Mass transfer & diffusion hari

MASS TRANSFER & DIFFUSION
PRESENTED BY:-
HARERAM MISHRA
GCT/1830155
EMAIL ID:-harirammishra18@gmail.com
DEPARTMENT OF CHEMICAL ENGINEERING
SANT LONGOWAL INSTITUTE OF ENGG.
&TEC & TECHNOLOGY
LONGOWAL,SUNGRUR,{PUB.},148106

CONTENTS:-
1.INTRODUCTION TO MASS TRANSFER
2.MECHANISM OF MASS TRASFER
3.CLASSIFICATION OF MASS TRANSFER
4.CONVECTIVE MASS TRANSFER
5.MASS TRANSFER COEFFICIENT
6.THEORIES OF MASS TRANSFER
7.MOMENTUM ,HEAT AND MASS TRANSFER ANALOGIES
8.INTERPHASE MASS TRANSFER
9.DIFFUSION & MOLECULAR DIFFUSION
10.KNUDSEN ,SURFACE&SELF DIFFUSION

INTRODUCTION TO MASS TRANSFER:-
DEFINITION:- “The process of transfer of mass as result of concentration
difference of a component in a mixture or two phase in contact is called mass
transfer”.
Ex:-Evaporation of water from a pool of water In to a stream of air flowing
over the water surface.
Absorption of oxygen of air into blood occurs in the lungs of a animal in
the process of respiration.
In mass transfer operations, mass
transfer may occur:-
(a)In one direction e.g Gas Absorption.
(b)In opposite direction e.g Distillation.
(c)With simultaneous heat transfer e.g
-Drying & crystallisation.
(d )With simultaneous chemical reaction
e.g absorption of CO2 in an aqueous
solution of KOH.
(e)With a exchange of one or more components.

The phenomena those must exit in a mass transfer operation are:-
(a)At least two phases must come in contact with each other
(b)Materials must flow from one phase to other.
(c)A part of total flow of material from one phase to other must occur by
molecular diffusion.
MECHANISM OF MASS TRANSFER:-
(a)Equilibrium between the phases is attained after a sufficiently long time of
phase contact between them.
(b)Material transfer is caused by the combined effect of molecular diffusion
and turbulence.
(c)There is no resistance to mass transfer at the phase interphase(because
of the existence of equilibrium at the interphase).
(d)Rate of mass transfer is evaluated by deviation/departure from equilibrium
concentration.

CLASSIFICATION OF MASS TRANSFER OPERATIOS:-
 ACCORDING TO THE PHASE IN CONTACT:-
PHASES IN
CONTACT
MASS TRANS.
OPERATIONS
DRIVING
FORCE
DEGREEOFF-DOM
F=C-P+2(P-RULE)
VARIALES OF
OPERATIONS
Liquid-vapour
(gas)
Distillation
(fractionation)
Vapour
pressure diff.
F=2-2+2=2
Liquid-gas Gas absorption
Stripping
Humidification
Dehumidification
Solubility diff.
Solubility diff.
Conc. diff.
Conc. difference
F =3-
2+2=3
Liquid-solid Crystallisation
Leaching
Adsorption
Liquid-liquid Extraction Diff. in solubil-
Ity of solute
Solid-vapour Sublimation
Solid-gas Adsorption

CONVECTIVE MASS TRANSFER:-
INTRODUCTION:-
Mass transfer occurring under the the
influence of motion in a fluid medium is called the
‘convective mass transfer’.
Ex:- A simple mechanism of dissolution of sugar crystals
in a stirred cup of water.
The water just in contact with a crystals i.e. liquid at solid-
liquid interface, get saturated with the sugar almost
instantly. The dissolved sugar diffuses from the interface to the bulk of liquid through a thin layer
or film of the solution adhering the crystals. More sugar dissolves in the liquid in the liquids
simultaneously the interface.
The thickness of the film decreases, if the stirring rate is more rapid, which results in fast or
quicker rate of dissolution or rate of mass transfer.
TYPES OF CONVECTIVE MASS TRANSFER:-
Similar to the convective heat transfer, there are
two types of convective mass transfer:-
1. forced convection mass transfer 2. free or natural convection mass transfer

1.Free convection mass transfer:-
In this
mechanism of mass transfer,the motion or
flow in the medium is caused by the diff.
in density.
2. forced convection mass transfer:-
This
type of mass transfer involves that the
motion in the medium is poroduced by
an external agency such as pump, blower
and agitator.
Convective mass transfer is strongly influenced by the flow field,If the flow field is well
defined,then mass transfer rate can be determined by mathematical analysis of mass and
momentum balance sach as in the case of- Dissolution of the solid coated on a plate in a fluid
flowing over it, the absorption of solute gas in a laminar liquid film falling down a wall.
For more complex situation sach as the dissolution of solid in a mechanically stirred
vessel,theor. calculation is difficult bcz flow field is complex.

