Screw Compressors- Mathematical Modelling — And Performance Calculation Patched
Screw compressors are a cornerstone of modern industrial systems, ranging from refrigeration to high-pressure air production. Their effectiveness is largely defined by their internal rotor geometry and the thermodynamic efficiency of the compression cycle. 1. Mathematical Modelling of Geometry
- Geometry and Kinematics: The geometry of the screw compressor, including the rotor profiles, housing, and clearances, is defined. The kinematics of the compressor, including the rotation of the screws and the movement of the fluid, are also analyzed.
- Thermodynamic Processes: The thermodynamic processes occurring within the compressor, such as compression, heating, and cooling, are modeled using equations of state and thermodynamic relations.
- Fluid Dynamics: The fluid dynamics of the compressor, including the flow of the fluid through the compressor, are simulated using computational fluid dynamics (CFD) techniques or simplified models.
4. Thermodynamic Modelling
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. Screw compressors are a cornerstone of modern industrial
The Story of Screw Compressors: Unveiling the Secrets of Mathematical Modelling and Performance Calculation Geometry and Kinematics : The geometry of the
): The instantaneous volume of the compression chamber is calculated as a function of the rotation angle ( ). over/under compression is avoided (optimal efficiency).
4. Mechanical & Friction Losses
- Mechanical power input P_in = P_comp + P_mech_losses + P_parasitic.
- Frictional losses:
If built-in $V_i$ matches system pressure ratio, over/under compression is avoided (optimal efficiency).
- Flow Equation: Flow is typically modelled using compressible flow through an orifice/nozzle equation: $$ \dotm_leak = C_d A \sqrt\frac2kk-1 P_1 \rho_1 \left[ \left(\fracP_2P_1\right)^\frac2k - \left(\fracP_2P_1\right)^\frack+1k \right] $$ Where $C_d$ is the discharge coefficient and $A$ is the clearance area.