ADSORBENT CHARACTERIZATION AND PERFORMANCE
OUR PREVIOUS APPROACH.
Originally we had adopted the well-known Langmuir formula, J.A.C.S.,38,2221,(1916), which describes the adsorption isotherms for single adsorbates versus the various adsorbents.
Originally we had adopted the well-known Langmuir formula, J.A.C.S.,38,2221,(1916), which describes the adsorption isotherms for single adsorbates versus the various adsorbents.
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- Subscript i: related to adsorbate "i",
- XE: total adsorbed quantity,
- XM: maximum adsorbable quantity,
- P: adsorbate pressure,
- B: constant (inverse of pressure),
- H: (differential) heat of adsorption.
this formula may be rewritten in the modified form:
In a socalled Langmuir plot, the value of Pi/XEi versus Pi should produce a straight line. The degree by which measured data, if presented in this way deviate from a straight line should give an indication of the suitability of the Langmuir formula. From the intersection at Pi=0 and the slope of the line, the constants XMi and Bi are established. For a multi-component system the adsorption of each of the n components may then be described by the extended Langmuir formula according Markham and Benton, J.A.C.S.,53,497,(1931), using the constants XMi and Bi as established for each component as a single adsorbate.
Although these expressions look simple and attractive to use, many measured data if presented in the modified Langmuir formula deviated too much from a straight line for some practicle range of pressures and no trustworthy results could be produced by using the extended Langmuir formula.
OUR PRESENT APPROACH
Dramatic improvements could be achieved by assuming adsorbents to consist of a mixture of (at least) two distinct formulations in recognition of its heterogeneity, the adsorption on each formulation to be described by a single Langmuir formula. By assuming an adsorbent to consist of two different formulations, the overall adsorption should be described by simply adding two single Langmuir formulas to obtain the dual Langmuir (D-L) formula as follows for component "i" on an adsorbent:
- XE: total adsorbed quantity,
- XM: maximum adsorbable quantity,
- P: adsorbate pressure,
- B: constant (inverse of pressure),
- H: (differential) heat of adsorption.
this formula may be rewritten in the modified form:
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OUR PRESENT APPROACH
Dramatic improvements could be achieved by assuming adsorbents to consist of a mixture of (at least) two distinct formulations in recognition of its heterogeneity, the adsorption on each formulation to be described by a single Langmuir formula. By assuming an adsorbent to consist of two different formulations, the overall adsorption should be described by simply adding two single Langmuir formulas to obtain the dual Langmuir (D-L) formula as follows for component "i" on an adsorbent:
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- Subscripts number 1 and 2: related to low and high activity formulation
respectively.
By finally using a special least square regression program, these two formulations can be distinguished from oneanother quite easily for establishing the constants XM1i, XM2i, B1i and B2i at the specified temperature of the isotherm. In case of two isotherms at two different temperature levels, the least square regression analysis of the two isotherms is conducted in mutual conjunction. From these, the constants of the D-L formula are established at a standard temperature of 40 deg. C, while the heat of adsorption for a component versus each of the two formulations of adsorbent is established as well. The magnitude of the heat of adsorption on each of the two formulations are usually very different, for the higher activity formulation (strong adsorption, low capacity) invariably appearing to be the highest. The latter means that the overall net heat of adsorption depends on the adsorbate distribution over the lower and the higher activity formulation and therefore on the total amount being adsorbed, i.e. the higher this total, the more prominent the lower activity formulation in absolute terms and therefore the lower the incremental heat of adsorption.
In comparison to the single Langmuir formula a far better match is obtained with the D-L formula when measured isotherm data are used for establishing their parameters. In the table below, further relevant data are given for adsorption of carbon dioxide on activated carbon BPL from Calgon Corporation.
Generalized adsorption correlations.
Without sufficient knowledge of adsorption and the availability of data for a realistic description, it is impossible to construct the necessary algorithms for a trustworthy simulation program, and to include any possible option which would otherwise be impossible to investi-
gate. Hence no innovation without a reliable model.