MASS TRANSFER COEFFICIENT:-
 The mass transfer coefficient is defined as:-
 Rate of mass transfer ∞ conc. Driving force
 Rate of mass transfer ∞ area of contact between the phases
 WA ∞ a∆CA W A= kca∆CA Where kc = mass transfer coefficient
 And,- WA = aNA = kCa∆CA So, NA = kc∆CA
 Mass transfer coefficient,kC = NA / ∆CA Or kC = molar flux/ conc. DV
 For the purpose of comparison-
 Heat transfer coefficient, h=Heat flux/temp.driving force.
 Mass transfer resistence ∞ 1/mass transfer coefficient
 Local flux/local driving force=local mass transfer coefficient.
 Avg.flux/Avg.driving force=Avg. mass transfer coefficient
 Units of k = unit of molar flux/ unit of driving force
 = kmol/m.m.s/unit of driving force
 If driving force is taken as conc. Diff.,then-unit of k =m.m/s(as unit of vel.)
 According to the units of D.F,units of mass trans.coff.(k ) will be changed.

types of mass transfer coefficient:-
Different types of mass transfer coefficients have been defined depending on:-
(A)Whether mass transfer occurs in gas phase or in liquid phase
(B)The choice of driving force ,and (C) Whether it is a case of diffusion of A through non- diffusing B or a case of
counter-diffusion.
the case of counter/concurrent diffusion
Differents types of mass transfer coefficients are:-
Diffusion of A through non-diffusing B Equimolar counter diffusion of A&B Units of mass
Transfer coff.
Flux,NA Mass transfer coefficient Flux,NA Mass transfer coefficient
Gas-phase mass transfer coff. kG(pA1-pA)
kG=DABP/RTδpBM
Ky(yA1-yA2) ky=DABP2/RTδpBM
Kc(CA1-CA2) kC=DABP/ δpBM
k’G(pA1-pA2) k’G=DAB/δR
K’y(yA1-yA2) k’y=DABP/δRT
k’C(CA1-CA2) k’c=DAB/δ
Mol/t(A)∆pA
Mol/t(A)∆yA
Mol/t(A)∆CA
Liquid-phase mass transfer coff.
kL(CA1-CA2) kL=DAB/δxBM
kX(xA1-xA2) kx=CDAB/δxBM
k’L(CA1-CA2) k’L=DAB/δ
k’x(xA1-xA2) k’x=CDAB/δ
Mol/t(A)∆CA
Mol/t(A)∆yA
Conversion:-
kGRT=RTkY/P=kc; kL=kx/Cav k’C=k’GRT=RTk’y/P; k’L=k’x/Cav

DIMENSIONLESS GROUPS IN MASS TRANSFER:-
There are most important dimensionless groups in mass transfer are:-
Dimensionless groups and their physical significance in
mass transfer
Analogous groups in heat transfer and their physical
significance
Reynolds number,Re=interia/viscous force
=lvρ/µ=lv/ν
The same
Schmidt number,Sc=momentum/mol. Diffu.
=µ/ρD=ν/D
Pr=cP/k=(µ/ρ)/(k/ρcP)=v/α=momentum
diffusivity/thermal diffusivity
Sherwood number,Sh=convective mass flux/diffusive flux
across a layer of thick.l
=kLl/D=kL∆C/(D/l)∆C
Nu=hl/k=convective heat flux/conduction heat flux
across a thickness of l
Stanton number,StM=convective mass flux/flux due to
bulk flow of a medium
=Sh/(Re)(Sc)=kL/v=kL∆C/v∆C
StH=Nu/(Re)(Pr)=h∆T/v∆T=convective heat flux/heat
flux due to bulk flow
Peclet number,PeM=flux due to bulk flow of
medium/diff. flux across a layer of l
=(Re)(Sc)=lv/D=v∆C/(l/D)∆C
PeH=(Re)(Pr)=(vρcP)∆T/(k/)∆T=heat flux due to bulk
flow/conduction heat flux across a thickness l
Colburn factor,jD=StM(Sc)2/3=Sh/(Re)(Sc)1/3 The same
Grashof number,Gr=l3∆ρg/µν The same

