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1
Part I
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1. Introduction to Mass Transfer and Diffusion
2. Molecular Diffusion in Gasses3. Molecular Diffusion in Liquids
4. Molecular Diffusion in Biological Solutions and Gels
5. Molecular Diffusion in Solids6. Unsteady State Diffusion
Part I
Part II
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7. Convection Mass Transfer Coefficients
8. Mass Transfer Coefficients for various geometries
9. Mass Transfer to Suspensions of Small Particles
Part III
Part II
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1. Introduction to Mass Transfer and Diffusion1.1 Fick’s law for molecular diffusion
2. Molecular Diffusion in Gasses2.1 Equimolar counterdiffusion in gases2.2 General case for diffusion of gases A
& B plus convection2.3 Special case for A diffusing through
stagnant, nondiffusing B
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2.4 Diffusion through a varying cross-sectional area
2.5 Diffusion coefficient for gases3. Molecular Diffusion in Liquids
3.1 Introduction3.2 Equation for diffusion in liquids3.3 Diffusion coefficients for liquids3.4 Prediction of diffusivities in liquids
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Mass transfer occurs:
Water evaporates into still air.
Sugar dissolves & diffuses to the
surrounding solution.
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Mass transfer occurs:
Distillation
Drying
Liquid-liquid extraction
Crystallization
Adsorption
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Molecular diffusion (transport): the transfer or movement of
individual molecules through a fluid by means of the random, individual movements of the molecules.
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Figure 6.1-1: Schematic diagram of molecular diffusion process.High
concentration
Low concentration
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Consider: the diffusion of molecules when the
whole bulk fluid is not moving but is stationary.
due to a concentration gradient. The Fick’s law equation:
dz
dxcDJ A
AB
*
AZ
Total concentration of A &B (kg mol/m3)
The mole fraction of A in mixture of A & B
Distance, m
The molecular diffusivity of the molecule A in B, m2/s
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If c is constant, then cA=cxA,
cdxA = d(cxA)=dcA
For constant total concentration:
dz
dcDJ A
AB
*
AZ
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Example 6.1-1
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Example 6.1-1
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Example 6.1-1
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2.1 Equimolar Counterdiffusion in Gases Two gases A & B at constant total
pressure P in two large chambers connected by a tube where molecular diffusion at steady state is occurring.
Figure 6.2-1: Equimolarcounterdiffusionof gases A and B
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Stirring in each chamber keeps the concentrations in each chamber uniform.
The partial pressure pA1>pA2 and pB2>pB1. Molecules of A diffuse to the right and
B to the left. Since the total pressure P is constant
throughout, the net moles of A diffusing to the right must equal to the net moles of B to the left.
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This means that
Writing Fick’s law for B for constant c,
Now since P=pA+pB = constant, then
Differentiating both sides,
*
B
*
A JJ
dz
dcDJ B
BA
*
B
BA ccc
BA dcdcdc
BA dcdcdc=0,
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Equating the equations,
Finally,
This shows that for a binary gas mixture of A & B, the diffusivity coefficient DAB for A
diffusing into B is the same as DBA
for B diffusing into A.
dz
dcD)(J
dz
dcDJ B
BA
*
BA
AB
*
A
BAAB DD Molecular diffusivity
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Example 6.2-1
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Example 6.2-1
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The diffusion flux J*A occurred because of the concentration gradient. The rate at which moles of A passed a
fixed point to the right (positive flux). This flux can be converted:
where vAD is the diffusion velocity of A (m/s).
3
2
m
Akgmol
s
mcv)m.s/Amolkg(J AAD
*
A
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Consider when the whole fluid is moving in bulk or convective flow to the right.
The molar average velocity of the whole fluid relative to a stationary point is vM
m/s. Component A is still diffusing to the
right, but now its diffusion velocity, vAd is measured relative to the moving fluid.
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To a stationary observer A is moving faster than the bulk of the phase, since its diffusion velocity vAd is added to that of the bulk phase vM.
The velocity of A relative to the stationary point is the sum of the diffusion velocity & the average or convective velocity:
Where vA – velocity of A relative to a stationary point.
MAdA vvv
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Multiplying by cA,
Let N be the total convective flux:
vA
vAd vM
MAAdAAA vcvcvcNA
(kgmol A/s,m2) J*A
BAM NNcvN
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So,
Since J*A is Fick’s law,
BAA*
AA NNc
cJN
BAAA
ABA NNc
c
dz
dxcDN
BABB
BAB NNc
c
dz
dxcDN
Equimolarcounterdiffusion
NA = -NB
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In the evaporation of a pure liquid (e.g. benzene (A) at the bottom of a narrow tube, where a large amount of inert or nondiffusing air (B) is passed over the top.
Figure 6.2-2a: Diffusion of A
through stagnant, nondiffusing B: (a)
benzene evaporating into air
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The benzene vapor (A) diffuses through the air (B) in the tube.
The boundary at the liquid surface at point 1 is impermeable to air, since air is insoluble in benzene liquid.
Air (B) cannot diffuse into or away from the surface.
