# Stokes law

At infinite dilution, the assumption of a constant relaxation time is reasonable and, using Stokes law as well, we have [c.573]

In the Smoluchowski limit, one usually assumes that the Stokes-Einstein relation (Dq//r7)a = C holds, which fonns the basis of taking the solvent viscosity as a measure for the zero-frequency friction coefficient appearing in Kramers expressions. Here C is a constant whose exact value depends on the type of boundary conditions used in deriving Stokes law. It follows that the diffiision coefficient ratio is given by [c.850]

A physical value for 7 for each particle can be chosen according to Stokes law (with stick boundary conditions) [c.234]

In the derivation of both Eqs. (9.4) and (9.9), the disturbance of the flow streamlines is assumed to be produced by a single particle. This is the origin of the limitation to dilute solutions in the Einstein theory, where the net effect of an array of spheres is treated as the sum of the individual nonoverlapping disturbances. When more than one sphere is involved, the same limitation applies to Stokes law also. In both cases contributions from the walls of the container are also assumed to be absent. [c.590]

The premise which underlies Eq. (9.48) is that all segments experience the flow as if it were undisturbed by other segments in the chain. This is the free-draining limit for the friction factor and, of course, assumes no interference from neighboring polymer molecules either. This result can be criticized on two grounds applying Stokes law to individual polymer segments and ignoring that there is ordinarily overlap of the hydrodynamic effect caused by neighboring units. The latter is the more serious of the two. The Kirkwood method of dealing with this overlap effect is to modify Eq. (9.48) as follows [c.612]

Rigid, unsolvated spheres. Stokes law, Eq. (9.5), provides a relationship between f and the radius of the particle. Since this structure is a reasonable model for some protein molecules, experimental D values can be interpreted, via f, to yield values of R for such systems. Note that this application can also yield a value for M, since M = N pj [(4/3)ttR ], where pj is the density of the unsolvated material. [c.625]

A force of viscous resistance is proportional to the stationary-stage velocity Vj according to Stokes law [c.636]

Eig. 5. Target efficiency of spheres, cylinders, and ribbons. The curves apply for conditions where Stokes law holds for the motion of the particle (see also N j ia Table 5). Langmuir and Blodgett have presented similar relationships for cases where Stokes law is not vaUd (149,150). Intercepts for ribbon or [c.393]

It is possible to set up a force balance on a given size particle in a precipitator and hence calculate its trajectory. Three forces tend to move a particle toward the collecting plate the charge on the particle, the integrated average field strength through which the particle moves, and the electric wind. This last results from the steady flow of ions from the discharge electrodes to the collecting plates the motion is similar to thermal convective currents. Particle movement is resisted by drag on the particle. In solving a force balance, it is convenient to define the particle migration velocity as the average velocity with which the particle moves toward the collecting surface. This velocity is a function of particle size, field strength, and such particle material properties as electrical conductivity and dielectric constant. The factors affecting migration velocity for conductive particles larger than 1 p.m that are in the Stokes law region can be expressed as (141) [c.400]

There are some general principles that can serve as guidelines for initial screening in terms of both flocculant chemistry and molecular weight. In general, the large floes formed by high molecular weight polymers tend to settle faster than smaller ones. In the upper part of a thickener or settling basin, the settling rate of an individual floe is governed by Stokes law. Using a spherical model to approximate an individual floe, at the terminal velocity the viscous drag is equal to the gravitational force on the floe. The downward velocity is given by equation 1 where g is the gravitational constant, a is the radius of the particle, is the density of the particle, d is the density of the Hquid, and Tj is the viscosity of the Hquid. [c.35]

Table 8 shows how much gas may be evolved from a typical amber soda—lime container glass batch. Fining agents are employed that generally react at higher temperatures than are needed for melting thus the fining reactions continue after dissolution and volatilisation have taken place. Only materials that can release gases without delay through the formation of boiling bubbles can act as fining agents. As the gases are released, the bubbles rise to the surface roughly according to Stokes law. Each bubble, during its trip through the glass melt, attracts new quantities of gas by diffusion from neighboring layers and by coalescence with other bubbles. High temperatures make the glass more fluid and increase the diffusivity which speeds up the fining process greatly. The most common fining agents are the sulfates, followed by sodium or potassium nitrates in combination with arsenic or antimony trioxides. Arsenic trioxide is used for higher melting glasses, 1450—1500°C, and antimony for lower melting glasses, 1300—1400°C. As the glass cools, dissolution fining may occur, ie, oxygen bubbles are removed by reaction with the arsenic or antimony trioxide to form the pentoxide. [c.305]

