Experiment-based Comparison of Prediction Methods for Pump Head Degradation with Viscous and Power-law Fluids

Although several methods are known to calculate pump performance with highly viscous and non-Newtonian fluids, research has not yet determined all the key parame - ters of these predictions. It is unclear how these parameters depend on the pump geome - try and the delivered fluid rheology, which can vary widely in the chemical industry. In our study, the performance curves of a radial centrifugal pump with a viscous Newtonian glycerol solution and a non-Newtonian power-law fluid were experimentally compared. The head degradation of the pump was also presumed with the ANSI/HI and the Ofuchi methods, which are evident and commonly used for viscous Newtonian fluids, but not for non-Newtonians. The required constants were estimated based on experimental data for both models, and the Ofuchi method was adapted to power-law fluid. Based on our results, the Ofuchi method proved to apply for head degradation prediction with the ex - amined power-law fluid.


Introduction
There are many applications in various fields of industry where highly viscous and non-Newtonian fluids are delivered by pumping. Some of them can be characterised with power-law model 1 , for instance, juices 2,3 and liquid egg yolk 4,5 in the food industry, crude oils in the petroleum industry 6 , fresh concrete in the materials industry 7 , activated sludge in wastewater treatment 8 , and polymer solutions 9 in the chemical industry. Moreover, the rheology of the fluid can also change during the technological process 10,11 , which influences the system's operation and efficiency.
When pumping highly viscous and non-Newtonian materials, the pipeline system's curve and the pump's performance curves are also required for providing energy-efficient and safe operation. It is known that the system's characteristic curve can change because the pressure losses of its elements with power-law fluids differ from those with Newtonian fluids 2,[12][13][14][15] . The manufacturer gives the baseline performance curves of the centrifugal pump with water, i.e., the head-volume flow rate and efficiency-volume flow rate curves. A question arises of how the non-Newtonian rheology of the delivered fluid changes (degrades) the performance curves of the pump.
There are two groups of methods in the literature for estimating the curve alterations for a highly viscous Newtonian fluid. Both methods predict correction factors, by which the points of the original performance curve can be modified to obtain the degraded characteristic curve point-by-point. The most commonly used methods are available for pumping viscous fluids; some are modified to apply to non-Newtonian materials.
The first group of the known methods works with experimentally determined factors. The American standard (American National Standard Institute/Hydraulic Institute -ANSI/HI) was made for highly viscous materials 16 . The relationships required to calculate the correction factors in this method (referred to as ANSI/HI method) are based on the determination of parameter B, which includes the kinematic viscosity of the fluid. In addition, the data of the Best Efficiency Point (BEP) are required for the calculations. In the same group, the KSB method is also empirical and feasible for viscous materials. The pump manufacturing company developed a simple graphical method 17 , and it is widely used in engineering practice.
Recently, one of the leading research directions has been adapting the ANSI/HI method to the non-Newtonian material properties. The difficulty of the work lies in the fact that, while the calculation formulae contain one viscosity value, the vis-cosity is a function of the shear rate for non-Newtonian materials, thus using apparent viscosity value is recommended 18 . The relevant shear rate can be different at each point of the performance curve. On the other hand, determining an average value of the viscosity is not trivial, as the shear rate changes over a wide range inside the pump 19 .
Walker and Goulas compared the ANSI/HI method with experiments for Bingham plastic fluids 20 . They determined the apparent viscosity and the pump Reynolds number for the shear rates corresponding to 2 · n, where n, was the rotational speed. Sery et al. 21 refined their method: a mean shear rate was estimated in the pump impeller according to Metzner and Otto 22 . In their work, the correlation between the shear rate and the pump's rotational speed was experimentally determined with a Herschel-Bulkley fluid. Pullum and Graham 23 stated that the shear rate is not constant inside the impeller. Therefore, an equivalent pipe diameter was introduced to substitute the pump. This quantity is a function of the impeller diameter and parameter w. For a given volume flow rate with the equivalent diameter, the characteristic velocity can be calculated. The shear rate can be expressed with the Rabinowitsch-Mooney equation 24 in the laminar case. In the case of turbulent flow, the apparent viscosity value for the shear rate of 4000 s -1 is recommended.
The limitation of their method is choosing the value of w to determine the equivalent pipe diameter. Pullum and Graham suggest 25 % of the impeller diameter 23 . Furlan et al. proposed 13.7 % and 15.8 % impeller diameter for two different pumps they had tested 25 . By experiments, other researchers found an up to 3.1-37.5 % span in the value of w parameter for a given type of pump, varying with the fluid 26 . Kalombo et al. concluded that it is impossible to predict w analytically so that it should be determined empirically for each pump with a new fluid 27 .
The second group of methods for calculating performance curves for highly viscous materials is based on dimensional analysis. Stepanoff showed that the introduction of dimensionless numbers provides a tool for calculating the correction factors 28 . The dimensionless groups that characterise a pump are the pressure number Using the data of the BEP, these values can be normalised not to include the pump geometry. A simple correlation between the head correction factor and the volume flow rate correction factor calculated for a given normalised specific speed was presented 28,29 . The relationship between the head correction factor and the announced modified Reynolds number was estimated from experimental results by Gülich et al. 30 and Ofuchi et al. 31 Ofuchi suggested a new model predicting head degradation with highly viscous fluids 31 . Although Csizmadia et al. showed that this method (referred to as the Ofuchi method) could be useful with one power-law fluid 32 , the dependence of the evaluating constants on pump type, impeller geometry, and material properties is still an open question. Furthermore, research to date has not yet revealed how to convert this method for other non-Newtonian fluids.
Our main goal was to predict and experimentally verify the head degradation of a centrifugal pump with the ANSI/HI and the Ofuchi method with viscous Newtonian and a power-law test fluid.

