PPT03-notes_probs - ME4 Polymer Processing Technology 3...

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3 Isothermal flow of polymer melts Even in rheometers, our initial assumptions (isothermal, incompressible, time- independent, laminar, fully developed, no wall slip) are not fully justiFed. Nevertheless, the power-law ±ow model can be used to obtain analytical solutions good enough for useful estimates. These notes begin by summarising and extending isothermal solutions. Some other simple situations are then analysed to deFne dimensionless groups. In complicated ±ows, the numerical values of these groups can be calculated to estimate whether an assumption (e.g. isothermal) is realistic or not. Learning outcomes After completing this section you should be able to… 1 Use a power-law representation to calculate shear stresses and heat generated in a pseudoelastic polymer melt. 2 Make broad approximations about the nature of complex mixed ±ows with heat transfer, by estimating the following dimensionless groups: dimensionless pressure gradient, ±ow rate and temperature; Reynolds number; ²ourier number; Graetz number; Gri³th number; Brinkman number; Biot number. 3 Demonstrate understanding of the following analyses, recognise situations in which they can be applied (in a slightly modiFed way, if necessary) and manipulate them as required: Pressure/±ow-rate relationship for isothermal pressure ±ow of a Newtonian or power-law melt along the axis of a circular, parallel or slowly tapering channel of wide and ±at, or circular cross sections, and of a disc cavity. Pressure/±ow-rate relationship for the metering section of a plasticating screw in terms of drag and back-pressure contributions. Temperature rise for adiabatic pressure or drag ±ow. 4 Demonstrate understanding and make appropriate use of the following terminology: isothermal ±ow, jetting, lubrication approximation, shear heating. ME4 Polymer Processing Technology PSL 21 October 2009
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3.1 Steady isothermal flow in channel networks For melt fow through a channel o± length L under isothermal conditions, the analysis and rheological models o± §3 lead to relationships between fow rate Q and the pressure drop Δ P o± the ±orm Δ P = L ± material properties, channel geometry () Q n Examples: 1 Power-law melt fowing through a parallel circular channel. From Problem 2.2: Δ P = LR ± 3 n + 1 2 ² 1 ± ³ 1 n ± 1 1 + 3 n ´ n Q µ · ¸ ¹ º n (3.1) 2 Parallel, fat, wide channel: Q = ± 2 n 1 + 2 n ± 1 n ± 1 1 Δ p L ´ µ · ¸ ¹ 1/ n H 2 º » ¼ ½ ¾ ¿ 1 + 2 n / n W (3.2) Δ P = L ± 3.2 The lubrication approximation This allows non-parallel circular (e.g. sprues) or fat channels to be dealt with. Flow through any section is treated as locally equivalent to fow through a parallel channel o± the same local dimensions. Thus a developing fow can be treated as locally ±ully developed.
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This note was uploaded on 04/07/2010 for the course MECH ENG 207 taught by Professor Levers during the Fall '10 term at Imperial College.

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PPT03-notes_probs - ME4 Polymer Processing Technology 3...

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