Mathematical Modeling of Air Distribution in Mines Considering Different Ventilation Modes

Abstract

The calculation of air distribution in large mine ventilation networks is usually carried out by numerically solving a system of equations representing Kirchhoff’s circuit laws. This formulation of the problem traditionally only considers the frictional air resistance of straight sections of mine airways. However, when analyzing the changing ventilation modes, it is also important to correctly consider the shock losses, which sufficiently affect the redistribution of airflows. The reason is that the shock loss factor strongly depends on the airflow rates at the junctions of mine airways. This paper presents a mathematical model of a ventilation network that considers shock losses. The model considers steady-state air distribution as well as natural draft. The significance of the shock was confirmed with a practical example; we describe the application of the proposed mathematical model for the analysis of air distribution in the ventilation network of a potash mine during planned airflow reversal.

1. Introduction

In recent decades, the intensity of mining has been constantly increasing; easily accessible resources have gradually been exhausted; and mining enterprises have been switching to the development of deep deposits of mineral resources [1]. All this inevitably leads to the expansion of mined areas of deposits and the expansion and complication of the structure of mine ventilation networks. In turn, this makes implementing the two main tasks of mine ventilation more difficult [2,3]:

• Provide the required amount of air in all working areas to remove harmful impurities (gas and dust).
• Minimize energy consumption for mine ventilation.

The mine ventilation network is a complex system with many degrees of freedom and many random factors, both human-made and natural [4]. As a result, today, when analyzing ventilation networks as a whole, there is more and more of a transition from the classical calculation of air distribution to the analysis of the sensitivity of the ventilation network in relation to the variation in its parameters [5,6] and the intelligent monitoring and control of ventilation [7,8]. Therefore, in addition to the classical concepts of ventilation on demand [9] and automated ventilation control systems [3], the concepts of RPOD [10] and IoT-based mine ventilation [8] have been introduced.

It is important to note here that the calculation of ventilation in large, branched networks with thousands of mine airways is usually carried out using 1D models, taking into account a single spatial coordinate along the axis of the mine airway. This is due to two reasons:

• Even 1D calculation is time-consuming for large ventilation networks.
• The addition of spatial dimensions does not add accuracy to the simulation results due to the error in the initial data of the model and the presence of unaccounted-for random factors.

At the same time, the analysis of individual sections of ventilation networks is carried out using CFD methods. Most often, this is the analysis of the dynamics of gases and dust in dead-end mine excavations and longwall faces [11,12], the non-uniform distributions of airflows near fans [13], and air movement in the worked-out space of longwalls [14]. A hybrid approach is also used when part of the mine airways is simulated within the framework of the 1D approach and part within the framework of the 3D approach [15].

In this paper, we will discuss 1D approaches to modeling ventilation networks as a single interconnected structure. The mine ventilation network is represented as a directed graph, each branch of which is an individual mine excavation (or group of mine excavations). The calculation of air distribution in such a network is usually carried out by solving a non-linear system of equations based on Kirchhoff’s first and second laws [16].

$${\displaystyle \sum }_{i\in V_j}{Q}_i{D}_i^{\left(j\right)}=0$$

(1)

$${\displaystyle \sum }_{i\in L_k}\left(Q_i\left|Q_i\right|R_i-H_i-F_i\right)=0$$

(2)

where $Q_i$ is the airflow in mine airway No. i, m³/s; $D_i^{\left(j\right)}$ is the direction of air movement in mine airway No. i with respect to node No. j (1 if the flow enters the node and –1 if the flow leaves the node in the airway); $R_i$ is the frictional (Atkinson) air resistance of mine airway No. i, N·s²/m⁸; $H_i$ is the pressure difference of the fan, Pa; $F_i$ is the term that includes other body forces acting on the airflow (gravity [17], inertia [18], etc.), Pa; $V_j$ is the set of mine airways (branches) adjacent to node No. j; and $L_k$ is the set of mine airways (branches) included in loop No. k.

Air resistance $R_i$ is determined from the reference literature or the actual measurements of airflow velocity $V_i$, cross-section area of the airway $S_i$, and pressure drop $\mathsf{\Delta }{P}_i$ according to the following formula:

$$R_i=\frac{\mathsf{\Delta }{P}_i}{{\left(V_i{S}_i\right)}^2}=\frac{\mathsf{\Delta }{P}_i}{Q_i^2}$$

(3)

Obviously, the second approach is appropriate only if the measured values of pressure drops and air velocities are much higher than the corresponding instrumental errors. That is, to calculate the air resistance of remote mine airways with low air velocities (the order of 0.1 m/s) and small pressure drops (the order of 1 Pa), the second approach is not applicable.

