Formulas for cylinders

Fig 14 (Ref 24) shows that the modified Gurney formula for cylinders (Eqn 23a) gives quite good agreement with expt when c/m approaches 1 agreement is less satisfactory for c/m< 1/2... 

The surface area of a given size pore was given by the usual formula for cylinders Eq. 3.4-7 ... 

The decay constants for overall motion (or tumbling) can be obtained from hydrodynamic estimates of the diffusion tensor. While this is a complicated topic, especially for hydrated, semiflexible molecules, there are useful exact analytic formulas for spheres and ellipsoids, approximate analytic formulas for cylinders, and numerical methods for more complicated shapes. Using these approaches, the rigid body second-rank correlation times (i.e., T2) for medium-sized proteins can be estimated to be in the 5-10 ns range. Hence, internal motions are typically well separated in time-scale from tumbling for most proteins, and can be assumed uncoupled. For smaller molecules, tumbling can be faster than isomerization (e.g., butane ), or the two can be highly coupled (octane ). 

Pipe-Wall Thickness. Once the design pressure and temperature have been established and the pipe material and size selected, the wall thickness is calculated using the appropriate section of the code. In rare cases, a thin pipe must be made thicker to withstand handling. Occasionally the thickness is affected by external loads or vibrations. All codes prescribe essentially the same design formula for metallic hoUow circular cylinders under internal pressure ... 

However, as the Grashof and Prandtl numbers can be different to determine, the following formula, which gives a close approximation, can be used for cylinders freely exposed to air ... 

Ellipsoid If the base of a vessel is one-half of an oblate spheroid (the cross section fitting to a cylinder is a circle with radius of D/2 and the minor axis is smaller), then use the formulas for one-half of an oblate spheroid. 

Runaway will occur when the calculated delta (8) exceeds the critical delta (8cr) which depends on the shape of the reaction mixture 0.88 for a plane slab, 2.00 for an infinite cylinder, 2.78 for a right cylinder with 1/d equal to 1, and 3.32 for a sphere. Bowes [133] provides formulas for calculation of 8cr for other geometric shapes and structures. In this model, heat is lost by conduction through the material to the edge, where the heat loss rate is infinite relative to the conduction rate. In this model, there is a maximum temperature in the center as shown in Figure 3.20 Case B. 

Use the formula for the volume of a cylinder, replacing the Uwith the number of cubic feet. The radius of a circle is half the diameter, so, if the height is equal to the diameter, then the height is equal to twice the radius. Replace the h in the volume formula with 2r and solve for r. 

Jacobs (Ref 16) has attempted to explain why y/2E values measured at NOL were consistently lower than those obtained elsewhere (see Ref 10). He points out certain possible shortcomings in the Gumey assumptions but eventually uses an empirical approach (and not very successfully, in this writer s opinion) to modify the Gumey formula for Comp B-filled cylinders to ... 

Table 4.3-2 shows the adequate formulae for closed and long enough cylinders during internal pressure p, the external pressure staying ambient. In this case the internal pressure is inducing an axial stress by acting on the covers resp. the end-pieces. 

TABLE 18.5. Heads and Horizontal Cylinders Formulas for Partially Filled Volumes and Other Data... 

The thin-wall bellows element should be designed for membrane stresses to conform to code-allowable stresses. The sum of membrane and secondary bending stresses should not exceed 1.5 times the yield stress in order to prevent the collapse of the corrugations caused by pressure. Bellows subjected to external pressure can be analyzed in a manner similar to a cylinder, utilizing an equivalent moment of inertia. The fatigue life can be estimated based on the sum of deflections and pressure stresses as compared to S/N curves based on bellows test data or using the curves in B31.3 Appendix X, Metal Bellows Expansion Joints. Formulas for the stress analysis of bellows are available in the Expansion Joints Manufacturing Association (EJMA) Standards (37). 

Refs l)H.Kast,SS 8,65-8 88-9(1913) (Calcn of brisance by Kast s formula and brisance values for some expls detd by Cu Cylinder Compression Test) la)C.Herlin,SS 8,448-52 (1913) (Kast s formula for calcn of brisance is... 

The dense packings of equal spheres around a cylinder have been examined by Erickson (1973) who derived useful formulae for their generation. These can be used... 

A porous catalyst for chemical reactions has an internal surface area of 800 m2 per cm3 of bulk material. Fifty percent of the bulk volume consists of the pores (holes), while the other 50% of the volume is made up of the solid substance. Assume that the pores are all cylindrical tubules of uniform diameter d and length /, and that the measured internal surface area is the total area of the curved surfaces of the tubules. What is the diameter of each pore Hint Find the number of tubules per bulk cm3, n, in terms of l and d, by using the formula for the volume of a cylinder V = 0.25 ird2l. Then apply the surface-area formula, S = itdl, to the cylindrical surfaces of n tubules.)... 

A plumber installs a pipe that has a diameter of 10 cm and a length of 2.4 m. Calculate the volume of water (in cm3) that the pipe will hold. Express your answer to the correct number of significant digits. Note The formula for the volume of a cylinder is V = nr2h, where r is the radius and h is the height or length. 

Step 1 Use the formula for the volume of a cylinder (nr2h) to approximate the volume of the gas inside. 

Compute required impeller pumping capacity. To determine the required pumping capacity, the bulk velocity (0.4 ft/s) must be multiplied by the appropriate cross-sectional area. Since a square batch is assumed for the design basis of bulk velocity, an equivalent tank diameter Teq is computed by rearranging the formula for the volume of a cylinder with the height equal to the diameter, that is,... 

For a perfectly 2D material the usual assumption of a constant field independent chemical potential has to be revised. If in an increasing field the FS is just between two successive Landau levels the chemical potential has to shift with the Landau cylinders and finally jump back for increasing field to the next lower level. An analytical formula for the dHvA effect of a 2D electron gas taking into account finite temperature and fields is given in [326]. The applicability of this formula could not be tested quantitatively. Qualitatively, however, the predicted increased harmonic content and a drastically changed shape of the oscillations was observed. 

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