The value of the diffusion coefficient. Diffusion coefficients

Decor elements 20.09.2019

Decor elements

Summer! The sun, heat, the smell of freshly cut grass in the air. The grass is far away from here, but you can smell it. This is physics, a similar phenomenon is called diffusion. And it is defined as the penetration of particles of adjoining substances into each other. For example, as now, the grass is in contact with the air, and the smell from the cut grass is carried through the air far from the place of mowing. And such a process is usually characterized by such a value as the diffusion coefficient.

We can say that everything that happens around us is largely connected with the processes of interpenetration. Thanks to this phenomenon, we even live. What's so amazing about that? penetration of oxygen or nutrients into the blood - this is the real diffusion, one substance penetrates into another. Diffusion is much more widespread in nature than we think. Such a phenomenon is not something exotic, but rather widely represented in the surrounding world. Even the fact that the air exhaled by a person does not accumulate around, but dissipates in space, is also due to diffusion.

Interpenetration can occur in the most different bodies- liquid, gaseous, solid. Its cause is the chaotic movement of the molecules of matter. By the way, such a phenomenon as diffusion is considered in the molecular-kinetic theory as one of the factors confirming it. So, returning to the state of aggregation of the body: the rate of penetration of a substance depends on it and on such a value as the diffusion coefficient.

What is this mysterious concept? This is the name of the quantitative characteristic of the rate of transfer of molecules of one substance to another. The diffusion coefficient, the formula of which is quite specific, makes it possible to estimate the amount of a substance that has passed through a unit area ( square meter) per unit of time (second).

In practice, it has been established that in gases, mutual penetration occurs with top speed, while in solids the penetration rate is minimal. The diffusion coefficient is influenced by body temperature and the mutual concentration of nearby substances. With an increase in temperature, the rate of interpenetration increases, and with an increase in the concentration of a substance, too.

Thus, the phenomenon of interpenetration is due to the so-called concentration gradient or temperature gradient. By type, diffusion is divided into free and forced. Forced arises under the influence external forces. Depending on their type, forced diffusion is defined as thermo-, electro-, baro- and ascending.

Diffusion is widely used in engineering. One of the most typical examples is diffusion welding. The essence of this technology is simple: two different bodies are connected (let it be two metals), and then they are subjected to pressure and temperature. The latter is smaller in magnitude than the melting point of substances.

The result is a combination of two dissimilar materials. This technology is widely used in instrument making and in the manufacture of large parts. complex shape, in experimental and small-scale production. Such welding can be carried out in different conditions, up to vacuum conditions, everything is determined by the specific requirements for finished product.

Another, no less common, use of the diffusion phenomenon is its application to obtain semiconductor structures. One of the technologies creating a district transitions is based on the phenomenon of interpenetration. Under the influence of high temperatures approaching in the crystal structure, areas with the desired concentration of impurities are obtained.

Consideration of the concept of "diffusion coefficient" made it possible to verify the extremely wide distribution of the phenomenon of interpenetration of substances in nature, as well as in various options for its use in technology.

LAB #3

INVESTIGATION OF THE PROCESS OF DIFFUSION OF IMPURITIES INTO A SEMICONDUCTOR

Purpose of work: study of models of the process of diffusion of impurities into a semiconductor for various technological conditions.

METHODOLOGICAL INSTRUCTIONS FOR PREPARATION FOR WORK

The content of the work: to understand the task, to get acquainted with the laws of diffusion - the Fick equations and the solutions of the diffusion equation for various special cases, which are widely used in carrying out diffusion processes in practice. To get acquainted with the methodology for working with a computer program that calculates the parameters of the diffusion process and graphic construction diffuser profile.

1.1. Basic information from the theory

Diffusion is the movement of atoms of a substance due to thermal motion in the direction of decreasing their concentration.

The basis of the mathematical description of diffusion processes are two Fick differential equations. The first equation (Fick's first law) is written as follows:

(3.1)

where J is the flux density of the diffusing substance, i.e. the amount of a substance passing per unit time through a unit surface area perpendicular to the direction of substance transfer;

N is the concentration of impurity atoms;

D - diffusion coefficient.

The transfer rate is proportional to the concentration gradient, and the diffusion coefficient is introduced as a proportionality factor. The minus sign on the right side of (3.1) indicates that diffusion occurs in the direction of decreasing concentration. In other words, diffusion occurs due to the desire of the system to achieve physical and chemical equilibrium. The process will continue until the chemical potentials of the components of the entire system become equal.

In the macroscopic representation, the diffusion coefficient determines the flux density of a substance at a unit concentration gradient and is thus a measure of the rate of concentration gradient equalization. The dimension of the diffusion coefficient is m 2 / s (in practice, the dimension cm 2 / s is more often used). In general, diffusion is anisotropic and the diffusion coefficient depends on the crystallographic direction.

The diffusion coefficient at the diffusion temperature is determined using the well-known expression in the form of the Arrhenius equation

where is the pre-exponential factor D 0 (diffusion constant) - diffusion coefficient at an infinitely high temperature, cm 2 / s;

Δ E - diffusion activation energy, eV;

k - Boltzmann's constant; T is the process temperature in degrees Kelvin.

Diffusion parameters of various elements in silicon, obtained by various researchers, are given in Table. 3.1

Table 3.1

	Factor D 0 , cm 2 /s	Activation energy , eV	Limiting solubility at 1200 0 C, cm -3	Conductivity type

Aluminum








				amphoteric

When the concentration of a substance changes in only one direction (one-dimensional diffusion) and when diffusing in an isotropic medium (diffusion coefficient is a scalar), the first Fick equation has the following form:

(3.2)

In the simplest analysis of structures and in the simplest models of doping processes in the technology of manufacturing ICs, such diffusion conditions are assumed.

The second diffusion equation (Fick's second law) is obtained by combining the first law and the principle of conservation of matter, according to which the change in the concentration of a substance in a given volume must be equal to the difference in the flows of this substance at the entrance to the volume and the exit from it.

In the general case, the second diffusion equation has the form

(3.3)

For one-dimensional diffusion in an isotropic medium, equation (3.3) can be written

(3.4)

Fick's second law characterizes the process of changing the concentration of a diffusing impurity over time at various points in the medium and is a mathematical model of the non-stationary (developing) state of the system (describes the period of time from the beginning of the process to the establishment of a stationary state).

At a constant diffusion coefficient D equation (3.4) is simplified

(3.5)

The assumption of a constant diffusion coefficient is valid in most cases implemented in IC technology.

The diffusion equations are purely phenomenological, i.e. they do not contain any information about the mechanisms of diffusion - about the diffusion process at the atomic level. In addition, equations (3.1) - (3.5) do not contain information about the charge state of diffusing particles.

Diffusion processes used to fabricate integrated structures are usually analyzed using partial solutions of equation (3.5) since, unlike (3.2), it contains an important parameter - the time of establishment of some analyzed state of the system .