Adsorption potential and volume filling
Based on fundamental work by Polanyi. (Verhandl. deutsch.physik Ges. 1914,16,1012 & 1916,18,55 & Zeitschrift Elektrochem. 1920,26,370 & Trans. Faraday Soc.,1932,28,316) satisfactory results could be obtained for components which are significant to the PSA process.
An adsorbent, loaded with adsorbate has performed work to compress gas molecules into a smaller volume, leading to a free energy difference of:
respectively.
By finally using a special least square regression program, these two formulations can be distinguished from oneanother quite easily for establishing the constants XM1i, XM2i, B1i and B2i at the specified temperature of the isotherm. In case of two isotherms at two different temperature levels, the least square regression analysis of the two isotherms is conducted in mutual conjunction. From these, the constants of the D-L formula are established at a standard temperature of 40 deg. C, while the heat of adsorption for a component versus each of the two formulations of adsorbent is established as well. The magnitude of the heat of adsorption on each of the two formulations are usually very different, for the higher activity formulation (strong adsorption, low capacity) invariably appearing to be the highest. The latter means that the overall net heat of adsorption depends on the adsorbate distribution over the lower and the higher activity formulation and therefore on the total amount being adsorbed, i.e. the higher this total, the more prominent the lower activity formulation in absolute terms and therefore the lower the incremental heat of adsorption.
In comparison to the single Langmuir formula a far better match is obtained with the D-L formula when measured isotherm data are used for establishing their parameters. In the table below, further relevant data are given for adsorption of carbon dioxide on activated carbon BPL from Calgon Corporation.
| Higher activity formulation | Lower activity formulation |
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| Generalized adsorption correlations. Adsorption potential and volume filling for adsorption by activated carbon and silica gel. Statistical thermodynamics for adsorption by zeolites. |
Generalized adsorption correlations.
Without sufficient knowledge of adsorption and the availability of data for a realistic description, it is impossible to construct the necessary algorithms for a trustworthy simulation program, and to include any possible option which would otherwise be impossible to investi-
gate. Hence no innovation without a reliable model.
Adsorption potential and volume filling
Based on fundamental work by Polanyi. (Verhandl. deutsch.physik Ges. 1914,16,1012 & 1916,18,55 & Zeitschrift Elektrochem. 1920,26,370 & Trans. Faraday Soc.,1932,28,316) satisfactory results could be obtained for components which are significant to the PSA process.
An adsorbent, loaded with adsorbate has performed work to compress gas molecules into a smaller volume, leading to a free energy difference of:
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[kJ/kmole] |
The fugacity f is related to the adsorbate pressure P, while the fugacity fs is consistent with the saturation pressure Ps of the condensed liquidlike state at adsorption temperature T. The consistent molar gas constant R=8.3144 kJ/kmole/K. It is convenient to express the adsorbed quantity in said condensed liquidlike state in m3/kg of adsorbent; this requires the knowledge of its density Dm in m3/kmol for a given adsorbent loading XE in kmole/kg. An energy function ENF or Adsorption Potential is now defined:
The adsorption potential ENF is reduced to zero, if all the micropores of the adsorbent are filled and therefore the value of f has reached its maximum, equal to fs.
It follows that the Adsorption Potential is related to the volume of adsorbed and condensed adsorbate Dm*XE in m3/kg of adsorbent. Hence Dm*XE is maximum at ENF=0.
Plottingln(Dm*XE) versus ENF appears to produce an almost straight line. In the diagram shown below, ln(Dm*XE)=-7.4 at ENF=0, which means that the adsorbate in that case occupies the maximum available volume, equal to EXP(-7.4)=6.1E-4 m3/kg, or 0.61 ml/g of adsorbent.
