Models or correlations for mixed-gas adsorption are ...kexhu.people.ust.hk/ceng521/521-3.pdf · Ideal Adsorbed Solution Theory (IAST) ... given set of gas concentrations p(k) (k=1,

Professor Xijun Hu CENG 5210 Advanced Separation Processes

52

Equilibrium Adsorption of Gas Mixtures

Models or correlations for mixed-gas adsorption are

crucial to the design of adsorptive separation

processes. They should be capable of predicting the

equilibrium adsorbed amount from pure gas

isotherms.

Extended Multicomponent Langmuir isotherm

For species i, the extended multicomponent

Langmuir isotherm equation is

q qb p

b pi si

i i

j jj

NC

11

(1)

where NC is the number of components.

The derivation follows the same assumptions of

single-component Langmuir isotherm. The

system is defined as containing partial pressure

p1, p2, ..., pNC in the gas phase, which is in

equilibrium with coverages 1, 2, ..., NC on the

surface. The rate of condensation for gas i, ri, is

given by:

r k pi ai i jj

NC

11

(2)


53

The rate of evaporation for gas i is kdii.

At equilibrium,

k p kai i jj

NC

di i11

(3)

By defining b=ka/kd, we have

b pi i jj

NC

i11

(4)

Summing Eq. (4) over all components (NC)

gives

b pj jj

NC

jj

NC

jj

NC

1 1 1

1 (5)

so

jj

NCj j

j

NC

j jj

NC

b p

b p

1

1

1

1

(6)

substituting Eq. (6) into (4) to give

ii i

j jj

NC

b p

b p

11

(7)

which is the extended Langmuir isotherm

(i=qi/qsi), & can be used for liquid adsorption.


54

The extended Langmuir equation is

thermodynamically consistent only when the

maximum adsorption capacities (qsi) are the

same for all species. If the maximum adsorption

capacity differs significantly between different

component, the usage of extended Langmuir

isotherm may cause errors.

Loading Ratio Correlation

As in the extended Langmuir equation, the

hybrid Langmuir-Freundlich equation can also

be extended to an n-component mixture.

q qb p

b pi si

i in

j j

n

j

NC

i

j

1

1

1

1

/

/ (8)

The ratio qi/qsi is referred to as loading ratio,

hence the term loading ratio correlation (LRC).


55

Ideal Adsorbed Solution Theory (IAST)

From thermodynamics we have the Gibbs adsorption

isotherm for species i

Ad n dsi i (9)

where is the spreading pressure, which defines the

lowering of the surface tension at the solid-fluid

interface upon adsorption.

Because the mixture isotherm is specified at constant

temperature and spreading pressure, the reduced

spreading pressures of various components

(i=1,2,...,NC) in their pure-gas standard states, *(i),

are equal to each other and to the reduced spreading

pressure of the adsorbed mixture *.

* * * *(1) = (2) =...= (NC) = (10)

By assuming that the gas phase is ideal, the fugacity

can be replaced by the pressure, so for pure-gas

adsorption we have

Ad q RTd pi i 0 0ln


56

hence, the reduced spreading pressure of i-th

component in its pure-gas standard state (i)* can be

calculated from the following equation:

*

0

(i) =(i)A

T= d

R

q

pp

pi

ii

i0 0

00

(11)

where A is the surface area, and qi0 is the adsorbed-

phase concentration in its pure-gas standard state and

is related to the gas-phase concentration pi0 by the

single-component adsorption isotherm:

q f pi i i0 0= (12)

Since the adsorbed solution is assumed to be ideal,

one can apply Raoult's law to the mole fraction in the

adsorbed phase xi and the partial pressure in the gas

phase pi, i.e.

p p xi i i 0 (13)

with pi0 as the pressure of the i-th component in its

standard state.


