Articles
All articles | Recent articles
Plots of Surface Temperature and Mass of a Heated, Large Limestone Sample in Terms of Regression Splines and the Programming Language R
R. Lech1, S. Sado2
1 AGH University of Science and Technology in Cracow
2 Zakłady Magnezytowe „ROPCZYCE” S.A. Research and Development Centre of Ceramics
received August 3, 2021, received in revised form July 27, 2022, accepted August 3, 2022
Vol. 13, No. 2, Pages 65-82 DOI: 10.4416/JCST2021-00015
Abstract
The use of the surface temperature and the mass of a limestone sample or sample conversion degree as variables present in various models of limestone calcination is shown in the paper. The linear model of the thermal dissociation rate of a large limestone sample is derived using the R program and the base of B - splines. The statistical significance of the formulated model is demonstrated. The example of the model recording of the thermal dissociation rate of a large limestone sample as a function of heating conditions, including the sample heating time, is shown. The values of the basis functions contained in the regression matrix have been identified. Moreover, the formulas for non-zero cubic basis functions in the selected span knot of the explanatory variable are derived.
Download Full Article (PDF)
Keywords
Limestone, thermal dissociation, thermal analysis, statistical modelling, regression splines
References
1 Zawadzki, J., Bretsznajder, S.: Über den Verlauf der Reaktionen zwischen CaO und CO2 sowie zwischen CaO und SO2, Bull. Acad. Pol. Sci. A., 271 – 286, (1932).
2 Zawadzki, J., Bretsznajder, S.: Zur Kenntnis der heterogenen Reaktionen vom Typus , I. Abweichungen von der Konstanz des Gleichgewichtsdruckes, Scheinbare Gleichgewichte und deren Deutung, Z. Phys. Chem., B.22, 60 – 78, (1933).
3 Zawadzki, J., Bretsznajder, S.: Zur Kenntnis der heterogenen Reaktionen vom Typus , II. Kinetik der Carbonatbildung und Zersetzung, Z. Phys. Chem., B.22, 79 – 96, (1933).
4 Zawadzki, J., Bretsznajder, S.: A contribution to the kinetics of reactions in which solid phases take part, Bull. Acad. Pol. Sci. A., 60 – 64, (1946).
5 Bretsznajder, S.: O przebiegu reakcji typu Aciało stat + Bgaz ←→ Cciało stat, w/in Błasiak E. (et al. eds.), Kataliza i katalizatory, PWT, Warszawa, 1952.
6 Hills, A.W.D.: The mechanism of the thermal decomposition of calcium carbonate, Chem. Eng. Sc., 23, 297 – 320, (1968).
7 Pigoniowa, J., Gumiński, K., Statyka chemiczna, w/in Bielański A. et al. eds., Chemia fizyczna, PWN, Warszawa, 1980.
8 Yagi, S., Kunii, D.: Studies on combustion of carbon particles in flames and fluidized beds, Fifth Symposium (International) on Combustion, Van Nostrand Reinhold, New York, 1954.
9 Verma, C.L.: Simulation of lime shaft kilns using mathematical modelling, Zement – Kalk – Gips, 43, [12], 576 – 582, (1990).
10 Khinast, J., Krammer, G.F., Brunner, Ch., Staudinger, G.: Decomposition of limestone: the influence of CO2 and particle size on the reaction rate, Chem. Eng. Sc., 51, 623 – 634, (1996).
11 Boynton, R.S.: Chemistry and technology of lime and limestone. John Wiley & Sons, Inc., New York, Chichester, Brisbane, Toronto, 1980.
12 Oates, J.A.H.: Lime and Limestone. Chemistry and Technology, Production and Uses, Wiley-VCH, Weinheim, New York, Chichester, Brisbane, Singapore, Toronto, 1998.
13 Locher, G.: Mathematische Modelle zum Prozess des Brennens von Zementklinker, T.1: Reaktionen und Grundoperationen, ZKG Int., 55, [1], 29 – 38, (2002).
14 Locher, G.: Mathematische Modelle zum Prozess des Brennens von Zementklinker, T.2: Vorwärmer, Calcinator und Bypass, ZKG Int., 55, 39 – 50, (2002).
15 Martins, M.A., Oliveira, L.S., Franca, A.S.: Modellierung und Simulation der Kalksteincalcination im Drehofen, Part I, ZKG Int., 55, [4], 76 – 87, (2002).
16 Martins, M.A., Oliveira, L.S., Franca A.S.: Modellierung und Simulation der Kalksteincalcination im Drehofen, Part II, ZKG Int., 55, [5], 74 – 83, (2002).
17 Gordon, Y.M., Blank, M.E., Madison, V.V., Abovian, P.R.: New technology and shaft furnace for high quality metallurgical lime production, AsiaSteel 2003, India, 1, 1.b1.1 – 1.b.1.6, 2003.
