mass transfer - suranaree university of...

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


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


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


Area A

af

aw

)(hA

mafawm

a

Mass transfer coefficient


If the mixture density is constant, we can write

)ww(hA

mafawm

a

Mass fraction

( - )

3m

kg2m.s

kgs

m


Let us look at boundary-layer for mass transfer

a

aw

V a

aw

)ww(hA

maawm

a

)(hA

maawm

a


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


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


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


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.



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


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


Simultaneous heat and mass transfer

airT

V

a

wT

aw

Wet bulb thermometer


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


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


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


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


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.

mass transfer - suranaree university of...

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