Catalytic Reaction Engineering...Reaction Engineering Professor of Industrial Chemistry Department...

CatalyticReaction Engineering

Professor of Industrial ChemistryDepartment of Chemical and Metallurgical EngineeringSchool of Chemical TechnologyAalto UniversityEmail: yongdan.li@aalto.fiKemistintie 1, E404

Yongdan Li

Nov-Dec, 2018

8. Overall Diffusion

8.1 Evaluation of Diffusion Limitations

A quick and dirty estimate of diffusion limitations

2. Mears criterion for external diffusion

Observed reaction rates are used

1. Weisz-Prater criterion for internal diffusion

3

8.1 Weisz-Prater Criterion for Internal Diffusion

1)coth(3

112

1

η

(1st order, spherical particles)

)1coth(3 11

2

1 η

2

1WPC η

Weisz-Prater criterion is written as Cwp:

Rearranging internal effectiveness factor:

(8-2)

(8-1)

4

η = Effectiveness factor of internal diffusion

Ф1= Thiele modulus of first order reaction

Cwp = Weisz Prater parameter

8.1 Weisz-Prater Criterion for Internal Diffusion

Internal effectiveness factor and Thiele modulus have already

been determined earlier

A

As

( )'

'

r obsη

r

2 22 As As

1

As As

'' '

a c c

e e

r S ρ R r ρ R

D C D C

2 22 As AA

1

As As As

( )( )' ''

c cWP '

e e

r ρ R r obs ρ Rr obsC

r D C D C

Weisz-Prater parameter (1st order reaction):

R=catalyst particle radius, m

ρc=solid density of catalyst pallet, kg/m3

CAs=surface reactant concentration, mol/m3

Sa=internal surface area/mass of catalyst, m2/g

De=Effective diffusivity, m2/s

(8-3) (8-4)

(8-5)

5

8.1 Weisz-Prater Criterion for Internal Diffusion

All terms are measured or known, if limited by internal

diffusion

22 A

1

As

( )'

cWP

e

r obs ρ RC

D C

If CWP << 1 => no internal diffusion resistance

If CWP >> 1 => internal diffusion limits the reaction

(8-5)

6

8.1 Mears Criterion for External Diffusion

A b

Ab

( )'

c

r obs ρ RnMR

k C

External diffusion is evaluated with MR

ρb=bulk density of catalyst bed, kg/m3

=(1-ɛ)ρc (ɛ=voidage)

n=reaction order

R=catalyst particle radius, m

ρc=solid density of catalyst pallet, kg/m3

CAb=bulk reactant concentration, mol/dm3

kc=mass transfer coefficient, m/s

kc can be calculated e.g. by Thoenes-

Kramers correlation (for flow throu-

gh a packed bed)

Others can be measured

MR＜0.15

External diffusion is neglected(8-6)

7

8.2 Overall Effectiveness Factor

What happens if both external AND internaldiffusion exist simultaneously…?

8

8.2 Overall Effectiveness Factor

Useful for first order reactions

External and internal diffusion in packed catalyst beds

In steady state, molar flow of reactant A to the catalyst surface can be expressed

MA= molar flow of reactant A to the catalyst surface

ac = external surface area of catalyst particles per unit volume of the reactor

V = the volume

A A Δr cM W a V (8-7)

9

8.2 Overall Effectiveness Factor

Because often Sab >> ac (porous particle)

Molar rate of A transfers to the surface, MA, is equal to the net (total) rate

of reaction on and within the pellet

A A (external area internal area)''M r

A A b( Δ Δ )''

c aM r a V S ρ V

ρb=bulk density of catalyst bed, kg/m3

=(1-ɛ)ρc (ɛ=voidage)Sa=internal surface area/mass of catalyst, m2/g

A A b A A b( )'' ''

A r c c a r c aM W a V r a S ρ V W a r S ρ

(8-8)

(8-9)

10

8.2 Overall Effectiveness Factor

For external mass transfer

WAr ac = kc(CAb- CAs) ac

Since internal resistance affects the system, CAs cannot be neglected in this case

(as earlier)

''

A As A 1 As

'' '' ''r ηr r ηk C (8-11)

