Screw Compressors- Mathematical Modelling And Performance Calculation -
The working chamber (the "lobe pocket") changes volume as the rotors turn.
The displacement volume (Vth) per revolution is:
[ V_th = z_1 \cdot A_flow \cdot L ]
Where:
The built-in volume ratio (R_v) is:
[ R_v = \fracV_sucV_dis ]
Where:
This ratio is fundamental. If the external system pressure ratio (discharge/suction) does not match the built-in ratio, under- or over-compression occurs, reducing efficiency.
At pressure ratio = 4.5, speed = 3000 rpm:
- Volumetric efficiency = 82.3%
- Adiabatic efficiency = 76.1%
- Leakage fraction: blowhole = 8.2%, radial = 5.4%
Would you like a sample MATLAB/Python code structure for implementing this feature, or a mathematical derivation of the leakage model?
Screw Compressors: Mathematical Modelling and Performance Calculation Nikola Stosic, Ian K. Smith, and Ahmed Kovacevic
is a seminal English-language text that provides a rigorous analytical framework for designing and optimizing twin-screw machines. Springer Nature Link Core Content and Structure
The work is structured into five distinct parts that bridge the gap between abstract mathematical theory and industrial application: Amazon.com Part 1: Historical and Technical Review
: Provides context on recent developments in design and manufacturing, such as the shift from symmetric to asymmetric rotor profiles which significantly reduced internal leakage. Part 2: Rotor Geometry
: Presents a generalized mathematical definition for rotor lobes. It details how to derive complex shapes that maintain contact while minimizing the "blow-hole" area. Part 3: Thermodynamics and Fluid Mechanics
: Establishes the differential equations for the compression and expansion processes. It covers mass and energy conservation, heat transfer, and the impact of oil injection in flooded machines. Parts 4 & 5: Practical Application
: Demonstrates how to apply these analytical models to real-world twin-screw compressors. It includes examples of multi-variable optimization
to find the ideal rotor size, speed, and injection positions for specific duties. Key Strengths
Screw Compressors: Mathematical Modelling and Performance Calculation a foundational engineering textbook written by Nikola Stosic Ahmed Kovacevic . Originally published by
in 2005, it serves as a comprehensive guide for the design, analysis, and optimization of twin-screw machines. Amazon.com Core Content & Structure
The book is structured into three primary parts that bridge the gap between theoretical geometry and practical thermodynamic performance: Rotor Design & Geometry Reviews recent developments in screw compressor design. The working chamber (the "lobe pocket") changes volume
Presents a generalized mathematical definition of screw machine rotors. Detailed analysis of asymmetric rotor profiles
, which significantly improved efficiency since their introduction in 1973. Thermodynamic & Fluid Modeling
Treats the thermodynamics and fluid mechanics of the compression and expansion processes. differential equations for conservation of mass and energy based on the Reynolds transport theorem
Models the internal working chamber as an open thermodynamic system with time-varying mass flow. Performance & Optimization Addresses issues like clearance management , rotor configuration, and scale.
Calculates pressure forces, torque, and rotor bending deflections.
Details methods for optimizing geometrical parameters (e.g., wrap angle, built-in volume ratio) to minimize power consumption and maximize efficiency. Better World Books Key Technical Concepts
Mathematical modelling and performance calculation are the cornerstones of modern screw compressor design, transitioning the industry from empirical "trial-and-error" methods to precise computer-aided engineering
. This analytical approach is essential for optimizing complex rotor profiles and predicting performance across varying operating conditions. Springer Nature Link 1. Geometric Modelling
The foundation of any screw compressor model is the geometric definition of the rotors and their intermeshing cycle. Screw Compressors - Springer Nature 14 Oct 2010 —
Mathematical modelling and performance calculation of screw compressors involve a multi-layered approach that integrates complex rotor geometry with thermodynamic and fluid flow principles . The primary goal is to predict key performance characteristics—such as volumetric efficiency, power consumption, and discharge temperature—by simulating the compression cycle within the machine's changing control volumes . 1. Geometric Modelling
The foundation of any screw compressor model is the accurate mathematical definition of the rotor profiles . Profile Generation: This involves defining the
coordinates of the main and gate rotor lobes, often using rack-generation techniques or analytical curves to ensure seamless meshing .
Volume Curves: The model calculates the instantaneous volume of the working chamber as a function of the rotation angle (
Clearance Areas: Critical for performance, the model must define leakage paths—including interlobe, radial, and end clearances—as these are the primary sources of efficiency loss . 1476.pdf - Purdue e-Pubs
Screw compressors are a cornerstone of modern industrial systems, ranging from refrigeration to high-pressure air production. Their effectiveness is largely defined by their internal rotor geometry and the thermodynamic efficiency of the compression cycle. 1. Mathematical Modelling of Geometry
The core of any screw compressor model is the geometric definition of the rotor profiles. Traditionally, rotors were designed using empirical curve fitting, but modern models use the mathematical theory of gearing for precise development.
Rotor Profile Generation: Contemporary designs often utilize asymmetric rotor profiles, which can reduce the "blow-hole" area (a major source of internal leakage) by up to 90% compared to older designs.
Coordinate Transformations: Mathematical equations describe the position of any point on the rotor in a coordinate system as a function of the rotation angle Volume Calculation: The working chamber volume
is defined as a function of the rotation angle. As the rotors mesh, this volume decreases, which results in the physical compression of the trapped gas. 2. Thermodynamic Process Modelling Use for moderate heat transfer; n fit from
Performance prediction relies on the conservation laws of mass and energy applied to the varying chamber volume.
