Pressure Change in a Duct with a Flow of a Homogeneous Gaseous Substance in the Presence of a Point Mass and Momentum Sink of Gas

Abstract

The flow characteristics of homogeneous gases in complex systems are an important issue in many areas, including underground mines. The flow in mine excavations and ventilation systems is described by known mathematical relationships that could be applied to various cases. In this paper, a flow in a duct with a local sink of mass and momentum for multiple variants of cooperation of a mechanical fan was analyzed. The relationships for the total and static pressure of air in the duct were derived. In the next stage, a calculation example of how the mass flow rate of air, and the total and static pressure of the flowing air will change in the tested sections for the duct with and without a sink, is presented. The derived formulas and calculated values for the considered calculation case allow the verification of the obtained relationships at the measurement station. Analyzing the results of the examples presented in the article, it can be concluded that the total and static pressure at the sink point differ depending on the equation of motion used. In the case of the classic equation, the value of total pressure is lower than the value calculated from the new equation of motion, and the difference between them is about 20 Pa. In the case of static pressure, this difference is about 46 Pa. Qualitative differences in the static pressure distribution at the release location were also demonstrated. Depending on the applied approach, positive or negative changes in the static pressure are noticed. The presented form of the equation of motion made it possible to determine the flow characteristics in the duct with a point mass and momentum sink in the case of the operation with and without a fan.

1. Introduction

Networks transporting specific media through ducts or pipes are often analyzed in fluid mechanics and topics related to various ventilation systems [1,2,3,4,5,6]. These ducts connected in a network are usually tight. The flow is forced by an operating mechanical source, which can be positioned in different places of the ducts. Such systems operate in a steady state, ensuring the network’s constant flow (transport) parameters and stable operating conditions of the mechanical energy source [7]. The fluid movement in networks is induced by the operating of flow machines, such as compressors or fans. They aim to convert the engine’s mechanical energy into gas movement [6]. Compressors are used in methane drainage systems and compressed air installations [2]. Fans are used to ventilate corridor workings in streamlined and separate ventilation [3,6]. The pressure generated by the fans overcomes the resistance of the network of workings, ventilation devices, and air ducts. Therefore, air movement in ducts is a case of flow with the exchange of mass, momentum, and energy between the streams inside the duct, pipe, or excavation [8].

Mathematical models of fluid flows are widely used in computational analyses for the optimization and increase in the efficiency of industrial installations [9,10,11,12]. The flow characteristics in complex systems are an issue commonly discussed in the context of gas transition pipeline networks such as natural gas pipelines [13,14,15]. Sidarto et al. [13] proposed a new approach to determining the gas pressure distribution in a pipeline network. The analysis was possible by solving a set of nonlinear equations of node pressures and edge flow rates. The nonlinear equation based on the topological relation and the conservation laws of mass, energy, and momentum for the ventilation network was derived by Zhu et al. [16]. The simulation was then established and calculated in MATLAB Simulink. Simplified mathematical models are often used in simulations and optimization processes of gas networks. A comparative analysis of models used for hydraulic calculations of gas networks was presented, for example, in the work of Osiadacz and Gburzyńska [17]. Herrán-González et al. [18] present two simplified models based on the set of partial differential equations governing the dynamics of the process for gas ducts. The authors took into account the slope of the ducts. In this paper, two numerical schemes for the integration of such models are presented and implemented using the MATLAB Simulink tool. A gas drainage network model based on graph theory principles was proposed by Zhou et al. [19]. A gas–air mixed flow model includes the state, parameters, flow laws, and binary boundary conditions of the mixture.

The literature also contains research regarding new algorithms and calculation approaches describing the flows of mine ventilation networks [20,21,22,23,24,25,26,27]. Mathematical models are often used for this purpose, allowing for increasing the accuracy of the flow characteristics analysis in the excavations and simplifying the complex calculation by adding some necessary assumptions. It can also be a good method for optimizing ventilation networks, allowing the assessment of many operating scenarios [22,23]. The paper of Sereshi et al. [24] presents a summary of manual and computerized methods for the mine ventilation networks analysis, including, among others, the Hardy Cross method, Newton–Raphson technique, linear analysis, and non-linear programming. The simplified mathematical model of the Q-H graph for the mine ventilation network based on the node adjacency matrix theory was presented by Jia et al. [25]. As the authors have proven, this approach can effectively improve the analysis speed and, at the same, time reflect the overall features of the network and describe a local situation in detail. A mathematical model of air distribution for large mine ventilation networks was proposed by Semin and Levin [26]. The authors noted that the commonly used approach, where calculations are carried out by solving equations representing Kirchhoff’s circuit laws, does not always give good results. The paper proposes a mathematical model of a ventilation network that considers shock losses. The subject of ventilation networks of small mines was addressed in the work of Novella-Rodriguez et al. [27]. The authors proposed a mathematical air distribution model in the mine ventilation network, considering the change in ventilation mode. As the analysis showed, the theoretical calculations agree quite well with the experimental data.

Computer software is commonly used during the flow modeling of mine ventilation networks. Among the tools often used in this type of simulation are such software as Ventsim DESIGN, VnetPC, or VentGraph [28,29,30,31,32,33,34,35]. Maleki et al. [32] presented simulation results using VENTSIM software and a mathematical programming model. The paper aimed to optimize the ventilation system in terms of cost reduction. The authors presented the results of manual design and simulation using VENTSIM software, and then the optimization process using mathematical programming. In the article of Nukić and Delić [33], the simulation results of a ventilation network were presented. The authors used the VnetPC software package and the CFD software Fluent. Commonly applied in mining for calculations of steady states of ventilation networks is VentGraph [34,35]. This software uses a mathematical model based on the classical equation of motion. The software can take into account external inflow in two ways, but still, the classical equation of motion is used in all branches of the ventilation network.

The article focuses on general ventilation networks made of tight ducts considering the point leak. In the theoretical models used so far in this type of issue, this was analyzed using the classical equation of motion. The article analyzes various cases of flow with and without a mass sink to check the differences between the classical approach [1,2,3,6] and the new approach proposed in the article to the issue of gas transport through a duct in the presence of a point gas loss of the same density and diversified momentum. The article [36] discusses the problem of gas transport through ducts in the presence of a point gas sink of the same density and different momentum. The authors showed that the mass and momentum loss (source) in the duct affect the form of the equation of motion for such a duct. Therefore, further analysis of various cases of flow with mass loss was undertaken to check the differences between the approach with the classical equation of motion [1,2,3,6] and the new one. It is expected that the mentioned differences will concern the values of mass flows and total fan pressures and the distributions of static pressures along the length of the duct under study.

2. Flow With and Without Mass Sink—Analyzed Cases

The theoretical foundations of designing such systems use the principle of mass conservation and the principle of momentum conservation, which for such fixed problems can be written in the form of a system of differential equations dependent on the distance $x$ traveled by a fluid stream of constant density $\rho$, the same as in the external environment of the tested duct. For flows without sources and sinks of mass and momentum, the above-mentioned principles lead to the system of Equation (1) [2]:

$$\left\{\begin{array}{c}{\displaystyle \frac{d\dot{m}\left(x\right)}{dx}}=0\\ {\displaystyle \frac{1}{2 \rho F^2}} {\displaystyle \frac{d{\dot{m}\left(x\right)}^2}{dx}}+{\displaystyle \frac{dp\left(x\right)}{dx}}+\rho g{\displaystyle \frac{dz\left(x\right)}{dx}}+r_{zas}^{\ast }\left(x\right)·\dot{m^2}\left(x\right)={∆p}_c\left(\rho \right)\delta (x-x_w) \end{array}\right.$$

(1)

where:

• $\dot{m}\left(x\right)$—mass flow rate of fluid, kg/s,
• p(x)—absolute static pressure of the fluid (the air stream), Pa,
• $F$—cross-sectional area of the duct, m²,
• ρ—density, kg/m³,
• g—gravity, m/s²,
• z(x)—height of the conduit at the coordinate, m,
• $r_{zas}^{\ast }\left(x\right)$—duct’s equivalent resistance per unit, kg⁻¹ m⁻²,
• ∆p_c(ρ)—total pressure increase as a function of density for a working fan, Pa,
• $\delta (x-x_w)$—Dirac delta function distribution, 1/m,
• x_w—coordinates of the location of the mechanical energy source, m.

