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Gas-Liquid Reactions on Solid Catalysts:
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Trickle Beds, Slurry Reactors, and Three-Phase Fluidized Beds
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Chemical Reaction Engineering
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extension.
CSU – IGA
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By:
ARELLANO, MARIA ELOISA ANGELIE G.
MARTINEZ, CHERRYL A.
NICOLAS, DARREN I.
COE – IGA
Innovative Thinking
Synthesis
Personal
Responsibility
June 2019
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Research Skill
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1
Table of Contents
I.
II.
Catalyst
A. Definition
4
B. Solid Catalyst
4
C. Properties of Solid Catalyst
4
Gas-Liquid Reactions: Mass Transfer Approach
5
A. Theories for Analysis of Transport Effects in Gas-Liquid Reactions
5
1. Two-Film Theory
5
(a) Assumptions
5
(b) Derivation
6
2. Penetration Theory
7
(a) Assumptions
8
(b) Derivation
8
3. Surface Renewal Theory
(a) Assumptions
9
(b) Derivation
9
4. Boundary Layer Theory
(a) Derivation
III.
IV.
V.
9
10
10
Understanding the Gas-Liquid-Solid Catalyst Reactions
11
A. The General Rate Expression
11
B. Two Simplifying Extremes
16
C. Testing the Extremes
17
Performance Equations for an excess of B
17
A. Mixed flow G/ any Flow (excess of B)
18
B. Plug flow G/ Any Flow L (excess B)
18
C. Mixed Flow G/Batch L (excess of B)
19
D. Plug Flow GI Batch L (excess of B)
20
E. Special Case of Pure Gaseous A (excess of B)
22
Performance Equation for an Excess of A
22
A. Plug Flow LI Any Flow G (tower and packed bed operations)
22
B. Mixed Flow LI Any Flow G (tank operations of all types)
23
C. Batch L/Any Flow G
23
2
VI.
VII.
Kinetics, Design and Modeling
23
A. Slurry Reactors
23
1. Advantages of Slurry Reactors relative to other types of reactions
24
2. Modeling
24
(a) Rate of Gas Adsorption
26
(b) Transport to catalyst
26
(c) Diffusion and Reaction in the Catalyst Pellet
26
(d) The Rate Law
27
3. Determining the Limiting Step
27
4. Limiting conditions
34
B. Trickle Bed Reactors
35
1. Fundamentals
36
2. Diffusion and Reaction in the Pellet
38
(a) Transport of B from Bulk Liquid to Solid Catalyst
39
(b) Diffusion and the Reaction of B Inside the Catalyst Pellet
39
3. Limiting Situations
39
4. Evaluating the transport coefficient
40
References
41
3
Gas-Liquid Reactions on Solid Catalysts: Trickle Beds, Slurry Reactors, and ThreePhase Fluidized Beds
The multiphase reactors to be discussed in this are those in which gas and liquid phases are
contacted with a solid catalyst. The reaction generally takes place between the gas and the
liquid on the catalyst surface. However, in some reactions, the liquid phase is an inert medium
for the gas to contact the solid catalyst. The latter situation arises when a large heat sink is
required for highly exothermic reactions. In many cases the catalyst life is extended by these
milder operating conditions.
I.
Catalyst
A. Definition
A catalyst is a substance that affects the rate of a reaction but emerges from the
process unchanged. A catalyst usually changes a reaction rate by promoting a different
molecular path or mechanism for the reaction. For example, gaseous hydrogen and
oxygen are virtually inert at room temperature, but reacts rapidly when exposed to
platinum. The platinum acts as a solid catalyst.
B. Solid Catalyst
Many solid catalysts are used in the form of porous granules. The size is a compromise
between the need for large specific surface and ease of handling. Fluidized bed
processes employ particles in the range 20 to 200 microns. Both fixed and moving bed
operations are restricted by pressure drop considerations to larger sizes; 4 to 8 mesh is
common but may be 1 to 20 micron. Slurry process may employ powders of 400 mesh
or so, the limitation being filterability of the slurry after the reaction is complete.
C. Properties
The most important properties are the specific surface, particle diameter, porosity and
pore diameter. Specific surface ranges from 1 to 1000 m2/g. pore diameter of common
4
catalyst ranges from 10 to 200 Angstroms. Porosity of a bed of particles consists of the
space between their exteriors and of the space within them.
II.
Gas –Liquid Reactions
The simplest conceptualization of the gas-liquid transfer process is attributed to
Nernst (1904). Nernst postulated that near the interface there exists a stagnant film.
This stagnant film is hypothetical since we really don’t know the details of the velocity
profile near the interface.
A. Theories for Analysis of Transport Effects in Gas-Liquid Reactions
1. Two-film theory
This interphase can represent any point of the gas absorption equipment where the
gas contracts the liquid. In the two-film model Figure 1. a stagnant film is assumed
at both sides of the interface and all resistance to mass transport is localized in
these two films. This means that concentration gradients will only develop in these
films. It is further assumed that no, resistance to transport occurs at the interface
itself, so that the interface concentration of the gaseous component in the solution
is related to the interfacial partial pressure by Henry's law:
p A  kCA
(E1)
Where:
p A  partial pressure of the solute
k  Henry' s Law Constant
C A  concentrat ion of the solute in the solution
(a) Assumptions
i.
A stagnant layer exists in both the gas and the liquid phases.
ii.
The stagnant layers or films have negligible capacitance and hence a local
steady-state exists.
iii.
Concentration gradients in the film are one- dimensional.
iv.
Local equilibrium exists between the gas and liquid phases as the gas-liquid
interface
v.
Local concentration gradients beyond the films are absent due to
turbulence
5
Figure 1. Interphase Between the Gas Phase and the Liquid Phase
(b) Derivations
Figure 2. Schematic Diagram of Concentration Profile with Stagnant Film
Steady-state mass balance is done over an elementary volume of thickness ∆Z.
Rate of input of solute at Z  N A ,Z
Rate of output of solute at Z  Z  N A ,ZZ
Rate of accumulation=0= (rate of input-rate of output)
At steady state, N A z  N A
Z Z
lim z0
NA z  NA
N A
0
Z
z
z  z
0
(E2)
(E3)
6
   D AB C A 

