From the Cleveland Clinic Foundation, Department for Biomedical Engineering, Cleveland, Ohio.

Submitted for consideration June 2001; accepted for publication in revised form December 2001.

Reprint requests: Markus Lorenz, The Cleveland Clinic Foundation, Department for Biomedical Engineering, 9500 Euclid Avenue, Cleveland, OH, 44195.

The design and performance prediction of hydraulic machinery in general, and of rotodynamic pumps in particular, has been and still is frequently empiric. Despite the extensive theoretical knowledge of fluid dynamics, the design process for pumps and hydraulic turbines still relies heavily on experimental data. Pumps are often developed using two major techniques. One common technique is to model the design of a new pump based on the designs of already existing pumps using the laws of similarity. ^{1} Another technique frequently used is to scale up production pumps from laboratory prototypes that have been refined to the desired performance in extensive experiments, using the same laws. Both approaches to pump design are usually considered highly reliable, although it is reflected throughout the literature that under certain circumstances these methods do not produce satisfactory results. ^{2}

One of the assumptions made for the derivation of the similarity laws by Wislicenus ^{3} is that the flow through the machine is fully turbulent. This does not always hold true, especially in machines of small size that pump fluids of high viscosity. Furthermore, the quality of the performance prediction by means of scaling can be affected by the different relative roughnesses of the model and the new design; and by inaccurate scaling of clearances and tolerances, especially if large scale factors are involved. The most significant drawback to the currently used techniques seems to be that the correction factors, which account for these effects, are usually only given for the best efficiency point (bep) of the pump, which makes off-design point performance prediction very difficult. A specific design process in which all these effects play a major role is the design of blood pumps. Blood pumps are used as heart assist or replacement devices. Compared with usual pumping applications, these pumps are very small in physical size and output low flows while pumping a fluid of considerable viscosity (blood has a normal viscosity of approximately 3.6 cP, but may vary significantly depending on the hematocrit). Blood pumps are usually designed to pump approximately 3–5 L/min against a pressure of approximately 100 mm Hg but frequently have to run off-design, depending on the patient’s condition (remaining cardiac activity, patient activity level, patient position horizontal or upright), while high efficiency is needed to minimize battery size and weight.

This study investigates how the scale factor used for a model blood pump affects the accuracy of the performance prediction, and the effectiveness of using water test data to predict the performance when higher viscous fluids are pumped. An attempt is made to derive a correction factor, which enables off-design performance prediction from geometrically similar pumps of different scale, tested with fluids of different viscosities.

Materials and Methods
Nomenclature
The following are definitions for terms used throughout the article.

a, b, c Coefficients for correction factor
d Pump Diameter, Coefficient for correction factor
e Coefficient for correction factor
f Linear Dimension Scale Factor
i Coefficient for correction factor
n Pump Speed
r Radius
s Coefficient for correction factor
t Number of data points
E_{Re} Reynolds Number Error
E_{λ} Average scaling Error
F_{Ψ} Nondimensional Pressure Correction Factor
Re Reynolds Number
T Pump Torque
Pump Flow
X_{1} Coefficients for correction factor
X_{2} Coefficients for correction factor
X_{3} Coefficients for correction factor
Δp Differential Pressure
η Liquid Viscosity
ρ Liquid Density
Φ =JOURNAL/asaio/04.02/00002480-200207000-00017/ENTITY_OV0312/v/2017-07-29T043104Z/r/image-png Nondimensional Flow 2 · π ·r ^{3} ·n
ψ =Δp Nondimensional Pressure 4 · ρ · π^{2} ·r ^{2} ·n ^{2}
Subscripts:

