# Simplified Numerical Model for Transient Flow of Slurries at Low Concentration

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Governing Equations

_{i}are the i-element components of the retarded strain in the N-element Kelvin-Voigt model. The time derivative of any i-th component ε

_{i}is equal to [18,24]:

_{i}are creep compliances, τ

_{i}—retardation time values (i = 1,2 … N) and:

_{0}is initial (steady state) pressure, D is internal pipe diameter, e is the pipe wall thickness, and c

_{1}is the coefficient dependent on pipeline fitting.

_{f}represents the time-dependent friction force per unit mass, usually calculated as the sum of the quasi-steady friction losses Δh

_{s}, and the term Δh

_{u}represents the influence of unsteadiness [25]:

_{m}, average mixture velocity v

_{m}) [5,9]. Such an approach is defined as the mixture model and is relatively often used in a variety of practical applications [4,26].

## 3. Slurries Parameters

#### 3.1. Density of the Slurry Mixture

_{L}and solid particles ρ

_{s}, the amount of solid phase in the slurry (which may be expressed by the volume concentration C

_{V}) and the flow velocity. The last two factors mentioned above may additionally vary both in time and space (along the pipeline and within the cross-section), which makes the problem of the mixture density determination not trivial.

_{m}, ρ

_{L}and ρ

_{S}are the densities of the mixture, pure liquid and solids, respectively, and C

_{V}is the volume concentration of solids. If the slurry is non-homogeneous, however, the relation describing the density is additionally affected by the flow velocity and the size of the pipeline. Moreover, all phases observed in the flow (liquid, solids and solid bottom layer, if observed) are characterized by different flow velocities and different friction, which makes the analysis even more complex. Thus, the question arises as to what extent the mixture density in simplified transient flow modeling may be defined by means of one representative and constant value defined by Equation (6). The quality (accuracy) of the results obtained with the use of such an approach becomes essential.

#### 3.2. Transient Wave Celerity

_{m}expressed by Equation (6) and the mixture bulk modulus K

_{m}are used.

_{m}denotes the density of the mixture expressed by Equation (6), K is the bulk modulus of a liquid phase, and E

_{S}is a solid bulk modulus.

_{A}is a fraction of a pipe cross-section area A occupied by a settled bed and C is a volume fraction of solids in the settled slurry bed.

## 4. Experimental Analysis

#### 4.1. Experimental Setup

#### 4.2. General Course and Exemplary Results of Measurements

^{3}, and the densities of the mixtures ranged from 1012 kg/m

^{3}to 1067 kg/m

^{3}, which corresponded to the volume concentrations C

_{V}from 0.007 to 0.039.

_{S}was estimated based on the literature data, assuming relative elastic modulus E

_{S}/E

_{L}for limestone after [2], as equal to 43.0.

_{av}(~0.3 m/s) for water (ρ = 1000 kg/m

^{3}) and slurries of two different concentrations (slurry 1: C

_{V}= 0.007, ρ

_{m}= 1012 kg/m

^{3}; slurry 2: C

_{V}= 0.039, ρ

_{m}= 1067 kg/m

^{3}) are presented in Figure 3.

#### 4.3. Results and Discussion

_{1}, a

_{2}, a

_{3}and a

_{4}were calculated with the use of the Equations (8)–(11) and compared with their experimental equivalents a

_{emp}. Additionally, for a better comparison of theoretical values in a wider range of concentrations, a few hypothetical slurries were also considered. Exemplary values are presented in Table 1 and in Figure 4.

_{V}. That suggests that the actual course of analyzed phenomena is much more complex than the models adopted for deriving theoretical formulas.

## 5. Numerical Simulations

#### 5.1. Numerical Model

- Wave celerity during each episode of water or slurry hammer remains constant;
- Viscoelasticity of the pipeline material is described with a one-element Kelvin-Voigt model with two parameters—retardation time and creep compliance;
- Initial values of pressure and velocity along the pipeline were calculated based on the steady flow equations, with boundary conditions defined by the measured values of steady-state discharge and pressure in the reservoir;
- Boundary conditions were defined as constant pressure values in the reservoir (upstream condition) and the valve closure function (downstream condition).

