高等輸送二 — 質傳

Lecture 2Diffusion in concentrated solutions

郭修伯 助理教授

Dilute vs. concentrated (solution)

• Any mass flux include both convection and diffusion (Maxwell, 1860).

• Diffusion causes convection.– Dilute solution: convection caused by diffusion is small and

can be neglected.– Concentrated solution: both convection and diffusion have to

be considered.

• Difference between Heat transfer and Mass transfer– Heat conduction can occur without convection– Diffusion and convection always occur together

Mass transported by convection

Mass transported by diffusion

Total mass transported =

))((

1

timearea

mass

n 111 vn c

w.r.t fixed coordinate

Average solute velocity

Local concentration

aaaa ccc vjvvvn 111111 )(

Convective reference velocity???

Convective reference velocity av

How to choose its value ?Our goal: choose va so that va is zero as frequently as possible!

aa c vjn 111 The mass transfer reduced to “diffusion” only.

va can be:

• the molar average velocity• good for ideal gases where the molar concentration is constant

• the mass average velocity• good for constant-density liquid

• the volume average velocity• good for constant-density liquid and for ideal gas

What are those?

nitrogen hydrogen

The temperature and pressure are such that the diffusion coefficient is 0.1 cm2/sec. Find the molar average velocity v*, the mass average velocity v, the volume average velocity v0 at the average concentration in the system.

The volume in this system does not move, so v0 = 0. If the gases are ideal, the molar concentration is constant, so v* = 0.

For nitrogen at an average concentration of 0.5 c

)( 110111 lccl

Dvcj Diffusion in the thin-film

sec/02.0)5.0

01(

10

1.0)(

1

1101 cm

c

cc

l

Dv l

For hydrogen at an average concentration of 0.5 c

sec/02.0)5.0

10(

10

1.0)(

2

2202 cm

c

cc

l

Dv l

933.0)2(5.0)28(5.0

)28(5.0~~

~

2211

111

McMc

Mcw

067.0)2(5.0)28(5.0

)2(5.0~~

~

2211

221

McMc

Mcw

Mass fraction of nitrogen

Mass fraction of hydrogen

sec/017.0)02.0(067.0)02.0(933.02211 cmwvwvv

z

Fast evaporation by diffusion and convection

A mass balance on a differential volume A z gives:

Solute transported out at z + z

Solute transported in at z

Solute accumulated in

volume Az=

zzz AnAnzcAt

111

Dividing A zz 0

z

n

t

c

11

l

aa c vjn 111

z

n

t

c

11

choose volume average velocity v0

)( 22211111

1 vVcvVccdz

dcD n

s.s

If the solvent vapor is stagnant

1111

1 nVcdz

dcD n

111 vcn

cp

RTV

11 Total molar concentration

111

1 nnc

c

dz

dcD

111

1 nn ydz

dyDc

contribution of both diffusion and convection and it is constant.

111

1 nn ydz

dyDc

B.C.

.

0

11

101

const

lzyy

zyy

l

1n

lz

l

y

y

y

y/

10

1

10

1

1

1

1

1

10

11 1

1ln

y

y

l

Dc ln

dz

dyDc 1

1 j

10

1

/

10

1101 1

1ln

1

11

y

y

y

y

l

yDc l

lz

lj

The diffusion flux is smallest at the bottom of the capillary and rises to a maximum value at the top of the capillary.

n1 = constant

Exponential concentration profile

lz

l

y

y

y

y/

10

1

10

1

1

1

1

1

10

11 1

1ln

y

y

l

Dc ln

10

1

/

10

1101 1

1ln

1

11

y

y

y

y

l

yDc l

lz

lj

Concentrated solution

Dilute solution

...)(11 110101 lyyl

zyy ll cc

l

Dyy

l

Dc11011011 jn

or

l

zcccc l )( 101101

Linear concentration profile

Fast Diffusion into a Semiinfinite slab

Fast evaporation by diffusion and convection

A mass balance on a differential volume A z gives:

Solute transported out at z + z

Solute transported in at z

Solute accumulated in

volume Az=

zzz AnAnzcAt

111

Dividing A zz 0

z

n

t

c

11

z

l, >>

aa c vjn 111

z

n

t

c

11

choose volume average velocity v0

)( 222111111 vVcvVccz

cD

zt

c

111 vcn

z

VnVnc

z

cD

t

c

)( 22111

21

21

222 vcn

)()( 22112211 VnVnz

VcVct

Continuity equation

z

cVnVn

z

cD

t

c

1

221121

21 )(

=1independent of z

z

cVnVn

z

cD

t

c

1

221121

21 )(

z

cVn

z

cD

t

cz

1

01121

21 )(

02 n Solvent gas is insoluble

011101

01

zzz nVcz

cDn

z

cnVc

z

cDV

z

cD

t

czz

1

011101

121

21 )(

z

cnVc

z

cDV

z

cD

t

czz

1

011101

121

21 )(

z

cnVc

z

cDV

z

cD

t

czz

1

011101

121

21 )(

B.C.

0,,0

)(,0,0

0,0,0

1

11

1

czt

satcczt

czallt

0)(2 121

2

d

dc

d

cd

Dt

z

4

)(2

101110

11 zz nVc

z

cDV

0,

)(,0

1

11

c

satcc

erf

erf

satc

c

1

)(1

)(1

1

How important is the convection term?

The vapor pressure of benzene at 6ºC is about 37 mmHgThe vapor pressure of benzene at 60ºC is about 395 mmHg

z

l

Total flux

049.0760

37)(1161

p

satp

c

cy

C

520.0760

395)(11601

p

satp

c

cy

C

10

11 1

1ln

y

y

l

Dc ln lyyl

Dc1101 n

Diffusion + convection Diffusion

6 ºC

049.01

01ln1 l

Dcn

60 ºC

52.01

01ln1 l

Dcn

0049.01 l

Dcn

052.01 l

Dcn

2 %

40 %

General form of the mass balance equation

x

z

y

x

z

yA

C

B

D

E

F

G

H

Input rate through ABCD zynx

Input rate through ADHE zxny

Input rate through ABFE yxnz

Output rate through EFGH xzynx

zyn xx

Output rate through BCGF yzxny

zxn yy

Output rate through CDHG zyxnz

yxn zz

input - output + generation = accumulation

zyxt

czyxrzyxn

zyzxn

yxzyn

x zyx

1

t

crn

zn

yn

x zyx

1t

cr

1n

t

cr

1n

00 vjn c Diffusion and the convection term

102 rccD

t

c

v

General equation include the effects of chemical reaction, convection, and concentration-driven diffusion

1rnz

ny

nxt

czyx

1

11rn

zn

rrn

rrt

czr

12

2 sin

1sin

sin

11rn

rn

rnr

rrt

cr

Combine with Fick’s law

Find differential equations for calculating the drop in flux caused by the concentration polarization (i.e., by the salt accumulation near the membrane surface)

Fresh water

Fresh water

Salt water

membrane

x

y

1rnz

ny

nxt

czyx

s.s. no reaction2D00 vjn c

21

2

21

2

11 y

c

x

cDvc

yvc

x yx

Responsible to concentration polarization

Usually much smaller than the convection term

Membrane example

The dissolution rate is diffusion-controlled. Calculate the rate at which the disc dissolves.

s.s. no reaction00 vjn c

21

2

21

2

21101

010 11

z

cc

rr

cr

rrD

z

cv

c

r

v

r

cv zr

Spinning disc example

Solute from dissolving disc

flow

1

11rn

zn

rrn

rrt

czr

Angularly symmetricAngularly symmetricDisc infinitely wideDisc infinitely wide

21

210

dz

cdD

dz

dcvz

B.C.

