# Numerical Reliability Study Based on Rheological Input for Bingham Paste Pumping Using a Finite Volume Approach in OpenFOAM

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Framework

#### 2.1. Numerical Methodology

#### 2.2. Numerical Regularisation

#### 2.3. Numerical Control

#### 2.4. Numerical Cases

#### 2.5. Mesh Independence

## 3. Experimental Framework

#### 3.1. Rheometry

#### 3.1.1. Rotational Rheometry

#### 3.1.2. Sliding Pipe Rheometry

#### 3.2. Small-Scale Pumping

## 4. Reliability Irrespective of Rheological Input

#### 4.1. Comparison with the Literature

^{®}, with the re-simulated ones performed in this work by OpenFOAM. No under-relaxation was considered, and a more strict regularisation was applied ($GNAR\approx 1500$). The error is presented relative to the expected Buckingham–Reiner pressure loss.

^{®}.

#### 4.2. Comparison with the Theory

## 5. Reliability Dependent on Rheological Input

#### Comparison with Experiments

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Simulation Settings

Discretisation | Field | Scheme |
---|---|---|

ddtSchemes | default | Gauss linear |

gradSchemes | default | Gauss linear |

grad(p) | leastSquares | |

divSchemes | default | none |

div(phi,U) | Gauss linear corrected | |

laplacianSchemes | default | none |

laplacian(nu,U) | Gauss linear corrected | |

laplacian(1|A(U),p) | Gauss linear corrected | |

interpolationSchemes | default | linear |

snGradSchemes | default | corrected |

Application | nonNewtonianIcoFoam | |
---|---|---|

Pressure-Velocity Coupling | PISO | |

nCorrectors | 5 | |

nNonOrthogonalCorrectors | 0 | |

Fields | ||

Solver Settings | p | U |

Solver | GAMG | smoothSolver |

Smoother | GaussSeidel | symGaussSeidel |

Tolerance | 1 × 10^{−6} | 1 × 10^{−6} |

Relative Tolerance | $0.1$ | 0 |

Final Relative Tolerance | 0 | |

Relaxation Factor | $1.0$ | $0.95$ |

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**Figure 2.**Illustrations of the numerical outcome. Examples of the kinematic pressure field p (m${}^{2}$/s${}^{2}$) and the x-component velocity field ${U}_{x}$ (m/s) are shown for both the DN100 and DN25 pipe cases. (

**a**) Pressure field p for a DN100 pipe. (

**b**) Velocity field ${U}_{x}$ for a DN100 pipe. (

**c**) Pressure field p for a DN25 pipe. (

**d**) Velocity field ${U}_{x}$ for a DN25 pipe.

**Figure 3.**Hexagonal cross-sectional mesh configuration, with an inner square (${n}_{yA}$×${n}_{zA}$) with diagonal $2b$ and outer circle segments (${n}_{zA}$×${n}_{zD}$) to pipe radius R. (

**a**) Cross-sectional view. (

**b**) Three-dimensionally rendered mesh view of a DN100 pipe. (

**c**) Rescaled mesh of experimental pumping circuit, with an equal radial mesh resolution for a DN25 pipe.

**Figure 4.**Rheometer test set-up configurations: (

**a**) rotational parallel plate rheometer configuration (Anton Paar MCR-102 with a 1 mm gap); (

**b**) sliding pipe rheometer (Sliper).

**Figure 5.**Sliper pressure loss recorded as a function of the discharge for all conducted experiments (mix design A2 to A4). No feasible results could be obtained for mix design A1. Rheological parameters were obtained from the linear regressions.

**Figure 6.**Example of a pressure sensor being screwed in a T-connector as close as possible to the virtual pipe wall.

**Figure 7.**Layout of a small-scale pumping circuit. Each number corresponds to a pressure sensor position. At the inlet, the paste was injected via a pressure barrel, for which the outlet discharge ${Q}_{out}$ was measured by a load cell.

**Figure 8.**Literature comparison of the numerical concrete pipe flow simulations performed by Tichko [13], relative to the expected Buckingham–Reiner theory. The results indicate that OpenFOAM performs at least as good as, if not better than, commercial CFD software ANSYS FLUENT

^{®}.

**Figure 9.**A good agreement is obtained between the simulated pressure loss and the expected theoretical Buckingham–Reiner theory.

**Figure 10.**Using a dimensionless form of the Bingham Poiseuille flow, again, a good agreement is achieved between the simulated pressure loss and the expected Buckingham–Reiner theory. Indeed, the simulations map onto a single Bingham discharge curve.

**Figure 11.**Significantly accurate simulations can be obtained in comparison with the Buckingham–Reiner theory. For higher discharges, a relative pressure loss error below $1\%$ can be obtained. Simulations with lower discharges or higher yield stresses can be more influenced by the unregularised viscoplastic problem [25,43].

**Figure 12.**The relative simulation pressure loss error (compared to Buckingham–Reiner) as a function of the pressure number $Pn$ reveals the influence of the flow regime or unregularised viscoplastic problem. The accuracy decreases for lower pressure numbers or for more yield stress dominant flow regimes.

