## 1 Introduction

In many applications a low flow and pressure pulsation is needed to obtain good results. For example, it is crucial to determine the delay between the visual detection and the separation of the cell types in flow cytometry. By reducing the flow pulsation of the used pump, the uncertainty in the calculation of the delay caused by the changing volume flow can be minimised. By increasing the mean flow, the pump can deliver the same volume flow while running at a lower rotation speed. This would result in less wear, less noise and reduced pulsation. Every positive displacement pump has a flow pulsation, which is mainly caused by its displacement device and the ripple of the pump drive. Within the group of continuously working positive displacement pumps, a gerotor pump is one of the pumps with the lowest pulsation. Nevertheless, some new approaches to generate gerotor profiles with low pulsation and less wear were made in the last years. In order to compare these new approaches to the older ones, a MATLAB toolbox was created.

## 2 Methods

All gerotor pumps consist of an inner rotor with N teeth and an outer rotor with N+1 teeth. The rotors are mounted eccentrically to each other. The geometry of both rotors will describe N+1 dynamically-changing volumes, of which N will take part in the volume displacement of a full rotation of the inner rotor.

Depending on the approach, the inner and outer rotors are either calculated separately or one of the rotors is calculated first and the other rotor is derived from this by e.g. the envelope theory.

All approaches to calculate either the first or both rotors use a roulette of one geometric shape or a combination of two or more roulettes. The calculation of the mean flow and the pulsation is carried out by using the given analytic functions.

The new MATLAB toolbox GPT (Gerotor Pump Tool) uses a different approach to calculate the mean flow, the flow pulsation and the pressure angle of gerotor profiles, which are all very difficult to describe analytically. A reason for this difficulty could, for example, be due to the fact that the point cloud data of a manufactured gear set is obtained via a microscope or a measuring machine and thus contains measurement errors, which would have to be considered before using an analytical approach.

The main menu of the GPT is divided into three sections as shown in Figure 1.

The first section, “Gerotor Profile Library”, shown in Figure 2, is used to generate a point cloud from one of the following profiles from literature:

Figure 1

Main menu of the GPT with the different tool boxes.

– GT1: Standard Trochoid [1, 2],

– GT2: Hypotrochoid [3–5],

– GT3: Epitrochoid-Circle-Hypotrochoid [6, 7],

– GT4: Ellipse [8–10],

– GT5: DF-Method [11–13],

– GT6: Ellipse-Involute-Ellipse [14].

The point clouds are calculated via the given analytic functions and the user given parameters, which are specific for each profile type.

Figure 2

Gerotor Profile Library function (Example: GT1 – Standard Trochoid).

After the generation of a point cloud from the Profile Library, the data can be passed on to the next section called “Flow Rate Analysis Tool”. This tool calculates the chamber area “A” between the profile of the inner and outer tooth for a whole turn using a step width given by the angle “*φ*”, which the user must specify. This is done by searching the minimum distance between the rotor profiles in a given area of the point cloud. This area contains all data between the centre and the upper as well as the lower end of a defined section. This section is centred at one inner rotor tooth tip and has a whole width of 360° divided by the amount of teeth of the inner rotor. The result is an A(*φ*)-plot and -chart. This tool can also import profile data provided via Excel-Worksheets to analyse a reverse engineered data set.

Afterwards, the data can be passed to the “Flow Rate Irregularity and Comparison” tool. The tool can be used to calculate the flow ripple and the mean flow by specifying the height of the rotors and the rotation speed. To do so, the A(*φ*)-data from the previous toolbox is first smoothed and then deviated by the angle *φ*.

