Accurate Effective Diffusivities in Multicomponent Systems
Abstract
:1. Introduction
2. New Effective Diffusivity Model
2.1. Derivation of Model Equations
2.2. Calculation Procedure
- Using tabulated experimental data or empirical correlations, collect the binary diffusion coefficients at infinite dilution of all pairs of components, . These are equal to the infinite dilution binary Maxwell–Stefan (MS) diffusion coefficients, .
- Compute the MS diffusion coefficients, , for the specific mixture composition using the following mixing rule:
- Calculate the elements of the matrix, via Equation (2b,c), and compute its inverse, .
- Compute the matrix by applying Equation (2d), which requires an appropriate thermodynamic model to describe the nonideal behavior of the mixture. The partial derivatives can be computed numerically using, for instance, central finite differences. The increments in the mole fraction of a component j are absorbed by negative increments in the nth component in order to maintain the sum of all mole fractions equal to 1. For instance:
- Obtain matrix and its inverse .
2.3. Effective Diffusivity for Ideal Mixtures
3. Examples of Application
3.1. Liquid Phase Reaction: Ethyl Acetate Synthesis
- Obtain K or from the literature.
- Make an initial guess for the extent of reaction at equilibrium, .
- Compute equilibrium compositions () for the assumed (via Equation (23)) and then the respective activity coefficients, .
- Calculate the equilibrium constant via Equation (24), .
- Compute the square of the deviation .
- Repeat steps 2–5 until the squared error is below a predetermined tolerance.
3.2. High-Pressure Gas Phase Reaction: Methanol Synthesis
- For a given temperature, pressure and initial mixture composition calculate the final values of the extents of reaction, and , as described in Section 3.1.
- Calculate the corresponding for each in the span of by solving Equation (38) numerically:
- Once has been determined as function of (over ), the derivatives can be calculated numerically using finite differences, for instance.
- The effective diffusivities can then be evaluated following the procedure delineated in Section 3.1.
4. Conclusions
Supplementary Materials
Author Contributions
Funding
Institutional Review Board Statement
Data Availability Statement
Conflicts of Interest
Nomenclature
a | Activity |
B | Coefficients defined by Equation (2b,c), s/cm2 |
Total concentration, mol/cm3 | |
D | Diffusion coefficient, cm2/s |
Ð | Maxwell–Stefan diffusion coefficient, cm2/s |
EoS | Equation of state |
h | Finite difference step size |
Molar diffusion flux, mol/(cm2 s) | |
K | Equilibrium constant |
Binary interaction parameter | |
MS | Maxwell–Stefan |
Molar flux, mol/(cm2 s) | |
n | Number of moles, mol, or number of components in a mixture |
P | Pressure, MPa |
PC-SAFT | Perturbed-Chain Statistical Associating Fluid Theory |
PR | Peng–Robinson |
r | Reaction rate, mol/(cm3 s) |
T | Temperature, K |
x | Mole fraction in the liquid phase |
y | Mole fraction in the gas phase |
Greek Letters | |
Element of matrix as defined by Equation (2d) | |
Activity coefficient | |
Kronecker function | |
Stoichiometric coefficient | |
Extent of reaction | |
Solvent association factor of Wilke–Chang equation | |
Fugacity coefficient | |
Subscripts | |
0 | Initial condition |
eff | Effective |
eq | Equilibrium |
ij | Refers to the pair of components i and j |
i, j, k, n | Arbitrary component identification |
T | Total |
Superscripts | |
Infinite dilution or Standard State | |
Reaction identification | |
calc | Calculated value |
Burghardt and Krupiczka effective diffusivity model | |
Ideal (Bird et al. [4]) effective diffusivity model | |
Element of inverse matrix | |
K | Kubota et al. [5] effective diffusivity model |
Kato et al. [6] effective diffusivity model | |
New effective diffusivity model | |
