Technical Note: Influence of surface roughness and local turbulence on coated-wall flow tube experiments for gas uptake and kinetic studies

Guo Li, Hang Su, Uwe Kuhn, Hannah Meusel, Markus Ammann, Min Shao, Ulrich Pöschl, Yafang Cheng 5 1 Multiphase Chemistry Department, Max Planck Institute for Chemistry, Mainz, Germany 2 Institute for Environmental and Climate Research, Jinan University, Guangzhou, China 3 Laboratory of Environmental Chemistry, Paul Scherrer Institute, 5232 Villigen, Switzerland 4 College of Environmental Sciences and Engineering, Peking University, Beijing, China


Motivation
Coated-wall flow tube reactors have been extensively employed for investigations of uptake and reaction kinetics of gases with reactive solid/semi-solid surfaces (Kolb et al., 2010). To simulate various heterogeneous or multiphase reactions relevant to atmospheric chemistry, these coated reactive surfaces can span a broad scale including inorganic salts (Davies and Cox, 1998;Chu et al., 2002;Qiu et al., 2011), organic acids and sugars (Shiraiwa et al., 2012;Steimer et al., 2015), proteins (Shiraiwa et 5 al., 2011), soot (McCabe andAbbatt, 2009;Khalizov et al., 2010;Monge et al., 2010), mineral dust (El Zein and Bedjanian, 2012;Bedjanian et al., 2013), ice (Hynes et al., 2001;Hynes et al., 2002;Bartels-Rausch et al., 2005;Fernandez et al., 2005;McNeill et al., 2006;Petitjean et al., 2009;Symington et al., 2012) and soils (Stemmler et al., 2006;Wang et al., 2012;Donaldson et al., 2014a;Donaldson et al., 2014b;VandenBoer et al., 2015;Li et al., 2016). Due to uptake or chemical reactions of gases at coated tube walls, radial concentration gradients are established within the tube and radial diffusion can be 10 significant. It is therefore necessary to account for this gas-diffusion effect on gas-surface interactions. The most commonly utilized methods for evaluation and correction for gas-diffusion in flow tubes include the Brown method (Brown, 1978), CKD method (Murphy and Fahey, 1987) and a more recently developed simple KPS method (Knopf et al., 2015). All of these methods are derived based on the assumption that gas flow in flow tubes should be well-developed laminar to ensure that the flow velocity profile is parabolic and that the radial transport of gas reactants is solely caused by molecular diffusion.

15
It is well known that the flow conditions in a tube depend on the Reynolds number, Re (Eqn. 1), where ρ is density of the fluid passing through the tube, V avg is the average velocity of the fluid (i.e., the volumetric flow rate divided by the cross sectional area of the tube), d is diameter of the tube, µ and ν are dynamic viscosity and kinematic viscosity 20 of the fluid, respectively. A laminar flow can be expected when Re is less than ~ 2000 (Murphy and Fahey, 1987;Knopf et al., 2015). Here, the expression of Re quantifies the nature of the fluid itself (i.e., ρ, V avg , µ and ν) and the tube geometry (i.e., d), but it does not account for the effects of surface roughness.
Surface roughness effects on flow conditions were firstly discussed by Nikuradse (1950). Based on his work, the Moody diagram 25 has been extensively used in industry to predict the effects of surface roughness (roughness height ε or relative roughness ε/d) on flow characteristics (in terms of friction factor). According to the Moody chart, when the surface roughness is small enough (i.e., ε/d ≤ 5%), the roughness effects within low Reynolds number regime (Re < 2000, characteristic of laminar flow) is negligible.
Recent experimental and theoretical studies, however, have found significant effects of surface roughness on laminar flow characteristics (e.g., fraction factor, pressure drop, critical Reynolds number and heat transfer etc.) in micro-channels and pipes 30 even under conditions of ε/d ≤ 5% (Herwig et al., 2008;Gloss and Herwig, 2010;Zhang et al., 2010;Zhou and Yao, 2011). This is because not only the ratio of ε and d but also other factors, such as shape of roughness elements (Herwig et al., 2008;Zhang et al., 2010) and spacing between different roughness elements , may determine the influence of surface roughness on the flow conditions.

