Positive displacement pumps, compressors and blowers

Reciprocating and rotary positive displacement pumps (discharge, slip, power, safe operation), and gas movers with isothermal, adiabatic and multistage compression work and clearance effects.

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Why it matters

Centrifugal pumps cannot do every job. Metering a catalyst or additive at a precise rate, pumping polymer melts or slurries, delivering very high pressure at low flow, or moving gas against a large pressure ratio all call for positive displacement machines or compressors. Choosing between a gear pump, a diaphragm metering pump, a Roots blower or a two-stage reciprocating compressor, and estimating its power, is everyday process-engineering work.

Key ideas

Positive displacement (PD) pumps. They trap a fixed volume of liquid and push it into the discharge. Delivery per cycle is nearly fixed, so flow is roughly proportional to speed and almost independent of discharge pressure (apart from slip, internal leakage that grows with pressure and falls with viscosity).

  • Reciprocating: piston, plunger and diaphragm pumps. High pressure capability; pulsating flow (single-acting worst; double-acting and multiplex designs smoother; air vessels or pulsation dampeners used). Diaphragm pumps isolate the liquid from moving parts — good for corrosive, toxic or abrasive liquids — and are used for metering.
  • Rotary: gear (external and internal), lobe, screw, vane and progressive-cavity pumps. Smooth flow, excellent for viscous liquids (efficiency often improves with viscosity because slip falls); gear pumps need clean liquids.

PD pump behaviour versus centrifugal.

  • The H–Q curve is nearly vertical: flow is set by speed, pressure by the system.
  • Never start or run against a closed discharge valve; pressure rises until something fails. A pressure relief valve on the discharge is mandatory.
  • Flow control by speed variation or recirculation (bypass), not by throttling the discharge.
  • Self-priming in most designs; NPSH still matters (reciprocating pumps need extra margin for acceleration head).

Reciprocating pump relations. Theoretical discharge Q_th = A·L·N/60 (single-acting) or about twice this for double-acting (minus the piston-rod area on one side). Coefficient of discharge C_d = Q_actual/Q_th, slip = 1 − C_d. C_d > 1 (negative slip) can occur at high speeds with long suction lines.

Gas movers: fans, blowers and compressors. Classified roughly by pressure ratio: fans (up to about 1.1), blowers (about 1.1–2), compressors (above about 2). Types: centrifugal and axial (dynamic, large flows); Roots (twin-lobe) blowers and screw, vane and reciprocating compressors (positive displacement). A Roots blower does not compress internally — gas is pushed against the discharge pressure — so it is efficient only at low ratios.

Compression work. For an ideal gas compressed from p₁ to p₂ in a steady-flow machine (neglecting kinetic and potential energy):

  • Isothermal (minimum work, the ideal of perfect cooling): W = p₁V₁·ln(p₂/p₁).
  • Adiabatic (isentropic): W = [γ/(γ − 1)]·p₁V₁·[(p₂/p₁)^((γ−1)/γ) − 1]; discharge temperature T₂ = T₁(p₂/p₁)^((γ−1)/γ).
  • Polytropic: replace γ by n.
  • Multistage with perfect intercooling (gas cooled to T₁ between stages): the work is minimised when each stage has the same pressure ratio, r_stage = (p₂/p₁)^(1/s) for s stages. Multistaging saves power and limits discharge temperature (important for lubricants and seals).
  • Isothermal or adiabatic efficiency = ideal work / actual shaft work.
  • In reciprocating compressors, the gas left in the clearance volume re-expands and reduces volumetric efficiency: η_v = 1 + c − c·(p₂/p₁)^(1/n), where c is clearance volume / swept volume.

