CBME 5210 Advanced Separation Processeskexhu.people.ust.hk/ceng521/521-1.pdf · 2019-02-08 · constant follows the vant Hoff equation: 2 0 2 ln ' 0; ln RT U dT d K RT H dT d K (3)

Professor Xijun Hu CENG 5210 Advanced Separation Processes

1

CBME 5210 Advanced Separation Processes

Instructor: Prof. Xijun Hu (Room 4559; Tel: 2358 7134;

Email: kexhu; web: http://kexhu.people.ust.hk)

TA: Mr. Xuanhao Mei, Lab 7104, email:

[email protected]

Aims:

This subject is to enable students to understand the principles and

processes of adsorption, chromatography and membrane

separation; to design an adsorber, a membrane unit to achieve a

specified separation; to optimize a chromatographic system.

Textbook:

D.D. Do, “Adsorption Analysis: Equilibria and Kinetics”,

Imperial College Press, 1998.

References: R.T. Yang, “Gas Separation by Adsorption Processes”, Butterworths,

Boston, 1987.

D.M. Ruthven, “Principles of Adsorption and Adsorption

Processes”, John Wiley & Sons, New York, 1984. S.D. Faust and O.M. Aly, “Adsorption Processes for Water

Treatment”, Butterworths, Boston, 1987.

R.L. Grob, “Modern Practice of Gas Chromatography”, 3rd ed., John

Wiley & Sons, Inc., New York, 1995.

C.J. Geankoplis, “Transport Processes and Unit Operations”, 3rd ed.,

Prentice Hall, Englewood Cliffs, New Jersey, 1993.

Assessments:

Assignments: 20% Final Exam: 80%

http://kexhu.people.ust.hk/

mailto:[email protected]


2

What to do if you have questions/problems:

Email me: [email protected]

Visit me at my office: Room 4559

Contact the TA

Encourages:

Ask & answer questions in the class

Discuss the course materials & homework after class

Preview the course materials before the class

Disciplines:

Turn off all mobile phones in the class

No talks between students in the class

Do not copy other’s homework (both people copied &

being copied will be penalized)

Do not cheat in the examinations

mailto:[email protected]


3

Lecture Outlines Week Lecture Content

2 Introduction Adsorption processes - why?

how?

Practical adsorbents;

Forces of adsorption

2,3 Adsorption

Equilibrium

(single

component)

Ideal Langmuir and BET

models;

Gibbs adsorption isotherm and

related models;

Dubinin - Polanyi theory

4 Practical

Approaches of

Pure Component

Adsorption

equilibrium

Energy distribution;

Pore size distribution

5 Adsorption

Equilibrium

(multi-

component)

Extended multicomponent

Langmuir;

Ideal adsorbed solution theory

(IAST)

6 Adsorption

kinetics

- fundamentals

Continuum diffusion;

Knudsen diffusion;

Surface diffusion

7 Adsorption

Kinetics

in a single

particle

Resistances to mass transfer in

practical systems;

Principles of diffusion in

porous media;

Uptake rate in batch systems


4

Experimental measurement of

intraparticle diffusivities

8 Adsorption

dynamics:

bed profiles and

breakthrough

curves

Equilibrium theory: analogy

with kinematic wave equation

Breakthrough curve;

Bohart-Adams model (Bed

Depth-Service Time, BDST)

9 Cyclic Gas

Separation

Processes by

Adsorption

Regeneration methods;

Thermal swing, pressure swing

and displacement systems;

Skarstrom Cycle

10 Membrane

Processes

Introduction, classification of

membrane processes;

Dialysis, gas permeation;

Flow patterns

11 Membrane

Processes

Reverse osmosis, ultrafiltration

12 Chromatographic

Separation

Basic principles

Optimizing separations in gas

chromatography

13 Final

Examination

7 May


5

Adsorption

In adsorption processes one or more components of a gas

or liquid stream (adsorbate or solute) are adsorbed on the

surface of a solid (adsorbent) and a separation is

accomplished.

Adsorption vs Distillation

Distillation has advantages of simplicity and scalability

so it is a standard against which other processes are

measured. However, distillation is an energy-intensive

process. The ease of separation by distillation is

determined by the relative volatility, which for an ideal

binary mixture is simply the ratio between the vapor

pressures. If the relative volatility becomes close to

unity, the number of stages and reflux ratio required to

achieve the specified separation will be much higher, and

so the costs of equipment & energy.