1 Original Size Pump
2 Scaled Pump
max maximum
min minimum
Re Reynolds number at test point
Re_{max} Maximum Reynolds number
Pumps
Figure 1 shows the three geometrically similar rotodynamic pumps with a scaling factors f of 1 (original scale), 3.2, and 6.4, respectively. The pumps were manufactured by a stereolithographic process, and their geometry was derived from an early design version of a nonpulsatile left ventricular assist device (LVAD) that has been developed by the Department of Biomedical Engineering at the Cleveland Clinic Foundation. The radial flow pump impeller has seven blades and a volute pump housing. The original scale pump has an outer impeller diameter d _{1} = 31.7 mm and is designed to pump blood at a flow of JOURNAL/asaio/04.02/00002480-200207000-00017/ENTITY_OV0312/v/2017-07-29T043104Z/r/image-png _{1} = 6 L/min against a pressure of Δp _{1} = 100 mm Hg at a speed of n _{1} = 3,100 rpm. The maximum expected flow rates and pressure rise for the original scale pump are Δp _{1,max} = 180 mm Hg and JOURNAL/asaio/04.02/00002480-200207000-00017/ENTITY_OV0312/v/2017-07-29T043104Z/r/image-png _{1} _{max} = 15 L/min at a speed of n _{1,max} = 3,588 rpm with a maximum torque of T _{1,max} = 8 Nmm. Figure 2 shows a computer study of the complete pump assembly of the 3.2 scale pump. The 6.4 scale pump was similarly assembled.

Figure 1: Model pumps.

Figure 2: Computer generated model of 3.2 scale pump.

Test Loop
To measure the performance of the original and scaled pumps, a test loop was set up. It consists of two main circuits. A schematic of the loop is shown in Figure 3 .

Figure 3: Test loop.

The booster pump is capable of pumping up to 80 L/min (Teel Co.) through the test loop, driven by a 1,120 W AC motor (GE Motor and Industrial Systems). This pump overcomes the test system losses, permitting the experimental pump to be exercised over a wider range of parameters. Two valves control flow through the pump under test: one controls the back pressure on the booster pump, and the second regulates flow back into the main tank. To dampen fluctuations in the booster pump’s flow, the test fluid is passed through a compliance chamber before it enters the test pump. The secondary circuit is designed for fluid conditioning. The fluid is pumped from the main tank, which has a volume of 120 liters, through the secondary circuit, with a small 200 W rotodynamic pump (Little Giant Pump Co.). It is filtered with a porous filter (Bonded filter 5 microns, Cole Parmer Instrument Co.) and sterilized by an ultraviolet sterilizer (Pura UV-20, Pura, Inc.). For temperature control, the fluid can be cooled by a heat exchanger (HE-30-Gold, Baxter Healthcare Co.) or heated with an electric heater (1,000 W, Omega, Inc.), controlled by an adjustable switching temperature controller (Omega CN 350, Omega, Inc.). Samples for viscosity and density measurements are taken from a sample valve. The viscosity is measured with a cone and plate viscometer (Model LVTDV, Brookfield, Inc.) with an accuracy of ± 0.01 cP. The density is measured with a laboratory scale (PR 5003, Mettler, Toledo, OH) and a pipette (10 ml, Falcon).

Data Acquisition and Sensors
The pump’s flow rate is measured using an ultrasonic flow meter (T110, Transonic Systems, Inc.) with a C-Series clamp on probe. The probe has a maximum range of 120 L/min, a resolution of 50 ml/min, and an absolute accuracy of ± 7%. The pressure rise is measured using a variable reluctance differential pressure sensor (Validyne Engineering Co.) with interchangeable diaphragms for different ranges. The accuracy is ± 0.25% of the full scale including nonlinearity, hysteresis, and nonrepeatability. A model CD15 carrier–demodulator provides the excitation signals and amplifies the output to ± 10V. The fluid temperature is read in the main tank and directly before the fluid enters the test pump using an Omega 2 Channel Microprocessor Thermometer and two Omega T-Type probes. Motor speed is determined using a chopper-disc with an optical pick-up. The signal from the optical pick-up is fed through a frequency to voltage converter, which produces 1V/47.76 RPM with an accuracy of ± 5 RPM. The data are recorded with a National Instruments data acquisition system (AT-MIO-16XE, National Instruments Co.) at a sampling frequency of 1,000 Hz. One data point consists of an average of 60,000 samples. Power data for the pump could not be collected, because the input current/output torque calibration of the motor, which was intended for this purpose, proved to be unreliable.

