# An Investigation of Parallel Post-Laminar Flow through Coarse Granular Porous Media with the Wilkins Equation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}

^{2}/m

^{2}) are the Darcy and Non-Darcy coefficients, respectively.

_{1}is a coefficient; γ is an exponent having a value between 0.5 and 1.0, depending on the nature of the regime; and μ is the dynamic viscosity of the fluid. For a constant viscosity, Equation (3) can be written as:

## 2. Experiments and Methodology

^{3}/s). The flow rate is measured as the average of three volumetric flow measurements (m

^{3}/s) with an accuracy of ±2.35%. The measured volumetric flow rate m

^{3}/s) is divided by the cross sectional area (m

^{2}) of the permeameter to obtain the superficial velocity (m/s). For every flow rate, eight separate piezometric head differences are noted from tapings placed vertically over the permeameter at a regular interval of 50 mm. The average hydraulic gradient calculated from the recorded head differences is used for further analysis with an accuracy of ±1.89%. Such an arrangement eliminates any error due to the non-uniformity of packing.

^{2}/s); d

_{k}= characteristic length, defined as d(1 − f)α

_{s}/4f; d and α

_{s}are the volume diameter (m) and shape factor of the media, respectively.

_{0})) is used as the characteristic length while calculating the Wilkins coefficients since it includes both parameters (porosity and media size) that influence the flow. The average surface areas and volumes which are prerequisites to calculate the specific surface (surface area per unit volume) are measured as described by Banerjee et al. [23] and presented in Table 1. After measuring the surface area and volume, the volume diameter (diameter of the sphere having the same volume of the media) and the specific surface is calculated.

## 3. Correction Factors

#### 3.1. Wall Correction

_{w}is the corrected velocity after wall correction, D is the diameter of the permeameter section (mm), and d is the diameter of the porous media used.

#### 3.2. Tortuosity Correction

#### 3.3. Porosity Correction

_{ν}is the pore velocity and f is the porosity.

_{c}and i

_{c}are the corrected velocity and hydraulic gradients, respectively.

## 4. Data Analysis

_{c}) and corrected hydraulic gradients (i

_{c}) are plotted in Figure 2 and a relationship in the form of a power law is obtained. This type of equations are also referred as Izbash equation [27] or Missbach equation [20] in the literature:

## 5. The Behavior of the Wilkins Coefficients with Independently Varying Media Size and Porosity

## 6. Numerical Modelling

#### 6.1. Turbulence Modelling

_{i}is the velocity component in the x

_{i}direction:

_{t}) and turbulent kinetic energy (k) [29].

#### 6.2. Model Description

^{th}(x, y or z) momentum equation; $\left|v\right|$ is the magnitude of the velocity; D and C are the prescribed matrices. For a homogeneous media:

_{2}is the inertial resistance factor simply specifying D and C as diagonal matrices; and μ and ρ are the dynamic viscosity and the density of the fluid, respectively. The viscous and inertial resistance are measured based on the pressure loss observed from the experimental setup [29,36]. Furthermore, the model is solved by the finite volume method with the velocity pressure coupling done by the SIMPLE algorithm. The governing equations are discretized using the second order upwind differencing scheme. Convergence is attained when the scaled residuals are less than 10

^{−4}times of their initial values.

#### 6.3. Meshing

_{1}, h

_{2}, h

_{3}) simulations are carried out to estimate the key variables ($\varphi $); in the present study these variables are Hydraulic gradient, Velocity (m/s) at 0.285 m and Velocity (m/s) at 1.11 m. The apparent order (p) of the method is then calculated using the following equations:

_{1}, h

_{2}and h

_{3}in Table 4. A similar analysis is performed for other selected grid sizes. Numerical uncertainty in the fine grid solution is observed to be 1.09%, 0.56%, and 0.02% respectively for the hydraulic gradient, the velocity at a vertical distance of 0.285 m from the entrance of the permeameter, and the velocity at a vertical distance of 1.11 m from the entrance of the permeameter (Table 4). The GCI values indicate a negligible discretization error and therefore a grid independent solution for the selected grid. Considering the simulation time and precision, a grid system of 1,288,576 elements is adopted for the model.

#### 6.4. Boundary Conditions

_{h}); and the inlet velocity(V) [38]. Equation (26) is used to calculate the turbulent intensity at the core of a fully developed duct [29]:

## 7. Results and Discussion

#### 7.1. Comparison between the Experimental and Simulation Data and Statistical Validation of the Simulation

#### 7.2. Validation of the Simulated Data Using the Z-Test

_{1}and μ

_{2}are the population mean of the experimental and simulation result. In order to validate the simulated result, the null hypothesis, given by H

_{0}, must be accepted.

