maths question

[PlacentaJuan]

i need to find an angle (a)

i have a standard butterfly type throttle of diameter (x) that i suppose is 0 degrees when closed and 90 degrees when open.

what angle (a) does the butterfly need to open to equal an open area of (y)?

this equation is quite beyond me, i know there are smarter poeople here than can probably do it in their heads and hope you can help me.

sorry i dont have real figures, but the algebraic equation would be good.

thanks

[Hash_ra23]

y = PI*(x/2)^2 - PI*(x/2)^2*cos(a)

[oldcorollas]

angle of closed throttle is not 0 either... a few degrees off...

[PlacentaJuan]

yea thats what i thought, its actually not a throttle and im pretty sure that this particular butterfly is copmletely closed, although it might have a few degrees in it if its not perfectly round to seal.

either was there is no gap when closed.

[Hash_ra23]

ohhh..well my calculation was for an ideal theoretical throttle....i.e. at 0 degrees fully closed...90degrees fully open. And the throttle valve has no thickness.

[PlacentaJuan]

hey what is that equation with relation to a?

ie a (angle) = .....

[Hash_ra23]

just gotta rearrange equation....

a = inversecos[ 1 - ( y / PI*(x/2)^2 )]

i think thats it....

[SuperDave]

I get the same.

[The Last Streetfighter]

Atot = Total Area = Pi * r^2
N = %open = 1 - cos (alpha) {alpha = 0.. 1 - cos (0) = 0; alpha = 90.. 1 - cos (90) = 1: there is your range from 0% open to 100% open. At 45degrees the thing is 29% open}
Aopen = Open Area = Atot * N = Pi * r * r * (1 - cos(alpha))

Alpha Radians Cos (alpha) 1 - cos (alpha)
0 0.00 1.00 0.00 Fully Closed
10 0.17 0.98 0.02
20 0.35 0.94 0.06
30 0.52 0.87 0.13
40 0.70 0.77 0.23
45 0.79 0.71 0.29
50 0.87 0.64 0.36
60 1.05 0.50 0.50
70 1.22 0.34 0.66
80 1.40 0.17 0.83
90 1.57 0.00 1.00 Fully Opened

[amichie]

I get the same.

Me too.

Area of a circle - area of the ellipse.