THEORIES OF MASS TRANSFER:-
There are no. of theories of mass transfer which aim at visualizing the mechanism and developing the expression
for mass transfer coefficient theoretically.In fact,any such theory is based on a conceptual model for mass transfer.
These important theories are:-
1.The Film theory (film model)
2.The Penetration Theory
3.The surface renewal theory
THE FILM THEORY (FILM MODEL):-
This theory was developed by Whitman,in 1923.According to this theory,when the mass transfer occurred from a
solid surface to a flowing fluid,even though the bulk liquid is in turbulent motion,the flow near the wall may be
considered to be laminar.
The concentration of dissolved solid (A) will decreses from Cai,at the solid liquid interphase to Cab at the bulk of
liquid. In reality the concentration profile will be very steep near the solid surface,where the effect of turbulence
is practically absent.
Molecular diffusion is responsible for mass transfer near the wall while convection dominates a little away from
it.
This theory is based on following assumptions such as:-

(a) Mass transfer occurs by purely molecular diffusion through a stagnant fluid layer at the phase boundary. Beyond
this film,the fluid is well mixed having a concentration which is the same at that of bulk fluid.
(b) Mass transfer through a film occurs at steady state.
(c)The bulk flow term [(NA+NB)CA/C] is small.
For the many practical situations, this assumption is satisfactory.
According to the above figure,let us consider that an elementary volume of thickness
∆Z and of unit area normal to the z-direction. Making a steady-state mass balance over this element located at
the position z.

Rate of input of solute at z = NA│Z
Rate of output of solute at z+∆z = NA│z+∆z
Rate of accumulation = 0 (at steady state)
By the mass balance over the elementary volume-
Input + accumulation = output NA│z- NA│Z+∆Z =0
Dividing by ∆z throughout and taking the limit ∆z→0, we get,-
-dNA/dz = 0, and putting NA= -DABdCA/dz DABd2CA/dz2=0, or- d2CA/dz2 = 0;
integrating and using the following boundary conditions (1) & (2)-
(1) at the z = 0 (i.e. the wall or the phase boundary or the interphase ), CA = Cai
(2) at the z = δ (i.e. the other end of film of thickness,δ), CA = CAb,
then,-
CA = CAi – (CAi – CAb) z/δ, according to this-
The theorectical concentration profile is linear as shown in figure.whereas the true concentration profile is curve
type.
At the steady state condition, the mass transfer flux through the film is constant.
And, NA = -DAB [dCA/dz]Z=0 = DAB / δ (CAi –CAb ) on
and also, NA = kL (CA1 – CA2) comparing kL = DAB / δ
Important points :-
1. The film theory has been extremely useful in the analysis of mass transfer accompanied by a chemical reaction.
2. The film thickness, is the thickness of the stagnant layer that offers a mass transfer resistence equal to the actual
resistence of mass transfer offered the fluid

3.The film theory predicts linear dependence of kL upon the diffusivity, i.e. D ∞ kL.
4. According to the experimental data on diverse system,- coefficient of mass transfer to a turbulent fluid varies as
(DAB)n , where n varies between 0 – 0.8.
THE PENETRATION THEORY:-
This theory was developed by Higbie.This theory can be
explained by a simple phenomenon of mass transfer from a
rising gas bubbles-for example, absorption of oxygen from
an air bubble in a fermenter.
According to the figure as shown- as the bubble rises,the
liquid element from the bulk of fluid reach the top of bubble
move along its spherical surface, reach its bottom and then deteached
detached from it. The detached liquid element eventually
get mixed up with the bulk liquid.
Absorption of oxygen in a small liquid element occurs as long
as the elements remains at gas-liquid interface (i.e. in contact with gas ).
Thus the above theory concluded that- in the case of mass transfer at the phase boundary, an element of the liquid
reaches the interface (by any mechanism whatsoever) and stay there for a short while when it receives some solute
from other phase. At the end of its stay at the interface, the liquid element moves back into the bulk liquid carrying
with it the solute it picked up during its brief stay at the interface.
In this process, the liquid element makes room for another liquid element, fresh from bulk, on the surface of
bubble.

