Someone on the Alpine list, in the last few days, had asked about horsepower
vs road speed. Since I am an old (stress "old") aero engineer, I thought I
would take a stab at it. There is a very simple formula for calculating
horsepower as a function of forward velocity. Forgive me if the "formatting"
does not come out quite right on the screen. The equations come from "Airplane
Aerodynamics" published in 1961, para 6:8.
HP = (Drag)(Velocity)/550
Drag = (Rho)( V*V)(Cd)(S)/2
Units of pounds (lb)
or
HP = (Rho)(V*V*V)(Cd)(S)/1100
where Rho is Std Air Density (0.00237691999 Slugs/cu.ft.)
V is velocity in ft/sec
Cd is drag coefficient
S is frontal area in sq ft.
Several things are apparent: Horsepower needed is a function of the cube of
the forward
velocity, Cd is probably hard to determine, but estimable, and S can be
grossly figured from knowing the dimensions of the car. Also, this is rear
wheel at the ground horsepower, not flywheel horsepower. Since our cars are
pretty slippery looking with the down sloped hood, rounded under valance, and
of course fins. The "S" in the equation is calculated to be about 18 sq ft.
using dimensions from "Tiger An Exceptional Motorcar". One thing you will have
noticed is the absense or rolling friction. We will cover that a little later
(it is pretty small). In the referenced old text book (yeah, I kept them all)
is a figure, figure 6:3 which has several shapes and the drag coefficient at
very low Mach numbers. As a SWAG I selected one which ia a cross between one
that looks a little like a 45 cal slug (0.295) and a bass ackward wedge (1.14)
giving me a drag coefficient of 0.5. Why did I select this one? Just because.
So, now everything is known and it is a simple matter to work the equation. I
use Excel 4.0. Of course you can play with the equation and use various Cd, S
and air densities to see how you fare in your neck of the woods. Note that Cd
varies from a worst of 1.28 (flat plate normal to the wind) to .01 type
numbers for really slick shapes (not our beamers, tho). Also bear in mind that
this is for straight and level cruising, no climb, no acceleration. Just what
it takes to get you down the road. As you can see, the horsepower to keep you
moving is pretty small. And you can see the effects of increased speed on fuel
economy: as speed increases, fuel consumption increases cubically. The effects
of rolling
friction from the rear axle bearings can be calculated in a similar fashion
but will add only small horsepower requirements. Excessive toe-in/out causes
drag also. Even if you estimate the rolling friction and toe-in to be as much
as 10 percent, it is still a small amount of horsepower. The table below only
factors wind resistance. The mechanical efficiency to the flywheel might be
70% so just divide the numbers below by 0.70 to get an estimate of SAE
horsepower (engine dyno) power needed.
A better question might have been: "How much horsepower to accelerate to
driving speeds in X seconds?" This is not much harder to do and it involves
Sir Isaac Newton's equation: F = m a and some derived equations. Shall I do
it?
Table 1
Horsepower
Vel (mph) Sea Level 2000 ft 4000ft
5 0.007675 0.007236 0.00681
10 0.06 0.06 0.05
15 0.21 0.20 0.18
20 0.49 0.46 0.44
25 0.96 0.90 0.85
30 1.66 1.56 1.47
35 2.63 2.48 2.34
40 3.93 3.70 3.49
45 5.60 5.27 4.96
50 7.68 7.24 6.81
55 10.22 9.63 9.06
60 13.26 12.50 11.77
65 16.86 15.90 14.96
70 21.06 19.85 18.69
75 25.90 24.42 22.98
80 31.44 29.64 27.89
85 37.71 35.55 33.46
90 44.76 42.20 39.71
100 61.40 57.88 54.48
110 81.72 77.04 72.51
120 106.10 100.02 94.14
130 134.90 127.17 119.69
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