Leipziger Symposium on dynamic sorption 2019 May 14 th 2019 Adsorption modelling as a tool to estimate transport properties Moises Bastos-Neto Universidade Federal do Ceará
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:
Thank you for your attention!