I3. MASS TRANSFER AND DIFFUSION I3.1. INTRODUCTION The movement of one type of molecules through other types of molecules is influenced by the concentration gradient, the physical and molecular properties of the participating species and the external forces. These factors affect the rate of transfer of the molecules. This molecular interaction is the basis of determining the rate of mass transfer, which is important in the design of mass transfer equipment such as gas absorbers, humidifiers, distillation columns, and others. To simplify the discussion, only binary system will be considered in this presentation. There are two types of diffusion that will be considered. One is molecular diffusion, which is highly influenced by concentration gradient, and the other is eddy or turbulent diffusion, which is influenced not only by concentration gradient but also by the movement or mixing of the material due to some external applied force.
I3.2. MOLECULAR DIFFUSION DIFFUSION Consider a binary system where a certain species A is moving at an average velocity u A in a bulk of material containing species B moving at a t an average velocity of u B. Let us assume that the mixture is moving at a bulk velocity uo referred to a stationary observer. Then the molar fluxes of A and B may be determined by the Fick’s Law of diffusion I OA c A u A uO DAB
dc A
I OB cB u B uO DBA
dcB
dZ
dZ
(I3 – 1) (I3 – 1)
(I3 – 2) (I3 – 2)
where I oA oA is the molar flux of A through a plane moving at uO and c A and c B are the concentrations of species A and B while D while D AB is the diffusivity of A relative to B and D BA is the diffusivity of B relative to A. The diffusivities are transport properties which may be determined experimentally or estimated from empirical equations in terms of the physical and molecular properties of the diffusing components. In design calculations, what is more important is the diffusion flux, not relative to the movement of the bulk but relative to a stationary observer. These diffusion fluxes, N A and NB are given by N A = c Au A
(I3 – (I3 – 3) 3)
N B = c B u B
(I3 – 4) (I3 – 4) 1
while the total flux of the entire bulk, N is given by N = N A + N B = m uo
(I3 – 5)
If uA, uB and uo are eliminated from Eqs. (I3 – 1) and (I3 – 2), we get the equations I OA N A
I OB N B
c A m c B m
( N A N B ) DAB
( N A N B ) DBA
dc A dZ
dcB dz
(I3 – 6)
(I3 – 7)
If we add Eqs. (I3 – 6) and (I3 – 7), it can easily be seen that I OA + I OB = 0
(I3 – 8)
D AB = D BA = Dv
(I3 – 9)
and
That is, the sum of the molar fluxes relative to the movement of the bulk is zero and, for binary system, the diffusivity of A relative to B is the same as the diffusivity of B relative to A. Here, 2 2 we will just refer to this as the volumetric or mass diffusivity, Dv with units of m /s or ft /hr. It is important to note that the diffusivity is based on the movement of the entire bulk and not on a stationary position. For gases, the diffusivity can also be expressed in terms of molar units, Dm defined by Dm Dv m
Dv P T RT
(I3 – 10)
where the units of Dm is in moles/time-length and m is in moles per unit volume. Solving for the molar flux relative to a stationary observer, N A from Eq. (I3 – 6), we get N A Dv
dc A dz
cA m
( N A N B )
(I3 – 11)
It is seen that the diffusion flux, N A is composed of two terms, the molecular diffusion flux as given by Fick’s Law and another type of flux which we can consider here as convective flux or phase drift. The differential equation presented in Eq (I3 – 11) may be solved by considering two ideal steady state diffusion models. These are Equimolar Counter Diffusion and Unicomponent Diffusion. An example of the former is encountered in the rectification of volatile components where both can co-exist in both phases such as ethanol-water system. An example of the latter 2
is in the absorption of a soluble component from an inert gas that is insoluble in the solvent where the soluble component is able to penetrate the solid-liquid interface while the inert gas becomes stagnant since it cannot diffuse to the liquid pha se.
