In a polymer extrusion process, some cross-sectional shapes of the extruded polymers are shown. Cross-sections of available dies are also shown. Which ONE of the following options CORRECTLY matches the extruded cross section with the die opening that most likely generated it? Note: The cross-sections are in a plane orthogonal to the extrusion direction. Figures are not to scale.
Show Hint
Think about die swell (the Barus effect): flat faces swell outward more than corners. A circular die stays circular; a square profile needs a pin-cushion shaped die to counteract uneven swelling.
Step 1: Understanding the Question:
The figure shows two extruded polymer cross sections, a small solid circle labeled P and a small solid square labeled Q.
Four possible die openings are shown, labeled 1 to 4, and we must pick which die opening produced each extruded shape.
The key physical idea we need is die swell, also called the Barus effect, which happens because polymer melts are viscoelastic.
Step 2: What die swell does to a shape:
When a polymer melt is forced through a die, the molecules get stretched and aligned along the flow direction.
Once the melt leaves the die and the confining walls are gone, the stored elastic energy relaxes and the extrudate swells outward.
This swelling is not always uniform. Near a flat straight edge the shear rate is roughly constant, so that edge bulges outward evenly.
Near a sharp corner the material is restrained from two directions at once, so the corner swells much less than a flat edge does.
Step 3: Matching shape P (circle) to a die:
A circle has no corners at all, every point on its boundary behaves the same way under shear.
So a circular die, shown as die 1 with a plain round hole, swells outward uniformly in every direction.
A uniformly swelling circle stays a circle, just a bigger one, which is exactly the round profile P shown in the figure.
So P must come from die 1.
Step 4: Matching shape Q (square) to a die:
If we used a plain square die (die 2), the middle of each flat side would swell out more than the corners.
That would turn the extrudate into a shape with bulging, barrel like sides instead of a true flat sided square.
Since profile Q in the figure is a clean, straight sided square, the die must already compensate for this uneven swelling.
Die 4 has concave, pin cushion type sides, meaning the die opening is pulled inward at the middle of each face.
When the melt exits die 4, the middles swell out more than the corners, and this extra swelling at the middle straightens the concave sides into flat ones.
The net result is a true square extrudate, so Q must come from die 4.
Step 5: Checking the given options:
Option (A) says P-1, Q-4, which matches our reasoning exactly.
Option (B) says P-1, Q-2, but a plain square die 2 would give a barrel shaped extrudate, not a true square, so this is wrong.
Option (C) says P-3, Q-2, but die 3 has an oval hole which cannot produce a perfect circle P, so this is wrong.
Option (D) says P-3, Q-4, but again die 3 is oval and cannot give the circular profile P, so this is wrong.
Final Answer:
The correct pairing is P with die 1 and Q with die 4.
\[ \boxed{P{-}1,\ Q{-}4\ (\text{Option A})} \]
