Design and numerical simulation of elliptical claw-type hydrogen circulation pump rotor

The claw-type hydrogen circulation pump has the advantages of smooth gas delivery, wide range of adapted working conditions, high efficiency, etc., which is the key equipment of the fuel cell system. In this paper, a new type of elliptical claw rotor is proposed by the combination of elliptical line, circular arc and pendulum line, and the equation of the line is calculated and deduced, and the distribution law of the flow field in the working chamber is obtained based on numerical simulation, and the internal flow characteristics, the working process, the change rule of the pressure, speed and the resisting moment are analysed. The results show that the elliptical claw rotor is subjected to a smaller gas resisting moment compared with the traditional cycloidal hydrogen circulating pump.


Introduction
Increasing the proportion of clean energy and renewable energy is an effective way and a key part of China's efforts to realize the goal of 'double carbon' .[1] In the context of global carbon neutrality, hydrogen energy will play an important role in the process of deep decarbonization in transportation, industry and other fields.As an environmentally friendly vehicle using hydrogen as fuel, the main advantages of hydrogen vehicles include zero emission, high energy density, fast refueling, and long range.Among them, the performance of the fuel cell is crucial to the power performance of hydrogen vehicles, and the hydrogen circulation pump is the heart of the fuel cell, which is responsible for transporting the unreacted hydrogen to the anode of the fuel cell in the hydrogen circulation system so that the next round of reaction can take place.At present, the types of hydrogen circulation pumps are Roots type, claw type and centrifugal type.[2,3] Compared to Roots pumps without built-in compression, claw pumps can achieve higher efficiencies at high pressure ratios while retaining the advantages of compactness and simplicity of Roots pumps.[4] Kerry Dong [5] et al. numerically simulated the internal flow characteristics of claw-type hydrogen circulation pumps, and the results showed that the leakage resulted in higher pressure in the working chamber with smaller volume.Li [6] et al. investigated the effect of pressure ratio on the internal flow characteristics of hydrogen circulation pumps, and the results showed that differential pressure-induced gap leakage was an important cause of the flow turbulence in the pump.Zhao Xin [7] et al. proposed a new elliptical arc-type straight claw rotor, and the results showed that the residual clearance volume and closed volume of the elliptical arc-type rotor were reduced compared with the sharp point-type rotor and the arc-type rotor, and the internal volume ratio and the area utilization were higher.Pan [8] et al. proposed a new type of three-claw rotor and outlet, and the results showed that this new type of rotor had the advantages of high internal volume ratio, small relative carryover, and would not carryover the process.Dong [9] et al. investigated the effect of thermal deformation on the leakage gap of a claw-type hydrogen circulating pump for fuel cells, and the results showed that the total leakage gap increased firstly and then decreased with the increase of the angle.Wang [10] et al. proposed a new type of eccentric involute claw-type rotor, and the results showed that the new type of eccentric involute clawtype rotor greatly improves the performance of the pump, and effectively mitigates the negative effects of the mixing process.Gu [11] et al. analysed the effect of dependencies such as rotational speed, inlet pressure, axial clearance, and compression ratio on the optimisation of volumetric efficiency and shaft power of a hydrogen recirculation pump.J Willie [12] used a 0-D thermodynamic chamber model and 3D Computational Fluid Dynamics (CFD) to predict the performance of a claw vacuum pump.Gu [13] et al. performed a 3D transient simulation considering radial and axial clearance leakage.Dynamic meshes are often used in the numerical simulation of claw vacuum pumps, and a set of high-quality meshes can improve the reliability of numerical simulations.Based on common profiles, parameters such as volume utilisation of the rotor profiles can be further improved by the design improvement of the profiles.[14] This paper selects the two-claw claw vacuum pump for research, and in order to improve its performance, a new type of elliptic-circular-arc-cycloid claw rotor profiles are put forward, and its profiles are deduced and solved through the complex vector method, and the physical model of the rotor is obtained, and the internal flow characteristics, pressure pulsation characteristics, and the change rule of pressure are analyzed based on numerical simulation, which provides a good solution for the design of hydrogen circulation pump.The design of hydrogen circulation pump provides certain reference significance.

