We present closed-form solutions for high Schmidt number mass transfer in a hydrodynamically fully developed turbulent flow. Governing equations for the near- and far-field are developed for a large class of boundary conditions (BCs) for which the mass flux is a function of the concentration at the wall. We show that for this class of BCs, which includes nonlinear wall reactions, the mass transfer coefficient is independent of the BC and the Sherwood correlation is therefore universal. For Dirichlet, Neumann and Robin BCs, the far-field solutions are in good correspondence with the method of separating variables and near-field solutions are in good agreement with numerical simulations. However, in contrast with the far-field solutions, the Sherwood correlation in the near-field depends on the specific BC. As an example of nonlinear BCs, solutions for a second-order wall reaction are derived which are compared with numerical simulations and found to be in excellent agreement.
The exchange of mass and/or heat of a turbulent flow with its bounding surface occurs in several areas of engineering. One classical example is the corrosion of pipe walls owing to the presence of a corrosive species in water, such as carbon dioxide (Sydberger & Lotz 1982, Sarin et al. 2004) or chlorine (Rossman et al. 1994; Clark & Haught 2005; Al-Jasser 2007). Another classical example is the heat transfer through conduit walls, as is relevant, for example heat exchangers (Bergmann & Fiebig 1999) and district heating (Webb & Kim 2005). In this paper, we will concentrate on turbulent mass transfer problems, although the results are equally applicable to heat transport problems where buoyancy effects are negligible.
A typical quantity of interest is the realized mass (heat) flux at the wall, represented by the mass (heat) transfer coefficient and in dimensionless form by the Sherwood number Sh (Nusselt number). They will depend on the Reynolds number Re and the Schmidt number Sc (Prandtl number), the former representing the ratio between inertial and viscous forces, and the latter the ratio between kinematic viscosity and molecular (thermal) diffusivity. These relations often are presented as power-laws Sh=b1Reb2Scb3, where b1,b2 and b3 are coefficients. Determination of these coefficients has been the objective of many experimental, numerical and theoretical investigations and a selection of Sh correlations for high Sc is presented in the electronic supplementary material. The power-law exponents depend critically on the near-wall behaviour of the eddy diffusivity and several theoretical models based on theory of turbulent boundary layers have been developed over the years (Kader & Yaglom 1972; Aravinth 2000).
The mass transport equation (see §3) is a linear partial differential equation with variable coefficients. Many analytical methods therefore make use of the method of separating variables (Sleicher et al. 1970; Notter & Sleicher 1971, 1972; Biswas et al. 1993; Weigand et al. 2001; Weigand 2004), and with great success: the classical power-law relationship Sh=0.016Re0.88Sc0.33 proposed by Notter & Sleicher (1972) remains widely used today (Rossman et al. 1994). However, there are some drawbacks and limitations to this method. First, the determination of each of the eigenfunctions and eigenvalues has to be done numerically because of the variable coefficients in the problem. Second, the method of separating variables is applicable to linear boundary conditions (BCs) only; obtaining solutions for nonlinear BCs, as of importance, for example, biofilm growth (Munavalli & Mohan Kumar 2004; Noguera & Morgenorth 2004) is generally not possible.
In this paper, we will focus on the development of asymptotic solutions for turbulent mass transport at high Sc and Re for linear (notably Dirichlet, Neumann and Robin) and nonlinear BCs. Closed-form solutions for the near- and far-field will be presented, where the near-field is the region where the concentration boundary layer is developing and the mass transfer coefficient will be dependent on the streamwise direction. The far-field is defined as the region where the mass transfer coefficient has become constant.
There is a long history of asymptotic solutions for developing concentration boundary layers. Analytical solutions for a thermal boundary layer in a fully developed turbulent flow were developed by Linton & Sherwood (1950). The derivation involved neglecting the streamwise advective, diffusive transport and all turbulent transport, thereby essentially reducing the problem to that solved by Lévêque (1928) in the context of heat transfer in a laminar boundary layer. By assuming self-similarity, closed-form solutions for the boundary layer growth and concentration profile and heat transfer could be provided. Electrochemical experiments of high Sc and Re mass transfer in the entrance region (Shaw et al. 1963; Berger & Hau 1977) showed good agreement between the predictions from the asymptotic solutions and the measurements. Kestin & Persen (1962) developed asymptotic solutions for the heat transfer across a developing boundary layer over a flat plate, including viscous entrance effects. Several solutions were presented, including a transition from a laminar to a turbulent boundary layer and stepchanges in the wall temperature.
