WO2019223628A1

WO2019223628A1 - Minimum flow unit-based method for calculating nitrogen charging pressure loss of goaf

Info

Publication number: WO2019223628A1
Application number: PCT/CN2019/087503
Authority: WO
Inventors: 陈世强; 姜文; 王海桥; 鲁义; 王鹏飞; 田峰; 于琦
Original assignee: 湖南科技大学
Priority date: 2018-05-21
Filing date: 2019-05-18
Publication date: 2019-11-28
Also published as: KR20200032155A; CN108733923A; KR102307223B1; CN108733923B

Abstract

Disclosed is a minimum flow unit-based method for calculating a nitrogen charging pressure loss of a goaf. The method comprises the following steps: (1) establishing a minimum flow model of a goaf, and determining a minimum flow unit of the minimum flow model; (2) taking the minimum flow unit as a study object, dividing a flow process into four different stages according to different relational expressions of a cross-sectional area changing along with a flow distance, and determining an overflow cross-sectional area expression of each stage; and (3) determining, according to the Navier-Stocks equation, the pressure loss value of nitrogen when flowing in the minimum flow unit. By means of the calculation method, the pressure loss of nitrogen in the flow process can be rapidly calculated, thereby providing theoretical support for the initial pressure needed by the nitrogen when nitrogen is injected; moreover, a charging range can be accurately calculated, and the calculation process can be simple and convenient.

Description

Calculation method of nitrogen filling pressure loss in goaf based on minimum flow unit

Technical field

The invention belongs to the technical field of disaster prevention and mitigation of mine ventilation and goaf, and in particular relates to a method for calculating pressure loss of nitrogen filling in goaf based on minimum flow unit.

Background technique

More than 70% of coal mine fire accidents occurred in the mined-out area adjacent to the mining area. The spontaneous combustion of coal in goaf and its fire extinguishing has been a hot issue in the industry. The use of nitrogen injection to treat spontaneous combustion is the most common and effective fire extinguishing measure. However, the existing calculation methods and detailed implementation rules cannot accurately calculate the flow distance of the injected nitrogen for the firefighting in the goaf, and thus cannot accurately predict the range that the injected nitrogen can affect, resulting in waste of liquid nitrogen and difficulty in fire suppression.

