A Statistical Approach to Determine Optimal Models for IUPAC-Classified Adsorption Isotherms
Abstract
:1. Introduction
2. Adsorption Isotherm Models
2.1. Dubinin–Astakhov (D–A) Model
2.2. Modified Dubinin–Astakhov (D–A) Model
2.3. Tόth Model
2.4. Langmuir Model
2.5. Modified Langmuir Model
2.6. Freundlich Model
2.7. Hill Model
2.8. Mahle Model
2.9. Brunauer–Emmet–Teller (BET) Model
2.10. Modified BET Model
2.11. Guggenheim–Anderson–De Boer (GAB) Model
2.12. Sun and Chakraborty Model
2.13. Hybrid Model (Henry + Sips)
2.14. Ben Yahia Model
2.15. Universal Isotherm Model
3. Error Evaluation Function
3.1. Root Mean Square Deviation (RMSD)
3.2. Hybrid Fractional Error Deviation (HYBRID)
4. Statistical Tools
4.1. Information-Based Criterion for Model Selection
4.2. Bootstrap Approach
4.3. Bootstrap p-Value and Confidence Interval (CI) of Residual Sum of Squares (RSS)
4.4. Proportion Tests
4.4.1. Chi-Squared Test for Equality of Proportions
4.4.2. Pairwise Test: Multiple Comparisons
5. Results and Discussion
5.1. Type-I(a) Adsorption Isotherm
5.1.1. Bootstrap Error Analysis and Model Selection Using Information Criteria
5.1.2. Overall and Pairwise Proportion Tests
5.2. Type-I(b) Adsorption Isotherm
5.2.1. Bootstrap Error Analysis and Model Selection Using Information Criteria
5.2.2. Overall and Pairwise Proportion Tests
5.3. Type-II Adsorption Isotherm
5.4. Type-III Adsorption Isotherm
5.5. Type-IV(a) Adsorption Isotherm
5.6. Type-IV(b) Adsorption Isotherm
5.7. Type-V Adsorption Isotherm
5.8. Type-VI Adsorption Isotherm
6. Conclusions
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
Abbreviations
ABIC | adjusted bayesian information criterion |
ACF | activated carbon fiber |
ACP | activated carbon powder |
AHT | adsorption heat transformation |
AIC | akaike information criterion |
AICc | corrected akaike information criterion |
BET | brunauer-emmet-teller |
BIC | bayesian information criterion |
BIDC | benzimidazole-derived carbons |
CI | confidence interval |
D-A | dubinin-astakhov |
GAB | guggenheim-anderson-de boer |
GRG | generalized reduced gradient |
IC | information criterion |
IRMOF | isoreticular metal-organic framework |
IUPAC | international union of pure and applied chemistry |
M-AC | mangrove based activated carbon |
mAIC | modified akaike information criterion |
MgO | magnesium oxide |
PBA | polymer based adsorbent |
PCB | poorly crystalline boehmite |
PVDC | polyvinylidene chloride |
RMSD | root-mean-square deviation |
RSS | residual sum of squares |
