## 274 Geometry

The main elements of a cooling tower shell in the form of a hyperboloid of revolution are shown in Figure 27.10. This form falls into the class of structures known as thin shells. The cross-section as shown depicts the ideal profile of a shell generated by rotating the hyperboloid R—f (Z) about the vertical (Z) axis. The coordinate Z is measured from the throat while z is measured from the base. All dimensions in the R-Z plane are specified on a reference surface, theoretically the middle surface of the shell but possibly the inner or outer surface. Dimensions through the thickness h(Z) are then referred to this surface. There are several variations possible on this idealized geometry such as a cone-toroid with an upper and lower cone connected by a toroidal segment, two hyperboloids with different curves meeting at the throat, and an offset of the curve describing the shell wall from the axis of rotation. Classically, the columns are arranged as V- or X-frameworks, the latter for very high air inlets; vertical (meridional) columns now popular and were used for the world's highest tower shown later in Figure 27.17.

FIGURE 27.10 Hyperbolic cooling tower.

Referring to Figure 27.10, the equation of the generating curve is given by

where dT is the diameter at the throat and b is a characteristic dimension of the shell that may be evaluated from the diameters dH and dU by b = ¿tZh/^(dH - dT) (27.2)

if the upper and lower curves are different. The dimension b is related to the slope of the asymptote of the generating hyperbola (see Figure 27.10) by b = 2dr/c (27.4)

For a cooling tower shell where the axis of the generating hyperboloid is offset from the axis of the cooling tower shell middle surface by a constant radial distance Ro, Equations 27.1-27.4 are modified slightly. However, the former equations may be used directly if the dimension b in Equations 27.2 and 27.3 is calculated using the values of dH and dT referred to the axis of the hyperboloid. Then, Ro is added to the value of R calculated by Equation 27.1 to determine the total radial distance from the axis of the cooling tower to the middle surface of the shell at any vertical coordinate Z. Such a shell is known in Figure 27.3, where the value of Ro = — 1.0370 m for the part of the shell below the throat.

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