Composition of rotor curves
In order to solve the two meshing rotor profiles, it is necessary to establish the corresponding coordinate system, as shown in figure 1, where X 1O1Y1 and X2O2Y2 are the static coordinate systems corresponding to the right rotor and the left rotor, respectively, and X1O1Y1 and X2O2Y2 are the dynamic coordinate systems fixed to rotor 1 and rotor 2, respectively, rotating with the rotation of the rotor.Where a is the center distance between the two rotors,  1 and  2 are the angles of rotation of the dynamic coordinate systems 1 and 2 respectively.Let the vector diameter of the meshing point M be   ( 1 ) in the static coordinate system 1,   ( 2 ) in the static coordinate system 2, and   ( 1 ) and   ( 2 ) in the two dynamic coordinate systems respectively.
When the two rotors mesh, we can represent the trajectory equations of the meshing point M in the four coordinate systems established above and thus solve them associatively.The coordinate systems as above are related to each other by the following transformation: The profiles of the pair of elliptical two-jaw rotors studied in this paper are engaged and different from each other, and since they are two-jaw rotors, their profiles are centrosymmetric about the rotor center.As shown in figure 2, rotor 1 consists of a combination of elliptic lines, circular arcs, and pendulum lines, and rotor 2 is conjugated with them everywhere, and its specific composition is as follows: Segment AB is a claw top arc with radius R1 and center on O1, conjugated to the claw ground arc of segment ab of rotor 2. segment BC is an elliptic line with the long half-axis on the Y1 axis and axis length R1, and the short half-axis on the X1 axis and axis length R2, and center on O1, conjugated to the curve of segment bc of rotor 2. segment CD is a claw bottom arc with radius R2, and center on O1, conjugated to the arc of the claw top arc of segment cd of rotor 2. The DE segment is a pendulum line, conjugated to the claw tip point d of the rotor 2. The claw tip point E is conjugate to the pendulum line of the de segment of rotor 2.

Derivation of curve equations for each segment of rotor
During the meshing of the curves, the meshing point M will move from the meshing start point of the curve to the meshing termination point, so the corresponding rotor profile can be obtained by finding the trajectory of the meshing point M in the dynamic coordinate system.

Section AB
As shown in figure 3, when the section AB is engaged, by the meshing theorem can be seen, the engagement point M in the O1O2 line, and in the static coordinate system under the position is unchanged.
It is easy to know that the point M in the static coordinate system under X1O1Y1 under the trajectory is According to the coordinate change relationship, from the equation (2.1) to get M point in the dynamic coordinate system  2 in the trajectory, that is, the rotor 2 on the corresponding conjugate curve vector equation, by the equation ( 2

Section BC
As shown in figure 4, when the section BC involved in meshing, by the meshing theorem, the meshing point M and the node P of the line that is the normal of the ellipse, where the point P for the speed of the two rotors instantaneous center (the two rotor origin line center point).
Rotor 1 section BC meshing.It is easy to know that the point P under  1 vector diameter is So the coordinates of point P under  1 is (− •  1 ， •  1 ), and set the coordinates of point M under  1 as( 0 ， 0 ), which can be obtained from the geometrical relationship: Equation (2.9), (2.10), to find So the vector diameter of M point under the moving coordinate system  1 is According to the coordinate transformation relationship, from the equation (2.3) to get M point in  2 under the trajectory equation 12) The following solves the relationship between the turning angle  1 of the moving coordinate system and the intermediate angle : From the M point and P point line for the BC ellipse normal to the M point can be obtained Associated (2.9), (2.10), (2.13) to get (2.14)

Section CD
As shown in figure 5, when the section CD engaged, by the meshing theorem can be seen, the meshing point M in the O1O2 line, and in the static coordinate system under the position is unchanged.
. Rotor 1 section CD engagement.It is easy to know that the M point in  1 under the vector diameter is

Section DE
As shown in figure 6, Section DE is the pendulum, the section DE is conjugate to the point of the claw tip on the rotor 2. 1 . Rotor 1 segment DE engagement.In the dynamic coordinate system  2 , M point to get the trajectory for the From the equation (2.4) can be obtained, the engagement point M in the dynamic coordinate system  1 trajectory as follows When the two rotor claw center point coincides, by the geometric relationship can be seen So when the claw point on rotor 2 engages the DE pendulum on rotor 1, the dynamic coordinate system  2 and static coordinate system  1 to get the angle  1 to change the range of

