Monday, 21 May 2012

On Horsepower and Hull Speed

An anonymous reader (I feel like I’m doing science again – responding to reviewer comments !) pointed out that the original 18 HP engine for the Dehler is a bit underpowered, and that we might consider upgrading to something stronger.  We had always used a simple rule of thumb to judge the appropriate size of the motor:  the horsepower of the motor, multiplied by 2, should be greater than the length of the boat in feet.  So an 18 HP engine gives a result of 36, which is greater than the 34 feet of the boat, indicating that the engine is about right for the boat.  This method, you realize, is total nonsense, since you don’t push length, you push weight.  I decided to get to the bottom of all this.

A brief tour of the web in both French and English provided a better rule of thumb:  for a sailboat, you need approximately 5 HP of power for every ton.  The Dehler’s displacement is about 4 tons, so we need 20 HP to push it, which suggests that the original 18 HP engine is, indeed, a bit on the small side. 

A nice example of a wave of a boat at hull speed.
Being a bit more rigorous, I decided to estimate the theoretical hull speed and then calculate the power required to reach that speed.  Let me explain.  As a sailboat moves through water, it has to push the water out of its way, and thus creates a bow wave that moves along the side of the boat as the water is displaced.  As the boat gains speed, this wave gets “stretched” and becomes longer.  When the boat reaches its hull speed, the wave length is equal to the length of the boat, creating a lovely sine wave* with a crest starting at the bow, a trough in the middle, and the end crest at the stern. If you were to go beyond this speed, you would stretch the wave to the point where the trough that was once in the middle of the boat would now be at the stern.  This literally means that the boat’s stern would be in a trough and that the boat would now have to go uphill, requiring an excessive amount of energy to overcome.  Sailboats under sail alone (at least monohull displacement boats) don’t have the power to do this. 
* okay, physical oceanography friends, I know that ocean waves are actually trochoids and not sine waves, but let’s keep it family friendly here, shall we?

The method for calculating theoretical hull speed and the power required to reach it are, of course, riddled with assumptions and highly criticized.  But let’s throw caution to the wind and judge the value of the method depending on whether or not we like the result it gives.  (Gee, now I REALLY feel like I’m back doing science again !).  Here are the equations:

(1) Hull speed = 1.34 x square root (length at waterline in feet).     (result is in knots)
  
(2) Power required to reach hull speed = Displacement (in pounds) / ((150)^2 / (hull speed)^2)     (where the symbol ^2 means to square the expression in parentheses).

Plug-n-chug the numbers for the Dehler and you get a theoretical hull speed of about 7 knots (significant figures would be total fantasy here), and a power requirement of 20 HP.

So the good news is that this answer jives with the (much simpler) rule of thumb of 5 HP per ton and suggests once again that the 18 HP on the Dehler is underpowered, or at least not powerful enough to reach the boat’s theoretical hull speed.  Now, most sailors, when faced with the theoretical hull speed of their craft, take on an indignant tone and announce that they have routinely sailed in excess of this speed.  This illustrates one of the criticisms of the estimation.  The “1.34” constant in Equation 1 is for a boat that moves by displacing water, as is the case for most sailboats.  Naval architect Dave Gerr has recently revised this constant, and now proposes 1.49 instead of 1.34, which gives us a speed of almost 8 knots for the Dehler.  This new number, however appealing, still doesn’t take into consideration the true motion of modern boats through water.  Modern design has made the sterns wider and flatter, giving the boat (and the crew) a more comfortable motion when wind and waves are from astern.  This also means that the boat has the ability to plane, or surf, as it slides down the waves, and when it does this, a fraction of the boat’s motion is no longer via displacement, but rather by planing, which requires a different calculation.  But even I don’t want to go there, and even if I did, I don’t have the numbers required to calculate this for my boat.    

Others cast aspersions on Equation 2, noting that propeller efficiency is never considered here.  What you calculate is just the energy required to turn the propeller shaft, but this doesn’t make allowances for how that motion is translated into push by different blades and blade-angle considerations.  The other problem is that this number only tells you the power required to reach a certain speed for a certain weight on a flat calm body of water with no wind.

So what have we learned?  I am now convinced that the 18 HP on the Dehler is not sufficient for us to reach the theoretical hull speed.  But now comes the real conundrum.  Is that something I really want to do?  As with most problems, the key is to appropriately formulate the question.  What is it I really want to know?  Am I asking “what is the biggest engine I can put on my boat without incurring structural damage or doing back-flips?” or “what size engine should I have to get me out of sticky situations (no wind, rapid currents) but that will not penalize me in terms of weight or fuel efficiency for the other 99% of the time when I am not in a sticky situation?”

Having recently read Lin and Larry Pardey’s excellent book, “The Self-Sufficient Sailor” where they extol the virtues of doing everything by sail and using the empty motor compartment to install a homemade bathtub, I’m leaning more towards the “less-is-more” approach.  In any case, this whole discussion is a bit academic for us, since we don’t have the money to install a new engine right now. 

But the real question that I should be asking, the one that will really get to the bottom of everything in a useful manner is, “What does my INSURANCE AGENT think is the biggest engine I can put on my boat?”.

Now I’m really in a pickle.  (“In a pickle” is another hillbilly expression meaning in a big mess.)  My insurance agent will simply look at the “maximum HP rating” for the boat, and, here in France, will officially allow a tolerance of 15% over this number.  If I look on the constructer’s plates, I see that the maximum power for my boat is 9.8 kilowatts.  With 1 HP = 0.746 kW, that gives me a maximum HP of (gulp) 13 !  Even with the 15% tolerance, the original 18 HP Dehler engine is already “too powerful” !?  I’m bamboozled.  Either I’m missing a conversion somewhere or there is some sort of interpretation of this number that escapes me.*

With a quivery chin, I broke down and decided I needed help on this one, so I sent an email to a friend who is a retired naval architect and spent many years as an expert boat inspector for an insurance company.  He estimates that with the 18 HP motor between 2/3rd – 3/4th  full throttle on the Dehler, I should be able to do 6 knots, which, in his opinion, is sufficient to get out of sticky situations, even if it wouldn’t allow us to go anywhere we want, anytime we want (although he also notes that our folding 2 blade propeller will reduce our efficiency somewhat).  A quick survey of the available French boating forums also suggests that the insurance companies really don’t care what size motor you put on the boat as long as it’s “reasonable” and that it is declared with them before you have an accident involving excessive speed. 

In the end, I like my friend’s summary of the situation, in part because it makes sense and in part because it accords with the limitations of our bank account at the moment:  when discussing the need to go beyond 6 knots with the motor, he simply replied, “After all, it’s a SAIL boat.”


*Addendum:  Now that we have the original papers for the boat, I see that the maximum HP rating is 19.8 kilowatts, not 9.8 !  Apparently, the “1” etched on the metal builders plate has not stood the test of time.  19.8 kilowatts gives 26.5 HP, which makes more sense.  The Dehler 34s were originally delivered with either 18HP or 27HP.  

For further and more comprehensive reading on the subject:
  1. The most excellent article by Bryon Anderson on The Physics of Sailing in Physics Today, now available on-line for free.
  2. Excellent posts on the forum of www.boatdesign.net
  3. The great post on Pocket Yacht Cruising entitled “Sailboat Math"


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