# Mathematical Modeling and Mechanism of VUV Photodegradation of H2S in the Absence of O2

Authors: Jian-hui Xu; Bin-bin Ding; Xiao-mei Lv; Shan-hong Lan; Chao-lin Li; Liu Peng
Journal: International Journal of Photoenergy
 (2018)

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## Abstract

The existence of H2S has limited the biogas energy promotion. The traditional photodegradation of H2S is usually conducted in the presence of O2, yet this is unsuitable for biogas desulfurization which should be avoided. Therefore, the ultraviolet degradation of H2S in the absence of O2 was investigated for the first time in the present study from a mathematical point of view. Light wavelength and intensity applied were 185 nm and 2.16 × 10−12 Einstein/cm2·s, respectively. Firstly, the mathematical model of H2S photodegradation was established with MATLAB software, including the gas flow distribution model and radiation model of photoreactor, kinetics model, mass balance model, and calculation model of the degradation rate. Then, the influence of the initial H2S concentration and gas retention time on the photodegradation rate were studied, for verification of the mathematical model. Results indicated that the photodegradation rate decreased with the increase in initial H2S concentration, and the maximum photodegradation rate reached 62.8% under initial concentration of 3 mg/m3. In addition, the photodegradation rate of H2S increased with the increase in retention time. The experimental results were in good accordance with the modeling results, indicating the feasibility of the mathematical model to simulate the photodegradation of H2S. Finally, the intermediate products were simulated and results showed that the main photodegradation products were found to be H2 and elemental S, and concentrations of the two main products were close and agreed well with the reaction stoichiometric coefficients. Moreover, the concentration of free radicals of H• and SH• was rather low.


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## Body

## 1. Introduction

During anaerobic digestion which is considered as one of the most important biomass-based renewable energy techniques to reclaim clean fuel of biogas [1], hydrogen sulfide (H2S) is also produced in addition to CH4 and CO2, with content of 0.3%–0.4% [2]. H2S is a foul acid gas, and it can result in the serious corrosion of pipeline, instruments, and equipment. In addition, H2S endangers human health and causes environmental pollution. When the biogas containing H2S is utilized as energy (such as burning, power generation, etc.), H2S will be converted into SO2 and cause serious air pollution [3]. Therefore, the existence of H2S has limited the biogas energy promotion, and effective approaches are required for H2S removal from biogas.H2S treatment methods include physical, chemical, biological, and combinatorial technologies. The direct degradation of H2S for production of hydrogen and sulfur has been the research focus of domestic and foreign researchers, since it can effectively control the H2S pollution produced during oil, gas, coal, and mineral processing and also achieve the recycling of hydrogen energy. The main methods of H2S degradation for hydrogen and sulfur production include thermal degradation [4, 5], electrochemical degradation [6], photocatalytic degradation [7–9], and plasma degradation [10, 11]. Herein, photocatalytic degradation of H2S is the most promising technology due to the high treatment efficiency and reaction rate.Currently, researches on photocatalytic degradation of H2S are mostly conducted under the condition of O2 [12–14]. Nevertheless, there is potential of explosion with O2 presented during photocatalytic degradation of H2S from biogas. Moreover, a by-product of ozone (O3) is produced and further treatment was required. Photocatalytic degradation of H2S with the absence of O2 can avoid the abovementioned problems, which is quite suitable for H2S removal from biogas and other gas treatments requiring anaerobic conditions. In addition, conventional UV light sources for degradation, such as mercury vapor lamps and high-pressure xenon lamps, are incapable of producing intense light near UV or deep UV lights. The self-made high-frequency discharge vacuum ultraviolet (VUV) lamp has lots of advantages compared with the conventional UV lamps and microwave discharge electrodeless lamps, such as high efficiency, high radiation intensity, high ratio of 185 nm light, and long service life [12, 15].Therefore, in the present study, the photodegradation of H2S with VUV lamp in the absence of O2 was studied. Firstly, the photodegradation model was built using the MATLAB software, and then the effect of initial H2S concentration and retention time on H2S degradation performance was investigated and compared with the modeling result. Moreover, the reaction kinetics and mechanism of H2S degradation in the absence of O2 were discussed. We hope the research result can help the effective H2S removal under anaerobic conditions.


## 2. Materials and Methods

### 2.1. Reactor and Experiments

Figure1 schematically depicts the H2S photodegradation reactor used in this study. The reactor was designed as cylindrical in shape to avoid the dead space for photodegradation and the potential uneven gas distribution within the reactor. A cylindrical VUV lamp was placed on top of the reactor, and its connection with the reactor was sealed with corrosion-resistant and high temperature-resistant silica gel. A high-frequency generator was connected with the VUV lamp via an external circuit. A porous plate-like gas distribution device was installed at the gas inlet, and thus, the incoming gas can evenly flow upwards. The gas outlet was installed at the top of the reactor. To avoid short flow, the gas inlet and outlet were placed on different sides of the reactor. For the sake of shielding the high-frequency electromagnetic radiation and preventing the corrosion of the reactor caused by H2S, the body of the reactor was made of SUS304 stainless steel.Figure 1
H2S photodegradation reactor. (1) High-frequency generator, (2) electrodeless VUV lamp, (3) gas distribution device, (4) gas outlet, and (5) gas inlet.The diameter and height of the reactor were 15 and 14 cm, respectively. The total volume of the reactor was 2.5 L, with the effective volume of 2.0 L when the cylindrical UV lamp with a height of 5.8 cm and diameter of 4 cm was placed.Figure2 is diagrammatic sketch of the cylindrical photoreactor with a cylindrical light source for radiation field modeling, where Ir,z is the light intensity at point Pr,z within the reactor, SL,λ is the UV intensity at wavelength λ, and r and z are the horizontal distance and vertical distance of a random particle from the center of the UV lamp, respectively. L is the length of the UV lamp. μλ is the absorption coefficient of the medium in the reactor at the wavelength of λ. R1 and R2 are the radii of the cylindrical UV lamp and the cylindrical reactor, respectively.Figure 2
Cylindrical photoreactor with a cylindrical light source.


