Diaphragm Gas Meter

What is modelled

Each diaphragm splits a chamber into a front and a rear half. A slide valve lets gas into the half that is filling and out of the half that is emptying, so the crank turns once for every four chamber volumes that pass. The meter therefore measures volume by counting turns. The model adds the valve-timed flow ripple, a pressure drop that grows with flow, a small leak past the diaphragms, and the correction from the dial volume to the volume at base conditions.

Equations

• Chamber volume vi = (Vs/2)(1 + cos(θ - φi)) with phases 0, 180, 90 and 270 degrees and Vs = Vc/4.
• Outflow with ideal valves q(θ) = ω (Vs/2)(|sin θ| + |cos θ|); mean (4/π) ω Vs/2.
• Crank speed n = Q / (Vc), so 2.4 m3/h with 4.0 L per turn is 10 turns per minute (the 3D page shows 100 rpm because its clock runs 10 times faster).
• Pressure drop Δp = Δp0 + K Q2, leak qleak = k Δp, register error -qleak/Q.
• Base conditions Vb = V (p/p0)(T0/T) from the ideal gas law.

Checks run in the page

• The outflow integrated over one turn equals the displacement per cycle.
• The ripple ratio is the square root of two and the mean is 4/pi.
• Counting turns for an hour matches the integrated flow to within one cycle.
• The corrected volume holds the same number of moles as the dial volume.

Honest simplifications

The valve is ideal (no overlap or dead volume), the diaphragm leak is a single linear conductance, and the pressure drop constants are made-up round numbers of the right size, not data for a real meter. Gas compressibility and the temperature of the meter body are ignored. This is a teaching model, not a calibration tool.

• Choose Pilot flame: the error turns strongly negative because the starting loss and leak matter more at low flow.
• Choose Worn diaphragms and see the error leave the plus or minus 2 percent band.
• Choose Cold high pressure: the same dial volume holds more gas at base conditions.