Consumption rate for a first order reaction

It is not possible to measure CAs=> elimination

(8-10)

11

k1’’= Rate constant per unit area

CAb ≠ CAs ≠ CA and only CAb can be measured

By combining equations (8-9)，(8-10) and (8-11):

''

Ab As 1 b As(C )c c ak a C ηk S ρ C

For CAs

AbAs ''

1 b

c c

c c a

k a CC

k a ηk S ρ

8.2 Overall Effectiveness Factor

(8-12)

(8-13)

12

8.2 Overall Effectiveness Factor

By substituting CAs in first order rate equation:

'' ''Ab1''

1 b

c cA

c c a

k a Cr ηk

k a ηk S ρ

''

1 Ab''

1 b1 ( )a c c

ηk C

η k S ρ / k a

Where is internal effectiveness factor

As s

Actual overall (observed) reaction rate

reaction rate if entire interior surface were exposed to

the conditions of external surface ,

η

C T

(8-14)

(8-15)

13

8.2 Overall Effectiveness Factor

Overall effectiveness factor is defined as

Actual overall (observed) reaction rate

reaction rate if whole particle were exposed to the conditions of bulk , Ab bC T (8-16)

Based on the definitions of Ω and equation (8-14):

A Ab( )'' ''r Ω r ''

1 Ab''

1 b1 ( )a c c

ηk C

η k S ρ / k a

''

1 b1 ( )a c c

ηΩ

η k S ρ / k a

A

''r

So the overall effectiveness factor is

''

1 b1 ( )a c c

η

η k S ρ / k a

Ab( )''r

(8-19)

(8-17) (8-18)

14

8.2 Overall Effectiveness Factor

''

1 b1 ( )a c c

ηΩ

η k S ρ / k a

Large flow rate results in large external transfer coefficient kc

Neglect

Therefore, in the case without external diffusion

Ω

Overall effectiveness factor approaches internal effectiveness factor

(8-20)

(8-21)

15

8.3 Mass Transfer and Reaction in a Packed Bed

Isomerization reaction in a packed bed reactor

z z+z z = L

FAb

Ac

z = 0

V

Plug flow assumed

Constant volumetric flow rate 0 (U = 0/Ac)

Ac=cro-sectional area of the tube, dm2

CAb=bulk gas concentration of A, mol/dm3

0=volumetric flow rate, dm3/s

U=superficial velocity, dm/s

FAb=molar flow, mol/s

(8-22)

16

Z = the length of the packed bed, m

8.3 Mass Transfer and Reaction in a Packed Bed

AAb Ab Δ( Δ ) 0'

b cz z zF F r A z

10'Ab

A b

C

dFr ρ

A dz

Molar balance for A in steady state:

Dividing by Acz and taking the limit by z→0

in - out + generated = 0

(8-24)

(8-23)

17

8.3 Mass Transfer and Reaction in a Packed Bed

A Ab

' 'r r Ω

Ab Ab Ab''' ''

a ar r S k C S

A Ab'''

ar Ωk C S

-r’A = real reaction rate

Assuming first order isomerization reaction A → B

Substituting into Equation (25)

(8-25)

(8-26)

(8-27)

18

k’’ = Rate constant per unit area

8.3 Mass Transfer and Reaction in a Packed Bed

Rate equation is substituted in molar balance equation

Ab b

1''Ab

a

C

dFΩk C S ρ

A dz

Constant volumetric flow rate

→ concentrations can be used in the balance equation

U = 0/Ac 1/Ac = U/0

Ab b Ab b

0

'' ''Ab Aba a

dF dCUΩk C S ρ U Ωk C S ρ

dz dz

(8-28)

(8-30)

(8-22) (8-29)

19

8.3 Mass Transfer and Reaction in a Packed Bed

Ab b''Aba

dCU Ωk C S ρ

dz

The final form of the balance equation

Ab bAb Ab

b

''

''

a

a

dC Ω ρ k SC a C

dz U

Ω ρ k Sa

U

Initial conditions (integration limits): CAb = CAb0 when z = 0

Ab

Ab0

AbAb Ab0

Ab 0

ln ln

C z

C

dCadz C C a z

C

(8-30) (8-31)

(8-32)

