adsorption modelling
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ADSORPTION - MODELING AND SIMULATION
Based on the operation mode
Liquids
Gases
Volumetric Gravimetric
Dynamic Fixed bed adsorption: occurs in an open system where adsorbate solution continuously passes through a column packed with adsorbentPulse chromatographic column: measures the response of a chromatographic column to a pulse in adsorbate concentration for determination of isotherms
Types of Adsorption
Static/BatchA closed system containing a desired amount of adsorbent contacted with a certain volume of adsorbate solution
Inside the particle/ Intraparticle
Intraparticle diffusionis more complex and diverse,is the keystone of modeling dynamic adsorption
Mechanisms involved in intraparticle transfer simultaneously:Pore diffusion (Macropore / or Micropore)Surface diffusionAdsorption
Set of equations are used to consider all the possible or required mechanisms
Particle Modeling
Rp
r
drAccumulation = Inlet – Outlet
rc
rrc
Dtc pp
pp 2
2
2
cp = conc inside particleDp = Diffusion coefficientɛp = Particle porosity
Without Adsorption
rc
rr
Drt
c pp
p 22
1
With Adsorption
Accumulation + Adsorption = Inlet – Outlet
rc
rrc
Dtq
tc pp
ppppp
p21 2
2
Batch Adsorption
Adsorption Isotherm Diffusion Coefficients
Equilibrium Kinetic
Linear Isotherm
Langmuir Isotherm
Freundlich Isotherm
Kcq
KcKcqq m
1
nKcq1
Batch Adsorption ModelingMass balance of bulk fluid
prr
ppp
p rc
Drt
c
31
Boundary Condition (t>0)
00
r
p
rc
)(tccprr
p
Initial Condition (t<0)
0rp
c
Linear Isotherm
Langmuir Isotherm
Freundlich Isotherm
Kcq
KcKcqq m
1
nKcq1
Equilibrium relation
0cc •Estimate diffusivity by fitting the exp data
•Predict diffusivity and use in the model
mp
p
DD
Constant Diffusivity Diffusivity = f(bulk conc)
Monodisperse Pore Diffusion model with local Equilibrium•Volumetric adsorption study•Linear adsorption isotherm
D=exp(-a+bc)
Parallel Diffusion model with local Equilibrium
Very often that when dealing with adsorption of many gases and vapours in high surface area solids such as activated carbon and silica gel that surface diffusion can contribute significantly to the overall uptake
rq
rrqD
rc
rrc
Dtq
tc
sppp
ppppp
p2121 2
2
2
2
Boundary Condition (t>0)
Initial Condition (t<0)
00
r
p
rc
)(tccprr
p
0rp
c0cc
Bi-dispersed solids ex. zeolites • In these solids, we have macropore diffusion in the void space between the grains and micropore diffusion in the channels within the grain.
Micropore diffusivity can have a strong concentration and temperature dependence
cr
cc
ppppp
pp r
qDrR
cRR
cD
tc
132
2
2
Bi-dispersed solids with local Equilibrium
rq
rrqD
tq
c2
2
2
pKcq
At r = rp (microcrystal radius)
At R = Rp (particle radius)
)(tccpRRp
Modeling of Batch Adsorption
Film diffusionThe flux film diffusion can be expressed in linear form by multiplying its driving force and the film mass transfer coefficient
Jf is the mass transfer fluxa is the volumetric surface areacs is the adsorbate concentration at the exterior surface of adsorbent kf is the film diffusion coefficient
The symmetry condition at the center of the particles and continuity condition on the external surface of the adsorbent bed are expressed as:
At t > 0
)( sff ccakJdtdq
00
r
p
rc
prr
pppsf rc
Dcck
)(
The fixed bed adsorption processes utilize a solid mass separating agent/ adsorbent packed inside a column to effect separation of one or more components from a mixture in a gas or liquid stream as it flows through the packed bed.
