1. Overview
The pressure-volume (PV) diagram is the fundamental representation of the thermodynamic cycle in a reciprocating compressor. It plots cylinder pressure against piston displacement volume throughout one complete revolution. The area enclosed by the PV curve represents the work performed per cycle, making it essential for power calculation, performance assessment, and troubleshooting.
Compression
Process 1-2
Valves closed; P*V^k = const
Discharge
Process 2-3
Discharge valve open; gas expelled
Re-expansion
Process 3-4
Clearance gas expands; both valves closed
Suction
Process 4-1
Suction valve open; fresh gas enters
PV Diagram Uses
| Application | Information Obtained | Method |
|---|---|---|
| Power calculation | Indicated horsepower (IHP) | Integrate area of measured PV card |
| Capacity verification | Actual vs design volume flow | Compare suction volume to theoretical |
| Valve diagnostics | Leaking, broken, or late valves | Shape analysis of PV curve |
| Ring condition | Piston ring blow-by | Slope of compression/expansion curves |
| Rod load calculation | Instantaneous pressure forces | Pressure at each crank angle |
| Performance optimization | Clearance, valve timing | Compare theoretical to actual |
Fundamental relationship: Work per cycle = area enclosed by PV diagram. Clockwise traversal (compression and discharge) represents work input to the gas. Power = W * RPM / 33,000 (for IHP in horsepower).
2. Theoretical PV Diagram
The theoretical (ideal) PV diagram assumes isentropic compression/expansion, instantaneous valve action, no pressure drop through valves, and no gas leakage.
Four Processes of the Ideal Cycle
Process 1-2: Isentropic Compression
Both valves closed. Piston moves from BDC toward TDC.
P * V^k = constant
P_2 = P_1 * (V_1/V_2)^k
Process 2-3: Constant-Pressure Discharge
Discharge valve opens when P_cyl = P_discharge.
Gas is pushed out at constant pressure P_d.
Volume decreases from V_2 to V_3 (clearance volume V_cl).
Process 3-4: Isentropic Re-expansion
Both valves closed. Piston moves from TDC toward BDC.
Clearance gas expands: P * V^k = constant
P_4 = P_3 * (V_3/V_4)^k
Re-expansion continues until P_cyl drops to P_suction.
Process 4-1: Constant-Pressure Suction
Suction valve opens when P_cyl = P_suction.
Fresh gas enters at constant pressure P_s.
Volume increases from V_4 to V_1 (full stroke + clearance).
Key Volumes
Cylinder volumes:
V_swept = (pi/4) * D^2 * Stroke (swept volume)
V_cl = Cl * V_swept (clearance volume)
V_total = V_swept + V_cl = V_swept * (1 + Cl)
Volume at each corner:
V_1 = V_total = V_swept * (1 + Cl) (BDC, suction)
V_2 = V_cl * (P_d/P_s)^(-1/k) + ??? (start of discharge)
V_3 = V_cl (TDC, clearance)
V_4 = V_cl * (P_d/P_s)^(1/k) (end of re-expansion)
Actual suction volume (gas delivered):
V_actual = V_1 - V_4
= V_swept * [1 - Cl * (r^(1/k) - 1)]
= V_swept * eta_v (volumetric efficiency)
Note: V_1 - V_4 is the actual new gas drawn in per stroke,
which determines the compressor capacity.
Theoretical Work per Cycle
Work = Area enclosed by PV diagram
W = integral(P dV) around the cycle
For ideal gas (isentropic compression):
W = [k/(k-1)] * P_1 * V_actual * [(P_2/P_1)^((k-1)/k) - 1]
Indicated Horsepower:
IHP = W * RPM / (33,000 * 12)
(W in in-lbf; 12 converts inches to feet; 33,000 ft-lbf/min per HP)
Mean Effective Pressure (MEP):
MEP = W / V_swept (psi)
IHP = MEP * V_swept * RPM / (33,000 * 12)
3. Actual PV Diagram
Actual PV diagrams differ from theoretical in several important ways. Understanding these differences is critical for performance assessment and diagnostics.
Differences from Theoretical
| Effect | Appearance on PV Card | Impact | Magnitude |
|---|---|---|---|
| Suction valve loss | Suction line drops below P_suction | Reduced effective suction P, lower capacity | 2-10 psi typical |
| Discharge valve loss | Discharge line rises above P_discharge | Increased work, higher discharge T | 5-20 psi typical |
| Valve flutter | Oscillations during suction/discharge | Noise, valve fatigue | Variable |
| Channel resonance | Pressure spikes at valve open/close | Valve plate impact, noise | 5-50 psi spikes |
| Piston ring leakage | Compression slope flatter than k | Reduced capacity, higher power | 1-5% capacity loss |
| Valve leakage | Rounded corners, slope changes | Recirculation, capacity loss | 2-15% capacity loss |
| Gas pulsation | Wavy suction/discharge lines | Capacity variation, valve impact | Variable |
Valve Loss Effects
Suction valve pressure drop:
P_cyl_suction = P_line_suction - delta_P_sv
The suction process occurs at a pressure lower than line pressure.