Fig. 14. Drag coefficient for terminal settling velocity correlation (single particle) where A represents Stokes law B, intermediate law and C, Newton s |

Elimination of between equations 1, 2, and 3 gives another form of Stokes law, as shown in Figure 1 as a straight line. [c.317]

Settling of Suspensions. As the concentration of the suspension iacreases, particles get closer together and iaterfere with each other. If the particles are not distributed uniformly, the overall effect is a net iacrease ia settling velocity because the return flow caused by volume displacement predominates ia particle-sparse regions. This is the weU-known effect of cluster formation which is significant only ia nearly monosized suspensions. For most practical widely dispersed suspensions, clusters do not survive long enough to affect the settling behavior and, as the return flow is more uniformly distributed, the settling rate steadily declines with increasing concentration. This phenomenon is referred to as hindered settling and can be theoretically approached from three premises (4) as a Stokes law correction by iatroduction of a multiplying factor by adopting effective fluid properties for the suspension different from those of the pure fluid and by determination of bed expansion using a modified Carman-Kozeny equation. These three approaches yield essentially identical results [c.317]

Separation by Density Difference. A single soHd particle or discrete Hquid drop settling under the acceleration of gravity in a continuous Hquid phase accelerates until a constant terminal velocity is reached. At this point the force resulting from gravitational acceleration and the opposing force resulting from frictional drag of the surrounding medium are equal in magnitude. The terminal velocity largely determines what is commonly known as the settling velocity of the particle, or drop under free-fall, or unhindered conditions. For a small spherical particle, it is given by Stokes law [c.396]

Stokes law can be readily extended to a centrifugal field [c.396]

As an additional guide, the values are correlated with the equivalent spherical particle diameter by Stokes law, as in equation 1. A density [c.405]

Fouling. Fouling occurs when insoluble particulates suspended in recirculating water form deposits on a surface. Fouling mechanisms are dominated by particle—particle interactions that lead to the formation of agglomerates. At low water velocities, particle settling occurs under the influence of gravity. Parameters that affect the rate of settling are particle size, relative Hquid and particle densities, and Hquid viscosity. The relationships of these variables are expressed by Stokes law. The most important factor affecting the settling rate is the size of the particle. Because of this, the control of fouling by preventing agglomeration is one of the most fundamental aspects of deposition control. [c.271]

Vjs Stokes law terminal velocity m/s ft/h [c.590]

The drag coefficient for rigid spherical particles is a function of particle Reynolds number, Re = d pii/ where [L = fluid viscosity, as shown in Fig. 6-57. At low Reynolds number, Stokes Law gives 24 [c.676]

Wall Effects When the diameter of a setthng particle is significant compared to the diameter of the container, the settling velocity is reduced. For rigid spherical particles settling with Re < 1, the correction given in Table 6-9 may be used. The factor k is multiplied by the settling velocity obtained from Stokes law to obtain the corrected set- [c.680]

The differing extent of hydration shown by the different types of ion can be detennmed experimentally from the amount of water carried over with each type of ion. A simple measurement can be carried out by adding an electrolyte such as LiCl to an aqueous solution of sucrose in a Flittorf cell. Such a cell consists of two compartments separated by a narrow neck [7] on passage of charge the strongly hydrated Li ions will migrate from the anode to tlie cathode compartment, whilst the more weakly hydrated Cr ions migrate towards the anode compartment the result is a slight increase in the concentration of sucrose in the anode compartment, since the sucrose itself is essentially electrically neutral and does not migrate in the electric field. The change in concentration of the sucrose can either be detennined analytically or by measuring the change m rotation of plane polarized light transmitted tln-ough the compartment. Measurements carried out in this way lead to hydration numbers for ions, these being the number of water molecules that migrate with each cation or anion. Values of 10-12 for Mg, 5.4 for K, 8.4 for Na and 14 for Li are clearly m reasonable agreement with the values inferred from the Stokes law arguments above. They are also in agreement with the measurements carried out using large organic cations to calibrate the experiment, since these are assumed not to be hydrated at all. [c.573]

