Isophotonic reactor for the precise determination of quantum yields in gas, liquid, and multi-phase photoreactions

Photocatalytic processes are an emerging field with a multitude of potential applications ranging from waste and wastewater treatment over fine chemical production to artificial photosynthesis. Knowing the quantum yield in a photoreaction is thereby essential to both


Motivation
Photocatalytic processes are an emerging field with growing interest and a multitude of potential applications ranging from waste [1] and wastewater treatment [2,3] over fine chemical production [4] to artificial photosynthesis [5,6].The economic viability of most photocatalytic processes stands and falls with the achieved photocatalytic efficiency [6][7][8][9], the efficiency metric of the process relating the number of production events to the number of incident photons in a specific optical band, Eq. (1) [10].This especially holds for solar driven processes.η = #of production events #of incident photons in specif ic band (1) The overall photocatalytic efficiency of a photocatalytic process is thereby predominantly shaped by its three essential elements: the photocatalyst, the photoreactor and the light source employed [8,9].The light source may be regarded as part of the photoreactor as is the case in the present work.Both, the selection of suitable photocatalysts and the design of optimal photoreactors depend on the knowledge of the quantum yield, the efficiency metric of the photocatalyst relating the number of production events to the number of photons absorbed by the catalyst at a specific wavelength, Eq. (2) [10].Of utmost importance for the design of photoreactors is the dependency of the quantum yield on operating conditions such as temperature (T), reactants' concentration (c), wavelength (λ), and local volumetric rate of photon absorption (L a p,λ ), meaning the number of photons at a specific wavelength absorbed per unit time and unit volume.Still, this detailed information on quantum yields is seldomly reported.

Φ =
#of production events #of absorbed photons at specif ic wavelength = f On a side note, the definition of the quantum yield can be extended according to the IUPAC recommendation for terms in photocatalysis to differentiate between an integral and a differential quantum yield [10].Whereas the integral quantum yield relates time-integral values, the differential quantum yield relates rates (reaction rates and photon absorption rates) to each other.In systems in which, for instance, a distinct dependency of the quantum yield on the reactant concentration is observed, integral and differential quantum yield differ from each other.
Most literature works on quantum yields report apparent quantum yields that are derived from experiments that are conducted under conditions in which the number of incident photons can be assumed to equal the number of photons absorbed by the photocatalyst, see for instance [11][12][13][14].Typical setups are simple irradiated glass vessels or reaction chambers with windows like described in [14].Noteworthy, photon losses through scattering events in the photocatalyst bed are not considered in these approaches.The latter entails a systematic underestimation of the quantum yield of the examined photocatalysts.Black body photoreactors, like introduced for instance in [15], address this issue by placing the light source in a cavity surrounded by the employed photocatalyst.This setup ensures full photon absorption even in highly scattering photocatalysts by guaranteeing that scattered photons are redirected towards the photocatalyst.Importantly, in both approaches, the layer thickness of the catalyst must be chosen sufficiently high to guarantee that there are no significant transmissive losses.Such systems, however, inherently depict high gradients in the field of the local volumetric rate of photon absorption or process intensity respectively.A particular lack of knowledge is therefore present for the dependency of the quantum yield on the local volumetric rate of photon absorption.For the experimental examination of this dependency, photoreactors with zero or small gradients in the field of the local volumetric rate of photon absorptionisophotonic photoreactorsare highly desirable.This is in analogy to isothermal reactors that are highly desirable if not mandatory for the experimental examination of the temperature dependency of thermochemical reactions.
Within the contribution presented herein, we therefore propose a novel approach for the reliable determination of quantum yields in gas, liquid, and multi-phase photocatalytic reactions under isophotonic conditions.The 3D-printable photoreactor we employ allows for the control of temperature, reactants' concentration, wavelength, and the local volumetric rate of photon absorption.The detailed report includes the validation of our method via the determination of the quantum yield of the standard potassium ferrioxalate actinometer and comparison of the obtained data to literature data.Further, results of an exemplary application of our approach to a titania-based methanol reforming photocatalyst delineates the subtle and complex nature that quantum yields present.The concluding discussion sets into perspective the influences of the interdependencies of the quantum yield and operating conditions onto high efficiency production photoreactor design and thereby highlights the value of the proposed approach.

Description of the approach
Quantum yield measurements require experimental access to the number of eventsin photoredox reactions the number of transferred electronsand experimental or simulative access to the number of photons absorbed by the photocatalyst at a specific wavelength, Eq. ( 2) [10].Noteworthy in case of a heterogeneously catalysed photoreaction, 'photocatalyst' refers to the particulate photocatalyst bed.In case of a homogenous photoreactionwith or without dissolved photocatalyst -'photocatalyst' refers to the mixture of reactants.
In most cases, the number of events can easily be derived experimentally by determining the number of product molecules via standard analytical methods, for instance via UV-vis spectroscopy or liquid or gas chromatography.The major challenge arises from the determination of the number of absorbed photons at a specific wavelength [10].This holds especially in heterogeneous photocatalysis where photon scattering by the photocatalyst bed significantly influences the radiation transport problem in the system [10].In the introduced approach the number of photons absorbed by the photocatalyst is derived via spectrally resolved 3D optical simulations conducted using a MATLAB®based implementation of a Monte Carlo ray tracing (MCRT) method.For details on the method and implementation, see Appendix A, including  number of transferred electrons per hydrogen molecule ṄN2 nitrogen feed rate, mol/s q emitted total flux, W q λ emitted spectral flux, W/nm q p emitted total photon flux, einstein s − 1 q p,λ emitted spectral photon flux, einstein s flow chart.The simulations are conducted considering the precise threedimensional geometry of a tailored isophotonic photoreactor, Fig. 2, and the specific optical properties of the reactor components as well as the photocatalyst to be characterized.For details on the simulation case setup, see Appendix A. The optical properties assumed for the reactor components and the employed photo catalysts are determined prior to any system simulation in a set of optical experiments.For details on the experimental proceeding, evaluation and resulting optical properties, see Appendix B. The simulations deliver the spectral radiation transport efficiency (η T,λ ) being defined as the ratio between the number of photons at a specific wavelength absorbed in the reaction volume by the photocatalyst considered and the number of photons sent into the reactor by the employed light source at the same specific wavelength, Eq. ( 3).

