Dimensionless flux

Accordingly, to convert from the dimensionless properties supplied, the conversion factor for the dimensionless flux value V". As previously derived,... 

The modeling procedure can be sketched as follows. First an approximate description of the velocity distribution in the turbulent boundary layer is required. The universal velocity profile called the Law of the wall is normally used. The local shear stress in the boundary layer is expressed in terms of the shear stress at the wall. From this relation a dimensionless velocity profile is derived. Secondly, a similar strategy can be used for heat and species mass relating the local boundary layer fluxes to the corresponding wall fluxes. From these relations dimensionless profiles for temperature and species concentration are derived. At this point the concentration and temperature distributions are not known. Therefore, based on the similarity hypothesis we assume that the functional form of the dimensionless fluxes are similar, so the heat and species concentration fluxes can be expressed in terms of the momentum transport coefficients and velocity scales. Finally, a comparison of the resulting boundary layer fluxes with the definitions of the heat and mass transfer coefficients, indiates that parameterizations for the engineering transfer coefficients can be put up in terms of the appropriate dimensionless groups. 

Dimensionless flux distributions may be obtained by multiplying the neutron density distributions by the neutron speed ratio v/vo ... 

Since our model system is in dimensionless units, we cannot find the flux, j, directly, but we can find the dimensionless flux at point X, Jxi, defined as... 

When we run a simulation, we record the dimensionless flux, J, as a function of the dimensionless potential, 9. To obtain a real voltammogram, the results of this simulation are simply scaled by the real values of c. Da and e. As a consequence, the results of a single simulation will apply to any set of these three values, so we don t need to run another simulation each time we change, for example, the concentration. This is exactly why we introduced dimensionless variables in the first place. 

To obtain a dimensionless voltammogram, we then record the dimensionless flux of species A, Ja, at the electrode surface as it varies with the dimensionless potential, 9 ... 

Using the same procedure as we used for the microdisc in Section 9.1.3, we find that the dimensionless flux across the band cross-section is... 

Mahajani and Kolah (1996) successfully extend the approach of Doherty and co-workers to packed columns, where liquid phase backmixing is totally absent. This approach relies on the performance of mass balances over differential elements of length dz in both rectifying and stripping sections, which can be re-written as functions of dimensionless numbers e.g. Da, HTU, dimensionless flux and ratio of characteristic times in the vapor and liquid sides f3 = x x k ))). 

Dimensionless flux during deliquoring averaged over the deliquoring time... 

For times, when exp(x) is much greater than the mass flux, it can be seen that the second derivative in time of the dimensionless flux in Equation (9.93) can be neglected compared with the first derivative. The resulting expression is the familiar expression for surface flux from the Fourier parabolic governing equation for constant wall concentration in a semi-infinite medium and is given by... 

We also introduce a dimensionless flux parameter y given by... 

It would be usefiil to transform Equation 8.50 using dimensionless variables. Having defined the dimensionless flux, J, thickness, X, and electromigration flux, Jem. as... 

The dimensionless fluxes of the fast and slow demicellization processes, denoted by ([) , respectively, can be expressed as follows [150] ... 

Figure 3.1.13 shows the enhancement and decrease, respectively, of the (dimensionless) flux compared to the isothermal case for different values of T2 for a fixed value of Ti (873 K). As expected, the flux from plate 1 to plate 2 is enhanced for Ti > T2 and decreases for T < T2. 

On the other hand, for Pe < 10 the role of electrostatic interaction becomes negligible and the dimensionless flux is given by the limiting SL law, that is,... 

The dimensionless variable 4> accounts for the additional flux of bromide ions due to the light [39] and is assumed to be proportional to the light intensity. It is known that this additional flux of bromide ions completely suppresses oscillations within the sample if the light intensity exceeds some critical value, which can be experimentally determined [39]. In our simulations below, we define 4>c as a critical value of the dimensionless flux of bromide ions (that corresponds to the respective critical value of the light intensity), above which the chemical oscillations are completely suppressed [39]. Generally, 4>c depends on the reaction parameters and the physical properties of the gel. Using the simulations, we determined that for a system with the above parameters and the size provided below this value is 4>c = 3.2 X 10 . Finally, we note that if we set 4> = 0 in the above equation, we recover our expression for the system in the absence of light, that is, Equation 3.5. Our simulation box is 100 x 10 x 10 nodes in the x-, y-, and z-directions, respectively, unless specified otherwise. We assume no-flux boundary conditions for u at the surfaces of the gel [18]. 

Prandtl number, ii/k, dimensionless flux of heat transfer, FL/L a constant rate of total beat removal, FL/0 const... 

E FCCH2OH/ Dimensionless flux when concentrations and distanc [33,51] F=r//DCo... 

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