The main goal of solving the equation is to find the impurity distribution N ( x , t ) in a semiconductor after diffusion for a certain time t at various conditions implementation of the process.

The general solution of equation (3.5) for an infinite rigid body with a given in general form the initial impurity distribution N ( x ,0) = f ( x ) can be found by separation of variables. It looks like

, (3.6)

where ξ - current integration coordinate.

The presented expression allows one to find the impurity distributions in a solid under any initial conditions. The solution of a particular problem is reduced to substituting into (3.6) the initial conditions corresponding to the situation, followed by, as a rule, very cumbersome transformations. In practice, when creating semiconductor ICs, three special cases are of interest: diffusion from a semi-infinite space, diffusion from permanent source and diffusion from an infinitely thin layer.

Diffusion from semi-infinite space(diffusion from the concentration threshold ).

The diffusing impurity (diffuser) enters the semi-infinite body through the plane x = 0 from the second semi-infinite body (source) with uniform distribution impurities. The impurity concentration in the source is N 0 . It is assumed that the body receiving the diffusant does not contain the impurity under consideration.

The initial distribution of concentrations for this case is given as

for x < 0,

for x > 0.

The solution of equation (3.6) for this case is the expression

, (3.7)

where erf z - called Gaussian error integral or error function ( error function ) Gaussian argument z . By abbreviation, this distribution is called erf – distribution:

. (3.8)

In mathematics, it is often used as an independent and other function

, (3.9)

which is called the complement of the error function to one or the additional error function - error function complement. Both functions are tabulated.

 Value
has the dimension of length and is called the diffusion length or diffusion length. The physical meaning of this parameter is the average distance traveled by diffusing particles in the direction of leveling the concentration gradient over time t .

The considered solution can be used as the simplest model representing the impurity distribution at the epitaxial film–substrate interface.

Diffusion from a permanent source.

The diffusant enters the semi-infinite body through the plane x = 0 from a source that provides a constant impurity concentration N 0 at the interface solid- source for any time. Such a source is called endless or source of infinite power. It is assumed that the body receiving the diffusant does not contain the impurity under consideration.

The initial distribution of concentrations and the boundary conditions for this case are given as

for x = 0,

for x >0.

The solution to equation (3.6) for these conditions is the expression

(3.10)

If in the volume of the semiconductor material before diffusion there was an impurity of the opposite type with respect to the diffusing one, this impurity is distributed uniformly over the volume and its concentration is equal to N ref , then in this case, an electron-hole transition is formed in the semiconductor. Its position (depth) x p - n is determined by the condition N ( x , t ) = N ref , where

(3.11)

and
(3.12),

where is the entry erfc -1 denotes an argument z functions erfc.

The considered model of a diffusion process with a constant source describes the process of diffusion doping of a semiconductor material from a gas or vapor phase. This process is used to create heavily doped diffusion layers (for example, emitter layers) with surface concentrations N 0 close to the values of the limiting solid solubility of the impurity N before in this semiconductor material.

Ddiffusion from an infinitely thin layer into a semi-infinite body with a reflective boundary on the surface. An example of the diffusion of an impurity from a thin layer into a semi-infinite body with a reflective boundary is diffusion into a silicon wafer from an epitaxial, implanted, or diffusion layer and coated with a layer of silicon dioxide. SiO 2 or silicon nitride Si 3 N 4 . The boundary between the plate and the film can be assumed to be reflective with a high degree of plausibility, since the diffusion coefficients of most impurities in silicon are several orders of magnitude greater than in silicon dioxide and silicon nitride.

The solution of the diffusion equation under these conditions is in the form

(3.13)

The above expression is the Gaussian distribution .

When solving this problem, it is necessary to know the amount of impurity Q accumulated in a solid during diffusion over time t . This value is determined by the formula

(3.14)

where J (0, t ) - diffusant flow into the volume through the plane x=0 .

Attention should be paid to the increasing value of the impurity accumulated in the diffusion layer with time during diffusion with given boundary conditions. p - n -transitions by the method of two-stage diffusion.

At the first stage of the process, short-term diffusion (at low temperatures) from a constant source is carried out, after which the impurity distribution is described by expression (3.10). Meaning N o in this case, it is large and is determined either by the solubility limit of a given impurity in the semiconductor material or by the impurity concentration in the glassy layer on the semiconductor surface. This stage is often referred to as "driving" .

After the end of the first stage, the plates are placed in another oven for subsequent diffusion, usually at higher temperatures. There is no impurity source in this furnace, and if it was created at the first stage in the form of a glassy layer on the surface of the plates, it is preliminarily removed. Thus, a thin layer of a doped semiconductor obtained at the first stage is a source of a redistributed impurity during the second stage of the process. To create a reflective boundary, the second step (often referred to as "stretching") is carried out in an oxidizing atmosphere. At the same time, a layer grows on the surface SiO 2 .

There is a noticeable discrepancy between the distribution of the impurity in the source, formed during the driving, with the one declared when deriving expression (3.14) - stepwise. This discrepancy should affect the accuracy of describing the real impurity distribution after the second stage of diffusion by expression (3.14).

When modeling two-stage diffusion and analyzing the results of the process, it is assumed that expression (3.14) corresponds quite accurately to the real one, provided that the value of the product D 1 t 1 for the first stage of the alloying process is much less than D 2 t 2 for the second - D 2 t 2 >> D 1 t 1 . This is a condition for the rapid depletion of the source. In this case, taking into account that the amount of impurity accumulated at the first stage is determined by the relation

from (3.14) we get

(3.15)

Quantities D 2 and t 2 belong to the second stage of diffusion.

A thin layer on the surface of a semiconductor wafer is a source that is depleted very quickly. Continuous diffusion in this case leads to a constant decrease in the surface impurity concentration in the semiconductor. This feature of this process is used in semiconductor technology to obtain controlled values of low surface impurity concentration, for example, to create the base regions of silicon transistor structures of discrete devices or ICs.

DESCRIPTION OF THE LABORATORY SETUP

V laboratory work To study the theoretical aspects of the process of diffusion of an impurity into a semiconductor, a special program is used that runs in the WINDOWS environment and an IBMPC personal computer. The program allows you to calculate the parameters of the diffusion process and plot impurity distribution graphs for various parameters: impurity type and source, impurity concentration, temperature and diffusion time

WORK PROCEDURE

3.1. Familiarize yourself with the methods of conducting diffusion processes to form p - n -transition and get the initial data from the teacher. Prepare and agree with the teacher a work plan.

3.2. Turn on the computer.

3.3. Run the program by double-clicking the left mouse button on the "LR 3" icon located on the desktop, and then "ChemicGraph.exe".

3.4.Press the "Settings" button and select "Values" in the drop-down menu. In the window that opens, enter the initial data for the diffusion process: D t , number of intervals (not less than 100), diffusion time t 1 , t 2 and t 3 .

Select the desired diffusion option at the bottom of the panel by checking the appropriate box. Enter impurity concentration value N 0 , cm -3 .