#of absorbed photons at specif ic wavelength #of photons sent into the reactor at specif ic wavelength
(3) Consequently, to derive the number of photons absorbed by the photocatalyst placed in the reaction volume at a specific wavelength via the spectral radiation transport efficiency derived from simulations, the number of photons sent into the reactor by the employed light source must be derived experimentally via radiometric characterization of the latter.
All steps for the determination of quantum yields with the proposed approach are graphically summarized in Fig. 1.
The unique isophotonic photoreactor employed in the approach is optimized to provide a low gradient field of the local volumetric rate of photon absorption in the reaction volume.The main components of the employed photoreactor are a quartz glass tube that encloses the reaction zone, a 3D-printed, diffuse reflecting optics module placed around the quartz glass tube which disperses light into the reaction volume and lastly an LED light source module that precisely shines its light into the optics module, Fig. 2.  4), moveable heating brackets (5), a 3D-printed and silver-coated diffuse reflecting optics module (6) and an LED light source module, made up of a 3D-printed cooler body (7a), a cooler fan (7b) and two times six high power LEDs (7c) with two different peak wavelengths, in the presentation 365 nm, facing up (being idle) and 385 nm, facing down into the reaction chamber.The temperature in the reaction zone is controlled with a heating cartridge in the optics' module heating cartridge bore (8).The black scale bar in the zoom in corresponds to 5 mm.

Design and fabrication of isophotonic photoreactor
The focus in the reactor design process lies on the LED light source modules, more precisely the axial position and orientation of the six symmetrically aligned LEDs.Firstly, the LEDs are alternatingly inclined for 20 • to prevent direct illumination of the reaction zone which would lead to hotspots in the field of the local volumetric rate of photon absorption.Secondly, the axial positions of the LEDs are numerically optimized in a way that the axial course of the mean value of the local volumetric rate of photon absorption shows the lowest possible axial gradients.This simulation-based optimization task is accomplished with surrogate-based standard optimization tool surrogateopt from the MATLAB® global optimization toolbox.For the calculation of the objective function in the optimization, the reaction volume is divided in twenty equally thick slices, for illustration, see Supplementary Figure S3.In each of the slices, the mean value for the local volumetric rate of photon absorption is calculated.The objective function value subject to minimization finally equals the variance of the twenty calculated mean values.
The LED position optimization depends on optical properties of the employed components.Of special interest is the dependency of the optimal LED alignment on the optical properties of the reaction volume/ the employed photo catalyst respectively.Only if the optimal positions of the LEDs on the LED light source module do not significantly depend on the optical properties of the reaction volume, a general design can be derived that is applicable to various photocatalysts.To check for a possible influence of the optical properties of the reaction volume on the optimal LED alignment, the optimization is conducted with optical properties of the reactor components at 405 nm and for different optical property sets of the reaction volume.
The derived optimal LED positions are realized practically by designing small, inclined pillars in the positions derived by the optimization onto a cooler structure equipped with a cooler fan, see zoom-in in Fig. 2. In this work, all reactor and light source structure components, namely the reactor body as depicted in Fig. 2, the optics module and the LED cooler body were printed in 316L stainless steel using a Realizer SLM 125 3D printer.The homogeneous and fine roughness of the 3D printed stainless steel substrate ensures Lambertian reflection characteristics as desired.To increase the reflectivity of the optics module after the printing process, the raw stainless steel part needs to be optically coated.In the present work, the optics module was galvanically silver coated by C. Jentner GmbH.
Lastly the LEDs of the LED light source modules must be electrically connected and mounted onto the pillar structures on the cooler body.The employed high-power LEDs with peak wavelengths of 365 nm, 385 nm, 395 nm, 405 nm, and 450 nm from by WÜRTH Electronics SE employed in this work were soldered to wires and subsequently glued onto the pillar structures on the printed LED cooler body with thermally conductive glue HERNON 746 from ALUTRONIC Kühlkörper GmbH & Co. KG.

