Lehrstuhl für Mechatronik in Maschinenbau und Fahrzeugtechnik (MEC)

Control of Population Balance Systems


M.Sc. Iqra Batool
Gottlieb-Daimler-Str. 42
67663, Kaiserslautern
Phone: +49 (0)631/205-3702
Fax: +49 (0)631/205-4201


State of Rhineland-Palatinate


  • Processes involving particulate materials are of great importance in various engineering applications and industries, including chemical engineering, food industry, systems biology, pharmaceutical engineering, milling, geology, etc.
  • Most of such  applications involve simultaneously various particulate phenomena such as nucleation, growth, filtering, dissolution, aggregation (also referred to as agglomeration) and breakage in the dispersed phase.
  • Batch crystallization represents a class of particulate processes which are widely applied, e.g., in particle or population manufacturing.
  • At the core of our research lies the development of an efficient and robust semi-analytical ODE schemes for modeling and control of population balance systems, involving various state dependent phenomena, with a particular emphasis on agglomeration and breakage.
  • The approximate polynomial method of moments (AMOM) represents an analytical extension to the standard method of moments.
  • It relies on analytical approximations of specific terms appearing in the population balance equation (PBE), such as size-dependent growth rate and filtering term functions in terms of orthogonal polynomials.


  • Population balance systems in batch crystallization are typically described by integro-differential equations, where integral terms describe the macroscopic kinetics and partial differential equations (typically, of hyperbolic type) describe the evolution of various local phenomena in dispersed phase, including advection, nucleation, fines dissolution, agglomeration and breakage.
  • The distribution function f(t,x) describes the number density of particles with a property x at time t.
  • Fundamental conditions in modeling the crystallization processes are given by the energy and mass balance.
  • Assuming an ideally mixed solution, the both balancing conditions are coupled to the state of the solid phase by the crystallization enthalpy and the solute consumption or release due to growth or dissolution, respectively.
  • To account for variation of the crystal shape distributions over time, a morphological population balance is formulated by means of a population balance equation (PBE) with one or more internal coordinates. The general population balance equation (PBE) for a batch process is represented by first order, one-dimensional, hyperbolic partial differential equation of the form:


  • the first and  second terms on the right side of the equation represent the growth and nucleation of the particles, with G(t,x) and B(t) representing the correspding rates of a particle with the property (i.e., size) x at time t. The last term h(x)f(t,x) models the mechanism of fines removal or dissolution.
  • the first integral term in the second line refers to the birth of particles of size x, which results from the agglomeration of two particles with sizes  x' and (x3-x'3)1/3, respectively. The second integral term also called "death term", describes the loss of particles of size x by aggregation with other particles of any size. The aggregation kernel β((x3-x'3)1/3,x') characterizes the dynamics of aggregation process and throughout our proposal, it will be assumed to be time invariant.
  • the third integral term represents the formation (birth) of particles of size x from breakage of particles of size x'. The term b(x,x') represents the probability function for the breakage of particles of size x' in the formation of daughter particles of size x. The breakage kernel S(x) illustrates the rate of breakage of particles of size x. The right-most term represents the sink term due to the breakage of particles of size x.

Crystallization kinetics

  • Growth: Mass transfer of the solute from the liquid to the solid phase

    where kg and g are empirical (positive) parameters.

  • Nucleation: In parallel to particle growth, nucleation occurs, i.e., solute molecules gathering into stable cluster, constituting new nuclei

    where kb and b are empirical (positive) parameters and μ3 is the 3rd moment.

  • Supersaturation: It is a measure of the difference between the solution concentration C and saturation concentration Csat fixed by the temperature T

    where a0, a1 and a2 are given empirical parameters.

  • Mass-balance: The solute consumption is described by the mass-balance equation

    where ρ is the mass density of crystal particles.