Applications include air purification gas dehydration solvent or hydrocarbon vapor recovery water purification, and many others
Fixed Bed Adsorption
Breakthrough curve provides the basic but predominant information for the design of a column adsorption system
Scale of a column adsorption for practical application
Breakthrough curve determinationdirect experimentationmathematical modeling
Experimental methods provide a direct and concise breakthrough curve of a given system time-consumingeconomically undesirable process, particularly
for the trace contaminants and long residence time
greatly depends upon the experimental conditions, such as ambient temperature and
residence time
Mathematical modeling Simple no experimental apparatus
Why there is a need to model Fixed Bed Adsorption Column?
Fixed Bed Adsorption Modeling
(1) Mass transfer including convective
mass transfer and molecular diffusion
(2) Interface diffusion between bulk and the
exterior surface of the adsorbent (i.e., film
diffusion)
(3) Intraparticle mass transfer involving pore
diffusion
(4) Adsorption
Fixed Bed Adsorption Modeling Mass balance over the bed
p
ppp
pL rr
drdc
Drdz
dcudzcdD
dtdc
13
2
2
tq
zcu
zcD
tc
L
)1(
2
2
Neglecting Dispersion
tq
zcu
tc
)1(
p
ppp
p
rrdrdc
Drdz
dcudtdc
13
Assumptions:(1)Isothermal process(2) The packing material is made of porous particles that are spherical and uniform in size(3) The bed is homogenous and the concentration gradient in the radial direction of the bed
is negligible(4) The flow rate is constant and invariant with the column position(5) The activity coefficient of each species is unity
Fixed Bed Adsorption Modeling
Boundary Condition (t>0)
0,0,0 ctcudztdcDL
0, tLdzdc
Initial Condition (t<0)
00, zc
0,0 ctc Step input
Fixed Bed Adsorption Modeling Parameters used:Length of the column = 6 mDiameter of column = 0.5 mAmount of adsorbent = 795 kgFlowrate = 2.058 m3/hrSuperficial Velocity = 10.44 m/hrDiffusivity of oleic acid = 2.95 × 10-6 m2/hr Dispersion coefficient = 2.16 × 10-6 m2/hrInitial concentration of feed = 0.0165 kg/m3
Equations used:Bed
pp
ppp
L rrrc
Drz
cuzcD
tc
13
2
2
Particle
rc
rr
Drt
qtc p
pp
pp
p2
2)1(
Equilibrium relation
pd
pdm
cKcKq
q
1
Fixed Bed Adsorption Modeling
Fixed Bed Adsorption Modeling
•Equilibrium capacity shows saturation period after about 3 years (1.17 kg/kg capacity- 795 kg bed-0.034 kg/hr feed)
•FFA Adsorption bed simulation shows the bed will get completely saturated after about 14000 hr ≈ 583 days ≈ 1 year 7 months
•Regeneration will be required after 1 year 7 months
Pulse Chromatography
Initial ConditionFixed bed adsorptionStep input
0,0 ctc
Initial ConditionChromatographic columnPulse input
itt
ectc
0,0
Input Pulse
Input Step
Why Pulse Chromatography?