WPT | waste palm trunk |
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Adsorption Pair | Type | Best Fitted Model | Reference |
---|---|---|---|
(i) Carbon K4-700/N2 at 77 K (ii) BIDC-1-700/Ar at 87 K (pore size less than 1 nm for both (i) and (ii)). | Type-I(a) | Fitted model is not available | (i) Hu et al. [18] (ii) Cychosz et al. [19] |
(i) Maxsorb III/ethanol (ii) Silica gel/water (iii) Carbon based composite/CO2 (iv) Carbon based composite/CO2 (v) WPT-AC/ethanol (vi) M-AC/ethanol (vii) Zeolite/water at 25 °C (viii) (SRD 1352/3, FR 20, ATO and AP4-60/COCL-1200)/ N2 * (ix) Ionic liquid binder based composite/ethanol (x) Silica gel based composite/water | Type-I(b) | (i) D–A model (ii) Tόth (iii) Tόth (iv) D–A and Tόth (v) D–A and Tόth (vi) D–A and Tόth (vii) Tόth (viii) D–A (ix) D–A (x) Tόth | (i) El-Sharkawy et al. [20] (ii) Rahman et al. [21] (iii) Pal et al. [22] (iv) Berdenova et al. [23] (v) Pal et al. [24,25] (vi) Pal et al. [24,25] (vii) Wang et al. [26] (viii) Brancato et al. [27] (ix) Pal et al. [28] (x) Younes et al. [29] |
(i) PBA/water at 25 °C (ii) Alumina/water at 20 °C (iii) PCB/water at 22 °C | Type-II | (i) GAB model (ii) BET model (iii) Not fitted | (i) Sultan et al. [30] (ii) Naono et al. [31] (iii) Wang et al. [32] |
(i) ACP/water at 30 °C (ii) Dried fruits/moisture | Type-III | (i) D–A model (ii) GAB model | (i) Sultan et al. [30] (ii) Maroulis et al. [33] |
(i) Oxidized carbon/water (ii) PVDC/water at 35 °C iii) Tripolis/water at 21 °C | Type-IV(a) | (i) Do et al. model (ii) Do et al. model (iii) BET model | (i) Do et al. [34] (ii) Do et al. [34] (iii) Rakitskaya et al. [35] |
(i) IRMOF-74-V-hex/argon at 87 K (ii) IRMOF-74-V-hex/nitrogen at 77 K (pore size less than 4 nm) | Type-IV(b) | The fitted model is not available | (i) Cho et al. [36] (ii) Cho et al. [36] |
(i) ACF/water at 30 °C (ii) Hydrophobic carbon/water (iii) AQSOA-Z01/water and AQSOA-Z02/water (iv) Ferroaluminophosphate/water (v) AQSOA zeolite/ water (vi) AlPO-18, FAPO-34, SAPO-34/water | Type-V | (i) D–A model (ii) Do et al. model (iii) D–A, modified Langmuir (iv) Hybrid model (v) D–A, modified Langmuir, Sun and Chakraborty (vi) D–A model | (i) Sultan et al. [30] (ii) Do et al. [34] (iii) Kayal et al. [37] (iv) Kim et al. [38] (v) Teo et al. [39] (vi) Brancato et al. [40] |
(i) MgO (100)/methane (ii) Exfoliated graphite/methane | Type-VI | (i) Ng et al. model (ii) Yahia et al. model | (i) Ng et al. [16] (ii) Yahia et al. [41] |
Model | Mean RMSD | CI of RMSD | Mean HYBRID | CI of HYBRID |