Far-field asymptotic solutions for high Sc solutes, however, seem to have escaped attention until recently. Garcia-Ybarra & Pinelli (2006) used matched asymptotic expansions to derive a closed-form solution for the concentration profile for a fixed concentration (Dirichlet) BC at high Sc. Sookhak Lari et al. (2010) independently arrived at the same closed-form solution by observing that the scalar flux was approximately constant across the concentration boundary layer in a study on the decay of residual chlorine in pipes as a result of a first-order reaction with the wall (Robin BC). Despite the different BCs, both studies report the same Sh correlation.
The aim of this paper is to generalize the work of Garcia-Ybarra & Pinelli (2006) and Sookhak Lari et al. (2010) to much more general BCs and to provide simple closed-form solutions for the mass-transfer, decay coefficients and concentration profiles. After a discussion on the appropriate velocity and turbulent diffusivity profiles (§2), the governing equations for the far-field are derived (§3). It will become apparent that the Sh correlation reported in Garcia-Ybarra & Pinelli (2006) and Sookhak Lari et al. (2010) is in fact representative for a very large class of BCs (§4). Closed-form solutions for Dirichlet, Neumann and Robin BCs are presented in §5a. We then use the Von Karman–Pohlhausen method to pose the governing equation for the near-field and present solutions in §6. As the solution method is not limited to linear BCs, closed-form solutions for a second-order wall reaction are presented (§§5b and 6). Concluding remarks are made in §7.
2. Near-wall profiles of velocity and eddy diffusivity
Consider a fully developed turbulent flow field. When Sc≫1, as is the case for many mass transfer problems (and heat transfer in e.g. heavy oils), the scalar diffuses much slower than momentum. The associated layer of thickness δm near the wall where molecular diffusion dominates over turbulent transport, hereafter referred to as the mass transfer boundary layer (MTBL) will then be entirely nested in the viscous sublayer (Pope 2000; Schlichting & Gersten 2000; Garcia-Ybarra & Pinelli 2006; Sookhak Lari et al. 2010). Outside the MTBL, the turbulence causes sufficient mixing to assume a uniform concentration. The two fundamental parameters governing the viscous wall region are the kinematic viscosity ν and the friction velocity , where τw is the wall shear stress and ρ is the fluid density. In the viscous sublayer, the turbulent stress is negligible, and therefore the average velocity profile is given by (Pope 2000; Schlichting & Gersten 2000; Bird et al. 2002) 2.1where u+=u/uτ and y+=y/δv, y represents the distance from the wall and δv=ν/uτ is the viscous length scale.
A second property of the viscous sublayer is that the turbulent momentum flux , and therefore the eddy-viscosity νT, has a cubic dependence on the wall distance 2.2which can be shown using Taylor expansions (Antonia & Kim 1991; Bird et al. 2002). The prefactor b has been approximated experimentally and numerically, and takes the value b≈9.5×10−4 (Bird et al. 2002). Using (2.2), the ratio of the turbulence diffusion coefficient DT to the molecular diffusion coefficient D is given by 2.3where Sc=ν/D is the Schmidt number and ScT=νT/DT the turbulence Schmidt number. The effective exponent for DT is crucial for high Sc mass transfer as it influences the Sh correlation: DT∼ym implies that Sh∼Sc1/m. Many laboratory experiments (Harriott & Hamilton 1965; Mizushina et al. 1971; Dawson & Trass 1972; Berger & Hau 1977; Zhao & Trass 1997) find that Sh∼Sc0.32−0.35, thereby indirectly confirming (2.3). Shaw & Hanratty (1977) report a slightly lower Sc dependence Sh∼Sc0.29, although it is not entirely clear what the cause is for the deviations between this and the other experiments. More information about Sh correlations for high Sc mass transfer, including the range of Sc and Re considered, can be found in the electronic supplementary material.