0
Z 
Z

(E4)
 D AB  2 C A
0
Z 2
(E5)
 2C A
0
Z 2
(E6)
Integrating (E5) for the following boundary conditions:
C A  C Ai when Z  0
C A  C Ab when Z  
We have now,
C A  C Ai  C Ai  C Ab 
Z

(E7)
Hence, according to film theory, concentration profile in stagnant film is linear.
Molar flux through film, NA:
  D AB C A 
NA  
  0 when Z  0
Z


NA 
D AB C Ai  C Ab 

kL 
D AB

(E8)
(E9)
Film theory is useful in the analysis of mass transfer with chemical reaction. For
turbulent flow, the mass transfer coefficient has much smaller dependency
compared to laminar flow. In the turbulent flow, the mass transfer coefficient is
proportional to DnAB where n may be zero to 0.9, depending upon the operating
conditions. Although the film theory offers some explanation of the mechanism
of mass transfer in fluid media, it does not explain the estimation thickness of
the film. Due to this disadvantage, application of the model is restricted to
mass transfer in a diffusion cell.
2. Penetration Theory
Most of the industrial processes of mass transfer are unsteady state process. In
such cases, the contact time between phases is too short to achieve a stationary
state. This non stationary phenomenon is not generally taken into account by the
film model. In the absorption of gases from bubbles or absorption by wetted-wall
7
columns, the mass transfer surface is formed instantaneously and transient
diffusion of the material takes place.
(a) Assumptions
i.
Unsteady state mass transfer occurs to a liquid element so long it is in
contact with the bubbles or other phase
ii.
Equilibrium exists at gas-liquid interface
iii.
Each of liquid elements stays in contact with the gas for same period of
time
(b) Derivation
Figure 3. Schematic of Penetration Model
Under these circumstances, the convective terms in the diffusion can be
neglected and the unsteady state mass transfer of gas (penetration) to the
liquid element can be written as:
c
 2c
 D AB 2
t
Z
(E10)
The boundary conditions are: t = 0, Z > 0 : c = cAb and t > 0, Z = 0 : c = cAi. The
term cAb is the concentration of solute at infinite distance from the surface and
cAi is the concentration of solute at the surface. The solution of the partial
differential equation for the above boundary conditions is given by the
following equation:


c Ai  c
Z

 erf 
2 D t 
c Ai  c Ab
AB 

(E11)
8
Where erf(x) is the error function defined by
x


2
erf x  
exp  Z 2 Z
0

(E12)
If the process of mass transfer is a unidirectional diffusion and the surface
concentration is very low (cAb~0), the mass flux of component A, NA [kg m–2 s–1],
can be estimated by the following equation:
NA 
 D AB
1  c Ab
 c 
 c 
    
