Leipziger Symposium on dynamic sorption 2019 May 14th 2019

Adsorption modelling as a tool to estimate

transport properties

Moises Bastos-NetoUniversidade Federal do Ceará

Advanced Sorbent Materials

on the Way to Application

Outline

Introduction – Concepts Aims and basics

Adsorption

Adsorbents

Column dynamics

Modelling – Theoretical Background Definitions and terminology

Momentum, Material and Energy Balances

Equilibrium theory

Adsorption kinetics

Assessing mass transfer Simple fit to breakthrough curves

From uptake curves

From calorimetry

Final remarks

Outline

Introduction – Concepts Aims and basics

Adsorption

Adsorbents

Column dynamics

Modelling – Theoretical Background Definitions and terminology

Momentum, Material and Energy Balances

Equilibrium theory

Adsorption kinetics

Assessing mass transfer Simple fit to breakthrough curves

From uptake curves

From calorimetry

Final remarks

Designing a process

0 200 400 600 800 10000.0

0.2

0.4

0.6

0.8

1.0

time / s

rela

tiv

e c

on

ce

ntr

ati

on

H2

N2

CO

CH4

CO2

feed gas

H2/N2/CO/

CH4/CO2

desorption

Tailgas

Column dynamics

What is your aim?

Synthesize better sorbents?

•Material

•Shape

•Properties

Design process units?

• Size

• Material

• Control

Optimize operations?

• Costs

• Maintenance

• Facilities

Understand the phenomena?

• Effects

• Simplifications

• Improvements

Necessary knowledge

• Which column size?

• Which flowrate?

• Is it reversible?

• How long do cycles last?

• Should columns be thermostated?

• Which operating conditions maximize purity, recovery

from the feed, and minimize energy /solvent

consumption?

etc

Basics

Conservation equations(mass, energy, momentum, electric charge)

Equilibrium laws at the interface(s)

Constitutive laws

Kinetic laws of heat/mass transfer and reaction

Initial and boundary conditions

Optimization criterion

Classification of systems

• Nature of equilibrium relationship• Linear isotherm• Favorable isotherm• Unfavorable isotherm

• Thermal effects• Isothermal• Near isothermal

• Concentration level• Trace systems• Nontrace systems

• Flow model• Plug flow• Dispersed flow

• Complexity of kinetic model• Negligible transfer resistance• Single transfer resistance• Multiple transfer resistance

Adsorbents

• Types

• Structures

• Homogeneous

• Porous

• Bidisperse

• Properties

• Adsorption capacity

• Selectivity

• Kinetics

• Stability

• Mechanical

• Thermal

• Chemical

Outline

Introduction – Concepts Aims and basics

Adsorption

Adsorbents

Column dynamics

Modelling – Theoretical Background Definitions and terminology

Momentum, Material and Energy Balances

Equilibrium theory

Adsorption kinetics

Assessing mass transfer Simple fit to breakthrough curves

From uptake curves

From calorimetry

Final remarks

An accurate process simulator is an important tool for learning, designing and optimization purposes.

Alírio E. Rodrigues

Definitions and terminology

• Concentration profiles – Ci(z) at a given t

• Concentration histories – Ci(t) at a given z

Alírio E. Rodrigues (2014)

Definitions and terminology

• Overall balance

Alírio E. Rodrigues (2014)

Definitions and terminology

• Concentration profile at t = tbt

Alírio E. Rodrigues (2014)

In general, one is interested in re-using the adsorbent for a relativelylarge numbers of cycles. Industrial sep processes alternate twosteps:

Adsorption: fluid phase is enriched with the weakly adsorbed species (raffinate)

Desorption: fluid phase is enriched with the strongly adsorbed components (extract) and the adsorbent is regenerated to be used in another cycle (by temperature, pressure, pH or concentration swings)

Adsorption Desorption

A + BA (+B)

B (+A) A (+B)

Adsorption-based processes

Breakthrough of mixtures

Modelling adsorption processes

Simulation Model

Equilibrium

Kinetics

Heat data

Hydrodynamics

Co-adsorption

Feed

Cycle time

Adsorberdimensions

Purity

Modelling adsorption processes

Modelling a fixed bed

Transport Phenomena

To model the dynamic behavior of an adsorption column is a problem far from trivial.