Press the "Intermediate values" button and in the window that opens:

Enter z 1 – 0,z 2 = 3…5 and step for calculating the function erfz and click the "Calculate" button to get the calculated values;

Enter x 1 ,x 2 and a step for calculating the profile of the distribution of impurities in depth and click the "Calculate" button.

Copy the calculated data to the report.

3.5. Close the Intermediate Values window and click OK on the Values panel to plot.

Click "Settings" and select "Chart Settings" from the drop-down menu. In the window that opens, by selecting the scale along the "X" and "Y" axes, achieve the optimal image of the graph on a linear scale.

3.6. In the Graph Setup window, check the Logarithmic Scale (Y-Axis) checkbox and click OK to plot semi-log scale.

3.7. Turn off your computer.

REPORT DESIGN

The report should contain: a) statement of the research problem; b) tables of calculation results; c) graphs of impurity distribution in a semiconductor wafer for t 1 , t 2 and t 3 ; d) analysis of the obtained data.

CONTROL QUESTIONS

1. What are the methods of introducing an impurity into a semiconductor?

2. What process parameters are included in Fick's first law?

3. What process parameters are linked by Fick's second law?

4. What is the dependence of the impurity distribution during diffusion from a constant source?

5. What is the dependence of the impurity distribution during diffusion from an infinitely thin layer?

6. What dependence does the impurity distribution have during diffusion from a semi-infinite body to a semi-infinite one?

7. What kind of addictions erfz and erfcz ?

8. What parameters are taken into account when choosing a diffuser?

9. What are the diffusion mechanisms?

10. How is the local introduction of an impurity carried out during diffusion?

Diffusion coefficients

Parameter name	Meaning
Article subject:	Diffusion coefficients
Rubric (thematic category)	Sport

Mixing path for gas content and gas streams

When solving problems of the dynamics of turbulent flows, the concept of a mixing path for an impulse is used. L. Prandtl defined this path as the distance traveled by a fluid particle before losing its individuality due to mixing with the surrounding turbulent flow. The mixing path characterizes the mixing capacity of the flow. This concept is also used in the theory of gas transfer. Keeping in mind that in diffusion processes the process of leveling the content is the main one, the mixing path is defined as the distance, ĸᴏᴛᴏᴩᴏᴇ passes a particle of a gas-air mixture until a significant change in the content of the diffusing gas contained in it due to mixing with environment. In this case, the expression ʼʼloss of identityʼʼ is interpreted as the particle losing its gas content, and the mixing path is commonly referred to as the content mixing path.

If the air flow is represented as a collection of spherical particles, then the mixing path can be considered as a turbulent analog of the free path of molecules, which, together with the speed of their movement, determines the intensity of gas molecular diffusion.

In general, the mixing paths for the momentum and for the content are not equal to each other, although until recently the latter was assumed to be equal to the mixing path for the momentum. Such an assumption should be taken as a first approximation only for a passive impurity.

The mixing path for the content is an important gas-dynamic characteristic that determines the main indicator of the intensity of the turbulent diffusion process - the turbulent diffusion coefficient.

Each of the four existing mechanisms for the spread of gaseous impurities in the ventilation flow (convective, diffusion molecular and diffusive turbulent transfers and the propagation of impurities by displacement) is characterized by a certain amount of gas carried by the gas flow through a unit area per unit time. According to the noted propagation mechanisms, there are convective, molecular diffusion, turbulent diffusion gas flows and "expansion flow".

If through a surface area S air is moving at an average speed u, then the vector of its flow through this surface, Q in = US. When the gas content With in volume Q into the vector of its flow through the considered surface due to convective transfer by the air flow Q r = WithQ in and vector of convective gas flow

J k = Q G / S = WithU , (6.5)

and its components along the coordinate axes

j k x = cu; j k y = cv; j k z = cw; (6.6)

where i, v, u- respectively, the components of the absolute velocity vector U.

When determining the molecular diffusion flow of a gas, they proceed from its proportionality to the gas content gradient (Fick's first law):

j m = - D m grad c, (6.7)

where D m - proportionality coefficient, called the coefficient of molecular diffusion.

Components of the molecular diffusion flow:

j m x=-D m ds/dh; j m y=-D m ds/du; j m z=-D m ds/dz.(6.8)

D m does not depend on the coordinates.

The minus sign in formula (6.7) means that the direction of the molecular diffusion gas flow is opposite to the content gradient vector, ᴛ.ᴇ. flow directed in the direction of falling content.

A turbulent diffusion gas flow can be expressed in a similar way to a convective one, using, however, the pulsating rather than average velocity vector u n and not the average, but the fluctuation value of the content With P . Then the vector of the instantaneous turbulent diffusion gas flow will be equal to With P u n, and time-averaged

The bar means time averaging. The components of this flow along the coordinate axes are:

where and P , v P , w n - components of the instantaneous pulsation velocity vector.

The turbulent diffusion flow, according to Boussingesk's idea of momentum transfer, is determined similarly to the molecular one, with the only difference that the coefficient of proportionality between the flow and the content gradient will be the turbulent diffusion coefficient D T , depending on its direction:

j t = - D t grad c, ; (6.11)

j T x=-D T x· ds/dh; j T y=-D T y· ds/du; j T z=-D T z· ds/dz,(6.12)

where D T x, D T y, D T z are the components of the turbulent diffusion coefficient.

The expansion flow is a convective flow. If a certain volume of a gas-air mixture with an average gas content With expands due to the introduction of additional quantities of the same gas into it, then the components of the expansion flow:

j R x=cu R ; j R y=cv R ; j R z = cw R ; , (6.13)

where u R ; v R ;w p are the expansion rate components.

The expansion flow must be positive (gas release occurs in the volume under consideration) and negative (gas absorption occurs in the volume under consideration).

Total gas flow at a point

j 0 =j k +j m +j T +j p (6.14)

Specific gravity each of the four gas flows in the overall balance of gas transfer in the development is determined by specific conditions. In the core of a turbulent air flow moving at a sufficiently high average velocity, the convective gas flow is usually predominant, followed by the turbulent diffusion flow. The molecular flow and expansion flow can be neglected in these cases. At low average velocities of the air flow (for example, chambers of large cross sections), the turbulent diffusion flow can become predominant in its core. At its solid boundaries, where the average and pulsation velocities are close to zero, the role of the molecular diffusion gas flow increases. Directly on a solid boundary, gas transfer is determined only by the mechanisms of molecular diffusion and expansion (in the case of gas release into the mine or its absorption). In the core of an air flow with developed turbulence, turbulent transport occurs hundreds and thousands of times more active than molecular transport.

