Numerical Characterization of the Performance of Fluid Pumps Based on a Wankel Geometry

The performance of fluid pumps based on Wankel-type geometry, taking the shape of a double-lobed limac¸on, is characterized. To the authors’ knowledge, this is the first time such an attempt has been made. To this end, numerous simulations for three different pump sizes were carried out and the results were understood in terms of the usual scaling coefficients. The results show that such pumps operate as low efficiency ( < 30%) valveless positive displacements pumps, with pump flow-rate noticeably falling at the onset of internal leakage. Also, for such pumps, the mechanical efficiency varies linearly with the head coefficient, and, within the onset of internal leakage, the capacity coefficient holds steady even across pump efficiency. Simulation of the flow field reveals a structure rich in three-dimensional vortices even in the laminar regime, including Taylor-like counterrotating vortex pairs, pointing towards theutilityofthesepumpsinmicrofluidicapplications.Giventheplanargeometryofsuchpumps,theirapplicationsasmicroreactorsandmicromixersarerecommended.


Introduction
The present study is part of a larger effort aimed at exploring applications of fluid pumps based on Wankel-type geometry and is focused on the performance characterization of the simplest of such pumps.
Such Wankel-type pumps are essentially rotary positive displacement pumps, which operate by having an inner rotor orbit inside a chamber. The rotor path, determined by the chamber profile, creates a trapped fluid volume which is displaced through the chamber. In contrast to rotodynamic pumps, the trapped fluid is continually compressed to a high pressure without being imparted high kinetic energies. As a positive displacement pump, it has characteristics similar to the reciprocating positive displacement pump and, hence, would generate the same flow at a given speed (RPM) regardless of the discharge pressure, that is, a flat H-Q curve. However, a rotary pump is more susceptible to internal flow leakages especially at high pump heads, leading to a significant reduction in efficiency. The advantages of rotary pumps are that, as well as being able to deliver a flow that is less pulsatile compared with reciprocating piston pumps, they are more compact in design and capable of valveless operation.
There are a number of rotary pump types that have been well established and have found industrial application, such as the Gear Pump, Lobe Pump, Sliding-Vane Pump, Screw Pump, and Progressive-Cavity Pump [1], but so far, to the authors' knowledge, the Wankel-type design for fluid pump applications has not been exactly established. Although rarely reported, it is known that applications of fluid pumps based on Wankel-type geometry are not new. For example, Monties et al. [2] developed a valveless blood pump comprising a double-lobed limaçon-shaped chamber and an elliptically shaped rotor rotating on an eccentric gear. It had a single inlet and outlet and delivered pulsatile flow.
Mathematically, the perimeter of Wankel-type geometry is an epitrochoid, the parametric form of which is given by = ⋅ sin ( ) + sin . geometry is obtained by setting = 2 (see Figure 1), which is therefore the simplest of such geometries. (The well-known Wankel rotary engine chamber is generated by setting = 3.) In the present study, numerous computer simulations for three different pump sizes were carried out using commercially available computational fluid dynamics (CFD) codes (after calibrating a typical setup against experimental data). To the authors' knowledge, it appears that, to date, there has been no such previous attempt for these Wankel-type pumps, although there are numerous studies incorporating computational fluid dynamics (CFD) analyses, reported in the open literature, on particular and general aspects of various other pump types, including both rotodynamic and positive displacement pump types.
Numerical results generated in the present study were expressed in terms of the usual scaling coefficients. To conclude, we present details of the flow structure within the chamber for a pump sized for microfluidic applications.

Numerical Method
The simulations were carried out in a proprietary finite volume CFD code, ANSYS CFX v 15, using the immersed solid method to model the motion of the rotor.