I3.3. EQUIMOLAR COUNTER DIFFUSION When the molar flux of A and B are moving at equal rates and in opposite direction, N A = - N B
or
N A + N B = 0
(I3 – 12)
Equation (I3 – 11) reduces to N A Dv
dc A dz
(I3 – 13)
This equation may be integrated for the total molar rate of diffusion, N TA, if the diffusion area, A, perpendicular to the direction of motion is constant,
c c Dv A1 A2 A z2 z 1
NTA
(I3 – 14)
It is to be noted that for constant area, the concentration profile is linear across the direction of diffusion. If the diffusion area is not a constant, it must be expressed in terms of z and the differential equation solved applying the limits from z1 to z2.. For ideal gases, the diffusion equation may be expressed in terms of partial pressure, p A, that is, c A
p A RT
(I3 – 15)
or NTA A
Dv dp A RT dz
(I3 – 16)
I3.4. UNICOMPONENT DIFFUSION For unicomponent diffusion of A through a stagnant component B, then N B = 0. Equation (I311) becomes, dc c N A Dv A A N A (I3 – 17) dz m since c A + c B = m, the above equation may be converted to 3
c dc Dv 1 A A A c B dz
NTA
(I3 – 18)
If this is expressed in terms of the mole fractions of A and B, that is x A and x B, the above equation can be integrated in the form of NTA A
Dv
c A1 c A2
z2 z1
x B ln
(I3 – 19)
where x Bln is the logarithmic mean of the mole fraction of B at point 2 and point 1. For ideal gases, Eq. (I3 – 18) may be expressed in terms of p A,, that is, NTA A
Dv P T dp A RTp B dz
(I3 – 20)
if the diffusion area is constant, with p B = P T – p A, the above equation can be integrated to give NTA A
Dv P T RT z
ln
PT p A2
PT p A1
(I3 – 21)
It is noted that the concentration profile for this case is non-linear but logarithmic.
I3.5. EVALUATION OF DIFFUSIVITIES The volumetric diffusivity, Dv for gases and liquids may be determined experimentally or from empirical correlations based on the kinetic theory of gases. Some of the more important equations are presented here.
I3.5.1. From Empirical Equations 1. For gases, Chen and Othmer Equation. (McCabe and Smith, 1976)
Dv
0.5
1 1 0.01498T 1.81 M A M B pTCA TCB
0.1405
V
0.4 CA
0.4 2 CB
V
(I3 – 22)
2. For gases, Gilliland Equation. (Brown, et al., 1950))
4
3
DG
0.0166T
2
1 1 V A 3 V B 3 P
1 2
M A
1
M B
(I3 – 23)
3. For gases, Chapman and Engskog Equation (Geankoplis, 1997)) D AB
1/ 2
1.8583 x107 T 3/ 2 1 2 D , AB P AB
1
M A M B
(I3 – 24)
4. For liquids, Stokes-Einstein Equation (Geankoplis, 1997)) D AB
9.96 x1016 T
V A1/ 3
(I3 – 25)
3. For Liquids, Wilke and Chang (Treybal, 1968)
D AB
0.5
7.4 10 8 M B T 'V A0.6
(I3 – 26)
Other empirical equations maybe found from literature. The nomenclature used in these equations is found in the Appendix.
I3.5.2. From Experimental Data Sources of diffusivity data can be found in Perry and Green (1984), Green, et al. (1997), McCabe, et al. (2001), Geankoplis (1995) and other textbooks. If the diffusivity is given at a particular reference temperature, say 273K and 1 atm, it is possible to estimate the diffusivity at a desired temperature and pressure by making use of the empirical equations as the basis. If the calculation is based on Chen and Othmer correlation, the equation becomes
T 1.81 Dv f p
(I3 – 27)
or 1.81
Dv T P 1
T Dv 273,1atm 273
1 p
(I3 – 28)
I3.6. TURBULENT DIFFUSION 5
The equation for molecular diffusion may be modified and applied to turbulent diffusion by introducing a correction M referred to as the turbulent or eddy mass diffusivity. Thus, Eq. (I3 1) may now be written as I oA DAB M
dc A dz
(I3 – 29)
I3.7. MASS TRANSFER COEFFICIENTS For equimolal counter diffusion, I oA = N A. The above equation can therefore be integrated across a film thickness of ( z 2 – z 1), to give N A
D AB M z2 z 1
(c A1 c A2 )
(I3 – 30)
This equation is then simplified by expressing it in terms of a convective mass transfer coefficient, kc’ based on the movement of the entire bulk phase.