Formulas

  • Q_th = A·L·N/60 — single-acting reciprocating pump (m³/s); A piston area (m²), L stroke (m), N (rpm).
  • C_d = Q_actual/Q_th, slip = 1 − C_d.
  • P = ρ·g·Q·H/η — pump shaft power (W).
  • W_iso = p₁·V̇₁·ln(p₂/p₁) — isothermal power (W); p (Pa), V̇₁ suction volumetric flow (m³/s).
  • W_ad = [γ/(γ − 1)]·p₁·V̇₁·[(p₂/p₁)^((γ−1)/γ) − 1] — single-stage adiabatic power (W); γ = c_p/c_v.
  • T₂ = T₁·(p₂/p₁)^((γ−1)/γ) — adiabatic discharge temperature (K).
  • W = s·[γ/(γ − 1)]·p₁·V̇₁·[(p₂/p₁)^((γ−1)/(sγ)) − 1] — s stages, equal ratios, perfect intercooling.
  • η_v = 1 + c − c·(p₂/p₁)^(1/n) — volumetric efficiency with clearance c.

Worked examples

Example 1 (standard). A single-acting reciprocating pump has a 150 mm bore and 300 mm stroke and runs at 60 rpm. It delivers 5.0 L/s of water against a total head of 20 m with an overall efficiency of 80 %. Find Q_th, C_d, slip and shaft power.

  1. A = (π/4)(0.15)² = 0.01767 m².
  2. Q_th = A·L·N/60 = 0.01767 × 0.3 × 60/60 = 5.30 × 10⁻³ m³/s.
  3. C_d = 5.0/5.30 = 0.943; slip = 5.7 %.
  4. P = 1000 × 9.81 × 0.005 × 20 / 0.8 = 1226 W. Q_th ≈ 5.30 L/s, C_d ≈ 0.943, slip ≈ 5.7 %, P ≈ 1.23 kW.

Example 2 (GATE level). Air (γ = 1.4) at 1 bar and 300 K, 0.1 m³/s at suction, is compressed to 6 bar. Compare the ideal power for isothermal, single-stage adiabatic and two-stage adiabatic compression with perfect intercooling, and find the discharge temperatures.

  1. Isothermal: W = 10⁵ × 0.1 × ln 6 = 17.92 kW.
  2. Single-stage adiabatic: (γ − 1)/γ = 0.2857; 6^0.2857 = 1.668. W = 3.5 × 10⁴ × (1.668 − 1) = 23.40 kW. T₂ = 300 × 1.668 = 500.6 K.
  3. Two stages: intermediate pressure √(1 × 6) = 2.449 bar; per-stage ratio 2.449, 2.449^0.2857 = 1.2917. W = 2 × 3.5 × 10⁴ × 0.2917 = 20.42 kW. Each stage discharges at 300 × 1.2917 = 387.5 K. Isothermal 17.9 kW; single-stage 23.4 kW (T₂ ≈ 501 K); two-stage 20.4 kW (T₂ ≈ 388 K). Two-stage compression saves about 13 % of the power and cuts the discharge temperature by over 110 K.

Common mistakes

  • Throttling the discharge of a PD pump to control flow.
  • Using discharge-condition volume instead of suction volume in compression-work formulas.
  • Using bar or kPa with V in the wrong units: keep p in Pa and V̇ in m³/s for watts.
  • Using °C instead of K in the temperature-ratio formula.
  • Assuming a Roots blower compresses gas internally.
  • Calling isothermal work the actual work; it is the ideal minimum.

For GATE CH

Expect reciprocating pump discharge, slip and power numericals, compressor work (isothermal, adiabatic, multistage with optimum intermediate pressure), discharge temperature, clearance and volumetric efficiency, and conceptual questions comparing PD and centrifugal pumps or choosing a pump for viscous, corrosive or metering duty.

Quick check

  1. What is the optimum intermediate pressure for two-stage compression from 1 bar to 9 bar?
  2. Why must a PD pump have a relief valve?
  3. Single-acting pump: A = 0.01 m², L = 0.2 m, N = 120 rpm. Find Q_th.
  4. Which compression process requires the least work?

Answers: 1. 3 bar. 2. It keeps delivering against a closed or blocked line and would over-pressurise it. 3. 0.01 × 0.2 × 2 = 0.004 m³/s. 4. Isothermal.

Interview questions

All Fluid Mechanics interview questions

Try answering each one aloud before you open it.

