# Experimental Investigation of Fluid Flow through Zinc Open-Cell Foams Produced by the Excess Salt Replication Process and Suitable as a Catalyst in Wastewater Treatment

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### Flow Laws

_{C}) is defined by Equation (1)

^{−3}) in this study, v is the superficial velocity (known also as seepage velocity or Darcy velocity) (m s

^{−1}), l is the characteristic length (D

_{C}in this case) (m), μ is the dynamic viscosity of air (kg m

^{−1}s

^{−1}), and ν is the kinematic viscosity of air (m²s

^{−1}).

^{2}) is the Darcy’s permeability of the foam of length L (m). It is obvious that this relation is linear in terms of v but only for lower velocities (v < 1 m s

^{−1}); then permeability can be deduced knowing μ. Creeping flows are difficult to sustain in the laboratory, and measurements of velocity and pressure drop are challenging, as these quantities may be too small to capture with good accuracy using common instruments, as stated by Dukhan [20], which is the case in this study.

^{−1}), and ρ is the fluid (air) density in (kg m

^{−3}). The coefficient β represents the blockage of the internal structure (ligaments, nodes, and occasional closed faces of open-cell metal foam) to the flow.

^{2}), C is the weak inertia factor or dimensionless C—cubic factor (-). Weak inertia is a regime in which the inertial force is of the same order as the viscous force.

^{3}, claimed that the necessity of using a cubic transition equation in the weak inertia regime, where the Forchheimer equation is often observed to fail, was quite possibly due to the fact that 𝛾 > 1 for certain pore space morphologies (while 𝛾 was a free parameter dictating the effective width of inertial transition).

^{3}m

^{−4}). They found that the permeability and the inertia coefficients were the same as the ones obtained interpolating the low velocity range. Thus, they concluded that the fitting of the high-velocity range was affected only by the cubic term.

_{1}and C

_{2}are dimensionless experimental parameters.

- The flow is horizontal and uniform all along the main flow z-direction (Figure 2);
- Air is treated as a monophasic Newtonian fluid;
- The dense wall effect in our case is neglected because the tested foams are relatively dense [2];
- The static gas pressure was constant throughout the porous medium and was equal to the difference between the inlet and outlet pressures of the test section.

## 3. Experimental Process

^{−3}.

^{−1}depending on the flow sensors threshold. This is the range used in most practical applications [45].

## 4. Results and Discussion

#### 4.1. Description of Air Flow through ESR Foams

_{C}), length (L) in the main flow direction, and porosity (ε). Thus, the choice of each sample with a special combination of parameters (D

_{C}, L, and ε) was made on purpose. The pressure loss decreases with increased salt grain size (D

_{C}) and porosity (ε) of the samples [48], which supposes that permeability also increases in this case. However, when length (L) increases, the pressure drop increases as well. This was also confirmed by Paek et al. [49], who showed that as the cell size D

_{C}of a foam metal decreases, the surface-area-to-volume ratio increases, and this causes additional flow resistance, thus resulting in an increased pressure drop. The latter is substantially influenced by the void fraction as well as the shape of the cell of the metal foam. The effect of length (L) on pressure drop behaviour is quite clear between ZkS35D and ZkS35T, which both belong to the same sample and have only a 2% difference in porosity. This effect is detailed later.

#### 4.2. Identification of Flow Regimes

#### 4.3. Laws Governing Air Flow through ESR Foams

^{2}, was greater than 0.9980. However, maximum likelihood estimation: AIC, AICc, and BIC (calculated according to [56]) better confirmed this. For example, for ZkS35D (in Figure 6), the linear fitting of experimental results of reduced pressure drop gave an AIC, an AICc, and a BIC of 542.9676, 543.1782, and 544.0122, respectively. However, the quadratic fitting gave −849.7924, −849.1257, and −847.7033, respectively, which are quite lower than those of the linear fitting.

#### 4.4. Residuals Analysis

^{3}. However, the full cubic law residuals were close to random noise, which means that it is the best law describing the air flow through the ESR foams.

_{FC}and inertia coefficient C

_{1}and C

_{2}, which are then discussed in comparison to that obtained from the Forchheimer equation (K

_{Forch}and β).

#### 4.5. Air Flow Properties of ESR Foams

_{Forch}) and that fitted with full cubic law (K

_{FC}), defined by Equation (16)

^{−10}and 11.36 × 10

^{−10}considering the Forchheimer equation (knowing that their compression ratio was 7 or 14), but there are small discrepancies from one regime to another.

