Kumykova Т., Kumykov V.

(East Kazakhstan State Technical University, Ust-Kamenogorsk, Kazakhstan)

NATURE OF THE MOVEMENT OF COMPRESSED AIR ON SHAFT PIPELINES

Movement of compressed air in a coal shaft network obeys the laws of the turbulence of compressible media. Moreover, the main factor determining the behavior of turbulent flow, and in particular the laws of resistance, is the roughness of the pipeline, depending on the material of which the pipe is made of.

Since pneumatic energy complex of underground mines are complex systems, experiments with such systems in order to determine their optimal parameters are costly, time-consuming, and demand violation of production flow.

Mathematical modelling is based on the identity of the equations describing the processes of the model and of the studied phenomenon. The mathematical model of pneumatic energy complex, given the explicitly of a temporary nature of processes occurring in a pneumatic system, can be represented in the form of differential equations, reflecting the change in the basic parameters over time.

The problem of turbulence, which attracts the attention of scientists over the years which have passed since the appearance of O. Reynolds’ studies, currently is still far from its full resolution, despite the large number of published studies and its increasing practical value.

The main characteristic of turbulent flow is a messy, chaotic nature of turbulent fluctuations. Since the turbulent velocity fluctuations, rapidly changing from one point to another and from one instant to the next, are too complicated to be studied in detail, it should be enough to study only some averaged variables.

While studying the turbulent flow of a compressible medium, in addition to the correlation between the velocity components, the relationship between speed and density and the ratio between pressure and velocity also should be investigated. This greatly complicates the analysis.

According to the law of hydrodynamics of a compressible medium, the flow of compressed air in a pipeline is fully described by the equations of motion, energy, continuity and state. Basic equations of hydrodynamics for a viscous compressible medium / 1 / are as follow:

(1)

, (2)

, (3)

, (4)

, (5)

Р=r×R×Т (6)

where

=

dissipation function, (7)

,

- Velocity vector with components u, w and J along the axes x, y and z, respectively;
T - the instantaneous temperature;
r - instantaneous value of density;
p - instantaneous pressure;
m - dynamic viscosity coefficient;
Н - enthalpy per unit mass of compressed air (Н= С_р× Т, where Cp - specific heat of air at the isobaric process);

R - universal gas constant variable;
Pr - Prandtl number of medium.

Equations (1) - (3) represent the equations of motion, (4) - the energy equation (5) - the continuity equation, (6) - the equation of state.

Studying turbulence in a compressible viscous medium, we assume that the instantaneous measures – velocity components u, w and J, of pressure p, of density r, and of temperature T satisfy the fundamental equations (1)-(6).

Osborne Reynolds first introduced basic statistical concepts when studying the turbulent flow. In his theoretical study of turbulence, he admitted that the instantaneous velocity can be divided into the average velocity and turbulent fluctuation of velocity.

Therefore we can assume that

u = `u + u¢ , Т = `Т + Т¢ etc. , (8)

where the bar over the symbol denotes the average value, and stroke - turbulent fluctuations.

Direct substitution of equation (8) in the fundamental equation (1) - (6) does not provide any simple and direct conclusions about the field of perturbation u¢, Т¢ etc. due to the nonlinearity of these equations.

To reveal the nature of the fields with very small fluctuations in a viscous compressible medium, it should be resorted to some simplifying assumptions. The first obvious assumption - it is the assumption of small pulsations, allowing linearizing fundamental equations.

In order to show that small pulsations in a viscous compressible medium can be divided into three independent form, Kovazhny /2/ made the following assumptions: - Specific heat of air С_P and С_V, coefficient of viscosity m, coefficient of thermal conductivity of the medium are constant;

- Prandtl number Р_r = С_Р × m / À equals 3/4;

- The considered region of space and time is a finite region G. There are no solid boundaries inside the region or on its borders;
- Coordinate system is chosen so that the average rate in G is equal to zero;
- The speed is small compared to an average speed of sound in G;
- Pulsations of pressure р¢, density r¢ and absolute temperature T ¢ are small compared with the corresponding average values of these quantities р, `r and T in region G.
With these assumptions and introducing the following dimensionless quantities:

(9)

And (10)

Where

- Ratio of specific heats,
the fundamental equations (1) - (6) can be simplified and put into the following form:
equation of motion

(11)

where grad – is the gradient operator;
       div – is divergence operator;
energy equation

, (12)

continuity equation

, (13)

equation of state

, (14)

Where and .

Equations (11) - (14) are general in nature and as such can not be applied to study the interaction parameters of the shaft pneumatic network.

In each theory when solving a task some assumptions are made. Therefore, comparing the theoretical predictions with experimental results, it is necessary to trace how assumptions made in the theory are satisfied in the experimental setup.

Taking as a theoretical basis for research Reynolds equation (1), (2), (3), it is necessary to know the experimental values that characterize the mean motion. Indications of anemometer give no direct value of the average velocity, but a mean value of the function J (x, t) (such as J² and others). The value of velocity and its direction in the horizontal plane are usually measured.

According to the equation of Darcy-Veysbakh in the case of the flow of compressed air along the x-axis the equation of motion can be represented as:

(15)

where P (x, t) - function of the pressure of compressed air;

J (x, t) - function of the velocity of compressed air;

r (x, t) - function of density of compressed air;

l - coefficient of hydraulic resistance of pipeline;

      D - diameter of the pipeline sector;

      x - axis, located along the axis of the pipeline.

In practice of calculation of shaft pneumatic networks, instead of function of velocity, such concepts as volume or mass velocity are used.

Given the fact that

(16)

(17)

Where`V (х,t) - function of the volume velocity of the compressed air;`

G (х,t) - function of mass velocity of the compressed air;`
F_Т - cross-sectional area of the pipeline
expression (15) is transformed into

(18)

Where Т (х,t) – function of absolute temperature of compressed air;

- universal gas constant.

The equation of continuity according to / 1 / has the form:

(19)

Where c - speed of sound in air;

t – time.

Given these relationships, equation (19) can be transformed into

(20

Since compressed air flows along the x-axis, and the amount of heat entering a unit volume of compressed air per unit time due to radiation or any other reason, than the thermal conductivity is zero, as well as the fact that

(21)

energy equation (4) can be transformed to the form

(22)

Condition of air (6) on a linear plot can be represented as a function

Р(х,t) = R×r(х,t) × T(х, (23)

For practical use of the expression (23) in the problem it is advisable to differentiate it in respect to x

(24)

Thus, as a result of mathematical modelling a system of differential equations with partial derivatives: (18), (20), (22) and (24) was derived, which fully describes the distribution and interrelationship of the main parameters of compressed air along the axis of the pipeline in time. The boundary conditions of application of this model are as follows:

throughout the pipeline;

according to the technical conditions of the mine

(D_min =50 мм; D_max = 400 мм; Р_min ³ P_пасп. ³ 0,5 МПа; Р_max £ P_кс = 0,7 МПа).

Air consumption through the sectors of pneumatic network = 1 м: Q_i = 0,1¸4,5 м³/с.

References:

1. Jaworski B.M., Detlaf A.A. Handbook of Physics. - M.: Nauka, 1984. - 430 pp.

2. Bai Shi Yi Turbulent Flows of Liquids and Gases / / Translation from English Ph.D. MG Morozov / Edited by Dr. K.D. Voskresensky. - Moscow: Foreign Literature, 1982. - 344 pp.