DEVELOPMENT OF A MATHEMATICAL MODEL FOR SIMULATION OF COAL CRUSHING IN A HAMMER MILL

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1 Proceedngs of the XI Internatonal Semnar on Mneral Processng Technology (MPT-200) Edtors: R. Sngh, A. Das, P.K. Baneree, K.K. Bhattacharyya and N.G. Goswam NML Jamshedur, DEVELOPMENT OF A MATHEMATICAL MODEL FOR SIMULATION OF COAL CRUSHING IN A HAMMER MILL Avnash Trath, Ashsh Agrawal and V.K. Guta Deartment of Fuel and Mneral Engneerng Indan School of Mnes, Dhanbad , Inda ABSTRACT Usng the basc sze-mass balance sze reducton model for a contnuous crushng oeraton and nvokng the concet of a erfectly mxed system, a mathematcal relatonsh has been obtaned between the feed and roduct sze dstrbutons for a hammer mll. The model arameters breakage dstrbuton functon for dfferent sze fractons of a Prme Cokng Coal and an morted coal were determned by conductng arorate tests n a laboratory hammer mll. Usng data generated n the Rourkela Steel Plant on a Producton hammer mll, varaton of the second set of model arameters, absolute rate of breakage of artcles of each sze class, wth coal feed rate to the mll was establshed. Analyss of the data generated n the lant has shown that only +0 mm artcles of the morted coal broke faster than the PCC artcles of the same sze. The develoed mathematcal model can be used to stmulate the erformance of the crushng lant n resect of the effect of feed sze dstrbuton and feed rate, and for takng control acton for keeng the roduct fneness constant by adustng the coal feed rate to the hammer mll. Keywords: Hammer mll, Coal crushng, Mathematcal model, Sze-mass balance, Product sze dstrbuton control. INTRODUCTION At steel lants, for roducton of coke n the coke ovens the coal receved from coal mnes s crushed to a desred degree of fneness. Sze dstrbuton of the coal charged to the coke ovens has a sgnfcant effect on the qualty of the coke roduced as both the oversze and undersze artcles are detrmental to coke qualty. Thus, control of the coal crushng oeraton s a task of consderable mortance. In Inda, generally hammer mlls are used for crushng of coal. And, the sze dstrbuton of the crushed coal s secfed as: % assng 3 mm (80 82%) and % assng 0.5 mm (30 35%). Determnaton of the otmum values of the oeratng arameters through smulaton of the rocess erformance wth the hel of a mathematcal model s now a well establshed ractce. In vew of these facts, a mathematcal model s resented n ths aer for smulatng crushng of coal n hammer mlls. MODEL DEVELOPMENT [, 2] Breakage knetcs n most mlls can be descrbed by a frst order reacton rate exresson:, dh / dt = r h + b r h () 343

2 AVINASH TRIPATHI, ASHISH AGRAWAL and V.K. GUPTA where, r : breakage rate constant for artcles of sze class. t : tme h : weght fracton of mll holdu of solds (coal) n the seve sze nterval at any gven tme t. b, : breakage dstrbuton arameter, weght fracton of artcles of sze class that reorts to sze class after undergong breakage. For convenence, let us defne a modfed breakage dstrbuton functon, b,, known as ractcal breakage dstrbuton functon n the lterature, as: [3] b b ( b, =,, ) (2) where, b, : modfed breakage dstrbuton arameter-weght fracton of artcles leavng sze class that reorts to sze class after undergong breakage. b, : breakage dstrbuton arameter- weght fracton of artcles of sze class that reorts to sze class after undergong breakage. The corresondng breakage rate constant s defned as:, r = r ( b ) (3) where r s modfed breakage rate constant for artcles of sze class, hr. Thus,, rb, rb = (4) For a steady state oeraton we can wrte: Rate of entry Rate of breakage Rate of Producton Rate of ext = 0, Ff Fh Hr h + Hr b, h = 0 = where, F : feed rate of coal to hammer mll, t/hr. H : total weght of coal artcles n the mll. f : weght fractons of sze artcles n the feed to the hammer mll. (5) Notng that for a fully mxed mll, as s the case wth a hammer mll, we have h = (6) Ff F Hr + Hr b, = 0 = (7) where s weght fracton of artcles of sze class n the mll dscharge (roduct). Rearrangng we obtan, 344