Integrating the system of Equation (1) circularly over a closed path (with $\rho \left(x\right)=\rho =const$—density equal to the density of the medium in the external environment of the duct), Equation (2) is obtained, in the form:

$$\left\{\begin{array}{c}\dot{m}\left(x\right)={\dot{m}}_{01} \\ {R^{\ast }}_{zas}\left(\rho \right) · {{\dot{m}}_{01}}^2={∆p}_c\left(\rho \right) H\left(x-x_w\right) \end{array}\right.$$

(2)

where:

• $R_{zas}^{\ast }\left(\rho \right)$—equivalent resistance of the entire duct, (kg·m)⁻¹,
• $\mathcal{H}\left(x-x_w\right)$—Heaviside’s function of unit stroke,
• ${\dot{m}}_{01}$—mass flow rate of fluid at the inlet to the main duct, kg/s.

This means that the same mass flow rate of air ${\dot{m}}_{01}$ flows along the flow path shown in Figure 1, which includes two sections of the duct (1-2) and (2-3) connected in series.

The loss of mechanical energy (pressure) $W_{L\left(1\text{-}2\text{-}3\right)}$ over the entire path (1-2-3) is defined by Formula (3) and is balanced by the total pressure of the installed source of mechanical energy ${∆p}_c\left(\rho \right)$, regardless of the location of this source in the duct.

$$W_{L\left(1\text{-}2\text{-}3\right)}=W_{L\left(1\text{-}2\right)}{+W}_{L\left(2\text{-}3\right)}={R^{\ast }}_{\left(1\text{-}2\right)}·{{\dot{m}}^2}_{01}+{R^{\ast }}_{\left(2\text{-}3\right)}·{{\dot{m}}^2}_{01}=R_{zas\left(1\text{-}3\right)}^{\ast }·{\dot{m^2}}_{01}$$

(3)

Hence, the commonly used interpretation of the so-called Second Kirchhoff’s Law, which states that the algebraic sum of all frictional pressure drops around any closed mesh, less any fan and natural ventilation pressure, is equal to zero [6]. This statement could be written in the form of Equation (4) considering the possibility of several sources of mechanical energy (i) occurring in the duct.

$$W_{L\left(1\text{-}2\text{-}3\right)}=\sum ^i∆p_{ci}\left(\rho \right)$$

(4)

The mass flow rate ${\dot{m}}_{01}$ is determined from Equations (3) and (4), where usually the total pressure ${∆p}_c\left(\rho \right)$ is determined for a specific energy source using the approximation Formula (5):

$$∆p_c\left(\rho \right)=\left(a·{\dot{m}}_{01}^3+b·{\dot{m}}_{01}^2+c·{\dot{m}}_{01}+d\right)·\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)$$

(5)

where the coefficients a, b, c, and d can be determined for the fan catalog characteristics specified for air density ${\rho }_0=1.2 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. If a gas with other density $\rho \ne {\rho }_0$ is to flow through the fan, the fan pressure should be determined from the Formula (5) [36]. In the event that the loss of mechanical energy (pressure) is determined using the catalog value of the specific resistance of the duct $R_{zas}$, kg/m⁷ or the catalog value of the unit distributed resistance $r$, kg/m⁸, it should be remembered that they were determined for density ${\rho }_0=1.2 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$. To use Formula (3), the resistances should be recalculated according to the following formula (with the additional assumption that there are no local resistances in the duct):

$$R_{zas\left(1\text{-}2\right)}^{\ast }\left(\rho \right)=R_{zas\left(1\text{-}2\right)}^{\ast }\left({\rho }_0\right){\displaystyle \frac{{\rho }^2}{{\rho }_0^2}}$$

(6)

$$r_{zas\left(1\text{-}2\right)}^{\ast }\left(\rho \right)=r_{zas\left(1\text{-}2\right)}^{\ast }\left({\rho }_0\right){\displaystyle \frac{{\rho }^2}{{\rho }_0^2}}={r\left({\rho }_0\right)}_{\left(1\text{-}2\right)}{\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^2$$

(7)

where:

• $r_{zas}^{\ast }\left(x\right)$—duct’s equivalent resistance per unit, kg⁻¹ m⁻².

Regarding the above considerations for the case shown in Figure 1, where $r_{\left(1\text{-}2\right)}^{\ast }\left({\rho }_0\right)=r_{\left(2\text{-}3\right)}^{\ast }\left({\rho }_0\right)=r^{\ast }\left({\rho }_0\right)$ and $R_{\left(1\text{-}3\right)}^{\ast }\left(\rho \right){\dot{·m}}_{01}^2=R_{\left(1\text{-}3\right)}^{\ast }\left({\rho }_0\right){·\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^2{\dot{·m}}_{01}^2$, the Formula (8) could be written.

$$R_{zas\left(1\text{-}2\right)}^{\ast }\left(\rho \right){·\dot{m}}_{01}^2+R_{zas\left(2\text{-}3\right)}^{\ast }\left(\rho \right){\dot{m}}_{01}^2=r^{\ast }\left({\rho }_0\right)(x_w-x_0)·{\dot{m}}_{01}^2·{\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^2$$

(8)

The second formula in the system of Equation (2) can be written as:

$$\left(a·{\dot{m}}_{01}^3+b·{\dot{m}}_{01}^2+c·{\dot{m}}_{01}+d\right){\displaystyle \frac{\rho }{{\rho }_0}}·\mathcal{H}\left(x-x_w\right)-R_{\left(1\text{-}3\right)}^{\ast }\left({\rho }_0\right)·{\left({\displaystyle \frac{\rho }{{\rho }_0}}\right)}^2·{\dot{m}}_{01}^2=0$$

(9)

Knowing the values of $a$, $b$, $c$, $d$, $\rho$, $R_{\left(1\text{-}3\right)}^{\ast }\left({\rho }_0\right)$, Equation (9) can be solved, and thus the mass flow rate ${\dot{m}}_{01}$, which will be established in the duct (1-3) during the operation of a given source of mechanical energy, can be determined. The operating point $\mathrm{P}$ of the fan has coordinates $\mathrm{P}$($\dot{m}={\dot{m}}_{01},; ∆p_c=\left(a·{\dot{m}}_{01}^3+b·{\dot{m}}_{01}^2+c·{\dot{m}}_{01}+d\right){\displaystyle \frac{\rho }{{\rho }_0}}$). For the assumptions made that $\rho \left(x\right)=const$ along the entire length of the duct, the operating point $\mathrm{P}$ of the fan will be the same regardless of the location of the fan installed in the duct, except for the location of the fan at both ends of the duct.

If the fan operates as a suction fan at the end of the duct ($x_w=x_k$), then the total pressure is equal to:

$${∆p}_c\left(\rho \right)=\left(a·{\dot{m}}_{01}^3+b·{\dot{m}}_{01}^2+c·{\dot{m}}_{01}+d-{\displaystyle \frac{{{\dot{m}}_{01}}^2}{2 \rho {F_{wyl}}^2}}\right){\displaystyle \frac{\rho }{{\rho }_0}}$$

(10)

where:

• ${\mathrm{F}}_{\mathrm{w}\mathrm{y}\mathrm{l}}$—cross-sectional area of the fan outlet (diffuser), m².

When the fan is at the beginning of the duct and works as a pressure fan, then its total pressure is described by the Formula (11):

$${∆p}_c\left(\rho \right)=\left(a·{\dot{m}}_{01}^3+b·{\dot{m}}_{01}^2+c·{\dot{m}}_{01}+d+{\displaystyle \frac{{{\dot{m}}_{01}}^2}{2 \rho {F_{wl}}^2}}\right){\displaystyle \frac{\rho }{{\rho }_0}}$$

(11)

where:

• ${\mathrm{F}}_{\mathrm{w}\mathrm{l}}$—cross-sectional area of the fan inlet (suction), m².

When the fan is built inside the duct, then the total pressure is described by the polynomial (12):

$${∆p}_c\left(\rho \right)=\left(a·{\dot{m}}_{01}^3+b·{\dot{m}}_{01}^2+c·{\dot{m}}_{01}+d\right){\displaystyle \frac{\rho }{{\rho }_0}}$$

(12)

When there is no mass sink in the duct ($\dot{D}=0—$mass flow rate of local source or outflow of mass in kg/s), as in Figure 1 at $\rho ={\rho }_0$, then such a case is described by the system of Equation (13). Knowing the values of $a$, $b$, $c$, $d$, $F_{wyl}$ (fan parameters) and $R^{\ast }$ (duct parameters), the value of ${\dot{m}}_{01}$ can be determined.

$$\left\{\begin{array}{c}\dot{m}\left(x=0\right)={\dot{m}}_{01} \\ R^{\ast }·{\dot{m^2}}_{01}=∆p_{c}H\left({x-x}_w\right)=a·{\dot{m}}_{01}^3+b·{\dot{m}}_{01}^2+c·{\dot{m}}_{01}+d- {\displaystyle \frac{{{\dot{m}}_{01}}^2}{2 \rho {F_{wyl}}^2}}\end{array}\right.$$

(13)

where:

• $R^{\ast }$—specific resistance of the duct, (kg·m)⁻¹.