5
Although the surface roughness effects can be potentially important, there has been a long-lasting debate on whether the coating surface roughness could disturb the fully developed laminar flow in flow tube kinetic experiments (Taylor et al., 2006;Herwig et al., 2008) and its effects were usually not well-quantified in most of the previous gas uptake or/and kinetic studies (Davies and Cox, 1998;Chu et al., 2002;McCabe and Abbatt, 2009;Khalizov et al., 2010;El Zein and Bedjanian, 2012;Shiraiwa et al., 10 2012;Wang et al., 2012;Bedjanian et al., 2013;Donaldson et al., 2014a;Donaldson et al., 2014b;VandenBoer et al., 2015). It is, however, conceivable that as the roughness of the coating surfaces increases it would eventually distort the steady laminar regime near tube walls and small-scale eddies would evolve from roughness elements giving rise to local turbulence, and hence corrupt the application of Brown/CKD/KPS methods for the derivation of uptake coefficient. The extent of these effects may depend on the coated film thickness and its surface roughness. It means that the roughness effects on flow conditions to a great 15 extent rely on the various coating techniques applied by different operators, leading to disagreement of the experimental results.
In the present study, the surface roughness effects on laminar flow are quantitatively examined. In view of the special laminar

30
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 δ 35 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 no changes of velocity along the axial direction. Then, a fully developed velocity profile is formed and this velocity profile is Atmos. Chem. Phys. Discuss., doi: 10.5194/acp-2017-232, 2017 Manuscript under review for journal Atmos. Chem. Phys. Discussion started: 28 April 2017 c Author(s) 2017. CC-BY 3.0 License.
parabolic, which is characteristic of well-developed laminar flow (Mohanty and Asthana, 1979;White, 1998). The development and formation of this velocity profile is illustrated in Fig. 2. Normally, for coated-wall flow tube experiments a pre-tube is employed to function as developing a well-developed laminar flow before it entering into the coated tube section.
As demonstrated in previous studies using micro-channels and pipes (Herwig et al., 2008;Gloss and Herwig, 2010;Zhang et al., 5 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 welldeveloped laminar flow patterns can be achieved artificially by roughness elements of the tube coating. However, there is a critical height δ c within which the roughness effects can become ignorable (Achdou et al., 1998).
10 Figure 3 shows a schematic of the structure of the δ c and its related flow conditions in a coated-wall flow tube. When a roughness height ε (here in Fig.3, the roughness height ε equates to the coating thickness ε max , see Sect. 3.1 for explanation) is larger than the critical height δ c , local eddies may occur in the spaces between the neighboring roughness elements (i.e., case 1 in Fig. 3A). 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 gaseous reactants and the application of 15 diffusion correction methods for the determination of γ (Brown, 1978;Murphy and Fahey, 1987;Knopf et al., 2015).
Nevertheless, when a roughness height comes into the critical height δ c where viscous effects overwhelmingly dominate, the flow very near the rough wall will tend to be Stokes-like or creeping, shown as Case 2 in Fig. 3B. This Stokes-like flow adjacent to the rough surfaces can eventually avoid local turbulence between the roughness elements and guarantee perfect laminar flow regime (i.e., only molecular diffusional transport of gas reactants to rough reactive coatings at the flow tube wall) formation 20 throughout the whole flow tube volume. Thus Case 2 satisfies the prerequisite for the diffusion correction methods used for flow tube experiments, i.e., ε/δ c < 1. In the next section, we will show how to derive δ c . Achdou et al., (1998) proposed effective boundary conditions for a laminar flow over a rough wall with periodic roughness elements, and observed that when ε/L c < Re -1/2 (ε: roughness height; L c : characteristic length, for a tube the characteristic length 25 L c = 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 L c Re -1/2 . Within the boundary layer, they found that local turbulence could occur between the roughness elements until ε/L c < 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 is also in the order of Re -3/4 , even though this theory only applies to 30 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 δ c can be expressed as:

δ c derivation
where d is the diameter of the flow tube, Re is the Reynolds number, V avg and v are the average velocity and the kinematic viscosity of the fluid, respectively.
With Eqn.
(2), for a specified experiment configuration (i.e., flow tube diameter, flow velocity and fluid properties etc.) the critical height δ c can be determined, and therefore the effects of coating roughness on laminar flow can be estimated provided the roughness height ε is known.

Error estimation with modified CKD method
The potential effects of coating roughness on laminar flow are described and classified into two cases in Fig. 3  can be applied to obtain accurate γ from flow tube experiments. Case 1, however, can be quantitatively simulated because local turbulence is constrained into the scale of the roughness height ε (see Fig. 3A) and the turbulence effects could be quantified by assuming a proper turbulent diffusion coefficient D t within ε.