Approach validation -Potassium ferrioxalate actinometry
) undergoes photoreduction when irradiated with photons with a wavelength smaller than 550 nm [16].One photon thereby triggers two reduction events, Eq. (4).Since the first publications from Hatchard and Parker in 1956 reporting detailed information on the quantum yield for this reaction [17], numerous literature contributions addressed the various aspects of the potassium ferrioxalate photoreduction reaction as a chemical actinometer [18][19][20], which makes it a well-suited reference for the validation of the introduced approach.
The experimental proceeding employed in this work for the determination of the number of production events is adapted from the work of Lehóczki et al [20].In the experiments 2 mM potassium ferrioxalate solution (potassium ferrioxalate, Alfa Aesar, in 0.05 M sulfuric acid, Fisher Scientific) was pumped through the photoreactor using a HITEC ZANG SyrDos TM tandem syringe pump.The concentration of the reaction product Fe II+ was determined in an Fe II+ 1,10-phenanthroline assay.In this assay 8 mL of collected sample were mixed with 3 mL 1,10-phenanthroline solution (0.1 g in 100 mL of deionized water), 4 mL of sodium acetate buffer solution (1 M sodium acetate, Arcos Organics, in deionized water mixed with 0.5 M sulfuric acid, REAGECON, in a volume ratio of 100:63), and 5 mL deionized water.The absorbance of the formed Fe II+ -1,10-phenathroline complex at 510 nm determined with an Agilent 8453 UV-Vis spectrometer was used to derive the desired Fe II+ concentration.The calibration of the assay linking the measured absorbance of the 1,10-phenathroline complex and the desired Fe II+ concentration was conducted with Mohr's salt solutions (Arcos Organics), see Supplementary Figure S5.In addition to the standard way via the Fe II+ assay, the actinometer conversion was determined online with a USB4000 UV-Vis spectrometer equipped with a FIA-Z-SMA-ULM flow cell (both Ocean Optics Inc.) that is included in the piping downstream of the photoreactor.The redundant determination increases the trustworthiness and reliability of the evaluated data set.The calibration linking the absorbance of the converted potassium ferrioxalate solution at 375 nm in the online measurement with the potassium ferrioxalate conversion was executed with synthesized actinometer solutions with different conversions prepared with potassium ferrioxalate and the commercially available reaction products iron-(II)oxalate (Alfa Aesar) and potassium oxalate (Arcos Organics), see Supplementary Figure S5.
To properly estimate the average volumetric production rate of Fe II+ in the photoreactor ( ṙFe II+ ), the actinometry experiment was conducted with three different flow rates (2.0 mL/min, 2.6 mL/min and 3.7 mL/ min).The three different derived Fe II+ concentrations at the reactor outlet were drawn over the corresponding space time.According to a simple plug flow reactor design equation, Eq. ( 5), and assuming the reaction rate was constant in the reaction volume, the slope of the resulting line in this plot corresponds to the volumetric production rate of Fe II+ .The assumption of a constant reaction rate is valid if the local volumetric rate of photon absorption and quantum yield are constant throughout the reaction volume ( ṙFe II+ = ΦL a p,λ ).
The LED emission power in the reported experiments was chosen in a way, that the mean volumetric rate of photon absorption equalled approximately 30 mEinstein m − 3 s − 1 .Considering the reaction volume of approx.0.23 mL, the potassium ferrioxalate concentration of 2 mM, the lowest feed rate employed of 2 mL/min and a quantum yield close to unity, a mean local volumetric rate of photon absorption of 30 mEinstein m − 3 s − 1 guarantees that the maximum conversion in the experiments is smaller than 20 %.The latter avoids precipitation of reaction products and limits conversion induced variations of the optical properties over the reactor length which would not be mapped in the optical simulations.
The average quantum yield (Φ) in the narrow emission band in which the employed LED light source module emits photons is derived via Eq.( 6).Thereby V corresponds to the volume of the reaction zone in the quartz glass tube, λ to the wavelength, and q p,λ to the total spectral photon flux sent into the reactor aperture by the LED light source module.

Φ =
ṙFe II+ V ∫ η T,λ q p,λ dλ (6) However, the quantum yield is defined at a specific wavelength, see Eq. ( 2), and not in an optical band.Even though the LED emission band is narrow, it is not strictly monochromatic, see Fig. C.1.Therefore, in addition to the mean values calculated by Eq. ( 6), estimated values of the quantum yield in narrow bands are reported.These values are derived via a least square estimation based on the assumption, that the experimentally determined mean quantum yields can be approximated via Eq.( 7).Thereby q p,i is the photon flux in each of the narrow bands considered in the estimation, Φ i the quantum yield in the band centre and q p the total photon flux of the employed LED lights source module.The least square estimation minimizes the sum of residual squares between experimental mean quantum yields recorded with different LED light source modules and the quantum yields estimated by Eq. (7).