It is noted in [19,20] that in addition to the frictional air resistance of straight sections of mine airways, the airflow also overcomes shock losses at flow turns, merging, and divisions at the junctions of mine airways. The shock losses can be comparable to the frictional resistances of straight sections of mine airways or exceed them [21]. The pressure drop caused by shock losses can be determined with the following common formula [22]:

$$R_{sh}=\xi \frac{\rho V^2}{2}$$

(4)

where $V$ is the air velocity after the shock loss, m/s; $\rho$ is the airflow density, kg/m³; and $\xi$ is the shock loss factor. The value of $\xi$ depends on the type of shock loss, the geometric parameters of the corresponding mine airways, and airflow velocities.

Model (1)–(2) does not explicitly consider shock losses; therefore, when it is parameterized according to experimental data from real mines, we obtain effective air resistances $R_i^{\left(e\right)}$, which implicitly include shock losses. Sometimes, this is taken into account by introducing an equivalent length in the formula for frictional resistance [23]. However, the shock loss factor in the general case depends on the air velocities at the junctions of mine airways. This means that the shock loss factor changes when the distribution of airflow rates in neighboring mine airways changes. This indicates that model (1)–(2) only works well for one mine ventilation mode, in which the experimental measurements of the airflow parameters and the determination of effective resistances $R_i^{\left(e\right)}$ are made. When the ventilation mode changes, the effective resistances change, and the model of the ventilation network may not correspond to reality.

Shock losses are also important for the correct calculation of ventilation networks with diagonal connections. It is known that the air flow rate in the diagonal branches of the ventilation network depends on the ratio of the air resistances of adjacent branches [24]. Too simplistic a calculation of this ratio can lead to the fact that the air flow in the diagonal branches is incorrectly predicted by the model.

In our previous work [25], an approach was proposed to consider the shock losses; we introduced an additional term into Equation (2) as follows:

$${\displaystyle \sum }_{i\in L_k}\left(Q_i\left|Q_i\right|R_i-H_i-F_i\right)+{\displaystyle \sum }_{j\in L_k}{H}_{jk}^{*}=0$$

(5)

where $H_{jk}^{*}$ is the pressure drop due to the shock loss during the passage of the airflow through node No. j in loop No. k.

The system of Equations (1) and (5) was implemented numerically in the Aeroset program. The numerical solution was carried out using the loop flow method adapted for mine ventilation problems in [26].

In most cases, engineers and researchers develop mine ventilation networks to explore various ventilation options, model the development of the ventilation network in the future, and select optimal technical solutions. This means that taking into account shock losses, which change with the change in ventilation parameters, is important. Another essential aspect is the calculation of emergency ventilation modes. In this case, the air distribution in the ventilation network may be subject to the greatest change, which also leads to a change in shock losses. An example is the reversal of the airflow, considered in [27,28,29]. In this case, the airflows in the entire mine change their direction of movement to the opposite direction. At the same time, the influence of shock losses was not studied in sufficient detail in the mentioned works. In [27,28], more attention was paid to the dynamics of accumulation and the removal of harmful gases, and in [29], the thermal effects in the ventilation shaft during the cold season were studied.

In this work, for the first time, we show the influence of shock losses on air distribution in mine ventilation networks when the main fan operation parameters are changed. In addition to shock losses, the influence of natural drafts is analyzed. The analysis was carried out on the example of the planned reversal of the airflows in a potash mine of Verkhnekamskoye deposits of potassium–magnesium salts. The research plan included several stages: experimental measurements of the aerothermodynamic parameters in mine airways, analysis of measurement data, and theoretical interpretation of changes in air distribution using a parameterized model of the ventilation network.

2. Methodology

2.1. Experimental Section

A practical example of a potash mine of the Verkhnekamskoye potassium–magnesium salt deposit was considered. The mine has a U-tube ventilation layout. In the normal ventilation mode, fresh air enters the mine through shaft No. 1. It flows down to the shaft inset and then becomes distributed to ventilate the service chambers (car garage, warehouses, etc.) and working areas located in the far sections of the main directions of the ventilation network. Air is supplied to the working areas through the transport and conveyor drifts of the main directions. The outgoing air stream is removed along the panel, and the main ventilation drifts to shaft No. 2, through which it is then released to the surface. The mine is ventilated by a TAF 47.3/28.2-1 axial fan located on the surface near shaft No. 2. Hereafter, we call this fan the main fan. The real mine ventilation network as of June 2022 is shown in Figure 1 as a 3D directed graph. Its schematic representation with an indication of the main airways can be found in Figure 2.

