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Modelling non-homogeneous flow and residence time distribution in a single-screw extruder by means of Markov chains

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Abstract

Non-homogeneous velocity distribution of the flow in the channel of a single-screw extruder is taken into account by a new model developed on the basis of the Markov chains. This model allows calculating the Residence Time Distribution (RTD) as well as the influence of the operating conditions on the process at any velocity distribution in the channel. It has been used to represent experimental results on mass flow rate and RTD previously obtained by extrusion of an acrylic polymer, Eudragit E100, at different temperatures and screw rotation speeds. The diffusion coefficient is the only adjusting parameter of the model. It was shown that it does not depend on the screw rotation speed and a correlation between this diffusion coefficient and the barrel temperature was found. The model provides global understanding of the transport kinetics of the flowing material through the extruder according to its behaviour and better describes the progress of the polymer flow all along the barrel from the hopper to the die.

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Abbreviations

a :

Coefficient of the homographic equation (m/s)

b :

Coefficient of the homographic equation (m2/s)

B :

Distance between flanks (m)

d :

Diameter of the screw (m)

d z :

Transition probability along the axis z (–)

d y :

Transition probability along the axis y (–)

D :

Diffusion coefficient (m 2/s)

D b :

Internal diameter of the barrel (m)

e :

Width of the screw flank (m)

E(i):

Calculated residence time distribution (–)

E exp(t):

Experimental residence time distribution (–)

g :

Coefficient of the homographic equation (m)

g z :

The down channel pressure gradient (Pa/m)

H :

Cross channel depth (m)

j :

The transition number (–)

L :

Length of the screw shaft (m)

L f :

Length of the die (m)

L s :

Length of the channel (m)

L t :

Total path length (m)

m :

Row number (–)

M :

Mass of material in the extruder (kg)

n :

Column number (–)

P :

Matrix of transition probabilities (–)

P ik :

Matrix of transition probabilities from kth to ith columns

Q :

Mass flow rate (kg/s)

r :

Throttle ratio (–)

S :

State vector (–)

t :

Time at which the calculated results are obtained (s)

T :

Temperature of the barrel (°C)

v :

Transition probability describing convection (–)

V z :

Average velocity along the axis z (m/s)

V ex :

Free volume of the extruder (m 3)

W :

Channel width (m)

Z :

Zero matrix (–)

Δt :

Time interval (s)

Δt exp :

Experimental time of sampling (s)

Δy :

Height of a cell (m)

Δz :

Length of a cell (m)

ρ :

Average density of the material (kg/s)

μ :

Viscosity (Pa s)

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Authors and Affiliations

  1. Ecole des Mines d’Albi, RAPSODEE Research Centre, CNRS, Université de Toulouse, 81013, Albi, France

    Denis Ponomarev, Elisabeth Rodier, Martial Sauceau, Clémence Nikitine & Jacques Fages

  2. CPE-LGPC, UMR 2214, Université de Lyon, 69100, Villeurbanne, France

    Clémence Nikitine

  3. Department of Applied Mathematics, Ivanovo State Power Engineering University, Rabfakovskaya 34, 153003, Ivanovo, Russia

    Denis Ponomarev & Vadim Mizonov

Authors
  1. Denis Ponomarev
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  2. Elisabeth Rodier
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  3. Martial Sauceau
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  4. Clémence Nikitine
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  5. Vadim Mizonov
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  6. Jacques Fages
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Corresponding author

Correspondence to Jacques Fages.

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Ponomarev, D., Rodier, E., Sauceau, M. et al. Modelling non-homogeneous flow and residence time distribution in a single-screw extruder by means of Markov chains. J Math Chem 50, 2141–2154 (2012). https://doi.org/10.1007/s10910-012-0022-x

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  • DOI: https://doi.org/10.1007/s10910-012-0022-x

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