Die designs are often verified in lab trials and sometimes involve simulation studies. In addition, we like to utilize outside consultants to keep up with and be introduced to other concepts/approaches. Please enjoy this white paper contribution from one of more trusted sources, IPlastic’s Randy Brown.
The Science of pressure flow of fluids through pipes and other non-round conduits has been around for a long time. Hagen and Poiseuille described the basic equation for flow of Newtonian fluids through a pipe back in 1838.[1] This basic equation given below describes the pressure needed to push any fluid in laminar flow through any enclosed channel as:
Where Loss in energy forcing the fluid to flow through the length L of pipe described as pressure drop
π = pi
μ = dynamic viscosity
L = length of flow
Q = volumetric flow rate
A = cross-sectional area of the channel
The pressure needed to push the fluid through a full length L of pipe of constant radius R is given by:
Of course any of the variables can be calculated if all of the other variables are known. As written, the pressure needed to pump a fluid of known viscosity through a known radius of pipe of length L and a given output rate can be calculated. However, the viscosity of any fluid can be calculated if the force (or pressure) is known to push a given flow rate through a pipe of known radius and length.
The next important point is that the pressure needed to push the fluid through different sections of different lengths or cross sectional areas is additive. The pressure and resulting flow can be calculated through a very complex die by breaking it down into simple sections and adding the pressure drop through each simple section.
Unlike simple fluids like water, many fluids of interest like polymer melts and food stuffs have complex viscosities that vary with shear rate and temperature. The equation requires a constant viscosity of fluid for the section of flow, however, if the viscosity is known or can be calculated for each of the many different sections of the flow channel, the pressure drop for the flow path through each simple section can be calculated and summed up for all sections making up the flow channel.
Thus for non-Newtonian fluids like polymer melts or food stuffs the equation can still be used if the complex geometry can be broken down into simple, calculatable sections. This leads to a finite elemental analysis of the flow. However, it has been demonstrated by the author that using the cells of a spreadsheet as individual elements, an estimate of the flow of almost any desired accuracy can be calculated.
The viscosity of most polymer melts depends on the shear rate during flow. This can be handled quite simply by breaking the equation into two parts. First the Shear Rate γ is calculated from the equation:
γ = C1Q/A
where C1 is a constant dependent on the cross sectional area of the channel.
Then the viscosity of the polymer melt at that Shear Rate (and Temperature) can be obtained from experimental data or a descriptive equation. The Pressure Drop can then be calculated from the following equation:
= C2γμL/R
where C2 and R will vary dependent on the cross sectional shape of the channel.
This is the basis for most die design programs that are currently on the market, however, they often go into more complex mathematical models that include Finite Element Analysis to calculate pressure drop and flow with improved accuracy and greater versatility.
The original description of this methodology was presented at the SPE ANTEC in 1979[2]. And a simplified version for profile die design was published in 1981[3]
Probably the best example of summing up the pressure through different sections in a die to balance the flow rate across the die is given by that of a sheet die. The critical requirement is to spread the polymer or other fluid sideways for the desired width of sheet which typically would be 5 or 6 feet, but could be as much as 20 feet. This is most often done with a manifold type design where a large opening running roughly perpendicular to the desired flow direction allows the fluid to take the easier path (lower pressure drop) down the manifold channel to distribute the flow across the die. The die is balanced by use of a pre-land that gets shorter moving away from the center (entrance) of the die to compensate for the pressure drop going down the manifold. A secondary manifold is often used to allow some side flow to correct any imbalance in the flow channel. Then a uniform land length of constant length and gap opening is used to define the final flow and surface appearance. As shown below:
The extruder builds pressure to force the fluid through the die, starting with the entrance, and then splitting in equal parts through the manifold toward each end of the die. As material leaves the manifold and flows over the Pre-Land toward the die exit, the flow rate of material moving down the manifold decreases so the cross-sectional area of the manifold typically decreases to compensate. The goal is to have the pressure drop be the same for any path that the fluid takes so that the flow rate is the same for each location across the die width. This is balanced by adjusting the Pre-Land length and gap to yield a pressure drop complementary to the pressure drop through the manifold length taken to reach that position of the Pre-Land. This assures an equal pressure drop and therefore, flow rate for any path that the fluid takes.
More detail and other examples are given in Design of Extrusion Dies by Kostic and Reifschneide.[4] Question? Should we add links to other die design programs?
Current sheet die manufacturers and all of the die design programs will balance the flow rate across the die face. That is their purpose and they do a very good job. However, most do not take into consideration the residence time spent in the die. For some polymers, like PVC, and most food stuffs that are temperature sensitive, long residence time can cause degradation of the material. In a wide sheet die using the above equations, balancing the flow can be relatively straightforward, but it takes a long time to flow perpendicular down the manifold to get to the end of the die. Without consideration for the dwell time the ends of the die can stagnate and burn or cause varying degrees of cure or degradation. In addition, the long residence time for the edges of the sheet will lead to long change over times for color changes or changes in material.
Therefore it is recommended to take a look at a Novel Sheet Die Design Technology that also takes into account the dwell time across the die. This technology is Excel based and utilizes the simple Rheological equations covered above. It allows for wide variations in geometry of flow channels and viscosity that is shear rate and temperature dependent. In addition to balancing the flow across the die, it also allows balancing the residence time across the die to eliminate degradation at the ends of the die and faster change-over times.
Both EDI and Cloeren’s design promoted more flow to the ends of the die in order to help prevent edge burning of the PVC. As a result the die temperatures on the ends of the dies had to be reduced to slow down the flow of material at the edges. This causes an uneven heat history of the material across the die. The Novel design has a much more even flow rate across the die and so the die can be set at a uniform temperature across the die. The flow rate varies by about 30% between the center of the die and the edges for the other dies, while the Novel die was balanced to within 4% and could be balanced closer if desired.
But even more dramatic is the difference in dwell time between these commercial dies and the Novel design approach:
In these examples, the dwell time at the end of the Cloeren die is 4 times the high dwell time in the center of the die. The dwell time at the end of the EDI die is 5 times the dwell time at the center. Whereas, the Novel die has a much more even dwell time of only 1.4 times the dwell time for this 5 foot wide sheet die.
The advantages of this Novel approach should be apparent especially for temperature sensitive materials. As long as the viscosity of the material to be extruded can be characterized with respect to temperature and shear rate, the die can be optimized for that material.
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