---|---|---|---|---|
K4-700/N2 pair | ||||
Tόth | 0.0745 | (0.045, 0.0956) | 1.884 | (1.24, 2.152) |
D–A | 0.0838 | (0.047, 0.0971) | 1.912 | (1.55, 2.923) |
Mod. BET | 0.0638 | (0.050, 0.0723) | 1.871 | (1.52, 1.983) |
Ng et al. | 0.1066 | (0.032, 0.3254) | 1.913 | (1.43, 3.851) |
Mahle | 0.1019 | (0.042, 0.2321) | 4.146 | (3.654, 5.123) |
BIDC-1-700/Ar pair | ||||
Tόth | 0.1047 | (0.045, 0.126) | 3.6648 | (2.24, 4.152) |
D–A | 0.0936 | (0.047, 0.0971) | 8.9234 | (6.55, 10.923) |
Mod. BET | 0.0872 | (0.050, 0.0986) | 3.5624 | (2.52, 4.783) |
Ng et al. | 0.1245 | (0.032, 0.3254) | 12.354 | (8.43, 15.851) |
Mahle | 0.1154 | (0.042, 0.2921) | 4.8132 | (2.65, 7.123) |
Model | Error Mean | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|---|
K4-700/N2 pair | ||||||||
Using RMSD error mean | ||||||||
Tόth | 0.0745 | -223.16 | −214.78 | −222.43 | −199.16 | −219.16 | −210.78 | −227.36 |
D–A | 0.0838 | −218.08 | −211.79 | −217.65 | −206.08 | −215.08 | −208.79 | −221.23 |
Mod. BET | 0.0638 | −234.48 | −228.20 | −234.06 | −222.48 | −231.48 | −225.20 | −237.64 |
Ng et al. | 0.1066 | −195.68 | −181.02 | −193.53 | −111.68 | −188.68 | −174.02 | −203.04 |
Mahle | 0.1019 | −204.38 | −196.00 | −203.65 | −180.38 | −200.38 | −192.00 | −208.58 |
Using HYBRID error mean | ||||||||
Tόth | 1.884 | −29.33 | −20.96 | −28.61 | −5.33 | −25.33 | −16.96 | −33.54 |
D–A | 1.912 | −30.47 | −24.19 | −30.05 | −18.47 | −27.47 | −21.19 | −33.63 |
Mod. BET | 1.871 | −31.78 | −25.49 | −31.35 | −19.78 | −28.78 | −22.49 | −34.93 |
Ng et al. | 1.913 | −22.44 | −7.78 | −20.29 | 61.56 | −15.44 | −0.78 | −29.80 |
Mahle | 4.146 | 17.96 | 26.34 | 18.69 | 41.96 | 21.96 | 30.34 | 13.76 |
BIDC-1-700/Ar pair | ||||||||
Using RMSD error mean | ||||||||
Tόth | 0.1047 | −69.02 | −63.69 | −67.28 | −45.02 | −65.02 | −59.69 | −76.12 |
D–A | 0.0936 | −74.15 | −70.16 | −73.15 | −62.15 | −71.15 | −67.16 | −79.48 |
Mod. BET | 0.0872 | −76.14 | −72.14 | −75.14 | −64.14 | −73.14 | −69.14 | −81.47 |
Ng et al. | 0.1245 | −58.17 | −48.84 | −52.57 | 25.83 | −51.17 | −41.84 | −70.60 |
Mahle | 0.1154 | −66.29 | −60.96 | −64.55 | −42.29 | −62.29 | −56.96 | −73.40 |
Using HYBRID error mean | ||||||||
Tόth | 3.6648 | 30.54 | 35.86 | 32.27 | 54.54 | 34.54 | 39.86 | 23.43 |
D–A | 8.9234 | 53.45 | 57.45 | 54.45 | 65.45 | 56.45 | 60.45 | 48.12 |
Mod. BET | 3.5624 | 27.74 | 31.74 | 28.74 | 39.74 | 30.74 | 34.74 | 22.41 |
Ng et al. | 12.354 | 70.56 | 79.89 | 76.16 | 154.56 | 77.56 | 86.89 | 58.12 |
