Volume Equations

[soho54]

Does anyone have the equations for the volume of a hyperbolic, and conical horn on hand.

I'm taking about the horns volume, not one of the chamber volumes.

Thanks.

[tb46]

Hi soho54,

For cones I have always used the approximation: volume = (area1+area2)/2 * height.

Here is a reference, this can get difficult pretty fast:

Solid of Revolution -- from Wolfram MathWorld

Hope this helps,
Regards,

[wakibaki]

Volume of a cone: 1/3 * pi * r^2 * h

I'm trying to lay my hands on a book that has the hyperbolic.

w

[wakibaki]

What is the equation of the hyperbolic?

w

[soho54]

There should be a direct way to do it. I'm trying not do to it with the slice method tb46 linked to.

This is the equation for the volume of an Exponential Horn
V=(St/k)*(e^(k*x)-1)

It is very easy to use as I already have St, K, and x from the cross-sectional area equations. Mo=St*e^(k*x)

The parabolic I can't remember right now, but it is similarly easy.



Conical cross-sectional area is:
Mo=St*((x+Xo)/Xo)^2

Hyperbolic cross-sectional area is:
Mo=St*((COSH(x/Xo)+(m*SINH(x/Xo)))^2

[wakibaki]

In the general form, volume of a rotation, its: the integral from a->b of pi()*(y^2)dx

You substitute for y^2 and perform the integration.

For a sphere, for example, the equation of the boundary curve is x^2 + y^2 = r^2

Re-arranging: y^2 = r^2 -x^2

Substituting, the volume of a hemisphere is: the integral from 0 -> r of pi()*(r^2 - x^2)

Which evaluates to [0 -> r] [pi()*(r^2*x - x^3/3)]

=pi()*(r^3-[r^3/3])

= 2/3 pi()*r^3

therefore volume of a sphere = 4/3 pi()*r^3

You just need the equation of your hyperbolic in terms of y, then you can work out y^2 and substitute, integrate.

vol_of_sphere.JPG

w

[Ron E]

Originally Posted by soho54

There should be a direct way to do it. I'm trying not do to it with the slice method tb46 linked to.

Conical cross-sectional area is:
Mo=St*((x+Xo)/Xo)^2

Hyperbolic cross-sectional area is:
Mo=St*((COSH(x/Xo)+(m*SINH(x/Xo)))^2

Look for the theorems of Pappus Guldinus - if you can do an integral and find the centroid, you are all set.

[soho54]

Conical horn volume
V =(1/3)*PI()*x*((r1^2)+(r2^2)+(r1*r2))

r1=sqrt(St/PI())
r2=sqrt(Mo/PI())
x=distance
St=throat Sd
Mo=mouth Sd

[tb46]

Hi soho54,

That looks correct, naturally I made a typo in Post #2, obviously it's divide by 3 not 2.

Hope you find an easy solution for the expo/hypex horns.

Regards,

[wakibaki]

Originally Posted by soho54

Conical horn volume
V =(1/3)*PI()*x*((r1^2)+(r2^2)+(r1*r2))

r1=sqrt(St/PI())
r2=sqrt(Mo/PI())
x=distance
St=throat Sd
Mo=mouth Sd

No, that just can't be right.

The formula is obtained by calculating the volume of the cone as it would be if extended to a point (the original formula I gave), and subtracting the volume contained by the cone which is 'missing'

(1/3 * pi * r1^2 * h1)-(1/3 * pi * r2^2 * h2)

where r2 is the radius of the throat, r1 is the radius of the mouth, h1 is the distance from the mouth to the point of the cone as it would be if extended to a point and h2 is the distance from the throat to the point of the cone.

By inspection it is obvious that the second formula,

(1/3 * pi * r1^2 * h1)-(1/3 * pi * r2^2 * h2)

rearranged,

Can never contain this term

(r1*r2)

which is in the formula you have given. The formula you have given, moreover has no provision for calculating the value of h1 or h2.

Taking x as the distance from throat to mouth since h1 = x + h2 we have:

(1/3 * pi * r1^2 * [x+h2])-(1/3 * pi * r2^2 * h2)=

1/3 *pi * [(r1^2 * [x+h2])-(r2^2 * h2)]

(x+h2)/h2=r1/r2 (similar triangles)
x+h2=h2*r1/r2

so in terms of h2 we have

1/3 *pi * [(r1^2 * [h2*r1/r2])-(r2^2 * h2)]
=1/3 *pi * h2 * [(r1^2 * [r1/r2])-(r2^2)]
=1/3 *pi * h2 * [(r1^3/r2)-(r2^2)]

but

h2=h1-x
h1=h2*r1/r2

therefore

h2=h2*(r1/r2)-x
x=[(h2*r1)/r2]-h2
x=h2[(r1/r2)-1]
h2=x/[(r1/r2)-1]

hence in terms of x

1/3* pi * x/[(r1/r2)-1] * [(r1^3/r2)-(r2^2)]

w

It's a bit late. I'll look at it again in the morning