Turbulence Model.
To solve for the flow field, the shearstress transport (SST) turbulence model of Menter [17] was invoked to close the Reynolds averaged continuity and momentum equations (in standard Cartesian tensor notation): Details of the specific implementation of the SST model can be found in the ANSYS CFX manual [18]. Very briefly, the SST model calculates the Reynolds stresses, − = (( / ) + ( / )) − (2/3) , by solving a transport equation for the turbulent kinetic energy, and another transport equation of the specific dissipation rate, , the ratio of energy dissipation rate, , to the turbulent kinetic energy, : The terms and represent, respectively, the production of the turbulent kinetic energy, , and the specific dissipation rate, , while the terms and represent the dissipation of the turbulent kinetic energy, , and the specific dissipation rate, . The cross-diffusion term, , arises from the transformation of the − model into the − form. The effective diffusivities, Γ = + ( / ) and Γ = + ( / ), are obtained from turbulent Prandtl numbers for and given by = 1/( 1 / ,1 + (1 − 1 )/ ,2 ) and = 1/( 1 / ,1 + (1 − 1 )/ ,2 ) (where 1 is a blending function; and ,1 , ,2 , ,1 , and ,2 are some constants). The value of the turbulent viscosity, , finally calculated from = ( / ), is limited by an expression, , containing another blending function, 2 .

The Immersed Solid
Technique. The immersed solid technique, implemented within the finite volume code of CFX [18], treats the presence of a solid object by applying a source term, , to the momentum equation, in the fluid volume geometrically occupied by the solid object in the fluid domain. To track the movement of the immersed solid, the solver updates the mesh positions of the immersed solid at the beginning of each time step and applies the immersed solid sources, , to the fluid nodes that lie within the immersed solid in an attempt to match the fluid velocity with the immersed solid velocity.

Simulation Calibration.
The simulation setup for the pump performance characterization work was calibrated against the experimental data from [2,19,20]. To the authors' knowledge, it appears that, to date, these are the only sources of experimental data in the open literature for such pump types.  Direct comparison was not possible as the exact dimensions of the blood pump and fluid properties were not reported. By trial and error and by examination of the information available, the parameters of = 0.039 m and = 0.0114 m (which generated the size of the pump as shown in Figure 1) and a chamber thickness of 0.015 m were estimated.
A mesh grid sensitivity study was carried out by comparing instantaneous pump flow-rates obtained for different grid cell sizes. As shown in Figure 2, a cell size of 0.0015∼0.0020 m would give reasonably accurate results. Using a time step of at least 1/500th of a period of shaft revolution and a mesh cell size of 0.002 m (yielding a total cell count of 55,000), we achieved reasonable agreement with the experimental data of Monties et al. [2], in terms of the instantaneous flow-rate time profile and magnitude, as indicated in Figure 3.
The uncertainty of the numerical results could not be assessed as Monties et al. [2] did not furnish information on the accuracy of and did not include error bounds on their experimental data. The mesh grids for the rotor and pump cavity for the calibration simulation are shown in Figure 4. A blood viscosity of 0.0035 Pa s and a density of 1.06 3 kg/m 3 [21] were assumed. A first order upwind scheme was used to discretize the convective terms. Temporal discretization was performed with the first order backward Euler scheme performing 10 outward iterations per time step. Collected data were analyzed with the aid of the following dimensional groups: capacity coefficient, = / 3 ; head coefficient, = / 2 2 ; efficiency, = / ; device Reynolds number, Re = 2 / ; and tip clearance ratio, = Δ / .

