Advanced Heat and Mass Transfer by Amir Faghri, Yuwen ...

1 Chapter 1: Introduction1.4 Fundamentals of Momentum, Heat and Mass Transfer

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell

Mass Transfer

When there is a species concentration difference in a multicomponent mixture, mass transfer occurs.

Two modes of mass transfer: Diffusive - random molecular motion at the

microscopic level, can occur in a solid, liquid or gas. Convective - combination of random molecular

motion at the microscopic level and bulk motion at macroscopic level, can occur in liquid or gas.

The species concentration in a mixture, , is defined as the mass of species i per unit volume of the mixture (kg/m3).

iρ

1.4.2 Momentum, Heat and Mass Transfer



The density of the mixture equals the sum of the concentrations of all N species, i.e.,

(1.83)

The concentration of the ith species can also be represented by the mass fraction of the ith species, defined as

(1.84)

It follows from eq. (1.83) that

(1.85)

1

N

ii

ρ ρ=

= ∑

ii

ρωρ

=

1

1N

ii

ω=

=∑




The concentration of the ith species can also be represented by molar concentration defined as molar number of the ith species per unit volume, , which is related to the mass concentration by

(1.86)

The molar concentration of the mixture equals the sum of the molar concentrations of all N species

(1.87) The molar fraction of the ith species is defined as

(1.88)

which leads to (1.89)

ii

ic

Mρ

=

1

1N

ii

x=

=∑

3(kmol/m )ic


iicxc

=

1=

= ∑N

ii

c c



The mean molecular mass of the mixture can be expressed as

(1.90)

The mass fraction is related to the molar fraction by

(1.91)

1

N

ii

ixM Mcρ

=

= = ∑

1

N

jj

i i i ii

jx

x M x MMM

ω

=

= =∑



The molar fraction is related to the mass fraction by

(1.92) The local mass-averaged velocity of all species is defined as

(1.93) The molar-averaged velocity can be defined in a similar manner:

(1.94) The mass flux and molar flux relative to stationary coordinate axes

are defined as (1.95)(1.96)

1

/

/N

jj

i ii

j

MxM

ω

ω=

=∑

1 1

1

1

N N

i i Ni i

iNi

ii

i i

i

Vρ ρω

ρρ

= =

=

=

= = =∑ ∑

∑∑

VV V%

1

1

*

N

i Ni

ii

i

i

cx

c=

=

= =∑

∑V

V V%

i i iρ′′ =m V&

i i ic′′ =n V&



The fluxes defined in eqs. (1.95) and (1.96) are related by

(1.97) Applying eqs. (1.95) and (1.96) into eqs. (1.93) and (1.94), the total

mass flux and molar flux are obtained.(1.98)

(1.99) The mass flux relative to the mass-averaged velocity is

(1.100)and the molar flux relative to the molar averaged velocity is

(1.101) According to eqs. (1.93) and (1.94), we have

(1.102)

ii

iM′′′′ = mn&&

1

N

ii

ρ=

′′ ′′= =∑m m V%& &

*

1

N

ii

c=

′′ ′′= =∑n n V%& &

( )i i iρ= −J V V%

* *( )i i ic= −J V V%

*

1 10

N N

i ii i= =

= =∑ ∑J J



According to eqs. (1.95) – (1.101), the following relationships between different fluxes are valid:

(1.103)(1.104)

For a binary system that is uniform with all aspects except concentration, i.e., no temperature or pressure gradient, the diffusive mass flux can be obtained by Fick’s law:

(1.105)or alternatively

(1.106) The mass and molar-flux relative to a stationary coordinate axes are

(1.107)(1.108)

1

N

i i i i j ij

ρ ω=

′′ ′′= + = +∑m V J m J%& &

* * *

1

N

i i i i i ii

c x=

′′ ′′= + = +∑n V J n J%& &

1 12 1Dρ ω= − ∇J

*1 12 1cD x= − ∇J

1 1 1 2 12 1( ) Dω ρ ω′′ ′′ ′′= + − ∇m m m& & &

1 1 1 2 12 1( )x cD x′′ ′′ ′′= + − ∇n n n& & &



Figure 1.11 shows an isothermal and isobaric steady-state one-dimensional binary system.

Surface (y=0) is impermeable to species 2.