BASIC ASSUMPTIONS OF PENETRATION THEORY:-
1.Unsteady state mass transfer occurs to a liquid element so long as it is in contact with the bubble (or other phase )
2.equilibrium exists at the gas- liquid interface.
3.Each of the liquid element stays in contact with the gas(or other phase)
for the same period of time.
Consider the mass transfer of gas bubble of the diameter db rising at velocity ub, the contact time of a liquid element
during this period of time can be described by partial differential equation as-
∂CA/dt = DAB∂2CA/∂z2
The appropriate boundary and initial conditions are:-
Boundary condition 1:- t > 0, z =0; CA = CAi
Boundary condition 2:- t > 0, z = ∞; CA = CAb
Initial condition:- t = 0, z ≥ 0; CA = Cab
The initial condition implies that in a fresh liquid element coming from the bulk, the concentration is uniform and
is equal to the bulk concentration.
The boundary condition 1 assumes that “interfacial equilibrium” exits at all time.
The last condition implies that if the contact time of an element with the gas is small, the ‘deapth of the
penetration’ of the solute in the element should also be small and effectively the element can be considered to be
of ‘infinite thickness’ in relative sense.

Due to subjected to initial and boundary conditions, the equation can be solved for the transient concentration
distribution of the solute in the element by introducing a ‘similarity variable’ η :-
CA-CAb/CAi-CAb = 1 – erfη; where η = z/2√DABt
The mass flux to the element at any time t is-
NA(t) = -DAB[∂CA/∂z]z=0 = √DAB(CAi-CAb)/Πt
The avg. mass flux over the contact time tC is-
NA,avg = 1/tC ∫0
tNA(t)dt = 2√DAB(CAi-Cab)/Πtc
On comparing-
Instantaneous mass transfer coefficient, kL=√DAB/Πt
Average mass transfer coefficient, kL,avg=2√DAB/Πtc
IMPORTANT POINTS:-
1. The flux decreases with the time because of a gradual build-up of solute concentration within the element and
resulting decreases in the driving force.
2. At a large time, the element becomes nearly saturated and the flux becomes vanishingly small.
3. According to this theory, the mass transfer coefficient is proportional to the square root of diffusivity, i.e. kL ∞ DAB.
4. On comparing with film theory, the contact time tc is a model parameter like the film thickness δ in the film
theory.

THE SURFACE RENEWAL THEORY:-
One of the major drawback of penetration theory is the assumption that the contact time or ‘age’ of the liquid
elements is same for all.
In the turbulent medium, it is much more probable that some of the liquid elements are swept away, while still
young, from the interface by eddies while some others, unaffected by eddies for the time being, may continue to be
in contact with the gas for longer times.
As a result, there will be a ‘distribution of age’ of the liquid element present at the interface at any moment.
This is how danckwert (1951) visualized the phenomenon. This theory contains the following assumptions:-
1. The liquid element at the interface are being randomly displaced by the fresh elements from the bulk.
2. At any moment, each of the liquid elements at the surface has the same probability of being replaced by the fresh
element.
3. Unsteady state mass transfer occurs to an element during its stay at the interface.
The Danckwert’s theory is thus called the surface renewal theory.
The model parameter is the fractional rate of surface renewal (s, the fraction of surface area renewed in unit time).
The equation for the mass transfer coefficient is:-
KL=√DABs

IMPORTANT POINTS:-
1. Surface Renewal theory also predicts the square root dependence of mass transfer coefficient kL on the
diffusivity, i.e. kL∞√DAB.
2.It has also similar dependence on the fractional rate of surface renewal, s.
3. Increasing the turbulence in medium causes more brick surface renewal (i.e. larger s) thereby increasing the
mass transfer coefficient.
THE BOUNDARY LAYER THEORY:-
Mass transfer in the boundary layer flows
occurs in a way similar to that of heat tran.
If the plate is located with a soluable subs.
And the liquid (or gas ) flows over it, two
boundary layers are formed- the ‘velocity
boundary layer’ and the ‘concentration’
or ‘mass boundary layer’.
It maybe recalled that in boundary layer flow
over a heated plate, a thermal boundary layer
is formed along with the velocity or moment-
um boundary layer (thickness = δ ).
The concentration distribution in the boundary layer is a function of position,
CA = CA(x,y), and its thickness is a function of distance from the leading edge,
δ = δm(x). The relative thickness of the velocity and the concentration boundary layer depends on the value of
Schmidt number, Sc.

