Centrifugal pumps being used for transporting slurries in the industry. The main parts of a Centrifugal pump are rotating element (impeller) with blades and a stationary element (volute casing). The transport of multiphase flow particularly the slurry which is mixture of liquid and solid is widely encountered in the petroleum, mining and chemical industries. The impeller-rotation reduces the pressure at the pump-inlet resulting in fluid flow into the pump, and the fluid gets kinetic energy. The volute casing collects the fluid from the impeller and helps converting the kinetic energy into the pressure energy. The pumps are generally used for high discharge with low head applications.

The performance of a centrifugal pump delivering slurry depends on the solid particle size, concentration, density and rheological property of slurry. The Hydraulic Institute (HI) method is well established for the de-rating performance prediction of centrifugal pumps for viscous Newtonian fluids. For non-Newtonian fluids, the viscosity is a function of shear rate; and the performance prediction based on constant average viscosity with varying shear rate is a difficult task.

Walker and Goulas (1984) used the Bingham plastic viscosity in HI method to predict the performance of a CENTRIFUGAL pump handling non-Newtonian fluid mixtures of coal-water and kaolin-water. Later, similar approach was used for different slurries (Sery and Slatter 2002; Kabamba 2006). Pullum, Graham, and Rudman (2007) estimated the apparent viscosity from equivalent pipe diameter of the pump, and predicted the performance of non-Newtonian fluids using the HImethod. The apparent viscosity approach showed a better head prediction; and the Bingham plastic viscosity resulted better prediction in efficiency (Kalombo et al. 2014) predict the pump performance for non-Newtonian fluids.

A quadratic polynomial regression (PR) model was developed to predict the de-rating head of the pump because of the non-Newtonian behavior of working fluid. The PR models for head and efficiency are the functions of slurry viscosity, speed and discharge of the pump.

## EXPERIMENTAL SET-UP

The test set up for evaluating the performance of the centrifugal pump is a closed loop system developed at the Department of Ocean Engineering, IIT Madras. It consists of a 36 mm pipe connected to the tank, and an assembly of a pump and a motor (see **Figure 1**). The water and bentonite slurry were stored in a rectangular tank of 160 l capacity. The pump suction was flooded to avoid cavitations. A gate valve controlled the flow, and two bourdon-tube pressure gauges of 0 to 1 kg/cm2 range were used to measure the pressure at the suction and the delivery ends. A dead weight pressure apparatus calibrated the pressure gauges; and the accuracy of ±1% was obtained.

A pre-calibrated wall mounted ultrasonic flow meter (Adept UFM 6710) with a velocity measurement range of 0–12 m/s was used to measure the flow rate. The accuracy of the instrument was ±1% of reading at rates ≥0.5 and ±0.005 m/s of reading at rates ≤0.5 m/s. The flow meter is suitable for measuring electrical conductive viscous and slurry fluid flows. The flow meter was recalibrated with a measuring tank having a graduated scale. The speed of the pump connected to a three-phase motor was controlled by a variable frequency drive (VFD). Pressure head, flow rate and input power were measured at a constant speed and for different valve openings. A tachometer with accuracy ±1 rpm measured the pump-speed.

## PREPARATION OF NON-NEWTONIAN SLURRY AND STUDY OF RHEOLOGY

The real fluids can be classified into Newtonian and non-Newtonian Newtonian fluids are fluids which obeys Newton’s law of viscosity. Newtonian fluids like water, alcohol or glycerin shows a linear relationship between the applied shear stress and the shear rate. The non- Newtonian fluid such as paint, slurry or higher order hydrocarbons has a non-linear relationship and opposes the Newton’s law of viscosity. The types of non-Newtonian fluids include shear thinning and shear thickening fluids. Viscosity decreases with increase in shear rate in shear thinning fluid, while it increases in shear thickening fluid.

To evaluate the performance of a CENTRIFUGAL pump, the non-Newtonian slurries were prepared and the pump was tested. The slurry was prepared by mixing bentonite and water which is a simplest drilling fluid. Bentonite of 200-mesh size, soda ash and water were mixed, stirred for an hour, and the colloidal solution or the non-Newtonian fluid was prepared. The apparent viscosity of the bentonite slurry increased with time and attained stability by 24 hours, the pump test was carried out after that.