57

In addition, the sum of the mole fractions in adsorbed

phase must be equal to 1:

i=1

NCx(i) = 1 (14)

The total amount of adsorbed gases, qT, is calculated

from the amounts adsorbed in the standard state, qi,

and the mole fractions in the adsorbed phase:

1

= x

i=1

NC

q qT

i

i

0 (15)

and the amount adsorbed by the species i is then

calculated from:

q q xi T i (16)

The choice of single-gas isotherm equations used to

calculate the spreading pressure is arbitrary. The only

requirement is that the chosen isotherm equation is in

good agreement with experimental data, especially in

low-pressure regions where the integrand q pi i0 0/ of

the spreading pressure equation (Eq. 11) has the

highest weight.


58

Methods of solution for IAST

In the IAST equations, for given partial pressures, pi,

(i=1, 2, ..., NC) in the gas phase, we have 3NC+1

unknowns, xi, pi0 , qi

0 , and *. To solve for these

variables, we need 3NC+1 equations. Equations (10),

(12) and (13) provide 3NC equations, and Equation

(14) gives the additional equation.

In order to calculate the spreading pressure, one needs

to do the integration on the isotherm function q pi i0 0/

to obtain the spreading pressure. With some isotherm

equations, like Langmuir, the spreading pressure can

be integrated analytically q b psi i iln 1 0 , while for

some other isotherm equations it must be done

numerically. An orthogonal polynomial can be used

to approximate the isotherm function and a Gaussian

quadrature can be used for the integration of the

integral.

The detailed algorithm to solve the multicomponent

equilibrium equations is given below.


59

Subroutine to calculate adsorbed phase

concentrations from gas concentrations by IAST

Output solution: x =

p

p =

1

x

q

; q = q xii

iT

i=1

NCi

i

i T i0

0

; q

Input No. of component (NC), isotherm type and

parameters and gas concentration for each component

Initial guess:

p = p = p

p(p )T

i=1

NC

i*

i=1

NCi

Tio

T ;

inverse function p = p ( )

q = f p

S1 =p

p= x(i) 2 =

p

p q

= (S1-1) / S2 = +

io

i*

i i i

i=1

NCi

i i=1

NC

i=1

NCi

i i

* *

0

0 0

0 0 0

;

;

S


60

Function subroutine for calculating the inverse

function p pi i0 0 ( )* of equation (11).

Diagram to calculate the gas concentration from the

reduced spreading pressure.

A program (iso-pre.exe) available in KESU2 can be used to

calculate the multicomponent adsorption equilibrium

Function C ( )pio *

Initial guess:

p =q

K(e )i

si

i

/q*si

0 1

q = f p

G = (p ) -

= Gp / q

p = p -

i i i

*i

*

i i

i i

0 0

0

0 0

0 0

Return C (i)po


61

Effect of surface energetic heterogeneity on

adsorption isotherms

Since the Langmuir theory is only applicable to ideal

(homogeneous) surfaces, a lot of efforts have been

made to propose more sophisticated models to better

describe the adsorption equilibria. Among these

attempts, the surface energetic heterogeneity concept

is the most popular in the study of adsorption

equilibrium isotherms of gases. In this approach, an

energy distribution is assumed to describe the spread

of adsorbent-adsorbate interactions. Next, a local

adsorption isotherm is assumed for a specified energy

site and then the observed adsorption isotherm is

simply an integral of the local adsorption isotherm

over the whole energy distribution range.


62

In the attempts to predict the multicomponent

equilibria using single component isotherm

information for heterogeneous systems, to simplify the

calculation, we assume a local multicomponent

isotherm (usually the extended Langmuir isotherm) at

a single site and integrate it over the full site energy

distribution (uniform energy distribution).

The local adsorption isotherm for the species k at a

given site with an adsorption energy E(k) is assumed

to follow the extended Langmuir equation of the

following form:

q q

p(k)

p(j)

k,E(k) = (k)b k,E(k)

1+ b j, E(j)s

j=1

NC

(17)

where q[k,E(k)] and b[k,E(k)] are the adsorbed phase

concentration and the affinity for adsorbate k for a

given site of energy E(k), respectively, and the affinity

is correlated to the interaction energy by the following

equation:

b k,E(k) = b (k)expE(k)

T0

R

(18)

with b0 being the affinity at the zero energy level.