In addition to the relaxation effect, the theory of Debye, Hiickel and Onsager also takes into acconnt a second effect, which arises from the Stokes law discussed above. We saw that each ion travelling tln-ough the solution will experience a frictional effect owing to the viscosity of the liquid. However, this frictional effect itself depends on concentration, since, with increasing concentration, enconnters between the solvent sheaths of oppositely charged ions will become more frequent. The solvent molecules in the solvation sheaths are movmg with the ions, and therefore an individual ion will experience an additional drag associated with tlie solvent molecules in the solvation sheaths of oppositely charged ions this is tenned tlie electrophoretic effect. [c.584]

This result is often called the Stokes-Einstein formula for the difflision of a Brownian particle, and the Stokes law friction coefficient 6iiq is used for [c.689]

The work required in order to move the xenon atom was computed by the slow-change method, according to (6). Results for 100 ps extractions are shown in Fig. 2. Near the starting point, i.e. in the bound state, the xenon is moved relatively easily, considerable work is required to move it through llic layer of hydrophobic side chains, and there is evidence for a small energy liarrier before the xenon leaves the protein at approximately 0.8 nm the work fjc formed beyond that point averages to that required according to Stokes law to drag a sphere of radius 0.22 nm through a continuum with the viscosity of water. [c.143]

A somewhat similar problem arises in describing the viscosity of a suspension of spherical particles. This problem was analyzed by Einstein in 1906, with some corrections appearing in 1911. As we did with Stokes law, we shall only present qualitative arguments which give plausibility to the final form. The fact that it took Einstein 5 years to work out the bugs in this theory is an indication of the complexity of the formal analysis. Derivations of both the Stokes and Einstein equations which do not require vector calculus have been presented by Lauffer [Ref. 3]. The latter derivations are at about the same level of difficulty as most of the mathematics in this book. We shall only hint at the direction of Lauffer s derivation, however, since our interest in rigid spheres is marginal, at best. [c.587]

Since f is a measurable quantity for, say, a protein, and since the latter can be considered to fail into category (3) in general, the friction factor provides some information regarding the eilipticity and/or solvation of the molecule. In the following discussion we attach the subscript 0 to both the friction factor and the associated radius of a nonsolvated spherical particle and use f and R without subscripts to signify these quantities in the general case. Because of Stokes law, we write [c.626]

The foregoing discussion is incomplete, since it offers an interpretation of f/fQ while presenting, through Eq. (9.79), a method for evaluating f alone. What is needed to be able to take advantage of this approach is a value for the friction factor of a nonsolvated spherical particle having the same mass as the species under consideration. Fortunately, this reference value fQ can be calculated if the molecular weight of the solute is known, since we are comparing f and fQ for particles of the same mass. The mass of a molecule divided by its (unsolvated) density gives the volume of an equivalent spherical particle. From this volume Rq can be evaluated, and this, in turn, allows fQ to be calculated from Stokes law. The following example illustrates this computational procedure. [c.627]

Terminal settling velocity can be calculated for spherical particles (eq. 1) in streamline flow (Stokes law region). Small particles fall faster then predicted as they tend to "sHp" between gas molecules, and the Stokes-Cunningham correction factor (eq. 2) must be appHed as indicated in Table 5. Figure 4 gives terminal settling velocities of spherical particles in air. For nonspherical particles, multiplication of equation 1 by a sphericity correction constant = 0.843log ( //0.065) has been recommended (145). The sphericity / (146) is defined as the ratio of surface area of a sphere (of volume equal to the particle) to the surface area of the particle. References 147 and 148 give further refinements when sphericity is less than 0.67. [c.392]

Because of high buoyancy and frequently large si2e, gas bubbles rising in a Hquid can deviate greatly from spherical shapes. Figure 6 illustrates the behavior observed. In pure Hquids small bubbles rise faster than would be predicted from the drag correlations developed for soHd spheres because internal circulation permits a higher fluid velocity at the surface and less drag than for the corresponding soHd. Very large bubbles rise much more slowly than do undeformed spheres of the same volume. This is caused by deformation to a shape of large frontal area but small thickness. Very small bubbles obey Stokes law for soHd (immobile) surfaces. This behavior is reflected also in the mass-transfer coefficients for such bubbles, which are lower than would be [c.92]

