# Influence of Local Gas Sources with Variable Density and Momentum on the Flow of the Medium in the Conduit

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

^{7}] or as R* [1/(kg·m)]. The usage of the specific drag of the duct R* (this is proposed by the authors and not found in the literature of fluid mechanics) is necessary to explain the influence of local mass efficiency of the gas sources (sinks) $\dot{D}$ with different density on the main mass flow gas without heat exchange along the length of duct (heat depression). From the literature [3,5,9] it is known that heat depression in the duct is defined for loops in which the source of mechanical energy $-{\displaystyle \oint}\frac{dp}{\rho}$ does not exist. It is concluded that at constant density of gas along the length of the duct, the natural depression is equal to zero. This proposition applies when no local mass and energy sources occur in the duct, and there is no heat exchange with the environment [3].

_{L}[N/m

^{2}] can be used to determine the gas transport efficiency [7]. This paper determines the gas flow rates in the duct, the gas energy losses in the duct due to the forces opposing motion, and the work done by the mechanical source of energy for various gas-source parameters, including the gas-source location. The gas transport efficiencies were determined for the considered cases involving the flow of gaseous media, with densities varying between duct sections.

_{f}—Figure 5) the energetic impact of the local source or sink of gas with different density on its flow in the duct. The aim of the research is to indicate that the reason of gas flow in the duct (in the absence of a mechanical fan in the duct) may not only be the heat depression (the density change as a result of heat exchange), but also an occurrence of a local source or sink of the gas stream with different density and momentum, or the lack thereof. The authors proposed the usage of the concept of specific drag of the duct (R*), and in the case of mechanical ventilation (suction), the influence of density change of gas flowing through the fan on its characteristic were included.

^{2000}, which carry out the calculations for the steady air flow state, which is based on the Cros method. Only some of them, including MineFire, Vuma-fire, VnetPC MineFire, and Mivena conduct additional calculations for unsteady air flow state, similar to the VentGraph program [12,13]. The described problem can also be applied to predict flows in industrial ventilation using multiple regression or mathematical modeling with the use of computational fluid dynamics (CFD) [14,15,16,17,18,19].

## 2. Considered Flow Cases and Calculation Results

^{3}] forced by a fan working at the end of the duct (at a point with the coordinate x = x

_{w}[m]). A gas mass source is present in the duct, with a gas density of ρ

_{d}[kg/m

^{3}], (different from density ρ) and specific mass efficiency $\dot{D}$ [kg/s]. As in [7], it is understood that the mass source flow rate is lower than the mass flow rate of gas flowing through the duct ${\dot{m}}_{0}$ [kg/s]. The local source is at a point with the coordinate x

_{A}[m]. The fan installed in the duct can work in the suction or blowing mode. The investigated systems are demonstrated in Figure 1.

_{a}and equal to 1 for x ≥ x

_{a}[4]. The function plotted in Figure 2 and Figure 3 shows the graph of the equation ρ(x).

_{sr}—thus:

_{c}(ρ) can be approximated with a cubic polynomial due to variable ${\dot{m}}_{w}$, so Equation (8) can be written as:

_{0}≤ x ≤ x

_{A}and x

_{A}≤ x ≤ x

_{w}.

_{1}from x

_{0}≤ x < x

_{A}, on the path C

_{2}from x

_{A}≤ x < x

_{w}, and on the path C

_{3}by external atmosphere (with the drag equal to zero) from x

_{w}≤ x < x

_{0}. On the path C

_{3}by the external atmosphere of: ρ(x) = ρ and $\dot{m}\left(x\right)={\dot{m}}_{0}$, which stems from the atmosphere’s infinite capacity. The circular integral, which is the last component on the left side of Equation (4), can be substituted with:

_{a}to x

_{w}has a density of ρ

_{sr}, which should be taken into account in the calculation of the duct’s drag:

_{a}–x

_{w}) and considering Equation (11), the following can be written for the fan working in the suction mode:

_{w}, z

_{a}—spot heights at a point of the current coordinates x

_{w}and x

_{a}; $\Delta \rho =\frac{\dot{D}\left({\rho}_{d}-\rho \right)}{\left({\dot{m}}_{0}+\dot{D}\right)}$ and Δz = z

_{w}− z

_{a}for the suction mode of the fan.

