# Different Experimental and Numerical Models to Analyse Emptying Processes in Pressurised Pipes with Trapped Air

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## Abstract

**:**

## 1. Introduction

## 2. Numerical Modelling in Irregular Pipelines

#### 2.1. Mathematical Modelling in Irregular Pipelines

#### 2.2. Two-Dimensional CFD Model

#### 2.3. 2D CFD Model vs. Mathematical Model

## 3. Numerical Modelling in Single Pipelines

#### 3.1. Mathematical Model of Single Pipelines (Cases 1 and 2)

#### 3.2. Three-Dimensional CFD Model

#### 3.3. 3D CFD Model vs. Mathematical Model

#### Comparison of 2D/3D CFD Models

## 4. Discussion

## 5. Conclusions

- Mathematical models based on the rigid column model are a fundamental tool for understanding the physical evolution of several hydraulic and thermodynamic variables based on differential equations that can be manipulated by the user according to the hydraulic scenario that is to be studied. These models have been validated in different investigations, showing good agreement with experimental measurements, and, additionally, they have been applied in large-scale hydraulic scenarios [19,36];
- A 2D CFD model allows study of the emptying process of an irregular pipeline with an air valve. The presence of air valves facilitates the air admission process during the water drainage process and mitigates subatmospheric pressures. In this sense, 2D CFD models are suitable for a simplified analysis of the interaction between air and water, with acceptable numerical and spatial resolution. The use of 2D CFD models is useful in cases where numerical information associated with physical parameters, such as velocity, pressure and heat transfer at different points of the pipeline, and visual information about the interaction between air and water are needed. These models require less computational time than a 3D CFD model and must be used with restrictions for the study of the emptying of pipes with air valves, when section changes occur (contractions, reductions), with the purpose of adjusting the mass flow conditions by means of a geometric aspect ratio defined in the literature for the sizing of orifices in pipes;
- A 3D CFD model allows simulation of the drainage of a single pipe in two cases (without air valves and with air valves). In Case 1, adverse subatmospheric pressures are generated due to entrapped air pockets and water flow oscillations. On the other hand, in Case 2, fewer critical pressure oscillations are observed compared to Case 1, due to the influence of the air admission orifices. All CFD model results in both cases were compared with the patterns recorded by the mathematical model, generally proposed by Fuertes-Miquel et al. [18]. It was observed that 3D CFD models showed hydraulic–thermodynamic phenomena that cannot be obtained in 2D CFD models, such as velocity distribution in pipe cross-sections, transient flows in pipe cross-sections, small air pockets created due to backflow air effect and their influence on water drainage flow transitions. In addition, there is the presence of a phenomenon that has not been studied in the literature, which is the temperature gradients between water and air phases, considering that the air pockets present a decrease in temperature due to the subatmospheric pressures and the water phase influences in heat transfer.

#### Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Notation

A | pipe cross section (m${}^{2}$) |

${C}_{adm}$ | admission coefficient (–) |

${d}_{adm}$ | admission orifice diameter (m) |

${D}_{k/\omega}$ | diffusivity terms for k and/or $\omega $ (m${}^{2}$/s) |

e | specific energy (J/kg) |

f | friction factor (–) |

${F}_{s}$ | surface tension (kg/s${}^{2}$) |

g | gravitational acceleration (m/s${}^{2}$) |

$\mathbf{g}$ | gravitational acceleration vector (m/s) |

k | turbulent kinetic energy (m${}^{2}$/s${}^{2}$) |

${L}_{iap}$ | initial air pocket length (m) |

${L}_{ap}$ | air pocket length (m) |

${L}_{e}$ | water column length (m) |

m | mass (kg) |

p | static pressure (N/m${}^{2}$) |

$\mathbf{q}$ | heat flux vector (w/m${}^{2}$) |

Q | flow rate (m${}^{3}$/s) |

${R}_{e}$ | Reynolds number (–) |

${R}_{v}$ | ball valve resistance coefficient (m s${}^{2}$/m${}^{6}$) |

R | universal gas constant (kg·m${}^{2}$s${}^{-2}$K${}^{-1}$mol${}^{-1}$) |

${S}_{k/\omega}$ | source term of k or $\omega $ |

T | temperature (°C) |

${t}_{m}$ | ball valve opening time (s) |

t | time (s) |

u | velocity (m/s) |

$\mathbf{u}$ | velocity vector (m/s) |

${u}_{r}$ | velocity source (m/s) |

${u}_{e}$ | water column velocity (mathematical model) (m/s) |

V | volume (m${}^{3}$) |

${\alpha}_{w}$ | phase fraction of water (–) |

$\kappa $ | polytropic coefficient (–) |

$\mu $ | dynamic viscosity (kg/(ms) |

$\nu $ | kinematic viscosity (kg/ms) |

$\rho $ | density (kg/m${}^{3}$) |

$\varphi $ | inner pipe diameter (m) |

$\tau $ | ball valve opening degree (%) |

$\omega $ | dissipation frequency (s${}^{-1}$) |

Subscripts | |

a | refers to air phase (e.g., air density) |

$adm$ | refers to air admission orifice (e.g., cross section of air admission orifice) |

$atm$ | refers to atmospheric conditions (e.g., atmospheric pressure) |

$nc$ | refers to normal conditions (e.g., air density in normal conditions) |

w | refers to water phase (e.g., water density) |

SST k–$\omega $ coefficients | |

$C{D}_{k\omega}$ | closure coefficient of k and $\omega $ |

$\gamma $ | refers to air phase (e.g., air density) |

$\beta $ | mass and energy transfer constant |

${\beta}^{*}$ | turbulence transport constant |

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**Figure 3.**Comparison of pressure patterns of (

**a**) Test 1-1, (

**b**) Test 1-2, (

**c**) Test 1-3 and (

**d**) Test 1-4 (Experimental model, 2D CFD model and mathematical model).

**Figure 4.**Comparison of velocity patterns of (

**a**) Test 1-1, (

**b**) Test 1-2, (

**c**) Test 1-3 and (

**d**) Test 1-4 (Experimental model, 2D CFD model and mathematical model).

**Figure 10.**Comparison of pressure patterns of (

**a**) Test C1-1, (

**b**) Test C1-2, (

**c**) Test C2-1 and (

**d**) Test C2-2 (3D CFD model vs. mathematical model).

**Figure 11.**Comparison between water flow patterns during emptying processes of a single pipe. (

**a**) Test C1-1, (

**b**) Test C1-2, (

**c**) Test C2-1 and (

**d**) Test C2-2.

**Figure 14.**Heat transfer between the water phase and the air phase. (

**a**) Test C1-1 and (

**b**) Test C1-2.

**Figure 16.**Comparison of air–water interaction of Test C2-1: (

**a**) 2D CFD Model [23] and (

**b**) 3D CFD Model.

**Figure 17.**Plane for evaluation of velocity profile in single pipe—Test C2-1: (

**a**) Normal 3D CFD model and (

**b**) Cut-out of 3D CFD model.

**Figure 18.**Comparison of velocity contours at upper end of Test C2-1: (

**a**) 2D CFD Model [23] and (

**b**) 3D CFD Model.

Parameter | Test 1-1 | Test 1-2 | Test 1-3 | Test 1-4 |
---|---|---|---|---|

${L}_{iap}$ (m) | null | 0.54 | 0.92 | 1.32 |

Mass Oscillation No. 1 | $\frac{d{u}_{e,1}}{dt}$ = $\frac{{p}_{a}-{p}_{atm}}{{\rho}_{w}{L}_{e,1}}$ + $\mathit{g}\frac{\Delta {z}_{e,1}}{{L}_{e,1}}$ − ${f}_{1}\frac{{u}_{e,1}\left|{u}_{e,1}\right|}{2\varphi}$ − $\frac{{R}_{v}g{{A}_{1}}^{2}{u}_{e,1}\left|{u}_{e,1}\right|}{{L}_{e,1}}$ |

Emptying Column No. 1 | $\frac{d{L}_{e,1}}{dt}$ = $-{u}_{e,1}({L}_{e,1,0}-{\int}_{0}^{t}\phantom{\rule{-0.166667em}{0ex}}{u}_{e,1}\phantom{\rule{0.166667em}{0ex}}dt)$ |