10
Hence, for Case 1, in order to estimate the potential error of the effective uptake coefficient (γ eff ) derived from moleculardiffusion-correction using the conventional methods aforementioned (here we adopt the CKD-B method proposed in our previous study Li et al., 2016), we further developed a modified CKD method (M-CKD, illustrated in Fig. 4) to account for local turbulence and therefore derive the real uptake coefficient (γ). In the M-CKD method, the molecular diffusion coefficient D of the gas reactant of interest is used in the main free-stream region (above the rough coating thickness ε max ) while a turbulent 15 diffusion coefficient D t is used in the roughness region to account for local turbulence between coating roughness elements (Fig.   4). The assumption of a whole turbulence layer (no laminar layer) in the roughness region represent an upper limit for the influence of turbulence, which corresponds to a largest uncertainty introduced to the calculation of γ eff . More details about derivation of γ eff and γ by CKD-B and M-CKD can be found in Appendix A.1 and A.2.

20
The turbulent diffusion coefficient D t can be approximately estimated by the following equation (Taylor, 1922;Roberts and Webster, 2002): where V is the flow velocity at the top edge of ε max (blue dashed line in Fig. 4). V is calculated according to the parabolic velocity profile (in tube radial direction) of the main laminar flow in the flow tube. Here the rough coating thickness ε max reflects the 25 largest scale to which a local eddy can develop, because a roughness height ε has the range of 0 ≤ ε ≤ ε max . Half of ε max is used as a characteristic diffusion distance in the turbulence-occurred region (Fig. 4).

Design of coated-wall flow tube experiments
The introduction of the critical height δ c , into the field of gas uptake or reaction kinetic studies using coated-wall flow tubes, and atomic force microscope etc.) are available for surface roughness examination (Poon and Bhushan, 1995). To simplify the discussion, here, we take the thickness of the coating film ε max as a maximum of its surface roughness (sometimes this case can happen), and use the comparison between ε max and δ c 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 calculating the coating film volume (coated mass divided by density) or by means of scanning electron microscope technique, 5 and the condition of ε max /δ c < 1 can definitely ensure the case of ε/δ c < 1.
Larger δ c would allow a wider range of coating thickness ε max without surface roughness effects. Based on Eqn.
(2), larger δ c can be achieved either by increasing the tube diameter d or by decreasing the fluid average velocity V avg . In most cases, constant residence time of gaseous reactants inside flow tubes is needed to allow for enough uptake/reactions within the coated flow tube 10 volume. This requirement can also be fulfilled by adjusting the coated tube length L, that is, to achieve larger δ c the influence of decreasing V avg on residence time can be balanced by reducing L.  Fig. 5C). These coatings may have a potential influence on laminar flow pattern and local turbulence may occur within the roughness-constructed spaces (see Fig. 4).

30
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 (e.g., 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 35 the application requirements for diffusion correction methods. For example, to ensure ε max /δ c < 1 for a thick coating (large ε max ), we can increase δ c by increasing the tube diameter or decreasing the flow velocity as shown in Fig. 5.
The conditions of ε max < δ c (constraining a coating thickness within the critical height of δ c , Case 2 in Fig. 3)  design, however, some exceptional circumstances can still be foreseen (as in Case 1 in Fig. 3). For example, one may encounter the conditions in Case 1 (Fig. 3A) due to the limit of coating techniques or other specific considerations. Then it is critical to make a priori evaluation of potential error of γ due to coating roughness effects in the design of flow tube experiments. Figure 6 shows the deviation of calculated uptake coefficient γ eff against the real uptake coefficient γ for Case 1. There, three 5 different cases of ε max /R 0 are presented with all the rest experimental configurations being kept the same (see Table A.1). For higher ε max /R 0, the deviation of γ eff is also larger, indicating that a thick coating will result in larger error of the calculated γ eff .
Meanwhile, this error is also closely related to the magnitude of γ: at γ < 10 -5 there is almost no difference between γ eff and γ, but at γ beyond 10 -5 the error is apparent and considerably increases. In the case where local turbulence cannot be avoided (Case 1), Fig. 6 can be used to estimate the error of calculated γ eff . With this error range in mind, the selection of coating techniques or/and 10 parametrization of other experimental conditions can be better constrained, for example, if γ can be assumed to be smaller than 10 -5 the coating roughness effects become negligible.