### 2.2. Experimental Methods

In order to investigate the degradation efficiency of H2S with only the high-frequency electrodeless VUV lamp without addition of O3, OH, and photocatalyst, the reactor was firstly dried, and Ar was flushed into the reactor with a 10 L/min flow rate for an hour to expel the residual O2 and H2O in the pipeline. During the whole experiment, Ar gas was continuously flushed to exclude the effects of O3 and OH on H2S degradation.


### 2.3. Analytical Methods

H2S concentration was determined with methylene blue spectrophotometry under a wavelength of 660 nm [16].


## 2.1. Reactor and Experiments

Figure1 schematically depicts the H2S photodegradation reactor used in this study. The reactor was designed as cylindrical in shape to avoid the dead space for photodegradation and the potential uneven gas distribution within the reactor. A cylindrical VUV lamp was placed on top of the reactor, and its connection with the reactor was sealed with corrosion-resistant and high temperature-resistant silica gel. A high-frequency generator was connected with the VUV lamp via an external circuit. A porous plate-like gas distribution device was installed at the gas inlet, and thus, the incoming gas can evenly flow upwards. The gas outlet was installed at the top of the reactor. To avoid short flow, the gas inlet and outlet were placed on different sides of the reactor. For the sake of shielding the high-frequency electromagnetic radiation and preventing the corrosion of the reactor caused by H2S, the body of the reactor was made of SUS304 stainless steel.Figure 1
H2S photodegradation reactor. (1) High-frequency generator, (2) electrodeless VUV lamp, (3) gas distribution device, (4) gas outlet, and (5) gas inlet.The diameter and height of the reactor were 15 and 14 cm, respectively. The total volume of the reactor was 2.5 L, with the effective volume of 2.0 L when the cylindrical UV lamp with a height of 5.8 cm and diameter of 4 cm was placed.Figure2 is diagrammatic sketch of the cylindrical photoreactor with a cylindrical light source for radiation field modeling, where Ir,z is the light intensity at point Pr,z within the reactor, SL,λ is the UV intensity at wavelength λ, and r and z are the horizontal distance and vertical distance of a random particle from the center of the UV lamp, respectively. L is the length of the UV lamp. μλ is the absorption coefficient of the medium in the reactor at the wavelength of λ. R1 and R2 are the radii of the cylindrical UV lamp and the cylindrical reactor, respectively.Figure 2
Cylindrical photoreactor with a cylindrical light source.


## 2.2. Experimental Methods

In order to investigate the degradation efficiency of H2S with only the high-frequency electrodeless VUV lamp without addition of O3, OH, and photocatalyst, the reactor was firstly dried, and Ar was flushed into the reactor with a 10 L/min flow rate for an hour to expel the residual O2 and H2O in the pipeline. During the whole experiment, Ar gas was continuously flushed to exclude the effects of O3 and OH on H2S degradation.


## 2.3. Analytical Methods

H2S concentration was determined with methylene blue spectrophotometry under a wavelength of 660 nm [16].


## 3. Results and Discussion

### 3.1. Mathematical Modeling of H2S Photodegradation

#### 3.1.1. Gas Flow Distribution Model

The rate of gas flow was determined by both the annular space of the reactor and the retention time. As the diameter of the reactor and the UV lamp were 15 cm and 4 cm, respectively, the rate of the gas flow was calculated as 20–40 L/min when the retention time was set as 3–6 s. Considering that the concentration of H2S in the gas influent was low, the Re constant could be calculated based on the physical properties of the incoming gas. The Re calculated at room temperature with Ar as carrier gas was 203–406.The velocity (v) of a particle at the radius of R away from the center axis of the annular reactor can be determined according to the following equation [13]:
(1)v=2UR22−R12/lnR2/R1lnR/R1−2UR2−R12R12+R22−R22−R12/lnR2/R1,where U is the known average velocity and R1, R2, and R are the radius of the UV lamp, the radius of the reactor, and the horizontal distance between the particle and the center axis of the annular reactor, respectively.Based on (1), the simulated maximum velocity was no more than 5.8 m/s (as shown in Figure S1), and then the calculated Re was also no more than 609. The above Re values were all far below the critical value of laminar flow (2300) and indicated a typical laminar flow pattern in the reactor (the Re calculation was provided in the Supplementary Materials).


#### 3.1.2. Radiation Field Model

The light intensity (Ir,z) at point Pr,z within the reactor can be expressed with (2) [17].
(2)Ir,zλ=SL,λπ2∫0Lrr2+z−x23/2×exp−μλr2+z−x21/21−R1rdx.The simulated three-dimensional model of the light intensity by the cylindrical VUV lamp at the center of the photoreactor with diameter and height of 15 cm and 14 cm, respectively, is shown in Figure3. The white area was occupied by the VUV lamp with diameter and height of 4 cm and 5.8 cm, respectively, as scaled in the figure coordinate axis. According to Figure 3, light intensity decreased rapidly with the increase in the distance from the center of the light source.Figure 3
The light intensity distribution in the photoreactor.In order to verify the result of the light intensity distribution model in the reactor, a UV radiation meter was applied to measure the UV254 intensity at 5–35 cm away from the lamp wick of the cylindrical lamp. In the meantime, the light intensity at the same position was calculated based on the UV light radiation field model and then was compared with the measured value to verify the results of the model. As shown in Figure 4, the results indicated that the calculated value and the measured values were close, indicating that (2) was feasible to predict the radiation intensity distribution.Figure 4
The experimental data and simulation data of the UV intensity with the change of space.


