Standard reporting of Electrical Energy per Order (E_EO) for UV/H₂O₂ reactors (IUPAC Technical Report)

2.1 The amount of H₂O₂ used in the process

H₂O₂ concentration is a term in both the numerator and the denominator (as part of the sum of the scavenging compounds) of Eq. 1 describing the steady-state hydroxyl radical concentration. As a result, [HO˙]_SS has a non-linear relationship with [H₂O₂]. As [H₂O₂] increases, [HO˙]_SS increases, but the incremental increase in [HO˙]_SS becomes smaller with each incremental increase of [H₂O₂] until [HO˙]_SS eventually plateaus. A sensible range of H₂O₂ mass concentration (γ_H2O2) used in full-scale applications is (5–40) mg L⁻¹. The E_EO can be greatly affected by γ_H2O₂ within this range. For example, consider a hypothetical reactor delivering 1 mW cm⁻² average irradiance and treating water with 2 mg L⁻¹ of total organic carbon (TOC) and 50 mg L⁻¹ of alkalinity as CaCO₃. Using sucralose as a hypothetical target compound C (k_HO,C=1.56×10⁹ L mol⁻¹ s⁻¹[17]), at γ_H2O₂=(10 and 40) mg L⁻¹, the residence times to achieve one order-of-magnitude decrease in sucralose in the specified reactor are 70 min and 23 min, respectively. These times are directly proportional to the E_EO, since the energy input is constant over time. Therefore, the E_EO of the same reactor treating the same compound in the same water matrix will be 3 times higher if tested with γH₂O₂=10 mg L⁻¹ than if tested with γ_H2O₂=40 mg L⁻¹. These calculations are based on Eq. 1 used to calculate [HO˙]_SS and then used to determine the first order reaction rate in the reactor using Eq. 3. The reactor is assumed to have plug-flow conditions typical for UV reactors.

In general, the lower the concentration of H₂O₂, the more effect its variation will have on the E_EO. In the hypothetical reactor described above, the effect of the H₂O₂ concentration on the required residence time to achieve one order-of-magnitude decrease in the contaminant concentration is shown in Fig. 1. The effect of H₂O₂ concentration on the E_EO has been previously discussed and experimentally demonstrated [6].

Fig. 1:

Residence time as a function of γ_H2O₂ for a plug-flow reactor with 1 mW cm⁻² average irradiance treating water with 2 mg L⁻¹ of TOC and 50 mg L⁻¹ of alkalinity as CaCO₃.

2.2 Water quality parameters

Bicarbonate alkalinity and background organic matter are of particular importance in drinking water and wastewater applications, with k_S=8.5×10⁶ L mol⁻¹ s⁻¹[18] and 3.6×10⁸ L mol⁻¹ s⁻¹[19], respectively. For organic matter, mol L⁻¹ in the units of the constant refers to molar concentration of organic carbon, and the molecular weight of 12 g mol⁻¹ can be used for conversions. Bicarbonate alkalinity and organic matter constitute the bulk of the collective HO˙ scavenging rate, k_S[S] of Eq. 1. The HO˙ scavenging rate has units of s⁻¹. Within the typical range of water quality parameters for drinking water (approximately (1–5) mg L⁻¹ of TOC and (50–200) mg L⁻¹ of alkalinity as CaCO₃), the scavenging rate can vary from under 50 000 s⁻¹ to almost 200 000 s⁻¹ (a factor of 4). The corresponding residence time in a plug-flow reactor delivering 1 mW cm⁻² average irradiance and using γ_H2O₂=10 mg L⁻¹ is shown in Fig. 2. The residence time was calculated for one order-of-magnitude decrease in sucralose concentration as in the previous section and is directly proportional to the E_EO. The scavenging rate in Fig. 2 takes into account the scavenging by H₂O₂ (7900 s⁻¹ for γ_H2O₂=10 mg L⁻¹) and sucralose (2000 s⁻¹ at 5 μg L⁻¹). As can be seen in Fig. 2, depending on the water quality, the E_EO of the reactor can vary by over a factor of 4. It is important to understand that the E_EO is affected by the composition of the water matrix for which it was tested. Often, the E_EO is calculated in pure water, where only H₂O₂ and the HO˙ probe (a chemical added to indirectly measure [HO˙]_SS via monitoring the chemical reaction between the probe and HO˙) are the HO˙-scavenging compounds. In order to estimate the process energy consumption for practical applications, it is important to understand how this value will change once applied to a real water matrix.

