Mass Transfer With Kinetics

transcript

Introduction to Multiphase ReactorsBasics of mass transfer with chemical reaction

Dr. Anand V. PatwardhanProfessor of Chemical EngineeringInstitute of Chemical Technology

Nathalal M. Parikh RoadMatunga (East), Mumbai-400019

av.patwardhan@ictmumbai.edu.in; avpuict@gmail.com; avpiitkgp@gmail.com

Objective: to ascertain the effect of chemical reaction on specific rate (flux) of mass transfer

Consider G-L, L-L, S-L reactions (absorption+reaction, extraction+reaction, leaching+reaction, respectively)

Assumptions:

Reaction occurs exclusively in liquid phase (Bphase)

Solute A (gas or liquid) slightly soluble in B phase

Mass transfer resistance confined to B phase

Reactive species B considered nonvolatile

IRREVERSIBLE REACTIONS:

Z ... (1)

A B productsth thm Aw.r.t. , order w.r n B.t.

Depending on the relative rates of diffusion and chemical reaction, four “regimes”:

Regime 1: very slow reactions

Regime 2: slow reactions

Regime 3: fast reactions

Regime 4, instantaneous reactions

REGIME 1: VERY SLOW REACTIONS

Reaction rate << rate of transfer of A into B phase

⇒ B phase saturated with A at any given moment

⇒ Rate of formation of products determined by true kinetics of homogeneous chemical reaction

Diffusional factors are unimportant in this regime

⇒ The transfer rate of A, is given by

( )m nR a k C C A L mn Ai

..= ε

Regime 1: very slow reactions

CAiCAi

G/L/Sphase

Liquid phase

Diffusionfilm

Condition for validity of Regime 1:

( )m nk aC k C CL Ai L mn Ai B

. 3>> ε

{ } Volumetric rateVolumetric rate of of homogeneousmass transfer chemical reactio

kmol kmol 3 3s nm s m

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

REGIME 2: SLOW REACTIONS

Reaction rate faster than rate of transfer of A into B phase

Reaction occurs uniformly throughout B phase, but,

Rate determined by rate of transfer of A into B phase.

The concentration of dissolved A in the bulk of phase B is zero.

( )R a k aC A L Ai

... 4=

Regime 2: slow reactions

C 0Ab =

Liquid phase

G/L/Sphase CAi

CB0 CB0

Diffusionfilm

Also, the amount of unreacted dissolved A that reacts in the diffusion film compared to that which reaches the bulk of B phase should be negligible. The condition for this to happen is,

Condition for validity of Regime 2:

( )m nk aC k C CL Ai L mn Ai B

. 5<< ε

{ } Volumetric rateVolumetric rate of of homogeneousmass transfer chemical reactio

kmol kmol 3 3s nm s m

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪⎩ ⎭

1 22 m 1 nD k C Cm 1 A mn Ai Bb 1 ...

⎛ ⎞−⎜ ⎟+⎝ ⎠ <<

REGIME BETWEEN 1 and 2 (Regime 1-2)

For some systems condition (6) satisfied but condition (5) is not satisfied

Dissolved A concentration in bulk phase, CA0 is finite, and CA < CAi

In such a case,

( ) ( )

... 7m nR a k C CA L mn A0 Bb

R a k a C C A L Ai A0

C R aAb A

m nk C C k a C CL mn Ab Bb L Ai A

Simultaneous solution of Equations (7) and (8) gives

stfor (if

k aC CL Ai AiC n nA0 k C k a k C

reaction is 1 ord

mn Bb L L mn Bb 1k

.r.t. )

ε = −

= =ε + ε

SPECIAL CASES:

For m = 1, Equations (7) and (8) are linear in [A]. Eliminating [A0] gives,

C 1 1Ai ... nR a k a k CA L L 1n Bb

R a aC ... A Ai

1 1 1 ... nk a k a

then, 11k CLR L L 1n B

Plot of versus will be a straight line

with slope and

1 1 nk a CLR

Y-interce

L Bb1 1

For m = n = 1:

Plot of versus will be a straight line

1 1 1 ... k a k a k C

LR L L 1n Bb1 1

k a CLR L Bb

th slope and Y-interck

2 Lept

Zero order reaction w.r.t. A (m = 0):

Volumetric rate of reaction (Equation (7)):

( )nR a k C A L 0n B

.. 13b

. = ε

( )nk CR a

L 0n BbAC C C k a k aAb Ai Ai

ε= − = −

Provided there is sufficient amount of dissolved A in the bulk of liquid !

