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Duty and geometry
The built-in volume ratio fixes the pressure p_i = p1 Vi^kappa at which the pocket opens to the discharge port. Efficiency is highest when the system pressure p2 equals p_i. Throttling the inlet lowers p1 and the flow.
Model checks
The science
What is modelled
Each pocket between the lobes traps air at the inlet pressure, shrinks by the built-in volume ratio Vi as the rotors turn, and then opens to the discharge port. The gas is compressed polytropically with no heat exchange. If the system pressure differs from the built-in pressure, the port opens too early (under-compression, gas rushes back in) or too late (over-compression, work is wasted).
Equations
p Vκ = const, sopi = p1 Viκ,κ = 1.4.- Isentropic work per kg:
wis = κ/(κ-1) R T1 [(p/p1)(κ-1)/κ - 1]. - Flow:
Q = n Vd ηv, mass flowρ1 Q, with a slip-type volumetric efficiency that falls with clearance, pressure ratio and low speed. - Under-compression adds
(p2 - pi) V2of irreversible work.
Checks run in the page
- Numerical integral of p dV against the closed form.
- Matched duty gives efficiency 1.
- RK4 integration of the polytropic law.
- Ideal-gas consistency of the end state and mass flow.
Honest simplifications
Linear volume reduction over the stroke, no oil or water injection, no heat transfer, a made-up but plausible slip law for the volumetric efficiency, a constant 96 percent mechanical efficiency and an inlet pressure that falls linearly with the valve closing. The rotors are drawn as simple lobed discs, not true screw profiles. Not a compressor design tool.
What to try
- Press Matched and read an adiabatic efficiency of 100 percent when p2 equals p_i.
- Press Over-compression: the pocket reaches p_i, which is well above the line pressure, and the efficiency drops.
- Press Worn rotors: the clearance leak cuts the flow.