Fig. 2:

Residence time as a function of background scavenging rate for a plug-flow reactor with 1 mW cm⁻² average irradiance treating water with 10 mg L⁻¹ of H₂O₂ and 5 μg L⁻¹ of sucralose.

It must be noted that the linear relationship between HO˙ scavenging rate and the E_EO presumes that changes in water quality that affect the scavenging rate do not affect the average irradiance in the reactor. In reality, this is an oversimplification. Increasing TOC, for example, will increase the absorbance of the water and will lower the average irradiance (Eq. 2) in the reactor, which was assumed to be constant in this illustration. The effect of water absorbance on average irradiance depends on the average path length in the reactor, which depends on the reactor hydraulics.

2.3 Test compound

Several aspects of the test compound identity can affect the E_EO:

1. The absorbance of the test contaminant affects the fraction of photons that can reach H₂O₂, which in turn affects the [HO˙]_SS that can be generated in the process.

2. The concentration of the test contaminant and its reaction rate with HO˙ affects [HO˙]_SS that can be generated in the process by contributing to HO˙ scavenging.

3. The susceptibility of the contaminant to direct photolysis affects the residence time needed to achieve certain level of removal in different reactors depending on average irradiance and H₂O₂ concentration.

The test compound is commonly added at a concentration that allows monitoring the compound by analytical determination without any additional processing, such as solid phase extraction. This typically means concentrations on the order of μg L⁻¹ or mg L⁻¹. For example, 4-chlorobenzoic acid, commonly used in research for measuring [HO˙]_SS, has a molar absorption coefficient on the order of 3000 L mol⁻¹ cm⁻¹ at 254 nm (emission wavelength of low pressure mercury vapor lamps common in UV AOP reactors). This translates to an absorption coefficient of approximately 0.02 cm⁻¹ at 1 mg L⁻¹. Depending on the background matrix of the water being evaluated, the addition of a probe compound could result in a significant decrease in the average irradiance delivered to the water. For example, most drinking waters have 85–95% transmittance at 254 nm wavelength, which translates to an absorption coefficient of (0.071–0.022) cm⁻¹. Using 1 mg L⁻¹ of 4-chlorobenzoic acid can double the absorption coefficient of the water with the original transmittance of 95%, which can significantly affect the average irradiance E and the E_EO as the result. The effect is most prominent for high transmittance (low absorbance) waters (Fig. 3). This can be even more prominent with medium pressure mercury vapor lamps. Many organic chemicals have higher absorbance at lower wavelengths and could even more dramatically affect the fraction of photons that reach H₂O₂ and the resulting HO˙ formation.

Fig. 3:

Effect of incremental changes in absorbance on average irradiance in reactors with different average UV path length. Average irradiance is directly proportional to the E_EO.

The contribution from a probe compound (also a HO˙ scavenger) to the overall scavenging rate is another important factor. The reactor is often used to address contaminants at trace levels that are part of the overall background organic matter. Adding a probe increases the overall organic content. Often, the probe is more reactive than the bulk organic matter and can change the scavenging rate considerably. For example, in a study by Keen et al. (2014) [20], the authors used methylene blue at 5 μmol L⁻¹ to measure the background HO˙ scavenging rate of wastewater effluent samples and reported that the probe compound accounted for 25–50% of the overall HO˙ scavenging rate in the sample (corresponding to approximately 30–100% change in overall scavenging rate). The scavenging rate is higher in wastewater effluent samples than in drinking water samples; therefore, adding 5 μmol L⁻¹ of methylene blue as a probe to a drinking water matrix would result in an even higher percent change. As it can be seen in Fig. 2, doubling the scavenging rate would approximately double the measured E_EO.