Mahle | 4.8132 | 38.17 | 43.50 | 39.91 | 62.17 | 42.17 | 47.50 | 31.06 |
Test | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|
Carbon K4-700/N2 | |||||||
Overall | 2.0 × 10−15 (2.3 × 10−14) | 2.1 × 10− (2.1 × 10−13) | 2.2 × 10−15 (2.0 × 10−14) | 2.0 × 10−13 (2.2 × 10−12) | 2.1 × 10−15 (2.2 × 10−13) | 2.0 × 10−14 (2.2 × 10−14) | 2.1 × 10−15 (2.2 × 10−13) |
Mod. BET vs. Tόth | 2.0 × 10−4 (2.1 × 10−5) | 2.1 × 10−5 (2.0 × 10−4) | 2.2 × 10−3 (2.1 × 10−4) | 1.4 × 10− (1.2 × 10−3) | 1.8 × 10−4 (2.1 × 10−3) | 1.9 × 10−3 (2.1 × 10−3) | 2.1 × 10−4 (2.2 × 10−3) |
Mod. BET vs. D–A | 1.5 × 10−8 (1.9 × 10−8) | 2.1 × 10−9 (2.0 × 10−7) | 1.1 × 10−8 (2.1 × 10−8) | 2.0 × 10−7 (2.0 × 10−7) | 2.0 × 10−8 (1.9 × 10−7) | 1.7 × 10−9 (2.2 × 10−9) | 2.1 × 10−11 (1.9 × 10−12) |
BIDC-1-700/Ar pair | |||||||
Overall | 2.0 × 10−4 (2.3 × 10−3) | 2.2 × 10−3 (2.1 × 10−3) | 2.1 × 10−4 (2.3 × 10−3) | 2.4 × 10−3 (2.1 × 10−4) | 2.1 × 10−4 (2.3 × 10−5) | 2.5 × 10−3 (2.1 × 10−4) | 2.2 × 10−4 (2.3 × 10−3) |
Mod. BET vs. Tόth | 0.0235 (0.0351) | 0.0351 (0.0421) | 0.2363 (0.5324) | 0.0425 (0.0354) | 0.0573 (0.0784) | 0.2381 (0.1465) | 0.0715 (0.0471) |
Mod. BET vs. D–A | 1.6× 10−3 (1.7× 10−4) | 1.1 × 10−4 (2.4 × 10−3) | 1.5 × 10−3 (2.3 × 10−3) | 1.4 × 10−4 (2.2 × 10−3) | 2.2 × 10−3 (1.5 × 10−4) | 1.9 × 10−4 (2.1 × 10−3) | 2.2 × 10−3 (1.44) |
Model | Mean RMSD | CI of RMSD | Mean HYBRID | CI of HYBRID |
---|---|---|---|---|
D–A | 0.05624 | (0.0457, 0.0650) | 0.4893 | (0.3432, 0.6382) |
Mod. D–A | 0.01532 | (0.0125, 0.0179) | 0.1096 | (0.0745, 0.1438) |
Tόth | 0.01435 | (0.0122, 0.0165) | 0.0373 | (0.0276, 0.0473) |
Freundlich | 0.05178 | (0.0451, 0.0583) | 0.4353 | (0.3282, 0.5444) |
Langmuir | 0.01839 | (0.0157, 0.0209) | 0.0724 | (0.0516, 0.0960) |
Hill | 0.04871 | (0.0451, 0.0583) | 0.3816 | (0.2740, 0.4965) |
Model | Error Mean | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|---|
Using HYBRID error mean | ||||||||
D–A | 0.4893 | −56.61 | −51.70 | −55.90 | −44.61 | −53.61 | −48.70 | −61.08 |
Mod. D–A | 0.1096 | −114.43 | −107.88 | −113.22 | −90.43 | −110.43 | −103.88 | −120.38 |
Tόth | 0.0373 | −157.53 | −150.98 | −156.32 | −133.53 | −153.53 | −146.98 | −163.49 |
Freundlich | 0.4353 | −63.29 | −60.01 | −62.95 | −59.29 | −61.29 | −58.01 | −66.27 |
Langmuir | 0.0724 | −133.03 | −128.11 | −132.32 | −121.03 | −130.03 | −125.11 | −137.50 |
Hill | 0.3815 | −66.56 | −61.65 | −65.85 | −54.56 | −63.56 | −58.65 | −71.03 |
Using RMSD error mean | ||||||||
D–A | 0.0562 | −143.15 | −138.24 | −142.44 | −131.15 | −140.15 | −135.24 | −147.62 |