Mass Oscillation No. 2 | $\frac{d{u}_{e,2}}{dt}$ = $\frac{{p}_{a}-{p}_{atm}}{{\rho}_{w}{L}_{e,2}}$ + $g\frac{\Delta {z}_{e,2}}{{L}_{e,2}}$ − ${f}_{2}\frac{{u}_{e,2}\left|{u}_{e,2}\right|}{2\varphi}$ − $\frac{{R}_{v}g{{A}_{2}}^{2}{u}_{e,2}\left|{u}_{e,2}\right|}{{L}_{e,2}}$ |

Emptying Column No. 2 | $\frac{d{L}_{e,2}}{dt}$ = $-{u}_{e,2}({L}_{e,2,0}-{\int}_{0}^{t}\phantom{\rule{-0.166667em}{0ex}}{u}_{e,2}\phantom{\rule{0.166667em}{0ex}}dt)$ |

**Table 3.**Mathematical equations for boundaries of the CFD model [26].

Variable | Inlet | Outlet | Walls |
---|---|---|---|

p (Pa) | ${p}_{inlet}$ = ${p}_{atm}$ − 0.5$\rho {u}^{2}$ | ${p}_{outlet}$ = ${p}_{atm}-0.5\rho {u}^{2}$ − $\rho g\Delta h$ | ${p}_{walls}$ = ${p}_{atm}$ |

u (m/s) | ${u}_{inlet}$ = ${u}_{inlet,0}$ − $\frac{{p}_{inlet}-{p}_{inlet,0}}{\rho {u}_{inlet,0}}$ | ${u}_{outlet}$ = ${u}_{outlet,0}$ − $\frac{{p}_{outlet}-{p}_{outlet,0}}{\rho {u}_{outlet,0}}$ | ${u}_{walls}$ = 0 |

**Table 4.**Maximum water drainage velocities and Reynolds number obtained in the emptying tests of an irregular pipe (${u}_{max}$ in m/s).

Test | ${\mathit{u}}_{\mathbf{max}}$ (Exp.) | ${\mathit{R}}_{\mathit{e}}$ (Exp.) | ${\mathit{u}}_{\mathbf{max}}$ (Math.) | ${\mathit{R}}_{\mathit{e}}$ (Math.) | ${\mathit{u}}_{\mathbf{max}}$ (CFD) | ${\mathit{R}}_{\mathit{e}}$ (CFD) |
---|---|---|---|---|---|---|

1-1 | 0.391 | 20,097 | 0.398 | 20,467 | 0.412 | 21,198 |

1-2 | 0.317 | 16,294 | 0.358 | 18,395 | 0.358 | 18,410 |

1-3 | 0.313 | 16,088 | 0.325 | 16,681 | 0.323 | 16,622 |

1-4 | 0.261 | 13,415 | 0.286 | 14,705 | 0.282 | 14,486 |

Parameter | Test C1-1 | Test C1-2 | Test C2-1 | Test C2-2 |
---|---|---|---|---|

${L}_{iap}$ (m) | 0.45 | 0.205 | 0.205 | 0.45 |

${t}_{m}$ (s) | 0.50 | 0.30 | 0.50 | 0.40 |

$\tau $ (%) | 12.0 | 6.0 | 24.5 | 13.4 |

${d}_{adm}$ (mm) | – | – | 3.0 | 3.0 |

Mass oscillation | $\frac{d{u}_{e}}{dt}$ = $\frac{{p}_{a}-{p}_{atm}}{{\rho}_{w}{L}_{e}}$ + $g\frac{\Delta {z}_{e}}{{L}_{e}}$ − $f\frac{{u}_{e}\left|{u}_{e}\right|}{2\varphi}$ − $\frac{{R}_{v}g{A}^{2}{u}_{e}\left|{u}_{e}\right|}{{L}_{e}}$ |

Water column length | $\frac{d{L}_{e}}{dt}$ = $-{u}_{e}({L}_{e,0}-{\int}_{0}^{t}\phantom{\rule{-0.166667em}{0ex}}{u}_{e}\phantom{\rule{0.166667em}{0ex}}dt)$ |

**Table 7.**Maximum water drainage velocities and Reynolds numbers obtained in emptying tests of a single pipe (${Q}_{w,max}$ in units of L/s).