For homologues like paraffines or olefins, all adsorp-
tion data can be represented by a single line, which means that irregularities in measured adsorption data are easily detected. Besides, since the lines are independent of temperature, such line, if constructed from isotherm data measured and available at only one single temperature, enables the calculation of isotherm data at any other temperature. The latter in turn enables the estimation of the heat of adsorption Ha through an Arrhenius type equation:
Equilibrium calculations
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[kJ/m3] |
It follows that the Adsorption Potential is related to the volume of adsorbed and condensed adsorbate Dm*XE in m3/kg of adsorbent. Hence Dm*XE is maximum at ENF=0.
Plotting
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tion data can be represented by a single line, which means that irregularities in measured adsorption data are easily detected. Besides, since the lines are independent of temperature, such line, if constructed from isotherm data measured and available at only one single temperature, enables the calculation of isotherm data at any other temperature. The latter in turn enables the estimation of the heat of adsorption Ha through an Arrhenius type equation:
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Equilibrium calculations
The concept is based on the Ideal Adsorbed Solution Theory.(IAST) LIT, (1): A.L. Myers & J.M. Prausnitz - A.I.Ch.E. Journal 1965,11,121, (2): A.L. Myers - A.I.Ch.E. Journal 1968,5,45.
The Gibbs' adsorption isotherm for a two-dimensional adsorbed phase is expressed by the spreading pressure as follows:
At a certain temperature "T", in stead of the Spreading Pressure "
", the value for "A/RT" for a single component "i", also written as (i)*, may be calculated.
An attractive aspect of using the D-L formula is the fact that the integral of above equation (1) can be readily solved, resulting into the simple expression:
At a fixed temperature and pressure, the spreading pressure of the mixture and of the pure adsorbates are the same:
The molar area of an adsorbed mixture "a" is considered as the total of the molar areas of each pure adsorbate "a()" at the same temperature and spreading pressure times its molar fraction "x()":
The total area "A" is occupied by XE moles, hence:
Since A/a=XE, A/a(0)°=XE(0)° ect., dividing by A × XE yields:
The loading of a component "i" in the mixture is given by:
The Ideal Adsorbed Solution Theory considers the adsorbed mixture as an ideal solution, analogously to liquid solutions. Consequently, as for liquid mixtures Raoult's law is likewise applied for adsorbed solutions:
P(i)° is the pressure of the pure adsorbate if adsorbed at the same temperature and spreading pressure.
Problem analysis
For calculating adsorbed quantities of "1+c" adsorbate components in a mixture at a given gas composition "YE(i)", a temperature "T" and a pressure "P", the basic conditions to meet are equations (3), (8) and as specified below under (9):
Remark: adsorbate components are numbered from 0 to c, giving 1+c components.
Distinguishing between the component numbers "i", the expression for the spreading pressure (2) is written as:
The equations (3) and (8) can now be solved for x(i) and P(i)° by two nested trial and error loops. Following calculations of XE(i)° using D-L isotherm equations at pressures P(i)°, values for XE and XE(i) are obtained through equations (6) and (7).
Shortcut procedure
Each time meeting the requirements of equation (3) by two nested iterations takes so much computer time, that a shortcut procedure has been developed, eliminating one iteration to shorten calculation time:
Relative ideal pure adsorbate pressures
Before the actual PSA simulation program is started, a preparatory program is executed where the relations
are established at fixed values for(i)* at two fixed temperatures TPOL and TPOH.
At any of the two temperatures the ideal pure adsorbate pressure P(i)° is given as a function of the ideal pressure P(0)° of pure adsorbed hydrogen. (Hydrogen is compon-
ent number 0)
Since equation (11) has been evaluated at constant spreading pressure, it automatic-
ally meets the requirement of equation (3).