In the research progress of determining the method and measures of nitrogen injection parameters in the goaf, Li Dong et al. Proposed a method for preventing and extinguishing fire by nitrogen injection in the full-face curtain of the U-shaped ventilation working face in the goaf, which improved the efficiency of nitrogen inertization. Zhu Hongqing invented an automatic control rotary traction type nitrogen injection fire prevention and extinguishing device, which realizes the spatial continuity of the nitrogen injection point in the goaf and improves the inertization effect of nitrogen injection in the goaf. Qin Botao and Lu Yi put forward a method to effectively treat the spontaneous combustion of coal remaining in large-scale goafs in shallow buried coal seams, integrating leakage control and rapid inertial cooling to reduce the spontaneous combustion of residual coal. Guo Junliu discussed the observation of the "three zones" of the goaf and the practical methods of nitrogen injection and fire suppression, which improved the safety and economic benefits of coal mining. According to the mass, momentum, and composition conservation equations, Zhu Hongqing established a theoretical fluid mechanics model for the change of gas volume fraction in the goaf, and determined the optimal nitrogen injection position and amount through numerical simulation through initialization, boundary value assignment, and iterative calculation. Li Zongxiang combined with Duanwang Coal Mine to perform a "two-in-one, one-back" simulation of nitrogen injection and fire suppression in a goaf of a complex stope, and determined the optimal nitrogen injection amount and location. He Junzhong used the SF ₆ tracer gas to determine the air leakage in the 4405 goaf of the Hongyan Coal Mine, and determined the main air leakage direction. Based on this, the nitrogen injection process in the goaf was optimized. Zhang Qi combined with the actual situation of the coal gangue in Datong Coal Mine Group Company, and studied the fire hazard management technology in the fully mechanized top coal caving face through the research of the comprehensive mining top coal caving face. Fire hazard management. Aiming at high gas-prone spontaneous coal seams, Luo Xinrong developed a computational fluid dynamics code with extraction holes based on a fully mechanized mining face in a certain mine. Liu Xingkui used Fluent software to simulate the change of the spontaneous combustion zone range before and after nitrogen injection, and analyzed the effect of nitrogen injection position and nitrogen injection amount on the distribution of the oxidation zone position in the goaf, and fitted the best nitrogen injection from it. parameter. In order to determine the optimal process parameters for continuous nitrogen injection and fire suppression in the goaf of the 5-2S-2 fully mechanized caving face, Dong Jun constructed a mathematical and physical model of the seepage field in the goaf according to the basic theory of computational fluid mechanics. The oxygen concentration distribution and "three-zone" division of the mined-out area were numerically studied to obtain the optimal nitrogen injection parameters. The above research is an important basis for preventing spontaneous combustion in goafs. However, there are still problems in applying these methods and devices to actual projects. First, because the physical parameters and production process parameters of goafs in different coal mines are different, the method of determining nitrogen injection parameters by numerical simulation is time-consuming, technically difficult, and difficult to popularize. Second, the above studies cannot clarify the nitrogen flow process. The quantitative relationship between the intermediate pressure loss and the nitrogen injection parameters makes it impossible to quickly determine the specific parameters of nitrogen injection. In order to overcome the above-mentioned shortcomings, the present invention proposes a method for optimizing nitrogen filling parameters in a goaf based on a minimum flow unit.

Summary of the Invention

In order to solve the above technical problems, the present invention provides a method for calculating a pressure loss of nitrogen filling in a goaf based on a minimum flow unit.

The technical scheme adopted by the present invention is: a method for calculating the pressure loss of nitrogen filling in the goaf based on the minimum flow unit, including the following steps:

1) Establish the minimum flow model of the goaf and determine its minimum flow unit;

2) Taking the minimum flow unit as the research object, divide the flow process into four different stages according to the relationship between the change of the cross-sectional area with the flow distance, and determine the expression of the cross-flow cross-sectional area at each stage;

3) Determine the flow rate of nitrogen according to the expression of the cross-flow area and the continuity equation. The initial velocity is u _0. The cross-flow area at the inlet of the minimum flow unit is A _0. The total volume of nitrogen flowing is A ₀ u. ₀ , the velocity of the downstream section is obtained by dividing the nitrogen volume flow rate by the area of the overcurrent section;

4) According to the Navier-Stocks equation, determine the pressure loss value of nitrogen flowing in the minimum flow unit. The specific operation is as follows:

a) Based on the steady flow of nitrogen when injecting nitrogen, and the only force the minimum flow unit receives along the z-axis is pressure and viscous force, so the Navier-Stocks equation becomes:

In the formula: the three terms on the left side of the equation are the inertial force components in the X, Y, and Z directions, and the two terms on the right side of the equation are the static pressure gradient term and the viscous force term. U _x , u _y , and u _z are X, Y, Z position changes, x, y, and z are length variables of the X, Y, and Z axes, respectively,

with

Partial derivatives of X-axis, Y-axis, and Z-axis, respectively, ρ is nitrogen density, and P is static pressure.

