Calculation of the volume of a chamber in a rotary vane pump

Question

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

• A circular rotor offset in a larger circular stator with a number of sliding sealing vanes extending between the rotor and the chamber wall.

• Sealed chambers are formed between adjacent vanes and the rotor and stator walls.

• 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.

• 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.

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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:

• The chamber volume of interest is shown in grey, bounded by vane1, vane2, and rotor and stator walls.

• 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.

• 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.

Answer 1

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.

Answer 2

One useful piece of information is how long each vane is at a given angle of rotation.

In the figure above, as in the question, $C_o$ is the center of the circular chamber of of radius $R_o,$ $C_i$ the center of the circular rotor of radius $R_i,$ and $d_o = C_oC_i$ is the distance between centers. The vane $PQ$ is parallel to the radial $C_iT$ of the rotor but offset by a distance $C_iM = \newcommand{dv}{d_{\mathit{voffs}}}\dv$. We suppose that the radial $C_iT$ is at an angle $\theta$ clockwise from the segment $C_iC_o$.

Then $MP = \sqrt{R_i^2 - \dv^2}$. Moreover, $C_oN = \dv + d_o \sin\theta$ and $$ PQ = MN + NQ - MP = d_o \cos\theta + \sqrt{R_o^2 - (\dv + d_o\sin\theta)^2} - \sqrt{R_i^2 - \dv^2}. $$

We can assume that $d_0,$ $R_o,$ and $R_i$ are all fixed values. In fact $\dv$ is a fixed value too for one surface of the vane, but since the vane has a front side and a back side at different offsets from the center of the rotor, we can get a little more accuracy in the calculations by considering these as too possible values of $\dv.$ So we write the exposed length of the vane as a function of $\dv$ and $\theta$: $$ L(\dv,\theta) = d_o \cos\theta + \sqrt{R_o^2 - (\dv + d_o\sin\theta)^2} - \sqrt{R_i^2 - \dv^2}. $$

If we consider $R_i$ a constant then the angle $\angle MPC_i$ is a function of $\dv$; let $\phi(\dv) = \angle MPC_i$, so $$ \phi(\dv) = \arcsin(\dv / R_i). $$

So now we are ready to look at the space between two vanes.

In this figure, $PQ$ is the front surface of a rotor, which has offset $d_1$ from $C_i$, while $P'Q'$ is the rear surface of the next rotor, which has offset $d_2$ from $C_i,$ with $d_2 < d_1$ since the rotor has some thickness.

If we let $\alpha$ be the angle between vanes (really the angle between the radials that each vane is parallel to), and define $\beta$ so that the angle $\angle PC_iP' = 2\beta,$ then $$ \beta = \frac12(\alpha + \phi(d_2) - \phi(d_1)). $$

Supposing again that the radial parallel to $PQ$ is at an angle $\theta$ clockwise from $C_iC_o,$ it follows that the radial parallel to $P'Q'$ is at an angle $\theta + \alpha$ clockwise from $C_iC_o.$ Then $PP' = 2R_i\sin\beta,$ $PQ = L(d_1,\theta),$ and $PQ = L(d_1,\theta+\alpha).$ This gives us three sides of the quadrilateral $QPP'Q'.$ We also have two of the angles: \begin{align} \angle QPP' &= \frac\pi2 + \beta + \phi(d_1),\\ \angle Q'P'P &= \frac\pi2 + \beta + \phi(d_2). \end{align}

These three sides and two angles completely determine the quadrilateral $QPP'Q',$ so it is possible to compute from them the area of the quadrilateral and the length of the side $QQ'.$ I would do this by assuming Cartesian coordinates with the $x$ axis parallel to $PP',$ but there are other ways.

Now from the radii $R_i$ and $R_o$ of the circles and the lengths of the chords $PP'$ and $QQ',$ you can compute the areas of the circular segments cut off by the two chords. The area of the chamber is then the area of the quadrilateral $QPP'Q'$ plus the area of the segment of the circular chamber ($QQ'$) minus the area of the segment of the rotor ($PP'$).

If you want to be really accurate I suppose you might want to account for the small gaps between the ends of the vanes and the chamber wall, because the flat front and back surfaces of the vane cannot simultaneously touch the chamber wall at all times while the rotor rotates. But this depends on the exact shape of the ends of the vanes, and it would be a relatively small correction.