Theoretical analysis of the mass transfer in laminar boundary layer is given by following correctional:-
Shx = kL,xx/DAB=0.332(Rex)1/2(Sc)1/3
where,-
x =distance of a point from leading age of the plate
kL,x= local mass transfer coefficient
Rex=local Reynold number
If ‘l’ is the length of the plate then ‘average Sherwood number’ is-
Shav=kL,avl/DAB=0.664(Rel)1/2(Sc)1/3
where,-
kL,av= mass transfer coefficient averaged over the length of the plate,
Rel = ρV∞l/µ = ‘plate Reynold number’ based on the length of the plate,l.
IMPORTANT POINTS:-
1. Boundary layer theory predicts that the mass transfer coefficient kL varies as (DAB)2/3, which reasonably matches
the experimental findings in many cases.
2. If the Schmidt number is greater than unity, the thickness of momentum boundary layer at any location on the
plate is more than the concentration boundary layer,it can be shown as,--
δ/δm =(Sc)1/3

COMBINED STUDY OF MASS TRANSFER THEORY ALONG WITH MASS TRANSFER COEFFICIENT:-
THEORY STEADYOR
UNSTEADYSTATE
EXPRESSIONFOR MASS
TRANSFERCOEFFICIENT,KL
Dependence
on
diffusivity
MODEL
PARAMETER
(UNIT)
FILM THEORY Steady state kl = d/δ Kl ∞d δ(m)
PENETRATION
THEORY
Unsteady state kl,ins=[d/Πt]1/2
Kl,avg=2[D/Πtc]1/2
Kl ∞ d1/2
Kl ∞ d1/2
tc(s)
SURFACERENEWAL
THEORY
Unsteadystate Kl,avg=[Ds]1/2 Kl ∞ d1/2 s(s-1)
BOUNDARY LAYER
THEORY
Steadystate Boundary layer correlation Kl ∞ d2/3

CORRELATIONS FOR THE CONVECTIVE MASS TRANSFER COEFFICIENT:-
A large number of mass transfer coefficient exist in literature,which is developed by researchers on the basis of large
volume of experimental data:-
DESCRIPTION RANGE OF
APPLICATION
CORRELATIONS
Laminar flow through a circular tube
Turbulent flow through a tube
Boundary layer flow over a flat plate
Flow through a wetted-wall tower
Gas-phase flow through a packed bed
Liquid flow through a packed bed
d=tubediameter;Rel,l=characteristics length
V’’=Superfacial velocity of fluid
(i.e,velocity based on the bed cross-section)
Re≤2100
4,000≤Re≤60,000;
0.6≤Sc≤3000
Rel<80,000//Rel>5*105
3,000<Re’<40,000
0.5<Sc<3
10≤Re’’≤2500
Re’’<55
3<Re’’<10,000
Sh=kld/D=1.62[(Re)(Sc)
(d/l)]1/3
Sh=0.023(Re)0.83(Sc)0.33
jD=0.664(Rel)-0.5
//jD=0.037(Rel)-0.2
jD=0.038(Re’)-0.3
1.17(Re’’)-0.415
jD=1.09(Re’’)-2/3
Sh=2+1.1(Re)0.6(Sc)0.33

MOMENTUM,HEAT AND MASS TRANSFER ANALOGIES:-
Transport of momentum, heat and mass in a medium in laminar motion are all diffusional
processes and occurs by similar mechanisms.
The three basic law in this connection—
1. Newton’s Law of viscosity
2. Fourier’s Law of Heat Conduction
3. Fick’s Law of Diffusion
Newton’s Law:- Momentum flux, ς = -µdux/dz ς = -vd(ρux)/dz
Fourier’s Law:- Heat flux, qz= -kdT/dz qz= -αd(ρcPT)/dz
Fick’s Law:- Mass flux, NA= -DABdCA/dz
v = µ/ρ, momentum diffusivity and ρux=volumetric conc. Of momentum in x-direc.
α = k/ρcP,thermal diffusivity and ρcpT = volumetric conc. Of thermal energy in x-di
These three equations concluded that:-
“Flux is proportional to the gradient of the quantity transported (momentum,heat energy and
mass), and proportionality constant is the corresponding diffusivity.”
A negative sign is included in each equation to indicate that transport occurs in the direction of
decreasing concentration (of momentum,heat and mass).

