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.