A Taylor expansion confirms that the turbulent scalar flux is indeed expected to vary as the cubic on wall distance for fixed concentration (Dirichlet) BCs (Antonia & Kim 1991; Bird et al. 2002; Garcia-Ybarra & Pinelli 2006). However, for flux (Neumann) BCs, a second-order dependence of the turbulent scalar flux (and therefore DT) on the wall distance is obtained. As the DT profile at high Sc has not been reported as yet for flux BCs, it is not known how dominant the second-order term is. The only available data are from simulations for the heat transfer across a fluid layer and a solid wall with finite thermal conductivity (Tiselj et al. 2004; Bergant & Tiselj 2007). The simulations show that ScT decreases very close to the wall, as expected for a quadratic DT profile. However, Bergant & Tiselj (2007) report that the influence on the mean temperature profiles and the heat transfer coefficients is almost negligible.
For Dirichlet BCs, studies performed with Direct Numerical Simulation (DNS), show that ScT is indeed constant when Sc is of order unity (Antonia & Kim 1991; Kawamura et al. 1998; Schwertfirm & Manhart 2007). At Sc>10, an increase of ScT is observed very close to the wall (y+<1), which becomes more pronounced for higher Sc (Na & Hanratty 2000; Crimaldi et al. 2006; Bergant & Tiselj 2007; Schwertfirm & Manhart 2007; Kozuka et al. 2009). Garcia-Ybarra (2009) used data from numerical simulations to show evidence that the fourth-order term overwhelms the cubic term a bit further away from the wall, which suggests that the effective exponent for DT is larger than three. This is consistent with the experiments of Shaw & Hanratty (1977) but not with the other studies mentioned above. Further laboratory experiments and/or DNS at higher Reτ and Sc will be required to settle this issue.
In what follows we will use the classical assumption (Kader 1981; Bird et al. 2002) that DT is a cubic and that ScT is constant. Even though this excludes some of the phenomena described above, the net effect of y+ variation of ScT (the modification of the mass transfer coefficient, etc.) can be incorporated by tuning of the parameter b/ScT which is discussed in appendix A. The procedure maps the actual profile for DT onto a cubic which has the same boundary layer thickness δm (defined below), thereby ensuring that integral quantities be predicted accurately. Note that if the profile for DT differs significantly from a cubic, the parameter b/ScT will become dependent on Sc and Re, the consequences of which will be described in §7.
The properties of the viscous sublayer and MTBL for a high Sc solute are depicted in figure 1. Figure 1a,b shows the dependence of u+ and DT/D on the wall distance, and figure 1c shows a typical concentration profile valid for e.g. a first-order reaction of chlorine with the wall (Robin BC) (Sookhak Lari et al. 2010). We follow Kader (1981) and define the typical thickness δm of the MTBL as the distance from the wall at which D=DT: 2.4Equation (2.4) clearly demonstrates that the MTBL will be nested in the viscous sublayer for Sc≫1 because δm/δv∝Sc−1/3.
Equations (2.1) and (2.3) will be used for the asymptotic solutions and are formally only valid very close to the conduit wall. To compare the predictions of the asymptotic solutions to the solutions to the full behaviour of the system, a model which accurately describes u and DT throughout the entire conduit is required. Over the years, a multitude of models have been developed (Reynolds 1975; Weigand 2004); here a modified Van Driest mixing-length model has been selected. This turbulence model accurately reproduces the flow and turbulent diffusion in the inner layer, including the cubic dependence of DT on y very close to the wall. For more details, see Hanna et al. (1981) and Sookhak Lari et al. (2010, 2011).
3. Derivation of far-field equations
Consider the transport of a high Sc solute through a conduit at high Re which exchanges mass with the conduit walls. For fully developed flow through a pipe with radius R, the governing equation is the axisymmetric Reynolds-averaged, steady-state mass transport equation (Bird et al. 2002) 3.1where x and r are the streamwise and radial directions, and C(x,r) is the (Reynolds-averaged) mass concentration. Streamwise diffusion has been neglected, which is permitted for a flow for which the Péclet number Pe=ReSc is very large. As stated before, we consider a hydrodynamically fully developed flow; u and DT are therefore functions of r only. The axisymmetric coordinate system is used for convenience of presentation; the approach is equally valid for non-circular cross sections as long as the viscous wall region (∝δv) is much thinner than the local surface curvature.
Equation (3.1) is supplemented by a wall BC of the form 3.2where Cw=C(x,R), ∂C/∂r|w=∂C/∂r(x,R) and G(Cw) is a generic function which depends on the wall concentration. Most BCs can be captured by equation (3.2), including the standard (linear) Dirichlet, Neumann and Robin BCs, but also higher order reactions and other BCs for which the wall-flux depends nonlinearly on the wall concentration. We will show that for BCs which satisfy (3.2), the Sherwood number Sh, which is the dimensionless mass flux, is universal and consistent with classical correlations of Sh(Sc,Re). The other two BCs are symmetry in the centre, ∂C/∂r|r=0=0, and a constant concentration at the entrance: C(0,r)=C0.