Momentum balance

2

3

2

23

17511150

p

g

p

g

ddz

p

.

Pressure drop in packed beds:

Blake-Kozeny equation Burke-Plummer equation

Laminar Flow Turbulent Flow

Material balance

0

tzstztzC

tT ,,, F

g

T

T CV

m

V

mC

z

CDC T

axT

F

z

CDC

g

axg

F

t

Qs

T

AP

V

mmQ

A

A

P

PP

A

A

A

P

V

m

V

m

V

m

V

mQ 11

infinitesimal cross sectional cut of the adsorbent column

Continuity – General Form:

Considering the interparticle volume:

Flux – Accounting convective and dispersive effects: or

Material “removal” rate: where Q is the amount of the species leaving the bulk phase in the control volume

Concentration inside the particle:

1TA VV PAP VV Defining the adsorbent (particle) volume: and the pore volume:

Then:

Material balance

01

2

2

t

q

t

C

z

CD

z

C

t

Cs

g

P

g

ax

gg

A

S

sV

mDefining the particle density:

qCm

mCQ sgP

S

AsgP

11Thus:

qCt

s sgP

1We get:

AV

A

dVtrqV

tq0

1,Where is defined as the average specific amount adsorbed:

Deriving to obtaint

Qs

q

01

qC

tz

CDCC

tsgP

g

axgg Substituting equations:

Three reasonable assumptions are very often made:

(i) bed porosity is homogeneous and constant along the bed

(ii) particle porosity is the same for every adsorbent particle and

(iii) the gas flows in only one dimension – axially

Energy Balance

0

tzstztzE

t,,, F

TcCTcV

mE

T

ggggg

T

g

g TcCTcV

mE

~~

sssss

T

s

s TcTcV

mE ˆˆ 1

infinitesimal cross sectional cut of the adsorbent column

Analogously to the Continuity:

Volumetric sensible heat in the control volume:

For the gas phase:

For the solid:

Temperature changes in the given control volume is represented by the temperature changes in the gas and in the

solid phases

Energy Balance

t

Tc

t

TcCEE

t

E s

ss

g

ggsg

ˆ~

1

t

TccC

t

E g

ssgg

ˆ~

1

sg TT

z

TTcCEE

g

gggdispconv

~F

2

2

z

T

z

TC

z

CT

zTCctz

gg

g

g

gggg

~,F

Summing up and differentiating:

Considering identical temperature profiles for the fluid and

solid phase in the adsorbent column operating at cyclic

steady state:

Then:

Heat is transported through the adsorbent bed along with the fluid flow and dispersed analogously to the mass.

The dispersion term can be simplified and evaluated by applying Fourier’s method of separation of variables. Thus,

the energy flux can be written as:

Applying the same assumptions as before:

(i) bed porosity is homogeneous and constant along the bed

(ii) particle porosity is the same for every adsorbent particle and

(iii) the gas flows in only one dimension – axially

Energy Balance

04

1

1

2

2

wg

i

ws

g

g

gggg

g

ssgg

TTd

h

t

qH

z

T

z

TC

zTCc

t

TccC

~ˆ~

011

42

2

t

qH

t

TccC

TTd

h

z

TC

zTCc

z

T

s

g

ssgg

wg

i

wg

gggg

g

ˆ~

~

TTUTTh

t

Tc wgwLwgww

www ˆ

wg

i

ws TT

d

h

t

qHs

41

Heat is generated in the system through adsorption and removed by conduction through the walls and later by

convection with the environment.

Substituting and arranging:

or

Additional equation - heat transfer from the wall to the environment:

0

1 ,

2

,

2

,

z

Cu

z

CD

t

q

t

C igig

axi

b

ig

Material Balance

accumulation dispersion convection

A.M. Ribeiro, Chem. Eng. Sci. 63 (2008)

D.M. Ruthven, Principles of Adsorption (1984)

D. Bathen and M. Breitbach, Adsorptionstechnik (2001)

Summarizing

Energy Balances

0

TTU

t

TCTTh wgwL

wpwwwgww

gas to wall

accumulation

wall to environment

accumulationgeneration convection transfer to wall

0411

2

2

wg

i

wg

PGg

g

PGgPSb

g

b TTd

h

z

Tcu

t

Tcc

z

T

t

qH

dispersion

• No gradients in the radial direction?