#### 3.1.3. Photochemical Reaction Kinetics Model

The photochemical reaction kinetics equation involving componenti can generally be presented as the following equation.
(3)dCidt=Φi⋅I⋅1−10εbCi,where Ci is the molar concentration of component i (mol/L), Φi is the quantum efficiency, ε is the molar absorption coefficient (L/mol·cm), b is the optical length (cm), and I is the light radiation intensity (Einstein/cm2·s). It should be noted that the VUV light absorption by other components of the initial biogas, including the typical CH4 and CO2, could be neglected [18].Equation (3) has an exponential term, making the calculation complicated. When εbCi<0.02, (3) can be reasonably approximated to the following expression:
(4)dCidt=2.303⋅ΦiIεbCi.Asili and De Visscher [13] proposed another equation for calculating the photochemical reaction rate ri (molecule/cm3·s):
(5)−ri=Φi⋅Ci⋅σi⋅Ep,where Ci is the molar concentration of component i (molecule/cm3), Φi is the quantum efficiency, σi is the absorption cross section of component i, and Ep is the photon flux (photon/cm2·s).The comparison of (4) and (5) revealed that both equations reflect the same reaction behavior, though (4) describes the concentration change macroscopically while (5) describes it microscopically. Both equations describe the chemical changes that occur when the reacting components absorb a certain amount of energy.


#### 3.1.4. Mass Balance Model

Firstly, the Peclet numberPe in the photoreactor was calculated, revealing the numerical zone of 132–263. Pe represented the relative proportion of convection to diffusion. When Pe is larger than 40, the dominant mass transfer type was convection at the airflow direction. As the main gas flow pattern was a laminar flow, the main mass transfer pattern at the vertical direction was diffusion. According to the empirical equation, for H2S-air mixed gas, the diffusion coefficient at the vertical direction can be expressed as [19]
(6)DH2S‐air=0.7914T1.75/P,where DH2S‐air is the diffusion coefficient (cm2/s), T is the gas temperature (K), and P is the gas pressure.For a random componenti of the photochemical reaction system in the annular photoreactor, its partial differential equation of mass conservation can be represented as
(7)v⋅∂Ci∂L=Di⋅∂2Ci∂R2+DiR⋅∂Ci∂R+ri,where v is the velocity of the gas molecule at current position (cm/s), Ci is the molar concentration of the component i (mole/cm3), L is the vertical distance between a random position and the gas inlet (cm), R is the horizontal distance between a random position and the linear light source, Di is the diffusion coefficient of the component i at the radical direction (cm2/s), and ri is the photochemical reaction rate (mole/cm3·s).The boundary condition of the partial differential equation (7) was assumed as the no-flow boundary, and the inlet gas concentration was considered as constant.


#### 3.1.5. Calculation Model of Degradation Rate

For a random componenti in the photochemical reaction system, the degradation degree ηi can be expressed as
(8)ηi=1−C¯iCi,initial,where C¯i is the random component i in the photochemical reaction system at the gas outlet and Ci,initial is the initial concentration of i at the inlet. For a tubular reactor, the average concentration of the random component i at the gas outlet can be expressed as
(9)C¯i=∫R1R2v⋅2πR⋅dR∫R1R2v⋅2πR⋅dR,where v is the velocity of the gas molecule at the present position (cm/s), R1 and R2 are the radius of the VUV lamp and the reactor (cm), and R is the horizontal distance between a random position and the linear light source.The velocity of the componenti in the reactor can be calculated using (1) and (2) and can be used to acquire the light intensity of a random position in the reactor. Then, the acquired data can be used in (4) and (7).Equation (4) is an ordinary differential equation (ODE). Although (7) is a partial differential equation (PDE), it can be converted into a series of ODEs using MATLAB ODE15s for equation solution. Then, (8) and (9) are used for solving the degradation rate of the target component and correlate with experimental data.


### 3.2. Major Influence Factors on H2S Photodegradation

#### 3.2.1. Initial H2S Concentration

The influence of the initial H2S concentration on the degradation rate under retention time of 6 s is profiled in Figure 5. Results showed that the degradation rate continuously decreased with the increase in the initial H2S concentration, which was similar to the situation with O2 present [12]. As shown, when the initial H2S concentration was 3 mg/m3, the maximum degradation rate was about 62.8%. When the initial H2S concentration reached 25 mg/m3, the degradation rate decreased to 36.2%. UV light can only pass a relatively short distance in vacuum, and also there were no oxidative free radicals to assist degradation [20]; therefore, the degradation rate was comparatively lower than that with the presence of O2 [12, 21]. Moreover, no SO42− in the photodegradation products of H2S was detected although it was confirmed as the main product in a previous study [22], and H2 and elemental sulfur were the possible products. Yet, solid S was also not observable throughout the whole experiment. The first reason was the rather low particle settling velocity calculated by Stokes formula, and another reason was the very low H2S concentration and high airflow velocity. Therefore, it was difficult for the sulfur particle to settle on the internal reactor walls and no deposition of elemental sulfur was observed.Figure 5
Influence of initial concentration of H2S on the degradation rate.


