Screw Compressors- Mathematical Modelling And Performance Calculation Official

| 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 |

Assumes uniform pressure and temperature at each time step. Most common for preliminary design.

Governing equations (for a control volume within a working chamber):

Continuity: $$ \fracdmd\theta = \dotmin - \dotmout + \sum \dotm_leaks $$

Energy (First Law): $$ \fracdUd\theta = \dotQ - \dotW + \dotmin hin - \dotmout hout + \sum (\dotmleak hleak) $$

Where:

Equation of state (real gas): $$ P v = Z(P,T) R T $$

The rotation is discretized into small steps ( \Delta\theta ) (e.g., 0.5° to 1.0°). At each step:

For many gases (especially refrigerants like R134a or hydrocarbons), ideal gas law fails. A real gas equation like Peng-Robinson or NIST REFPROP correlations is used:

[ p = \fracRTv - b - \fraca(T)v(v+b) + b(v-b) ]

Where ( v ) is specific volume, ( a(T) ) and ( b ) are fluid-specific parameters.

No model is complete without experimental validation. Key calibration steps:

A well-calibrated model predicts volumetric efficiency within ±2% and power within ±3%. | Parameter | Formula | Typical Range |

The working chamber is treated as an open thermodynamic system (control volume). The governing equations are derived from the conservation laws of mass and energy.

A. Conservation of Mass: $$ \fracdmd\phi = \fracd\dotmsucd\phi - \fracd\dotmdisd\phi + \fracd\dotmleak,ind\phi - \fracd\dotmleak,outd\phi $$

Where:

B. Conservation of Energy: The First Law of Thermodynamics for a control volume is applied: $$ \fracd(mu)d\phi = -P\fracdVd\phi + \sum \dotminhin - \sum \dotmouthout + \fracdQd\phi $$

Where:

Oil-injection improves sealing and cooling. Additional terms in energy equation: $$ \fracdUd\theta = \dotQgas + \dotQoil - \dotW + \dotmin hin - \dotmout hout + \dotmoil cp,oil (T_oil - T_gas) $$