Let the energy distribution in the adsorbed phase be

F[k,E(k)], the macroscopically observed isotherm at a

given set of gas concentrations p(k) (k=1, NC) is:


63

q q

p(k)

p(j)s(k) = (k)

b k,E(k)

1 + b j,E(j)

F k,E(k) dE(k)0

j=1

NC

(19)

where the energy distribution function satisfies the

following equation:

0

F k,E(k) dE(k) = 1

(20)

The uniform energy distribution will be used:

F k,E(k) =1

E (k) - E (k) for E (k) E < E (k)

max minmin max

(21)

and F[k,E(k)] = 0 elsewhere. By assuming that the

ordering of the energy sites from low to high is the

same for different adsorbates and the cumulative

energy distribution function is the same for all species

in the mixture, the correlation of energies of different

species for a uniform energy distribution is

E(i) - E (i)

E (i) - E (i)=

E(j) - E (j)

E (j) - E (j)

min

max min

min

max min

(22)

where the whole energy distribution region of one

species is matched to that of another species.

The solution is obtained by numerical integration.


64

Effect of isotherm fitting method

Example: The adsorption equilibrium isotherms

of ethane and propane on activated carbon have

been experimentally determined by a volumetric

measurement technique as follows (Hu et al.,

AIChE Journal, 39(2), 249-261 (1993)):

Adsorption equilibrium isotherm of ethane on activated carbon

10oC

p

(kPa)

q

(mmol/g

carbon)

0.96 0.621

2.89 1.12

6.04 1.62

10.57 2.09

16.49 2.55

26.66 3.14

40.12 3.63

54.78 4.06

75.51 4.5

100.4 4.92

125.4 5.29

30oC

p

(kPa)

q

(mmol/g

carbon)

0.68 0.275

2.14 0.592

5.76 1.06

11.05 1.49

18.13 1.92

26.74 2.31

40.96 2.8

57.75 3.22

74.99 3.6

97.59 4.01

124.3 4.4

60oC

p

(kPa)

q

(mmol/g

carbon)

0.786 0.124

1.81 0.249

5.07 0.524

12 0.916

20.83 1.28

31.4 1.62

43.45 1.93

62.09 2.34

82.59 2.72

102.9 3.03

128.3 3.38


65

Adsorption equilibrium isotherm of propane on activated carbon

(C3-60C)

(C3-30C)

(C3-10C)

1. Determine the Langmuir constants for ethane

at three temperatures by using the linear format of

Langmuir equation. Calculate the sum of error

squares (qmodel - qexp)2.

2. Determine the Langmuir constants for the two

gases at three temperatures by fitting the original

Langmuir equation to the data by non-linear

regression (use program iso.exe). Calculate the

10oC

p

(kPa)

q

(mmol/g

carbon)

0.213 1.13

0.64 1.74

1.39 2.28

3.03 2.89

5.67 3.37

12.66 3.96

31.99 4.58

44.79 4.8

62.45 5.05

81.41 5.27

106.1 5.51

126.4 5.68

30oC

p

(kPa)

q

(mmol/g

carbon)

0.6 1.12

1.71 1.71

3.55 2.23

7.13 2.79

12.08 3.22

22.57 3.72

45.85 4.26

59.77 4.48

78.28 4.71

98.03 4.92

123.5 5.15

60oC

p

(kPa)

q

(mmol/g

carbon)

2.03 1.09

5.16 1.63

9.69 2.09

17.02 2.56

25.74 2.91

39.89 3.33

67.07 3.8

82.17 4

102 4.22

122.5 4.44


66

sum of error squares (qmodel - qexp)2 (= the

function value computed by iso.exe). Compare

the results of ethane adsorption with those in Q1

and comment on the performance of linear and

non-linear regression methods. Plot the results of

two methods in the same graph for ethane.

3. Calculate the Henry constant for ethane at

30oC, the heat of adsorption (Q) of ethane and

propane using b=b0exp(Q/RT).

