Leveque approximation

A further simplification (known as the Leveque approximation [57]), linearises the profile in the vicinity of the surface (where a local coordinate x, perpendicular to the surface, is taken) ... 

For hydrodynamic electrodes, in order to solve the convective-diffusion equation analytically for the steady-state limiting current, it is necessary to use a first-order approximation of the convection function(s) (such as the Leveque approximation for the channel). These approximate expressions for the steady-state mass transport limited currents were introduced in Section 4 (see Table 5). 

Leveques approximation Concentration BL is thin. Assume velocity profile is linear. High mass velocity. Fits liquid data well. 

Assuming both diffusion in the direction of convective flow and diffusion-al side-edge effects to be negligible, Levich arrived at the equation (under the Leveque approximation)... 

Finite difference methods have been used bpth to test the assumptions made in the derivation of eqn. (27) under the Leveque approximation [35] and to solve electrochemical diffusion-kinetic problems with the full parabolic profile [36-38]. The suitability of the various finite difference methods commonly encountered has been thoroughly investigated by Anderson and Moldoveanu [37], who concluded that the backward implicit (BI) method is to be preferred to either the simple explicit method [39] or the Crank-Nichol-son implicit method [40]. 

At the channel electrode, this problem has been treated by Matsuda, initially employing the reaction-layer approximation [101], but subsequently the full coupled convective-diffusion equations were solved [102] (at the level of the Leveque approximation). It was shown that, to within 1%, the equation... 

In addition, the so-called Leveque approximation [Leveque, 1928] has also been extended and applied to power-law fluids. The key assumption in this approach is that the temperature boundary layer is confined to a thin layer of the fluid adjacent to tube wall. This is a reasonable assumption for high flow rates and for short tubes, i.e. large values of Gz. A linear velocity gradient can then be assumed to exist within this thin layer ... 

The use of the Leveque approximation in this instance leads to the following expression for the mean Nusselt munber over the entry length [Bird, 1959] ... 

Attention was then turned to the electrochemical behavior shown in Figure 24. Because of the well-defined and known hydrodynamics of the cell, the dependence of the transport-limited current on the flow rate can be calculated for both ECE and DISPl processes using analytical theory and the Leveque approximation. The predicted behavior is governed by a normalized rate... 

Obviously, this approximate treatment fails when <5, > R, as then the Leveque linearisation is clearly unapplicable. For an approximation to account for lateral effects of nonactive walls in box-like channels, see ref. [46]. 

For the validation purpose, the data from Ref. [100] are used. In this study, absorption experiments were carried out using a baffled vessel operated batch-wise with respect to liquid, and the experimental results were compared with an approximate analytical solution based on the Leveque model. The authors proposed a two-reaction-plane model and achieved a good agreement between theoretical and experimental absorption rates (see Section 9.5.4.5). 

B. Tubes, approximate solution k d A 1/3 Nshzt = = 1.077Pj (NM k A / J 1/3 JVs4,g = = 1.615 1 (NM [T] For arithmetic concentration difference. w >400 p Dx Leveque s approximation Concentration BL is thin. Assume velocity profile is linear. High mass velocity. Fits liquid data well. [141] p. 166... 

Levich [32] solved eqn. (1) for a channel electrode by invoking an approximation originally introduced in 1928 by Leveque in his theory of heat transfer in pipes [33]. In the present context, this simplification can be written as... 

Thus, replacing x with y from Eq. 10.126 into the diffusion term, we have the approximate expression (after Leveque 1928) ... 

If it is assumed that the velocity profile is flat as in rodlike flow, the solution is more easily obtained (SI). A third solution, called the approximate Leveque solution, has been obtained, where there is a linear velocity profile near the wall and the solute diffuses only a short distance from the wall into the fluid. This is similar to the parabolic velocity profile solution at high flow rates. Experimental desigrv equations are presented in Section 7.3D for this case. 

Lev que s problem was extracted from the rescaled mass balance in Equation 8.28. As can be seen, this equation is the basis of a perturbation problem and can be decomposed into several subproblems of order 0(5 ). The concentration profile, the flux at the wall, and consequently the mixing-cup concentration (or conversion) can all be written as perturbation series on powers of the dimensionless boundary layer thickness. This series is often called as the extended Leveque solution or Lev jue s series. Worsoe-Schmidt [71] and Newman [72] presented several terms of these series for Dirichlet and Neumann boundary conditions. Gottifredi and Flores [73] and Shih and Tsou [84] considered the same problem for heat transfer in non-Newtonian fluid flow with constant wall temperature boundary condition. Lopes et al. [40] presented approximations to the leading-order problem for all values of Da and calculated higher-order corrections for large and small values of this parameter. 

The approximations given by Equations 8.35 are the solution to Leveque s problem given in Equation 8.30 with a linear wall reaction. Since the formulation of the problem leads to a linearized velocity profile in a planar boundary layer, laminar flows (parabolic velocity profiles) in curved channels are more susceptible to present higher deviations from these results. For a fully developed flow in a round tube, the error associated with Equation 8.35b is 1.4 and 0.13% for aPe ,lz equal to 100 and 1000, respectively. Lopes et al. [40] observed that these differences are visible mainly for Da — 00 and calculated corrections to account for these effects. It was shown that in the mass transfer-controlled limit. 

Ghez R. Mass transport and surface reactions in Leveque s approximation. International Journal of Heat and Mass Transfer 1978 21 745-750. 

The earliest studies were those of Tien [49], Suckow et al. [50], and Crozier et al. [51]. The work of Tien involved taking the approximate velocity profile found by Schecter [52] and then using it to solve the energy equation (assuming constant physical properties and no viscous dissipation) to yield average temperature and the Nusselt number. Suckow et al. used the exact velocity distribution instead of the approximate profile to solve the energy equation. Crozier and co-workers used the Leveque technique to find appropriate equations for heat transfer. 

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