# Bladeless Heart Pump Design: Modeling and Numerical Analysis
## 2 Theoretical Analysis of Heart Pump Parameters
According to Equation (16), the inner disk radius \(R_{1}\) Equation (22) is a function of the pressure increment \(\Delta p\), the acceptable shear stress \(\tau_{\theta z}=150\) Pa, and the distance between the disks \(h=1\) mm. \[\rho\frac{\partial v_{r}}{\partial t}+\rho\Bigg{(}v_{r}\frac{\partial v_{r}}{ \partial r}+\frac{v_{\theta}}{r}\frac{\partial v_{\theta}}{\partial\theta}- \frac{v_{\theta}^{2}}{r}+v_{z}\frac{\partial v_{r}}{\partial z}\Bigg{)}=\rho f _{r}-\frac{\partial p}{\partial r}+\mu\bigg{[}\frac{1}{r}\frac{\partial}{ \partial r}(rv_{r})\bigg{)}+\frac{1}{r^{2}}\frac{\partial^{2}v_{r}}{\partial \theta^{2}}+\frac{\partial^{2}v_{r}}{\partial z^{2}}-\frac{2}{r^{2}}\frac{ \partial v_{\theta}}{\partial\theta}\bigg{]} \tag{3}\]1  
The inner radius \(R_{1}=9.1\) mm is calculated for the typical operating condition. Any pressure increase above the designed pressure of \(\Delta p=65\) mmHg would result in shear stresses greater than those allowed at the bottom of the disk (\(R_{1}=9.1\) mm). The inner rotor radius is designed in a way that the shear stress increases to its acceptable value at pressures 30% greater than design value (\(\Delta p=85\) mmHg). The inner radius \(R_{1}=12\) mm is calculated for the operating conditions defined above. The maximum value of the outer radius \(R_{2}=21.6\) mm is calculated using Equation (21). With such large disks the pump can achieve a required nominal flow of \(Q=5\) L/min with one pair of disks. In the \(z\) direction the minimal pump height is defined by the diameter of inlet and outlet pipe (10 mm). The space in the \(z\) direction allows six disks to be embedded. The rotor with six disks can have the same nominal flow with a smaller outer radius. A reduction in the rotor size decreases the pump volume and decreases the residence time (time that blood spends in the pump). The rotor with an outer radius \(R_{2}=14.8\) mm with six disks achieves a nominal flow of \(Q=5\) L/min. Based on the performed analysis for the blackeless heart pump, an inner radius of \(R_{1}=12\) mm and an outer radius of \(R_{2}=15\) mm is adopted. The main geometrical and hydrokinetics parameters are defined using a relatively simplified flow model. The pump operating point is \(\Delta p=65\) mmHg, \(Q=5.43\) L/min and the disk angular velocity is \(\omega=6000\) rpm. An inner radius of \(R_{1}=12\) mm and an outer radius of \(R_{2}=15\) mm is adopted. The rotor has six disks, where each disk is 1 mm thick and the distance between the disk is \(h=1\) mm. The shear stress at the bottom of the disks is \(\tau_{\theta z}=\)115 Pa and at the top of the disks is \(\tau_{\theta z}=92\) Pa. The power required to rotate the disks is \(P=1.324\) W Equation (19) and the power transferred to the fluid is 0.785 W, with pump efficiency at 59%. The part of mechanical energy converted to the heat is 0.538 W, resulting in the fluid temperature increase of 0.0014 K. No significant impact on the blood is expected due to such a small temperature increase [32]. These values for the parametersrepresent the final range of the applied model. Additional analysis must be performed using numerical simulation to further improve the pump design.