RESEARCH ARTICLE
An existence-uniqueness result for the pure binary collisional breakage equation
Jayanta Paul,
Jitendra Kumar,
Corresponding Author
Jayanta Paul
Department of Mathematics, Indian Institute of Technology, Kharagpur, India
Correspondence
Jayanta Paul, Department of Mathematics, Indian Institute of Technology, Kharagpur, India.
Email: [email protected]
Search for more papers by this authorJitendra Kumar
Department of Mathematics, Indian Institute of Technology, Kharagpur, India
Search for more papers by this authorJayanta Paul,
Jitendra Kumar,
Corresponding Author
Jayanta Paul
Department of Mathematics, Indian Institute of Technology, Kharagpur, India
Correspondence
Jayanta Paul, Department of Mathematics, Indian Institute of Technology, Kharagpur, India.
Email: [email protected]
Search for more papers by this authorJitendra Kumar
Department of Mathematics, Indian Institute of Technology, Kharagpur, India
Search for more papers by this authorAbstract
The study of collision-induced breakage phenomenon in the particulate process has much current interest. This is an important process arising in many engineering disciplines. In this work, the existence of continuous solution of the pure collisional breakage model is developed beneath some restrictions on the breakage kernels. Furthermore, the mass conservation and uniqueness of solution are investigated in the absence of “shattering transition.” The underlying theory is based on the compactness result of Arzelà-Ascoli and Banach contraction mapping principle.
REFERENCES
- 1Krapivsky PL, Ben-Naim E. Shattering transitions in collision-induced fragmentation. Phys Rev E. 2003; 68(2): 021102.
- 2Kunii D, Levenspiel O. Fluidization Engineering. Boston:Elsevier; 2013.
- 3Fries L, Antonyuk S, Heinrich S, Dopfer D, Palzer S. Collision dynamics in fluidised bed granulators: a DEM-CFD study. Chem Eng Sci. 2013; 86: 108-123.
- 4Crüger B, Salikov V, Heinrich S, et al. Coefficient of restitution for particles impacting on wet surfaces: an improved experimental approach. Particuology. 2016; 25: 1-9.
- 5Lee KF, Dosta M, McGuire AD, et al. Development of a multi-compartment population balance model for high-shear wet granulation with discrete element method. Comput Chem Eng. 2017; 99: 171-184.
- 6Srivastava RC. Parameterization of raindrop size distributions. J Atmospheric Sci. 1978; 35(1): 108-117.
- 7Srivastava RC. A simple model of particle coalescence and breakup. J Atmospheric Sci. 1982; 39(6): 1317-1322.
- 8Kudzotsa I. Mechanisms of aerosol indirect effects on glaciated clouds simulated numerically. PhD thesis: University of Leeds; 2013.
- 9Yano JI, Phillips VT, Kanawade V. Explosive ice multiplication by mechanical break-up in ice–ice collisions: a dynamical system-based study. Q J Roy Meteorol Soc. 2016; 142(695): 867-879.
- 10Fuerstenau DW, De A, Kapur PC. Linear and nonlinear particle breakage processes in comminution systems. Int J Miner Process. 2004; 74: S317-S327.
- 11Capece M, Bilgili E, Dave RN. Emergence of falsified kinetics as a consequence of multi-particle interactions in dense-phase comminution processes. Chem Eng Sci. 2011; 15; 66(22): 5672-5683.
- 12Brosh T, Kalman H, Levy A, Peyron I, Ricard F. DEM–CFD simulation of particle comminution in jet-mill. Powder Technol. 2014; 31; 257: 104-112.
- 13Johnson C. Effect of wave collision on fragmentation, throw, and energy efficiency of mining and comminution. Energy Efficiency in the Minerals Industry. Cham: Springer; 2018: 55-70.
10.1007/978-3-319-54199-0_4 Google Scholar
- 14Kelly EG, Spottiswood DJ. Introduction to Mineral Processing. New York:John Wiley & Sons; 1982.
- 15Chakraborty J, Ramkrishna D. Population balance modeling of environment dependent breakage: role of granular viscosity, density and compaction. Model formulation and similarity analysis. Ind Eng Chem Res. 2011; 50(23): 13116-13128.
- 16Spampinato A, Axinte DA. On modelling the interaction between two rotating bodies with statistically distributed features: an application to dressing of grinding wheels. Proc Royal Soc A Math Phys Eng Sci. 2017; 473(2208):20170466.
- 17Bilgili E, Capece M, Afolabi A. Modeling of milling processes via DEM, PBM, and microhydrodynamics. In: Predictive Modeling of Pharmaceutical Unit Operations. Amsterdam, Netherlands: Elsevier; 2016: 159-203.
- 18Laurenċot P, Wrzosek D. The discrete coagulation equations with collisional breakage. J Stat Phys. 2001; 104(1): 193-220.
