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<br />NOVEMBER 1978 <br /> <br />LARRY VARDIMAN <br /> <br />2169 <br /> <br />between crystal type i, size j and crystal type k, <br />size l; E is the collision efticiency; C;j is the concen- <br />tration of crystal type i, size j; C kl is the concentra- <br />tion of crystal type k, size I; A ijkl is the collision cross <br />section between crystal type i, size j and crystal <br />type k, size l; and Vij-Vkl is the relative fall velocity <br />between crystal type i, size j and crystal type k, <br />size l. Because the majority of crystals studied in <br />this research are large enough to be outside the r3;nge <br />of interaction effects and those that are in the range <br />of wake effects have such small relative fall velocities, <br />which contribute little or nothing to secondary particle <br />generation, E will be assumed equal to 1. <br /> <br />2) FRAGMENT GENERATION FUNCTIONS <br /> <br />In general, the number of fragments generated in <br />a collision is a function of the type and size of both <br />crystals involved in the collision. However, the task <br />of obtaining the necessary information to predict the <br />number of fragments generated for all of the various <br />combinations of types and sizes of crystals in a cloud <br />is' prohibitive. It is far simpler and should not be <br />too bad an assumption to find a predictor for the <br />number of fragments generated as a function of some <br />single parameter of the collision. The most appro- <br />priate parameter would seem to be the maximum <br />force exerted during a conision. The number of frag- <br />ments generated in a given. collision should be pro- <br />portional to the maximum force exerted. However, <br />the maximum force exerted in a 'collision is a function <br />oimany parameters of a collision, such as the <coeffi- <br />cient of restitution, the masses of the two particles, <br />the relative velociti~s, the contact time of the col- <br />lision and the collision force as a function of time. <br />Most of these parameters could ,be taken into account <br />except for the contact time and the collision force. <br />Unfortunately, these parameters are unknown and <br />probably vary greatly. Even if the contact time is <br />assumed constant for all collisions and the collision <br />force is assumed to be normally distributed ".ith <br />time, the error in the estimated maximum force would <br />still be extrenie. Two other logical parameters come <br />to mind-the kinetic energy of impact and the change <br />in momentum. Neither of these parameters is as basic <br />as the maximum force but they have the advantage <br />that most of the variables upon which they depend <br />can be reasonably treated. Hobbs and Farber (1972) <br />used kinetic energy as a parameter since the elastic <br />properties of a theoretical study naturally lend them- <br />selves to this treatment. However, for this study the <br />change in momentum due to the collision between <br />particles was used because it is conservative, while <br />kinetic energy is nonconservative. This is true since <br />momentum is conserved even in a nonelastic collision <br />while energy is not. Crystal collisions are necessarily <br />inelastic if fracturing is to occur since the fracturing <br />process requires the absorption of energy. <br /> <br /> <br />The change in momentum is similar to the maxi- <br />mum force because it is the integral of force over <br />time. It cannot distinguish a short, hard collision <br />from a long, soft collision. However, if the contact <br />time is similar for all collisions, it will approximate <br />the maximum force quite well. <br />The change in momentum of a particle when hit <br />by another particle can be obtained by solving two <br />equations with two unknowns. The two equations are <br />the equation for the conservation of momentum for <br />the system and the equation for the coefficient of <br />restitution. From Sears and Zemansky (1957) these <br />equations are <br /> <br />mij(V{j-vB) = ~mkl(vkl-v21), <br /> <br />(2) <br /> <br />Vh-Vkl <br />e=-- <br />, <br />Vg-V21 <br /> <br />(3) <br /> <br />where mij and mkl are the masses of the two particles, <br />vg and V~l are the initial velocities, v~ and vii are the <br />final velocities, and e is the coefficient of restitution. <br />The two unknowns are v{j and vii given that the coeffi- <br />cient of restitution e is known. Since only two particles <br />are involved, the change in momentum of one particle <br />must equal the change in momentum of the other. <br />Therefore, <br /> <br />b.Mijkl=mij(v~-Vi~)= -mkl(v{l-V~J, (4) <br /> <br />where b.Mijkl is the change in momentum of a particle <br />of type i, size j when hit by a particle of type k, size l. <br />Solving Eqs. (2) and (3) simultaneously for v~ gives <br /> <br />1 <br />~~h= {mijvg+mklv21+emkl(v21-vB)}. (5) <br />mij+mkl <br /> <br />Substituting v{j into Eq. (4) and simplifying gives <br /> <br />mijmkl 0 0 <br />b.Mijkl= {1+e}{vkl-Vij}. (6) <br />mij+mkl <br /> <br />This change in momentum b.Mijkl is the maximum <br />possible change because Eq. (6) assumes that the <br />partides collide center-to-center. However, in a real <br />cloud the particles have an equal probability of col- <br />liding in all configurations between center-to-center <br />and a grazing blow. The most likely change in mo- <br />mentum is then the expected value of b.Mijkl obtained <br />by integrating over all possible configurations. The <br />most likely change in momentum can be shown to <br />be that given in Eq. (6) multiplied by 71"/4. Therefore, <br />the statistical change in momentum of a particle hit <br />by another becomes <br /> <br />7I"{ mijmkl } <br />b.Mijkl=- (l+e) (Vk/-Vij) , <br />4 mij+mkl <br /> <br />(7) <br />