Review of Mass Transfer and Radiation Notes PDF

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CN2125 Review #3 (April 2020) Mass Transfer Steady-state Diffusion (WWWR pg 452-474) Fick’s Equation

N A, z = −cD AB

General Differential Equation

.N A +

dy A + y A (N A, z + N B, z ) dz

cA − RA = 0 t

Bulk contribution can be ignored when diffusion through stagnant medium, diffusion through solids, equimolar counter-diffusion, dilute situations. (i) Diffusion through stagnant gas film p A,1 − p A,2 DAB P N A ,z = RT (z 2 − z1 ) p B ,lm =

 1 − y A ,2  cDAB cDAB ln  ( y A ,1 − y A ,2 ) = (z 2 − z 1 )  1 − y A ,1  zyB ,lm

(ii) Equimolar counter-diffusion N A, z = N B, z N A, z =

DAB (c − c ) (z2 − z1 ) A,1 A,2

(iii) Pseudo steady-state Assume steady-state for calculating flux Then use mass balance to equate flux with change in volume  z (accumulation) (output) N A ,z = A, l M A t (iv) Chemical reaction Heterogeneous reaction if reaction outside diffusion zone, RA = 0, RA represents a homogenous reaction

Unsteady-state Diffusion (WWWR pg 496-512) When the boundary conditions change with time, or when the concentration profile changes with time.

Fick’s 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) at t = 0, cA = cAo for all z - boundary conditions at t > 0, at z = 0, c A = c AS at z =  , cA = cA

cA − cAo = 1 − erf ( ) c AS − c Ao cAS − cA = erf ( ) - General solutions: c AS − c Ao z where  = 2 DABt

(ii) Transient diffusion into a defined geometry with negligible surface resistance - when you need to evaluate throughout the whole object at t = 0, cA = cAo for all z

at t > 0, at z = 0, c A = c AS - boundary conditions: at z = L, cA = cAS dcA = 0 because of symmetry) dz c −c - define dimensionless concentration change, Y = A AS c Ao − c AS D t L and relative time, X = AB2 , where x1 = x1 2 - use Heissler charts (Appendix F) x with Y and X, and relative position n = x1 D and relative resistance m = AB (m = 0 for negligible surface resistance) kc x1 - Y = YaYbYc if more than one dimension. (at z = L/2,

Convective Mass Transfer (WWWR pg 517-545)

N A = k c (c As − c A ) Important numbers:

Schmidt, Sc =



D AB kL Sherwood, Sh = c DAB Lv Reynolds, Re =



1. From exact analysis of laminar flow next to flat plate, no reaction, steady-state, incompressible, using Blasius solution to solve

Shx =

kc x = 0.332 Rex0.5 DAB

When Sc ≠ 1,

 = Sc1/3 c

 Sh x = 0.332 Rex0.5 Sc1/3 1/3 and Sh L = 2Sh x x = L = 0.664 Re 0.5 x Sc

2. For turbulent flow, Sh x = 0.0292 Rex Sc 4/5

1/3

L

3. For flow with both laminar and turbulent regions, k c

 k dx =  dx c

0

L

0

1 1 1 4 4 D D k C = 0.664 AB Re tr2 Sc 3 + 0.0365 AB Sc 3  ReL 5 − Retr 5     L L Re tr = 2  10 5

=



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

Cf h k = c =  C pv  v  2 Cf =

s

 ( v2 / 2 )

(b) Chilton-Colburn analogy, for Pr and Sc ≠ 1; laminar and turbulent regimes; valid for flat plates, cylinders, circular pipe and annulus etc k jD = c ( Sc2 / 3 ) v C jD = jH = f 2 Convective Mass Transfer between Phases (WWWR pg 551-563) 2 resistance theory - applies for gas-liquid and liquid-liquid systems - follow the equilibrium relations (Raoult’s law, Dalton’s law, Henry’s 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 =

- correlation of coefficients:

1/ k L 1/ k G or 1/ KL 1/ KG

1 1 m = + K G kG k L 1 1 1 = + K L mkG k L

- Refer to my summary notes on Gas➔Liquid, Liquid➔Gas, Liquid1➔Liquid2, Liquid2➔Liquid1

Radiation Definitions (ID pg 724-770) 1. Intensity: I ,e ( , , ) 

dq dA1 cos   d   d 

2. Emission: - spectral hemispherical is E ( ) = 

2

0



/2

0

I , e( ,  ,  ) cos  sin  d d



- total hemispherical is E =  E ( ) d 0

- for diffuse emitter, the spectral and total is E  ( ) =  I  ,e ( ) and E =  Ie 3. Irradiation: - spectral hemispherical is G ( ) = 

2

0



 /2

0

I , i( ,  ,  ) cos  sin  d d



- total hemispherical is G =  G (  ) d  0

- for diffuse incident irradiation, the spectral and total is G ( ) =  I ,i ( ) and G =  I i 4. Blackbodies: - ideal surface, absorbs all radiation, no other surface can emit more energy, is diffuse (a) Planck’s distribution 2hc 2o  5  exp(hco /  kT ) − 1 (b) Wien’s Displacement Law E ,b ( ,T ) =  I ,b ( , T ) =

maxT = C3 = 2897.8 m  K (c) Stefan-Boltzmann Law Eb =  T 4 Ib =

Eb



(d) Band emission - fraction of emission for the wavelength interval or band

-

F(0 →)

  



0  0

E , b d  E ,b d 

=

T 0

F( 1 → 2) = F(0 → 2) − F(0 →1)

E , b

 T5

d  = f ( T )

- 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) E ( , T )   ( ,T )   E  ,b ( ,T ) - emissivity:  E (T ) 0   ( , T ) E , b( , T ) d =  (T )  Eb (T ) Eb (T )

  ( )  - absorptivity:



G  ( )

Gabs G

 ( ) 

G ,ref (  )

- reflectivity:



G ,abs ( )

G  ( )

Gref G

  ( )  - transmissivity:



G, tr ( ) G ( )

Gtr G

- by balance,  +  +  = 1  +  +  =1

E1 (Ts ) - Kirchoff’s law:

1

=

E 2 (Ts )

2

=

=

b

s

)

= - 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 if spectral emissivity is not given and vice versa.  = 

© copyrighted material, not for circulation.

© copyrighted material, not for circulation.

CN2125 Heat and Mass Transfer 20192020; Final Examination Open-Book Examination: 1. Textbooks and all references 2. Homework and Tutorial Solutions. 3. Certified calculators.

Answer all questions.

I. (1), (2), (3). II. (4), (5), (6). III. (7), (8).

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; TemperatureTime 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.

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

IV. (9), (10) V. (11), (12)

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 –Black body, calculating energy loss from surface...