Conjugate curve of E point on rotor 2
When the E point and rotor 2 on the pendulum line engagement, the engagement point M in the dynamic coordinate system  1 trajectory is From equation (2.3), the engagement point M in the dynamic coordinate system  2 trajectory is When the two rotor jaws coincide at the top of the point, from the geometric relationship can be seen So when the claw point on the rotor 2 is engaged with the conjugate curve on the rotor 1, the dynamic coordinate system  2 and the static coordinate system Above is the process of solving the curve of each section on the rotor 1 and rotor 2, due to the claw type of the two-jaw rotor for the center symmetric, the single claw type curve about the rotor origin for the center symmetric can be obtained to get a closed complete elliptic two-jaw rotor type profile, the specific parameters are detailed in Table 1.
After obtaining the equation of the rotor profile, the use of Matlab software to write the elliptical two-claw rotor type profile calculation program, the results of Matlab's calculations into the geometric modeling software for stretching, adding the inlet and outlet runners, that is, to obtain the rotor's threedimensional geometric model, as shown in figure 7.

Elliptic type in the theoretical working process of rotor pumps
Elliptical rotor pump adopts radial air intake and radial air exhaust working mode, and its working process is shown in figure 8.   8 (c) in the end of the inlet, the left rotor and the right rotor were formed with the working chamber of the two unequal closed working chamber, the gas enters the process of isotropic conveying; figure 8 (d) in the exhaust port is about to be opened, the left working chamber is first into the exhaust stage; figure 8 (e) the left working chamber has been connected with the exhaust port, rotor 1 and rotor 2 form a compression working chamber, and the first time in the two engagement points to form a clearance volume; figure 8 (f) with the movement of the rotor, the right working chamber is about to be connected to the exhaust port, and the two rotors for the second time to form a clearance volume; figure 8 (g) the left and right working chambers of the gas gathered in the exhaust port; figure 8 (h) the two rotors continue to compress the gas until the discharge chamber.
Figure 9 shows the volume change of the working chamber formed by the rotor and the chamber in one working cycle of the elliptical two-jaw pump.As shown in the figure 9, during the suction stage, the intake working chamber increases continuously from 5457.54 mm 3 to 33280.57mm 3 .When the rotation angle reaches 172°, the left rotor is closed to the chamber, forming a left working chamber of 19843.64 mm 3 , and the left rotor starts to enter the isotropic conveying process with an angle of 172° to 247°, so that the volume of the left working chamber remains unchanged in the range of this angle.At this time, the right rotor has not yet formed a closed working chamber with the cavity, still connected with the suction chamber.When the angle reaches 180°, the right rotor is closed to the chamber, forming a right working chamber of 13559.24mm 3 , and the right rotor starts to enter the isovolumetric conveying process with an angle of 180° to 332°.When the left rotor rotates to 247°, it is about to connect with the exhaust cavity, and the volume of its working cavity increases abruptly, and the volume decreases gradually with the rotation, and when the rotation angle reaches 262°, the left and right rotors appear two meshing points, forming a residual gap volume of 2336.17mm 3 .With the further rotation of the two rotors, the clearance volume gradually decreases until it is connected with the suction chamber.When the rotation angle reaches 332°, the right rotor is about to connect with the exhaust chamber into the final compression stage, compression process, its volume from 26693.65mm 3 reduced to 18025.69mm 3 , the rotor continues to rotate until the gas is discharged, completing a working cycle.

Numerical simulation
The fluid domain of the claw-type hydrogen circulation pump includes three parts: the inlet fluid domain, the outlet fluid domain, and the working cavity fluid domain.In order to avoid the interference of the rotor in the working process, there is a gap between the left rotor and the right rotor, and between the rotor and the cavity, and the value of the gap is 0.1 mm.
The elliptical rotor pump has a complex rotor shape, and the fluid domain is non-regular shape with a small gap, which makes the internal mesh distortion of the fluid domain serious during the simulation process, and it is easy to produce a negative volume.Therefore, this paper adopts unstructured mesh to mesh the fluid domain of the working cavity, the working cavity fluid domain mesh is shown in Fig.
The minimum mesh is, the inlet and exhaust port fluid domain mesh is shown in figure 10.The total number of meshes is 1.51 million.