Approach application -Methanol reforming
Titania-based photocatalysts are well-known to catalyse the endothermal photo reforming reaction of alcohols [21,22], for instance methanol [23,24], at ambient conditions resulting in hydrogen as target molecule, Eq. ( 8).Many of the literature reported photocatalysts are straightforward to synthesize from commercial precursors and reported to be stable in operation, which makes them a suitable exemplary system for the present study.
In the methanol photo reforming experiments reported, a palladium impregnated titania catalyst with 0.025 wt-% palladium loading was used.For details on the catalyst preparation see Appendix D. To ensure optical properties that allow isophotonic operation of the catalyst bed, meaning an absorption and scattering coefficient that allow penetration of photons into the reaction volume of approx. 2 mm thickness, the photocatalyst powder needs to be optically dilute.In the present work, the fine photocatalyst powder was therefore mixed with a translucent silica aerogel powder.The loading applied was 10 mg photocatalyst per mL reaction volume resulting in approximately 2.3 mg of catalyst in the reaction volume in the conducted experiments.For details on the catalyst support synthesis, powder preparation and mixing, see Appendix D.
The catalyst needs to be precisely placed in the foreseen reaction volume.The aerogel supported catalyst employed therefore was held in position in the quartz glass tube of the photoreactor by quartz glass wool plugs.The temperature of the catalyst bed in the isophotonic reactor is controlled with the optics module heating cartridge (see Fig. 2) and measured with a thermocouple axially inserted into the catalyst bed.The reactantsmethanol vapour and steamare supplied via a carrier gas stream.The carrier gas is dosed with mass flow controllers and passed through a saturator filled with a liquid methanol water mixture prior to entering the reactor.The saturator is equipped with a precise temperature control that allows the control of the methanol vapour and steam content of the carrier gas.The reactor effluent passes a pressure regulating valve and is subsequently fed into a gas chromatograph analysing the reaction products.The gas chromatograph was calibrated prior to the reported experiments for methanol, water, hydrogen, carbon dioxide, and methyl formate.All piping is electrically heated to prevent condensation of any species.Details on the test rig components and the piping and instrumentation diagram are part of the Supplementary information.
To check a possible influence of mass transport hinderances in the experiments, two control experiments with helium and nitrogen as carrier gas under high local volumetric rates of photon absorption were executed.The experiment idea is based on the approx.three times higher binary diffusion coefficient of methanol vapour in helium (approx.3.7 10 -5 m 2 s − 1 , estimated via Chapman-Enskog-Wilke-Lee model) compared to the binary diffusion coefficient of methanol vapour in nitrogen (approx.1.3 10 -5 m 2 s − 1 , estimated via Chapman-Enskog-Wilke-Lee model).If mass transport hinderancesboth internal or externalconsiderably influence the apparent reaction rate, the higher diffusivity of methanol vapour in helium compared to nitrogen as carrier gas leads to a significant difference in the observed apparent reaction rates in the experiments with helium and nitrogen.If no such influence can be observed mass transport hinderances can be neglected in the considerations.
For each of the employed LED light source modules, the LED light source emission power was varied through a drive current variation in five levels to vary the volume average local volumetric rate of photon absorption between zero (dark reference) and approximately 100 µEinstein g Cat -1 s − 1 .On a side note, the local volumetric rate of photon absorption can be given volume or catalyst mass specific.The two metrics are connected via the catalyst volumetric loading in the reaction volume.The reactor temperature was varied in three levels from 60  of approx.3 mol-% and approx.4 mol-%.The system pressure and carrier gas feed rate were kept constant at 0.5 bar relative and 5 mL N min − 1 , respectively.The experiment design comprised a full factor variation resulting in 30 operating points per LED light source module, Fig. 3A.Each operating point was measured three times before the next operating point was set.
To ensure stable reactant feed supply, the saturator was refilled with fresh methanol water mixture before each experiment set with one of the five LED light source modules.Additionally, after each saturator refill, a bypass measurement was conducted to check the reactant feed concentration.
To verify the catalyst is not deactivated, a reference measurement with the 365 nm LED light source module, at 60 • C reactor temperature, a reactant feed concentration of approx.3 mol-% and a volume average local volumetric rate of photon absorption of approximately 20 µEinstein g Cat -1 s − 1 was executed before each experiment with a new LED light source module.A flow chart of the whole experimental proceeding including all LED light source module changes, saturator refill procedures as well as bypass and reference measurements is depicted in Fig. 3B.
The average quantum yield (Φ H2 ) in the emission band of the employed LED light source module is derived based on the number of reduced hydrogen atoms.The latter corresponds to the number of produced hydrogen molecules derived from the experimentally determined hydrogen molar fraction (y H2 ) multiplied by the number of two electrons per hydrogen molecule (n e − ,H2 ), Eq. ( 9).Thereby ṄN2 corresponds to the carrier gas feed rate and y N2 to the molar fraction of nitrogen in the reactor effluent.

Optimal LED positions
The LED positions derived in the alignment optimization assuming different optical properties of the reaction volume result in the LEDs next to the front and back face of the LED light source module being slightly closer to each other compared to the LEDs in the centre, see Table 1.For an illustration of this clustering of LEDs near the front and back face of the LED light source module, see zoom in in the CAD model rendering in Fig. 2.
As evident from the optimal LED alignment parameter values listed in Table 1, the optimal LED alignment only slightly depends on the optical properties of the reaction volume.Since the deviations lie in the range of the manufacturing accuracy of the LED light source modules (roughly ± 400 µm), the LED light source modules do not need to be adjusted to a specific photocatalyst with its individual optical properties.However, a verification of the design criterion of an isophotonic reaction volume is advised for each photocatalyst considered.This especially holds since radial gradients that might occur as a consequence of a radiation transport hinderance in the photocatalyst bed are not addressed in the optimization.The verification should be based on the density distribution of the local volumetric rate of photon absorption as described in Section 3.1.2.
In the presented work, the realized optimal centre positions of the first, second, and third LED counted from the front face of the 54 mm long optics module were 1.7 mm, 6.5 mm, and 20.2 mm, respectively.For an illustration of the positioning, see yellow circles in Fig. 4. With the clustering of LEDs near the front and back face, the optimization algorithm compensates photon losses over the front and back face opening of the optics module resulting in an almost constant axial course of the mean local volumetric rate of photon absorption throughout the reaction volume, Fig. 4. On a side note, non-ideal LED alignments significantly deteriorate the homogeneity of the local volumetric rate of photon absorption, see axial absorption profiles resulting under nonideal LED alignments in Supplementary Figure S4.

Photocatalyst simulations
The simulations mapping the radiation transport problem in both, the validation experimentthe potassium ferrioxalate actinometryand the exemplary use casethe photo methanol reforming experiment assumed the accordantly derived optimal LED alignment.The simulations give access to the spectral radiation transport efficiency needed for the quantum yield calculations following Eqs.( 6) and (9), Fig. 5. Additionally, they reveal the density distributions of the local volumetric rate of photon absorption needed for the assessment of the homogeneity of the field of the local volumetric rate of photon absorption, insets Fig. 5.
The similar characteristic course of the spectral radiation transport efficiency in the two simulation sets presented in Fig. 5 with a significantly higher spectral radiation transport efficiency in the UV band compared to the Vis band is induced by a significantly higher absorption

Table 1
Derived optimal axial LED positions resulting in a minimal axial gradient in the radial mean value of the local volumetric rate of photon absorption.Values are indicated for different absorption and scattering coefficients of the reaction volume.The employed LED numbering is graphically highlighted in Fig. 4   .Whereas the design goal of an isophotonic reaction volume is precisely fulfilled for the potassium ferrioxalate system, the methanol reforming system shows a slightly broadened density distribution of the local volumetric rate of photon absorption especially for wavelengths below approx.375 nm.