Mod. D–A | 0.0153 | −193.17 | −186.62 | −191.96 | −169.17 | −189.17 | −182.62 | −199.12 |
Tóth | 0.0143 | −195.78 | −189.23 | −194.57 | −171.78 | −191.78 | −185.23 | −201.74 |
Freundlich | 0.0517 | −148.45 | −145.18 | −148.11 | −144.45 | −146.45 | −143.18 | −151.43 |
Langmuir | 0.0183 | −187.86 | −182.95 | −187.16 | −175.86 | −184.86 | −179.95 | −192.33 |
Hill | 0.0487 | −148.90 | −143.99 | −148.19 | −136.90 | −145.90 | −140.99 | −153.37 |
Test | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|
Overall | 2.1 × 10−16 (2.2 × 10−12) | 2.1 × 10−15 (2.2 × 10−13) | 2.1 × 10−16 (2.2 × 10−14) | 2.2 × 10−14 (2.2 × 10−13) | 2.2 × 10−16 (2.2 × 10−14) | 2.0 × 10−13 (2.2 × 10−16) | 2.2 × 10−16 (2.2 × 10−15) |
Tόth vs. Mod. D–A | 2.0 × 10−12 (2.0 × 10−16) | 2.0 × 10−11 (2.0 × 10−16) | 2.0 × 10−11 (2.0 × 10−16) | 1.0 × 10−10 (1.0 × 10−10) | 2.0 × 10−13 (2.0 × 10−16) | 2.1 × 10−12 (2.0 × 10−16) | 2.0 × 10−13 (2.0 × 10−16) |
Mod.D–A vs. Langmuir | 1.9 × 10−15 (2.0 × 10−16) | 2.0 × 10−16 (2.0 × 10−16) | 2.1 × 10−15 (2.0 × 10−16) | 2.0 × 10−16 (2.0 × 10−16) | 2.0 × 10−16 (2.0 × 10−16) | 2.0 × 10−16 (2.0 × 10−16) | 2.0 × 10−16 (2.0 × 10−16) |
Alumina/Water Pair | Boehmite/Water Pair | ||||
---|---|---|---|---|---|
Model | Mean Error | 95% CI of Error | Model | Mean Error | 95% CI of Error |
Mod. BET | 0.2739 | (0.1476, 0.4204) | Mod. BET | 0.0079 | (0.0028, 0.0141) |
Ng et al. | 0.8420 | (0.3389, 1.5064) | Ng et al. | 0.0311 | (0.0121, 0.0588) |
D–A | 1.0182 | 0.1739, 2.5075) | D–A | 0.0357 | (0.0132, 0.0743) |
Tόth | 1.9345 | (1.2436, 2.6768) | Tόth | 0.6532 | (0.2401, 1.2636) |
Redlich–Peterson | 1.6072 | (0.9935, 2.3879) | Langmuir | 1.4360 | (0.5127, 2.7516) |
Model | Mean Error | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|---|
Adsorption of water onto alumina | ||||||||
Mod. BET | 0.2739 | −41.32 | −37.44 | −40.28 | −29.32 | −38.32 | −34.44 | −46.76 |
Ng et al. | 0.8420 | −3.00 | 6.07 | 2.89 | 81.00 | 4.00 | 13.07 | −15.68 |
D–A | 1.0182 | −5.87 | −1.98 | −4.82 | 6.13 | −2.87 | 1.02 | −11.30 |
Tόth | 1.9345 | 13.46 | 18.65 | 15.28 | 37.46 | 17.46 | 22.65 | 6.22 |
Redlich–Peterson | 1.6072 | 8.46 | 13.64 | 10.28 | 32.46 | 12.46 | 17.64 | 1.21 |
Adsorption of water onto poorly crystalline boehmite | ||||||||
Mod. BET | 0.0079 | −70.43 | −68.11 | −68.43 | −58.43 | −67.43 | −65.11 | −77.30 |
Ng et al. | 0.0311 | −40.48 | −35.07 | −26.48 | 43.52 | −33.48 | −28.07 | −56.50 |
D–A | 0.0356 | −46.31 | −44.00 | −44.31 | −34.31 | −43.31 | −41.00 | −53.18 |