4. Refit the data simultaneously for three

temperatures by choosing option 9 (Langmuir) in

the program of iso.exe. This will give a unique qs

for different temperatures. Compare the isosteric

heat of adsorption (parameter Q in option 9) with

the results in Q3. Explain why. Do this for both

ethane and propane.


67


temperatures by choosing option 11 (Uniform

energy distribution & local Langmuir isotherm,

Unilan equation) in the program of iso.exe. Do

this for both ethane and propane. The fits should

be much better than Q4. Why? Write down the

parameters.

6. The binary adsorption equilibrium of ethane

and propane on activated carbon at 30oC has been

experimentally determined as:

p (ethane):

kPa

p

(propane):

kPa

q (ethane):

mmol/g

q

(propane):

mmol/g

5 10 0.17 3.0

10 10 0.37 2.80

20 10 0.72 2.70

Use program iso-pre.exe to calculate the model

predictions and compare the results with the


68

experimental data. The models should be used

are: extended Langmuir (parameters from Q2),

IAST-Langmuir (parameters from Q2), IAST-

Unilan (parameters from Q5). Which model is

the best? Comment on the performance of

various model predictions.

Solution

1. Determine the Langmuir constants for ethane

at three temperatures by using the linear format of

Langmuir equation. Calculate the sum of error

squares (qmodel - qexp)2.

By rearranging the Langmuir Isotherm Equation,

we get

pbqqq ss

1111 (1)

Plotting 1/qs vs 1/p should give a linear line with

an intersect on y-axis of 1/qs, an intersect on x-

axis of -b, and a slope of 1/bqs. The data are

plotted in Figure 1 and the extracted parameters


69

are listed in Table 1. The sum of error squares

(qmodel - qexp)2 is calculated by using iso.exe

with the obtained parameters. The sum of error

squares is given in Table 1 as F.

Table 1. Isotherm parameters of ethane

adsorption in activated carbon by linear

regression.

T1=283.2K T2=303.2K T3=333.2K

q s

mmol/g

3.7010 2.8588 2.3810

b, 1/kPa

F

(mmol/g)2

0.1988

7.0437

0.1515

6.3105

0.0686

3.1585

2. Determine the Langmuir constants for the two

gases at three temperatures by fitting the original

Langmuir equation to the data by non-linear

regression (use program iso.exe). Calculate the

sum of error squares (qmodel - qexp)2 (= the


70

function value computed by iso.exe). Compare

the results of ethane adsorption with those in Q1

and comment on the performance of linear and

non-linear regression methods. Plot the results of

two methods in the same graph for ethane.

By using program iso.exe, the isotherm

parameters of gas adsorption in activated carbon

were obtained by non-linear regression. The

results were listed in Tables 2 & 3 for ethane and

propane, respectively, together with the values of

the sum of error squares F= ( )mod expq qel 2 . It is very

clear that the errors between the model fits and

the experimental data are much smaller by using

the non-linear regression method. This is so

because the original error has been changed in

the linear format of the Langmuir equation. By

linear regression, the objective function to be

minimised is F’= ( )mod exp

1 1 2

q qel

, which is

F= ( )mod expq qel 2 in the non-linear regression.

Therefore, the non-linear regression to the

original Langmuir equation gives much better

fits to the experimental isotherm data. This is


71

more apparent in Figure 2 which shows the

adsorption isotherm of ethane in activated

carbon, with experimental data as symbols, non-

linear regression fits as solid lines, and the linear

regression fits as dashed lines.

Table 2. Isotherm parameters of ethane

adsorption in activated carbon by non-linear

regression.

T1=283.2 T2=303.2 T3=333.2

q s mol/g

b, 1/kPa

F, (mmol/g)2

5.7443

0.0516

0.5825

5.2269

0.0320

0.2962

4.7832

0.0167

0.0874

Table 3. Isotherm parameters of propane

adsorption in activated carbon by non-linear

regression.

