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Theoretical Considerations

In the early days, the flow of polymeric materials was not well understood. However, due to the increasing demands on plastic materials and molding machines, it is becoming essential to be able to make reliable quantitative predictions about performance.

Mechanism of Flow in Extruder

The plastic melts as it moves along with the screw. As the screw rotates inside the barrel, the movement of the plastic along the screw is dependent on whether or not it adheres to the screw and barrel. In theory, there exist two extremes:

• The material sticks to the screw only and therefore the screw and material rotate as a solid cylinder inside the barrel- This produces zero output and is undesirable.
• The material slips on the screw and has a high resistance to rotation inside the barrel- This produces a purely axial movement of the melt and is the ideal situation.

In general, the material adheres to both the screw and the barrel.

The useful output from the extruder is the result of:

1. A drag flow due to the interaction of the rotating screw and stationary barrel. This is equivalent to the flow of a viscous liquid between two parallel plates when one plate is stationary and the other is moving.
2. A pressure flow due to the pressure gradient which is built up along the screw. Since the high pressure is at the end of the extruder, the pressure flow will reduce the output.
3. A back flow (called leakage) due to the clearance between the screw and the barrel. This clearance allows material to leak back along the screw and effectively reduces the output. This leakage will be worse when the screw becomes worn.

The thermal processes in the extruder fall somewhere between two ideals:

• Adiabatic process: The system is fully insulated to prevent heat gain or loss from or to the surroundings. This ideal state is reached if the work done on the melt produces just the right amount of heat without the need for heating or cooling.
• Isothermal process: The temperature at all points is the same. This requires immediate heating or cooling from the barrel to compensate for any loss or gain of the heat in the melt.

Analysis of Flow in Extruder

We present a Newtonian analysis of the flow of material during the extrusion process. It is important to note that most of polymer melts are Non-Newtonian, and that the assumption of constant viscosity can lead to serious errors in some cases. The purpose of this simplified discussion is to illustrate the approach to the problem with little mathematical complexity. For further information, we refer the reader to Crawford in his book " Plastics Engineering."

Fig 1. Fluid element ABCD in the flow between a stationary plate and a moving plate

Drag Flow

For the small element of fluid ABCD the volume flow rate dQ is given by:

If the velocity gradient is assumed to be linear, we get:

Substituting in the expression for the volumetric flow and integrating over the channel depth H, we get the total drag flow:

Hence,

We are interested in the fluid flow in the extruder as it is dragged along by the relative movement of the screw and barrel.

Fig 2. Details of extruder screw

The terms in the expression for Qd relevant to the extruder dimensions are:

Vd = DN cos

Where N is the screw speed, and

T = (D tan - e) cos

So

Qd = (1/2).( D tan -e).[DN (cos )2] H

If e is small compared to ( D tan), we get

Qd = (1/2) 2D 2 NHsincos

Pressure Flow

Let P be the pressure and d be the shear stress acting on the fluid element ABCD in fig1. Hence, the forces acting on that element are:

F2 = P. dy dx

F3 = ddz dx

For steady flow, the element is in static equilibrium and the balance of forces gives:

Substituting for F1 ,F2 and F3 we get the following differential equation:

Integrating the above equation to give the shear stress at any distance y from the centerline:

Hence,

For a Newtonian fluid, the shear stress, y , is related to the viscosity, , and the shear rate, , by the equation:

Substituting,

and integrating,

Hence,

Now, the volume flow rate is given by:

Substituting for V and integrating to get the pressure flow, Qp:

For the case shown in fig2, where the fluid element is between the two flights, assuming e is small, T is approximated by:

T = D tan cos

Also,

So,

Finally, the expression for Qp becomes:

Source: Crawford, Plastics Engineering.