• Plug flow with axial mass dispersion?

• Mass transfer into the particle in accordance to the linear driving force

(LDF) model?

• Thermal equilibrium between the gas and the adsorbent?

• Adiabatic operation?

• Constant heat transfer coefficients?

• Homogeneous porosity along the bed?

• No pressure drop?

Assumptions and Simplifications

A.M. Ribeiro, Chem. Eng. Sci. 63 (2008)

D. Bathen and M. Breitbach, Adsorptionstechnik (2001)

Equilibrium law

At interfaces: igCfqi ,

*

02

*2

,

igdC

qd i02

*2

,

igdC

qd i 02

*2

,

igdC

qd i

favorable unfavorable linear

rectangular with an inflection

D.M. Ruthven, Principles of Adsorption (1984)

W. Kast, Adsorption aus der Gasphase (1987)

Influence of the equilibrium

pK

pKqq

1max

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

rela

tive

co

nce

ntr

atio

n

time / min

qmax0

qmax0

+ 20%

qmax0

- 20%

Max. Adsorption Capacity

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

rela

tive

co

nce

ntr

atio

n

time / min

K0

K0 + 20%

K0 - 20%

Adsorption Affinity

Primary influence

The shape and nature of the breakthrough curve are strongly influenced by the equilibrium

Influence of the equilibrium

Bastos-Neto et al., Chem. Ing. Tech., 83 (2011)

Increasing the partial pressure

Flow [Norm] P y Pi Partial Flow

mL/min bar % bar mL/min

100 10 10 1.0 10

150 15 10 1.5 10

200 20 10 2.0 10

300 30 10 3.0 10

0 20 40 60 80 100 1200.0

0.2

0.4

0.6

0.8

1.0 10 bar

15 bar

20 bar

30 bar

rela

tive

co

nce

ntr

atio

n

time / min

methane

AC CarbTech D 55/2 PSA

300 K

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.50.0

0.5

1.0

1.5

2.0

2.5

3.0

10°C

20°C

40°C

am

ount

adsorb

ed /

mm

ol.g

-1

pressure / bar

Methane

AC CarboTech D 55/2 PSA

Equilibrium theory

“Simple is beautiful (and useful)”

Isothermal operation

Equilibrium reached instantaneously in each point of the bed:

Plug flow

Negligible pressure drop

Negligible dispersion and mass transfer effects

01 *

,,

t

q

t

C

z

Cu iigig

i

01

,

2

,

2

,,

t

q

t

C

z

CD

z

Cu

t

Cs

ig

P

ig

ax

igiig

ii qq *

becomes

the material balance

Equilibrium theory

it results in

considering

0)('1

1,

,

,

t

CCf

z

Cu

ig

ig

ig

i

t

ig

z

ig

c

z

C

t

C

t

z

,

,

)( ,

*

igCfq

then

since

A.E. Rodrigues and D. Tondeur, Percolation Processes: Theory and Applications (1981)

D. DeVault, J. Am. Chem. Soc. 65 , 532 (1943)

De Vault’s Equation

The velocity of propagation of a concentration C, i.e. uc, is inversely proportional to the local slope of the isotherm f’(C)

Adsorption as a wave phenomenon

)('1

1 ,ig

i

c

c

Cf

u

t

zu

Equilibrium theory

unfavorable isotherms

As Ci the local slope of the isotherm f’(Ci) and uc

Concentration profiles Ci (z) at a given t

Higher concentrations travel at lower velocities

Dispersive Front

Equilibrium theory

favorable isotherms: “shock wave”

