Other Die Geometries

For other die geometries it is necessary to use the appropriate form of equation (4.12). The equations for a capillaiy and a slit die are derived in Chapter 5. For other geometries it is possible to use the empirical equation which was developed by Boussinesq. This has the form 

Using equation (4.14) it is possible to modify the expression for the operating pressure to the more general form 

Example 4.1 A single screw extruder is to be designed with the following characteristics. 

screw speed = 100 rev/min, screw diameter — 40 mm flight depth (metering zone) = 3 mm. 

If the extruder is to be used to process polymer melts with a maximum melt viscosity of 500 Ns/m, calculate a suitable wall thickness for the extruder barrel based on the von Mises yield criterion. The tensile yield stress for the barrel metal is 925 MN/m and a factor of safety of 2.5 should be used. 

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For the other die geometry, that of a annular space between two cylinders shown in Figure 13.24, we have the foDowing momentum balance ... 

To illustrate one possible way of translating capillary die swell measurements to some other die geometry we consider the swell of extrudate leaving an annular die. In the swell of polymer extruded from an annular die as shown back in Figure 3.1 (this figure is associated with Design Problem II), there is swell of the diameter as well as the thickness of the extrudate. The two most eommon swell parameters are the diameter swell, Bi, and the thiekness swell, B2, defined, respectively, as... 

The results presented here illustrate the complexity in trying to extend die swell measurements from a capillary to other die geometries. As an initial approximation one can use the relations between Bi, B2, and B given in Eqs. 7.13, 7.14, and 7.15. However, one must be aware of the fact that when significant strain hardening arises in the extensional behavior, the data will deviate more dramatically from these... 

Die mathematical significance of rh has been established for a number of other pore geometries (Everett, 1958), but with most real systems it is not possible to arrive at an unambiguous evaluation of f p or ivp or a usefiil interpretation of rK. For example, with an assemblage of packed spheres, as already noted, the porosity is dependent on the packing density as well as the particle size. Similarly, in the case of a network of intersecting pores, the value of rh is dependent on both the pore radius and the lattice spacing of the intersections. 

The triethylaluminum or triethylborane ate complexes (12) of the (isopropylthio)allyl carbanitm react with carbonyl compounds at the a-position (equation 10). In the reactions with carbonyl compounds, very high regioselectivity (for example, butanal 93 5, 3-methylbutanal 99 1, cyclohexanone 92 8 and acetophenone 95 5) was achieved by using the aluminum ate complex. On the other hand, the a-tegio-selectivity with ketones decreases if die boron ate complex is used (cyclohexanone 72 28, acetophenone 45 55). It is noteworthy that the stereoselectivity of the a-adduct fitmi an aldehyde is low. Presumably die geometry of the double bond in the ate complex (12) is not homogeneous. ... 

Other types of instabilities may exist, for example, a problem has been observed in feedblock coextrusion of axis5mimetric sheet (27). A wavy interface is also characteristic of this instability, but the wave pattern is more regular when viewed from the surface. The instability originates in the die, well ahead of the die land, and internal die geometry influences both the severity and pattern. For a given die geometry, the severity of instability increases with structure asymmetry and some polymers are more susceptible to unstable flow than others. 

Left to right The binding of glucose molecule (red) to hexokinase (an enzyme in the metabolic pathway). Note how the region at the active site closes around glucose after binding. Frequently, die geometries of both the substrate and the active site are altered to fit each other. 

There are two other points worth noting about this test. Firstly the flow data is produced using a capillary die so that its use on channels of a different geometry would require a correction factor. However, in most cases of practical interest, the factor is not significantly different from 1 and so there is no justification for the additional complication caused by its inclusion. 

Flow through a convergent channel, either two-dimensional or axisymmetric, are probably the most investigated geometries (due to their connection with the technical die entry flow problem). Most of the other convergent flows could be derived from the abrupt contraction flow which is depicted schematically in Fig. 25. 

It is often of interest to calculate force constants from observed vibrational frequencies. However, it is not generally possible to derive analytical expressions for the force constants as functions of the frequencies and the molecular geometry. The calculation is necessarily an iterative one. Starting with a set of assumed force constants - usually obtained by analogy with similar bonds in other molecules - the values are refined until a suitable set is found. The set that yields the best agreement between calculated and observed frequencies constitutes the accepted force field for die molecule. 

The selection rules for the QM harmonic oscillator pennit transitions only for An = 1 (see Section 14.5). As Eq. (9.47) indicates diat the energy separation between any two adjacent levels is always hm, the predicted frequency for die = 0 to n = 1 absorption (or indeed any allowed absorption) is simply v = o). So, in order to predict die stretching frequency within the harmonic oscillator equation, all diat is needed is the second derivative of the energy with respect to bond stretching computed at die equilibrium geometry, i.e., k. The importance of k has led to considerable effort to derive analytical expressions for second derivatives, and they are now available for HF, MP2, DFT, QCISD, CCSD, MCSCF and select other levels of theory, although they can be quite expensive at some of the more highly correlated levels of theoiy. 

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