Is the partial derivative of the static pressure on the Z axis, υ is the kinematic viscosity of nitrogen,

Is the Laplace operator symbol,

Is the viscous force divergence;

b) The Z-axis direction is the normal direction of the cross section of the nitrogen gas inlet. The initial coordinate axis of the Z-axis is used to determine the X-axis and Y-axis directions according to the Cartesian coordinate system guidelines. as a symmetrical axis of symmetry, so that X, equal to the current velocity in the Y direction; there _{dx = u x dt, dy =} u y dt, dz = u z dt, u x = tgθ 1 u z and u _y = tgθ ₂ u _z, Where: dx, dy, and dz are the length components of the X, Y, and Z axes, dt is the time component, and θ ₁ and θ ₂ are the angles between the velocity component u _x and the velocity component u _z , The angle between the velocity component u _y and the velocity component u _z , and substitute it into the Euler fluid mechanics

It can be obtained that tgθ ₁ ≈tgθ ₂ ＜< 1. Then, it is obtained that the pressure loss in the X-axis and Y-axis directions is much smaller than the pressure loss in the Z-axis direction, so only the flow process in the Z-axis direction is considered;

c) using the concept of average speed,

Calculate the average velocity of the smallest flow unit at each stage and substitute it into the Navier-Stocks equation to obtain the pressure loss calculation formula per unit length, and then calculate the pressure loss in the flow process;

Where

Is the average value of u _z in the finite volume V _z , dV is the differential component of the volume, and A _z is the cross-sectional area of the flow at the position of the coordinate value z.

In the above-mentioned calculation method for the pressure loss of nitrogen filling in the goaf based on the minimum flow unit, the minimum flow model in step 1) includes eight spheres with a diameter of d, and one diameter is

Small diameter spheres and 6 diameters are

Small diameter hemisphere; 8 spheres with a diameter d are located at the eight vertices of a cube with a side length d, and each sphere with a diameter d is tangent to an adjacent sphere with a diameter d; the diameter is

The small-diameter sphere is located at the center of eight spheres with a diameter of d and is tangent to the eight spheres with a diameter of d;

The small diameter hemispheres are located at the center of the six faces of the cube, with a diameter of

The trailing hemisphere is tangent to the surrounding sphere.

In the above-mentioned calculation method for the pressure loss of nitrogen filling in the goaf based on the minimum flow unit, the determination of the minimum flow unit in step 1) includes the following steps: the center point of each edge of the cube with the side length d is cut Point, the minimum flow model is divided into eight equally divided small cells, and each small cell consists of a sphere with a diameter of d and four 1 / 8-diameter spheres. The cut small cell is regarded as the minimum flow model Smallest flow unit.

In the above method for calculating the pressure loss of nitrogen filling in the goaf based on the minimum flow unit, the specific operation of step 2) is as follows:

For the smallest flow unit, in the Z-axis direction, the origin of the coordinates is the center of the small-diameter sphere at the center of the smallest flow model, and the area of the cross-section at the point of z = 0 is

Where: d is the diameter of the sphere, and π is the circumference;

When constraints are met

, Z is the coordinate of the overcurrent section on the Z axis. As z increases, the area of the overcurrent section is equal to the square section with side length d minus 1 small-diameter semicircle area that changes with z and 2 changes with z Area of the large diameter semicircle, ie:

Similarly: when

Time,

when

Time,

when

Time,

In one cycle, the change of the overcurrent section is composed of the above-mentioned four phases; when the sectional relationship formulas of formula (1) (2), (3), or (4) are respectively satisfied, they are called the first flow phase, The second, third, or fourth flow phase.

In the above-mentioned calculation method for the pressure loss of nitrogen filling in the goaf based on the minimum flow unit, the pressure loss calculation formula for each flow stage obtained in step c) in step 4) is as follows:

The formula for calculating the pressure loss per unit length of the smallest flow unit in the first flow stage is:

J = Au ₀ + Bu ₀ ² , where:

In the formula: J is the pressure gradient, A and B are combined variables, u ₀ is the average cross-sectional flow velocity along the Z axis at the centroid of the smallest flow unit (z = 0), α ′ = α ₁ + α ₂ , α ₁ , Α ₂ are the ratio of the viscous force of the X-axis and the Z-axis, the ratio of the viscous force of the Y-axis and the Z-axis, β ′ = β ₁ + β ₂ , and β ₁ and β ₂ are the inertial force ratio of the X-axis and the Z-axis, respectively. Ratio of inertial force of Y axis and Z axis, μ is nitrogen dynamic viscosity;