Method validation -Potassium ferrioxalate actinometry
The mean quantum yields derived via Eq.( 6) for the potassium ferrioxalate system are already in good agreement with literature data [25] and thereby indicate the validity of the proposed approach, dark and light grey circles in Fig. 6.Additionally, the estimated quantum yields in four 20 nm wide bands between 350 nm and 410 nm support the interim conclusion of a valid approach, red circles in Fig. 6.The applicability of the proposed approach for the determination of quantum yields under isophotonic conditions is thereby successfully validated and the scheme presented in Fig. 1 can be applied to any photocatalyst.Noteworthy, the outlier at the lower end of the experimentally covered optical band in Fig. 6 can be attributed to the unfortunate device-specific inaccuracy of the determination of the LED light source module emission at wavelengths below 370 nm, discussion see Appendix C. The high standard deviations in the band from 390 nm to 410 nm results from lower emission power of the employed LED light source modules in this band and a thereby induced lower sensitivity of the estimation model towards Fig. 5. Simulated spectral radiation transport efficiencies in the optical band from 350 nm to 550 nm (primary graphs) and density distributions of the local spectral volumetric rate of photon absorption in the optical band from 350 nm to 450 nm (insets) for both, the potassium ferrioxalate actinometry experiments (A) and the methanol reforming experiment (B).The spectral resolution in the simulations was 2.5 nm.Fig. 6.Mean quantum yields determined via Fe II+ -assay and online UV-Vis spectroscopy at a mean local volumetric rate of photon absorption of approximately 30 mEinstein m − 3 s − 1 (solid circles, dark and light grey) together with estimated quantum yields in 20 nm wide bands between 350 nm and 410 nm (solid circles, red) in comparison with literature data [25] (open circles).Indicated error bars of experimental data are errors propagated from the determination of the Fe II+ reaction rate and thereby represent the experimental error.Indicated error bars of the estimated data correspond to one standard error derived in the quantum yield estimation in narrow bands.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)Fig. 7. Proposed reaction network in the methanol reforming system mapping oxidation of methanol (CH 3 OH) to carbon dioxide (CO 2 ) with the two intermediates formaldehyde (CH 2 O) and formic acid (HCOOH) and methyl formate as side product.The yellow box depicts the photon induced charge carrier dynamics that precedes the depicted chemical reactions including both charge carrier generation via photon absorption and charge carrier recombination in non-radiative processes.The reaction arrows highlighted in green and red indicate thermally activated processes that increase (green) or decrease (red) the quantum yield with increasing temperature, respectively.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)the quantum yield in this band.Higher wavelength specific resolution, or narrower bands for the quantum yield estimation respectively, would only be possible with more employed light source modules with peak wavelengths differing from those already employed.