3. Analysis of Air Mass Flows and Fan Depression in a Duct

3.1. A Duct with Mass and Momentum Sink and a Suction Fan at Its End—A New Form of the Motion Equation

In the work [36], a duct with a local mass sink located at coordinate xa was examined. The mass sink with a constant mass flow rate $\dot{D}$, and a medium with the same density $\rho$ and momentum as the one flowing in the duct section before the sink, was considered. This situation is illustrated in Figure 2.

For such a case, a new form of the equation of motion was given in [36] and then, after integrating the system of equations describing this case along a circular path, the system of Equation (14) is obtained:

$$\left\{\begin{array}{c}\dot{m}\left(x\right)={\dot{m}}_0-\dot{D}H\left({x-x}_a\right) \\ {\displaystyle \frac{1}{\rho F^2}}\left[-2{\dot{m}}_0·\dot{D}\mathcal{H}\left({x-x}_a\right)+\dot{D^2}{\mathcal{H}}^2\left({x-x}_a\right)\right]+W_{L\left(1\text{-}3\right)}={∆p_c}_{\left({\dot{m}}_0\right)}H\left({x-x}_w\right) \end{array}\right.$$

(14)

The second formula in the system of equations can be written in the form (15):

$$\begin{gathered} R^{\ast }{{\dot{m}}_0}^2-\left[2\left(1-n\right)R^{\ast }·\dot{D}+{\displaystyle \frac{2\dot{D}}{\rho F^2}}\right]{\dot{m}}_0+\left[\left(1-n\right)R^{\ast }-{\displaystyle \frac{1}{\rho F^2}}\right]{\dot{D}}^2 \\ =a·{\left({\dot{m}}_0-\dot{D}\right)}^3+b·{\left({\dot{m}}_0-\dot{D}\right)}^2+c·\left({\dot{m}}_0-\dot{D}\right)+d-{\displaystyle \frac{{\left({\dot{m}}_0-\dot{D}\right)}^2}{2\rho {F_{wyl}}^2}} \end{gathered}$$

(15)

where the parameter $\mathrm{n}$—ratio of the resistance 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 < n < 1).

Knowing the values of $R^{\ast }, n, \dot{D}, \rho , F, F_{wyl}, a, b, c$ and $d$, it is possible to calculate the value of ${\dot{m}}_0$, as well as the fan operating point, which will have coordinates [${\dot{m}}_0; a·{\left({\dot{m}}_0-\dot{D}\right)}^3+b·{\left({\dot{m}}_0-\dot{D}\right)}^2+c·\left({\dot{m}}_0-\dot{D}\right)+d-{\displaystyle \frac{{\left({\dot{m}}_0-\dot{D}\right)}^2}{2\rho {F_{wyl}}^2}}=∆p_c\left({\dot{m}}_0\right)$].

3.2. A Duct with Mass and Momentum Sink and a Blowing Fan at Its End—A New Form of the Motion Equation

The next case analyzed is a duct with a mass and momentum sink and a blowing fan at its end using a new form of the equation of motion. This variant is presented in Figure 3.

For such a case, the equivalent of Equation (14) is Formula (16):

$$\begin{gathered} R^{\ast }·{{\dot{m}}_0^{\prime }}^2+\left[2\left(1-n\right)R^{\ast }·\dot{D}+{\displaystyle \frac{2\dot{D}}{\rho F^2}}\right]{\dot{m}}_0^{\prime }+\left[\left(1-n\right)R^{\ast }-{\displaystyle \frac{1}{\rho F^2}}\right]{\dot{D}}^2 \\ =a·{\left({\dot{m}}_0^{\prime }+\dot{D}\right)}^3+b·{\left({\dot{m}}_0^{\prime }+\dot{D}\right)}^2+c·\left({\dot{m}}_0^{\prime }+\dot{D}\right)+d+{\displaystyle \frac{{\left({\dot{m}}_0^{\prime }+\dot{D}\right)}^2}{2\rho {F_{wl}}^2}} \end{gathered}$$

(16)

and ${\dot{m}}_0^{\prime }$ means the mass stream of air pumped behind the fluid release, kg/s.

Knowing the values of $R^{\ast }, n, \dot{D}, \rho , F, F_{wl}, a, b, c$ and $d$, it is possible to calculate the value of ${\dot{m}}_0^{\prime }$, as well as the fan operating point, which will have coordinates [${\dot{m}}_0^{\prime }; a·{\left({\dot{m}}_0^{\prime }+\dot{D}\right)}^3+b·{\left({\dot{m}}_0^{\prime }+\dot{D}\right)}^2+c·\left({\dot{m}}_0^{\prime }+\dot{D}\right)+d+{\displaystyle \frac{{\left({\dot{m}}_0^{\prime }+\dot{D}\right)}^2}{2\rho {F_{wl}}^2}}=∆p_c\left({\dot{m}}_0^{\prime }\right)$].

3.3. A Duct with Mass and Momentum Sink and a Suction Fan at Its End—The Classic Form of the Motion Equation

If there is a sink in the duct with a suction fan working at its end (Figure 4) and the classical equation of motion is used for the analysis, then after integrating the system of equations analogous to the system (14), Formula (17) is obtained.

$$\left\{\begin{array}{c}\dot{m}\left(x\right)={\dot{m}}_{02}-\dot{D}H\left({x-x}_a\right)\\ {n·R^{\ast }·\dot{m}}_{02}+\left(1-n\right)R^{\ast }{\left({\dot{m}}_{02}-\dot{D}\right)}^2=a·{\left({\dot{m}}_{02}-\dot{D}\right)}^3+b·{\left({\dot{m}}_{02}-\dot{D}\right)}^2+c·\left({\dot{m}}_{02}-\dot{D}\right)+d-{\displaystyle \frac{{\left({\dot{m}}_{02}-\dot{D}\right)}^2}{2\rho {F_{wyl}}^2}}\end{array}\right.$$

(17)

where:

• ${\dot{m}}_{02}$—mass flow rate in the additional source, kg/s.

By substituting the values of $R^{\ast }, n, \dot{D}, a, b, c, d$, and $F_{wyl}$ into Equation (17), it is possible to calculate the value of ${\dot{m}}_{02}$ and determine the operating point of the fan, which will have the coordinates [${\dot{m}}_{02}; a·{\left({\dot{m}}_{02}-\dot{D}\right)}^3+b·{\left({\dot{m}}_{02}-\dot{D}\right)}^2+c·\left({\dot{m}}_{02}-\dot{D}\right)+d-{\displaystyle \frac{{\left({\dot{m}}_{02}-\dot{D}\right)}^2}{2\rho {F_{wyl}}^2}}=∆p_c\left({\dot{m}}_{02}\right)$].

The determined values of ${\dot{m}}_{01}, {\dot{m}}_0, {\dot{m}}_0^{\prime }$, and ${\dot{m}}_{02},$ as well as the corresponding total fan pressures for the duct with a sink, were collected in the system of Equation (18):

$$\left\{\begin{array}{c}a·{\left({\dot{m}}_{02}-\dot{D}\right)}^3+b·{\left({\dot{m}}_{02}-\dot{D}\right)}^2+c·\left({\dot{m}}_{02}-\dot{D}\right)+d-{\displaystyle \frac{{\left({\dot{m}}_{02}-\dot{D}\right)}^2}{2\rho {F_{wyl}}^2}}\\ a·{\left({\dot{m}}_0-\dot{D}\right)}^3+b·{\left({\dot{m}}_0-\dot{D}\right)}^2+c·\left({\dot{m}}_0-\dot{D}\right)+d-{\displaystyle \frac{{\left({\dot{m}}_0-\dot{D}\right)}^2}{2\rho {F_{wyl}}^2}}\\ a·{\left({\dot{m}}_0^{\prime }+\dot{D}\right)}^3+b·{\left({\dot{m}}_0^{\prime }+\dot{D}\right)}^2+c·\left({\dot{m}}_0^{\prime }+\dot{D}\right)+d+{\displaystyle \frac{{\left({\dot{m}}_0^{\prime }+\dot{D}\right)}^2}{2\rho {F_{wl}}^2}}\end{array}\right.$$

(18)

The first relation in the system of Equation (18) determines the total pressure of the suction fan operating on the duct with a relief valve using the classical equation of motion. The second equation determines the same parameter but for the new equation of motion. The third equation is the total pressure of the force fan mounted in the duct with a sink, determined based on the new equation of motion.

From the above considerations, it follows that the mass flow volume in the duct without a sink depends on the parameters of the mechanical fan and its location. If the fan is located at the beginning (end) of the duct and works as a blowing one, then the mass flow of air flowing through such a duct is the largest. The flow in the duct will be slightly smaller if the fan is installed inside the duct, and the smallest size of the flow in the duct will be if the fan is installed at the end of the duct and works as a suction one. Such results, obtained by the derived formulas, are confirmed by observations in the field of mine ventilation and industrial ventilation.

