Appendix 12

CN2125 Review #3 (April 2015)

Mass Transfer

Steady-state Diffusion (WWWR pg 452-474)

Ficks Equation , , ,( )A

A z AB A A z B z

dyN cD y N N

dz

General Differential Equation . 0AA Ac

N Rt

Bulk contribution can be ignored when diffusion through stagnant medium, diffusion

through solids, equimolar counter-diffusion, dilute situations.

(i) Diffusion through stagnant gas film

,1 ,2

,

2 1 ,

,2

,1 ,2

2 1 ,1 ,

( )

1ln ( )

( ) 1

A AABA z

B lm

AAB ABA A

A B lm

p pD PN

RT z z p

ycD cDy y

z z y zy

(ii) Equimolar counterdiffusion

, ,

, ,1 ,2

2 1

( )( )

A z B z

ABA z A A

N N

DN c c

z z

(iii) Pseudo steady-state

Assume steady-state for calculating flux

Then use mass balance to equate flux with change in volume

(output),

,

A l

A z

A

zN

M t

(accumulation)

(iv) Chemical reaction

Heterogeneous reaction if reaction outside diffusion zone, RA = 0

Homogeneous reaction if reaction within diffusion zone, RA non zero.

If first order reaction,

1

2

120

A A

AAB A

R k c

d cD k c

dz

Integrate to concentration profile of

1 11 2

1

1

1

, 0

1

( ) cosh 2 sinh 2

/Hatta Number

tanh( /

/

tanh( /

A

AB AB

AB

AB

ABAB Ao

A z z

AB

k kc z c c

D D

k D

k D

k DD cN

k D

Unsteady-state Diffusion (WWWR pg 496-512)

When the boundary conditions change with time, or when the concentration profile

changes with time.

Ficks 2nd law, 1-dimension, no bulk contribution, and no reaction; solve by separation of variables or Laplace transforms.

(i) Transient diffusion into semi-infinite medium

- semi-infinite when one of the boundaries is infinite and does not change

(concentration constant with time)

- boundary conditions

at t = 0, for all z

at t > 0, at z = 0,

at z = ,

A Ao

A AS

A A

c c

c c

c c

- General solutions: 2

2

3

0

0

0

1 ( )

( )

where 2

2 2( ) ( ) if 0.5

3

1 1 if 1

( )

A Ao

AS Ao

AS A

AS Ao

AB

AS AoA

zAB

ABAS AoAz z

c cerf

c c

c cerf

c c

z

D t

erf e d

e

c cdc

dz D t

DN c c

t

(ii) Transient diffusion into a defined geometry with negligible surface resistance

- when you need to evaluate throughout the whole object

- boundary conditions:

at t = 0, for all z

at t > 0, at z = 0,

at z = ,

(at z = L/2, 0 because of symmetry)

A Ao

A AS

A AS

A

c c

c c

L c c

dc

dz

- define dimensionless concentration change, A AS

Ao AS

c cY

c c

and relative time, 121

, where 2

ABD t LX xx

- solutions:

2

2

( / 2)

1

( / 2)

,

1

4sin , n = 1,3,5...

4( ) cos

n X

n

n XA ABA z AB AS Ao

n

n zY e

L

dc D n zN D c c e

dz L L

- or use Heissler charts (Appendix F)

with Y and X, and relative position 1

xn

x

and relative resistance 1

(m = 0 if negligible surface resistance)AB

c

Dm

k x

but must obey (a) Ficks 2nd law, (b) uniform initial condition, (c) new condition constant with time

- Y = YaYbYc if more than one dimension.

Convective Mass Transfer (WWWR pg 517-545)

( )A c As AN k c c

Important numbers:

-

3

AB 2

Schmidt, Sc

Lewis, Le

Sherwood, Sh

Reynolds, Re

Grashof, Gr

AB

p AB

c

AB

A

D

k

C D

k L

D

Lv

L g

1. From exact analysis of laminar flow next to flat plate, no reaction, steady-state,

incompressible, using Blasius solution to solve 2

2

x

0.5

0

,

and 0

with boundary conditions:

at y = 0, 0, 0v

at y = , 1, 1

has solution of:

0.332( ) Re

0

A A Ax AB

y

x A As

A As

x A As

A As

Ay A As x

A y AB

c c cv D

x y y

vv

x y

v c c

c c

v c c

v c c

dcc c

dy x

N D

0.5

0.5

.332 Re( )

Sh 0.332 Re

xA As

cx x

AB

c cx

k x

D

When Sc 1,

1/3

0.5 1/3

0.5 1/3

Sc

Sh 0.332Re Sc

and Sh 2Sh 0.664Re Sc

c

x x

L xx x L

2. For turbulent flow, 4/5 1/3Sh 0.0292Re Scx x

3. For flow with both laminar and turbulent regions, 0

0

L

c

c L

k dxk

dx

4. Using Von-Karmans integral analysis by mass balance over the boundary layer

2 2

2 2

boundary conditions:

at y = 0, 0, 0

at y = , ,

at y = , 0, ( ) 0

at y = 0, 0, ( ) 0

x A As

x A As A As

xA As

xA As

v c c

v v c c c c

vc c

y y

vc c

y y

Assuming a velocity of concentration profile of: 2 3A Asc c a by cy dy ,

The solution by applying the boundaries is: 1/ 2 1/3

4/5 1/3

Sh 0.36Re Sc for laminar region

Sh 0.0292Re Sc for turbulent region

x x

x x

5. By analogies, 5 conditions must be met,

(a) Reynolds analogy, for laminar flow, Sc = 1 and Pr = 1, no drag, only skin friction

2

2

/ 2

fc

p

sf

Ckh

C v v

Cv

(b) Prandtl analogy, for flows with laminar and turbulent regimes

/ 2

1 5 / 2(Sc 1)

( / 2) ReScSh

1 5 / 2(Sc 1)

fc

f

f

f

Ck

v C

C

C

(c) Von-Karman analogy, for flows with laminar, turbulent and buffer layer

/ 2

51 5 / 2 Sc 1 ln 1 (Sc 1)

6

( / 2) ReScSh

1 5 / 2 Sc 1 ln (1 5Sc)/6

fc

f

f

f

Ck

vC

C

C

(d) Chilton-Colburn analogy, for Pr and Sc 1; laminar and turbulent regimes; valid for flat plates, cylinders, circular pipe and annulus; 0.6 < Sc < 2500.

