Oil absorbs a massive amount of compression heat due to its high specific heat capacity, making the compression process near-isothermal.
Mathematical modelling serves two primary purposes:
Once the numerical integration settles into a steady-state cycle, integral parameters are calculated to evaluate the overall compressor performance. Indicator Diagram (
In oil-injected screw compressors, lubricating oil is sprayed directly into the working chamber during the compression process. This alters the mathematical model significantly due to multi-phase interactions.
| Parameter | Formula | Typical Range | |-----------|---------|----------------| | Volumetric efficiency | $ \eta_v = \frac\dotm del\rho_s \dotV th$ | 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\dotW ind\dotW shaft$ | 0.92 – 0.98 | | Total efficiency | $ \eta_total = \eta_v \cdot \eta_is \cdot \eta_m$ | 0.50 – 0.80 | Oil absorbs a massive amount of compression heat
Actual mass flow: m_dot = η_v × m_dot_th
Twin-screw compressors primarily consist of a male rotor with convex lobes and a female rotor with concave flutes contained within a close-fitting housing. The continuous compression cycle operates across four distinct sequential phases:
$$ P v = Z(P,T) R T $$
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 , power consumption , and discharge temperature —by simulating the compression cycle within the machine's changing control volumes . 1. Geometric Modelling This alters the mathematical model significantly due to
V(ϕ1)=∫0z(ϕ1)A(z,ϕ1)dzcap V open paren phi sub 1 close paren equals integral from 0 to z open paren phi sub 1 close paren of cap A open paren z comma phi sub 1 close paren space d z
The working chamber (the "lobe pocket") changes volume as the rotors turn.
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.
[ \dotm = A_leak \cdot p_u \sqrt\frac2\kappa(\kappa-1)RT_u \left[ \left( \fracp_dp_u \right)^\frac2\kappa - \left( \fracp_dp_u \right)^\frac\kappa+1\kappa \right] ] The primary goal is to predict key performance
As the rotors turn, the mesh line moves, creating a shifting cavity. The chamber volume
This ratio is fundamental. If the external system pressure ratio (discharge/suction) does not match the built-in ratio, occurs, reducing efficiency.
Why do we care about the math? Because it directly dictates the Performance Calculation
The indicated power represents the rate of work delivered by the rotors to the gas, calculated from the closed diagram loop:
Where ( n ) is the polytropic index. Excessive ( T_dis ) (over 225°C for air) risks lubricant coking.