The above observations can be used for verification by measurements, but the differences in the calculated mass flow values can be small because they also depend on the slope of the total pressure characteristic of the mechanical fan. The steeper the branch of the fan characteristic on which it operates is, the smaller will be the measured differences in air flows in the duct resulting from the cases considered above. Then, it seems easier to compare the position of the fan operating points and their depressions, resulting from calculations and their measurements, for different cases of airflow in the duct without and with a sink, using the classical form of the equation of motion and its new form.

The above considerations allow for a more precise determination of air mass flows in individual sections of the tested duct. Thanks to this, it will be possible to confirm the accuracy of the new form of the equation of motion and the determined influence of mass and momentum sink in the duct on the parameters of the mechanical fan operating in the duct by laboratory tests.

Without prejudging the results of the above verification, an additional contribution to this topic is presented below.

3.4. A Duct with Mass and Momentum Sink Without a Mechanical Fan

Next, the case of a duct with a point mass sink but with no mechanical energy source is considered. This is shown in Figure 5.

In such a case, using a system of equations containing the continuity equation and the classical equation of motion, it is found that airflow in such a duct does not occur. In the classical equation of motion for this case, there is no source of mechanical energy or natural energy that would force such motion. In practice, however, such motion does occur and is caused by the kinetic energy of mass release, which depends on the parameters of the external environment of the tested duct. By applying the new equation of motion to such a case, we can determine the mass flows of air on both sections of this duct. We can distinguish two closed loops:

• duct on section <0, a> with air mass flow ${\dot{m}}_{03}$ (kg/s) and a sink with kinetic energy equal to ${{\displaystyle \frac{1}{\rho F^2}}\dot{D}}^2{\mathcal{H}}^2(x-x_a)$;
• duct on section <w, a> with air mass flow $\dot{D}-{\dot{m}}_{03}$ and a sink with kinetic energy equal to ${{\displaystyle \frac{1}{\rho F^2}}\dot{D}}^2{\mathcal{H}}^2(x-x_a)$.

Using the integrated new equation of motion, which is part of the system of Equation (14), in the studied problem ${\dot{m}}_0=0$ and for both closed loops we can write the following relations (19)–(22):

$$W_{L\left(0\text{-}a\right)}=W_{L\left(w\text{-}a\right)}$$

(19)

in which $W_L$—loss of mechanical energy due to resistance to movement on the wire section, from its beginning ($x$ = 0) to the coordinate $x$ = a or $x$ = w, Pa.

$$n·R^{\mathrm{*}}·{{\dot{m}}_{03}}^2=\left(1-\mathrm{n}\right){\mathrm{R}}^{\ast }{\left(\dot{D}-{\dot{m}}_{03}\right)}^2$$

(20)

$${\displaystyle \frac{{\dot{m}}_{03}}{\dot{D}-{\dot{m}}_{03}}}=\sqrt{{\displaystyle \frac{\left(1-\mathrm{n}\right)}{n}}{\mathrm{R}}^{\ast }}$$

(21)

$${\dot{m}}_{03}=\dot{D}{\displaystyle \frac{\sqrt{\frac{\left(1-\mathrm{n}\right)}{n}{\mathrm{R}}^{\ast }}}{\left(1+\sqrt{\frac{\left(1-\mathrm{n}\right)}{n}{\mathrm{R}}^{\ast }}\right)}} , 0<n<1$$

(22)

As can be seen, the use of the new form of the equation of motion for this case made it possible to determine the mass flow rates in the tested duct, which could not be achieved using the classical form of the equation of motion.

The influence of mass and momentum sink on the distribution of total and static air pressure in the duct is determined in the further part of the article.

4. Distribution of Total and Static Pressure in a Duct

4.1. A Duct Without Mass and Momentum Sink

4.1.1. Case with a Suction Fan at the End of the Duct

The analysis of the distribution of total and static air pressure in the duct is first explained in the case of a duct without a sink and with a suction fan operating at the end of the duct. This is shown in Figure 1 and Figure 6.

When there is no sink in the duct, then the same value of the mass flow ${\dot{m}}_0$ flows through the entire duct.

From the system of Equation (2) containing the continuity equation and the classical equation of motion, it follows that for the tested duct and the location of the mechanical fan at the end, the value of ${\dot{m}}_0$ is calculated from the equation resulting from the circular integration of the aforementioned system of equations, as given:

$$R^{\mathrm{*}}{{\dot{m}}_0}^2=∆p_c·\mathcal{H}(\mathrm{x}-x_w)$$

(23)

$$∆p_c=d+c·{\dot{m}}_0+b·{{\dot{m}}_0}^2+a·{{\dot{m}}_0}^3-{\displaystyle \frac{{{\dot{m}}_0}^2}{2\rho ·F_{wyl}^2}}$$

(24)

The value of ${\dot{m}}_0$ is determined from the solution of Equation (25):

$$-d-c·{\dot{m}}_0-{{\dot{m}}_0}^2\left(R^{\mathrm{*}}-b+{\displaystyle \frac{1}{2\rho ·{F_{wyl}}^2}}\right)-a·{{\dot{m}}_0}^3=0$$

(25)

After calculating ${\dot{m}}_0$, the total fan pressure can be determined at the same time. Then, to determine how the air pressure changes during its flow in the duct as a function of the variable x, which at the air inlet to the duct takes the value $x_0=0$, and at the end of the duct, i.e., $x=x_w=x_k$, the classical differential equation of motion [1,2,3] should be used. For the tested duct and the adopted simplifications $r^{\ast }\left(x\right)=r^{\ast }=const ($resistance of the duct section, kg⁻¹ m⁻²), $\rho$ = const, it has the form (26).

$${\displaystyle \frac{1}{2\rho F^2}}{\displaystyle \frac{d{\dot{m_0}}^2}{dx}}+{\displaystyle \frac{dp\left(x\right)}{dx}}+\rho g{\displaystyle \frac{dz}{dx}}+r^{\mathrm{*}}{\dot{m_0}}^2=∆p_c\delta \left(x-x_w\right)$$

(26)

The duct is horizontal, so ${\displaystyle \frac{dz}{dx}}=0$, and using (24) it could be written that:

$${\displaystyle \frac{d}{dx}}\left({\displaystyle \frac{{\dot{m_0}}^2}{2\rho F^2}}+p(x)\right)+r^{\mathrm{*}}{\dot{m_0}}^2=\left[d+c·{\dot{m}}_0-\left(b-{\displaystyle \frac{1}{2\rho ·F_{wyl}^2}}\right){{\dot{m}}_0}^2-a·{{\dot{m}}_0}^3\right]\delta \left(x-x_w\right)$$

(27)

The notation in brackets on the left side of Equation (27) is equal to the total pressure of the flowing air stream with density $\rho$, at the point with coordinate $x$, which can be written as:

$${\displaystyle \frac{{\dot{m_0}}^2}{2\rho F^2}}+p\left(x\right)=p_c(x)$$

(28)

where the component ${\displaystyle \frac{{\dot{m_0}}^2}{2\rho F^2}}$ is called the dynamic pressure of the flow. Equation (27) takes the form:

$${\displaystyle \frac{d}{dx}}{p}_c\left(x\right)+r^{\mathrm{*}}{\dot{m_0}}^2=∆p_c\delta \left(x-x_w\right)$$

(29)

After integrating both sides of Equation (29) concerning the variable $x$, it is obtained:

$$p_c\left(x\right)-p_c\left(x=0\right)+r^{\mathrm{*}}(x-0){\dot{m_0}}^2=∆p_c\mathcal{H}\left({x-x}_w\right)$$

(30)

or

$$p_c\left(x\right)=p_0-r^{\mathrm{*}}x {\dot{m_0}}^2+\left[d+c·{\dot{m}}_0+\left(b-{\displaystyle \frac{1}{2\rho ·F_{wyl}^2}}\right){{\dot{m}}_0}^2+\mathrm{a}·{{\dot{m}}_0}^3\right]\mathcal{H}(\mathrm{x}-x_w)$$

(31)

The total pressure $p_c\left(x=0\right)$ at the initial point of the duct is equal to the static pressure of the duct environment $p_0$.

