Estimating the optimal value of diameter of the inlet to the impeller of a centrifugal blood pump

. The article is devoted to the optimization of the design of the inlet zone of a centrifugal blood pump. One of the disadvantages of such pumps is the generation of turbulence in the flow of the pumped fluid, which contributes to high blood hemolysis. The reason for turbulence lies in a combination of geometric and kinematic factors, in particular, a 90 degree turn of the flow in the inlet part of the impeller channels and a high circumferential speed of rotation of its channels. The hydrodynamic consequence of this is an increase in the hydraulic resistance of the pump flow path and separation of the boundary layer from the streamlined surface. As a result, the non-uniformity of the flow velocity field increases and separation flow zones appear, stimulating vortex formation and turbulence of the blood flow. The article provides the derivation of an equation for calculating the optimal diameter of the impeller D 1 , which does not have a bushing in the inlet. This design is typical for centrifugal blood pumps. As a criterion for the optimal value of D 1 , the minimum level of hydraulic resistance in the inlet zone of the pump was considered. Based on the derived formula, a graph of D 1 values was constructed in the range of parameters: blood volume flow 3...6 liters per minute, speed 4000-10000 rpm. With an increase in blood flow and a decrease in the number of revolutions of the impeller, the diameter of the inlet to the impeller must be increased from 6.4 mm to 13.8 mm.


Introduction
Pumps for cardiopulmonary bypass and assisted circulation according to the method of converting mechanical energy into hydraulic energy are divided into pulsating and nonpulsating blood pumps [1]. The principle of operation of pulsating pumps is based on a volumetric method of pumping blood, similar to the work of the human heart, which is a contracting muscle chamber. For 70 years of human life, with an average heart rate of 60 times per minute, the heart makes more than 9 billion pulsations (contractions).
The principle of operation of non-pulsating pumps is based on the mechanism of converting the energy of rotation of the rotor into kinetic, and then into static energy of the pumped blood. Despite the name non-pulsating, at the outlet of these pumps, the blood also pulsates, but at a high frequency. One source of pulsation, called the turn frequency, is equal to the angular frequency of rotation of the rotor, and the second source of pulsation is the blade frequency, which is equal to the turn frequency times the number of impeller blades.
As a result, the blood pulsation frequency at the pump outlet is several kilohertz and cannot be measured with a tonometer. Pumps of this type do not occur in nature.
A legitimate question arises. Why develop an artificial heart based on a non-pulsating pump, the analogue of which is not found in nature? The answer is that the problem lies in the complex mechanism of the heart, which is functionally divided into 2 parts: a drive device and a pump. Both functions in the heart are performed by myocardial cells. As a driving device, the myocardium converts chemical energy into mechanical energy of muscle contraction. As an elastic wall of the chamber, myocardial cells form a volumetric pump cavity in the form of the right and left ventricles of the heart. Artificial hearts differ from the human heart in that the functions of the driving device and the pumping function in them are performed by separate devices. In a pulsating design, the pumping part is similar to the ventricle of the heart. This is a chamber with deformable walls. The drive device is a hydraulic or pneumatic drive, consisting of pipelines and a compressor, the dimensions and power consumption of which are significantly higher than those of the human heart.
In a non-pulsating design, the pump part is similar to a vane pump, axial [2][3][4] or centrifugal [5][6][7][8], the main part of which is the impeller. Electromagnetic DC motors are used as driving devices. The dimensions of the pump depend on the number of rpm of the electric motor n. Upon reaching n=(6...10)103 rpm, the dimensions of the non-pulsating pump are reduced to the size of a natural heart.
According to their parameters: blood volume flow 3...6 liter per minute and the size of the impeller, these pumps are classified as low-flow, small-sized centrifugal pumps [6][7][8][9]. As a rule, their impeller does not contain a sleeve protruding from the impeller inlet, as is seen in industrial pumps.
The disadvantage of such pumps is the turbulence of the pumped liquid, which contributes to high blood hemolysis [10][11][12][13][14]. The hydrodynamic cause of increased flow turbulence is the hydraulic resistance of the pump flow path and separation of the boundary layer from the streamlined surface. In this regard, it is advisable to clarify the formula for calculating the diameter of the inlet D1, see Figure 1 into the impeller without bushing.
This formulation of the problem is due to the fact that the existing methods for calculating centrifugal superchargers consider the diameter of the sleeve and its cluttering effect in the inlet of the pump.  where u1 is the circumferential speed, w1 is the relative speed, c1 is the absolute speed, assuming zero swirl (c1x=0) of the flow at the impeller inlet, index "1" indicates the input section.
With an orthonormal basis, it follows from the velocity triangle that, see Figure 2b: The condition for reducing the degree of turbulence of the blood flow is the minimum level of hydraulic resistance and losses Llos, including losses in the impeller Lim and outlet Lout: The impeller losses can be expressed in terms of the square of the relative speed w1: where the loss factor im in the wheel takes values in the range im=0.3…0.5 [15]. Obtaining the minimum velocity head of the flow in relative motion makes it possible to ensure acceptable energy efficiency of a centrifugal pump in the process of varying its operating parameters.
Given that Lim w1 2 , and the relative speed w1 depends on the geometry of the impeller at the inlet, consider the criteria for choosing the diameter of the inlet to the impeller D1, see Figure 1.

