# center of flotation calculation and implications?

Discussion in 'Boat Design' started by capt vimes, Jan 7, 2010.

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### rwatsonSenior Member

Something that might intrigue the mind - Archimedes theorem can be much more accurately defined with the principle of "hydrostatic pressure "

The story I remember is a physics question - "if you built a bathtub a centimetre larger than a battleship (all around) - could you float the battleship ?"

This would take much less water than the weight of the battleship, apparently defying Archimedes principle.

The answer is yes, it would float. Interesting discussion here at

Not just a technical point, as some structures do "float" in very confined spaces. If my memory serves me well, I think the huge telescope at Palomar Observatory floats in a sea of oil only millimetres thick.

Likewise, if you wanted to bring your houseboat home, the pond in the backyard would only have to be a few inches bigger than the boat itself, rather than big enough to hold the boat plus the amount of water equivalent to the boats weight.

I have often thought a pond that size might be a cheaper way to support a house than foundations in difficult soil - or a convenient way to have a movable house.

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### Eric SponbergSenior Member

Ok, here is another interesting Archimedes Principle riddle (and I may have posted this once before on the forum somewhere else):

A man is sitting in a small boat in a pond. In his boat he has a large rock. He lifts the rock up, tosses it over the side, and the rock sinks to the bottom of the pond. What happens to the water level in the pond: Does the water level go up, does it go down, or does it stay the same?

Eric

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### FanieFanie

The level will go down - the weight of the stone displaces more in the boat than what it's volume will displace when dunked. The stone is heavier than water.

The amount would be equal to the stone's weight in water volume less the displacement of the stone.

Very small pond if you can see that

Here's an add on to that -

If the guy pulls the drain plug and the boat fills with water to where it just remains afloat, the boat would be filled with water, and the boat would weigh much more on a scale, but the pond level would remain the same.

So if you throw the anchor and chain out the sea level will go down a bit.
Now if every one throws their anchor's out.... So that's where tides come from

Same question but now if the same rock hangs from the boat on a rope, by how much would the pond level rise

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### FanieFanie

Rwatson,

What saves your battleship in the 10mm of water is the pressure. Of course when it does sink due to getting holed the crew can just walk out

Water pressure is a function of depth, and not volume. If you measure the water pressure in a 100mm wide tube at 10m below surface it would be the same as measuring it in a 1000m tube at the same depth below the surface and might I add, the friend of sailors and the enemy of submarines

What keeps a boat afloat is displacement, or pressure over the submerged area. If you use your imagination you can see the water pushing up...
Wide hull, beeg area pushing up, thin hull, small area pushing up.

The nice thing about the metric system is it's easy - if the displacement is approx a cubic meter you are floating a ton or 1000kg. If you stand besides a relative small boat that would measure 5m x 2m x 0.5m which is about the size of a bass boat, and a wave dumps a 1000 liters of water in it, the draft would decrease only 100mm. Doesn't sound like much, but it is more than what small cars weigh. A Toyota Carolla 1600 weighs about 900kg's. If the boat weighs 900kg's then you can just barely float four Toyota's in it.

While the contraption would float, gravity comes into play and this is where Eric's center of flotation comes into play again. Where to park the Toy friggin yota so the rig remains balanced. Are we going in circles here

Remember to add center of gravity to this also.

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### rwatsonSenior Member

Hmmm - perhaps their is a subtle definition point here in what we can mean as volume, as volume *can* increase water pressure.

As you rightly point out, the total volume of the body of liquid (eg water tank or milk bottle) *may* make no difference. If they both have only three inches of water in the bottom - the water pressure at the bottom of both containers will be identical.

But water pressure can be increased by volume if the volume increases the depth of the water .

If you put another 3000 litres of water in your pond, the water pressure at the bottom WILL increase, IF the depth of the pond has increased.

So, Submarines wont be able to get as close to the bottom of the ocean as they used to, once the oceans have risen a metre from melted glaciers. (notice I didnt say "from melted Ice caps and IceBergs" - wow, I am soooo subtle) as the VOLUME of the ocean increases.

The best summary of the formulae I have found so far is at
http://hyperphysics.phy-astr.gsu.edu/hbase/pflu.html#fp

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### FanieFanie

I don't think I will like submarines. People will throw their anchors on you.