As Ci the local slope of the isotherm f’(Ci) and uc

Higher concentrations travel at higher velocities

Compressive (shock) Front

Concentration profiles Ci (z) at a given t

Equilibrium theory

favorable isotherms: “shock wave”

ig

i

i

c

sh

C

q

u

t

zu

,

11

st

csh

t

Lu

ig

icst

C

q

Q

Vt

,

11

“ideal” situation

real situation

Influence of the equilibrium

02

*2

,

igdC

qd i02

*2

,

igdC

qd i 02

*2

,

igdC

qd i

favorable unfavorable linear

CiCi

If non-idealities are present

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

Dax,1

< Dax,2

< Dax,3

Dax,3

Dax,2

time / min

rela

tive

co

nce

ntr

atio

n

Axial Dispersion Coefficient

Dax,1

Axial dispersion

If non-idealities are present

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

rela

tive

co

nce

ntr

atio

n

time / min

kLDF

kLDF

x 0.2

kLDF

x 5.0

Mass Transfer Coefficient

Mass transfer resistances

For favorable isotherms, the

concentration front disperses up

to a certain extent and assumes a

CONSTANT PATTERN BEHAVIOR

For unfavorable and linear

isotherms, the concentration front

disperses continuously as it moves

along the bed and hence follows a

PROPORTIONATE PATTERN

BEHAVIOR

If non-idealities are present

Axial dispersion and mass transfer resistances are to be considered

What about kinetics?

How to assess the phenomenon?

How relevant can it be to the process?

Intraparticle kinetics

0

'1

2

2

t

q

t

C

z

CD

z

C

t

Cs

g

P

g

ax

gg

Remembering the material balance

r

qr

rrD

t

qh

2

2

1

t

q

t

qs

'

Homogeneous

particle

Porous

particle

r

qr

rrD

t

q

r

Cep

p 2

2,

1

massadsorbent

amountadsorbedq '

volumeadsorbent

amountadsorbedq

Intraparticle kinetics

Adsorbed amount as a function of radius and time

qqkt

qLDF

*

AV

A

dVtrqV

tq0

,1

trfq ,

Averaging:

E. Glueckauf, Trans. Far. Soc., 51(11), (1955)

Linear Driving Force (LDF) approach:

q

0 RpRp

LDF model

The LDF model works in practice!

Adsorptive separation process models require several sets ofaveraging of local kinetic properties, which are often lost during aseries of integration processes.

The overall adsorption kinetics for a heterogeneous adsorbent canbe described by a heterogeneous-LDF model, even though thekinetics in each adsorption site is Fickian.

S. Sircar, Ind. Eng. Chem. Res., 45 (16), (2006)

Outline

Introduction – Concepts Aims and basics

Adsorption

Adsorbents

Column dynamics

Modelling – Theoretical Background Definitions and terminology

Momentum, Material and Energy Balances

Equilibrium theory

Adsorption kinetics

Assessing mass transfer Simple fit to breakthrough curves

From uptake curves

From calorimetry

Final remarks

Estimating kLDF

“Simple Fit” - Experiment vs Simulation

0.0 1.5 3.0 4.5 6.0 7.5 9.00.0

0.2

0.4

0.6

0.8

1.0

4 MPa

2 MPa

1 MPa

rela

tive

co

nce

ntr

atio

n (

C/C

0)

time / 103 s

KÖSTROLITH 5ABF

nitrogen

model

0.1 MPa

0.5 MPa

0

'1

2

2

t

q

t

C

z

CD

z

C

t

Cs

g

P

g

ax

gg

qqkt

qLDF

*

Estimating kLDF

From uptake curves

Convenience: during the measurement of equilibrium isotherms

• Continuous measurement of mass variation for each pressure step

• Mass and energy balances are used to estimate the mass transfer coefficient

• Relatively simple, but reliable

• Restricted to “non-instantaneous” adsorption

• LDF approach

From uptake curvesEquilibrium model

1/

,*

1/ 

( )

( )

