9.1 Compressors: Reciprocating, Rotary, Scroll, Screw & Centrifugal

9.1 Compressors: Reciprocating, Rotary, Scroll, Screw & Centrifugal

The compressor is the mechanical heart of the vapor-compression refrigeration cycle. It performs two thermodynamic functions: maintaining the low-pressure evaporating environment by continuously removing vaporized refrigerant from the evaporator, and elevating the refrigerant vapor pressure and temperature so that thermal energy can be rejected to an ambient heat sink (air or water). On the PE Mechanical: HVAC and Refrigeration exam, compressor problems evaluate thermodynamic efficiency derivations, clearance volume re-expansion, volumetric displacement calculations, aerodynamic operating boundaries, and capacity modulation techniques.

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1. Fundamental Classification of Refrigeration Compressors

Refrigeration compressors are categorized into two fundamental operational classes:

+---------------------------------------------------------------------------------------------------------+
|                                 REFRIGERANT COMPRESSOR CLASSIFICATION                                   |
+----------------------------------------------------+----------------------------------------------------+
| 1. POSITIVE DISPLACEMENT                           | 2. DYNAMIC (AERODYNAMIC TURBOMACHINERY)            |
| Entraps successive volumes of vapor in a closed    | Imparts continuous angular momentum to incoming   |
| cavity and increases pressure by reducing volume.  | vapor via high-speed impeller blades and converts   |
| Flow rate is directly proportional to speed (RPM). | kinetic energy to static pressure in a diffuser.   |
+----------------------------------------------------+----------------------------------------------------+
| • Reciprocating: Piston-cylinder with valves       | • Centrifugal: Radial outflow impeller (single     |
| • Rotary / Rolling Piston: Eccentric rotor & blade |   or multi-stage, 50 to 3,000+ tons)               |
| • Scroll: Interleaving fixed & orbiting scrolls    | • Axial Flow: Multi-stage axial blading            |
| • Screw (Helical Rotary): Meshing twin rotors      |   (specialized industrial applications)            |
+----------------------------------------------------+----------------------------------------------------+

Hermetic, Semi-Hermetic & Open Configurations

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2. Positive Displacement Compressors: Mechanics & Performance

A. Reciprocating Compressors

Reciprocating compressors utilize a crankshaft, connecting rod, and piston reciprocating within a honed cylinder. Suction and discharge reed or ring valves open and close automatically based on differential pressure.

   Pressure (P)
        ^
     Pd +-------------3=============2 (Discharge valve open, mass expelled)
        |            /              |
        |           /               | Compression (1 -> 2: P*V^n = C)
        |          / (Re-expansion  |
        |         /   3 -> 4)       |
     Ps +--------4==================1 (Suction valve open, mass drawn in)
        |        |                  |
        +--------+----+-------------+--------> Volume (V)
        0        Vc   V4            V1 = Vc + Vd
                 <---> <------------>
                Clearance   Effective
                 Volume    Suction Vol

Clearance Volume & Volumetric Efficiency

Because the piston cannot make physical contact with the cylinder head at Top Dead Center (TDC), a small clearance volume ($V_c$) remains. High-pressure vapor trapped in $V_c$ at state 3 must re-expand polytropically to the suction pressure ($P_s$) before the suction valve can open at state 4:

$C = \frac{V_c}{V_d} = \frac{V_c}{V_1 - V_c}$

Where $C$ is the clearance volume fraction (typically $0.02$ to $0.07$), $V_d$ is the piston displacement volume, and $V_c$ is the clearance volume. Assuming polytropic re-expansion ($P V^n = \text{constant}$, where $n \approx k = c_p / c_v$):

$\eta_{v,\text{clearance}} = 1 + C - C\left(\frac{P_d}{P_s}\right)^{\frac{1}{n}} = 1 - C\left[\left(\frac{P_d}{P_s}\right)^{\frac{1}{n}} - 1\right]$

