Sternwheel Steamboat Thrust

What is modelled

Each blade that is under the surface acts like a flat plate pushed through water at the speed of the blade relative to the water. The blades push water backwards, the reaction drives the boat forwards, and the boat speeds up until the hull drag, which grows with the square of speed, equals the blade thrust. The paddle slip is the fraction of the blade speed that the boat does not make up.

Equations

• Mean blade speed u = 2 π N (R − d/2) / 60, immersed blade count n = 2 acos((R − d)/R) / (2 π / Nb).
• Thrust T = ½ ρ Cb n b d (u − v)², hull drag D = ½ ρ (CdS) v².
• Balance T = D gives v = u / (1 + √(CdS / (Cb n b d))).
• Efficiency η = v / u, shaft power P = T u.

Checks run in the page

• A time integration of the boat against the direct balance.
• The closed-form speed law and the doubling of speed with rpm.
• The power balance and slip loss.
• A count of immersed blades against the geometry.

Honest simplifications

Real wheels have feathering or fixed floats, entry and exit shock, wheel-hull interaction and a trimming hull, so the blade drag coefficient and the hull drag area are fixed illustrative numbers, not measurements. The coal figure assumes a flat 6 % plant efficiency. It is a teaching model, not a ship design tool.

• Press Slack water, then Flood current, and watch the speed over ground fall while the speed through the water barely changes.
• Raise the wheel rpm and see the efficiency curve: more rpm gives more speed but also more slip.
• Press Shallow loaded hull and see how much more power the same speed costs.