2.4 Pressure-Enthalpy (P-h) Diagrams & System Operating Analysis

2.4 Pressure-Enthalpy (P-h) Diagrams & System Operating Analysis

The Pressure-Enthalpy (P-h) Diagram (also known as the Mollier diagram for refrigerants) is the primary engineering tool used to quantify, visualize, and troubleshoot the vapor compression refrigeration cycle. While standard manifold gauges and thermometers measure surface conditions, plotting those parameters on a P-h chart reveals exact thermodynamic properties including specific enthalpy (h), entropy (s), specific volume (v), and vapor quality (x).

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1. Anatomy of the P-h Diagram

                    PRESSURE-ENTHALPY (P-h) DIAGRAM

  LOG
PRESSURE (P)
  ▲
  │                         CRITICAL POINT
  │                               ▲
  │                              / \
  │                             /   \
  │   SUBCOOLED                /     \              SUPERHEATED
  │     LIQUID                /   ▲   \                VAPOR
  │     REGION               /    │    \               REGION
  │                         /   SAT.    \
  │                        /    DOME     \
  │                       /  (2-Phase)    \
  │                      /                 \
  │    Saturated Liquid /                   \ Saturated Vapor
  │    Line (x = 0.0)  /                     \ Line (x = 1.0)
  │                   /                       \
  └──────────────────┴─────────────────────────┴─────────────────────►
                                                      SPECIFIC ENTHALPY (h)
                                                           (BTU / lb)

Diagram Axes & Coordinate Geometry

• Vertical Y-Axis (Pressure - P): Plots Absolute Pressure (P_psia or bar) on a logarithmic scale. The logarithmic scale expands the low-pressure evaporator region and compresses the high-pressure condenser region so both can be analyzed accurately on a single sheet.
• Horizontal X-Axis (Specific Enthalpy - h): Plots specific enthalpy in BTU/lb (or kJ/kg) on a linear scale. Specific enthalpy represents the total heat energy contained in one pound of refrigerant above an arbitrary reference state (0 BTU/lb at -40°F). Moving horizontally to the right indicates heat absorption; moving left indicates heat rejection.

The Saturation Dome & Boundaries

• Saturated Liquid Line (Bubble-Point Line): The left curve of the dome (x = 0.0). Any point on this line represents 100% liquid at saturation temperature.
• Saturated Vapor Line (Dew-Point Line): The right curve of the dome (x = 1.0). Any point on this line represents 100% vapor at saturation temperature.
• Critical Point: The apex where the saturated liquid and saturated vapor lines meet. Above the critical temperature and pressure (approximately 160°F (71.3°C) and 711 psia for R-410A), liquid and vapor merge into a single supercritical fluid phase.
• The Three Diagram Zones:
  1. Subcooled Liquid Region (Left of Dome): T_actual < T_sat.
  2. Two-Phase Saturation Dome (Under Dome): Saturated boiling/condensing mixture (0.0 < x < 1.0).
  3. Superheated Vapor Region (Right of Dome): T_actual > T_sat.

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2. Six Families of Property Lines

To extract data from a P-h diagram, a contractor must recognize the six sets of intersecting lines:

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3. Mapping the 4-State Cycle on the P-h Diagram

                 4-STATE CYCLE ON THE P-h DIAGRAM

  PRESSURE (P)
     ▲
     │                  CONDENSER (Desuperheat & Condense)
     │              State 3 ◄────────────────────── State 2
     │              (Subcooled)                   (Discharge Gas)
 P_cond ────────────┐                              ▲
     │              │                              │
     │    EXPANSION │                              │ COMPRESSION
     │    (Throttling)                             │ (Isentropic)
     │              │                              │
 P_evap ────────────▼──────────────────────────────┘
     │              State 4 ─────────────────────► State 1
     │              (Flash Gas)                   (Superheated Suction)
     │                      EVAPORATOR (Boiling)
     └───────────────────────────────────────────────────────────►
                    h3=h4          h1             h2         ENTHALPY (h)

