Solution to Design Problem

AP/L is replaced by -dp/dz and R by Eq. 2.108. This gives a first order differential equation for finding p  

This approximation is probably adequate for tapers of less than 30°, but for more abrupt contractions the viscoelastic nature of polymeric fluids may make the pressure drop higher (see Section 7.2). However, for the most part the lubrication approximation will be extremely useful in the design of extrusion processes (i.e., the design of extrusion dies and screw design). 

This would lead to the following form of the equation of 

If one evaluated dx jdz relative to the other term, we would And that for small amounts of taper this term would be small compared to the other term. One would follow the same analysis as in the previous example to show the derivative is small. Hence, one neglects 9%/9z and solves the following equation  

This is just the equation that would be solved for a straight tube. Hence, the solution to this equation is 

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In product development cycles, stakeholders constantly deliberate about alternative solutions to design problems. This observation stresses the importance of decision-making processes for the overall effectiveness and efficiency. Requirement specifications serve as a reference forjudging the available decision alternatives. As such, they have a large impact on the course of development cycles. 

In conclusion, the future of pultrusion is very promising as one of the most cost-effective continuous composites manufacturing methods available to produce composite solutions to design problems (Sumerak and Martin, 1987). Pultrusion will not only remain a primary production process for secondary applications but also will flourish with the development of more and more novel products. 

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The solution to Design Problem I is presented in this section. The lubrication approximation is used first to obtain a solution. This is followed by a numerical approach in which the die is broken into a series of annuli. 

In this section the analogy between heat and mass transfer is introduced and used to solve problems. The specific estimation relationships for permeants in polymers are discussed in Section 4.2 with the emphasis placed on gas-polymer systems. This section provides the necessary formulas for a first approximation of the diffusivity, solubility, and permeability, and their dependence on temperature. Non-Fickian transport, which is frequently present in high activity permeants in glassy polymers, is introduced in Section 4.3. Convective mass transfer coefficients are discussed in Section 4.4, and the analogies between mass and heat transfer are used to solve problems involving convective mass transfer. Finally, in Section 4.5 the solution to Design Problem III is presented. 

FIGURE 7.2 Exit face of the die used to produce the parison in Figure 7.1. The die opening is an elliptically shaped annulus of major axis a and minor axis b. The thickness, a, and b are determined in the solution to Design Problem VI. 

Section 7.6. Finally, in Section 7.7, the solution to Design Problem VI is presented. 

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