A shearer cuts a web of coal along a panzer conveyor, the supports advance behind it. Cutting speed, the power limit, turn-around time and stoppages decide the output of a shift.
The science
Output. Each pass removes a slice of coal of volume V = L H w (face length L, cut height H, web depth w) and mass m = ρ L H w. While the shearer hauls at speed v the output rate is ρ H w v. Raising v raises output only until the motors reach their limit.
Power limit. Cutting power is the specific energy times the mass flow: P = SE · 60 ρ H w v (v in m/min), with SE in kWh/t. With installed power Pmax the fastest sustainable haulage speed is vP = Pmax / (60 SE ρ H w) (P in kW, v in m/min, SE in kWh/t; the 60 converts minutes to hours), so the actual speed is min(vset, vP). Harder coal or a thicker web lowers vP.
Cycle. One pass takes tp = L/v + tturn; the shearer cuts in both directions (bi-directional). In a shift with effective time T = 480 − stoppages minutes, the output is n m for n full passes plus the part of the next one already cut. The face advances by w per pass, and the roof supports advance behind the shearer (the goaf, collapsed rock, fills the space behind).
Numerical checks. (1) A time-stepped (0.5 s) state machine of the shearer (cutting, turning) must reproduce the closed-form shift output. (2) Energy: the integral of drawn power over the shift equals SE times the coal produced. (3) Geometry: face advance × L × H × ρ equals the coal mass counted from the mass flow.
Honest simplifications. Constant speed along the whole face, a single uniform web and seam height, no cleaning pass or sumping, no coal-flow or conveyor limit, stoppages lumped as a fixed time, and values of SE (0.4–3 kWh/t) and power chosen as typical ranges, not data for any specific machine. Ground control, gas and dust management, and the safety systems that dominate real longwall operation are outside this model.