A Computationally Efficient Surrogate-Based Reduction of a Multiscale Comill Process Model

  • Nirupaplava Metta
  • Rohit Ramachandran
  • Marianthi IerapetritouEmail author
Original Article



Particle breakage in milling operations is often modeled using population balance models (PBMs). A discrete element method (DEM) model can be coupled with a PBM in order to explicitly identify the effect of material properties on breakage rate. However, the DEM-PBM framework is computationally expensive to evaluate due to high-fidelity DEM simulations. This limits its application in continuous process modeling for dynamic simulation, optimization, or control purposes.


The current work proposes the use of surrogate modeling (SM) techniques to map mechanistic data obtained from DEM simulations as a function of processing conditions. To demonstrate the benefit of the SM-PBM approach in developing integrated process models for continuous pharmaceutical manufacturing, a comill-tablet press model integration utilizing the proposed framework is presented.


The SM-PBM approach is in excellent agreement with the DEM-PBM approach to predict particle size distributions (PSDs) and dynamic holdup, with a maximum sum of square errors of 0.0012 for PSD in volume fraction and 0.93 for holdup in grams. In addition, the time taken to run a DEM simulation is in the order of days whereas the proposed hybrid model takes few seconds. The SM-PBM approach also enables comill-tablet press model integration to predict tablet properties such as weight and hardness.


The proposed hybrid framework compares well with a DEM-PBM framework and addresses limitations on computational expense, thus enabling its use in continuous process modeling.


Surrogate model Reduced model Milling Discrete element method Population balance model Model integration 



specific energy, J/kg


force, N


velocity, m/s

M(x, t)

particle mass of size x and at time t, kg


rate of formation of particles, kg/s


rate of depletion of particles, kg/s

\( {\dot{M}}_{in} \)

rate of particles entering the mill, kg/s

\( {\dot{M}}_{out} \)

rate of particles exiting the mill, kg/s


breakage distribution function, dimensionless


daughter particle volume, mm3


parent particle volume, mm3


breakage distribution function parameter, dimensionless


log normal distribution standard deviation, mm3


breakage distribution function parameter for mass conservation, mm6


breakage kernel, 1/s


material strength parameter, kg/J m


size-independent threshold energy, J m/kg


collision frequency per particle per second, 1/s


threshold energy, J/kg


size-dependent parameter to define screen model, dimensionless


critical screen size, μm


critical screen size ratio


screen size, μm


impeller speed, rpm

vimp, min

minimum impeller speed, rpm


critical size ratio coefficient, dimensionless


critical size ratio exponent, dimensionless


time interval, s

\( \widehat{f}\left({x}^i\right) \)

kriging model predictor for a d dimensional input xi

\( \widehat{\mu}\left({x}^i\right) \)

regression term in the kriging model predictor

\( \widehat{\varepsilon}\left({x}^i\right) \)

error term in the kriging model predictor


correlation model in kriging, size mxm


vector of kriging correlation model parameters [θ1, θ2, θ3, θ4]


kriging regression model parameter, constant


kriging model correlation factors, size mx1


kriging model variance parameter


kriging model scaling factors for design sites, size 2 × 3


kriging model scaling factors for responses, size 2 × 1


vector representing untried point, size 3 × 1


scaled vector representing untried point, size 3 × 1

bI →  H

ANN model biases, input to hidden layer, size 10 × 1

wI →  H

ANN model weights, input to hidden layer, size 10 × 3

bH →  O

ANN model bias, hidden to output layer, size 1 × 1

wH →  O

ANN model weight, hidden to output layer, size 10 × 1


maximum likelihood function


sum of square errors of steady state particle size distribution defined by ns bins


sum of square errors of dynamic holdup computed for pre-defined time intervals


granule bulk density, kg/m3


bulk density regression model parameters, i = 0,1,2,3,4


granule true density, kg/m3


relative density of material in the die, kg/m3


critical relative density of material in the die, dimensionless


diameter of the tablet, m


length of the tablet, m


fill depth in the die, m


tablet hardness, N


maximum tablet hardness, N



The authors would like to thank Stephen Cole for insights into DEM model setup. The authors also gratefully acknowledge support from Process Systems Enterprise and EDEM solutions for providing academic licenses.

Funding Information

This work is supported by the US Food and Drug Administration (FDA), through grant 11695471, and a Consortium Agreement between Janssen Pharmaceutica, University of Ghent, and Rutgers University.


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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  • Nirupaplava Metta
    • 1
  • Rohit Ramachandran
    • 1
  • Marianthi Ierapetritou
    • 1
    Email author
  1. 1.Department of Chemical and Biochemical EngineeringRutgers UniversityPiscatawayUSA

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