Be fun if your anchor does get stuck on one of those things. They want to go down and you want to go up, or at least stay up

Made a little error in post #33... it is now corrected

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### hoytedowOld Woodbutcher

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### mcollins07Senior Member

Eric,
I want to thank you for your explanation of CF. What I found unusual about your explanation is that you not only gave us a way to look at CF, but HOW TO use it in hull design. I read most every naval architecture book that I can get my hands on; however, few authors give details of their methods for hull design of sailing hulls. Your explanation as to why we want CF to stay the same or move forward for sailing hulls was excellent.

Could you expand on the block coefficient in regard to how you use that number in hull design? Perhaps what an appropriate value for a sailboat would be, and why that is different for a cargo ships? Why it is more important for large ships?

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### Eric SponbergSenior Member

Block coefficient, Cb, isn't as useful in sailing yacht design because the Cb values are so small, say 0.4 to 0.45, that they don't really mean anything. This is because the draft dimension in the Cb equation is driven by the depth of the keel, yet the keel has very little volume. You can have drastic differences in keel shape and volume, yet that still has little effect on Cb. So Cb becomes a meaningless number. The prismatic coefficient, Cp, is much more sensitive to variation and analysis, and I'll take that up in another post.

One could calculate Cb without the keel, but again, Cp, usually taken without the keel, is still more sensitive and so is better suited for analysis and comparison between designs and power prediction.

Hull resistance, and therefore speed, is directly related to Cp. But you want Cp to be within a certain range, depending on how the boat is going to be used. I get ahead of myself here, and will explain further in a later post.

In ship design, Cb is useful because a lot of ships are what's called "High Block" ships, meaning Cb is very large, on the order of 0.7 ot 0.8. Slight variations in hullform will have measureable and meaningful effects on Cb and, therefore, hull resistance and speed. Cp, also will be similarly affected and so is still also used in ship design, analysis, and performance prediction.

In motoryacht design, those which don't plane anyway--that is, remain in displacement mode when moving, Cp is useful for comparison of hull designs and for power prediction, again, more so than Cb because it is more sensitive to changes in volume and displacement.

In planing powerboat design, Cb and Cp become meaningless because hydrodynamic forces lift the hull higher out of the water and totally change the shape and volume of the submerged part of the hull. Cb and Cp are simply not used. Analysis for planing powerboats, therefore, is based on dynamic lift and drag forces and their related coefficients, not on gravity forces so much which relate back to Cb and Cp.

Eric

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### capt vimesSenior Member

thank's - again - eric... that explains it quite good - for me at least...

that would increase the pressure at 'the former' designed depth (new depth above ground + 1 meter) of a submarine by 0.1 bar (1 meter of water-depth equals 0.1 bar at any given depth) ... not that big of a difference considering that the design depth might be some 1000 meters equaling in some 100 bar of pressure - isn't it?
should be covered by the standard safety factors easily...

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### mcollins07Senior Member

Yes, thanks again Eric, very nice.

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### m20b25Junior Member

Wow what a useful few pages. I'm going to watch this one! Come in useful with the design of my 6mr :s !

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### LandlubberSenior Member

Thanks again Eric, I am saving these as we get them, I know many people will appreciate your time in explaining these to us all in full.

Ta, John

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### ancient kayakeraka Terry Haines

- that goes for me too. Looking forward to returning to Prof. Eric's class ...

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### Eric SponbergSenior Member

Midship Area and Prismatic Coefficients

Today we take up two coefficients: Midship Area Coefficient, which we’ll label Cmc, and Prismatic Coefficient, labeled Cp. See the diagrams attached.

Cmc is the ratio of the area of largest midship section of the submerged portion of the hull to the area of the box that bounds it. The dimensions of the box are the Beam at the Waterline at the largest section, Bwl, and the Draft at that section, Do. The largest section area may not be at the exact mid-length of the hull, or at the maximum beam or draft of the hull. See the first figure below. The easiest way to determine the maximum size midship section size is to plot the Sectional Area Curve, and the maximum area will be at the peak of the curve. Unfortunately, I don’t have an example of my own sectional area curve to show, but you can see one in Principles of Yacht Design (Larsson & Eliasson, 3rd ed.) Figure 4.4, pg. 35.

At the fore and aft location of the peak of the sectional area curve, measure the Bwl and the Do. The box bounding area for the largest midship section, therefore, is Bwl x Do. Midship Section Coefficient, Cmc, then is the Actual Midship Section Area divided by the Bounding Box area, as shown in the diagram below. The concept and the equation are shown in the second figure below.

