Powering and Resistance

Part B

1. Ship Drive Train and Power Hierarchy

Power flows from the engine to the hull, experiencing losses at each stage. The relative magnitude follows: $BHP > SP > DP > TP > EP$

    graph TD
    BHP[Brake Horsepower - Engine Output] -->|Reduction Gear Losses| SP[Shaft Power - After Gears]
    SP -->|Bearing & Seal Losses| DP[Delivered Power - At Propeller]
    DP -->|Propeller Losses| TP[Thrust Power - Generated by Screw]
    TP -->|Hull-Propeller Interaction| EP[Effective Power - Bare Hull Resistance]

Definitions & Key Equations

Effective Power (EP): Power required to tow/move the bare ship hull at speed $V_s$ without propeller action. $EP = R_t \times V_s$
Brake Horsepower (BHP): Power output at the engine shaft before reduction gears.
Shaft Power (SP): Power output coming out of the reduction gears.
Delivered Power (DP): Power delivered directly to the propeller. $DP = SP - \text{losses in shafting, bearings, and seals}$
Thrust Power (TP): Power created by the propeller. $TP = DP - \text{propeller losses} = T \times V_a$

2. Propulsion Efficiencies

Losses along the drive train are quantified using efficiencies ( $\eta$ ):

Gear Efficiency: $\eta_{gear} = \frac{SP}{BHP}$ (Typical: $\sim 98 - 99\%$ )
Shaft Transmission Efficiency: $\eta_{shaft} = \frac{DP}{SP}$ (Typical: $\sim 97 - 98\%$ )
Propeller Efficiency: $\eta_{propeller} = \frac{TP}{DP}$ (Typical: $\sim 70 - 75\%$ )
Hull Efficiency: $\eta_{H} = \frac{EP}{TP}$ (Well-designed: $\eta_H > 1$ ; Poorly-designed: $\eta_H < 1$ )
Propulsive Coefficient (PC): Overall efficiency combining bearing, guide, and propeller losses. $\eta_{p} = PC = \frac{EP}{SP}$ (Typical range: $55\% - 75\%$ )

3. Total Hull Resistance and Components

Total hull resistance ( $R_t$ or $R_T$ ) is the force opposing ship motion, broken down into three principal components: $R_T = R_{AA} + R_W + R_V$

    graph TD
    RT[Total Hull Resistance Rt] --> RAA[Air Resistance R_AA - Superstructure Calm Air]
    RT --> RW[Wave-Making Resistance R_W - Gravity & Speed Dependent]
    RT --> RV[Viscous Resistance R_V - Skin Friction & Form Drag]

Resistance Components & Formulas

Air Resistance ( $R_{AA}$ ): Caused by airflow over the superstructure (typically $4\% - 8\%$ of total resistance).
Wave-Making Resistance ( $R_W$ ): Caused by surface waves generated by the moving hull. Affected by beam-to-length ratio, displacement, hull shape, and Froude number.
Theoretical wave length: $L_{wave} = \frac{2\pi V^2}{g}$
Hull speed limit: $V = 1.34 \sqrt{L_s}$ (resistance increases sharply when ship speed approaches hull speed).
Viscous Resistance ( $R_V$ ): Combines skin friction and viscous pressure drag.
Total Resistance Coefficient: $C_T = C_V + C_W + C_A = C_F(1 + K) + C_W + C_A$
Tangential skin friction coefficient (ITTC 1957 line): $C_F = \frac{0.075}{(\log_{10} R_n - 2)^2}$
Reynolds Number: $R_n = \frac{L V_s}{\nu}$ , where kinematic viscosity $\nu = 1.140 \times 10^{-6} \, m^2/s$ (fresh water) or $1.190 \times 10^{-6} \, m^2/s$ (salt water).
Form Factor ( $K$ ) for normal component multiplier: $K = 19 \left[ \frac{\nabla}{L B T} \frac{B}{L} \right]^2$
Total Resistance equation: $R_t = 0.5 \rho S V_s^2 \cdot C_T$
Correlation Allowance ( $C_A$ ): Accounts for full-scale hull surface roughness, paint texture, and corrosion.

4. Model Testing and Similitude (Froude’s Law)

Because analytical modeling of 3D viscous and wave-making flow is complex, towing tanks use geometrically and dynamically similar models.

Scale Factor ( $\lambda$ ): $\lambda = \frac{L_s}{L_m}$
Area scale: $\lambda^2 = \frac{S_s}{S_m}$
Volume/Displacement scale: $\lambda^3 = \frac{\nabla_s}{\nabla_m}$
Law of Corresponding Speeds (Froude Similarity): Dynamic similarity for wave-making requires equal Froude numbers. $\frac{V_s}{\sqrt{L_s}} = \frac{V_m}{\sqrt{L_m}} \implies V_m = V_s \lambda^{-1/2}$

5. Screw Propellers and Performance Mechanics

Terminology & Geometry

Diameter ( $D$ ): Tip-to-tip circle diameter.
Pitch ( $P$ ): Theoretical distance a propeller moves forward in one revolution through an unyielding fluid.
Pitch Angle ( $\phi$ ): Blade angle relative to incoming flow, calculated as $\tan \phi = \frac{P}{2\pi r}$ .
Speed of Advance ( $V_a$ ): Ship speed accounting for the wake velocity field ( $V_w$ ): $V_a = V_s (1 - W_t)$ , where $W_t$ is Taylor’s wake fraction ( $W_t \approx 0.5 C_b - 0.05$ ).
Slip:
Apparent Slip Speed = $V_t - V_s$ , where theoretical speed $V_t = P \times N$ .
Real Slip Speed = $V_t - V_a$ .

Propeller Forces & Coefficients

Thrust Force ( $T$ ): Estimated by momentum change through the propeller disc: $T = \rho A P^2 N^2 s$ (where $s$ is real slip, $A$ is effective disc area, $N$ is revs/sec).
Thrust Coefficient ( $K_T$ ): $K_T = \frac{T}{\rho n^2 D^4}$
Thrust Loading Coefficient ( $C_t$ ): $C_t = \frac{T}{0.5 \rho V_a^2 A_o}$
Ideal Propeller Efficiency: $\eta_{propeller} = \frac{2}{1 + \sqrt{1 + C_t}}$

Cavitation

Definition: Formation and collapse of vapor bubbles when local blade pressure drops below the vapor pressure ( $P_v$ of water).
Consequences: Loss of propeller efficiency, severe mechanical erosion, noise, and structural vibration.
Mitigation: Smooth surfaces (free of fouling/nicks), gradual RPM adjustments, proper pitch control, and increased hydrostatic pressure (e.g., deeper submarine diving depth).

Features of Propellers Resistance and Powering of Ships