3 f + (f )b, = = + r τ (8) where, τ = H/F, s known as the mean resdence tme of coal artcles n the mll. For smulaton uroses, the task can now be defned as: quantfcaton of the effect of feed rate on the mathematcal model arameters b and r'. As shown by a number of researchers, [4] generally b arameters are not affected arecably by the feed rate (or more correctly, the hold u of the solds n the mll, whch ncreases wth feed rate). Further, t should be onted out that t was not ossble to measure the mll hold-u weght of coal, only the roduct r H or r τ could be estmated (.e. each r value multled by a constant H or τ), not the r value alone. However, the term r H has the secal sgnfcance as t reresents the absolute rate of breakage of sze artcles n the mll n tons/hour. [5, 6] For these two ont arameters we have the followng relatonshs: or, (f ) + (f )b, = r τ= (9) F r H = (f ) (f )b +, (0) = EXPERIMENTAL Plant tests Tests were conducted on one of the two secondary hammer mlls of the mllon ton coal handlng faclty of the Rourkela Steel Plant. [7] Samles of the mll feed and dscharge were collected only durng those erods when lant was observed to be runnng under near steady state condtons (as reflected from the load and the sze dstrbuton of coal on the conveyor carryng the mll dscharge). For better accuracy n the results, samles collecton was done at two locatons on the conveyor. Deendng on the feed rate, the weght of each test samle vared from 20 to 30 kg. The weghts of the feed and dscharge samles were found to be nearly same as would be exected under steady state condtons. In some cases two samles of feed were taken one before takng the dscharge samle and the other afterwards. The dfference n the sze dstrbutons of two samles was found to be ractcally neglgble. In ths way, t was ensured that near steady state condtons revaled durng the tests. For the Prme Cokng Coal (PCC), four sets of the feed and roduct (dscharge) sze dstrbuton data could be generated corresondng to feed rates: 73, 93, 60 and 269 t/hr. Data on Imorted coal corresonded to 95 and 224 t/hr. 345

4 Laboratory exerments AVINASH TRIPATHI, ASHISH AGRAWAL and V.K. GUPTA To determne the breakage dstrbuton functon for two tyes of coal, exerments were carred out n a laboratory hammer mll (2 nch dameter 3n. wdth) As the mll had bg rectangular slots all along ts erhery, t was argued that when fed slowly, feed artcles wll undergo breakage only once durng ther assage through the mll. Thus, f only one selected sze fracton of coal s fed to the hammer mll, the mll dscharge sze dstrbuton should corresond to the breakage dstrbuton functon, b, of ths sze fracton. For ths urose, test samles of crushed PCC and morted coal were screened to obtan +0, 6/0, 3/6 and /3 mm sze fractons. About 400 g of each sze fracton was slowly fed to the mll searately. The mll dscharge thus obtaned was screened usng the same seres of screens. b values could be calculated from ths data as dscussed above. RESULTS Breakage dstrbuton functons, b,, for PCC and Imorted coals as obtaned from the laboratory hammer mll test results and eq. (2) are gven n Tables and 2. Varaton of r H wth feed rate s shown n Fg. for all the four sze classes under consderaton: () + 0 mm, (2) 6 0 mm, (3) 3 6 mm, and (4) 3 mm. It wll be seen that deendng on the artcle sze, the absolute rate of breakage ntally ether decreases (for +0 mm artcles) or ncreases ( for the remanng three fner sze fractons) wth ncrease n the feed rate, and beyond about 75 t/hr feed rate there s ractcally no change,.e. r H = K, F > 75 t/hr () where K s a mll-materal constant for sze fracton. Table : Breakage dstrbuton matrx, b,, for PCC Coal Sze b, nterval Feed sze (mm) Table 2: Breakage dstrbuton matrx, b,, for morted coal Sze b, nterval (mm) Feed sze

5 As can be seen n eq. (8), t s the arameter r τ that drectly relates to f. Esecally, n the case of to sze (+0 mm) and the second sze (for whch b 2, = 0) eq. (8) smlfes to, = f ( + r τ ) (2) 2 = f 2 (+ r 2τ ) (3) Ths means that a fracton /( + r τ) of feed of sze class remans unbroken. And, smlarly a fracton /( + r2τ) of sze class 2 remans unbroken. Fgure 2 shows varaton of r τ wth feed rate for the four sze fractons under consderaton. It wll be seen that: () r τ wth feed rate qute sharly (sharer than the rate of declne observed n case of rh), () r 2 τ decreases wth ncrease n feed rate, though at a much slower rate than that observed n case of r τ (whle r 2 H was observed to ncrease wth feed rate), () r 3 τ vares lttle feed rate, and (v) r 4 τ ncreases wth feed rate (but at a slower rate than that observed n case of r 4 H). These results show that +0 mm artcles are broken much more effectvely at lower feed rates. Partcles n the sze class 6 0 mm also break better at lower feed rate, though the feed rate effect s comaratvely less ronounced. Breakage of artcles of sze class 3 6 mm s not affected by feed rate. And, artcles n the sze range 3 mm break better at hgher feed rates (.e. when hammer mll s loaded wth more materal, because mll hold u of solds s exected to ncrease wth feed rate). An nsecton of curves shown n Fgures and 2 shows that t s not ossble to descrbe these curves n terms of smle mathematcal functons. The best that can be done s to descrbe r H curves ecewse, n two arts. The frst art corresonds to low values of feed rate. Ths art s not of ractcal nterest. The second art corresonds to nearly constant absolute rate of breakage,.e. r H = K As r τ = r H/F, corresondng exressons for r τ follows mmedately from eq. (). It can be seen n Fgs. and 2 that +0 mm artcles of morted coal break much faster rate than the same sze PCC artcles. However, the remanng three fner sze fractons of morted coal break at ractcally the same rate as the PCC artcles of these szes. Ths s an mortant ont to be noted. Fg. : Varaton n the absolute rate of breakage r H,' wth feed rate for dfferent sze of coal. 347