The static pressure of air flowing through the duct as a function of path $x$ is described by the equation:

$$p\left(x\right)=p_c\left(x\right)-{\displaystyle \frac{{\dot{m_0}}^2}{2\rho F^2}}=p_0-r^{\mathrm{*}}x·{\dot{m_0}}^2-{\displaystyle \frac{{\dot{m_0}}^2}{2\rho F^2}}+\left[d+c·{\dot{m}}_0+\left(b-{\displaystyle \frac{1}{2\rho ·F_{wyl}^2}}\right){{\dot{m}}_0}^2+a·{{\dot{m}}_0}^3\right]\mathcal{H}({x-x}_w)$$

(32)

The course of pressure $p_c\left(x\right)$ and static air pressure $p\left(x\right)$ as a function of $x$ according to Formulas (30) and (32) is shown in Figure 7.

4.1.2. Case with a Blowing Fan at the Beginning of the Duct

When the fan is located at the inlet to the tested duct (without a sink), the air mass flow rate is marked by m to indicate that this is a different value than for the previous case. The total pressure pc of the flowing air is described by a differential equation, analogous to Equation (29), as in Formula (33):

$${\displaystyle \frac{d}{dx}}{p}_c\left(x\right)+r^{\mathrm{*}}{{\dot{m}}_{01}}^2=∆p_c\delta \left(x-{\mathrm{x}}_0\right)$$

(33)

After its integration over the variable x, the equation takes the form according to Formula (34):

$$p_c\left(x\right)=p_0-r^{\mathrm{*}}x {{\dot{m}}_{01}}^2+\left[d+c·{\dot{m}}_{01}+\left(b+{\displaystyle \frac{1}{2\rho ·F_{wlt}^2}}\right){{\dot{m}}_{01}}^2+a·{{\dot{m}}_{01}}^3\right]\mathcal{H}({x-x}_0)$$

(34)

In Equation (34), the quantity ${\displaystyle \frac{{\dot{m_{01}}}^2}{2\rho F_{wlot}^2}}$ determines the dynamic pressure on the suction part of the fan, where ${\mathrm{F}}_{\mathrm{w}\mathrm{l}\mathrm{t}}$ is the area of the fan suction nozzle. Consideration of this component in the total pressure of the fan results from the location of the blowing fan at the beginning of the tested duct. Given its location (and the modified formula for its pressure), the formula for $\dot{m_{01}}$ in the duct is determined from Equations (35) and (36):

$${W_L-∆p}_c=0$$

(35)

$$r^{\mathrm{*}}\left(x_k-x_0\right){{\dot{m}}_{01}}^2-\left[d+c·{\dot{m}}_{01}+\left(b+{\displaystyle \frac{1}{2\rho ·F_{wlt}^2}}\right){{\dot{m}}_{01}}^2+a·{{\dot{m}}_{01}}^3\right]=0$$

(36)

The value of ${\dot{m}}_{01}$ determined from Formula (36) is different from the value $\dot{m_0}$ determined for a duct without a sink with a fan at the end of the duct working as a suction source. Considering that $p_c\left(x\right)=p\left(x\right)+{\displaystyle \frac{{{\dot{m}}_{01}}^2}{2\rho F^2}}$, the static pressure as a function of the variable x for such a duct is defined by the Equation (37).

$$p\left(x\right)=p_0-r^{\mathrm{*}}{{\dot{m}}_{01}}^{2}x-{\displaystyle \frac{{\dot{m_{01}}}^2}{2\rho F^2}}+\left[d+c·{\dot{m}}_{01}+\left(b+{\displaystyle \frac{1}{2\rho ·{F_{wlt}}^2}}\right){{\dot{m}}_{01}}^2+a·{{\dot{m}}_{01}}^3\right]\mathcal{H}(0)$$

(37)

where $∆p_c\left({\dot{m}}_{01}\right)$ is the total pressure of the blowing fan at the beginning of the duct described by the Equation (38).

$$∆p_c\left({\dot{m}}_{01}\right)=\left[d+c·{\dot{m}}_{01}+\left(b+{\displaystyle \frac{1}{2\rho ·{F_{wl}}^2}}\right){{\dot{m}}_{01}}^2+a·{{\dot{m}}_{01}}^3\right]$$

(38)

The graphical distribution of the change in the total and static pressure in the duct with the blowing fan located at its beginning according to Equations (34) and (37) is presented in Figure 8.

4.1.3. Case with a Fan Inside

The case of a duct without a sink and a source of mass with a fan operating inside the duct (excluding extreme locations) is shown in Figure 9. The fan is built into the cross-section of the duct.

For the example shown in Figure 9, a mechanical fan located at coordinate $x_w$ has a total pressure characteristic approximated by the polynomial (39) [7].

$${∆p}_c\left({\dot{m}}_{02}\right)=d+c·{\dot{m}}_{02}+b·{{\dot{m}}_{02}}^2+a·{{\dot{m}}_{02}}^3$$

(39)

The mass flow rate ${\dot{m}}_{02}$ is determined from the Equation (40).

$$r^{\mathrm{*}}\left(x_k-x_0\right)·{\dot{m}}_{02}-\left[d+c·{\dot{m}}_{02}+b·{{\dot{m}}_{02}}^2+a·{{\dot{m}}_{02}}^3\right]=0$$

(40)

The total air pressure as a function of path $x$ is defined for this case by the Formula (41).

$$p_c\left(x\right){=p_0-r}^{\mathrm{*}}·x·{\dot{m}}_{02}+\left[d+c·{\dot{m}}_{02}+b·{{\dot{m}}_{02}}^2+a·{{\dot{m}}_{02}}^3\right]\mathcal{H}\left(x-x_w\right)$$

(41)

By contrast, the static air pressure as a function of path $x$ is defined for this case by the Equation (41).

$$p\left(x\right){=p_0-r}^{\mathrm{*}}·x·{\dot{m}}_{02}+\left[d+c·{\dot{m}}_{02}+b·{{\dot{m}}_{02}}^2+a·{{\dot{m}}_{02}}^3\right]\mathcal{H}\left({x-x}_w\right)-p_{dyn}$$

(42)

or

$$p\left(x\right){=p_0-r}^{\mathrm{*}}·x·{\dot{m}}_{02}+\left[d+c·{\dot{m}}_{02}+b·{{\dot{m}}_{02}}^2+a·{{\dot{m}}_{02}}^3\right]\mathcal{H}\left({x-x}_w\right)-{\displaystyle \frac{{\dot{m_{02}}}^2}{2\rho F^2}}$$

(43)

A graphical representation of the total and static air pressure change according to Formulas (41) and (43) is shown in Figure 10.

The three variants of fan location in the duct considered above without a mass sink or source made it possible to present the methodology of determining the total and static air pressure in the duct, which in the next steps were applied to the ducts with a sink. The obtained curves of the mentioned pressures in the duct, as functions of the path $x$, presented in Figure 7, Figure 8 and Figure 10, are presented in the literature [7,36] and are a good reference for analogous cases for the duct with a sink analyzed below.

4.2. A Duct with Mass and Momentum Sink

4.2.1. Case Without a Fan

As a first example, a duct system with a sink was considered in the absence of a mechanical fan. From the previous considerations, it follows that the new form of the equation of motion derived in [36] made it possible to determine the mass flows of air flowing through two sections of the duct in the direction of the mass sink. This is illustrated below in Figure 11.

In the absence of a fan, i.e., when $∆p_c=0$, the new form of the equation of motion integrated along the circular path leads to Equation (44):

$${\displaystyle \frac{1}{\rho F^2}}\left[-2{\dot{m}}_0\dot{D}\mathcal{H}\left(x-x_a\right)+{\dot{D}}^2{\mathcal{H}}^2\left(x-x_a\right)\right]+w_L=0$$

(44)

Without a fan, the mass flow ${\dot{m}}_0$ is equal to zero, then the above equation of motion has the form:

$${\displaystyle \frac{{\dot{D}}^2{\mathcal{H}}^2\left({x-x}_a\right)}{\rho F^2}}+w_L=0$$

(45)

Airflow in the duct will occur, then the energy loss $w_L>0$, because ${\displaystyle \frac{{\dot{D}}^2{\mathcal{H}}^2\left({x-x}_a\right)}{\rho F^2}}$ is different from 0. This component is a source of mechanical energy, defined by the value of the kinetic energy of the stream flowing out at the location with coordinate $x_a$ and related to the unit volume flow of the flowing air. The location of a point release of air kinetic energy inside the duct causes the air to flow from both ends of the duct to the release location point. We can distinguish two closed loops (I and II), in which the mentioned component ${\displaystyle \frac{{\dot{D}}^2{\mathcal{H}}^2\left(x-x_a\right)}{\rho F^2}}$ exists. The point mass sink energy for these loops can be represented by the equations:

$${\displaystyle \frac{{\dot{D}}^2{\mathcal{H}}^2\left(x-x_a\right)}{\rho F^2}}+w_{L1oczko}=0$$

(46)

$${\displaystyle \frac{{\dot{D}}^2{\mathcal{H}}^2\left(x-x_a\right)}{\rho F^2}}+w_{L2oczko}=0$$

(47)

where:

• $w_{L1oczko}$—loss of mechanical energy of the closed loop 1, Pa,
• $w_{L2oczko}$—loss of mechanical energy of the closed loop 2, Pa.

