8
A study, by continuous monitoring of particle size in the cyclone overflow, of factors influencing run-of-mine mill performance SYNOPSIS by A. H. MOKKEN*, B.Sc. (Eng.) (Rand) (Fellow), G. K. I. BLENDULFt M.Sc. (Stockholm) (Member), K. A. G. BLENDULFt, Dipl. (Min & Met) (Visitor), and N. R. COWINt, B.Sc. (Min. Tech.) (Visitor) An account is given of an investigation of the use of a continuous particle-size monitor - the PSM-IOO. The monitor system, in addition to showing marked variations in grinding fineness over short periods of time, particu- larly after stopping and re-starting the feed to the mill, has indicated weaknesses in accepted mill operating practice. The main potential of the system is the opportunity it has provided for almost complete automation of the milling circuit. SAMEVATTING Daar word verslag gedoen oor 'n ondersoek na die gebruik van 'n deurlopende partikelgroottemonitor - die PSM-IOO. Benewens die feit dat die monitorstelsel duidelike variasies in die maalfynheid oor kort periodes, veral na die staking en hervatting van die toevoer na die meul, getoon het, het dit ook swak punte in die aanvaarde meulbedryfpraktyk aangedui. Die hoofvoordeel van die stelsel is die geleentheid wat dit vir die byna algehele outomatisering van die maalkring gebied het. INTRODUCTION The performance of a run-of-mine milling circuit is sensitive to a number of variables such as rate of feed, rock size and hardness, mill load, mill dilution, and classifi- cation efficiency. Present automatic control is limited to new feed rate. Williamson1 and Digicon controllers are used to effect this control, and are based on the power drawn by the mill, which, in turn, is essentially a function of mill load. Simple control units also regulate the level of the pump sump and dilution to the cyclones, but final control rests with the operator who depends on his 'feel' and skill to get the most efficient operation from the mill, i.e., the highest possible tonnage at the finest possible grind. When a continuous particle-size monitor (PSM) was installed at Kinross, it was realized that this instrument could become a useful indicator to the operating staff, and perhaps even take over some of the control functions. It would also provide an opportunity for doing more basic research into run-of-mine milling. *Union Corporation, Limited. tUnion Corporation Group Research Laboratories. :J:Kinross Mines Limited. THE KINROSS MILLING CIRCUIT The dilution of water to the mil discharge sump and the steady- head tank are regulated by ball valves to give a constant pulp level. In the mill discharge sump, the object is mainly to prevent pump surges, and, in the steady-head tank, constant level assures a reason- Milling is done in a single stage in three 4,27m by 6,lOm mills, each operating in closed circuit, with two 1,O7m hydro cyclones in parallel. The complete circuit is shown in Fig. 1. Plate I-The final installation JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY MA Y197S249

A study, bycontinuous monitoring ofparticle ... · special launder for the continuous sampling of cyclone overflow. The launder is shown in Fig. 2. The perforated plate curtains near

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Page 1: A study, bycontinuous monitoring ofparticle ... · special launder for the continuous sampling of cyclone overflow. The launder is shown in Fig. 2. The perforated plate curtains near

A study, by continuous monitoring of particlesize in the cyclone overflow, of factorsinfluencing run-of-mine mill performance

SYNOPSIS

by A. H. MOKKEN*, B.Sc. (Eng.) (Rand) (Fellow),G. K. I. BLENDULFt M.Sc. (Stockholm) (Member),K. A. G. BLENDULFt, Dipl. (Min & Met) (Visitor),and N. R. COWINt, B.Sc. (Min. Tech.) (Visitor)

An account is given of an investigation of the use of a continuous particle-size monitor - the PSM-IOO. Themonitor system, in addition to showing marked variations in grinding fineness over short periods of time, particu-larly after stopping and re-starting the feed to the mill, has indicated weaknesses in accepted mill operating practice.The main potential of the system is the opportunity it has provided for almost complete automation of the millingcircuit.

SAMEVATTING

Daar word verslag gedoen oor 'n ondersoek na die gebruik van 'n deurlopende partikelgroottemonitor - diePSM-IOO. Benewens die feit dat die monitorstelsel duidelike variasies in die maalfynheid oor kort periodes, veralna die staking en hervatting van die toevoer na die meul, getoon het, het dit ook swak punte in die aanvaardemeulbedryfpraktyk aangedui. Die hoofvoordeel van die stelsel is die geleentheid wat dit vir die byna algeheleoutomatisering van die maalkring gebied het.