_{D}is the total number of data points (sample size) and σ

_{1}, σ

_{2}are the standard deviations for the experimental and simulation results.

_{0}is accepted, validating the simulation results with the experimental results. Result from the statistical analysis along with the percentage deviation between the simulation and experimental data signify the accuracy of the CFD model used.

## 8. Conclusions

- The Wilkins equation can be satisfactorily used to represent post-laminar flow through porous media.
- The Wilkins coefficients are found to have a non-deviating nature with varying hydraulic radius. The obtained results from the present study are similar to the results reported in the literature.
- When subjected to variation in media size, the coefficients of the Wilkins equation are constant, given that the porosity is constant. However, variations in the porosity result in small variations of the coefficient W.
- The flow condition inside the experimental set up is simulated with a CFD model in the ANSYS FLUENT software. Trends similar to the experimental ones are obtained from the simulation results. The percentage deviation between the simulation and experimental results are within the acceptable range.
- For further validation, the experimental results are statistically compared with the simulation results using the standard Z-test. The values of Z calculated are found to be within the acceptable region for all the experimental results.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**The experimental set-up with the parallel flow permeameter. (

**a**) the front view; (

**b**) the side view; and (

**c**) the schematic diagram.

**Figure 2.**The variation of corrected hydraulic gradient with a corrected velocity for the media size being (

**a**) 29.8 mm; (

**b**) 34.78 mm; and (

**c**) 41.59 mm.

**Figure 4.**Thevariation of W with (

**a**) varying volume diameters of the media with constant porosities and (

**b**) with varying porosity with constant media size.

**Figure 6.**The values of (

**a**) the total pressure and (

**b**) the velocity in the direction of the flow subjected to different discharges for 29.8 mm media packed with 43.34% porosity.

**Figure 7.**The values of (

**a**) the total pressure and (

**b**) the velocity in the direction of the flow subjected to different discharges for 34.78 mm media packed with 44.70% porosity.

**Figure 8.**The values of (

**a**) the total pressure and (

**b**) the velocity in the direction of the flow subjected to different discharges for 41.59 mm media packed with 43.62% porosity.

**Figure 9.**Comparison of simulation and experimental results for 29.8 mm media packed with (

**a**) 40.59%; (

**b**) 43.34%; and (

**c**) 45.69% porosities.

**Figure 10.**Comparison of simulation and experimental results for 34.78 mm media packed with (

**a**) 41.72%; (

**b**) 44.70%; and (

**c**) 46.34% porosities.

**Figure 11.**Comparison of simulation and experimental results for 41.59 mm media packed with (

**a**) 41.03%; (

**b**) 43.62%; and (

**c**) 46.15% porosities.

**Figure 12.**The variation of the percentage deviation between the experimental and simulation results with flow velocities for (

**a**) 29.8 mm; (

**b**) 34.78 mm; and (

**c**) 41.59 mm media.

**Figure 13.**The Z

_{calculated}values on the binomial distribution curve for (

**a**) 29.8 mm; (

**b**) 34.78 mm; and (

**c**) 41.59 mm media sizes.

Passing and Retaining Sieve Sizes (cm) | Volume (cm^{3}) | Volume Diameter (cm) | Avg. Surface Area (cm^{2}) | Specific Surface (/cm) |
---|---|---|---|---|

2.50–3.15 | 14.46 | 2.98 | 44.68 | 2.88 |

3.15–3.75 | 23.10 | 3.48 | 58.44 | 2.53 |

3.75–5.00 | 38.55 | 4.16 | 82.11 | 2.13 |

Media Size (mm) | Porosity (%) | W (m-s) | β | γ |
---|---|---|---|---|

29.80 | 40.59 | 6.15 | 0.39 | 0.54 |

43.34 | ||||

45.69 | ||||

34.78 | 41.72 | 5.52 | 0.38 | 0.56 |

44.70 | ||||

46.34 | ||||

41.59 | 41.03 | 5.55 | 0.38 | 0.55 |

43.62 | ||||

46.15 |

Proposed by | Media | Volume Diameter (mm) | Porosity (%) | W (m/s) | β | γ |
---|---|---|---|---|---|---|

