Subject: Re: [baidarka] Superiority of Native paddles
From: Michael Daly (michaeldaly@home.com)
Date: Mon Jun 18 2001 - 21:14:21 EDT
From: "Peter A. Chopelas" <pac@premier1.net>
> Hi all,
>
> Therefore just for grins I have listed a mathematical proof that you
> technical types might find interesting.
Derivation - not proof.
> The aspect ratio (AR) of a surface is the span squared divided by the area
> of the surface: AR=b^2/s where b=span s=surface area
>
> This is used to accommodate all shapes such as ellipse or round plan forms,
> notice that for rectangular surfaces the AR simply becomes the length
> divided by the width (or chord length c) or AR=b/c
The chord is measured streamwise. Below you use the width of the paddle,
which in typical usage is normal to the flow. Why are you not using the thickness
of the blade?
If you use the thickness, you'll find the difference in aspect ratios between Euro
and native paddles not so great because if the considerable mean thickness of
the native paddles.
> So this relationship between the
> induced drag and the AR is just a natural phenomenon that is both observed
> and mathematically valid as well.
Assuming you use the definition of AR for which the equation is derived.
> Since the efficiency we are looking for is the power-out (i.e. forward
> thrust) divided by the power-in (resistance at the paddle handle) we have
> to covert these forces (in lbs. for example) to units of power by
> multiplying them by the velocity of the paddle through the water. so the
> equation will reduce to the following:
>
> efficency=P-out/P-in = TxV/DxV = T/D since the velocity cancels
>
There are several problems with this assumption.
One - I haven't got any paddles with handles. Mine all use a shaft to transmit
the force to the blade and the forces in the shaft are derived from a moment
produced by my torso's muscles (I don't change the angle of my elbows much
and thus don't use arm muscles for generating thrust). This puts the paddle
blade on a roughly circular path. The path is about an eighth of a circle,
possibly more. I'll go along with your assumption of multiplication of velocity
for small angles, but an eighth of a circle isn't a small angle.
Two - A long native blade can easily be shown to have a tip velocity about
twice the velocity where the blade is at the surface; perhaps more. Again,
I think that a multiplication is an oversimplification. Think convolution.
Three - Why is the "handle" velocity the same as the blade velocity? Given
the mechanics of paddling, I'm having a hard time mapping your "handle"
to a real paddle in such a way that I can see the equality.
> But for low speed cursing
I may paddle slow, but I curse fast! (sorry, couldn't resist!)
> typical kayak at 5 knots (about 5 LB of drag, therefore 5 Lbs. of thrust at
> five knots need to be generated) I came up with the following for different
> paddle AR but with the same blade area:
>
> for typical Euro touring blade it has a blade approx. 6"x18" and AR= 3:1
>
> a typical native style blade 3"x36" an AR=12:1
>
> hypothetical very high AR paddle of 2"x54"...AR=27
>
> the following sustained power at the handle would be required from the
> paddler:
>
> AR=3:1 P=0.4 hp
> AR=12:1 P=0.22 hp
> AR=27: P=0.15 hp
Compare this to Matt Broze's figures in Sea Kayaker: 0.09 hp at 5 knots.
(August 1998, page 43)
>
> Useful Power-out: 5 lb drag at 5 knots = 5 LB x 5 x 1.688 feet/sec/knot =
> 42.2 foot-lb/min = 0.001279 hp
>
> efficiency of converting power-in to forward movement: 0.001279/0.025 =
> 0.051 or about 5 percent
>
> which is about what I suspected. The rest of your input goes to other
> things like heating water, lifting the weight of the paddle, etc.
I can think of few things, other than those powered by infernal combustion
engines, which would survive in the real world with such pitiful efficiencies.
Your initial definition of efficiency, once the velocity is factored out, says
that the efficiency is the ratio of the forces in and out. That means that
5 lb out requires 100 lb in. I think you should check your math.
Consider another gross check on your numbers. I can easily paddle my
Solstice GTH with an Aquabound Expedition paddle at 7 km/hr for at least
an hour (measured twice in different sea conditions). That's faster
than I walk (everyday, home from work)! Now try to convince me that I can
do that with such poor efficiency. If that were true, arm-powered bikes would
whomp the leg-powered variety any day. My partner works with disabled
athletes, some of whom use arm-powered bikes. I can verify that this doesn't
happen.
> From this you can see that smooth foil shaped, high aspect ratio blades are
> best for conserving energy over long hauls.
>From this I can see that you have to try again. Also note that John Winters has
published, in The Shape Of The Canoe, a graph of his experimental results that
show that efficiency _decreases_ with aspect ratio. You'll have to explain why
your hypothesis is more valid than his experiments with real paddles.
Note that I'm not saying anything about whether Euro or native paddles are better.
I'm just looking for a derivation I can put some faith in.
Mike
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