Quasi-steady Situation Research Articles

Particle interactions in electrophoresis: III. Axisymmetric motion of multiple spheres

A semianalytic study is presented for the axisymmetric electrophoretic motion of a finite chain of rigid spheres along their line of centers. The spheres may differ in radius and in zeta potential and they are allowed to be unequally spaced. The thin-double-layer assumption is employed. Using a collocation technique, the electrostatic and hydrodynamic governing equations are solved in the quasisteady situation and the particle interaction effects are calculated for various cases. The numerical solutions on convergence can be obtained to any desired degree of accuracy by the increase of the collocation points on each sphere surface. For the electrophoretic motion of two-sphere systems, our results for the particle interaction coefficients agree very well with the exact calculations using spherical bipolar coordinates. For the cases of two or three spheres touching one another, the numerical calculations for the particle interaction effects, which compare quite favorably with the formulae analytically derived, illustrate that the large spheres dominate the migration of the linear cluster of spheres. All of our data also demonstrate the fact that the electrophoretic velocity of each particle is unaffected by the presence of the others if all of the particles have the same zeta potential.

A theoretical and experimental study of carbon combustion in stagnation flow

The combustion of solid carbon in the stagnation flow of an O 2CO 2inert gas stream is theoretically and experimentally studied for the quasi-steady situation of constant surface temperature. In the theoretical phase chemical reactions considered include the surface CO 2 and CCO 2 reactions and the gas-phase COO 2 reaction. By using generalized species-enthalpy coupling functions derived without any limit or near-limit behavior, effects of gas-phase Damköhler number, surface Damköhler numbers, surface temperature, CO 2 mass fraction in the oxidizer flow, flow configuration, and the incompressibility assumption on the mass burning rate of carbon are investigated. Calculated results emphasize the intimate coupling between the gas-phase and the surface reactions, and the fact that the combustion mechanism is intermediate of those assuming frozen mode and attached or detached flame modes. It is also shown that the density variation in the combustion field exerts a suppressing effect on the mass burning rate in the kinetically controlled regime, while it exerts a promoting effect in the diffusion controlled regime. By using a definite set of kinetic and thermophysical parameters, it is further demonstrated that the theoretical calculations agree well with both existing experimental data as well as those determined in the present study.

Thermocapillary motion of a fluid droplet normal to a plane surface

The thermocapillary motion of a fluid droplet in a constant applied temperature gradient perpendicular to a plane surface is analyzed. The exact solution for the temperature and velocity fields in the quasisteady situation is obtained by using spherical bipolar coordinates, and the asymptotic formulas for the droplet velocity are derived from a method of reflections. The boundary effect on the thermocapillary motion is found to be weaker than that on the motion driven by a gravitational force. Even so, the interaction between the plane surface and the droplet can be very strong when the gap thickness approaches zero. For the motion of a droplet normal to a solid plane, the effect of the plane surface is to reduce the migration velocity of the droplet. For the case of droplet migration toward a free fluid surface due to thermocapillarity, the droplet velocity can be either greater or smaller than that which would exist in the absence of the plane surface, depending on the relative thermal conductivity of the droplet and its relative distance from the plane. In general, a free surface exerts less influence on droplet movement than a solid surface docs.

The axisymmetric thermocapillary motion of two fluid droplets

An exact analytical study is presented for the thermocapillary motion of two spherical droplets in a constant applied temperature gradient along their line of centers. The droplets may be formed from different fluids and have arbitrary radii. The appropriate energy and momentum equations are solved in the quasisteady situation using spherical bipolar coordinates and the droplet velocities are calculated for various cases. The interaction between droplets can be very strong when the surface-to-surface spacing approaches zero. The influence of the interaction, in general, is stronger on the smaller droplet than on the larger one. For the thermocapillary motion of two identical liquid droplets, both migrate faster than the velocity they would possess if isolated. For the specific case of two gas bubbles with equal radii, there is no particle interaction for all separation distances. A comparison between our exact results and predictions from a method of reflections is made. The asymptotic formula for the droplet velocities up to O(r 12 −6) , where r 12 is the center-to-center distance between the droplets, is found to underestimate the effect of droplet interactions; the error can be significant when the droplet surfaces are less than a quarter of the sum of the radii apart.

Particle interactions in electrophoresis: I. Motion of two spheres along their line of centers

The electrophoretic motion of two charged colloidal spheres with very thin electrical double layers in a constant applied electric field along their line of centers is considered. The particles may differ in radius and in zeta potential at the surface. The electrostatic and hydrodynamic governing equations are solved in the quasi-steady situation using bipolar coordinates and the electrophoretic velocities of particles are calculated for various cases. The interaction effect between particles can be very significant when the distance between particle surfaces gets close to zero. The particle with smaller zeta potential is speeded up by the motion of the other, which is retarded at the same time by the motion of the former one, if the two spheres have unequal zeta potentials of the same electrical sign. For two particles of different signs in zeta potential, motions of both are hindered by each other. The influence of the interaction between particles in general is stronger on the smaller one than on the larger one. For the special case of two electrophoretic spheres with identical zeta potentials, there is no particle interaction for all particle sizes and separations.

Quasi-steady and transient combustion of a carbon particle: Theory and experimental comparisons

The combustion of a carbon particle in an O 2 /CO 2 /Inert environment is theoretically studied for the quasi-steady situation of constant particle temperature and the transient situation with particle heating and radiative heat transfer. Chemical reactions considered include the surface C−O 2 and C−CO 2 reactions and the gas-phase CO−O 2 reaction. Generalized species-enthalpy coupling functions are derived for this chemically-complex system without assuming any limit or near-limit behavior, thereby providing useful insight into the appropriate conserved quantities in carbon combustion as well as facilitating further theoretical developments. Calculated results emphasize the intimate coupling between the gas-phase and surface reactions, the importance of radiative heat loss, and the fact that the particle combustion mechanism is intermediate of those assuming frozen and equilibrium gas-phase reactions. By using a restrictive set of kinetic and thermophysical parameters, it is further demonstrated that the theoretical calculations agree well with existing experimental data for both the quasi-steady and transient combustion situations.

Heat and mass exchange of a droplet in a polyatomic gas

Lees’ method, combined with a generalized Grad’s moment procedure, is used to calculate the heat and mass transport to and from a droplet in a polyatomic gas. A near-equilibrium and quasisteady situation is assumed. The results are represented by the transport coefficients of a symmetric Onsager matrix. The coefficients depend on the collision numbers of the internal degrees of freedom. Simple results are obtained by performing approximations analogous to the Mason and Monchick analysis for the case of the heat conductivity.