Wall-roughness-induced error of γ eff in Case 1: for previous flow tube studies
Local turbulence caused by rough coatings had not been well-quantified in previous studies. Nevertheless, it may indeed happen and therefore introduce errors in the calculated uptake coefficient derived from the Brown/CKD/KPS methods (i.e., effective 15 uptake coefficient γ eff ). It is therefore meaningful too, for previous flow tube designers, to have an estimation of the potential error of the measured γ eff if their experiment conditions match Case 1. We show here an example illuminating how this estimation can be accomplished, by means of simulation under the pre-defined experimental configurations identical with those adopted in the above section. 20 Figure 7 shows the maximum relative errors that can be expected for a series of γ eff under the three different cases of ε max /R 0 shown in Fig. 6. Similar to the dependence of γ eff /γ on γ (Fig. 6), the increase of γ eff and ε max /R 0 are also accompanied by the increase of γ eff /γ in Fig. 7. The experiment example with coating thickness matching Case 1 (soil coating in Fig. 5B) is examined and the estimated maximum error (i.e., γ eff /γ, γ eff is 5.5 × 10 -5 , an average value from the measurements by Li et al., 2016) is ~ 1.5 (red solid circle), implying that the coating roughness has a small effect on laminar flow in their case. But for larger γ eff , γ eff /γ 25 versus a wide range of γ eff (red dotted line in Fig. 7) also gives high values, for example, when γ eff > 10 -3 . This simulation further highlights the need to inspect the possible errors of previously measured high γ eff (e.g., > 10 -3 ) if their coating roughness is accord with Case 1.

Conclusions
In this study, a new criterion is proposed to eliminate/minimize the potential effects of coating surface roughness on laminar flow maximum relative error of the effective uptake coefficient γ eff . The error estimation demonstrates that a smaller positive bias of γ eff can be expected for experimental configurations employing gas reactants with lower uptake efficiency or/and smaller relative ratio of ε max /R 0 .

Data availability
The underlying research data and Matlab code for the M-CKD and CKD-B methods can be accessed upon contact with Yafang 5 Cheng (yafang.cheng@mpic.de), Hang Su (h.su@mpic.de) or Guo Li (guo.li@mpic.de).

A.1 Evaluation of the modified CKD method
To have an intuitive feeling of the change of concentration profiles due to local-turbulence-induced enhancement of the uptake coefficient, the CKD-B and the M-CKD have been applied to the experiment configuration of the exampled Case 1 in Fig. 7 (see 10 HCHO in Table A.1), as shown in Fig. A.2. Comparison between Fig. A.2 (A) and (B) shows that local turbulence within the roughness thickness can enhance radial transport of the gas reactant and thus increase the effective uptake coefficient γ eff .

A.2 Derivation procedure of γ eff /γ versus γ or γ eff
Derivation of γ eff /γ versus γ or γ eff is based on a combination of a modified CKD method (M-CKD), assuming that roughnessinduced local turbulence occurs within the domain of 0.5ε max (simulation of Case 1), and the CKD-B method (a CKD-based 15 method using Matlab) which was described in our previous study (Li et al., 2016). The respective derivation procedures are shown in Fig. A.3. For one specific experiment configuration, both CKD-B and M-CKD can generate a correlation table with its first column being concentration transmittance (C/C 0 ) and the second column the corresponding uptake coefficient (γ), and their one-to-one correspondence is indicated by the same subscripts (e.g., k, n, j, m etc.), as shown in Fig

30
Due to the different algorithms employed, the CKD method (Murphy and Fahey, 1987) and the KPS method (Knopf et al., 2015) could derive contrasting γ when turbulence occurs (see Fig. A.4). As shown in Fig. A.4, with ideal laminar flow (without any local turbulence, Case 2) the KPS (with diffusion correction) and CKD show perfect agreement for the derived γ in the C/C 0 range of 0.548 to 1 (shaded area). If C/C 0 is smaller than the critical transmittance value (< 0.548), e.g., because of enhanced Atmos. Chem. Phys. Discuss., doi:10.5194/acp-2017-232, 2017 Manuscript under review for journal Atmos. Chem. Phys. Discussion started: 28 April 2017 c Author(s) 2017. CC-BY 3.0 License. mass transport towards the coated-wall due to local turbulence in laminar flow, the KPS results in negative γ (for details, see Knopf et al., 2015) while the CKD has no solutions. For C/C 0 larger than 1, both methods derive negative γ implying emissions of gas reactants from the coating.

Acknowledgments
This study was supported by the Max Planck Society (MPG) and National Natural Science Foundation of China (41330635).   (Wang et al., 2012); circle (Li et al., 2016) and star in (B) (Steimer et al., 2015); blue triangle (McNeill et al., 2006) and light red triangle (Petitjean et al., 2009) in (C). 10 15 Atmos. Chem. Phys. Discuss., doi:10.5194/acp-2017-232, 2017 Manuscript under review for journal Atmos. Chem. Phys. Discussion started: 28 April 2017 c Author(s) 2017. CC-BY 3.0 License. Figure 6. Maximum error of the effective uptake coefficient (γ eff ) relative to the real uptake coefficient (γ) versus γ, for three cases with different ratio of the coating thickness to tube radius (ε max /R 0 ). The choices of ε max /R 0 cover the general ratio range in previous studies. The curves cannot be further extended due to reaching the limits of diffusion correction methods (see Appendix A.3).