  1. 1.What is a positive displacement pump and how does it differ from a centrifugal pump?Concept

    A positive displacement pump moves fluid by trapping a fixed amount and forcing (displacing) that trapped volume into the discharge pipe. It differs from a centrifugal pump, which imparts velocity to the fluid to increase its pressure. Positive displacement pumps are typically used for high-viscosity fluids and provide a constant flow regardless of pressure, whereas centrifugal pumps are used for low-viscosity fluids and their flow rate varies with pressure.

  2. 2.Explain the working principle of a reciprocating compressor.Concept

    A reciprocating compressor uses a piston within a cylinder to compress gas. As the piston moves down, it creates a vacuum that draws gas into the cylinder through an intake valve. When the piston moves up, it compresses the gas, increasing its pressure, and forces it out through a discharge valve. This cycle repeats to continuously compress gas.

  3. 3.What are the main components of a blower and how do they function?Concept

    A blower typically consists of an impeller, a casing, an inlet, and an outlet. The impeller rotates to draw air in through the inlet, increasing its velocity. The casing directs the high-velocity air to the outlet, where it is expelled at a higher pressure. Blowers are used to move air or gas with moderate pressure increases.

  4. 4.Why are positive displacement pumps preferred for handling viscous fluids?Application

    Positive displacement pumps are preferred for viscous fluids because they can maintain a consistent flow rate regardless of the fluid's viscosity. Unlike centrifugal pumps, which lose efficiency with high-viscosity fluids, positive displacement pumps can handle thick fluids without significant loss of performance.

  5. 5.What happens if a positive displacement pump is operated with a closed discharge valve?Application

    If a positive displacement pump is operated with a closed discharge valve, it can cause excessive pressure build-up, leading to potential damage to the pump or the system. Unlike centrifugal pumps, positive displacement pumps do not have a shut-off head, so they will continue to build pressure until something fails or a relief valve opens.

  6. 6.How does a change in gas density affect a blower's performance?Application

    A blower or fan essentially moves a fixed volume per revolution (positive displacement) or produces a fixed head in metres of gas (centrifugal), so efficiency itself changes little with density. For a centrifugal blower at fixed speed, the pressure rise and the shaft power both scale roughly in proportion to gas density. Hot or high-altitude (lighter) air gives less pressure rise and less mass flow for the same volumetric flow. Blowers are therefore rated at standard conditions and corrected for actual inlet density, and motors are sized for the densest, coldest start-up case.

  7. 7.Calculate the theoretical flow rate of a positive displacement pump with a displacement volume of 0.01 m³ per revolution operating at 1500 rpm.Numerical

    Theoretical flow = displacement per revolution × speed = 0.01 m³/rev × 1500 rev/min = 15 m³/min, or 0.25 m³/s. The actual delivery is lower by the slip (internal leakage back to suction), expressed as a volumetric efficiency, typically 0.9–0.98. Slip grows with discharge pressure and falls with liquid viscosity.

  8. 8.A reciprocating compressor has a cylinder diameter of 0.1 m and a stroke length of 0.2 m. Calculate the swept volume per cycle.Numerical

    The swept volume (V) per cycle can be calculated using the formula V = π/4 × D² × L, where D is the diameter and L is the stroke length. Here, V = π/4 × (0.1 m)² × 0.2 m = 0.00157 m³. Therefore, the swept volume per cycle is 0.00157 cubic meters.

  9. 9.Explain why compressors are used in refrigeration systems.Application

    Compressors are used in refrigeration systems to increase the pressure of the refrigerant gas, which raises its temperature. This allows the refrigerant to release heat when it passes through the condenser. After releasing heat, the refrigerant cools and condenses into a liquid, which can then absorb heat from the environment in the evaporator, completing the refrigeration cycle.

  10. 10.What are the advantages of using a screw compressor over a reciprocating compressor?Application

    Screw compressors have several advantages over reciprocating compressors, including smoother and quieter operation, higher efficiency at full load, and lower maintenance requirements due to fewer moving parts. They are also better suited for continuous operation and can handle larger volumes of gas, making them ideal for industrial applications.

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