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This week i’ll post a video montage i had made of taking the boat back to Puerto Rico.

Turned out that our arrangements in Vieques made it very difficult to actually get the necessary modifications made to the boat since we lived on one side of the island and the only safe harbours are on the other side, making me waste far too much time going back and forth.

The changes the boat needed were a cabin, some basic accomodations and changing the rig from racing to something more suitable for cruising. But the most pressing thing to attend to was the weak chainplates; since they had been changed from the original outside the hull setup to the present inside the hull arrangement, it created a leak point and the constant passage of rainwater was very detrimental for the structure in that area.

The boy in the video is my son, and he is feeling noticeably green towards the beginning of the trip. It is Dia de los Reyes, which is why the presents at dawn.

If you watch the video, you’ll notice i did not put up the mainsail, despite the weather being perfect. This was because i was worried about putting too much stress on the chainplates. In fact, as it was, the planks were flexing inwards every time the boat rolled heavily due to the inertia of the mast. It’s the sort of inevitable thing that comes with a “new for you” boat, even one that is ready to sail, which this one was. In fact, the problem had been creeping up for some time, as was evident by the patch-up work done above the chainplates where blocking had been added to help hold the plate supports down, in turn shifting the loads from hull to deck. The thought of these guys racing like this was scary enough, but was also a testament to how sometimes pretty improbable things work. At any rate, i was not in the mood to take any additional chances.

The jib had a little rip in it too which held up fine downwind, but the moment i rounded up for the last leg up the bay it ripped the stitching the rest of the way, par for the course with “new for you” boats. No big deal, even like that the boat climbed up to windward well enough to make the last couple mile beat.

The best part of course never got taped, precisely because it was exciting. There is a shortcut into that huge bay, which cuts some fifteen miles off the deep anchorage if approaching from windward; it is called “La boca del infierno” (hell’s mouth). It is a cut between two of the barrier islands with 3.3 meters of water if you cut it at the right place, but with a bit of swell running becomes a very ugly bit of surf over the coral heads indeed. Now with 2.2 meters of draft that does not give a comfortable margin, but conditions seemed good enough, so i cut through with my wife being my second eyes up front and the corals flashing by underneath so close you could see the veins on the brain coral.

At one time it had occurred to me that it may be worthwhile to make a video documentary of this supposedly “impossible” voyage to windward to Brasil from the Caribbean and it had also passed through my mind that just when it may be interesting to film, everyone is busy dealing with the boat. Therefore, there has to be a person on board dedicated solely to the camera.

Now it so happens that a while ago already, i decided that i will no longer take men on board small boats with me. At first my wife was ok with this but eventually jealous feelings cropped up, so the whole idea was ditched. And that is how the best part got lost.

——–*——–

In other news,

Dmitry Orlov over at cluborlov.blogspot.com wrote a post about about moding his boat with a permanent auxiliary rudder, in order to facilitate a suitable self steering method. He is of course completely correct about the absurdity of wheel steering in small boats. I would go further and say that wheels in anything under several dozen tons is for fashion, not for any practical benefits.

It reminds me a little bit of Eric Sponberg’s moding of another sailboat’s rudder. Although considerably more sophisticated (and expensive) the concept is somewhat similar and was also a great improvement.

It is very important for rudders to have enough power, and unfortunately, this is something that seems to be rather neglected in a lot of designs. I have plenty to add to that, and rudder issues in general, but it will have to wait for a future post.

As promised, here is a short article about scaling laws. Scaling is a topic that does not get much coverage outside of specialized engineering forums so i think it is worthwhile to explain the concept here more generally.

The basic concept is that you cannot simply take an existing design and scale it up or down without affecting the design in other ways than the linear dimensions. That may seem obvious, but apparently it wasn’t obvious to the not insignificant number of people who have attempted to do just this and ended up with boats that were floating disasters.

We will introduce a variable here, the linear scaling factor; a
Such that                                         a = \dfrac {L_{OA_n}}{L_{OA_o}}

Where   L_{OA_n}    is   the new linear dimension    Length Over All new (say)  and   L_{OA_o} is the old corresponding linear dimension Length Over All old, and similarly for all other linear dimensions.