T1=283.2 T2=303.2 T3=333.2

qs, mmol/g 5.2265 4.9083 4.5639

b, 1/kPa 0.4705 0.2169 0.0863

F, (mmol/g)2 1.850 1.017 0.4374


72

Pressure (kPa)

0 20 40 60 80 100 120 140

Ad

sorb

ed

am

ou

nt (m

mo

l/g)

0

1

2

3

4

5

6

non-linear regression

10oC

30oC

60oC

linear regression

Figure 2. Adsorption equilibrium isotherm of

ethane in activated carbon.


73

3. Calculate the Henry constant for ethane at

30oC, the isosteric heats of adsorption of ethane

and propane using the results of Q2.

The Henry constant for the Langmuir isotherm is

K’=bqs (2)

Using the isotherm parameter of ethane

adsorption, the Henry constant for ethane at 30oC

is

K’=0.0320 (1/kPa) x 5.2269 (mmol/g)

=0.167 (mmol/g/kPa)

The isosteric heats of adsorption can be

determined by the following equation:

b=b0exp( H

RT) (3)

lnb=lnb0 - H

RT (4)

By plotting lnb vs 1/T, we should obtain a

straight line with a slope of -H R/ and an intersect

on y-axis of lnb0. From Question 2, we have

Ethane adsorption

b lnb T 1/T

0.0516 -2.9642 283.2 3.531E-03

0.0320 -3.442 303.2 3.298E-03


74

0.0167 -4.0923 333.2 3.001E-03

H -17719 J/mol, b0=2.80110-5

Propane adsorption

b lnb T 1/T

0.4705 -0.7540 283.2 3.531E-03

0.2169 -1.5283 303.2 3.298E-03

0.0863 -2.4499 333.2 3.001E-03

H -26567 J/mol, b0=5.85710-6

So the isosteric heats of adsorption are 17.719

kJ/mol for ethane adsorption, and 26.567 kJ/mol

for propane adsorption.


75


temperatures by choosing option 9 (Langmuir) in

the program of iso.exe. This will give a unique qs

for different temperatures. Compare the isosteric

heat of adsorption (parameter Q in option 9) with

the results in Q3. Explaine why. Do this for both

ethane and propane

The extracted parameters by simultaneously

fitting the model to the isotherm data at three

temperatures are as follows:

ethane:

qs=5.555 (mmol/g) bo=0.2216E-05 (1/kPa)

Q = 23857 J/mol F=1.119 (mmol/g)2

propane:

qs=5.033 (mmol/g) bo=0.2783E-06 (1/kPa)

Q=34054 J/mol F=3.810 (mmol/g)2

It should be noted that the F value here is the

sum of error squares over three temperatures.

When concluding the goodness of fitting, this


76

value should be compared with the sum of F

values at three temperatures in Question 2, which

are 0.9661 for ethane, and 3.304 for propane.

From the F values we can see that the fitting here

are similar to that in Question 2.

By reducing the original van Hoff equation to

Eq. (3) to calculate the isosteric heat of

adsorption, we have made an implicit assumption

that the maximum adsorption capacity qs remains

unchanged for different temperatures. This is

true by using option 9 in the iso.exe program. If

the qs value changes with temperature, then this

factor should be taken into account in the

computation of isosteric heat of adsorption.

Because the goodness of fits in Q2 and Q4 are

similar, the results given in this part, with the

restriction of the same maximum adsorption

capacity for different temperatures, should be

more reliable.


temperatures by choosing option 11 (Uniform

energy distribution & local Langmuir isotherm)


77

in the program of iso.exe. Do this for both ethane

and propane. The fits should be much better than

Q4. Write down the parameters.