INTRODUCTION

The performance of a run-of-minemilling circuit is sensitive to anumber of variables such as rateof feed, rock size and hardness,mill load, mill dilution, and classifi-cation efficiency.

Present automatic control islimited to new feed rate. Williamson1and Digicon controllers are used toeffect this control, and are based onthe power drawn by the mill, which,in turn, is essentially a functionof mill load. Simple control unitsalso regulate the level of the pumpsump and dilution to the cyclones,but final control rests with theoperator who depends on his 'feel'and skill to get the most efficientoperation from the mill, i.e., thehighest possible tonnage at thefinest possible grind.

When a continuous particle-sizemonitor (PSM) was installed atKinross, it was realized that thisinstrument could become a usefulindicator to the operating staff,and perhaps even take over someof the control functions. It wouldalso provide an opportunity fordoing more basic research intorun-of-mine milling.

*Union Corporation, Limited.tUnion Corporation Group Research

Laboratories.:J:Kinross Mines Limited.

THE KINROSS MILLINGCIRCUIT

The dilution of water to the mildischarge sump and the steady-head tank are regulated by ballvalves to give a constant pulp level.In the mill discharge sump, theobject is mainly to prevent pump

surges, and, in the steady-head

tank, constant level assures a reason-

Milling is done in a single stagein three 4,27m by 6,lOm mills, eachoperating in closed circuit, withtwo 1,O7m hydro cyclones in parallel.The complete circuit is shown inFig. 1.

Plate I-The final installation

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY MA Y197S249

Page 2: A study, bycontinuous monitoring ofparticle ... · special launder for the continuous sampling of cyclone overflow. The launder is shown in Fig. 2. The perforated plate curtains near

P.S.M.f

'"To thickeners.

Autosampler

Cyclones

Run-of-mine ore---

ably constant inlet pressure to thecyclones.

Cyclone underflow and overfloware automatically sampled every24 hours and are screened on a74 p.m screen, but the results of thescreen analysis are available onlyon the following day.

The mill tonnage is regulated by aDigicon mill-feed controller, whichkeeps the motor power near maxi-mum by stopping and starting thefeed belt. The motor power, feed.belt running time, and feed rateare continuously recorded on charts.

To cope with increased through-put, 100 mm steel balls are addeddaily to the mills, the additionbeing about 1 kg of steel per tonneof ore feed. The steel load representsabout 5 to 6 per cent of the totalmill volume.

THE PSM SYSTEM 100

The PSM-lOO consists of threesections: the air eliminator, thesensor section, and the electronicssection. '

The air eliminator draws a samplefrom the process flow stream andremoves entrained air bubbles. Thede-aerated pulp then passes betweenthe sensors, where measurements ofparticle size and percentage solidsare taken by ultrasonic means, andthe pulp returns to the main stream

250 MAY 1975

Dilution (auto-level)

Steady-headtank

Dilution

I

(semi-auto)

AutolSampler

Mill~

Fig. I-Milling circuit at Kinross Mines Limited

below the sampling point. Theelectrical signals from the Sensorsare processed in the electronicssection, where they are converted tooutputs representing particle sizeand percentage solids. These outputsare displayed on direct-readingmeters and are also recorded.

Particle size is shown on themeter and on the recorder in unitsof cumulative percentage passinga 74 p.m standard screen. Solids areshown as percentage solids by massand as water-solids ratio. Additionaldetail& on the design and operationof the PSM System 100 are availablefrom previously published papers2,S.

Installation

The Kinross mill was chosen forthe investigation because of its highproduction rate and also becauseit consists of three identical unitsoperated under similar conditions.This arrangement provided a readymeans for direct comparison be-tween the performance of the millfitted with the PSM recorder andthat of the two remaining mills.