Figure 10. Mesh division of elliptical rotor.
A dynamic mesh method is used to simulate the rotational process of the rotor, and the rotational speed of the rotor is defined by a user-defined function.The operating medium is hydrogen, using a density-based solver, RNG k-ε turbulence model, and coupled implicit solution method.The inlet pressure Pin = 110 kPa, the inlet temperature is 293 K, the outlet pressure Pout = 132 kPa, the rotational speed is 3000 r/min, and the time step is set to 2.0 × 10 -6 s.
The rotor's rotational motion is defined by the UDF code written in the dynamic mesh, the outer surface of the rotor is set as a rigid body, and the upper and lower surfaces of the pump chamber are set as deforming regions, and the mesh is adjusted by using the spring approximation model and local reconstruction method.The spring approximation model and local reconstruction method are used to adjust the mesh.When the rotor rotates, the volume starts to change at the same time, which causes the volume change and the mesh is calculated automatically.

Analyses of pressure distribution
The pressure distribution in the working chamber is shown in figure 11.   11 (a) shows the starting point of the suction process, and the volume of the suction chamber increases gradually with the rotation of the two rotors.Figure 11 (b) for the mixing stage, the high pressure gas in the residual gap volume is mixed with the suction chamber inlet gas, because the residual gap volume is less gas, so the pressure change is not obvious.Figure 11 (c) for the fixed capacity delivery stage, the left rotor and the right rotor and the chamber to form the left and right two working chambers, due to the profile of the claw in the top arc of the circle of the centroid angle of 75 °, the rotor and the chamber between the gap between the channel is long, sealing is strong, gas leakage is small, so the fixed capacity delivery stage of the working chamber of the gas pressure change is not significant.Figure 11 (d) for the mixed compression stage, the left and right working chambers are connected to the exhaust chamber, with the rotation of the rotor, the volume of the exhaust chamber continues to decrease, the gas continues to be compressed until it is discharged from the chamber.
The pressure change in the residual gap volume is shown in Figure 12.When the rotor rotation angle reaches 303.84 °, the gas pressure in the residual gap volume is the highest, reaching 126030Pa.with the rotation of the rotor, the gap channel between the residual gap volume and the exhaust chamber becomes narrower and longer, and the gap channel with the suction chamber becomes wider and wider, so that the high-pressure gas in the residual gap volume passes through the gap and mixes with the gas in the suction chamber, and the pressure decreases.When the rotor rotation angle reaches 306.72°, the gas pressure in the clearance volume decreases to 113,585Pa.

Analyses of velocity distribution
The velocity distribution in the working chamber is shown in figure 13.
. Velocity distributions of the elliptical rotor.Figure 13 (a) for the gas velocity distribution in the suction stage.Since the boundary of the suction chamber is composed of the concave pendulum line of the left rotor and the convex claw-top arc of the right rotor, the gas enters the suction chamber and flows preferentially along the pendulum line of the left rotor, and vortexes are formed above the meshing point of the two rotors.As the rotor continues to rotate, the suction chamber volume increases, the asymmetry is weakened, the vortex gradually disappears.Figure 13 (c) shows the mixing process between the left working chamber and the exhaust chamber, and the high-pressure gas in the exhaust chamber flows into the chamber at high speed from the gap between the tip point of the left rotor and the chamber, and develops into a vortex below the top arc of the claw of the left rotor.Figure 13 (d) for the right working chamber and the exhaust cavity of the mixing process, the maximum speed at the radial gap, with the left rotor, the same reason, the right rotor under the pendulum line due to the inflow of high-pressure gas from the exhaust cavity to form a high-speed vortex.

