Parameter Estimation of Breakthrough Curve Models in the Adsorption Process of H2S and CO2 Using the Markov Chain Monte Carlo Method
Abstract
:1. Introduction
2. Analytical Models of Breakthrough Curves
2.1. Thomas Model
2.2. Yoon–Nelson Model
2.3. Adams–Bohart Model
2.4. Yan Model
3. Methodology
Markov Chain Monte Carlo (MCMC)
4. Results and Discussion
5. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Initial Concentration (mg/L) | H2S | CO2 | |||
---|---|---|---|---|---|
1.327 | 2.577 | 18 | 29 | 35 | |
Adsorbent mass (g) | 18.4052 | 18.3893 | 230 | ||
Maximum adsorption capacity (mg/g) | 20.61 | 7.26165 | 10.43037 | 12.14676 | |
Flow rate (L/min) | 1.8 | 5 |
Models | Equations | Parameters | Assumptions | References |
---|---|---|---|---|
Thomas | —Initial concentration of adsorbate (mg/L); —Maximum adsorption ca-pacity (mg/g); t—Flow time (min); Q—Flow rate (L/min); W—Adsorbent mass (g); —Thomas velocity constant (L/min·mg). |
| [23] | |
Yoon– Nelson | —Time required to reach 50% breakthrough (min); —Flow time (min); —Yoon–Nelson kinetic constant (1/min). |
| [24] | |
Adams– Bohart | —Initial concentration of adsorbate (mg/L); N0—Maximum volumetric adsorption capacity (mg·L−1); v—Interstitial velocity (L/min·cm2); t—Flow time (min); H—Bed height (cm); kba—Adams–Bohart constant representing adsorption rate (L/min·mg). |
| [21,26] | |
Yan | —Initial concentration of adsorbate W—Mass of adsorbent in the bed (g); —Feed flow rate of the column (L/min); — t—Flow time (min); —Yan model constant. |
| [28] |
Models | Parameters |
---|---|
Thomas | PT = [kth qs] |
Yoon–Nelson | PT = [] |
Adams–Bohart | PT = [kba N0] |
Yan | PT = [ay qs] |
Models | Parameters | (qs, N0) Estimated (Present Work) | (qs, N0) Deterministic [29] | ||
---|---|---|---|---|---|
Thomas | C0 (mg/L) | 1.327 | 2.577 | 1.327 | 2.577 |
qs (mg/g) | 18.9861 | 17.8316 | 20.61 | 20.61 | |
kth (L/min·mg) | 0.0306 | 0.0204 | 0.0278 | 0.0168 | |
R2 | 0.9937 | 0.9700 | 0.9746 | 0.9488 | |
R2Adjusted | 0.9932 | 0.9673 | 0.9728 | 0.9443 | |
BIC | 249.1613 | 1.0900 × 103 | 1.2473 × 103 | 1.9488 × 103 | |
Yoon– Nelson | kyn (min) | - | - | 0.0412 | 0.0534 |
τ (min) | - | - | 145.7883 | 69.9294 | |
R2 | - | - | 0.9936 | 0.9698 | |
R2Adjusted | - | - | 0.9732 | 0.9672 | |
BIC | - | - | 251.3955 | 1.0958 × 103 | |
Adams–Bohart | kba (105) | 3.0840 | 2.0333 | 2.4615 | 1.4473 |
N0 (10−4) | 4.8053 | 4.5456 | 5.5729 | 5.5729 | |
R2 | 0.9937 | 0.9701 | 0.9331 | 0.9293 | |
R2Adjusted | 0.9932 | 0.9675 | 0.9284 | 0.9232 | |
BIC | 249.2162 | 1.0845 × 103 | 3.6145 × 103 | 2.7708 × 103 | |
Yan | qs | 18.5658 | 16.9291 | 20.61 | 20.61 |
ay | 5.7419 | 3.3525 | 5.8802 | 3.3550 | |
R2 | 0.9982 | 0.9899 | 0.9671 | 0.9479 | |
R2Adjusted | 0.9981 | 0.9890 | 0.9648 | 0.9434 | |
BIC | 11.2732 | 316.4672 | 1.6493 × 103 | 1.9835 × 103 |
Models | Parameters | (qs, N0) Estimated (Present Work) | (qs, N0) Deterministic [30] | ||||
---|---|---|---|---|---|---|---|
Thomas | C0 (mg/L) | 18 | 29 | 35 | 18 | 29 | 35 |
qs (mg/g) | 8.3114 | 12.7682 | 13.2753 | 7.2617 | 10.43037 | 12.1468 | |
kth (L/min·mg) | 0.0632 | 0.0801 | 0.0699 | 0.0205 | 0.0099 | 0.0213 | |
R2 | 0.9811 | 0.9935 | 0.9955 | 0.6660 | 0.7059 | 0.8223 | |
R2Adjusted | 0.9799 | 0.9934 | 0.9951 | 0.6451 | 0.6863 | 0.8062 | |
BIC | 550.4370 | 274.7252 | 153.2467 | 1.6583 × 104 | 2.4228 × 104 | 1.0195 × 104 | |
Yoon–Nelson | kyn (min) | - | - | - | 1.1780 | 2.4075 | 2.4467 |
τ (min) | - | - | - | 21.2342 | 20.2541 | 17.4458 | |
R2 | - | - | - | 0.9809 | 0.9939 | 0.9955 | |
R2Adjusted | - | - | - | 0.9797 | 0.9935 | 0.9951 | |
BIC | - | - | - | 557.1840 | 275.0701 | 153.3209 | |
Adams– Bohart | kba | 6.4991 × 10−5 | 8.7282 × 10−5 | 7.0016 × 10−5 | 8.8822 × 10−6 | 5.7670 × 10−6 | 4.9604 × 10−6 |
