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 justified. Nevertheless, the power-law flow 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 define dimensionless groups. In complicated flows, 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 flows with heat transfer, by estimating the following dimensionless groups: dimensionless pressure gradient, flow rate and temperature; Reynolds number; Fourier number; Graetz number; Griffith number; Brinkman number; Biot number. 3 Demonstrate understanding of the following analyses, recognise situations in which they can be applied (in a slightly modified way, if necessary) and manipulate them as required: Pressure/flow-rate relationship for isothermal pressure flow of a Newtonian or power-law melt along the axis of a circular, parallel or slowly tapering channel of wide and flat, or circular cross sections, and of a disc cavity. Pressure/flow-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 flow. 4 Demonstrate understanding and make appropriate use of the following terminology: isothermal flow, 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 flow through a channel of length L under isothermal conditions, the analysis and rheological models of §3 lead to relationships between flow rate Q and the pressure drop Δ P of the form Δ P = L f material properties, channel geometry ( ) Q n Examples: 1 Power-law melt flowing 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, flat, wide channel: Q = ± 2 n 1 + 2 n ( ) ± ² 1 n ± 1 ³ 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 flat channels to be dealt with. Flow through any section is treated as locally equivalent to flow through a parallel channel of the same local dimensions. Thus a developing flow can be treated as locally fully developed.
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