For ethane:

bo=0.361796E-05 (1/kPa) E min = 0.214

J/mol E max =27212 J/mol

qs=15.4025 mmol/g (10oC) qs=14.7583

mmol/g (30oC) qs=14.3822 mmol/g (60oC)

F=0.2237 (mmol/g)2

For propane:

bo=0.104557E-06 (1/kPa) E min =195.5

J/mol E max =44303.9 J/mol

qs=13.884 mmol/g (10oC) qs=14.076

mmol/g (30oC) qs=14.696 mmol/g (60oC)

F=0.05748 (mmol/g)2

Comparing the F values of Q5 with Q4, it is

quite clean that the Langmuir-Uniform

distribution isotherm is much better than the

Langmuir isotherm. This is so because here the

adsorbent energetic heterogeneity has been taken

into account.


78

6. The binary adsorption equilibrium of ethane

and propane on activated carbon at 30oC has been

experimentally determined as:

p (ethane):

kPa

p

(propane):

kPa

q (ethane):

mmol/g

q

(propane):

mmol/g

5 10 0.17 3.0

10 10 0.37 2.80

20 10 0.72 2.70

Use program iso-pre.exe to calculate the model

predictions and compare the results with the

experimental data. The models should be used

are: extended Langmuir, LRC (not in the

program), IAST-Langmuir, IAST-Unilan,

heterogeneous extended Langmuir (option 12),

IAST-Langmuir-uniform energy distribution

(option 18). Which model is the best? Comment

on the performance of various model predictions.


79

Since the isotherm parameters for the hybrid

Langmuir-Freundlich equation were not

available for any single component, iso.exe was

first used to extract these parameters, which are

shown below.

ethane: qs=10.7844 mmol/g b=0.03739

(1/kPa) 1/n=t=0.6028

F=0.0034 (mmol/g)2

propane: qs=7.2327 mmol/g b=0.2425

(1/kPa) 1/n=t=0.4725

F=0.016 (mmol/g)2

The calculated qmodel are listed in the following

table.

p kPa q mmol/g qmodel mmol/g

exp. data exp. data Extended Langmuir LRC IAST-Langmuir

E P E P ethane propane ethane propane ethane propane

5 10 0.17 3.0 0.2512 3.1980 0.5850 2.8630 0.2617 3.1879

10 10 0.37 0.28 0.4794 3.0513 0.8641 2.7846 0.5000 3.0324

20 10 0.72 2.7 0.8782 2.7950 1.2600 2.6735 0.9125 2.7618

F= (qmodel -qexp)2 0.1550 0.7278 0.1555

p Kpa q mmol/g qmodel mmol/g

exp. data exp. data IAST-unilan option 12 option 18

E P E P ethane propane ethane propane ethane propane

5 10 0.17 3.0 0.1813 2.9425 0.1906 2.981 0.1975 2.975

10 10 0.37 0.28 0.3546 2.8752 0.3701 2.891 0.3832 2.878

20 10 0.72 2.7 0.6789 2.6977 0.700 2.726 0.7237 2.703

F= (qmodel -qexp)2 0.0110 0.0101 0.0077


80

From the above table, it can be concluded that

the predictions by the IAST-Langmuir-uniform

energy distribution (option 18) are the best,

followed by the heterogeneous extended

Langmuir (option 12), IAST-Unilan, IAST

Langmuir or extended Langmuir. The derivation

between the experimental data and the predictions

from the LRC is the largest, which suggestion

that LRC is not a good prediction model for

multicomponent adsorption equilibrium, although

the Hybrid Langmuir-Freundlich equation can

well fit the single component isotherm data.

The predictions from the extended Langmuir and

the IAST-Langmuir are very close to each other.

This is physically expected since the maximum

adsorption capacity of ethane (5.2269 mmol/g) is

nearly the same as that of propane (4.9083

mmol/g). These two models become identical

when the maximum adsorption capacity is the

same for different components.

The IAST-Unilan equation gives better results

than IAST-Langmuir, because the Unilan

equation better fits the single component isotherm

data. Therefore, in using the IAST, it is


81

important to properly select the pure gas isotherm

equation.

Documents

Models or correlations for mixed-gas adsorption are ...kexhu.people.ust.hk/ceng521/521-3.pdf · Ideal Adsorbed Solution Theory (IAST) ... given set of gas concentrations p(k) (k=1,