Naval Architecture & Towage Engineering, ASD Tugboat Static Bollard Pull & Ducted Nozzles
Tugboat Static Bollard Pull & Ducted Nozzle Thrust Formula
Calculate Total Swept Propeller Disk Area (m²), Certified Continuous Bollard Pull (tonnes), Specific Thrust Efficiency (kg/kW), and Storm Towline Reserve Margin (%).
Static tugboat thrust scales as Power^(2/3)—so surrounding twin propellers with airfoil nozzles adds 39% MORE bollard pull without bigger diesel engines!
At zero forward speed (`V_a = 0` static pier trial!), classical propeller efficiency `T * V_a / P_D` is identically zero—so naval architects govern **Harbor & Offshore Tugboat Bollard Pull** by **Rankine-Bendemann Static Actuator-Disk Momentum Theory** in Seawater (`rho = 1,025` kg/m³)! Across `N_p` propellers of Diameter `D` (`m`, Swept Disk Area per Screw `A_1 = (PI() / 4) * D^2` m²) absorbing Total Shaft Power `P_D` (`kW` -> `P_1 = (P_D * 1,000) / N_p` W per shaft) with **Bendemann Figure of Merit `FM`** (`~0.54–0.58` for open propellers where slipstream vena-contracta contracts to `A / 2`, vs **`0.75–0.82` for accelerating airfoil ducted nozzles** that suppress tip-vortex contraction and carry `~40%` of thrust directly on the nozzle lip!), **Continuous Bollard Pull `BP = (N_p * FM * (2 * 1,025 * A_1 * P_1^2)^{1/3}) / 9,806.65`** (`tonnes-force`)!
Verify on a `4,200` kW (`5,632` bhp) Twin-Z-Drive (`ASD`) Harbor Assist Tug—Total_Delivered_Power `P_D` = 4,200 kW (B2), Azimuth_Thrusters `N_p` = 2 (B3), Propeller_Diameter `D` = 2.80 m (B4 -> Total Swept Disk Area = `12.32` m² in E2), Ducted-Nozzle_Figure_of_Merit `FM` = 0.78 (B5), and Required_Storm_Escort_Pull = 48.0 tonnes (B6): Continuous Bollard Pull hits **`60.74` tonnes** (E3), Specific Efficiency reaches `14.46` kg/kW (E4), providing a `+26.5%` storm reserve margin (E5)!
| ASD Harbor Tug Bollard-Pull KPI (4,200 kW, Twin 2.80 m Screws: Ducted FM=0.78 vs Open FM=0.56) | Formula | Verified Output |
|---|---|---|
| Total Swept Disk Area (m²) | Prop_Count * (PI() / 4) * (Prop_D_m ^ 2) | 12.32 m² |
| Continuous Bollard Pull (tonnes) | (Np * FM * (2*1025*A1*P1_W^2)^(1/3)) / 9806.65 | 60.74 t (43.61 t @ open propeller!) |
| Specific Bollard Efficiency (kg/kW) | (Bollard_Pull_tonnes * 1000) / Shaft_Power_kW | 14.46 kg/kW (10.38 kg/kW @ open screw) |
| Storm Tow Reserve Margin (%) | ((Bollard_Pull_t - Req_Tow_t) / Req_Tow_t) * 100 | +26.5% (-9.2% DEFICIT without nozzles!) |
Bendemann Static Actuator-Disk Thrust (`P^(2/3) * A^(1/3)`) & Tow Margin Formula
Validate `Shaft_Power P_D > 100` kW, integer `Propellers N_p` in `[1, 4]`, `Diameter D` in `[0.8, 6.0]` m, `Figure_of_Merit FM` in `[0.40, 0.90]`, and `Required_Tow > 1` t!
=LET(pd_k, B2, np_c, B3, dp_m, B4, fm_b, B5, rt_t, B6, a1_m, (PI() / 4) * (dp_m ^ 2), p1_w, (pd_k * 1000) / np_c, ti_n, (2 * 1025 * a1_m * (p1_w ^ 2)) ^ (1 / 3), bp_t, (np_c * fm_b * ti_n) / 9806.65, IF(OR(pd_k <= 100, np_c < 1, np_c > 4, dp_m < 0.8, dp_m > 6, fm_b < 0.4, fm_b > 0.9, rt_t <= 1), "Invalid Tug Input", ROUND(((bp_t - rt_t) / rt_t) * 100, 1)))Un-Ducted Open-Wheel Propeller (`FM = 0.56`) Storm-Tow Deficit Check (Set FM B5 = 0.56)
Remove the accelerating shroud nozzles (`FM` drops to open-wheel `B5 = 0.56` on the identical `4,200` kW diesels): Bollard Pull E3 plunges `-28.2%` to **`43.61` tonnes** (`10.38` kg/kW in E4), flipping Storm Tow Reserve E5 negative (`-9.2%`)—requiring `6,800+` kW of fuel-thirsty main engines to match what `2.80` m nozzles achieve at `4,200` kW!
=IF(ISTEXT(E5), E5, LET(a1_m, (PI() / 4) * (B4 ^ 2), p1_w, (B2 * 1000) / B3, ti_n, (2 * 1025 * a1_m * (p1_w ^ 2)) ^ (1 / 3), ROUND((B3 * B5 * ti_n) / 9806.65, 2)))Try the example
Use the original values or change them to test your understanding.