It must be noted that methylene blue is highly reactive with hydroxyl radicals, with a reaction rate constant of 2.1×10¹⁰ L mol⁻¹ s⁻¹[18]). The reactivity of the probe determines how much it affects the E_EO measurement. For example, Fig. 4 shows the calculated time it takes to decrease by one order of magnitude the concentration of the algal toxin microcystin-LR in a hypothetical reactor with 1 mW cm⁻² average irradiance, 2 mg L⁻¹ of TOC and 50 mg L⁻¹ of alkalinity as CaCO₃ using 10 mg L⁻¹ of H₂O₂ and 500 μg L⁻¹ of HO˙ probe concentration. The probe options are either using the target compound microcystin-LR (high HO˙ reactivity, k_HO,M-LR=2.3×10¹⁰ L mol⁻¹ s⁻¹[21]) or using an alternative lower-reactivity probe, for example sucralose (k_HO,sucralose=1.56×10⁹ L mol⁻¹ s⁻¹). For the given theoretical reactor, using microcystin-LR spike as a probe, the time to oxidize 90% of the compound would be measured as 7.0 min. Based on the [HO˙]_SS that this reactor can generate when the scavenging rate excludes the probe compound (the actual operating conditions), 90% oxidation of microcystin-LR is achievable in 4.8 min. Because the residence time in the reactor for one order-of-magnitude decrease in a contaminant is directly proportional to the E_EO of the reactor for the given contaminant and water quality, the test with microcystin-LR would result in a measured E_EO that is 46% higher than the actual E_EO. With sucralose used to determine [HO˙]_SS, the calculated residence time required for 90% oxidation of microcystin-LR is 4.9 min, only a 2% difference from the actual value.

Fig. 4:

Residence time in a hypothetical reactor (1 mW cm⁻² average irradiance, 2 mg L⁻¹ of TOC and 50 mg L⁻¹ of alkalinity as CaCO₃ using 10 mg L⁻¹ of H₂O₂ and 500 μg L⁻¹ of HO˙ probe concentration) to decrease the concentration of microcystin-LR by one order of magnitude: actual (no probe) and measured by spiking a probe with relatively low or high HO˙ reactivity.

It is possible to back-calculate the performance of the reactor by factoring out the contribution of the probe to the scavenging rate, but an estimation of the average irradiance to be used in Eq. 1 would be necessary, and this can be a complex task for a full-scale reactor. Therefore, the use of a lower-reactivity probe can provide a good estimate of the actual E_EO for a given reactor, water quality, and target contaminant. The times to achieve one order-of-magnitude removal for the probe compound and for the target compound are related through their second-order reaction rate constants with HO˙ (Eq. 4), and thus the E_EO can be calculated for the specific target compound relative to the probe used.

The third property of the probe compound relevant for evaluating the E_EO of the reactor is its susceptibility to direct photolysis. The photolysis rate constant for a compound is determined from the observed first-order decay in the reactor. It is reported in either time-based units (e.g. s⁻¹) or fluence-based units (e.g. cm² mJ⁻¹). The time-based rate constant can be divided by the average irradiance to convert it to the fluence-based rate constant [22]. The influence of photolysis is particularly relevant for determining the effect of the parameters that affect the HO˙ production on the E_EO of the reactor (e.g. H₂O₂ concentration or background HO˙ scavenging rate). Depending on the relative contribution of direct photolysis vs. HO˙ oxidation to the contaminant decrease in the reactor, this phenomenon may or may not have a pronounced effect on the E_EO estimation. A prime example of a compound for which this caution would be relevant is N-nitrosodimethylamine (NDMA). NDMA is highly susceptible to photolysis (k_phot,NDMA=2.3 cm² J⁻¹), but has a comparatively low reaction rate constant with HO˙ (k_HO,NDMA=3.3×10⁸ L mol⁻¹ s⁻¹) [23]). For example, the effect of increasing the H₂O₂ concentration on the E_EO of the UV/H₂O₂ AOP will not be the same for NDMA as, for example, for microcystin-LR, a compound much more reactive with HO˙ and about as susceptible to photolysis as NDMA (k_HO,M-LR=2.3×10¹⁰ L mol⁻¹ s⁻¹; k_phot,M-LR=3.65 cm² J⁻¹[21], [24]). This concept is illustrated in Fig. 5: the E_EO for treating microcystin-LR would decrease rapidly with H₂O₂ addition, but it would stay relatively unaffected by the H₂O₂ for NDMA.

