#### 3.1. Empirical Model

The sulfate and nitrite concentrations were almost constant during the experiments; the sulfate concentration was 8.9 ± 1.6 gS-SO

_{4}·L

^{−1} and nitrite concentration was between 0.1 and 10 mgN-NO

_{2}^{−}·L

^{−1}. The H

_{2}S

RE obtained during the experimentation carried out to obtain the empirical model is shown in

Figure 3.

As expected, a high H

_{2}S

RE was found at low

F_{G} (lower H

_{2}S

IL). Therefore, the best results were for an

F_{G} of 1 m

^{3}·h

^{−1}, where the H

_{2}S

IL was 35.1 ± 1.5 gS·m

^{−3}·h

^{−1} and the

RE between 97.3 and 99.5%. Under these conditions, the effects of the nitrate concentration and

TLV were negligible. However, at higher biogas flow rates the decrease in the nitrate concentration led to a lower H

_{2}S

RE. In anoxic biofiltration nitrate (or nitrite) is the electron acceptor, so its concentration must be a significant factor on the BTF performance, and this is even more important considering that the use of a nitrate feed controlled by ORP [

16] leads to a decrease in the nitrate concentration until depletion.

The

RE versus the

TLV values for the three

F_{G} (

EBRT of 600, 200 and 120 s) are listed in

Table 2. When the nitrate concentration was not limiting, the

TLV effect on H

_{2}S

RE was only notable for an

F_{G} equal to or greater than 3 m

^{3}·h

^{−1} (i.e. for H

_{2}S

IL of 109.1 ± 11.7 and 172.4 ± 3.4 gS·m

^{−3}·h

^{−1}). Thus, for an

F_{G} of 3 m

^{3}·h

^{−1}, the H

_{2}S was between 84.7 and 97.6% and for 5 m

^{3}·h

^{−1} the range was between 79.3 and 92.1%. The improvement observed could be explained by various effects: a higher wetted area [

21], an increase in the hold-up liquid (6.4, 8.5 and 10.6 L for 5.1, 10.2 and 15.3 m·h

^{−1}) and a higher area in contact with the flowing liquid, as proposed by Almenglo et al. [

15].

Consequently, the influence of the nitrate concentration in the recirculating liquid on the H

_{2}S

RE was dependent of two factors: the H

_{2}S

IL and the

TLV. A higher

TLV level supplies a higher nitrate availability in the biofilm.

TLV has usually been kept constant in anoxic BTFs between 10 [

22] and 15 [

23] m·h

^{−1}, but for a high H

_{2}S

IL it would be interesting to study the effect of this parameter. As in aerobic BTFs, where

TLV is a key operational variable, in aerobic BTFs the regulation of

TLV improves the oxygen mass transfer along the packed bed [

24]. Fernández et al. [

6] studied the effect of the

TLV (2−20.5 m·h

^{−1}) on H

_{2}S

RE in an anoxic BTF, for H

_{2}S

ILs from 93 to 201 gS·m

^{−3}·h

^{−1}, packed with open pore polyurethane foam (the same support material as used in this study). It was found that there was no discernable influence for

TLV values higher than 5 m·h

^{−1} for H

_{2}S

IL values below 157 gS·m

^{−3}·h

^{−1}. However, at an H

_{2}S

IL of 201 gS·m

^{−3}·h

^{−1} it was observed that

TLV values below 15 m·h

^{−1} produced a significant decrease in the H

_{2}S

RE from 92 to 85% at 4.5 m·h

^{−1}. On using polypropylene Pall rings [

25] the optimal

TLV was also 15 m·h

^{−1} at high H

_{2}S

IL (>201 gS·m

^{−3}·h

^{−1}) although

TLV did not have any effect at low H

_{2}S

IL (< 78.4 gS·m

^{−3}·h

^{−1}). Zeng et al. [

3] studied the effect of

TLV between 2.63 an 9.47 m·h

^{−1} (H

_{2}S

IL < 86.92 gS·m

^{−3}·h

^{−1}) and achieved an efficient removal of H

_{2}S for the lowest

TLV, probably due to the larger height-diameter (

H/D) ratio (10.9) and the higher

EBRT (342 s). A high

H/D ratio and a low

EBRT improve the gas-liquid mass transfer [

26] but increase the installation cost due to the higher pressure drop [

27] and the higher volume of the packed bed.

The statistical results for the ‘

C model’ show the significance and high predictability of the regression model. The R-squared was 95.77%, the residual standard deviation was 0.1784 and the mean absolute error was 0.1224. The Durbin–Watson statistic was 1.00088 (p-value = 0.0001) and this shows a possible autocorrelation in the sample with a significance level of 5.0%. Moreover, the plot of residual versus predicted values (

Figure 4a) does not show any patterns and we can assume a good correlation between the model prediction and the experimental results. The second-order polynomial model fitted with calibration data is represented by Equation (11).