1

i

i

n

m i i

i n

i

q b Pq

b P

Adsorption kinetics

, aveˆ ( ) ( )s p s s

T qm c m H h A T T

t t

Energy Balance

128H kJ mol

Clausis-Clapeyron Natural convection coefficient

70 ²h W m K

Estimating kLDF

qqkt

qLDF

*

LDF 2

Dk

r

From uptake curves

film and macropore resistances are negligible

Estimating kLDF

LDFkt

q

From uptake curves

CO2 and N2 uptake curves

Estimated coefficients correspond to the minimum value fitting to experiments

Estimating kLDF

R.M. Siqueira et al., Chem. Eng. Technol. 41(8), (2018)

From uptake curves

Adsorbent mass [kg] 0.155

Bed density 𝜌𝐿 [kg m-3

] ads colm V

Bed porosity 𝜀𝐿 [-] ˆ ˆ1 p s LV V

Particle density 𝜌𝑝 [kg m-3

] 1L L

Particle porosity 𝜀𝑝 [-] poˆ ˆ ˆ1 s sV V V

Heat transfer coefficient [W m² K] 100

Solid specific heat 𝑐 𝑝 ,𝑠 [J kg-1

K-1

] 820

Wall specific heat 𝑐 𝑝 ,𝑤 [J kg-1

K-1

] 477

Wall density 𝜌𝑤 [kg m-3] 786

Axial mass dispersion 𝐷𝑎𝑥 [m² s-1

] a)

2 i p

ax

u rD

Pe ;

1 0.70.5

Pe ReSc

Axial heat dispersion 𝑎𝑥

[W m-1

K-1

] (7 0.5 )ax

g

Pr Rek

1

Adsorbent mass [kg] 0.155

Bed density 𝜌𝐿 [kg m-3

] ads colm V

Bed porosity 𝜀𝐿 [-] ˆ ˆ1 p s LV V

Particle density 𝜌𝑝 [kg m-3

] 1L L

Particle porosity 𝜀𝑝 [-] poˆ ˆ ˆ1 s sV V V

Heat transfer coefficient [W m² K] 100

Solid specific heat 𝑐 𝑝 ,𝑠 [J kg-1

K-1

] 820

Wall specific heat 𝑐 𝑝 ,𝑤 [J kg-1

K-1

] 477

Wall density 𝜌𝑤 [kg m-3

] 786

Axial mass dispersion 𝐷𝑎𝑥 [m² s-1

] a)

2 i p

ax

u rD

Pe ;

1 0.70.5

Pe ReSc

Axial heat dispersion 𝑎𝑥

[W m-1

K-1

] (7 0.5 )ax

g

Pr Rek

1

Model parameters used for fixed bed simulations

Breakthrough curves experiments were measured to validate the simulation model

using the kLDF values estimated from gravimetric experiments.

Estimating kLDF

From uptake curves

Simulated results using the estimated coefficients showed good agreement with the

experimental breakthrough data.

Estimating kLDF

R.M. Siqueira et al., Chem. Eng. Technol. 41(8), (2018)

From uptake curves

Different coefficient values were used to evaluate the influence of the kLDF on the

relative concentration curve shape and the temperature history.

Estimating kLDF

R.M. Siqueira et al., Chem. Eng. Technol. 41(8), (2018)

From uptake curves

Temperature histories are important to cross-check the method to estimate kLDF

Estimating kLDF

R.M. Siqueira et al., Chem. Eng. Technol. 41(8), (2018)

Using adsorption-related heat effects and heat transport to estimate mass transfer

Estimating kLDF

Tian-Calvet microcalorimeter

Δℎ𝑎𝑑𝑠,𝑇,𝑛 =𝑑𝑄𝑟𝑒𝑣𝑑𝑛𝜎

𝑇,𝐴

+ 𝑉𝑑𝑑𝑝

𝑑𝑛𝜎𝑇,𝐴

Estimating kLDF

Discontinuous procedure:

,,

,rev

ads

T AT A

T ndQ dp

Vc hdn dn

Vcalorimetric cellnadsorbed

D.A.S. Maia et al., Chem. Eng. Res. Des. 136 (2018)