The mass flux of species 1 at the surface (y=0) is

(1.109)

Figure 1.11 One-dimensional mass diffusion.

y

L

0


1ym′′&

1( )yω

1ω

12 11

11yDm

yρ ω

ω∂′′ = −

− ∂&



For a multicomponent low-density gaseous mixture, the following Maxwell-Stefan relation can be used to relate the molar fraction gradient of the ith component and the molar flux by:

(1.110)

The Maxwell-Stefan relation can also be rewritten in terms of mass fraction and mass flux

(1.111)

( )1( )

1( )

( )

1 , 1,2, , 1

Ni j

i i jijj j i

N

j i i jijj j i

x xx

D

x x i NcD

= ≠

= ≠

∇ = − −

′′ ′′= − − = −

∑

∑

V V

n n& & L

1( )

(ln ) , 1,2, , 1N

i j j ii i

ij jj j i

MM i ND M

ω ωω ω

ρ = ≠

−∇ + ∇ = = −∑ J J

L



The diffusive mass fluxes for a system that contains N components can be obtained by solving a set of N – 1 equations

(1.112)

which is referred to as the generalized Fick equation, which is applicable to a system with concentration, temperature, and pressure gradients.


1

ln , 1,2, ,N

Ti i i ij j

j

D T i Nρ=

= − ∇ + =∑J d LD



The second term is diffusion caused by all other driving forces, including concentration gradient, pressure, and body force, where is the multicomponent Fick diffusivity. is obtained from

(1.113)where

(1.114) The multicomponent Fick diffusivities are symmetric ( )

and satisfy . Equation (1.112) can be rearranged to express the driving force, di,

in terms of mass flux, i.e.,

(1.115)

ijDijD

1( )

ˆ ˆN

i ij ij n inn n i

D Dω ω= ≠

= − ∑D

ˆ i jij ij

i jD D

x xω ω

=

ijD ij ji=D D1

1N

i iji

ω=

=∑ D

1( )

1( )

(ln )

, 1,2, ,

TTNi j ji

iij i jj j i

Ni j ji

ij i jj j i

x x DD TD

x xi N

D

ρ ρ

ρ ρ

= ≠

= ≠

= − − ∇

− − =

∑

∑

d

JJ L



The diffusional driving force di is obtained by

(1.116) For an ideal gas mixture, the generalized driving force becomes

(1.117) Substituting eq. (1.117) into eq. (1.112) and after some

manipulations, the final form of mass flux in the Fick form is obtained as

(1.118)

Similarly, substituting eq. (1.117) into eq. (1.115), the final form of the Maxwell-Stefan equations becomes

(1.119)

1

1 1 1lnN

u i ii i j j

ji i i

cR T g hT T pT M M

ρρ ρ ρ =

= ∇ + ∇ − ∇ − +

∑d X X

1

N

u i i i i i i j jj

cR T p pω ρ ω ρ=

= ∇ − ∇ − + ∑d X X

1 1

lnN N

T ii i ij i j j j j n n

u j n

D T p pcR T

ρ ω ρ ω ρ= =

= − ∇ + ∇ − ∇ − +

∑ ∑J X XD

1( )

1 1( )

(ln ), 1,2, ,

Ni j j i

u i i i iij j ij j i

T TN Ni j j i

i j j uij j ij j j i

x xcR T p p

D

x x D DcR T T i ND

ω ρρ ρ

ω ρρ ρ

= ≠

= = ≠

− = ∇ − ∇ −

+ − − ∇ =

∑

∑ ∑

J J X

X L



The multicomponent thermodiffusivity for species i is expressed as

(1.120)

where is the thermal diffusion ratio. For applications that involve mass diffusion in a

binary mixture containing species 1 and 2, the multicomponent thermodiffusivity becomes

(1.121)


1 212 122

T Ti

M MD D kM

ρ=

21( )

Ni jT T

i ij ijj j i

M MD D k

Mρ

= ≠

= ∑Tijk




Table 1.6 Thermal diffusion ratios for selected liquids and low-density gases

12TkComponents T(K) x1

Liquids C2H2Cl4 – n-C6H14 298 0.5 1.08

C2H4Br2 – C2H4Cl2 298 0.5 0.225

C2H2Cl4 – CCl4 298 0.5 0.060

CBr4 – CCl4 298 0.09 0.129

CCl4 – CH3OH 313 0.5 1.23

CH3OH – H2O 313 0.5 -0.137

cyclo-C6H12 – C6H6 313 0.5 0.100

Ne-He 330 0.2 0.05310.6 0.1004

0.294 0.05480.775 0.06630.10 0.01450.50 0.04320.90 0.0166


Advanced Heat and Mass Transfer by Amir Faghri, Yuwen Zhang, and John R. Howell(1.122)

The components of matrices A and B can be found below.

(1.123)

(1.124)(1.125)

(1.126) Equation (1.122) is in a form that can be very easily programmed

and it can be simplified for a variety of cases. For a binary mixture – a simplest mixture, A and B are both a single value.