Wilkins (1956) [21] | Crushed stone | 51.00 | 40.00 | 5.24 | 0.50 | 0.54 |

Garga et al. (1990) [22] | Crushed stone | 24.60 | 47.00 | 5.39 | 0.50 | 0.53 |

Pradeep Kumar (1994) [27] | Crushed stone | 13.10 | 47.00 | 4.94 | 0.51 | 0.52 |

20.10 | 45.88 | |||||

28.90 | 48.73 | |||||

39.50 | 48.26 |

**Table 4.**Sample calculations of numerical uncertainty using the GCI method [37].

Parameter | Hydraulic Gradient | Velocity (m/s) at 0.285 m | Velocity (m/s) at 1.11 m |
---|---|---|---|

h_{1} | 0.005216 | 0.005216 | 0.005216 |

h_{2} | 0.006937 | 0.006937 | 0.006937 |

h_{3} | 0.009226 | 0.009226 | 0.009226 |

r_{21} | 1.33 | 1.33 | 1.33 |

r_{32} | 1.33 | 1.33 | 1.33 |

${\varphi}_{1}$ | 0.01992 | 0.01083 | 0.01156 |

${\varphi}_{2}$ | 0.01984 | 0.01080 | 0.01159 |

${\varphi}_{3}$ | 0.01973 | 0.01077 | 0.01096 |

P | 1.32772 | 1.30788 | 10.5924 |

${\varphi}^{21}{}_{ext}$ | 0.02010 | 0.01088 | 0.01156 |

${\varphi}^{32}{}_{ext}$ | 0.02010 | 0.01088 | 0.01163 |

${e}_{\alpha}^{21}$ | 0.40% | 0.20% | 0.27% |

${e}_{\alpha}^{32}$ | 0.59% | 0.29% | 5.45% |

${e}_{ext}^{21}$ | 0.86% | 0.45% | 0.01% |

${e}_{ext}^{32}$ | 1.26% | 0.65% | 0.28% |

$GC{I}_{fine}^{21}$ | 1.09% | 0.56% | 0.02% |

Media Size (mm) | Porosity (%) | Range of Velocity (m/s) | $\overline{{\mathit{x}}_{1}}$ | $\overline{{\mathit{x}}_{2}}$ | ${\mathit{\sigma}}_{1}$ | ${\mathit{\sigma}}_{2}$ | ${\mathit{N}}_{\mathit{D}}$ | Z Value |
---|---|---|---|---|---|---|---|---|

29.80 | 40.59 | 0.01–0.757 | 0.643 | 0.599 | 0.540 | 0.359 | 250 | −0.940 |

43.34 | 0.01–0.757 | 0.586 | 0.606 | 0.497 | 0.517 | 250 | 0.429 | |

45.69 | 0.01–0.757 | 0.554 | 0.535 | 0.464 | 0.446 | 250 | −0.459 | |

34.78 | 41.72 | 0.01–0.757 | 0.565 | 0.554 | 0.463 | 0.452 | 250 | −0.276 |

44.70 | 0.01–0.757 | 0.527 | 0.510 | 0.435 | 0.420 | 250 | −0.425 | |

46.34 | 0.01–0.757 | 0.496 | 0.471 | 0.407 | 0.383 | 250 | −0.694 | |

41.59 | 41.03 | 0.01–0.757 | 0.511 | 0.489 | 0.422 | 0.406 | 250 | −0.574 |

43.62 | 0.01–0.757 | 0.464 | 0.440 | 0.391 | 0.368 | 250 | −0.684 | |

46.15 | 0.01–0.757 | 0.439 | 0.418 | 0.369 | 0.349 | 250 | −0.661 |

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**MDPI and ACS Style**

Banerjee, A.; Pasupuleti, S.; Singh, M.K.; Kumar, G.N.P.
An Investigation of Parallel Post-Laminar Flow through Coarse Granular Porous Media with the Wilkins Equation. *Energies* **2018**, *11*, 320.
https://doi.org/10.3390/en11020320

**AMA Style**

Banerjee A, Pasupuleti S, Singh MK, Kumar GNP.
An Investigation of Parallel Post-Laminar Flow through Coarse Granular Porous Media with the Wilkins Equation. *Energies*. 2018; 11(2):320.
https://doi.org/10.3390/en11020320

**Chicago/Turabian Style**

Banerjee, Ashes, Srinivas Pasupuleti, Mritunjay Kumar Singh, and G.N. Pradeep Kumar.
2018. "An Investigation of Parallel Post-Laminar Flow through Coarse Granular Porous Media with the Wilkins Equation" *Energies* 11, no. 2: 320.
https://doi.org/10.3390/en11020320