_{FC}of all the ESR samples in the turbulent regime were the lowest compared to those of the remaining regimes whatever the salt grain size, except in the case of ZkS35T because of its important length in the flow direction. However, they were the highest values in the Forchheimer zone compared to those of other regimes, except the permeability of ZkS40D. From the macroscopic point of view of air flow, the full range permeability was identical to that of the transition F–T for ZkS25D, the transition D–F for ZkS40D and ZkS45T, and to the Forchheimer for ZkS35T. However, it was never influenced by the turbulent regime. It is believed that this influence is due to foams’ structural properties such as porosity, length in the air flow direction, and the salt grain size (subject of an ongoing research paper), as open-cell foams generally have a random anisotropic cellular structure that has a significant effect on their permeability, which was demonstrated many times (e.g., [62,63]). The permeability is additionally influenced by the surface hardness of the pores and is extraordinarily impacted by the quantity of closed cells in ESR foams [1].

_{1}. Thus, if C

_{1}is high, it means that an important blockage exists due to a very complicated pore structure that may have closed faces (as ZkS35T is suspected to be).

_{1}. The same remark was observed in ESR foam, for example, between ZkS35T and ZkS45T. This increase is attributed to the higher specific surface area generated by the smaller pore size, which was confirmed by the corresponding global nonlinear cubic coefficient C

_{2}, which is −2.59/ZkS35T compared to −0.10/ZkS45T in this example.

_{2}is rather low and occasionally negative or positive depending on the velocity range considered (regime), but overall it is negative, which is likely due to the fact that the internal geometry of the micro-channels where the air flows is tortuous (according to Panfilov et al. [57]). This was an intuitive affirmation from Mei and Auriault [58], who expected that to maintain the same seepage velocity, a higher pressure gradient is needed if fluid inertia becomes increasingly important. Thus, C

_{2}should be non-positive.

_{2}. They considered a generalized Darcy’s law, including two additional nonlinear terms (quadratic and cubic) with respect to the mean flow velocity. The quadratic term described the pure inertia effect, which was caused by an irreversible loss of kinetic energy due to flow acceleration. The cubic term corresponds to a cross inertia–viscous effect, which consists of an additive viscous dissipation caused by the streamline deformation due to inertia forces. As inertia forces tend to straighten the flow streamlines, reducing the viscous dissipation, the inertia–viscous parameter should be negative. However, the formation of secondary flows, such as reverse jets, may invert this tendency and may thus change the sign of the inertia–viscous parameter. For a high amplitude of wall corrugation, the local inertia forces become somewhat greater than the global inertia. Therefore, the total viscous dissipation decreases, and the inertia–viscous parameter becomes negative. Note that the negative viscous–inertia dissipation increment is equivalent to a reduction in effective permeability, and the wall corrugation of the fracture corresponds to the tortuosity in the ESR foams. This is another important piece of information on the cell internal structure of these foams.

#### 4.6. Sample Length Effect on Airflow Regimes

_{C}(as a characteristic length) was calculated by Equation (1), and values are presented in Table 6. A dimensionless number NCLD is defined by Equation (13), where NCL and NCD are the number of cells by sample length and the number of cells by sample diameter, respectively. This number linked external macroscale properties and internal microscale properties of the ESR samples. The Reynolds number based on the salt grain diameter represents fluid flow information. Lower and upper limits of the Reynolds number are calculated using the lower and upper values of superficial velocity of each regime.

#### 4.7. Qualitative description of the ESR foams

_{p}used by some researchers was not considered because of the confusing definition of the pore; it is defined sometimes as the cell diameter and sometimes as the window diameter [67].

**.**

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**Experimental setup of the air channel used for permeability measurements of the different zamak 5 foams.

**Figure 3.**Measured pressure gradient normalized by the thickness versus air superficial velocity across all zamak 5 samples.

**Figure 5.**Delimitation of different regimes identified when air flows through ESR open-cell foams. It seems that all samples present nonlinear turbulence trends even for low velocity (v~0.4 m/s).

**Figure 6.**Linear and quadratic fitting of experimental results of reduced pressure drop versus superficial air velocity for each sample.

**Figure 8.**Dimensionless analysis of the effect of sample length on airflow limits using the number of cells per length normalised by the number of cells per sample diameter (NCLD). It is clear that each extension of the regime is deeply influenced by the length of the sample.