This means less gas mass enters the cylinder per stroke.
Discharge valve pressure drop:
P_cyl_discharge = P_line_discharge + delta_P_dv
The compression must reach a higher pressure before the
discharge valve opens, requiring more work.
Effect on power:
Additional work = (delta_P_sv + delta_P_dv) * V_delivered
Typical: 3-8% power increase over theoretical
Effect on capacity:
Effective ratio: r_eff = (P_d + delta_P_dv) / (P_s - delta_P_sv)
r_eff > r_nominal, so eta_v is lower than calculated.
Compression Exponent
Theoretical compression follows PV^k = constant.
Actual compression follows PV^n = constant.
The actual exponent n can be determined from two points
on the measured PV card:
n = log(P_2/P_1) / log(V_1/V_2)
Interpretation:
n = k: Ideal isentropic compression (no losses)
n > k: Heat addition or gas leakage into cylinder
n < k: Heat removal (cooling) or gas leakage out
Typical values for natural gas (k=1.27):
New, well-sealed: n = 1.25-1.30
Normal wear: n = 1.20-1.27
Significant ring wear: n = 1.10-1.20
Severe blow-by: n < 1.10
4. Indicator Card Analysis
Modern electronic indicator systems capture cylinder pressure vs crank angle or volume at high sampling rates. The resulting "indicator card" is the measured PV diagram used for performance analysis.
Measurement System Components
| Component | Function | Specification |
|---|---|---|
| Pressure transducer | Measure cylinder pressure | 0-5,000 psi, 0.25% accuracy, 10 kHz |
| Crank angle encoder | Correlate pressure to position | 360-3,600 pulses/rev, TDC reference |
| Data acquisition | Record synchronous P-theta data | 12-16 bit, 50+ kHz sample rate |
| Analysis software | Compute PV, power, rod loads | Real-time display and trending |
Calculations from Indicator Cards
Indicated Horsepower (per end):
IHP = [Sum(P_i * delta_V_i)] * RPM / (33,000 * 12)
Where the summation is the numerical integration of the
PV diagram area using measured pressure and calculated
volume at each crank angle.
Volume from crank angle:
V(theta) = V_cl + (pi/4)*D^2 * x(theta)
x(theta) = R*(1-cos(theta)) + (R^2/L)*(1-cos(2*theta))/2
Where:
R = Crank radius = Stroke/2
L = Connecting rod length
theta = Crank angle (0 = TDC)
Indicated power (double-acting):
IHP_total = IHP_HE + IHP_CE
Brake horsepower:
BHP = IHP / eta_mechanical
eta_mechanical = 0.93-0.97 typical
Key Metrics from PV Analysis
| Metric | Definition | Normal Range | Concern If |
|---|---|---|---|
| Compression exponent n | log(P2/P1)/log(V1/V2) | 0.95k to 1.05k | n < 0.9k (ring wear) |
| Expansion exponent n_e | Same, for re-expansion | 0.95k to 1.05k | Differs from compression n |
| Suction valve loss | P_line - P_cyl during suction | 1-3% of P_s | > 5% (dirty/damaged valve) |
| Discharge valve loss | P_cyl - P_line during discharge | 2-5% of P_d | > 8% (spring tension, deposits) |
| Capacity (actual) | V_1 - V_4 from card | Within 5% of design | > 10% deviation |
| IHP per end | Integrated PV area | Per design curves | > 10% above normal |
5. Diagnostic Interpretation
The shape of the actual PV diagram provides diagnostic information about compressor condition. Comparing actual to theoretical reveals specific mechanical problems.