2/3

2

cD

f

D H

kj Sc

v

Cj j

Convective Mass Transfer between Phases (WWWR pg 551-563)

2 resistance theory

- applies for gas-liquid and liquid-liquid systems

- follow the equilibrium relations (Raoults law, Daltons law, Henrys law, Distribution law)

- equilibrium established instantly at the interface only, no resistance at interface

- mass transfer resistance is the inverse of mass transfer coefficient

- overall resistance = 1 / KG or 1 / KL

- individual phase resistance = 1 / kG or 1 / kL

- percent resistance in a phase = 1/1/

or 1/ 1/

GL

L G

kk

K K

- correlation of coefficients:

1 1

1 1 1

G G L

L G L

m

K k k

K mk k

Liquid-Liquid convective mass transfer

Radiation

Definitions (ID pg 724-770)

1. Intensity: ,1

( , , )cos

e

dqI

dA d d

2. Emission:

- spectral hemispherical is 2 / 2

,0 0

( ) ( , , )cos sineE I d d

- total hemispherical is 0

( )E E d

- for diffuse emitter, the spectral and total is ,( ) ( )eE I and eE I

3. Irradiation:

- spectral hemispherical is 2 / 2

,0 0

( ) ( , , )cos siniG I d d

- total hemispherical is 0

( )G G d

- for diffuse incident irradiation, the spectral and total is ,( ) ( )iG I and iG I

4. Radiosity:

- spectral hemispherical is 2 / 2

,0 0

( ) ( , , )cos sine rJ I d d

- total hemispherical is 0

( )J J d

- for diffuse emitter and reflector , the spectral and total is ,( ) ( )e rJ I and e rJ I

5. Blackbodies:

- ideal surface, absorbs all radiation, no other surface can emit more energy, is diffuse

(a) Plancks distribution

2

, , 5

2( , ) ( , )

exp( / ) 1

ob b

o

hcE T I T

hc kT

(b) Wiens Displacement Law

max 3 2897.8T C m K

(c) Stefan-Boltzmann Law 4

b

bb

E T

EI

(d) Band emission

- fraction of emission for the wavelength interval or band

-

1 2 2 1

, ,0(0 ) 50

,0

( ) (0 ) (0 )

( )Tb b

b

E d EF d f T

TE d

F F F

- use Table 12.1 in ID or Table 23.1 in WWWR

6. Surface emission

- for REAL surfaces, we define ratios to IDEAL surfaces

- can assume diffuse (or averaged over all directions, and thus is hemispherical)

- emissivity: ,

,0

( , )( , )

( , )

( , ) ( , )( )( )

( ) ( )

b

b

b b

E TT

E T

T E T dE TT

E T E T

- absorptivity:

, ( )( )

( )

abs

abs

G

G

G

G

- reflectivity:

, ( )( )

( )

ref

ref

G

G

G

G

- transmissivity:

, ( )( )

( )

tr

tr

G

G

G

G

- by balance, 1

- Kirchoffs law:

1 2

1 2

( ) ( )( )s s b s

E T E TE T

- If the spectral distribution of absorptivity or emissivity is given then we can equate them to

determine the other quantity. If spectral distribution of absorptivity is given then we can equate

them to spectral emissivity is spectral emissivity is not given and vice versa.

copyrighted material, not for circulation.

copyrighted material, not for circulation.

CN2125 Heat and Mass Transfer 2014-

2015; Final Examination

Answer all questions.

I. (1), (2), (3).

II. (4), (5), (6).

III. (7), (8).

-------------------------------

IV. (9), (10)

V. (11)

Hot topics:

1. Steady-state diffusion pseudo-steady-state, calculating flux and concentration profile

2. Unsteady-state diffusion calculating concentration, time or position

3. Convective mass-transfer calculating flux, coefficients

4. Radiation Solid angles, black body, calculating

energy loss from surface

Open-Book Examination:

1. Textbooks and all references 2. Homework and Tutorial Solutions. 3. Certified calculators.

Hot Topics:

(i) Steady Heat Conduction: Basic definitions;

Differential equations and boundary conditions;

Thermal resistor models for composite walls. Critical

thickness of insulation. Uniform and non-uniform heat

generation and the resulting temperature profiles in

different coordinate systems. (ii) Unsteady Heat

Conduction: Lump parameter analysis; Temperature-

Time charts for simple geometrical shape (1-D)

(iii) Energy- and Momentum Transfer Analogies:

Application to pipe flow. (iv) Natural Convection:

Correlations for spheres and cylinders. (v) Natural

convection for vertical and horizontal cylinders.

Forced Convection: Laminar and Turbulent Pipe

flows. Cross flow past through spheres. (vi) Boiling

and Condensation: Nucleate and film boiling; Film

condensation on vertical plate; (vii) heat exchangers;

(viii) Mass Transfer Fundamentals: Estimation of gas

and liquid phase diffusivities. Pore diffusion.