**Figure 9.**(

**a**) Photography of the salt used in the ESR process: the particle had an irregular cuboid form. (

**b**) SEI image magnification by JOEL JSM-6360 of cuboid cell shape replicas of the salt grain used and the interconnection windows of ESR foam from [12]. In this study, the nominal cell diameter (D

_{C}) considered was the diameter of the internal sphere surrounding the cuboid cell and having the same salt grain circumscribing sphere diameter determined by sieving.

**Figure 10.**Description of the relationship between full cubic permeability normalised by the square cell diameter in terms of porosity of the ESR sample.

Foam | L (mm) | Φ (mm) | D_{C} (mm) | ε (%) | NCL | NCD | NCLD |
---|---|---|---|---|---|---|---|

ZkS25D | 18.30 | 37.50 | 2.50 | 64.77 | 7 | 15 | 0.47 |

ZkS35D | 15.10 | 38.18 | 3.50 | 57.65 | 4 | 11 | 0.36 |

ZkS35T | 68.83 | 38.18 | 3.50 | 59.73 | 19 | 11 | 1.73 |

ZkS40D | 15.30 | 38.09 | 4.00 | 62.56 | 3 | 9 | 0.33 |

ZkS45T | 38.97 | 38.28 | 4.50 | 60.00 | 8 | 8 | 1.00 |

**Table 2.**Percentage of change of ESR foams between the permeability fitted with Forchheimer equation and that fitted with full cubic law.

Sample | Transition D–F | Forchheimer | Transition F–T | Turbulent | Full Range |
---|---|---|---|---|---|

ZKS25 D | 3.32 | −31.09 | −48.61 | −47.53 | −48.84 |

ZKS35 D | - | −8.13 | −47.61 | −22.73 | −39.06 |

ZKS35 T | 7.64 | −4.17 | 38.36 | −8.43 | −18.65 |

ZKS40 D | −0.50 | −11.79 | −38.21 | 101.43 | −35.72 |

ZKS45 T | 0.00 | −13.59 | −44.50 | −61.60 | −52.94 |

**Table 3.**The permeability fitted by the full cubic law and Forchheimer equation for all flow regimes of air flow through ESR foams.

Full Cubic Permeability K_{FC} (m²) | Forchheimer Permeability K_{Forch} (m²) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Sample | Transition D–F | Forchheimer | Transition F–T | Turbulent | Full Range * | Transition D–F | Forchheimer | Transition F–T | Turbulent | Full Range * |

ZkS25 D | 1.58 × 10^{−9} | 1.96 × 10^{−9} | 1.83 × 10^{−9} | 6.31 × 10^{−10} | 1.83 × 10^{−9} | 1.63 × 10^{−9} | 1.35 × 10^{−9} | 9.41 × 10^{−10} | 3.31 × 10^{−10} | 9.35 × 10^{−10} |

ZkS35 D | - | 4.75 × 10^{−11} | 4.49 × 10^{−11} | 2.15 × 10^{−11} | 3.60 × 10^{−11} | - | 4.36 × 10^{−11} | 2.35 × 10^{−11} | 1.66 × 10^{−11} | 2.19 × 10^{−11} |

ZkS35 T | 3.23 × 10^{−11} | 3.32 × 10^{−11} | 2.49 × 10^{−11} | 2.73 × 10^{−11} | 3.49 × 10^{−11} | 3.48 × 10^{−11} | 3.18 × 10^{−11} | 3.45 × 10^{−11} | 2.50 × 10^{−11} | 2.84 × 10^{−11} |

ZkS40 D | 2.18 × 10^{−9} | 2.34 × 10^{−9} | 2.48 × 10^{−9} | 4.65 × 10^{−10} | 2.19 × 10^{−9} | 2.17 × 10^{−9} | 2.07 × 10^{−9} | 1.53 × 10^{−9} | 9.37 × 10^{−10} | 1.41 × 10^{−9} |

ZkS45 T | 2.36 × 10^{−10} | 2.84 × 10^{−10} | 2.03 × 10^{−10} | 1.78 × 10^{−10} | 2.35 × 10^{−10} | 2.36 × 10^{−10} | 2.46 × 10^{−10} | 1.12 × 10^{−10} | 6.83 × 10^{−11} | 1.11 × 10^{−10} |

**Table 4.**The inertia coefficient fitted by the full cubic law and the Forchheimer equation for all regimes of air flow through ESR foams.