N A kc' (cA1 cA2 )
(I3 – 31)
For mass transfer of A in a non-diffusing B, Equation (I3-19) may be modified to give N A
( D AB M ) c A1 c A2 z2 z1
x B ln
(I3 – 32)
which may be simplified to N A
k c' x B ln
(c A1 c A2 ) kc (c A1 c A2 )
(I3 – 33)
The mass transfer coefficients kc’ and kc have a unit of m/s or ft/hr. It is possible to express these coefficients in terms of other units depending on the driving forces used in the defining mass transfer equation. Examples are
N A kG ( pA1 pA2 ) k y ( y A1 y A2 ) k x (xA1 xA2 )
(I3 – 34)
I3.8. EVALUATION OF MASS TRANSFER COEFFICIENTS
I3.8.1. Dimensionless Numbers 6
The dimensionless numbers obtained by the usual procedure of dimensional analysis that are important in mass transfer operations are the following: Reynolds Number, N Re
Du
Schmidt Number , N Sc
Dv
Sherwood Number, N Sh
kc' L D AB
inertia forces
viscous forces
(I3 – 35)
momentum
(I3 – 36)
mass diffusivity
turbulent diffusion molecular diffusion
(I3 – 37)
The mass transfer coefficient is correlated as a dimensionless, J D factor given by J D
kc' v
( N Sc )
2/ 3
kc' PT
v m
N Sh
1/ 3 N Re N Sc
(I3 – 38)
I3.8.2. Mass, Heat and Momentum Transfer Analogies The transport mechanism of mass, heat and momentum have similarities that could be used to relate the three mechanisms especially in determining approximate values of the transfer coefficients in the absence of a more reliable experimental data. The more common analogies are presented here. Reynolds Analogy ( N Sc = N Pr = 1.0)
f 2
h c p G
k c' uav
(I3 – 39)
Chilton-Colburn Analogy
f 2
J H
h c p G
( NPr )
2/ 3
JD
k c' uav
( N Sc )2 / 3
(I3 – 40)
I3.8.3. Mass Transfer Coefficients A. F or F low I nside Pipes
For Laminar flow, refer to Fig. 7.3 – 2 (Geankoplis, 1995) For Turbulent Flow, for NSc of 0.6 to 3000
7
N Sh kc'
D D AB
0.83 0.023N Re N Sc0.33
(I3 – 41)
B. F or F low Outside Soli d Sur faces
1. Parallel Flat Plates N Sh kc'
L D AB
0.5 1/ 3 0.664 N Re, L N Sc
(I3 – 42)
2. Flow Past Single Spheres For gases, NSc = 0.6 to 2.7 and NRe = 1 to 48,000 0.53 1 / 3 NSh 2 0.552NRe N Sc
For liquids,
(I3 – 43)
NRe = 2 to 2000 0.5 1/ 3 NSh 2 0.95NRe N Sc
(I3 – 44)
NRe = 2000 to 17,000 0.62 1 / 3 NSh 0.347 NRe N Sc
(I3 – 45)
C. F or Packed Beds
For Gases through spheres with NRe = 10 to 10,000 J D J H
0.4545
0.4069 N Re
(I3 – 46)
For Liquids with NRe = 0.0016 to 55 and NSc = 165 to 70,600 J D
1.09
2 / 3 N Re
(I3 – 47)
For Liquids with NRe = 55 to 1500 and NSc=165 to 10,690 J D
0.250
0.31 N Re
(I3 – 48)
8
The representative equations given above are obtained from Geankoplis(1995). Many more correlations are available in Green, et al . (Perry’s Handbook, 1997) and other references. D. Penetrati on Th eory of M ass Tr ansf er
For cases where surface renewal rather than film theory applies, for equimolal diffusion, the individual mass transfer coefficient is given by k
2 M
Dv
t L
(I3 – 49)
where t L is the average time the fluid elements remain at the interface. This is dependent on the fluid velocity, fluid properties and the geometry of the system.