For the problem under consideration, the concentration is expected to be uniform, except within the MTBL. A dimensional analysis of the advective and diffusive terms of equation (3.1) in the MTBL results in 3.3where is a typical concentration and is the typical length scale for the streamwise variations. The typical velocity in the MTBL was estimated by evaluating equation (2.1) at y=δm.
The central premise of the approximation is that streamwise variations occur on much longer length scales than changes in the wall-normal direction, i.e. that (Notter & Sleicher 1972; Garcia-Ybarra & Pinelli 2006; Sookhak Lari et al. 2010). It follows from the estimates above that advection will be negligible relative to diffusion if the ratio satisfies 3.4where Reτ=uτR/ν is the shear Reynolds number. The validity of this assumption will be established at the end of this section.
When (3.4) holds, equation (3.1) will no longer depend on x. Indeed, by neglecting the advective term, changing coordinates to mass transfer wall units η=(R−r)/δm, assuming that δm≪R, and using the cubic for DT given in (2.3), the following linear partial differential equation is obtained 3.5where c=C/C0. The equation above is equivalent to Garcia-Ybarra & Pinelli (2006, eqn 30) and Sookhak Lari et al. (2010, eqn 29). One BC is provided by equation (3.2) which in dimensionless form is given by 3.6where g=δmG/C0. The second BC used is , which states that c tends to the bulk concentration cb far away from the MTBL. Here, the dimensionless streamwise coordinate ξ is defined as . With these BCs, equation (3.5) admits the following closed-form solution (Garcia-Ybarra & Pinelli 2006; Sookhak Lari et al. 2010) 3.7where the wall concentration cw(ξ) and F(η) are defined as 3.8and 3.9The function F increases monotonically from F(0)=−1 to , and .
To complete the approximation, an equation is required which governs the behaviour of cb. Such an equation can be obtained by averaging equation (3.1) over the cross section 3.10where is the average streamwise mass flux. Because c is assumed constant throughout the cross section except in the MTBL, we can approximate 〈uc〉≈Ucb, where U is the average velocity. This results in 3.11where rh=R/2 is the hydraulic radius and Jw is the wall mass flux per unit area 3.12Substituting equation (3.12) into (3.11) results in 3.13and 3.14Here, we have used the standard pipe Reynolds number definition Re=2UR/ν. The typical length-scale can be interpreted as the distance for which the solute mass in the entire cross section is depleted by the flux through the wall. Equation (3.14) can be used to check the validity condition (3.4), which now takes the form of b2/3ReSc2/3≫1. Even for very low values of Re=2000 and Sc=100, b2/3ReSc2/3≈466 and therefore, condition (3.4) is satisfied for high Sc compounds in turbulent flows.
Equations (3.7), (3.9) and (3.13) comprise a set of coupled equations which can be used to construct asymptotic far-field solutions for the concentration profile and the mass transfer at the wall. Because the derivation does not rely on linear techniques such as separating variables, this includes nonlinear BCs. Examples will be discussed §5.
4. A universal Sherwood number equation
A universal expression for the Sherwood number Sh can be derived for BCs satisfying equation (3.6). The Sherwood number is defined by Sh=2kfR/D, where kf [LT−1] is the mass transfer coefficient (Bird et al. 2002) 4.1Substituting equation (3.9) into equation (4.1), and using that Jw=g(cw)DC0/δm results in 4.2i.e. kf is independent of the type of BC. Using equation (4.2), the Sh equation for high Sc compounds is 4.3The universality is a direct consequence of the linear dependence of the wall concentration gradient and the concentration difference between wall and bulk, as is evident from (3.9). The underlying reason for this is the invariance of (3.5) to scaling because of its linearity.
