It’s really important to note that to do this right you need to know the specific fuel consumption (usually found in the OEM manual or commissioning data) and the percent load of the engine at those speeds.
Fuel consumption may or may not be fairly linear compared to load, but higher speed may require significantly more load and therefore fuel.
Normally the approximate rate of daily fuel consumption is assumed to be proportional to the cube of the speed. However, researchers have found that this figure should be between 2.7 and 3.3 for container ships. Similarly, they showed that it is equal to 3.5 for the feeder container ships and equal to 4 for medium sized container ships and even 4.5 for very large sizes. So there is a fuel dependency on the displacement of the ship. If you think about it this is a logical conclusion.
For those interested in this theory there is a research article “A new method for calculating fuel consumption and displacement of a ship in maritime transport” that gives more information on this subject.
For a rough estimate. And with consideration to what @Dutchie mentioned below.
If I’m looking for an accurate estimate to provide the office for a transit I’m still going to use specific consumption curve data gathered from actual running of the specific engines, as I’ve yet to see one match up to the formula.
Most definitely. Which is why engine load and sp-consumption is a good component for this calculation. A ship only in ballast but empty of cargo would presumably require less engine load for a given speed (and thus less fuel per kt) than one fully laden.
In principle you could calculate your own exponent of the base number with on board gathered data. For instance calculate the daily average speed and distance. These days with a Doppler log or especially with a GPS on board this can be done with a high accuracy. I suppose that also the fuel consumption can be accurately measured.
A problem could be that you will have to steam at least one day, preferably a couple of days at a slower speed to get sufficient data to carry out the calculation of the exponent. See also this website.
There probably also is a much smaller draft dependency as the underwater hull form can change with the draft. Same for the trim.
As a follow-up, if you ever really did want to use the first question as a basis for the second, that is figuring out what speed will get you that distance with the same fuel and taking into account the consumption at different speeds, here is the interpolation:
I just put the formulas in an excel spreadsheet so it’s easy to see each step. Then just copied the formulas down to start showing fuel burn for each given speed. Once finding that 15kt looked close I put in the 15.x decimal speeds to find the closest one.