PRANDTL ANALOGY:-
StH= (f/2)/1+5√f/2(Pr-1) and, StM= (f/2)/1+5√f/2(Sc-1)
The prandtl analogy reduces to Reynold analogy if Pr = 1 or Sc = 1.
COLBURN ANALOGY:- Colburn related the mass transfer coefficient to the friction factor by
proposing the well known ‘colburn analogy’.He introduced the colburn j-factor and suggested the
following analogies for transport in pipe flow.
jH= StH Pr2/3 = Nu/Re.Pr1/3 = 0.023Re-0.2
jD= StM Sc2/3= Sh/Re.Sc1/3 = 0.023Re-0.2
The importance of the analogies lies in the fact that if the heat transfer coefficient is known at a
particular hydrodynamic condition characterized by the Reynold Number, the mass transfer
coefficient in a system having similar geometry and at a similar hydrodynamic condition can be
determined:-
jH = jD

In the turbulent flow,-
The rate of transport except that an ‘eddy diffusivity’ is included to account for the contribution
of eddy transport. Thus,--
Momentum Flux, ς = -(v + Ev) d( ρµX)/dz
Heat Flux, qz= -(α + EH)d(ρcPT)/dz
Mass flux (at a low conc.) NA= -(DAB + EM)dCA/dz
Where EV, EH, EM, are the eddy diffusivities of momentum,heat and mass respectively. Since
turbulence transport is a much faster process than the molecular transport, all the eddy
diffusivities are larger the their molecular counterparts by about two order of magnitude.
ANALOGY IN MASS TRANSFER ALONG WITH FLOW OF FLUID:-
REYNOLD ANALOGY:-
StH= Nu/Re.Pr = f/2 and StM= Sh/Re.Sc = f/2
where, StH = Staton number for Heat Transfer
StM = Staton number for Mass Transfer
Nu = Nusselt Number involving the mass transfer coefficient
f = friction factor for the fluid flowing through a pipe.

INTERPHASE MASS TRANSFER:-
INTRODUCTION:-
Mass transfer of the molucules from one point to another
in a single phase or in a homogeneous medium occurs by
Diffusion mechanism. However in practice, most of the mass
transfer operations involve transport of one or more solute
from one phase to another.
Ex:-In the sulphuric acid plant, the air supplied to the sulphur
burner should be moist-free, drying of air is done in contact
with concentrated sulphuric acid in a ‘packed column’.
Ammonia is removed from ammonia-air mixture by using H2O.
Each of the above is a case of ‘Interphase mass transfer.’
How can we calculate the rate of mass transfer in two – phase system?
In the case of interphase mass transfer, the driving force is not merely the difference of concentrations of the solute
in two phases, but the driving force is rather measured by how far the phases are away from equilibrium.
For the purpose of comparing-
In the heat transfer, equilibrium means that the phases are at the same temperature ,so that driving force is zero.
But in the case of mass transfer ,the equality of concentration of two phases does not necessarily ensure about the
equilibrium.

EQUILIBRIUM BETWEEN THE PHASES IN MASS TRANSFER:-
Equilibrium between two phases in contact means a state
in which there is no net transfer of solute from one phase to
to other. At the equilibrium, the chemical potentials of
the solute in two phases are equal. Which can be
explained by two phase system of absorption of SO2 from
air into water.
Sulphur dioxide being soluabe in water, will be absorbed
in it, when contacted with water.
However some SO2 molecules will simultaneously leave
the water phase and re-enter the gas phase but necessa-
rily at the same rate. The vessel is maintained at constant
temperature and this process of absorption and desorption
(stripping) continues.
Eventually a time comes, at which the rate of absorption becomes equal to the rate of desorption.
The parital pressure of SO2 in air and its concentration in water will no longer change, thus the system is said to be at
‘equilibrium’.
If some sulphur dioxide gas is fed into the vessel and sufficient time is allowed thereafter the system will reach a
new equilibrium state and a new sets of equilibrium value (xA
*,pA) is obtained.
Sets of equilibrium values at constant temp. and at a constant total pressure is called the ‘Equilibrium Data’.
A plot of these data is called the ‘Equilibrium line’ or ‘Equilibrium curve’.