- 19Walker C. Coalescence and breakage processes. Math Methods Appl Sci. 2002; 25(9): 729-748.
- 20Cheng Z, Redner S. Scaling theory of fragmentation. Phys Rev Lett. 1988; 60(24): 2450-2453.
- 21Kostoglou M, Karabelas A. A study of the nonlinear breakage equation: analytical and asymptotic solutions. J Phys A: Math Gen. 2000; 33(6): 1221.
10.1088/0305-4470/33/6/309 Google Scholar
- 22Cheng Z, Redner S. Kinetics of fragmentation. J Phys A: Math Gen. 1990; 23(7): 1233-1258.
10.1088/0305-4470/23/7/028 Google Scholar
- 23Kostoglou M, Karabelas AJ. A study of the collisional fragmentation problem using the Gamma distribution approximation. J Colloid Interface Sci. 2006; 303(2): 419-429.
- 24Ernst MH, Pagonabarraga I. The nonlinear fragmentation equation. J Phys A: Math Theor. 2007; 40(17): F331-F337.
- 25Stewart IW, Meister E. A global existence theorem for the general coagulation–fragmentation equation with unbounded kernels. Math Methods Appl Sci. 1989; 11(5): 627-648.
- 26Stewart IW. On the coagulation-fragmentation equation. Z Angew Math Phys (ZAMP). 1990; 41(6): 917-924.
- 27Stewart IW. A uniqueness theorem for the coagulation-fragmentation equation. Math Proc Cambridge Philos Soc. 1990; 107: 573-578.
- 28Dubovskii P, Stewart IW. Existence, uniqueness and mass conservation for the coagulation-fragmentation equation. Math Methods Appl Sci. 1996; 19(7): 571-591.
- 29McLaughlin D, Lamb W, McBride A. Existence and uniqueness results for the non-autonomous coagulation and multiple-fragmentation equation. Math Methods Appl Sci. 1998; 21(11): 1067-1084.
- 30Laurenċot P, Mischler S. From the discrete to the continuous coagulation-fragmentation equations. Proc R Soc Edinb Sec A: Math. 2002; 132(5): 1219-1248.
- 31Banasiak J. On a non-uniqueness in fragmentation models. Math Methods Appl Sci. 2002; 25(7): 541-556.
- 32Lamb W. Existence and uniqueness results for the continuous coagulation and fragmentation equation. Math Methods Appl Sci. 2004; 27(6): 703-721.
- 33Fasano A, Rosso F. Dynamics of droplets in an agitated dispersion with multiple breakage. Part II: uniqueness and global existence. Math Methods Appl Sci. 2005; 28(9): 1061-88.
- 34Oukouomi Noutchie SC. Existence and uniqueness of conservative solutions for nonlocal fragmentation models. Math Methods Appl Sci. 2010; 33(15): 1871-1881.
- 35McGuinness GC, Lamb W, McBride AC. On a class of continuous fragmentation equations with singular initial conditions. Math Methods Appl Sci. 2011; 34(10): 1181-1192.
- 36Brilliantov N, Krapivsky PL, Bodrova A, et al. Size distribution of particles in Saturn's rings from aggregation and fragmentation. Proc Natl Acad Sci. 2015; 112(31): 9536-9541.
- 37Camejo CC, Warnecke G. The singular kernel coagulation equation with multifragmentation. Math Methods Appl Sci. 2015; 38(14): 2953-2973.
- 38Saha J, Kumar J. The singular coagulation equation with multiple fragmentation. Z Angew Math Phys (ZAMP). 2015; 66(3): 919-41.
- 39Vledouts A, Vandenberghe N, Villermaux E. Fragmentation as an aggregation process: the role of defects. Proc Royal Soc A: Math Phys Eng Sci. 2016; 472(2185):20150679.
- 40Rachah A, Noll D, Espitalier F, Baillon F. A mathematical model for continuous crystallization. Math Methods Appl Sci. 2016; 39(5): 1101-1120.
- 41Breschi G, Fontelos MA. A note on the self-similar solutions to the spontaneous fragmentation equation. Proc Royal Soc A: Math Phys Eng Sci. 2017; 473(2201):20160740.
- 42Banasiak J, Joel LO, Shindin S. Analysis and simulations of the discrete fragmentation equation with decay. Math Methods Appl Sci. 2017: 1-16.
- 43Peterson TW. Similarity solutions for the population balance equation describing particle fragmentation. Aerosol Sci Technol. 1986; 5(1): 93-101.
- 44Austin LG. A treatment of impact breakage of particles. Powder Technol. 2002; 126(1): 85-90.
- 45Sommer M, Stenger F, Peukert W, Wagner NJ. Agglomeration and breakage of nanoparticles in stirred media millsa comparison of different methods and models. Chem Eng Sci. 2006; 61(1): 135-148.