 Z  Z0
 Z  Z0
(E13)
Substituting (E11) into (E13) the rate of mass transfer at time t is given by the
following equation:
N A t   
D AB
c Ai  c Ab 
t
(E14)
Then the mass transfer coefficient is given by
k L (t) 
D AB
t
(E15)
The average mass transfer coefficient during a time interval tc is then obtained
by integrating (E11) as:
t
k L ,av
c
D AB
1

k t t  2
tc o
t c

(E16)
So from the above equation, the mass transfer coefficient is proportional to the
square root of the diffusivity. This was first proposed by R. Higbie in 1935 and
the theory is called Higbie’s penetration theory.
3. Surface renewal theory
For the mass transfer in liquid phase, Danckwert (1951) modified the Higbie’s
penetration theory. He stated that a portion of the mass transfer surface is replaced
with a new surface by the motion of eddies near the surface.
(a) Assumptions
i.
The liquid elements at the interface are being randomly swapped by fresh
elements from bulk
9
ii.
At any moment, each of the liquid elements at the surface has the same
probability of being substituted by fresh element
iii.
Unsteady state mass transfer takes place to an element during its stay at the
interface
(b) Derivation
Average molar flux, NA,av
N A ,av  C Ai  C Ab  s  D AB
(E17)
k L ,av  s  D AB
(E18)
and
where s is fraction of the surface renewed in unit time, or the rate of surface
renewal [ s 1 ].
4. Boundary layer theory
Boundary layer theory takes into account the hydrodynamics/flow field that
characterizes a system and gives a realistic picture of the way mass transfer at a
phase boundary.
(a) Derivation
A schematic of concentration boundary layer is shown in Figure 4.
Figure 4. Schematic of Concentration Boundary Layer
10
When   x  u  U  and when    m x   u  .99U  distance over which
solute concentration drops by 99% of (CAi-CAb).
Sh x 
Where:
xk L ,x
D AB
 .332Re 0.5 Sc0.33
(E19)
x is the distance of a point from the leading edge of the plate
kL,x is the local mass transfer coefficient
Sh av 
Where:
III.
lk l,x
D AB
 0.664Re 0.5 Sc0.33
(E20)
l is the length of the plate.
Understanding the Gas-Liquid-Solid Catalyst Reactions
Reaction occurring at the surface of the catalyst
•
A Reactant in the gas phase
•
B
Non-volatile reaction in the liquid phase
Number of steps
•
Transport of A from bulk gas phase to gas-liquid interface
•
Transport of A from gas-liquid interface to bulk liquid
•
Transport of A&B from bulk liquid to catalyst surface
•
Intraparticle diffusion in the pores
•
Adsorption of the reactants on the catalyst surface
•
Surface reaction to yield product
A. The General Rate Expression
Consider the following reaction and stoichiometry


 mol B 
catalyst
A g dissolve

 l  Bl  on
solid


 products...b  

 mol A 
(R1)
 rA'''  k 'A'' C A C B
(E21)
 rB'''  k 'B'' C A C B
(E22)
11
Where :
 rA'''
 rB''' mol A

;
b m 3 cat  s
 k 'A'' 
 k 'B''
mol 6
;
b mol B  m 3 cat  s
Mechanism
Gas reactant must first dissolve in the liquid, then both reactants must diffuse or move
to the catalyst surface for reaction to occur. Thus the resistance to transfer across the
gas-liquid interface and then to the surface of solid, both enter the general rate
expression.
Figure 5. Various Ways of Running Gas-Liquid Reactions Catalyzed by Solids
The overall local rate of reaction is given as
 1

