The rate of change of gas mass within the cavity is the difference between leakage into and out of the cavity:
Screw Compressors: Mathematical Modelling and Performance Calculation
$$ \eta_v = \frac\dotm delivered\rho suction \cdot \dotV_theor $$
The foundation of any performance calculation model is the precise geometric definition of the rotors and the working chamber. Rotor Profile Generation The rate of change of gas mass within
| Path | Description | Significance | |------|-------------|--------------| | Blow-hole | Triangular gap at rotor end | High | | Seal line | Between rotor lobes | Medium | | Radial gap | Between rotor tip and casing | Medium | | End face gaps | Between rotor face and housing | Low |
The mathematical representation of a screw compressor typically involves two primary layers: geometric modelling and thermodynamic process simulation. 1. Geometric Modelling
ηv=ṁactualṁtheoretical=ṁactualρsuction⋅Vmax⋅zm⋅n60eta sub v equals the fraction with numerator m dot sub a c t u a l end-sub and denominator m dot sub t h e o r e t i c a l end-sub end-fraction equals the fraction with numerator m dot sub a c t u a l end-sub and denominator rho sub s u c t i o n end-sub center dot cap V sub m a x end-sub center dot z sub m center dot n over 60 end-fraction end-fraction is the rotor speed in RPM. 2. Indicated Power ( Windcap W sub i n d end-sub : Heat transfer rate between gas
If built-in $V_i$ matches system pressure ratio, is avoided (optimal efficiency).
Performance is typically calculated by solving the conservation laws for an open system (the compression chamber). : Internal energy. : Enthalpy of inlet/outlet gas. : Heat transfer rate between gas, rotors, and casing. : Work done by the piston-like action of the rotors. Mass Conservation (Continuity): : Instantaneous mass in the chamber.
Pressure (p) ▲ │ /─► Discharge Port Opens │ / │ │ / │ │ / │ │ / │ │ / │ │ ┌─────┘ │ │ │ │ └───┴─────────────┴────────► Volume (V) Suction Compression Volumetric Efficiency ( ηveta sub v is avoided (optimal efficiency).
The power calculated based on the P-V diagram (the area inside the pressure-volume loop).
If you are interested in a deeper analysis, I can provide information on specialized software used for simulation (such as SCORG ) or focus on specific applications like refrigeration or high-pressure gas.