Introduction
Mass transfer
This is a basic, introductory treatment There is a significant extent of similarity between processes of heat
transfer and those of mass transfer Many relations are also similar Heat transfer and mass transfer occur simultaneously in many
important situations.
Introduction
Mass transfer
What will we study? Fick’s law of diffusion One dimensional problems in a stationary medium Mass transfer in a moving medium Convective mass transfer - analogy between heat transfer and mass
transfer Simultaneous heat and mass trasfer
Mass transfer
We have learned that heat is transferred if there is a temperature difference in a
medium. Similarly, if there is a difference in the concentration of some chemical species in a mixture, mass transfer must occur
• Air moving by fan• Water being flowed through a pipe
Are the following examples mass transfer?
How about
• Dispersion of oxide of sulfur release from power plant• Transfer of vapour into dry air• Drying of material
Mass transfer
Mass diffusion occurs in liquids and solids, as well as in gases. However, since mass transfer is strongly influenced by molecular spacing, diffusion occurs more readily in gases than in liquids and more readily in liquids than in solids.
Mass transfer
Mixture composition
Mass transfer
Mass transfer
R = 8.3145 J/mol∙K
Mass transfer
Mass transfer
Mass transfer
Fick’s law of diffusion
Mass transfer
Analogous of Fourier’s law of heat conduction Valid gas for gas mixtures, and liquid solutions and solid solutions Empirical law, based on experimental evidence
Concentration of “A”
x Diffusion of “A”
Mass transfer
Gas A diffuses into gas B:
n
w
A
m aa
Mass flux of Adensity
Mass fraction of A
sm
kg
m
s/kg22
3m
kg
m
mixtureofkg
Aofkg
Mass transfer
If a = mass density of gas A, andb = mass density of gas B, then
ba
Then
aaw
3m
Aofkg
3m
Bofkg
Mass fraction of A
3m
kg
bbw Mass fraction of B
Mass transfer
The constant of proportionality is called the diffusion coefficient, Dab
If the density of the mixture is uniform, then we can write
n
wD
A
m aab
a
nD
A
m aab
a
Mass transfer
In Cartesian coordinates, Fick’s law can be written as:
x
wD
A
m aab
x
a
y
wD
A
m aab
y
a
z
wD
A
m aab
z
a
xz
y
Mass transfer
In vector notation, Fick’s law can be written as:
aab
awD
A
m
We can express the Fick’s law in any suitable system of coordinates. The expressions are very similar to the expressions for Fourier’s law of heat conduction
Mass transfer
Often, the concentration of gas A is expressed in terms of its mole fraction (xa) and molar density (ca). Fick’s law can be expressed in terms of these as:
n
xcD
n
)c/c(cD
A
N aab
aab
n
a
Molar flux of “A”
sm
Aofmolekg2
Molar density
3m
molekg
Molar density of A3m
Aofmolekg
Mole fraction of A
mixtureofmolekg
Aofmolekg
s
m 2
Mass transfer
If ac = molar density of gas A, andbc = molar density of gas B, then
ba ccc
If the molar density c of the mixture is uniform, then we can write
3m
Aofmolekg
3m
Bofmolekg
andc
cx a
a
n
cD
A
N aab
n
a
2m
Aofmolekg
3m
Aofmolekg
m
s
m 2
mass diffusivity or diffusion coefficient
Mass transfer
If the gas mixture is made of components which obey the ideal gas law, then in terms of the partial pressure of component A, pa, we have:
n
)p/p(D
RT
p
A
N aab
n
a
n
)RT/p(D
A
N aab
n
a
and
Mass transfer
We can write expression for diffusion of component B.
n
wD
A
m bba
b
Fick’s law can also be used for multicomponent gas mixtures. It is applicable to liquid solutions and solid solutions as well.
Mass transfer
Diffusion coefficient
Diffusion coefficient of some gases in air at 1 atm total pressure and 25 ºC
Gas/vapour Dab x 10-5 (m2/s)
Ammonia 2.80
Carbon dioxide 1.64
Hydrogen 4.10
Naphthalene 0.62
NO 1.80
Oxygen 2.06
Mass transfer
Diffusion coefficient
Variation of diffusion coefficient of water vapour in air with different temperature at 1 atm total pressure
Gas/vapour Dab x 10-5 (m2/s)
200 2.12
300 2.54
325 3.00
350 3.49
375 4.03
400 4.61
Mass transfer
Diffusion coefficient
For a gas or vapour, diffusing through a gaseous medium, the diffusion coefficient increases with temperature, but decreases as the pressure increases. In many such cases, over a small range of T and p, we have
p
TD
n
ab
When n is a number between 1.5-2.
Mass transfer
Steady-state mass diffusion in a stationary medium
Typical problem:Steady-state diffusion of a gas through a stationary, large, isothermal ‘slab’ of width b. The mass fraction of the gas on the two faces of the slab is wa1 and wa2.