And hence:

$$w_{L1oczko}=w_{L2oczko}$$

(48)

Using Equations (46)–(48) and the values of ${\dot{m}}_{03}$ and $\left(\dot{D}-{\dot{m}}_{03}\right)$ calculated for this case, for the section of the duct in the loop I, it can be written that:

$${\displaystyle \frac{{\dot{m}}_{03}}{2\rho F^2}}+p_{Ioczko}\left(x\right)=p_{cIoczko}\left(x\right)$$

(49)

$$p_{Ioczko}\left(x\right)-p_{cIoczko}\left(x=0\right)+r^{\mathrm{*}}\left(x-0\right)·{{\dot{m}}_{03}}^2={\displaystyle \frac{{\dot{D}}^2\mathcal{H}\left({x-x}_a\right)}{\rho F^2}}$$

(50)

After transforming the total air flow as a function of path $x$ in the part of the duct at the distance $x\in <0,x_a>$ it has the form:

$$p_{cIoczko}\left(x\right)=p_0-r^{\mathrm{*}}\left(x-0\right)·{{\dot{m}}_{03}}^2+{\displaystyle \frac{{\dot{D}}^2\mathcal{H}\left({x-x}_a\right)}{\rho F^2}}$$

(51)

The static air pressure as a function of path $x$ in the loop I could be obtained from Equation (52).

$$p_{Ioczko}\left(x\right)=p_{cIoczko}\left(x\right)-{\displaystyle \frac{{\dot{m_{03}}}^2}{2\rho F^2}}=p_0-r^{\mathrm{*}}\left[x-{\displaystyle \frac{1}{2\rho F^2}}\right]·{{\dot{m}}_{03}}^2+{\displaystyle \frac{{\dot{D}}^2\mathcal{H}\left(x-x_a\right)}{\rho F^2}}$$

(52)

For the section of the duct $<x_k,x_a>$ forming part of the loop II, Formula (53) should be used.

$${\displaystyle \frac{{\left(\dot{D}-\dot{m_{03}}\right)}^2}{2\rho F^2}}+p_{IIoczko}\left(x\right)=p_{cIIoczko}\left(x\right)$$

(53)

After further transformations, it could be written that the total pressure as a function of path x calculated from the inlet $x_k$ to $x_a$ is expressed by Equation (54).

$$p_{cIIoczko}\left(x\right)=p_0\left(x=k\right)-r^{\mathrm{*}}\left(x_k-x\right)·{{\dot{m}}_{03}}^2={\displaystyle \frac{{\dot{D}}^2\mathcal{H}\left({x-x}_a\right)}{\rho F^2}}$$

(54)

The relationship for static pressure as a function of path has the form of Equation (55).

$$p_{IIoczko}\left(x\right)=p_{cIIoczka}\left(x\right)-{\displaystyle \frac{{\left(\dot{D}-{\dot{m}}_{03}\right)}^2}{2\rho F^2}}=p_0\left(x=k\right)-r^{\mathrm{*}}\left[\left(x_k-x\right)+{\displaystyle \frac{1}{2\rho F^2}}\right]{\left(\dot{D}-{\dot{m}}_{03}\right)}^2+{\displaystyle \frac{{\dot{D}}^2\mathcal{H}\left({x-x}_a\right)}{\rho F^2}}$$

(55)

Figure 12 shows the change of the total and static air pressures in the duct without a mechanical fan and with a mass sink at point $x_a$.

4.2.2. Case with a Suction Fan at the End of the Duct

The case of a duct with a fan operating in suction mode at the end of it and with a point mass sink is shown in Figure 8. For such a configuration, the new equation of motion integrated over a closed path has the form:

$${\displaystyle \frac{1}{\rho F^2}}\left[-2{\dot{m}}_0\dot{D}\mathcal{H}\left({x-x}_a\right)+{\dot{D}}^2{\mathcal{H}}^2\left(x-x_a\right)\right]+W_L=∆p_c\mathcal{H}\left(x-x_w\right)$$

(56)

where $W_L$ is the loss of mechanical energy due to resistance to motion in the entire tested duct, Pa.

The characteristic of the total pressure of the fan located at the end of the duct has the form (57).

$${∆p}_c=d+c·\left({\dot{m}}_0-\dot{D}\right)+b{\left({\dot{m}}_0-\dot{D}\right)}^2+a{\left({\dot{m}}_0-\dot{D}\right)}^3-{\displaystyle \frac{{\left({\dot{m}}_0-\dot{D}\right)}^2}{2\rho F_{wyl}^2}}$$

(57)

The loss of mechanical energy due to resistance to motion in the duct (pressure) has the form of Equation (58).

$$W_L=n·R^{\mathrm{*}} ·{{\dot{m}}_0}^2+\left(1-n\right)R^{\mathrm{*}}·{\left({\dot{m}}_0-\dot{D}\right)}^2$$

(58)

After removing the Haveside function and rearranging, Formula (59) was obtained.

$$\begin{array}{ll}(d-a·{\dot{D}}^3& +{\dot{D}}^2\left(b-{\displaystyle \frac{1}{2\rho F_{wyl}^2}}-\left(1-n\right)R^{\mathrm{*}}-{\displaystyle \frac{1}{\rho F^2}}\right)-c·\dot{D})\\ & +{\dot{m}}_0\left(c+3a·{\dot{D}}^2-2\dot{D}\left(b-{\displaystyle \frac{1}{2\rho F_{wyl}^2}}-\left(1-n\right)R^{\mathrm{*}}-{\displaystyle \frac{1}{\rho F^2}}\right)\right)\\ & +{\dot{m_0}}^2\left(b-R^{\mathrm{*}}-{\displaystyle \frac{1}{2\rho F_{wyl}^2}}-3a·\dot{D}\right)+a·{\dot{m_0}}^3=0\end{array}$$

(59)

By solving Equation (59) numerically, the mass flow rate ${\dot{m}}_0$, which flows in the initial section of the duct with a suction fan at its end, can be calculated.

After calculating the flow rate, the operating point of the mechanical fan with such a location of the outlet and with the assumed outlet flow rate $\dot{D}$ can be determined. After integrating over $x$ the new equation of motion given in [36], the relationship (60) for the horizontal duct can be written.

$$d{p}_c\left(x\right)+{\displaystyle \frac{1}{\rho F^2}}\left[-2{\dot{m}}_0\dot{D}\mathcal{H}\left({x-x}_a\right)+{\dot{D}}^2{\mathcal{H}}^2\left(x-x_a\right)\right]+w_L(x)=∆p_c\mathcal{H}\left({x-x}_w\right)$$

(60)

The total pressure of the fan ${∆p}_c\left(x\right)$ installed at the end of the duct ($x_k=x_w$) is defined by Formula (10). Then, Equation (60) can be written as Formula (61):

$$\begin{array}{ll}{p}_c\left(x\right)-p_c\left(x=0\right)& +{\displaystyle \frac{1}{\rho F^2}}\left[-2{\dot{m}}_0\dot{D}\mathcal{H}\left(x-x_a\right)+{\dot{D}}^2{\mathcal{H}}^2\left(x-x_a\right)\right]+w_L(x)\\ & =\left[d+c·\left({\dot{m}}_0-\dot{D}\right)+b{\left({\dot{m}}_0-\dot{D}\right)}^2+a{\left({\dot{m}}_0-\dot{D}\right)}^3-{\displaystyle \frac{{\left({\dot{m}}_0-\dot{D}\right)}^2}{2\rho F_{wyl}^2}}\right]\mathcal{H}\left({x-x}_k\right)\end{array}$$

(61)

Since $p_c\left(x=0\right)=p_{c0}=p_0$, Equation (61) can be written as a function of the path $x$ in the form (62).

$$p_c\left(x\right)=p_0-w_L\left(x\right)+{\displaystyle \frac{1}{\rho F^2}}\left[-2{\dot{m}}_0\dot{D}\mathcal{H}\left({x-x}_a\right)+{\dot{D}}^2{\mathcal{H}}^2\left(x-x_a\right)\right]+∆p_{c \left({\dot{m}}_0-\dot{D}\right)}\mathcal{H}\left({x-x}_w\right)=0$$

(62)

where $∆p_{c \left({\dot{m}}_0-\dot{D}\right)}\mathcal{H}\left({x-x}_w\right)$ is the total pressure of the fan installed at coordinate $x_w$ with a mass flow $\left({\dot{m}}_0-\dot{D}\right)$, Pa.