Equation (8) gives,

Second order reaction w.r.t. A (m = 2):

Equation (8) gives,

2 nk C CR aL 2n Ab BbAC C C

k a k aAb Ai AiL L

2nk C C C CL 2n Ai Bb Ab Ab 1 0

k a C CL A

... 15

ε= − = −

ε ⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟+ − =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝

⇒⎠

For a general order reaction (mth order) w.r.t. A, the concentration of A in the bulk (A0) is found by trial or error method or any suitable numerical method, from the following equations:

( )nk C k a CL mn Bb L Ai

kmoVolumetric rate Volumetric rateof homogeneous of

chemical reaction mass transfer

l kmol3

mC CAb A

3s m s m

nk C k a k aCL mn Bb L L Ai

mC CAb Ab

⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪⎨

ε = −

ε + =

⎬ ⎨ ⎬⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩ ⎭

ENHANCEMENT FACTOR FOR MASS TRANSFER:

The “enhancement factor for mass transfer” (or simply, enhancement factor), φ, is defined as:

rate of mass transfer with chemical reactionRate coActual

Regime 2 slow reaction regimenforming to

... 1R a

FAST REACTIONS(Regime 3 and Regime between 2 and 3)

Under certain condition, diffusion and reaction are parallel steps. This condition is given by:

1 22 m 1 nD k C Cm 1 A mn Ai Bb 1 ... 18

⎛ ⎞−⎜ ⎟+⎝ ⎠ >>

Regime 3: fast reactions

C 0Ab =

G/L/Sphase

Liquid phase

Increase inreaction rate

dCA 0dx

CBbDiffusion

filmBulk

Under certain conditions, there is no depletion of reactive species B in the film; the condition is given by,

1 22 m 1 nD k C C C Dm 1 A mn Ai Bb Bb B

k ZC DL Ai

⎛ ⎞−⎜ ⎟ ⎛ ⎞⎝ ⎠+ ⎜ ⎟<<⎜ ⎟⎝ ⎠

The experimental data are better correlated through (DB/DA)½, rather than (DB/DA).

at ... 2dC

B x 0, C C , 0dxA Ai

x , C 0A

at ... 22

= δ =

REGIME 3 (A reacts entirely in the Film) ...

Boundary conditions:

( )2d C

m n m ...AD k C C k C 2 20 ,

n A Bb m Adxnk k C

m mn Bb

Solution:

dCAR D

dxA Ax 0

1 22 m 1C D Cm 1Ai A Ai

nk Cmn B

2 m 1C D Cm 1Ai A

⎛ ⎞⎜ ⎟= −⎜ ⎟⎝ ⎠ =

⎡ ⎤⎛ ⎞ −= ⎜ ⎟⎢ ⎥+⎝ ⎠⎣ ⎦

For example:

Kinetics of absorption of carbon monoxide in aqueous solutions of sodium hydroxide and aqueous calcium hydroxide slurries. Anand V. Patwardhan; Man Mohan Sharma, Industrial & Engineering Chemistry Research 1989, 28, 5-9.

Kinetics of reactive absorption of carbon dioxide with solutions of aniline in non-aqueous aprotic solvents. Srikanta Dinda; Anand V. Patwardhan; Narayan C. Pradhan, Industrial & Engineering Chemistry Research2006, 45, 6632-6639.

Regime Overlapping 1, 2, & 3 (Generalised Derivation):

Reaction occurs partly in Film, partly in Bulk, and yet there may be a finite concentration of A in the bulk of the liquid phase (B-phase).

