An reference ship based power prediction method

Note: there may be words and definitions which might not be familiar to everyone in this article. Please take a look at the bucketeers glossary Bucketeers Glossary or contact acelanceloet on discord or by forum pm if any questions remain (which I will then answer and add to the bucketeers glossary)

One of the most difficult and complex parts of ship design is the prediction of required propulsive power. Various methods to calculate this power exist, such as simulations or the well-known Holtrop-Mennen method but these require a lot more information then is normally available at the very start of a ship design. Some use cases require a more simplified (and less accurate) method. One such a method is described below. This article is written by J. Scholtens / acelanceloet to help ship design enthousiasts to help with the realism of their ship designs.

In ship design, it is common to use reference ships both to check the accuracy of the calculations and to make first estimations early in the design process. Because of this, it is important to be able to compare characteristics of real ships and real ship designs to see if they match a new design. The accuracy of this method depends on a good choice of reference ship for the new design: the closer the reference ship(s) are to the new design, the more accurate this method becomes.

The formulas and their meaning

To compare reference ships, it is required to compare the different characteristics of a ship that influence the resistance. The below formulas are used for that purpose. To make using these formula’s as simple as possible, the formulas are brought together into one formula in which all the defining characteristics for this method are brought together.

The resistance formula

• R = ½ * ρ * C_{r * S * v²}

In which:

• ρ is the density of the medium (in our case either water or salt water, 1000 or 1025 kg / m3)
• Cr is the resistance coefficient (more on that later)
• S is the wetted surface in square meters
• v is the velocity in m/s

The effective power formula

To get a power requirement from the resistance, it is require to multiply this resistance with the speed. Resulting is the power that needs to be delivered to propel the ship forward at the given speed.

• P_effective = R * v

In which:

• R is the resistance from the above formula
• v is the velocity in m/s

The engine power formula

The effective power requirement is however not the same as the power a ships engines have to deliver. This is because no propulsion method is ever 100% efficient. For example, a 50% efficient propulsive system results in twice the engine power compared to the effective power.

• P_engine = P_effective * η

in which:

• P_effective is the power required in the above formula
• η is the efficiency of the propulsive system.

The completed formula

When all the above formulas are filled in according to their relation, this results in the following:

• P_engine = (½ * ρ * C_r * S * v²) * v * η

This can still be slightly simplified to the final formula.

• P_engine = η * ½ * ρ * C_r * S * v³

Using the formula for the comparison

The basis of this power prediction method is comparing this formula between 2 different designs. This allows us to look at each value separately and cross away the ones which do not differ or do not differ significantly. If ship one and ship two are compared, that would result in the following formula:

• η₁ * ρ₁ * C_r1 * S₁ * v₁³ = (η₂ * ρ₂ * C_r2 * S₂ * v₂³) * Power Coefficient

The power coefficient is a dimensionless factor between the required powers. For example: if the result is that ship 2 requires 10% more power, the Power Coefficient is 1.10. This daunting formula might still look very complicated, but is simplified easily when the reference ships are well chosen. For example: in many cases the density of the water and the propulsive efficiency will not differ a lot between the reference ships. In such a case, we could simply remove these from the formula because calculating them would not any value as they would just be equal on both sides. In that situation, the formula would be brought back to just the following:

• C_r1 * S₁ * v₁³ = (C_r2 * S₂ * v₂³) * Power Coefficient

The value of having similar reference ships is proven here: the closer they are, the more characteristics we can ignore and the smaller the influence of the remaining characteristics are. In essence, the above formula(s) describe the base of this method. However, to fully use the potential of this method one has to know how to estimate or define each of the characteristics used. The next few chapters will define each of the characteristics used, show what they are influenced by and provide some help to estimate and use them. An interesting side effect of using a comparison for the power prediction is that there is no requirement to use the SI system of measurement units. As long as the same units are used for the same characteristics on both sides of the “equal” sign they will not influence the outcome as the method uses a relative outcome. It is however recommended to use the SI system of measurement units.

Wetted surface

The wetted surface is, as the name suggests, the surface area of the hull that is under water. There are multiple methods to estimate or calculate this area. However, as this article is about comparing ships and not about designing them from scratch, this section will describe easy ways to compare wetted surfaces instead of calculating it's exact dimensions.

For these comparisons, it is easiest to assume wetted surface 1 has an area of 1 or 100%, and wetted surface 2 is increased or decreased as needed.

• When a ships hull is scaled by factor a, the wetted surface is scaled by the factor a cubed. So if the ships is scaled to 110% of the original, the wetted surface is increased by 1,1^2 = 1,21)
• When a ships hull is stretched (so scaled in 1 dimension, the shape does not change but is stretched out) by factor a, the wetted surface is scaled by the factor a.
• If the shape of a hull is changed, comparing it's wetted area becomes more complicated. However, it is easy to calculate maximum and minimum areas and estimate how close to either of those it would be, especially when using the block coefficient as a guide for that.
  • If a ship has a block coefficient of near 1, the wetted area will be close to that of a full box too. That area would be ( L * B ) (the bottom) + 2 * ( L * T ) (the sides). The flat areas fore and aft are ignored for this comparison. This can also be written down as "L * ( 2* T * B )"
  • If a ship has a block coefficient of near 0, it is a fact that both beam and draft decrease significantly on the ships ends compared to the midships that sets the "Block" dimensions (L*B*T). An good estimation of an wetted surface area in those conditions is to just take the smallest of either L*B or L*T. On most ships B is larger then T.
  • An formula for the wetted surface that gives an very rough estimation would, using the above, be ( L * B ) + ( 2 * Cb * L * T ) or L * ( B + (2 * Cb * T )). This estimation of wetted surface is good enough for the comparison of two ships, do NOT use it when doing calculations in other methods!

Efficiency

The efficiency is in this calculation a factor that describes how efficient the conversion from power to motion is. While the losses in this conversion can mostly be said to be in the propeller and the gear train or electric conversion, there are aspects which might make this more complicated when comparing 2 designs and/or conditions.

All components of a ship have ideal circumstances. For example an engine which has an optimal RPM (which is often coupled to a certain amount of torque and power) or the ideal speed for the ships propellers. When doing an comparison of two designs where nothing is set in stone, the easiest method is to say that both designs are as optimised for the condition that is being compared, so the efficiencies are the same. This will be true in most of the use cases of this method.

However, when aspects of the design, such as the type of the propellers or the choice of the engines are already set in stone and for example 2 speeds of a given ship are compared, it is important to take note of how running an engine at an less then optimal speed might create quite the drop in efficiency! The magnitude of that drop in efficiency differs between engine types so requires additional research to get exactly right.

Velocity

The velocity is the simplest factor in this method, it is simply the speed of the vessel. While the formula's are written for SI units, as long as you use the same unit on both sides of the equal sign, the comparison works. This method can be used for any speed of a given ship, be it cruising speed, top speed or anything below those. Be careful that the other factors may change when the speed differs a lot as for example the wavemaking resistance can be very different, but when comparing for example the power required for 20 and 21 knots it is not too far off to just compare the power factor for the 2 speeds.

Density

The density is the density of the medium the resistance is calculated in. As with the velocity, different units can be used as long as they are both the same on both sides of the equal sign. Commonly, 998 kg/m³ is used for fresh water and 1025 kg/m³ for seawater.