THE WETTED WALL COLUMN
The wetted wall column is the most popular apparatus used in experimentally determining the mass transfer coefficient of a system since the mass transfer area can be determined with reasonable accuracy. Correlations on the behaviour of the dimensionless numbers such as the Sherwood number, Reynolds number and Schmidt number under turbulent diffusion have been derived using this apparatus. Applying the material balance and the rate of mass transfer of component A around the differential area dA yields dN A = V’dY = k y(Y i - Y) dA
(I3 – 50)
Since, V’ = V(1-y) and
Y
1
and dY
1 y
dy
1 y
2
Substituting in Equation (I3 – 50)), we get V 1 y A
k y dA
o
V
dy
1 y
2
y 2
k y yi y dA
dy
(I3 – 51)
1 y y y i
y1
Under adiabatic conditions, the temperature of the liquid remains constant, thus the interfacial concentration, yi may be taken also as constant. Integrating Equation (I3 - 51), we get k y A V
1 yi 1
ln
1 y1 yi y2 ........ 1 y2 yi y1
(I3 – 52)
9
With the temperature, flow rate and concentrations measured experimentally, together with the surface area of contact between the gas and the liquid, the mass transfer coefficient of the diffusing component maybe determined. Several correlations have been derived for wetted-wall columns. An example is the GillilandSherwood Equation (McCabe and Smith, 1976) given by 0.81 0.44 N Sh 0.023 N Re N Sc
(I3 – 53)
which is very similar to Eq. (I3 – 41). The equation applies for NRe between 2,000 to 35,000; NSc from 0.6 to 2.5; and over a pressure range of 0.1 to 3 atm. A second correlation for wetted-wall columns, which shows the general analogy for momentum, heat and mass transfer, although less precise than the above equation, can be written as j M j H
f
2
0. 2 0.023N Re
(I3 – 54)
where f is the Fanning friction factor for flow in smooth pipes. The above equation is not applicable if form drag exists.
10
NOMENCLATURE Symbol A c A c D AB Dm Dv f G h I oA kc kc’ M N A N TA N Re N Sc N Sh P, P T p A R T Tc u V A Vc x A y z AB D m M
Description Area perpendicular to the moving species Concentration of species A heat capacity Diffusivity of A relative to B molal diffusivity volumetric diffusivity Fanning friction factor mass velocity heat transfer coefficient Molar flux of A relative to bulk motion mass transfer coefficient for unicomponent diffusion mass transfer coefficient for equimolar diffusion molecular weight Molar flux of A Total moles of A diffusing Reynolds Number Schmidt Number Sherwood Number total pressure partial pressure of A universal gas constant=8314.34 temperature critical temperature linear velocity solute molar volume at normal boiling point critical volume Mole fraction of species A in liquid phase mole fraction in the gas phase Distance in the direction of moving species porosity of bed viscosity association parameter of the solvent average collision diameter collision integral Molal density of mixture eddy or turbulent mass diffusivity
Units m kg-mole/m J/kg-K 2 m /s kg-mole/s-m 2 m /s [-] 2 kg/m -s 2 W/m -K 2 kg-mole/s-m m/s m/s kg/kg-mols kg-mol/m -s kg-mol/m -s
atm or Pa mm Hg or Pa J/kg-mol-K K K m/s 3 m /kg-mol m [-] [-] M [-] Pa-s [-] M [-] kg-mols/s-m m /s
11
References: Brown, George G., D. Katz, A.L. Foust and R. Schneidewind. (1950). "Unit Operations", John Wiley and Sons, New York Foust, A.S., L.A. Wenzel, C.W. Clump, L. Maus and L.B. Andersen. (1960) " Principles of Unit Operations", John Wiley and Sons, New York. rd Geankoplis, Christie J. (1995) “Transport Processes and Unit Operations” , 3 edition. Printice-Hall International ed., Green, Don W.(ed ) and James O. Maloney (asoc. ed ), (1997) “ Perry's Chemical Engineers' th Handbook, 7 edition", McGraw-Hill Book, New York McCabe, Warren L., Julian C. Smith and Peter Harriott,(2001) Unit Operations of Chemical th Engineering, 6 edition, McGraw-Hill International. th Perry, Robert H. and D. Green. (1984). " Perry's Chemical Engineers' Handbook, 6 edition", McGraw-Hill Book, New York. nd Treybal, Robert E., (1968), “ Mass Transfer Operations”, 2 edition, McGraw-Hill Kogakusha, Ltd., Tokyo
TABLES NEEDED: Diffusion Coefficients of Combination of Gases at 1 atm Diffusion Coefficients of a Gas in Air at 1 atm and 273K Atomic Diffusion Volumes Diffusion coefficients for Dilute Liquid Solutions Atomic and Molar Volumes at Normal Boiling Point o Diffusion Coefficients for Dilute Solutions of Gases in Water at 20 C
12