Test Results and Discussion
Impact of Scale on Nondimensional Performance Curves
To investigate the effects of scaling, the dimensionless performance data from the three pumps was investigated. All three pumps where tested over a range of three speeds and with fluids of three different viscosities. As a primary test fluid, water with a viscosity of 0.73 cps at 37°C was selected. The second test fluid was a mixture of water and glycerin at a viscosity of 3.6 cps, which is commonly used in blood pump testing to mimic the viscosity of blood. To cover a wider range of Reynolds Numbers, a third fluid with a viscosity of 7.2 cps was also used for testing. Test speeds were based on the design speed of the original scale pump, ranging from 2,500 to 3,588 rpm. To maintain Reynolds Number similarity between the scaled pumps and the original pump, the speeds for the scaled models was calculated by equation 1 . This form of the equation, using the impeller tip speed as the characteristic velocity, is commonly used in pump scaling calculations, for example in Balje. ^{4}

Table 1 shows the test matrix for the pumps at three viscosities and three speeds, leading to nine tested Reynolds Numbers ranging from 1,400 to 17,400. The speeds for the scaled models were calculated using equation 1 . To perform a consistent geometric scaling of the three pumps, clearances between pump housing and impeller where adjusted according to the scaling factor of the specific pump.

Table 1: Test Matrix

It is important to realize that the Reynolds Number for a complex geometry like a pump is not as straightforward as that for a simple geometry such as a pipe. At different locations in the pump, laminar to turbulent transitions may take place at different points during operation. A set of characteristic lengths and velocities could be chosen to represent a passage Reynolds Number or a Clearance Reynolds Number. Gap Reynolds Number similarity may take on particular importance if the power consumption must be effectively scalable. The Reynolds Number used here is conventionally accepted form that represents the overall ratio of the inertial and viscous forces in a pump.

Figure 4 shows the nondimensional test data for the pumps averaged over the whole range of tested pump Reynolds Numbers. Although the curves do not fully collapse to one single curve as the scaling theory for a perfectly scaled model suggests, the error made if data from one pump at a specific pump Reynolds Number were used to predict the performance of a pump of a different scale at that same pump Reynolds Number is small.

Figure 4: Nondimensional pressureversus nondimensional flow averaged Reynolds Number.

To quantify the dependency and magnitude of the scaling error on either relative surface roughness of the different pump combinations, their respective scaling factors, or the Reynolds Number itself, the scaling error averaged over the full measured range of the flow coefficient as defined by equation 2 was plotted against the Reynolds Number (Figure 5 ).

Figure 5: Average scaling error of different pump combinationsversus Reynolds Number.

The plot indicates that the performance of a scaled pump can be predicted with fairly good accuracy from a model pump of a different scale. For most of the investigated Reynolds Numbers, the error stays within an acceptable 5% range. The error is the lowest close to the bep Reynolds Number of Re = 3,300, indicating that the assumptions made for the deviation of the similarity laws by Balje ^{4} “machines that are geometrically similar, have similar velocity triangles at similar points in the flow path, and will have equal fluid dynamic characteristics, that is equal efficiencies” and Wislicenus ^{3} “the forces acting on the fluid must be similarly arranged, because otherwise, the fluid would be forced off the geometrically similar path.” are matched most closely here. The quality of the prediction does not show any apparent correlation to the scale ratio of the pumps.

Impact of the Reynolds Number on the Nondimensional Performance Curves
The influence of the Reynolds Number on the pump performance similarity can be evaluated by plotting the nondimensional performance curves for each Reynolds Number averaged for all three pumps (Figure 6 ).