1
1
R A  A* 



 k L a k S a P C wk 2 B l 
1
(E23)
12
Consider the reaction:
A (g )  B (l)  P(l)
(R2)
Gas Limiting Reactant (Completely Wetted Catalyst)
Kinetic rate
 mol 
   A (per unit catalyst volume)
k v A  3

 m cat v s 
Rate in Catalyst
mol 

k v P 1   B A S  3

 m react  s 
Transport rate
mol


 3
 (per unit volume)
 m react s 
Gas-Liquid
 Ag

k 1 a B 
 A 1 
 Ha

Liquid-Solid
k 1 a p A 1  A 2 
Overall Apparent Rate
mol


 3

 m react s 
(E24)
Ag
R A   o k v 1   B 
Ag
HA

HA
1
1
1


k 1 a B k s a P 1   B  p k v
(E25)
Figure 6. Gas – Liquid Solid Catalyzed Reaction
13
•
In catalytic reactor selection, the task to scale-up and design is to either maximize
volumetric productivity, selectivity or product concentration or an objective
function of all of the above.
•
The key to its success is the catalyst.
For each reactor type considered, we can plot feasible operating points.
Figure 7. Plot of Volumetric Productivity Versus Catalyst Concentration
Where:
 kgP 
  specific activity
S a 
 kg  Cat  h 
 kgcat  
x 3
  catalyst concentration
 m reactor 
•
 v max is determined by transport limitations and xmax by reactor type and
Clearly m
flow regime.
•
 v if we are not already transport limited.
Improving S a only improves m
•
Chemists or biochemists need to improve S a and together with engineers work on
increasing xmax .
•
 v max .
Engineers by manipulation of flow patterns affect m
In Kinetically Controlled Regime
v
m

x,
Sa
(E26)
xmax limited by catalyst and support or matrix loading capacity for cells or enzymes
14
In Transport Limited Regime
v
m

0 p
•
p
Sa ,
xp
(E27)
1
2
 v and set
Mass transfer between gas-liquid, liquid-solid etc. entirely limit m
m v max .
•
Changes in S a do not help, alternating flow regime or contact pattern may
help.
Figure 8. Resistances involved in the gas-liquid reaction on a catalyst surface
Considering (R1)


 mol B 
catalyst
A g dissolve

 l  Bl  on
solid


 products...b  

 mol A 
For A
 rA''' 
1
HA
HA
H
1


 ''' A
k Ag a i k Al a i k Ac a c (k A C B ) A f s
(E28)
15
For B
1
 rA''' 
1
k Bc a c

1
C Bl
(E29)
(k 'B'' C A ) B f s
Where:
 A  effectiveness factor for the first order reaction A with rate constant (k '''A C B )
 B  effectiveness factor for the first order reaction A with rate constant (k '''B C A )
Now either (E28) or (E29) should give the rate of reaction because even with all the
system parameters known (k, a, f, etc.) these expressions can’t still be solved without
trial and error because C B is not known in (E28) and C A is not known in (E29).
B. Two simplifying extremes
Extreme 1: C Bl » C Al In systems with pure liquid B and slightly soluble gas A we can
take
CBs  CB, within pellet t  CB ... same value everywhere
With C, constant the reaction becomes first order with respect to A overall and the
above rate expressions with their required trial and error all reduce to one directly
solvable expression
 rA''' 
Where:
1
HA
HA
H


 ''' A
k Ag a i k Al a i k Ac a c (k A C B  ) A f s
1
(E30)
(k 'A'' C B  )  first - order rate constant A
Extreme 2: CB  « C A  . In systems with dilute liquid reactant B, highly soluble A, and
high pressure, we can take
C Aι 
p Ag
HA
.....throughout the reactor
(E31)
16
The rate then becomes first order with respect to B and reduces to
1
 rA''' 
1
k Bc a c
Where:
 k 'B'' p A g