The formula for calculating the pressure loss per unit length of the minimum flow unit in the second flow stage is:

The formula for calculating the pressure loss per unit length of the smallest flow unit in the third flow stage is:

The formula for calculating the pressure loss per unit length of the smallest flow unit in the fourth flow stage is:

Compared with the prior art, the beneficial effects of the present invention are:

The invention can quickly calculate the pressure loss during the flow of nitrogen, provide theoretical support for the initial pressure required for nitrogen when injecting nitrogen, can realize accurate calculation of the filling range, and can make the calculation process simple and convenient.

BRIEF DESCRIPTION OF THE DRAWINGS

Figure 1 is a front view of the minimum flow model.

Figure 2 is a left view of the minimum flow model.

Fig. 3 is a top view of the minimum flow model.

FIG. 4 is a structural diagram of a minimum flow unit.

FIG. 5 is a graph showing the relationship between pressure loss and temperature during the first flow stage of nitrogen flowing through the minimum flow unit.

FIG. 6 is a graph showing the relationship between pressure loss and temperature of the second flow stage of nitrogen flowing through the minimum flow unit.

FIG. 7 is a graph showing the relationship between pressure loss and temperature of the third flow stage of nitrogen flowing through the minimum flow unit.

FIG. 8 is a graph showing the relationship between pressure loss and temperature of the fourth flow stage of nitrogen flowing through the minimum flow unit.

Figure 9 is a graph showing the relationship between pressure loss and temperature for a complete stage of nitrogen flow through the minimum flow unit.

Detailed ways

The invention is described in further detail below with reference to the drawings.

The invention includes the following steps:

1) Establish the minimum flow model and determine the minimum flow unit of the minimum flow model. For a cube with a side length of 2d, it can just fit 8 spheres with a diameter of d. The most loose arrangement is that the spheres are tangent between the spheres and the sides of the cube. This cube contains 8 spheres. The rate is 1-π / 6, which is about 0.4764. Then, there are 8 large-diameter spheres with diameter d and 4 diameters.

The minimum flow model porosity of the small-diameter sphere is 0.3737.

Therefore, the minimum flow model shown in Figure 1-3 is established. The minimum flow model includes 8 spheres with a diameter d and

Small diameter spheres and 6 diameters are

The trailing hemisphere is tangent to the surrounding sphere.

In order to simplify the theoretical derivation process, the minimum flow unit is cut based on the minimum flow model. The minimum flow unit is as follows:

Taking the center point of each edge of a cube with side length d as the cut point, the minimum flow model is cut into eight equally divided small units, and each small unit consists of a sphere with a diameter of d and 4 It consists of 1/8 small-diameter spheres. The small unit cut out is regarded as the smallest flow unit in the smallest flow model, as shown in Figure 4. The porosity of each smallest flow unit is 0.3737, which is equal to the model porosity of eight large and four small spheres. The flow of fluid on the smallest flow unit has a complete flow cycle.

2) The Z-axis direction is the normal direction of the nitrogen inlet section. Based on the Z-axis initial coordinate axis and according to the Cartesian coordinate system, the X-axis direction and the Y-axis direction are determined respectively. The center of the sphere establishes a coordinate system as the origin.

For the smallest flow unit, in the Z-axis direction, the cross-sectional area from the point of z = 0 is:

(A ₀ is the initial cross-sectional area of the flow, d sphere diameter, and π is the circumference).

When constraints are met

, Z is the coordinate of the cross section in the Z axis direction. As z increases, the area of the overcurrent cross section is equal to the square cross section with side length _d minus 1 area of small diameter semicircle that changes with z and 2 Large diameter semicircle area, ie:

In the formula, A _z is the cross-sectional area of the current at the position of the coordinate value z.