Exemplary application -Methanol photoreforming
The methanol photo reforming experiments conducted with the exemplary titania-based catalyst revealed significant hydrogen production rates up to 32 mmol H 2 g cat -1 h − 1 and a neglectable dark activity below the detection limit of the employed gas chromatograph (which equals approx.0.2 mmol H 2 g cat -1 h − 1 ).The carbonaceous reaction products detected in the experiments were carbon dioxide and methyl formate with traces of formaldehyde at certain operating points supporting the proposed reaction network depicted in Fig. 7. Importantly the catalyst did not show significant deactivation in the more than 500 h on stream with constantly changing operating conditions, see reference measurements Supplementary Figure S8.Furthermore, the data from the experiment conducted to check for a possible influence of mass transport hindrances in the kinetic measurements indicates these can be neglected in the considerations, see Supplementary Figure S7.On a side note, the photocatalyst bed temperature, detected with a thermocouple placed at the reactor inlet, was increased through the absorption of photons by maximum 1.5 K in the experiments.The mean quantum yields derived via Eq.( 9) for the methanol reforming system are with maximum values of almost 60 % astonishingly high for a simple, only slightly modified commercial catalyst system and obviously depend on all varied process conditions, Fig. 8.The nature of these dependencies is complex.The clearest influence however shows the conducted wavelength variation.Whereas the mean quantum yield recorded using the 365 nm LED light source module lies between 60 % and 30 %, the experiment with the 450 nm LED light source module did not depict any significant hydrogen production rate and thus resulted in quantum yields close to zero, Fig. 8A.This observation is to be expected since the estimated band gap of the employed titania-based catalyst system lies just above 400 nm.For details on the band gap estimation, see Appendix B. Photons from the 450 nm LED light source module, all having a wavelength larger than 400 nm, therefore, cannot excite electrons into the conduction band and consequently cannot induce any photoreaction.This finally leads to an apparently 'dead' catalyst in the experiments with the 450 nm LED light source module.The position of the band gap might also explain some of the UV-band Besides the wavelength dependency of the experimentally determined mean quantum yields, the influence of the mean local volumetric rate of photon absorption has the second strongest experimentally observed influence on the performance of the employed catalyst system, see Fig. 8A.Increasing the mean local volumetric rate of photon absorption reduces the observed mean quantum yield throughout all operating points with different LED light source modules, under all reaction temperatures and reactant feed concentrations examined in this work.This effect could be the consequence of an intensification of nonradiative relaxation processes of photon-induced charge carriers with increasing charge carrier generation density under intensified local photon absorption, red reaction arrow in yellow box Fig. 7.
The observed dependency of the determined mean quantum yields on temperature is less straight forward and less significant than the wavelength and local volumetric rate of photon absorption dependencies.The exemplary data set presented in Fig. 8B shows the mean quantum yield drawn over the catalyst bed temperature at different mean local volumetric rates of photon absorption and an LED light source module peak wavelength of 385 nm.At the depicted low and high mean local volumetric rates of photon absorption of 10 and 69 µEinstein g cat -1 s − 1 , the temperature courses indicate an optimum temperature between 70 • C and 80 • C at which the mean quantum yield peaks.In contrast to that, the temperature course of the mean quantum yield at an intermediate local volumetric rate of photon absorption of 40 µEinstein g cat -1 s − 1 depicts a continuous increase of the mean quantum yield with increasing temperature.This complex behaviour of the examined photocatalyst can be interpreted as an indication for multiple competing thermally activated processes in the underlying reaction network, coloured reaction arrows in Fig. 7.An increase of the mean quantum yield with increasing temperature can be explained with an intensification of thermally driven desorption processes of the reaction products and, of higher importance, with an intensification of thermally driven decomposition of the reaction intermediate formic acid on the palladium co-catalyst to carbon dioxide and hydrogen, green highlighted reaction arrows in Fig. 7.A decrease of the mean quantum yield with increasing temperature could be explained by a thermal intensification of non-radiative charge carrier recombination processes leading to less excited charge carriers that can drive chemical reactions.Additionally, the competitive thermal reaction of the reaction intermediate formic acid and the methanol feed to methyl formate reduces the mean quantum yield, red highlighted arrows in Fig. 7. Both, the decomposition reaction of formic acid and the side reaction of formic acid with methanol, are likely to be influenced by the process intensity or the local volumetric rate of photon absorption, because the concentration of the reaction intermediate formic acid is likely to be higher if the photoreaction of methanol to formaldehyde and further to formic acid is enhanced.In the exemplary data depicted in Fig. 8B, the appearance of a temperature optimum between 70 and 80 • C at low mean local volumetric rate of photon absorption could result from an intensification of the thermal side reaction of formic acid to methanol with increasing temperature.Increasing the mean local volumetric rate of photon absorption from 10 µEinstein g cat -1 s − 1 to 40 µEinstein g cat -1 s − 1 apparently favours the decomposition reaction of formic acid rather than the side reaction towards methyl formate leading to the observed monotonous increase of the mean quantum yield with increasing temperature at intermediate local volumetric rate of photon absorption.This is an indication for different substrate concentration sensitivities of the two reactions that formic acid undergoes with the decomposition reaction being slightly more sensitive towards the formic acid concentration.The appearance of a temperature optimum at high mean local volumetric rate of photon absorption of 69 µEinstein g cat -1 s − 1 depicted in Fig. 8B could be shaped by the temperature dependency of non-radiative charge carrier recombination processes being more relevant at higher charge carrier generation densities or local volumetric rates of photon absorption respectively.The influence of the methanol feed concentration on the mean quantum yield depicted in Fig. 8C, is a further indication of the influence of the two reactions of formic acid onto the mean quantum yield.At lower reactor temperatures and especially for lower mean local volumetric rates of photon absorption, higher methanol feed concentrations induce a slight drop of the determined mean quantum yield indicating that the side reaction towards methyl formate dominates the concentration dependency of the mean quantum yield.The picture inverts for high mean local volumetric rates of photon absorption and increased temperature, where increasing the methanol feed concentration results in a slight increase of the mean quantum yields indicating that the photoreaction towards formic acid and its subsequent decomposition determines the concentration dependency of the mean quantum yield.
Summarizing, the titania-based photo reforming system is complex in its nature due to multiple competing processes being influenced by all varied operation conditions, namely temperature, reactant concentration, wavelength, and local volumetric rate of photon absorption.Noteworthy the 'photon-related' operation conditions, wavelength, and local volumetric rate of photon absorption, show the strongest influence on the quantum yield.In the varied ranges, temperature and reactant concentration play a minor role.

