Question 4 of 6: Design Procedure for an Inverted-L Calender (PVC Film)
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format: Open-book, 3 hours; six numbered problems of equal value (20 points each), of which five constitute a complete paper (only the first five in the answer book are marked). All six problems are solved below so the set is complete for study.
Reference texts: Odian, Principles of Polymerization (4th ed., Wiley) — chain-growth & living/anionic kinetics, molecular-weight distributions; Rudin & Choi, The Elements of Polymer Science and Engineering (3rd ed., Academic Press) — dilute-solution rheology, MWD averages, capillary viscometry; Tadmor & Gogos, Principles of Polymer Processing (2nd ed., Wiley) — calendering, injection filling, die flow; Sperling, Introduction to Physical Polymer Science (4th ed., Wiley) — viscoelasticity; Young & Lovell, Introduction to Polymers (3rd ed.) — polyolefin processing; Middleman, Fundamentals of Polymer Processing — power-law tube/runner flow.
Question 4: Design Procedure for an Inverted-L Calender (PVC Film) (20 points)
Given. Film width W = 2 m; gauge 2hf = 0.1 mm; output 1200 kg/hr; rigid-PVC melt density ρ ≈ 1380 kg/m3; inverted-L four-roll stack.
Find. A rational design procedure that fixes line speed, roll diameters, nip gaps and thermal/operating conditions.
Approach. First close the mass balance to get line speed, then use Gaskell calendering theory to relate the final gauge to the minimum nip gap and to size the separating force and power, and finally layer on PVC-specific thermal and finishing constraints.
Fig. 4 — inverted-L four-roll calender: molten PVC is fed to the first nip, the sheet follows the lower rolls with decreasing gap, and the final nip sets gauge.
The design proceeds in five linked stages.
Line speed from the mass balance. The take-off speed is fixed by throughput, gauge and width:$$V=\frac{\dot m}{\rho\,W\,(2h_f)}=\frac{1200/3600}{1380\,(2)(1.0\times10^{-4})}=\boxed{1.21\ \text{m/s}\approx72.5\ \text{m/min}}$$This sets the surface speed of the last roll pair; all upstream rolls run at slightly different speeds to give controlled friction ratios (see step 4).
Roll diameter and face length. The face length must exceed the 2 m film width plus edge trim, so choose Lface ≈ 2.2–2.5 m. Roll diameter is set by deflection: at these forces a slender roll bends and thickens the sheet at the centre, so pick D so that L/D ≈ 2.5–3 (here D ≈ 0.7–0.9 m), then verify centre deflection by beam theory and compensate with roll crowning, roll bending or axis crossing.
Nip gap from Gaskell theory. In the calender model the sheet leaves where the pressure returns to ambient; for a Newtonian-like melt the exit half-thickness relates to the minimum half-gap by hf/h0 ≈ 1.226, so$$2h_0\approx\frac{2h_f}{1.226}=\frac{0.10}{1.226}=\boxed{0.082\ \text{mm}}$$the last nip is set near this value and the preceding nips are progressively larger to build the sheet gradually. (For the shear-thinning PVC the ratio shifts slightly, so the gap is trimmed on the running line.)
Separating force, power and friction ratio. From the calendering pressure profile the roll-separating force per unit width and the power are $F/W\sim\mu V R^{1/2}/h_0^{3/2}$ and $P\sim F V$; evaluate these with the PVC viscosity at the melt temperature to size bearings, roll-bending jacks and drive motors. A small speed differential (friction ratio 1.05–1.4) between adjacent rolls controls which roll the sheet follows and improves melt homogenisation.
Thermal and finishing conditions. Rigid PVC is calendered at roll-surface temperatures of about 160–200 °C with a rising profile from feed to last roll; stabiliser and lubricant packages prevent thermal degradation at the long residence time. Downstream, embossing/cooling rolls and tension control set surface finish and lock in the 0.1 mm gauge. Confirm the design by checking shear heating and the residence time against the PVC degradation window.