The relationship between turbulent and molecular flows is determined from expressions (5.11) and (5.7):

The ratio between the flow components is defined similarly. So, for the components transverse relative to the main movement of the air flow

Example. Let's evaluate the role of the expansion flow for generation as a whole. Let's consider a working area in underground mining with a length of 100 m, with a cross-sectional area of 10 m 2 . The specific gas release into the production in this area is 1.5 l / (min m 2). Then, with a seam thickness of 1 m and two outcrops, the total gas release in the considered section of the working will be 1.5 × 1 × 100 × 2 = 0.3 m 3 / min. Therefore, the expansion rate along the working in two directions and p \u003d 0.3: (10 2) \u003d 1.5 10 -2 m / min. If the average fractional gas content in the considered section of the development c = 0.005, then, in accordance with formula (5.13), the expansion flow along the working will be equal to 0.005 1.5 10 -2 = 7.5 10 -5 m 3 / (min m 2). With the existing values of the coefficients of molecular and turbulent diffusion and the longitudinal gradient of the content, corresponding to the accepted gas release and air velocity in the working of 1 m / s and equal to 0.5 10 -5 m -1, the longitudinal molecular diffusion flow will be of the order of 10 -8 m 3 / (min m 2), longitudinal turbulent -10 -5 m 3 / (min m 2).

In expressions for diffusion gas flows, the coefficients of molecular and turbulent diffusion are the only parameters that take into account the properties of the medium. Naturally, these quantities are of a complex nature, and their determination is one of the important tasks of the theory of diffusion processes.

Molecular diffusion coefficients. For gases with similar molecules (having almost equal masses and effective cross sections), Maxwell obtained the following expression for the molecular diffusion coefficient:

where is the mean free path of molecules; v m - the speed of their thermal motion; the bar means the mean value of the quantity. At normal conditions has the order of 10 -5 cm, v m = 10 -4 ÷10 -5 cm/s.

Due to the statistical homogeneity of molecular motion, the quantities and , and, consequently, the coefficient of molecular diffusion do not depend on the direction. The molecular diffusion coefficient weakly depends on the content of the diffusing gas. With increasing temperature, it increases proportionally T 1+ a , where T - absolute temperature of the environment, a - coefficient varying from 0.5 to 1. With increasing pressure, the coefficient decreases in inverse proportion.

It was noted above that under mine conditions, molecular diffusion is of subordinate importance in the process of gas transfer. At the same time, changes in the content of gases, temperature and air pressure in actively ventilated mine workings are relatively small. For this reason, when solving problems of gas transfer in mines, one can take D m = const.

It should be borne in mind that the coefficient of molecular diffusion of a gas into a medium is equal to the coefficient of molecular diffusion of a medium into a given gas. The average values of the coefficients of molecular diffusion of some gases are given below.

Gas Temperature, °С Diffusion coefficient, cm 2 /s

Ammonia in air 0 0.217

Hydrogen in the air - 0.634

Methane in the air - 0.196

Carbon monoxide in the air - 0.129-0.138

Carbon dioxide in the air 0 0.142

Turbulent diffusion coefficients . In the theory of turbulence, the coefficient of turbulent (or eddy) diffusion is introduced as a certain proportionality factor. At the same time, three fundamentally different approaches are used to express it.

In the first way turbulent diffusion coefficient is determined, following Boussin-esque, as a coefficient of proportionality between the gas flow and the content gradient in accordance with the formula (6.11) - j t = - D t grad c.

It is known that the product of a vector, which is the content gradient in formula (6.11), by a certain value [in expression (6.11) it is the turbulent diffusion coefficient D m ] can give a vector [in formula (6.11) this is the gas flow vector] only if this quantity is a scalar or a tensor. The coefficient of turbulent diffusion should not be a scalar due to the fact that in the case of equality of derivatives of content in directions, the components of gas flows in these directions would also be equal, which is impossible under conditions of a significantly inhomogeneous and nonisotropic turbulent air flow in workings due to the difference in the components of pulsation velocities .

Τᴀᴋᴎᴍ ᴏϬᴩᴀᴈᴏᴍ, it remains to be assumed that the coefficient of turbulent diffusion in a mine working is a tensor.
Hosted on ref.rf
It can be shown that under conditions of inhomogeneous and nonisotropic turbulence, the turbulent diffusion coefficient is a tensor of the second rank. Then the components of the gas flow will have the following expression:

(6.17)

(6.18)

(i,j = x,y,z) is the second rank turbulent diffusion coefficient tensor with components D T xx, D T xy,..., D T zz.

Expression (6.17) should be written in folded form

. (6.19)

where right part is the sum of three values , obtained if we fix i, a j sequent x, y, z(summation over double index).

Expression (6.19) is usually simplified by assuming that the axes Oh, oh, oz are the principal axes of the tensor. If the tensor is symmetric, then the turbulent diffusion coefficient is determined only by the diagonal components D T xx,D T yu , D T zz.

For homogeneous and isotropic turbulence, the spherical symmetry of gas flows takes place. Hence,

In this particular case, it can be considered as a scalar.

In expression (6.11) the vectors j t and grad With collinear*. Therefore, according to the definition, the direction of the vector grad With is the main direction of the tensor, and the coordinate axis corresponding to it is the main axis. Finding the main axes of the tensor of diffusion coefficients for working out is in some cases an uncertain task, since for this it is extremely important to know the surfaces of equal contents in the stream, ᴛ.ᴇ. the field of grades, which is usually the ultimate goal of research. Only in simple cases diffusion, the main directions are determined quite simply. For example, when gas is released from one wall grad With with some approximation can be taken as normal to this side and, therefore, the main axes of the tensor will be directed along the air flow and perpendicular to it. In more complex cases, the main axes of the tensor may have other directions.

It should be noted that the adoption of the tensor D t y symmetrical for the case of air movement in a mine working is also a certain assumption. It should be said that for a non-uniform and non-isotropic turbulent flow, such as the ventilation flow in a working, the turbulent diffusion coefficient tensor will be asymmetric. It is noted below that the components of the turbulent diffusion coefficient tensor can be expressed through the averaged product (correlation) of the instantaneous values of the pulsation velocity and n i and agitation paths for content (here i,j = x, y, z, and n i = and n; u P at = v u n at =w n). It is important to note that for a symmetric tensor the equalities must be observed, which leads to the relations . At the same time, for nonisotropic ventilation flows, the correlation is asymmetric with respect to i and j, but this does not correspond to the given equalities. The asymmetry of the turbulent diffusion coefficient tensor for mine ventilation flows is indirectly proved by the factor of different intensity of turbulent diffusion in different directions.

The noted approximations, which are used in solving practical problems of mine gas dynamics, are currently not evaluated. As applied to the conditions of diffusion in the surface layer of the atmosphere, the errors are insignificant (in some cases they amount to 15-20%). At the same time, the degree of anisotropy of mine ventilation flows is much higher than atmospheric flows, which can lead to the extremely importance of taking into account the fact of the asymmetry of the diffusion tensor.

The second way to determine turbulent diffusion coefficient is based on the use of the Prandtl theory of the mixing path, according to which the gas flow components can be defined as the sum of three terms:

. (6.21)

Here, just as it was accepted in expression (6.19) - , the summation is performed over the double index ( j); i = j = x, y, z;; Lc- shuffling path for content.

From expression it follows that the coefficient of turbulent diffusion is a tensor of the second rank

(6.22)

defined by nine components -

Comparing the methods for expressing the turbulent diffusion coefficient according to Boussingesk and Prandtl, we see that in the first case the turbulent diffusion coefficient remains undefined, in the second it is determined through the characteristics of turbulent motion ().

In the case of a flat flow ( ) the coefficient of turbulent diffusion in the direction transverse to the main movement is determined from expression (6.21):

In the case of isotropic turbulence, one can take L cx = L su, which leads to equality

ᴛ.ᴇ. in this particular case, the turbulent diffusion coefficient is a scalar.

If in equation (6.23) v n Express Prandtl in terms of the mixing path for momentum L then for a flat flow we obtain the expression

, (6.24)

where the root mean square value v p

a 1 - coefficient of proportionality between u n and v p. If we accept that

L / L c \u003d a 2 \u003d const,(6.25)

. (6.26)

The quantity for the case of gas diffusion is analogous to the Karman momentum mixing path (not the identical Prandtl mixing path).

From the equation It can be seen that, having any hypotheses regarding the values l C(), it is possible, by measuring in the flow, to determine the coefficient of turbulent diffusion. The simplest assumption is to identify l s by stirring for momentum l; in many cases this approximation gives quite satisfactory results.

The next step in this direction is the adoption of proportionality between l s and l; the value of the proportionality coefficient between them depends on the properties of the diffusing gas, the difference in the gas content in the diffusing volume and in the medium. This coefficient is reportedly greater than 1; for nitrogen it is ~, for helium ~. There are attempts to evaluate l s across l and the Richardson criterion, which characterizes the attenuation of turbulence under the action of volumetric (gravitational) forces during diffusion of the active gas.

Finally, the third way to determine the coefficient of turbulent diffusion is based on the representation of the diffusion process as a random movement of liquid particles, initially concentrated in a certain area. Batchelor showed that in this case, too, the turbulent diffusion coefficient is a tensor of the second rank. Its record (for the case of homogeneous turbulence), however, has a different form:

where y i , y j - Lagrangian coordinates of a liquid particle, random quantities that are a function of time.

The representation of the turbulent diffusion coefficient in the form of a tensor is mainly of theoretical importance. Today, almost nothing is known about the off-diagonal components of this tensor. The components of the diffusion tensor studied to any extent are the diagonal components D T xx,D T yu , D T zz, which will be considered further on. For ease of writing, we denote D T xx = D T X etc.

It should be noted that, in the general case, the coefficient of turbulent diffusion is a function of the coordinates. This can be seen, for example, from equation (6.24), where the quantities d and / d,v ´ n , l s for flows in mine workings are functions of transverse coordinates, and in some cases (change in section along the length of the working, free jets) - and longitudinal coordinates. The same quantities are also functions of the flow velocity (more precisely, the Reynolds number - Re* of the flow), which indicates the existence of a dependence of the turbulent diffusion coefficient on the Re number as well.

Data on the coefficients of turbulent diffusion in mine workings are scarce, which is largely due to the technical difficulties of their measurement. The available information is partly based on data on the turbulent exchange coefficient for momentum and the assumption that the turbulent diffusion coefficient is proportional to it.

K.M. Tumakova established the self-similarity of the transverse components of the relative coefficient of turbulent diffusion:

; - average flow rate; α – coefficient of aerodynamic resistance; r- flux density; H - working height) according to the Reynolds number, starting from Re = 13600, as well as the equality of the vertical and horizontal transverse components of the diffusion coefficient . Their values in the core of the flow were 0.02, and at a distance of 0.13 H and 0.8 H from the roof - 0.03.

In a number of cases, good results are obtained if the values of the turbulent diffusion coefficients are averaged over the height (width) of the generation.

The turbulent diffusion coefficient must be calculated from the gas dispersion characteristic. It is important to note that for the case of homogeneous and isotropic turbulence in a uniform air flow (without a velocity gradient), the distribution of the gas content in the gas torch behind the source of outgassing is described by a Gaussian error curve:

, (6.27)

where With- gas content at the point with coordinates x, y; z - distance from source downstream; y - distance from the point corresponding to the maximum gas content with max in the plane X= const, measured in the direction perpendicular to the direction of air movement; and - air flow rate.

If in formula (6.27) With expressed as a part with max, then from it we can determine D T . For example, assuming With= with max /2, we get

D T =, (6.28)

where - the distance from the axis of the gas torch to the point in its cross section at which c = with max /2.

All quantities in equation (6.27) are directly measurable: and and X measured directly at the place of the experiment͵ - according to the dependency graph s(y), built on the basis of content measurement at a distance from the source equal to X.

Since expression (6.27) is valid only for homogeneous and isotropic turbulence, then, by virtue of equality (6.20), the diagonal terms of the diffusion coefficient tensor, which do not depend on the coordinates, are determined from it.

It is known that the turbulence of mine ventilation flows is non-isotropic; it can be considered homogeneous only in the direction of the main flow (with unchanged cross-sectional shape, wall roughness, and air flow). For this reason, for mine conditions, expression (65.28) gives, firstly, inaccurate values D m and, secondly, only some average values of the transverse component of the tensor D T y. The errors will increase as the gas source approaches from the flow axis to the wall, since in this case the source enters the region of an ever greater velocity gradient, ᴛ.ᴇ. ever greater anisotropy of turbulence.

Taking into account the experimental confirmation of the Reynolds analogy for the processes of momentum transfer and passive impurities in near-wall flows, the diffusion coefficients of mine ventilation flows during the diffusion of passive gases in the first approximation can be taken equal to the turbulent exchange coefficient for momentum. For Reynolds numbers from 1.25 10 4 to 3.72 10 4, the relative values of the latter for a drift-shaped working of a rectangular cross section, fixed with frame support from round timber with a longitudinal caliber of 7.5, relative roughness in the direction of the vertical axis 8.9, horizontal (perpendicular to the main movement) 8.4 are shown in the graphs of fig. 6.1 and 6.2, where y- coordinate ͵ perpendicular to the sides of the working, z- roof and soil. The recalculation of the relative values of the turbulent momentum exchange into absolute ones is carried out according to the formula ε = ε *v*D, where D- characteristic linear size of the flow (for example, diameter). The data presented in the graphs correspond to the cross-sectional average absolute values of the turbulent exchange coefficients for momentum ε y and εz , about 5 10 -3 m 2 / s at an average air speed in the working u cf =1 m/s, coefficient of friction α \u003d 15 10 -3 N s 2 / m 4, air density r= = 1.22 kg / m 3, working diameter D= 2.5 m.

Rice. 6.1. Dependence on y* = = y/H (H is the working height)

Rice. 6.2. Dependence on z* = = z/B (B - working width)

Component values D T y 10 3 (m 2 / s) obtained for some types of workings are given below:

drift model, cross-sectional area 13.4 × 14.2 cm, average air jet velocity 0.25 m/s..........................................1,1

crosscut, fixed with anchors, cross-sectional area 24.5 m 2,

air jet speed 0.5-1.2 m/s................................................. ......2.4÷4.1

the same, cross-sectional area 23 m 2, air jet speed 1.1 m / s........6,8

crosscut without fastening, vaulted section, cross-sectional area 11.8 m 2, air jet speed 1.7 m/s.......................................................5,1

the same, cross-sectional area 7.5 m 2, air jet speed 0.8 m / s…...1,8

drift without fastening, vaulted section, cross-sectional area 10 m 2 , air jet velocity 0.27 m/s...............................................................0,8

To calculate the longitudinal D T x and transverse D T y component of the coefficient of turbulent diffusion of methane in air, you can use the formulas below.

For drift-shaped workings

; (6.29)

, (6.30)

where , and the Reynolds number is not calculated from .- cross-sectional area of the working, m 2.

According to these formulas for some average conditions ( u cf = 1 m/s; H= 2.5 m = 0.1 m/s; R= 1 m) component values D T x, D T y are about 10 -3 m / s.

Turbulent diffusion coefficient D characterizes the dispersion of gas in the flow due to the work of turbulent pulsations. In some cases, the movements of the diffusing gas are superimposed by stronger movements caused by the presence of a shift (gradient) in the flow velocity. It is to such flows - "flows with a shift" - that mine ventilation flows belong.

In 1951 ᴦ. V.N. Voronin showed that when a gas cloud moves along a working, its longitudinal deformation is determined by the velocity profile. In 1953 ᴦ. J. Taylor published a solution to the problem of longitudinal turbulent diffusion of impurities from an instantaneous source in round pipe. He showed that the longitudinal dispersion of impurities caused by the velocity gradient is much greater than the dispersion caused by turbulent velocity fluctuations. J. Taylor suggested estimating the total effect of longitudinal dispersion of an impurity relative to a plane moving at an average flow velocity by a coefficient, which is called the effective diffusion coefficient D

Coefficient D e must be determined at a point or be averaged.

Research by S.P. Grekov and A.E. Kalyussky made it possible to obtain the following expression for the effective diffusion coefficient of a roadway:

; (6.37)

according to I.F. Yarembashu

. (6.38)

Here v- kinematic coefficient of air viscosity, m 2 / s; u cf, - the average speed of the air flow, m/s; D- working diameter, m; α - coefficient of aerodynamic resistance of the mine, N·s 2 /m 4 ; S- cross-sectional area of the mine, m 2 ; r- air density, kg / m 3.

K.Yu. Laigna and E.A. Potter in his recent works* give the following expression for the cross-sectional average effective diffusion coefficient:

, (6.39)

where , and the Reynolds number is calculated from . Meaning D e can also be determined from the graphs shown in Fig. 6.3.

Rice. 6.3. Graphs for determining the effective coefficient of turbulent diffusion

Calculations according to the above formulas and graphs give the values D e, on the order of several m 2 /s.

J. Taylor and K.Yu. Laigna, investigating the effect of channel curvature on the diffusion coefficient, concluded that this factor can increase D up to two times.

To study gas-dynamic processes under the action of free jets, V.N. Voronin applied the turbulent diffusion coefficient k m, defining it as the ratio of the average gas content in the cross section of the core of the constant mass of the free jet With i to the average content on its border with gr:

Values k m depend on the free jet propagation conditions and vary from 0.3 to 0.9.

Based on the similarity of the fields of velocities and contents in the core of the constant mass of the free jet, V.N. Voronin obtained the following expressions for the coefficient of turbulent diffusion of pure (without gas in the initial section) free jets:

for the main section of the round jet

k t = 1÷1.84 A; (6.41)

for the main section of the flat jet

k t = 1÷1.44 A;. (6.42)

In the above formulas

(6.43)

, (6.44)

where R I AM - radius of the nucleus of constant mass; and - speed at a point with coordinates X, y; and 0 - axial speed; φ i is the relative coordinate of the boundary of the nucleus of constant mass;

a - free jet structure coefficient, depending on the initial turbulence and velocity profile (according to V.N. Voronin, for a round jet a= 0.044÷0.053, for flat a= 0.09÷0.12).

The values of the turbulent diffusion coefficients calculated by the above formulas are given in Figs. 6.4.

The above expressions are valid for free jets in an unbounded space. Under the conditions of mine workings, free jets often propagate in limited volumes, while the air exchange between the jet and the surrounding air is determined not only by the structure of the jet, but also by the structure of air flows in its environment, which in turn depends on the geometry of the bounding surfaces and their roughness, and in general case is different from that in unlimited volumes. As a result, the coefficients of turbulent diffusion of jets in bounded spaces differ from those in unbounded spaces. This was first noted by Yu.M. The first who proposed to take it into account by a corresponding change in the structure coefficient a.

Rice. 6.4. Dependence k T from for round (a) and al / b 0 for flat (b) jets (l - jet length, S - area of its initial section)

With considering P - the ratio of the width of the chamber to the width of the workings that supply air, according to Yu.M. Pervov:

for a jet emerging from a square smooth hole, at

n>2,33a = 0,077(n-0,5)(n + 1).

At P< 2.33 the structure coefficient does not depend on the degree of restriction and is equal to 0.42;

for a jet emerging from a round smooth tube, at n> 2,33

a= 0,062 (P-0,5)

At n<2,33 a = 0,034;

for a flat jet at n> 3,12

a = 0,2(n 3/2 - 1,25n + 0,25)/(n 3/2 - 1),

and at P<3,12

a = 0,085 .

Diffusion coefficients - concept and types. Classification and features of the category "Diffusion coefficients" 2017, 2018.

Mixing path for gas content and gas streams

When solving problems of the dynamics of turbulent flows, the concept of a mixing path for an impulse is used. L. Prandtl defined this path as the distance traveled by a fluid particle before losing its individuality due to mixing with the surrounding turbulent flow. The mixing path characterizes the mixing capacity of the flow. This concept is also used in the theory of gas transfer. Keeping in mind that in diffusion processes the content equalization process is the main one, the mixing path is defined as the distance that a particle of a gas-air mixture travels before a significant change in the content of the diffusing gas contained in it due to mixing with the environment. In this case, the expression "loss of individuality" is interpreted as the particle's loss of its gas content, and the mixing path is called the content mixing path.

If the air flow is represented as a collection of spherical particles, then the mixing path can be considered as a turbulent analogue of the free path of molecules, which, together with the speed of their movement, determines the intensity of gas molecular diffusion.

In general, the mixing paths for the momentum and for the content are not equal to each other, although until recently the latter was assumed to be equal to the mixing path for the momentum. Such an assumption can be taken as a first approximation only for a passive impurity.

Each of the existing four mechanisms for the spread of gaseous impurities in the ventilation flow (convective, diffusion molecular and diffusive turbulent transfers and the propagation of impurities by displacement) is characterized by a certain amount of gas carried by the gas flow through a unit area per unit time. According to the noted propagation mechanisms, there are convective, molecular diffusion, turbulent diffusion gas flows and "expansion flow".

J k = Q G / S = WithU , (6.5)

and its components along the coordinate axes

j k x = cu; j k y = cv; j k z = cw; (6.6)

where i, v, u- respectively, the components of the absolute velocity vector U.

When determining the molecular diffusion flow of a gas, one proceeds from its proportionality to the gradient of the gas content (Fick's first law):

j m = - D m grad c, (6.7)

where D m - proportionality coefficient, called the coefficient of molecular diffusion.

Components of the molecular diffusion flow:

j m x=-D m ds/dh; j m y=-D m ds/du; j m z=-D m ds/dz.(6.8)

D m does not depend on the coordinates.

The minus sign in formula (6.7) means that the direction of the molecular diffusion gas flow is opposite to the content gradient vector, i.e. flow directed in the direction of falling content.

The bar means time averaging. The components of this flow along the coordinate axes are:

where and P , v P , w n - components of the instantaneous pulsation velocity vector.

The turbulent diffusion flow, according to Boussinesq's idea of momentum transfer, is determined similarly to the molecular one, with the only difference that the coefficient of proportionality between the flow and the content gradient will be the turbulent diffusion coefficient D T , depending on its direction:

j t = - D t grad c, ; (6.11)

j T x=-D T x· ds/dh; j T y=-D T y· ds/du; j T z=-D T z· ds/dz,(6.12)

where D T x, D T y, D T z are the components of the turbulent diffusion coefficient.

The expansion flow is a convective flow. If a certain volume of a gas-air mixture with a volume-average gas content With expands due to the introduction of additional quantities of the same gas into it, then the components of the expansion stream:

j R x=cu R ; j R y=cv R ; j R z = cw R ; , (6.13)

where u R ; v R ;w p are the expansion rate components.

The expansion flow can be positive (gas release occurs in the considered volume) and negative (gas absorption occurs in the considered volume).

Total gas flow at a point

j 0 =j k +j m +j T +j p (6.14)

The share of each of the four gas streams in the total balance of gas transfer in the development is determined by specific conditions. In the core of a turbulent air flow moving at a sufficiently high average velocity, the convective gas flow is usually predominant, followed by the turbulent diffusion flow. The molecular flow and expansion flow can be neglected in these cases. At low average velocities of the air flow (for example, chambers of large cross sections), the turbulent diffusion flow can become predominant in its core. At its solid boundaries, where the average and pulsation velocities are close to zero, the role of the molecular diffusion gas flow increases. Directly on a solid boundary, gas transfer is determined only by the mechanisms of molecular diffusion and expansion (in the case of gas release into the mine or its absorption). In the core of an air flow with developed turbulence, turbulent transport occurs hundreds and thousands of times more active than molecular transport.

The relationship between turbulent and molecular flows is determined from expressions (5.11) and (5.7):

The ratio between the flow components is defined similarly. So, for the components transverse relative to the main movement of the air flow

Example. Let's evaluate the role of the expansion flow for generation as a whole. Let's consider a working area in underground mining with a length of 100 m, with a cross-sectional area of 10 m 2 . The specific gas release into the mine in this area is 1.5 l/(min m 2 ). Then, with a seam thickness of 1 m and two outcrops, the total gas release in the considered working area will be 1.5 × 1 × 100 × 2 = 0.3 m 3 /min. Therefore, the expansion rate along the working in two directions and p \u003d 0.3: (10 2) \u003d 1.5 10 -2 m / min. If the average fractional gas content in the considered section of the development c = 0.005, then, in accordance with formula (5.13), the expansion flow along the working will be equal to 0.005 1.5 10 -2 = 7.5 10 -5 m 3 / (min m 2). With the existing values of the coefficients of molecular and turbulent diffusion and the longitudinal gradient of the content, corresponding to the accepted gas release and air velocity in the working of 1 m/s and equal to 0.5 10 -5 m -1, the longitudinal molecular diffusion flow will be of the order of 10 -8 m 3 / (min m 2), longitudinal turbulent -10 -5 m 3 / (min m 2).

In expressions for diffusion gas flows, the coefficients of molecular and turbulent diffusion are the only parameters that take into account the properties of the medium. Naturally, these quantities are complex, and their determination is one of the important problems in the theory of diffusion processes.

where is the mean free path of molecules; v m - the speed of their thermal motion; the bar means the mean value of the quantity. Under normal conditions, it has the order of 10 -5 cm, v m = 10 -4 ÷10 -5 cm/s.

It was noted above that under mine conditions molecular diffusion is of subordinate importance in the process of gas transfer. In addition, changes in gas content, temperature and air pressure in actively ventilated mine workings are relatively small. Therefore, when solving problems of gas transfer in mines, one can take D m = const.

It should be borne in mind that the coefficient of molecular diffusion of a gas into a medium is equal to the coefficient of molecular diffusion of a medium into this gas. The average values of the coefficients of molecular diffusion of some gases are given below.

Gas Temperature, °С Diffusion coefficient, cm 2 /s

Ammonia in air 0 0.217

Hydrogen in the air - 0.634

Methane in the air - 0.196

Carbon monoxide in the air - 0.129-0.138

Carbon dioxide in the air 0 0.142

In the first way turbulent diffusion coefficient determined, following Boussinescu, as a coefficient of proportionality between the gas flow and the content gradient in accordance with the formula (6.11) - j t = - D t grad c.

It is known that the product of a vector, which is the content gradient in formula (6.11), by a certain value [in expression (6.11) it is the turbulent diffusion coefficient D m ] can give a vector [in formula (6.11) this is the gas flow vector] only if this quantity is a scalar or a tensor. The coefficient of turbulent diffusion cannot be a scalar due to the fact that in the case of equality of derivatives of content in directions, the components of gas flows in these directions would also be equal, which is impossible under conditions of a significantly inhomogeneous and nonisotropic turbulent air flow in workings due to the difference in the components of pulsation velocities .

Thus, it remains to be assumed that the coefficient of turbulent diffusion in a mine working is a tensor. It can be shown that under conditions of inhomogeneous and nonisotropic turbulence, the turbulent diffusion coefficient is a tensor of the second rank. Then the components of the gas flow will have the following expression:

(6.17)

(6.18)

(i,j = x,y,z) is the second rank turbulent diffusion coefficient tensor with components D T xx, D T xy,..., D T zz.

Expression (6.17) can be written in folded form

. (6.19)

where the right side is the sum of the three values , obtained by fixing i, a j sequent x, y, z(summation over double index).

Expression (6.19) is usually simplified by assuming that the axes Oh, oh, oz are the principal axes of the tensor. If the tensor is symmetric, then and, consequently, the turbulent diffusion coefficient is determined only by the diagonal components D T xx,D T yu , D T zz.