20CAb0 = Initial phase concentration, mol/dm3

8.3 Mass Transfer and Reaction in a Packed Bed

For bulk concentration as a function of reactor length

Ab Ab0ln lnC C a z Ab Ab0

azC C e

b ''

Ab Ab0

aΩ ρ k S z

UC C e

Conversion at the reactor outlet, z = L

b( '' )Ab

Ab0

1 1 aΩ k S L /UCX e

C

(8-32) (8-33)

(8-34)

(8-35)

21

8.4 Reaction Rate Dependency

External mass transfer-limited reactions in packed beds

Robert the Worrier (Example 14-4)

(8-36)

(8-37)

(8-38)(8-39)

(8-40)

6(1 ) / c pa d

22ɛ = Void of the bed dp = Particle diameter, m

8.4 Reaction Rate Dependency

Internal mass transfer-limited reactions

Large Thiele modulus

Surface-reaction-limited reactions

(8-41)

(8-42)

(8-43)

(8-44)(8-45)23

8.4 Reaction Rate Dependency

Limiting

parameter

Flow

rate

Catalyst

particle size

Temperature

External

diffusion

U1/2 (dp)-3/2 linear

Internal

diffusion

independent (dp)-1 exponential

Surface

reaction

independent independent exponential

Surface reaction

stronger dependent

24

8.4 Reaction Rate Dependency

How to determine diffusion limitations experimentally (in packed bed)?

Internal (pore)

diffusion

• Keep catalyst mass, mass flow rate

and reactant inlet concentration

constant

• Change catalyst particle size and

observe reaction rate/conversion

External (film)

diffusion

• Keep reactant inlet concentration and

catalyst mass to mass flow rate ratio

constant

• Change linear flow velocity and

observe reaction rate/conversion

25

8.4 Reaction Rate Dependency

How to determine diffusion limitations experimentally (in packed bed)?

Exp. Particle size Superfacial flow

rate

Reaction rate

I 1 high 3

II 3 low 1

III 3 high 1

• Experiments II and III: no external diffusion with large particles no external diffusion with small particles

• Experiments I and III:• strong pore diffusion

26

Summary

27

Evaluation of Diffusion LimitationsWeisz-Prater Criterion for Internal Diffusion

Mears Criterion for External Diffusion

Both external and

internal diffusion exist

internal effectiveness factor

Overall effectiveness factor Ω

Mass Transfer and Reaction in a Packed Bed

Reaction Rate Dependency

External mass transfer-limited reactions in packed beds

Internal mass transfer-limited reactions

Surface-reaction-limited reactions

How to determine diffusion limitations

28

8.5 Warming-up

EXAMPLE

The catalytic reaction

takes place within a fixed bed containing spherical porous catalyst X22. Figure E1 shows

the overall rates of reaction at a point in the reactor as a function of temperature for

various entering total molar flow rates, FT0 .

Figure E1:

Reaction rates in a catalyst bed.

Gas properties:

Diffusivity: 0.1 cm2/s

Density: 0.001 g/cm3

Viscosity: 0.0001 g/cm·s

Bed properties:

Tortuosity of pellet: 1.414

voidage=0.3

8.5 Warming-up

EXAMPLE

(a) Is the reaction limited by external diffusion?

(b) If your answer to part (a) was “yes,” under what conditions of those shown

(i.e., T, FT0) is the reaction limited by external diffusion?

(c) Is the reaction “reaction-rate-limited”?

(d) If your answer to part (c) was “yes,” under what conditions of those shown

(i.e., T, FT0) is the reaction limited by the rate of the surface reactions?

(e) Is the reaction limited by internal diffusion?

(f) If your answer to part (e) was “yes,” under what conditions of those shown

(i.e., T, FT0) is the reaction limited by the rate of internal diffusion?

(g) For a flow rate of 10 mol/h, determine (if possible) the overall effectiveness factor,

Ω, at 362 K.

(h) Estimate (if possible) the internal effectiveness factor, η, at 367 K

29

8.5 Warming-up

EXAMPLE

(i) If the concentration at the external catalyst surface is 0.01 mol/dm3, calculate (if

possible) the concentration at r = R/2 inside the porous catalyst at 367 K. (Assume a

first-order reaction.)