_{a}–x

_{0}) is the only section where there is a density difference Δρ, which, in this case, is equal to:

_{sr}= const. Density ρ

_{sr}(x) is constant along the duct section (x

_{a}÷ x

_{w}), where $\dot{D}\mathscr{H}\left({x}_{a}\right)\ne 0$ (fan suction work). When the fan is working in the blowing mode, Equation (23) is the same, but the density ρ

_{sr}(x) is constant along the duct section (x

_{a}÷ x

_{0}), where for this case $\dot{D}\mathscr{H}\left({x}_{a}\right)\ne 0$.

_{sr}is unknown.

_{sr}=const. Hence, ρ

_{sr}can be moved in front of the integral sign such that both considered working modes of the fan can be written as:

_{d}, ρ, z

_{(xw)}, z

_{(xa)}, ${R}_{\left({x}_{0}-{x}_{a}\right)}^{\ast}\left(\rho \right)$, ${R}_{\left({x}_{a}-{x}_{w}\right)}^{\ast}\left(\rho \right)$, and approximation ratios of the cubic polynomial (a, b, c, d) that defines the fan pressure increase Δp

_{c}(ρ).

_{sr}are the unknowns. Considering the ρ

_{sr}equation which is written as follows for the suction mode of the fan:

_{sr}− ρ, is the increase in gas density along a duct with a height difference between its entry and end point, determined using Δz and expressed in meters.

_{sr}= 1 and Δρ = 0, Equation (37) for the suction mode of the fan is written as:

## 3. Results of Numerical Calculations for the Mathematical Flow Models

_{sr}and the density difference Δρ, without knowing the mass flow rate $\dot{{m}_{0}}$. To this end (as the first calculation step), $\dot{{m}_{0}}$ in Formulas (31), (20), (32), and (21) was substituted with the corresponding value of the flow rate in the case considered by [7], where the inflow density was the same. For the values so determined, Equation (38) or (39) are calculated numerically by determining the mass flow rate $\dot{{m}_{0}}$ as the first calculation step. Once this value is known, the values ρ

_{sr}and Δρ are calculated again for the relevant fan mode, and the mass flow rate $\dot{{m}_{0}}$ is calculated as the second step. This procedure was repeated until the difference in the value ρ

_{sr}obtained by adjacent calculation steps was lower than 1·10

^{−4}kg/m

^{3}. The imposed condition was already satisfied in the third calculation step. After determining the value of the flow rate $\dot{{m}_{0}}$, which satisfies Equation (38) or (39), we calculate the loss of mechanical energy due to forces opposing motion, according to Equation (35) or (36), and the fan work parameters (${\dot{m}}_{w}$ and Δp

_{c}) and transport efficiency. The calculations included the same input data and flow variants as in [7], so that the results could be compared with the results obtained for inflows of gas with the same density. The following input data were assumed: R*(ρ) = 0.0165 (kg·m)

^{−1}, ρ = 1.2 kg/m

^{3}, ρ

_{d}= 1.6 kg/m

^{3}, F = 50 m

^{2}, $\dot{D}$ = {10, 30; 50} kg/s, n = {0, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0}, approximating polynomials of the fan characteristic curve (Formula (9)): a = −0.000095812, b = −0.0105393, c = 15.5984, d = 1963.75, Δz = 400 m (the same for the suction and blowing mode of the fan). In the calculations assuming a constant value Δz, regardless of where the local source of inflow is situated, it is easier to calculate and prepare input data. The results from the Mathematica program are illustrated in Figure 4 and Figure 5, arranged similarly to those in [7].

_{L}—loss of mechanical energy (total head), N/m

^{2}. The results of the calculations are shown in Figure 5.