In other words ;                            L_{OA_n} = aL_{OA_o}

Now, the sail area is a times longer and a times taller so a \times a = a^2 times as much

so                                              [SA]_n = a^2[SA]_o
And, the volume of displaced water is a   times longer, a wider and a deeper so  a \times a \times a = a^3 more voluminous

so, displacement;                                      \triangledown_n = a^3\triangledown_o

Therefore, sail area increases as the square of the linear scaling factor, whereas displacement increases as the cube of a.

———–*———–

Relative stability.

Furthermore, righting arm;        GZ_n = aGZ_o                        since GZ is linear
and righting moment;                 RM = GZgm                       where m is total boat mass (same materials means same densities but times volume, which scales to the cube, so mass also scales to the cube)
So                                                   RM_n = aGZ_o\times{a^3gm_o}

i.e.                                                  RM_n = a^4RM_o

On the other hand, heeling moment is heeling arm times heeling force; HA x equation of lift, V is wind speed

.                                                                    HM = HA\times(0.5\rho[SA]{C_L}V^2)

([SA] is sail area) or                                           = HA\times{[SA]\times(0.5\rho{C_L}V^2)}

So                                                     HM_n = aHA_o\times{a^2[SA]_o(0.5\rho{C_L}V^2)}

i.e.                                                       HM_n = a^3HM_o

So we see that righting moment increases to the fourth power of the linear scaling factor, whereas the heeling moment increases to the cube. This means that in a given wind a boat which is scaled up by a factor of two is sixteen times as stiff, but is subject to only eight times as much heeling force; so would heel approximately (since the righting moment curves are not linear but at small angles can be taken as approximately linear) half as much, for a given wind.

This also means that when scaling down, boats become excessively tender, so draft needs to be proportionately increased, and/or greater ballast ratios used and/or other means employed to offset the disproportionate loss of stability.

In the case of functional model sailboats – as opposed to scale model sailboats – this translates to what appears to the novice as absurdly disproportionate keels as well as extremely high ballast ratios. On the free sailing models i used to make standard dimensions were; 30 cM Loa , 15 cM draft and a ballast ratio that was around 95%.

————–*—————

Speed to length ratio.

The equation for the Froude number is  F_n = \dfrac{U}{\sqrt{gL}}                     (the Froude number is the dimensionless speed/length ratio) where U is boat speed, g is the acceleration due to gravity on the surface of Terra and L is waterline length.

So in order for the boat to operate at the same speed to length ratio, the speed needs to scale too, in such a way that the Froude number remains the same.

i.e.                                                              \dfrac{F_{n_n}}{F_{n_o}} = 1

so                                                      \dfrac{\dfrac{U_n}{\sqrt{gL_n}}}{\dfrac{U_o}{\sqrt{gL_o}}} = 1

but                                                         L_n = aL_o

so                                                          U_n = U_o\times{\sqrt{\dfrac{gL_n}{gL_o}}}

Thus                                                      U_n = \sqrt{a}U_o

——————*——————

Period of rolling and period of pitching.

The mass moment of inertia ( I ) is dependant on mass – which is proportional to volume – and distance squared so scales to the fifth power.

The equivalent equation for   F = ma    in rotation is    T = I\ddot\theta     where  T   is torque and  \theta   is heel or pitch angle (the corresponding values of RM and I need to be used depending on whether one is looking at pitch or roll motion).

Substituting T = RM we get

.                                                           \ddot\theta = \dfrac{RM}{I}

.                                                         \ddot\theta_n = \dfrac{a^4RM_o}{a^5I_o}

so                                                      \ddot\theta_n = a^{-1}\ddot\theta_o

i.e.  the angular acceleration scales to the reciprocal of a or the larger the boat the lesser the g forces and vice versa.

For small angles,                            RM \simeq -\phi \theta        where \phi is the slope of the righting moment per heel angle curve near the origin.

so                                                  -\phi \theta \simeq I\ddot\theta

or                                                  \ddot\theta + \dfrac{\phi}{I}\theta \simeq 0                                [1]

which we recognize as the standard differential equation for simple harmonic motion and which can be solved

by putting                                 \theta = Asin(\sqrt{\frac{\phi}{I}}t + t_1)

which gives                              \dot\theta = A\sqrt{\frac{\phi}{I}}cos(\sqrt{\frac{\phi}{I}}t + t_1)

and                                           \ddot\theta = -A\frac{\phi}{I}sin(\sqrt{\frac{\phi}{I}}t + t_1)    which, by inspection can be seen to indeed satisfy eqn [1] and gives an arbitrary starting time point t_1 and a roll frequency of   \frac{\phi}{I}   and an amplitude A. Both A and t_1 depend on initial conditions.

