Screw Compressors- Mathematical Modelling And Performance Calculation 〈2024〉

Screw Compressors: Mathematical Modelling and Performance Calculation

$$ \dotm leak = C_d \cdot A gap \cdot \sqrt \frac2R T_up \cdot \frac\kappa\kappa-1 \left[ \left( \fracP_downP_up \right)^\frac2\kappa - \left( \fracP_downP_up \right)^\frac\kappa+1\kappa \right] $$

Between the high-pressure rotor end-faces and the discharge bearing housing.

Presents a generalized mathematical definition of screw machine rotors. Detailed analysis of asymmetric rotor profiles

The ultimate goal of modelling is to quantify and predict performance. Performance is evaluated through a set of key metrics, which in turn require a clear calculation methodology. Performance is evaluated through a set of key

For air, the ideal gas law often suffices. However, for refrigerants or process gases, we must integrate real gas equations of state (like Peng-Robinson or NIST REFPROP) into the model to ensure accuracy in enthalpy and density calculations. 3. Fluid Flow and Leakage Modelling

: Accounts for the internal energy changes, work done by the rotors, and heat transfer between the gas, oil (in oil-injected models), and the casing.

, on the other hand, involve solving the mathematical models described earlier, using numerical methods such as:

is the mass flow rate of injected oil (for oil-injected compressors). Energy Conservation adiabatic (isentropic) process.

$$ \eta_is = 20.1 / 23.65 = 0.85 \text (85%) $$

Evaluates how thermal expansion and high gas pressures structurally deform the rotors. This helps engineers optimize clearance sizes without risking mechanical rotor-to-rotor contact.

ηs=ṁactual⋅(hdischarge,isentropic−hsuction)Wshafteta sub s equals the fraction with numerator m dot sub a c t u a l end-sub center dot open paren h sub d i s c h a r g e comma i s e n t r o p i c end-sub minus h sub s u c t i o n end-sub close paren and denominator cap W sub s h a f t end-sub end-fraction Computational Algorithm for Simulation

(for a control volume within a working chamber): and prone to vibration. In response

A typical model assumes the behaviour of an ideal gas and uses standard thermodynamic relations. The mass flow rate into or out of the working chamber through suction and discharge ports, as well as through leakage paths, is calculated using orifice flow equations. For the gas–oil mixture within the cavity, the equations of mass and energy can be written as:

[ \eta_v = \frac\dotV actual\dotV swept ]

ηs=PisentropicPactualeta sub s equals the fraction with numerator cap P sub i s e n t r o p i c end-sub and denominator cap P sub a c t u a l end-sub end-fraction 5. Oil Injection Modelling

It was the early 20th century, and the industrial world was in need of more efficient and reliable compressors to power their machinery. The reciprocating compressors of the time were cumbersome, noisy, and prone to vibration. In response, the screw compressor was born. Over the years, the design and performance of screw compressors have been shaped by mathematical modeling and performance calculation.

This measures how close the actual compression process is to an ideal, reversible, adiabatic (isentropic) process.