Soda ash, which adds ions in bentonite slurry, dissolved in water before adding bentonite to make the stable slurry. A pH strips gave the pH level of the slurry. The pH was maintained at a range of 8.5–9.5 by adding ~0.06% Na2CO3 (by weight). **Figure 3a **shows that the bentonite settles down from the slurry if a very small amount (2%, by weight) of bentonite is added, while a higher amount (3% and 5%) gives less settlement. Hence, the slurries with 3% and 5% bentonite were used for the pump experiments. In the further discussions: slurry-0 for pure water, slurry-A for 3% bentonite and slurry-B for 5% bentonite slurry terms were used.

A volume of 0.050 m^{3} (**Figure 3b**) slurry was prepared for the experiments and, a sample was taken from it to measure the properties of slurry. A density meter (Anton Paar) and a Viscometer (Brookfield) measured the density and the viscosity of the slurry, respectively. A water bath maintained the temperature at 25 ± 0.1ºC. The densities of slurry-A and slurry-B were 1,014 and 1,025 kg/m^{3}, respectively. In the Viscometer, the shear rate was decreased from 200 to 14.68s^{–}^{1} to get different shear stresses.

The commonly used rheological models in oil industry are Bingham plastic, power law and Newtonian model. Recently, a three variable Herschel-Bulkely model was replaced by Bingham-plastic and power-law models, which are two rheological parameter models.

The statistical approaches such as linear and nonlinear regressions were used to deduce rheological models (**Figure 3**, **Table 1**). The power law and the Herschel-Bulkely models were closer to the rheological experiment and have higher accuracies.

The Bingham plastic model is the simplest rheological model, but accuracy is lesser as compared to the power law and Herschel-Bulkely model. In the Bingham plastic model, a minimum shear stress (yield stress) is required to overcome the yield point so that the fluid starts flowing. Once the yield point has exceeded, the shear stress is directly proportional to shear rate with a slope called plastic viscosity. The plastic viscosity and yield stress increase with increase in the solid concentration. Increase in solid concentration causes increase in the particle- particle contact results in increase in plastics viscosity. The plastics viscosity for 3 and 5% bentonite slurry are 0.0075 and 0.013 Pa.s, respectively. The slurry B is more viscous than the slurry A.

In the power law model, when the value of *n *< 1 the fluid is shear thinning and *n *> 1 the fluid is shear thickening. The values of *n *for slurry-A and slurry-B are 0.34 and 0.13 respectively, defines shear thinning (pseudo plastic) characteristics. A smaller value of *n *(as in slurry-B) represents more shear thinning characteristics compared to slurry-A. Another constant in the power law model is a consistency index (*K*) which is the shear stress or viscosity of the fluid at 1 s1 shear rate. The effective viscosity of the fluid increases with increase in value of *K*. *K *= 0.38 Pa.sn for slurry-A and *K *= 4.26 Pa.sn for slurry-B, shows slurry-B is more viscous than slurry-A. The similar response can be seen in Herschel-Bulkely model is the combination of the Bingham plastic and the power law model.

## PARTICLE DISTRIBUTION OF BENTONITE POWDER

The scanning electron microscopy (SEM) image of bentonite powder at a magnification of 1000× and 500× are depcited in **Figures 4a **and **5a **respectively. SEM produces images of bentonite powder by scanning it with a focused electrons beam. The particles in the image are irregular in shape. Hence the particles are not consider as a circle and considered as an irregular particles. The irregular particle size can be represented by Feret’s diameter which is the longest distance between any two points along the boundary. The Feret’s diameter distribution for bentonite powder at a magnification of 1000× and 500× are shown in **Figures 4b **and **5b**. The SEM image analysis was done in ImageJ, a java based image processing open source program developed at the National Institutes of health (NIH), USA.

