Marker velocities, equations

In equation (2) Rq is the equivalent capillary radius calculated from the bed hydraulic radius (l7), Rp is the particle radius, and the exponential, fxinction contains, in addition the Boltzman constant and temperature, the total energy of interaction between the particle and capillary wall force fields. The particle streamline velocity Vp(r) contains a correction for the wall effect (l8). A similar expression for results with the exception that for the marker the van der Waals attraction and Born repulsion terms as well as the wall effect are considered to be negligible (3 ). 

The effective mobility, expressed by Equation 6.16, can be directly calculated from the observed mobility by measuring the electroosmotic mobility using a neutral marker, not interacting with the capillary wall, which moves at the velocity of the EOF. Accordingly, the effective mobility p of cations in the presence of cathodic EOF is calculated from p ts by subtracting p gf ... 

Equation 3.23 for the velocity of a local C-frame with respect to the E-frame is therefore the velocity of any inert marker with respect to the E-frame. The assumptions that fli and il2 are each constant throughout the material, and thus that there are no changes in overall specimen volume during diffusion, permit the use of Eq. 3.19 to derive the unique choice of the E-frame. 

Equation 3.23 gives the velocity of the local C-frame with respect to the V-frame (i.e., the velocity of local mass flow measured by the velocity of an embedded inert marker relative to the ends of a diffusion couple such as in Figs. 3.3 and 3.4). The measurement of and D at the same concentration in a diffusion experiment thus produces two relationships involving Di and D2 and allows their determination. In the V-frame, the diffusional flux of each component is given by a simple Fick s-law expression where the factor that multiplies the concentration gradient is the interdiffusivity D. In this frame, the interdiffusion is specified completely by one diffusivity. 

In a capillary tube, the applied electric field E is expressed by the ratio VILj, where V is the potential difference in volts across the capillary tube of length Lj (in meters). The velocity of the electro-osmotic flow, Veo (in meters per second), can be evaluated from the migration time t of (in seconds) of an electrically neutral marker substance and the distance L, (in meters) from the end of the capillary where the samples are introduced to the detection windows (effective length of the capillary). This indicates that, experimentally, the electro-osmotic mobility can be easily calculated using the Helmholtz-von Smoluchowski equation in the following form ... 

The computation starts with a predictor-corrector algorithm for the determination of velocity field at to + At. In the predictor stage of the solution algorithm, the pressure is replaced by an arbitrary pseudo-pressure P (which in most cases is set equal to zero at full cells), and tentative velocities are then calculated. A pseudo-pressure boundary condition is applied in surface cells to satisfy the normal stress condition. Since pressure has been ignored in the full cells, the tentative velocity field does not satisfy the incompressible continuity equation. The deviation fi om incompressibility is used to calculate a pressure potential field [J/, which then is used to correct the velocity field. In the final steps, the velocity boundary conditions are calculated, the new location of free surface is determined by tracking the markers, and the velocity boundary conditions associated with the new fluid cells are assigned (Fig. 5). 

On the other hand, according to the law of conservation of matter, the vector sum of components fluxes in the laboratory reference frame (connected with one of the ends of an infinite, on diffusion scale, diffusion couple) must be equal to zero. According to Darken, the alloy provides fluxes balancing at the expense of lattice movement as a single whole at some certain velocity, u, which is measured by inert markers ( frozen in lattice) displacement. Correspondingly in the laboratory reference frame, components fluxes acquire a drift component (similar to Galilean velocity transformation equations) as... 

As proven, in order to find K-planes as well as the concentration dependencies for partial coefficients, one should have the velocity curve. The usual experimental method, providing this, is the multilayer method. The diffusion couple consists of several thin layers (generally, 10-20 gm thick) with inert markers placed between them. After annealing, the displacement of each marker plane is measured, and the velocity of each plane is determined by the equation [34, 35]... 

The geometry and mesh arrangement in the fluid region are exactly the same as those of the steady-state subchannel analysis code. Figure 6.60 shows the entire algorithm. The momentum conservation equations for three directions and a mass conservation equation are solved with the Simplified Marker And Cell (SMAC) method [32]. In the SMAC method, a temporary velocity field is calculated, the Poison equation is solved, and then the velocity and pressure fields are calculated as shown in Fig. 6.61. The Successive Over-Relaxation (SOR) method is used to solve a matrix. 

For the MIEC membranes without external circuit, the overall charge balance is applied or Zz,-/,- = 0 and the local velocity of inert marker is negligible, 0 = 0. Accordingly, the transport flux of charged defects in the MIEC membrane at steady state can be derived (one-dimensional model) from Equations [Al] to [A3] as ... 

As to the marker-particle tracking analysis, the location of marker-particles is estimated by integrating the velocity vectors with respect to time under the assumption of massless particles. The time in which the marker-particles stay in the flow channel corresponds to the residence time. The stress the marker-particles undergo in the flow channel means the stress magnitude and is represented by the following equation. 

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