LAW OF EQUILIBRIUM:-
Two important law of equilibrium between the phases are Raoult’s Law and Henry’s Law.
Raoult’s Law:-
For the ideal gas-liquid or vapour-liquid system,the equilibrium relationship is given by Raoult’s Law-
Statement:-
“ The equilibrium partial pressure of a component in a solution at a given temp. is equal to the product
of its vapour pressure in pure state and its mole fraction in liquid phase”.
Mathmatically,-
pA = po
A xA
Henry’s Law:-
Equilibrium relationship for many non-ideal gas-liquid system at the low concentration is given br
Henry’s Law-
Statement:-
“ The mole fraction of a solute gas dissolved in a liquid is directly proportional to the equilibrium
partial pressure of the solute gas above the liquid surface”.
Mathmatically,-
xA = 1/HA pA
or,- pA = xA HA
Where HA = Henry’s Law contact , which is strongly depends on Temperture (Increases). The temperature
dependence of the Henry’s Law constant sometimes follows as- m = m0exp(-E/RT).

Equilibrium diagram of SO2 in water at different temp.
is shown in figure .Equilibrium lines become steeper
because the solubility of SO2 in water decreases with
increase with temperature.
IMPORTANT POINTS:-
1. In a system at the equilibrium, the number of phases p
P, the number of components C, and the number of
variants F, also called ‘The Degree of Freedom’ and are
related by ‘Phase Rule’-
F = C – P + 2
In the above case,-
F = 3 – 2 + 2 = 3
And, these three variants should be specified to define the system completely.
2. If two phase at equilibrium, there is no net transfer of solute from one phase to another. It rather means if a few
molecules go from phase-1 to phase-2, the same number of molecules move from phase-2 to phase-1 in order to
maintain the concentrations in the phases constant.This equilibrium is sometimes called ‘Dynamic Equilibrium’.
3. If two phase are not at equilibrium, mass transfer from phase-1 to phase-2 occurs so long as the concentration of
solute in phase-2 is lower than the equilibrium concentration.
“ The extent of Deviation from Equilibrium state is a measure of driving force”.
And for the system at the equilibrium, the driving force is zero.

MASS TRANSFER BETWEEN TWO PHASES:-
Concentration profile near the interface:- Mass transfer
between two phases involves the following sequential
steps:-
1. The solute (A) is transported from the bulk of the gas
phase (G) to the gas-liquid interface
2. The solute (A) is picked-up or absorbed by the liquid
phase (L) at the gas-liquid interface.
3. The absorbed solute (A) is transported from the
interface to the bulk of liquid (L).
The steps 1 & 3 are facilitated by the turbulence in the
fluid medium.
Interface means, geometrical plane or surface of contact between two phases.
From the figure, pA and CA are the function of distance (z) along the mass transfer path and z = 0
means the interface.
lim pA = pAi (z→0-) and lim CA = CAi (z→0+)
These are called ‘Interfacial concentration’. In most it is assumed that equilibrium exits at the interface as -
pAi = Θ (CAi)
Where Θ represents the dependence of pA on CA at the equilibrium. This function is also called ‘Equilibrium
relation’. If it so happens that the Henry’s Law is valid for solute-solvent pair, the equilibrium relation becomes linear
as- pAi = mCAi

TWO FILM THEORY:-
Lewis and Whitman (1924) visualized that in the case of mass transfer involving two phases in contact, there are two stagnant
fluid film exits on either side of interface and mass transfer through these films , in sequence, purely by molecular diffusion.
Beyond these films the concentration in a phase is equal to the bulk concentration.This is “Two film theory of lewis and
Whitman”. The two films and concentration distributions are schematically shown in the figure.
Determination of interfacial concentration:-
Interfacial concentration can be determined either
graphically or algebraically.
If the mass transfer occurs from a gas phase to a liquid
phase at the steady state, the mass flux of solute A from
the bulk gas to the interface must be equal to that from
the interface to the bulk liquid. This is because, at the steady
state, there cannot be any accumulation of solute anywhere.
In the term of gas and liquid phase coefficient-
NA = ky (yAb – yAi ) = kx(xAi-xAb)
Gas-phase flux to the liquid –phase flux
interface to interface
And from the figure- yAb – yAi /xAb – xAi = -kx/ky slope of the line PM

DIFFUSION:-
“It is the movement of molecules or individual component through a mixture from a region of higher concentration
to a region of lower concentration at a fixed temperature and pressure with or without the help of external force.
Ex:- A drop of ink released in water in a glass gradually spreads
to make the water uniformaly coloured.
The fragrance of a bunch of rose kept on the centre
table of drawing reaches out even to a remote corner of
the room.
Molecular Diffusion
DIFFUSION
Eddy or Turbulent Diffusion
Molecular Diffusion:-
Diffusion result from random
movement of Molecules.
Turbulent Diffusion:- When the movement of the molecule
occurs with the help of an external force such as
mechanical stirring and convective movement of fluid,then it is called ‘eddy or turbulent diffusion