Additionally, Figure 2 shows the measuring points at which air parameters were measured in the normal ventilation mode as well as in the reverse ventilation mode. In addition to these points, we measured the atmospheric air parameters on the surface. The fan reversal was planned and was not associated with any emergency. The mines in Russia and CIS countries must organize scheduled short-term reversal of the main fan once every six months. The measuring points are numbered. At points 1, 2, 3, and 11 near the mine shafts, the absolute air pressure, the average cross-sectional air velocity, the cross-sectional area of the excavation, the temperature, and the relative humidity of the air were measured. At points 4–10, the average cross-sectional air velocity and the cross-sectional area of the excavation were measured.

To determine the air velocity, an APR-2 portable mine explosion-proof anemometer was used. The absolute error, $\Delta V$ (m/s), of this instrument is

$$\Delta V=0.2+0.05V$$

(6)

where V is the measured air velocity, m/s.

To determine the cross-sectional area of an excavation, a Leica Disto L2 laser ruler was used. The absolute error of this instrument is 1.5 mm under normal conditions and 0.15L mm under unfavorable conditions, such as bright sunlight or when measuring very uneven surfaces (L is the measured length).

Based on the known cross-sectional area, $S$, and the average velocity, $V$, in the airway, the volumetric airflow, $Q=SV$, was calculated.

A DPI 740 barometer was used to determine pressure drops in individual sections and airways of the mine ventilation network. Using it, we measured absolute pressures at two points and calculated the pressure difference from these data. The absolute error of this barometer in the vicinity of atmospheric pressure is 15 Pa. When calculating the pressure difference, change $h$ in the elevation of the measuring points ($\mathsf{\Delta }p=\rho gh$) was also taken into account.

The determination of air temperature and relative humidity was carried out using a Fluke-971 thermal moisture meter. The absolute error of temperature measurement in the range from −20 °C to +60 °C is ±0.1 °C. The absolute error of relative humidity measurement in the range of 5–95% is 2.5%.

In the normal ventilation mode, the discharge of the main fan was maintained at 400 m³/s, and the impeller rotation speed was 500 rpm. At the beginning of the reversal, the impeller speed was reduced from 500 rpm to 150 rpm while the blades of the main fan were transferred to the reverse position, after which the main fan entered the reverse operating mode with fixed discharge of 265 m³/s and impeller rotation speed of 310 rpm. This procedure took 10 min. The reverse ventilation mode lasted about 1.5 hours. During this time, the air velocity and pressure at all measuring points managed to reach their stationary value, after which the necessary air measurements were taken in the reverse ventilation mode. Auxiliary fans in the working areas of the underground level were turned off throughout the entire measurement period.

2.2. Theoretical Section

The mathematical model of the ventilation mine network is a directed graph $G=\left\{E,V\right\}$, where $E$ is the set of its branches, $e_i\in E$, and $V$ is the set of its nodes, $v_j\in V$. The branches of this graph are straight or curved sections of mine airways, characterized by length $L$, cross-sectional area $S$, cross-sectional perimeter $\mathsf{\Pi }$, frictional air resistance $R$, and airflow $Q$.

The geometric parameters of the airways were considered known values; they were set as properties of the corresponding branches of the mine ventilation network. The frictional air resistance of mine airways was calculated with the following formula:

$$R=\frac{\alpha LP}{S^3}+R_{vs}$$

(7)

where $\alpha$ is the air resistance coefficient taken from tabular reference books or experimental data, N·s²/m⁴; and $R_{vs}$ is the empirically determined resistance of ventilation structures in the airway (doors, stoppings, etc.), N·s²/m⁸. For mine airways of underground levels of the Verkhnekamskoye deposit, $\alpha$ is, on average, equal to 0.000495 N·s²/m⁴.

The calculation of air distribution in all branches of the ventilation network was carried out by solving the system of Equations (1) and (5). Parameter $F_i$ was taken as equal to

$$F_i={\rho }_{i}g\mathsf{\Delta }{z}_i$$

(8)

where ${\rho }_i$ is the average air density in branch No. i, kg/m³; $g$ is the gravity acceleration, m/s²; and $\mathsf{\Delta }{z}_i$ is the height difference between the start and end nodes of branch No. i, m.

According to [30,31], when taking into account natural draft and other thermodynamic factors leading to a large difference in air temperatures in the ventilation network of a mine, it is necessary to consider the variability of air density with temperature.

$$\rho ={\rho }_0\left(1-b\left(T-T_0\right)\right)$$

(9)

where $T$ is the air temperature determined experimentally, °C; $T_{ref}$ is the reference temperature (0 °C); ${\rho }_{ref}$ is the reference density (1.292 kg/m³); and $b$ is thermal expansion coefficient of air (0.00335 °C⁻¹).