Actual volumetric efficiency ($\eta_v$) is lower than clearance volumetric efficiency due to suction valve pressure drop, cylinder wall heating (which expands incoming gas), and piston ring blow-by leakage:

$\eta_v = \frac{\dot{m} \cdot v_{\text{suction}}}{\dot{V}_{\text{disp}}}$

$\dot{V}_{\text{disp}} = N_{\text{cylinders}} \times \left(\frac{\pi}{4} D_{\text{bore}}^2\right) \times L_{\text{stroke}} \times \text{RPM}$

B. Scroll Compressors

Scroll compressors feature two interleaved Archimedean spiral scrolls: one fixed and one orbiting eccentric scroll. As the orbiting scroll moves, it forms progressive crescent-shaped gas pockets that travel radially inward toward the central discharge port, continuously compressing the refrigerant.

• Continuous Compression: Multiple compression pockets operate simultaneously, resulting in continuous suction and discharge with negligible torque pulsation and smooth sound profiles.
• Absence of Dynamic Valves: No suction or discharge reed valves are required, eliminating valve pressure throttling losses and valve flutter failures.
• Axial and Radial Compliance: Mechanical compliance allows the scroll wraps to separate momentarily if liquid refrigerant or small debris enters, making scroll compressors resilient against liquid slugging.
• Application Range: Dominant in light commercial DX rooftop units, split systems, VRF systems, and packaged scroll chillers from 2 to 40 tons per compressor.

C. Screw (Helical Rotary) Compressors

Twin-screw compressors utilize two intermeshing helical rotors: a male rotor (typically 4 lobes) driven by the motor, and a female rotor (typically 6 flutes) enclosed in a stationary housing. Refrigerant vapor enters axially at one end, is trapped between the rotor flutes and casing, and is compressed axially as the meshing lobes reduce the cavity volume toward the discharge port.

Built-in Volume Ratio ($V_i$) & Built-in Pressure Ratio ($P_i$)

Screw compressors have fixed port geometry that establishes a fixed internal built-in volume ratio ($V_i$):

$V_i = \frac{v_{\text{suction}}}{v_{\text{discharge, internal}}} = \frac{V_{\text{suction pocket}}}{V_{\text{discharge pocket}}}$

$P_i = (V_i)^k$

Where $k$ is the isentropic exponent ($c_p / c_v$) of the refrigerant.

Over-Compression (Pd < Pi):           Under-Compression (Pd > Pi):
Internal gas compressed above Pd.    Internal gas under-compressed.
Expands wastefully into discharge.   Discharge gas backflows into pocket.
Requires excess shaft power!         Increases compression work & noise!

  Pressure                             Pressure
     ^                                    ^
 Pi +---\                            Pd +---+===\
    |    \                              |   |    \
 Pd +-----+===                          Pi +---+     \
    |                                   |             \
    +------------> Volume               +-------------> Volume

• Over-Compression ($P_i > P_{\text{system discharge}}$): The gas is compressed internally to a pressure higher than condensing pressure. When the discharge port uncovers, gas expands into the discharge line, wasting mechanical shaft work.
• Under-Compression ($P_i < P_{\text{system discharge}}$): The gas is under-compressed when the discharge port uncovers. High-pressure gas from the condenser rushes back into the rotor pocket, requiring extra compressor work to push it back out.

Slide Valve Capacity Modulation

A sliding valve located axially along the rotor housing bypasses a portion of suction gas back to the compressor inlet before compression begins, allowing smooth, stepless capacity modulation from 100% down to 10% load.

Economizer Port Injection

Screw compressors feature an intermediate economizer injection port. Flash gas from an intermediate subcooler or flash tank is injected directly into the closed rotor pocket mid-compression, increasing refrigeration capacity by 15% to 25% with minimal incremental power.

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3. Dynamic Compressors: Centrifugal Compressors

Centrifugal compressors are dynamic turbomachines. Low-pressure refrigerant vapor enters axially through an inlet suction plenum and passes through Variable Inlet Guide Vanes (VIGVs) into the center (eye) of a high-speed rotating impeller.