Similarly, when

Time,

when

Time,

when

Time,

Different flow stages are divided according to the relationship of the cross-sectional area with z, and when the cross-sectional relationship of formula (1), (2), (3), or (4) is satisfied, it is called the first Flow phase, second flow phase, third flow phase, or fourth flow phase. In one cycle, the change of the cross-sectional flow is composed of the above four stages; on this basis, a micro-flow process with the smallest flow unit in one cycle is formed.

3) Determine the pressure loss of nitrogen flowing in the minimum flow unit according to the Navier-Stocks equation. The specific operation is as follows:

Since the law of motion of incompressible fluid conforms to the Navier-Stocks equation, there are:

In the formula: the three terms on the left side of the equation are the inertial force components in the X, Y, and Z directions, and the two terms on the right side of the equation are the static pressure gradient term and the viscous force term. U _x , u _y , and u _z are X, Y, Z position changes, t is time,

Is the partial derivative of time,

Is the partial derivative of u _z ,

with

Partial derivatives of the X-axis, Y-axis, and Z-axis, respectively, ρ is the nitrogen density, f _z is the volume force component of the Z-axis, and P is the static pressure.

Is the partial derivative of the static pressure on the Z axis, μ is the dynamic viscosity of nitrogen,

Is the Laplace operator symbol,

Is the viscous force divergence.

Because the flow of nitrogen is stable when nitrogen is injected, and the only force the minimum flow unit receives in the z-axis direction is pressure and viscosity, so

f _z = 0; and, according to the basic physical relationship between the physical parameters, there is μ = ρυ; then the Navier-Stocks equation of nitrogen, that is, formula (5) is simplified as:

In the formula, υ is the kinematic viscosity of nitrogen.

In formula (6), the left end of the equation is the inertia force action term, the first term at the right end of the equation is the pressure loss term, and the second term is the viscous force action term. As long as the terms in the equation are divided into the minimum flow unit by volume, the equation of motion of the fluid in a typical minimum flow unit can be obtained.

Create a four-phase microflow model:

a) Modeling of the first flow stage

In the minimum flow unit, when z = 0, it is assumed that the point velocity in the z-axis direction at the centroid is equal to the average flow velocity of the over-flow section, and is u ₀ ; where the area of the over-flow section is A ₀ ; along the Z-axis direction , The velocity of the nitrogen at the coordinate z is u _z , and the cross-sectional area of the flow is A _z , according to the continuity equation:

A ₀ u ₀ = A _z u _z (7);

When z = 0, the cross-sectional area of the overcurrent

The area of the cross-section at the coordinate z is

Then, the average velocity of the over-current section at the coordinate z is:

Therefore, the local acceleration along the Z axis is:

From equations (8) and (9), the migration acceleration is:

The components of the inertial force term and the viscous force term on the over-current section at the coordinate z are obtained. Since the actual flow can only occur in the pore portion, in order to obtain the resultant force of the entire cubic unit in the Z-axis direction, the force on the pore portion must be averaged to the cross-flow section of the entire smallest flow unit. Let the volume of the cube be V ₀ and the volume of the pores be V _z . According to the conservation of mass and conservation of momentum, we can get:

In the formula:

Represents the average value of the velocity u _z of nitrogen in different flow stages,

Changes in velocity per unit length due to different positions in the Z-axis direction at the same time in different flow stages

average of,

For viscous forces in the Z-axis direction at different flow stages

average of.

Substituting equation (8) into equation (12), we get:

The volume of the void portion in the first flow stage is

From formula (15), the average cross-section flow velocity in the first flow stage is:

Substituting equation (10) into equation (13), the average inertial force in the first flow stage is:

Substituting equation (11) into equation (14), the average viscosity force in the first flow stage is:

For the components of inertial and viscous forces in the X and Y directions, from the symmetry, we get:

In the formula:

Are

average of.