6

The analogous parameter for adsorption is the separation

factor defined by:

iji i

j j

X Y

X Y

/

/

where Xi and Yi are the equilibrium mole fractions of

component i in the adsorbed phase and fluid phase,

respectively.

The analogy is purely formal, there is NO quantitative

relationship between the separation factor and relative

volatility. For two given components the relative

volatility is fixed whereas the separation factor varies

widely depending on the adsorbent.

When will adsorption be favored?

Relative volatility is less than 1.5, such as the

separation of isomers, where the separation factor

for adsorption is infinity by using zeolites.

The bulk of feed is a low-value, more volatile

component, the product is in low concentration so

a large reflux ratio is required.

The two groups of components have overlapping

boiling ranges.

A low temperature and a high pressure are required

for liquefaction.

For small to medium throughput (<50,000 m3/h).


7

Categorizations of adsorptive separation processes

Based on method of adsorbent regeneration

- TSA (Temperature Swing Adsorption):

regenerated by heating

- PSA (Pressure Swing Adsorption):

regenerated by lowering the pressure

Based on feed composition

- Bulk separation: 10 weight percent or more of

the mixture is adsorbed.

- Purification: almost all impurities, which are

usually less than 10%, are adsorbed, pure gas at

purities higher than 99.999% can be obtained.

Based on mechanism of separation

- Steric effect: by using molecular sieve, only

small and properly shaped molecules can

diffuse into the adsorbent, other molecules are

totally excluded.

- Kinetic effect: by the differences in diffusion

rates of different molecules.

- Equilibrium effect: by the equilibrium

adsorption of the mixture.


8

Representative commercial gas adsorption

separation

Separation Adsorbent I. Gas bulk separation

Normal paraffins/iso-paraffins,

aromatics

N2/ O2

O2/ N2

CO, CH4, CO2, N2, A, NH3/H2

Acetone/vent streams

C2H4/vent streams

Zeolite

Zeolite

Carbon molecular sieve

Zeolite, activated carbon

Activated carbon

Activated carbon

II. Gas purification

H2O/olefin-containing cracked

gas, natural gas, air, synthesis

gas, etc.

CO2/C2H4, natural gas, etc.

Organics/vent streams

Sulfur compounds/natural gas,

hydrogen, liquefied petroleum

gas (LPG), etc.

Solvents/air

Odors/air

NOx/N2

SO2/vent streams

Hg/chlor-alkali cell gas effluent

Silica, alumina, zeolite

Zeolite

Activated carbon, others

Zeolite

Activated carbon

Activated carbon

Zeolite

Zeolite

Zeolite


9

Adsorbents

Good adsorbents should have

high surface area or micropore volume (for large

adsorption capacity)

large pore network for the transport of molecules to

the interior (for fast kinetics)

The porous solid must have small pore size (micropore)

with a reasonable porosity to satisfy the first

requirement and have a network of large pore size

(macropore) for the second requirement.

Micropore: d < 2 nm

Mesopore: 2 < d < 50 nm

Macropore: d > 50 nm


10

Adsorbent Manufacturing

method

Surface

area

(m2/g)

Pore

diamete

r (Ao

)

Usage

Activated

carbon

by thermal

decomposition

of coal, wood,

vegetable shells,

etc.

300-

2000

10-60 Removal of

organic

vapors;

H2

purification

Silica gel by acid

treatment of

sodium silicate

solution & then

dried

340-

800

20-140 dehydrate

gases &

liquids;

fractionate

hydrocarbon

Activated

alumina

hydrated

aluminum oxide

is activated by

heating to drive

off the water

200-

500

20-140 drying;

gas chromatograph

y

Molecular

sieve

zeolites

porous

crystalline

aluminosilicate

containing

precisely

uniform pores

300-

1200

3-10 drying,

separations

based on

molecular

size &

shape


11

Properties to characterize adsorbents:

Specific pore volume (cm3/g).

Specific surface area (m2/g).

Particle density(g/cm-3).

Average pore diameter (Ao ).

Pore size distribution


12

Physical Adsorption Chemisorption

Low heat of adsorption

(<2 or 3 times latent heat

of evaporation)

High heat of adsorption

(>2 or 3 times latent heat

of evaporation)

Non specific Highly specific

Monolayer or multilayer.

