Modeling, Simulation and Optimization of Process Chains

Abstract

Processes in the field of chemical engineering do not consist of one single step, but typically a high number of strongly interconnected unit operations linked with recycling streams. This inherent complexity further exacerbates when distributed particle properties, i.e., dispersity, must be considered, noteworthy being the case whenever particulate products are in focus. Out of all five possible dimensions of dispersity (size, shape, composition, surface and structure) particle size most often determines the efficiency of particulate products. Thus, its optimization is key to reach tailored handling and end product properties. In this work, a model-based optimization tool for particle synthesis was elaborated which is often the first step of a process chain. It is described by population balance equations relying on the method of characteristics for the numerical simulation and on the usage of gradient information to enhance the performance of the optimization. The presented scheme to optimize time-dependent process conditions in a time efficient manner is applicable for a wide range of particle syntheses.

Symbols Used

Abbreviations

ADJ	Adjoint equation	FBSG	Fluidized bed spray granulation
PBE	Population balance equation	PRI	Primal equation
PSD	Particle size distribution	RSD	Relative standard deviation

Greek Letters

$\gamma$	[a.u.]	Weight function in nonlocal term	$\mu$	[m]	Mean particle size
$\xi$	[a.u.]	Solution of characteristic equation	$\pi$	[–]	$3.1415\dots$
$\sigma$	[m]	Standard deviation	$\Phi$	[–]	Grade efficiency function
$\chi$	[–]	Characteristic function	$\Omega$	[–]	Domain of interest

Symbols

$\partial$	[–]	Partial derivative	$\tfrac{\,\mathrm {d}}{\,\mathrm {d}z}$	[–]	Total derivative (to z)
$\nabla$	[–]	Gradient	t	[s]	Time
$t_{\text {f}}$	[s]	Fnal time	n	[–]	Dimension of disperse properties
x	[m]	Particle radius	$\varvec{x}$	[a.u.]	Vector of disperse properties
$\mathrm {G}$	[a.u.]	One-dimensional growth function	$\mathbf{G}$	[a.u.]	Multi-dimensional growth function
$\mathrm {G}_{\text {Ost}}$	[–]	Ostwald ripening function	$u_{\mathrm {G}}$	[a.u.]	Time-dependent parameter in $\mathrm {G}$
$u_\mathbf{G }$	[a.u.]	Time-dependent parameter in $\mathbf{G}$	$u_{\text {in}}$	[–]	Time-dependent influx rate
$u_{\text {out}}$	[–]	Time-dependent extraction rate	$u_0$	[–]	Scaling factor of initial PSD
$\underline{u}$	[a.u.]	Lower bound on process condition u	$\overline{u}$	[a.u.]	Upper bound on process condition u
q	[a.u.]	PSD	$q_3$	[a.u.]	Mass-related PSD
$q_{\text {in}}$	[a.u.]	PSD of external nuclei	$q_{\text {in},3}$	[a.u.]	Mass-related PSD of external nuclei
$q_0$	[a.u.]	Initial PSD	$q_{\text {d}}$	[a.u.]	Desired PSD
$W_q$	[a.u.]	Nonlocal term in (1)	$\varvec{x}_{\min }$	[a.u.]	Vector of minimal disperse properties
$x_{\min }$	[m]	Minimal radius	p	[a.u.]	Solution of (3)
$Y_p$	[a.u.]	Nonlocal term in (3)	J	[a.u.]	Cost functional
$U_{\text {ad}}$	[–]	Set of admissible process conditions	$\hat{\varvec{\xi }}$	[m]	Semi-discr. of $\xi$
$\varvec{q}$	[a.u.]	Semi-discr. of q	$\varvec{p}$	[a.u.]	Semi-discr. of p
$\hat{W}_q$	[a.u.]	Approx. of $W_q$	$\hat{q}$	[a.u.]	Approximation of q
$\hat{Y}_p$	[a.u.]	Approx. of $Y_p$	$\hat{p}$	[a.u.]	Approximation of p
d(u)	[–]	Number of discretization points of u	d	[–]	Number of discretization points
K	[–]	number of grid points	$x_k$	[a.u.]	Grid point
$x_L$	[a.u.]	Largest grid point	$\mathrm {E}^n$	[a.u.]	n-th moment
$\mathrm {E}$	[a.u.]	First moment	$\mu _{\text {d}}$	[m]	Desired mean particle size
$\mu _{\text {in}}$	[m]	Mean particle size of entering PSD	$\mu _{\text {out}}$	[m]	Mean particle size of outgoing PSD
P	[W]	Milling power	$q_{\text {tot}}$	[m$^{-1}$]	PSD of total product
$q_{\text {crush}}$	[m$^{-1}$]	PSD of crushed particles	$q_{\text {crush},3}$	[m$^{-1}$]	Mass-related PSD of crushed particles
$x_{\text {cut}}$	[m]	Cut size	$\dot{m}$	[kg$\cdot \mathrm{s}^{-1}$]	Mass flow

Norms

$\Vert \cdot \Vert _{L^p ((0,t_{\text {f}}))}$	Norm on Lebesgue space of p-integrable functions on $(0,t_{\text {f}})$
$\Vert \cdot \Vert _{H^1 ((0,t_{\text {f}}))}$	Norm on Sobolev space of functions $f\in L^2((0,t_{\text {f}}))$ with $\dot{f}\in L^2((0,t_{\text {f}}))$