Steam Locomotive Boiler

What is modelled

Coal burns and sends hot gas through a bank of tubes that run through the boiler water. The water temperature fixes the steam pressure through the saturation curve. The cylinders draw steam in proportion to the wheel speed, the cylinder size and the cut-off, and the heat they remove, plus the cold feed water they replace, is balanced against the heat from the tubes. A safety valve vents any surplus. The pull at the rails comes from the mean pressure in the cylinders.

Equations

• Saturation pressure from the Antoine equation; latent heat hfg ≈ 2501 - 2.36 T kJ/kg (T in C).
• Tube heat transfer Q = Cg (1 - e-UA/Cg)(Tgas - Twater), with UA = U n π d L and gas flow Cg = coal rate x 14 x cp.
• Water heating Mw cw dT/dt = Q - ṁs(hfg + cw(T - Tfeed)).
• Steam draw ṁs = 4 nrev (π d²/4) s c ρsteam.
• Mean pressure p c (1 + ln(1/c)) - pback; tractive effort TE = pmean d² s / D.

Checks run in the page

• The saturation pressure against a numerical integration of the Clausius-Clapeyron equation.
• The gas temperature marched along the tube against the effectiveness formula.
• The time to raise steam from cold against the closed form.
• The indicator diagram integrated over the stroke against the closed form.

Honest simplifications

The heat transfer coefficient is a constant 50 W/m2K, the inlet gas temperature is fixed at 1100 C, the water is one well-mixed lump at saturation, steam is dry with no priming, and the cylinder is ideal: no clearance volume, no valve throttling and isothermal expansion after cut-off. Real locomotives turned 4 to 8 percent of the coal energy into work; the model does not predict that figure. The clock runs 40 times faster than real time. The 3D cutaway is illustrative and its slider labels are not calibrated to these units.

• Choose Starved fire: the heat cannot keep up with the steam draw and the pressure sinks.
• Choose Many small tubes: the heating surface grows and the same coal raises more steam.
• Choose Standing, safety valve and watch the surplus heat vent.