(v) P.ock conductivity k » 3.0 PTU/ft2 hr (°f/ft)

2(vi) Hock diffusivity D » 0.1 ft /hr.

?' 1 $ <r <* n '* 1 v '

„ tThe wet portions of the stcrpe were assuned to he wet

over the conolete cycle. The abscissa in Fig. (6.17) is the

l e n g t h froa the scatterwall alone the hkngingwall» followed

W the length down the stope fuce and then the length ?rcm

the scatterwall along the footvoll as shown in Fig. ,6 .17 ).

i .e . the total perimeter of tee stope tas been arranged so

that it foras a single straight line.

As expected the dry stope produces a symmetrical

tmperatur, distribution. The vet footvall produce, a

c o n s i d e r a b l e drop in the footvall « v r f « . t ^ a t u r o » *

little effect on the resnirfer of the store surface. chile

the vet face and vet foctvall produce a fairly substantial

drop in the hanginjr.mil surface t w e r o t ^ e as veil. The

sudden drop vhich oc.-urs at 9 ft alone the tangingvall is

thought to have been generated due to an unsatisfactory

-i-r- nh-sicrJL conditions, viz. Isxgc simulation of the p r e v a i l s nh sic.o.

- * ^r«+nr- c-cur at the corncr of the stope a^upt changes m teoperature c~car

where the condition of the surface changes fro* wet to drj.

F i g . (6 .1 M shows the dir.onsionless fnperasuro 2 ft

parameters as above. Here W t e n u r e profiles ar,.

A face -nd footvall again inducing a drop smoother, vith the vet face ^ ^

in teaoeraturc all around the stope as con,

1 case. Ao before the face a * h a ^ i n g ^

virtually uiaffected by the vet footvall co

VO

53

nost important conclusion which nay be drawn froa

these results is that wetting the stcmr faer- affects the tear.

orature distribution all around the stope, whereas wetting the

hangingwall or footwall only has a localised effect. It

Sja&ltt*.' .

would thus r-rpoor that the face is the sost significant part

-I.of the stone with regard to wetting. This iB a logical

-jostulate since the highest rock temperatures are always

.15 Conclusion

Saul'ev’ s (ADEP) net hod for solving a tvo dimensional

neah gives an accurate solution in a reasonable computing tine.

Variations in ppaco and tine intervals, conductivity, diffu-

sivitv, density, specific heat and surface heat transi er

coefficient throughout the nesh sake the nethed quite general.

The determination of the flux into the stope at ary

tine is not accurate since it depends on nunerical differentia-

tion, but the total flux over one cycle m y be accurately

detemino3. *by nunerical in’fcegrabion#

Also, the ADEP is suitable for deternining the dinen-

sionless surface tareneratitre of the stope vhic-. fcr^s the

has it for calculating vet and dry hulh air temperature changes

in a stope (see Chapter T )»

t ev* s net bod could also prove extremely useful in

solving the ccaplete nroblen of an advancing id r a f . The

adaption required f * » the stope cituntion is not too great,

although the boundary condition, describing the physical

situation are far nore co:

ions

'Hie development of e. mathematical made.stop®

Using tha basic equations fomulated by S t a r f i e l d ^ and

(7) .Jordan , together with the diaiens monies a tecroerature calculated

by saul’ ev'a method, a rsodel of a wet or dry advancing stone is

produced. The rsodel involves the use of discrete elements, wherer- »

the aaaller the element size ^and consequently the larger the

aunber of elements), the better the representation of the actual

physical situation. The capacity of the IBM 3o0 computer nade

possible the U30 of extremely snail elements, thus providing a

better simulation than was previously poseihle on the smaller IBM

1620 Model

The equations describing the physical situation in the

at tine t 3 n IT (from Chapter I) are

{a©2 + bd + c - w (n + v8 )} — (7.1)

8n n n

- ♦ ) + I I <9a . + ea . ) (V i " V i - l ' (T*2) “ i«*0 i i+ l

■ Sim

e * 9, - — ea 0 a

-

8s i *

_______ „_______ (7.5)(a - en ) + -------8 ̂ p cp Q

{a02 ♦ be ♦ c - v(u + ve ); —S S O

+ 2 DX(2d)

+ ae2 + be + c - v iv + V9.

where the solution nust. be built up in

which are all known quantities.

knovn constant since

Having solved this quadratic for 0* the values 01 9.* ,n n

v* and 0’ nav be obtained directly by substitution. Tho value n a '

nof the variables for tine step (n + l)DT is now found and so on

until the end of the cycle is rcached.

Application of the equations to a_ stope

The cross-sectional area of a stope nay be divided, uP

into any nunber of regular rectangular eoreas as in Fig. ,7 .1

/

/ / & ' +ve x direction

Dn °12 Si3 h k p i T l y

D21 L22 L23 D2k 2<6direction of advance

v stoping height

Fis .(’M ) Die atic : tion of a ?top

Each of tho^e areas D . . defines the reetingular ductij

which is forred when a finite length of face is considered.

The procedure for calculating vet and dry bulb tcuperature

m sm

increases in a stope 13 as follovs :

1. The cross-sectional area of the stope is divided into

| * r r j n M p t M m i M i suitable rectangular areaa.

2. Tho vet bulb toaporature 6 , the dry bulb tenperaturev

9 and the rock surface tecnvrctvre 0 are found at D * s

the storting point (i .e . at x » 0) over the couplete

cycle in finite tine intervals DT* by treating each of

i I ifef’ tthe ducta independently as in paragraph 7*1» lo this

E s i S w m Hway irregular vet portions of the stope nay be handled

since each duct fortis an independent entity,

The sane parameters are no’.r found at a length DX along

the stope. The value of P (perineter in the stated

equations) to be used for each duct depends on the

length of the perineter of the duct which has a rock

boundary.

i S2Vi,'

on t!