#### 3.2.2. Gas Retention Time

Under the initial H2S concentration of 12 mg/m3, the simulation of the H2S degradation profile based on modeling and the experimental data are compared in Figure 6. The experimental results indicated that the degradation rate was significantly affected when gas retention time in the reactor was less than 12 s. Specifically, when gas retention time was 3 s and 6 s, the degradation rate was 22.0% and 48.1%, respectively, indicating that the degradation efficiency proportionally increased with gas retention time. With higher gas retention time, the increasing rate of the degradation efficiency became slower. For instance, when gas retention time was 24 s, the degradation efficiency was 74% and only increased by 30% compared with that at gas retention time of 6 s. However, it still could not reach 100% degradation with the further increase in gas retention time.Figure 6
The relationship between H2S degradation rate and retention time of direct photolysis by UV with the absence of oxygen.The dead zone was placed in a position far away from the lamp, and so H2S was hard to be degraded thoroughly. Longer retention time was beneficial for increasing the degradation rate; however, the energy consumption was also increased. Therefore, the appropriate retention time should be selected considering the initial H2S concentration, equipment cost, and operating cost.The root-mean-square error (RMSE) for this model simulation was as small as 0.0622. At-test was conducted for comparing the simulation data and experimental data, and the test result accepted the null hypothesis with a possibility of 96.72%, indicating that the simulation data and the experimental data were consistent. This proves that the model to simulate the degradation efficiency in the photodegradation reactor was feasible.


### 3.3. Mechanism of H2S Photodegradation Using Only VUV Light

Wilson et al. [23] studied the direct degradation of H2S by photon of a near ultraviolet band. Results showed that a photon with a wavelength less than 270 nm could directly photodegrade H2S into H· and SH·, and SH· could be degraded into H· and S· with a photon wavelength less than 230 nm, as shown in (10) and (11). The possible reactions during the direct photodegradation of H2S are listed as follows:
(10)H2S+hvλ≤270nm→H·+SH·(11)HS·+hvλ≤230nm→H·+S(12)H·+H2S→H2+HS·(13)H·+H·→H2(14)H·+HS·→H2+S(15)HS·+HS·→S+H2SBy referring to the chemical dynamics database of the National Institute of Standards and Technology (NIST) and the related literatures, the rate constants can be acquired for the above reactions and are summarized in Table1. A photon participates in reactions (10) and (11), and the molar absorption coefficients for wavelengths of 185 nm and 254 nm can be obtained by converting the absorptivity data summarized in Table 1.Table 1
The chemical reaction of direct photolysis of H2S and the corresponding photolysis rate constants.
Reaction
Quantum efficiency
Light absorptivity at 185 nm (L/mol·cm)
Light absorptivity at 254 nm (L/mol·cm)
Rate constant (L/mol·s)
Reaction order
References
H2S + hv → H· + SH
1
1004.66
1.96
/
/
[24]
HS· +hv → H· + S
1
20.5
0
/
/
[24]
H· + H· → H2
/
/
/
2.498E + 10
2
[25]
H2S + H· → H2 + SH
/
/
/
4.786E + 08
2
[26]
H· + SH· → H2 + S
/
/
/
8.0E + 09
2
[27]
SH· + SH· → S + H2S
/
/
/
8.368E + 9
2
[28]Based on (10), (11), (12), (13), (14), and (15), the degradation rate equation of the various intermediates during hydrogen sulfide photodegradation in the absence of O2 could be established. As photon is presented in (10) and (11), let kobs=2.030Φiεb ((4)), denoting kobs1, kobs2, k3, k4, k5, and k6 as the rate constants for (10), (11), (12), (13), (14), and (15). The reaction rate equations for each component can be expressed as follows:
(16)dH2Sdt=−kobs1⋅H2S−k3⋅H2SH·+k6⋅SH·2,(17)dH·dt=kobs1⋅H2S+kobs2⋅SH·−k3⋅H2SH·−k4⋅H·2−k5⋅SH·H·,(18)dSH·dt=kobs1⋅H2S−kobs2⋅SH·−k5⋅SH·H·−k6⋅SH·2,(19)dH2dt=k3⋅H2SH·+k4⋅H·2+k5⋅H·SH·,(20)dSdt=kobs2⋅SH·+k5⋅SH·H·+k6⋅SH·2.Then, the reaction rate equations forH2S, H·, SH·, H2, and S (from (16), (17), (18), (19), and (20)) were integrated into the UV-photodegrading reactor model, and the H2S photodegradation effect without O2 was simulated under the following conditions: the initial H2S concentration was set as 12 mg/m3 with Ar as the carrier gas. The concentration evolution of H2S, H·, SH·, H2, and S were calculated 2 cm away from the UV light with MATLAB software and demonstrated by the logarithmic scale in Figure 7.Figure 7
The simulated concentration changes of components of H2S photodegradation by UV without the presence of oxygen.It was revealed that the final products ofH2S photodegradation were mainly H2 and S, and their concentrations were close to each other, which was in accordance with the stoichiometric coefficients of the photodegradation reaction. Moreover, the simulated concentrations of H· and SH· radicals were rather low, which were about two orders of magnitude lower than H2 and S.Finally, we tried to build the analytical equation connecting the rate ofH2S consumption and its concentration and light intensity using steady-state approximation for radicals. Firstly, let (16) be equal to zero, as shown in (21).
(21)dH2Sdt=0.Then, (21) and (22) could be obtained combining (16) and (21), as follows.
(22)H2Ss‐tkobs1+k3H·=k6SH·2,(23)H2Ss‐t=k6SH·2kobs1+k3H·.Then,kobs1 was substituted with (4), and the analytical equation connecting the rate of H2S consumption and its concentration and light intensity using steady-state approximation for the radical could be obtained as (24).
(24)H2Ss‐t=k6SH·22.303⋅ΦiIεbCi+k3H·.