Estimating kLDF

Heat peaks

𝑄𝑎𝑑𝑠 =

𝑖=1

8

න ሶ𝑄𝑑𝑡

𝑄𝑑𝑒𝑠𝑜𝑟 = න ሶ𝑄𝑑𝑡

𝐸𝑟𝑒𝑔𝑇 = 𝑄𝑎𝑑𝑠 − 𝑄𝑑𝑒𝑠𝑜𝑟

Estimating kLDF

Modelling – Defining the system

3 defined parts (volumes):

• Dosing cell

• Dead volume

• Calorimetric cell

Mass and Energy balances for each part

ሶ𝑛𝑒

Vv, Cv

TvPvT

P

ሶ𝑛𝑠

calorimetric cell with

temperature control

dosing cell dead

volume

Vd, Td, P

Estimating kLDF

Modelling – Assumptions

• Ideal gas behavior

• The dosing cell and the dead volume are under isothermal and

non-adiabatic operation

• The pressure in the dead volume is the same of the calorimetric

cell

• Two approaches for mass transfer: Linear Driving Force e

Diffusion in a spherical particle

Estimating kLDF

Dosing cell

𝑃𝑣 = 𝐶𝑣𝑅𝑇𝑣

𝑑𝐶𝑣𝑑𝑡

= −ሶ𝑛𝑠𝑉𝑣

𝐶𝑣𝑐𝑝𝑑𝑇𝑣𝑑𝑡

=ℎ𝑣𝐴𝑣𝑉𝑣

𝑇01 − 𝑇𝑣

(EOS)

Vv, Cv

TvPv

ሶ𝑛𝑠

Estimating kLDF

Dead volume

𝑃𝑑 = 𝐶𝑑𝑅𝑇𝑑 (EOS)

𝑉𝑑𝑐𝑝𝐶𝑑𝑑𝑇𝑑𝑑𝑡

− 𝑉𝑑𝑑𝑃

𝑑𝑡= ሶ𝑛𝑠𝑐𝑝 𝑇𝑣 − 𝑇𝑑 − ℎ𝑐𝐴𝑑 𝑇𝑑 − 𝑇01

𝑃𝑑 = 𝑃

𝑑𝐶𝑑𝑑𝑡

𝑉𝑑 = ሶ𝑛𝑠 − ሶ𝑛𝑒

ሶ𝑛𝑠 ሶ𝑛𝑒

dead volume

Vd, Td, P

Estimating kLDF

Calorimetric cell

𝑃 = 𝐶𝑅𝑇 (EOS)

𝑑𝐶𝑐𝑑𝑡

𝑉𝑐 = ሶ𝑛𝑒 −𝑚𝑠

𝑑ത𝑞

𝑑𝑡

𝑉𝑐𝑐𝑝𝐶𝑑𝑇

𝑑𝑡− 𝑉𝑐

𝑑𝑃

𝑑𝑡+ 𝑚𝑠𝑐𝑝𝑠

𝑑𝑇

𝑑𝑡+ 𝑚𝑠

𝑑ത𝑞

𝑑𝑡−∆𝐻 +𝑚𝑠𝑐𝑝 ത𝑞

𝑑𝑇

𝑑𝑡= ሶ𝑛𝑒𝑐𝑝 𝑇𝑑 − 𝑇 − ℎ𝑐𝐴𝑐 𝑇 − 𝑇𝑐(𝑅1)

𝑃 = 𝑓(𝑡)

ሶ𝑛𝑒

T

P

calorimetric cell with temperature control

Measured – Needed for the solution

Estimating kLDF

Calorimetric cell wall – Energy Balance

R1 < R < R2:

𝜌𝑐1𝑐𝑐2𝑑𝑇𝑐𝑑𝑡

=1

𝑅𝐾𝑐1

𝜕

𝜕𝑅𝑅𝜕𝑇𝑐𝜕𝑅

R2 <R < R3:

𝜌𝑐2𝑐𝑐2𝑑𝑇𝑐𝑑𝑡

=1

𝑅𝐾𝑐2

𝜕

𝜕𝑅𝑅𝜕𝑇𝑐𝜕𝑅

Boundary conditions:

𝑇𝑐 𝑡, 𝑅3 = 𝑇02ℎ𝑐 𝑇 − 𝑇𝑐(𝑡, 𝑅1) = −𝐾𝑐1𝑑𝑇𝑐𝑑𝑡

(𝑡, 𝑅1)

Initial condition: 𝑇𝑐 0, 𝑅 = 𝑇02

Estimating kLDF

Two approaches

1. Linear Driving Force

𝑑ത𝑞

𝑑𝑡= 𝑘𝐿𝐷𝐹 𝑞∗ − ത𝑞

2. Diffusion

𝑑𝑞𝑝

𝑑𝑡= 𝐷𝑐

𝜕𝑞𝑝

𝜕𝑟+2

𝑟

𝜕𝑞𝑟𝜕𝑟

𝜕𝑞𝑝

𝜕𝑟𝑡, 0 = 0 𝑞𝑝 𝑡, 𝑟𝑝 = 𝑞𝐸(𝑃)𝜌𝑝

𝑞𝑝 0, 𝑟 = 𝑞𝐸(𝑃𝑖)𝜌𝑝

Boundary conditions:

Initial condition:

Estimating kLDF

Heat flux and total heat

The heat flux out of the cell is given by

The total heat is calculated as follows

𝑄1 = −𝐾𝑐1𝐴𝑐0𝜕𝑇𝑐𝜕𝑅

𝑡, 𝑅2

𝑄𝑡𝑜𝑡𝑎𝑙 = න0

∞

𝑄1𝑑𝑡

Estimating kLDF

Experimental procedure

• Heat of adsorption is determined prior to each run according to the equation for the total heat

• The kinetic parameters are then fitted 𝒌𝑳𝑫𝑭 or𝑫𝒄

𝑹𝒄𝟐

Resistance transitions

Relationship between and the mass transfer resistances (film, macro and micropores)

1

𝑘𝐿𝐷𝐹.𝑖=

𝑅𝑝𝑞0

3𝑘𝑓,𝑖𝐶0+

𝑞0𝑅𝑝2

15𝜀𝑝𝐷𝑝,𝑖𝐶0+

𝑅𝑐2

15𝐷𝑐,𝑖

Estimating kLDF

AC Norit RB4 – CO2 adsorption

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

16

18

20

22

24

26

28

30

32

34

en

thalp

y o

f a

dsorp

tio

n (

kJ/m

ol)

adsorbed amount (mmol/g)

Peak 3

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

ad

sorb

ed

am

ou

nt

(mm

ol/g

)

pressure (bar)

experimental

Langmuir fit 0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

he

at

rate

(m

Wa

tt)

time (s)

experimental

simulation

kLDF

=0.13 s-1

0 500 1000 1500 2000 2500 30000.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

he

at

rate

(m

Wa

tt)

time (s)

experimental

simulation

Dc/R

2 = 0.0084 s

-1

Estimating kLDF

AC Norit RB4 – CO2 adsorption

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8

16

18

20

22

24

26

28

30

32

34

en

thalp

y o

f a

dsorp

tio

n (

kJ/m

ol)

adsorbed amount (mmol/g)

Peak 5

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

ad

so

rbe

d a

mo

un

t (m

mo

l/g

)

pressure (bar)

experimental

Langmuir Fit

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

he

at

rate

(m

Wa

tt)

time (s)

experimental

simulation

Dc/R

2= 0.009 s

-1

0 500 1000 1500 2000 2500 30000.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

he

at

rate

(m

Wa

tt)

time (s)

experimental

simulation

kLDF

=0.13 s-1

R.M. Siqueira et al., Chem. Eng. Technol. 41(8), (2018)

Estimating kLDF

AC Norit RB4 – Comparing with the uptake measurements

Sample Method kLDF (s-1)

