Screw Compressors- Mathematical Modelling And Performance Calculation -

Modern performance prediction relies heavily on translating this physical hardware into accurate mathematical equations. 2. Kinematic and Geometric Modeling

Applying the first law of thermodynamics to the transient control volume yields the temperature or internal energy change:

In many applications (especially refrigeration), oil or refrigerant liquid is injected to cool the compressor and seal gaps. Modelling this requires adding terms for two-phase flow and evaporation energy in the energy balance equation.

Models use differential equations to calculate changes in pressure and temperature relative to the rotation angle. Real Gas Effects:

The internal compression process is modeled using a non-steady open control volume approach. The first law of thermodynamics and the conservation of mass govern the state of the gas inside the chamber. Mass Conservation The change in gas mass ( Modelling this requires adding terms for two-phase flow

While 1D models are fast and effective for design, they often rely on empirical correlations for leakage and heat transfer. More advanced methods are now used to resolve the complex three-dimensional, transient, compressible flow inside the compressor. CFD can capture local effects like pressure non-uniformities, shock waves, and detailed jet interactions from oil injection, providing highly accurate data for validating and refining simpler models. A common CFD approach involves dividing the internal flow domain into three distinct fluid zones—inlet fluid, primitive volume fluid, and outlet fluid—and solving the Navier-Stokes equations on a dynamic mesh that moves with the rotating rotors. However, CFD is computationally expensive for design optimization studies.

) within a control volume results from mass fluxes across the suction port, discharge port, and leakage paths:

) required to compress the gas is calculated by integrating the work over the compression cycle:

[xfyf]=[cos(θm+θf)sin(θm+θf)−sin(θm+θf)cos(θm+θf)][xmym]+[a⋅cosθf−a⋅sinθf]the 2 by 1 column matrix; x sub f, y sub f end-matrix; equals the 2 by 2 matrix; Row 1: Column 1: cosine open paren theta sub m plus theta sub f close paren, Column 2: sine open paren theta sub m plus theta sub f close paren; Row 2: Column 1: negative sine open paren theta sub m plus theta sub f close paren, Column 2: cosine open paren theta sub m plus theta sub f close paren end-matrix; the 2 by 1 column matrix; x sub m, y sub m end-matrix; plus the 2 by 1 column matrix; Row 1: a center dot cosine theta sub f, Row 2: negative a center dot sine theta sub f end-matrix; is the center distance between the rotors. θmtheta sub m θftheta sub f The first law of thermodynamics and the conservation

The instantaneous state of the gas inside a single compressor chamber is tracked using a system of differential equations relative to the rotor rotation angle ( Mass Conservation Equation The rate of change of gas mass (

within a single compression chamber equals the sum of mass flows entering minus the mass flows leaving:

The rotor profiles determine the sealing lines and the volume curve. Modern profiles use asymmetric combinations of circles, involutes, and cycloids to minimize the blow-hole area (the leakage path between the rotor tips and the casing).

Screw compressors are positive displacement rotary machines widely used in refrigeration, air compression, and industrial processes. Optimizing their design requires a deep understanding of the interaction between rotor geometry and thermodynamic processes. This report outlines the fundamental approaches to mathematical modelling of screw compressors, focusing on the geometric definition of rotors, the thermodynamic chamber model, and the calculation of performance indicators such as volumetric efficiency and indicated power. the thermodynamic chamber model

[ T_dis = T_suc \cdot \left( \fracp_disp_suc \right)^\fracn-1n \cdot \frac1\eta_ad ]

is the heat transfer rate between the gas, rotors, and housing. represents the mechanical work done via volume reduction. 3. Modelling Internal Leakages and Clearance Paths

While lump-parameter thermodynamic models (zero-dimensional or one-dimensional models) provide quick performance calculations, complex engineering challenges require higher-fidelity simulations.