According to the proverbial boundary layer theory proposed by Prandtl (1904), when a fluid (normally a gas mixture, a gas reactant mixed with a carrier gas, in uptake kinetic studies) enters the inlet of a flow tube with a uniform velocity, a laminar boundary layer (i.e., velocity boundary layer) will form very close to the tube wall (Fig. 2). This buildup of laminar boundary layer is because of the non-slip condition of the tube wall and the viscosity of the fluid; that is, viscous shearing forces between fluid layers are felt and dominant within the laminar boundary layer (Mauri, 2015). The thickness of laminar boundary layer δl will continuously increase in the flow direction (axial direction in Fig. 2) until at a distance (from the tube entrance) where the boundary layers merge. Beyond this distance, the tube flow is entirely viscous, and the axial velocity adjusts slightly further until the velocity along the axial direction does not change anymore. Then, a fully developed parabolic velocity profile is formed, characteristic of well-developed laminar flow (Mohanty and Asthana, 1979; White, 1998). The development and formation of this velocity profile are illustrated in Fig. 2. Normally, for coated-wall flow tube experiments, a chemically inert entrance region with smooth surface is designed to ensure the development of laminar flow before the reactive gas enters into the coated-wall region.

As demonstrated in previous studies using microchannels and pipes (Herwig et al., 2008; Gloss and Herwig, 2010; Zhang et al., 2010; Zhou and Yao, 2011), the roughness elements on flow tube coatings can have non-ignorable effects on laminar flow conditions even if these coatings are entirely submerged into the laminar boundary layer. In other words, the disturbance on well-developed laminar flow patterns can be artificially achieved by roughness elements of the tube coating. However, there is a critical height δc within which the roughness effects can become ignorable (Achdou et al., 1998).

Figure 3 shows a schematic of the structure of the δc and its related flow conditions in a coated-wall flow tube. When a roughness height δr (here in Fig. 3, the roughness height δr equates to the geometric coating thickness δg; see Sect. 2.3 for explanation) comes into the critical height δc where viscous effects overwhelmingly dominate, the flow very near the rough wall will tend to be Stokes-like or creeping, denoted as the laminar flow (LF) regime in Fig. 3a. This Stokes-like flow adjacent to the rough surfaces can avoid local turbulence between the roughness elements and guarantee perfect laminar flow conditions (i.e., only molecular diffusional transport of gas reactants to rough reactive coatings) throughout the whole flow tube volume. Thus, the LF regime satisfies the prerequisite for the diffusion correction methods used for flow tube experiments, i.e., δr / δc < 1. Nevertheless, when a roughness height is larger than the critical height δc, local eddies may occur in the spaces between the neighboring roughness elements (i.e., the local turbulence (LT) regime in Fig. 3b). Local turbulence induced by these roughness elements will enhance local transport of air masses within the scales of the roughness heights, which invalidates the assumption of solely molecular diffusion of gas reactants and therefore the application of diffusion correction methods for the determination of γ (Brown, 1978; Murphy and Fahey, 1987; Knopf et al., 2015). In the next section, we will show how to derive δc.

Achdou et al. (1998) proposed effective boundary conditions for a laminar flow over a rough wall with periodic roughness elements and observed that when δr/Lc < Re-1/2 (δr: roughness height; Lc: characteristic length, for a tube the characteristic length Lc=d) the roughness elements could be contained in the boundary layer. This means that, for their case, the boundary layer thickness is in the order of LcRe-1/2. Within the boundary layer, they found that local turbulence could occur between the roughness elements until δr/Lc < Re-3/4, where the viscous effects became dominated in roughness elements and then the flow near the rough wall tended to be creeping. This result coincides with Kolmogorov's theory (Kolmogorov, 1991), in which the critical length ratios between small-scale and large-scale eddies are also in the order of Re-3/4, even though this theory only applies to turbulent flow with large Reynolds numbers. Here, we adopt this criterion to judge if local eddies could occur in the spaces between neighboring roughness elements. Thus, the critical height δc can be expressed as δc=d×Re-3/4=d1/4×Vavgν-3/4, where d is diameter of the flow tube, Re is the Reynolds number, and Vavg and ν are average velocity and kinematic viscosity of the fluid, respectively.

With Eq. (2), for a specified experiment configuration (i.e., flow tube diameter, flow velocity, and fluid properties) the critical height δc can be determined, and therefore the effects of coating roughness on laminar flow can be estimated provided the roughness height δr is known.

