Investigation on suitable design parameters for hydrodynamic bearing of LVADs

Left ventricular assist devices (LVADs) are crucial and helpful to support heart function for patients with heart failure. For the LVAD with hydrodynamic bearings, high reliability of the bearing is necessary to support the rotor and achieve the performance of LVADs. This research investigates the effect of various design parameters on the performance of a dislocated hydrodynamic bearing in LVAD based on numerical simulation, where Reynolds equation is solved to simulate the behaviour of the bearing under different design conditions. For the design of hydrodynamic bearing, several parameters are considered, such as wedge number, wedge depth, wrap angle, bearing clearance, length-diameter ratio, etc. The results show the effects of these parameters on friction loss, supporting capability, and so on. The optimal hydrodynamic bearing design is proposed so as to improve the performance of LVADs.


Introduction
Continuous flow left ventricular assist device (LVAD) based on the rotating blood pump (RBP) has been successfully applied in clinic [1,2].LVAD provides mechanical circulatory support for patients with heart failure, serving as a bridge to transplantation or destination therapy [3].
In RBP, the bearings that connect the motor and impeller play a crucial role.In the second generation LVAD, conventional mechanical bearings were used.However, the contact surfaces of mechanical bearings may damage blood cells, increasing the risk of hemolysis and thrombosis.To avoid direct surface contact and improve the biocompatibility of LVADs, the third generation LVADs mostly adopt suspension bearings, such as magnetic bearings and hydraulic bearings.
Magnetic bearings require additional sensors and suspension coils, making the structure of LVADs more complex and energy-consuming.On the other hand, fluid dynamic bearings have a simple structure and do not require additional power.However, LVADs using fluid dynamic bearings may have relatively poorer blood compatibility [4].Kataoka et.al [5] found that the blood compatibility of fluid dynamic bearings is related to their stability, as unstable motion of the rotor within the bearing can exacerbate hemolysis.
Therefore, in the development of LVADs, it is essential to study the impact of design parameters used for the bearing on suspension stability.Graefe et.al [6] investigated the influence of parameters such as diameter, length, clearance, speed, and load on hemolysis.Increasing the length of the bearing and reducing the clearance can enhance the load capacity, while an increased aspect ratio is favourable for improved stability.Well-designed multi-wedge bearings have been found to increase bearing stiffness and reduce blood cell damage, as demonstrated by Kazuya Yasui et.al using physiological experiments and numerical simulations [7].
This research explores the impact of various design parameters on the performance of dislocated fluid dynamic bearings in LVADs.Changes in fluid wedge number, wedge depth, wedge wrap angle, length-to-diameter ratio, and clearance are investigated to determine their effects on the bearing supporting capability, friction losses, and stability.The performance of the bearings under different design conditions is simulated using numerical solutions of the Reynolds equation.

Design of hydrodynamic bearing
An in-house centrifugal rotary blood pump (RBP) was developed as a Left Ventricular Assist Device in our lab [8].The rotary blood pump is designed to operate at 3500 rev/min, and has the flow discharge of 5 L/min.The RBP impeller was combined with the motor's rotor.The journal bearing film locates between the rotor and the pump casing as shown in Figure 1.  Figure 2 shows the sketch of journal bearing marked with the key design parameters.The bearing was designed as a multiarc dislocated bearing.The bearing clearance c is defined as the smallest clearance.When the load W acts on the rotating rotor with speed , the film can provide hydraulic pressure to support the rotor suspending with eccentric distance e.The eccentricity  of rotor is defined as e/c.

Governing equations
Reynolds equation was used to estimate the hydrodynamic bearing performance, which was derived from the Navier-Stokes equations.In radial hydrodynamic bearing, dimensionless Reynolds equation is as follows [9]: where P is dimensionless pressure:  is circumferential angle. is dimensionless length. is blood viscosity. is rotation speed.H is dimensionless film thickness, h/c.γ is length-diameter ratio, l/d. is dimensionless clearance, c/d.

Numerical simulation
Because the bearing journal is combined with the rotor of RBP, some parameters of hydrodynamic bearing are limited by the design of pump system.The rotary speed and diameter of bearing are dependent on the hydraulic design of pump,  = 3500 rev/min and d = 32 mm.The lubricating fluid is blood and blood viscosity hardly changed with temperature.Thus, the dimensionless clearance  and length-diameter ratio γ are dominant parameters for bearing performance.The parameters of fluid wedge, i.e. wedge angle , wedge depth b and wedge number N also affect the bearing performance.
In this study, fluid wedge parameters with different levels are investigated.Firstly, wedge number N with 4 levels (2, 3, 4, and 6) were investigated.Secondly, wedge depth b with 3 levels (50 m, 100 m and 150 m) were investigated.Thirdly, wrap angle with 5 levels (10°, 30°, 60°, 90°, 120°) were investigated.For each combination of wedge parameters, the dimensionless clearance  and lengthdiameter ratio γ were investigated varied from 0.001~0.1 and 0.1~1.0,respectively.1083 combinations of  and γ were simulated with each combination of wedge parameters.Noted that bearing performance was estimated with eccentricity  = 0.5.
The Reynolds equation was solved using MATLAB.Boundary conditions are set based on CFD simulation [10].Inlet pressure is 101200 Pa and outlet pressure is 89137 Pa.