Approximate method of moments (AMOM)

An early and popular solution scheme that provides an equivalent description of a simple class of infinite-dimensional system (i.e., PDE) by means of a finite-order system of ordinary differential equations (ODE) is the method of moments (MOM). The ODE structure of MOM is highly desirable for applying the control and optimization techniques but due to the closure issue, MOM is useful to the very limited class of PBEs involving nucleation and size-independent growth only. We have developed an approximate method of moments (AMOM) which represents a generalization of the standard method of moments for population balance models (PBM) with nucleation, growth and fines dissolution. Below are the two cases discussed, how closed ODE structure is achieved via AMOM

Size-dependent growth
  • Consider the PBE with size-dependent growth rate
  •  Assume the seperability condition G(x,t)=γ(x)G0(t) and introduce coordinate transformation
  • The PBE is transformed into the following with a new scaled density function 
  • Now consider polynomial approximation for factors xi in some polynomial basis {φk(λ)}

     and introduce the generalized moments

  • We get an approximating ODE structure (internal moment model)
  • Finally, the relationship of internal to the classical moments is given by
Size-independent growth
  • Consider the PBE with size independent growth and without fines dissolution term:

   and define the ith moment as:

  • Differentiating the latter equation w.r.t. time t yields the moment's model

    for i=1,2, ... and initial conditions

  • The ODE structure closing at the 3rd moment μ3 with σ as an input
  • Introducing time transformation yields
  • This reveals the closed ODE computational scheme in terms of standard moments (for the case, h(x)=0).

  • It is a simple chain structure of integrators and it does precisely resemble to standard moment model.

Optimal Control of PBM

  • Semi-analytical, closed ODE structure obtained from AMOM is then utilized for optimal control setting in the context of shaping the particle size distribution (PSD) and we compute optimal control (i.e., supersaturation or/and temperature) profiles for crystal shape manipulation.
  • We focus on controlling the growth process, where (a relatively narrow) distribution of seed crystals is to be relocated to a desired target position.
  • Several optimal control scenarios have been investigated by the author et al., such as the time-constrained shape manipulation with a minimal final mass of nuclei particles in the context of PBE and ODE models for univariate and bivariate particulate processes with size-independent and size-dependent growth rate.
Problem Formulation

In the univariate case, the problem formulates:

which means that we shift the part of the seed distribution function fseed(x) for a fixed particle size τf,c within a specified time interval [0, tf,c], while producing a minimal nuclei mass μ3n. Here U is the set of admissible supersaturation values.

In the bivariate case the optimal control problem reads:

where U is the set of admissible super- saturation values, σ is supersaturation, tf is the final time and VC,n the volume of the nuclei. It constitutes of the time-constrained shape manipulation with a minimal final mass of nuclei particle. The goal of the problem is to make particles with initial PSD to desired PSD. In the accompanying figure, f0(x) is the initial condition, h(x) is fines dissolution, fn(x,t) is the distribution function for nucleated particles and fs(x,t) is the distribution function of seed particles



  • Hofmann, S., Bajcinca, N., Raisch, J. and Sundmacher, K. (2017): Optimal control of univariate and multivariate population balance systems involving external fines removal. Chemical Engineering Science, 168, 101-123. ( pdf)
  • Eisenschmidt H., Soumaya M., Bajcinca N., Le Borne S., Sundmacher K.  (2017): Estimation of aggregation kernels based on Laurent polynomial approximation. Computers & Chemical Engineering. 103, 210-217. ( pdf)
  • Eisenschmidt H., Bajcinca N., Sundmacher K (2016): Optimal control of crystal shapes in batch crystallization experiments by growth-dissolution cycles. Crystal Growth & Design, 16(6), 3297-306. ( pdf)
  • Bajcinca, N., S. Hofmann, D. Bielievtsov, and K. Sundmacher (2015): Approximate ODE models for population balance systems. In Computers & Chemical Engineering, 74, 158-168. ( pdf)
  • Bajcinca, N., S. Hofmann, H. Eisenschmidt, and K. Sundmacher (2015): Generalizing ODE modeling structure for multivariate systems with distributed parameters. 9th IFAC Symposium on Advanced Control of Chemical Processes ADCHEM, Canada, 240-247. ( pdf)
  • Bajcinca, N., S. Hofmann, and K. Sundmacher (2014): Method of moments over orthogonal polynomial bases. Chemical Engineering Science, 119, 295-309. ( pdf)
  • Bajcinca, N. and S. Hofmann (2011): Optimal control for batch crystallization with size-dependent growth kinetics. In Proceedings of the 2011 American Control Conference, USA, 2558-2565. ( pdf)
  • Bajcinca, N., Menarin, H.A. and Hofmann, S. (2011): Optimal control of multidimensional population balance systems for crystal shape manipulation. IFAC Proceedings Volumes, 44(1), 9842-9849. ( pdf)
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