The system of Equations (1) and (5) was solved numerically using the loop flow method [32], which has been adopted by us to consider the additional term with shock losses. This is an iterative method for determining the set of airflow rates $\mathit{Q}=\left\{Q_1,\dots ,Q_N\right\}$, where $N$ is the number of branches in the mine ventilation network. First, a set of independent loops of the $L_I$ network was chosen. Next, the vector of independent airflows, $\mathit{q}=\left\{Q_i;i=1,\dots ,N; i\in A_I\right\}$, was set. This vector contains the airflows from set of branches $A_I$, forming a selected set of independent loops of the network $L_I$ without repetitions. Vector of approximate airflows ${\mathit{q}}^{\tau }$ at each iteration $\tau$ was calculated based on vector ${\mathit{q}}^{\tau -1}$ at previous iteration $\tau -1$ using the following formula:

$${\mathit{q}}^{\tau }={\mathit{q}}^{\tau -1}-{\left[\begin{array}{c}{\mathit{M}}^T\cdot \mathit{D}\\ \widehat{\mathit{A}}\end{array}\right]}^{-1}\cdot \left[\begin{array}{c}{\mathit{M}}^T\cdot \mathit{A}\\ \widehat{\mathit{A}}\end{array}\right]\cdot {\mathit{q}}^{\tau -1}.$$

(10)

$$D_{ik}=\left(2R_i\left|Q_i\right|-\frac{\partial P_i\left(Q_i\right)}{\partial Q_i}-\frac{\partial H_{ik}^{*}}{\partial Q_i}\right)\cdot {\delta }_{ik}.$$

(11)

$$A_{ik}=\left(R_i\left|Q_i\right|-\frac{P_i\left(Q_i\right)}{Q_i}-\frac{H_{ik}^{*}}{Q_i}\right)\cdot {\delta }_{ik}.$$

(12)

where $\tau$ is the quasi-time step of the iterative procedure; ${\mathit{M}}^T$ is the transposed incidence matrix of the network graph; $\mathit{D}$ is a diagonal matrix, the elements of which are equal to the pressure drop of branch No. i, included in loop No. k, when its airflow changes by one; $\mathit{A}$ is a diagonal matrix introduced to reduce the system of Equations (1) and (5) to the pseudolinear form of $\mathit{A}\left(\mathit{q}\right)\cdot \mathit{q}=0$; $\widehat{\mathit{A}}$ is the network graph incidence matrix; ${\delta }_{ik}$ is the Kronecker delta function; $R_i$ is the frictional air resistance of branch No. i; and $P_i\left(Q_i\right)$ is the draft source (fan pressure $H$, natural draft $F$, etc.).

Matrix $\mathit{M}$ is filled in as follows:

• $M_{ki}=1$ if the direction of branch No. i coincides with the direction of bypassing loop No. k.
• $M_{ki}=-1$ if the direction of branch No. i does not coincide with the direction of bypassing loop No. k.
• $M_{ki}=0$ if branch No. $i$ is not included in loop No. $k$.

The elements of the matrix $\widehat{\mathit{A}}$ were defined as follows:

• ${\widehat{A}}_{ij}=1$ if branch No. i starts at vertex No. j.
• ${\widehat{A}}_{ij}=-1$ if branch No. i ends at node No. j.
• ${\widehat{A}}_{ij}=0$ if branch No. i is not connected to node No. j.

Further, at the end of each iteration, the maximum discrepancy between the airflows in the loops was calculated as follows:

$${\epsilon }^{\tau }=\underset{k}{\max }\left|q_k^{\tau +1}-q_k^{\tau }\right|.$$

(13)

The iterative procedure continued until discrepancy ${\epsilon }^{\tau }$ became less than the specified accuracy, ${\epsilon }_0=0.001$ m³/s.