 HA


C Bl
1
 k 'B'' p A g

 HA

(E32)

 f
 B s


  first - order rate constant which is used to calculate 
B


C. Testing the Extremes
•
By the unequal signs » or « we mean two or three times as large.
•
More generally compare the rates calculated from (E30) and (E32) use the
smaller one. Thus,
'''
'''
if,  requation
4 «  requation 5 then Cl is in excess and extreme 1 applies.
'''
'''
if  requation
4 »  requation 5 , then (E32) gives the rate of reaction.
•
IV.
Nearly always does one or other of the extremes apply.
PERFORMANCE EQUATIONS FOR AN EXCESS OF B
All types of contactors-trickle beds, slurry reactors, and fluidized beds-can be treated
at the same time. What is important is to recognize the flow patterns of the contacting
phases and which component, A or B, is in excess. First consider an excess of B. Here
the flow pattern of liquid is not important. We only have to consider the flow pattern
of the gas phase. So we have the following cases
Figure 9. Gas bubbles swim around the bed; hence, gas is in mixed flow
17
A. Mixed flow G/ any Flow (excess of B)
Here we have the situation in Figure 9. A material balance about whole reactor gives
FAO X A 
Where:

1
FBO X B   rA''' V r
b

(E33)
Rate of loss of A FAO X A
1
FBO X B
b
Rate of loss of B

Rate of reaction  rA''' V r

Solution is straightforward so just add rate of A, rate of loss of B and reaction rate.
B. Plug flow G/ Any Flow L (excess B)
With a large excess of B, Cb stays roughly constant throughout the reactors shown in
Figure 10 even though the concentration of A in the gas phase changes as the gas flows
through the reactors. For a thin slice of reactors as shown in the Figure:
Figure 10. Contractors with Gas Rising in Plug Flow
We can write

FAO X A   rA''' V r
1  XA 

p A O   AO 
p 
p
dilute only A O dilute A
2
p AO 
p AO
p AO   p A 
(E34)
(E35)
18
 X A 
Where:
O  PAO p A
 p
p
dilute O A dilute A
2
PAO 
p AO
PAO   p A 
(E36)
  cons tan t
Overall, around the whole reactor
Vr

FAO
XA
 X A 
F
...with...FAO X A  BO X B
''' 
b
A 
   r
0
(E37)
C. Mixed Flow G/Batch L (excess of B)
With through flow of liquid, its composition stays roughly constant when B is in
excess. However, with a batch of liquid its composition slowly changes with time as B
is being used up, but B is roughly constant in the reactor at any time, as shown in
Figure 11. Here the material balance at any time t becomes
FAO X A ,exit
Vl  C B 
'''

  (rA )Vr
b  t 
(E38)
Figure 11. Gas Bubbles Through a Batch of Liquid, B in excess
Figure 12. Evaluation of the reaction time for batch of liquid
19
The general procedure for finding the processing time is as follows:
Then from II and III or I and III solve for t
V
t l
Vr

CBO
CB f
C
C B
Vl BO C B

(rA''' ) bFAO C X A ,exit

(E39)
Bf
as shown in Figure 12.
D. Plug Flow GI Batch L (excess of B)
As with the previous case, CB changes slowly with time; however, any element of gas
sees the same CB as it flows through the reactor, as shown in Figure 13.
Figure 13. Plug Flow of Gas Through a Batch of Liquid
Consider a slice of contactor in a short time interval in which CB is practically
unchanged. Then a material balance gives
FAO X A  (rA''' )
(E40)
20
Integrating gives the exit conversion of A
Vr

FAO
X A ,exit

0
X A
(rA''' )
(E41)
Considering B we may now write
FAO X A ,exit 
Vl  CB   mol 

,
b  t   s 
(E42)
and on integrating we find the processing time to be
Vl
t
bFAO
CBO
C B
X
CBf
(E43)
A ,exit
The procedure is as follows:
Then solve (E43) graphically to find the time as shown in Figure 14.
Figure 14. Evaluation of the Reaction in the Batch Liquid Reactor
Figure 15. Pure Gaseous A Can be Compressed and Recycled
21
E. Special Case of Pure Gaseous A (excess of B)
One often encounters this situation, especially in hydrogenations. Here one usually
recycles the gas, in which case pA and CA stay unchanged; hence, the preceding
equations for both batch and flow systems simplify enormously, see Figure 15. When
solving problems, it is suggested that one write down the basic material balances and
then carefully sees what simplifications apply, is pA constant? and so on.
Comments (other term)
The rate expressions used so far have been first order with respect to A and first order
with respect to B. But how do we deal with more general rate forms, for example
catalyst
A (g )  bB (l ) on
solid


