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It is a computational project to simulate cracking of plastics as a random process. It could be formulated as a population balance on large starting polymers for which the fragments were tracked in an array. A classical population balance is performed on a collection of particles either growing or shrinking in size, but the basic idea could be employed for molecules instead of particles and carbon number instead of particle size (Ci vs Di) and with carbon number decreasing with each time step according to equal probability of cracking at any of the bonds in the chain.

Thus, build a matrix with the first column representing Ci ranging from i=1 to, say, i=1000 at time step t = 0 and each subsequent column representing progressive time steps. In this way it could be started with, say, 100 molecules populating the C1000 cell of column 1 and “roll the dice” for the first time step for the cracking of one of the C1000 molecules, populating the next time step. For example, suppose a random number generator indicated the first of these C1000 molecules cracked to C100 and C900. Then the second column would have all zeros except for the 1000th row which would have 1 molecule and the second column would be all zeros except the 100th row would have 1 molecule, the 900th row would have 1 molecule, and the 1000th row would have 999 molecules. The next time step then would calculate the products for cracking a molecule in each of rows 100, 900, and 1000 in the previous time step column.

This should be straight forward in a population balance approach except in recognizing that each row has a distinct set of possible products. For example, molecules in row 1 cannot further crack; molecules in row 2 can only crack into C1+C1 with 100% chance; molecules in row 3 can crack into two possible product distributions (C1+C2, C2+C1) with 50% chance for each; molecules in row 4 can crack into three possible distributions (C1+C3, C2+C2, C3+C1) with 33% probability for each. Notice the probability is calculated as 1/(i-1) where i is the number of carbon atoms in the molecule’s chain.

Notice that this model assumes one molecule in each cell cracks at each time step and that we have not assigned a particular time to the time steps, just equal time intervals.

It can be started t with i=20 with maybe 50 time steps to ease diagnostics in building the model.

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