C_{1} (m^{−1}) (Full Cubic Law) | β (m^{−1}) (Forchheimer Law) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Sample | Transition D–F | Forchheimer | Transition F–T | Turbulent | Full Range | Transition D–F | Forchheimer | Transition F–T | Turbulent | Full Range |

ZkS25 D | 11,583.49 | 14,100.64 | 13,179.79 | 12,188.00 | 13,611.91 | 12,442.48 | 11,959.30 | 10,910.22 | 7561.98 | 10,770.95 |

ZkS35 D | - | 589,388.41 | 609,391.31 | 269,496.22 | 469,425.78 | - | 495,386.93 | 24,8916.45 | 137,176.77 | 206,951.25 |

ZkS35 T | 140,171.70 | 521,216.88 | 0.00 | 359,694.79 | 606,814.26 | 524,663.65 | 436,902.63 | 480,645.54 | 258,975.56 | 328,123.88 |

ZkS40 D | 13,047.41 | 13,472.89 | 13,350.30 | 7278.47 | 13,063.34 | 12,894.31 | 12,650.44 | 11,907.28 | 11,085.37 | 11,706.81 |

ZkS45 T | 34,851.68 | 100,090.94 | 88,252.99 | 82,705.81 | 92,231.81 | 34,851.68 | 84,141.38 | 48,054.40 | 27,310.87 | 44,912.40 |

**Table 5.**The cubic coefficient fitted using the full cubic law for all regimes of air flow through ESR foams.

C_{2} (-) (Full Cubic Law) | |||||
---|---|---|---|---|---|

Sample | Transition D–F | Forchheimer | Transition F–T | Turbulent | Full Range |

ZkS25 D | 0.0083 | −0.0045 | −0.0024 | −0.0037 | −0.0033 |

ZkS35 D | - | −1.0708 | −1.5207 | −0.3152 | −0.8304 |

ZkS35 T | 15.1539 | −1.3040 | 5.1841 | −0.7381 | −2.5935 |

ZkS40 D | −0.0023 | −0.0027 | −0.0020 | 0.0033 | −0.0016 |

ZkS45 T | 0.0000 | −0.1043 | −0.0970 | −0.0846 | −0.1008 |

Re Lower Limit | Re Upper Limit | |||||||
---|---|---|---|---|---|---|---|---|

Sample | Transition D–F | Forchheimer | Transition F–T | Turbulent | Transition D–F | Forchheimer | Transition F–T | Turbulent |

ZkS25 D | / | 209.24 | 889.48 | 1502.79 | 185.42 | 847.35 | 1395.07 | / |

ZkS35 D | / | 0 | 244.48 | 602.00 | - | 220.82 | 542.93 | / |

ZkS35 T | / | 73.59 | 156.36 | 191.86 | 61.74 | 144.56 | 168.15 | / |

ZkS40 D | / | 216.24 | 954.68 | 1973.65 | 189.27 | 889.61 | 1836.3 | / |

ZkS45 T | / | 85.87 | 543.98 | 1304.24 | 55.22 | 501.36 | 1231.52 | / |

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**MDPI and ACS Style**

Hassein-Bey, A.H.; Belhadj, A.-E.; Tahraoui, H.; Toumi, S.; Sid, A.N.E.H.; Kebir, M.; Chebli, D.; Amrane, A.; Zhang, J.; Mouni, L.
Experimental Investigation of Fluid Flow through Zinc Open-Cell Foams Produced by the Excess Salt Replication Process and Suitable as a Catalyst in Wastewater Treatment. *Water* **2023**, *15*, 1405.
https://doi.org/10.3390/w15071405

**AMA Style**

Hassein-Bey AH, Belhadj A-E, Tahraoui H, Toumi S, Sid ANEH, Kebir M, Chebli D, Amrane A, Zhang J, Mouni L.
Experimental Investigation of Fluid Flow through Zinc Open-Cell Foams Produced by the Excess Salt Replication Process and Suitable as a Catalyst in Wastewater Treatment. *Water*. 2023; 15(7):1405.
https://doi.org/10.3390/w15071405

**Chicago/Turabian Style**

Hassein-Bey, Amel Hind, Abd-Elmouneïm Belhadj, Hichem Tahraoui, Selma Toumi, Asma Nour El Houda Sid, Mohammed Kebir, Derradji Chebli, Abdeltif Amrane, Jie Zhang, and Lotfi Mouni.
2023. "Experimental Investigation of Fluid Flow through Zinc Open-Cell Foams Produced by the Excess Salt Replication Process and Suitable as a Catalyst in Wastewater Treatment" *Water* 15, no. 7: 1405.
https://doi.org/10.3390/w15071405