Where

Are

average of.

Because nitrogen flows from the pore belly to the pore throat or from the pore throat to the pore belly, the streamline is also continuously contracted or enlarged, and a boundary layer separation and vortex will occur near the spherical surface. Therefore, it is difficult to obtain the velocity gradient distribution function of the infiltration velocity along the X-axis and Y-axis directions, and the components of the inertial force and the viscous force in the X and Y directions cannot be directly derived by theory. However, it can be seen from the symmetry that the components of the inertial force and the viscous force in the Z-axis direction have a certain correlation with each other in the X- and Y-axis directions, and Irmay has proved this using mathematical estimation. Take a section along the Y-axis direction Δy, and set u _x and u _z as the partial velocities in the X and Z directions, respectively:

Where θ ₁ is the angle between the velocity component u _x and the velocity component u _z ;

Further, we get:

Based on the same principles of experimental fluid mechanics with dimensions, we get:

Assume:

In the formula, β ₁ and β ₂ are the inertial force ratio of the X-axis and the Z-axis, and the inertial force ratio of the Y-axis and the Z-axis, respectively.

Similarly, we can get:

Assume:

In the formula, α ₁ and α ₂ are the viscous force ratio of the X axis and the Z axis, and the viscous force ratio of the Y axis and the Z axis, respectively.

Let β '= β ₁ + β ₂ , then the average inertial force on the cross section of the current can be expressed as:

In the formula:

Represents the change in speed per unit length caused by different positions in the X, Y, and Z directions at the same time in different stages.

average of.

Let α '= α ₁ + α _{2 again} , the average viscous force term on the cross section can be expressed as:

Applying formula (6) to the overcurrent section, there are:

Where

Represents the viscous force divergence,

Is the average pressure gradient over the cross section, and P is the nitrogen pressure at the center of the cross section.

Shifting formula (29), we get:

Substituting μ = ρ · υ into formula (30), we have:

make

Substituting formulas (27) and (28) into formula (31), we can get:

In the formula: J is the pressure loss per unit length.

In addition, according to the aforementioned derivation process,

That is: the average flow rate of nitrogen in the first flow stage

It is about 1.8069 times the peritoneal abdominal flow rate u ₀ .

make:

Again:

Substituting formulas (33) and (34) into formula (32), we get:

J = Au ₀ + Bu ₀ ² (35);

b) Similarly, in the second flow stage, the pressure loss model per unit length is:

c) Similarly, in the third flow stage, the pressure loss model per unit length is:

d) Similarly, in the fourth flow stage, the pressure loss model per unit length is:

e) According to Euler fluid mechanics, there are:

According to Newton's classical mechanics,

dx = u _x dt

dy = u _y dt (49);

dz = u _z dt

According to formula (21), and similar definitions are:

In the formula, θ ₂ is an angle between the velocity component u _y and the velocity component u _z .

Putting formulas (49) and (50) into formula (48), we get:

Since the direction of the initial velocity u ₀ is consistent with the Z axis, the pressure caused by the nitrogen flow mainly acts on the Z axis direction and affects the flow in the Z axis direction, and tgθ ₁ ≈ tgθ ₂ ＜< 1, so it can be ignored at X, The pressure loss in the Y-axis direction only considers the pressure loss in the Z-axis in the mainstream direction, so the pressure loss obtained in step e) is the pressure loss in the Z-axis.

From the results of the previous derivation, in the case of the model determination, the ball diameter d and the initial velocity u ₀ are constant, and the dynamic viscosity μ and nitrogen density ρ are two variables that affect the pressure loss J. Taking the ball diameter d as 5mm and the initial velocity u ₀ as 1 m / s as an example, since the temperature change range of the goaf is concentrated between 20 ° C and 60 ° C, and the corresponding dynamic viscosity μ and nitrogen density ρ at different temperatures All are different, so the relationship between J and μ, ρ should be studied under different temperature conditions. Take the dynamic viscosity μ and density ρ at 20 ° C, 30 ° C, 40 ° C, 50 ° C, and 60 ° C into the equations of the four flow stages, and calculate the pressure loss J per meter under different viscosity coefficients and model four. The pressure loss of each flow stage, and the relationship between the kinematic viscosity υ and the pressure loss J is shown in a chart. The relationship between the pressure loss J and the temperature T per unit length in four different stages is shown in Figure 5-8. Show.

