Understanding Hull Coefficients – Types and Significance

Hull coefficients, also referred to as form coefficients, provide a comprehensive insight into the overall shape and underwater volume of the hull through unitless values.

This article provides a concise overview of five distinct hull coefficients, each serving a specific purpose in naval architecture. The coefficients are outlined as follows:

1. Block coefficient

2. Midship coefficient

3. Waterplane coefficient

4. Prismatic coefficient - horizontal

5. Prismatic coefficient - vertical

BLOCK COEFFICIENT

The block coefficient, denoted as Cb, represents the ratio of a vessel's underwater volume to the product of its length, breadth, and draft. 

Block Coefficient (Cb) = Underwater volume / (Length*Breadth*Draft)

Length = Length between perpendiculars

This dimensionless quantity ranges between 0 and 1. 

A vessel's Cb value offers insights into its hull nature: higher values indicate fuller hull forms, while lower values suggest finer hull forms.

Vessels with higher Cb values typically experience increased resistance, resulting in slower speeds and higher fuel consumption due to the additional energy required for propulsion. They also have higher usable volume within the hulls.

This trait is commonly found in tankers and bulk carriers. Conversely, vessels designed for high speeds often have lower Cb values.

For instance, a barge typically has a Cb value close to 1, as it closely resembles a rectangular cuboid in its hull design.

MIDSHIP COEFFICIENT

The midship coefficient, denoted as Cm, represents a dimensionless quantity in naval architecture, akin to the block coefficient in a two-dimensional context. It signifies the relationship between the area of the submerged midship section and the product of the vessel's draft and breadth, falling within the range of 0 to 1.

Midship coefficent (Cm) = Area of submerged midship portion / (Breadth*Draft)

As the hull of a vessel becomes fuller, the midship coefficient increases. This augmentation indicates a larger submerged midship section concerning the draft and breadth. 

Notably, a rise in the midship coefficient, independent of changes in the block coefficient, leads to an increase in frictional and separation resistance. This is a result of increased wetted surface area and unevenness in velocity distribution, although the resulting rise in resistance remains relatively minor.

Moreover, a higher midship coefficient lessens the necessity for significant plate curvature from the keel to the ship's main deck, thereby reducing the demand for frame bending. 

Additionally, vessels with a higher midship coefficient experience improved roll damping, leading to diminished rolling compared to those with lower midship coefficients.

Vessels boasting a high midship coefficient tend to accommodate greater displacement and cargo capacity. Conversely, ships featuring lower midship coefficients prioritize enhanced speed and fuel efficiency.

WATERPLANE COEFFICIENT

The waterplane coefficient, designated as Cw, describes the relationship between the waterplane area and the product of a vessel's length and breadth. This coefficient specifically pertains to the waterplane corresponding to the vessel's draft.

Waterplane coefficient = Area of waterplane / (Length * Breadth)

Length = Length between perpendiculars

The Cw value signifies how closely the waterplane resembles a rectangle with the vessel's length and breadth. A higher waterplane coefficient contributes to increased stability, enhanced maneuverability, and heightened frictional resistance.

Conversely, a lower waterplane coefficient suggests reduced frictional resistance and indicates finer ends. 

PRISMATIC COEFFICIENT

The prismatic coefficient (Cp) is the ratio of the underwater volume of a vessel's hull to the volume of a prism that has the same length as the vessel's waterline, the same cross-section area as the maximum cross-section of the vessel, and the same depth as the draft of the vessel.

LONGITUDINAL PRISMATIC COEFFICIENT

The longitudinal prismatic coefficient, known as Cpl, represents the ratio between a vessel's displacement volume and the product of its midship area and length. This coefficient serves as an indicator of how the underwater volume of the vessel is distributed along the ends versus the midship of the hull.

Longitudinal prismatic coefficient Cpl = Volume of displacement / (Area of submerged midship portion * Length)

Length = Length between perpendiculars

A lower prismatic coefficient suggests finer ends, while a higher coefficient indicates fuller ends. 

Notably, due to the smaller area of a ship's midship section compared to an ideal rectangular block derived from the beam and midship draft, the prismatic coefficient is consistently greater than the block coefficient specific to a particular vessel.

VERTICAL PRISMATIC COEFFICIENT

On the other hand, the vertical prismatic coefficient, denoted as Cpv, represents the ratio between the displacement volume and the product of the waterplane area and vessel draft.

Vertical prismatic coefficient Cpv = Volume of displacement / (Area of waterplane at waterline * Draft)

Unlike the longitudinal coefficient which showcases the lengthwise distribution of the underwater volume of the hull, the vertical prismatic coefficient demonstrates the vertical distribution of the underwater volume of the hull, spanning from the waterline to the keel.

These coefficients are instrumental in determining the vertical centre of buoyancy, aiding in understanding how the hull's volume is distributed in the vertical dimension, impacting a vessel's stability and flotation characteristics.

Relation between Longitudinal prismatic coefficient (Cpl), Block coefficient (Cb), and Midship coefficient (Cm)

Another approach to determine the Longitudinal Prismatic Coefficient (Cpl) involves obtaining it through the ratio of the Block Coefficient (Cb) and the Midship Coefficient (Cm). In this scenario, if you have the calculated values for Cb and Cm, you can use their ratio to find the Cpl value:

Therefore,