Fig. 5:

Residence time as a function of H₂O₂ concentration in a hypothetical reactor with 1 mW cm⁻² average irradiance treating water with 2 mg L⁻¹ of TOC and 50 mg L⁻¹ of alkalinity as CaCO₃. Microcystin-LR and NDMA are assumed to be at trace levels and do not affect the calculated [HO˙]_SS.

3.1 Selecting the optimal H₂O₂ concentration

Mathematically, the E_EO will decrease indefinitely as the H₂O₂ concentration increases. For practical purposes, the [HO˙]_SS will plateau. This concept is illustrated in Fig. 1. In treating sucralose in the hypothetical reactor as illustrated in the figure, an increase of H₂O₂ concentration from 5 mg L⁻¹ to 10 mg L⁻¹ would decrease E_EO by a factor of 1.9. A further increase of the H₂O₂ concentration by another 5 mg L⁻¹ from 10 to 15 mg L⁻¹ would drop the E_EO by an additional factor of 1.4. For every next incremental increase of 5 mg L⁻¹, the energy saving will become smaller. The cost of additional H₂O₂ will at some point exceed the cost of the saved energy. The E_EO vs. H₂O₂ concentration can be mapped out for the specific application (specific to the water quality and the contaminant) and the optimal H₂O₂ concentration can be selected with consideration of the costs for both H₂O₂ and electrical energy.

The optimal H₂O₂ concentration for contaminants not susceptible to photolysis is not affected by the k_HO of the specific contaminant, as the H₂O₂ concentration directly affects the [HO˙]_SS, and the E_EO for the given contaminant is directly related to the [HO˙]_SS. For contaminants that rely mainly on direct photolysis and less on the reaction with HO˙ (e.g. NDMA), the optimal H₂O₂ concentration would be different for each specific contaminant. The relative contribution of the direct photolysis and HO˙ reaction to the overall contaminant degradation will affect the incremental decrease in E_EO for every additional 5 mg L⁻¹ of H₂O₂. For example, a compound X, 10 times as reactive with HO˙ as sucralose (k_HO,X≈1×10¹⁰ L mol⁻¹ s⁻¹), but not susceptible to photolysis, would still have a 1.9 times decrease in the E_EO when the H₂O₂ concentration is increased from 5 to 10 mg L⁻¹ and a 1.4 times decrease as the H₂O₂ concentration goes from 10 to 15 mg L⁻¹, all other parameters being constant. For a substance like microcystin-LR, which is fairly susceptible to photolysis and has a fast reaction with HO˙, the E_EO decrease in the same reactor treating water with the same quality would be 1.5 and 1.3 for a H₂O₂ concentration increase from 5 to 10 mg L⁻¹ and from 10 to 15 mg L⁻¹, respectively. For NDMA, highly susceptible to photolysis and reacting slowly with HO˙, those values would be 1.0 for each increment (< 5% improvement), indicating that the E_EO would be virtually unaffected by any increase in the H₂O₂ concentration. These theoretical calculations for NDMA are confirmed by the experimental data from the literature [23].

