The design process of centrifugal pumps relies on experimental data. The most common technique is to model the design of new pumps on existing efficient pumps using the concept of fluid dynamic similarity, also called similitude or similarity law. This concept states that two machines have equal fluid dynamic characteristics (equal efficiencies) if they 1) are geometrically similar, 2) have similar velocity triangles at similar points in the flow path, 3) have the same ratio of gravitational to inertia forces acting in the flow path (Reynolds number, Re) and 4) operate with fluids with the same thermodynamic quality.
Specific speed versus specific diameter (n s − d s) graphs are widely used to model industrial pumps using previous designs of efficient pumps. Existing graphs for larger industrial pumps are not suitable for small sized heart-assist pumps as the operating conditions are different, resulting in different ratios of gravitational to inertial forces acting in the flow path (the Reynolds numbers are not similar).
In the 1950s, Cordier1 collected experimental data of different turbomachines and intended to correlate the data using nondimensional characteristics of the machines. The data points were plotted on a specific speed–specific diameter diagram, which are dimensionless quantities used to characterize the machine’s diameter with flow rate, pressure rise, and rotational speed. It was discovered that the data points of the machines could be fitted into curves based on their efficiency. The graph was later introduced as a practical guideline for the design and selection of turbomachines. This method was further developed by Balje2 in the 1980s by collecting extensive experimental data from numerous industrial turbomachine designers and manufacturers. Balje’s2 graph has been used as the first step for pump designers by relating the highest achievable efficiency of a turbomachine to its size, speed, pressure rise, and flow rate. On the basis of Balje’s2 method, centrifugal pumps could be designed to have an efficiency of up to 80% and the data apply to industrial machines working in the Reynolds number order of 108. Small-sized centrifugal impellers for ventricular assist devices (VADs) and mechanical circulatory support (MCS) devices operate with Re lower than 105. Therefore, Balje’s2 nondimensional graph is not suitable to design pumps for VADs and MCS devices and a new nondimensional graph needs to be developed for small-sized pumps in low Reynolds number regions. Several studies have been conducted on applying this method to small pumps to be used in MCS devices.
Smith et al.3 collected nondimendional data of 37 rotary dynamic blood pumps containing axial, mixed flow, and radial pumps with different number of blades, splitters, and shroud configurations and different types of discharge. The nondimensional data points are plotted on a Cordier diagram. From these data, a pump designer can make a first estimate at the size, speed, and performance of a blood pump. The study is a pioneering work, based on reliable experimental data, but for a broad range of pump types and not concentrated on one type of pump.
This study is focused on centrifugal impellers. A total of 100 impellers were designed based on existing nondimensional experimental data and conventional pump design methods. The pumps were modeled and studied with the aid of computational fluid dynamics (CFD) and the specific speed–specific diameter diagram of the pumps was produced. A single loop test rig was used to test a group of impellers to validate the numerical results. The following section in brief explains the process of design, computational analysis, and selection of the impellers.
Materials and Methods
The formal derivation of the similarity concept is based on the fact that any physical quantity Q 1 (efficiency for instance) is interrelated to other physical quantities Q 2 to Q n (significant variables) by
Because this equation must be dimensionally homogeneous each Q i term must be of the same dimension, or the Q i terms must be transformed in dimensionless form using the primary dimensions: length [L] = meters; time [T] = seconds; and mass [M] = kg. Because there are three primary dimensions, so the n quantities of Q i must be transformed in n-3 dimensionless π 1 quantities (nondimensional parameter groupings, or similarity parameters).
This is known as the Buckingham π-theorem,4,5 where each π and the resulting product must become dimensionless when each Q term is expressed in the primary dimensions. This argument can be used to determine the number of π terms, or parameter groupings (similarity parameters) needed to determine a physical process. Nine independent variables can be defined for turbomachines,2 so six independent dimensionless similarity parameters can be formulated. These are: efficiency; Reynolds number; Laval number; ratio of specific heats; specific speed and specific diameter. In the design process, efficiency, Reynolds number and the specific speed and diameter are the most important parameters to be considered (Table 1).
