부산대학교현규열및물질전달 1

부산대학교 화공생명공학부현 규 (Kyu Hyun)

열 및 물질전달(Heat and Mass Transfer)

(7)

부산대학교현규

Diffusivity or Diffusion coefficient (확산계수)

• 확산도는 분자량, 분자크기, 그리고 혼합물 내 분자상호간 인력과같은 분자성질에 의존한다.

• 기체 확산에 대해 매우 잘 맞는 이론이 존재하므로 이성분 기체혼합물의 확산도를 잘 추정할 수 있다.

• 액체의 경우 분자 크기로부터 분자간 힘을 추정하기가 매우 어려운점 등으로 인해 확산도의 추정이 수월하지 않다.

• 고체의 경우도 확산도를 예측하기란 쉽지않다.

• 대략적인 DAB의 값은 다음과 같다.

• 기체내 기체 : 0.1-1cm2/s = 10-5-10-4m2/s • 액체내 액체 또는 기체: 10-5cm2/s = 10-9m2/s • 고체내 기체: 10-6-10-10cm2/s = 10-10-10-14m2/s

물질전달 2

부산대학교현규

Diffusivity of gases (1)

• Theoretical expressions for the diffusion coefficient in low-density gaseous mixture as a function of the system’s molecular properties were derived by Sutherland, Jeans, and Chapman and Cowling based upon the Kinetic Theory of Gases.

• Lenard Jones Potential : the attractive and repulsive forces

d

Rigid, nonattracting spherical molecules

Diameter=dMass=m

물질전달 3

부산대학교현규

Diffusivity of gases (2)

• Lenard-Jones Potential

• Self Diffusion coefficient213

223

23

32

/

/

/

*

AAAA M

NP

TD P

DAA

1*

23/* TDAA

612 rB

rArVLJ )(

Lenard-Jones diameter of the spherical molecule, 실제로는분자의지름이되는데분자의지름보다크게된다.

d

d

물질전달 4

부산대학교현규

Diffusivity of gases (3)

• Hirschfelder et al. (1949) : gas pairs of nonploar, nonreacting molecules

• The Wilke-Lee modification of the Hirschfelder-Bird-Spotz method

21

232

110018580/

/.

BADABAB MM

TP

D

2.37T)f(kT/ε)(rp

)1/M1/M0.349(1.08410 3/2

AB2

ABt

BA4 21

11/

BAAB MM

D

collision integral (확산충돌적분): 실제분자가당구공과같지않음을고려해준다. Rigid molecules=1

물질전달 5

부산대학교현규

Diffusivity of Liquids (1)• Lack of a rigorous model• For dilute solute (A) of very large, rigid, spherical molecules and

stationary solvent (B) of small molecule:

• Stokes-Einstein equation:

• Wilke-Chang empirical equation: (for dilute solution)

tconsT

DNR

RTD BoAB

AAB

oAB tan

6

BookinunitSIv

TMxDAB

BBoAB ).()(.

.

/

44210311760

2118

물질전달 6

부산대학교현규

Diffusivity of Liquids (2)• The diffusivity in concentrated solutions differs from that in dilute

solutions: Change in viscosity & changes in the degree of nonidealityof the solution (ideal solution: Raoult’s law)

• For strong electrolytes dissolved in water, the diffusion rates are those of the individual ions, which move more rapidly than the large, undissociated molecules,

• Nernst-Haskell equation (a single salt in an aqueous solution)

).()lnln()()( , 4521 PT

B

AxB

oAB

xA

oBAA

BA DDD

).( 462AA

tA

AA

AA px

pypx

p

)]/1()/1[(F)]n/1()n/1[(RT)D( 2AB

물질전달 7

부산대학교현규

Similarity between Momentum, Heat, Mass transfer (1) – Driving Force

물질전달 8

-Driving force -Driving force -Driving force

-Initial state -Initial state -Initial state

부산대학교현규

Similarity between Momentum, Heat, Mass transfer (1) – Driving Force

물질전달 9

-Driving force -Driving force -Driving force

-Initial state -Initial state -Initial state

부산대학교현규

Mass Balance of component A (1)• In any element of fluid, the equation for a mass balance is

(Rate of mass Accumulation)=(Rate of mass flow In) –(Rate of mass flow Out)

+ Rate of Generation (or Consumption)

zyxr

yxnn

zxnn

zynnt

zyx

A

zzzAzzA

yyyAyyA

xxxAxxAA

)(

)(

)(

,,

,,

,,

vjvn AAAAA AofFluxmass )(

=mass of A produced/(volume)(time)

물질전달 10

부산대학교현규

Mass Balance of component A (2)• Dividing through ΔxΔyΔz gives

• Taking the limit as Δx, Δy, and Δz approach zero

• Vector form으로 정리하면 다음과 같다.

AzzzAzzAyyyAyyAxxxAxxAA r

znn

ynn

xnn

t

)()()()()()( ,,,,,,

AzAyAxAA r

zn

yn

xn

t

,,,

AAA r

t

n vjvn AAAAA AofFluxmass )(

Molecular diffusion

Convection

물질전달 11

부산대학교현규

Mass Balance of component A (3)• Mass flux를 convection term과 molecular diffusion term으로나누어서대입하면

• Molar quantity로다시정리하면

AAAA r

t

jv)(

-Net rate of addition of mass A per unit volume by convection

-Net rate of addition of mass A per unit volume by molecular diffusion

-Rate of production of mass A per unit volume by reaction

-Rate of increase of mass A per unit volume

AAAA Rct

c

JV)(

물질전달 12

부산대학교현규

Mass Balance of component B

AAA r

t

n

BBB r

t

n

)()()(BABA

BA rrt

nn

)( v

t

vvvnn BBAABA

BA rr

물질전달 13

부산대학교현규

Equation of Continuity for component A• Basic Equation of motion using Momentum Balance

• ρ & DAB (밀도와 확산도)가 일정

AAAA r

t

jv)(

AAAA Rct

c

JV)(

AAABAA rwDw

tw

2 v

AAABAA RcDct

c

2V

AAABA rwD

DtDw

2

AAABA RcD

DtDc

2

물질전달 14

부산대학교현규

Summary I• General Balance equation :

• General Balance equation in case of flow :

• Flux = the rate of flow of any quantity per unit area• Divergence = the volume density of the outward flux of a vector field

from an infinitesimal volume = output flux – input flux

Accumulation = Input- Output + generation - consumption

Rate of accumulation = rate of Input – rate of Output + rate of (generation – consumption)

div ( , , ) ( , , ) ( , , ).P x y z Q x y z R x y zx y z

F F

물질전달 (부산대학교현규) 15

부산대학교현규

• Fick’s law of diffusion

Summary II

• ρ & μ (밀도와 점도)가 일정

Momentum balance equation Mass balance equationV A

gτVVV

p

t][)()(

AAA r

t

n

gΦV

p

t)(

AAAA r

t

jV)(

The rate of addition quantity by convection and molecular term

• ρ & DAB 가 일정

gVV pDtD 2

AAABA rwD

DtDw

2

AABA wD jdy

dwDj AABAy

dydux

yx )( Tvv τ

• Newton’s law of viscosity

물질전달 (부산대학교현규) 16