No dissociation of

adsorbed species.

Only significant at

relatively low

temperatures.

Monolayer only

May involve dissociation.

Possible over a wide range

of temperature.

Rapid, non-activated,

reversible.

No electron transfer

although polarization of

sorbate may occur.

Activated, may be slow

and irreversible.

Electron transfer leading to

bond formation between

sorbate and surface.


13

Forces and Energies of Adsorption

1. Van der Waals forces (Dispersion-Repulsion).

This is always present. By combining the attraction

potential and the repulsion between two isolated

molecules, we have the Lennard-Jones potential

function:

4

12 6

r r

which is sketched in the following figure. The force

constants and are characteristics of the particular

molecular species and available in the literature.

2. Electrostatic interactions (polarization, dipole, and

quadrupole).

This is significant only for adsorbents having ionic

structure.


14

Entropy changes

Physical adsorption from the gas phase is

exothermic, why? Let us look at the thermodynamic

argument.

Since the adsorbed molecules is more regular,

so the disorder degree is lower

entropy is lower 0 gasads SSS

For the adsorption to happen, the free energy

change, G, should be negative, hence

0 STHG

H < 0

adsorption is exothermic


15

Adsorption equilibrium isotherm

(Single component)

This is the equilibrium relationship between the

concentration of a solute in the fluid phase and its

concentration on the solid phase at a given

temperature. Most isotherms can be classified

into five types, as shown below.


16

The numerous isotherm models are based on three

different approaches:

1. The Langmuir approach: This was given by

Langmuir in 1918, assuming the adsorption system is

in dynamic equilibrium, where the rate of evaporation

is equal to that of condensation. It is the most useful

for data correlation in separation processes.

2. The Gibbs approach: This one employs the Gibbs

adsorption isotherm:

0 dnAd s (1)

where is the spreading pressure, A is the surface

area, ns is the number of moles, and is the chemical

potential. An integration of the Gibbs equation results

in the desired isotherm. By assuming one equation of

state (like the ideal gas law), a corresponding isotherm

can be obtained.

3. The potential theory: First formalized by Polanyi in

1914, the adsorption system is viewed as a gradual

concentration of gas molecules toward the solid

surface due to a potential field.


17

Adsorption at low coverage:

Linear isotherm (Henry's law)

q Kc or q = K' p (2)

where q is the adsorbed concentration, c is the

solute concentration and p is the gas pressure. K

is called the Henry constant. This linear isotherm

is not common, but many systems follow this

relationship in the dilute region.

From the ideal gas law, K=K’RT.

The temperature dependence of the Henry

constant follows the vant Hoff equation:

20

20 ln

;'ln

RT

U

dT

Kd

RT

H

dT

Kd

(3)

where H0 and U0 are the changes in enthalpy

and internal energy during adsorption.

Irreversible (rectangular) isotherm

The adsorbed amount is independent of the solute

concentration (q = constant). It happens when the

adsorption affinity is extremely high (large

molecules).


18

Isotherms based on the Langmuir approach

The Langmuir isotherm is the simplest and still

the most useful equilibrium equation for both

chemical and physical adsorption. It is based on

the following assumptions:

1. Adsorption is at fixed number of definite,

localized sites.

2. Each site can hold only one adsorbate molecule

(monolayer).

3. All sites are equivalent.

4. No interaction between adsorbed molecules,

even on adjacent sites.


19

The Langmuir equation can be derived by the

kinetics of condensation and evaporation of gas

molecules at a unit solid surface. If is the

fraction of the solid surface covered by a

monolayer of adsorbate, then the rate of

evaporation from the surface (desorption) is

proportional to (i.e., kd). Similarly, the rate of

condensation of gas molecules onto the surface

(adsorption) is proportional to the fraction of free

sites remaining, (1- ), and the absolute gas

pressure, p, (i.e., kap(1- )). Equilibrium is

established when these two rates are equal.

kap(1-)=kd (4)

where ka and kd are the rate constants for

condensation (adsorption) and evaporation

(desorption). By rearranging Eq. (4), the fraction

of surface covered, , is obtained as:

bP

bp

Pkk

pk

ad

a

1 (5)


20

The parameter, b=ka/kd, is called affinity constant

& related to the heat of adsorption (Q) by

)/exp(0 RTQbb (6)

where b0 is a constant, R is the gas constant and T

the temperature in Kelvin. Since adsorption is

exothermic (Q > 0) b should decrease with

increasing temperature. The higher value of b, the

stronger the adsorption.