Figure 6: Nondimensional pressurevs. nondimensional flow averaged scale.

Here, too, in the ideal case of complete similarity, the plots should collapse. This is true only for the highest range of Reynolds Numbers. For lower Reynolds Numbers, the plots deviate significantly from each other, indicating that by changing the Reynolds Number to lower values the similarity conditions are no longer satisfied. To quantify the error that would be made using data obtained at one Reynolds number to predict the performance of a pump at a different Reynolds number the Reynolds Number error E_{Re} was calculated using equation 3 :

The value of the Reynolds Number error of the pumps as a function of the Reynolds Number ratio and flow coefficient is plotted in Figures 7–9 .

Figure 7: Reynolds Number error versus Reynolds Number ratio and nondimensional flow coefficient original size pump.

Figure 8: Reynolds Number errorversus Reynolds Number ratio and nondimensional flow coefficient 3.2 scale pump.

Figure 9: Reynolds Number errorversus Reynolds Number ratio and nondimensional flow coefficient 6.4 scale pump.

From the plots of the Reynolds Number error, it is evident that E_{Re} is not only a function of the Reynolds Number ratio, but also a function of the nondimensional flow coefficient Φ. It is postulated that this effect might be due to the commonly used definition of the Reynolds Number in equation 1 , which is based on the impeller tip speed. Using the pump’s impeller tip speed as the representative velocity for the calculation of the Reynolds Number is only a valid approach as long as this velocity represents the true velocity in the pump. At operating points far from the pump’s best efficiency point, this assumption might not hold true and a Reynolds Number based on passage velocity might be a better assumption. ^{4}

Because the Reynolds Number is also dependent upon a characteristic length, it is of interest whether the scaled pumps performed in a similar way. Figures 8 and 9 show that the error due to changes in Reynolds Number of the scaled-up models basically present the same behavior as the original size pump. Large Reynolds Number ratios and large flow coefficients result in a large E_{re} . The average E_{Re} values for nondimensional flow coefficients ranging from 0 to 0.15 for the three pumps are given in Table 2 .

Table 2: Maximum and Average Reynolds Number Errors

It can be concluded that Reynolds Number similarity is a necessary condition to achieve good performance prediction for the pump type and operating points investigated. However, it can sometimes become impractical or even impossible to follow this procedure. Large scaled-up models can lead to flows so high in magnitude and pressures so low in value, that accurate data acquisition becomes extremely difficult. On the other hand, very small scale models can lead to speeds that are difficult to produce. In these cases, a prediction of the pump performance at Reynolds Numbers other than the one used to obtain data can prove to be useful.

Reynolds Number Correction Factor
Inspection of the data suggests that a mathematical description of the correction factorMATH must be a function of the Reynolds Number and flow coefficient. Therefore, a first approach in the form of a product was attempted.MATH

Numerical experiments (essentially trial and error) on a and b show that a = 4 yields the highest R^{2} Value, but that the data vary with Re more than a power law relationship can accommodate. Using an exponential with a scaling factor instead, yields a relationship in the form of:MATH

Values of s = −1,064, i = 0.9913, a = 4, and c = 2,500 results in the plot in Figure 10 . The R^{2} value is 0.865. The plot (Figure 10 ) still shows residual scatter between the actual correction factors and the predicted values.

Figure 10: Reynolds Number/flow coefficient correction.

It is known that the relative surface roughness, or the roughness coefficient λ, is a factor in the calculation of fluid friction and performance losses. Further numeric experimentation reveals that writing the equation for the head correction FΨ as:MATH leads to a slightly better approximation of the data, with an R^{2} value of 0.885. The plot for this function with s = −499.48, a = 4, c = 2,225, d = 0.1, and i = 1 can be seen in Figure 11 .

Figure 11: Reynolds Number/flow coefficient correction/relative roughness correction.