Solution

(a) Is the reaction limited

by external diffusion?

30

8.5 Warming-up

EXAMPLE

Limiting

parameter

Flow

rate

Catalyst

particle size

Temperature

External

diffusion

U1/2 (dp)-3/2 linear

Internal

diffusion

independent (dp)-1 exponential

Surface

reaction

independent independent exponential

Reaction rate dependency

31

8.5 Warming-up

EXAMPLE

(a) Is the reaction limited

by external diffusion?

YES

(b) what conditions?

All temperautres, FT0=10

mol/h. The rate of reaction

changes with Flow rate and

increases linearly with

temperature.

32

8.5 Warming-up

EXAMPLE

(c) Is the reaction “reaction-rate-limited”?

(e) Is the reaction limited by internal diffusion?

33

8.5 Warming-up

EXAMPLE

Limiting

parameter

Flow

rate

Catalyst

particle size

Temperature

External

diffusion

U1/2 (dp)-3/2 linear

Internal

diffusion

independent (dp)-1 exponential

Surface

reaction

independent independent exponential

Reaction rate dependency

Surface reaction

stronger dependent

34

8.5 Warming-up

EXAMPLE

(c) Is the reaction limited

by surface reaction?

YES

(e) Is the reaction limited

by internal diffusion?

YES

(f) T＞367K, 1000, 5000 mol/h

(d) T＜367K, 1000, 5000 mol/hT＜362K, 100 mol/h

35

8.5 Warming-up

EXAMPLE

(g) For a flow rate of 10 mol/h, determine (if possible) the overall effectiveness factor,

Ω, at 362 K.

Actual overall (observed) reaction rate

reaction rate if whole particle were exposed to the conditions of bulk , Ab bC T

External and internal diffusion is eliminated

Reaction is “reaction-rate-limited”

Actual overall (observed) reaction rate, 10 mol/h, 362K

Rate of surface-reaction limited reaction, 362K

(8-16)

(8-46)

36

8.5 Warming-up

EXAMPLE

(g) For a flow rate of 10 mol/h, determine (if possible) the overall effectiveness factor,

Ω, at 362 K.

Ω =−𝑟𝐴(362𝐾, 10 𝑚𝑜𝑙/ℎ)

−𝑟𝐴(362𝐾, 5000 𝑚𝑜𝑙/ℎ)

Ω =0.26

0.70= 0.37

(d) T＜367K, 1000, 5000 mol/hT＜362K, 100 mol/hReaction limited

(8-47)

(8-48)

37

8.5 Warming-up

EXAMPLE

(h) Estimate (if possible) the internal effectiveness factor, η, at 367 K

As s

Actual overall (observed) reaction rate

reaction rate if entire interior surface were exposed to

the conditions of external surface ,

η

C T

Focus on 5000 mol/hAt 5000 mol/h, no external diffusion

No internal diffusion

Rate of surface-reaction limited reaction,

5000 mol/h, 367 K

(8-15)

38

8.5 Warming-up

EXAMPLE

Focus on 5000 mol/h， 367 K

T＜367K, 5000 mol/hReaction limited

η =−𝑟𝐴(𝑎𝑐𝑡𝑢𝑎𝑙)

−𝑟𝐴(𝑒𝑥𝑡𝑟𝑎𝑝𝑜𝑙𝑎𝑡𝑒𝑑)

η =1.2

1.4= 0.86

(8-49)

(8-50)

39

8.5 Warming-up

EXAMPLE

(i) If the concentration at the external catalyst surface is 0.01 mol/dm3, calculate (if

possible) the concentration at r = R/2 inside the porous catalyst at 367 K. (Assume a

first-order reaction.)

1

1

sinh λ1ψ

λ sinh

A

As

C

C

1λ =

2

r

R

1 12

1

3( coth 1)η

0.86η

1=1.60

CA

(8-1)

(8-51) (8-52)

40

CatalyticReaction Engineering

Professor of Industrial ChemistryDepartment of Chemical and Metallurgical EngineeringSchool of Chemical TechnologyAalto UniversityEmail: yongdan.li@aalto.fiKemistintie 1, E404

Yongdan Li

Nov-Dec, 2018