# Transport Efficiency of a Homogeneous Gaseous Substance in the Presence of Positive and Negative Gaseous Sources of Mass and Momentum

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Considered Flow Cases and Calculation Results

_{a}: $D\left(x,\tau \right)=\dot{D}\left(\tau \right)\text{}\delta \left(x-{x}_{a}\right)$, whose mass flow rate as a function of time is $\dot{D}\left(\tau \right)$, kg/s. It is assumed that the mass flow rate of such a source will be constant over time $\dot{D}=0$, which can be expressed as:

^{−1}, $\rho $ = 1.2 kg/m

^{3}, F = 50 m

^{2}. The approximation ratios of the cubic polynomial that describe the fan are a = −0.0000958, b = −0.010539, c = 15.59840, d = 1963.750. For the purposes of this paper, the reference value of the mass flow rate of gas flowing through the duct (no mass sources/sinks) is ${\dot{m}}_{0}$ = 350.076 kg/s.

#### 2.1. Case 1

_{a}), equal to 0 for x < x

_{a}and equal to 1 for x $\ge $ x

_{a}, [4]. As can be seen from (17), when $\dot{D}=0$ (no source), the system of Equation (17) is expressed as:

#### 2.2. Case 2

_{a}, with mass efficiency $\dot{D}$ that is constant over time. The mass leaving the duct (in contrast to Case 1) has a momentum, which means that this case involves a local mass and momentum sink. Local mass sources can occur at different points of the duct. The calculations below take this into account. At one end of the duct, a fan is installed as a mechanical source of mechanical energy, working in one of the two modes (as in Case 1): suction (Case 2a) or blowing (Case 2b). The sink of mass flow rate is independent of the thermal driving head created by the fan, although it is a factor affecting the resulting values ${\dot{m}}_{0}$. The value of momentum depends on the fan’s characteristic curve and the flow rate of mass leaving the duct. These cases are demonstrated in Figure 4.

## 3. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

$d$ | hydraulic diameter of the duct, m |

$D$ | internal source of mass flow, kg/(m·s) |

$\dot{D}$ | mass flow rate of local source of mass, kg/s |

$ef$ | energy efficiency for the observed medium’s flow, (m·s)^{−1} |

$F$ | cross-sectional area of the duct, m^{2} |

$g$ | gravity, m/s^{2} |

$i$ | hydraulic gradient per unit (dimensionless) caused by the linear resistance of the duct, |

${L}_{L}$ | duct length, m |

$\dot{m}$ | mass flow rate of fluid, kg/s |

${\dot{m}}_{0}$ | mass flow rate of gas entering the duct, the opposite end of which features a mechanical suction source (mass flow rate of gas leaving the duct, the opposite end of which features a mechanical suction source), kg/s |

${\dot{m}}_{w}$ | mass flow rate of gas flowing through the fan, kg/s |

$n$ | ratio of the drag of the duct section from its entry to the point where the local source of mass is located and the drag of the entire duct ($0\le n\le 1$) |

$p$ | absolute static pressure of the fluid, Pa |

$P$ | cross-sectional circumference of the duct, m |

$r$ | distributed drag per unit of the duct (a straight duct), kg/m^{8} |

$R$ | specific drag, Ns^{2}/m^{8} or kg/m^{7} |

R* | specific drag of the duct, 1/(kg·m) |

${r}_{zas}$ | duct’s equivalent drag per unit, kg/m^{8} |

$u$ | flow velocity of incoming mass aligned with the mass flow direction $v\left(x,\tau \right)$ in the duct, m/s |

$\dot{V}$ | volumetric flow rate of the flowing fluid, m^{3}/s |

${W}_{L}$ | loss of mechanical energy (total head), N/m^{2} |

${w}_{LL}$ | static-pressure drop at local resistance at a point with the coordinate ${x}_{L}$, N/m^{2} |