It seems logical that the magnitude of the correction factor should be related to the specific speed of the pump, because it is a measure of the proximity of the point of interest to the pump’s bep. At the bep, the correction should be smallest because all theoretical considerations are based on the bep performance. Further numeric experiments on the specific speed, n_{s} , show that using:MATH with s = −416.94, a = 4, c = 2050, d = 0.1, i = 0.977, and e = 0.25 has some benefit on the fit of the data as seen in Figure 12 ; the R^{2} value is 0.915.

Figure 12: Reynolds Number/flow coefficient correction/relative roughness/specific speed correction.

Benefits of the Correction Factor
Using the mathematical description of the correction factor to correct the pressure coefficient data, the Reynolds Number error was replotted. The maximum error for the 1:1 size pump was reduced from 30.9 to 7.5%, and the average error has been reduced from 5.2 to 2.6% (Figure 13 ). For the 3.2 scale pump, a reduction in maximum error from 45.4 to 14.1% and a reduction from 6.3 to 2.6% of the average error could be achieved (Figure 14 ). For the 6.4 scale pump, a reduction in maximum error from 37.5 to 9.6% and a reduction from 4.1 to 1.6% of the average error could be achieved (Figure 15 ).

Figure 13: Reynolds Number errorversus Reynolds Number ratio and nondimensional flow coefficient original size pump with correction.

Figure 14: Reynolds Number errorversus Reynolds Number ratio and nondimensional flow coefficient 3.2 scale pump with correction.

Figure 15: Reynolds Number errorversus Reynolds Number ratio and nondimensional flow coefficient 6.4 scale pump with correction.

Conclusion
The goal of this study was to investigate whether the use of scaling laws and scaling from models is a valid technique for the design and performance prediction of dimensionally small, relatively high viscosity blood pumps. A comparison and analysis of test data obtained from three geometrically similar pumps of different scale was used to validate the use of these techniques.

The use of scaling laws for the prediction of pressure flow performance of small size pumps from scaled models produces satisfactory results. Although the analysis revealed that a perfect match of the performance curves is not always achievable, the maximum and average error found using these laws are well within the limits usually accepted for engineering purposes. This can, of course, only be assumed true for the pump types, scale factors, and viscosities investigated in this project.

The investigation of the major parameters controlling the quality of the similitude showed that the effect of the Reynolds Number on the similarity of the nondimensional performance curves is stronger than the size effect. The magnitude of the size errors was shown to be highly dependent upon the Reynolds Numbers. A weak dependency on the relative surface roughness was found with the test methods used.

The Reynolds Number is usually assumed to have small effect on the similarity of the performance curves of conventionally sized turbulent flow pumps but has a major impact on these small, most probably laminar flow pumps. Whenever possible, it should be kept constant for performance measurement with scaled models of this size. The proposed correction factor has to be regarded as a preliminary description of the Reynolds Number effects and is only a starting point for further studies. The spacing of the model sizes, the limited database of only nine investigated Reynolds Numbers, and the restriction to only one impeller and housing geometry limit the usefulness of the derived equation for general use. However, it is a first attempt to predict the off-design performance of small pumps, pumping highly viscous fluids, and might prove to be useful for the design and further development of these pumps. Future experiments will have to investigate whether the pump power consumptions follow the scaling laws as closely as the pressure and flow characteristics.

References
1. Gülich JF: Kreiselpumpen: ein Handbuch für Entwicklung, Anlagenplanung und Betrieb. Berlin, Springer, 1999.

2. Stepanof AJ: Centrifugal and Axial Flow Pumps, 2nd edition. New York, Wiley, 1957.

3. Wislicenus GF: Fluid Mechanics of Turbomachinery, 2nd enlarged ed. New York: Dover Publications, 1965.

4. Balje OE: Turbomachines. New York, Wiley, 1980.

Copyright © 2002 by the American Society for Artificial Internal Organs