( )2d C

AD k C2A 1 A ..

dx. 32=

Boundary conditions:

B.C. 1 : at

B.C. 2 : at

x 0, C CA Ai

x , C CA Ab

= δ =

2 2d C d C kA A 1D k C C2 2 DA 1 A Adx dx A

2d C k2A 1c C ; c2 DAdx A

cx cxC h e h eA 1 2

x 0, C C A Ai

C h h CAi 1 2 Ai

.C. 1 : at gi

−= +

⇒ == +

x , C C A Ab

c cC h e h eAb 1 2

B.C. 2 : at g

C h e C eAb 1 Ai

C C C CAb Ai

c ce eh hc c c c1 2e

eAi Ab

cx cow,

x A h e h e1 2

= δ =

δ − δ= +

δ − δ−

− δ δ− −= =

δ − δ δ − δ− −

dC dCcx cxA Ac h e h e c h h

dx dx1 2 1 2x 0

dC kA 1D D h h

dx DA A 2 1Ax 0

R D k h hA A 1 2 1

c cC e C C C eAi Ab Ab AiR D k c c c cA A 1 e e e e

−⎡ ⎤ ⎡ ⎤= − = −⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦=

⎡ ⎤− = −⎢ ⎥⎣ ⎦=

⎡ ⎤= −⎢ ⎥⎣ ⎦

δ − δ⎡ ⎤− −⎢ ⎥= −⎢ ⎥δ − δ δ − δ⎣ ⎦− −

⇒ ⇒

c cC e C C C eAi Ab Ab AiR D k c cA A 1 e e

c cC e e 2CAi AbR D k c cA A 1 e e

Cc ce e AbR D k C c c c cA A 1 Ai e e e e2

δ − δ⎡ ⎤− − +⎢ ⎥= ⎢ ⎥δ − δ⎣ ⎦−

δ − δ⎧ ⎫⎡ ⎤+ −⎪ ⎪⎢ ⎥⎣ ⎦⎨ ⎬=δ − δ⎪ ⎪⎩ ⎭−

δ − δ⎡ ⎤⎛ ⎞+⎢ ⎥⎜ ⎟= −δ − δ δ − δ⎜ ⎟ ⎛ ⎞⎢ ⎥− −

⎝ ⎠ ⎜ ⎟⎢ ⎥⎜ ⎟⎢ ⎝ ⎠⎣ ⎦

( ) ( )

Substituting and gi

CCAbAiR D k

tanh c sinh cA A 1

k D1 A c ,

D kA L

CCAbAiR D k

A A 1 D k D kA 1 A 1tanh sinh

k kL L

⎡ ⎤⎢ ⎥= −

δ δ⎢ ⎥⎣ ⎦

= δ =

⎡ ⎤⎢ ⎥= −

⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜

⎟⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Under different appropriate conditions, Equation (33) reduces to Regime 1, 2, or 3.

( )C D k C D k

Ai A 1 Ab A 1R A D k D k

A 1 A 1tanh sinhk k

... 33=⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥

⎣ ⎦

Rearranging Equation (33):

[ ] [ ]

( ) [ ]( )

( )[ ]

reacted in

CCAbAiR k

D k AA 1 Mk A

filmwhere, Hatta number diffused

M M 1 limiAs , tanh M

tanh M si

M 1 li, sinh M

⎧ ⎫⎪ ⎪⎨ ⎬= −⎪ ⎪⎩ ⎭

⎡ ⎤⎢ ⎥= = =⎢ ⎥⎣ ⎦

→ ∞ →

→⎛ ⎞⎜ ⎟⎝

→⎠

Rearranging Equation (33):

CAiR k C

A L AbR k C M

A L AiD k

A 1R k CkA L Ai

LR C D k ... 23

... Regime 3 All reacts wi

thin the diffusionA Ai A 1

filA m

⎧ ⎫⎪ ⎪⎨ ⎬= −⎪ ⎪⎩ ⎭

[ ] [ ]

( )[ ]

CCAbAiR k

M 0 M 1 1

MAs , limittanh M

MAs , limitsinh M

which gives: ... Regime 1-2R k C C A L Ai Ab

No reacts within the diffusi

A0 C C

h M sinh

⎧ ⎫⎪ ⎪⎨ ⎬= −

⎛ ⎞⎜ ⎟⎝ ⎠⎛ ⎞⎜ ⎟⎝ ⎠

⎡ ⎤⎢ ⎥⎢ ⎥

⎪ ⎪⎩ ⎭

→ →

CAiR k C kA L Ai L 1 1

1R k C 1 ;