In Figure 5-8, the horizontal axis is υ × 10 ^-5 and the unit is 1 m ² / s; the vertical axis is the pressure loss over the unit length and the unit is Pa / m. By analyzing the specific implementation scheme, the rules are summarized as follows: ① During the flow of nitrogen in the model, the pressure loss and the initial speed meet:

The quadratic relationship. ② With the change of temperature, the dynamic viscosity and density of nitrogen will also change, and both of them affect the pressure loss per unit length. ③ As the kinematic viscosity increases, the pressure loss on the entire model will decrease. It can be seen from FIG. 9 that during the flow of nitrogen gas, the pressure loss in the first flow stage is the smallest, and the pressure loss in the second flow stage is the largest when the viscosity coefficient is unchanged. The pressure losses in the third and fourth flow stages are very close. The pressure loss of the four flow stages also increases with the increase of the kinematic viscosity coefficient. Compared with the existing method, the present invention can quickly calculate the pressure loss during the flow of nitrogen, can accurately calculate the filling range, and can make the calculation process simple and convenient.

Claims

A method for calculating pressure loss of nitrogen filling in a goaf based on a minimum flow unit includes the following steps: 1) establishing a minimum flow model in the goaf and determining its minimum flow unit;

2) Taking the smallest flow unit as the research object, the flow process is divided into four different stages according to the relationship between the change of the cross-sectional area with the flow distance, and the expression of the cross-flow area of each stage is determined;

3) Determine the flow rate of nitrogen according to the expression of the cross-flow area and the continuity equation. The initial velocity is u 0. The cross-flow area at the inlet of the minimum flow unit is A 0. The total volume of nitrogen flowing is A 0 u. 0 , the velocity of the downstream section is obtained by dividing the nitrogen volume flow rate by the area of the overcurrent section;

4) According to the Navier-Stocks equation, determine the pressure loss value of nitrogen flowing in the minimum flow unit. The specific operation is as follows:

a) Based on the steady flow of nitrogen when injecting nitrogen, and the only force the minimum flow unit receives along the z-axis is pressure and viscous force, so the Navier-Stocks equation becomes:

In the formula: the three terms on the left side of the equation are the inertial force components in the X, Y, and Z directions, and the two terms on the right side of the equation are the static pressure gradient term and the viscous force term. U x , u y , and u z are X, Y, Z position changes, x, y, and z are length variables of the X, Y, and Z axes, respectively,

with
Partial derivatives of X-axis, Y-axis, and Z-axis, respectively, ρ is nitrogen density, and P is static pressure.
Is the partial derivative of the static pressure on the Z axis, υ is the kinematic viscosity of nitrogen,
For Laplace operator symbol,
Is the viscous force divergence;

b) The Z-axis direction is the normal direction of the cross section of the nitrogen gas inlet. The initial coordinate axis of the Z-axis is used to determine the X-axis and Y-axis directions according to the Cartesian coordinate system guidelines. as a symmetrical axis of symmetry, so that X, equal to the current velocity in the Y direction; there dx = u x dt, dy = u y dt, dz = u z dt, u x = tgθ 1 u z and u y = tgθ 2 u z of formula Middle: dx, dy, and dz are the length components of the X, Y, and Z axes, dt is the time component, and θ 1 and θ 2 are the angles between the velocity component u x and the velocity component u z , The angle between the velocity component u y and the velocity component u z is substituted into the Euler fluid mechanics
It can be concluded that tgθ 1 ≈tgθ 2 ＜< 1, and then the pressure loss in the X-axis and Y-axis directions is much smaller than the pressure loss in the Z-axis direction, so only the flow process in the Z-axis direction is considered;

c) using the concept of average speed,
Calculate the average velocity of the smallest flow unit at each stage and substitute it into the Navier-Stocks equation to obtain the pressure loss calculation formula per unit length, and then calculate the pressure loss in the flow process;