Mass transfer
Diffusion through a slab
1aw
2aw
b
aw
x
Gradient of concentration of A
Diffusion of A
“Gas A”“Gas A”
Gas “B” slab
Mass transfer
Diffusion through a slab
1aw
2aw
b
aw
x
For steady state: amount of gas flowing to each plane is equal.
“Gas A”“Gas A”
“B”
Mass transfer
Since we have a steady state, and a one-dimensional situation, using Fick’s law, we get:
A
m a
dx
dwD a
abconstant
Assuming the density of the medium to be a constant, we get
A
m a
dx
dD a
ab
constant
Mass transfer
integration gives:
b
wwD
bD
A
m 2a1aab
2a1aab
a
Mass transfer
Since the diffusion coefficient is a constant, integration gives:
b
wwD
bD
A
m 2a1aab
2a1aab
a
and
b
x
ww
ww
2a1a
a1a
Mass transfer
Comparing heat and mass transfer
bD
A
m 2a1aab
a
b
TTk
A
q 12
Mass flux
Heat flux
Mass transfer
Diffusion through a hollow cylinder
)r/rln(
)ww(LD2m
io
aoaiaba
)r/rln(
)r/rln(
ww
ww
io
i
aiao
aia
ir
or
aiw
aow
Mass transfer
Diffusion through a hollow sphere
oi
aoaiaba
r1
r1
)ww(D4m
oi
o
aiao
aia
r1
r1
r1
r1
ww
ww
ir
or
aiw
aow
Mass transfer
ProblemDetermine the rate at which hydrogen will diffuse through a 4 cm thick steel plate having a face area of 2 m2 at a temperature of 400 K. The concentration of hydrogen at the two faces of the plate is 0.10 and 0.01 kg/m3, respectively. The value of diffusion coefficient of hydrogen is steel at 400 K is 1.6 x 10-11 m2/s.
Mass transfer
Diffusion in a moving medium Let us consider a one-dimensional steady-state situation. Mixture of two gases A and B Moving at different velocities Va and Vb in the x-direction. Molar densities ca and cb and mole fractions xa and xb vary with x Diffusion takes place in the direction of flow because of
concentration gradients.
Mass transfer
Diffusion in a moving medium The molar average velocity is defined as
The mixture will be considered to be stationary when
bbaaba
bbaam VxVx
cc
VcVcV
0V m
The molar flux of a species is made of two componentsa) that due to the molar average velocity (convection), and b) that due to diffusion of species
Mass transfer
The molar flux of species A
And of species B
dx
dxcDVc
A
N aabma
a
dx
dxcDVc
A
N bbamb
b
Mass transfer
If the molar density of the mixture (c) is constant , then we have
And of species B
dx
dcDVc
A
N aabma
a
dx
dcDVc
A
N bbamb
b
Mass transfer
Since
Hence
A
N
A
N
A
N ba
dx
dxcD
dx
dxcD b
baa
ab
But since ,1xx ba
dx
dx
dx
xd ba
Hencebaab DD
Total molar flux
Mass transfer
Since
In general
bbaaba
bbaa VwVwVV
V
bbaaba
bbaam VxVx
cc
VcVcV
0V m does not mean 0V
0V m does not mean0V
Mass average velocity
Molar average velocity
Mass transfer
Equimolar counter-diffusionThis is a special case of one-dimensional steady-state diffusion in a moving medium
Reservoir 1 Reservoir 2
P, T P, Tca1, cb1 ca2, cb2
ca1+cb2 = c ca2+cb2 = c
ca2
ca1 cb2
cb1
Assuming ideal gas
Concentration profile ?