## 4. Distribution of Static Pressure and Mechanical Energy Losses on the Resistance of Motion in the Duct

^{2}, which corresponded to an internal diameter d = 7.98 m. Developed turbulent flow was assumed and its resistance value R*(ρ) = 0.0165 (kg·m)

^{−1}was pointed. In this condition, the point of fan work was calculated and it guaranteed the stable fan operation in the recommended area of its characteristic. To determine the effects of pressure profile p(x) and mechanical energy losses on the resistance of motion ${W}_{L}\left(x\right)$ as a function of the current coordinate “x” of the duct, the duct length L should be determined by using the duct parameters from the article in [7]. The following dependencies have to be used:

^{3}). Additionally, inflows with different momentum values can be taken into account. Therefore, extensive research and designated profiles of the effects of pressure and mechanical energy losses on the resistance of motion can be easier to use when comparing the results to those of other authors. Mentioned profiles will be presented by the authors in the future, while the procedure of pressure and mechanical energy loss profiles are presented for the duct, which is located in the middle of the duct length (${x}_{a}=10000\mathrm{m}$), local source of mass gas inflow with density $\mathsf{\rho}=1.6$ kg/m

^{3}, and mass stream $\dot{D}=50$ kg/s.

- For section <${x}_{0}-{x}_{a}$> of duct:$$p\left(x\right)=p\left({x}_{0}\right)-{R}_{\left({x}_{0}-{x}_{a}\right)}^{\ast}\left(\rho \right)\xb7{\dot{m}}_{0}^{2}=p\left({x}_{0}\right)-{r}^{\ast}{\left(\rho \right)}_{\left({x}_{a}-{x}_{0}\right)}\xb7{\dot{m}}_{0}^{2}\xb7\left(x-{x}_{0}\right)$$

- For section <${x}_{a}-{x}_{w}$> of the duct, the dependence on the pressure profile is also the decreasing linear function:$$p\left(x\right)=p\left({x}_{a}{}^{+}\right)-{r}^{\ast}{\left(\rho \right)}_{\left({x}_{a}-{x}_{w}\right)}\xb7{\left(\frac{\rho}{{\rho}_{sr}}\right)}^{2}\xb7{\left({\dot{m}}_{0}+\dot{D}\right)}^{2}\xb7\left(x-{x}_{a}\right)$$

## 5. Conclusions

- If there is a local source of gas mass with a density of 1.6 kg/m
^{3}entering a duct containing gas flowing at a rate of 1.2 kg/m^{3}, the pressure increases in the fan are also loaded by the energy spent to provide the required kinetic energy to this gas mixture with the resultant density. - If there is a source of gas mass with a different (higher) density, the different duct sections will be passed through by gas with varying density (steady within a range) (Figure 3). If the duct fan is working in the suction mode, the fan will be passed through by gas with an average density of more than 1.2 kg/m
^{3}. - If the duct fan is working in the blowing mode, gas with a density of 1.2 kg/m
^{3}will be flowing through the fan, regardless of where the local source of gas with a higher density is situated. - Changes in the density of gas flowing through the fan require the recalculation of the fan characteristic curve from the catalogue density of 1.2 kg/m
^{3}. This means that if the fan is working in the suction mode, the local inflow of gas with varying densities causes changes in the parameters of the mechanical energy source (no such changes occur when the duct fan is working in the blowing mode). - Differences in gas densities in different duct sections do not change the relationship between the losses of mechanical energy due to forces of opposing motion. However, these differences do affect the values of these relationships by making it necessary to recalculate the drag values for these duct sections, as shown in Formulas (35) and (36). An increase in density of gas flowing through the duct causes a reduction in the duct’s existing drag.
- Such differences in gas density between duct sections due to the presence of local sources of gas with varying densities generate mechanical energy in the inclined parts of the duct through which average-density gas is flowing. This energy is associated with a change in the existing buoyancy in this section. This value is defined as natural head and is included for the suction mode of the fan in Formula (29). The local natural head can be negative or positive, depending on the height difference of a duct in which an average-density gas mixture is flowing. This value may differ between duct sections, due to an exhaust or forcing fan. For the same local source of gas mass with a steady flow density, the local natural head for the suction mode is slightly lower than for the blowing mode, as follows from Formulas (31), (32), (37) and (38) for ∆z = 0. The source of gas mass is located at the end of the duct, with the fan working in the suction mode, and conversely, if the source is located at the entry of the duct, with the fan working in the blowing mode, the local natural head is equal to zero.
- The gas transport efficiency is higher with the suction mode of the fan if the gas source of higher density is located closer to the beginning of the duct. Gas transport with greater efficiency requires a greater expenditure of mechanical energy of the fan.
- The presented mathematical model allows for the determination of the static pressure distribution and mechanical energy loss in the conduit.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Symbol | Parameter |