... Regime 1-2

CAiC kAb L 1 1

kA L Ai L 1 1k a

k a kL L 1R k C 1 k C

k k a k k aA L Ai L AiL 1 L L 1 L

⎧ ⎫⎪ ⎪= −⎪ ⎪ε⎨ ⎬

+⎪ ⎪⎪ ⎪⎩ ⎭

⎧ ⎫= −⎪ ⎪ε⎨ ⎬+⎪ ⎪⎩ ⎭

ε⎧ ⎫ ⎧ ⎫⎪ ⎪ ⎪ ⎪= − =⎨ ⎬ ⎨ ⎬ε + ε +⎪ ⎪ ⎪ ⎪⎩ ⎭ ⎩

k kL 1 L 1 k k a, R k C Ck a aL 1 L A L Ai Ai

R a k C A L 1 Ai

If then

... Regime 1

Regime controlled by pure kinet C CA

ics; b Ai

ε ε⎧ ⎫⎪ ⎪ε = =⎨ ⎬⎪ ⎪⎩ ⎭

⎡ ⎤⎢ ⎥⎣ ⎦

If then

... Reg

kL 1 k k a,

R k CkL 1 L A L Ai

R k CA L Ai

R a k aC A L Ai

me controlled by pure mass transfer

ε⎧ ⎫⎪ ⎪ε = ⎨ ⎬ε⎪ ⎪⎩

⎡ ⎤⎢ ⎥⎢

= ⎥⎦

If then

... Regime 2-3

Most A reacts within the diffusion film,and the rest r

C 0, Ab

C MAiR k 0

eacts in the bul

tanh M

k C ML AiR

liquid;

⎧ ⎫⎪ ⎪⎨ ⎬= −⎪ ⎪⎩ ⎭

⎡ ⎤ =⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

REGIME 4: INSTANTANEOUS REACTIONS

Reaction “potentially” so fast that A and B cannot coexist !

At a certain distance from the interface, “reaction plane” formed

The mass transfer rate (& hence reaction rate) governed by rate at which “dissolved A” and “reactant B” supplied to “reaction plane”.

Regime 4: instantaneous reactions

G/L/Sphase Liquid phase

0 δλ

Reactionplane

Condition for validity for Regime 4:

At steady state:

diffusion rate of dissolved A through region 0 < x < λ

= diffusion rate of B through region δ > x > λ

k ZC DL Ai

⎛ ⎞−⎜ ⎟ ⎛ ⎞+⎝ ⎠ ⎜ ⎟>>⎜ ⎟⎝ ⎠

( ) ( )

D CD CB BbA AiR

Z D C D CA Ai B BbZ D C

Z D C D C1 1 A Ai B BbZ D C

... 39

Z D C D Ck1 A Ai B BbLD Z D C

... 40

A A Ai

= =λ δ − λ

⎧ ⎫+⎪ ⎪= ⎨ ⎬λ δ ⎪ ⎪⎩ ⎭

⎧ ⎫+⎪ ⎪= ⎨

⇒ ⎬λ ⎪ ⎪⎩ ⎭

CDBbB1

D CA AiR

Z D C D CkA Ai B BbLR D C

D Z D CA A AiA A Ai

R k C ... 41

Asymptotic enhancement fa

C DBb B1

ZC DaAi A

ctor ... 42

+⎧ ⎫⎪ ⎪= ⎨ ⎬⎪ ⎪⎩ ⎭

=⎫⎪ ⎪+⎨ ⎬

⇒⎩

⎪⎭

For example:

Kinetics of absorption of oxygen in aqueous alkaline solutions of polyhydroxybenzenes. Anand V. Patwardhan; Man Mohan Sharma, Industrial and Engineering Chemistry Research 1988, 27, 36-41.

Transition between Regime 3 and Regime 4)(regime overlapping R3 and R4):

If ... 43

Depletion of reactant in the di

ffusion film

DL Ai A

⎛ ⎞−⎜ ⎟ ⎛ ⎞+⎝ ⎠ ∼

⎜ ⎟⎜ ⎟⎝ ⎠

The relevant differential equations are,

... 44

... 45

The boundary conditio

2d Cm nAD k C C 2A mn A Bdx

2d Cm nBD Z k C C 2B mn A Bdx

x 0, C C , C , A

ns are,

at ... Ai B

dCBC 0

dxBi... 45, C 0, C

= δ = =

Eliminating from (44) and (45),

... 46

Integration of (46) gives,

... 47

Second inte

2 2d C d CA BZ D

gration gives,

D2 2A Bdx dx

dC dCA BZ D D Z R

dx dxA B A

D Z RA AC C ZC

DBi B0 AiB

= + − δ

Equation (48) gives " " concentrationof in terms of

and are two variables,

interfacialB R

R CA Bi

2d Cm nAD k

we must have another equation.