Effect of Pressure Ratio on pulsation characteristics
The pressure pulsation data of 1 rotation cycle of the hydrogen circulation pump is selected, and 5 monitoring points are set along the circumference of the rotating runner of the cavity, As shown in figure 14.In this paper, the rated speed n of the elliptical two-jaw pump is 3000r/min, so the pump rotation frequency is 50Hz.the number of pump vanes Z is 2, so the leaf frequency is 100Hz.As can be seen in figure 15, the pressure ratio is γ=1.2, the main frequency of pulsation at each point of the flow channel is 84Hz, and the secondary frequency of pulsation is 294Hz, which is about 3 times of the leaf frequency.the amplitude of pressure pulsation in the M3 and M4 is higher, which is 10092pa and 9613pa.the amplitude of pressure pulsation in the M1 is higher, which is 10092pa and 9613pa.the pressure pulsation frequency distribution in the M3 and M4 is higher than that in the M1.The pressure pulsation amplitude at M1 is the lowest, which is 1494 pa.The main frequency of pulsation at each point in the flow path under pressure ratio γ=1.2 is 93 Hz.The main frequency of pulsation at each point in the flow path under pressure ratio γ=1.3 is 93 Hz, and the sub-frequency of pulsation is 186 Hz.The primary frequency of pulsation at each point in the flow path under pressure ratio γ=1.4 and pressure ratio γ=1.5 is 100 Hz, and the sub-frequency of pulsation is 200 Hz, and the pressure pulsation is about 3 times of leaf frequency.200Hz, and the amplitude of pressure pulsation increases with the increase of pressure ratio.
Hydrogen circulating pump rotary flow channel pressure of each monitoring point shows a cyclical law of change, in a rotary cycle, a total of 2 times the pressure pulsation, 0 ° ~ 60 ° for the compression effect of the gas is not obvious.From the 90° position, the rotor enters the compression process to make the pressure increase effect is obvious, in line with the compression principle of hydrogen circulation pump.

Comparison of Resistance Moment
A conventional cycloidal rotor with the same geometrical parameters and operating conditions as the elliptical double-jaw rotor is selected to compare and analyse the resisting moments of the two as shown in figure 16.The variation of gas resisting moment with rotor angle for elliptical rotor and conventional cycloidal rotor is shown in figure 16, and the gas resisting moment for both of them changes periodically with rotor angleThe average resistance moment of the pendulum rotor is -0.1408N•m for one rotation period; the average resistance moment of the elliptical claw rotor is -0.1171N•m for one rotation period; therefore, the elliptical rotor has less loss caused by gas force during the working process compared with the pendulum rotor.

Conclusion 1.
For the existing two-claw claw pump, this paper proposes a novel type of elliptic two-claw rotor profile, using a combination of elliptic lines, arc lines, pendulum lines of the curves, the use of complex vector method to solve the meshing curve, to obtain the designed rotor profile.
2. Numerical simulation of the working process of the elliptical two-jaw rotor is carried out to analyze the pressure change, speed change and pressure pulsation characteristics of the gas inside the working chamber.3.By comparing the simulation results with the cycloidal rotor, it is found that the elliptical rotor has a smaller gas resisting moment compared with the cycloidal rotor with the same geometrical parameters, opening angle of the exhaust port and boundary condition settings, which can effectively improve the working efficiency of the hydrogen circulation pump.

r 2 x 1 y 1 Figure 1 .
Figure 1.Schematic diagram of the dynamic and static coordinate system of the external meshing rotor.

Table 1 .
Geometric parameters of the rotor.Parameter categoryValue (mm) Ellipse long half-axis R1 Short half-axis of ellipse R2

Figure 8 (
Figure 8 (a) to figure 8 (b) gas from the inlet into the working chamber; figure8 (c) in the end of the inlet, the left rotor and the right rotor were formed with the working chamber of the two unequal closed working chamber, the gas enters the process of isotropic conveying; figure8 (d) in the exhaust port is about to be opened, the left working chamber is first into the exhaust stage; figure8(e) the left working chamber has been connected with the exhaust port, rotor 1 and rotor 2 form a compression working chamber, and the first time in the two engagement points to form a clearance volume; figure8 (f) with the movement of the rotor, the right working chamber is about to be connected to the exhaust port, and the two rotors for the second time to form a clearance volume; figure8(g) the left and right working chambers of the gas gathered in the exhaust port; figure8(h) the two rotors continue to compress the gas until the discharge chamber.Figure9shows the volume change of the working chamber formed by the rotor and the chamber in one working cycle of the elliptical two-jaw pump.