**Figure 13.**Plotting the relative simulation error as a function of the dimensionless discharge $\widehat{Q}$ also shows that low discharges or more yield stress dominated pipe flows impede numerical accuracy.

**Figure 14.**The relative pressure loss error as a function of the inverse pressure number is equivalent to the degree of plug formation. The ratio of plug radius ${R}_{p}$ to pipe radius R equals the inverse pressure number $1/Pn$.

**Figure 15.**Comparison of numerically simulated pressure loss with experimentally obtained pumping results, for four different mix designs (A1 to A4) and based on rheological input from the MCR-102 rheometer and the Sliper.

**Figure 16.**The relative simulation pressure loss error compared to the pumping experiment series indicates that the rheolocigal input from the Sliper is in better agreement than that based on MCR-102.

**Figure 17.**A close-up of the comparison between numerical simulations and pumping experiments shows that the rheological input based on Sliper is significantly better than that for MCR-102, with relative errors ranging from 0% to ca. 100%.

**Table 1.**Pipe flow boundary conditions of the velocity field U and the pressure field p. Herein, Q is the imposed discharge, uniformly distributed over the inlet surface A.

Boundary | Field | Type | Definition | |
---|---|---|---|---|

Inlet | U | Dirichlet | uniform value | $U=Q/A$ |

p | Neumann | zero gradient | $\nabla p=0$ | |

Wall | U | Dirichlet | noSlip | $U=0$ |

p | Neumann | zero gradient | $\nabla p=0$ | |

Outlet | U | Neumann | zero gradient | $\nabla U=0$ |

p | Dirichlet | zero value | $p=0$ |

**Table 2.**Conceptual overview of rheological mixture designs of the considered four Bingham model pastes (A1 to A4), as a function of the water (W), powder (P) and superplasticiser (SP) content.

Mix Designs | Plastic Viscosity $\mathit{\mu}$ | ||||
---|---|---|---|---|---|

Low | High | ||||

Yield Stress ${\mathit{\tau}}_{0}$ | Mix | $\mathit{W}/\mathit{P}$ | Mix | $\mathit{W}/\mathit{P}$ | $\mathbf{SP}/\mathit{P}$ |

Low | A1 | $0.40$ | A2 | $0.20$ | $0.175\%$ |

High | A3 | $0.33$ | A4 | $0.25$ | $0.100\%$ |

**Table 3.**Rheological overview of several mixture design samples, depicting the paste density $\rho $, Bingham yield stress ${\tau}_{0}$ and plastic viscosity $\mu $ obtained from different rheometers.

MCR-102 | Sliper | ||||
---|---|---|---|---|---|

Mixture | $\mathbf{\rho}$ | ${\mathbf{\tau}}_{\mathbf{0}}$ | $\mathbf{\mu}$ | ${\mathbf{\tau}}_{\mathbf{0}}$ | $\mathbf{\mu}$ |

Sample | [kg/L] | [Pa] | [Pa.s] | [Pa] | [Pa.s] |

A1-1 | 1.776 | 55.5 | 0.5 | N/A | N/A |

A1-2 | 1.786 | 69.7 | 0.7 | N/A | N/A |

A1-3 | 1.792 | 67.7 | 0.7 | N/A | N/A |

A1-4 | 1.794 | 94.1 | 0.9 | N/A | N/A |

A2-1 | 2.035 | 57.2 | 8.5 | 44.3 | 4.6 |

A2-2 | 2.075 | 65.7 | 9.1 | 32.2 | 3.6 |

A2-3 | 2.047 | 43.6 | 10.3 | 62.8 | 4.6 |

A3-1 | 1.903 | 150.5 | 2.6 | 73.4 | 0.6 |

A3-2 | 1.865 | 165.6 | 2.7 | 76.3 | 0.6 |

A3-3 | 1.896 | 181.1 | 2.8 | 79.0 | 0.5 |

A4-1 | 1.980 | 135.1 | 4.6 | 23.0 | 2.2 |

A4-2 | 1.979 | 129.3 | 4.8 | 67.3 | 2.3 |

A4-3 | 1.981 | 110.7 | 5.0 | 65.6 | 2.2 |

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**MDPI and ACS Style**

De Schryver, R.; El Cheikh, K.; Lesage, K.; Yardimci, M.Y.; De Schutter, G.
Numerical Reliability Study Based on Rheological Input for Bingham Paste Pumping Using a Finite Volume Approach in OpenFOAM. *Materials* **2021**, *14*, 5011.
https://doi.org/10.3390/ma14175011

**AMA Style**

De Schryver R, El Cheikh K, Lesage K, Yardimci MY, De Schutter G.
Numerical Reliability Study Based on Rheological Input for Bingham Paste Pumping Using a Finite Volume Approach in OpenFOAM. *Materials*. 2021; 14(17):5011.
https://doi.org/10.3390/ma14175011

**Chicago/Turabian Style**

De Schryver, Robin, Khadija El Cheikh, Karel Lesage, Mert Yücel Yardimci, and Geert De Schutter.
2021. "Numerical Reliability Study Based on Rheological Input for Bingham Paste Pumping Using a Finite Volume Approach in OpenFOAM" *Materials* 14, no. 17: 5011.
https://doi.org/10.3390/ma14175011