LITERATURE 1. 2e 3. 4.

CITED

O. Ya. Kokorin, Air-Conditioning Installations: Calculation and Design [in Russian], Mashinoatroenie, Moscow (1978). N. N. Koshktn (ed.), Refrigeration Machines [in Russian], Plshchevaya P r e m y s h i e n n o s ~ , Moscow (1973). Yu. A. Tseitlin, Air-Conditioning Installations f o r Mines [in Russian], Nedra, Moscow (1974). Mamtal on Use of Air-Conditioning Equipment in Deep Mines [in Russian], Izd. Mak. NII, Makeevka (1980).

DETERMINATION COEFFICIENT FOURIER

OF THE FOR

HEAT-TRANSFER

A ROCK MASS WITH

SMALL

NUMBERS

A. K. Yakovenko

and G. V. Averin

P r o b l e m s of nonsteady convective heat exchange between the r o c k m a s s and m i n e methane a r e associated with solution of a differential F o u r i e r thermal-conductivity ec~atlon; in this case, boundary conditions of the t h i r d kind (Newton' s law) in the f o r m q == = ( t ~ - t=), where twa and t a a r e the wall and a i r t e m p e r a t u r e s , respectively, in the mine excavation, a r e examined at the boundary between the r o c k m a s s and ventilating a i r . The h e a t - t r a n s f e r coefficient a depends on the aerodynamic ~m4 physical conditions in the excavation, and, in the general ease, is a function of t i m e and the length of the rock excavation. LykDv [1] indicates that in proDl e m e involving nonsteady heat exchange, the h e a t - t r a n s f e r coefficient ~ cannot be c o n s i d e r e d constant, since this m a y give r i s e to e r r o r s that a r e especially e|~n4fleant in the initial time i n t e r v a l s in the p r o c e s s of nonateady heat exchange. In this ease, theoretical determination of the heat-exchange coefficient of a r o c k m a s s with s m a l l F o u r i e r numbers, the value of which m a y be u s e d on the basis of solution of the r e l a t e d p r o b l e m of conve0tive heat exchange between the rock m a s s and m i n e methane, plays a significant r o l e . In this case, the general statement and method of solution of the related p r o b l e m of convective heat e x change a r e ~dareased in detail in [21. As we know, the problem mathematically recktces to solution of the following s y s t e m of equations:

#~=

~te

~-"~'+

+ PoT; i -

t #t.

T"R"

t +

(Vo>O;,t , ~ < = ) ;

| at

(1)

(Fo>O; 0 < ~ < t ;

o
For the initial and bou~a~y oonditions, Fo--O,

l - - t,.

l.tc;

t - t,;

at

~-o.

b T " O;

(6)

t . - tc: OIj

#1

- ~ - k~ ~ . ,

(3) (4) (5)

t . - t.

if)

Scientific-Research Institute mz Mining Tm~,stry Safety, Makeevka. T r a n s l a t e d f r o m Fiziko-Teknicheekie P r e b l e m y Razralmtki Poleznykh Iskgpaemykh, No. 1, pp. 63-6~, J = m , a r y - F e b r u a r y . 1984. Origin=l article submitted J m m a r y 25, 1983.

52

0038-5581/84/2001-0052508.50

0 1984 Plenum Publishing C o r p o r a t i o n

The following designations a r e adopted in s y s t e m of ec~aUons (1)"(~):

,,.-

,,- ,+.;

,.,.-

,,.-

-.,.... . ,.,_--,,. ,.

R and x a r e spatial cylindrical coordinates; ~, ~ , - e ; t s and t, t e m p e r a ~ r e s of the r o c k m a s s and air, r e s p e c tively, in the excavation; w0, a v e r a g e a i r t e m p e r a t u r e in the excavation; I~, equivalent radius of the excavation; a, ;~, ns, and ~s, t e m p e r a t u r e and t h e r m a l condnctivities of the a i r and rock m a s s , respectively; and ~ ( s t n r bulent t h e r m a l conductivity of the a i r in the excavation, the selection of which is based on investigations by Uzhaknv [~1. Proceeding from examination of two-layer flow in a cylindrical channel, the quantity ~(~) was selected in the following manner:. in the viscous l a y e r (the n e a r - t h e - w a l l region with a thtckness ~w, where moleottlar frit~ion a s s u m e s decisive significance, and the coefficient of turbulent t h e r m a l conductivity is equal to ~ r o ) , t - ~w~ ~ ~ t; ~w< t: ~ -A- 0 . in the turbulent core

(0 < ~ < I -- ~w):

)~ 0.) ~.

