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  "body": "\n## Example\n\nLet $n(v, t) \\ge 0$ be the number density of particles of volume $v \u003e 0$ at time $t$ in a well-mixed vessel.\nLet $G(v)$ be the growth rate of a particle of volume $v$, and let $\\beta(v, w) = \\beta(w, v) \\ge 0$ be the aggregation kernel.\nLet $S(v) \\ge 0$ be the breakage rate.\nLet $b(v \\mid w)$ be the density of fragments of volume $v$ produced when a particle of volume $w \u003e v$ breaks.\nA population balance equation with growth, aggregation and breakage is\n$$\\frac{\\partial n}{\\partial t} + \\frac{\\partial}{\\partial v}\\big(G(v) n\\big) = \\frac12 \\int_0^v \\beta(v - w, w) \\, n(v - w, t) \\, n(w, t) \\, dw - n(v, t) \\int_0^\\infty \\beta(v, w) \\, n(w, t) \\, dw + \\int_v^\\infty b(v \\mid w) \\, S(w) \\, n(w, t) \\, dw - S(v) \\, n(v, t).$$\nThe left side is a transport equation in the internal coordinate $v$.\nThe right side lists birth and death of particles by aggregation and by breakage.\nIt contains partial derivatives in $t$ and $v$ and integrals of $n$ in $v$, so it is an [[integro-partial differential equation|TH4NGC4016]].\n\nCheck in two special cases.\nWith $G = 0$ and $S = 0$ only aggregation remains, and the equation is the [[Smoluchowski coagulation equation|9JRNCRZ97G]] in the volume variable.\nWith $\\beta = 0$, $S = 0$ and constant $G = g$, the equation is $\\partial_t n + g \\, \\partial_v n = 0$.\nIt is solved by $n(v, t) = n_0(v - g t)$ for any initial profile $n_0$ that vanishes for $v \\le 0$.\nEvery particle grows at the same rate, and the profile moves to the right.\n\n## Remarks\n\nIn general the internal coordinate can be a vector of several properties, such as size, composition and age, and the particles can also move in physical space.\nMoment methods turn the equation into ordinary differential equations for the moments $\\int v^k n \\, dv$ when the kernels allow it.\nSuch equations are used for crystallization, emulsions, aerosols and cell populations.\n\n## Explanations\n\n### L1 human\n\nA population balance keeps track of how many particles of each size there are.\nParticles grow, which moves them to larger sizes; they stick together, which creates big particles from small ones; and they break, which creates small particles from big ones.\nGrowth gives a derivative in the size variable, while sticking and breaking give integrals over the sizes of the partners.\n",
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