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

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

Abstract

Purpose

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.

Methods

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.

Results

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.

Conclusions

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.

Keywords

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

Nomenclature

E

specific energy, J/kg

F

force, N

v

velocity, m/s

M(x, t)

particle mass of size x and at time t, kg

form

rate of formation of particles, kg/s

dep

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

b

breakage distribution function, dimensionless

u

daughter particle volume, mm3

w

parent particle volume, mm3

n

breakage distribution function parameter, dimensionless

σ

log normal distribution standard deviation, mm3

C

breakage distribution function parameter for mass conservation, mm6

K

breakage kernel, 1/s

fmat

material strength parameter, kg/J m

Econst

size-independent threshold energy, J m/kg

fcoll

collision frequency per particle per second, 1/s

Emin

threshold energy, J/kg

fd

size-dependent parameter to define screen model, dimensionless

Δ

critical screen size, μm

δ

critical screen size ratio

dscreen

screen size, μm

vimp

impeller speed, rpm

vimp, min

minimum impeller speed, rpm

ε

critical size ratio coefficient, dimensionless

α

critical size ratio exponent, dimensionless

Δt

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

R

correlation model in kriging, size mxm

θ

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

β

kriging regression model parameter, constant

γ

kriging model correlation factors, size mx1

σkrig

kriging model variance parameter

Sx

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

Sy

kriging model scaling factors for responses, size 2 × 1

xnew

vector representing untried point, size 3 × 1

xscale,new

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

L

maximum likelihood function

SSEPSD

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

SSEholdup

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

ρbulk

granule bulk density, kg/m3

ai

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

ρtrue

granule true density, kg/m3

ρr

relative density of material in the die, kg/m3

ρr,cr

critical relative density of material in the die, dimensionless

Dtablet

diameter of the tablet, m

Ltablet

length of the tablet, m

FD

fill depth in the die, m

H

tablet hardness, N

Hmax

maximum tablet hardness, N

Notes

Acknowledgements

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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