Dimensionless performance parameters
Performance of the hydrodynamic bearing is presented using dimensionless parameters, i.e. supporting capability G, loss coefficient , temperature change T and comprehensive stiffness Keq.
Supporting capability G presents the ratio of bearing supporting force and the weight of rotor, defined as: Loss coefficient  is defined as the ratio of bearing friction loss Pf and pump output power Pout.The pump output power was estimated with CFD simulation [10]: Temperature change T is defined as the temperature change of the blood flowing through the bearing clearance.Assuming that the blood absorbs all the heat generated by the bearing: Equivalent stiffness Keq presents the rotation stability of bearing-rotor system.Based on Routh-Hurwitz criterion, the bearing-rotor system is unstable when Keq < 0 [9].Noted that the rotor is affected by unbalanced magnetic pull because the rotor is combined with motor rotor.The unbalanced magnetic pull can be considered as a bearing with negative stiffness.Keq is defined as [11] where Kmotor is estimated by numerical simulation of the motor, Kmotor = -55133 N/m.  Figure 6 shows the influence of wedge number N on the equivalent stiffness Keq of the bearing.The equivalent stiffness of the bearing gradually increases from the top left to the bottom right, indicating that the equivalent stiffness of the bearing increases with an increase in γ and a decrease in .It should be noted that the gray area represents the negative equivalent stiffness region, where the rotor cannot operate stably.The stable operating region initially increases and then decreases with an increase in the wedge number.

Effect of wedge number
Figure 7 shows the influence of wedge number N on the temperature change T of the bearing.The temperature change of the bearing gradually increases from the top left to the bottom right, indicating that the temperature change of the lubricating fluid in the bearing increases with an increase in γ and a decrease in .With an increase in the wedge number, the temperature change slightly decreases.

Hydrodynamic bearing performance maps
Based on the above results, an optimal combination of bearing parameters can be obtained, namely N = 3, b = 100 m, and  = 90°.For this parameter combination, performance maps for γ and  can be obtained, as shown in figure 16.
In this study, it is considered that the bearing performance should satisfy the following criteria: bearing supporting force greater than 6 times the rotor weight, power loss less than 2 times the pump output power, temperature change less than 10°C, and equivalent stiffness ensuring stable rotor operation.Based on these criteria, a recommended range for the design parameters can be determined as shown in the blue part in figure 16.

Concluding remarks
In this study, the effect of various design parameters on the performance of a dislocated hydrodynamic bearing in LVAD based on numerical simulation, where Reynolds equation is solved to simulate the performance of the bearing under different design conditions.For the design of fluid wedges, wrap angle , wedge depth b and wedge number N are considered.For the basic bearing parameters, dimensionless clearance  and length-diameter ratio γ are investigated.Based on the present study, the following conclusions can be drawn: (1) The bearing supporting capability increases as the length-to-diameter ratio D increases and the dimensionless clearance  decreases, while the friction power loss, temperature change, and equivalent stiffness also follow the same trend.
(2) The wedge number N has a limited effect on bearing performance.Increasing wedge depth b reduces bearing supporting force, friction power loss, and temperature rise.However, it expands the stable operating region and lowers average equivalent stiffness.Decreasing wedge wrap angle  improves supporting capability but increases power loss and temperature rise.It also enlarges the stable operating region, then reduces it.The bearing with N = 3, b = 100 m, and  = 90° exhibits better performance.
(3) The performance maps for γ and  was obtained.The findings from this research can serve as valuable guidelines for engineers in the design and development of high-performance hydrodynamic bearings for LVAD.

Figure 4 Figure 4 .
Figure 4. Influence of wedge number N on bearing supporting capability G.

Figure 5
Figure5shows the influence of wedge number N on the bearing loss coefficient .It can be seen that the loss coefficient gradually increases from the top left to the bottom right, indicating that the power loss of the bearing increases with an increase in γ and a decrease in .It can be concluded that the wedge number has a relatively small effect on the power loss of the bearing.Figure6shows the influence of wedge number N on the equivalent stiffness Keq of the bearing.The equivalent stiffness of the bearing gradually increases from the top left to the bottom right, indicating

Figure 8 Figure 8 .
Figure 8. Influence of wedge depth b on bearing supporting capability G.

Figure 9 6 Figure 9 . 11 .
Figure 9 shows the influence of wedge depth b on the bearing loss coefficient .The bearing power loss coefficient gradually increases from the top left to the bottom right, indicating that the power loss

Figure 10
Figure 10 shows the influence of wedge depth b on the equivalent stiffness Keq of the bearing.The equivalent stiffness of the bearing gradually increases from the top left to the bottom right, indicating that the equivalent stiffness of the bearing increases with an increase in γ and a decrease in .With an

Figure 12 Figure 12 .
Figure 12 compares the influence of wrap angle  on the bearing supporting capability G.In this case, the wedge number N = 3, and the wedge depth b = 100 m.The bearing supporting capability gradually increases from the top left to the bottom right, indicating that the supporting capability increases with an increase in γ and a decrease in .With a decrease in oil wedge angle, the supporting capability increases.

Figure 13
Figure 13 compares the influence of wrap angle  on the bearing loss coefficient .The bearing power loss coefficient gradually increases from the top left to the bottom right, indicating that the power loss of the bearing increases with an increase in γ and a decrease in .With a decrease in wrap angle, the power loss increases.

Figure 14
Figure 14 compares the influence of wrap angle  on the equivalent stiffness Keq of the bearing.The equivalent stiffness of the bearing gradually increases from the top left to the bottom right, indicating that the equivalent stiffness of the bearing increases with an increase in γ and a decrease in .With a decrease in wrap angle, the stable operating region initially increases and then decreases.

Figure 15
Figure 15 compares the influence of wrap angle  on the temperature change T of the bearing.The temperature rise of the bearing gradually increases from the top left to the bottom right, indicating that the temperature change of the lubricating fluid in the bearing increases with an increase in γ and a decrease in .With a decrease in wrap angle, the temperature change of the bearing increases.