The average irradiance in the reactor is directly proportional to the [HO˙]_SS. Therefore, the discussion in the previous paragraph holds for any reactor treating water of a given quality. Water quality, however, would have an effect on the optimal H₂O₂ concentration. The greater the background scavenging rate, the greater will be the incremental improvement in the E_EO with every incremental increase in H₂O₂ concentration. However, the effect is not very dramatic within the range of water quality values. For example, for very low TOC and alkalinity water (0.5 mg L⁻¹ and 50 mg L⁻¹ as CaCO₃, respectively), the incremental decrease in E_EO would be 1.7 times and 1.3 times as the H₂O₂ concentration is increased from 5 to 10 mg L⁻¹ and from 10 to 15 mg L⁻¹. For a water matrix with higher amounts of HO˙ scavenging substances (5 mg L⁻¹ of TOC and 200 mg L⁻¹ of alkalinity as CaCO₃), those values are 2.0 and 1.5. The optimal H₂O₂ concentration for matrices with higher HO˙ scavenging potential is likely to be higher than for matrices with lower scavenging potential. However, the difference is not great within the range of water quality in the typical UV/H₂O₂ AOP process, and, depending on the costs of energy and H₂O₂, the range of optimal H₂O₂ concentrations is likely to be narrow and somewhere in the (10–30) mg L⁻¹ range for most applications.

3.2 Evaluating the energy use by a specific reactor for a specific purpose

This is the most straightforward application of the E_EO, and it should be reported with the specifics, such as water quality, H₂O₂ concentration used, and the target contaminant identity. This value would not be transferrable to other applications or comparable with the use of other reactors. This can be considered akin to the reactor validation used in disinfection, where a reactor is tested for a range of water transmittances and flow rates to span the range of those variables in the actual field placement of the reactor. For example, a utility can use this information to estimate the energy use for the process for a range of water quality fluctuations. The value can be adjusted to a desired level of contaminant removal. For example, the electrical energy required to achieve two orders of magnitude of contaminant degradation will be twice the electrical energy for one order of magnitude.

Another example where such evaluations may be useful is in determining the placement of the UV/H₂O₂ reactor in the treatment train or to evaluate the cost-effectiveness of different pre-treatment technologies for UV/H₂O₂. For example, the E_EO can be calculated for the oxidation of taste- and odour-forming compounds in drinking water applications, comparing sand filtration and membrane filtration as pre-treatment. The energy savings calculated based on the E_EO can then be compared with the energy expenditure/costs of the pre-treatment options.

3.3 Comparing reactor performance for different contaminants

The E_EO values for a range of potential contaminants can be calculated for a specific reactor with specific water quality and H₂O₂ concentration. However, it is important to understand that these relative E_EO values are affected by the water quality and the H₂O₂ concentration in a given reactor. For example, as shown in Fig. 5, in the same hypothetical reactor, the ratio of the E_EO values for NDMA and microcystin-LR varies from 2.4 to 5.1 as the H₂O₂ concentration increases from 5 to 30 mg L⁻¹. While these ratios are likely to be similar for different reactors based on the relative susceptibility of each compound to direct UV photolysis vs. HO˙ reaction, the results are not transferrable between reactors, and the practical utility of such information is limited.

3.4 Comparing the performance of various reactors

The electrical energy used by the reactor to achieve one order-of-magnitude decrease in the contaminant per unit volume (the definition of E_EO) is directly related to the ability of the reactor to transfer efficiently that energy to H₂O₂, which depends on the reactor hydraulics, configuration, reflective wall material, etc. This ability is quantified by the portion of Eq. 1 shown in Eq. 5 below. This quantity (ξ) has units of s⁻¹ and represents the rate by which H₂O₂ absorbs energy and converts it to HO˙.

To compare two reactors, the comparison should be essentially on their ξ, or, more specifically, on the efficiency of converting electrical energy to ξ. Comparison of reactors on their ability to convert electrical energy to E_λ is possible for monochromatic UV reactors emitting the same wavelength. Using ξ allows comparison between reactors using different wavelengths and polychromatic spectra. Energy use vs. ξ essentially shows the efficiency of converting electrical energy into HO˙.

