On an integro-PDE arising from the balance equation of crystal growth -- Analytical solutions and simplifications (Q1208803)

The authors consider the balance equation, originated from the field of industrial crystallization, written under the form of an integro-partial differential equation \[ (\partial n/\partial t)+G(\partial n/\partial x)+K(t) n-A(t)f(x) \int_ 0^{+\infty} \xi^ 2 n(\xi,t)d\xi=0 \] with the initial condition, describing the seed crystals, \(n(x,0)=n_ 0(x)\); \(n(x,t)= (\partial N/\partial x)(x,t)\) where \(N(x,t)\) is the number per unit volume of crystals at time \(t\) having the equivalent diameter smaller than \(x\); \(G\) is the linear crystal growth rate. For exponential size distribution of the birth rate, \(f(x)=\exp(-x/x_ n)\), explicit analytical solutions are given for two special cases. For \(f(x)\) as a given piecewise exponential function, the modification of the above solutions is shown. An appendix of this paper describes a solving method to obtain a solution \(n(x,t)\) of the form \[ n(x,t)=\exp\left(-\int_ 0^ t K(\tau)d\tau\right)(n_ 0(x-Gt)+\exp(-x/x_ n)S(t)), \qquad x>0, \] for the above equation.