Equation (11) takes the form of the following polynomial:
The Gibbs' adsorption isotherm for a two-dimensional adsorbed phase is expressed by the spreading pressure as follows:
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(1) |
| Where: | XE A R T P |
= spreading pressure = adsorbed quantity = specific surface area of adsorbent = gas constant = temperature = adsorbate pressure |
At a certain temperature "T", in stead of the Spreading Pressure "
", the value for "A/RT" for a single component "i", also written as (i)*, may be calculated.An attractive aspect of using the D-L formula is the fact that the integral of above equation (1) can be readily solved, resulting into the simple expression:
| A/RT = XM1×ln(P×BE1+1)+XM2×ln(P×BE2+1) |
(2) |
At a fixed temperature and pressure, the spreading pressure of the mixture and of the pure adsorbates are the same:
| * =(0)* =(1)* = ... =(c)* |
(3) |
The molar area of an adsorbed mixture "a" is considered as the total of the molar areas of each pure adsorbate "a()" at the same temperature and spreading pressure times its molar fraction "x()":
| a=a(0)° × x(0) + a(1)° × x(1) + ... + a(c)° × x(c) | (4) |
The total area "A" is occupied by XE moles, hence:
| A=XE × a=XE × a(0)° × x(0) + XE ×a(1)° × x(1) + ... + XE × a(c)° × x(c) | (5) |
Since A/a=XE, A/a(0)°=XE(0)° ect., dividing by A × XE yields:
| 1/XE=x(0)/XE(0)° + x(1)/XE(1)° + ... + x(c)/XE(c)° | (6) |
The loading of a component "i" in the mixture is given by:
| XE(i)=XE × x(i) | (7) |
The Ideal Adsorbed Solution Theory considers the adsorbed mixture as an ideal solution, analogously to liquid solutions. Consequently, as for liquid mixtures Raoult's law is likewise applied for adsorbed solutions:
| P × YE(i) = P(i)° × x(i) | (8) |
P(i)° is the pressure of the pure adsorbate if adsorbed at the same temperature and spreading pressure.
Problem analysis
For calculating adsorbed quantities of "1+c" adsorbate components in a mixture at a given gas composition "YE(i)", a temperature "T" and a pressure "P", the basic conditions to meet are equations (3), (8) and as specified below under (9):
 x(i)=1 and YE(i)=1 for i=0 to c |
(9) |
Remark: adsorbate components are numbered from 0 to c, giving 1+c components.
Distinguishing between the component numbers "i", the expression for the spreading pressure (2) is written as:
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(i)*= |
XM1(i) × ln(P(i)° × BE1(i)+1)+XM2(i) × ln(P(i)° × BE2(i)+1) | (10) |
The equations (3) and (8) can now be solved for x(i) and P(i)° by two nested trial and error loops. Following calculations of XE(i)° using D-L isotherm equations at pressures P(i)°, values for XE and XE(i) are obtained through equations (6) and (7).
Shortcut procedure
Each time meeting the requirements of equation (3) by two nested iterations takes so much computer time, that a shortcut procedure has been developed, eliminating one iteration to shorten calculation time:
| 1. | Generating polynomial fuctions, defining ideal pressures of pure adsorbates, relative to a reference adsorbate at identical spreading pressures. |
| 2. | Pedicting calculating results. |
Relative ideal pure adsorbate pressures
Before the actual PSA simulation program is started, a preparatory program is executed where the relations
| P(i)° = f(P(0)°) | (11) |
are established at fixed values for
(i)* at two fixed temperatures TPOL and TPOH. |
ent number 0)
Since equation (11) has been evaluated at constant spreading pressure, it automatic-
ally meets the requirement of equation (3).
Equation (11) takes the form of the following polynomial:
| ln(P(i)°);= | CIAS(i0,i1,i,iA) + CIAS(i0,i1+1,i,iA) × ln(P(0)°) + CIAS(i0,i1+2,i,iA) × P(0)° |
(12) |
| i0=0 | For T=TPOL |
| i0=1 | For T=TPOH |
| iA : | Adsorbent identification number |
| i1 : | Value of i1 defines validity of constants "CIAS(....)" in a certain range of P(0)°-values. The total number of ranges has been established at 6: 0 to 50, 50 to 600, 600 to 1E4, 1e4 to 1e5, 1e5 to 4E5, and 4E5 to 1E6. Values larger than 1E6 are made equal to 1E6. Depending on the actual P(0)°-value, the corresponding range and value of i1 assigned to this range is selected. |