$\dot{A}\left(\varphi \right)=\frac{\partial A\left(\varphi \right)}{\partial \varphi}$

After this step, only the positive values are taken into account, as these values describe the output volume flow of the pump. In order to calculate the flow of the gear set, these values are duplicated N times and each one is shifted by 360°/N to the one before. The angle wise sum of all these values multiplied with the rotor height “h” will be the volume change $\dot{V}\left(\varphi \right)$ generated by the gear set:

$\dot{V}\left(\varphi \right)=\text{h}*\sum _{i=0}^{N}{\dot{A}}_{i}\left(\varphi \right)$

From this, the maximum, minimum and mean flow, denoted by Q, can be calculated for a defined rotation speed “n”:

$\text{Q}=\text{n}*\sum _{\varphi =0}^{360}{\dot{V}}_{i}\left(\varphi \right)$

With these values, the flow ripple can be calculated by:

$\delta =\frac{{Q}_{max}-{Q}_{min}}{{Q}_{mean}}$

The tool is designed to read a list of input data files and compare them, after each one has been analysed. The result is also provided via a plot and a chart.

In order to check which profile type is the one with the lowest pulsation and the highest volume flow in a well-defined limitation size regime, a large set of different profiles (GT1, GT2, GT3, GT4) was generated by the “Gerotor Profile Library” tool. This was done by varying the parameters of the rotor profile within their mathematical limitations. These limitations were reached, if the profiles started to overlap with themselves or if one or more teeth had an undercut.

In addition, one reimported reference profile, which is an optimized GT1 profile, was included in the analysis. To make the results comparable, all gear sets have an inner rotor with 6 teeth and an outer rotor with 7 teeth. Furthermore, all gear sets are limited to the same maximal size. This was done by setting the outer rotor tooth root value to be equal or less than 3.64 mm.

All gear sets were analysed with a step width of 1° in the “Flow Rate Analysis Tool” and afterwards compared with a height of h=3.2 mm and a rotation speed of n=6000 rpm in the “Flow Rate Irregularity and Comparison” tool.

## 3 Results

In order to give a detailed view of the resulting data one generated profile is shown first, before all data sets are compared to each other. The profile which can be seen in Figure 3 is a GT4 profile with a robust tooth profile, as well as a low pulsation and a high mean flow.

Figure 3

Robust GT4 gear set with approx. 270 ml/min mean flow and a flow pulsation of 3.4%.

The shown gear set is very likely to be robust because of the wide and flat teeth. It is also very likely to have a high hydraulic efficiency and therefore low leakage, because of the extended areas and very low distances between the rotors at the upper three teeth of the inner rotor. Also, the amount of trapped fluid, seen at the bottom tooth of the inner rotor, is very low.

In Figure 4 all solution plots of this gear set are shown. The chamber area change A(*φ*) of the second toolbox is shown in the upper left corner. In the upper right corner, the deviated chamber area change $\dot{A}\left(\phi \right)$ is plotted. At the bottom, the resulting volume flow $\dot{V}\left(\phi \right)$ of this gear set is plotted.

Figure 4

Comparison of the volume flow and the flow pulsation of different gerotor gear sets.

In order to obtain a better overview, all results from the data sets are compared by their flow pulsation and their mean flow (Figure 5). From this, it was determined that the eccentricity must be maximized in order to gain a high mean flow. This can be seen by the groups of points along the same mean flow lines, which all share the same eccentricity, respectively. The eccentricity can therefore be regarded as the main parameter when aiming for a high mean flow.

Figure 5

Comparison of the volume flow and the flow pulsation of different gerotor gear sets.

As can be seen, the ellipse-based-profile GT4 is the profile type with the lowest pulsation at higher mean flow rates.

## 4 Conclusion

A group of toolboxes was presented, which enable the generation of gear sets for gerotors. Furthermore, a first analysis for four different gerotor profile types was shown. The results show that the ellipse-based gerotor profile sets deliver the best compromise between a low pulsation and a high mean volume flow within the given limitations. For that reason, this profile type is the most suitable for use in pharmaceutical industry and for biomedical applications, where a low pulsation is required.

Further investigation is needed to compare the other two remaining profiles types, as well as gear sets with different numbers of teeth.

## Funding

The authors like to thank the Ministerium für Wirtschaft, Bau und Tourismus MecklenburgVorpommern which provided funding of the project with financial resources by the Europäischer Fonds für regionale Entwicklung EFRE. (European Regional Development Fund ERDF).