The Feret’s diameter of bentonite powder ranges from 0.1 to 80 μm and distribution is not regular. The particle size of the bentonite powder has a direct effect on the apparent viscosity of the bentonite water slurry mixture. For a constant volume fraction of powder, the number of particles is more when the particle size decreases. Increase in number of particle cause more interaction between the particle which lead to resistance of the flow or increase in the apparent viscosity Some studies revealed contradiction to this work the apparent viscosity increase with increase in particle diameter because of inertial effect.

## POLYNOMIAL REGRESSION MODEL

Prediction of head and efficiency of CENTRIFUGAL pumps are essential for the design and optimization and the economic operation and maintenance of a transportation system. The viscosity of fluids changes the pump performance. The HI method is used for the performance prediction of the pumps for viscous Newtonian fluids. In case of non-Newtonian fluids, viscosity varies with shear rate and the performance prediction based on constant average-viscosity with varying shear rate is a difficult task.

There were several literatures available to predict the de-rated head and efficiency of pumps. PR models for head and efficiency have been developed as a function of discharge, slurry viscosity and rotational speed.

### Centrifugal Pump Performance Characteristics

The performance of pump varies, when it delivering slurry which is a multiphase compared to the single phase liquid. Slurry is a mixture of solids and liquid, generally water. The pump performance depends on the CENTRIFUGAL pump configuration and the operating conditions such as speed, viscosity and pressure head. The tests were conducted at *N *= 896, 1280 and 1490 rpm speeds with bentonite slurries and water. The pressure difference (*ΔP*), discharge (*Q*) and input power (*I*) were measured for each pump speed (*N*). To evaluate *H*–*Q *curves (or head-discharge curves), the gate valve was controlled for different *Q *and *H *at constant *N*.

**Figure 6a **shows the *H-Q *curve with *N *= 1490 rpm. At , *Q *= 0, *ΔP *become maximum at a particular speed. Since the CENTRIFUGAL blades are backward curved blades, increase in discharge causes reduction in pressure head for all working fluids. The head drops from slurry-A to slurry-B but the drop increases from slurry-A to slurry-0. At *Q *= 150 m3/day, slurry-A and slurry-B gives the pressure drops of 11.2 and 15.6% relative to slurry-0, respectively. The slurry-0 gives higher pump efficiency (**Figure 6b**). At *Q *= 150 m3/day, the efficiency drop was 25.46 and 35.51% for slurry-A and slurry-B relative to slurry-0, respectively.

The power consumption increases and head drops as the density and viscosity of the fluid increases because of the bentonite added in water. The presence of solid particle and increase in slurry viscosity produce friction losses in the pump impeller. Slury-0 consumes less power than the slurry A and slurry B (**Figure 6c**). For example, 18.26 and 29.33% extra power is required to lift slurry-A and slurry-B respectively, when compared to slury-0. The yield stress of the slurry increase with increase in solid concentration, for slurry A and B the yield stress are τy = 1.03 Pa and τ*y *= 6.28 Pa respectively. The initial starting power required for lifting the slurry is higher for higher yield stress slurries.

The plastic viscosities of slurry-A and slurry-B are 0.0075 and 0.013 Pa.s respectively. The plastic viscosity increases with the increase in bentonite concentration because of higher particle-particle collision. Increase in bentonite concentration causes reduction in pressure head and efficiency and increase in the power required to lift the slurry. The irregular particle shape of bentonite powder (**Figures 4 **and **5**) also cause reduction in efficiency of the pump when compared to rounded shape particles.

### Head and Efficiency Predicting Model

A Matlab code developed for the PR models and experimental data were used. The second-order model gave better fit as compared to the first-order model. The slurry viscosity values were taken from the Bingham plastic model. The regression coefficients are shown in the **Tables 2 **and **3**.

The degree of fitness and performance of the PR models can be observed in **Figures 7 **and **8 **and in **Table 4**. The head and efficiency PR models have a coefficient of determination (*R*^{2}) of 99.57 and 99.18%, respectively. The value of *R**2 *close to 100% implies that the curve is good fit. The relative error for head and efficiency PR models are 3.137 and 5.096%. A higher order polynomial gives higher accuracy, but the equation becomes complex and he number of regression coefficient increases. In the Appendix, the relevant expressions of the PR models are given.