The potential effects of coating roughness on laminar flow are described and classified into two regimes in Fig. 3 (Sect. 2.1), in which only the LF regime provides the ideal precondition ensuring that the diffusion correction methods (Brown/CKD/KPS methods) can be applied to obtain accurate γ through flow tube experiments. Regarding the LT regime, however, the roughness-induced effects can be quantitatively simulated, because local turbulence is constrained into the scale of the roughness height δr (Oke et al., 2017).

Hence, for the LT regime, in order to estimate the potential error of the uptake coefficient derived from molecular diffusion correction using the conventional CKD/KPS methods, we further develop modified CKD/KPS methods (denoted as CKD-LT/KPS-LT, illustrated in Fig. 4) to account for local turbulence impact. In the CKD-LT/KPS-LT methods, some basic assumptions are made: (1) the scale of a roughness element is much larger than the size of pores inside the bulk coating, and the macroscopic diffusion inside pores is not the domain of roughness-induced local eddies; (2) half of the surface roughness height is defined as the local-eddy-occurring region (i.e., 0.5δr=Rm – Rg); (3) the turbulent diffusion coefficient within the local-eddy-occurring region is infinitely large (i.e., the turbulent transport within it is extremely fast). When a coating is smooth, the mass-based coating thickness δm is equal to the geometric coating thickness δg. In this case, the radial molecular diffusion distance from the tube centerline is Rm, while with large surface roughness height, the radial molecular diffusion distance is reduced to Rg. With the CKD-LT/KPS-LT methods, derivation of the uptake coefficient using Rg rather than Rm reflects an upper limit for the influence of local turbulence, as the turbulent diffusion coefficient in the local-eddy-occurring region is assumed to be infinitely large and turbulent transport occupies its whole volume. More details about CKD and KPS, and the derivations of γCKD, γKPS, γCKD-LT, and γKPS-LT can be found in Appendices C–E.

The introduction of the critical height δc, into the field of gas uptake or reaction kinetic studies using coated-wall flow tubes, provides us the way for determining when the surface roughness effects can be negligible in flow tube experiments. That is, the roughness height δr of a coating film should be well within the domain of δc (LF regime in Fig. 3a). Only in this case, the free molecular diffusion of a gas reactant in the radial direction can be ascertained, and thus the Brown/CKD/KPS methods can be safely applied. Note that in real operations of flow tube coating design several techniques (stylus profiler, non-contact optical profiler, scanning electron microscopy and atomic force microscopy, etc.) are available for surface roughness examination (Poon and Bhushan, 1995). To simplify the discussion, here, we take the geometric thickness of a coating film δg as a maximum of its surface roughness and use the comparison between δg and δc as a reference for the design of flow tube coating thickness. Such treatment is more suitable for practical applications, because determination of coating film thicknesses can be simply achieved either by weighing the coating film mass (i.e., mass-based coating thickness δm) or by utilizing the scanning electron microscopy technique (i.e., geometric coating thickness δg), and the condition of δg / δc < 1 can definitely ensure the case of δr / δc < 1. As discussed in Sect. 2.3, for coatings with large surface roughness, their δg may be significantly larger than δm. In this case, the criterion of δg / δc < 1 is more appropriate to be adopted.

Figure 5 shows the calculated δc, with varying the tube diameter d and the average flow velocity Vavg. From Eq. (2), kinematic viscosity of a fluid (carrier gas in flow tubes) will affect δc. It is therefore necessary to classify the flow tube experiments according to the types of the utilized carrier gases, such as synthetic air (Fig. 5a), nitrogen (Fig. 5b), and helium (Fig. 5c). For future flow tube coating design, Fig. 5 can be used to eliminate the potential coating surface roughness effects. Figure 6 summarizes and evaluates the potential effects of surface roughness in previous flow tube experiments. To reflect the influence of inherent roughness of the inner surface of a flow tube wall itself, the mean wall roughness is also accounted for coating thickness calculation when using rough-wall flow tubes (e.g., sandblasted tubes), for example, in the protein coating experiment. As shown in Fig. 6, most of the coating thicknesses are well below the calculated values of δc (LF regime), implying that their surface roughness effects on laminar flow and on the calculated uptake coefficient are ignorable. A few coating thicknesses, however, are significantly larger than the calculated δc (LT regime), as shown by the solid symbols. As the thicknesses of these two coatings are reported in terms of geometric coating thickness δg (Li et al., 2016; McNeill et al., 2006), they may have had a potential influence on laminar flow pattern, and local turbulence may have occurred within the roughness-constructed spaces.