## 3.1. Mathematical Modeling of H2S Photodegradation

### 3.1.1. Gas Flow Distribution Model

The rate of gas flow was determined by both the annular space of the reactor and the retention time. As the diameter of the reactor and the UV lamp were 15 cm and 4 cm, respectively, the rate of the gas flow was calculated as 20–40 L/min when the retention time was set as 3–6 s. Considering that the concentration of H2S in the gas influent was low, the Re constant could be calculated based on the physical properties of the incoming gas. The Re calculated at room temperature with Ar as carrier gas was 203–406.The velocity (v) of a particle at the radius of R away from the center axis of the annular reactor can be determined according to the following equation [13]:
(1)v=2UR22−R12/lnR2/R1lnR/R1−2UR2−R12R12+R22−R22−R12/lnR2/R1,where U is the known average velocity and R1, R2, and R are the radius of the UV lamp, the radius of the reactor, and the horizontal distance between the particle and the center axis of the annular reactor, respectively.Based on (1), the simulated maximum velocity was no more than 5.8 m/s (as shown in Figure S1), and then the calculated Re was also no more than 609. The above Re values were all far below the critical value of laminar flow (2300) and indicated a typical laminar flow pattern in the reactor (the Re calculation was provided in the Supplementary Materials).


### 3.1.2. Radiation Field Model

The light intensity (Ir,z) at point Pr,z within the reactor can be expressed with (2) [17].
(2)Ir,zλ=SL,λπ2∫0Lrr2+z−x23/2×exp−μλr2+z−x21/21−R1rdx.The simulated three-dimensional model of the light intensity by the cylindrical VUV lamp at the center of the photoreactor with diameter and height of 15 cm and 14 cm, respectively, is shown in Figure3. The white area was occupied by the VUV lamp with diameter and height of 4 cm and 5.8 cm, respectively, as scaled in the figure coordinate axis. According to Figure 3, light intensity decreased rapidly with the increase in the distance from the center of the light source.Figure 3
The light intensity distribution in the photoreactor.In order to verify the result of the light intensity distribution model in the reactor, a UV radiation meter was applied to measure the UV254 intensity at 5–35 cm away from the lamp wick of the cylindrical lamp. In the meantime, the light intensity at the same position was calculated based on the UV light radiation field model and then was compared with the measured value to verify the results of the model. As shown in Figure 4, the results indicated that the calculated value and the measured values were close, indicating that (2) was feasible to predict the radiation intensity distribution.Figure 4
The experimental data and simulation data of the UV intensity with the change of space.


### 3.1.3. Photochemical Reaction Kinetics Model

The photochemical reaction kinetics equation involving componenti can generally be presented as the following equation.
(3)dCidt=Φi⋅I⋅1−10εbCi,where Ci is the molar concentration of component i (mol/L), Φi is the quantum efficiency, ε is the molar absorption coefficient (L/mol·cm), b is the optical length (cm), and I is the light radiation intensity (Einstein/cm2·s). It should be noted that the VUV light absorption by other components of the initial biogas, including the typical CH4 and CO2, could be neglected [18].Equation (3) has an exponential term, making the calculation complicated. When εbCi<0.02, (3) can be reasonably approximated to the following expression:
(4)dCidt=2.303⋅ΦiIεbCi.Asili and De Visscher [13] proposed another equation for calculating the photochemical reaction rate ri (molecule/cm3·s):
(5)−ri=Φi⋅Ci⋅σi⋅Ep,where Ci is the molar concentration of component i (molecule/cm3), Φi is the quantum efficiency, σi is the absorption cross section of component i, and Ep is the photon flux (photon/cm2·s).The comparison of (4) and (5) revealed that both equations reflect the same reaction behavior, though (4) describes the concentration change macroscopically while (5) describes it microscopically. Both equations describe the chemical changes that occur when the reacting components absorb a certain amount of energy.


### 3.1.4. Mass Balance Model

Firstly, the Peclet numberPe in the photoreactor was calculated, revealing the numerical zone of 132–263. Pe represented the relative proportion of convection to diffusion. When Pe is larger than 40, the dominant mass transfer type was convection at the airflow direction. As the main gas flow pattern was a laminar flow, the main mass transfer pattern at the vertical direction was diffusion. According to the empirical equation, for H2S-air mixed gas, the diffusion coefficient at the vertical direction can be expressed as [19]
(6)DH2S‐air=0.7914T1.75/P,where DH2S‐air is the diffusion coefficient (cm2/s), T is the gas temperature (K), and P is the gas pressure.For a random componenti of the photochemical reaction system in the annular photoreactor, its partial differential equation of mass conservation can be represented as
(7)v⋅∂Ci∂L=Di⋅∂2Ci∂R2+DiR⋅∂Ci∂R+ri,where v is the velocity of the gas molecule at current position (cm/s), Ci is the molar concentration of the component i (mole/cm3), L is the vertical distance between a random position and the gas inlet (cm), R is the horizontal distance between a random position and the linear light source, Di is the diffusion coefficient of the component i at the radical direction (cm2/s), and ri is the photochemical reaction rate (mole/cm3·s).The boundary condition of the partial differential equation (7) was assumed as the no-flow boundary, and the inlet gas concentration was considered as constant.


### 3.1.5. Calculation Model of Degradation Rate

For a random componenti in the photochemical reaction system, the degradation degree ηi can be expressed as
(8)ηi=1−C¯iCi,initial,where C¯i is the random component i in the photochemical reaction system at the gas outlet and Ci,initial is the initial concentration of i at the inlet. For a tubular reactor, the average concentration of the random component i at the gas outlet can be expressed as
(9)C¯i=∫R1R2v⋅2πR⋅dR∫R1R2v⋅2πR⋅dR,where v is the velocity of the gas molecule at the present position (cm/s), R1 and R2 are the radius of the VUV lamp and the reactor (cm), and R is the horizontal distance between a random position and the linear light source.The velocity of the componenti in the reactor can be calculated using (1) and (2) and can be used to acquire the light intensity of a random position in the reactor. Then, the acquired data can be used in (4) and (7).Equation (4) is an ordinary differential equation (ODE). Although (7) is a partial differential equation (PDE), it can be converted into a series of ODEs using MATLAB ODE15s for equation solution. Then, (8) and (9) are used for solving the degradation rate of the target component and correlate with experimental data.