x = 0 x = L
Mass transfer
We assume that the components A and B obey the ideal gas law.For reservoir 1:
)cc(RTppp 1b1a1b1a
For reservoir 2:
)cc(RTppp 2b2a2b2a
Hence ccccc 2b2a1b1a constant
Mass transfer
HenceMolar flow of A from left to right = molar flow of B from right to left
Hence, the molar average velocity of the mixture,
0V m
Hence, we have
dx
dcD
A
N aab
a
dx
dcD
A
N bba
b
and
Mass transfer
Using Dab = Dba, and integrating, we get
L
)pp(
RT
D
L
ccD
A
N 1a2aab1a2aab
a
A/NA/N baWe also have
L
)pp(
RT
D
L
ccD
A
N 1b2bba1b2bba
b
From which, we get
L
x
cc
cc
cc
cc
1b2b
1bb
1a2a
1aa
Mass transfer
L
x
cc
cc
cc
cc
1b2b
1bb
1a2a
1aa
Reservoir 1 Reservoir 2
P, T P, Tca1, cb1 ca2, cb2
ca1+cb2 = c ca2+cb2 = c
ca2
ca1 cb2
cb1
Assuming ideal gas
linear
A
B
Mass transfer
ProblemA large tank contains ammonia gas at 1 atm pressure and 25ºC. A long open tube, 1 m long and 5 mm in diameter, connect the tank to the air outside which is at the same pressure and temperature. Make any reasonable assumptions and calculate (a) the rate at which ammonia is lost through the tube (mºammonia) and (b) the rate at which air enters the tube (mºair) . Dab = 2.8 x 10-5 m2/s
In convective mass transfer, for the heat flow from a surface to a fluid surrounding it, we have
Convective mass transfer
In a similar fashion, we now seek to obtain, for mass transfer of a species A from a surface to the surrounding fluid
)TT(hA
qfw
)(hA
mafawm
a
Where hm is the mass transfer coefficient
Convective mass transfer
Area A
af
aw
)(hA
mafawm
a
Mass transfer coefficient
Convective mass transfer
If the mixture density is constant, we can write
)ww(hA
mafawm
a
Mass fraction
( - )
3m
kg2m.s
kgs
m
Convective mass transfer
Let us look at boundary-layer for mass transfer
a
aw
V a
aw
)ww(hA
maawm
a
)(hA
maawm
a
Convective mass transfer
The governing differential equation for mass convection (species A) in a boundary is
2a
2
aba
ya
x yD
yV
xV
Subject to the boundary conditions
aw;0y
a;y
Note the similarity with the boundary-layer for heat transfer
Convective mass transfer
The solution for the local mass (at any point) transfer coefficient is
2/1x
3/1
abab
m ReD
332.0D
xh
where
Sh, Sherwood numberab
m
D
xh
and
abDSc, Schmidt number
Kinematic viscosity =
NuSimilar to
Similar to
k
CPr p
Convective mass transfer
Thus we have, for the local mass transfer coefficient
3/12/1xx ScRe332.0Sh
And the average mass transfer coefficient
3/12/1LL ScRe664.0Sh
Convective mass transfer
Analogy between heat and mass transferFor laminar flow, we notice the similarity between the correlation for mass transfer (flat plate).
3/12/1LL ScRe664.0Sh
And that for heat transfer
3/12/1LL PrRe664.0Nu
This leads to an extension of the Colburn analogy between fluid flow and heat transfer to include mass transfer as well.
Convective mass transfer
Convective mass transfer
Analogy between heat and mass transferThe Colburn analogy, extended to mass transfer, lead to the relations
23232 /fPrStScSt //m
Where
V
h
ScRe
ShSt m
m
And
pCV
h
PrRe
NuSt
Stanton number for heat transfer
Stanton number for mass transfer
Mean velocity
Convective mass transfer
The Colburn analogy can be shown to be approximately valid for turbulent flows, and useful for both local as well as average mass transfer coefficients
60Pr6.0
It is applicable in the range
60Sc6.0 andIt will be more accurate if Pr and Sc is closed to 1 and these Pr and Sc close to each other.
A
m
A
m flowa
Also, good for
Convective mass transfer
Simultaneous heat and mass transfer
airT
V
a
wT
aw
Wet bulb thermometer
Convective mass transfer
Heat and mass transfer rate to/from the bulb are
)TT(hAq w
and
)(Ahm aawma
alsoamq
)(Ah)TT(hA aawmw 1)
s
kg.
kg
J
s
J
Latent heathence
Convective mass transfer
From the Coburn analogy,
Combining the two equation 1) and 2)
)()TT(D
C aaww
3/2
abp
3/2
p
3/2m PrCV
hSc
V
h
2)
Three unknown
Convective mass transfer
ProblemDry air at 1 atm and 20 ºC is blown on both sides of a flat plate in a direction parallel to the plate with a velocity of 2.5 m/s. The plate is 1 m long and 0.25 wide. It is at the same temperature as the air and its surface are wetted with a film of water. Calculate the rate of evaporation from the plate (kg/s).
http://www.engineeringtoolbox.com/air-properties-d_156.htmlAir properties
http://www.engineeringtoolbox.com/dry-air-properties-d_973.html
Vapor diffusion coefficient
http://web2.clarkson.edu/class/me310/Links/Thermodynamic_tables_SI.pdf
Convective mass transfer
ProblemThe inner surface of a circular pipe, 4 cm ID is coated with a thin layer of liquid water. Dry air at 300 K and 1 atm flows through the tube with a mean velocity of 1.9 m/s. Calculate the mass transfer coefficient for a fully developed flow using the Colburn analogy and the Gnielinski correlation. Compare the mass flux of water vapor with the mass flux of air flowing in the tube.
Vapor diffusion coefficient
Air properties http://www.engineeringtoolbox.com/dry-air-properties-d_973.html
http://www.thermexcel.com/english/tables/vap_eau.htm
Convective mass transfer
ProblemAir at 30ºC and 1 atm pressure flows with a velocity of 0.5 m/s across wet bulb thermometer, which reads 26ºC. Calculate the mass fraction of water vapor in the air steam and its relative humidity.