Tόth | 0.6532 | 2.24 | 5.33 | 5.87 | 26.24 | 6.24 | 9.33 | −6.91 |
Langmuir | 1.4359 | 12.84 | 15.16 | 14.84 | 24.84 | 15.84 | 18.16 | 5.98 |
Model | Mean Error | 95% CI of Error | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|---|---|
GAB model | 0.2932 | (0.1720, 0.4280) | −73.10 | −64.66 | −71.34 | −33.10 | −68.10 | −59.66 | −80.30 |
Mod. BET | 0.8758 | (0.3509, 1.5002) | −33.33 | −28.26 | −32.66 | −21.33 | −30.33 | −25.26 | −37.65 |
Ng et al. | 1.0613 | (0.6907, 1.4828) | −17.64 | −5.82 | −14.14 | 66.36 | −10.64 | 1.18 | −27.73 |
D–A | 1.2565 | (0.6192, 2.0008) | −18.89 | −13.82 | −18.22 | −6.89 | −15.89 | −10.82 | −23.21 |
Sun and Chakraborty | 1.3858 | (0.7615, 2.0736) | −12.97 | −6.22 | −11.83 | 11.03 | −8.97 | −2.22 | −18.73 |
Mod. Langmuir | 0.7541 | (0.3423, 1.2238) | −31.31 | −19.49 | −27.81 | 52.69 | −24.31 | −12.49 | −41.39 |
Test | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|
Overall | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 |
GAB vs. Mod. BET | 2.0 × 10−14 | 2.0 × 10−13 | 2.0 × 10−14 | 2.0 × 10−12 | 2.0 × 10−15 | 2.0 × 10−14 | 2.0 × 10−14 |
GAB vs. Mod. Langmuir | 2.0 × 10−15 | 2.0 × 10−15 | 2.1 × 10−14 | 2.2 × 10−15 | 2.2 × 10−15 | 2.0 × 10−15 | 2.1 × 10−15 |
PVDC/water | Mean Error | 95% CI of Error | Tripolis/Water | Mean Error | 95% CI of Error |
---|---|---|---|---|---|
Ng et al. | 0.08324 | (0.0486, 0.1210) | Ng et al. | 0.0127 | (0.0043, 0.0247) |
Yahia et al. | 2.69468 | (1.0143, 4.8338) | Yahia et al. | 0.0903 | (0.0341, 0.1678) |
Mod. BET | 0.51836 | (0.2596, 0.8222) | BET | 0.2576 | (0.0550, 0.5013) |
Mahle | 0.92660 | (0.1397, 1.8578) | Mod. BET | 0.1471 | (0.0613, 0.2460) |
Sun and Chakraborty | 1.12546 | (0.2890, 2.1980) | Do et al. | 0.4532 | (0.0043, 0.0247) |
Model | Mean Error | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|---|
Adsorption of water onto PVDC | ||||||||
Ng et al. | 0.08324 | −5.50 | 5.96 | −1.77 | 78.50 | 1.50 | 12.96 | −15.93 |
Yahia et al. | 2.69468 | 31.27 | 44.37 | 36.24 | 143.27 | 39.27 | 52.37 | 19.36 |
Mod BET | 0.51836 | 4.79 | 9.70 | 5.49 | 16.79 | 7.79 | 12.70 | 0.32 |
Mahle | 0.92660 | 12.59 | 19.15 | 13.81 | 36.59 | 16.59 | 23.15 | 6.64 |
Sun and Chakraborty | 1.12546 | 14.54 | 21.09 | 15.75 | 38.54 | 18.54 | 25.09 | 8.58 |
Adsorption of water onto Tripolis | ||||||||
Ng et al. | 0.0127 | −34.11 | −22.64 | −30.37 | 49.89 | −27.11 | −15.64 | −44.53 |
Yahia et al. | 0.0903 | −8.62 | 4.48 | −3.65 | 103.38 | −0.62 | 12.48 | −20.53 |
BET | 0.2576 | −8.03 | −4.76 | −7.69 | −4.03 | −6.03 | −2.76 | −11.01 |