IMPORTANT POINTS:-
1. Diffusion is a passive process.
2. The molecular diffusion is a slow process, whereas eddy diffusion is a fast process.
3. The molecular diffusion is a mechanism of stationary fluid i.e., a fluid at rest and flow is laminar
whereas in the case of fluid in the turbulent flow ,the mechanism of mass transfer is by eddy molecular transfer
eddy transfer.
ROLE OF DIFFUSION IN MASS TRANSFER:- Diffusion may occur in one phase or in both phases
in all the mass transfer operations.
In the case of Distillation-
The more volatile component diffuses through the liquid phase to
the interface between the phases and away from interface into the vapour phase, the less
volatile component diffuses in opposite direction.
In the case of gas absorption -
The solute gas diffuses through the gas phase to the interface
and then through liquid phase from the interface between the phases.
In the case of crystallization-
The solid solute diffuses through a mother liquor to the crystals and
deposit on the solid surface .
In the case of drying operation-
The liquid water (moisture) diffuses through a solid towards the
surface of solid,evaporates and diffuses as a vapour into gas phase (Drying medium).

FICK’S LAW OF DIFFUSION:-
The basic law of diffusion, called fick’s Law was enunciated by adolf Eugen Fick, a physiologist,
in 1885. According to this law , the molar flux of a species relative to an observer moving with the
molar average velocity is proportional to the concentration gradient of the species.
Mathematicaly,-
JA ∞ dCA /dz JA = -DAB dCA /dz
molar flux
molecular diffusivity concentration gradient in
or diffusion coefficient z-direction
The diffusional flux JA is a positive quantity by the convection. Since Diffusion occurs
spontaneously in the direction of decreasing concentration [(dCA/dz) < 0],negative sign. Is involved in the
mathematical representation of Fick’s law for diffusion.
And,
NA = NA + NB (CA / C) – DAB dCA / dz
NA + NB (CA /C) Bulk Flow
-DAB dCA /dz Molecular diffusion or NA = JA = -DAB dCA /dz (for small conc. Of A )
Diffusion velocity:-
Diffusing velocity of species A, vA,d = uA - U = JA / CA = -DAB dCA /dz

For the gas phase diffusion-
NA = (NA + NB )pA /P – DAB dpA /RTdz
In term of mole fraction-
NA = (NA + NB )yA –CDAB dyA /dz and JA = -CDAB dyA /dz
Average total molar concentration of mixture
If the gas mixture is ideal, the mutual diffusivities of A and B are equal, that is-
DAB = DBA
STEADY STATE MOLECULAR DIFFUSION IN A BINARY GAS MIXTURE-
Diffusion of A through non- diffusing B- For the steady state
diffusion of A through non-diffusing B,-
NA = constant and NB = 0
And, Molar flux- NA = DAB P ln (pA0 – pAl )/RTl*pBM
Where pBM = log mean partial pressure of B
or-NA = DABP ln (P – pA )/RTz*P-pA0
Equimolar counter current diffusion of A & B:- For the equimolar
counter-current diffusion of A and B-
NA = DAB (pA0 – pAl )/RTl

KNUDSEN DIFFUSION,SURFACE DIFFUSION AND SELF-DIFFUSION:-
(a)Knudsen Diffusion:-
If the gas diffusion occurs in a very fine pore,
particularly at a low pressure ,the mean free path of the molecules
may be larger than the diameter of passage. Then collision with the
wall becomes much more frequent than collision with other
molecules. The rate of diffusional transport of a species is now
governed by its molecular velocity, the dia. Of passage, and
Of course, the gradient of concentration or partial pressure .
This is called “Knudsen Diffusion” and becomes important if pore
size is normally below 50 mm.
Ex :- Intra-particle transport in a catalyst containing fine pores.
(b) Surface diffusion:-
Surface diffusion is the transport of
adsorbed molecules on a surface in the presence of a conc.
Gradient. Molecules adsorbed on a surface remain anchored
to the active sites, if the fractional surface converage is less than
unity,some of active sites remains vacant. The flux due to the surface diffusion is-
Js = -Ds dCs /dz
(c) Self-diffusion:- The molecules of a gas or a liquid continuously move from one position to
another as a result a mixture of uniform composition also diffuses but net rate is not changed.

Mass transfer & diffusion hari

Mass transfer & diffusion hari

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