Sphere frictional

Person 2 Estimate the nnsolvated Stake s sphere friction coefficient, fo, nsing the Stokes-Einstein relationship for spheres, fo = 6jrp,r. 

This viscometer allows determination of the viscosity of the liquid from the velocity regime of a falling sphere. When a spherical ball (radius r) falls in a viscous liquid under gravity it tends to approach a constant velocity Vi after a transient regime. This is an important hypothesis that has to be checked (Appendix D). The movement of such a ball can be obtained from classical mechanics assuming that the sphere friction can be described by Stokes law (Appendix D). The velocity of the ball is written as... 

Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

The spherical geometry assumed in the Stokes and Einstein derivations gives the highly symmetrical boundary conditions favored by theoreticians. For ellipsoids of revolution having an axial ratio a/b, friction factors have been derived by F. Perrin, and the coefficient of the first-order term in Eq. (9.9) has been derived by Simha. In both cases the calculated quantities increase as the axial ratio increases above unity. For spheres, a/b = 1. 

We shall see in Sec. 9.10 that sedimentation and diffusion data yield experimental friction factors which may also be described-by the ratio of the experimental f to fQ, the friction factor of a sphere of the same mass-as contours in solvation-ellipticity plots. The two different kinds of contours differ in detailed shape, as illustrated in Fig. 9.4b, so the location at which they cross provides the desired characterization. For the hypothetical system shown in Fig. 9.4b, the axial ratio is about 2.5 and the protein is hydrated to the extent of about 1.0 g water (g polymer)". ... 

This concludes our discussion of the viscosity of polymer solutions per se, although various aspects of the viscous resistance to particle motion continue to appear in the remainder of the chapter. We began this chapter by discussing the intrinsic viscosity and the friction factor for rigid spheres. Now that we have developed the intrinsic viscosity well beyond that first introduction, we shall do the same (more or less) for the friction factor. We turn to this in the next section, considering the relationship between the friction factor and diffusion. 

Rigid particles other than unsolvated spheres. It is easy to conclude qualitatively that either solvation or ellipticity (or both) produces a friction factor which is larger than that obtained for a nonsolvated sphere of the same mass. This conclusion is illustrated in Fig. 9.10, which shows the swelling of a sphere due to solvation and also the spherical excluded volume that an ellipsoidal particle requires to rotate through all possible orientations. 

The particle can be assumed to be spherical, in which case M/N can be replaced by (4/3)ttR P2, and f by 671770R- In this case the radius can be evaluated from the sedimentation coefficient s = 2R (p2 - p)/9t7o. Then, working in reverse, we can evaluate M and f from R. These quantities are called, respectively, the mass, friction factor, and radius of an equivalent sphere, a hypothetical spherical particle which settles at the same rate as the actual molecule. 

Frictional coefficients of f, solvated protein, and fo, nonsolvated sphere. 

Therefore, the inertia forces have an insignificant influence on the sedimentation process in this regime. Theoretically, their influence is equal to zero. In contrast, the forces of viscous friction are at a maximum. Evaluating the coefficient B in equation 55 for a = 1 results in a value of 24. Hence, we have derived the expression for the drag coefficient of a sphere, = 24/Re. 

Khan, A.R and Richardson, J.F., 1990. Pressure gradient and friction factor for sedimentation and fluidisation of uniform spheres in liquids. Chemical Engineering Science, 45, 255-265. 

Scheraga-Mandelkern equations (1953), for effective hydrodynamic ellipsoid factor p (Sun 2004), suggested that [rj] is the function of two independent variables p, the axial ratio, which is a measure of shape, and Ve, the effective volume. To relate [r ] to p and Ve, introduced f, the frictional coefficient, which is known to be a direct function of p and Ve. Thus, for a sphere we have... 

It was suggested by Sutherland [363] that for a sphere diffusing through a medium consisting of molecules of comparable size to the diffusant, the friction coefficient is equal to... 

For heterogeneous media composed of solvent and fibers, it was proposed to treat the fiber array as an effective medium, where the hydrodynamic drag is characterized by only one parameter, i.e., Darcy s permeability. This hydrodynamic parameter can be experimentally determined or estimated based upon the structural details of the network [297]. Using Brinkman s equation [49] to compute the drag on a sphere, and combining it with Einstein s equation relating the diffusion and friction coefficients, the following expression was obtained ... 

According to Eq. (18), the high polymer molecule should exhibit the frictional coefficient of an equivalent sphere (compare Eq. 15) having a radius proportional to the root-mean-square end-to-end distance (r ) (or to (s ) / ). Similarly, according to Eq. (23) its contribution to the viscosity should be that of an equivalent sphere (compare Eq. 16) having a volume proportional to (r ) / In analogy with Eq. (17 ), we might write... 

Theory presented earlier in this chapter led to the expectation that the frictional coefficient /o for a polymer molecule at infinite dilution should be proportional to its linear dimension. This result, embodied in Eq. (18) where P is regarded as a universal parameter which is the analog of of the viscosity treatment, is reminiscent of Stokes law for spheres. Recasting this equation by analogy with the formulation of Eqs. (26) and (27) for the intrinsic viscosity, we obtain ... 

If the frictional effects of the molecule in both translation and in shear were equivalent to those which would be exhibited by the same hard sphere of radius e, we should then be justified in eliminating between Eqs. (15) and (16), with the result... 

When a particle falls under the influence of gravity, it will accelerate until the combination of the frictional drag in the fluid and buoyancy force balances the opposing gravitational force. If the particle is assumed to be a rigid sphere, at this terminal velocity, a force balance gives3,4,7,8... 

In fact, the diffusion constant in solutions has the form of an Einstein diffusion of hard spheres with radius Re. For a diffusing chain the solvent within the coil is apparently also set in motion and does not contribute to the friction. Thus, the long-range hydrodynamic interactions lead, in comparison to the Rouse model, to qualitatively different results for both the center-of-mass diffusion—which is not proportional to the number of monomers exerting friction - as well as for the segment diffusion - which is considerably accelerated and follows a modified time law t2/3 instead of t1/2. 

Here, p is the density of the fluid, V is the relative velocity between the fluid and the solid body, and A is the cross sectional area of the body normal to the velocity vector V, e.g., nd1/4 for a sphere. Note that the definition of the drag coefficient from Eq. (11-1) is analogous to that of the friction factor for flow in a conduit, i.e.,... 

---