Eq. (6) can be used to calculate the heat of

adsorption. From the assumptions of identical

sites and no interaction between adsorbed

molecules, the heat of adsorption should be

independent of coverage ().

In general, equation (5) can be written as:

q qbp

bpq q

bc

bcs s

1 1

, or (7)

where q is the adsorbed phase concentration, qs is

the maximum (saturation) adsorption capacity, so


21

=q/qs. The parameters qs and b are obtained by

fitting the Langmuir model to experimental data.

The Henry constant is

obtained by taking the

limit p 0:

sp

bqp

qK

lim

0' (8)

By rearrangement of

equation (7) we get

pbqqq ss

1111 (9)

Therefore if we plot

1/q vs. 1/p, a linear

line should be

obtained with a slope

of 1

bqs

, an intersect on y-axis of 1

qs

, and an

intersect on x-axis of -b.

q

p

low T

high T

1/p

1/q

-b

1/q slope=1

bqs s

0


22

Langmuir isotherm on nonuniform surfaces: Because Langmuir isotherm doesn’t well fit the data

for the whole pressure region, other more rigorous

models have been developed.

Freundlich isotherm If the adsorption sites are not identical, the total

adsorbed amount is summed over all types of sites.

When an exponentially decaying energy distribution is

assumed, the Freundlich isotherm is obtained, which

was originally an empirical equation:

mn KpqorKpq /1 (10)

where K and n (m) are constants determined

experimentally. The isotherm is favorable when n<1,

linear if n=1, and unfavorable if n>1.

q

p

n=1 n<1

n>1

0

lnp

lnq

lnK

0

slope=n


23

Taking logarithm on both sides of equation (10) to give

Kpnq lnlnln (11)

So by plotting lnq vs. lnp we should obtain a linear

line with the slope being the exponential constant n,

and the intersect on y-axis being lnK.

Sips (Langmuir-Freundlich) isotherm

This is the hybrid Langmuir-Freundlich isotherm, it

takes into account the fragmentation of the

molecule so one molecule occupies 1/t sites.

t

t

sbp

bpqq

1 (12)

It is a three parameter isotherm (qs, b, t).

Unilan isotherm (UNIform energy distribution

& local LANgmuir equation)

qq

s

be p

be p

ss

s

2

1

1ln (13)

Three parameters are qs, b, s.

Toth isotherm

q q

b p

bps

t

t t

1

11

/

/ (14)

Three parameters are qs, b, t.


24

Multilayer adsorption:

BET (Brunauer-Emmett-Teller) isotherm

smonomonos p

p

cq

c

cqppq

p 11

)( (15)

where qmono is the monolayer saturation

concentration and ps is the saturation vapor

pressure. By using experimental data in the range

of p/ps = 0.05 to 0.35, the left-hand side of

equation (15) is plotted against the relative

pressure (p/ps), the values of qmono and k can be

obtained from the slope and intercept of the linear

line. Knowing the molecular area (eg., 16.2

Ao

2

/molecule for nitrogen at 77K), the value of the

surface area can be calculated directly from qmono.

0 p/p

p

q(p -p)

slope=mono

s

s

q1 c-1

monoqc c


25

Surface areas from the BET & Langmuir

equations

When qmono is obtained, we can calculate the

specific surface area of the adsorbent by the

following equation:

Sq g g solid)N A

M g molt

mono

w

(m / g solid)2 ( /

( / )

0 (16)

where Mw is the molecular weight, N0 is

Avogadro’s number (6.023x1023/mol), and A is

the surface area of one molecule, which is

16.2x10-20m2, for N2 at 77K.

The BET isotherm is only valid in the range of

p/ps = 0.05 to 0.35, beyond this region, BET is not

a good model because of the capillary

condensation (p/ps > 0.3) or the system fails to

form multilayer adsorption (p/ps < 0.05).

There is also a Langmuir surface area calculated

by substituting qmono with the Langmuir maximum

adsorption capacity parameter qs.


26

Isotherms based on the Gibbs approach

From thermodynamics, we have the Gibbs-Duhem

equation at equilibrium:

0 ndAdSdT (17)

where A is surface area, is spreading pressure,

and n is the number of moles of adsorbate per unit

volume of adsorbent.

