Which sails quicker the rowing boat "trainera" or "trainerilla" ?

The great mathematics magazine + Plus Magazine, in its May 2012 issue published an article entitled "Outer space: Canoeing and kayaking", authored by John D. Barrow, which raises the question of whether these boats run more or less when they double the crew, which we do not know in advance because if the propulsive power is doubled so does the displacement and therefore the hydrodynamic resistance.

+ Plus presents simple and reasonable equations to estimate the variation of this resistance and determine from it the necessary power increase, which compares with the increased power provided by the additional rowers. He gets a theoretical conclusion that  a double crew boat increases its speed by 8 to 10%, what matches well with the actual data recorded in kayacs and canoes races. This conclusion appears consistent with the intuitive impression we have that larger similar boats are faster than smaller ones. 

Kayak

The theoretical conclusions drawn by J.D. Barrow are confirmed by actual data recorded in canoes an kayaks, and I wondered if same conclusions could be derived from other rowing boats such as the spanish "trainera" and "trainerilla"

a popular rowing boats family, together with the "batel", formerly used for local fishing, that nowadays are racing boats in summer competitions, propelled by:

Trainera

• "trainera": 13 rowers and a skipper
• "trainerilla":  6 rowers an skipper
• "batel":  4 rowers and no skipper.

From data recorded in races of "trainera" and "trainerilla" I have compared actual results with the theoretical ones and arriving at similar conclusions as in the kayacs family but with a less diference produced by boat size.

I will first show the theoretical calculations and then compare with actual data from kayaks and "traineras" family.

Equations from + Plus Magazine

1) In these low speed vessels, the hydrodynamic resistance (Rh) is basically due to friction of the hull with the water and therefore proportional to a) the wet surface (which in turn is proportional to the length to the square L^2) and b) the square of the velocity V^2. By indicating the proportionality with the # sign we have 

Rh # L^2 * V^2

2) The power Ph required to push the boat at velocity V, is the product of the resistance Rh by the speed and thus we have

Ph # Rh * V = L^2 * V^3

3) Since the displacement (weight) of the vessel is proportional to the cube of the length (assuming that the relationship between the dimensions is kept constant), and also the weight of the boat plus crew is approximately proportional to the number N of these, we can say that L^3 is proportional to N, i.e. L # N^1/3

4) Therefore we can say that

Ph # L^2 * V^3 = N^2/3 * V^3

5) On the other hand the power supplied by rowers is proportional to their number N, and must be the same as that required to move the boat at speed V, i.e.

N^2/3 * V^3 # N

i.e. V # N^1/9

The corresponding elapsed times will vary obviously with the inverse N^-1/9

By doubling the crew (N = 2) the speed would increase (time would decrease)  by 2^1/9 = 1.08,   

Actual recorded data in kayacs

Actual data times, as shown in Barrow paper, recorded in the Beijing Olympics, are:

                                     N=1      N=2      N=4

Men 1000m  K times 206.323s  191.809s   175.714s

Women 500m K times 110.673s  101.308s     92.231s

The quotients between these times for 1/2 and 2/4 are, for men and women, 1,08 - 1,09 - 1.09 - 1,10 in very good agreement with the theoretical prediction of 1,08.

Actual recorded data in "trainera" family (Ref.2)

Data from records at Bizkaya races (regatas)

N= 6            N=13

Men  3.500 m  time            791s

                       speed       4,42 m/s  

Men  5.556 m time                              1.200s

                    speed                              4,63 m/s

For quotient N = 13/6 = 2.167 the theoretic speed quotient is N^1/9 = 1,089

Actual speed quotient is 4,63/4,42 = 1.048 

Correlation with theory is not so good in theses boats but a factor to consider is that as run distance and times are so diferent rowers become more tired in trainera (longer trip) what gives a comparatively lower speed than otherwise should be.

References
Ref.1 +Plus "Outer space: Canoeing and kayaking"
Ref.2 Federación Bilbaina de remo (Rowing association of Bilbao)