 .....  rA,,, 
 rA'''
 k 'A'' C nA C Bm
b
(E44)
To be able to combine the chemical step with the mass transfer steps in simple
fashion, we
must replace the above awkward rate equation with a first-order
approximation, as follows:

n 1
m

 rA'''  k 'A'' CnA CBm  rA'''  k 'A'' C A CB C A
(E45)
This approach is not completely satisfactory, but it is the best we can do. Thus,
instead of (E30) the rate form to be used in all the performance expressions will be
 rA''' 
V.
pA
1
1
1
1
1
HA



H A k Ag a i k Al a i k Ac a c k 'A'' C nA1 C Bm  A f s


(E46)
PERFORMANCE EQUATIONS FOR AN EXCESS OF A
Here the flow pattern of gas is of no concern. All we need to worry about is the flow
pattern of liquid
A. Plug Flow LI Any Flow G (tower and packed bed operations)
Making the material balances gives on integration
Vr

FBO
XB
X B
 r
0
'''
B
where 1 - X B 
CB
C BO
(E47)
22
B. Mixed Flow LI Any Flow G (tank operations of all types)
Here the performance equation is simply
Vr
X
C
 B''' where XB  1  B
FBO  rB
C BO
(E48)
C. Batch L/Any Flow G
Noting that CA  constant throughout time (because it is in excess), the performance
equation for B becomes
t
C
V
C B
 B  rB  r (rB''' )... or... t 
t
Vt
 rB
0

VI.
(E49)
Kinetics, Design and Modeling
The two types of three-phase reactors to be discussed are the slurry reactor and the
trickle bed reactor. In the slurry reactor the catalyst is suspended in the liquid and the
gas is bubbled through the liquid. The slurry reactor may be operated in either the
semi-batch or continuous mode. The trickle bed reactor essentially a vertical packed
fixed-bed reactor in which the liquid and gas flow concurrently down the reactor.
Trickle beds saw their first use in the removal of organic material from wastewater
steams. Here, aerobic bacteria would attach and contacted with air. Since this first
application the trickle bed reactor has been used for a wide variety of reactions.
A. Slurry Reactors
Slurry reactors are commonly used in situations where it is necessary to contact a
liquid reactant or a solution containing the reactant with a solid catalyst. To facilitate
mass transfer and effective catalyst utilization, the catalyst is usually suspended in
powdered or in granular form. This type of reactor has been used where one of the
reactants is normally a gas at the reaction conditions and the second reactant is a
liquid like in the hydrogenation of various oils. The reactant gas is bubbled through
the liquid, dissolves, and then diffuses to the catalyst surface. Obviously mass transfer
limitations can be quite significant in those instances where three phases, the solid
catalyst, and the liquid and gaseous reactants, are present and necessary to proceed
rapidly from reactants to products.
23
1. Advantages of slurry reactors relative to other modes of operation
•
Well-agitated slurry may be kept at a uniform temperature throughout,
eliminating "hot" spots that have adverse effects on catalyst selectivity.
•
The high heat capacity associated with the large mass of liquid facilitates
control of the reactor and provides a safety factor for exothermic reactions that
might lead to thermal explosions or other "runaway" events.
•
Constant overall catalytic activity can be maintained by the addition of small
amounts of catalyst with each reuse during batch operation or with constant
feeding during continuous operation.
2. Modelling
A schematic diagram of a slurry reactor is shown in Figure 12. In modelling the
slurry reactor we assume that the liquid phase is well mixed, the catalyst particles
are uniformly distributed, and the gas phase is in plug flow.
The reactants in the gas phase participate in five reaction steps:
i.
Absorption from the gas phase into the liquid phase at the blue surface
ii.
Diffusion in the liquid phase from the bubble surface to the bulk liquid
iii.
Diffusion from the bulk liquid to the external surface of the solid catalyst
iv.
Internal diffusion of the reactant in the porous catalyst
v.
Reaction within the porous catalyst
Figure 16. Slurry Reactor for the Hydrogenation of Methyl Linoleate.
24
The reaction’s products participate in the steps above but in reverse order v
through i. Each step may be thought of as a resistance to the overall rate of
reaction R. These resistances are shown schematically in Figure 16. The
concentration in the liquid phase is related to the gas-phase concentration through
Henry’s Law:
C i  Pi H'
(E49)
One of the things we want to achieve in our analysis of slurry reactors is to learn
how to detect which resistance is the largest (i.e., slowest step) and how we might
operate the reactor to decrease the resistance of this step and thereby increase the
efficiency of the reactor.
To illustrate the principles of slurry reactor operation, we shall consider the
hydrogenation of methyl linoleate, L, to form methyl oleate, O.
Methyl linoleate (l) + hydrogen (g)
L
+
H2
methyl oleate(l)
(R3)
O
Hydrogen is absorbed in liquid methyl linoleate, diffuses to the external surface of
the catalyst pellet, and then diffuses into the catalyst pellet, where it reacts
Figure 17. Steps in a slurry reactor
with methyl linoleate, L, to form methyl oleate, O. Methyl oleate then diffuses out
of the pellet into the bulk liquid.
25
(a) Rate of gas absorption
The rate of absorption of H2 per unit volume of linoleate oil is
R A  k b a b (C i  C b )
Where:
(E50)
kb = mass transfer coefficien t for gas absorption , dm/s
a b = bubble surface area, dm2/dm3 of solution
C i = H 2 concentration at oil - H 2 bubble interface, mol/dm3
C b = bulk concentration of H 2 in solution, mol/dm3
RA 
dm
dm2
mol
mol
(
)