Figure 9 .
Figure 9. Working process volume change.As shown in the figure9, during the suction stage, the intake working chamber increases continuously from 5457.54 mm 3 to 33280.57mm 3 .When the rotation angle reaches 172°, the left rotor is closed to the chamber, forming a left working chamber of 19843.64 mm 3 , and the left rotor starts to enter the isotropic conveying process with an angle of 172° to 247°, so that the volume of the left working chamber remains unchanged in the range of this angle.At this time, the right rotor has not yet formed a closed working chamber with the cavity, still connected with the suction chamber.When the angle reaches 180°, the right rotor is closed to the chamber, forming a right working chamber of 13559.24mm 3 , and the right rotor starts to enter the isovolumetric conveying process with an angle of 180° to 332°.When the left rotor rotates to 247°, it is about to connect with the exhaust cavity, and the volume of its working cavity increases abruptly, and the volume decreases gradually with the rotation, and when the rotation angle reaches 262°, the left and right rotors appear two meshing points, forming a residual gap volume of 2336.17mm3 .With the further rotation of the two rotors, the clearance volume gradually decreases until it is connected with the suction chamber.When the rotation angle reaches 332°, the right rotor is about to connect with the exhaust chamber into the final compression stage, compression process, its volume from 26693.65mm 3 reduced to 18025.69mm 3 , the rotor continues to rotate until the gas is discharged, completing a working cycle.

Figure 11 .
Figure 11.Pressure distributions of the elliptical rotor.Figure11(a) shows the starting point of the suction process, and the volume of the suction chamber increases gradually with the rotation of the two rotors.Figure11(b) for the mixing stage, the high pressure gas in the residual gap volume is mixed with the suction chamber inlet gas, because the residual gap volume is less gas, so the pressure change is not obvious.Figure11 (c) for the fixed capacity delivery stage, the left rotor and the right rotor and the chamber to form the left and right two working chambers, due to the profile of the claw in the top arc of the circle of the centroid angle of 75 °, the rotor and the chamber between the gap between the channel is long, sealing is strong, gas leakage is small, so the fixed capacity delivery stage of the working chamber of the gas pressure change is not significant.Figure11 (d)for the mixed compression stage, the left and right working chambers are connected to the exhaust chamber, with the rotation of the rotor, the volume of the exhaust chamber continues to decrease, the gas continues to be compressed until it is discharged from the chamber.The pressure change in the residual gap volume is shown in Figure12.

Figure
Figure 11.Pressure distributions of the elliptical rotor.Figure11(a) shows the starting point of the suction process, and the volume of the suction chamber increases gradually with the rotation of the two rotors.Figure11(b) for the mixing stage, the high pressure gas in the residual gap volume is mixed with the suction chamber inlet gas, because the residual gap volume is less gas, so the pressure change is not obvious.Figure11 (c) for the fixed capacity delivery stage, the left rotor and the right rotor and the chamber to form the left and right two working chambers, due to the profile of the claw in the top arc of the circle of the centroid angle of 75 °, the rotor and the chamber between the gap between the channel is long, sealing is strong, gas leakage is small, so the fixed capacity delivery stage of the working chamber of the gas pressure change is not significant.Figure11 (d)for the mixed compression stage, the left and right working chambers are connected to the exhaust chamber, with the rotation of the rotor, the volume of the exhaust chamber continues to decrease, the gas continues to be compressed until it is discharged from the chamber.The pressure change in the residual gap volume is shown in Figure12.

Figure 12 .
Figure 12.Pressure change of the clearance volume.

Figure 14 .
Figure 14.Location of monitoring points.In order to analyze the pressure pulsation frequency distribution law between the rotor of the hydrogen circulation pump and the rotating runner casing due to static-rotating interference, FFT technique is applied to transform the pressure pulsation time-domain information of each monitoring point into frequency-domain information.And four different pressure ratio schemes are proposed, as shown in table2.Table2.Numerical simulation schemes for different pressure ratios

5 Figure 15 .
Figure 15.Pressure pulsation at measuring points in the pump chamber under different pressure ratios.

Figure 16 .
Figure 16.Variation of moment.The variation of gas resisting moment with rotor angle for elliptical rotor and conventional cycloidal rotor is shown in figure16, and the gas resisting moment for both of them changes periodically with rotor angleThe average resistance moment of the pendulum rotor is -0.1408N•m for one rotation period; the average resistance moment of the elliptical claw rotor is -0.1171N•m for one rotation period; therefore, the elliptical rotor has less loss caused by gas force during the working process compared with the pendulum rotor.

Table 2 .
Numerical simulation schemes for different pressure ratios