## 3.1.1. Gas Flow Distribution Model

The rate of gas flow was determined by both the annular space of the reactor and the retention time. As the diameter of the reactor and the UV lamp were 15 cm and 4 cm, respectively, the rate of the gas flow was calculated as 20–40 L/min when the retention time was set as 3–6 s. Considering that the concentration of H2S in the gas influent was low, the Re constant could be calculated based on the physical properties of the incoming gas. The Re calculated at room temperature with Ar as carrier gas was 203–406.The velocity (v) of a particle at the radius of R away from the center axis of the annular reactor can be determined according to the following equation [13]:
(1)v=2UR22−R12/lnR2/R1lnR/R1−2UR2−R12R12+R22−R22−R12/lnR2/R1,where U is the known average velocity and R1, R2, and R are the radius of the UV lamp, the radius of the reactor, and the horizontal distance between the particle and the center axis of the annular reactor, respectively.Based on (1), the simulated maximum velocity was no more than 5.8 m/s (as shown in Figure S1), and then the calculated Re was also no more than 609. The above Re values were all far below the critical value of laminar flow (2300) and indicated a typical laminar flow pattern in the reactor (the Re calculation was provided in the Supplementary Materials).


## 3.1.2. Radiation Field Model

The light intensity (Ir,z) at point Pr,z within the reactor can be expressed with (2) [17].
(2)Ir,zλ=SL,λπ2∫0Lrr2+z−x23/2×exp−μλr2+z−x21/21−R1rdx.The simulated three-dimensional model of the light intensity by the cylindrical VUV lamp at the center of the photoreactor with diameter and height of 15 cm and 14 cm, respectively, is shown in Figure3. The white area was occupied by the VUV lamp with diameter and height of 4 cm and 5.8 cm, respectively, as scaled in the figure coordinate axis. According to Figure 3, light intensity decreased rapidly with the increase in the distance from the center of the light source.Figure 3
The light intensity distribution in the photoreactor.In order to verify the result of the light intensity distribution model in the reactor, a UV radiation meter was applied to measure the UV254 intensity at 5–35 cm away from the lamp wick of the cylindrical lamp. In the meantime, the light intensity at the same position was calculated based on the UV light radiation field model and then was compared with the measured value to verify the results of the model. As shown in Figure 4, the results indicated that the calculated value and the measured values were close, indicating that (2) was feasible to predict the radiation intensity distribution.Figure 4
The experimental data and simulation data of the UV intensity with the change of space.


## 3.1.3. Photochemical Reaction Kinetics Model

The photochemical reaction kinetics equation involving componenti can generally be presented as the following equation.
(3)dCidt=Φi⋅I⋅1−10εbCi,where Ci is the molar concentration of component i (mol/L), Φi is the quantum efficiency, ε is the molar absorption coefficient (L/mol·cm), b is the optical length (cm), and I is the light radiation intensity (Einstein/cm2·s). It should be noted that the VUV light absorption by other components of the initial biogas, including the typical CH4 and CO2, could be neglected [18].Equation (3) has an exponential term, making the calculation complicated. When εbCi<0.02, (3) can be reasonably approximated to the following expression:
(4)dCidt=2.303⋅ΦiIεbCi.Asili and De Visscher [13] proposed another equation for calculating the photochemical reaction rate ri (molecule/cm3·s):
(5)−ri=Φi⋅Ci⋅σi⋅Ep,where Ci is the molar concentration of component i (molecule/cm3), Φi is the quantum efficiency, σi is the absorption cross section of component i, and Ep is the photon flux (photon/cm2·s).The comparison of (4) and (5) revealed that both equations reflect the same reaction behavior, though (4) describes the concentration change macroscopically while (5) describes it microscopically. Both equations describe the chemical changes that occur when the reacting components absorb a certain amount of energy.


## 3.1.4. Mass Balance Model

Firstly, the Peclet numberPe in the photoreactor was calculated, revealing the numerical zone of 132–263. Pe represented the relative proportion of convection to diffusion. When Pe is larger than 40, the dominant mass transfer type was convection at the airflow direction. As the main gas flow pattern was a laminar flow, the main mass transfer pattern at the vertical direction was diffusion. According to the empirical equation, for H2S-air mixed gas, the diffusion coefficient at the vertical direction can be expressed as [19]
(6)DH2S‐air=0.7914T1.75/P,where DH2S‐air is the diffusion coefficient (cm2/s), T is the gas temperature (K), and P is the gas pressure.For a random componenti of the photochemical reaction system in the annular photoreactor, its partial differential equation of mass conservation can be represented as
(7)v⋅∂Ci∂L=Di⋅∂2Ci∂R2+DiR⋅∂Ci∂R+ri,where v is the velocity of the gas molecule at current position (cm/s), Ci is the molar concentration of the component i (mole/cm3), L is the vertical distance between a random position and the gas inlet (cm), R is the horizontal distance between a random position and the linear light source, Di is the diffusion coefficient of the component i at the radical direction (cm2/s), and ri is the photochemical reaction rate (mole/cm3·s).The boundary condition of the partial differential equation (7) was assumed as the no-flow boundary, and the inlet gas concentration was considered as constant.


