Conic section

This equation can be solved for y to give an equation for the line 

With the identification b = y — mx, this takes the standard form of Eq. (5.6). 

A right circular cone can be represented by the three-dimensional equation 

FIGURE 5.3 Mathematical definition of a cone and one possible physical realization. Each of the two sheets on opposite sides of the vertex is called a nappe. 

FIGURE 5.4 Intersections of the cone and plane showing how the different conic sections are generated. 

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Figure 4 Ferritic probe with a conic section.

A BrasiHan company has reportedly produced ammonium chloride from hydrogen chloride gas (16). Hydrogen chloride is mixed with air and iatroduced iato a saturated ammonium chloride suspension at 80 °C. Excess ammonia is added to a conical section of the saturator to maintain a pH of 8. The ammonium chloride is recovered from the suspension by thickening ia a hydroclone, foUowed by centrifugation and dryiag. Mother Hquor and the water used to scmb waste gases, are returned to the saturator. 

In order to pilot this process, SNPE has developed an annular ceU having a cone-shaped electrode section. The magnesium anode is placed into the conic section and fed like a pencil into a sharpener as it is consumed during electrolysis. The carbon dioxide is introduced into the electrolyte externally to the cell. 

Conic Sections The cui ves included in this group are obtained from plane sections of the cone. They include the circle, ehipse, parabola, hyperbola, and degeneratively the point and straight line. A conic is the locus of a point whose distance from a fixed point called the focus is in a constant ratio to its distance from a fixea line, called the directrix. This ratio is the eccentricity e. lie = 0, the conic is a circle if 0 < e < 1, the conic is an ellipse e = 1, the conic is a parabola ... 

Venturi Meters The standard Herschel-type venturi meter consists of a short length of straight tubing connected at either end to the pipe line by conic sections (see Fig. 10-15). Recommended proportions (ASME PTC, op. cit., p. 17) are entrance cone angle Oti = 21 2°, exit cone angle Cto = 5 to 15°, throat length = one throat diameter, and upstream tap located 0.25 to 0.5 pipe diameter upstream of the entrance cone. The straight and conical sections should be joined by smooth cui ved surfaces for best results. 

A final note is with regard to flat bottom cyclones which have gained acceptance in recent years. The installation of a flat bottom in place of the conical section will coarsen the separation by more than twofold. Additionally, the sharpness of the recovery curve will decrease significantly. As such, flat bottom cyclones should be restricted to those applications in which coarse separations are required. The flat bottom cyclone does produce a very clean underflow but at the expense of a large amount of misplace coarse solids in the overflow. An illustration of a flat bottom cyclone is shown in Figure 58. The reader should refer to the reference section of this chapter for citations that provide more in-depth coverage of this equipment, as well as design case studies and example. 

Determine the weight for the following free-water knockout. It is butt weld fabricated with spot x-ray and to be built to Division 1. A conical head (bottom of the vessel) is desired for ease in sand removal. Compare this weight to that of a vessel without the conical section and that to ti ves.sel with a 4-m. plate internal cone. 

The molded liners for the inlet head, cylinder section, and conical sections have Integral molded gaskets for sealing at the flanged joints. Molded liners and vulcanized linings are offered in gum rubber, polyurethane, nitrile rubber, butyl. Neoprene , Viton , Hypalon , and other liner materials can be supplied. 

In the conics section we discussed the property of parabolas. Since one of its conjugates is at infinity, it will form a perfect image from the infinite conjugate at focus. This is the basis for a single mirror telescope. This telescope is either used at this prime focus, or sometimes a folding flat mirror (Newtonian telescope) is used to fold the beam to a convenient location for a camera or the... 

The flow in the die hole is predominantly in the axial direction with the axial velocity, Vz, a function of both r and z position. The radial component of velocity, Vr, is significant only in the conical section of the die hole. However, Vj- is about two orders of magnitude smaller compared with the axial velocity, vz. Therefore, Vj- is estimated by forcing the continuity equation to be... 

A shell of revolution is the form swept out by a line or curve rotated about an axis. (A solid of revolution is formed by rotating an area about an axis.) Most process vessels are made up from shells of revolution cylindrical and conical sections and hemispherical, ellipsoidal and torispherical heads Figure 13.3. 

Conical sections (reducers) are used to make a gradual reduction in diameter from one cylindrical section to another of smaller diameter. 

A formed section would normally be used for the transition between a cylindrical section and conical section except for vessels operating at low pressures, or under hydrostatic pressure only. The transition section would be made thicker than the conical or cylindrical section and formed with a knuckle radius to reduce the stress concentration at the transition, Figure 13.11. The thickness at the knuckle can be calculated using equation 13.46, and that for the conical section away from the transition from equation 13.45. 

Design procedures for conical sections are given in the codes and standards. 

Consider a fluid flowing in a conical section, as illustrated in Fig. 5-P75. The mass flow rate is the same going in (through point 1) as it is coming out (point 2), but the velocity changes because the area changes. They are related by... 

At one end of the bowl a conical section is attached giving a smaller diameter. The liquid runs round the helical scroll and is discharged over weir plates fitted at the parallel end of the bowl. The solids are moved by the conveying action of the helical scroll up the gentle slope of the conical section, out of the liquid and finally out of the machine. Decanters of this type are known as solid bowl decanters. 

A variant of the decanting centrifuge is the screen bowl decanter, shown in Figures 9.13 and 9.14. In this unit a further perforated section is attached to the smaller diameter end of the conical section. This is known as the screen and allows further drying and/or washing of the solids to take place. 

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