## Author’s Statement

Author’s Statement: Conflict of interest: Authors state no conflict of interest. Material and Methods: Informed consent: Informed consent is not applicable. Ethical approval: The conducted research is not related to either human or animals use.

## References

[1] Kwon S M, Kim M S, Shin J H. Analytical Wear Model of a Gerotor Pump without Hydrodynamic Effect. *Journal of Advanced Mechanical Design Systems, and Manufacturing*, 2008, Vol. 2, No. 2.10.1299/jamdsm.2.230Search in Google Scholar

[2] Fabiani M, Mancň S, Nervegna N, Rundo M, Armenio G, Pachetti C, Trichilo R. Modelling and Simulation of Gerotor Gearing in Lubricating Oil Pumps. Society of Automotive Engineers, Inc., 1999: 99P-46410.4271/1999-01-0626Search in Google Scholar

[3] Kwon S M, Kang H S, Shin J H. Rotor profile design in a hypogerotor pump [J]. *Journal of Mechanical Science and Technology*, 2009, 23: 3459-3470.10.1007/s12206-009-1007-ySearch in Google Scholar

[4] Kwon S M, Kim C H, Shin J H. Optimal rotor design in hypotrochoidal gear pump using genetic algorithm. *J. Cent. South Univ. Technol*., 2011, 18: 718-725.10.1007/s11771-011-0753-zSearch in Google Scholar

[5] Kwon S M, Sim M, Nam H, Shin J H. Optimal Wear Design for a Hypotrochoidal Gear Pump without Hydrodynamic Effect, 10.3795/KSME-A.2009.33.12.1383.Search in Google Scholar

[6] Choi T H, Kim M S, Lee G S, Jung S Y, Bae J H, Professor Kim C. Design of Rotor for Internal Gear Pump Using Cycloid and Circular-Arc Curves. *Journal of Mechanical Design*, 2012, Vol. 134, 01100510.1115/1.4004423Search in Google Scholar

[7] Kim M S, Lee H W, Jung S Y, Kim C. Development of Rotor for Internal Gear Pump using Cycloid and Polycircular-arc Curves. *Journal of the Korean Society for Precision Engineering*, 2012, Vol. 29, No. 9, pp. 1003-1011.10.7736/KSPE.2012.29.9.1003Search in Google Scholar

[8] Jung S Y, Han S M, Cho H Y, Kim C. Automated design system for rotor with an ellipse lobe profile. *Journal of Mechanical Science and Technology* 23, 2009, 2928-2937.10.1007/s12206-009-0808-3Search in Google Scholar

[9] Moon H K, Jung S Y, Kim C, Han S M, Cho H Y. Development of an Automated Design System for Generating Ellipse Lobe Profile of Gerotor. Journal of the Korean Society for Precision Engineering, 2008.Search in Google Scholar

[10] Karamooz Ravari M R. Elliptical lobe shape gerotor pump design to minimize wear. *Front. Mech. Eng*., 2011, 6(4): 429-434.10.1007/s11465-011-0247-6Search in Google Scholar

[11] Warren S E. New Rotary Engine Designs by Deviation function Method, PhD dissertation. University of California, Los Angeles, 2012.Search in Google Scholar

[12] Tong S H, Yan J, Yang D C H. Design of deviation-function based gerotors. *Mechanism and Machine Theory* 44, 2009, 1595-1606.10.1016/j.mechmachtheory.2009.01.001Search in Google Scholar

[13] Yan J, Yang D C H, Tong S H. A New Gerotor Design Method With Switch Angle Assingability. *Journal of Mechanical Design*, 2009, Vol.131, 011006.10.1115/1.3013442Search in Google Scholar

[14] Jung S Y, Bae J H, Kim M S, Kim C. Development of a New Gerotor for Oil Pumps with Multiple Profiles. *International Journal of Precision Engineering and Manufacturing*, 2011, Vol. 12, No. 5, pp. 835-841.10.1007/s12541-011-0111-ySearch in Google Scholar

This article is distributed under the terms of the Creative Commons Attribution Non-Commercial License, which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.