W | Wilke effective diffusivity model |
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Component | CH3COOH | CH3CH2OH | CH3COOCH2CH3 | H2O | |
---|---|---|---|---|---|
Initial mole fractions | 0.500 | 0.500 | 0.000 | 0.000 | |
Calculated equilibrium mole fractions | 0.190 | 0.190 | 0.310 | 0.310 | |
Ratio | Reference Model | ||||
Ideal (Bird et al.) [4], Equation (6) | (1.379) 1.124 | (0.737) 0.904 | (1.000) 0.809 | (1.000) 1.384 | |
Wilke [3], Equation (5) | (1.536) 1.212 | (1.066) 1.029 | (1.000) 0.694 | - | |
Burghardt and Krupiczka [2], Equation (8) | (1.254) 1.188 | (0.793) 0.999 | (1.0000) 0.650 | - | |
Kato et al. [6], Equation (9) | (1.086) 1.048 | (0.780) 1.039 | (1.000) 1.625 | - |
Component | CO | CH3OH | H2 | H2O | CH4 | CO2 | |
---|---|---|---|---|---|---|---|
Initial mole fractions | 0.1224 | 0.0012 | 0.7084 | 0.0026 | 0.1490 | 0.0164 | |
Calculated equilibrium mole fractions | 0.0580 | 0.0897 | 0.6545 | 0.0054 | 0.1753 | 0.0170 | |
Ratio | Reference Model | ||||||
Ideal (Bird et al.) [4], Equation (6) | (1.011) 0.9961 | (0.9994) 0.9612 | (0.9436) 0.8708 | (0.8817) 0.9465 | (0.5557) 0.5118 | (1.2423) 1.0446 | |
Wilke [3], Equation (5) | (0.6340) 0.8241 | (0.9992) 0.9414 | (0.6826) 0.6808 | (1.3416) 1.1576 | (5.2983) 10.3318 | - | |
Kubota et al. [5], Equation (7) | (0.7529) 0.8717 | (1.0018) 1.1376 | (0.2793) 0.3119 | (1.2530) 1.0454 | - | - | |
Burghardt and Krupiczka [2], Equation (8) | (0.8063) 0.8912 | (1.0015) 1.1119 | (0.7403) 0.6922 | (1.3480) 1.1658 | (6.8448) 12.9857 | - | |
Kato et al. [6], Equation (9) | (0.7775) 0.8782 | (1.0023) 1.1822 | (0.7162) 0.7011 | (1.3473) 1.1656 | (6.7167) 12.7846 | - |
Component | CO | CH3OH | H2 | H2O | CH4 | CO2 | |
---|---|---|---|---|---|---|---|
Initial mole fractions | 0.1223 | 0.0012 | 0.7085 | 0.0025 | 0.1490 | 0.0165 | |
Calculated equilibrium mole fractions | 0.0592 | 0.0880 | 0.6557 | 0.0053 | 0.1748 | 0.0170 | |
Ratio | Reference Model | ||||||
Ideal (Bird et al.) [4], Equation (6) | (1.0100) 1.0007 | (0.9994) 0.9569 | (0.9424) 0.8695 | (0.8339) 0.9152 | (0.7685) 0.8996 | (1.2238) 1.0319 | |
Wilke [3], Equation (5) | (0.6525) 0.8441 | (1.0330) 0.9738 | (0.6962) 0.6976 | (1.2542) 1.1845 | (4.6316) 8.1610 | - | |
Kubota et al. [5], Equation (7) | (0.7749) 0.8963 | (1.0357) 1.1687 | (0.2848) 0.3231 | (1.1762) 1.1586 | - | - | |
Burghardt and Krupiczka [2], Equation (8) | (0.8298) 0.9148 | (1.0354) 1.1466 | (0.7551) 0.7100 | (1.2599) 1.1928 | (5.9837) 10.2637 | - | |
Kato et al. [6], Equation (9) | (0.7943) 0.8986 | (1.0362) 1.2237 | (0.7450) 0.7306 | (1.2591) 1.1923 | (5.8233) 10.0110 | - |
Component | CO | CH3OH | H2 | H2O | CH4 | CO2 |
---|---|---|---|---|---|---|
Initial mole fractions (PR) | 0.1224 | 0.0012 | 0.7084 | 0.0026 | 0.1490 | 0.0164 |
Initial mole fractions (PC-SAFT) | 0.1223 | 0.0012 | 0.7085 | 0.0025 | 0.1490 | 0.0165 |
Calculated final mole fractions (PR) | 0.0593 | 0.0880 | 0.6557 | 0.0053 | 0.1748 | 0.0170 |
Calculated final mole fractions (PC-SAFT) | 0.0592 | 0.0880 | 0.6557 | 0.0053 | 0.1748 | 0.0170 |
(0.9716) 0.9722 | (0.9672) 0.9678 | (0.9805) 0.9754 | (1.0697) 0.9796 | (1.1438) 1.2379 | (0.9140) 0.9343 |
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Rios, W.Q.; Antunes, B.; Rodrigues, A.E.; Portugal, I.; Silva, C.M. Accurate Effective Diffusivities in Multicomponent Systems. Processes 2022, 10, 2042. https://0-doi-org.brum.beds.ac.uk/10.3390/pr10102042
Rios WQ, Antunes B, Rodrigues AE, Portugal I, Silva CM. Accurate Effective Diffusivities in Multicomponent Systems. Processes. 2022; 10(10):2042. https://0-doi-org.brum.beds.ac.uk/10.3390/pr10102042
Chicago/Turabian StyleRios, William Q., Bruno Antunes, Alírio E. Rodrigues, Inês Portugal, and Carlos M. Silva. 2022. "Accurate Effective Diffusivities in Multicomponent Systems" Processes 10, no. 10: 2042. https://0-doi-org.brum.beds.ac.uk/10.3390/pr10102042