• Process 1 → 2: Compression (Suction Inlet to Discharge Outlet)
  Refrigerant vapor at State 1 (P_evap, h1, superheated) is compressed along a line of constant entropy to State 2 (P_cond, h2, high-temperature superheated gas). Work input adds enthalpy.
• Process 2 → 3: Condensation (Discharge Outlet to Liquid Line)
  High-pressure gas cools at constant pressure (P_cond): first desuperheats to the dew point, condenses along the horizontal isotherm inside the dome to the bubble point, and subcools to State 3 (h3). Enthalpy decreases from h2 to h3.
• Process 3 → 4: Expansion (Liquid Line through Metering Device)
  Subcooled liquid undergoes isenthalpic expansion (constant enthalpy throttling, vertical drop: h4 = h3) from P_cond down to P_evap. Flashing vapor instantly cools the mixture to State 4.
• Process 4 → 1: Evaporation (Metering Device Outlet to Suction Line)
  Refrigerant absorbs space heat at constant pressure (P_evap): boiling from quality x4 to 100% saturated vapor, then absorbing sensible superheat to State 1 (h1).

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4. Thermodynamic Performance Calculations

By obtaining the specific enthalpy values at each state point (h1, h2, h3, h4), all primary engineering metrics can be calculated.

Primary Thermodynamic Formulas

1. Net Refrigerating Effect (NRE): Useful cooling enthalpy absorbed in the evaporator: $\text{NRE} = h_1 - h_4 = h_1 - h_3 \quad (\text{BTU/lb})$
2. Heat of Compression (HOC): Compressor mechanical/thermal energy added to vapor: $\text{HOC} = h_2 - h_1 \quad (\text{BTU/lb})$
3. Total Heat of Rejection (THR): Total thermal energy rejected by the condenser: $\text{THR} = h_2 - h_3 = \text{NRE} + \text{HOC} \quad (\text{BTU/lb})$
4. Refrigerant Mass Flow Rate (m_dot): Pounds of refrigerant circulated per minute for a given total cooling capacity (Q_cooling in BTU/hr): $\dot{m} = \frac{Q_{\text{cooling}} \text{ (BTU/hr)}}{60 \times \text{NRE}} \quad (\text{lbs/min})$
5. Coefficient of Performance (COP): Theoretical cycle efficiency: $\text{COP} = \frac{\text{NRE}}{\text{HOC}} = \frac{h_1 - h_4}{h_2 - h_1}$
6. Energy Efficiency Ratio (EER): Direct relation to COP: $\text{EER} = 3.41214 \times \text{COP} \quad (\text{BTU/Watt}\cdot\text{hr})$

Worked Comprehensive Numerical Example

Operating Data for an R-410A 3.0-Ton (36,000 BTU/hr) System:

• Evaporating Pressure: 118.0 psig + 14.7 = 132.7 psia (T_sat = 45.0°F)
• Condensing Pressure: 335.0 psig + 14.7 = 349.7 psia (T_sat = 105.0°F)
• Suction Line Superheat: 10.0°F → T1 = 55.0°F → h1 = 122.5 BTU/lb
• Compressor Discharge Temp: 165.0°F → h2 = 142.0 BTU/lb
• Liquid Line Subcooling: 15.0°F → T3 = 90.0°F → h3 = 45.5 BTU/lb
• Metering Expansion: h4 = h3 = 45.5 BTU/lb

Calculations:

1. Net Refrigerating Effect (NRE): $\text{NRE} = h_1 - h_4 = 122.5 - 45.5 = 77.0\text{ BTU/lb}$
2. Heat of Compression (HOC): $\text{HOC} = h_2 - h_1 = 142.0 - 122.5 = 19.5\text{ BTU/lb}$
3. Total Heat of Rejection (THR): $\text{THR} = h_2 - h_3 = 142.0 - 45.5 = 96.5\text{ BTU/lb}$ Verification: THR = NRE + HOC = 77.0 + 19.5 = 96.5 BTU/lb.
4. Refrigerant Mass Flow Rate (m_dot): $\dot{m} = \frac{36,000\text{ BTU/hr}}{60 \times 77.0\text{ BTU/lb}} = \frac{36,000}{4,620} = 7.79\text{ lbs/min}$
5. Coefficient of Performance (COP): $\text{COP} = \frac{77.0\text{ BTU/lb}}{19.5\text{ BTU/lb}} = 3.95$
6. Theoretical EER: $\text{EER} = 3.41214 \times 3.95 = 13.48\text{ BTU/Watt}\cdot\text{hr}$