Batch adsorption systems masked with other rate limiting processes such as external film mass transfer resistance and /or heat dissipation
These effects are less severe in flow systems
With sufficiently high velocity, external resistances to heat and mass transfer can be reduced
Mainly used for diffusivity measurements
Measure exit concentration versus time
To analyze and interpret the data from chromatographic experiment a mathematical model is required
The equilibrium and kinetics is derived by matching experimental response curves to model predictions
Pulse Chromatographic Modeling
Mean residence time - measure of the affinity of adsorbate towards adsorbent
Variance/Spread - measure of its dynamic characteristics
Spread of the curve is a complex function of all dispersion forces 1. Axial dispersion2. Film resistance3. Pore diffusion resistance (macropore/mesopore/micropore)
Pulse Chromatographic Modeling
p
ppp
pL rr
drdc
Drdz
dcudzcdD
dtdc
13
2
2
Bed
Particle
cr
cc
ppppp
pp r
qDrR
cRR
cD
tc
132
2
2
rq
rrqD
tq
c2
2
2
Linear Isotherm
Langmuir Isotherm
Freundlich Isotherm
Kcq
KcKcqq m
1
nKcq1
Equilibrium relation
Pulse Chromatographic ModelingDesired and undesired Response Curves
Pulse Chromatography for analytical techniques (GC, HPLC)Desired : Symmetrical
Pulse Chromatography for diffusivity (micropore) measurementsDesired : Tailing (skewed)
Reasons of skewed response curve
Micropore diffusion control regime - TailingNon-linear isotherm – Tailing, Fronting
Pulse Chromatographic Modeling The elution peak travels in the form of waves through the chromatographic column at
a certain velocity The concentration wave velocity is the velocity that a particular value of
concentration will propagate through the system The velocity of the wavefront is a function of the type of isotherm Shape of the wave/ response curve is a function of the velocity
tc
cq
tq
Fluid mass balance
tq
zcu
tc
)1(
cq
uuz
)1(1
Velocity of wavefront
For Linear isotherm
Kcq constantcq
Shape of wave does not
change during displacement
Concentration velocity is retarded relative to the interstitial velocity
Pulse Chromatographic Modeling
For Non-linear isotherm constantcq
Langmuir Isotherm (Favorable isotherm)
Slope of isotherm reduces with increasing concentration
cq
uuz
)1(1
Velocity of wavefront
Wavefront velocity increases with increasing concentrations
Sharp front and tailing rear
ation)f(concentrcq
Pulse Chromatographic Modeling
w.r.t. Bed lengthFronting
w.r.t. timeTailing
Sharp front with diffuse rear
Pulse Chromatographic Modeling
For Non-linear isotherm constantcq
Freundlich Isotherm (Favorable isotherm)
Slope of isotherm reduces with increasing concentration and attains a constant value
cq
uuz
)1(1
Velocity of wavefront
Wavefront velocity is constant at higher conc. but will decrease at lower conc.
How will the response curve appear?
Pulse Chromatographic Modeling
For Non-linear isotherm constantcq
Unfavorable isotherm
Slope of isotherm increases with increasing concentration
cq
uuz
)1(1
Velocity of wavefront
Wavefront velocity decreases with increasing concentrations
Tailing front and sharp rear
Pulse Chromatographic Modeling
wrt Bed lengthTailing
wrt timeFronting
Diffuse front with sharp rear
Pulse Chromatographic ModelingWhat is the nature of isotherm?
Pulse Chromatographic Modeling
Tailing
Fronting
For Non-linear isotherm constantcq
•Symmetrical for Linear constantcq
•Tailing for Favorable
•Fronting for unfavorable
Pulse Chromatographic Modeling
The concentration velocity of an adsorbing or desorbing component is less than that of a component that has no interaction with the solid phase
How to ensure Linear Regime?
cq
utzu z
)1(1
Velocity of wavefront
cq
uLt
)1(1
For Linear isotherm
Kcq
Kcq
K
uLt )1(1
Mean retention time should not change with pulse size
Pore Diffusion model with linear adsorption kinetics
the pore diffusion model and the rate of mass exchange between the two phases is much faster than the diffusion rate. In this section we shall consider the case where such mass exchange is comparable in rate to the diffusion,
Absorption v/s Adsorption
Absorption - Fluid is dissolved by a liquid or a solid (absorbent)
Adsorption - Atoms, ions or molecules from a substance (it could be gas, liquid or dissolved solid) adhere to a surface of the adsorbent
Adsorption - a film of adsorbate is created on the surface -surface-based process
Absorption-involves the entire volume of the absorbing substance
Types of AdsorptionBasis of Separation
Steric separation - the porous solid has pores having dimension such that it allows small molecules to enter while excluding large molecules from entry
Equilibrium separation – based on the level of affinity of the adsorbent towards adsorbate so that the stronger adsorbing species is preferentially held by the solid
Kinetic separation - based on the different rates of diffusion of different species into the pore; the faster diffusing species is preferentially removed by the solid
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