Mathematical modeling of screw compressors is essential for optimizing energy efficiency, as these machines consume approximately 15–20% of global electrical power. By simulating thermodynamics and fluid mechanics, engineers can predict performance before physical prototyping. 1. Geometric Modeling
The foundation of any screw compressor model is the rotor geometry. The working chamber is formed by the meshing of helical lobes (typically male and female rotors) within a fixed housing.
Rotor Profiles: Modern designs use asymmetric profiles to minimize "leakage triangles" and improve efficiency. Volume Calculation: The instantaneous volume ( ) is a function of the rotation angle (
Kinematic Relationship: A differential equation describes the change in volume over time (
), which is critical for defining the suction, compression, and discharge phases. 2. Thermodynamic Modeling
The core of the performance calculation involves solving conservation equations for the working fluid. 1476.pdf - Purdue e-Pubs
Screw Compressors: Mathematical Modelling and Performance Calculation
Screw compressors are the workhorses of modern industry, providing reliable compressed air and gas for everything from food processing to large-scale refrigeration. While their exterior looks like a simple metal casing, the interior houses a complex dance of geometry and thermodynamics.
Understanding how to model these machines mathematically is essential for engineers looking to optimize efficiency, reduce noise, and predict performance under varying conditions. 1. The Geometric Foundation: Rotor Profiling
The heart of a screw compressor is the pair of helical rotors (male and female). Mathematical modelling begins with the rotor profile generation.
Rotor Geometry: The rotors must maintain a continuous line of contact to prevent leakage. This is typically defined using rack-generated profiles or "N" profiles.
Volume Curve: As the rotors turn, the space between the lobes (the working chamber) changes. We model this as a function of the rotation angle . The volume
starts at a maximum during suction and decreases to a minimum at the discharge port.
Sealing Lines and Blowhole: No seal is perfect. Mathematical models must calculate the length of sealing lines and the area of the "blowhole"—the tiny triangular gap where the two rotors and the housing meet. This is a critical factor in volumetric efficiency. 2. Thermodynamic Modelling: The Control Volume Approach
To calculate performance, we treat the compression chamber as a transient control volume. We apply the laws of thermodynamics to the fluid as it moves from suction to discharge. The Governing Equations
We use differential equations to track the state of the gas: Conservation of Mass:
This accounts for the main flow plus internal leakages (backflow) and oil injection. Conservation of Energy: is internal energy, is heat transfer, is work, and is enthalpy. Real Gas Effects
For air, the ideal gas law often suffices. However, for refrigerants or process gases, we must integrate real gas equations of state (like Peng-Robinson or NIST REFPROP) into the model to ensure accuracy in enthalpy and density calculations. 3. Fluid Flow and Leakage Modelling The displacement volume (Vth) per revolution is: [
Efficiency is largely dictated by what doesn't get compressed. Leakage paths include:
Leading/Trailing Edge Leaks: Gas escaping between the rotor tips and the housing.
Inter-lobe Leaks: Flow across the contact line between rotors.
Blowhole Flow: Flow through the aforementioned geometric gap.
These are typically modelled as isentropic nozzle flows with discharge coefficients ( Cdcap C sub d ) applied to account for friction and turbulence. 4. The Role of Oil Injection
Most screw compressors are "oil-flooded." Oil serves three purposes: sealing, lubrication, and cooling. In a mathematical model, the oil is treated as an incompressible fluid that exchanges heat with the gas.
Heat Transfer: The high surface area of oil droplets allows for nearly isothermal compression, which is much more efficient than adiabatic compression.
Sealing: The presence of oil in the gaps significantly reduces gas leakage rates. 5. Performance Calculation Metrics
Once the differential equations are solved (usually via numerical methods like Runge-Kutta), we can calculate the key performance indicators (KPIs): Volumetric Efficiency ( ηveta sub v
): The ratio of actual delivered gas to the theoretical displacement. Isentropic Efficiency ( ηseta sub s
): How close the process is to an "ideal" frictionless compression.
Specific Power: The power required per unit of flow rate (kW/m³/min). This is the ultimate "utility bill" metric for the end-user.
Discharge Temperature: Crucial for ensuring the oil and seals don't degrade. 6. Advanced Considerations: Porting and Dynamics
Modern modelling also looks at pressure pulsations. As the discharge port opens, there is often a "pressure mismatch" (over-compression or under-compression). This creates shock waves that lead to noise and vibration. Advanced models use CFD (Computational Fluid Dynamics) to optimize the shape of the discharge port to minimize these losses. Conclusion
Mathematical modelling of screw compressors has evolved from simple "black box" calculations to sophisticated simulations that account for micron-level clearances and complex fluid-structure interactions. By mastering these models, manufacturers can push the boundaries of energy efficiency, making industrial processes more sustainable and cost-effective.
Before any performance calculation begins, one must accurately define the rotor geometry. A twin-screw compressor consists of a male rotor (convex lobes) and a female rotor (concave flutes). The meshing of these rotors creates moving chambers that trap, reduce in volume, and discharge gas.
This feature calculates the instantaneous volumetric efficiency of a twin-screw compressor by dynamically modeling internal leakages (through rotor clearances, blowholes, and discharge gaps) and real-gas properties of the working fluid (e.g., refrigerants or process gases).
Modern modelling uses the rack generation method: the rotor profiles are defined by a series of analytic curves (circles, ellipses, cycloids) that satisfy the meshing condition. The centre distance ( C ) and rotor radii ( r_1, r_2 ) must satisfy:
[ C = r_1 + r_2 ]
The contact ratio ensures no leakage between lobes. Mathematically, the sealing line length is minimized to reduce blow-hole leakage.
From the thermodynamic model, the following performance parameters are extracted.