Experiments Rheology
Two test fluids were investigated: a Newtonian 25 % glycerol solution and a non-Newtonian jelly textured bath gel (referred to as 'gel'). An Anton Paar Physica MCR301 rotational viscometer was used to determine the rheological parameters of the test fluids, with thermoelectric temperature control via built-in Peltier elements. The measurement range was 0.1 -100 s -1 . The instrument was used with a cone-plate layout with a gap of 0.054 mm. Both liquids were measured three times, and the parameters of the rheograms were determined based on their mean. The gel was evaluated with a power-law rheology model and showed pseudoplastic behaviour. The fit is presented in Fig. 1. The describing equations of the test fluids were: was determined with a standard orifice meter. This method was also checked with a Fuji Electric (FSS-C1YY1) ultrasonic flowmeter and bucketing. The temperature of the fluid was checked with a tem-perature gauge in the tank. The input electrical power, which was used to estimate the efficiency, was measured with the built-in meter of the motor. The experimental setup is presented in Fig. 2.
The parameters of the BEP were: Q BEP = 18.25 m 3 h -1 , H BEP = 26.08 m at the nominal (baseline) rotational speed of n n = 3355 rpm determined with water.

Measurements
The reference performance curve was measured with water. Measurements were performed at least at eight operating points by each rotational speed. For glycerol solution, five different rota-  tional speeds were adjusted between the minimum (n 5 = 1000 rpm) and maximum (n 1 = 3355 rpm) values allowed by the pump control; while in the case of the gel -due to some unstable operation-only two speeds (n 3 = 2200 rpm and n 1 = 3355 rpm) were set. It took special care to avoid heating the fluids during the experiments, in order to keep the fluid at room temperature of 22 -23 °C.
The measured baseline head was modelled with a quadratic polynomial equation, and the curves for other rotational speeds with water were calculated with the affinity laws 28 . The measured head curves as the function of the volume flow rate are shown in Fig. 3.

ANSI/HI method
The ANSI/Hydraulic Institute method was used based on Pullum and Graham 23 . In this method, the parameter B is calculated from the BEP values at nominal rotational speed with the rheology as: The correction factors for the volume flow rate ( Q _ HI C ) and the head ( H _ HI C ) in the ANSI/HI method are defined as: The predicted values of volume flow rate and head are derived from those with water as There is a simple correlation between the correction factors defined in Eq. (11) and Eq. (12) proposed by Stepanoff 28  In practical application, the normalised specific speed with Eq. (9) and Eq. (10) can be calculated for each point of the baseline water curve. In the range of 0.6 ≤ ω n ≤ 1.25, the modified Reynolds number can be determined with Eq. (15) for any rotational speed and fluid given by the kinematic viscosity value. Eq. (11) specifies the C H_Of head correction factor, the one for the volume flow rate by Eq. (13). Finally, the points of the predicted degraded curves in the Ofuchi method were defined with Eq. (16) and Eq. (17): Our study aimed to verify the polynomial relationship between the correction coefficients C Q_Of and C H_Of in Eq. (13) based on the experimental results. For this, the factors were calculated from the measured points at selected normalised specific speed values of ω n = 0.6, 1, 1.25. As shown in Fig.  4, the difference between the experimental values and the Eq. (13) was less than 0.5 %. Thus, the correspondence between the head and volume flow rate factors was verified with our pump. In addition, the good match with the literature also confirmed the accuracy of the measurements.
The new model of Ofuchi et al. is available for Newtonian fluids, but in non-Newtonian adaptation, the kinematic viscosity in Eq. (15) is not apparent. Moreover, the actual values of the constants in Eq. (14) are needed to be found.

