Underground Mine Ventilation Network: Atkinson Resistance, Fan Curve, Regulator Doors and Methane Dilution

Presets

Live readouts

Graphs

Main fan curve (orange) and mine curve (blue): operating point

Methane at faces and in the return, last 15 minutes

Pressure along the two airflow routes (Pa)

Airflow in every airway (m3/s)

Numerical checks

running...

How to use

What to try: press Door left open and read the crosscut airflow. Throttle district 1 with its regulator and watch the split change. Switch to Gassy face 2 and see the gas alarm, then add the booster fan or raise the main fan. Fan failure shows what natural ventilation alone can do, and Winter NVP / Summer NVP show it helping or fighting the fan. Press Gas burst for a 30 s surge at face 1 and watch the slug travel to the return.

The science

Mine air moves because a fan (or a temperature difference between the shafts) creates a pressure difference, and it loses that pressure to friction and shock in every airway. The page solves the whole network at once.

Atkinson's law. The pressure lost in one airway is quadratic in flow:

H = R Q^2, R = k L P / A^3

k friction factor (kg/m3, about 0.004 for a smooth shaft, 0.012 for a roadway, 0.02 for a face with equipment), L length, P perimeter, A area. Halve A and R rises eightfold: that is the rock fall preset.

Kirchhoff's laws. At every junction the flows balance, sum Q = 0. Around any closed loop the pressures balance, sum (R Q|Q| - fan head) = 0. The page solves this with a damped Newton method on the junction pressures and checks it against an independent Hardy Cross loop solution dQ = - sum(R Q|Q| - H_f) / sum(2 R |Q|)

Fan. The fan curve is modelled as H_f = H0 s^2 - a Q^2 where s is relative speed (fan law: pressure scales with speed squared). The operating point is where it meets the mine curve H = R_mine Q2, which is what the first graph shows.

Regulators and doors. A louvre bank with open area fraction f adds a shock loss R_reg = rho / (2 A^2 C_c^2) * (1/f - 1)^2 with contraction coefficient C_c = 0.65. At f = 1 it adds nothing; a nearly closed door leaks a little (f is never below 0.4%).

Natural ventilation pressure. Cold surface air makes the intake shaft column heavier than the upcast column: NVP = rho g Z (T_up - T_in) / T with Z = 420 m. It helps in winter and opposes the fan in summer.

Methane. Gas released at a face at q (m3/s) is diluted by the airflow Q. Each airway is split into cells that carry the methane mixing ratio c downstream with an upwind scheme, V dc/dt = Q (c_in - c) + q and junctions mix inflows by flow weight. At steady state the face value is q/Q and the return value is sum(q)/Q_upcast; the page runs a separate copy of the transport model to steady state and compares.

Energy check (Tellegen). Power dissipated in all airways, sum R|Q|^3, equals the power injected by the fans and natural ventilation.

Thresholds. The colours use 1.0% (warning) and 1.5% (withdraw); methane is explosive between about 5% and 15% in air. Limits differ between countries and mines: these values are illustrative.

Honest simplifications. Steady incompressible flow per time step (the fan and doors move with first-order lags, the gas is transported in time); the fan curve is a smooth parabola without the stall region of a real fan; no leakage through stoppings, no air density change with depth, no heat or humidity, no gas layering at the roof, and the booster fan is not checked for recirculation. Geometry, resistances and fan data are plausible round numbers, not a real mine. This is an educational model, not a design or safety tool.

Live checks. (1) junction flow residual, (2) Newton vs Hardy Cross, (3) Tellegen power balance, (4) Atkinson law on the main drift, (5) closed-form solver self-test, (6) methane steady state vs q/Q.

This model is a generic teaching network; no real mine, company or operating procedure is represented.