N0 (mg/L) | 18065 | 27771 | 28866 | 5954.6 | 8552.9034 | 9960.3 | |
R2 | 0.9809 | 0.9940 | 0.9955 | 0.2020 | 0.3307 | 0.3665 | |
R2Adjusted | 0.9797 | 0.9935 | 0.9951 | 0.1522 | 0.2861 | 0.3089 | |
BIC | 557.3886 | 270.7467 | 153.3194 | 1.3121 × 105 | 1.1801 × 105 | 8.2006 × 104 | |
Yan | qs | 9.5851 | 11.3468 | 13.0378 | 7.2617 | 10.4304 | 12.1468 |
ay | 5.6501 | 5.0110 | 5.0643 | 7.4527 | 5.4479 | 5.4352 | |
R2 | 0.9750 | 0.9891 | 0.9937 | 0.9749 | 0.9891 | 0.9937 | |
R2Adjusted | 0.9734 | 0.9884 | 0.9932 | 0.9733 | 0.9884 | 0.9932 | |
BIC | 763.9940 | 559.1430 | 236.2257 | 766.4743 | 559.1664 | 236.2271 |
Models | References | Qmax (mg/g) | Parameters | Operating Conditions | |
---|---|---|---|---|---|
Thomas | H2S | [39] | 8.5 | - | Adsorbent: residual oil fly ash W = 1 g; C0 = 100 ppm; Q = 0.4 L/min |
[40] | 208.99 | Kth—0.001028 (L/min·mg) | Adsorbent: activated carbon xerogel from coconut shell W = 5 g; C0 = 25 ppm mol; Q = 2.5 L/min | ||
CO2 | [41] | 22.9 | Kth—0.2329 (mL/mg·s) | Adsorbent: activated carbon W = 1 g; C0 = 10%; Q = 30 mL/min | |
[42] | 16.257 (mL/min·mg) | Kth—9.12 × 10−6 (mL/min·mg) | Adsorbent: eggshell waste W = 25 g; C0 = 30.8%; Q = 0.03 m3/h | ||
Yoon–Nelson | H2S | [40] | 208.99 | Kyn—0.0257 (min−1) | Adsorbent: activated carbon xerogel from coconut shell W = 5 g; C0 = 25 ppm mol; Q = 2.5 L/min |
[39] | 8.5 | Kyn—0.1882 (min−1) τ—69.21 (min) | Adsorbent: residual oil fly ash W (g) = 1; C0 = 100 ppm; Q = 0.4 L/min | ||
CO2 | [41] | 22.9 | Kyn—0.0434 (s−1) τ—73.8 (s) | Adsorbent: activated carbon W = 1 g; C0 = 10%; Q = 30 mL/min | |
[43] | 126 | Kyn—4.24 (h−1) τ—2.37 (h) | Adsorbent: zeolite 4A W = 6.3 g; C0 = 20 mg/L; Q = 8 L/min | ||
Adams–Bohart | H2S | [44] | - | kba—0.9595 N0—571.49 (mg/L·min) | Adsorbent: pure ash W = 3.4432 g; C0 = 500 mg/L; Q = 25 mL/min |
[44] | 508.43 | kba—0.9595 N0—508.43 (mg/L·min) | Adsorbent: sol–gel method ash W = 1.9295 g; C0 = 500 mg/L; Q = 25 mL/min | ||
CO2 | [42] | 16.257 (mL/min·mg) | kba—2.2 × 10−4 (L/mg·min) N0—28.15 (mg/L) | Adsorbent: eggshell waste W = 25 g; C0 = 30.8%; Q = 0.03 m3/h | |
[41] | 2.9 | kba—0.2329 (mL/mg·s) N0—4515 (mg/L) | Adsorbent: activated carbon W = 1 g; C0 = 10%; Q = 30 mL/min | ||
Yan | H2S | [44] | 502.35 | —1.14 | Adsorbent: hydrothermal process ashes W = 1.4843 g; C0 = 500 mg/L; Q = 25 mL/min |
[44] | 571.49 | —10.58 | Adsorbent: pure ash W = 3.4432 g; C0 = 500 mg/L; Q = 25 mL/min | ||
CO2 | [41] | 22.9 | —3.0939 | Adsorbent: activated carbon W = 1 g; C0 = 10%; Q = 30 mL/min | |
[45] | 2.5 mmol/g | —19.27 | Adsorbent: modified polyacrylonitrile nanofibers C0 = 5%; Q = 100 mL/min |
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Lima, H.B.S.; Sousa, A.P.S.d.; Silva, W.B.d.; Costa, D.S.d.; Rodrigues, E.C.; Estumano, D.C. Parameter Estimation of Breakthrough Curve Models in the Adsorption Process of H2S and CO2 Using the Markov Chain Monte Carlo Method. Appl. Sci. 2024, 14, 6956. https://doi.org/10.3390/app14166956
Lima HBS, Sousa APSd, Silva WBd, Costa DSd, Rodrigues EC, Estumano DC. Parameter Estimation of Breakthrough Curve Models in the Adsorption Process of H2S and CO2 Using the Markov Chain Monte Carlo Method. Applied Sciences. 2024; 14(16):6956. https://doi.org/10.3390/app14166956
Chicago/Turabian StyleLima, Haianny Beatriz Saraiva, Ana Paula Souza de Sousa, Wellington Betencurte da Silva, Deibson Silva da Costa, Emerson Cardoso Rodrigues, and Diego Cardoso Estumano. 2024. "Parameter Estimation of Breakthrough Curve Models in the Adsorption Process of H2S and CO2 Using the Markov Chain Monte Carlo Method" Applied Sciences 14, no. 16: 6956. https://doi.org/10.3390/app14166956