0,

)(,0

1

11

cz

satccz

0

)()/1(

0

)()/1(

1

1

0

0

0

0

1)(

dre

dre

satc

cr

z

r

z

dssvD

z dssvD

0

0

1

13

3

1)( due

due

satc

cu

u

Levich, 1962

1

2/1

6/13/182.1

vD

z

0

0

1

13

3

1)( due

due

satc

cu

u

01

01

zz z

cDj

)(62.0)(62.0 1

31

21

2

16/1

2/13/2

01 satcD

d

d

Dsatc

v

Dj z

1

2/1

6/13/182.1

vD

z

Reynolds number Schmidt number

Independent of the disc radius: constant flux

Mass transfer versus Heat transfer

Diffusion-induced convection; chemical reaction only for mass transferRadiation only for heat transfer

J. Crank, (1975) The Mathematics of Diffusion, 2nd ed. Oxford: Clarendon Press(include reactions)

H.S. Carslaw, and J. C. Jaeger (1986) The Conduction of Heat in Solids, 2nd ed. Oxford: Clarendon Press(a more complete selection of boundary conditions)

Diffusion through a polymer film

Diaphragm （隔板） cell

Initial pressure zero

Polymer film

Initial pressure one atmosphere

Measure the ethylene concentration in the upper compartment as a function of time.

From mass balance and Fick’s law: 21

21

z

cD

t

c

Boundary conditions:

0,,0

,0,0

0,0,0

1

01

1

lHpclzt

Hpczt

czallt

l: film’s thickness H: Henry’s law coefficient

21

21

z

cD

t

c

0,,0

,0,0

0,0,0

1

01

1

lHpclzt

Hpczt

czallt

Crank, (1975)

222 /

10

1 )/sin(21 ltDn

n

en

lzn

l

z

Hp

c

Mole balance on the top compartment: lZz

cAD

dt

dp

RT

V

dt

dN

11

)1(

cos2 222 /

122

20 ltDn

n

en

nHlHDt

Vl

ARTpp

t = 0, p = 0

)1(

cos2 222 /

122

20 ltDn

n

en

nHlHDt

Vl

ARTpp

Large time

6

20 Hl

HDtVl

ARTpp

Time

Pressure From the intercept and the slope, we can obtain the equilibrium Henry’s law coefficient, H, and the diffusion coefficient, D, in a single experiment

A dissolving pill

Estimate the time required to produce a steady flux of drug pill in the gut ( 腸 ).

Assumption: the drug’s dissolution is controlled by diffusion into the stagnant contents of the gut.(i.e. , • The dissolution is diffusion-controlled• The surrounding s are stagnant )

The mass balance on a spherical shell:

12

2 sin

1sin

sin

11rn

rn

rnr

rrt

cr

Diffusion control

r

cr

rr

D

t

c 22

Boundary conditions:

0,,0

)(,,0

0,0,0

0

crt

satccRrt

crallt

r

cr

rr

D

t

c 22

0,,0

)(,,0

0,0,0

0

crt

satccRrt

crallt

Crank, (1975)

Carslaw and Jaeger (1986)r

cu

)4Dt

R-rerf-(1

)(00

r

R

satc

c

0Rrz

cDjn

The flux in a dilute solution:

Dt

R

R

satDcn

0

0

1)(

The time to reach steady flux: 10 Dt

R

Time ~ 80h is much longer than the experimental result (i.e., 10 min)Because: free convection driven by the density difference caused by the dissolution

Effective diffusion coefficients in a porous catalyst pellet

A porous catalyst pellet containing a dilute gaseous solution. Determine the effective diffusion of solute by dropping this pellet into a small, well stirred bath of a solvent gas and measuring how fast the solute appears in this bath.

A mass balance on a spherical pellet:

12

2 sin

1sin

sin

11rn

rn

rnr

rrt

cr

Diffusion control

r

cr

rr

D

t

c eff 122

1

Boundary conditions:

)(,,0

0,0,0

,0,0

110

1

101

tCcRrtr

crt

ccrallt

Bath concentration

A mass balance on the solute in the bath of volume VB

00

121

21 44 RreffRrB r

cDRnR

dt

dCV

t = 0, C1 = 0

r

cr

rr

D

t

c eff 122

1

)(,,0

0,0,0

,0,0

110

1

101

tCcRrtr

crt

ccrallt

122

02

101 )1(9

61

2

n n

tD

B BRB

eBV

B

cC

meff

Crank, (1975)Carslaw and Jaeger (1986)

220

00 3

3)tan(

n

nn BR

RR

30)3/4( R

VB B

Void fraction in the sphere