## 3.1.5. Calculation Model of Degradation Rate

For a random componenti in the photochemical reaction system, the degradation degree ηi can be expressed as
(8)ηi=1−C¯iCi,initial,where C¯i is the random component i in the photochemical reaction system at the gas outlet and Ci,initial is the initial concentration of i at the inlet. For a tubular reactor, the average concentration of the random component i at the gas outlet can be expressed as
(9)C¯i=∫R1R2v⋅2πR⋅dR∫R1R2v⋅2πR⋅dR,where v is the velocity of the gas molecule at the present position (cm/s), R1 and R2 are the radius of the VUV lamp and the reactor (cm), and R is the horizontal distance between a random position and the linear light source.The velocity of the componenti in the reactor can be calculated using (1) and (2) and can be used to acquire the light intensity of a random position in the reactor. Then, the acquired data can be used in (4) and (7).Equation (4) is an ordinary differential equation (ODE). Although (7) is a partial differential equation (PDE), it can be converted into a series of ODEs using MATLAB ODE15s for equation solution. Then, (8) and (9) are used for solving the degradation rate of the target component and correlate with experimental data.


## 3.2. Major Influence Factors on H2S Photodegradation

### 3.2.1. Initial H2S Concentration

The influence of the initial H2S concentration on the degradation rate under retention time of 6 s is profiled in Figure 5. Results showed that the degradation rate continuously decreased with the increase in the initial H2S concentration, which was similar to the situation with O2 present [12]. As shown, when the initial H2S concentration was 3 mg/m3, the maximum degradation rate was about 62.8%. When the initial H2S concentration reached 25 mg/m3, the degradation rate decreased to 36.2%. UV light can only pass a relatively short distance in vacuum, and also there were no oxidative free radicals to assist degradation [20]; therefore, the degradation rate was comparatively lower than that with the presence of O2 [12, 21]. Moreover, no SO42− in the photodegradation products of H2S was detected although it was confirmed as the main product in a previous study [22], and H2 and elemental sulfur were the possible products. Yet, solid S was also not observable throughout the whole experiment. The first reason was the rather low particle settling velocity calculated by Stokes formula, and another reason was the very low H2S concentration and high airflow velocity. Therefore, it was difficult for the sulfur particle to settle on the internal reactor walls and no deposition of elemental sulfur was observed.Figure 5
Influence of initial concentration of H2S on the degradation rate.


### 3.2.2. Gas Retention Time

Under the initial H2S concentration of 12 mg/m3, the simulation of the H2S degradation profile based on modeling and the experimental data are compared in Figure 6. The experimental results indicated that the degradation rate was significantly affected when gas retention time in the reactor was less than 12 s. Specifically, when gas retention time was 3 s and 6 s, the degradation rate was 22.0% and 48.1%, respectively, indicating that the degradation efficiency proportionally increased with gas retention time. With higher gas retention time, the increasing rate of the degradation efficiency became slower. For instance, when gas retention time was 24 s, the degradation efficiency was 74% and only increased by 30% compared with that at gas retention time of 6 s. However, it still could not reach 100% degradation with the further increase in gas retention time.Figure 6
The relationship between H2S degradation rate and retention time of direct photolysis by UV with the absence of oxygen.The dead zone was placed in a position far away from the lamp, and so H2S was hard to be degraded thoroughly. Longer retention time was beneficial for increasing the degradation rate; however, the energy consumption was also increased. Therefore, the appropriate retention time should be selected considering the initial H2S concentration, equipment cost, and operating cost.The root-mean-square error (RMSE) for this model simulation was as small as 0.0622. At-test was conducted for comparing the simulation data and experimental data, and the test result accepted the null hypothesis with a possibility of 96.72%, indicating that the simulation data and the experimental data were consistent. This proves that the model to simulate the degradation efficiency in the photodegradation reactor was feasible.


## 3.2.1. Initial H2S Concentration

The influence of the initial H2S concentration on the degradation rate under retention time of 6 s is profiled in Figure 5. Results showed that the degradation rate continuously decreased with the increase in the initial H2S concentration, which was similar to the situation with O2 present [12]. As shown, when the initial H2S concentration was 3 mg/m3, the maximum degradation rate was about 62.8%. When the initial H2S concentration reached 25 mg/m3, the degradation rate decreased to 36.2%. UV light can only pass a relatively short distance in vacuum, and also there were no oxidative free radicals to assist degradation [20]; therefore, the degradation rate was comparatively lower than that with the presence of O2 [12, 21]. Moreover, no SO42− in the photodegradation products of H2S was detected although it was confirmed as the main product in a previous study [22], and H2 and elemental sulfur were the possible products. Yet, solid S was also not observable throughout the whole experiment. The first reason was the rather low particle settling velocity calculated by Stokes formula, and another reason was the very low H2S concentration and high airflow velocity. Therefore, it was difficult for the sulfur particle to settle on the internal reactor walls and no deposition of elemental sulfur was observed.Figure 5
Influence of initial concentration of H2S on the degradation rate.


## 3.2.2. Gas Retention Time

Under the initial H2S concentration of 12 mg/m3, the simulation of the H2S degradation profile based on modeling and the experimental data are compared in Figure 6. The experimental results indicated that the degradation rate was significantly affected when gas retention time in the reactor was less than 12 s. Specifically, when gas retention time was 3 s and 6 s, the degradation rate was 22.0% and 48.1%, respectively, indicating that the degradation efficiency proportionally increased with gas retention time. With higher gas retention time, the increasing rate of the degradation efficiency became slower. For instance, when gas retention time was 24 s, the degradation efficiency was 74% and only increased by 30% compared with that at gas retention time of 6 s. However, it still could not reach 100% degradation with the further increase in gas retention time.Figure 6
The relationship between H2S degradation rate and retention time of direct photolysis by UV with the absence of oxygen.The dead zone was placed in a position far away from the lamp, and so H2S was hard to be degraded thoroughly. Longer retention time was beneficial for increasing the degradation rate; however, the energy consumption was also increased. Therefore, the appropriate retention time should be selected considering the initial H2S concentration, equipment cost, and operating cost.The root-mean-square error (RMSE) for this model simulation was as small as 0.0622. At-test was conducted for comparing the simulation data and experimental data, and the test result accepted the null hypothesis with a possibility of 96.72%, indicating that the simulation data and the experimental data were consistent. This proves that the model to simulate the degradation efficiency in the photodegradation reactor was feasible.