The adiabatic head of the pump is given by
where ΔP is the pressure difference (N/m2) from inlet to outlet of the pump and ρ is the density of the fluid (kg/m3).
The Reynolds number Re is a characteristic dimensionless number that compares the inertia forces to the viscous forces in the fluid and indicates whether and to what extent fluid flow is steady or turbulent and it is defined by
where μ and ν are dynamic (Pa·s) and kinematic viscosity (m2/s) of the fluid respectively, C is the velocity (m/s) based on the cross section area, and D is the hydraulic diameter (m).
Specific diameter and specific speed are characteristic dimensionless numbers indicative of the rotor diameter and rotative speed, respectively.
The specific speed is given by
and the specific diameter is given by
where N s is the rotor speed (rad/s), Q is the volumetric flow rate (m3/s), and ϕ and ψ are flow and pressure coefficients.
There are various parameters to be considered in the design of a centrifugal impeller; pump size, number of blades, blade thickness, inlet and outlet width, inlet and outlet diameters and angles. The velocity triangles and a combination of conventional turbomachinery pump design method6 and collected nondimensional experimental data3 were used to select suitable groups of parameters.
In order to find the optimum curvature, the meridional coordinate system was used as it is the most convenient system for axisymmetric flow. The meridional profile was divided into several streamlines and the data points of each one were achieved by using a cubic Bezier curve. The leading edge and trailing edge were defined by the inlet and outlet angles and the curvature data points were modified in order to achieve the largest possible radius at the bend to avoid flow separation.7–12 Small radius in meridional profile results in high velocity and flow separation and reduces the efficiency. The design procedure was developed as a MATLAB (MathWorks, Natick, MA) code that generates three-dimensional (3D) coordinates for the blade and the diffusion volute.
The 3D geometry model of the impellers and volutes was generated using SolidWorks software and then imported to ANSYS Workbench v15.0. ANSYS CFX (ANSYS Inc., Canonsburg, PA) was used to model the impellers. The 3D incompressible Navier–Stokes equations were used to predict the flow field through the centrifugal pump.
The model included two domains; rotating (impeller fluid zone) and stationary (volute fluid zone). The rotating fluid zone was defined by creating the fluid volume around the rotor surface and the volute fluid zone was defined by creating the fluid volume inside the volute section.
Boundary conditions were specified to define the rotational speed, inlet pressure, and outlet flow rate. At the inlet, a relative pressure of 0 Pa was defined; whereas at the outlet, a healthy human body blood flow rate (5 L/min) was imposed.
The model was classified into two sections: heart and circulation system. Figure 1 shows the schematic diagram of the system. A single loop test rig, which will be called O-loop hereafter, was developed for initial testing to find impellers characteristics of flow rate and pressure difference at various operating conditions.13,14 The circulation system consisted of impeller housing, a valve to change the resistance of the loop, a flow meter to measure flow and two pressure transducers for pressure measurements.
The power system consisted of a high-speed Brushless DC (BLDC) motor (Maxon EC 45, 45 mm diameter, brushless, 250 Watt), a controller to control the motor speed and a coupling mechanism to directly drive the rotor of the pump. The rotational speed of the motor drive was varied from a low value to a nominal maximum speed. The pressure difference versus flow rate characteristics of the pumps was investigated under a steady flow condition.
The pressures upstream and downstream of the impeller and the corresponding flow rates generated by the impeller were measured using the pressure transducers and the flow meter respectively. All the measurement process were repeated for a wide range of resistances from low (wide open, full flow) to high (closed tube, no flow).
Efficiency, pressure rise, specific speed and diameter, head and flow coefficients were measured or calculated at different operating conditions and were plotted in relevant graphs. Figure 2 is a Cordier diagram showing specific speed and specific diameter of 88 efficient pumps based on the highest efficiency points of each pump.
The numerical results were used to make an initial estimation at the size, rotational speed, shape and the performance (pressure rise and flow rate) of a group of impellers. A group of nine impellers with different characteristics were manufactured and tested in the O-loop. Figure 3 shows the profile of two representative models selected for illustration. Impeller A (31 mm diameter, 4 blades, 22.5° outlet angle, 4.2 mm outlet width) and impeller B (31 mm, 7 blades, 27.5°, 4.0 mm).
