3.4 Analytical RTD Predictions of the Damköhler number.
The previously published properties of the Embarras River (Sukhodolov et al., 2006) were used to normalize the residence times (\(\tau_{\text{rp}}\sim 1\) h and \(\tau_{\text{dn}}\sim 10\) h, respectively) using \(eq.\ 11\). Then, for different \(d_{b}^{*}\), the empirical \(\text{Da}_{\text{rp}}\) and \(\text{Da}_{\text{dn}}\) were calculated as the ratio between the empirical \(t_{50}\) and the normalized respiration and the denitrification time scales\(\tau_{\text{rp}}^{*}\) and \(\tau_{\text{dn}}^{*}\), of \(\sim 1\) h and \(\sim 10\) h respectively. The corresponding\(\text{Da}_{\text{rp}}\) and \(\text{Da}_{\text{dn}}\) were calculated for each of our four analytical representations of the hyporheic zone RTD (GAM, LN, FR, EXP), and then compared to the Da values estimated from the empirical RTD (which was assumed here to be the gold standard) (Figure \(6\)). It should be noticed that\(\text{Da}_{\text{rp}}\) and \(\text{Da}_{\text{dn}}\) are simply proportional to each other (because they differ only by the reaction timescale), so the comparison between empirical and analytical representations of Da does not depend on the specific reaction considered.
Damkhöler Numbers generated from the empirical RTD (asterisks in Figure 6) decline more-or-less monotonically with increasing dimensionless streambed depth. This pattern is best represented by GAM for\(d_{b}^{*}<1.2\), and by FR over the full range of \(d_{b}^{*}\)evaluated here. The LN and EXP distributions under- and over-estimate the Da for dimensionless depths\(d_{b}^{*}<1.0\) and\(d_{b}^{*}>\ 1.0\), respectively. This result–that FR provides the best estimate of the Damkohler Number over the two-order of magnitude change in dimensionless sediment bed depth evaluated here—is surprising given that this analytical distribution is not the best representation of the empirical RTD for \(d_{b}^{*}\) <\(3.1\)(see above). The explanation is that, even for shallow depths, the optimized FR CDF intersects the empirical CDF at a cumulative probability of 0.5 across all dimensionless depth ranges evaluated here (compare green curve and asterisks, left panels, Figures 2-6). Hence, the analytical distribution’s estimate for \(t_{50}\) (and hence the Damköhler Number) is accurate, even for shallow bed depths where FR is a relatively poor representation of the empirical RTD.