Elliptic sectors

ASP is called the true anomaly of the planet. It is found that, in nstronoinieal calculations, the true anomaly is not a very convenient angle with which to deal. Instead we use the mean anomaly, which is defined to be 271 times the ratio of the area of the elliptic sector ASP to the area of the ellipse. Another angle of significance is the eccentric anomaly, u, of the planet defined to be the angle ACQ where Q is the point in which the ordinate through P meets the auxiliary circle of the ellipse. 

Before closing this chapter, we feel that it is useful to list in tabular form some isothermal pressure-flow relationships commonly used in die flow simulations. Tables 12.1 and 12.2 deal with flow relationships for the parallel-plate and circular tube channels using Newtonian (N), Power Law (P), and Ellis (E) model fluids. Table 12.3 covers concentric annular channels using Newtonian and Power Law model fluids. Table 12.4 contains volumetric flow rate-pressure drop (die characteristic) relationships only, which are arrived at by numerical solutions, for Newtonian fluid flow in eccentric annular, elliptical, equilateral, isosceles triangular, semicircular, and circular sector and conical channels. In addition, Q versus AP relationships for rectangular and square channels for Newtonian model fluids are given. Finally, Fig. 12.51 presents shape factors for Newtonian fluids flowing in various common shape channels. The shape factor Mq is based on parallel-plate pressure flow, namely,... 

The symmetry of the pressure distribution in the two halves of the contact separated by the minor axis of the projected ellipse is apparent. Two pressure peaks are found at symmetrical locations on either side of the centre of the contact. The pressure gradients in the vicinity of the maxima are very large but the pressure decreases much more gradually as the elliptical boundary is approached. The angular location of the position of maximum pressure (equation (21)) was found to be in the sector between ( n /A) and (tt/2) radians from the appropriate minor axis. An increase in the radius ratio (a) caused this location angle to increase. 

Fig. 10.7.1. A fixed point when 71 = 2. The sectors between the separatrices are called elliptic.

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