The calculation of the pressure drop shock losses was carried out according to the formula obtained earlier in [25,33] based on the laws of mass, momentum, and energy balance for mixing flows in the 1D formulation.

$$\begin{array}{cc}{H}_{ji}^{**}=-{\beta }_i\hfill & {\omega }_i^{(out)}{\displaystyle \sum _s|}\frac{Q_s^{(in)}}{Q_{{\displaystyle \sum }}}|\frac{\rho {(V_s^{(in)}-V_s^{(out)})}^2}{2}\hfill \\ & -{\beta }_i{\omega }_i^{(out)}{\displaystyle \sum _s|}\frac{Q_s^{(in)}}{Q_{{\displaystyle \sum }}}|\frac{\rho (V_j^{{(in)}^2}-V_s^{{(in)}^2})}{2}\hfill \\ & -{\beta }_i{\omega }_i^{(out)}{\displaystyle \sum _s\rho }\frac{Q_s^{(in)}}{S_j^{(in)}}(V_j^{(in)}-V_s^{(in)})\hfill \\ & -2{\beta }_i{\displaystyle \sum _s\rho }|V_i^{(out)}{V}_s^{(in)}\cdot \frac{Q_s^{(in)}}{Q_{{\displaystyle \sum }}}|\cdot si{n}^2({\delta }_{si}/2).\hfill \end{array}$$

(14)

where ${\beta }_i$ is the wall roughness coefficient of branch No. i; ${\omega }_i^{(out)}$ is the stability coefficient; $Q_s^{(in)}$ is the airflow from branch No. s entering the junction, m³⁄s; $Q_i^{(out)}$ is the airflow leaving the junction and entering branch No. i, m³⁄s; $V_s^{(in)}$ is the air velocity of the stream flowing into the junction along branch No. s, m/s; $V_i^{(out)}$ is the air velocity of the stream flowing from the junction along branch No. i, m/s; $Q_{\Sigma }$ is the total airflow through the junction, m³⁄s; ${\delta }_{si}$ is the angle between the incoming stream from branch No. s and the outgoing stream in branch No. i; and $\rho$ is the air density, kg/m³.

Here, $j$ is the number of the branch along which the flow enters the junction and $i$ is the number of the branch along which the flow leaves the junction (see Figure 3). If both branches i and j are included in loop No. k, then value $H_{ji}^{**}$ is substituted with $H_{ik}^{*}$.

Wall roughness coefficient ${\beta }_i$ is a function of coefficient ${\alpha }_i$.

$${\beta }_i=0.95+280{\alpha }_i$$

(15)

Air resistance coefficient ${\alpha }_i$ in (15) depends on the type of mine airway and is most often set according to reference data. This parameter is similar to the friction factor in the Chézy–Darcy expression [34]. For the horizontal mine airways of potash mines of the Verkhnekamskoye salt deposit, it is usually taken equal to 0.000495 N s²/m⁴ [35]. It is assigned to branch No. i, along which the flow leaves the junction, because the stagnant flow zones with vortices usually form the outgoing branches.

Stability coefficient ${\omega }_i^{(out)}$ has the following form:

$${\omega }_i^{(out)}=\frac{1.05}{1+0.02Q_{\Sigma }/Q_i^{(out)}}$$

(16)

This coefficient was introduced to stabilize the numerical procedure in (10). For this, several conditions must be met [30]. Dependence (14) did not initially have multiplier ${\omega }_j^{(out)}$ and thus did not satisfy one of the conditions: when $V_j^{(out)}$ turns to zero, the pressure loss also vanishes, eliminating the pressure jump during reverse flow. At the same time, Shalimov [30] proposed a modification of the formula by adding a dimensionless artificial factor, $Q_j^{(out)}/Q_{\Sigma }$. However, this multiplier leads to significant changes in the magnitude of shock losses over the rest of the range of airflow rates. In addition, Shalimov [30] did not consider shock losses due to the airflow turns.

In our previous work [25], we proposed an artificial factor to satisfy condition No. 3 of the convergence of the method. This factor slightly differs from unity in the range of airflows $Q_j^{(out)}=\left(0.2÷1\right)\cdot Q_{\Sigma }$, i.e., Multiplier (16).

In Equation (14), the first term on the right is the pressure drop during the expansion/compression of the flow. The second and third terms characterize the pressure drop when the flows are mixed at the junction. Finally, the last term is the pressure drop due to flow turn.

3. Results

3.1. Experimental Section

Table 1. Measured airflows in normal and reverse ventilation modes.

No.	Location	Airflow, m3/s		RRN, %
		Normal Mode	Reverse Mode
1	Main fan drift	401.2	260.0	64.8
2	Ventilation shaft (elev. of −222 m)	305.4	181.8	59.5
3	Ventilation shaft (elev. of −258 m)	63.2	62.9	99.5
4	Northern wing	51.5	29.4	57.1
5	Main northeast direction	33.9	19.1	56.3
6	Main northwest direction	35.5	20.6	58.0
7	Southern wing	81.4	44.2	54.3
8	Main southeast direction	10.4	4.7	45.3
9	Main southwest direction	31.7	16.4	51.7
10	Main west direction	70.3	36.6	52.0
11	Air-supply shaft (elev. of −259 m)	324	214.9	66.3

presents the results of the experimental measurements of airflows in normal and reverse ventilation modes at the measuring points shown in Figure 2. The obtained data correspond to measurements in June 2022 (atmospheric air temperature of +20 °C). In addition, the ratios of reverse ($Q_r$) and normal ($Q_n$) airflows (RRN = $\frac{Q_r}{Q_n}$ · 100%) are displayed in the rightmost column.