To fit the instrument, the onlyalteration required to the existinginstallation was the provision of aspecial launder for the continuoussampling of cyclone overflow. Thelaunder is shown in Fig. 2. Theperforated plate curtains near thepulp surface were installed to break

I

Dilution

(auto-lev 1)

~

up large air bubbles, which couldinfluence the accuracy of the PSM.The adjustable gate in the lastbaffle was installed to provide ameans of obtaining a representativesample by controlling the silting.It was found, however, that themost representative sample was ob-tained with the gate fully closed.

To achieve the water pressurerequired for the vacuum aspirator(690 kPa), a small booster pump wasinstalled in the mill gland-servicesystem. However, it was foundlater, after the booster pump hadfailed, that the ordinary gland-service pressure of 500 kPa wasquite adequate to create the neces-sary vacuum, and the pump wastherefore removed.

In addition to the meter on theinstrument, a pen recorder indicat-ing particle size and pulp densitywas mounted on the regular millinstrument console for the benefitof the mill operators.

The complete installation is shownin Plate I.

Calibration

The PSM system was pre-calibrated in the factory by themanufacturer, using typical samplesof cyclone overflow from theKinross mill.

The site calibration was carriedout by the supplier, together with

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

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JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY MAY 1975 251

Page 4: A study, bycontinuous monitoring ofparticle ... · special launder for the continuous sampling of cyclone overflow. The launder is shown in Fig. 2. The perforated plate curtains near

Kinross personnel. This was doneby taking samples of pulp from thePSM discharge pipe and, at thesame time, noting the reading onthe instrument. The density of thesampled pulp was measured, aswas the percentage of minus 74 /Lmmaterial determined by screenanalysis, which was carried outwith extreme care.

The final calibration curve wasbased on 36 samples and is shownin Fig. 3.

Of the 36 samples taken, 23 werewithin 0,5 per cent of minus 74 /Lm,and all 36 were within 1,0 per centof minus 74 /Lm of the calibrationcurve. Once it was established thatthe PSM meter was showing thecorrect grading of the sample tested,the PSM was calibrated in relationto the grading of the whole pulpstream. This was done by takingsimultaneous samples from the PSMdischarge and the automatic sam-pler, which is installed just aheadof the sampling launder. This samp-ler is used for regular mill samplesand cuts the entire flow stream.

It was found that the best samp-

ling agreement was achieved whenthe PSM intake screen was placedin the centre of the launder andthe adjustable gate was fully closed.Mechanical Performance of the PSM

Owing probably to the very dilutepulp tested, which contains only10 to 15 per cent solids by mass,the wear rate on parts in contactwith the pulp is very low. Theimpeller has been inspected afterrunning times of 1000, 2000, and3000 hours and has shown very littlewear.

A more serious problem is thebuild-up of lime scale, especially onthe ultrasonic sensors. After onlythree days of operation, the PSMregisters about 2 per cent finergrading than actual. It has beenfound that acid treatment of thesensors with 25 per cent hydro-chloric acid removes the scale. Thisoperation takes less than 1 minuteand is carried out daily.

The system is calibrated withclean water once a week by the millpersonnel, which takes about 5minutes.

In addition, the sample intake

screen tends to blind with limescale and wood chips, and thereforehas to be cleaned and acid-treatedregularly.

Initial ObservationsAfter installation of the PSM

system, the first observations showedmarked variations in grinding fine-ness over short periods of time undercertain operating conditions. In par-ticular, the stopping and startingof the feed to the mill was foundto have a profound effect on thefineness of grind, as illustrated inFig. 4.

The interruption of feed to themill for a period of 5 minutesresulted in a drastic change in thefineness of the cyclone overflowproduct. In the case illustrated, itincreased rapidly from 61 per centminus 74 /Lm material to almost70 per cent minus 74 /Lm material.

After the feed to the mill hadbeen resumed, a period of over 1hour elapsed before the finenesshad returned to its original value.The pulp density followed the samepattern, but the feed rate before

252 MAY 1975

Fig. 4-lnfluence of interruptions in the feed

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

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JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

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MAY 1975 253

Page 6: A study, bycontinuous monitoring ofparticle ... · special launder for the continuous sampling of cyclone overflow. The launder is shown in Fig. 2. The perforated plate curtains near

and after the stoppage remainedunchanged.