+-----------------------------------------------------------------------------------------+
| CENTRIFUGAL COMPRESSOR ENERGY CONVERSION MECHANISM                                      |
+-----------------------------------------------------------------------------------------+
| 1. Impeller Eye: Suction vapor enters axially at radius r1.                             |
| 2. Impeller Blades: Rotational kinetic energy increases gas absolute velocity to V2.    |
| 3. Impeller Tip: Centrifugal force elevates static pressure across rotor passages.      |
| 4. Radial Vaned/Vaneless Diffuser: Decelerates high-velocity gas, converting dynamic   |
|    kinetic head (V^2 / 2g_c) into static pressure head according to Bernoulli theorem.  |
| 5. Volute / Collector: Channels high-pressure vapor to the condenser discharge nozzle.  |
+-----------------------------------------------------------------------------------------+

Euler Turbomachinery Equation

The theoretical work imparted to the fluid per unit mass is given by Euler's turbine equation:

$w_{\text{fluid}} = \frac{u_2 \cdot V_{\theta 2} - u_1 \cdot V_{\theta 1}}{g_c}$

Where $u = \omega r = \frac{\pi D N}{720}$ is the impeller tangential blade speed ($\text{ft/s}$ with $D$ in inches and $N$ in $\text{RPM}$), and $V_\theta$ is the tangential component of absolute fluid velocity. If entry flow has zero prerotation ($V_{\theta 1} = 0$, radial entry):

$w_{\text{fluid}} = \frac{u_2 \cdot V_{\theta 2}}{g_c} = \frac{\mu_s \cdot u_2^2}{g_c}$

Where $\mu_s$ is the impeller slip factor (typically $0.85$ to $0.92$).

Head, Pressure Ratio & Tip Speed

The adiabatic head generated by a centrifugal compressor is directly proportional to the square of the impeller tip speed ($u_2$):

$H_{\text{adiabatic}} = \frac{\mu_s \cdot u_2^2}{g_c} = \frac{k}{k-1} R T_1 \left[\left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} - 1\right]$

To achieve the required pressure lift for high-pressure refrigerants without exceeding blade sonic velocity limits, centrifugal chillers utilize multi-stage impellers or gear-driven high-speed single-stage impellers (10,000 to 30,000 RPM).

Aerodynamic Operating Boundaries: Surge, Stall & Choke

   Pressure Rise (Lift / Head)
        ^
        |             SURGE LINE (Limit of stable operation)
        |                /              Peak Efficiency
        |   UNSTABLE    /   .--------.   Island
        |    REGION    /  (  78% 82%  )
        |             /  (  85%  80%   )       STALL REGION
        |   Surge    /    '----------'      (Blade separation)
        |  Reversal /  N = 100% Speed
        |          /-----------------------\ 
        |         /   N = 80% Speed         \  CHOKE / STONEWALL
        |        /---------------------------\ (Sonic velocity at throat)
        +-------+-----------------------------+------------------------> Mass Flow Rate (m_dot)

1. Surge: The minimum stable flow limit. When system condensing pressure (lift requirement) exceeds the aerodynamic pressure head generated by the impeller, forward vapor flow ceases. The high-pressure gas from the condenser violently reverses flow through the impeller back into the evaporator. This causes rapid cyclical pressure oscillations, severe mechanical vibration, and destructive thrust bearing loading. Anti-surge protection utilizes hot gas bypass (HGBP) to maintain minimum mass flow across the impeller.
2. Rotating Stall: Localized boundary-layer flow separation on the impeller or diffuser blades that rotates asynchronously around the annulus. Stall causes low-frequency acoustic noise and serves as the precursor to full compressor surge.
3. Choke (Stonewall): The maximum flow limit. Occurs when fluid velocity reaches local sonic speed ($M = 1.0$) at the impeller throat or diffuser passages. At choke, no further increase in mass flow rate is possible regardless of lower discharge pressure.