... 44

... non-linear, hence, analytical

A mn A Bdlut

is NOT possible.

B.C. (45a) : constant in the IMMEDIATE

neighbourhood of interface.

So, restricting the solution of (44) in the IMMEDIATEneighbourhood of interfac

Simplifying as

e, can be

sumption:C C

replaced by C .CB Bi

( )2d C

Equati

k C C2A mn A Bid

on (44) then becomes,

( )( )

n 2k C M C CL Ai Bi B0R n 2A tanh M C C

Integrating (49) as usual,

... 50

When 3BiMC

2 2m 1 n

, DENOMINATOR 1

... 51R C D k C Cm 1A Ai A mn Ai Bi

⎡ ⎤⎣ ⎦=

⎡ ⎤⎢ ⎥⎣ ⎦

⎛ ⎞⎜ ⎟⎜ ⎟⎝

≈⎠

⎡ ⎤⎛ ⎞ −= ⎜ ⎟⇒ ⎢ ⎥+⎝ ⎠⎣ ⎦

( )( )

C R2Bi A

C k CB0 L Ai

nM.0 1ntanh M.

m 1 n 1

M 12 1 0 q q

C DB0 Bq .

Let , and enhancement factor ,

which gives,

... 52 ;

For and , Equation (48) becomes

... 53

where, ... 5C

⎛ ⎞⎜ ⎟ = η ϕ⎜ ⎟⎝ ⎠

ηϕ = < η <

⎡ ⎤η⎢ ⎥⎣ ⎦= =

⎛ ⎞η + η − + =⎜ ⎟

⎝ ⎠

Solution of (53):

... 55

M M 1 4 1 2q qq 2

M q 1 Ren , (NO depletion of in liquid film)

gime 3B

⎛ ⎞+ + +⎜ ⎟

⎝ ⎠η

When , (COMPLETE depletion of in liquid film)

and, for , ,Equation (53) can be solved to gi

M q 0 Regime 4B

C DB0 B1 1 q

ZC DaAi A

for m 1, n 1 ... 61

ϕ = + = +

η > ϕ = η

⎛ ⎞ϕ − ϕ⎜ ⎟η = = =ϕ −⎜

⎟⎝ ⎠

Hikita and Asai (J. of Chemical Engineering of Japan

1964, vol. 2, p.77)

have shown that the following equation holds

for a GENERAL ord

n 2n aE M M

er reaction

... 5 1

⎛ ⎞ϕ − ϕ⎜ ⎟= η =ϕ −⎜ ⎟

⎝ ⎠

Role of diffusion within the catalyst pellet(internal diffusion)

Reaction within a solid catalyst: reactant must first diffuse into it ⇒ lowering of reactant concentration in the inner regions of catalyst.

As A diffuses inward, it is also reacting to form the product, but at a progressively diminishing rate.

Observed rate = true or intrinsic rate multiplied by an effectiveness factor, which is a function of the true rate constant, diffusivity, and pellet shape and size.

Effectiveness factor is a co-determinant of the actual reaction rate, it is very important in the analysis and design of catalytic reactors.

Effectiveness factor (η) is defined as,

actual reaction rate inside catalystrate based on surface (bulk) concentration

Isothermal effectiveness factors (single pore model):

Consider a first-order reaction in a cylindrical catalyst pore, which is closed at one end (for example).

Reaction occurring on the inside surface of catalyst pore is first order w.r.t. A, irreversible, isothermal:

catalysA product

t⎯⎯⎯⎯→

CAS dC

With reference to the cylindrical pore, the following differential mass (mole) balance can be written:

( )dC dCA AD D k CeA eA 1 Adx dx

x x dx

dC dCA AD DeA eAdx

dx 2kx dx x 1 CAdx rP

2 2d C 2k d C 2A 1 AC c C ; A A2 2D rdx d

2 r dxP P

x reeA P A P

⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪⎢ ⎥ ⎢ ⎥− − − =⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪+⎣ ⎦ ⎣ ⎦⎩ ⎭

