Engineering
External gear pump
Run a gear pump into a load valve and close the valve: the pressure, the slip leakage and the motor’s torque rise together until the relief valve lifts and the motor slows on its torque line. Open the gears’ clearance and watch the volumetric efficiency fall at the same drive; without leakage the mean flow is exactly the displacement times the speed. The shaft’s work is booked against the useful flow, the leak, the relief and the friction.
What happens
A 10 cm³/rev gear pump on a motor with a straight torque-speed line (20 N·m stall, 300 rad/s no load) feeds a 0.2 L line with a load orifice and a relief valve. The gears deliver V_d·ω/2π; laminar slip through the clearances takes G·δ³·p/μ back; the relief poppet opens between 100 and 110 bar through a 5 ms lag; the line’s pressure follows the oil’s bulk modulus. The pump asks V_d·p/2π plus Coulomb and viscous friction from the shaft, so closing the valve raises the torque and slows the motor on its line. RK4 at 10 µs; the shaft’s work is found again in the flow’s work, the leak, the relief, the friction and the oil stored in the line.
Try this
Close the load valve slowly and watch the pressure, the leakage and the torque rise until the relief valve lifts; open the clearance to 60 µm and read the volumetric efficiency; set it to zero and the flow is the displacement times the speed.
The model and its limits
Lumped line volume, laminar slip leakage with a geometric constant, turbulent orifices with a constant coefficient, no flow ripple or trapped-volume effects, a motor without thermal limits.
Equations used in the module
Equations Q_th = V_d·ω/(2π), Q_leak = G·δ³·p/μ dp/dt = β/V·(Q_th − Q_leak − Q_out − Q_relief) T = V_d·p/(2π) + T_f + c·ω, T_m = T_s·(1 − ω/ω_0) η_v = 1 − Q_leak/Q_th, η_m = V_d·p/(2π·T)
