Modeling and Control of Roller Compaction for Pharmaceutical Manufacturing. Part I: Process Dynamics and Control Framework

Abstract

We derive a dynamic model for roller compaction process based on Johanson’s rolling theory, which is used to predict the stress and density profiles during the compaction and the material balance equation which describes the roll gap change. The proposed model considers the relationship between the input parameters (roll pressure, roll speed, and feed speed) and output parameters (ribbon density and thickness), so it becomes possible to design, optimize, and control the process using the model-based approach. Currently, the operating conditions are mostly found by trial and error. The simulation case studies show the model can predict the ribbon density and gap width while varying roll pressure, feed speed, and roll speed. The roll pressure influences the ribbon density much more than roll speed and feed speed, and the roll gap is affected by all three input parameters. Both output variables are very insensitive to the fluctuation of inlet bulk density. If the ratio of feed speed to roll speed is kept constant, neither ribbon density nor gap width change, but the production rate changes proportionally with feed speed. Based on observations from simulations, a control scheme is proposed. Furthermore, Quality by Design of the roller compactor can be achieved by combining this model and optimization procedure.

Glossary

A

Compact surface area

B

$\frac{1}{2}{\left( {\frac{\pi }{2} + \theta + \nu } \right)}$, a function of θ

C ₁

Pre-exponential coefficient in the model of material compression

dV

Differential volume

F

Roll-separating force per unit roll width

h ₀

Half of roll gap width

h_min, h_max

Minimum and maximum allowable values of h ₀

K

Compressibility factor

P _d

Hydraulic pressure (set point)

P _h

Hydraulic pressure (actual value)

P_min, P_max

Minimum and maximum allowable values of P _h

${\mathop m\limits^ \cdot }$

Production rate

N(a, b)

Normal distribution with mean a and variance b

R

Radius of the rollers

s

Complex argument in Laplace domain

s _in

standard deviation of inlet powder density

t

time

u _d

Feed speed (set point)

u _in

Feed speed (actual value)

u_min, u_max

Minimum and maximum allowable values of u _in

W

Roll width

x _in

Rsin θ_in

α

Nip angle

Δ

Effective angle of friction

θ

Angular position

θ _in

π/2 − ν

μ

$\frac{\pi }{4} - \frac{\delta }{2}$

v

$\frac{1}{2}{\left( {\pi - \sin ^{{ - 1}} \frac{{\sin \phi }}{{\sin \delta }} - \phi } \right)}$

ρ

Compact density

ρ _exit

Compact density at exit point

$\rho ^{*}_{{{\text{exit}}}}$

Desired ribbon density

ρ _in

Inlet powder density

$\overline{\rho } _{{{\text{in}}}}$

Average of inlet powder density

σ

Material stress, a function of θ

σ _exit

σ(θ = 0), material stress at exit point

τ _p

Time constant of the roll force response due to the change of hydraulic pressure

τ _u

Time constant of the feed speed response

τ _ω

Time constant of the roll speed response

Φ

Cost function

ϕ

Angle of wall friction

ω

Angular velocity of the rolls (actual value)

ω _d

Angular velocity of the rolls (set point)

ω_min, ω_max

Minimum and maximum allowable values of roll speed