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Solving equation

  1. Jul 9, 2010 #1
    Hello there,

    I am a mechanical engineer and am attempting to build an Excel document for calculations purposes.

    Currently I am stuck on orifice plate calculations.

    I was wondering if someone could help me solve the following equation for d2?

    [tex]\alpha[/tex] = [tex]\pi[/tex](d2/2)2 [tex]\sqrt{}[/tex]1/1-(d2/d1)4

    Any help would be appreciated.

    Thanks in advance.

    (Apologies if I have posted this in the wrong place)

    Matt
     
  2. jcsd
  3. Jul 9, 2010 #2

    Mentallic

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    Homework Helper

    Sure thing,

    [tex]\alpha = \pi\left(\frac{d_2}{2}\right)^2\sqrt{\frac{1}{1-\left(\frac{d_2}{d_1}\right)^4}}[/tex]

    square the entire equation to rid yourself of the square root:

    [tex]\alpha^2 = \pi^2\left(\frac{d_2}{2}\right)^4\left(\frac{1}{1-\left(\frac{d_2}{d_1}\right)^4}\right)[/tex]

    Multiply through by that denominator:

    [tex]\alpha^2\left(1-\left(\frac{d_2}{d_1}\right)^4}\right) = \pi^2\left(\frac{d_2}{2}\right)^4[/tex]

    Expand the left side, and move the right to the left side:

    [tex]\alpha^2-\alpha^2\left(\frac{d_2}{d_1}\right)^4} -\pi^2\left(\frac{d_2}{2}\right)^4=0[/tex]

    This can be more easily visualized as:

    [tex]\alpha^2-d_2^4\frac{\alpha^2}{d_1^4} -d_2^4\frac{\pi^2}{16}=0[/tex]

    Factorize out the required variable:

    [tex]\alpha^2-d_2^4\left(\frac{\alpha^2}{d_1^4} +\frac{\pi^2}{16}\right)=0[/tex]

    Well you can probably finish it from here, and you might want to manipulate some things so you don't have fractions in fractions.
     
  4. Jul 9, 2010 #3
    Check if it's
    [tex]d_2=\frac{1}{\sqrt[4]{\left(\frac{\pi}{4\alpha}\right)^2+\frac{1}{d_1^4}}}[/tex]
     
  5. Jul 9, 2010 #4
    Thankyou both for yor prompt reply.

    I have tested the two formulas based on existing figures and Mentallic's formula/equation gives the desired result.

    Gerenuk, you equation gives d1, not d2.

    Thank you very much for your assistance.

    Matt
     
  6. Jul 9, 2010 #5

    Mentallic

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    Homework Helper

    You're welcome :smile:
     
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