-

C (E -- ~'),

where C is a constant c~effielent [2]. Solution of edge p r o b l e m (I)--(?) for small F o u r i e r m~rnbers is possible using a two-~_mensional Laplace t r a n s f o r m in t e r m s of the v a r i a b l e s Fo and ~ with subsequent u s e of the method of quasiolassic approx/maUon to determine the solution of Eq. (2) in the domain of the r e p r e s e n t a t i o n s in [2]. After using asymptotic methods and an inverse Laplace t r a n s f o r m , we found the solution of s y s t e m (1)--(7) f o r s m a l l Fourier mmtbers (0 < Fo < 0.3) in the f o r m of rapidly converging s e r i e s . In this case. the rHrn,~nsio~less wall t e m p e r a t u r e of the r o c k excavation (0 wa) and the Klrpichev criterion (Ku) a s s u m e the f o r m ewa(V,,. ~l) " (l.)~.t -- t, = l -- 2

.~,

re--,,

qwalte

K.(Fo, q ) - ~.,(tc

( - - I)" ~' ~

{~Pop~

.,,,v(, + t

)'

(F,,)(.-,~ 2 (-- t)"+z -c ~- I-./~""E--/ t,) =" .-.., 9 + I"--'-~'

(8)

(9)

where d c are constant ooeffloients dependent on the physical p a r a m e t e r s characterizing the heat-exchange p r o cess, and on the dtmens/onless length ~ of the exoavaUon. DeterminaUon of the h e a t - t r a n s f o r coeffio/ent a , which is a function of Hme and the length of the rook e x cavation in the given case, is of i n t e r e s t f r o m the theoret/md standpoint. Th@ analytical relatiooah/p for the local h e a t - t r a n s f e r coefficient a a s s u m e s the f o r m K . (Po, I|)

~0)

., (Fo. ~) - 7~'" ewatVo, q) -.- Oav~Po.,1) '

where #av(Fo, 7) is the average ~mensionless a i r temperature over the section of the exoavsUon, which is equal A

to 0 ~ -

2~0~. @

Here,.

O--./-'.t'. I c -- I I 9

Let us dwell in greater detail on determination of the quantity ear- The two-di,~enslonal representation of the average dimensionless a i r temperatmm in the excavation, which is determ/ned from solution of system (1)-'(7) by the method d e s c r i b e d in [2], a s s u m e s the f o r m '

Vb.p . ; *~ V p - - " ~ ,

,

(xx)

" \ y ' + - x,tt;J|"/

*.l/~' I.(,b) I / . /

/,(*'l(~))~

53

/

/

-b=,Pm/sec

~

0.2 ---

mls~ ~1~

~ ~

i

~oo

,~

Fig. I .

Fig. 2.

in the case of l a r g e values of the p a r a m e t e r p (small F o u r i e r numbers), where

!'i/

I "'

:.....

~-t

A

I .- 4b kak~. + a

z' ~- Itp "- I'.. s.

F o r an approximate evaluation of the definite integral ~Jiz')) in 411), we can m a k e u s e of the fact that for l a r g e values of the p a r a m e t e r z', the neighborhood of points where g(~) > 0 m a k e s the basic contribution to the value of the integral, since in this ease the function Ie4zv g4~)) a s s u m e s the asymptotic representation: ,-.'~*,t,

(

T, (.-'g(~))-- : t ~

,

i+ ~

,,

)

+ ~ (--'t(~))-"+

. . . .

Consequently, the definite integral J(z') can be represented approximately as I

j, ainoe limg(~)--O andg(~) > O w h e n O < ~ <

1. H e r e , G ( ~ ) ~ / -

.