AC Norit RB4Uptake 0.1

Calorimetry 0.13

-0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1000 2000 3000 4000

KLD

F.Q

1

Time

KLDF.Q1 KLDF.Q1 KLDF.Q1

DcR2 = 0,0041 1/s

DcR2 = 0,0041/5 1/s

DcR2 = 0,0041/10 1/s

Estimating kLDF

AC Norit RB4 – Sensibility of the method

P. A. S. Moura et al., Adsorption 24, (2018)

Estimating kLDF

ACPX series – kLDF x PSD

Sample kLDF (1/s) Dc/R2 (1/s) Ratio

ACPX 22 0.075 0.004 19.7

ACPX 41 0.120 0.009 13.3

ACPX 76 0.136 0.009 14.9

J. A. C. Silva, K. Schumann, A. E. Rodrigues, Microporus Mesoporous Materials 158, (2012)

Estimating kLDF

Zeolite 13X

https://www.explainthatstuff.com/zeolites.html

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Hea

t R

ate

(m

Wa

tt)

Time (s)

Experiment

Simulation

KLDF

= 0.022 s-1

0 250 500 750 1000 1250 1500 1750 2000 2250 2500

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Hea

t R

ate

(m

Wa

tt)

Time (s)

Experiment

Simulation

Dc/R

2 = 0.0013 s

-1

Sample Method Dc/R2

binderless

13X zeolite

ZLC at 313 K 0.0009-0.0012

Calorimetry at 298 K 0.0013

Outline

Introduction – Concepts Aims and basics

Adsorption

Adsorbents

Column dynamics

Modelling – Theoretical Background Definitions and terminology

Momentum, Material and Energy Balances

Equilibrium theory

Adsorption kinetics

Assessing mass transfer Simple fit to breakthrough curves

From uptake curves

From calorimetry

Final remarks

IUPAC classification for condensable

vapors (e.g. N2 at 77 K)

PX isotherm

MIL-53 (Al)

Remy et al., 2011

CO2 isotherms

YO-MOF

Mulfort et al., 2010

What about breakthrough curves of systems with non-conventional adsorption isotherms?

Final remarks

Final remarks

A zoo of breakthrough curves

Final remarks

Xylenes adsorption in MOFs

Final remarks

MIL-47 (V) – Materiaux de l’Institute Lavoisier

Octahedral metallic clusters VO4(OH)2 connected by terephthalic acid linkers

Octahedral metalic cluster AlO4(OH)2 connected by terephthalic acid linkers

Final remarks

MIL-53 (Al) – Flexible MOF: “Breathing effect”

Finsy et al., Chem. Eur. J., 15, 7724 – 7731, (2009)

Final remarks

CS 1: OX/EB breakthrough curves in MIL-53

From Rietveld refinement of in-situ DRX of OX adsorption in MIL-53 (Al), it was found that…

Final remarks

ISOTHERM EQUATIONS

Final remarks

Fixed bed model equations (LDF)

Final remarks

Final remarks

CS2: MIL-47 (V) – a rigid MOF

• No breathing

• Similar isotherms, although PX shows an inflexion point at 10-3 and 10-2 bar

Final remarks

PX and MX isotherms in MIL-47 at 343 K

Same trick applied to provide a mathematical description of PX isotherm

Final remarks

Finsy et al., JACS (2008) 130, 7110

Remy et al., Langmuir (2011) 27, 13064

Final remarks

Finsy et al., JACS (2008) 130, 7110

Remy et al., Langmuir (2011) 27, 13064

Final remarks

Final remarks

Coming back to the equilibrium theory...

• Classical concepts such as phase equilibrium and transport phenomena have been revisited and applied to the description of adsorption dynamics in a fixed bed

• The correct analysis of batch adsorption data should provide scalable (design) parameters that will be useful not only for process design and optimization, but also to plan experiments in fixed bed in lab scale

Final remarks

We hope this is a small brick in bridging the gap between different approaches of adsorption scientists from more fundamental and more applied backgrounds

Final remarks

Post-Doc Fellows

Diana Azevedo Eurico TorresCélio Cavalcante Jr. Enrique Vilarrasa-GarciaMoises Bastos-Neto

Our team

D Sc

M Sc

Faculty

Ba

Laboratório de Pesquisa em Adsorção e Captura de CO2

(Laboratory of Adsorption Research and CO2 Capture)

Universidade Federal do Ceara

Campus do Pici – bloco 731

Fortaleza - Brazil

email:

mbn@ufc.br

Thank you for your attention!