International Journal of Mineral Processing, 4 (1977) 7--32

В© Elsevier Medical Publishing Organization, Amsterdam -- Printed inside the Netherlands

T H Elizabeth BACK-CALCULATION OF SPECIFIC RA T Electronic S OF B Ur E A E A G E AND NON-NORMALIZED B R E A K A G Elizabeth D We ST L I BU T IO N G A L A Meters E Capital t E R S VIA BATCH G R We N D I N G INFO

R. R. KLIMPEL and L. G. AUSTIN

Mathematics Division, Physical Research Lab, The Dow Chemical Firm, Midland, Meine person. 48640 (U. S. A. ) Office of Material Sciences, The Philadelphia State School, University Recreation area, Pa. 16802 (U. T. A. )

(Received Dec 2, 1974; revision recognized July 30, 1976)

SUMMARY

Klimpel, 3rd there’s r. R. and Austin, M. G., 1977. The back-calculation of particular rates of breakage and non-normalized damage distribution guidelines from batch grinding data. Int. M. Miner. Method., 4: 7--32. This daily news describes a mathematical approach for establishing parameters relating specific prices of breakage and break products distribution from batch grinding a known feed for several mincing times. Knowledge of such variables leads to superior equipment design criteria and consistent working correlations. The approach entails the use of non-linear optimization with appropriate statistical tests and share parameter estimates close to those gained coming from direct trial and error measurements. A lot of results and experiences using the technique happen to be summarized.

LAUNCH Mill and mill outlet simulation simply by solution of grinding equations (Mika and Fuerstenau, year 1971; Austin, 1971--72; Luckie and Austin, 1972), offers the possibility o farrenheit more accurate signal design, operation and control. As lately discussed (Austin, 1973; Austin texas et approach., 1976), nevertheless , the appropriate data is only slowly and gradually becoming offered. The analysis o f grinding assessments in terms of specific rates o f damage, S, and breakage circulation functions, B, is boring experimentally, even though the information obtained is considerably m ore comprehensive than th a t by empirical grindability tests. Farreneheit or some years, a and u meters b elizabeth r of research groupings (Klimpel and Austin, 1970; Herbst ainsi que al., 1971; Gardner and Sukanjnajtee, 1972; Snow, 1973) have been aiming to develop a mathematical technique n o 3rd there’s r back-computing S and W parameters through the size distributions obtained simply by batch running a t n to w and feed for several grinding moments. Since this capital t y g e o farreneheit test is definitely done and it is n to t to o to time-consuming, it will be possible, after that, to run the tests essential to determine the parameters being a funct my spouse and i on of

ball, mill and particle diameter (for ball milling, for example) and as a function of work operating circumstances. This paper describes a plan capable of calculating both S and B ideals from straightforward experimental info even when W is non-normalized, for first-order batch running. It demonstrates its make use of oil experimental data, and compares the results to those from direct experimental measurement (Austin and Bhatia, 1971--72). The batch-grinding equation is conveniently expressed ( Austin, 1971 --72) as: i actually 1

dwi (t)/dt = ~

j=l i: > l

Sibijwi(t)-S~wi(t )

(1)

where wg(t) is the fat fraction of total impose present in size i at time t; Si is the selection-for-breakage or specific-rate-of-breakage of size i, with devices of capital t i meters e - 1; and b~j is a breakage circulation of size j in smaller size i; bi~ = N i, y - B i. one particular, /where Drone, j is the cumulative damage distribution function. MATHEMATICAL WAY

The Reid (1965) way to the finite-size-interval, time-continuous form of the first-order batch milling equation is: wl (t) = wl (0)e -sl t S1 b21 w2 (t) sama dengan w~ (0)

i

--

Sz-SI

e-S, t + w2(O)

-----

SI b21 S2-$1

w~ (O)

1

e s~t

-

wi(t) = ~

j=l

aije -sit

~0 where a U =

my spouse and i

fori l

It can be seen that the stability on the ith size span introduces we parameters (Si; bi, 1, bi, a couple of • • • bi. i-1 ) which have certainly not been presented previously. Within a total of n periods, the major number of new...