63

!"kF5 uu'vaJl0Q8» each of the ducts . inherits

l j thS y of the duct vhich It replaces B f l

aseani:* oach dact to be of »idth DL ft, and sapiw.ina the

stope to advance by 2 DL ft erery blast, then immediatelyt at i o

after blast D . . 7 , * r would irihr,v!+- +v<* «. .10 j » 1-..U toulc inherit the tonperature history

" £ ‘ o"'f ^1*0+2 j ® 1 , . . U* Gnd Di tj j * 5* \ would advance into

rea just blasted out.«

........I ! I ■At a point DX along tl.a stope face the parameters will

he define* for each duct. In order to obtain starting rrlues

for the next step, a certain amount of Fixing of thlfilir nay be

ainulated by

ture

the average vet bulb and dry bulb t.anpera-

tne ducts for each tine interval. Once aore is

known of tocrperature distributions frou the face to t?^ scetter-

wall, better approxinations ray be uade. The linited

I M m m m Sexperimental data available seens to indicate that in nany

Rc-pes nost of the hot air rssains closc to th-3 face, while the

(57]cooler air passes close to the scatterwall .

Using these average valuer as a starting point, the

variables nay now be calculated at 2 DX and so on as far as

required.

Zm

In order t.j nake use of Duhojzol's theorea, Starfield

assuned that the stope would advance into l ^ k at VRT regard­

less of the rate of advance. The error in the dinen*ionlea«

W H 3 P S S & m t . . .surface te^orature mt,rod';

W M

(as predicted by Saul’ ev*s

(7 .2 ).

.need a t the face by t h i3 assumption

nethod o f solution) is sfiown in Fig.

6U

It cay ‘>e see- that for typical rates of advance (l tc

2 ft per day) » a fairly large error in introduced and obviously

the greater the rate of advance the lees the error involved.

A heat transfer coefficient H = 2,7 BHJ/ft2 hr °F , which

represents a high air velocity in a atope 12 ft x h ft, was

used. By inspection it say be noen that the lower the air

velocity (and hence E ), the higher the t riperature of the rock

. (7.2) prooably represents a

reasonable uppr-? liait on «ne errors likely to he incurred jy

the asrunption of VRT ahead of any stope. The error shown in

Pig. (7 .2) is the error ir the diuenaionless surface tencsra-

■ H

surrounding the atope. Thus

S O I

ture and the error introduced i * the final wet and dry hulh

resvteooeratures 23 a result of this will not necessarily be the

saae.

Fig. (7 .3 ) gives a conporiaon between the on3 and two

din^nsiocal solutions for the dinensionless surface teripera-

ture as used in the stope prograa, fcr points alonj the length

m

of the stone fron the acatterwall to the face. Curvas are

sho’jn at 0 , U and 2U hours after blast reap actively fcr a

£,tope which advances ot a rate of 3 ffc eve-y **0 hours, v, r f .'~ i

a heat traasfor coefficient H “ 1.0 BTU/ft hr F (which an a

.stope U ft x 12 ft , representa aa air velocity of about 100 ft./

n in ,). Fret Fig. (7*3) it m y be seer. that over the grater

nart cf stope surface the two solutions agree- reasonably

well. At the face, the assumption of VRT imediately after

blast in the oneavar-dinensional caae tends to prod’ice an :

termerature. However, sinceestimate of the diaansionlosb tenpera.

the dinensionless tenraratures at the face Ia r > , -1C

relative percentage differ ones between the t ^ o l * :

only/

jtiens is

Byr.iVii3dW3i_____i__

00

BDVd

w

(N U)cr: x

UJ<N ozgo<

IX.o

<a:

u.

i s

D4-*il' u.

Ui 9 °

it M

X >

only about 10# in this case. Obviously as the rate of

advance decreases this margin will widen.

Frcea the above it nay be seen that in certain cases,

the assumption that the stope advances into rock at ,TRT,

regardless of the rate of advance, is quite inaccurate. In

an attempt to o' c:-ear.ie this difficulty, it was assumed that

inmedir.tely after VCLast, the surface cf the rock at the face,

had already undergone such cooling a? was required in crder to

bring the face to it1 s actual temperature. Cooling the face

from VRT to it ’ s actual temperature imediately after blast,

with dry air at a constant temperature, gave, "by means of the

heat balance equations, fictitious air temperatures 9^ for the

period of time - mDT to 0 , where m is the number of tine inter­

vals required to move from the face to rock at VKr. . This

process in fact means that the stope is initially considered

as extending as far as the position at which the rock attains

VRT, finding 0r for the elements of the "stope” ahead of the

actual stop** face and using these values 9^ as 'che initial

part of the temperature history required by Duhcnel’ s theorem.

The air ventilating t ’ stope at x a KDX in the period - mDT

to zero, was assumed to behave exactly as the ordinary \entel­

ating air as the solution progressed up the stope face to x =

(K + 1) DX.

Irt prictice it was found that using an imaginary ventil­

ating air temperature close in value to the initial dry bulb

temperature of the inlet air waB most satisfactory, since

using a very different value resulted in a severe step at

time zero, sometimes giving rise +o negative values of 9ft.

Author Gould Michael John

Name of thesis The determination of parameters concerned with heat flow into underground excavations. 1968

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