Solving Orifice Plate Calculation Equation - Help Needed

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A mechanical engineer seeks assistance with an orifice plate calculation equation for d2, specifically α = π(d2/2)²√(1/(1-(d2/d1)⁴)). A user provides a step-by-step solution, including squaring the equation to eliminate the square root and rearranging terms to isolate d2. The engineer confirms that the provided formula yields the correct results when tested against existing figures. Another user points out that one equation mistakenly calculates d1 instead of d2. The discussion highlights collaborative problem-solving in engineering calculations.
mattaddis
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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?

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

Any help would be appreciated.

Thanks in advance.

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

Matt
 
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Sure thing,

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

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

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

Multiply through by that denominator:

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

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

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

This can be more easily visualized as:

\alpha^2-d_2^4\frac{\alpha^2}{d_1^4} -d_2^4\frac{\pi^2}{16}=0

Factorize out the required variable:

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

Well you can probably finish it from here, and you might want to manipulate some things so you don't have fractions in fractions.
 
Check if it's
d_2=\frac{1}{\sqrt[4]{\left(\frac{\pi}{4\alpha}\right)^2+\frac{1}{d_1^4}}}
 
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
 
You're welcome :smile:
 
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