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥−⎢ ⎥ ⎢ ⎥+ ⎣ ⎦⎣ ⎦ =

⇒=⇒ = =

The above differential equation can be solved for the following two boundary conditions:

B.C. 1: At pore inlet ,

B.C. 2: At pore's closed end ,

General solution:

B.C. 1:

x 0 C CA ASdCAx L 0

dxcx cxC h e h eA 1 2

x 0 C C A

h A h2 s 1

AS At pore inlet , gi

C h h AS 1

−= +

⎡ ⎤+ ⇒ = −⎣ ⎦

( )( )

( )B.C. 2: At pore's close

cx cxC h e h eA 1 2dC cx cxA c h e h e1 2dx

dCAx L 0 dx

cL cL0 c h e h e1 22cLh h e2 1

2cLC h h

d end , gives,

eAS 1 1

−= +

−= −

− =⇒

2cLC C eAS ASh h1 22cL 2cLe 1 e 1dC cx cxA c h e h e1 2dx

dCA c h h1 2dxx 0dCAD D h heA eA 1 2dx

x 02D keA 1R h hA 2 1r

2k1D reA

= =+ +

−= −

= −⎡ ⎤⎣ ⎦=

− = −

⇒ ⇒

⇒ −⎡ ⎤⎣ ⎦=

= −⇒ ⎡ ⎤⎣ ⎦

2cL2D k C e CeA 1 AS ASRA 2cL 2cLr e 1 e 1P2cL2D k C e CeA 1 AS ASRA 2cLr e 1P2cL2D k e 1eA 1R CA AS 2cLr e 1PcL cL2D k e eeA 1R CA AS cL cLr e eP

⎡ ⎤⎢ ⎥= −⎢ ⎥+ +⎣ ⎦

⎡ ⎤−⎢ ⎥= ⎢ ⎥+⎣ ⎦

⎡ ⎤−⎢ ⎥=⎢ ⎥+⎣ ⎦

−⎡ ⎤−⎢ ⎥=−⎢

+ ⎦⇒

2k1D reA P

Thiele mo

2D keA 1R C tanh LA AS rP

2k1L 1D reA Pdulus

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟=⎢ ⎥⎜ ⎟⎣ ⎦

⇒⎠

An effectiveness factor (η) for pore is defined as,

actual reaction rate within catalyst porerate based on surface concentration throughout pore

( )( )

2D k 2keA 1 1C tanh LAS r D rP eA Pk C1 AS

2D k 2keA 1 1tanh Lr D rP eA P

k12k1tanh L

D r tanheA P 12k 11L

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦η =

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣

⎦η =

⎛ ⎞⎜ ⎟⎜ ⎟ ϕ⎝ ⎠η = =

( )( )

tanh 1As , limit

As , l

tanhimi 11 1

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

ϕϕ → →

ϕ →∞ ϕ →

n 12k Cn ASL1 D reA P

−ϕ =

Expression for Thiele modulus (single pore model) for an irreversible reaction, which nth order w.r.t. A is given by:

Squaring both sides:

( )r CPn 1n 1 2 L k C2k C n AS2 2 n ASL1 D r D r LeA

ASr CP AP A P Se

−ϕ =

n2 L k Cn AS21

maximum possible surface reaction ratein absence of any Diffusion resistance2

1 maximum possible diffusion ratein absence of any Reac

OBSERVED re

ϕ =⎛ ⎞⎜ ⎟⎝ ⎠

⎧ ⎫⎨ ⎬⎩ ⎭ϕ =

⎫⎨ ⎬⎩ ⎭

ϕ =η

⇒action rate

maximum possible diffusion rate

Porous Catalyst Particles

The results for a single pore can approximate the behavior or particles of various shapes – spheres, cylinders, etc. For these systems:

1. Use of the proper diffusivity: Replace the effective diffusivity for single pore by the effective diffusivity of fluid in the porous structure.