If $\left({\dot{m}}_0-\dot{D}\right)>0$, then the air flows in the same direction through the entire duct. Therefore, it could be written that the loss of mechanical energy of the air flow in the duct is additive, which is written using Formula (63).

$$w_L\left(x\right)=w_{L1}\left(x\right)+w_{L2}\left(x\right)$$

(63)

where:

$$w_{L1}\left(x\right)=r^{\mathrm{*}}x·{{\dot{m}}_0}^2 \mathrm{for} 0\le x\le x_a$$

(64)

$$w_{L1}\left(x\right)=w_{L1}\left(x_a\right) \mathrm{for} x_a\le x\le x_k$$

(65)

$$w_{L1}\left(x\right)=0 \mathrm{for} {x>x}_k$$

(66)

$$w_{L2}\left(x\right)=r^{\mathrm{*}}(x-x_a){·\left({\dot{m}}_0-\dot{D}\right)}^2 \mathrm{for} x_a<x\le x_k$$

(67)

$$w_{L2}\left(x\right)=0 \mathrm{for} x_k<x\le x_a$$

(68)

The quantities $w_{L1}\left(x\right)$ and $w_{L2}\left(x\right)$ are functions defined only in closed intervals of the variable $x$, but as additive functions over the entire length of the duct, i.e., from $x=0$ to $x=x_k$ they transfer their final values from the intervals. Below are the mathematical forms of these quantities that meet these requirements.

$$w_{L1}\left(x\right)=r^{\mathrm{*}}x·{{\dot{m}}_0}^2·\mathcal{H}\left(x_a-x\right)+r^{\mathrm{*}}{x}_a·{{\dot{m}}_0}^2·\mathcal{H}\left(x-x_a\right)$$

(69)

where $\mathcal{H}\left(x_a-x\right)$ is the unit step function (Heaviside function) and it could be expressed as (70).

$$\mathcal{H}\left(x_a-x\right)=\left\{\begin{array}{c}1 for x\le x_a\\ 0 for x>x_a\end{array}\right.$$

(70)

$$w_{L2}\left(x\right)=r^{\mathrm{*}}\left(x-x_a\right)·{\left({\dot{m}}_0-\dot{D}\right)}^2·\mathcal{H}\left(x-x_a\right)·\mathcal{H}\left(x_k-x\right)$$

(71)

where:

$$\mathcal{H}\left(x-x_a\right)=\left\{\begin{array}{c}1 for x\ge x_a\\ 0 for x<x_a\end{array}\right.$$

(72)

$$\mathcal{H}\left(x_k-x\right)=\left\{\begin{array}{c}1 for x\le x_k\\ 0 for x>x_k\end{array}\right.$$

(73)

It is also possible to write using the symbols adopted for the duct according to Formula (74).

$$n={\displaystyle \frac{x_a}{x_k}}$$

(74)

Using the above, Formula (62) can be given in the form of Equation (75).

$$\begin{gathered} \begin{array}{cc}\hfill p_c\left(x\right)=& p_0-r^{\mathrm{*}}x·{{\dot{m}}_0}^2·\mathcal{H}\left(x_a-x\right){-r}^{\mathrm{*}}{x}_a·{{\dot{m}}_0}^2·\mathcal{H}\left(x-x_a\right)-r^{\mathrm{*}}\left(x-x_a\right)·{\left({\dot{m}}_0-\dot{D}\right)}^2·\mathcal{H}\left(x-x_a\right)·\mathcal{H}\left(x_k-x\right)\hfill \\ & +{\displaystyle \frac{2}{\rho F^2}}{\dot{m}}_0\dot{D}\mathcal{H}\left({x-x}_a\right)-{{\displaystyle \frac{1}{\rho F^2}}\dot{D}}^2{\mathcal{H}}^2\left({x-x}_a\right)\left[d+c·\left({\dot{m}}_0-\dot{D}\right)+b{\left({\dot{m}}_0-\dot{D}\right)}^2+a{\left({\dot{m}}_0-\dot{D}\right)}^3- \\ {\displaystyle \frac{{\left({\dot{m}}_0-\dot{D}\right)}^2}{2\rho F_{wyl}^2}}\right]\mathcal{H}\left(x-x_w\right)\hfill \end{array} \end{gathered}$$

(75)

Formula (75) shows the changes in the total pressure of the airflow in a duct with a mass sink and a source of mechanical energy at its end as a function of the path. It is presented graphically in Figure 13.

Depending on components 5 and 6 of the right-hand side of Equation (75), a step in total pressure occurs at the point with coordinate $x_a$ and on the entire section $<x_a,x_k>$. This step will be positive if:

$${\displaystyle \frac{{2·\dot{m}}_0·\dot{D}}{\rho F^2}}>{{\displaystyle \frac{1}{\rho F^2}}\dot{D}}^2\Rightarrow {\displaystyle \frac{{2·\dot{m}}_0}{\rho F^2}}>{\displaystyle \frac{\dot{D}}{\rho F^2}}\Rightarrow {2·\dot{m}}_0>\dot{D}$$

(76)

Then, on the second section of the duct, an increase in total pressure $p_c$ will occur by the value of the algebraic sum of components 5 and 6 of the aforementioned Formula (75).

The distribution of static air pressure in the tested duct is determined by the relationship (77).

$$p_0=p_c\left(x\right)-p_{dyn}\left(x\right)$$

(77)

where $p_{dyn}\left(x\right)$ is the dynamic pressure of the airflow in the duct, Pa, and is expressed by the Formula (78).

$$p_{dyn}\left(x\right)={\displaystyle \frac{{{\dot{m}}_0}^2\left(x\right)\rho }{2{\rho }^2{F}^2}}={\displaystyle \frac{{{\dot{m}}_0}^2\left(x\right)}{2\rho F^2}}$$

(78)

The value of the dynamic pressure is equal to:

• for the duct section $<0,x_a>$ according to Equation (79):

  $$p_{dyn}\left(x\right)={\displaystyle \frac{{{\dot{m}}_0}^2}{2\rho F^2}}$$

  (79)
• for the duct section $<x_a,x_k>$ according to Equation (80):

  $$p_{dyn}\left(x\right)={\displaystyle \frac{{\left({\dot{m}}_0-\dot{D}\right)}^2}{2\rho F^2}}$$

  (80)

Considering Formulas (79) and (80) in Equation (75), the relationship for static air pressure in the duct $p\left(x\right)$ has the form:

$$\begin{array}{ll}p\left(x\right)=p_0& -\left(r^{\mathrm{*}}x·{{\dot{m}}_0}^2+{\displaystyle \frac{{{\dot{m}}_0}^2}{2\rho F^2}}\right)\mathcal{H}\left(x_a-x\right)-\left(r^{\mathrm{*}}{x}_a·{{\dot{m}}_0}^2+{\displaystyle \frac{{{\dot{m}}_0}^2}{2\rho F^2}}\right)\mathcal{H}\left(x-x_a\right)\\ & -\left(r^{\mathrm{*}}\left(x-x_a\right) {\left({\dot{m}}_0-\dot{D}\right)}^2+{\displaystyle \frac{{\left({\dot{m}}_0-\dot{D}\right)}^2}{2\rho F^2}}\right)\mathcal{H}\left(x-x_a\right)·\mathcal{H}\left(x_k-x\right)+{\displaystyle \frac{2}{\rho F^2}}{\dot{m}}_0\dot{D}\mathcal{H}\left(x-x_a\right)\\ & -{{\displaystyle \frac{1}{\rho F^2}}\dot{D}}^2{\mathcal{H}}^2\left(x-x_a\right)\\ & +\left[d+c·\left({\dot{m}}_0-\dot{D}\right)+b{\left({\dot{m}}_0-\dot{D}\right)}^2+a{\left({\dot{m}}_0-\dot{D}\right)}^3-{\displaystyle \frac{{\left({\dot{m}}_0-\dot{D}\right)}^2}{2\rho F_{wyl}^2}}\right]\mathcal{H}\left({x-x}_w\right)\end{array}$$

(81)

The rest of the article contains sample calculations for a duct with a sink and a suction fan operating at its end. It also provides a graphical representation of the changes in the total pressure for the duct tested in the example and the static pressure as a function of the path $x$, according to Formulas (80) and (81).

Due to the excessive volume of this article, the calculation example does not include formulas and courses of the tested air pressures in the duct with a sink and a different location of the mechanical fan. Implementation of this methodology for different configurations should not be complicated, because the calculation procedures given in this article will still be relevant.

