Kinematics table

The kinematics table, shown in Table 7.1, introduces a method (referred to within this chapter as the table method) for efficiently managing the mathematics involved in analyzing multibody and multiple coordinate system problems, and can be used in either the Lagrangian or the Newton-Euler approach. A schematic diagram, which defines an inertial or body-fixed coordinate system, must accompany every kinematics table. The purpose of the schematic is to identify the point on the body at which the absolute velocity and acceleration is to be determined. The corresponding schematic for Table 7.1 is shown in Fig. 7.7. 

The kinematics table is divided into three sections. 

For a multibody system, each body would require a kinematics table and a corresponding schematic. The following examples illustrate the steps required for solving problems by the table method. Note that one example includes the expressions for acceleration to demonstrate the use of the table method with the Newton-Euler approach, while all other examples consider only the velocity. 

FIGURE 7.7 Schematic to accompany the kinematics table (Table 7.1). 

TABLE 7.2 Kinematics Table for the Single Rigid Body in an Inertial Coordinate System... 

Table 7.4 is the kinematics table for the single elastic body pendulum shown in Fig. 7.3. The absolute velocity of the center of mass G is required to complete the Lagrangian approach, and the Newtonian approach utilizes die absolute acceleration of point <3 Eq. (7.86), in F = mao, for problems of constant mass. 

TABLE 7.4 Kinematics Table for the Single Elastic Body Pendulum... 

The e, 62, 63 coordinate system is defined to generalize the discussion of the angular velocity derivations and represents the inertial frame of reference. The 3-1-2 transformation follows an initial rotation about the third axis, Cj, by an angle of to yield the e[, 2, ej coordinate system. Then a second rotation is performed about the e, axis by an angle of yielding the e", e, e, system. Finally, a third rotation is performed about the axis by to yield the final e", e" body frame of reference. This defines the transformation from the ej, C2,63 system to the ef, e, e" system. To supplement the kinematics tables, an expression for the angular velocity vector is defined from this transformation as... 

The heavy fuel should be heated systematically before use to improve its operation and atomization in the burner. The change in kinematic viscosity with temperature is indispensable information for calculating pressure drop and setting tbe preheating temperature. Table 5.20 gives examples of viscosity required for burners as a function of their technical design. 

A number of arbitrary viscosity units have also been used. The most common has been the Saybolt Universal second (SUs) which is simply the time in seconds required for 60 mL of oil to empty out of the cup in a Saybolt viscometer through a carefully specified opening. Detailed conversion tables appear in ASTM D2161, approximation of kinematic viscosity V in mm /s(= cSt) can be made from the relation shown in equation 8 ... 

If it is necessary to calculate kinematic viscosities from efflux times, such as in a caUbration procedure, equation 20 should be used, where /is the efflux time and k and K are constants characteristic of the particular viscosity cup (see Table 5) (158,159). 

In most cases it is sufficient to be able to convert from one viscometer value to another or to approximate kinematic viscosities with the help of charts or tables Hterature from manufacturers is useful. 

Rheology. PVP solubihty in water is limited only by the viscosity of the resulting solution. The heat of solution is — 16.61 kJ/mol (—3.97 kcal/mol) (79) aqueous solutions are slightly acidic (pH 4—5). Figure 2 illustrates the kinematic viscosity of PVP in aqueous solution. The kinematic viscosity of PVP K-30 in various organic solvents is given in Table 13. 

Rizzuti et al. [Chem. Eng. Sci, 36, 973 (1981)] examined the influence of solvent viscosity upon the effective interfacial area in packed columns and concluded that for the systems studied the effective interfacial area a was proportional to the kinematic viscosity raised to the 0.7 power. Thus, the hydrodynamic behavior of a packed absorber is strongly affected by viscosity effects. Surface-tension effects also are important, as expressed in the work of Onda et al. (see Table 5-28-D). 

Table 3.3 summarizes the history of the development of wave-profile measurement devices as they have developed since the early period. The devices are categorized in terms of the kinetic or kinematic parameter actually measured. From the table it should be noted that the earliest devices provided measurements of displacement versus time in either a discrete or continuous mode. The data from such measurements require differentiation to relate them to shock-conservation relations, and, unless constant pressures or particle velocities are involved, considerable accuracy can be lost in data processing. 

This formula gives accurate values only when the kinematic viscosity of the liquid is about 1.1 centistokes or 31..5 SSU, which is the case with water at about OOF. But the viscosity of water varies with the temperature from 1.8 at 32F to. 29 centistoke.s at 212F. The tables are therefore subject to this error which may increa.se the friction loss as much as 20% at 32F and decrease it as much as 20% at 212F. Note that the tables may be used for any liquid having a viscosity of the. same order as indicated above. 

Kinematics is based on one-dimensional differential equations of motion. Suppose a particle is moving along a straight line, and its distance from some reference point is S (see Figure 2-6a). Then its linear velocity and linear acceleration are defined by the differential equations given in the top half of Column 1, Table 2-5. The solutions... 

The kinematic viscosity of MEM containing aqueous electrolytes at different concentrations of MEM and ZnBr2 and at different temperatures has been studied [68] (see Table 8). 

Table 8. Kinematic viscosity (m2sH ) of aqueous electrolyte containing MEM and 3 mol L l ZnBr2 (taken from Ref. [68])...

Dynamic similarity requires geometric and kinematic similarity in addition lo force ratios at corresponding points being equal, involving properties of gravitation, surface tension, viscosity and inertia [8, 21]. With proper and careful application of this principle scale-up from test model lo large scale systems is often feasible and quite successful. Tables 5-... 

Experimental values of diffusivities are given in Table 10.2 for a number of gases and vapours in air at 298K and atmospheric pressure. The table also includes values of the Schmidt number Sc, the ratio of the kinematic viscosity (fx/p) to the diffusivity (D) for very low concentrations of the diffusing gas or vapour. The importance of the Schmidt number in problems involving mass transfer is discussed in Chapter 12. 

Table II. Kinematic Conditions for the Formation of Tertiary Ions by Various Collision Mechanisms...

By using a liquid with a known kinematic viscosity such as distilled water, the values of Ci and Cj can be determined. Ejima et al. have measured the viscosity of alkali chloride melts. The equations obtained, both the quadratic temperature equation and the Arrhenius equation, are given in Table 12, which shows that the equation of the Arrhenius type fits better than the quadratic equation. 

The physical and chemical properties of hazardous dense solvent compounds are given in Tables 18.8 and 18.9, in which the absolute viscosity and kinematic viscosity are expressed in cen-tipoises and centistokes, respectively. 

As mentioned before in Eq. (3), the most common source of SGS phenomena is turbulence due to the Reynolds number of the flow. It is thus important to understand what the principal length and time scales in turbulent flow are, and how they depend on Reynolds number. In a CFD code, a turbulence model will provide the local values of the turbulent kinetic energy k and the turbulent dissipation rate s. These quantities, combined with the kinematic viscosity of the fluid v, define the length and time scales given in Table I. Moreover, they define the local turbulent Reynolds number ReL also given in the table. 

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