The measured cross-sectional areas of mine airways were used when calculating the airflows. They range from 15 to 20 m². The sectional shape of the airways is arched. The exceptions are the mine shafts, which have a diameter of 8 meters and circular cross-sections.

The percentage of air reversal on the main fan was approximately 65%. This is due to the fact that the fixed performance of the main fan is set in this way in normal and reverse ventilation modes. We mention this in the above section of the paper. The following two points are noteworthy:

• Almost all other measurement points had a reversal percentage lower than 65%.
• There was a point where the percentage of reversal was unusually high, close to 100%.

We formulated the main hypotheses that may account for this phenomenon: (1) shock losses and (2) changes in the internal leakages in the mine. The theoretical interpretation of the presented experimental data is described in the following sections of the paper.

During the experimental survey, the parameters of the main fan and the air resistance of the underground part of the mine in normal and reverse ventilation modes were also determined. They are presented in

Table 2. Measured parameters of the main fan.

Parameter	Normal Mode	Reverse Mode	Change, %
Discharge, m3/s	401 (375)	260 (255)	64.8 (68)
Head, Pa	2380 (1700)	800 (950)	27.3 (55.8)
Air resistance of the underground part of the mine, N·s2/m8	0.0148 (0.0121)	0.0118 (0.0147)	64.8 (121.5)

. The data correspond to measurements in June (atmospheric air temperature of +20 °C) and October (atmospheric air temperature of +2 °C). The latter values are given in brackets.

The small difference in the performance of the main fan (no more than 6%) was due to instrumental error. Additionally, the most interesting thing here is that the air resistance of the underground part of the mine turned out to be different in all studies cases. Thus, we formulated hypotheses that can explain this phenomenon: (1) a change in the natural draft, (2) a change in internal leakages, and (3) a change in shock losses at shaft junctions with fan drift and airways of the underground level.

Table 3. Air temperatures and relative humidity at measuring points near the shafts.

Measuring Point	Normal Mode		Reverse Mode
	$\mathbf{Temperature},℃$	Humidity, %	$\mathbf{Temperature},℃$	Humidity, %
Atmosphere	20.0	50.0	21.6	41.0
1	7.0	30.0	19.8	47.5
2	15.8	—	16.4	—
10	12.3	—	11.5	—
11	19.1	50.8	16.2	60.4

presents the data of measurements of air temperature and relative humidity at measuring points near the shafts in normal and reverse ventilation modes. The data correspond to measurements in June (atmospheric air temperature of +20 °C). They were used for further theoretical analysis of air distribution.

It is important to note that the air resistance of the underground part of the mine was calculated based on the actual discharge and head on the main fan. Thus, this air resistance included a term caused by the action of the natural draft due to the difference in the weights of the air columns in the shafts. For this reason, the air resistance of the underground part of the mine should be called effective air resistance. The influence of the natural draft is seen in the data in Table 2; at different times of the year, air with different temperatures and relative humidity enters the mine, and the calculated air resistance turns out to be different both in normal and reverse ventilation modes.

3.2. Theoretical Section

This section describes the theoretical analysis of the obtained experimental measurement data using the model described above in Section 2.2. It is important to begin by saying that the mathematical model of the mine ventilation network was parameterized according to the data of experimental studies in a given normal ventilation mode; the correspondence between the theoretical and experimental airflows in the normal ventilation mode was achieved by setting the actual (effective) air resistances of underground structures, $R_{vs}$, as well as by adjusting air resistance coefficients $\alpha$ in the main airways. This procedure is not explained in detail here because it is of little interest.

In the following paragraphs, we carry out a comparative analysis of the results of simulation and experimental measurements according to two criteria. The first criterion is the percentage of reversal (RRN—Ratio of Reverse and Normal airflows), determined by the following formula:

$$RRN=100\%\cdot \frac{Q_r}{Q_n}$$

(17)

where $Q_r$ and $Q_n$ are airflows in reverse and normal ventilation modes, respectively, m³/s.

The second criterion is the effective resistance of the underground part of the mine, which is determined based on discharge $Q$ and head $H$ of the main fan.

$$R_u=\frac{H}{Q^2}$$

(18)

Table 4. RRN values according to simulation data and experimental measurements.