(1.127)

(1.128)

[ ] [ ]1 1 lnTD TMρ − − = ∇ − ∇ J A B ω

1( )

N jiii

j j ii N iN i j ijA

M M D M M Dωω

= ≠

= − +

∑

1 1ij i

i j ij i N iNA

M M D M M Dω

= −

( )1ii i i i NB M Mω ω= − +

( )ij i N jB M Mω= −

1 2 12

1AM M D

= −

2 2 1 1 1 22 1 1 2 1 2

x M x M M MB M M M MM M M

ω ω= + = + =



Therefore, the diffusion mass flux of a binary mixture is:(1.129)

For a mixture of several species that are all very dilute in species N ( ), the B matrix is approximately

(1.130)

and the A matrix is approximately

(1.131)

( )12 lnTi i iJ D D Tρ ω= − ∇ − ∇

1, for 1,2,..., 1 and 1i Ni Nω ω= − ≈=1

2

1

0 00

00 0 N

MM

M −

≈

B

KO M

M O OK

( )( )

( )

11 1

12 2

11 -1,

0 0

00

0 0

N N

N N

N N N N

M M D

M M DA

M M D

−

−

−−

…

≈

…

O MM O O



The mixture molecular mass is also approximately equal to the molecular mass of the Nth component, . Therefore the diffusion flux for each component is:

(1.132) The mass flux of species 1 due to ordinary pressure, body force,

and thermal diffusion for a binary system becomes

(1.133)

One can reduce the generalized Maxwell-Stefan equation to the following form by neglecting pressure, body force, and thermal diffusion effects and assuming the mean molecular mass is constant

(1.134)

NM M≈

( )1 lnTi N i iJ D D Tρ ω= − ∇ − ∇

1 2 12 11 12 1

1 2 12 1 21 2 1 ( ) (ln )T

u

M M DD pp

M M D D TR T

ωρ ω

ω ω

= − ∇ + ∇

+ − − ∇

J

X X

1( )

, 1,2, , 1N

i j j ii

ij jj j i

M i ND M

ω ωω

ρ = ≠

−∇ = = −∑ J J

L



In order to solve for the mass flux J, eq. (1.134) can be rearranged to get

(1.135)where

(1.136)

(1.137)

(1.138)

The second mode of mass transfer, convective mass transfer, may be expressed in a manner analogous to eq. (1.74):

(1.139)where the species mass flux is again exemplified by transport from a

flat surface to a vapor stream flowing over that surface.

1

,1

, 1,2, , 1N

i eff ij jj

D i Nρ ω−

=

= − ∇ = −∑J L

1,[ ]eff ijD −= F

1( )

Ni k

iiiN N ik kk k i

M MFD M D M

ω ω

= ≠

= + ∑1 1 , ij iij j iN N

F M i jD M D M

ω

= − ≠

1 1, 1, 1, 1,( ) ( )m w m wm h hρ ρ ρ ω ω∞ ∞′′ = − = −&




Table 1.7 Summary of fundamental laws in momentum, heat and mass transfer




Table 1.7 Summary of fundamental laws in momentum, heat and mass transfer cont’d.



Sherwood number (1.140)

Three rates of diffusion transport equations for mass, momentum, and heat for 1-D and constant properties: Fick’s law (binary diffusion)

(1.141) Newton’s law viscosity (viscous fluid shear)

(1.142) Fourier’s law (heat conduction)

(1.143)

12

mh LShD

=


12, 12y

dm Ddyωρ′′ = −&

( )yx

d udyρτ ν= −

( )py

d c Tq

dyρ

α′′ = −



All the rate equations are of the same form on flux equals a constant times potential gradient.

The proportional constant is a function of method involved in the transport process, thus a transport phenomenon can be predicted on the base of knowledge of another transport phenomenon.

The convective heat and mass transfer coefficients for laminar and turbulent can be related by the following equation

(1.144)


2 / 31/ 3

12

LeLep

m m

h h kch h D

ρ= = =



Example 1.2:Show that the general Maxwell-Stefan equation (1.110)for the mass flux in a multicomponent gas system can be simplified for a binary system to Fick’s law, eq. (1.106). Solution: For a binary system, the number of component, N, is equal to 2. The Maxwell-Stefan equation, (1.110), for the first component (i=1) becomes

(1.145)


( )1 2 1 1 212

1x x xcD

′′ ′′∇ = − −n n& &



The molar fluxes of component 1 relative to stationary coordinate axes can be obtained by eq. (1.104), i.e.,

(1.146)which can be rearranged to

(1.147)Since , eq. (1.147) can be rearranged to

(1.148)Substituting eq. (1.148) into eq. (1.145), one obtains

(1.149)which is identical to the Fick’s law, eq. (1.106).


*1 1 1 2 1( )x′′ ′′ ′′= + +n n n J& & &

*1 1 1 2 1(1 )x x′′ ′′− − =n n J& &

2 11x x= −*

2 1 1 2 1x x′′ ′′− =n n J& &

*1 12 1cD x= − ∇J

Advanced Heat and Mass Transfer by Amir Faghri, Yuwen ...

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