For most cases of flow tube experiments design, a coating layer cannot be thin enough due to requirements of reaction kinetics (bulk diffusion and surface reactions can both play important roles), and the thickness of a coating layer had been found to have an influence on gases uptake until a critical threshold was reached (Donaldson et al., 2014a; Li et al., 2016). This means that there is a need to comprehensively consider all the parameters (coating thickness, tube diameter, tube length, flow velocity, etc.) and a compromise of each parameter for the others is necessary to finally ensure both the unaffected laminar flow conditions and the specific requirements for an individual flow tube design. Larger δc would allow a wider range of coating thickness δg without surface roughness effects. Based on Eq. (2), larger δc can be achieved either by increasing the tube diameter d or by decreasing the fluid average velocity Vavg. Under the conditions of fast uptake kinetics, relatively short residence time of gas reactants inside the coated-wall region is needed to allow for distinguishable penetration C/C0 (i.e., the flow tube outlet concentration divided by the inlet concentration; see Fig. C1 for details). This requirement can be fulfilled by optimizing flow tube design. One can increase d or decrease Vavg to achieve larger δc, but this operation will inevitably extend the residence time of gas reactants. Then, this effect can be offset by reducing L, which could be easily achieved by adjusting the position of a movable injector inside the flow tube apparatus as in previous studies (Howard, 1979; Jayne et al., 1997; Pöschl et al., 1998; Kolb et al., 2010; VandenBoer et al., 2015).

Normally, through coated-wall flow tube experiments, a penetration C/C0 can be measured, and therefore an effective uptake coefficient γeff can be experimentally determined (see Eq. C1 in Appendix C) under the assumption that the loss process on the wall is first order. As discussed above, without roughness-induced local turbulence, the radial concentration gradient can give rise to molecular diffusion limitations of the gas reactant, which need to be corrected using the diffusion correction methods (i.e., Brown/CKD/KPS) to derive the real uptake coefficient γ. Thus, the deviation between γeff and γ is only caused by molecular diffusion effects under ideal laminar flow conditions (LF regime).

With roughness-induced local turbulence (LT regime), the preconditions of conventional molecular diffusion correction methods can be corrupted. Figure  7 displays a schematic of different types of uptake coefficients and their divergences due to molecular diffusion and local turbulence effects. For the LT regime, the conventional CKD or KPS may cause overcorrection of γeff, i.e., γCKD ≥ γ or γKPS ≥ γ (upper limit indicated in red in Fig. 7). In this case, the derived γCKD-LT or γKPS-LT (blue in Fig. 7) using our proposed CKD-LT or KPS-LT methods may serve as a lower limit of γ (see Sect. 2.3 for explanation), thus defining the uncertainty range of γ, as shown in Fig. 7.

To have a general cognition of the quantified divergence among the different types of uptake coefficients, we further present Figs. 8 and 9 as examinations of two specific experimental configurations. Each figure has two panels: Figs. 8a and 9a show the uptake coefficient corrected by the CKD and CKD-LT methods, and Figs. 8b and 9b by the KPS and KPS-LT methods. The derivation of γeff is based on Eq. (C1). For Fig. 8, the experimental configuration of the soil coating case (Li et al., 2016) in Fig. 6 (solid circle) is used as input parameters for the diffusion correction, while an assumed configuration with higher volumetric flow rate F and larger relative roughness height δr / R0 (see caption for details) is adopted for Fig. 9. As shown in both figures, the uncertainty range of γ can be constrained by γCKD and γCKD-LT, or γKPS and γKPS-LT. In general, larger divergence, which corresponds to larger molecular diffusion and/or local turbulence effects, can be found at higher uptake coefficient magnitudes. The experimental configuration used for Fig. 8 results in a smaller difference of γCKD against γCKD-LT and γKPS against γKPS-LT than that for Fig. 9. This indicates that, for experiment design with rough coating, higher volumetric flow rate and/or larger relative roughness height will make the coating surface roughness effects more prominent. The higher values of the uptake coefficient derived using CKD and CKD-LT than those using KPS and KPS-LT, respectively, can be due to the different algorithms employed for CKD and KPS (see Appendix C). At last, it should be noted that the whole discussion about surface roughness and the way the different diffusion correction methods are applied are linked to the assumption that first-order reaction kinetics are granted, as mentioned upfront.