The spreading pressure is a measure of the

tendency of a liquid phase

to spread (complete wetting) on a second, liquid

or solid phase. The spreading pressure S is the

difference between the work of

adhesion W1,2 between the phases and the work of

cohesion W1,1 of the phase under consideration:

S = W1,2 - W1,1

Equally, the spreading pressure can be expressed

as the difference between the surface

tensions σ1and σ2 and the interfacial tension σ1,2:

S = σ2 - σ1 - σ1,2

If the spreading pressure is positive, the phase

under consideration spreads; if it is negative,

wetting is not complete.

https://www.kruss-scientific.com/services/education-theory/glossary/spreading/

https://www.kruss-scientific.com/services/education-theory/glossary/wetting/

https://www.kruss-scientific.com/services/education-theory/glossary/work-of-adhesion/

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https://www.kruss-scientific.com/services/education-theory/glossary/work-of-cohesion/

https://www.kruss-scientific.com/services/education-theory/glossary/work-of-cohesion/

https://www.kruss-scientific.com/services/education-theory/glossary/surface-tension/

https://www.kruss-scientific.com/services/education-theory/glossary/surface-tension/

https://www.kruss-scientific.com/services/education-theory/glossary/interfacial-tension/


27

At constant temperature, Eq. (17) yields the Gibbs

adsorption isotherm:

constant)=(T 0 ndAd (18)

At equilibrium, the chemical potential of the

adsorbed phase is equal to that of the gas phase:

pRTgas ln0 (19)

Tconstant at )(ln pRTdd (20)

RTA

n

pd

d

T

ln

(21)

This is the relationship between the adsorption

isotherm and the corresponding equation of state.

: Joule/m5; A: m2; n: mol/m3;

R: (Joule)/(mol K)

By integrating Eq. (21), the spreading pressure, ,

is found:

p dpp

q

A

RT0 (22)

Henry’s law:

For the ideal gas law, pV= nRT, so A= ns RT.

RT

A

n

pd

d s

Tln (23)

pKpdd

' );(ln

(24)


28

RT

AKKp

RT

pAK

RT

Anq s

'=K ;

'

(25)

which is the Henry’s law.

Volmer isotherm:

If the equation of state is taken as:

nRTA (26)

where is the area occupied by n molecules. This

is analogous to P(V-b)=nRT:

2

A

nRT

d

d

T

(27)

From the Gibbs isotherm [Eq. (21)], we have

2

A

Ad

p

dp (28)

22

)()(])[(

A

Ad

A

Ad

A

dA

p

dp (29)

Integrate to get

AAp )ln(ln or

AAp )](ln[

AAp exp)( (30)


29

The fractional loading, , is simply the area

occupied by n molecules divided by the total area:

A

(31)

Eq. (30) becomes

1exp

)1(p (32)

Let b=, we have

1exp

1bp (33)

This is the Volmer isotherm to describe the

adsorption on surfaces where the mobility of

adsorbed molecules is allowed, but no interaction

among the adsorbed molecules.


30

Isotherms based on the potential theory

Dubinin-Polanyi theory:

For the adsorbed fluid under the pressure of

adsorption forces, the free energy change is

G RTf

fRT

p

p

s

s

ln ln (34)

is a function of the volume of adsorbed fluid

(W). It is independent of temperature for

dispersion forces:

T W

0 (35)

W W 02exp (36)

where W0 is the specific micropore volume.

The rearrangement of Eq. (36) gives the Dubinin-

Radushkevich equation:

q q Dp

ps

s

exp ln

2

(37)

where qs, D are the model parameters, and ps is

the saturation vapor pressure which can be found

from literature or also as a parameter to be

optimized (extracted) from experimental data.


31

Heat of adsorption

The isosteric heat of adsorption (-H) is

determined from the Clausius-Clapeyron equation

applied to adsorption isotherm:

2

ln

RT

H

T

p

q

(38)

Integrate assuming H independent of T,

tconsRT

Hp tanln

(39)

A plot of lnp vs 1/T at constant q gives a linear

line with a slope of H/R.

For the Langmuir isotherm, we have

s

s

q

bp

dT

dq

RT

Q

RT

H

122

(40)

where we allow the maximum adsorbed capacity,

(qs) to vary with temperature.

s

s

q

bp

dT

dqRTQH

12

(41)

However, the isosteric heat of adsorption (-H) is

constant for the Langmuir equation if the

saturation capacity is independent of temperature,

but changes with the surface loading for other

isotherms.


