adsorption modelling

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