3
3
3
s dm of solution dm
(dm of solution) s
(E50)gives the rate of H2 transport from the gas-liquid interface to the bulk
liquid.
(b) Transport to catalyst
The rate of mass transfer of H2 from the bulk solution to the external surface of
the catalyst particles is
R A  k c a c m(C b  C s )
Where:
(E51)
kc= mass-transfer coefficient for particles, dm/s
ac=external surface area particles, dm2/s
m= mass concentration of catalyst (g of catalyst/dm3 of
solution); the parameter m is also referred to as the catalyst
loading
RA (
g
dm dm2
mol
mol
)(
)(
)

3
3
3
s
g
dm of solution dm
dm of solution  s
(c) Diffusion and reaction in the catalyst pellet
Reaction, -r’A to the rate -r’A that would exist if the entire interior of the pellet
were exposed to the reactant concentration at the external surface, Cas.
26
Consequently, the actual rate of reaction per unit mass of catalyst can be
written
r' A  (rA ,s )
(E52)
Multiplying by the mass of catalyst per unit volume of solution, we obtain the
rate of reaction per volume of solution:
R A  m(r' A ,s )
RA 
(E53)
g of catalyst 1
mol
mol
( )

3
3
dm of solution 1 g of catalyst  s dm of solution  s
(d) The rate law
The rate law is first order in hydrogen and first order in methyl linoleate.
However, since the liquid phase is essentially all linoleate, it is in excess and its
concentration, CL, reminds virtually constant at its initial concentration, CL0,
for small to moderate reaction times:
r' A,s  kCLD C  kC
(E54)
The rate of reaction evaluated at the external pellet surface is
 r' A,s  kC
(E55)
Where:
C s = concentration of hydrogen at the external pellet surface,mol/dm3
k = specific reaction rate, dm3 /g cat  s
3.
Determining the limiting step
Since, at any point in the column, the overall rate of transport is at steady state, the
rate of transport from the bubble is equal to the rate of transport to the catalyst
surface, which in turn is equal to the rate of reaction in the catalyst pellet. Then
R A  k ba b (Ci  Cb )  k cmac  (Cb  Cs )  m' (r' A,s )
(E56)
(E50) through (E55) can be rearranged in the form
RA
 Ci  Cb
k ba b
(E57)
27
Rb
 C b  Cs
k ba b
(E58)
RA
 Cs
mk
(E59)
Adding the equations above yields
RA  (
1
1
1


