# Precise Method to Estimate the Herschel-Bulkley Parameters from Pipe Rheometer Measurements

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## Abstract

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## 1. Introduction

## 2. Herschel-Bulkley Flow in a Pipe

#### 2.1. Practical Estimation of the Herschel-Bulkley Parameters

#### 2.2. Laboratory Flow Loop

^{3}at 25 °C. This is a non-Newtonian fluid showing no signs of thixotropy and is well described by the Herschel-Bulkley model.

#### 2.3. Rheometer and Data Correction

## 3. Pipe Rheometer

#### 3.1. Background

#### 3.2. Experimental Determination of $\frac{d\mathrm{ln}\left({\dot{\gamma}}_{N,w}\right)}{d\mathrm{ln}\left({\tau}_{w}\right)}$

#### 3.3. Results

## 4. Discussion

## 5. Conclusions

- It is possible to estimate the Herschel-Bulkley rheological behavior parameters utilizing differential pressure measurements along a circular pipe made at different volumetric flow rates. The advantage of using pressure gradients to obtain information about the viscous properties of non-Newtonian fluid is that it does not put any specific requirements on the transparency of the fluid nor the possible negative side effects of diffractions when attempting to measure the fluid velocity field when the fluid contains large proportions of solid particles.
- The method described by Mullineux [14] to calibrate the Herschel-Bulkley parameters based on rheometer measurements, i.e., a series of pairs of shear rate and shear stress at the wall could also be transposed to the context of calibrating the parameters of Herschel-Bulkley fluid utilizing a series of pairs of volumetric flow rate and pressure gradients. The method has the advantage of being also precise at a low shear rate which is not the case of the method based on a logarithm development of the difference of the shear and yield stresses [12].
- The obtained precision of the calibrated parameters is of the same order of magnitude as the one obtained with a scientific rheometer.
- The calibration method based on the method from Mullineux is simple enough to be implemented on real-time computer systems such as single-board computers or programmable logic controllers, therefore allowing the possibility to devise real-time equipment that can measure continuously the rheological behavior of non-Newtonian fluids that follow the Herschel-Bulkley rheological behavior.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

$\frac{dP}{dL}$ | pressure gradient [ML^{−2}T^{−2}] (Pa/m) |

$K$ | consistency index [ML^{−1}T^{n−2}] (Pa.s^{n}) |

${K}_{M}$ | consistency index obtained with the Mullineux method based on the flow-loop measurements [ML^{−1}T^{n−2}] (Pa.s^{n}) |

${K}_{L-M}$ | consistency index obtained with the Levenberg-Marquart optimization [ML^{−1}T^{n−2}] (Pa.s^{n}) |

${K}_{Rheo}$ | consistency index obtained with the Mullineux method based on the scientific rheometer measurements [ML^{−1}T^{n−2}] (Pa.s^{n}) |

$m$ | number of measurements |

$n$ | flow behavior index (dimensionless) |

${n}_{M}$ | flow behavior index obtained from the Mullineux method based on the flow-loop measurements (dimensionless) |

${n}_{L-M}$ | flow behavior index obtained with the Levenberg-Marquart optimization (dimensionless) |

${n}_{Rheo}$ | flow behavior index obtained with the Mullineux method on the scientific rheometer measurements (dimensionless) |

$p$ | number of bins in the non-flowing condition histogram |

$Q$ | volumetric flowrate [L^{3}T^{−1}](m^{3}/s) |

$R$ | pipe radius [L](m) |

$S$ | least square error [M^{2}L^{−2}T^{−4}] (Pa^{2}) |

$\dot{\gamma}$ | shear rate [T^{−1}](1/s) |

${\dot{\gamma}}_{i}$ | measured shear rate [T^{−1}](1/s) |

${\dot{\gamma}}_{N,w}$ | wall shear for a Newtonian fluid [T^{−1}](1/s) |

$\epsilon $ | shear stress span for a bin in the non-flowing condition histogram [ML^{−1}T^{−2}] (Pa) |

