Prandtl layer

Initially it was assumed that no solution movement occurs within the diffusion layer. Actually, a velocity gradient exists in a layer, termed the hydrodynamic boundary layer (or the Prandtl layer), where the fluid velocity increases from zero at the interface to the constant bulk value (U). The thickness of the hydrodynamic layer, dH, is related to that of the diffusion layer ... 

FIGURE 1-6 The hydrodynamic boundary (Prandtl) layer. Also shown (as dotted line), is the diffusion layer. 

In consideration of the hydrodynamic problem alone, it is usually attempted to characterize the studied system by three quantities, the characteristic length / (e.g. the length of the plate in the direction of the flowing liquid or the radius of a rotating disk), the velocity of the flowing liquid outside the Prandtl layer V0 and the kinematic viscosity v. 

The Cartesian coordinates of the system, x, y and z, are referred to the characteristic length / and the components of the velocity vx> vy and vz within the Prandtl layer are referred to the characteristic velocity V0>... 

It holds approximately for the thickness of the Prandtl layer that... 

An important concept in fluid mechanics is the hydrodynamic boundary layer (also known as Prandtl layer) or region where the effective disturbance... 

Potentiostatic techniques, 787, 1115, 1118 and impurities on electrodes. 1120 potential interval measurements, 1121 p-polarized light, 802 Potentiodynamic techniques, 1423, 1438 vs. potentiostatic techniques, 1426 Potentiostatic transients, 1414 difficulties in, 1415 double layer charging, 1416 radicals in, 1416 IR drop in, 1416 Prandtl layer, 1228... 

The structure of turbulence in the transition zone from a fully turbulent fluid to a nonfluid medium (often called the Prandtl layer) has been studied intensively (see, for instance, Williams and Elder, 1989). Well-known examples are the structure of the turbulent wind field above the land surface (known as the planetary boundary layer) or the mixing regime above the sediments of lakes and oceans (benthic boundary layer). The vertical variation of D(x) is schematically shown in Fig. 19.8b. Yet, in most cases it is sufficient to treat the boundary as if D(x) had the shape shown in Fig. 19.8a. 

Helmholtz layer contains the second water molecule layer. From the Helmholtz double layer toward the bulk electrolyte are the diffusion layer and the hydrodynamic layer. In the diffusion layer, the concentration of species changes from that of the bulk electrolyte to that of the electrode surface. The diffusion layer does not move, but its thickness will decrease with increasing bulk electrolyte flow rate to allow higher reaction rates. The diffusion layer thickness is inversely proportional to the square root of the flow rate. The hydrodynamic layer or Prandtl layer has the same composition as the bulk electrolyte, but the flow of the electrolyte decreases from that of the bulk electrolyte to the stationary diffusion layer. 

The atmospheric boundary layer may be divided into two horizontal layers. The lowest layer, extending not more than 100 m above the surface, is the surface boundary layer (or the Prandtl layer). Here the vertical fluxes of momentum, heat, and water vapor may be assumed independent of height, and the atmospherie structure is primarily determined by characteristics of the earth s sm-face, thermal stratification, and the variation of wind with height. 

Dynamic properties of i.s.e.s. differ greatly for various electrode types and constructions. When the capacitance of analyte/active surface interface is the only cause of response delay, then relaxation time (or time constant of first-order step-response characteristic) is in the order of milliseconds. When the transport of ions across the dynamic Prandtl layer to the surface of the i.s.e. is the main factor (i.e., this transport is the slowest process of equilibrium reinstallation), for a mixing velocity of about lOcm/s a relaxation time of several seconds occurs. This is typical of solid-membrane electrodes with the exception of glass ones. On the other hand, the limited rate of the exchange process in the liquid membrane, the small diffusion flux of the tested ions into the membrane, the slow dynamics for the creation of diffusion potential and the solubility of the active component of the membrane in the testing solution are the main reasons for the slow response of liquid ion-exchanger electrodes (time constants 10-30 s or even more). 

If the solution flow is parallel to the electrode surface and perpendicular to the diffusion direction, the Prandtl layer thickness, 5pj, and the Nemst diffusion layer thickness, 5n, in a steady state are related as follows [2] ... 

Figure 7.5 illustrates a physical meaning of both the Prandtl layer and the Nemst diffusing layer. 

The Prandtl layer represents the hydrodynamic layer where the flow speed of the solution is changing, and the Nemst diffusion layer represents the layer where the concentration of electrochemically active species is changing. At the distance below 8pp the velocity of solution is gradually reducing down to zero, and at the distance below 8n, the concentration of an electrochemically active species is gradually reducing down to c ... 

The reader may want to better understand Figure 7.4 and Equation 7.16 by calculating the ratio of the Nemst diffusion layer to the Prandtl layer for an aqueous solution species at a given temperature and pressure using the viscosity and density of water from [Chapter 10, Table 10.22] and an electrolyte diffusion coefficient from [Chapter 10, Table 10.12]. The result of such calculations should show that at ambient conditions 8 is about 10 times smaller than 8pr. 

FIGURE 7.5 Illustration of the Nernst diffusion layer, 5, and the Prandtl layer, Sp, n is the solutions velocity at (theoretically) infinite distance from the electrode, and c" is the huUc solution concentration. 

Electrochemical kinetics significantly depends on the thickness of the Nemst diffusion layer, while hydrodynamics provides the thickness of the Prandtl layer. The ratio of the Nemst diffnsion layer to the Prandtl layer can be calculated using the properties of the solntion. 

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