Magnetic Bearing Centrifugal Technology

Modern oil-free centrifugal chillers utilize active magnetic bearings (AMB). Permanent magnets with digital electromagnet feedback loops levitate the rotor shaft in a frictionless magnetic field at 15,000 to 40,000 RPM. Eliminating lubricating oil eliminates oil management systems (separators, heaters, pumps, coolers), prevents oil film fouling inside heat exchangers (which degrades heat transfer by 5% to 15%), and drastically reduces part-load parasitic drag.

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4. Compressor Performance Metrics & Energy Equations

Theoretical & Actual Compression Work

For steady-state adiabatic compression, the ideal isentropic work is:

$\dot{W}_{s} = \dot{m} (h_{2s} - h_1)$

$w_s = \frac{k}{k-1} P_1 v_1 \left[\left(\frac{P_2}{P_1}\right)^{\frac{k-1}{k}} - 1\right]$

Where $P_1$ and $P_2$ must be in absolute pressure units ($\text{psia}$ or $\text{kPa}$). Compressor isentropic efficiency ($\eta_s$) accounts for aerodynamic friction and turbulence:

$\eta_s = \frac{\dot{W}_s}{\dot{W}_{a,\text{fluid}}} = \frac{h_{2s} - h_1}{h_{2a} - h_1}$

$\dot{W}_{\text{shaft / brake}} = \frac{\dot{W}_{a,\text{fluid}}}{\eta_{\text{mech}}} = \frac{\dot{m}(h_{2s} - h_1)}{\eta_s \cdot \eta_{\text{mech}}}$

$\dot{W}_{\text{electrical}} = \frac{\dot{W}_{\text{shaft}}}{\eta_{\text{motor}}} = \frac{\dot{m}(h_{2s} - h_1)}{\eta_s \cdot \eta_{\text{mech}} \cdot \eta_{\text{motor}}}$

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5. Comparative Engineering Matrix: Refrigerant Compressors

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6. Worked Example: Reciprocating Compressor Sizing & Performance

Problem: A commercial 4-cylinder reciprocating compressor operates with R-410A at a suction pressure of $P_s = 130.0\text{ psia}$ ($h_1 = 181.50\text{ Btu/lbm}$, $v_1 = 0.4850\text{ ft}^3/\text{lbm}$) and discharge pressure $P_d = 390.0\text{ psia}$. At $390.0\text{ psia}$, the isentropic enthalpy is $h_{2s} = 203.20\text{ Btu/lbm}$. The refrigerant behaves with an isentropic exponent of $k = 1.18$. The compressor has a bore $D = 2.50\text{ in.}$, stroke $L = 2.00\text{ in.}$, runs at $1,750\text{ RPM}$, and has a clearance volume ratio of $C = 0.050$ (5%). The isentropic efficiency is $\eta_s = 0.76$ and the combined mechanical-electrical motor efficiency is $\eta_m = 0.90$.

Find:

1. The clearance volumetric efficiency ($\eta_v$).
2. The total volumetric displacement ($\dot{V}_{\text{disp}}$) in $\text{CFM}$ and the actual refrigerant mass flow rate ($\dot{m}$) in $\text{lbm/min}$.
3. The actual electrical power input in kilowatts ($\text{kW}$) and the cooling capacity in tons assuming liquid leaves the condenser with enthalpy $h_4 = 62.00\text{ Btu/lbm}$.