Measuring Point	Measured RRN, %	Simulation
		Calculated RRN (Shock Losses), %	Deviation from Meas., %	Calculated RRN (Shock Losses + Leakages), %	Deviation from Meas., %
Main fan drift	64.8	61.9	−4.4	61.2	−5.5
Ventilation shaft (elev. of −222 m)	59.5	57.7	−3.1	56.9	−4.4
Ventilation shaft (elev. of −258 m)	99.5	86.7	−12.8	85.4	−14.2
Northern wing	57.1	62.3	9.1	61.2	7.2
Main northeast direction	56.3	56.4	0.2	52.9	−6.0
Main northwest direction	58.0	67.6	16.6	62.6	7.9
Southern wing	54.3	58.9	8.4	57.9	6.6
Main southeast direction	45.3	58.1	28.2	48.4	6.8
Main southwest direction	51.7	57.6	11.3	52.8	2.1
Main west direction	52.0	60.3	15.9	55.2	6.2

Bold font shows the measuring points where the relative deviation is higher than 15% for the model that considers only shock losses at mine airways junctions.

shows a comparative analysis of the results of the simulation and experimental measurements according to the first criterion. Several calculations were carried out. First of all, we calculated the air distribution in the ventilation network of the mine without taking into account shock losses. However, at all measuring points, the same RRN value was obtained, equal to the set reversal percentage on the main fan. This case turned out to be of no interest for further analysis. Next, we took into account shock losses according to the approach described in Section 2.2. The resulting reversal percentages are presented in Table 4 in columns 3 and 4. However, this model did not explain the small reversal percentages at the three measurement points (highlighted in bold). For this reason, we also considered the factor of changes in the shock losses of ventilation structures in the model. This resulted in a small change in internal leakages in reverse mode and made it possible to achieve the desired result (relative error less than 15%; see columns 5 and 6). It should be noted that natural draft $F_i$ was also taken into account in the analysis of criterion No. 1, but it did not have any effect on the distribution of airflows at the underground level. This is due to very small variations in the elevations of the vortices of the underground level (except for a small number of airways near shaft No. 2). The simulation was carried out for atmospheric air parameters in June.

We would especially like to note that the model of shock losses turned out to be able to explain, with sufficient accuracy (error of no more than 15%), the anomalous increase in the percentage of reversal at measuring point 3 at the lower mark of the shaft No. 2. Importantly, the shock losses of air path section A–B (see Figure 4) in normal ventilation mode $R_n$ were higher than in reverse mode $R_r$. A reverse air stream at point A, directed down the shaft, is more likely to go down to point B than to point C because, in this case, it does not need to make a turn of 90°. We did not obtain a more accurate (error < 14%) correspondence between the theoretical and measured values at point 3 due to the local features of the junction of the shaft with horizontal airways at the level of −258 m (complex geometry of the shaft sump).

Table 5. Effective air resistances of the underground part in reverse ventilation mode.

Month	Experiment	Simulation
		Without Natural Draft	Deviation, %	Including Natural Draft	Deviation, %
June	0.0118	0.0132	11.9%	0.0122	3.3%
October	0.0147	0.0132	10.2%	0.0145	1.3%

shows a comparative analysis of the results of the simulation and experimental measurements according to the second criterion. Table 5 shows that, without considering the natural draft, the change in the effective air resistance of the underground part of the mine remained practically unchanged after the air stream was reversed. Moreover, the factor of shock losses did not significantly impact the change in the ventilation network’s resistance when the airflow direction changed. If the natural draft is considered, the ventilation network’s calculated resistance dropped in summer (June) and increased in autumn (October). Moreover, the discrepancy with the experiment was no more than 4%.

Previously, we also formulated a hypothesis about the effect of changing internal leakages on changes in the air resistance of the underground part of the mine. This hypothesis was not confirmed. This is because the main part of the pressure loss falls on the main ventilation units, i.e., shafts and their interfaces with ventilation ducts and main airways of the underground level. Therefore, increasing internal leakages in relatively remote sections of the ventilation network does not lead to a significant decrease in the total air resistance of the ventilation network.

4. Discussion

In Section 2.2, we interpreted the experimental data of the airflow reversal using the ventilation network model. The model took into account changes in shock losses and natural draft when changing the direction of the air streams. Changes in shock losses were taken both at turns and junctions of mine airways, and at ventilation doors and stoppings. All of these factors had to be considered to achieve consistency between the measured and calculated airflows at the 11 chosen measuring points.