Cmc is used as a gauge to judge the fineness of the midship section. It is useful for comparison between different designs, or to judge how a design is being developed. Say, for example, that you are designing a new hull, and you want the displacement to be within a certain range. On your first pass at developing the lines, you see that the displacement is too large. Maybe the turn of the bilge is a little too sharp. So then you develop a second set of lines and find that the displacement is too low; maybe now the turn of the bilge is not sharp enough. You can compare the shapes of the midship sections, because they are directly related to displacement, and compare the Cmc ratios of each. Analyzing these features will lead you to where you want the final Cmc to be (somewhere in between), and the geometry of the sectional area will show you where to make adjustments in the shape (again, somewhere in between). Cmc is used as an analysis tool, therefore, for developing a new design or comparing two different designs.

This leads us to Prismatic Coefficient, Cp. Cp is like Cmc, except that we take the calculation one step further with a third dimension--length. Cp is a comparison of volumes, not areas. Cp is the ratio of the volume of displacement of the hull divided by the volume of a prism which is the maximum section area multiplied by the Length on the Waterline, Lwl. See the third figure below. The figure shows the equation. Obviously, the prism volume is always larger than the actual displaced volume, so Cp is always going to be less than 1.0, by definition.

Looking at the equation, we see an interesting thing happen. In the denominator, we see Lwl x midship area. We know from the Cmc above, that if we turn its equation around, Midship Area = Bwl x Do x Cmc. So if we substitute these parameters into the Cp equation, we get that Cp = Vol/(Lwl x Bwl x Do x Cmc). But we know the part Vol/(Lwl x Bwl x Do) is equal to Block Coefficient, Cb. At least it will equal it perfectly if Do is also the maximum draft, D. If Do is not the maximum draft, then the calculation will be a little off. But basically, this all reduces to the fact that Cp is the ratio between Cb and Cmc, as shown in the figure. In words, this is: “Prismatic coefficient is the Block Coefficient divided by the Midship Area Coefficient.”

It turns out, after some 150 years or so of analysis, that performance is closely related to Cp. That is, there is an optimum range of Cp for various speeds of the boat traveling through the water. You can see a table of speed/length ratios versus optimum Cp in Skene’s Elements of Yacht Design (by Francis Kinney, 5th ed.) pg.284, which I repeat below:

Speed/Length ratio Cp
1.0 0.52
1.1 0.54
1.2 0.58
1.3 0.62
1.4 0.64
1.5 0.66
1.6 0.68
1.7 0.69
1.8 0.69
1.9 0.70
2.0 0.70

Larsson/Eliasson shows a similar range in their book on page 83, Fig. 5.22, in which they plot optimum Cp against Froude Number. Froude Number is very similar to Speed/Length ratio, and if you convert Froude Number to Speed/Length ratio, you will find that Larsson/Eliasson’s curve is a bit lower than Skene’s curve tabulated above. As is true with many things, therefore, there is some wishy-washiness in the guidelines. Nothing is hard and fast.

Nevertheless, what this tells you is that most displacement boats travel most of the time at Speed/Length ratios of at least 1.0 and slightly above, so you need enough volume to support the hull at those speeds. If volume is either too much or too low—that is if Cp is too big or too small—your hull drag is going to go up. Either the boat is going to have to push too much water out of the way (Cp too big) or it is going to sink into its own waves (Cp too small).

Usually, in sailboat design, the keel and it’s draft is left out of the calculation of volume. This is because, as in the Cb calculation, the keel tends to make Cp less sensitive. So we ignore the keel for calculation of Cp. In powerboat design, we do not do this. If we are designing a trawler or lobster boat, for example, we keep the keel in the calculation because it is a major portion of the hull.

Interestingly, I typically design my sailboats with a Cp of about 0.60. I did the same with my Moloka’i Strait motoryachts. This is just below hull speed, Speed/Length ratio = 1.34. (We can take that up in another post, if you wish). You can also see that approaching planing speeds (Speed/Length ratio => 2.50), Cp reaches 0.70. This goes along with very long and narrow hulls—that is, being still in displacement mode at S/L = 2.0, you need a high Cp. This is why catamarans and trimarans (which have long narrow hulls) have very high Cp ratios.

That’s a lot of material to digest, so I’ll leave it there and wait for questions.

Eric

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