For example, reactor A is reported to have an E_EO of 7.1 kWh m⁻³ per order of magnitude change in the concentration of a target component when tested with caffeine concentration=20 mg L⁻¹ and γ_H2O₂=25 mg L⁻¹[25]. Because caffeine is susceptible to direct photolysis, it is important to know the photolysis rate in the units of s⁻¹ and the overall observed decay rate in the same units. If the rates are reported in units of cm² mJ⁻¹, average irradiance in cm² mW⁻¹ should be reported as well. In the cited example, the respective values for caffeine were 0.006±0.003 and 0.19±0.02 min⁻¹. The observed first-order decay rate is therefore (0.19−0.01) min⁻¹=0.18 min⁻¹=0.0030 s⁻¹. From here, the [HO˙]_SS achieved in the reactor can be calculated knowing the k_HO for caffeine (4.1×10⁹ L mol⁻¹ s⁻¹ as measured by using competition kinetics with 4-chlorobenzoic acid [25]). Based on this, it can therefore be determined that the [HO˙]_SS generated in the reactor was 0.0030 s⁻¹/4.1×10⁹ L mol⁻¹ s⁻¹=7.3×10⁻¹³ mol L⁻¹. The quantity described in Eq. 5 can now be calculated using the experimental data ([HO˙]_SS, [H₂O₂], k_HO,H2O₂, [caffeine], k_HO,caffeine) and Eq. 1. ξ is thus determined to be 4.4×10⁻⁴ s⁻¹. The ξ for the reactor is directly related to the ability of the reactor to generate HO˙. It is also directly related to the photolysis rate if the contaminant is susceptible to it, as the E in the ξ term directly affects the time it takes to achieve a specific UV dose in a given reactor. Therefore, the E_EO is directly related to energy use divided by ξ.

Another hypothetical reactor B was tested with γ_sucralose=0.5 mg L⁻¹ and γ_H2O₂=10 mg L⁻¹, and the E_EO was calculated to be 5.5 kWh m⁻³ per order. It is impossible to directly compare the two reactors A and B, as they were tested with different compounds under different conditions. Although the E_EO of the hypothetical reactor B is lower, it is not necessarily a better reactor than reactor A described above. Knowing the time-based observed (reactor-specific) first-order reaction rate constant and the reaction rate constant for direct photolysis (not applicable for sucralose), ξ for the reactor can be calculated. That value in this hypothetical reactor B was calculated to be 1.43×10⁻³ s⁻¹. This can now be used with other target compounds and H₂O₂ concentrations to calculate [HO˙]_SS, which is directly related to the E_EO.

To properly compare the two reactors (A and B), caffeine should be considered as a potential reactant in the hypothetical reactor B, or sucralose can be modelled as a probe in the reactor A, described in [25]. Because sucralose has no photolysis component that needs to be considered, it is easier to compare the two reactors by modelling the performance of the reactor A described in [25] for sucralose and comparing the results with the results of the other (hypothetical) reactor B. Using the H₂O₂ and sucralose concentrations from the hypothetical reactor B, with ξ from the reactor A, we can estimate that, under the hypothetical test conditions of γ_H2O₂=10 mg L⁻¹ and γ_sucralose=0.5 mg L⁻¹, the reactor A described in [25] would achieve [HO˙]_SS=1.3×10⁻¹¹ mol L⁻¹ (Eqs. 1 and 5). Using the k_HO,sucralose, the observed first-order reaction rate constant k_obs can be calculated to be 0.020 s⁻¹. If in the original experiment, the E_EO for caffeine with k_obs=0.19 min⁻¹=0.0032 s⁻¹, and the E_EO was calculated to be 7.1 kWh m⁻³ per order, then for sucralose with k_obs=0.020 s⁻¹, the E_EO will be proportionately less, that is, if k_obs for sucralose is 6.25 times higher than k_obs for caffeine, the E_EO for sucralose will be 6.25 times lower, namely 1.1 kWh m⁻³ per order. This rests on the fact that the log[C]/[C₀], which goes into the E_EO calculations, has a linear relationship between the first order reaction rate constant and the residence time in the reactor. Therefore, even though the E_EO for reactor A, tested with caffeine, was 7.1 kWh m⁻³ per order, and the value for the hypothetical reactor B, tested with sucralose, was 5.5 kWh m⁻³ per order, reactor A had better energy efficiency, because when adjusted to sucralose, the E_EO was calculated to be 1.1 kWh m⁻³ per order.