2. Use of proper measure of particle size: To find the effective distance penetrated by reacting fluid to get to all the interior surfaces, a characteristic size of particle is defined as:

If the reaction rate in the single pore is based on the “pore volume” (volumetric rate) instead of inner pore surface area, then we get following expressions:

Thiele modu

R C D k tanh LA AS eA 1

k1L 1e

lusD A

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟=⎢ ⎥⎜ ⎟

⎝ ⎠⎣

( )( )

C D k tanh LAS eA 1

k C1 AS

k1tanh L

D tanheA 1k 11L

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟

⎝ ⎠⎣ ⎦η =

⎡ ⎤⎛ ⎞⎢ ⎥⎜ ⎟⎢ ⎥⎜ ⎟ ϕ⎝ ⎠⎣ ⎦η = =

ϕ⎛ ⎞⎜ ⎟⎜ ⎟⎝

... any particle volume of pelletexterior surface availablefor reactant penetration

flat slab pellet... open ONLY on

two sides

cylindrical pellet... open ONLY on

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

=⎛ ⎞⎜ ⎟⎜

⎟⎝ ⎠=

...spherical pelletR3

⎧⎪⎪⎪⎪⎪⎪

⎪⎨⎪

⎛ ⎞⎪⎜ ⎟⎪ ⎜ ⎟⎪ ⎝ ⎠

⎪⎪⎪⎩

thickness

Flat slab

Sphere

...any particle shvolume of pelletex

apeterior surface available

for reactant penetration

area thicknessL2 area two sides

thickness

flat slab pellet... open ONLY on

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

⎧⎪⎪⎪⎪ ⎛ ⎞⎨= ⎜ ⎟=

⎪ ⎜ ⎟⎝ ⎠⎪

⎪⎪⎩

thickness area through whichreactant penetrates

area through whichreactant penetrates

...any particle shvolume of pelletexterior surface availablefor reactant penetration

2R LengthL2 R Length curved sideR

cylindrical pellet... open ONLY

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

π× ×=

×π× ×

⎧⎪⎪⎪⎪ ⎛ ⎞⎨= ⎜ ⎟⎪ ⎜ ⎟

⎝ ⎠

⎪⎪⎪

RA AA A

Areathroughwhich

reactantspenetrate

...any particle shapevolume of pelletexterior surface availablef

spherical pellet... with all sur

or reactant penetration4

face3R

3Lpor24 R

⎛ ⎞= ⎜ ⎟⎜ ⎟⎝ ⎠

× π×=

⎧⎪⎪⎪⎪

⎛ ⎞⎨= ⎜ ⎟⎪ ⎜ ⎟⎪ ⎝ ⎠π

× ×⎪⎪

3. Measure of reaction rate: In catalytic systems the rate of reaction can be expressed in many equivalent ways. For example, for FIRST-ORDER KINETICS,

mol A reactedBased on oid , olume in reactors

mol A reactedBased on eight , of catalyst

dN1V Ar kCV A A 3V dt m voidV

dN1W Ar' k 'CA AW dt k pellets s

Based on catalyst urface

dN1 Ar'' k 'S A S dtarea

= ⎡ ⎤⎢ ⎥⎣ ⎦

⎡ ⎤⎢ ⎥

= ( )mol A reacted, s

'CA 2m cat s urf

⎡ ⎤⎢ ⎥⎣ ⎦

mol A reactedBased on olume , of catalyst elletss

mol A reactedBased on total

dN1V Ar k CP vA v A 3V dt m catP

dN1 Ar'''' k ''''CR V A , eactor A 3V dt m reacto olume

⎡ ⎤⎢ ⎥⎣ ⎦

⎡= ⎤⎢ ⎥⎣ ⎦

For porous catalyst particles: rates based on unit mass and on unit volume of particles, (r’ and r’’’) are the useful measures.

4. Similar to a single cylindrical pore, Thiele (1939) and Aris (1957) related η (pellet effectiveness factor) with φ (pellet Thiele modulus) for various pellet shapes as:

( )( )

tanh 1 1

I 21 1 1 I 21 0 1

I & I Bessel functions1 01 1 1

... flat slab pellet

... cylindrical pellet

... spherical pellet

where,

Thiele

h 3 31 1 1

k stvL 11 DeAmodulus for order k

⎧ ϕ⎪=

ϕ⎪⎪

ϕ⎪⎪=⎪ ϕ ϕη = ⎨⎪

=⎪⎪

⎛ ⎞⎪= −⎜ ⎟⎪ ϕ ϕ ϕ⎜ ⎟⎪ ⎝ ⎠⎩

ϕ = = inetics

For a first order irreversible isothermal reaction given by A → R, then actual rate is then given by:

} ( )r k CvAmol A reactedBased on volume , of catalyst pelle v A 3m catts s

⎡= ⎤η ⎢ ⎥⎣ ⎦

Effectiveness factor versus Thiele modulus1 1η = ϕ

Strong porediffusion effects

SphereCylinder

Flat plateNo resistance topore diffusion

Thiele modulus L k D1 v eAϕ = →

0.1 0.2 0.3 0.4 1 2 3 4 5 10 200.05

0.40.5

5. Finding pore resistance effects from experiments: another modulus is defined which includes only observable and measurable quantities. This is known as the Weisz modulus φ2.

actual rate intrinsic rate2 2L L2 D C D CeA AS eA AS

⎡ ⎤ ⎡ ⎤ϕ = = η⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Pore resistance limits: When reactant fully penetrates the pellet and covers all its surfaces, then pellet is in the diffusion-free (kinetic) regime (φ1<0.4 or φ2<0.15).

When the pellet is starved for reactant and is unused then the particle is in the strong pore resistance regime (diffusion regime) (φ1>4 or φ2>4).

Thiele modulus φ1 and Weisz modulus φ2 →Doraiswamy (2001)

tor η→

φ1<0.4φ2<0.15

Kinetic control Diffusion control

φ1,φ2>4.0

ThieleWeisz

6. Particles of different sizes: Comparing the behavior of particles of size R1 and R2, the diffusion-free regime is found out.

( )( )

mol A reactedBased on we r' k 'CA A kg ight , of catalyst pellets s

r' k 'CA 1 A 1r' k 'CA A2

⎡ ⎤η ⎢ ⎥⎣ ⎦

η η⇒ =

In the regime of strong diffusion resistance:

( )( )

r' LA 1 1 2r' LA 2

12 1 1

η= = =

Rate is inversely proportional to the particle size.

7. Mixture of particles of various shapes and sizes: For a catalyst bed consisting of a mixture of particles of various shapes and sizes, Aris (1957) proved the correct mean effectiveness factor as:

f f f ...mean 1 1 2 2 3 3η = η + η + η +

8. Arbitrary reaction kinetics: If the Thiele modulus is generalised as follows [Froment and Bischoff (1962)],

equilibrium concentrati

r LvA1 1 2CA

2D r dCeA vA ACAeCAe on

−ϕ =

⎡ ⎤⎢ ⎥−∫⎢ ⎥⎢ ⎥⎣ ⎦

then the η versus φ1 curves for all forms of rate equation closely follow the curve for the 1st order reaction. This generalised modulus becomes:

kvL1 D XeA Aeqϕ =

for first-order reversible reactions:

n 1k Cn 1 v ASL1 2 DeA

for nth order irreversible reactions:

Combining the nth order rate with the generalised Thiele modulus gives:

( )( )

n nr k C k CvA v AS v AS

D1 2 neAr k CvA v ASn 1L n 1 k Cv AS

n 1 2r

1 2k D2 v eA2n 1 L

kv,obser

CvA AS

n 1 2r CvA Aed Sv

⎛ ⎞⇒ ⎜ ⎟⎜ ⎟+

− = =

− =−+

+− =⎠

⇒ +=

That is, in strong pore diffusion regime, an nth order reaction behaves like a [(n+1)/2]th order reaction.

1 2k D2 v eAkv,observed 2n 1 L

⎛ ⎞= ⎜ ⎟⎜ ⎟+⎝ ⎠

Also, the temperature dependency of reactions is affected by strong pore resistance (diffusion resistance).

Taking logarithms and differentiating w.r.t. temperature and (reaction rate and to a lesser extent the diffusional process are T-dependent):

( ) ( ) ( )dln k dln k dln D1v,observed v eAdT 2 dT dT

⎛ ⎞⎜ ⎟= +⎝ ⎠

The Arrhenius temperature dependence for reaction and diffusion are expressed as:

E Etrue Diffuk exp D expk and Dv,v eART RT

E Etrue D

iffuEobserved 2

A,0− −⎛ ⎞ ⎛ ⎞

= =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Etrue for gas-phase reactions is high (~ 80-240 kJ/mol), and Ediff is small (~ 5 kJ/mol at room temperature or 15 kJ/mol at 1000 0C). Therefore,

EtrueEobserved 2≈

Best wishes for end-semester examinations,

andtimely DECLARATION OF

RESULTS !

Mass Transfer With Kinetics

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