Solving Orifice Plate Calculation Equation - Help Needed

[mattaddis]

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

 

[Mentallic]

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.

 

[Gerenuk]

Check if it's
$$d_2=\frac{1}{\sqrt[4]{\left(\frac{\pi}{4\alpha}\right)^2+\frac{1}{d_1^4}}}$$

 

[mattaddis]

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

 

[Mentallic]

You're welcome