The initial increase in fineness isexplained by a drop in drivingforce 'at the inlet of the mill follow-ing interruption of new feed supply.This results in a sudden drop inthe volume of pulp discharged fromthe mill and entering the cyclones.The drop in volume is compensatedby additional dilution to the steady-head' tank and mill sump underautomatic control, which results ina finer split in the cyclone.

In the meantime, the grindingprocess continues on the materialleft in the mill and on the circulatingload, resulting in a build-up ofmaterial in the circuit, which isfiner than during normal continuousfeeding. On resumption of new feedsupply, this finer material leaves thecircuit via the cyclone overflow.

It should be pointed out herethat the two mill-feed controllersmentioned previously regulate themill load by stopping and startingthe feed belt. Further test work willbe done to determine more fullythe effect of stopping and startingof the feed belt on the overall

efficiency of the grinding operation.

It happens occasionally that, be-cause the density of the mill pulpis too high, part of the mill loadrevolves with the mill. This pheno-menon is generally referred to as'centrifuging'. To study what hap-pens to the cyclone overflow gradingduring such a period, the mill wasrun with reduced water addition.The results are shown on the re-corder charts in Fig. 5. The circularchart shows the power drawn bythe mill motor.

During periods of centrifuging,a rapid decrease in the power isnoticeable. When the centrifugingmass collapses, the power immedi-ately reverts to normal, followed by anew period of centrifuging. The stripchart shows the condition in thecyclones during the same period.It is noted that violent oscillationsoccur, both in the density andparticle size in the cyclone overflow.Centrifuging conditions show upmuch more pronounced on thePSM than on the power chart, thusaiding the operator in taking cor-rective measures more promptly.

PSM for Density Control of CycloneOverflow

As there appeared to be a directrelationship between cyclone over-flow density and the percentageminus 74 (Lm material (Fig. 4), anattempt was made to control thefineness of grind directly by means ofthe water addition to the cyclone.When this failed owing to anincurable oscillation of the controlloop, the controller input wasswitched to the pulp density signalrather than the particle size.

The controller used was a PhilipsPCS, the output of which wasconnected to a Foxboro current-to-air transducer. This transducer oper-ated the two diaphragm valves thatcontrolled the dilution water tothe steady-head tank.

Mter a suitable setting of thecontroller had been found, theresults of Fig. 6 were achieved.

It will be noted that, in spite ofthe very steady pulp density, thereis still a long-term fluctuation inthe fineness of grind. It has_notyet been possible to establish anyrelationship between this variationand any specific operational feature.

254 MAY 1975

Fig. 6-Automatically controlled cyclone dilution

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY

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CONCLUSION

The desirability of continuousmonitoring of the fineness of milledpulp has taxed the innovative in-genuity of mill men for many years.

In the gold-mining industry, inwhich the continuous monitoringof the gold content of plant residuescan be accepted as a practicallyimpossible task, a measure of thefineness of grind of milled pulpbecomes an indirect but importantalternative in view of the closerelationships that have been foundto hold, over many years of opera-tion, between fineness of grind andundissolved gold content of residues.By the use of this relationship, fora particular mine, inefficiences inmill operation can be translated intoequivalent gold loss.

The PSM system installed atKinross, in addition to its functionof monitoring the fineness of grind,has already shown up weaknessesin accepted operating practice. Itsmain potential, however, lies in theopportunities it has provided forthe almost complete automation ofthe milling circuit. Developmentstowards this end are being pursued.

REFERENCES

1. WtLUAMSON, J. E. The automaticcontrol of grinding medium in pebblemills. J. S. Air. Inst. Min. Metall.,vol. 60. 1960. pp 333.345.

2. HATHAWAY,R. E. A proven on-streamparticle size monitoring system forautomatic grinding circuit control.Mining and Metallurgy Symposium ofthe Instrument Society of America,Phoenix, Arizona, April 24, 1972.

3. BASSAREAR, J. H., and MCQUIE,G. R. On-stream analysis of particlesizes. Mining Show of the AmericanMining Congress, Las Vegas, Nevada,October 11, 1971.