Conclusion and perspective
The introduced approach for the determination of quantum yields in gas, liquid, and multi-phase photoreactions could successfully be validated via the determination of the quantum yield of the well-known potassium ferrioxalate actinometer.Further the approach's adaptability could be demonstrated with an application to an exemplary aerogel-supported titania-based methanol photo reforming catalyst system.The novel isophotonic photo reactor employed allowed to precisely vary wavelength, local volumetric rate of photon absorption, temperature, and reactants' concentration in the conducted photo reforming experiments.
Thereby, from the chemical engineering point of view, the observation of a decreasing quantum yield with increasing local volumetric rate of photon absorption is of utmost importance.The local volumetric rate of photon absorption translates to the process intensity and thereby dictates the size of the reactor and the amount of catalyst needed to achieve a desired production in a production photoreactor.With the quantum yield of a photocatalytic system decreasing with increasing local volumetric rate of photon absorption, challenging optimisation issues arise.They demand a trade-off between high achieved photocatalytic efficiency and low capital invest for smaller apparatuses comprising less catalyst.Additionally, in a slightly broader view, the finding of decreasing quantum yields with increasing local volumetric rate of photon absorption also indicates that the design of high efficiency production photoreactors not only demands for efficient photon transport from the light source or reactor aperture into the reaction volume, but also an adequate photon distribution in the reaction volume to avoid hot spots in the field of the local volumetric rate of photon absorption to a reasonable extend.
Further, from a material science point of view, the dependency of the quantum yield on the local volumetric rate of photon absorption is of outstanding importance.When reporting quantum yields or comparing different photocatalysts with each other the local volumetric rate of photon absorption should be indicated according to the findings depicted above.This claim is in line with recent reports on the quantum yield of the potassium ferrioxalate system indicating that the quantum yield drops under very intense photon absorption [19].It is also in good agreement with reports of observed photocatalytic efficiencies in photocatalytic processes being dependent on the intensity of the employed light source [23].In best case, quantum yields would be determined using photoreactors providing isophotonic reaction conditions in analogy to the requirement that thermal reaction kinetics should be determined in isothermal reactors.
Further development of the proposed approach should address the LED light source modules that currently show narrow but not monochromatic emission spectra.To achieve higher wavelength resolution and access quantum yield data directly without the detour over mean quantum yields with subsequent quantum yield estimations, as showcased with the data from the potassium ferrioxalate system, filters should be used to reduce the band width of the photons sent into the reactor.
For a maximum reliability of the necessary optical simulations, detailed knowledge of the radiation transport properties is essential.In the present work, radiation transport properties were experimentally determined prior to any simulation.Thereby, the assumption of isotropic scattering characteristics of the photocatalyst bed is a simplification.Particles typically depict complex scattering characteristics depending on the particle shape and material.In future work, this issue should be addressed in more detailed optical characterisations of the photocatalysts employed.
To facilitate the adaptation and development of the approach by researchers in the field, all STEP files needed for 3D-printing of the reactor components as well as a MATLAB® toolbox needed for the optical simulations are published alongside this publication via KITO-penData [31] .The published MATLAB® toolbox comprises the employed MCRT simulation environment with an easy-to-use function to derive all needed simulation data for quantum yield estimations according to the proposed approach.It can also be used to setup and run user-defined optical simulations.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
CAD print files for the reported 3D-printable photoreactor and a MATLAB® toolbox with tools for Monte Carlo ray tracing and reproduction of the reported simulations are provided alongside the publication.Further data will be made available on request.P. Kant et al. for their great technical support, the provision of their lab and fruitful discussion regarding the radiometric characterization of the LED light source modules.Further, the authors express their gratitude towards Christoph Klahn (KIT, Institute of Mechanical Process Engineering and Mechanics) and the colleagues at the Institute for Micro Process Engineering (IMVT) at KIT for their efforts and constructive advice in the design and printing process of all 3D-printed components employed in the photoreactor and LED light source modules.The authors also thank Professor Peter Pfeifer from IMVT, KIT for his support in the design of the methanol reforming experiments especially his input regarding the considerations regarding mass transport hindrances.Further the authors want to thank Qihao Jin from LTI, KIT for his support and provision of instruments for the determination of optical properties of diverse optical coatings and catalyst powders.The authors also express their gratitude towards Luca Heim and Marius Riedinger who developed and established in their Bachelor thesis at IMVT the synthesis routes towards the titania-based photocatalyst and the aerogel photocatalyst support employed in this work.Lastly, the authors thank WÜRTH Electronics SE for the generous provision of various sets of high-power UV and VIS SMD LEDs, C. Jentner GmbH for their support with silver coatings on printed 316 L and QSIL GmbH for their support in the determination of optical properties of Ilmasil PN quartz glass.
The second target metric reported, the density distribution of the local volumetric rate of photon absorption, is derived from absorbed ray counts in voxels that together shape the cylindrical reaction volume in the quartz glass tube.The local volumetric rate of photon absorption in one voxel is thereby calculated via relating the number of absorbed rays in the voxel to the number of simulated rays and the voxels' volume.Weighting with the spectral emission of the LED light source modules finally delivers the local volumetric rate of photon absorption.
The number of rays needed to achieve convergence in the spectral radiation transport efficiency is chosen in a way that the relative standard deviation in five repeat simulations is well below one percent in an exemplary case.This number equalled two million rays per case.The computation time per case under these conditions on a workstation equipped with an AMD Ryzen Threadripper TM 3960X CPU is between 60 s and 120 s per simulation case.

Appendix B. . Optical property determination
The reflectivity of the LED cooler body's 316 L surface and the optics module's silver coating, Fig. B.1A and B, were derived from diffuse reflectance measurements using a Perkin Elmer Lambda 1050 UV-Vis spectrometer equipped with an integrating sphere assembly.For accurate measurements, the data was recorded relative to a Zenith Polymer diffuse reflection standard from Sphere Optics GmbH and corrected accordingly.
The Napierian absorption coefficient and refractive index of the employed Ilmasil PN quartz glass reaction tube purchased from QSIL GmbH, Fig. B.1C, were derived from manufacturer indications on the transmittance of quartz glass samples with thicknesses of 1.0 mm, 1.5 mm, and 2 mm.In the data evaluation the glass samples were assumed to be plane-parallel windows under perpendicular irradiation with a collimated beam.The phase boundary reflectivity was modelled with the Fresnel Eqs.(A.3).Absorption in the window's volume was modelled with Lambert Beer's law, Eq. (A.1).Ten window-internal reflections were considered.Scattering was neglected.
The refractive index of the 2 mM potassium ferrioxalate solution employed in the actinometry experiments was assumed to equal the refractive index of pure water and taken from literature [26].The Napierian absorption coefficient of the 2 mM potassium ferrioxalate solution was derived from transmittance measurements of a sample in an optical quartz glass (QS) cuvette with an optical thickness of 10 mm using an Agilent 8453 UV-vis spectrometer, Fig. B.2A.The measurements were taken relative to a reference water sample in the same QS cuvette.The data evaluation was assuming perpendicular irradiation with a collimated beam, the optical path length being equal to the layer thickness of the cuvette and Lambert Beer's law (A.1) to model the ray attenuation.
The refractive index of the aerogel supported titania-based catalyst powder can be assumed to be close to the refractive index of air and therefore was set to unity in all simulations.The Napierian absorption and scattering coefficients were derived from diffuse reflectance and transmittance measurements of a powder sample in a Spectrosil® quartz glass cuvette (QS) with an optical thickness of 1 mm, Fig. B.2B.To prepare the catalyst sample, 10 mg of activated titania catalyst powder were mixed with 1 mL of aerogel support in a 5 mL borosilicate vessel with a snap on cap by rigorous shaking.The measurements were conducted with a Perkin Elmer Lambda 1050 UVVIS spectrometer equipped with an integrating sphere assembly.For precise measurements of the diffuse reflectance, the measurements were taken relative to a Zenith Polymer diffuse reflection standard from Sphere Optics GmbH and corrected accordingly.The powder sample was measured three times with random agitation of the cuvette between the repeat measurements to check influences of the random order of the aerogel and titania particles in the quartz glass cuvette on the derived transmittance and reflectance values.In the data evaluation the sample cell filled with catalyst powder was mapped in the MATLAB®-based Monte Carlo ray tracing simulation environment considering absorption and scattering in the catalyst bed volume and refraction and partial reflection on the phase boundaries between environment and glass and glass and catalyst bed.The incoming light was assumed to be perpendicular to the cuvette surface and ideally collimated.
From the onset of the determined absorption coefficient of the aerogel supported titania-based photocatalyst, in Fig. B.2B, the band gap can be estimated to lie at approx.403 nm which equals 3.08 eV and is in good agreement with literature data for titania [27].