Common PV Card Abnormalities
| Symptom | PV Card Appearance | Likely Cause | Action |
|---|---|---|---|
| Low compression slope | Flat compression line (n << k) | Ring blow-by, valve leak | Replace rings or valves |
| Rounded suction corner | Gradual transition to suction | Suction valve leakage or late opening | Inspect suction valves |
| Rounded discharge corner | Gradual transition from discharge | Discharge valve leakage | Inspect discharge valves |
| Suction line below normal | Deep dip during suction stroke | Plugged suction valve, high velocity | Clean/replace valve, check sizing |
| Discharge line above normal | High plateau during discharge | Plugged discharge valve, deposits | Clean/replace valve |
| Oscillations on suction line | Wavy pattern during intake | Pulsation, valve flutter | Check pulsation bottles, valve springs |
| Pressure spike at TDC | Sharp peak at minimum volume | Liquid in cylinder (hydraulic) | Emergency: check for liquids |
| Unequal HE/CE cards | Different areas for each end | Ring wear, valve issue on one end | Compare to identify affected end |
Leakage Detection from PV Cards
Valve leakage test (stationary method):
1. Block in the cylinder (close suction and discharge block valves)
2. Pressurize to discharge pressure
3. Record pressure decay over time
4. Leakage rate = V_cyl * (dP/dt) / P_avg
From running PV cards:
Suction valve leak: Re-expansion curve slope differs from compression
n_expansion < n_compression (gas escaping during expansion)
Discharge valve leak: High pressure during suction stroke
P_min_suction > P_line_suction significantly
Piston ring blow-by indicators:
- Both compression and expansion n values lower than k
- HE and CE cards show opposite effects
(blow-by goes from high-P end to low-P end)
- Capacity loss without valve leakage signs
Trending is critical: A single PV card provides a snapshot. Trending PV parameters over time (IHP, valve loss, compression exponent) reveals degradation patterns that enable predictive maintenance and prevent unplanned failures.
6. Worked Examples
Example 1: Theoretical PV Diagram Construction
Given:
Bore = 10 in, Stroke = 8 in, Clearance = 15%
P_suction = 200 psia, P_discharge = 600 psia
k = 1.27
Step 1: Volumes
V_swept = (pi/4) * 10^2 * 8 = 628.3 in^3
V_cl = 0.15 * 628.3 = 94.2 in^3
V_total = 628.3 + 94.2 = 722.5 in^3
Step 2: Corner points
Point 1 (BDC, suction): V = 722.5 in^3, P = 200 psia
Point 3 (TDC, discharge): V = 94.2 in^3, P = 600 psia
Point 2 (end of compression):
P_2 = 600 psia
V_2: 200 * 722.5^1.27 = 600 * V_2^1.27
V_2 = 722.5 * (200/600)^(1/1.27) = 722.5 * 0.333^0.787
V_2 = 722.5 * 0.399 = 288.0 in^3
Point 4 (end of re-expansion):
P_4 = 200 psia
V_4 = V_cl * (P_d/P_s)^(1/k) = 94.2 * (600/200)^(1/1.27)
V_4 = 94.2 * 3.0^0.787 = 94.2 * 2.510 = 236.4 in^3
Step 3: Actual suction volume
V_actual = V_1 - V_4 = 722.5 - 236.4 = 486.1 in^3
eta_v = 486.1 / 628.3 = 0.774 (77.4%)
Step 4: Work per cycle (numerical)
W = [k/(k-1)] * P_1 * V_actual * [(r)^((k-1)/k) - 1]
W = [1.27/0.27] * 200 * 486.1 * [3.0^0.213 - 1]
W = 4.704 * 200 * 486.1 * [1.270 - 1]
W = 4.704 * 200 * 486.1 * 0.270
W = 123,470 in-lbf per stroke
Step 5: IHP (at 720 RPM)
IHP = 123,470 * 720 / (33,000 * 12) = 224.5 HP
Example 2: Diagnosing Valve Leakage
Given: Measured PV card shows:
Compression exponent: n_comp = 1.25 (near k=1.27 -- OK)
Re-expansion exponent: n_exp = 1.18 (lower than k -- abnormal)
Suction valve loss: 8 psi (4% of P_s -- slightly high)
Discharge valve loss: 15 psi (2.5% of P_d -- normal)
Analysis:
1. Compression n is near k: Piston rings are sealing well
2. Re-expansion n < k: Gas is leaking OUT during expansion
- If n_exp < n_comp, gas escapes as clearance gas expands
- This indicates discharge valve leakage
- High-pressure gas leaks back through discharge valve
during re-expansion, lowering the expansion curve
3. Suction valve loss is moderately high:
- Could be deposits on valve seats
- Or undersized valve (insufficient flow area)
Diagnosis: Discharge valve leakage
- Gas leaks backward through discharge valve during expansion
- This explains n_exp < n_comp
- Also explains slightly high suction valve loss
(more gas tries to enter during suction to compensate)
Recommendation:
Inspect discharge valves. Look for:
- Damaged valve plates or sealing surfaces
- Worn valve seats
- Broken or weak springs
- Deposits preventing full closure
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