Equivalent radius

This allows for the equivalence between crossed cylinders and the particle on a plane problem. Likewise, the mechanics of two spheres can be described by an equivalently radiused particle-on-a-plane problem. The combination of moduli and the use of an effective radius greatly simplifies the computational representation and allows all the cases to be represented by the same formula. On the other hand, it opens the possibility of factors of two errors if the formula are used without realizing that such combinations have been made. Readers are cautioned to be aware of these issues in the formulae that follow. 

Fig. 2.7 Construction of equivalent radius r = 2 rule, nj), that is l -order in time from a radius r = 1, 2" -order rule,...

This allows us to define a thermodynamically effective equivalent radius Rf by replacing the actual sphere radius of a hard sphere by R j< which gives... 

Together with Eq. (24) this gives a relationship for the interpenetration function in terms of this equivalent radius ... 

If T is based on volume-equivalent radius, rather than equatorial radius as used here, E has almost no effect on the trajectory for prolate spheroids (LI). However, this definition of t obscures the effect of shape for oblate particles. 

Equation (4.69) for self-diffusion can be arrived at in a similar way through proper derivation of the friction coefficient. Eqnation (4.70) does not take into account hydro-dynamic interactions. It is also necessary to come np with an equivalent radius, r, for a polymer chain, which can be difficnlt, especially when the conformation is such that the chain is extended, and does not form a sphere at all. Nonetheless, a radius of gyration, rg, is often nsed to characterize polymer chains in solntion, and the resulting friction coefficient is = Anfxrg. 

Calibration is invariably based on spheres, which means that an instrument can be used with confidence only for such particles. Of course, a response will duly be recorded if a nonspherical particle passes through the scattering volume. But what is the meaning of the equivalent radius corresponding to that response Is it the radius of a sphere of equal cross-sectional area Or equal surface area Or equal volume Or perhaps equal mean chord length Answers to these questions depend on the particular instrument and nonspherical particle comprehensive answers do not come easily because it is difficult to do calculations for nonspherical particles, even those of regular shape. 

When polydispersity is present, we have no problem labeling the calculated equivalent radius an average . We have used quotation marks around this label to alert us to the possibility that this average may be something other than the mean. 

The radius of curvature of sharp points or protuberances on the particles has a larger effect on the solubility of irregular particles than the equivalent radius of the particles themselves. 

One should note that for non-spherical geometries an equivalent radius, Re, should be used instead of R in equations 1,3 and 4. 

The equivalent radius of the capillaries is then found from a similar experiment using a liquid which completely wets the solid - i.e. 

Let C be the circumference of the cross section of the pores and R be the equivalent radius of a circle having the same circumference, i.e.,... 

First, consider the case where the flow is parallel to the cylinders. It is assumed that the fluid is moving through the annular space between the cylinder of radius a and the fluid envelope of equivalent radius b, as shown in Fig. 7.14. Assume that the fluid motion is in the creeping flow regime so that inertia terms can be omitted from the Navier-Stokes equations. Thus, in cylindrical coordinates, we have... 

A> Area, dimension, defined by b Equivalent radius of flow cell... 

Bubble deformation in shear flow increases mass transfer because of the increase in surface area and because of convection. The latter brings volatile-rich liquid to the bubble surface. Favelukis et al. (39) studied the (identical but experimentally easier) reverse problem of dissolution of a gas bubble in a sheared liquid, both theoretically and experimentally, and they confirmed the increase of mass transfer with increasing shear rate. They also showed that the rate of dissolution, da/dt, where a is the equivalent radius of the bubble, is given by... 

Equivalent radius Hydraulic radius Mean radius/diameter Sedimentation... 

In which RQ Is termed "equivalent radius" and Is the radius of a circle having the same area as that of the particle profile, and n Is the order of the coefficient. 

Particle Size. The equivalent radius R0 may also be reported out as the equivalent particle diameter 2R0. The size may be reported out as mean and standard deviation as shown in Table II or as a histogram as shown in Figure la,b. The size template in Figure 2 illustrates the principle underlying the RQ term. 

Equivalent radius mean equivalent radius, standard... 

In this regard, Washburn [147,150] supposed that for cylindrical pores of equivalent radius, r, wetted with a liquid of contact angle, 8 [149] (see Figure 4.66), a liquid/gas interface enclosed inside a pore will at equilibrium take on the shape of the uniform average curvature. That is, in a uniform cylindrical pore of radius r or in a parallel-sided slit of width r, the mean curvature, 1/r, is equal to [148]... 

For sufficiently short times, eq 8 holds for any particle size. In this case, R is understood as an equivalent radius being equal to three times the ratio between the particle volume and the external surface. 

This equivalent radius rm is comparable in magnitude to atomic radii. Table 8.3 also lists the ratio of rm to the Bohr15 radius (Eq. 3.1.21) a0 = h2/ meez (cgs) = 47re0fz2/meez (SI) = 0.529177A = 0.0529177nm. 

There it forms two slightly asymmetric H-bonds with neighboring water molecules, and survives for about 1 ms at pH 7. During its brief lifetime it exists in an s-like (near-spherical) ground state. After formation, the equivalent radius of the hydrated electron is about 200 pm. Ultimately it decomposes by reaction with a proton to create a hydrogen atom ... 

Here, Sh is the Sherwood number based on the equivalent radius of the packing material, Rpacking material, ReL = l/i.i /vL(l — hG) (where vL is the kinematic viscosity of the liquid), Sc is the Schmidt number, and m is the thickness of the liquid layer on the packing obtained from the relation... 

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