Now \phi scales to the a^4 , while I scales to the a^5

so                                       \theta_n = Asin(\sqrt{\frac{a^4\phi}{a^5I}}t + t_1)

giving                                \theta_n = Asin(a^{-0.5}\sqrt{\frac{\phi}{I}}t + t_1)

and                                    \ddot\theta_n = -A\frac{a^4\phi}{a^5I}sin(a^{-0.5}\sqrt{\frac{\phi}{I}}t + t_1)

giving                                \ddot\theta_n = -A\frac{\phi}{aI}sin(a^{-0.5}\sqrt{\frac{\phi}{I}}t + t_1)

So we can see, that, as determined earlier, the magnitude of the acceleration is inversely proportional to the scaling factor a, and we see that the frequency scales as the inverse root of a. Or that the period scales as the root of a, since period is the inverse of frequency.

At larger angles the approximation of linearity is no longer valid. However, the dimensional analysis still holds up the same.

————–*—————

Apparent speed

Time taken to travel one boat length is \tau = \dfrac{L_{OA}}{U}

so                                                                 \dfrac{\tau_n}{\tau_o} = \dfrac{\dfrac{L_{OA_n}}{U_n}}{\dfrac{L_{OA_o}}{U_o}}

thus, substituting appropriately;   \tau_n = \tau_o{\dfrac{\dfrac{aL_{OA_o}}{\sqrt{a}U_o}}{\dfrac{L_{OA_o}}{U_o}}}

giving                                                \tau_n = \tau_o\times{\dfrac{a}{\sqrt{a}}}

which is                                            \tau_n = \sqrt{a}\tau_o

so if a video of the a times scale version of a boat is replayed at \sqrt{a} times the original speed it will simulate the appearance of speed of the original boat. And, happily, since roll and pitch period scale at a rate of a^{0.5} as well, the same time stretch factor will also give the correct visual impression of the boat’s oscillatory movements.

Film producers everywhere; you’re welcome.

———————–*————————

Scale wind.

This brings us to the potential utility of scale models for prototyping full scale boats. There is certainly immense value in these kinds of trials which can be done at a fraction of the cost of finding a flaw in the full sized boat. However, as i hope is now pretty clear, considerable attention needs to be given to the subtleties of scaling laws .

The first consideration is to establish the scale wind; the amount of wind that will affect the model in a comparable way to the original boat.

Wind force is proportional to wind speed squared, as given by the equation of lift;

.                                                                                     L = 0.5\rho{SC_L}V^2

We need to find the scaling relation between the new wind speed V_n and the original wind speed V_o such that

.                                                                  \dfrac{HM_n}{RM_n} = \dfrac{HM_o}{RM_o}

heeling arm is HA

so                              \dfrac{aHA_o(0.5\rho(a^2SA_o)C_L)V_n^2}{a^4RM_o} = \dfrac{HA_o(0.5\rho(SA_o)C_L)V_o^2}{RM_o}

eliminating                                                   V_n^2 = aV_o^2

thus                                                               V_n = a^{0.5}V_o

So we see that in order for the boat to behave similarly the wind speed needs to be scaled by root a .  This explains why large sailboats have greater difficulties in light winds. If a is 10, say, then the wind needs to blow ~3.16 times as fast to have the same effect.

————–*—————-

Reynolds number.

There is still the issue of the Reynolds number, which is the dimensionless ratio of inertial to viscous forces in the fluid stream. This number is important to distinguish what type of fluid flow regime the boat is operating at and which governing fluid flow simplifications can be reasonably applied.

.                                                            \mathbb{R}_e = \dfrac{UL}{\nu}        where \nu is the fluid’s kinematic viscocity

Which can be seen to scale to the  a^{1.5}   so it is impossible to keep both the Froude number and the Reynolds number correct as one scales a boat, unless different special fluids are used, but this topic is beyond the scope of the present essay…

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Feedback on if the maths symbols do not parse, along with which formula is giving problems would be highly appreciated , this is the first time i try “Latex” mathematical notation typesetting in wordpress.

planet finite diameter

Non infinite planet

Enough with reminiscing for now, lets look instead at the present and the future. This post is not one I have been looking forward to since the topic invariably raises peoples intellectual defense mechanisms, so I’ll try to keep it short and straight to the point. Besides, there is an enormous amount of information on the subject of oil, energy and modern civilzation on the net already. The point of the post is more of a summary of the situation and its intersection with boats.

