# Bladeless Heart Pump Design: Modeling and Numerical Analysis
## 4 Results and Discussion  
The design parameters used are the inner radius \(R_{1}=12\) mm, the outer radius \(R_{2}=15\) mm, the distance between the disks \(h=1\) mm, and the number of disks \(n=6\). The final design of bladeless continuous flow heart pump is presented in Figure 4. For the  
Figure 4: Flow separator—cross section A-A. defined computational domain, the streamlines, colored by velocity magnitude, are depicted for the pump operating point (\(\Delta p=65\) mmHg, \(Q=5.43\) L/min, and \(\omega=6000\) rpm). The boundary layer is very difficult to display in the \(xy\) plane (floor plan). The fluid next to the upper disk moves at the speed of the disk. Therefore, the streamlines in the \(xy\) plane always have a dominant representation of the fluid flow close to the top plate (Figure 5a). The boundary layer is nicely depicted in the isometric view (Figure 5b). The wall shear stress (WSS) in the rotor is within the range of 46-108 Pa, depicted in Figure 6. There is a small area near the follow separator where the wall shear stress exceeds the defined acceptable value of 150 Pa. The acceptable amount of wall shear stress is achieved in more than 95% of the pump volume [26]. The residence time in the pump is 0.0194 s, while the residence between two disks is 0.0127 s. The exposure residence time in the area with higher values of the wall shear stress is 5 \(\times\) 10\({}^{-4}\) s. According to Hellums shear stress-exposure time [26], for exposure residence time of 5 \(\times\) 10\({}^{-4}\) s, the acceptable  
Figure 5: Streamlines in bladeless heart pump in: (a) \(xy\) plane, (b) 3D space. value of shear stress is 700 Pa. The calculated maximum shear stress is 308 Pa, and it is smaller than Hellums. The scalar shear stress (SSS) for the computational domain is depicted in Figure 7. The SSS in the rotor is within the range between 4.8 \(\times\) 10\({}^{-5}\) and 667.85 Pa. Figure 6 depicts the SSS on the multiple planes between two disks with different offset from the disk (\(z=0\) mm (Figure 7a), \(z=0.05\) mm (Figure 7b), \(z=0.2\) mm (Figure 7c), and \(z=0.5\) mm (Figure 7d)). The first disk surface is at plane \(z=0\) and the second disk surface is at plane \(z=h\). Figure 6: Contours of wall shear stress (Pa). Figure 7: Cont. The low SSS is in a range of 0-10 Pa, mid SSS 10-100 Pa, and high SSS >100 Pa. The high SSS is in 10% of the volume, mid SSS is in 10% of the volume, and the low SSS is in the remaining 80% of the volume between disks. Furthermore, the \(\Delta p\)-\(Q\) characteristic curve is presented in Figure 8. The numerically calculated \(\Delta p\)-\(Q\) characteristic curve for the k-omega and the k-epsilon turbulence model are depicted in the diagram, together with the theoretical \(\Delta p\)-\(Q\) characteristic (Equation (21)). The theoretical and numerical models concur, as shown in Figure 8. The numerical \(\Delta p\)-\(Q\) characteristic curve is calculated considering flow in the computational domain (with flow separator and connection tubes), while the theoretical \(\Delta p\)-\(Q\) characteristic curve has considered only disks. This results in the discrepancy between theoretical and numerical \(\Delta p\)-\(Q\) characteristic curves. The \(\Delta p\)-\(Q\) characteristic curve obtained with the theoretical model has a slightly flatter slope. The slope is flatter because the original assumption is that 3/4 of the disk is active and 1/4 is inactive in the flow separator (Figure 1). The final pump design has 9/10 of the disk as active and 1/10 as inactive in the flow separator, as shown in Figure 5. Therefore, the \(\Delta p\)-\(Q\) characteristic curve obtained with the numerical model has a steeper slope (Figure 8). The pump hemocompatibility assessment is only possible through clinical studies and comparison with the existing heart pump. The quality of a blackeless heart pump cannot be fully assessed without experimental research. The only thing that is possible, at this stage of research, is to compare parameters from numerical simulation between two heart pumps.