32

Measurement of adsorption isotherm

(Batch adsorption)

The adsorption isotherm is often obtained by

using a batch reactor. In liquid adsorption, this is

conducted in a stirred tank. A certain volume of

liquid (V) with known concentration (c0) is fed

into the tank, then a known mass of adsorbent

particles (m) is added to the solution. After

sometime (usually overnight to a few days,

depending on the particle size) the system

becomes steady state. The final concentrations in

the liquid phase (cf) and in the solid phase (qf) are

in equilibrium. By taking a material balance we

have

q m c V q m c Vf f 0 0 (42)

So qf can be calculated as

qq m c V c V

mf

f 0 0 (43)

Now we get one isotherm point (cf, qf), more

isotherm data can be obtained by changing

relative amount of solution and solid, or the initial

liquid concentration and repeat the above

procedure.


33

Example: A wastewater solution having a volume

of 1.0 m3 contains 0.21 kg phenol/ m3 of solution

(0.21 g/L). A total of 1.40 kg of fresh granular

activated carbon is added to the solution, which is

then mixed thoroughly to reach equilibrium.

Using the isotherm in the figure below, what are

the final equilibrium values, and what percent of

phenol is extracted?


34

Solution 1: The given values are c0=0.21 kg

phenol/ m3, V=1.0 m3, m=1.40 kg carbon, and q0

is assumed zero. Substituting into eq. (11)

qf(1.40)+ cf(1.0)=0(1.40)+0.21(1.0) This is a linear line, which is plotted in the figure

below together with the isotherm. The equilibrium

values are obtained from the intersection:

cf=0.062 kg phenol/ m3

qf=0.106 kg phenol/kg carbon

The percent of phenol extracted is

% extracted

c c

cf0

0

1000 210 0 062

0 210100 70 5( )

. .

.( ) .


35

Solution 2: From the mass balance equation (eq.

20), we have

qf(1.40)+ cf(1.0)=0(1.40)+0.21(1.0)

1.4qf+ cf=0.21

Fitting the Freundlich equation to the isotherm

data, we obtain

qf = 0.194cf1/4.4848

Solving the mass balance and isotherm equations

together, we get the solution.

cf=0.062 kg phenol/ m3

qf=0.106 kg phenol/kg carbon

The percent of phenol extracted is

% extracted

c c

cf0

0

1000 210 0 062

0 210100 70 5( )

. .

.( ) .


36

Volumetric measurement rig for gas

adsorption isotherm

It has two separate chambers. One has a supply

bomb and the other contains an adsorption cell.

Each chamber has a transducer & a thermocouple

to record the pressure & temperature. The volume

of each chamber is also known.


37

Before an equilibrium experiment starts, a

molecular drag pump is used to evacuate the

system to 10-5 mmHg. The operation procedure is

described below.

1. Activated carbon was crushed to about 0.1

mm and dried in an oven at 200oC for three hours

to remove excess moisture. The carbon was then

weighted and loaded into the adsorption cell.

2. The whole system was first evacuated to

vacuum using an Alcatel molecular drag pump and

the activated carbon particles in the adsorption cell

were heated up to 300oC and kept at this

temperature and under vacuum for overnight to

degas any possible adsorbed species.

3. Pure gas was then dosed into the supply

bomb and the pressure and temperature were

recorded when constant readings were reached.


38

4. A small amount of pure gas was dosed

from the supply bomb into the adsorption cell

which was kept isothermal by using a water bath.

After the pressure in the adsorption cell becomes

constant (typically about two hours), the pressures

and temperatures in both supply bomb and

adsorption cell were recorded.

5. The amount adsorbed which was in

equilibrium with the pressure in the gas phase of

the cell was calculated from the amount supplied

from the supply bomb via the ideal P-V-T

relationship.

6. Steps 4 and 5 were repeated to get the

equilibrium isotherm data at higher pressures until

a full isotherm curve was obtained.

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CBME 5210 Advanced Separation Processeskexhu.people.ust.hk/ceng521/521-1.pdf · 2019-02-08 · constant follows the vant Hoff equation: 2 0 2 ln ' 0; ln RT U dT d K RT H dT d K (3)