Where
Is the average value of u z in the finite volume V z , dV is the differential component of the volume, and A z is the cross-sectional area of the flow at the position of the coordinate value z.
The method for calculating the pressure loss of nitrogen filling in the goaf based on the minimum flow unit according to claim 1, wherein the minimum flow model in step 1) includes 8 spheres with a diameter of d, and a diameter of
Small diameter spheres and 6 diameters are
Small diameter hemisphere; 8 spheres with diameter d are located at the eight vertices of a cube with side length d, each sphere with diameter d is tangent to an adjacent sphere with diameter d;
The small-diameter sphere is located at the center of eight spheres with a diameter of d and is tangent to the eight spheres with a diameter of d;
The small diameter hemispheres are located at the center of the six faces of the cube, with a diameter of
The trailing hemisphere is tangent to the surrounding sphere.
According to the method for calculating the nitrogen filling pressure loss in the goaf based on the minimum flow unit according to claim 2, determining the minimum flow unit in step 1) includes the following steps: the center point of each edge of the cube with the side length d It is the cut point, and the minimum flow model is divided into eight equally divided small units. Each small unit consists of a sphere with a diameter of d and four 1 / 8-diameter spheres. Is the minimum flow unit of the minimum flow model.
According to the method for calculating the pressure loss of nitrogen filling in the goaf based on the minimum flow unit according to claim 3, the specific operation in step 2) is as follows:

For the smallest flow unit, in the Z-axis direction, the origin of the coordinates is the center of the small-diameter sphere at the center of the smallest flow model, and the area of the cross-section at the point of z = 0 is
Where: d is the diameter of the sphere, and π is the circumference;

When constraints are met
, Z is the coordinate of the overcurrent section on the Z axis. As z increases, the area of the overcurrent section is equal to the square section with side length d minus 1 small-diameter semicircle area that changes with z and 2 changes with z Area of the large diameter semicircle, ie:

Similarly: when
Time,

when
Time,

when
Time,

In one cycle, the change of the overcurrent section is composed of the above-mentioned four phases; when the sectional relationship formulas of formula (1) (2), (3), or (4) are respectively satisfied, they are called the first flow phase, The second, third, or fourth flow phase.
The method for calculating the pressure loss of nitrogen filling in the goaf based on the minimum flow unit according to claim 4, wherein the pressure loss calculation formula of each flow stage obtained in step c) in step 4) is as follows:

The formula for calculating the pressure loss per unit length of the smallest flow unit in the first flow stage is:

J = Au 0 + Bu 0 2 , where:
J is the pressure loss per unit length, A and B are combined variables, u 0 is the average cross-sectional flow velocity along the Z axis at the centroid of the smallest flow unit (z = 0), α ′ = α 1 + α 2 , α 1 and α 2 are the viscous force ratio of the X-axis and the Z-axis, and the viscous force ratio of the Y-axis and the Z-axis, respectively, β ′ = β 1 + β 2 , and β 1 and β 2 are the inertial forces of the X-axis and the Z-axis, respectively. The ratio, the ratio of the inertial force of the Y-axis and the Z-axis, μ is the dynamic viscosity of nitrogen; the formula for calculating the pressure loss per unit length of the minimum flow unit in the second flow stage is:

The formula for calculating the pressure loss per unit length of the smallest flow unit in the third flow stage is:

The formula for calculating the pressure loss per unit length of the smallest flow unit in the fourth flow stage is: