DISTRIBUTION

OF THE

IN TURBULENT

FLOW ALONG

I. F.

Yarembash

CONCENTRATION

and

OF TOXIC

GASES

A WORKING

L. G .

Dashkovskii

UDC 622.457

The development of the mining industry involves longer workings driven at greater speeds. The amount of simultaneously detonated explosives is increasing, so that large amounts of toxic gases are formed. Such workings are v e n t i l a t e d by the forced method, i . e . , by feeding fresh air through ventilation tubes into the gassy part of the working. The gassy zone is i n i t i a l l y ventilated partly by the free turbulent flow of fresh air coming from the v e n t i lation tube (the zone of mixing); the rest of the working is v e n t i l a t e d by expansion of the toxic cloud as it moves along the working and mixes with the pure air. The face area is v e n t i l a t e d much more rapidly than the working as a whole [I]. The toxic gases must be diluted to a safe concentration before the gas cloud reaches the men present (who must be at a certain distance from the site of the explosion and must not get into a g a s - a i r current containing a dangerous concentration) or before the toxic gas cloud mixes with fresh air streams which m a y flow across the face. Men are not allowed at the face of the working before the toxic gases are diluted to the permissible l e v e l over a given length of the working (until there are no more dangerous concentrations), and therefore it is most i n t e r esting to study the ventilation process, not of the face area, but of some m i n i m u m length of working along which the toxic gas concentration is reduced to the l e v e l p e r m i t t e d by the safety rules. It has b e e n found that the gas-air mixture expands as it moves along the working, and mixes with fresh air not only where it meets it in front of and behind the cloud, but also within it. In addition, the toxic gases are partly absorbed by moisture in the air and adsorbed on the surface of the working. The process of r e m o v a l of toxic gases from the mixing zone has been studied in some d e t a i l [1, 2]; however, owing to our incomplete knowledge of the physics of the process of dilution of the toxic gases as they move along the working, the existing formulas for the amount of air required to c l e a r the working in a given period give large deviations from the actual data; and the m a x i m u m length of working over which the gases are diluted to safe c o n centrations is determined without allowance for the d y n a m i c a l and physical parameters of the air current along the whole of the working. The theories do not take accurate account of the boundary and i n i t i a l conditions in solving the p a r t i a l differential equation for the concentration of toxic gases, C(x ' t), along the whole length of the working [3]. In this a r t i c l e we give a f u n d a m e n t a l l y new determination and reassessment of the boundary and i n i t i a l c o n :ditions, in solving the equation for the variation of the toxic gas concentration in the turbulent flow along the working, we take account of absorption of toxic impurities along the entire path of the g a s - a i r cloud. As the toxic gas is diluted, its concentration varies owing to the presence of a concentration gradient C(x ' t ) , regarded as a function of the t i m e t and coordinate x (x being the distance from the site of the explosion); thus the problem concerns o n e - d i m e n s i o n a l diffusive dilution of gas. The gas cloud is entrained by the air flow, at the m e a n v e l o c i t y v averaged over the cross section of the working, p a r a l l e l to the axis of the latter. In this case we obtain a p a r t i a l differential equation of the parabolic type for C(x ' t) [4]. Using the coefficient of absorption 7 [2], this takes the form

Donets Polytechnic Institute. Translated from F i z i k o - T e k h n i c h e s k i e Problemy Razrabotki Poleznykh Iskopaemykh, No. 5, pp. 88-94, S e p t e m b e r - O c t o b e r , 1969. Original article submitted May 16, 1968.

I @1970 Consultants Bureau, a division o/ Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This artic!e cannot be reproduced for any purpose whatsoever [ without permission ef t,~c publis,~er. ,4 copy of t,~iS article is available fro..,: t,he publisher for $!5.00.

564

OC = Ot

D d2C Ox ~

v OC Ox

~ C,

(1)

where 7 is the coefficient of absorption of toxic gases in the working and D is the coefficient of m i x i n g . Mixing is due to the presence of turbulence and m o l e c u l a r diffusion, and therefore we can assume that D ~- D m +

Do,

(2)

where D m is the coefficient of m o l e c u l a r diffusion and D O is the coefficient of mixing due to turbulence. For suffuciently small values of C, for Dm we have the well-known Einstein formula [4], D m --

k r 6~lr"

(3)

where k is Boltzmann's c o m t a n t , T is the absolute temperature of the diffusing mixture, 0 is the k i n e m a t i c visvocisty of the air, r is the radius of the spherical diffusing particles. In this case r ,~ 10 "a c m , the radius of the gas m o l e c u l e s . We see that D m does not depend on either the t i m e t, the coordinate x, or the v e l o c i t y v of the d i f fusing cloud. From qualitative considerations it is obvious that to a first approximation D o wiU depend on v and the cross section S of the working. Then by the theory of dimensions we have: D,, ~- [

t

when v ~

0

9 ~OV S -

vr

(4)

when v > Vr,

where v T is the c r i t i c a l v e l o c i t y for transition from l a m i n a r to turbulent flow and $ is a dimensionless coefficient, depending on the coefficient of frictional resistance and the p e r i m e t e r , and c h a r a c t e r i z i n g the cross-sectional shape of the working; it can be found e x p e r i m e n t a l l y . From (4) we see that Do can also be regarded as constant, i.e., as independent of x and t at a given velocity. If v >> VT, D m can be n e g l e c t e d in (2). In this problem, to solve (1) we must use the c o m b i n e d boundary conditions. The boundary is L-shaped for the given semiinfinite problem. The boundary is formed by the r a y s ot and ox bounding the first quadrant, t > 0 >0

On the part of the boundary where t > 0, we use the boundary conditions in the Neumann form,

o C = 0,

(8)

Ox

and on the rest of the boundary x _> 0 we use the Dirichlet boundary conditions C ( x , 0 ) = f(x), which can be found by solving the equation 0p

__Ds

Ot

020

(6)

dx~

with the simplest boundary and i n i t i a l conditions

OOtx.,O x' dx"

=0 =o

,

(7)

where ptx, t~ is the density of toxic gas at t i m e t and point x ' , D B is the coefficient of mixing during the explosion, 6(x ) is the s o - c a l l e d d e l t a - f u n c t i o n [8], and 15 is the normalizing factor 9 w

x

,

z

In fact, at the beginning of the explosion the toxic gas can be regarded as concentrated at the point where the charge is sited, x' = 0, and then its density p ( x l , 0) will be given by the delta function. The normalization c o e f ficient i s found from the condition S S Ptxk. 0 ) d x " - - A b Po, 0

(8)

565

v,

where S is the unobstructed cross section of the working in square meters, p 0 is the density of the toxic gas at temperature T o and pressure P0 in the face area after the explosion (we assume that they have returned to the same values as before the explosion), A is the number of kilograms of simultaneously detonated explosives in kg, b is the gas factor of the explosives in liters per kilogram, and Abpo is the mass of toxic gas.

vz

V2>V,

Fig. 1. 1) Boundary of turbulent current; 2)zone of molecular-viscous motion; 3) zone with uniform turbulent field.

Substituting for P(Xl', 0) from (7) and (8) and using the well-known property of the delta-function that

S ~ (x') dx"

1, we obtain

I

(9)

= A b P o - $- .

For the semiinfinite problem with the given boundary and initial conditions (7), we obtain the following solution [3] to (6):

P(x,t)

_

1

2 1 / t O , toS

( x[ -D~ - x ' to ) =4 ] q - e x p

i p(x,, o) {exp [

(x + Dax')~ to ]}

(10)

0

Substituting for P(x[,0) from (7) in (10) and using (9), we get

A b Po e x p ( P(x, t) - - Vf = Ds to S

x~ ). 4D8 to

(11)

Then the initial conditions for (1) will appear as follows:

Abpo

C(x, o ) . = {

x*

exp(

4 D , to

] / ~ DB to Sp~

)

when O - . < x \ < . lo .

(12)

whenl o < x < 0C

Ox Ix

-

0

(13)

--0.

Here the time is reckoned from the moment t0, where to is the time taken by the toxic gases to spread along the expansion zone l 0. In (12) we used a formula linking the volumetric concentration of toxic gases with their density, C(x ' t) = P(x, t)/PB + P(x, t)' where PB is the density of the air in the working. If PB >> P(x, t) (this condition is always satisfied at to, when the toxic gases are already spread along the whole of the expansion zone and their partial pressure is vanishingly small),

P(x, t) C(x, 0 :

(14)

PB

From (11) and (14) we get condition (12). Since at the same pressure P and temperature T, p/PB = P/PB, where PB is the molecular weight of air and p is the molecular weight of the toxic gas, (12) finally takes the form

0) =

V

b; to s 0

566

4 D, to

when 0 -.< x < l o , whenl o
oo.

(15)

We can estimate D B as follows [3]:

D , ~-, ulo,

(16)

where u is the change in the speed of the gas-air mixture along the working from the explosion until it has c o m pletely stopped along the expansion zone, i.e., this change coincides with the initial velocity of the explosion current. Assuming that the gas-air mixture is uniformly retarded, we can express to in terms of u and l0 as follows: to _

2 l,

(lq)

U

By (16) and (17), D,

4 ~" ~

(18)

l0~,

where w is a proportionality coefficient, w =const. Putting'(18) into (15), we finally get

C(x, o) =

exp

Slo C x ~ 0

4 ~ t~

when 0 ~.< x ~

lo ,

(19)

~8 whenlo~x
":

The coefficient w can be can be found experimentally. The extra term ?,C appears in (1) in the presence of absorption proportional to the concentration. easily verify that the solution of (1) with this term takes the form

We can

(2 o)

CT (*, o = Co o,, o exp ( - -~ t),

where C0(x, t) is the solution to the equation 0 C / 0 t = D(02C/0x 2) - - v ( 0 C / 0 x ) and is obtained from (20) without the absorption term, ?, = 0. We can verify that (20) is indeed a solution of (1) by direct substitution of (20) into (1). The solution of (1) with boundary and initial conditions (13) and (19) is

C(., t) --

C(x, o) exp

2 kf - ~ - t

+ exp

4Dr

(21)

0 Putting (19) into (21) and making some simple transformations, we finally get

A b ~ exp (-- I t) exp [ " 4(--D-Tt~:/o2)(x--vt)2] C(x, t} --

2s~.VTV

~/~ )(

lo

D t + ~12o

D t ~120

o , + 0,r

x t]+ o[+r lg

Dt +~t 2

(x

vt)

D t +tol~o

o t.l~ ,

(22)

7r

where @(x) is the Laplace function ~(x)

l f - ~0

56"/

,,r 0'05t

o4t ffo,o31 o.ot o,01t

I

0

230

,00

330

4bdL. m

Fig. 2. l) Theoretical curve with allowance for absorption; 2) the same without allowance for absorption; 3) points at which toxic gas concentration was measured.

From (1) we see that the absorption coefficient ~, has the dimensions of the reciprocal of t i m e . This cgefficient characterizes the rate of absorption of toxic gases per unit time, and depends on the amount of moisture present, the area of contact of the toxic gases with the absorbing surface, the latter's adsorptive c a p a c i t y , and the speed of the air through the working. A specially important part is here played by t h e speed of the air, which characterizes the turbulence of the current and the possibility of repeated contact of the whole mass of toxic gases with the absorbing surface. Figure 1 shows the structures of a turbulent current in a working at different speeds. From qualitative considerations it is clear that

= a f (v, P.),

(23)

where a is a constant coefficient depending on the humidity of the air, the surface area of the working, and other factors, and PB is the perimeter of the cross section of the working. As v increases, 7 will also increase, because the zone of molecular-viscous motion which isolates the turbulent current from the absorbing surface decreases (see Fig. 1). The molecular-viscous zone itself makes almost no contact with the absorbing surface, because the motion in this zone is nearly l a m i n a r and the tramverse components of the gas velocity are negligible. From (22) we see that the m a x i m u m value of C(x ' t), as x varies with fixed t, is reached when x = vt. Let us call the plane x = vt the front of the air current carrying the toxic gas (the front of the current). Then we c a n say that the m a x i m u m concentration is preserved at the front of the current which is at the face of the working at time t = 0:

Cmax (x) - -

'

(24)

---- + tol~

To verify (24) in field conditions, we studied the variation of the m a x i m u m concentration of toxic gases moving along a working in the following conditons: A = 15 kg, b = 60 liter/kg, S = 7.4 m 2, v = 0.5 m / s e c , P/PB ~ 1, l0 = 27 m, ca =0.02, D = 5.5 mZ/sec, 7 = 1/300 sec -1. The concentration of toxic gases was measured at six points, 60, 100, 150, 200, 300, and 400 m from the face. From three measurements we compiled a system of three equations (24) with three unkonws D, 7, ca, which made it possible to determine the values of these coefficients. Figure 2 gives the theoretical and e x p e r i m e n t a l curves of the m a x i m u m concentration in the current of gasair mixture. The theoretical curve of Cma x agrees with the e x p e r i m e n t a l l y measured points. CONCLUSIONS Equation (24) characterizes the variation of the m a x i m u m concentration of toxic gases in a current of gas-air mixture throughout the length of the working. For given x, Cma x is independent of the speed of the air along the

568

working, provided that v > v T and y = 0, i . e . , in the absence of absorption. In this case, increase of the air speed (other conditions being constant) does not reduce the value of Cma x for the toxic gases, although the t o t a l time required to clear the working is reduced. If 0 < v < v T (laminar flow), the concentration of toxic gases changes p r a c t i c a l l y only as a result of absorption. For given x, the m a x i m u m concentration is not proportional to the quantity of simultaneously detonated e x plosives (which is not the case for other values of C, which must increase according to (22)), because 10 = 5 AS -~ Equation (24) enables us to c a l c u l a t e the value of v (or the quantity of air Q mS/sec) at whichthe m a x i mum permissible concentration of toxic gases is reached for given v e n t i l a t i o n t i m e t and given distance x, or the m i n i m u m length x0 of the working in which the m a x i m u m permissible concentration Cma x is attained for a given value of Q m3/sec. LITERATURE lo 2.

4.

CITED

V. N. Voronin, Principles of Mine Aerogas-Dynamics [in Russian], Ugletekhizdat, Moscow (1951). V. V. Skobunov, nTurbulent diffusion of impurities from instantaneous sources i n the face area of a tunnel working, ~ F i z . - T e k h . Probl. Razrabotki Polezn. Iskop., No. 4 (1967). V. N. Osipov and S. P. Orekov, ~Analysis of ventilation of blind workings by turbulent flow with periodic gas sources, ~ in: Planning and Construction of Coal Enterprises [in Russian],No. 7, Izd. TsNIITEIUglya, Moscow (1967). A. D. Landau and E. M. Lifshits,The Mechanics of Continuous Media [in Russian], Gostekhizdat, Moscow (1954).

569