Contribution to the above paperby G. J. Rees*

Mr Mokken and his team are tobe congratulated on the publicationof their results and the descriptionof equipment that gives a continuousmeasure of the proportion of finesless than 74 fLm in milled pulp.Continuous measurement of fineparticle sizes is certainly anextremely difficult problem, whichbas taxed minds over many decades.When it is realized that the efficiencyof a milling process is of the order

JIoBoart Research Laboratory.

of 1 per cent, i.e. only 1 per centof the energy supplied to the millis responsible for the production ofnew surface, and that approxi-mately 6 per cent of the cost ofgold-mining operations is consumedby the milling process, the import-ance of control of the grindingprocess becomes apparent, and theneed for investigations to improvethe performance of mills. But, inorder to do this, a simple fast on-linemeans of controlling particle-sizeanalysis is required, and the PSMSystem 100 seems to fulfil this needfor the gold-extraction process asindicated by the results in the paper.I am sure that, with the PSM-lOO,the point at which further grindingbecomes uneconomic or inefficientwill be found, and substantial costsavings will result. Furthermore, atypical throughput for a large single-tube mill is of the order of 90 tonnesper hour, and the PSM-lOO is usedfor the control of fineness of rockquantities of this magnitude.

Very few methods of determiningparticle size are adaptable for con-tinuous on-line measurement. Ex-amination of a table of availablemethods for particle-size analysestin the sub-sieve range indicates thatonly ultrasonic and centrifugal tech.niques and the electrical resistancechange method (as employed in theCoulter Counter) are suitable foron-line operation.

Since that time, the optical on-line method developed by Talbotand his colleagues at the Chamber ofMines Physical Sciences Labora-tory2 has received attention, andfurther speakers will probably com-pare the advantages and disadvan-tages of this technique with thoseof the PSM-lOO system. The prin-ciples employed by the CoulterCounter3, in which a stream ofparticles is drawn between twoelectrodes, producing a change inelectrical resistance proportional toparticle volume, would appear tobe amenable to particle-size analysisof mill pulp, but our experiencewith this instrument is that, al-though the values obtained maybe easily correlated with othermethods of particle-size analysis, thephysical significance of the para-meter measured is difficult tointerpret.

tlOUR~AI. OF THE SOUTH AFRICAN JNSTITUn OF MINING AND M!;TALI.URGY

The PSM-lOO technique effectivelygives a continuous single-valuemethod of particle-size analysis,and this is certainly a good start forthe control of grinding mill efficiency.I am sure, however, that metal-lurgists employed in the gold-miningindustry have several well-defined(and possibly divergent) views onthe importance of particle-size distri-bution (as opposed to a single-valuesize parameter) to the gold-extraction process, and I would liketo ask the authors of the paperwhether the technique they usedcan be adapted for complete part-icle-size analysis at well-defined timeintervals, say every 30 minutes,and whether these investigationsform part of their further plans.Certainly, particle-size distributionis of importance in all surface-chemistry processes, and I am surethat further improvements in theefficiency and economics of thegold-recovery process would resultif a rapid on-line method for thecontinuous monitoring of particle-size distribution was available.

The results given in the paperindicate a high precision for thePSM.lOO system. Have the authorsany further comments on the abcur-acy of the results1 For example, arethe results checked at regular inter-vals against those yielded by screen-ing tests1 Our experience with fineparticle-size analysis indicates thatregular tests with standard samplesare necessary to prevent drifts inresults occurring over periods oftime from our equipment, whichuses physical methods of measure.ment.

My view is that the UnionCorporation team have made aninteresting contribution to themethods for continuous particle-sizeanalysis. This problem certainly is ofgreat interest to me personally, be-cause, in the industry in which Iam engaged, we have the problemof trying to continuously monitorthe particle-size distribution ofmultiphase slurries in which thelargest particles are always lessthan 10fLm in size, and, with certaincompositions, about 60 per cent bymass of the slurry may be less thanIfLm in size. Both the size and sizedistribution affect the performanceof our sintered products. Over the

MAY 1975 255

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years, we have examined severaltechniques for continuous on-lineparticle-size measurement, and todate have found no reliable methodthat is suitable. Mr Mokken andhis team are therefore to be thankedfor their contribution, for it is onlyby continuously searching for im-

proved methods of particle-sizeanalysis that the efficiencies andquality control in our respectiveindustries can be improved.