x | distance, m |

z | above-sea-level spot height of a given duct location, m |

Greek symbols | |

$\alpha $ | distributed-resistance coefficient of the duct, kg/m^{3} |

$\delta \left(x-{x}_{a}\right)$ | Dirac delta function distribution, 1/m |

$\Delta {p}_{c}$ | total pressure increase for a working fan, N/m^{2} |

$\lambda $ | dimensionless distributed resistance coefficient of the duct, |

$v$ | average gas velocity, m/s |

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

$\tau $ | time, s |

## References

- Pawiński, J.; Roszkowski, J.; Strzemiński, J. Przewietrzanie Kopalń; Śląskie Wydawnictwo Techniczne: Katowice, Poland, 1995. [Google Scholar]
- Ptaszyński, B. Charakterystyka przepływowa przewodu transportującego różne media. In Proceedings of the 8 Szkoła Aerologii Górniczej. Sekcja Aerologii Górniczej: Nowoczesne Metody Zwalczania Zagrożeń Aerologicznych w Podziemnych Wyrobiskach Górniczych, Jaworze, Poland, 13–16 October 2015. [Google Scholar]
- Wacławik, J. Mechanika Płynów i Termodynamika; Wydawnictwo AGH: Kraków, Poland, 1993. [Google Scholar]
- Dziurzyński, W.; Krach, A.; Pałka, T. Airflow Sensitivity Assessment Based on Underground Mine Ventilation Systems Modeling. Energies
**2017**, 10, 1451. [Google Scholar] [CrossRef] - Zmrhal, V.; Boháč, J. Pressure loss of flexible ventilation ducts for residential ventilation: Absolute roughness and compression effect. J. Build. Eng.
**2021**, 44, 103320. [Google Scholar] [CrossRef] - Sleiti, A.K.; Zhai, J.; Idem, S. Computational fluid dynamics to predict duct fitting losses: Challenges and opportunities. HVAC&R Res.
**2013**, 19, 2–9. [Google Scholar] [CrossRef] - Kodali, C.; Idem, S. Modeling flow and pressure distributions in multi-branch light-commercial duct systems. Sci. Technol. Built Environ.
**2021**, 27, 240–252. [Google Scholar] [CrossRef] - Semin, M.A.; Levin, L.Y. Stability of air flows in mine ventilation networks. Process Saf. Environ. Prot.
**2019**, 124, 167–171. [Google Scholar] [CrossRef] - Onder, M.; Cevik, E. Statistical model for the volume rate reaching the end of ventilation duct. Tunn. Undergr. Space Technol.
**2008**, 23, 179–184. [Google Scholar] [CrossRef] - Auld, G. An estimation of fan performance for leaky ventilation ducts. Tunn. Undergr. Space Technol.
**2004**, 19, 539–549. [Google Scholar] [CrossRef] - Akhtar, S.; Kumral, M.; Sasmito, A.P. Correlating variability of the leakage characteristics with the hydraulic performance of an auxiliary ventilation system. Build. Environ.
**2017**, 121, 200–214. [Google Scholar] [CrossRef] - McPherson, M.J. Subsurface Ventilation and Environmental Engineering; Chapman&Hall: London, UK, 1992. [Google Scholar]
- Wacławik, J. Wentylacja Kopalń, Tom I, II; Wydawnictwo AGH: Krakow, Poland, 2010. [Google Scholar]
- Yamaguchi, H. Engineering Fluid Mechanics; Springer Netherlands: Dordrecht, The Netherlands, 2010. [Google Scholar]
- Bergander, M.J. Fluid Mechanics Volume 1. Basic Principles; AGH University of Science and Technology Press: Krakow, Poland, 2010. [Google Scholar]

**Figure 1.**Gas flow chart for Case 1 under consideration: (

**a**) fan working in the suction mode, (

**b**) fan working in the blowing mode.

**Figure 4.**Gas flow chart for Case 2 under consideration: (

**a**) fan working in the suction mode, (

**b**) fan working in the blowing mode.

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**MDPI and ACS Style**

Ptaszyński, B.; Kuczera, Z.; Życzkowski, P.; Łuczak, R.
Transport Efficiency of a Homogeneous Gaseous Substance in the Presence of Positive and Negative Gaseous Sources of Mass and Momentum. *Energies* **2022**, *15*, 6376.
https://doi.org/10.3390/en15176376

**AMA Style**

Ptaszyński B, Kuczera Z, Życzkowski P, Łuczak R.
Transport Efficiency of a Homogeneous Gaseous Substance in the Presence of Positive and Negative Gaseous Sources of Mass and Momentum. *Energies*. 2022; 15(17):6376.
https://doi.org/10.3390/en15176376

**Chicago/Turabian Style**

Ptaszyński, Bogusław, Zbigniew Kuczera, Piotr Życzkowski, and Rafał Łuczak.
2022. "Transport Efficiency of a Homogeneous Gaseous Substance in the Presence of Positive and Negative Gaseous Sources of Mass and Momentum" *Energies* 15, no. 17: 6376.
https://doi.org/10.3390/en15176376