It’s no secret that modern civilization relies on petroleum, but it’s easy through habituation to forget at what point.

For instance; there’s a high chance that the clothes you’re wearing are at least partially made from synthetic fibres, made from petroleum. Even if you’re wearing all cotton or linen clothes, those crops were undoubtedly nourished with synthetically made nitrogen, sprayed with pesticides made from petroleum, the fields mechanically tilled by machinery that runs on petroleum. Then the fibres are irrigated, harvested, processed, transported, woven and sewn with yet more machinery all ultimately running on fossil fuel energy.

The computer you’re reading this on is made from metals mined with machinery running on petroleum and plastics made from petroleum. A comparable amount of energy goes into building the computer as it will use during its lifetime.

Cars are much the same except worse.

Of greatest concern of all, food has become extremely reliant on petroleum. Depending on which study you read, between 10 and 14 calories of fossil fuel energy goes into producing one calorie of food. This is a less than parity EroEI (energy returned on energy invested) which is clearly untenable, but which has permitted, temporarily, for us to vastly exceed the earth’s normal carrying capacity.

Indeed, there are very few things left in modern society that does not depend in some way on fossil fuels.

rate pop growth vs per capita oil

Rate of world population growth
upper graph aligned with
per capita world oil extraction rates
lower graph

A useful mental exercise is visualizing how far your car can go with just one litre of fuel. Then imagine you have to push the same car that same distance, up and downhill. There is a lot of energy in that litre of fuel!

Then think that we burn more than 70 million barrels every day to power our industrial society… This is a staggering amount of added energy. It is only because of this that we have an industrial society at all.

There is a tremendous amount of unrealistic wishful thinking that goes on in the attempt to believe otherwise; that we will be able to transition to alternative energy sources and carry on with all the energy hungry modern technological marvels, but this does not stand up to closer analysis. A typical reaction is “I’m sure they’ll think of something” which is about the least useful response possible. This not only downplays the gravity of the situation but also reveals humanity’s hubris and arrogance.

The main issue is that of exergy. Fossil fuels have remarkably high levels of exergy, as the mental exercise of pushing the car illustrates. There is ample total energy in renewable, but the exergy is nowhere near high enough to be a reasonable replacement for fossil fuels.

Simply put, the only energy that we can use sustainably in the long term, is a small fraction of the total solar energy throughput that falls on our orb. And fossil fuels are nothing other that stored solar energy accumulated through millions of years of minutely imbalanced biological energy flows. Energy that we are using in a blink of an eye compared to the time it took to accumulate. There we come to the crux of the matter; we have become accustomed to a situation which is by definition unsustainable, and no amount of denial will make that fact go away.

Energy can neither be created or destroyed, only transformed from one form to another form and this is crucial to understand. What happens instead is that as any process is performed by high exergy energy the energy gets transformed into different forms always of lower exergy. This is the conclusion of the second law of thermodynamics. In other words, the same amount (almost exactly) of energy shines onto the daylight side of the planet from Sol as our planet radiates away on the night side. The big difference though, is that the exergy of the outflowing energy is always lower than that of the inflowing energy.

Our planet is finite and thus contains finite amounts of every resource. The rate of usage of every non renewable resource (and even renewable resources whenever the rate of consumption is sufficiently greater than the rate of renewal) tends to follow a roughly bell shaped curve. Over half of the world’s oil producing countries have already peaked and have declining production rates. Conventional global crude peaked around 2005 which caused a persistent increase in the price of crude, allowing previously economically non-viable fields to be exploited. This then has created a certain plateau to world oil production, at the expense of future decline rates. Cantarell oil fields in Mexico is a good example of this process at work at smaller scale. They were able to extend the peak using unconventional means, but when the decline finally came it was precipitous.

The EIA and IEA Have been quietly revising their projected date of expected world peak oil for the last few years, reflecting the dawning realization that the day has come and it is us, rather than the not yet born that will have to deal with this.

Oil production vs price

Oil production vs price

The above plot should be reminder enough that economics is a subset of environment, not the other way around… No matter how high the price goes, physical laws refuse to concede to economic fanatsies.

When the predominant form of energy peaks it tends to put pressure on other sources of non renewables making them peak sooner in turn.

