Calculation =link= - Screw Compressors- Mathematical Modelling And Performance

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

3.2 Flow Through Clearances

For small gaps, flow is modelled as compressible, isentropic flow through a nozzle or viscous slit flow. The mass flow rate ( \dotmleak ) for a given clearance area ( Aleak ) and upstream/downstream pressures ( p_u, p_d ): Simplified empirical leakage fraction: φ_leak ≈ k_leak ×

3. The energy equation: The energy equation for a compressible fluid is:

• Set p(0) = p1, T(0) = T1, m_gas = p1 V1 / (R T1).

2.1 Governing Differential Equation

Applying the first law of thermodynamics to a chamber of volume ( V(\theta) ) (function of male rotor rotation angle ( \theta )): The energy equation : The energy equation for

References:

5. Numerical Implementation – A Practical Example

5.1 Input Data Required

• Rotor geometry files (profile coordinates)
• Rotor length, lead, wrap angle
• Clearance values (typically 0.05–0.2 mm)
• Suction pressure/temperature, discharge pressure
• Rotational speed
• Gas property tables or EOS parameters

2. The Thermodynamic Control Volume (The "Cell" Method) We don’t model the whole machine at once. Instead, each trapped gas pocket between rotor flutes is a moving control volume. Set p(0) = p1