Particles in a Stagnant Medium

Spherical particle. At Pe = 0, problem (5.3.1), (5.3.2) admits an exact closed-form solution for a first-order volume reaction, which corresponds to the linear function /v = c. In this case, we have 

The mean Sherwood number for the solution (5.3.3) is given by the formula 

For an arbitrary dependence of the kinetic function on the concentration, the mean Sherwood number for a spherical particle in a stagnant fluid can be calculated [360] by using the expression 

Formula (5.3.5) guarantees an exact asymptotic result in both limit cases fcv - 0 and kv - oo for any function /v(c). For a first-order volume reaction (/v = c), the approximate formula (5.3.5) is reduced to the exact result (5.3.4). The maximum error of formula (5.3.5) for a chemical volume reaction of the order n = 1 /2 (/v = fc) in the entire range of the dimensionless reaction rate constant fcv is 5% for a second-order volume reaction (/v = c2), the error of (5.3.5) is 7% [360], The mean Sherwood number decreases with the increase of the rate order n and increases with kv. 

Nonspherical particles. For nonspherical particles in a stagnant medium with the first-order volume chemical reaction taken into account, the mean Sherwood number can be calculated by using the approximate expression 

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When a spherical particle exists in a stagnant, suspending gas, its velocity can be predicted from viscous fluid theory for the transfer of momentum to the particle. Perhaps no other result has had such wide application to aerosol mechanics as Stokes (1851) theory for the motion of a solid particle in a stagnant medium. The model estimates that the drag force 2) acting on the sphere is... 

Here Sho is the Sherwood number corresponding to mass transfer of a particle in a stagnant medium without the reaction. Each summand in (5.3.6) must be reduced to a dimensionless form on the basis of the same characteristic length. The value of Sho can be determined by the formula Sho = all/S , where a is the value chosen as the length scale and, is the surface area of the particle the shape factor II is shown in Table 4.2 for some nonspherical particles. 

The dimensionless time, t, for Sh to come within 100x% of the steady value indicates the duration of the unsteady state for Pe = 0, Tq.i == 31.8, and — 2.35. Diffusivities in gases are of order 10" times diffusivities in liquids hence, for particles with equal size and equal exposure, transient effects in a stagnant medium are much more significant in liquids. 

The effect of flow depends solely on a Peclet number formed using the conductance in a stagnant medium, k A/Q), as the characteristic length. Equation (4-60) has wider generality it is valid for a fluid or solid particle of any shape at any Re so long as Pe 0 and the stream far from the particle is uniform. This expression gives a good prediction of the conductance ratio for k/k < 1.2. Equation (3-45) is the special case of Eq. (4-60) for spheres. The next term in the series expansion depends explicitly upon the shape and the orientation of the particle. 

Statement of the problem. Following [367, 368], let us consider stationary diffusion to a particle of finite size in a stagnant medium, which corresponds to the case Pe = 0. We assume that the concentration on the surface of the particle and remote from it is constant and equal to Cs and C), respectively. The concentration field outside the particle is described by the Laplace equation... 

Figure 4.1. Shape factor ratio against perimeter-equivalent factor for particles of various shape in a stagnant medium 1, circular cylinder 2, oblate ellipsoid of revolution 3, prolate ellipsoid of revolution 4, cube...

Suppose that at the initial time t = 0 the concentration in the continuous medium is constant and is equal to C and that a constant surface concentration Cs is maintained for t > 0. The transient mass exchange between a particle and a stagnant medium is described by the equation... 

For non-moving submerged particles (with rigid or mobile surface) in a stagnant medium, mass transfer occurs only by radial diffusion. Re = Gr = 0. whence it can be shown that ... 

In order to include the coupling between the rugged laminar flow in a porous medium and the molecular diffusion, Horvath and Lin [50] used a model in which each particle is supposed to be surrounded by a stagnant film of thickness 5. Axial dispersion occurs only in the fluid outside this stagnant film, whose thickness decreases with increasing velocity. In order to obtain an expression for S, they used the Pfeffer and Happel "free-surface" cell model [52] for the mass transfer in a bed of spherical particles. According to the Pfeffer equation, at high values of the reduced velocity the Sherwood number, and therefore the film mass transfer coefficient, is proportional to... 

The external force based particle velocity in a stagnant fluid medium leads to anumher flm itp, of the particles across any given cross-sectional area. If the total numher of particles per unit volume of the fluid is N, (see equation (2.4.2c)) and the terminal velocity of the particles is Upp then the particle flux tip in terms of numbers of particles crossing a surface area perpendicular to Up, is given by (see Figure 3.1.3A)... 

Consider the gravitational force acting on a spherical particle of mass ntp, radius tp, density pp falling vertically downward in a stagnant liquid of density p, (< pp). (All of our considerations are also valid if the medium is gaseous... 

Limestone dissolution in throwaway scrubbing can be modeled by mass transfer. The mass transfer model accurately predicts effects of pH, Pcc>2> temperature, and buffers. For particles less than 10-20 pm, the mass transfer coefficient can be obtained by assuming a sphere in an infinite stagnant medium. This model underpredicts the absolute dissolution rate by a factor of 1.88, probably because it neglects agitation and actual particle shape. 

In sparged yegsels with a stagnant liquid medium a minimum suspension gas velocity can be defined at which all solid particles are just suspended. Three different experimental techniques have been used to obtain an operational definition of minimum suspension gas velocities ... 

For the case with porous particles, the pore fluid can be treated as a mass transfer medium rather than a separate phase thus enabling it to be combined with the bulk fluid in the overall mass balance. Under plug flow transfer conditions, at the end of each time increment, the pore fluid was assumed to remain stagnant, and only the bulk fluid was transferred to the next section. Based on these assumptions and initial conditions, the concentrations of the polypeptide or protein adsorbate in both liquid and solid phase can be calculated. The liquid phase concentration in the last section C , is the outlet concentration. The concentration-time plot, i.e., the breakthrough curve, can then be constructed. Utilizing this approach, the axial concentration profiles can also be produced for any particular time since the concentrations in each section for each complete time cycle are also derived. 

The question is what the expression for the Magnus force is when the ambient medium is not stagnant (initially) but also exhibits vorticity of its own. The idea is that in such a case it is the particle rotational velocity relative to the fluid vorticity, or the relative sHp, that drives the Magnus lift force. This idea found its way into the textbook by Crowe et al (1998) and can also be found in, e.g., Yamamoto et al (2001) and Derksen (2003). The most general expression for the Magnus Hft force then runs as follows ... 

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