The universality of equation (4.3) is confirmed by recent work with Robin BCs (Sookhak Lari et al. 2010) and Dirichlet BCs (Garcia-Ybarra & Pinelli 2006). It also compares favourably with experimental data. The Fanning friction factor f is defined as (Bird et al. 2002, eqn 6.1-4a). Substitution into equation (4.3) results in 4.4which corresponds well to the established correlation (Bird et al. 2002, eqn 14.2-5) upon substituting b=9.5×10−4 and ScT=1. By applying the Blasius formula f=0.0791Re−0.25 (Bird et al. 2002, eqn 6.2-12), we obtain Sh=0.016Re0.88Sc1/3 which is in good agreement with the Sh correlations presented in the electronic supplementary material.
The Stanton number St is defined as the ratio of mass transfer coefficient to average velocity: 4.5Note that St is closely related to by . The value of St will therefore immediately give an indication of the appropriateness of neglecting the streamwise advection.
5. Far-field solutions
(a) Linear boundary conditions
We will now provide closed-form far-field solutions for Dirichlet, Neumann and Robin BCs, by considering a linear BC of the form 5.1Here, α,β and γ are constants. Using equations (3.6), (3.9) and (5.1), the wall concentration and gradient are given by 5.2Substituting (5.2) into (3.13) and solving for cb results in 5.3
The specific solutions for Dirichlet (α=1, β=0), Neumann (α=0 and β=1) and Robin (α=−σ, β=1 and γ=0) BCs are presented in table 1 as solution AS-D, AS-N and AS-R, respectively. The Neumann solution required performing a Taylor series expansion around ξ=0 and taking the limit of . Solution AS-R is equivalent to the solution derived in Sookhak Lari et al. (2010). AS-N is documented to a large extent in Bird et al. (2002, pp. 411–414), but that solution still contains an integral which needs to be approximated numerically. Garcia-Ybarra & Pinelli (2006) derived F(η) and (4.5) for Dirichlet BCs, but did not solve for cb (solution AS-D).
The far-field asymptotic solution will be compared with solutions of equation (3.1) obtained with the method of separating variables. To allow comparison to the full solution, the method retains the cylindrical coordinate system and uses the realistic velocity and diffusivity profiles provided by the modified Van Driest Mixing length model. An expansion of the form 5.4is used, where h(η) is a function which maps (5.1) onto a homogeneous BC. The eigenvalues kn and associated eigenfunctions Y n(η) can be found by solving a Sturm–Liouville problem. The ordinary differential equation (ODE) governing Xn can be solved analytically and takes the form of a damped exponential if kn>0 and a linear function if kn=0. The eigenvalues and eigenfunctions do not have closed-form solutions and are determined numerically using a shooting method. Details of the method and implementation are discussed in the electronic supplementary material.
The solutions for Dirichlet, Neumann and Robin BCs using the method of separating variables are presented in table 1 and are denoted by SV-D, SV-N and SV-R, respectively. The parameter Rem=R/δm represents the distance to the centre of the pipe in mass-transfer units.1 It is clear from table 1 that the structure of the two solution methods is very similar.
All results presented in this paper are for Reτ=2000, Sc=1000 and ScT=1 unless stated otherwise. The prediction for AS-D is , which compares well with SV-D, which predicts k1=0.8510. For AS-R with σ=2, , against SV-R which predicts k1=0.5970. The small difference can be traced back to differences in the eddy diffusivity profile. The wall damping employed in the modified Van Driest mixing length model is purely empirical, and only satisfies cubic behaviour very close to the wall. Between 1<y+<5, DT is up to 30 per cent higher than a pure cubic. This will make the MTBL a bit thinner, which in return results in a slightly higher decay coefficient. This small difference could have been avoided altogether by adopting a modified value for b/ScT using the calculation method outlined in appendix A, but presenting the slight differences was deemed more instructive.
It is clear from (3.7) that (c−cw)/(cb−cw)=1+F(η) regardless of the BC. For the method of separating variables (c−cw)/(cb−cw) in the far-field can be obtained by setting . The equations are presented in the third column of table 1 and are plotted in figure 2a. For SV-D and SV-R, (c−cw)/(cb−cw) is closely related to the first eigenfunction and the correspondence to the far-field asymptotic solution is excellent. The small differences originate again from the differences in turbulence model, and could be avoided using a modified value for b/ScT (appendix A). For SV-N, (c−cw)/(cb−cw) shows deviations very close to the wall which are caused by a truncation of the infinite sum. The convergence to the asymptotic solution can be seen to be quite slow as is evident from the profile for N=25 modes (plus symbols) and N=100 modes (squares).