At point 2 the partial pressure pA2=0, since a large volume of air is passing by.
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In the absorption of NH3 (A) vapor which is in air (B) by water. The water surface is impermeable to
the air, since air is only very slightly soluble in water.
Figure 6.2-2: Diffusion of A through stagnant, nondiffusingB. (b) ammonia in air being absorbed into water.
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thus, since B cannot diffuse, NB = 0.
Keeping the total pressure P constant, substituting
0AAA
ABA Nc
c
dz
dxcDN
Convective flux of A
RT
Pc Pxp AA
P
p
c
c AA
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Then,A
AAABA N
P
p
dz
dp
RT
DN
dz
dp
RT
D
P
pN AABA
A 1
2
11
2
1
A
A
p
p A
AAB
z
z
APp
dp
RT
DdzN
1
2
12 A
AABA
pP
pPln
zzRT
PDN
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A log mean value of the inert B is defined:
Then,
2211 BABA ppppP
11 AB pPp 22 AB pPp
12
21
12
12
AA
AA
BB
BBBM
pPpPln
pp
ppln
ppP
21
12
AA
BM
ABA pp
pzzRT
PDN
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EXAMPLE 6.2-2
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EXAMPLE 6.2-2
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EXAMPLE 6.2-3
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EXAMPLE 6.2-3
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So far, the cross-sectional area A m2
through which the diffusion occurs has been constant with varying distance z. In some situations the area A may vary.
At steady state, will be constant but not A for a varying area.
A
NN A
A
kg moles of A / s
AN
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2.4.1 Diffusion from a sphereThe evaporation of a drop of liquid, the
evaporation of a ball of naphthalene, and the diffusion of nutrients to a sphere-like microorganism in a liquid.
Figure 6.2-3a: A sphere of fixed
radius, r1 (m) in an infinite gas
medium.
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Since this is a case of A diffusing through stagnant, nondiffusing B:
Note that dr was substituted for dz. Integrating between r1 and some point r2
a large distance away:
24 rπ
N
A
NN AA
A
drPp
dp
RT
D
rπ
NN
A
AABAA
14 2
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r2>>r1, 1/r2≈0.
drPp
dp
RT
D
r
dr
π
N A
A
p
p A
AAB
r
r
A2
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2
1
2
1
2
21
11
4 A
AABA
pP
pPln
RT
PD
rrπ
N
BM
AAABA
A
p
pp
RTr
DN
rπ
N 21
1
12
14
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If pA1 is small compared to P (a dilute gas phase), pBM≈P,
2r1=D1 (diameter), cA1=pA1/RT
This equation can be used for liquids, where DAB is the diffusivity of A in the liquid.
21
1
1
2AA
ABA cc
D
DN
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Example 6.2-4
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Example 6.2-4
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If the sphere is evaporating, the radius r of the sphere decreases slowly with time.
The time it takes for the sphere to evaporate completely can be derived by assuming pseudo-steady state and by equating the diffusion flux equation, where r is now a variable, to the moles of solid A evaporated per dt time and per unit area as calculated from a material balance.
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21
2
1
2 AAABA
BMAF
ppPDM
RTprpt
Density of the sphere
Original sphere radius
Molecular weight
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2.5.1 Experimental determination of diffusion coefficients
To evaporate a pure liquid un a narrow tube with a gas passed over the top (see Fig 6.2-2a).
The fall in liquid level is measured with time and the diffusivity calculated:
21
2
1
2 AAFA
BMAAB
ppPtM
RTprpD
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The common method is the two-bulb method (N1).
Pure gas A is added to V1 and pure B to V2
at the same pressures.
Figure 6.2-4: Diffusivity
measurement of gases by the
two-bulb method.
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The valve is opened, diffusion proceeds for a given time, and then the valve is closed and the mixed contents of each chamber are sampled separately.
Assumptions:Neglecting the capillary volume &
assuming each bulb is always of a uniform concentration.
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Quasi-steady-state diffusion in the capillary,
The rate of diffusion of A going to V2 is equal to the rate of accumulation in V2:
L
ccD
dz
dcDJ AB
AB
*
A12
Concentration of A in V2 at
time t
Concentration of A in V1 at
time t
dt
dcV
L
AccDAJ AB*
A2
212
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The average value cav at equilibrium can be calculated by a material balance from the starting composition c1
0 and c20 at
t=0:
A similar balance at time t gives,
0
22
0
1121 cVcVcVV av
221121 cVcVcVV av
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Rearranging & integrating between t=0 & t=t,
if c2 is obtained by sampling at t, DAB
can be calculated.
12
21
0
2
2
VVAL
VVDexp
cc
cc AB
av
av
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Some typical data are given in Table 6.2-1, Perry & Green (1984) and Reid et al., (1938).
Table 6.2-1: Diffusivity coefficient of gases at 101.32 kPa Pressure
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The diffusivity of a binary gas mixture in the dilute gas region, that is, at low pressure near atmospheric, can be predicted using the kinetics theory of collision with another molecule, which implies that momentum is conserved.
