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 ( m ) = mass inside chamber (
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. the volume between lobes decreases
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.
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: Mathematical Modelling and Performance Calculation
Modern industrial systems rely heavily on screw compressors for efficient gas compression in applications ranging from refrigeration to natural gas processing. The transition from intuitive design to high-performance machinery was driven by sophisticated mathematical modelling and performance calculation. 1. Mathematical Foundations of Rotor Geometry reduce in volume
The performance of a screw compressor is fundamentally dictated by its rotor profile. Mathematical modelling begins by defining the coordinate systems for the male (lobe) and female (groove) rotors.
Coordinate Systems: A right-handed system is typically attached to each rotor ( -axis along the rotor axis, -axis perpendicular).
Profile Generation: Modern asymmetric rotor profiles are designed using enveloping theory to minimize the "blow-hole" area—the primary source of internal leakage. Volume Calculation: The instantaneous working volume is a function of the rotation angle
. This volume decreases as the rotors mesh, leading to compression. 2. Thermodynamic Modelling of the Compression Process
The core of performance calculation involves solving a set of coupled differential equations derived from the conservation of mass and energy. Screw Compressors - Springer Nature
Applying the first law of thermodynamics to a chamber of volume ( V(\theta) ) (function of male rotor rotation angle ( \theta )):
[ \fracd(mu)d\theta = \dotmin \cdot hin - \dotmout \cdot hout + \dotQ \cdot \fracdtd\theta - p \fracdVd\theta ]
Where:
A screw compressor consists of two mating helical rotors (male and female) enclosed in a casing. As rotors rotate, the volume between lobes decreases, compressing the trapped gas.
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.