As can be seen in

Figure 4b, the most influential factor on the outlet H

_{2}S concentration was the H

_{2}S

IL (p-value 2.26·10

^{−8}). Moreover, the

TLV and nitrate concentration had a negative effect on the outlet H

_{2}S concentration with p-values of 5.26·10

^{−14} and 5.34·10

^{−4}, respectively. It is interesting to note that the interaction AB (

IL and

TLV) was significant (p-value 1.18·10

^{−5}) and therefore the effect of one variable depended on the value of another. This behavior can be seen in

Figure 3, where the

TLV had an effect on the

RE at high H

_{2}S

IL but not at low ones.

The response surface for the ‘

C model’ is shown in

Figure 5. The model can be used to predict the factor limits to achieve a desired H

_{2}S outlet concentration. Depending on the combustion engine company, the inlet H

_{2}S limit is in the range 100–500 mg·Nm

^{−3} (72–359 ppm

_{V} at 25 °C) [

28]. Therefore, for a nitrate concentration of 35.5 mgN-NO

_{3}^{−}·L

^{−1} and

TLV of 15.27 m h

^{−1} the maximum H

_{2}S

IL would be 66.72 and 119.75 gS·m

^{−3}·h

^{−1} for outlet H

_{2}S concentrations of 72 and 359 ppm

_{V}, respectively.

The statistical results for the ‘

EC model’ show a higher significance and predictability of the regression model when compared with the ‘

C model’. The R-squared was 99.63%, the residual standard deviation was 2.545 and the mean absolute error was 1.646. The Durbin–Watson statistic was 1.7041 (p-value = 0.0626), although the p-value was higher than 5% there were no traces of autocorrelation—as verified by checking the residual plot (

Figure 6a). The second-order polynomial model fitted with calibration data is represented by Equation (12).

As can be seen in

Figure 6b, the most influential factor on the

EC was the H

_{2}S

IL (p-value 1.81·10

^{−28}). Moreover, the

TLV and nitrate concentration had a positive effect on the

EC with p-values of 2.37·10

^{−11} and 1.38·10

^{−5}, respectively. In this case, the interactions AB (

IL and

TLV) and BC (

IL and nitrate concentration) and the quadratic term (BB or

IL^{2}) were more significant than the nitrate concentration. Therefore, H

_{2}S

IL and

TLV had a greater effect on the

EC than the nitrate concentration.

The response surface for the ‘

EC model’ is shown in

Figure 7. As expected, for high H

_{2}S

IL (>109.1 ± 11.7 gS·m

^{−3}·h

^{−1}) or high biogas flow rate (

F_{G} > 3 m

^{3}·h

^{−1}) an increase in

EC was observed when the nitrate concentration and

TLV were increased.

A sensitivity analysis was performed by calculating the partial derivative in both models [

29]. The maximum and minimum values for the variation of estimated variables corresponding to each factor are provided in

Table 3. For the ‘

C model’ the maximum negative effect corresponded to a

TLV of –0.1579 (gS·m

^{−3})/(m·h

^{−1}) and the maximum positive effect was for an H

_{2}S

IL of 0.0177 (gS·m

^{−3})/(gS·m

^{−3}·h

^{−1}). However for the ‘

EC model’ the maximum effects were due to

TLV values of –1.252 (gS·m

^{−3})/(m·h

^{−1}) and 4.3303 (gS·m

^{−3})/(m·h

^{−1}).

#### 3.2. Ottengraf’s Model

The concentration profiles along the bed height were analyzed using Ottengraf’s model, without nitrate concentration limitations, for H

_{2}S

IL values between 33 and 176 gS·m

^{−3}·h

^{−1} and

TLV values between 5.09 and 15.27 m·h

^{−1}. The linear adjustments are provided in

Table 4 according to the following simplifications: controlled by zero-order diffusion, zero-order kinetic and first-order kinetic. The behavior of the concentration profile for 5.09 m·h

^{−1} and 130 and 170 gS·m

^{−3}·h

^{−1} could be explained using a zero-order simplification for diffusion or kinetic. However, for all other conditions the first-order kinetic simplification can be applied. The kinetic constant was in the range between 0.0025 and 0.0092 s

^{−1}.

To our knowledge, Ottengraf’s model has not be applied to biogas desulfurization, although studies on H

_{2}S removal from air have been modeled using Ottengraf’s model [

30,

31,

32,

33].

Jin et al. [

32] found that the zero-order kinetic limitation described the outlet H

_{2}S concentration (

TLV of 0.62 m·h

^{−1} and a maximum H

_{2}S

IL around 30 gS·m

^{−3}·h

^{−1}). Oyarzún et al. [

30] applied zero-order diffusion equations for an inlet H

_{2}S concentration below Ks (Monod saturation constant) and zero-order kinetic equation and inlet concentration above Ks. The microbial kinetic of the biotrickling filter presented in this work can be described by a Haldane model [

15], with an affinity constant for sulfide (Ks) of 8.4 gS·m

^{−3} and a gas concentration in equilibrium of 1.28 gS·m

^{−3}. This concentration is considerably lower than the minimum inlet concentration employed in this work (5.55 gS·m

^{−3}). Therefore, the study was carried out at an inlet H

_{2}S concentration higher than Ks with a first-order kinetic as obtained by Oyarzún et al. [

30].