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Math Help - Draining a conical tank

  1. #1
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    Draining a conical tank

    I am working on a paper at the moment which has to do with draining tanks. I have already set up a differential equation which explains the drain from a tank where the cross-sectional areas of the container and of the outflow are constants. But now I have to set up one which explains the drain from a conical tank where the cross-sectional area of the container varies. And that is my problem. I am finding it difficult to make this particular differential equation. I need it to be a dh/dt equation. I have got the following set up so far: (I got no clue if I am on the right track or not).

    From Bernoullis equation I got:

    (p1/Pg) + (v12/2g) + h1 = (p2/Pg) + (v22/2g) + h2, where P is the density of the fluid, v is the velocity and g is the gravitational constant, p is the pressure and h is the height. One must assume that the pressure is the same throughout the flow, therefore p1 = p2 = 0 and the parts of the equation containing p is removed:
    (v12/2g) + h1 = (v22/2g) + h2

    The velocity at the outflow (v2) can be defined as the change in height to the time dt:
    v2= -(dh/dt)
    Furthermore v2<< v1. Then v1 can be defined as follows:
    v1 = (2g*(h2-h1))^(1/2)

    From the continuity equation we have that: v1A1 = v2A2. Which means that the velocity at cross-sectional area 1 is different from the velocity at cross-sectional area 2. It also means that the smaller the cross-sectional area is the fluid has a lower velocity and vice versa.
    The cross-sectional area of a circle is: A = pi * r^2 but then the radius has to be known. The radius of the circle can be defined as: r = h*tan(a), where a is the angle of the cone. This radius is valid for all heights h. Then by inserting in the continuity I get:

    (2g*(h1-h2))^(1/2) * pi * r^2 = -(dh/dt) * pi * h^2 * tan(a)^2

    What do I do now?

    Thanks in advance, Thure.
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  2. #2
    MHF Contributor ebaines's Avatar
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    Re: Draining a conical tank

    First I would make the variables consistent. On the left hand side you use the term h_1-h_2, but on the right hand side you just use h. I would stick with just h, so you have:

     \pi R^2 \sqrt {2gh} = - \pi h^2 \tan^2(a) \frac {dh}{dt}

    Note that 'R' here is a constant - namely the radius of the outlet hole.

    You can rearrange to get a differential equation:

    \frac {dh}{dt} = - \sqrt {2g} R^2 \tan^2(a) h^{-3/2}

    If we set the cionstant K = \sqrt {2g} R^2 \tan^2(a) this can be written as \frac {dh}{dt} = - K h^{-3/2}. Rearrange:

     h^{3/2} dh = -Kdt, and integrate to get \frac 2 5 h^{(5/2)}= Kt +C, or  h(t) = [\frac 5 2 (-Kt+C)]^{(2/5)} The value for C can be found from the initial conditions:  C = \frac 2 5 h_0 ^{5/2}
    Last edited by ebaines; December 14th 2012 at 06:05 AM.
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