Estimated parameters
For the glycerol solution, the ANSI/HI method was directly applicable. In addition, the required parameter w was evaluated as a percentage of the impeller diameter in the ANSI/HI method for the power-law fluid. The best agreement between the measured and calculated curves for the actual pump with the power-law gel was obtained at 48 % of the impeller diameter, so the parameter was w = 0.48 · D imp . This value matched neither those proposed by Furlan et al. 25 nor the 25 % suggested by Pullum and Graham 23 , but confirmed the findings of Kalombo et al. 27 that it could be different for each pump with each fluid.
To adapt the Ofuchi method to non-Newtonian fluids, we assumed that an average shear rate could be used by estimating the gel's viscosity based on Metzner-Otto 22 , as seen in Buratto et al. 26 We sup-posed a linear correlation between the average shear rate and the actual rotational speed of the pump 32 .
The assumption was tested at the baseline rotational speed n n . The best fit was defined at the minimum of the mean square deviation between the measured and predicted curve, and re-  sulted in constant c = 3.01 with a coefficient of determination of R 2 = 0.9971. This determined actual shear rate was used to estimate the apparent viscosity of the power-law fluid as in Eq. (4), and the derived kinematic viscosity was applied in Eq. (15). The actual pump data with the different fluids at fixed normalised specific speed values were used to evaluate parameters a and b in Eq. (14) by the nonlinear least squares method, using MatLab. The values of the regression were found as a = 5.513 and b = 0.705 with the coefficient of determination of R 2 = 0.9691, which were very close to those suggested by Gülich 30 (a Gülich = 6.7 and b Gülich = 0.735), and Ofuchi et al. 31 (a Ofuchi = 4.462 and b Ofuchi = 0.695). The regression curve is presented in Fig. 5.

Predicted curves
The prediction with ANSI/HI method was performed in the entire range of the measured volume flow rates, but for the Ofuchi method, ω n < 1.25 is a mathematical upper limit of the application. It was also mentioned by Ofuchi et al. 31 that 0.1·Q BEP instead of 0.6 < ω n as lower limit, yields good results in certain conditions, so we also used this value.
In the case of 25 % glycerol, the performance curves' shape changes were not remarkable, as shown in Fig. 6. The head curves predicted with both methods were in good agreement with the measured ones. Only the ANSI/HI model at rotational speed n 1 = 3355 rpm with high volume flow rate and both approaches at n 5 = 1000 rpm were outside the ±5 % error margin of the experimental data. The latter can be explained by the uncertainty of the measurements at low volume flow rates.
With the power-law gel, the measured head degradation was significant, and the shape of the experimental head curve differed from the baseline curve, as presented in Fig. 7. The ANSI/HI estimation did not result in performance curves similar to those measured. However, it was still within the ±5 % error band in that volume flow rate range, where the Ofuchi method is also valid.
The Ofuchi approach proved accurate at n 1 = 3355 rpm, and provided a good match with the experimental data with the average relative difference of 2 %. The Ofuchi curve was qualitatively better at the lower rotational speed than the ANSI/HI curve. Still, the mean relative difference was 10 %, and the highest value was 31 % between the experimental and calculated heads in the investigated range. It can also be noted that the particular degradation by the zero flow head cannot be modelled with this method due to the lower limit of application, as the method is valid only for Q > 0.1 · Q BEP 31 .
F i g . 6 -Measured (black curves with ±5 % error) and predicted performance curves with glycerol at different rotational speeds with measured baseline curve with water as reference (thick blue with ◊). Green □: predicted with ANSI/HI method, orange Δ: predicted with Ofuchi method.

Conclusions
The following conclusions can be drawn from the present study, which shows the results of comparing two prediction methods based on experiments carried out with a two-stage radial centrifugal pump with the specific speed of n q = 35.
-With the viscous Newtonian test fluid, both the ANSI/HI and Ofuchi methods could estimate the head degradation of the radial centrifugal pump with sufficient accuracy of ±5 %. -With the examined power-law fluid, the ANSI/HI model failed to replicate the measured curves, even though the key parameter w was estimated based on the experimental results. -The characteristic parameters a and b of the Ofuchi method were successfully determined for our pump. To extend the prediction for non-Newtonian applications, the average shear rate was introduced to assess the apparent viscosity of the power-law gel. -The predicted head degradation agreed quite well with the experiments at the nominal rotational speed with the mean difference of 2 %. The Ofuchi method predicted the performance curve qualitatively well, even at the lower rotational speed. N o m e n c l a t u r e a, b -constants in the Ofuchi method B -parameter of the ANSI/HI method BEP -best efficiency point c -constant in the ANSI/HI method C H_HI -head correction factor in the ANSI/HI method C H_Of -head correction factor in the Ofuchi method C Q_HI -volume flow rate correction factor in the ANSI/HI method C Q_Of -volume flow rate correction factor in the Ofuchi method -density, kg m -3 τ -shear stress, Pa ω n -normalised specific speed ω s -rotational specific speed φ -flow number ψ -pressure number