The experiments were repeated three times for each impeller in order to minimize the uncertainty and verify the repeatability of the experiments.15
The nondimensional approach of the study makes the results applicable to different performance characteristics rather than particular values. Q/Q design shows the ratio of volumetric flow rate to its value at the design point and H/H max is the ratio of pressure rise to the maximum pressure produced in that particular case.
Figures 4 and 5 show the numerical and experimental efficiency and head ratio (H/H max) versus flow ratio (Q/Q design). Uncertainty is the standard error and the error bars are shown only in specific data points for clarity.
Figure 6 plots the numerical and experimental efficiency versus specific speed, and Figure 7 shows the flow and head coefficients relation for impellers A and B. On the basis of the location of the data points, second and third order polynomial regression lines were fitted to the data as they were the lowest orders that provided a good fit to the data points.
Figure 8 is a comparison between steady and transient flow conditions for impeller A, where the transient flow conditions were simulated using the cardiovascular flow emulator rig presented in Rezaienia et al.,16 Shi and Korakianitis,17 and Rezaienia et al.18
A blood analog solution (65% water, 35% glycerol, by volume) was used as the working fluid in the O-loop to obtain values of density and viscosity of working fluid representative of those of human blood at 37°C. Using a higher viscosity fluid instead of water will result in higher shear stress, higher friction and therefore slightly higher input power and lower efficiency.
Friction loss as a source of head loss can be defined as the loss of pressure occurring in the system due to the effect of the fluid’s viscosity near a surface (shear stress). Two sources of friction loss are present in this experiment; friction between the fluid and the surface of the pipes, and between the fluid and the impeller surface.
Although it is obvious that blood is a non-Newtonian fluid, for the numerical simulations, blood was considered as an incompressible Newtonian fluid because of the high shear rates of centrifugal blood pumps. Experimental results conducted by Kim et al.19 confirmed that when the shear rate is higher than 100/s the non-Newtonian properties of blood such as shear thinning and viscoelasticity are negligible.
By considering blood as a Newtonian fluid, the shear stress on each boundary is proportional to the strain rate in the fluid and is defined by
where μ is the dynamic viscosity of blood, u is the velocity, and y is the height above the boundary. Based on the values of linear velocity on the surface of the impeller (4–8 m/s) and inside the pipes (≈0.2 m/s), the shear stress in the pipe is negligible compared with that on the impeller surface (20 to 40 times larger on the impeller).
There are three main types of volute design—singular, circular, and double (or split). The major effect of using different types is the hydraulic radial forces on the impeller. This force is extremely important in case the impeller is suspended within the casing by contactless means. Higher radial force will result in a higher force to balance the impeller and therefore using more power and lower efficiency. Based on an experimental study conducted by Boehning et al.20 on three types of volute, the single volute had the lowest radial force acting on the impeller at the design point (≈0 N), whereas the circular volute yielded the highest (≈2 N) and the double volute resulted in a force of approximately 0.5 N. The study showed that the double volute will be a better choice for contactless impellers in off design conditions, but the single volute will have a higher efficiency at the design point.
In this study, the impellers are driven by an external electric motor through a shaft. The main purpose is to find the highest achievable efficiency for each individual pump even if it is just a peak point. Therefore, a single volute was the most convenient type to use. A single volute was designed in order to keep a constant angular momentum through the path, with a slight modification to resolve the friction losses at the walls.
The comparison between the experimental results and the expected numerical curves shows a consistent difference pattern for all nine models and is shown in Figure 4 for impellers A and B. The highest experimental efficiency of 72.28 ± 0.41% was obtained for impeller A at Q/Q design = 1.19. This value was close to the numerical efficiency of 75.17% at the same flow rate (3.84% error). The difference between numerical and experimental efficiencies reaches a maximum (≈7%) at Q/Q design = 0.8 for impeller A.
The difference between the numerical and experimental efficiencies is mainly because of the frictional head losses in the experiments. These losses are composed of the frictional losses in the suction and discharge piping system, which are the sum of frictional losses caused by the water flowing through the inlet and outlet pipes, fittings and measuring equipment, two bearings used on the shaft which are not modeled in the numerical simulations. Also, the impeller and the volute are manufactured with the finest possible surface precision, but are not as smooth as simulated models. This surface roughness effect is another factor contributing to this difference.