Recommendations: Austin, M. G., 1971--72. A review summary of the information of grinding as a level process. Powdered Technol., 5: 1--17. Austin texas L. G., 1973. Understanding ball work sizing. Ind. Eng. Chem. Proc. Kklk. Develop., doze: 121--129. Austin L. G. and Bhatia, V. K., 1971--72. Experimental methods for milling studies in laboratory ball mills. Powdered Technol., your five: 261--266. Austin texas L. G. and Luckie, P. To., 1971--72. Options for determination of breakage division parameters. Powdered Technol., your five: 215--222. Austin texas L. G. and Luekie, P. Capital t., 1972. Appraisal of non-normalized breakage syndication parameters via batch milling. Powder Technol., 5: 267--277. Austin D. G., Shoji, K. and Everell, M. D., 1973. An explanation of abnormal damage of large particles in clinical mills. Natural powder Technol., six: 3--8. Austin tx L. G., Shoji, T., Bhatia, Sixth is v. K. and Aplan, Farreneheit. F., mid 1970s. Extension in the empirical Alyavdin equation for representing group grinding data. Int. J. Miner. Process., 1: 107-123. Austin, D., Klimpel, L., Shoji, K., Bhatia, Sixth is v., Jindal, Sixth is v. and Savage, K., 1976. Some results on the explanation of size reduction as a rate method in various mills. Ind. Eng. Chem., Method Des. Develop., 15: 187--196. Bard, Y., 1974. Nonlinear Parameter Estimation. Academic Press, New York, D. Y. Bhatia, V. E., 1971. Mincing Studies in Laboratory Ball Mills. Thesis, Dep. Material Sciences Pa State Univ., Dec. year 1971. Blau, G., Klimpel, R. and Steiner, E., 1972a. Equilibrium frequent estimation and model distinguishability. Ind. Eng. Chem. Fund., 11: 324--332. Blau, G., Kiimpel, 3rd there’s r. and Steiner, E., 1972b. Parameter evaluation and version distinguishability of physicochemical versions at substance equilibrium. May. J. Chem. Eng., 40: pp. 399--409. Draper, In. R. and Smith, They would., 1966. Used Regression Examination. John Wiley and Kids, New York, In. Y. Gardner, R. L. and Sukanjnajtee, 1972. A combined dire and back-calculation method for deciding particulate damage functions in ball milling: Part I actually. Powder Technol., 6: sixty five. Goldfarb, Deb. and Lapidus, L., late 1960s. A rapid method of linearly restricted non-linear optimization. Ind. Eng. Chem. Pay for., 7: 142--151. Herbst, J. A., Grandy, G. A., Mika, T. S. and Fuerstenau, D. W., the year of 1971. An approach to the estimation from the parameters of lumped parameter grinding types from on-line measurements. Proc. 3rd Western european Symp. about Size Reduction (H. Rumpf and T. Schonert, Editors), pp. 475--514.

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Himmelblau, D. M., 1970. Procedure Analysis by simply Statistical Strategies. John Wileyand Sons, New York, N. Sumado a. Kelsall, G. F., Reid, K. M. and Restarick, C. L., 1967--68. Continuous grinding in a small wet bed mill: Component I, A study of the influence of ball diameter. Natural powder Technol., one particular: 291. Klimpel, R. and Austin, M. G., 70. Determination of selection-for-breakage capabilities in the batch grinding equation by non-linear optimization. Ind. Eng. Chem. Fund., being unfaithful: 230--237. Luckie, P. Capital t. and Austin, L. G., 1972. An evaluation Introduction to the Solution of the Milling Equations simply by Digital Computation. Miner. Sci. Eng., 5: 24--51. Mika, T. and Fuerstenau, G., 1971. The Transient Patterns of a Sent out Parameter Mill Model. Proc. 3rd Western Symp. on Size Decrease (H. Rumpf and K. Schonert, Editors), pp. 389--434. N. B. S., 1964. h a n m b u o e of Mathematical Functions. (M. Abramowitz and i also. A. Stegon, Editors). Nationwide Bureau of Standards, U. S. A., p. 932. Reid, E., 1965. A solution to the batch grinding formula. Chem. Eng. Sci., 20: 953. Shoji, K. and Austin, L. G., 1974. A model intended for batch fly fishing rod milling. Dust Technol., 12: 29--35. Snow, R. They would., 1973. Running mill simulation and scale-up of ball mills. Proc. 1973 Conf. on Molecule Technology, IITRI, Chicago, Unwell., 1973. Stewart, P. H. B. and Restarick, C. J., year 1971. A comparison from the mechanism of breakage completely scale and laboratory scale grinding mills. Proc. Australas. Inst. Min. Metall., 239: 81. Taut~, W. M., Meyer, S. and Austin tx, L. G., 1973. Proc. IFAC Symp. on Programmed Control in Mining, Vitamin and Material Processing, Sydney, Australia, pp. 11--19.

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