# Spherical curves for the design of conical Moineau pumps

## Jens Gravesen

Department of Mathematics

Technical University of Denmark

for Grundfos

## Background

The Moineau pump is an invention from 1931 (US-Patent no. 1 892 217) by the French engineer René Moineau.

In the photo to the right models of the classical 2:1 hypo-cycloid construction and the 3:2 epi- hypo-cycloid construction are shown. In both cases the stator is to the left and the rotor is to the right. The oddly shaped object in front of the 2:1 hypocycloid construction is a model of the pump chamber.

The rotor is moved inside the stator by an eccentric motion that can be generated by letting a cylinder of radius n-1 roll inside a cylinder of radius n.

During the motion a series of closed pump champers are moved up (or down) and in contrast to centrifugal pumps the speed and thereby the flow can be as small as you like. The pump chambers have constant volume, and this makes the pumps ideally suited for incompressible fluids.

## The classical cylindrical Moineau pumps

The geometry of the pumps seems quite complicated but it helps that each horizontal section looks exactly the same. A horizontal plane intersect the pump in two closed curves that are moved relative to each other by the motion generated by letting a circle of radius n-1 roll inside a circle of radius n.

The (green) stator is the envelope generated by the motion of the (red) rotor. The two closed curves forms n chambers that during the motion open and close.
The classical designs are based on hypo- and epi-cycloids and are described in the table below.

Curves Number of "teeths" on the stator and rotor
Hypo-cycloids 2:1 3:2 4:3
Epi-cycloids 2:1 3:2 4:3
Hypo- & epi-cycloids 2:1 3:2 4:3

## The general design of cylindrical Moineau pumps

By looking at a horizontal section the design of a cylindrical Moineau pump is reduced to the design of two closed planar curves that are moved relative to each other by the motion generated by letting a circle of radius n-1 roll inside a circle of radius n. If the rotor is given then the stator can be determined as the envelope given by the motion of the rotor. In fact, the stator can be determined by the motion of a single convex arc of the rotor and the concave parts of the rotor can be found as envelope of the stator during the opposite motion. The whole proces is described here.

The determination of the envelope is a non-linear problem and can normally only be done numerically. But by using the support function to represent planar curves, see [2], the equations can be solved analytically and it is possible to find new designs in closed form, see [1,3].

## Conical Moineau pumps

In the photo to the right a model of a concial Moineau pump is shown (a 2:1 hypo-cycloid construction). The cylindrical motion is replaced by a conical motion. That is, the motion is generated by letting two circular cones roll on each other.

We now intersect the construction with spheres centered at the commen apex of the cones defining the motion. A pair of closed spherical curves that are moved relative to each other by the motion generated by letting two spherical circles roll on each other.

As we move away from the apex of the cones the size of the curves grows so in order to have constant volume of the pump chambers the offset is decreased.

In order to have a periodic motion the circumference of the two circles has to be in the proportion n : n-1. If the top angles of the two cones are 2v and 2w, respectively, then

(n-1)sin(w) = nsin(v).

If the 2:1 hypo-cycloid construction (n = 2) is intersected by the unit sphere there will be a gab of the size

gab = w - 2v = Arcsin(2sin(v)) - 2v = v3 + O(v5),

see the figure on the right.

As the pumps in practise are equipped with some sort of rubber sealing this is not a serious problem for small angles, but it would of course be preferable to have a mathematical watertight construction.

## The problem

• Is it possible to construct a conical Moineau pump which is mathematical watertight?
• If not, then what design has the minimal possible gab (for a given angle v)?
• Is it possible to find a representation of spherical curves that allows exact calculations similar to the planar case?

## References

1. Report from the 57th European Study Group with Industry, Kgs. Lyngby, Denmark, 2006, http://www2.mat.dtu.dk/ESGI/57/report.
2. Zbyněk Šír, Jens Gravesen, and Bert Jüttler, Curves and surfaces represented by poynomial support functions, Theoretical Computer Science 392, 141-157, 2008.
3. Jens Gravesen, The geometry of the Moineau pump, submitted to Computer Aided Geometric Design