Test | ${\mathit{Q}}_{\mathit{w},\mathbf{max}}$ (Math.) | ${\mathit{R}}_{\mathit{e}}$ (Math.) | ${\mathit{Q}}_{\mathit{w},\mathbf{max}}$ (CFD) | ${\mathit{R}}_{\mathit{e}}$ (CFD) |
---|---|---|---|---|

C1-1 | 0.286 | 8682 | 0.223 | 6760 |

C1-2 | 0.088 | 2677 | 0.110 | 3335 |

C2-1 | 0.636 | 19,285 | 0.632 | 19,183 |

C2-2 | 0.490 | 14,871 | 0.516 | 15,660 |

**Table 8.**Advantages and disadvantages for model development for simulation of emptying processes of pipes with trapped air.

Model | Advantages | Disadvantages |
---|---|---|

Mathematical model (1D) | Provide accurate and adequate numerical information, where variables such as (i) entrapped air pocket pressure, (ii) water drainage velocity, (iii) water flow rate, (iv) air admission flow rate (in the case of presence of air valves) and (v) temperature can be studied through numerical values that vary as a function of time. | These models do not represent different events that occur during emptying processes, such as (i) backflow air effects, (ii) deformation of the air–water interface and (iii) velocity and temperature gradients. |

These models are easy to implement and require fewer computational resources; their results are obtained in a few seconds. | The water column is considered a rigid uniform column; water velocity is equal at different points in the water column. | |

The modelling equations can be manipulated. | ||

2D CFD Model | The simplicity of these models allows the study of physical parameters associated with fluids such as pressure, velocity and temperature with adequate numerical accuracy. | Two-dimensional solution restricts the possibility to analyse hydraulic parameters such as water and/or air-flow rate (in the case of pipes with air valves). |

A 2D CFD model shows the variations in physical quantities (pressure, temperature, velocity) at different points of the hydraulic system by means of contours. | Contractions and expansions in pipes must be adjusted by geometric aspect ratios to obtain a guarantee of equivalent mass flow [22,34]. | |

Allows the visualisation of physical phenomena associated with water and air-flows, such as (i) deformation of the air–water interface and (ii) backflow air [7,20]. | Simulations require more computational resources in comparison with mathematical models. | |

The effect of opening drain valves can be simulated by means of a dynamic mesh [21]. | ||

On multi-core processors, simulation times are significantly reduced. | ||

3D CFD Model | The 3D CFD models show complete information on the physical, hydraulic and thermodynamic phenomena associated with transient flows in pipes with entrapped air during emptying events, simulating real conditions. | The definition of mesh quality, boundary conditions and numerical schemes are more difficult to calibrate, and require an independent analysis of the numerical results on the spatial discretisation conditions. |

In addition to the backflow air and the deformation of the air–water interface, this model allows obtaining new information such as (i) water flow rates, (ii) transient flows in the pipe cross-sectional, (iii) velocity profiles in different planes, (iv) heat transfer between water and air and (v) analysis of flow parameters with contour view in cross-sections [8,24]. | Simulation times on high-performance processors are on the order of hours or days. | |

The simulation of the drain valve approximates the real conditions in more detail. | These models require more significant computational resources than the other models. |

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## Share and Cite

**MDPI and ACS Style**

Paternina-Verona, D.A.; Coronado-Hernández, O.E.; Espinoza-Román, H.G.; Fuertes-Miquel, V.S.; Ramos, H.M.
Different Experimental and Numerical Models to Analyse Emptying Processes in Pressurised Pipes with Trapped Air. *Appl. Sci.* **2023**, *13*, 7727.
https://doi.org/10.3390/app13137727

**AMA Style**

Paternina-Verona DA, Coronado-Hernández OE, Espinoza-Román HG, Fuertes-Miquel VS, Ramos HM.
Different Experimental and Numerical Models to Analyse Emptying Processes in Pressurised Pipes with Trapped Air. *Applied Sciences*. 2023; 13(13):7727.
https://doi.org/10.3390/app13137727

**Chicago/Turabian Style**

Paternina-Verona, Duban A., Oscar E. Coronado-Hernández, Hector G. Espinoza-Román, Vicente S. Fuertes-Miquel, and Helena M. Ramos.
2023. "Different Experimental and Numerical Models to Analyse Emptying Processes in Pressurised Pipes with Trapped Air" *Applied Sciences* 13, no. 13: 7727.
https://doi.org/10.3390/app13137727