#### 5.2. Numerical Results and Discussion

_{1}÷ a

_{4}, estimated on the base of theoretical formulas Equations (8)–(11), were applied to conduct the numerical solutions for each analyzed case. The exemplary pressure characteristics compared to experiments are shown in Figure 5.

_{V}. The differences between the characteristics obtained for different theoretical formulas for the wave speed differ to a negligible degree only and were irrelevant to the quality of the solution in the considered case. None of the analyzed theoretical formulas leads to a satisfactory result.

_{emp}were conducted. The exemplary results are shown in Figure 6.

_{eq}was applied in transient flow equations instead of mixture density ρ

_{m}. It could be treated as a conceptual parameter representing all the additional mechanisms affecting the pressure wave oscillations, including the existence of the bottom layer. The value of ρ

_{eq}was thus dependent on the type and size of the particles, slurry concentration and flow velocity. It increased with the concentration increase and steady-state velocity decrease. In the considered slurry hammer cases, the values of ρ

_{eq}were estimated during model calibration as equal to 1076 kg/m

^{3}and 1790 kg/m

^{3}for the slurry of C

_{V}, equal to 0.007 and 0.039, respectively. The exemplary results are shown in Figure 7.

## 6. Summary and Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 3.**Measured pressure characteristics of transient flow: (

**a**) water; (

**b**) slurry, C

_{V}= 0.007; (

**c**) slurry, C

_{V}= 0.039.

**Figure 5.**Pressure characteristics: measured (black line) and calculated with different theoretical values of wave celerity: (

**a**) water; (

**b**) slurry, C

_{V}= 0.007; (

**c**) slurry, C

_{V}= 0.039.

**Figure 6.**Pressure characteristics: measured (blue line) and calculated (with ρ = ρ

_{m}, a = a

_{emp}) (red line): (

**a**) water; (

**b**) slurry, C

_{V}= 0.007; (

**c**) slurry, C

_{V}= 0.039.

**Figure 7.**Pressure characteristics: measured (blue line) and calculated (red line) with use of equivalent density concept: (

**a**) water, ρ

_{eq}= 1000 kg/m

^{3}; (

**b**) slurry, C

_{V}= 0.007, ρ

_{eq}= 1076 kg/m

^{3}; (

**c**) slurry, C

_{V}= 0.039, ρ

_{eq}= 1790 kg/m

^{3}.

Fluid | ρ_{m} (kg m^{−3}) | C_{V} | Wave Celerity (m/s) | ||||
---|---|---|---|---|---|---|---|

a_{1} | a_{2} | a_{3} | a_{4} | a_{emp} | |||

Water | 1000 | 0.000 | 452.2 | 452.2 | 452.2 | 452.2 | 401.0 |

Slurry 1 | 1012 | 0.007 | 449.6 | 449.7 | 451.4 | 450.8 | 385.0 |

Slurry 2 | 1067 | 0.039 | 437.8 | 438.6 | 447.4 | 444.1 | 300.0 |

Slurry 3 * | 1172 | 0.100 | 417.8 | 419.7 | 439.7 | 431.0 | — |

Slurry 4 * | 1686 | 0.400 | 348.3 | 354.8 | 398.3 | 356.8 | — |

Slurry 5 * | 1858 | 0.500 | 331.8 | 339.6 | 382.9 | 327.3 | — |

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**MDPI and ACS Style**

Kodura, A.; Weinerowska-Bords, K.; Kubrak, M.
Simplified Numerical Model for Transient Flow of Slurries at Low Concentration. *Energies* **2022**, *15*, 7175.
https://doi.org/10.3390/en15197175

**AMA Style**

Kodura A, Weinerowska-Bords K, Kubrak M.
Simplified Numerical Model for Transient Flow of Slurries at Low Concentration. *Energies*. 2022; 15(19):7175.
https://doi.org/10.3390/en15197175

**Chicago/Turabian Style**

Kodura, Apoloniusz, Katarzyna Weinerowska-Bords, and Michał Kubrak.
2022. "Simplified Numerical Model for Transient Flow of Slurries at Low Concentration" *Energies* 15, no. 19: 7175.
https://doi.org/10.3390/en15197175