(b) Nonlinear boundary conditions
Approximate analytical solutions can be constructed for nonlinear BCs. As a demonstration, a second-order wall reaction 5.5is studied, where a is a dimensionless reaction coefficient. Substituting (5.5) into (3.9) gives 5.6Substitution of (5.6) and (5.5) into (3.13) results in 5.7The change of variables simplifies equation (5.7) to 5.8
which has a solution 5.9where , and W0 is the Lambert W function (Corless et al. 1996).
The far-field asymptotic solution in equation (5.9) was compared with the solution of a finite-volume approximation of the full three-dimensional axisymmetric partial differential equation (3.1) and mass transfer BC (5.5). As for the method of separating variables, the modified Van Driest mixing-length model was used to determine velocity and turbulent diffusivity profiles. The advective term was discretized using a first-order upwind scheme, which allows for explicit marching in the x-direction. The r-direction is discretized using second-order central scheme which is solved using direct matrix inversion. The nonlinearity of the BC (5.5) is incorporated using a simple iterative method.
The problem was solved for a=10−1, 100 and 101. Grid convergence was observed at Nx=1200 and Nr=600, although logarithmic spacing was required because of strong variations very close to the wall and near the entrance. The cell-sizes vary up to eight orders of magnitude. A conservative method such as the finite volume method is crucial for such extreme stretching (Mathias & van Reeuwijk 2009). The Grid Convergence Index (GCI; Roache 1994) for these simulations is GCI<1.2% in the far-field based on . Note that in the calculation of the GCI, we assumed that the method is entirely first order; the reported value for the GCI is therefore a conservative estimate.
The asymptotic solution for cb (5.9) and the three-dimensional axisymmetric simulations are in excellent agreement (figure 3a), even though the asymptotic solution does not take into account entrance effects. In §6, we will show that this is the case because the entrance length is so small that it does not influence cb.
6. Near-field solutions
In the near-field, streamwise advection will not be negligible. In dimensionless variables, (3.1) is given by 6.1Here, (2.1), (2.3) were used for the velocity and eddy diffusivity profiles, respectively. The small parameter ϵ is given by 6.2Hence, a suitable near-field coordinate is ξe=ξ/ϵ, for which (6.1), (3.13) become 6.3and 6.4The equations above are parameter-free which suggests universal behaviour, although it should be noted that the BCs may still introduce a parameter dependence. As ϵ≪1, (6.4) immediately results in cb(ξe)≈1. The entrance length-scale is given by 6.5Note that depends on flow properties only. This can be understood by realizing that is related to the time TD it takes for the mass in the MTBL to deplete: . During this time, the boundary layer section moves at a typical velocity δm/δvuτ, which is the velocity at the edge of the MTBL. The entrance length-scale can therefore be estimated by 6.6Note that the validity condition (3.4) for the approximation can be expressed using and as 6.7
In order to obtain closed form solutions for (6.3), the Von Karman–Pohlhausen integral method (Lighthill 1950; Spalding 1954; Schlichting & Gersten 2000) will be used. This method is not exact as it involves substituting the assumed concentration profile F(η). However, (6.3) does not admit self-similar solutions, because (i) the BC (3.6) is nonlinear and (ii) the total diffusion (1+η3) does not allow power-law behaviour for the boundary layer thickness.
By introducing the concentration deficit 6.8and integrating from η=0 to λδ(ξ), (6.3) becomes 6.9here, δ(ξ) is the typical boundary layer thickness and λ>1 is a coefficient.
The concentration is assumed to be of the form c(ξe,ζ)=cb+(cb−cw)F(ζ), where ζ=η/δ. In terms of the concentration deficit, the assumed profile is 6.10
Equation (6.14) is a first-order nonlinear ODE. For the general BC (3.6), g′=g′(cw) which introduces another dependence on δ. It may therefore be impossible to derive closed-form solutions for complicated BCs. For the linear BC (5.1) however, g′=−α/β evaluates to a constant. In table 2, the solutions to (6.14) are presented for Dirichlet (α=1, β=0), Neumann (α=0, β=1) and Robin (α=−σ, β=1, γ=0) BCs. The Dirichlet and Neumann BCs have the classic x1/3 dependence (Linton & Sherwood 1950; Kestin & Persen 1962; Shaw et al. 1963; Berger & Hau 1977). The Robin BC is more complex because g′ is finite, but essentially behaves like a Neumann BC when σ≪1 and like a Dirichlet BC when σ≫1.
