Screw Compressors- Mathematical Modelling And Performance Calculation

Models flow along the rotor axis, capturing pressure waves and velocity distribution. Used for high-speed compressors.

For refrigerants or high-pressure gas, compressibility factor $Z(P,T)$ must be used. Equations like Peng-Robinson or Soave-Redlich-Kwong are applied.

| Parameter | Formula | Typical Range | |-----------|---------|----------------| | Volumetric efficiency | $ \eta_v = \frac\dotmdel\rho_s \dotVth$ | 0.75 – 0.98 | | Isentropic efficiency | $ \eta_is = \frach_dis,is - h_sh_dis - h_s$ | 0.70 – 0.88 | | Mechanical efficiency | $ \eta_m = \frac\dotWind\dotWshaft$ | 0.92 – 0.98 | | Total efficiency | $ \eta_total = \eta_v \cdot \eta_is \cdot \eta_m$ | 0.50 – 0.80 | Models flow along the rotor axis, capturing pressure

The thermodynamic model simulates the change in gas properties (Pressure $P$, Temperature $T$, Mass $m$) inside the working chamber as a function of the rotation angle.

The instantaneous volume of a working chamber depends on the rotation angle $\theta$. The thermodynamic model simulates the change in gas

For a symmetric profile: $$ V(\theta) = V_max \cdot \left[ 1 - \frac\theta\theta_w \cdot (1 - \frac1V_i) \right] $$

More precisely, the male rotor volume variation for ideal profiles: $$ V(\theta) = A_s \cdot L - A_int(\theta) \cdot L $$ Models flow along the rotor axis

Where $A_s$ is suction port area and $A_int(\theta)$ is interlobe area.

Volume derivative: $$ \fracdVd\theta = - \omega \cdot \dotV $$ Where $\omega$ is rotational speed and $\dotV$ is volumetric displacement rate.