Question proper: What is an expression for the area shown in grey in the diagram below?
I wish to ascertain the volume of a chamber in a RVP (Rotary Vane Pump).
(Volume is proportional to area in the 2D view shown below).
RVPs are used as both vacuum and pressure pumps for compressible fluids.
A basic RVP consists of
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A circular rotor offset in a larger circular stator with a number of sliding sealing vanes extending between the rotor and the chamber wall.
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Sealed chambers are formed between adjacent vanes and the rotor and stator walls.
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As the rotor turns the offset stator-rotor relationship results in a progressive reduction in chamber volume, which results on pressurisation of the pumped fluid.
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RVPs may have a number of vanes and chambers. The photo and diagram below show a 3 vane / 3 chamber pump, but 2 vane and 4 to 6 vane RVPs are common and more vanes might be used in specialist applications. 1 vane pumps are used in some applications with the necessary high pressure seal being formed by extremely low rotor to stator clearances at the closest point.
Much more complex chamber shapes and other refinements may be used but this model matches many real-world pumps well.
The photo below shows a typical 3 vane RVP and the diagram shows the relationship that I wish to be able to analyse.
Note that the diagram shows vanes extending along radii while the photo shows them extending along lines parallel to radii but offset by a distance of say Dvoffs. An analysis which includes this offset would be useful if available. I will add a second version with this condition to the question as soon as I can (3 am here now) but the question as currently framed is useful to me.
The circular rotor (blue) turns in a clockwise direction within a larger circular stator (orange).
Fluid (air usually) is drawn in via a port typically at upper right and exhausted at a higher pressure via a port at far left.
This is important in practice but irrelevant to the current analysis.
Three vanes / 3 chambers are shown but an analysis based on an arbitrary inter-vane angle will allow any number of vanes and intermediate state situations to be accommodated.
Volume of interest:
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The chamber volume of interest is shown in grey, bounded by vane1, vane2, and rotor and stator walls.
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The chamber occupies angle A1 of arc – here = 360/3 = 120 degrees but desired to be able to be specified in an analysis as any angle for specified angles A2 and A0.
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The vanes are assumed to be of zero radial thickness.
I can see that the required expression is liable to be "not overly complex" but have not been able to see how to deal with the non-normal angles that the vanes make with the stator. These are more important at some vane angles than at others.
A chamber could lie above and below the line A-B but if an expression for volume is made too complex by this then an analysis could readily be carried out seperately for the portions above and below line A-B and the results summed.
R0 … Radius of Stator (outer)
Ri … Radius of rotor (inner)
Do .. Offset between Rotor and Stator
A …. Wall clearance between rotor and stator
(This is usually maintained at close to zero for practical reasons but may be non zero)
A = R0 – Ri – Do
Co … Centre of outer / stator
Ci … Centre of inner / rotor
It can be seen that:
Line Co-F is normal to outer but not to inner.
Line Ci-E is normal to inner but not to outer.
Best Answer
This answer uses slightly different parameters than the ones in your question
Here, radius of outer circle $O = R$ and radius of inner circle $C = r$. Distance between the centres of the circles is $d$.
The area of the shaded region $A = \text{area(large sector) - area(small sector) - area(AOC) - area(BOC)}$
Let $BC = x$ and $AC = y$. Applying cosine rule in $\triangle BOC$, we have $$\begin{gather} \cos \beta = \frac{x^2 + d^2 - R^2}{2xd} \\ \implies x = d\cos \beta + \sqrt{R^2 - d^2\sin^2 \beta} \end{gather}$$
Thus, $\text{area}(\triangle BOC) = \frac 12 xd\sin \beta$. Using the same method, we can find $y = d\cos \alpha + \sqrt{R^2 - d^2\sin^2 \alpha}$ and $\text{area}(\triangle AOC) = \frac 12 yd\sin \alpha$. The area of smaller sector is $A_s = \frac 12 r^2(2\pi -\alpha -\beta)$.
Finding the area of the larger sector requires some more calculation. Using sine rule this time, $\sin(\angle COB) = \frac{x\sin \beta}{R} \implies \angle COB = \arcsin\left(\frac{x\sin \beta}{R}\right)$. Similarily, $\angle COA = \arcsin\left(\frac{y\sin \alpha}{R}\right)$
The area of the shaded region is thus $$A = \frac 12 \left( R^2 \left(\arcsin\left(\frac{y\sin \alpha}{R}\right) + \arcsin\left(\frac{x\sin \beta}{R}\right) \right) - r^2(2\pi -\alpha -\beta) - xd\sin \beta - yd\sin \alpha \right)$$
where $x = d\cos \beta + \sqrt{R^2 - d^2\sin^2 \beta}$ and $y = d\cos \alpha + \sqrt{R^2 - d^2\sin^2 \alpha}$.
The volume is easy to find once the area has been found. $V = Ah$, where $h$ is the height of the enclosing cylinder in question.