18 Gordon, Y.M., Shvidkiy, V., Yaroshenko, Y.: Optimization of the design and operating parameters of shaft furnaces, METEC Congress, 3rd ICSTI, Düsseldorf, 2003.
19 Stanmore, B.R., Gilot, P.: Review – calcination and carbonation of limestone during thermal cycling for CO2 sequestration, Fuel Process. Technol., 86, 1707 – 1743, (2005).
20 Georgakis, Ch., Chang, C.W., Szekely, J.: A changing grain size model for gas – solid reactions, Chem. Eng. Sci., 34, [8], 1072 – 1075, (1979).
21 Bes, A.: Dynamic process simulation of limestone calcination in normal shaft kilns, Thesis (PhD), Otto-von-Guericke-Universität Magdeburg, 59 – 62, 79 – 80, (2006).
22 Clark, J.B., Hastie, J.W, Kihlborg, L.H.E., Metselaar, R., Thackeray M.M.: Definitions of terms relating to phase transitions of the solid state, Pure & Appl. Chem., 66, [3], 592, (1994).
23 Specht, E.: Wärme- und Stoffübertragung in der Thermoprozesstechnik. Grundlagen, Berechnungen, Prozesse, Vulkan Verlag GmbH, Essen, 198 – 205, 484 – 502, (2004).
24 Hallak, B., Herz, F., Specht, E., Gröpler, R., Warnecke, G.: Simulation of limestone calcination in normal shaft kilns – mathematical model, ZKG, 9, 66 – 71, (2005).
25 Hallak, B., Herz, F., Specht, E., Gröpler, R., Warnecke, G.: Simulation of limestone calcination in normal shaft kilns – Part 2: Influence of process parameters, ZKG, 10, 46 – 50, (2015).
26 Hallak, B., Herz, F., Specht, E., Gröpler, R., Warnecke, G.: Simulation of limestone calcination in normal shaft kilns – Part 3: Influence of particle size distribution and type of limestone, ZKG, 3, 64 – 68, (2016).
27 Piringer, H.: Lime shaft kilns, INFUB-11th European Conference on Industrial Furnaces and Boilers, INFUB-11, Energy Procedia 120, 75 – 95, 2017.
28 Brown, M.E.: Introduction to thermal analysis. Techniques and application, 2.Ed., Kluwer Academic Publishers, New York, Boston, Dordrecht, London, Moscow, 1 – 12, (2004).
29 Hastie, T., Tibshirani, R., Friedman, J.: The Elements of Statistical Learning: Data Mining, Inference, and Prediction, 2nd edit., New York: Springer, 139 – 161, 186 – 189, 203 – 232, (2009).
30 Trzęsiok, J., Trzęsiok, M.: Nieparametryczne metody regresji, w/in Walesiak M., Gatnar E. (Sci.Ed.), Statystyczna analiza danych z wykorzystaniem programu R, Wyd.Nauk. PWN, Warszawa, 2002.
31 R Core Team, R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria, 2020.
32 Kozłowski, S.: Surowce skalne Polski, Wyd. Geologiczne, Warszawa, 1986.
33 Lech, R.: Thermal decomposition of limestone: Part 1 – Influence of properties on calcination time, Sil. Ind., 71, [7 – 8], 103 – 109, (2006).
34 Lech, R.: Termiczny rozkład wapieni: Transport masy i ciepła, Polska Akademia Nauk, Oddział w Krakowie, Ceramika, 105, 29 – 53, (2008).
35 Aczel, A.D., Sounderpandian, J.: Statystyka w zarządzaniu, Wyd. Nauk. PWN SA, Warszawa, 2008.
36 Neter, J., Wasserman, W., Whitmore, G.A.: Applied Statistics, 3 ed., Allyn & Bacon, Inc., Newton, 1988.
37 Faraway, J.J.: Extending the linear model with R: Generalized Linear, Mixed Effects and Nonparametric Regression Models, 2 ed., CRC Press, Taylor & Francis Group: A Chapman & Hall Book, 2016.
38 Hastie, T., Tibshirani, R.: Generalized additive models, Stat Sci., 1, [3], (1986).
39 Lis, A.: Uogólnione modele addytywne z parametrem połoŻenia, skali i kształtu, Praca magisterska, UW WMIM, 2011.
40 Carl de Boor, C.: A practical guide to splines, New York: Springer-Verlag, 87 – 108, 2001.
41 Davies, T.M.: The book of R. A first course in programming and statistics, No Starch Press, San Francisco, 2016.
42 Hurvich, C.M., Tsai, C.-L.: Regression and Time Series Model Selection in Small Samples, Biometrika, 76, 297 – 307, (1989.
Copyright
Göller Verlag GmbH