Pump Performance Scaling Studies
Time averaged values of flow-rate and reactive torque (over one cycle) were used in calculating the dimensional groups' values.
From the plots of capacity coefficient, , versus head coefficient, , for various tip clearance ratios, , at a particular device Reynolds number, Re = 1 6 ( Figure 5), we observe that the capacity coefficient, , remains constant up to a certain value of the head coefficient, , beyond which it begins to drop noticeably, particularly for the largest tip clearance ratio, . These are typical of positive displacements pumps, where internal leakage would cause a deterioration of pump flow-rate at high pump heads. For a particular clearance ratio, we find that the pump efficiency varies fairly linearly with the head coefficient, , across the three pump sizes studied here. See Figure 6.
It appears therefore that the capacity coefficient, , holds quite steady over variations in pump efficiency, , as well.
To confirm this, we ran simulations at a low and a higher efficiency and indeed the flow-rate, , scales linearly with 3 on a single line at these two disparate efficiencies ( ∼ 0.008 and ∼ 0.26) as shown in Figure 7

Pump Chamber 3D Flow Field
While the results of the above scaling studies are useful for a preliminary assessment of a suitable pump size for a particular application, detailed examination of the flow field in the pump chamber would be necessary to see if such pumps could be exploited for certain biomedical, microfluidic, or microreactor applications.
Hence, in this section, we report on a 3D simulation of pump scales to = 0.01 m (which would be typical of an active microfluidic device) operating at a shaft speed of 200 rad/s on water against zero heads.
By plotting surface streamlines on a horizontal plane bisecting the thickness of the pump chamber as well as on a vertical plane along the axis of symmetry of the pump chamber for various rotor positions, as shown in Figure 8, it can already be observed that the flow pattern within the pump chamber is very complex.
Delving deeper, surface streamlines at various vertical planes across the pressure side of the pump chamber, for a particular rotor position, were then plotted. As shown in Figure 9, pairs of lateral vortices appear to first emerge near the tip of the rotor as indicated on plane P1. As the flow traverses the pump chamber, the vortices develop further while being stretched due to an increase in cross-sectional area in the main flow direction, from plane P2 to P5. By the time the flow reaches the outlet, the vortices appear to have more or less disappeared. A velocity vector plot on one of the planes, plane P2, as shown in Figure 10 shows that these vortices resemble counterrotating Taylor vortices. The suction side of the pump chamber, on the other hand, is dominated by a lateral vortex in the horizontal plane (see again Figure 8). In essence, as the flow is drawn into the pump chamber, it sees a backward facing step, and the well-recognized recirculation zone is produced. A vector plot on a horizontal plane, as shown in Figure 11, confirms this observation.
Thus as the flow transits from the suction to the pressure sides, it is being continuously stretched from a pattern characterized by a dominant, horizontal lateral vortex to one characterized by pairs of vertical, lateral Taylor-like counterrotating vortices that are continuously deformed as the flow is being evacuated. The resulting complex flow pattern, rich in threedimensional vortices, suggests that pumps based on Wankel geometries have a good potential for mixing applications.

Concluding Remarks
We conclude this technical brief with the following salient points.  Figure 10: Velocity vectors plotted on one of the cut-planes in Figure 9, confirming the existence of Taylor-like counterrotating vortex pairs. (ii) As with positive displacement pumps in general, the capacity coefficient, , falls noticeably at the onset of fluid internal leakage.
(iii) For a particular set of dimensionless groups, pump efficiency, , varies linearly with head coefficient, .
(iv) Within the onset of internal leakage, the capacity coefficient, , holds steady even across pump efficiency, , thus scaling pump flow-rate, , with shaft rotor speed, , and the cube of rotor size, (rotor centre-to-tip distance).
(v) Maximum pump efficiency is in the region of 30%.
(vi) Even in the laminar flow regime, flow structure within the pump chamber is complex and very rich in threedimensional vortices.
(vii) Detailed examination of the flow structure reveals a transition from a structure dominated by a backward facing step type recirculation zone to one characterized by Taylor-like counterrotating vortex pairs that are continuously stretched, as the trapped fluid traverses from the suction side to the pressure side of the pump chamber.
(viii) It appears that a fluid pump based on the simplest Wankel geometry is best suited for applications in which pump efficiency is not an overriding issue; valveless operation is an advantage and complex flow patterns in the pump chamber are exploited to serve a particular function. Given the planar geometry of such pumps, their application as microreactors and micromixers is recommended.