Although the Sh correlation for the far-field is the same for all BCs satisfying (3.6), the near-field correlation Shx(ξe) is different. Indeed, the mass transfer coefficient is given by kf=DF′(0)/(δmδ) and therefore 6.15
The coefficient λ controls, via B, the growth rate of the boundary layer. As the integral (6.12) is divergent for , λ has to be tuned to results from three-dimensional axisymmetric simulations of the full problem. The resolution of the simulations is Nx=1200 and Nr=600 and the GCI<1% based on . Good agreement is found for λ=1.5 and therefore B=0.4. The near-field and far-field solutions can be combined using . The correspondence of the equation for Shx with three-dimensional axisymmetric solutions for Dirichlet, Neumann and Robin BCs is good (figure 2b). Note that Shx for the Dirichlet and Neumann BCs are scaled by a factor 100 and 10, respectively. Also shown are the solutions obtained using the method of separating variables using the first 100 modes. There is good agreement with both the asymptotic solutions and the simulations, although more modes are required to describe the behaviour for ξe<10−2.
Using the same substitutions as for the Sh correlation, the Shx correlation for Dirichlet BCs is given by 6.16This correlation is in good agreement with electrochemical mass transfer experiments which report Shx=0.184Re0.58Sc1/3(x/2R)−1/3 (Shaw et al. 1963; Berger & Hau 1977).2 For the asymptotic solution for Neumann BCs, the correlation is identical to (6.16) but the prefactor is 0.22.
For nonlinear BCs, g′ depends on cw. For the second-order wall reaction (5.5), substitution into (6.13) results in a quadratic in Δw, of which the physically relevant root is given by 6.17Therefore, g′ is given by 6.18Substitution of the equation above into (6.14) and solving the ODE results in 6.19The good agreement of Shx/Sh based on the equation above with the numerical simulation is shown in figure 3b for three different values of a. Note that B was kept the same value as for the linear BCs.
7. Concluding remarks
In this paper, asymptotic solutions for high Sc scalars for both linear and nonlinear BCs were developed, which provide simple closed-form solutions and predict accurately the concentration profile and mass transfer in the near- and the far-field. It was shown that in the far-field, the mass transfer coefficient kf and associated dimensionless Sherwood number Sh is independent of the specific wall BC. This result is valid for all BCs satisfying (3.6), which include virtually all known BCs including those of Dirichlet, Neumann and Robin type. This is a remarkable result which emphasizes that for high Sc turbulent flows, the mass transfer coefficient kf (4.2) depends only on the molecular diffusivity D and the thickness of the MTBL δm.
The method presented in this paper assumed that the eddy diffusivity DT is a cubic in the distance to the wall. As discussed in §2, the precise near-wall behaviour of DT is subject to some uncertainty. However, the universality of the mass transfer coefficient is independent of the exact profile for DT. Indeed, making use of a different profile for DT, for example, the fourth-order polynomial suggested in Garcia-Ybarra (2009), would have led to the same conclusion.
The solutions reported in this paper can be used even when the actual DT profile is not cubic by calculating the effective value for b/ScT using the procedure described in the appendix. This procedure matches δm from the actual DT profile to that of the assumed cubic, thereby ensuring that the integral parameters are predicted accurately. Note that this would make the parameter b/ScT dependent on Re and Sc. As discussed in §2, the DT profile may be different for specific BCs. In that case, b/ScT would have a different Sc and Re dependence for each BC and therefore the Sh correlation (4.3) would not be universal. It is therefore desirable that further research focuses on the near-wall profile of DT at high Sc turbulent mass transfer, in particular the influence of different BCs.
Appendix A. Inferring the value of b/ScT from turbulence models
The coefficient b has a strictly defined physical meaning stated in (2.2). However, it can also be treated as a free parameter representing a measure for the ‘conductivity’ of the MTBL, in which case the value can be determined from the turbulence model employed. We start from (3.1), neglect horizontal advection and change to plus-units using the change of variables r=δv(Reτ−y+). The result is A1
Integrating twice and using (3.2) results in A2and the mass transfer coefficient kf is therefore given by A3Many theoretical studies on mass transfer approximate the integral on the right-hand side to develop Sh correlations (Kader & Yaglom 1972; Aravinth 2000).
- Received September 6, 2011.
- Accepted January 23, 2012.
- This journal is © 2012 The Royal Society