First-order volume

Comparisons with experimental data have been effected by Batchelor. In careful experiments, Kops-Werkhoven and Fijnaut (1981) obtained —6 + 1 for the coefficient of the first-order volume fraction term in the sedimentation velocity [cf. Eq. (4.9)] and 1.3 0.2 for the corresponding term in the diffusion coefficient [cf. Eq. (4.12)]. The latter term is derived from light-scattering experiments. For monodisperse suspensions, these results agree reasonably well with Batchelor s predictions. 

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... 

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... 

Moderate Peclet Numbers. For spherical particles, drops, and bubbles (under limiting resistance of the continuous phase), in the case of a first-order volume reaction, the mean Sherwood number can be calculated [358] by the formula... 

Maximum errors of formula (5.3.8) and of the cubic equation (5.3.9) for different flows past particles, drops, or bubbles at high Peclet numbers in the presence of a first-order volume chemical reaction... 

For a first-order volume reaction at Pe = 0, the exact closed-form solution of our problem reads... 

The method of asymptotic analogies (see Section 4.1) permits one to generalize formulas (5.4.2)—(5.4.4) to cavities of an arbitrary shape. In the special case of a first-order volume reaction, we obtain the formula... 

For a first-order volume reaction, the mean concentration over the volume is calculated as follows ... 

In Figure 5.2, the dependence of the mean Sherwood number on the dimensionless parameter kv is shown for a first-order volume chemical reaction in the problem of quasi-steady-state mass transfer within a drop for the extreme values Pe = 0 (formula (5.4.2)) and Pe = oo (formula (5.4.9)) of the Peclet number. The dashed line corresponds to the rough upper bound (5.4.8). For moderate Peclet numbers (0 < Pe < oo), the mean Sherwood number gets into the dashed region bounded by the limit curves corresponding to Pe = 0 and Pe = oo. One can see that the variation of the parameter Pe (for fcv = 0(1)) only weakly affects the mean influx of the reactant to the drop surface, i.e., one cannot achieve a substantial increase in the Sherwood number by any increase in the Peclet number. In the special case fcv = 10, the maximum relative increment of the mean Sherwood number caused by the increase in the Peclet number from zero to infinity is only... 

Figure 5.2. The mean Sherwood number against the dimensionless rate constant of a first-order volume chemical reaction for the inner problem (continuous lines the lower line corresponds to Pe = 0, the upper line, to Pe = oo). The dashed line corresponds to a zero-order reaction...

For large rate constants kv of the volume chemical reaction, a thin diffusion boundary layer is produced near the drop surface its thickness is of the order of ky1//2 at low and moderate Peclet numbers, and the solute in this layer has time to react completely. As the Peclet number is increased further, because of the intensive liquid circulation within the drop, there is not enough time to complete the reaction in the boundary layer. The nonreacted solute begins to get out of the boundary layer and penetrate into the depth of the drop along the streamlines near the flow axis. If the circulation within the drop is well developed, a complete diffusion wake is produced with essentially nonuniform concentration distribution that pierces the entire drop and joins the endpoint and the origin of the diffusion boundary layer. In case of a first-order volume chemical reaction, an appropriate analysis of convective mass transfer within the drop for Pe > 1 and kv > 1 was carried out in [150,151]. It should be said that in this case, in view of the estimate (5.4.8), which is uniform with respect to the Peclet number, the mass transfer intensity within the drop is bounded by the rate of volume chemical reaction. 

This equation with T replaced by C and 6 > 0 is encountered in mass transfer problems with a first-order volume reaction. 

First, a strong volume change can be excited by a large spectrum of different physical and chemical factors such as temperature, electrical voltage, pH, concentration of organic compounds in water, and salt concentrations. The possibility of a first-order volume phase transition in gels was suggested by K. Dusek and... 

Certain polyelectrolyte hydrogels display first-order volume phase transitions, with hysteresis, in response to external stimuli 2,7). Hydrophobic polyelectrolyte hydrogels, in particular, may undergo discrete transitions in response to changes in external pH. Such hydrogels have been considered as chemically-sensitive mechanical switches 8). 

Gas (or gas with homogeneous catalyst) heat of reaction endothermic reaction rate, fast capacity 0.001-200 L/s good selectivity for consecutive reactions and irreversible first order volume of reactor 1-10000 L OK for high pressures or vacuum. For temperatures < 500 °C. For temperatures > 500 °C use fire tube. For example, used for such homogeneous reactions as acetic acid cracked to ketene. Liquid (or liquid with homogeneous catalyst) heat of reaction endothermic reaction rate, fast or slow capacity 0.001-200 L/s good selectivity for consecutive reactions volume of reactor 1-10000 L OK for high pressures. For temperatures... 

Dukhin et al. [83-85] have performed the direct calculation of the CVI in the situation of concentrated systems. In fact, it must be mentioned here that one of the most promising potential applicabilities of these methods is their usefiilness with concentrated systems (high volume fractions of solids, 4>) because the effect to be measured is also in this case a collective one. The first generalizations of the dynamic mobility theory to concentrated suspensions made use of the Levine and Neale cell model [86,87] to account for particle-particle interactions. An alternative method estimated the first-order volume fraction corrections to the mobility by detailed consideration of pair interactions between particles at all possible different orientations [88-90]. A comparison between these approaches and calculations based on the cell model of Zharkikh and Shilov [91] has been carried out in Refs. [92,93],... 

KEL 06] Kelly J.V., Gleeson M.R., Close C.E. et al., Temporal response and first order volume changes during grating formation in photopolymers . Journal of Applied Physics, vol. 99, no. 113105, 2006. 

On cooling through Tc in the range 0.10 < x < 0.30, there is a first-order volume contraction associated with a discontinuous increase in the volume... 

On cooling through Tc in the range 0.10 < x < 0.30, there is a first-order volume contraction associated with a discontinnons increase in the volume fraction of the more conductive FV and FM phases (Radaelli et al., 1996 Huang et al., 1998) where electrons tunnel from Mn(III) to Mn(IV) neighbors in a time tj, <. In contrast, the transition from the paramag-... 

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