In particle-size measurement, gravity sedimentation at low soHds concentrations (<0.5% by vol) is used to determine particle-size distributions of equivalent Stokes diameters ia the range from 2 to 80 pm. Particle size is deduced from the height and time of fall usiag Stokes law, whereas the corresponding fractions are measured gravimetrically, by light, or by x-rays. Some commercial instmments measure particles coarser than 80 pm by sedimentation when Stokes law cannot be appHed. [c.316]

The coalescence of internal phase droplets can be further decreased by raising the viscosity of the external contiauous phase through additioa of gums or syathetic polymers, for example, ceUulosic gums such as hydroxypropylmethyl-ceUulose [9004-65-3] fermentation gums such as xanthan gum [11138-66-2] or cross-linked carboxyvinyl polymers such as carbomer [39007-16-3J. The iacreased viscosity also counteracts changes ia the emulsion resultiag from differeaces ia the specific gravity of the two phases as mandated by Stokes law. An advance ia cosmetic emulsificatioa technology has resulted from the development of cross-linked carboxyvinyl polymers, ia which some of the carboxylic acid residues are esterified with various fatty alcohols. These polymers possess the ability to act as primary emulsifiers and thicken the system when some of the remaining carboxylic groups are neutralized with alkali (see Carboxylic acids). [c.294]

Recirculating Cooling Water. Water used to cool plant processes and buildings contains contaminants that can accelerate corrosion of metal surfaces and leave scale or particle deposits on pipes and heat-transfer surfaces. To prevent corrosion and scale, cooling water treatment formulations contain corrosion inhibitors, biocides, phosphonates, and dispersants. Dispersants aid in the prevention of inorganic fouling (silt, iron oxide), scaling (calcium carbonate, calcium sulfate), and corrosion. The dispersant minimises settling of inorganic foulants by adsorbing on particles to increase their mutual repulsion (20). Dispersants prevent scale by adsorbing on active sites on growing crystals, ideally minimising their particle size to less than the wavelength of visible light, making them invisible to the unaided eye. Small particles settle out on pipes and surfaces less easily, following Stokes law. In addition, the relatively large surface area of these particles favors their redissolution. Dispersants also restrict the particle size of corrosion-inhibiting agents such as calcium phosphonate and zinc. In alkaline cooling water treatment, corrosion inhibitors form films at high pH cathodic areas on a corroding metal surface. By helping to minimize particle size of these corrosion inhibitors in the bulk water, dispersants enable the inhibitors to precipitate preferentially at cathodic surfaces as a thin film of closely packed particles in the form of a hydroxide salt or complex (21). Specialty dispersants have been developed to prevent specific inorganic scales and particles from fouling cooling water surfaces. For example, calcium phosphate and iron are controlled with copolymers of acryflc acid—nonionic monomers (22,23), acryflc acid—sulfonate monomers (24), or acryflc acid—sulfonate—nonionic terpolymers (25). [c.151]

A. Solid particles suspended in agitated vessel containing vertical baffles, continuous phase coefficient -2 + 0.6Wi f,.Wi D Replace Osi p with Vj = terminal velocity. Calculate Stokes law terminal velocity [S] Use log mean concentration difference. Modified Frossling equation K, -< T.d,P. [97] [146] p.220 [c.616]

In the Stokes law region (Re < 0.3), n = 4.65 and in the Newton s law region (Re > 1,000), n = 2.33. Equation (6-242) may be applied to particles of any size in a polydisperse system, provided the volume frac tion corresponding to 1 the particles is used in computing terminal velocity (Richardson and Shabi, Trans. Inst. Chem. Bng. [London], 38, 33—42 [I960]). The concentration effect is greater for nonspherical and angular particles than for spherical particles (Steinour, Ind. Eng. Chem., 36, 840-847 [1944]). Theoretical developments for low-Reynolds number flow assemblages of spheres are given by Hap-pel and Brenner Low Reynolds Number Hydrodynamics, Prentice- [c.678]

See pages that mention the term

**Stokes law**:

**[c.688] [c.404] [c.587] [c.613] [c.628] [c.392] [c.316] [c.317] [c.317] [c.319] [c.401] [c.131] [c.616] [c.679]**

Molecular modelling Principles and applications (2001) -- [ c.388 ]

Industrial ventilation design guidebook (2001) -- [ c.1479 ]