5. Calculation Example

As a calculation example, a ventilation duct with a mass sink (or without) and a suction fan placed at its end was used. This duct corresponds to the parameters of a mine working. A 200 m long straight duct with a cross-sectional area of 25 m² and resistance $R^{\ast }$ = 0.0552 kg⁻¹·m⁻¹ was selected for calculations. It is assumed that a mechanical fan is installed at the end of the duct and operates as a suction fan. The cross-sectional area of the outlet opening (diffuser) from the fan is 12 m². The cross-sectional area of the suction part of the fan is 10 m². The air density was assumed to be ${\rho }_0$ = 1.2 kg/m³. The fan total pressure characteristic was given as a third-degree polynomial according to Equation (1), in which the appropriate coefficients are $d$ = 200; $c$ = 2.005; $b$ = 0.408; $a$ = −0.0025. The fan and duct characteristics are shown in Figure 14. The course of the total fan pressure characteristic is marked in blue, while the characteristic of the analyzed duct is highlighted in yellow.

Since the fan in the analyzed example operates in suction mode at the end of the duct, its total pressure characteristic ${∆p}_{c1}\left(m\right)$ is described by the relationship (10), and these characteristics as a function of the air mass flow are shown in Figure 15.

To better illustrate the fan characteristic curves, the area of Figure 15 has been enlarged and is presented in Figure 16 below.

The point of intersection of the duct characteristic with the suction fan characteristic has been replaced, resulting in a lower air mass flow rate than the total pressure characteristic given by the fan manufacturer.

Figure 17 shows the distribution of the total and static pressure in the analyzed duct without a mass sink, determined according to the relations (31) and (32).

Figure 15, Figure 16 and Figure 17 refer to a fan operating in a suction system in a duct considered as tight. For such a case, the classical equation of motion is correct and commonly used.

In the next step of the analysis, a mass sink located 50 m from the air inlet to the duct was assumed. The mass flow rate of the sink was adopted at the level of 50 kg/s. Using Formula (61), the mass airflow rate in the initial section of the duct (before the sink) was calculated. Equation (61) considers the form of the equation of motion derived in the paper [36], which includes the occurrence of a mass sink in the duct. Using the above and Kirchhoff’s II law for a closed loop containing the tested duct, the determination of the expected mass flow of air as 195.252 kg/s was possible. For such a case, the remaining section of the duct, i.e., from the sink to the suction fan installed at the end of the duct, a mass flow of 145.252 kg/s was obtained. The total pressure of the fan is 1376.856 Pa.

Using the relations derived in this article in Equations (75) and (81), the total and static air pressure as functions of the path in the tested duct with a sink was calculated and presented graphically in Figure 18.

Figure 19 presents the comparison of the obtained distribution of total pressure for a duct in which a mass sink occurred with the total pressure for the duct without a sink.

An analogous comparison for static pressure distributions is shown in Figure 20.

Additionally, an analysis was performed using the classical form of the equation of motion for the duct with a mass sink. Using Kirchhoff’s second law for a closed loop (59) and the relationship (58) for the pressure loss using the classical equation of motion, the mass flow rate of air (194.844 kg/s) that will flow through the first section of the tested duct (before the sink) was determined. In this case, the total pressure of the suction fan operating at the end of the duct was equal to 1392.469 Pa. The calculations show that the difference in the mass flow rate at the beginning of the duct is 0.408 kg/s. For the new motion equation, the calculated mass flow rate is higher than when using the classic motion equation. The difference in the total fan pressure is 15.6 Pa, but the obtained value is lower than in the classic motion equation. This means that the fan operating point according to the new motion equation is located lower on the fan characteristic than the operating point determined based on the classic motion equation. By using the new motion equation in the duct, better fan operating parameters were obtained. Using relationships (75) and (81), the values of the total and static air pressure were determined as a function of the path $x$ in the duct (using the classical form of the equation of motion). These values are graphically presented in Figure 21 and Figure 22 as functions of the length of the duct. In Figure 21, the courses of both values of the total pressure at the initial section of the tested duct are similar to each other. From the location of the mass sink to the end of the duct, the values of the total pressures are noticeably different. The total pressure from the sink point calculated from the model based on the new equation of motion is higher than that determined from the classical motion equation. A positive change in the value of the total pressure at the sink point is also noticeable for the model using the new equation of motion. Figure 22 presents the courses of the static pressure along the length of the pipe, obtained from the classical and new equation of motion. Similarly to Figure 21, at the initial section of the duct (from the inlet to the location of the sink), the static pressures obtained from both models are similar to each other. The figure shows that the static pressure calculated using the classical equation of motion calculated at the sink point shows a negative change in its value. On the other hand, the change in this pressure calculated based on the model based on the new model is positive at this point. For a better illustration of these quantities in the considered pipe sections, the differences in total and static pressures determined according to the new and classical equation of motion as a function of the pipe length are presented in Figure 23 and Figure 24.

Due to the relatively small differences between the compared values and to increase the readability of the presented results, additional calculations were introduced. The differences between the appropriate pressures obtained using classic and new forms of motion equations were calculated. Their results are presented graphically as functions of the path $x$ in Figure 23 and Figure 24.

These drawings show that in the first section of the duct (from the inlet to the sink point), the total and static pressure calculated according to the model with the new equation of motion is lower at every point of this section than the corresponding values calculated according to the classical equation of motion. The pressure differences are minimal. The differences in these pressures for the sink location point are significant for measurement purposes. The total pressure determined from the model with the new equation of motion is about 20 Pa higher than the pressure at this point determined with the classical equation of motion. The static pressure at the sink point calculated from the model with the new equation of motion is about 46 Pa higher than the static pressure at this point determined from the model with the classical equation of motion. The determined differences in these pressures at the sink location in the duct are large enough that it seems possible to use them to verify the equation of motion used. Over the entire length of the final section of the duct, the values of these pressures decrease linearly, but for the static pressure, they still exceed 40 Pa. This may have practical implications for safety in mines (e.g., related to gas emissions or in the issues of passive combating of contained underground fires).

6. Conclusions

This study is an extension of the earlier work [36] about the influence of a point sink and source of mass and momentum on pressure distribution in a duct with a mechanical fan. A new form of the equation of motion was derived, which affects the loss of mechanical energy in the entire duct and, therefore, the fan’s operating point.

This study shows the consequences of the new form of the equation of motion on the mass airflow in a duct with a point sink of mass and momentum in the case of the absence of a fan and its presence. In the absence of a mechanical fan in such a duct, it was confirmed that air flows directed from the ends of the tested duct to the location of the sink. A formula was derived for the size of these flows, which depends on the mass flow rate of the sink D, resistance R, and the location in the sink inside the duct (value “n”). These results cannot be obtained using the classical form of the known equation of motion. Due to this, the proposed new form of the equation of motion has a wider application than the classical equation of motion. Nevertheless, it requires confirmation by measurements in the case of a duct with a mass sink without a mechanical fan.

The case of a duct with a mass and momentum sink discussed in the research, at the end of which the fan operates in suction mode, confirmed that the values of the air mass flows depend on whether the mechanical fan is installed at the end (beginning) of the duct and what is its operating mode (suction or blowing), or inside the duct. Different values of the mentioned mass airflow also correspond to different values of the air mechanical energy loss on both sections of the duct. This results in obtaining different operating points of the installed fan. This research also showed the effect of the sink on the total and static air pressure as a function of the path $x$ in the duct. Mathematical relationships for such functions were derived and applied to the calculation example. The values calculated for this example show that, in the case of the model with the new equation of motion, a greater mass flow of air flowing into the duct with sink was obtained at a lower total pressure of the working suction fan. The difference in mass flow is small (equal to about 0.4 kg/s) but the difference in total pressure is already noticeably significant and amounts to about −15 Pa. Such a difference in total pressures can be easily verified in laboratory tests. The obtained graphical courses of these pressures are interesting. For comparison, appropriate pressure courses were performed for the same data but using the known classical equation of motion.

The presented analyses allow us to state that the occurrence of a mass sink in the duct causes a step change in the total air pressure at the location of the sink. In the case of the tested mass release, it was a step increase in the total pressure (by approx. 20 Pa), the value of which, depending on the tested case of the resistance of the duct, fan, or mass flow of the sink, can be significant for measurement purposes.

The occurrence of a mass sink in the duct also causes a step change in the static air pressure at the location of the sink. In the case of the tested mass release, it was a step increase in static pressure greater than in the case of the total pressure and amounted to about 46 Pa. The demonstrated differences in static pressures in the duct from the location of the mass sink to its outlet may have an impact on physical phenomena occurring in underground workings (emission of gases from the rock mass, migration of gases from or to the zone of a contained fire). For this reason, they may be of significant importance in practice.

The authors plan to conduct laboratory tests to verify the proposed new form of the equation of motion. The results of the obtained tests will be published in the next paper. Further studies will also consider the pressing nature of the mechanical fan operation in the pipe with a vent and the occurrence of a point source of mass in the duct instead of a mass sink.