d | hydraulic diameter of the duct, m |

$\dot{D}$ | mass efficiency of the gas source (steady value), 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} |

$\dot{m}$ | mass flow rate of gas, 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 a suction source), kg/s |

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

p | absolute static pressure, Pa |

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

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

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

${r}_{ZAS}^{\ast}$ | duct equivalent drag per unit, 1/(kg·m^{2}) |

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

x | current coordinate measured along the duct’s axis, m |

z_{a} | spot heights at a point of the current coordinates x_{a}, m |

z_{w} | spot heights at a point of the current coordinates x_{w}, m |

δ(x–x_{a}), δ(x–x_{w}) | Dirac delta function distribution, 1/m |

δ(x–x_{L}) | Dirac delta function distribution at the point of local resistance x_{L}, 1/m |

Δp_{c}(ρ) | fan’s pressure increase when gas with the density ρ is flowing through the fan, Pa |

Δp_{c}(ρ_{w}) | fan’s total pressure increase with the fan flow rate of gas with density ρ, ρ(x) = ρ(x_{w}), Pa |

λ | dimensionless coefficient of distributed resistance, - |

ξ | dimensionless coefficient of local resistance, - |

ρ_{d} | density of local-source gas stream, kg/m^{3} |

ρ(x) | gas density at a point with the current coordinate x, kg/m^{3} |

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**Figure 1.**Gas flow chart for Example 1 under consideration of a (

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

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

**Figure 2.**Plot of the function $\dot{m}\left(x\right)$ for Example 1: (

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

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

**Figure 3.**Plot of the function ρ(x) for Example 1: (

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

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

**Figure 4.**Results of numerical calculations for analyzing examples: (

**a**)—mass flow rate of gas entering the duct ${\dot{m}}_{0}$, (

**b**)—mass flow rate of gas flowing through the fan ${\dot{m}}_{w}$.

**Figure 5.**Duct gas transport efficiency in the presence of a local source of gas with different densities for examples 1a and 1b.

**Figure 6.**The profile of pressure and mechanical energy loss in the duct for the case of mass gas inflow with zero momentum.

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**MDPI and ACS Style**

Ptaszyński, B.; Łuczak, R.; Kuczera, Z.; Życzkowski, P.
Influence of Local Gas Sources with Variable Density and Momentum on the Flow of the Medium in the Conduit. *Energies* **2022**, *15*, 5834.
https://doi.org/10.3390/en15165834

**AMA Style**

Ptaszyński B, Łuczak R, Kuczera Z, Życzkowski P.
Influence of Local Gas Sources with Variable Density and Momentum on the Flow of the Medium in the Conduit. *Energies*. 2022; 15(16):5834.
https://doi.org/10.3390/en15165834

**Chicago/Turabian Style**

Ptaszyński, Bogusław, Rafał Łuczak, Zbigniew Kuczera, and Piotr Życzkowski.
2022. "Influence of Local Gas Sources with Variable Density and Momentum on the Flow of the Medium in the Conduit" *Energies* 15, no. 16: 5834.
https://doi.org/10.3390/en15165834