## 3.3. Mechanism of H2S Photodegradation Using Only VUV Light

Wilson et al. [23] studied the direct degradation of H2S by photon of a near ultraviolet band. Results showed that a photon with a wavelength less than 270 nm could directly photodegrade H2S into H· and SH·, and SH· could be degraded into H· and S· with a photon wavelength less than 230 nm, as shown in (10) and (11). The possible reactions during the direct photodegradation of H2S are listed as follows:
(10)H2S+hvλ≤270nm→H·+SH·(11)HS·+hvλ≤230nm→H·+S(12)H·+H2S→H2+HS·(13)H·+H·→H2(14)H·+HS·→H2+S(15)HS·+HS·→S+H2SBy referring to the chemical dynamics database of the National Institute of Standards and Technology (NIST) and the related literatures, the rate constants can be acquired for the above reactions and are summarized in Table1. A photon participates in reactions (10) and (11), and the molar absorption coefficients for wavelengths of 185 nm and 254 nm can be obtained by converting the absorptivity data summarized in Table 1.Table 1
The chemical reaction of direct photolysis of H2S and the corresponding photolysis rate constants.
Reaction
Quantum efficiency
Light absorptivity at 185 nm (L/mol·cm)
Light absorptivity at 254 nm (L/mol·cm)
Rate constant (L/mol·s)
Reaction order
References
H2S + hv → H· + SH
1
1004.66
1.96
/
/
[24]
HS· +hv → H· + S
1
20.5
0
/
/
[24]
H· + H· → H2
/
/
/
2.498E + 10
2
[25]
H2S + H· → H2 + SH
/
/
/
4.786E + 08
2
[26]
H· + SH· → H2 + S
/
/
/
8.0E + 09
2
[27]
SH· + SH· → S + H2S
/
/
/
8.368E + 9
2
[28]Based on (10), (11), (12), (13), (14), and (15), the degradation rate equation of the various intermediates during hydrogen sulfide photodegradation in the absence of O2 could be established. As photon is presented in (10) and (11), let kobs=2.030Φiεb ((4)), denoting kobs1, kobs2, k3, k4, k5, and k6 as the rate constants for (10), (11), (12), (13), (14), and (15). The reaction rate equations for each component can be expressed as follows:
(16)dH2Sdt=−kobs1⋅H2S−k3⋅H2SH·+k6⋅SH·2,(17)dH·dt=kobs1⋅H2S+kobs2⋅SH·−k3⋅H2SH·−k4⋅H·2−k5⋅SH·H·,(18)dSH·dt=kobs1⋅H2S−kobs2⋅SH·−k5⋅SH·H·−k6⋅SH·2,(19)dH2dt=k3⋅H2SH·+k4⋅H·2+k5⋅H·SH·,(20)dSdt=kobs2⋅SH·+k5⋅SH·H·+k6⋅SH·2.Then, the reaction rate equations forH2S, H·, SH·, H2, and S (from (16), (17), (18), (19), and (20)) were integrated into the UV-photodegrading reactor model, and the H2S photodegradation effect without O2 was simulated under the following conditions: the initial H2S concentration was set as 12 mg/m3 with Ar as the carrier gas. The concentration evolution of H2S, H·, SH·, H2, and S were calculated 2 cm away from the UV light with MATLAB software and demonstrated by the logarithmic scale in Figure 7.Figure 7
The simulated concentration changes of components of H2S photodegradation by UV without the presence of oxygen.It was revealed that the final products ofH2S photodegradation were mainly H2 and S, and their concentrations were close to each other, which was in accordance with the stoichiometric coefficients of the photodegradation reaction. Moreover, the simulated concentrations of H· and SH· radicals were rather low, which were about two orders of magnitude lower than H2 and S.Finally, we tried to build the analytical equation connecting the rate ofH2S consumption and its concentration and light intensity using steady-state approximation for radicals. Firstly, let (16) be equal to zero, as shown in (21).
(21)dH2Sdt=0.Then, (21) and (22) could be obtained combining (16) and (21), as follows.
(22)H2Ss‐tkobs1+k3H·=k6SH·2,(23)H2Ss‐t=k6SH·2kobs1+k3H·.Then,kobs1 was substituted with (4), and the analytical equation connecting the rate of H2S consumption and its concentration and light intensity using steady-state approximation for the radical could be obtained as (24).
(24)H2Ss‐t=k6SH·22.303⋅ΦiIεbCi+k3H·.


## 4. Conclusions

In the present study, the mathematical model of ultraviolet degradation ofH2S with the absence of O2 was established, and the influence of the initial H2S concentration and gas retention time on the photodegradation rate were studied and for verification of the model. The main findings were as follows.
(1)
The photodegradation rate decreased with the increase in initialH2S concentration, and the maximum photodegradation rate was about 62.8% under initial concentration of 3 mg/m3.
(2)
The photodegradation rate increased with the increase inH2S retention time.
(3)
Experimental results were in good accordance with the modeling results.
(4)
The main photodegradation products wereH2 and elemental S based on mathematical modeling. Concentrations of both products were close and agreed well with the reaction stoichiometric coefficients.



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