For homogeneous and isotropic turbulence, the spherical symmetry of gas flows takes place. Hence,

In this particular case, it can be considered as a scalar.

In expression (6.11) the vectors j t and grad With collinear*. Therefore, according to the definition, the direction of the vector grad With is the main direction of the tensor, and the coordinate axis corresponding to it is the main axis. Finding the principal axes of the diffusion coefficient tensor for a working is, in some cases, an uncertain task, since for this it is necessary to know the surfaces of equal contents in the flow, i.e. the field of grades, which is usually the ultimate goal of research. Only in simple cases of diffusion can the principal directions be determined quite simply. For example, when gas is released from one wall grad With with some approximation can be taken as normal to this side and, therefore, the main axes of the tensor will be directed along the air flow and perpendicular to it. In more complex cases, the main axes of the tensor may have other directions.

It should be noted that the adoption of the tensor D t y symmetrical for the case of air movement in a mine working is also a certain assumption. For a non-uniform and non-isotropic turbulent flow, such as the ventilation flow in a working, the turbulent diffusion coefficient tensor will be asymmetric. It is noted below that the components of the turbulent diffusion coefficient tensor can be expressed in terms of the averaged product (correlation) of the instantaneous values of the fluctuating velocity and n i and agitation paths for content (here i,j = x, y, z, and n i = and n; u P at = v u n at =w n). For a symmetric tensor, the equalities must be observed, which leads to the relations . However, for non-isotropic ventilation flows, the correlation is asymmetric with respect to i and j, but this does not correspond to the given equalities. The asymmetry of the turbulent diffusion coefficient tensor for mine ventilation flows is indirectly proved by the factor of different intensity of turbulent diffusion in different directions.

The noted approximations, which are used in solving practical problems of mine gas dynamics, are currently not evaluated. As applied to the conditions of diffusion in the surface layer of the atmosphere, the errors are insignificant (in some cases they amount to 15-20%). However, the degree of anisotropy of mine ventilation flows is much higher than that of the atmosphere, which may lead to the need to take into account the fact that the diffusion tensor is asymmetric.

The second way to define turbulent diffusion coefficient is based on the use of the Prandtl theory of the mixing path, according to which the gas flow components can be defined as the sum of three terms:

. (6.21)

Here, just as it was accepted in expression (6.19) - , the summation is performed over the double index ( j); i = j = x, y, z;; Lc- shuffling path for content.

From expression it follows that the coefficient of turbulent diffusion is a tensor of the second rank

(6.22)

defined by nine components -

Comparing the methods for expressing the turbulent diffusion coefficient according to Boussinesq and Prandtl, we see that in the first case the turbulent diffusion coefficient remains undefined, in the second it is determined through the characteristics of turbulent motion ().

In the case of a flat flow ( ) the coefficient of turbulent diffusion in the direction transverse to the main movement is determined from expression (6.21):

In the case of isotropic turbulence, one can take L cx = L su, which leads to equality

those. in this particular case, the turbulent diffusion coefficient is a scalar.

If in equation (6.23) v n Express Prandtl in terms of the mixing path for momentum L then for a flat flow we obtain the expression

, (6.24)

where the root mean square value v p

a 1 - coefficient of proportionality between u n and v p. If we accept that

L / L c \u003d a 2 \u003d const,(6.25)

. (6.26)

The quantity for the case of gas diffusion is analogous to the Karman momentum mixing path (not the identical Prandtl mixing path).

where y i , y j - Lagrangian coordinates of a liquid particle, random quantities that are a function of time.

The representation of the turbulent diffusion coefficient in the form of a tensor is mainly of theoretical importance. At present, almost nothing is known about the off-diagonal components of this tensor. The components of the diffusion tensor studied to any extent are the diagonal components D T xx,D T yu , D T zz, which will be considered further on. For ease of writing, we denote D T xx = D T X etc.

It should be noted that, in the general case, the coefficient of turbulent diffusion is a function of the coordinates. This can be seen, for example, from equation (6.24), where the quantities d and / d,v ´ n , l s for flows in mine workings are functions of the transverse coordinates , and in some cases (change in the section along the length of the working, free streams) - and the longitudinal coordinates. The same quantities are also functions of the flow velocity (more precisely, the Reynolds number - Re* of the flow), which indicates the existence of a dependence of the turbulent diffusion coefficient on the Re number as well.

K.M. Tumakova established the self-similarity of the transverse components of the relative coefficient of turbulent diffusion:

In a number of cases, good results are obtained if we use the mean height (width) values of the workings of the turbulent diffusion coefficients.

The turbulent diffusion coefficient can be calculated from the gas dispersion characteristic. For the case of homogeneous and isotropic turbulence in a uniform air flow (without a velocity gradient), the distribution of gas content in the gas plume behind the source of outgassing is described by a Gaussian error curve:

, (6.27)

If in formula (6.27) With expressed as a part with max, then from it we can determine D T . For example, assuming With= with max /2, we get

D T =, (6.28)

where - the distance from the axis of the gas torch to the point in its cross section at which c = with max /2.

All quantities in equation (6.27) are directly measurable: and and X measured directly at the site of the experiment, - according to the dependency graph s(y), built on the basis of content measurement at a distance from the source equal to X.

It is known that the turbulence of mine ventilation flows is non-isotropic; it can be considered homogeneous only in the direction of the main flow (with unchanged cross-sectional shape, wall roughness, and air flow). Therefore, for mine conditions, expression (65.28) gives, firstly, inaccurate values D m and, secondly, only some average values of the transverse component of the tensor D T y. The errors will increase as the gas source approaches from the flow axis to the wall, since in this case the source enters the regions of an increasing velocity gradient, i.e. increasing anisotropy of turbulence.

Taking into account the experimental confirmation of the Reynolds analogy for the processes of momentum transfer and passive impurities in near-wall flows, the diffusion coefficients of mine ventilation flows during the diffusion of passive gases in the first approximation can be taken equal to the turbulent exchange coefficient for momentum. For Reynolds numbers from 1.25 10 4 to 3.72 10 4, the relative values of the latter for a drift-shaped working of a rectangular cross section, fixed with frame support from round timber with a longitudinal caliber of 7.5, relative roughness in the direction of the vertical axis 8.9, horizontal (perpendicular to the main movement) 8.4 are shown in the graphs of fig. 6.1 and 6.2, where y- coordinate perpendicular to the sides of the working, z- roof and soil. The recalculation of the relative values of the turbulent momentum exchange into absolute ones is carried out according to the formula ε = ε *v*D, where D- characteristic linear size of the flow (for example, diameter). The data presented in the graphs correspond to the cross-sectional average absolute values of the turbulent exchange coefficients for momentum ε y and εz , about 5 10 -3 m 2 / s at an average air speed in the working u cf =1 m/s, coefficient of friction α \u003d 15 10 -3 N s 2 / m 4, air density r= = 1.22 kg / m 3, working diameter D= 2.5 m.

Rice. 6.1. Dependence on y* = = y/H (H is the working height)