Results
Let us set the condition: the value of D1 at any combination of the flow rate of the working fluid Q and the angular frequency of rotation of the impeller =n/30 (sec -1 ) should provide the minimum speed level w1. Representing the vectors c1 and u1 as a linear combination of the basis vectors, and taking into account the collinearity of the vectors u1 and i (u1у=0), from equation (1) we obtain: w1=u1xi + c1yj.

(4)
Knowing from (4) the coordinates of the vector w1, we find the square of the modulus of the relative velocity: Using the geometric and energy parameters of the pump, we specify the content of the components of the right side of (5): The value of c1y, taking into account the constraint of the flow, is set by the approximate ratio: where c0x is the axial flow velocity before entering the impeller 1=1-(1/t1) -coefficient of constraint at the entrance to the wheel, as a rule, 1=0.8…0.9; 1 -blade thickness at the inlet in the circumferential direction; t1 is the pitch of the blades at the inlet.
The axial speed c0x is found by the formula: where Q is the flow rate of the working fluid through the pump; D0 -inlet pipe diameter; сconsumption efficiency (usually for small-sized pumps, с=0.8…0.85 is accepted).
Taking the values c1y=1.2c0x/1 and D1=0.95D0, typical for the designs of small-sized centrifugal pumps [15], we obtain the formula for an approximate estimate c1y: In expanded form, equation (5) takes the form: We investigate the function w1 2 =f(D1) for an extremum, realizing the condition: The value of the argument at the critical point of the first kind will be denoted as D1*. From the solution (11), the value D1* is found by the formula: Expression (12), regulating the optimal value of D1 of centrifugal pumps, makes it possible to simultaneously predict the nature and magnitude of the change in the inlet diameter for various combinations of flow rate and angular speed.

Discussion
Formula (12) makes it possible to estimate the range of variation of the impeller diameter at the inlet D1 for different combinations of the number of revolutions n and blood flow Q and constant values of the consumption efficiency c, as well as the flow constraint coefficient 1, which evaluates the degree of obstruction of the inlet section of the impeller blades.
In Figure 3 plotted the value field of the function D1=f(n), with fixed values c =0.8 and 1=0.9. The value of the number of revolutions n changed in the range n=4000...10000 rpm. D1 values were found for Q= 3 liters per minute and Q= 6 liters per minute.
At constant speeds, in accordance with the nature of dependence (12), a 2-fold increase in blood flow through the pump from 3 liters per minute to 6 liters per minute leads to an increase in the diameter D1 by 63%. The values of the inlet diameter for the indicated ranges of the combination of Q and the number of revolutions n change in the range D1=(6.39…13.76)10 -3 m. 13.7610 -3 m, and at n =10000 rpm from D1= 6.3910 -3 m to D1= 10.1410 -3 m.