~ '

~':"

and l' is some s m a l l n u m b e r

g /

(j' 9 0). I . ~ us evaluate integral (12) asymptotically using the Laplace method [4l. The funcCt.on g4~) attains its own c l e a r l y e x p r e s s e d m a x i m u m value in the vicinity of point ~ = 1 (the viscous l a y e r n e a r the wall with a thickn e s s 6w). This is apparent f r o m Fig. 1, where the c h a r a c t e r i s t i c form of the fanction g(~) is p r e s e n t e d s c h e m a t i c a l l y . F o r l a r g e values of the p a r a m e t e r z', oonsequently, only the value of the flmction under the integral exists in 412) when 1 - 6 w ~ ~ ~; 1, w h e r e g4~) m b; ~ 4~) ~ 1; G4~) ~ 1. Making use of the basic positions of the Laplace method, we obtain an a s y m p t o t i c expansion of the integral J ( ~ ) in the f o r m

J" 4z') ~, ~ ,"'

[l + .p..J_~ ~T" ,e,,~,,,-* + . . . . ,]

(13)

where l~ - ~-~k,. Taking (13) into account, e x p r e s s i o n 411) can be written in the approximate f o r m

y +a-

9p y p+--gl~,, + , . i / ; g W ; § ~,1/; f o r l a r g e v a l u u of the p a r a m e t e r z ' . Proceeding Into the domain of originals In t e r m s of the p a r a m e t e r e and m a k i n g u s e of the method of o b a l ~ r e ~ m i t e Iolutions f o r s m a l l Fo m m b e r e on the basis of s i m F l l f l ~ t t o n of the a p p r o p r i a t e r e p r e s e n 9_-~on f o r l a r g e values of the p a r a m e t e r p of the Laplace t r a n s f o r m [41, we obtain t h e a v e r a g e a i r t e m p e r a t u r e o v e r the section of the mine ex~vafloB, which f o r Fo > ~ / P e and T/ 9 0 Is r e p r e s e n t e d in the f o r m of the s e r i e s :

54

.o0

L I I

~f2

,ii i II, O,OOf

~ , F~O

- .

I o

.ooo

J'OO o

;,,oo

~00

Fig. 3.

(14) where a t , .... ~n a r e constant coefficients, for example, a t , ~ , and ors a s s u m e the form

01|

Knowing the a v e r a g e a i r _b~.nerature threughout the sect/on in the exoavation, it is po~aible to d e t e r m i n e the overall qumzUty of h e ~ 1/berated from the rock m a s s p e r length x, wh/ch is equal to Q -c,pws

t,) -

c,o~J(:c- t,)on,

(is)

where Cp and p a r e the spec/flr i s o b a r heat capacity and density of the air, respect/vely, mad S Is the e r o s s - s e o tional a r e a of the m i n e excavation. According to (15), the a v e r a g e quamdty of heat l/berated f . ~ m the rook m a s s p e r unit surface of r o c k e x onvzd~on p e r u ~ t t i m e will be

q- ~

o,, ( , ~ - tJl, -

t'p~P~ Ogv i ~ Z q vc--l,),

where P is the p e r i m e t e r of the m i n e exoavation. Let u s intreduce an analoSy of the coefficient of nonsteady heat exchange ~T, w h l o h / s equal to the a v e r a g e specific heat flux between the nuum and ventilating a i r as applies to a unit t e m p e r a h t r e differenoe At = I (to t~ = x) [41

The a v e r a g e d!menaloelees a i r t e m p e r a t u r e 0av a o r o u the sect/on of the m/he excavation and the h e a t - t r a n s f e r ooofflciant a w e r e oomputed f r o m Eqs. (14) and (I0) f o r kx = 0.023, Ira = 5.62, Pe = 0.313.10 T, and Re - 0.85510 e. Results of the Oar ocmputat/on f o r d / f f e r ~ t a v e r a g e a i r speeds in the exoavailon a r e premmted in Fig. 2 as a funoflon of the F o u r i e r n u m b e r Fo and the dimensionless longitudinal ooord/naie ~. It is apparent f r o m

