Abstract
Adsorption isotherm data of methanol, ethanol, and dimethyl carbonate were determined via headspace experiments on two activated carbons (RB 4 from CABOT Norit and SC 40 from SilCarbon) at different temperatures. However, dimethyl carbonate (DMC) showed a significant decomposition to methanol and CO2 during the adsorption experiments. Hence, the measured isotherm data for DMC were always two-component isotherm data assuming CO2 as non-adsorbing component. Thus, ideal adsorbed solution theory (IAST) was inverted to determine the single-component isotherm of DMC using two-component isotherm data and the known single component isotherm of methanol. The inverse IAST was supported by the calculation of the adsorption equilibrium distribution. This calculation was done with the expectation maximization method and results were used to determine the most probable isotherm equations to initialize the inverse IAST calculation. The methodology was validated for mixtures of methanol and ethanol where both single-component isotherms were known. Finally, single-component isotherms were calculated for DMC on RB 4 and SC 40 at different temperatures.
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Abbreviations
- AED:
-
Adsorption energy distribution
- AEqD:
-
Adsorption equilibrium distribution
- DMC:
-
Dimethyl carbonate
- EM:
-
Expectation maximization
- EMC:
-
Ethyl methyl carbonate
- EtOH:
-
Ethanol
- IAST:
-
Ideal adsorbed solution theory
- MeOH:
-
Methanol
- b i :
-
Langmuir coefficient of binding site i, L mmol−1
- c :
-
Coefficient for power function describing separation factor, Ld mmol-d
- d :
-
Coefficient for power function describing separation factor, 1
- E A :
-
Adsorption energy, J mol−1
- f :
-
Adsorption equilibrium distribution, mmol g−1
- H i :
-
Henry coefficient of binding site i, L g−1
- I i :
-
Integral, mmol g−1
- K 0 :
-
Pre-exponential factor, L mmol−1
- K :
-
Binding coefficient, L mmol−1
- m AC :
-
Mass of adsorbent, g
- M :
-
Molar mass, g mol−1
- n add :
-
Initial amount of adsorptive, mol
- R :
-
Universal gas constant, J mol−1 K−1
- R ′2 :
-
Fictive regression coefficient, 1
- S X :
-
Weighed linear deviation between experimental and calculated loading, 1
- S I :
-
Weighed linear deviation between the IAST integrals, 1
- S :
-
Sum of SX and SI, 1
- TSS X :
-
Total sum of squares of deviation between experimental and calculated loading, 1
- TSS I :
-
Total sum of squares of deviation between the IAST integrals, 1
- TSS :
-
Sum of TSSX and TSSI, 1
- T :
-
Temperature, K
- V vial :
-
Volume of the vial, m3
- y i :
-
Mole fraction of component i in the gas phase, \({\text{mol}}_{i} \,{\text{mol}}_{\text{total}}^{ - 1}\)
- Y i :
-
Gas loading of component i, mmol L−1
- Y′i :
-
Corrected gas loading of component i, mmol L−1
- Y min :
-
Minimum gas loading of an experimental isotherm, mmol L−1
- Y max :
-
Maximum gas loading of an experimental isotherm, mmol L−1
- Y overall :
-
Overall gas loading, mmol L−1
- x i :
-
Mole fraction of component i in the adsorbed phase, \({\text{mol}}_{i} \,{\text{mol}}_{\text{total}}^{ - 1}\)
- \(x_{i}^{0}\) :
-
Mole fraction of pure component i in the adsorbed phase if no other substance was present, \({{{\text{mol}}_{i}}} \,{{{\text{mol}}_{\text{total}}}^{ - 1}}\)
- X i :
-
Adsorbent loading of component i, mmol g−1
- X inverse IAST :
-
Adsorbent loading of pure component calculated by inverse IAST, mmol g−1
- X cal :
-
Calculated loading, mmol g−1
- X mon :
-
Monolayer capacity, mmol g−1
- X IAST :
-
Overall loading of the adsorbent by IAST, mmol g−1
- \(X_{i}^{0}\) :
-
Pure component loading if no other substance was present, mmol g−1
- X′i :
-
Corrected adsorbent loading of component i, mmol g−1
- α :
-
Separation factor, mol mol−1
- π :
-
Spreading pressure, kg m−1 s−2
- ρ A :
-
Buoyancy density, kg m−3
- ϑ :
-
Temperature, °C
- ϑ B :
-
Boiling temperature, °C
- θ :
-
Local isotherm model, 1
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Acknowledgements
We gratefully acknowledge receipt of activated carbon from CABOT Norit and SilCarbon Aktivkohle GmbH and financial support from the German Federal Ministry for Environment, Nature Conservation and Nuclear Safety through the project LithoRecII (Project Grant Number: 16EM1025).
Funding
This work was supported by the German Federal Ministry for Environment, Nature Conservation and Nuclear Safety through the project LithoRecII (Project Grant Number: 16EM1025).
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Stehmann, F., Hartig, D. & Scholl, S. Inverse ideal adsorbed solution theory for calculation of single-component adsorption equilibria from mixture isotherms supported by adsorption equilibrium distribution. Adsorption 25, 1475–1486 (2019). https://doi.org/10.1007/s10450-019-00136-z
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DOI: https://doi.org/10.1007/s10450-019-00136-z