Velocity angular

The procedure Split selects the internal displacement coordinates, q, and momenta, tt, (describing vibrations), the coordinates, r, and velocities, v, of the centers of molecular masses, angular velocities, a>, and directional unit vectors, e, of the molecules from the initial Cartesian coordinates, q, and from momenta, p. Thus, the staring values for algorithm loop are prepared. Step 1 Vibration... 

The quaternions obey coupled differential equations involving the angular velocities tJi, tbody frame (i.e. tJi represents the angular velocity about the first axis of inertia, etc.). These differential equations take the form... 

Other orientational correlation coefficients can be calculated in the same way as tf correlation coefficients that we have discussed already. Thus, the reorientational coiTelatio coefficient of a single rigid molecule indicates the degree to which the orientation of molecule at a time t is related to its orientation at time 0. The angular velocity autocorrelatio function is the rotational equivalent of the velocity correlation function ... 

Next, suppose we consider the tangential velocity v of segment i in a polymer molecule. The segment is located a distance r from the center of mass of the molecule and possesses an average angular velocity co. The situation is sketched in Fig. 2.12a. Since v = rco, it follows that the x and y components of the velocity are given by... 

Since the radial acceleration functions simply as an amplified gravitational acceleration, the particles settle toward the bottom -that is, toward the circumference of the rotor-if the particle density is greater than that of the supporting medium. A distance r from the axis of rotation, the radial acceleration is given by co r, where co is the angular velocity in radians per second. The midpoint of an ultracentrifuge cell is typically about 6.5 cm from the axis of rotation, so at 10,000, 20,000, and 40,000 rpm, respectively, the accelerations are 7.13 X 10, 2.85 X 10 , and 1.14 X 10 m sec" or 7.27 X 10, 2.91 X 10, and 1.16 X 10 times the acceleration of gravity (g s). 

A larger sedimentation force can be developed by centrifuging the suspension. At an angular velocity of CO radians pet second the centrifugal... 

Fig. 21. Dynamic viscoelastic properties of a low density polyethylene (LDPE) at 150°C complex dynamic viscosity Tj, storage modulus G and loss modulus G" vs angular velocity, CO. To convert Pa-s to P, multiply by 10 to convert Pa to dyn/cm, multiply by 10.

Fig. 27. Concentric cylinder viscometer. R and R are the radii of the inner and outer cylinder, respectively, and Q is the relative angular velocity.

The relationship between viscosity, angular velocity, and torque for a Newtonian fluid in a concentric cylinder viscometer is given by the Margules equation (eq. 26) (21,146), where M is the torque on the inner cylinder, h the length of the inner cylinder, Q the relative angular velocity of the cylinder in radians per second, T the radius of the inner cylinder wall, the radius of the outer cylinder wall, and an instmment constant. 

Therefore, the viscosity can be determined from the torque and angular velocity. However, the viscosity is usually calculated from the shear rate and shear stress, which can be obtained from the Margules equation. The shear rate is given by equation 27, where r is any given radius. 

In addition to non-Newtonian flow, the main correction necessary for concentric cylinder measurements is that on account of end effects. Because the inner cylinder is not infinitely long, there is drag on the ends as well as on the face of the cylinder. The correction appears as an addition, to the length, b. The correction is best deterrnined by measuring the angular velocity and torque at several values of b, that is, at various depths of immersion. The data are plotted as M/Q vs b, and extrapolation is made to a value of at M/H = 0. The quantity (/i + h ) is substituted for b in the various equations. 

Specific Commercial Rotational Viscometers. Information on selected commercial rotational viscometers can be found ia Table 7. The ATS RheoSystems Stresstech rheometer is an iastmment that combines controlled stress as well as controlled strain (shear rate) and oscillatory measurements. It has a torque range of 10 to 50 mN-m, an angular velocity range of 0 to 300 rad/s, and a frequency range of seven decades. Operation and temperature programming (—30 to 150°C higher temperatures optional) are computer controlled. 

The mathematical model chosen for this analysis is that of a cylinder rotating about its axis (Fig. 2). Suitable end caps are assumed. The Hquid phase is introduced continuously at one end so that its angular velocity is identical everywhere with that of the cylinder. The dow is assumed to be uniform in the axial direction, forming a layer bound outwardly by the cylinder and inwardly by a free air—Hquid surface. Initially the continuous Hquid phase contains uniformly distributed spherical particles of a given size. The concentration of these particles is sufftcientiy low that thein interaction during sedimentation is neglected. 

Equation 26 is accurate only when the Hquids rotate at the same angular velocity as the bowl. As the Hquids move radially inward or outward these must be accelerated or decelerated as needed to maintain soHd-body rotation. The radius of the interface, r, is also affected by the radial height of the Hquid crest as it passes over the discharge dams, and these crests must be considered at higher flow rates. 

Angle, Angular Velocity, and Angular Acceleration. Thek SI units are rad, rad/s, and rad/s, respectively. Because the radian is here taken to be dimensionless, the units 1, 1/s, and 1/s are also used where appropriate. 

Angular Momentum (Moment of Momentum). Angular momentum is linear momentum (kg-m/s) times moment arm (m). Its SI unit is kg-m /s. For a rotating body the total angular momentum is equal to the moment of inertia I (kg-m ) times the angular velocity CO (rad/s or 1/s). 

Centripetal Acceleration. Centripetal acceleration, /r or CO r, where is the tangential linear velocity (m/s), rthe radius (m), and CO the angular velocity (rad/s), is, like any other linear acceleration, measured in SI units m/s. Centripetal force, equal to mass times centripetal acceleration, is, like any force in SI, measured in newtons. 

Rotary atomisation produces the most uniform atomisation of any of the aforementioned techniques, and produces the smallest maximum particle sise. It is almost always used with electrostatics and at lower rotational speeds the electrostatics assist the atomisation. At higher rotational speeds the atomisation is principally mechanical in nature and does not depend on the electrical properties of the coating material. If the viscosity of a coating material is sufficiendy low that it can be deUvered to a rotary atomiser, the material can generally be atomised. The prime mover is usually an ak-driven turbine and, provided that the turbine has the requked power to accelerate the material to the angular velocity, Hquid-dow rates of up to 1000 cm /min can be atomised using an 8-cm diameter beU. 

Equation 65 gives the pressure at any point within the centrifuge, (r), as a function of the coordinate r, the pressure at the axis (0), the angular velocity of the centrifuge, and the temperature and mol wt of the gas. Should the centrifuge contain not a single pure gas, but a gas mixture, equations of the above forms could be written for each species present. In particular for the case of a binary gas mixture, consisting of species M and B. 

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