---|---|---|
| All pockets closed | 12% | 83.5% | 100% |
| HE FVP open (+8%) | 20% | 73.1% | 87.5% |
| CE FVP open (+8%) | 20% | 73.1% | 87.5% |
| Both FVPs open | 28% | 62.8% | 75.2% |
| HE unloaded (SA) | 12%* | 83.5%* | ~48% |
| HE unloaded + CE FVP | 20%* | 73.1%* | ~42% |
*eta_v applies to CE only when HE is unloaded. Capacity is CE fraction of total.
Values for r = 3.0, k = 1.27.
Speed Control Effect
Variable speed does NOT change eta_v significantly:
eta_v is nearly independent of speed because:
- Clearance re-expansion is pressure-driven (not speed-dependent)
- Valve losses increase slightly at higher speed
- Ring leakage is approximately constant
Capacity is proportional to speed:
Q = PD * eta_v = (pi/4) * D^2 * Stroke * RPM * eta_v
Q(RPM_new) / Q(RPM_design) = RPM_new / RPM_design
(approximately, with minor eta_v correction)
Advantage of speed control:
Continuous capacity variation without eta_v penalty.
Power tracks linearly with speed (vs. clearance control
where power/unit flow increases at reduced capacity).
Efficiency comparison: Speed control maintains specific power (HP/MMSCFD) nearly constant across the range. Clearance control increases specific power at lower capacity because the cylinder does work compressing and re-expanding clearance gas without delivering it. Speed control is 5-15% more efficient at 50% capacity.
5. Field Measurement
Measuring actual volumetric efficiency in the field provides the most accurate performance data and enables condition monitoring.
Measurement Methods
| Method | Accuracy | Equipment | Application |
|---|---|---|---|
| Flow meter (orifice) | +/- 2-3% | Orifice plate, DP transmitter | Continuous monitoring |
| Flow meter (ultrasonic) | +/- 1-2% | Clamp-on or inline ultrasonic | Non-intrusive, temporary |
| PV card analysis | +/- 3-5% | Electronic indicator system | Periodic performance test |
| Nozzle test | +/- 1% | Calibrated nozzle, P, T | Shop or acceptance test |
Calculating eta_v from Field Data
From measured flow rate:
eta_v = Q_actual / PD
Where:
Q_actual = Measured flow in ACFM at suction conditions
PD = Piston displacement (known from cylinder geometry)
Converting measured flow to suction ACFM:
If flow is measured downstream at different P, T:
Q_suction = Q_measured * (P_measured / P_suction)
* (T_suction / T_measured) * (Z_suction / Z_measured)
From PV cards:
eta_v = (V_1 - V_4) / V_swept
Where V_1 and V_4 are read directly from the measured
PV diagram at suction pressure intersections.
Trending:
Plot eta_v vs time to track degradation:
- Gradual decline: Ring wear, valve deposit buildup
- Sudden drop: Valve failure, ring breakage
- Step change: Operating condition change
Typical eta_v Degradation Rates
| Component | eta_v Loss Rate | Inspection Interval | Replacement Trigger |
|---|---|---|---|
| Piston rings | 0.5-2% per year | 8,000-16,000 hours | eta_v drops > 5% from baseline |
| Valve plates | 1-3% per year | 4,000-8,000 hours | eta_v drops > 3%, or noise increase |
| Valve seats | 0.5-1% per year | 16,000-24,000 hours | Leakage test failure |
| Packing rings | Minimal on eta_v | 4,000-8,000 hours | Visible leakage, emissions test |
6. Worked Examples
Example 1: Calculate eta_v and Capacity
Given:
Cylinder: Bore = 8 in, Stroke = 6 in, Rod = 2.5 in
Double-acting, Speed = 1,000 RPM
Clearance: Cl = 14% (both ends average)
P_suction = 250 psia, P_discharge = 750 psia
Gas: Natural gas, k = 1.27, Z_s = 0.93, Z_d = 0.88
Machine condition: Good (L_total = 0.05)
Step 1: Compression ratio
r = 750 / 250 = 3.0
Step 2: Clearance re-expansion term
r^(1/k) = 3.0^(1/1.27) = 3.0^0.787 = 2.510
Z correction: (Z_s/Z_d) = 0.93/0.88 = 1.057
Corrected: (Z_s/Z_d) * r^(1/k) - 1 = 1.057 * 2.510 - 1 = 1.653
Step 3: Volumetric efficiency
eta_v = 1 - 0.14 * 1.653 - 0.05
eta_v = 1 - 0.231 - 0.05 = 0.719 (71.9%)
Without Z correction: eta_v = 1 - 0.14*1.510 - 0.05 = 0.739 (73.9%)
Z correction reduces eta_v by 2.0 percentage points.
Step 4: Displacement
A_HE = (pi/4) * 8^2 = 50.27 in^2
A_CE = (pi/4) * (64 - 6.25) = 45.36 in^2
PD_HE = 50.27 * 6 * 1,000 / 1,728 = 174.5 CFM
PD_CE = 45.36 * 6 * 1,000 / 1,728 = 157.5 CFM
PD_total = 174.5 + 157.5 = 332.0 CFM
Step 5: Actual capacity
Q_actual = 332.0 * 0.719 = 238.7 ACFM at suction
Convert to SCFM:
SCFM = 238.7 * (250/14.696) * (520/560) / (1.0/0.93)
SCFM = 238.7 * 17.01 * 0.929 * 0.93
SCFM = 3,509 SCFM = 5.05 MMSCFD
Example 2: Clearance Pocket Sizing for Turndown
Given:
Same cylinder as Example 1
Design capacity: 5.05 MMSCFD at eta_v = 71.9%
Required reduced capacity: 3.5 MMSCFD (69.3% of design)
Step 1: Required eta_v at reduced capacity
eta_v_target = 0.693 * 71.9% = 49.8%
Step 2: Required total clearance
0.498 = 1 - Cl_total * 1.653 - 0.05
Cl_total * 1.653 = 0.452
Cl_total = 0.452 / 1.653 = 0.274 (27.4%)
Step 3: Required pocket clearance
Cl_pocket = 0.274 - 0.14 = 0.134 (13.4% additional)
Step 4: Pocket volume
V_swept = (pi/4) * 8^2 * 6 = 301.6 in^3 (HE)
V_pocket = 0.134 * 301.6 = 40.4 in^3
Use a fixed volume pocket of approximately 40 in^3,
or a variable volume pocket adjustable from 0 to ~60 in^3
for continuous control.
Verify rod load reversal at this clearance before finalizing.
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