The mathematical modeling of the natural draft is less complicated and has been repeatedly previously studied [26,36]. Much more interesting is that the example described above shows the importance of shock losses. Shock losses turned out to be a significant factor that led to the redistribution of airflows among the airways of the underground level when the ventilation mode was changed. The fact is that the shock losses of mine airway junctions are not symmetrical with respect to the change in the direction of the airflow. This fact follows from the particular example in Figure 4 and the general formula exemplified in Equation (14).

We considered the case of reverse mine ventilation, but this is not the only case when the ventilation system is changed. We believe that the described features of air distribution in underground mine airways are relevant in case of any significant changes in the ventilation system—for example, with a decrease in air consumption in certain main directions and levels of the mine during maintenance shifts and shutdowns (this applies to the implementation of ventilation on demand and other automatic mine ventilation systems).

It is noteworthy that the relative contribution of shock losses to the total resistance of the ventilation network becomes higher as the cross-sectional area of the mine airways (assuming constant airflow) becomes larger. As is known, frictional air resistances are $R_{fr}~S^{-2.5}$, while shock losses are $R_{sl}~S^{-2}$, according to Equation (14), as well as other formulas and approaches to estimating the magnitude of shock losses [21,22,30]. The mine airways of the underground level of the potash mine we considered are not very large (the cross-sectional area is 15–20 m²). There are examples of mines with areas of the main airways greater than 100 m² [37].

The contribution of shock losses is that the stronger the shock losses are, the higher the air velocity is. Therefore, it is not surprising that the shock losses of high-speed sections of the ventilation network have been studied in many works [20,21,38,39], where the pressure drops on shock losses were determined and the geometry of the flow channels was optimized. However, here, for the first time, we clearly show the role of shock losses in the redistribution of airflows in a complicated system of mine airways at an underground level with relatively low total air resistance. The geometrical parameters of mine airways (transport, conveyor drifts, and main return airways) are usually selected based on the convenience of mining operations, the delivery of people, materials, and ore. Therefore, the junctions of mine airways are asymmetrical from the point of view of aerodynamics. This means that a change in the ventilation modes of the mine also leads to a change in shock losses at these junctions and different redistribution of flows in the ventilation mine networks.

An effective way to calculate the air distribution in small systems of mine airways is 3D simulation using CFD methodology. This allows one to obtain a more accurate and physically clear result than that of the 1D approach described here. However, when analyzing large mine ventilation networks, the only option for calculating air distribution in a given time period is to use a 1D approach with explicit consideration of shock loss dependencies of the type given in Equation (14). Dependence (14) is quite simple and does not consider many subtleties of real geometries of mine airways and their junctions. For example, the axes of mine airways with a common junction may not intersect at one point. In addition, the cross-section of the junction may not correspond to the cross-sections of the branches included in it but may be a separate parameter that greatly affects the resulting pressure drop. The interaction of flows entering the junction can be more complex and does not satisfy the hypothesis about the instantaneous mixing of flows. All this precludes a certain error present in Equation (14). However, the advantage of Equation (14) is its simplicity and versatility with respect to the type of junction (the number of mine airways in the junction and their geometric and aerodynamic parameters). Equation (14) makes it possible to quickly assess the nature of the change in air distribution in the ventilation network when changing the parameters of ventilation structures and fans. In our opinion, its accuracy agrees with the accuracy of models of mine ventilation networks in general.

In the future, it would be of interest to obtain some modifications of Equation (14) to take into account the local features of mine airway junctions, which can affect the shock losses. It is important to determine a reasonable degree of detail and complexity of analytical dependencies. Not all features should be taken into account in 1D air distribution models, and when considering extremely exotic cases, it is better to use CFD modeling in a 3D formulation. In addition, it would be interesting to conduct more extensive validation of the mathematical model described here. To do so, more experimental data should be collected on the change in air distribution when the main fan is reversed. Particular attention should be paid to direct measurements of local features of mine airway junctions and direct measurements of changes in the aerodynamic parameters of ventilation stoppings when the sign of the pressure drop changes.

5. Conclusions

The conducted study of air distribution in normal and reverse mode of the main fan at a potash mine allowed us to obtain the following main results:

• Shock losses significantly affected the air distribution in the system of mine airways of the underground level. At the same time, the total air resistance of the mine was weakly dependent on the variation in shock losses. However, it also changed due to changes in natural draft.
• The changes in the distribution of air in the ventilation network were associated both with changes in the shock losses of mine airway junctions and with changes in the losses of ventilation structures at the connections between the main air supply and return airways.
• A mathematical model is proposed that can describe the air distribution in the mine ventilation network when the ventilation mode is changed. The theoretical calculations agree quite well with experimental data for the case of planned reversal of the main fan of a potash mine (relative error of no more than 15%).