$\tau $ | shear stress [ML^{−1}T^{−2}] (Pa) |

${\tau}_{i}$ | measured shear stress [ML^{−1}T^{−2}] (Pa) |

${\tau}_{\gamma}$ | yield stress [ML^{−1}T^{−2}] (Pa) |

${\tau}_{\gamma ,M}$ | yield stress obtained with the Mullineux method based on the flow-loop measurements [ML^{−1}T^{−2}] (Pa) |

${\tau}_{\gamma ,L-M}$ | yield stress obtained with the Levenberg-Marquart optimization [ML^{−1}T^{−2}] (Pa) |

${\tau}_{\gamma ,rheo}$ | yield stress obtained from scientific rheometer measurements [ML^{−1}T^{−2}] (Pa) |

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**Figure 1.**Schematic description of the laboratory flow loop (courtesy of Cayeux and Leulseged in [16]).

**Figure 2.**Comparison of the effect of utilizing the Newtonian-fluid shear rate conversion instead of the non-Newtonian shear rate conversion of the rheometer speed for a rheogram obtained with an aqueous solution of Carbopol 980 having a density of 997 kg/m

^{3}at 25 °C.

**Figure 3.**Graph of $\mathrm{ln}\left({\dot{\gamma}}_{N,w}\right)$ as a function of $\mathrm{ln}\left({\tau}_{w}\right)$, shown in red dots where ${\dot{\gamma}}_{N,w}$ is expressed in reciprocal seconds and ${\tau}_{w}$ in Pascals. The blue line is a linear regression of the 25% highest ${\tau}_{w}$ data points $\left({\tau}_{w}\ge 11.2Pa\right)$. The slope is 1.67, therefore the estimated $n=0.60$, which is an acceptable initial approximation considering that the value obtained with the scientific rheometer was $0.64$.

**Figure 4.**Estimations of ${\tau}_{\gamma}$ looking at pressure losses at a low flow rate. (

**a**) Pressure losses at a low flow rate are not steady since it depends on the flow history of the fluid. (

**b**) This histogram shows the percentage of occurrences of non-flowing state at a given wall shear stress ${\tau}_{w}$ up to the stress overshoot value.

**Figure 5.**Estimations of $K$ using 10% of the measured $\left(Q,\frac{dP}{dL}\right)$ pair data. The final estimation is obtained by taking the median of the calculations. Here, $K=0.219\mathrm{Pa}.{\mathrm{s}}^{\mathrm{n}}$, which is an acceptable initial value considering that the consistency index measured with the scientific rheometer was $0.272\mathrm{Pa}.{\mathrm{s}}^{\mathrm{n}}$.

**Figure 6.**Measured and calculated pressure losses as a function of the flow rate. Curves named Rheometer, Levenberg-Marquart, and Mullineux are the result of a calculation using the H-B parameters estimated by the technique referring to their name and inverting Equation (8).

**Figure 7.**Estimation of $\frac{d\mathrm{ln}\left({\dot{\gamma}}_{N,w}\right)}{d\mathrm{ln}\left({\tau}_{w}\right)}$ using Equation (12) and the method corresponding to the legend for the first three curves and using a polynomial fit of $\frac{\mathrm{ln}\left({\dot{\gamma}}_{N,w}\right)}{\mathrm{ln}\left({\tau}_{w}\right)}$ in the order mentioned in the legend. ${\dot{\gamma}}_{N,w}$ is expressed in reciprocal seconds and ${\tau}_{w}$ in Pascals.

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**MDPI and ACS Style**

Magnon, E.; Cayeux, E.
Precise Method to Estimate the Herschel-Bulkley Parameters from Pipe Rheometer Measurements. *Fluids* **2021**, *6*, 157.
https://doi.org/10.3390/fluids6040157

**AMA Style**

Magnon E, Cayeux E.
Precise Method to Estimate the Herschel-Bulkley Parameters from Pipe Rheometer Measurements. *Fluids*. 2021; 6(4):157.
https://doi.org/10.3390/fluids6040157

**Chicago/Turabian Style**

Magnon, Elie, and Eric Cayeux.
2021. "Precise Method to Estimate the Herschel-Bulkley Parameters from Pipe Rheometer Measurements" *Fluids* 6, no. 4: 157.
https://doi.org/10.3390/fluids6040157