6 AVINASH TRIPATHI, ASHISH AGRAWAL and V.K. GUPTA Fg. 2: Varaton n the arameter r τ wth feed rate for dfferent sze fractons of coal. DISCUSSION Wth a vew to determnng the otmum feed rate to get the desred sze dstrbuton of 80% assng 3 mm screen, some smulaton exerments were carred out usng the develoed mathematcal model. Two feed sze dstrbutons were selected: () Feed- corresondng to the average feed observed for PCC and () Feed-2 corresondng to the average feed observed for Imorted coal. Usng arorate equatons, mll dscharge sze dstrbutons were calculated. It was observed that for a tycal PCC feed (Feed- sze dstrbuton) the wt. % assng 3 mm screen decreases from 69.2 to 62.2 as the feed tonnage was ncreased from 75 to 225 t/hr. Even at 75 t/hr feed rate, however, the dscharge was not fne enough as the wt.% assng 3 mm screen was only It was observed that f somehow feed to the secondary hammer mll s made as fne as feed- 2, the wt.% assng 3 mm screen can be as hgh as 79.4 at 75 t/hr feed rate. Moreover, even f feed rate s ncreased to 225 t/hr, ths ercentage reduces only margnally to 76%. In ths context t s nterestng to ont out that the mll dscharge at the feed rate of t/hr for feed- has ractcally the same sze dstrbuton as feed-2. Ths observaton suggests that n order to get the roduct of desred sze dstrbuton ether a tertary hammer mll should be used or both the rmary and secondary mlls should be made more effectve through sutable modfcatons n ther desgn. It was observed that n case of Imorted coal the desred sze dstrbuton s acheved to a reasonable degree of closeness even when the feed rate s as hgh as 275 t/hr. CONCLUSION. By carryng out a dynamc sze-mass balance around the hammer mll and by nvokng the concet of erfectly mxed system, a mathematcal relatonsh has been obtaned between the mll feed and roduct sze dstrbutons. 2. Usng the actual lant data, varaton of the model arameters r H (absolute rate of breakage of artcles of sze class n the hammer mll) wth feed rate has been establshed. Thus, t has now become ossble to smulate the erformance of the crushng lant n resect of the effect of feed sze dstrbuton and feed rate. 3. At about 200 t/hr feed rate the value of these arameters for 6 0, 3 6 and 3 mm sze fractons of Imorted coal were found to be ractcally the same as those obtaned for PCC. 348

7 However, for the +0 mm sze fracton of Imorted coal the absolute rate of breakage was found to be about 2.2 tmes hgher than, that observed n the case of PCC. These results ndcate that only coarse +0 mm artcles of Imorted coal break faster than the same sze PCC artcles. 4. The tests carred out n a small laboratory hammer mll for establshng breakage dstrbuton functons have shown that all coarse sze fractons of Imorted coal roduce about 78% mm materal as they undergo breakage n the hammer mll. As exected, ths ercentage value s lower for PCC (72%). 5. The degree of reducton (elmnaton due to breakage) for artcles smaller than 0 mm was found to be very low (9 30 % only). To break these artcles more effectvely, the desgn of the hammer mll needs to be mroved. REFERENCES [] Austn, L.G., Klmel, R.R. and Lucke, P.T., 984, In: Process Engneerng of Sze Reducton: Ball Mllng, SME, New York,. 7. [2] Guta, V.K. and Kaur, P.C., 976, In: Zerklenern, Proc. 4th Eur. Sym. on Commnuton, H. Rumh and K. Schonert (Eds.), Verlag Cheme, Wenhem, [3] Whten, W.J., 974, Chemcal Engneerng Scence, 29, [4] Austn, L.G., Klmel, R.R. and Lucke, P.T., 984, In: Process Engneerng of Sze Reducton: Ball Mllng, SME, New York,. 4. [5] Sho, K. et al., 982, Powder Technology, 3,. 2. [6] Austn, L. G., Klmel, R.R. and Lucke, P. T., 984, In: Process Engneerng of Sze Reducton: Ball Mllng, SME, New York,. 90. [7] Reort No. R&D: 67 : 02 : 228 : 93, 993, Research and Develoment Centre for Iron & Steel, Steel Authorty of Inda Lmted, Ranch. 349