Appendix C. . Light source characterization
The total spectral photon emission of employed the LED light source modules was derived in goniometer measurements using a RIGO801-300 photogoniometer from TechnoTeam GmbH.The goniometer was equipped with a calibrated UV-vis spectrometer measuring the irradiance (E λ ) in positions on the spherical surface scanned by the goniometer arm.The spherical cap scanned in the measurement covers an azimuth angle (φ) from 0 • to 360 • (full round) and a polar angle (θ) from 0 • (perpendicular view on the LED light source modules) to 110 • and had a radius (r) equal to 310 mm.The step width in both azimuth and polar angle in the measurements was 5 • .From the irradiance data recorded in the roughly 3200 positions on the scanned spherical cap, the total spectral emission (q λ ) was calculated over a sphere surface integral, Eq. (C.1).
To reduce the measurement time, the full scan of the sphere cap was only conducted for the highest of all LED drive current settings.The characteristic linking the LED drive current (I LED ) to the total emission of the LED light source modules was derived from irradiance measurements in one position (φ = 180 • , θ = 0 • ) and Eq.(C.2).  6) and ( 9) is executed with Eq. (C.3).Thereby h and ν correspond to the Planck constant and the frequency of photons at the specific wavelength, respectively.q p,λ = q λ hν (C.3) P. Kant et al.
photocatalytic efficiency

Fig. 1 .Fig. 2 .
Fig. 1.Graphical summary of the proposed approach for the determination of quantum yields with isophotonic photoreaction technology.Experimental steps are highlighted in blue, simulation-based steps are highlighted in green and evaluation steps are highlighted in grey.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

P
.Kant et al.

Fig. 3 .
Fig. 3. A: Experiment routine executed automatically with one LED light source module in place varying reactant concentration, reactor temperature and mean local volumetric rate of photon absorption.B: Schedule of the total kinetic measurement using five different LED light source modules, including saturator refill procedures ensuring stable reactant supply and bypass and reference measurements conducted to check the catalyst for possible deactivation.

Fig. 4 .
Fig. 4. Simulated axial course of the mean local volumetric rate of photon absorption in the reaction volume after optimization of the LED positions on the LED light source modules at a wavelength of 405 nm with the exemplary optical properties of the reaction volume (absorption coefficient of 200 m − 1 and scattering coefficient of 450 m − 1 ).The yellow circles on the x-axis indicate the derived optimal axial positions of the six LEDs of the LED light source module.The Arabic numbers refer to the LED numbering in Table1.(For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8 .
Fig. 8. Determined mean quantum yields in the methanol photo reforming experiments.A: Dependency of the mean quantum yield on the mean local volumetric rate of photon absorption at different LED light source module peak wavelengths between 365 nm and 450 nm, a reactor temperature of 60 • C and methanol and steam feed concentrations of 3.1 % and 3.3 % respectively.B: Dependency of the mean quantum yield on the reactor core temperature at three different mean photon absorption rates between 10 and 69 µEinstein g cat -1 s − 1 , a wavelength of 385 nm and methanol and steam feed concentrations of 3.1 % and 3.3 %.C: Dependency of the mean quantum yield on the methanol feed concentration for different mean local volumetric rates of photon absorption between 10 and 50 µEinstein g cat -1 s − 1 at three different reactor temperatures between 60 • C and 85 • C and a fixed peak wavelength of 385 nm.Error bars are derived via error propagation of the standard deviations of the measured species concentrations based on three repetition measurements.

Fig. A. 1 .
Fig. A.1.Code flow chart of the implemented Monte Carlo ray tracing method highlighting the repeatedly executed steps of ray generation, intersection search and interaction modelling on phase boundaries and in volumes.The flow chart is drawn in accordance with DIN66001.

Fig. B. 1 .
Fig. B.1.Optical properties measured and assumed for the simulations for all reactor components.A: printed 316L cooler of the LED modules, B: cavity silver coating, C: quartz glass tubing.

Fig. B. 2 .
Fig. B.2. Optical properties of the 2 mM ferrioxalate solution employed in the actinometry experiments (A) and catalyst bed employed in the methanol reforming experiments (B).

Fig. C. 1 .
Fig. C.1.Measured emission spectra for all five employed LED light source modules normalized to have an area equal to unity (A).Current total emission characteristics measured for all LED light source modules (B).

2 )
For convenient data handling in the quantum yield evaluation in this contribution, the determined LED light source module emission spectra are normalized to their area and the LED characteristic expressed as relation between the LED drive current and the total emission power, Fig. C.1A and B. To yield the total emission spectrum at a specific drive current setting the total emission power read from the characteristic (Fig. C.1B) acts as linear scaling factor for the corresponding normalized total emission spectrum (Fig. C.1A) to yield the desired total emission spectrum at the specific drive current setting.The conversion of the energy-based data in Fig. C.1 to the photon number-based data required for the evaluations according to Eqs. ( for clarification.