Mathematic modeling of polymer extrusion process

 Q-3. Briefly explain the mathematical modelling of any one of the manufacturing processes.

Ans: 

Mathematical model of polymer extrusion process:

Polymer material consists of long chain molecules. In extrusion of thermoplastic polymers, important parameters are

• Material flow and type of flow

• Heat transfer in flow

• Residence time

• Mixing of particles in flow in multiple polymers, etc.

To analyse these parameters, mathematical models are useful. Using governing equations and boundary conditions (B.C), a model is created and a solution is obtained for particular B.C.

Governing equations: 

To model any flow and thermal transport in the manufacturing process, conservation of mass, conservation of  energy and the force momentum balance equation is used. The equations are as follow,

Where, ρ is density,  t is time, T is temperature, V is the velocity vector, μ is dynamic viscosity, F is body force, p is pressure, Cp is specific heat at constant pressure, β is coefficient of volumetric thermal expansion and μΦis viscous dissipation.

Mathematical model to calculate residence time for extrusion process: 

The residence time means time spent by fluid particles (melted polymer particles) in the extruder part from hopper to die. Using more or less residence time material can be over or under processed. The characteristic of the product depends upon the residence time distribution in process, because change in polymer structure and chemical reaction are time dependent phenomena. 

To mathematically obtain residence time, a finite amount of material flow simulation is used. Flow is simulated between the hopper to die and time is calculated. Solving governing equations leads to velocity fields. Axial velocity is found by solving the momentum equation in axial direction. 

The flow simulation is shown in figure-(a). Particles near the barrel and screw take longer time to come out and particles near the extruder axis take least time to come out. 

Dye flow rate is shown in figure-(b). In extrusion, in a shorter interval of time, most of the die emerges. Extended region represents material near the screw root and barrel.

The function F(t) indicates the cumulative fraction of the total amount of dye emerging up to time t.

Here, f(t) dt is the amount of material that has a residence time within time t and t+dt.

Te is average residence time.

Te= VeQe

where, Ve= total volume of extruder, Qe= volume flow rate. 

In figure-(c), F(t) is plotted with the distributions for a few common flows. The residence time distribution (RTD) is affected by the nature of flow and fluid. RTD is slightly affected by barrel temperature.