## Water Boundary Layer Resistance

Exact models for mass transfer through the WBL exist only for some simple flow arrangements, such as the flow through ducts and pipes, and parallel flow along an absorbing flat plate [17-20]. Starting at the leading edge of the plate, the momentum of the water that is immediately adjacent to the plate is reduced due to surface friction. As the water moves along the plate, this retarded water layer in turn attenuates the momentum of the water layers at larger distance from the surface, which results in the development of a viscous sublayer, with a thickness that increases with distance downstream of the leading edge. Similarly, analytes are removed from a layer with a thickness that increases downstream, leading to the development of a concentration boundary layer. With increasing thickness of this layer, transport by eddy diffusion becomes increasingly important, since turbulent diffusion coefficients increase with increasing distance from the surface [19,21]. At large distances from the leading edge, a steady-state concentration profile is established that no longer depends on the distance along the plate. Equations for the short-plate limit (growing concentration boundary layers) and the long-plate limit (distance-independent concentration boundary layers) have been given by Opdyke et al. [22] for hydrodynamically smooth flows (i.e. flows along surfaces where the roughness elements are embedded in the viscous sublayer). The (surface averaged) mass-transfer coefficients for the short-plate limit are given by [22,23]

where n is the kinematic viscosity of the water, L is the length of the plate, and u* is the friction velocity, which is frequently used in the literature on hydrodynamics to parameterise the shear stress (t)

where p is the density of water. In turbulent flows, u* can be interpreted as the characteristic eddy velocity relative to the main stream [24,25]. The friction velocity for an essentially laminar flow along a flat surface is related to the free-stream velocity (U) by [18]

Equation (7.24) is arranged so as to stress that it is dimensionally consistent (i.e. u* has the same dimension as the main stream velocity U, and v/(UL) is dimensionless). The transition from laminar to turbulent flow takes place at values of UL/v>4 x 106 when special precautions are taken to reduce the turbulence intensity of the main flow [17]. When no such precautions are taken, the transition to turbulent flow takes place at lower values, i.e. UL/v> 350,000 to 500,000, depending on the turbulence intensity of the main flow [17].

In the long-plate limit, the mass-transfer coefficients are given by [19,22]

and u* (for fully developed turbulent flow) may be estimated from the free-stream velocity by [18]

For more complex scenarios, such as mass transfer for cylinders and packed bed reactors, empirical correlations have been established of the form [18,26]

ReSc1/3

where the (dimensionless) Sherwood (Sh), Reynolds (Re) and Schmidt (Sc) numbers are defined by k d

Dw where d is a conveniently chosen characteristic length scale and u a characteristic velocity. The constant B in Eq. (7.27) is of the order 1 and mx — 0.5 (range—0.3 to —0.7). For the case of mass transfer to a cylinder with its main axis perpendicular to the flow, d equals the diameter of the cylinder, B — 0.6 and m — —0.487, which is valid for the range 100 < Re <3500 and 1000 <Sc< 3000. It follows from Eq. (7.27) that mass-transfer coefficients are proportional to D2/3 and to the flow velocity U0 5.

Equations (7.22) and (7.25) could be used for passive samplers with a planar configuration. It should be realised, however, that in many situations, the flow near the sampler surface may vary in both time and space. The sampler may be mounted in a protective cage in a zigzag or twisted configuration, and the main flow may generate vortices when passing through ventilation holes or over sharp edges. Furthermore, the sampler surface may bend, twist or vibrate depending on flow velocity, angle of incidence, sampler material. In addition, the flow velocity may vary along the sampler surface, where even dead spots may exist as a result of the mounting pattern. Similarly complex hydrodynamics may exist around samplers with a cylindrical configuration. Despite the complexity of the hydrodynamics near passive samplers, some general conclusions remain, however. First, the number of variables in experiments on mass transfer through the boundary layer may be reduced by correlating the appropriate dimensionless numbers Sh, Re and Sc for a given sampler geometry. Second, a wide number of such empirical correlations from the engineering literature suggests that Sh typically is proportional to Sc1/3, indicating that kw be proportional to D2/3 [18,27]. This in turn indicates that the effective boundary layer thickness increases with increasing diffusion coefficient according to dw^D1/3. Third, the effective WBL thickness, though useful for visualising the extent to which the concentration gradient penetrates into the main flow, should not be misinterpreted as the thickness of physically unrealistic entities like a stagnant film or an unstirred boundary layer. Fourth, for a given geometry and flow, the kw values for small samplers can be expected to be larger than for large samplers. Fifth, kw increases with flow velocity, for a given PSD geometry, but its absolute value is difficult to predict.

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