55

Fig. 2 that the average a i r t e m p e r a t u r e a c r o s s the section i n c r e a s e s less vigorously with increasing average a i r speed in the excavation. Analysis of the results of oomputaflon of the h e a t - t r a n s f e r coefficient a indicated that a depends on both the Fo number and the longitudinal coordinate ~. Curves showing the dependence of ~ on Fo and 77 f o r an a v e r age a i r speed w 0 = 3 m / s e c in the exoavation a r e p r e s e n t e d in Fig. 3a. It is apparent f r o m these curves that the h e a t - t r a n s f e r coefficient can be considered virtually constant, i r r e s p e c t i v e of t i m e and the length of the rock excavation only when ~ > 100, o r x > 1001~, i.e., the h e a t - t r a n s f e r coefficient a a s s u m e s a c l e a r l y e x p r e s s e d variable c h a r a c t e r for all 0 < x <100R 0 in the case in question. The dependence of the h e a t - t r a n s f e r coefficient ~ on the coordinate ~ for different a v e r a g e a i r speeds in the excavation when Fo = 0.001 is p r e s e n t e d in Fig. 3b. It follows frmn what we have stated above that the h e a t - t r a n s f e r coefficient a I s a v a r i a b l e quantity in p r o b l e m s of nonsteady heat exchange with small Fo n u m b e r s ; this m u s t be considered in heat computations. LITERATURE le

2. 3. 4.

A. V. Lykov, Heat and Mass T r a n s f e r (Handbook) [in Russian], ~.nerglya, Moscow (1971). A. K. Yakovenko and G. V. Averin, "Related p r o b l e m of convective heat exchange in benches of deep shafts," Flz.-Tekh. Probl. Razrab. Polezn. Iskop., No. 6 (1982). K. Z. Ushakov, Aeromechanics of Ventilating Flows in Mine Excavations [in Itusslan], Nedra, Moscow (1975). A. V. Lykov, Heat C o n ~ c U v i t y Theory [in Rasaian], Vyeshaya ~ k o l a , Moscow (1967).

ESTIMATING FOR

CITED

THE

VOLUME

A DEVELOPMENT

ON T H E

AMOUNT

OF VENTILATION

WORKING

OF GAS FORMED

FACE

AIR

BASED

DURING

BLASTING

V. I. B u g r i m o v

The existing method f o r calculating the required quantity of ventilation a i r f o r development workings based on the amount of gas f o r m e d during explosions [I, 2] m a k e s use of the following p r e v i c u s l y published formula:

Qdw" 7

V

,~p

Lm~J

11)

Thls method has basic shortcomings: I . Expression (I) oon*~In~ in the ~ r under the root sign the a i r l e a k ooefflolent k l p that c o r responds to total length of the plpeUne, although, quite often, the p/pellne may be m u c h longer than the length of the face pert/on of the worklnl~ so that not all the a i r leaking f r o m the pipeline w o r k s to dilute the toxic gas wlth/n the ventilated dsadend face; tlds o v e r e s t h n a t / o n of the leak ratio reducos the resulting r a t e d amount of ventflat/on a i r . 2. The a i r volume estimated by (I) with the 9 orit/oal length = In the m u n e r l t o r of the t e r m u n d e r the root sign m a k e s u s e of the quantity I c r < It, but tim a i r l e a k ratio is lnvariant a n d c o r r e s p o n d s t o t o t a l p l p e l i n e length. slthough the a i r leak on the near-face a r e a is only a fraotion of total leak; o v e r r a t i n g the leak also r e s u l t s in an u n d e r e s ~ m s t e d required a i r volume.

3. In eva~ating the ~eriUoal lensth;" the caloulaUon begins wtth the formula . . . .

!t == lZ.;)

n,,...~,~

~, 1",

(2)

B ~ m c h I m ~ t u t e of the Ministry of the Coal Industry of the USSR, Kemerovo. T r a n s l a t e d f r o m Ftz/kgT.drhnl~'~skte P r o b l e m y ~ Polesnykh I s k n p a m n y k ~ No. 1, pp. 68-71, J a ~ t a r y - F e b r u a r y , 1984. Orig-

/hal a.-'t/ole submJtted ,Tmmary 14, 1962.

56

0038-5581/84/2001-0056508.50 Q 1984 lZ.enum Publtsh/~g CoriporaUon