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Polymers & Plastics Processing formula solvers

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ABEIMPRSW

Apparent Wall Shear Rate

γ˙app=4QπR3\dot{\gamma}_{app} = \frac{4Q}{\pi R^{3}}

Polymers & Plastics ProcessingThe shear rate at the wall of a round channel, worked as if the fluid were Newtonian. It is the x-axis of every capillary-rheometer flow curve and the first number to reach for when asking whether a gate is shearing a melt too hard — and it is called APPARENT because for a real polymer melt it is not the true wall shear rate until Rabinowitsch has corrected it.

Blow-Up Ratio and Blown Film Gauge

BUR=DbDdt=hdBUR⋅DDR=hd DdDb DDR\mathrm{BUR} = \frac{D_b}{D_d} \qquad t = \frac{h_d}{\mathrm{BUR} \cdot \mathrm{DDR}} = \frac{h_d\, D_d}{D_b\, \mathrm{DDR}}

Polymers & Plastics ProcessingA blown-film bubble is a mass balance you can watch. The melt leaves an annular die of gap h_d and diameter D_d, the internal air inflates it to a bubble diameter D_b — that ratio is the blow-up ratio — and the nip pulls it up faster than it left, which is the draw-down ratio. Whatever the two stretches multiply to, the gauge divides by.

Extruder Net Output

Q=Qd−QpQ = Q_d - Q_p

Polymers & Plastics ProcessingWhat actually leaves the die: the drag the barrel puts into the melt, less what the die pressure pushes back. Plotted against head pressure it is the SCREW CHARACTERISTIC — a falling straight line — and where it crosses the die's own rising line is the operating point the machine settles at.

Injection Moulding Clamp Force

F=pcav AprojF = p_{cav} \, A_{proj}

Polymers & Plastics ProcessingPressure inside the cavity acting on the area the parting line sees, which is the force trying to blow the mould open. It decides which press the job goes on, and it is the single most common reason a tool flashes: the cavity pressure was underestimated, or the projected area forgot the runner.

Injection Moulding Cooling Time

t=h2π2α ln⁡ ⁣[4π⋅Tm−TwTe−Tw]t = \frac{h^{2}}{\pi^{2}\alpha}\, \ln\!\left[\frac{4}{\pi}\cdot\frac{T_m - T_w}{T_e - T_w}\right]

Polymers & Plastics ProcessingThe one-term solution to transient conduction in a slab cooled from both faces, written the way a moulder needs it: how long the part must sit in the tool before it is stiff enough to eject. The wall thickness is SQUARED, which is why it dominates every other term on the page and why halving the wall quarters the cooling time.

Mould Shrinkage

S=Lmould−LpartLmouldS = \frac{L_{mould} - L_{part}}{L_{mould}}

Polymers & Plastics ProcessingThe part comes out smaller than the steel it was made in, and this is by how much — expressed as a fraction of the MOULD dimension, which is the convention the tool is cut to. Getting the reference wrong, or the axis wrong, is how a first-off comes back out of tolerance on a tool that was cut exactly to the drawing.

Power-Law Viscosity Ratio

ηηref=(γ˙γ˙ref) n−1\frac{\eta}{\eta_{ref}} = \left( \frac{\dot{\gamma}}{\dot{\gamma}_{ref}} \right)^{\,n-1}

Polymers & Plastics ProcessingThe Ostwald–de Waele power law, written as a ratio between two points on the same flow curve instead of against a consistency index. Give it one measured viscosity at one measured shear rate and the index n, and it gives the viscosity anywhere else on the straight part of the curve. Written this way every term has an honest unit — which the textbook form cannot manage.

Rabinowitsch Shear-Rate Correction

γ˙true=γ˙app 3n+14n\dot{\gamma}_{true} = \dot{\gamma}_{app}\, \frac{3n + 1}{4n}

Polymers & Plastics ProcessingRabinowitsch's 1929 correction turns the apparent wall shear rate — the one worked as if the melt were Newtonian — into the true one. A shear-thinning fluid has a blunter velocity profile than a Newtonian fluid, so the real velocity gradient at the wall is steeper than 4Q/πR³ admits, and the factor (3n+1)/4n is exactly how much steeper.

Single-Screw Drag Flow

Qd=π2D2NHsin⁡φcos⁡φ2Q_d = \frac{\pi^{2} D^{2} N H \sin\varphi \cos\varphi}{2}

Polymers & Plastics ProcessingRowell and Finlayson's 1928 result, unchanged since: the melt in a screw channel is dragged forward by the barrel moving over it, exactly as fluid is dragged between a moving plate and a fixed one. This is the forward half of a single-screw extruder's output, and it is proportional to screw speed and to channel depth — a straight line through the origin on an output-against-rpm plot.

Single-Screw Pressure Flow

Qp=πDH3ΔPsin⁡2φ12ηLQ_p = \frac{\pi D H^{3} \Delta P \sin^{2}\varphi}{12 \eta L}

Polymers & Plastics ProcessingThe die at the end of an extruder builds pressure, and that pressure drives melt BACK up the screw channel against the drag. This is the same Poiseuille flow that runs a pipe, written for a shallow rectangular channel wrapped round a screw — and it is why an extruder's output falls as the die is made more restrictive.

Wall Shear Stress in a Capillary

τw=ΔP R2L\tau_w = \frac{\Delta P \, R}{2L}

Polymers & Plastics ProcessingA force balance and nothing else: the pressure pushing melt down a round channel has to be carried by shear at the wall, and this is what that shear works out to. Paired with the apparent shear rate it gives one point on a flow curve, and the slope of that curve is the power-law index.