Mod BET | 0.1471 | −12.76 | −7.85 | −12.05 | −0.76 | −9.76 | −4.85 | −17.23 |
IRMOF-74-V-hex/argon | Mean Error | 95% CI of Error | IRMOF-74-V-hex/nitrogen | Mean Error | 95% CI of Error |
---|---|---|---|---|---|
Ng et al. | 0.95400 | (0.8126, 0.9967) | Ng et al. | 0.18451 | (0.1124, 0.2435) |
Mod. BET | 2.71755 | (1.9571, 3.1684) | Mod. BET | 1.19510 | (0.5642, 2.9856) |
Mahle | 4.24836 | (3.2145, 6.2541) | Mahle | 1.18360 | (0.4265, 3.2113) |
Mod. Langmuir | 2.68435 | (1.2563, 4.1256) | Mod. Langmuir | 0.53406 | (0.0613, 2.1560) |
Tόth | 4.34000 | (3.5671, 5.3461) | Tόth | 1.65845 | (0.4125, 2.6571) |
Model | Mean Error | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|---|
Adsorption of argon onto IRMOF-74-A-hex | ||||||||
Ng et al. | 0.95400 | −48.49 | −37.03 | −44.76 | 35.51 | −41.49 | −30.03 | −58.92 |
Mod. BET | 2.71755 | −1.01 | 3.90 | −0.31 | 10.99 | 1.99 | 6.90 | −5.48 |
Mahle | 4.24836 | 24.67 | 31.22 | 25.88 | 48.67 | 28.67 | 35.22 | 18.71 |
Mod. Langmuir | 2.68435 | 6.34 | 17.80 | 10.07 | 90.34 | 13.34 | 24.80 | −4.09 |
Tόth | 4.34000 | 25.80 | 32.35 | 27.01 | 49.80 | 29.80 | 36.35 | 19.84 |
Adsorption of N2 onto IRMOF-74-A-hex | ||||||||
Ng et al. | 0.18451 | −135.57 | −124.1 | −131.84 | −51.57 | −128.5 | −117.1 | −145.9 |
Mod. BET | 1.19510 | −44.55 | −39.64 | −43.84 | −32.55 | −41.55 | −36.64 | −49.02 |
Mahle | 1.18360 | −19.79 | −13.24 | −18.58 | 4.21 | −15.79 | −9.24 | −25.75 |
Mod. Langmuir | 0.53406 | −79.24 | −67.78 | −75.51 | 4.76 | −72.24 | −60.78 | −89.66 |
Tόth | 1.65845 | −25.18 | −18.63 | −23.97 | −1.18 | −21.18 | −14.63 | −31.14 |
Model | Mean Error | CI of HYBRID | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|---|---|
D–A | 0.5370 | (0.3538, 0.7506) | −43.21 | −38.46 | −42.46 | −31.21 | −40.21 | −35.46 | −47.83 |
Mahle | 0.9121 | (0.4822, 1.4353) | −22.14 | −15.81 | −20.85 | 1.86 | −18.14 | −11.81 | −28.30 |
modified Langmuir | 0.2172 | (0.1225, 0.3370) | −67.79 | −56.71 | −63.79 | 16.21 | −60.79 | −49.71 | −78.58 |
GAB | 2.6254 | (1.9507, 3.3515) | 13.92 | 18.67 | 14.67 | 25.92 | 16.92 | 21.67 | 9.30 |
Sun and Chakraborty | 0.1453 | (0.0741, 0.2364) | −88.27 | −81.94 | −86.98 | −64.27 | −84.27 | −77.94 | −94.44 |
Test | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|
Overall | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 | 2.2 × 10−16 |
Sun and Chakraborty vs. Mod. Langmuir | 2.0 × 10−16 | 2.0 × 10−16 | 2.0 × 10−16 | 2.0 × 10−16 | 2.0 × 10−16 | 2.0 × 10−16 | 2.0 × 10−16 |