In **Figure 9a**, the head PR model predict the *H *for different viscosities at *N *= 1490 RPM. The slurry viscosity ranges from 0.001 to 0.012 Pa.s and the head drops as the viscosity increases. With increase in speed, the pump produced higher pressure head and discharge (**Figure 9b**).

The *H-Q *curve for different viscosities and speed are shown in **Figure 9**. The efficiency PR model predicts efficiency for different discharges, viscosities and speed ranges. As earlier, the drop in efficiency occurs with increase in viscosity at a constant speed (**Figure 8a**). The efficiency, at a particular slurry viscosity, increases with the increase in the speed and discharge (**Figure 8b**).

The generalized equations (Equations 8 and 9) produced a surface, which indicates the relationships among the depending parameter (**Figure 11**). In **Figure 11a**, the effect of slurry viscosity and discharge in pressure is plotted. Increase in slurry viscosity and discharge cause reduction in the pressure head. In **Figure 11b**, it is observed that the efficiency increases with the increase in the discharge and reduces with increase in slurry viscosity.

## CONCLUSION

The performance characteristics of a centrifugal pump delivering non-Newtonian slurry fluid have been studied through experimental analyses. The fluid was prepared by mixing bentonite, Na2CO3 and water, and the centrifugal pump experiments were carried out in the laboratory. An approximation model was also developed to estimate the pressure head and efficiency as a function of discharge, slurry viscosity and rotational speed. The conclusions of this work are:

- Bingham plastic, power law and Herschel-Bulkely models were used to find the rheological characteristics of the bentonite.
- Increase in concentration of bentonite results in increase of viscosity and density of the slurry mixture than the clear water.
- The presence of solid particle and increase in slurry viscosity produce more friction losses in the system when compared to the clean water. This result in drop in head and efficiency and also increase in the power input of the pump.
- Increase in rotational speed increases the pressure head, discharge, input power and efficiency for all the viscosity range and speed range.
- The PR models can predict head and efficiency as a function of discharge, speed and slurry viscosity with negligible error.

### Nomenclature

MSE *Mean square error*

PR *Polynomial regression*

RMSE *Root mean square error*

RPE *Relative percentage error*

RPM* Revolution per minute*

SEM *Scanning electron microscopy*

SSE *Sum of squares due to error*

SSR *Sum of squares due to regression*

* *

**Table 1. **Rheological constants for different models

Rheological model | y, (Pa) |
ηpl, Pa.s |
K, Pa.sn |
n |
||||

Slurry A | Slurry B | Slurry A | Slurry B | Slurry A | Slurry B | Slurry A | Slurry B | |

Bingham plastic model | 1.03 | 6.28 | 0.0075 | 0.0133 | ||||

Power law model | 0.38 | 4.26 | 0.34 | 0.13 | ||||

Herschel-Bulkely model | 0.350 | 0.79 | 0.20 | 3.55 | 0.44 | 0.15 |

** **

**Table 2. **Regression constants for head-PR model

B0 | B1 | B2 | B3 | B4 |

19.288 | -2.939 | -5.803 | 39.506 | -5.007 |

B5 | B6 | B7 | B8 | B9 |

-0.023 | -6.525 | -10.851 | 4.297 | 9.865 |

**Table 3. **Regression constants for efficiency-PR model

B0 | B1 | B2 | B3 | B4 |

0.362 | 91.767 | -6.652 | 3.761 | -26.641 |

B5 | B6 | B7 | B8 | B9 |

15.008 | -1.02 | -41.43 | 8.325 | -4.408 |

**Table 4. **Comparison of first- and second-order PR model

Parameters | Regression model | |

Head model | Efficiency model | |

Sum of squares due to error (SSE) | 94.54 | 148.20 |

Sum of squares due to regression (SSR) | 16,090.29 | 17,993.41 |

Total sum of squares (SST) | 16,158.44 | 18,141.50 |

Coefficient of determination (R^{2}) |
99.57% | 99.18% |

Adjusted R-square (R_{adj}^{2}) |
0.991 | 0.983 |

Mean square error (MSE) | 1.688 | 1.164 |

Root mean square error (RMSE) | 1.299 | 2.694 |

Relative percentage error (RPE) | 3.137% | 5.096% |