Nuclear power needs uranium or other unstable elements which itself will peak soon (uranium ores are already becoming very poor) and mining uranium itself requires petroleum. Additionally, nuclear power plants take over a decade to bring online and at enormous monetary and energy cost. In fact, the only way nuclear power is economically feasible is through government subsidies and by working symbiotically with the nuclear arms industry.

Besides, de we really want the next ~100 000 generations curse us for leaving behind impossible to dispose of toxic waste just for the sake of running AC units etc for ours and maybe one more generation?

Nuclear fusion I think can be safely relegated to the aptly named “vaporware” closet, after the better part of a century spent investing enormous resources into researching nuclear fusion, the solving of the immense technical difficulties remain perpetually over the horizon..

Coal will peak too in the not too distant future, and dirtily.

Gas the same but cleaner.

Tidal energy is interesting, although it also requires a large energy investment, and may well prove to be a false avenue once fossil energy subsides are no longer available or are being used on other more urgent applications.

Geothermal can also be useful, but only in places where the subsurface temperature gradient is steep enough for the energy investedto be a sufficiently small fraction of the energy payback over the lifetime of the heat well.

Biomass, wind energy, solar power, hydro and wave energy are all ultimately derived from the energy of the sun. As such, there are very definite limits to how much can be extracted, nevertheless, this is about our only viable long term strategy. Remember that biomass means without any fossil fuel inputs, otherwise it is just fossil energy transformed, and that usually at a loss! In practice, and for most applications, the most efficient use of cropland is to fuel humans with food. Solar panels are massively over rated. Their EroEI is quite poor and can really only be justified for specialized applications, such as boats where space and weight are very limited. Wherever weight and space are not at such a premium (industrial, residential rather than on board transport) it makes much more economical sense to use solar thermal rather than photovoltaic.

It is important not to be fooled by claims to the contrary; there is no way we can maintain present day energy use into the future.

That is why this is a predicament, not a problem. Problem implies solution.

However, that is not to say that there aren’t plenty of worthwhile mitigating strategies that one can apply, and despite the inevitable process of technological triage in the decades and centuries to come, there are useful and low energy technologies which can and ought to be preserved moving forward.

——————-***——————

So what does all this have to do with boat design?

A lot actually.

There are a great many ‘modern’ boats which will be seen as curious relics of a by gone age within a generation or two. The first to go extinct will be the highest energy consuming boats, of course, but a lot of standard design and material choices which are commonplace nowadays will soon feel the squeeze too.

wake monster

Only with plenty of cheap fuel can something like this exist

Here is a short list of the embodied energies per mass of some common boat building materials in MJ/kg. The ranges represent the spread depnding on source of information and their method of calculation.

_________________________________________________________

Titanium                                                       361 – 745

Aluminium sheet, virgin, sheet                 160 – 217

Aluminium sheet, recycled, sheet            14.8 – 27.8

Epoxy resin                                                         ~125

Fiberglass cloth                                                  ~55

GRP composite  approx                                  100

Bronze                                                                 77

Stainless Steel                                                    57

Lead                                                                  35.1

Steel , virgin                                                    25 – 35

Plywood                                                           10 – 14

Timber, softwood air dried, roughsawn    0.3 – 7.4

Timber, hardwood air dried, roughsawn   0.5 – 7.8

_________________________________________________________

These numbers agree to a certain extent with the graph of per capita oil above, when one considers when the different materials became commonplace boat building materials.

The other side of energy use is energy used to operate during lifetime.

Here the first line of adaptation will be towards greater efficiency. Huge motors pushing motorboats that have more than a passing hydrodynamic resemblance to household appliances will get rarer with each passing decade. Most cargo ships are already very efficient, but yachts and most ferries can be tremendously improved in terms of efficiency. Another thing will be the trend towards slower speeds. Resistance is roughly proportional to speed squared, while passage time is inversely proportional to speed. Fuel consumption is proportional to power neededwhich is speed times resistance, so fuel consumption is proportional to speed cubed. Therefore a lot less fuel gets used when going slowly, despite spending more time powering. We can already see this trend with large commercial ships; the so called ‘slow steaming’ approach.

It’s pretty much a given that sailing will become more and more used as it’s comparative economic viability improves.

Eventually motors themselves will become rare, although this will take a long time still and potentially heat engines will never completely disappear evn if they are limited to special uses where the energy equations make sense despite there being no more fossilized sunlight to exploit.

Welcome to the new paradigm.