MgO/Methane | Mean Error | 95% CI of Error | Graphite/Methane | Mean Error | 95% CI of Error |
---|---|---|---|---|---|
Yahia et al. | 0.1237 | (0.0699,0.1835) | Yahia et al. | 0.562948 | (0.3545,0.7757) |
Ng et al. | 0.6007 | (0.4107,0.7994) | Ng et al. | 7.848831 | (3.4595,13.359) |
D–A | 6.6061 | (3.4087,10.022) | D–A | 15.06557 | (9.0459,22.345) |
Mod. Langmuir | 21.165 | (12.654,30.267) | Mod. Langmuir | 21.242790 | (12.848,30.937) |
Mahle | 11.430 | (6.3491, 17.249) | Mahle | 12.461560 | (7.7736,17.705) |
Model | Mean error | AIC | BIC | AICc | mAIC | AIC3 | CAIC | ABIC |
---|---|---|---|---|---|---|---|---|
Adsorption of methane onto MgO | ||||||||
Yahia et al. | 0.1237 | −89.70 | −69.73 | −77.70 | 174.30 | −77.70 | −57.73 | −107.27 |
Ng et al. | 0.6007 | −26.06 | −4.43 | −11.50 | 285.94 | −13.06 | 8.57 | −45.10 |
D–A | 6.6061 | 47.45 | 52.44 | 48.13 | 59.45 | 50.45 | 55.44 | 43.05 |
Mod. Langmuir | 21.1652 | 100.86 | 112.50 | 104.47 | 184.86 | 107.86 | 119.50 | 90.60 |
Mahle | 11.4302 | 70.83 | 77.48 | 72.00 | 94.83 | 74.83 | 81.48 | 64.97 |
Adsorption of methane onto graphite | ||||||||
Yahia et al. | 56.29480 | 159.73 | 181.14 | 169.79 | 423.73 | 171.73 | 193.14 | 143.53 |
Ng et al. | 784.88310 | 277.66 | 300.86 | 289.80 | 589.66 | 290.66 | 313.86 | 260.12 |
D–A | 1506.55700 | 286.35 | 291.71 | 286.95 | 298.35 | 289.35 | 294.71 | 282.31 |
Mod. Langmuir | 2124.27900 | 309.47 | 321.96 | 312.58 | 393.47 | 316.47 | 328.96 | 300.03 |
Mahle | 1246.15600 | 280.00 | 287.14 | 281.03 | 304.00 | 284.00 | 291.14 | 274.61 |
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Rahman, M.M.; Muttakin, M.; Pal, A.; Shafiullah, A.Z.; Saha, B.B. A Statistical Approach to Determine Optimal Models for IUPAC-Classified Adsorption Isotherms. Energies 2019, 12, 4565. https://doi.org/10.3390/en12234565
Rahman MM, Muttakin M, Pal A, Shafiullah AZ, Saha BB. A Statistical Approach to Determine Optimal Models for IUPAC-Classified Adsorption Isotherms. Energies. 2019; 12(23):4565. https://doi.org/10.3390/en12234565
Chicago/Turabian StyleRahman, Md. Matiar, Mahbubul Muttakin, Animesh Pal, Abu Zar Shafiullah, and Bidyut Baran Saha. 2019. "A Statistical Approach to Determine Optimal Models for IUPAC-Classified Adsorption Isotherms" Energies 12, no. 23: 4565. https://doi.org/10.3390/en12234565
APA StyleRahman, M. M., Muttakin, M., Pal, A., Shafiullah, A. Z., & Saha, B. B. (2019). A Statistical Approach to Determine Optimal Models for IUPAC-Classified Adsorption Isotherms. Energies, 12(23), 4565. https://doi.org/10.3390/en12234565