REFERENCES

1. MITCHE, R., FISCHMEISTER, H. F., andEXNER, H. E. Berg-und Hutten-

mannsche Monatshefte 777 (3) 138(1966).

2. CHAMBER OF MINES OF SOUTH AFRICA:Project 11/511/66 Research ReportNumber 1/69.

3. EXNER, H. E., FISCHMEISTER, H. F.,and LINDELOF, G. Modern develop-ments in powder metallurgy. vo!. 1.(ed. H. H. Hausner) Plenum Press,New York (1966) p. 87.

Company AffiliatesThe following members have beenadmitted to the Institute as Com-pany Affiliates.

AE & Cl Limited.Afrox/Dowson and Dobson Limited.Amalgamated Collieries ofS.A. Limit-

ed.Apex Mines Limited.Associated Manganese Mines of S.A.

Limited.Blackwood Hodge (S.A.) Limited.Blyvooruitzicht G.M. Co. Ltd.Boart & Hard Metal Products S.A.

Limited.Bracken Mines Limited.Buffelsfontein G.M. Co. Limited.Cape Asbestos South Africa (Pty) Ltd.Compair SA (Pty) Limited.Consolidated Murchison (Tvl) Gold-

fields & Development Co. Limited.Doornfontein G.M. Co. Limited.Durban Roodepoort Deep Limited.East Driefontein G.M. Co. Limited.East Rand Prop. Mines Limited.Free State Saaiplaas G.M. Co. Limited.Fraser & Chalmers S.A. (Pty) Limited.Gardner-Denver Co. Africa (Pty) Ltd.Goldfields of S.A. Limited.The Grootvlei (Pty) Mines Limited.

Harmony Gold Mining Co. Limited.Hartebeesfontein G.M. Co. Limited.Hewitt-Robins-Denver (Pty) Limited.Highveld Steel and Vanadium Corpo-

ration Limited.Hudemmco (Pty) Limited.Impala Platinum Limited.Ingersoll Rand Co. S.A. (Pty) Ltd.James Sydney & Company (Pty)

Limited.Kinross Mines Limited.Kloof Gold Mining Co. Limited.Lennings Holdings Limited.Leslie G.M. Limited.Libanon G.M. Co. Limited.Lonrho S.A. Limited.Loraine Gold Mines Limited.Marievale Consolidated Mines Limit-

ed.Matte Smelters (pey) Limited.Northern Lime Co. Limited.O'okiep Copper Company Limited.Palabora Mining Co. Limited.Placer Development S.A. (Pty) Ltd.President Stern G.M. Co. Limited.Pretoria Portland Cement Co. Limit-

ed.Prieska Copper Mines (Pty) Limited.Rand Mines Limited.

Rooiberg Minerals Development CoLimited.

Rustenburg Platinum Mines Limited(Union Section).

Rustenburg Platinum Mines Limited(Rustenburg Section).

St. Helena Gold Mines Limited.Shaft Sinkers (Pty) Limited.S.A. Land Exploration Co. Limited.Stilfontein G.M. Co. Limited.The Griqualand Exploration and Fi-

nance Co. Limited.The Messina (Transvaal) Develop-

ment Co. Limited.The Steel Engineering Co. Ltd.Trans-Natal Coal Corporation Limit-

ed.Tvl Cons. Land & Exploration Co.Tsumeb Corporation Limited.Union Corporation Limited.Vaal Reefs Exploration & Mining Co.

Limited.Venterspost G.M. Co. Limited.Vergenoeg MiningCo. (Pty) Limited.Vlakfontein G.M. Co. Limited.Welkom Gold Mining Co. Limited.West Driefontein G.M. Co. Limited.Western Deep Levels Limited.Western Holdings Limited.Winkelhaak Mines Limited.

Institute for Publicity and Exchange of Computer ProgramsThe document on which we based

our earlier notice (page 180 of theFebruary Journal) has been amendedby the above Institute.

The third sentence of our notice

256 MAY 1975

read as follows:Scientists and engineers who havedeveloped, tested, and fullydocumented computer programs,especially those related to mathe-

matical modelling, are invitedto submit their problems to theInstitute in the required format.The word problems should read

programs.

JOURNAL OF THE SOUTH AFRICAN INSTITUTE OF MINING AND METALLURGY