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The final relation for predicting the diffusivity of a binary gas pair of A & B molecules is:
21
2
237 111085831/
BAAB,DAB
/
ABMMζP
Tx.D
Ω
Average collision diameter
A collision integral based on the Lennard-Jones
potential
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The above equation is relatively complicated to use (σAB is not available or are difficult to estimate).
23131
217517 1110001/
B
/
A
/
BA
.
ABννp
MMTx.D
ΣΣ
Sum of structural volume increments
Note: DAB α 1/P, DAB α T1.75, DAB α T1.75/P
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Table 6.2-2: Atomic diffusion volumes for use with Fuller et al., method.
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Example 6.2-5
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Example 6.2-5
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3.1 IntroductionDiffusion of solutes in liquids is
important in many industrial processes:
Solvent extraction
Gas absorption Distillation
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Diffusion in liquids also occurs in nature:
Oxygeneration of rivers and lakes by
the air
Diffusion of water in blood
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It should be apparent that the rate of molecular diffusion in liquids is considerably slower than in gases. The molecules of the diffusing solute
A will collide with molecules of liquid B more often and diffuse more slowly than in gasses.
The diffusion coefficient in a gas will be on the order of magnitude of about 105 times greater than in a liquid.
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The flux in a gas is being only 100 times faster, since the concentrations in liquids are
considerably higher than in gases.Since the molecules in a liquid are
packed together much more closely than in gases, the density & resistance to diffusion in a liquid are much greater. the attractive forces between
molecules play important role in diffusion.
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Diffusion of liquids: the diffusivities are often dependent on the concentration of the diffusing components.
3.2.1 Equimolar counterdiffusion An equation similar to gases at steady
state [NA=-NB]:
12
21
12
21
zz
xxcD
zz
ccDN AAavABAAAB
A
Kg mol A/s.m2 m2/s
Concentration of A (kg mol A/m3)
at point 1.
Mole fraction of A at point 1.
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cav defined by
The average value of DAB may vary some with concentration, &
the average value of c may vary with concentration Linear average of c is usually used.
22
2
1
1
M
ρ
M
ρ
M
ρc
av
av
Average total concentration of A+B (kg mol/m3)
Average molecular weight of the solution at point 1
(kg mass / kg mol)
Average density of
the solution(kg/m3)
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A dilute solution of propionic acid (A) in a water (B) solution being contacted with toulene.Only the propionic acid (A) diffuses
through the water phase (B), to the boundary & then into the toluene phase.The toulene-water interface is a
barrier to diffusion of B and NB = 0. Substituting
RT
Pcav
RT
ρc A
A1
1P
px BM
BM
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For liquids at steady state:
Note xA1+xB1=xA2+xB2=1.0 For dilute solution, xBM is close to 1.0 c is essentially constant.
21
12
AA
BM
avABA xx
xzz
cDN
)xxln(
xxx
BB
BBBM
12
12
12
21
zz
ccDN AAAB
A
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Example 6.3-1
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Example 6.3-1
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3.2.3.1 Experimental determination of diffusivities
A relatively dilute solution & a slightly more concentrated solution are placed in chambers on opposite sides of a porous membrane of sintered glass (see Fig6.3-1).
Figure 6.3-1: Diffusion cell for determination of diffusivity in a
liquid
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Quasi-steady-state diffusion in the membrane is assumed:
ηδ
'ccDεN ABA
Concentration in the lower chamber at a time, t
Concentration in the upper chamber
Fraction of area of the glass open to diffusion
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Combining & integrating:
tDVηδ
Aεln AB'cc
cc ' 200
Initial concentrations
Final concentrationsCell constant
Note: For liquids, unlike gases, the diffusivity DAB does not equal DBA.
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Table 6.3-1: experimental diffusivity data for binary mixtures in the liquid phase are given.
The diffusivity values are quite small and in the range of about 0.5x10-9 to 5x10-9
m2/s for relatively nonviscous liquids. Diffusivities in gases are larger by a
factor of 104-105.
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Table 6.3-1: experimental
diffusivity data for binary
mixtures in the liquid phase are
given.
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The Stokes-Einstein equation was derived for a very large spherical molecule (A) diffusing in a liquid solvent (B) of small molecules.Stokes’ law was used to describe the
drag on the moving solute molecule.
31
1610969/
A
ABVμ
Tx.D
Viscosity, Pa.s or kg/m.s
Solute molar volume at its normal boiling
point, m3/kgmol
KDiffusivity, m2/s
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The Wilke-Chang correlation can be used for most general purposes, where the solute (A) is dilute in the solvent (B).
60
2116101731.
AB
/
BABVμ
TMθx.D
Molecular weight of solvent B
Viscosity of B, Pa.s or kg/m.s
Solute molar volume at the boiling point (Table 6.3-2)
Association parameter
of the solvent
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The association parameter (φ):
φ φ
Water 2.6 Benzene 1.0
Methanol 1.9 Ether 1.0
Ethanol 1.5 Heptane 1.0
Unassociated solvents
1.0
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Example 6.3-2
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Example 6.3-2