Such a difference in the E_EO values for a single reactor with two different test compounds is not surprising, as the scavenging rate in each test is entirely different and greatly affects the test outcome, as discussed in Section 2.2. The background HO˙ scavenging rate due to 20 mg L⁻¹ of caffeine and 25 mg L⁻¹ of H₂O₂ is 440 000 s⁻¹, while the scavenging rate due to 0.5 mg L⁻¹ of sucralose and 10 mg L⁻¹ H₂O₂ is only 9900 s⁻¹. For practical purposes, it is important that the background scavenging rate is as close to the actual application as possible.

Alternatively, the E_EO of the hypothetical reactor B could be calculated for the caffeine test conditions reported for reactor A. In this case, the k_obs would need to be estimated based on its two components: the first-order reaction of photolysis (k_phot) and the first-order reaction with HO˙ (k_radical). k_radical is equal to k_HO,caffeine[HO˙]_SS and is easily calculated using the reactor ξ and the H₂O₂ and caffeine concentrations (see Eq. 1 and Eq. 5). k_phot is more difficult to transfer between reactors and requires that both studies report the conversion between the time units and the fluence in mJ cm⁻². In the paper describing reactor A [25], that conversion can be obtained from the fluence-based rate constants reported in units of cm² mJ⁻¹ along with the time-based counterparts. k_phot in units of cm² mJ⁻¹ should be converted to units of s⁻¹ using the specifics of the hypothetical reactor. Since in previous sections the hypothetical reactor was discussed to have the average irradiance of 1 mW cm⁻², the conversion for the k_phot for caffeine is (0.02×10⁻³ cm² mJ⁻¹)×(1 mW cm⁻²)=0.02×10⁻³ s⁻¹.

Therefore, it can be concluded that the following parameters need to be reported along with the E_EO for comparing values for reactors tested under different conditions: H₂O₂ concentration, probe compound and concentration, second-order k_HO of the probe (with source referenced), HO˙ scavenging by the water matrix (if tested in natural water), the observed overall reaction rate for the test compound, and the reaction rate for direct photolysis, both in units of cm² mJ⁻¹ and in units of s⁻¹ (or other time-based units). This approach does not take into account the change in absorbance with each compound and its corresponding effect on ξ. Any comparison between the E_EO values reported in different sources is only approximate. Use of the ξ term allows for the comparison of reactors that use different wavelengths, as well as polychromatic sources.

3.5 Evaluating the process at bench scale for full-scale application

The ability to compare bench-scale results with full-scale results relies on the ability to match the average irradiance in the bench-scale reactor with that of the full-scale reactor. If such conditions can be achieved, then the relative (not absolute) E_EO values measured at bench scale for various contaminants, H₂O₂ concentrations, and water quality parameters should be transferrable to the full scale. Transmittance of water and average path length also need to be considered and can lead to significant overestimation of E_EO from bench to full scale if not taken into account [22]. Additionally, the actual E_EO for the full-scale reactor would still have to be calculated with respect to a reference E_EO measured at full scale. For example, if the ratio of the E_EO at 5 and 10 mg L⁻¹ of H₂O₂ is equal to 1.7 in a bench-scale reactor, the same ratio will be observed at these H₂O₂ concentrations in a full-scale reactor, as long as the reactors have the same average irradiance.

3.6 Common mistakes when applying the E_EO parameter