Step-by-Step Solution:

Step 1: Calculate clearance volumetric efficiency ($\eta_v$). $\text{Pressure Ratio: } r_p = \frac{P_d}{P_s} = \frac{390.0\text{ psia}}{130.0\text{ psia}} = 3.000$

$\eta_v = 1 + C - C\left(r_p\right)^{\frac{1}{k}} = 1 + 0.050 - 0.050\left(3.000\right)^{\frac{1}{1.18}} = 1.050 - 0.050\left(3.000^{0.84746}\right)$ $3.000^{0.84746} = 2.5385$ $\eta_v = 1.050 - 0.050(2.5385) = 1.050 - 0.1269 = 0.9231 \implies 92.31\%$

Step 2: Calculate compressor displacement and mass flow rate. $\text{Piston Area: } A_p = \frac{\pi}{4} (2.50\text{ in.})^2 = 4.9087\text{ in.}^2$ $\text{Displacement per revolution: } V_{\text{rev}} = 4 \text{ cylinders} \times 4.9087\text{ in.}^2 \times 2.00\text{ in.} = 39.270\text{ in.}^3/\text{rev}$ $\dot{V}_{\text{disp}} = \frac{39.270\text{ in.}^3/\text{rev} \times 1,750\text{ RPM}}{1,728\text{ in.}^3/\text{ft}^3} = \frac{68,722.5}{1,728} = 39.770\text{ CFM}$

$\text{Actual Volumetric Intake: } \dot{V}_{\text{actual}} = \eta_v \times \dot{V}_{\text{disp}} = 0.9231 \times 39.770\text{ CFM} = 36.712\text{ CFM}$

$\text{Mass Flow Rate: } \dot{m} = \frac{\dot{V}_{\text{actual}}}{v_1} = \frac{36.712\text{ ft}^3/\text{min}}{0.4850\text{ ft}^3/\text{lbm}} = 75.695\text{ lbm/min}$

Step 3: Calculate electrical power input and cooling capacity. $\text{Isentropic Work: } w_s = h_{2s} - h_1 = 203.20 - 181.50 = 21.70\text{ Btu/lbm}$ $\text{Actual Gas Work: } w_a = \frac{w_s}{\eta_s} = \frac{21.70\text{ Btu/lbm}}{0.76} = 28.553\text{ Btu/lbm}$ $\dot{W}_{\text{fluid}} = \dot{m} \times w_a = 75.695\text{ lbm/min} \times 28.553\text{ Btu/lbm} = 2,161.32\text{ Btu/min}$

$\dot{W}_{\text{electrical}} = \frac{\dot{W}_{\text{fluid}}}{\eta_m} = \frac{2,161.32\text{ Btu/min}}{0.90} = 2,401.47\text{ Btu/min}$ $\text{Convert to kW (1 kW = 56.87 Btu/min): } \dot{W}_{\text{elec}} = \frac{2,401.47\text{ Btu/min}}{56.87\text{ Btu/(min}\cdot\text{kW)}} = 42.23\text{ kW}$

$\text{Refrigeration Effect: } q_e = h_1 - h_4 = 181.50 - 62.00 = 119.50\text{ Btu/lbm}$ $\dot{Q}_{\text{evap}} = \dot{m} \times q_e = 75.695\text{ lbm/min} \times 119.50\text{ Btu/lbm} = 9,045.55\text{ Btu/min}$ $\text{Cooling Capacity (Tons): } \text{Capacity} = \frac{9,045.55\text{ Btu/min}}{200\text{ Btu/(min}\cdot\text{ton)}} = 45.23\text{ Tons}$

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7. NCEES Reference Handbook Navigation & Exam Tips

• Thermodynamics & Refrigeration Sections: Look up isentropic work relations $w_s = \frac{k}{k-1} P_1 v_1 [(P_2/P_1)^{(k-1)/k} - 1]$ and steady-flow energy balance equations.
• Absolute Pressure Reminder: Always use absolute pressures ($\text{psia} = \text{psig} + 14.696$) in the compression ratio $r_p = P_d / P_s$. Calculating $r_p$ with gauge pressure causes severe errors.
• Units of Power: Remember $1\text{ Ton} = 12,000\text{ Btu/hr} = 200\text{ Btu/min} = 3.51685\text{ kW}$. $1\text{ HP} = 2,544.4\text{ Btu/hr} = 42.418\text{ Btu/min} = 0.7457\text{ kW}$.