In all cases, it was observed that the numerical efficiency is closer to the real tested values at higher flow rates and higher speeds. This could be the result of using the k-ε turbulence model in the computational analysis, which improves the accuracy and is closer to the real value at higher Reynolds numbers.
Figure 5 shows the head ratio versus flow ratio for impellers A and B. In the numerical model, the highest pressure is at no flow condition, as expected from theory. However in the experiments, pressure slightly increases by flow rate until it gets to a maximum value at flow rate of about half the design point and then decreases moderately. The reason is the slip between the fluid and the blades, which is higher at very low flow rates, because of higher resistance in the system. This difference is normally smaller in impellers with lower number of blades because of lower contact area and therefore lower slip factor.21
Figure 6 plots the numerical and experimental hydraulic efficiency versus specific speed for impellers A and B. Data points are deleted for clarity and third order curves are fitted, with R 2 values presented on the graph. The gap between the numerical and experimental efficiency gets large when n s ≥ 1.5 and this applies to all tested models. The reason is higher rotational speed and flow rate and therefore lower pressure rise due to no or low resistance in the system. This leads to frictional head losses being the only factor in the difference between numerical and experimental pressure rise. Neglecting high specific speeds, the maximum difference between numerical and experimental efficiency is 7.1% for impeller A.
Figure 7 shows the numerical and experimental head and flow coefficients for impellers A and B. The second order curves are fitted to the data points, with R 2 values presented on the graphs. This graph in addition to η − n s and n s − d s graphs is useful for preliminary design and finding the optimum diameter and rotational speed for a desired range of flow rate and pressure rise. The maximum difference between numerical and experimental head coefficients at the design point is 0.05, which is 9.1%.
Figure 8 is presenting a typical H-Q curve for impeller A in steady conditions of the O-loop in comparison with transient flow in a cardiovascular simulator.16–18 The top axis relates to the steady flow curve with a pressure of 90.84 mm Hg at 5 L/min. The bottom axis is related to the transient flow at 5 L/min with maximum and minimum pressures of 115 and 78 mm Hg and an average of 90.72 mm Hg.
Smaller clearance between the impeller and housing results in higher efficiency but also higher shear stress. An optimum gap needs to be determined for each individual impeller in order to maximize the efficiency while keeping the shear stress under the critical value. According to a numerical model on the platelet activation based on theory of damage,22 a shear stress higher than 150 Pa indicates hemolysis. The shear stress modeled and calculated in the numerical analysis for impeller A has a maximum value of 79.3 Pa showing this impeller will not induce hemolysis at the simulated conditions.
Hemocompatibility could be studied for each individual impeller by calculating the shear stress and index of hemolysis separately, after the design and selection process. A pump designer can use the data from this graph to select parameters based on high efficiency points and then vary different parameters in order to achieve a hemocompatible design while keeping the nondimensionals in the selected range. The efficiency will remain high as long as the specific speed and specific diameter are in the same range.
Combined experimental and numerical techniques were carried out on an extensive number of centrifugal impellers to characterize the performance and were verified by experiments on nine models with various design characteristics. The consistency of the gap between numerical and experimental results for all models and the small justifiable errors show a good validation of the numerical analysis.
Every geometric parameter has an optimum value for each individual pump. The best way to study the effect of parameters on the performance is to find an optimum range for each parameter using optimization methods and extensive experiments. Hopefully, this work will be a useful tool to design and test more impellers with different characteristics and study the effect of each parameter on the performance thoroughly.
The highest efficiency point of each impeller was plotted on a Cordier diagram in Figure 2. Although the number of data points are high and shows the efficient regions, it is still insufficient to predict the pattern and produce isoefficiency curves. If MCS pump developers analyze and report pump performance in nondimensional n s − d s form in the future, hopefully a more precise and extensive version of this diagram will be produced and updated.
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Keywords:Copyright © 2016 by the American Society for Artificial Internal Organs
ventricular assist device; MCS; specific speed; specific diameter