)  Ci
k b ab k c ac m km
(E60)
Rearranging, we have
Ci
1
1
1
1

 (
 )
R A k b a b m k c a c k
(E61)
Each of the terms of the right-hand side can be thought of as a resistance to the
overall rate of reaction such that
Ci
1
 rb  (rc  rr )
RA
m
(E62)
Ci
1
 rb  rcr
RA
m
(E63)
or
Where:
rb 
1
, resistance to gas absorption , s
k b ab
rc 
gs
1
, resistance to transport to surface of catalyst pellet,
k c ac
dm3
rr 
gs
1
, resistance to diffusion and reaction within the catalyst,
k
dm3
rcr  rr  rc , (combined resistance to internal diffusion,
reaction and external diffusion),
gs
dm3
For reactions other than first order
rr 
Cs
(r' A ,s )
(E64)
28
We see from equation (E63) that a plot of Ci/RA as a function of the reciprocal of
the catalyst loading, (1/m) should be a straight line. The slope will be equal to the
combined resistance rcr and intercept equal to the gas absorption resistance Rb.
Consequently, to learn the magnitude of the resistance, one varies the
concentration of catalyst, m, and measure the corresponding overall rate of
reaction Figure 18. The ration of gas absorption resistance to diffusional resistance
Figure 18. Plot to Delineate Controlling Resistances
to and within the pellet at a particular catalyst loading m is
absorption resistance int ercept  m
rb
=
=
1
diffusion resistance
rcr ( 1 m)
rcr ( )
m
(E65)
Suppose it is desired to change the catalyst pellet (to make them smaller, for
example). Since gas absorption is independent of catalyst particle size, the
intercept will remain unchanged. Consequently, only one experiment is necessary
to determine the combined diffusional and reaction resistance rcr. As the particle
size is decreased, both the effectiveness factor and the mass-transfer coefficient
decreasing slope in Figure 16.(a). In Figure 16.(b), we see that as the resistance to
gas absorption increases the intercept increases. The two extremes of these
controlling resistances are shown in Figure 17. Figure 17.(a) shows a large intercept
(rb) and a small slope (rc + rr), while Figure 17.(b) shows a large slope (rc + rr) and a
small intercept. To decrease the gas absorption resistance, one might consider
changing the sparger to produce more gas bubbles of smaller diameter.
29
Now that we have shown how we learn whether gas absorption rb or diffusion
reaction (rc + rr) is limiting by varying the catalyst loading, we will focus on the
case when diffusion and reaction combined are limiting. The next step is to learn
how we can separate rc and ri to learn whether

External diffusion is controlling,

Internal diffusion is controlling, or

Surface reaction is controlling.
To learn which of these steps controls, one must vary the particle size. After
determining rcr from the slope of Ci / R vesus 1/m at each particle size, one can
construct a plot of rcr vesus particle size, dp.
Figure 16. (a) Effect of Particle Size; (b) Effect of Gas Absorption
Figure 17. (a) Gas Absorption Controls; (b) Diffusion and Reaction Control
rcr 
1
1

k c a c k
(E66)
30
Very small particles: shows that as the particle diameter becomes small, the
surface reaction controls and the effectiveness factor approaches 1.0. For small
values of k (reaction control)
rcr 
1
k
(E67)
Consequently, rcr and rr are independent of particle size and a plot of in rcr as a
function of in dp should yield a zero slope for this condition of surface reaction
limitations.
Small to moderate-size particles: For large values of the Thiele modules we have
shown that
1

De 2
3 6

(
)
 d p kpS a
(E68)
Then
rr 
1
  1 dp
k
(E69)
We see that internal diffusion limits the reaction if a plot of rcr vesus dp is linear.
Under these conditions the overall rate of reaction can be increased by decreasing
the particle size. However, the overall rate will be unaffected by the mixing
conditions in the bulk liquid that would change the mass-transfer boundary layer
thickness next to the pellet surface.
Moderate to large particles: External resistance to diffusion was given by the
equation
rc 
1
k ca c
(E70)
The external surface area per mass of catalyst is
ac 
area
d 2 p
6
,

 3
mass
( )d pp d pp
6
(E71)
Next we need to learn the variation of the mass-transfer coefficient with particle
size.
31
Case 1: No Shear Stress between Particles and Fluid. If the particles are sufficiently
small, the move with the fluid motion such that there is no shear between the
particle and the fluid. This situation is equivalent to diffusion to a articles in a
stagnant fluid. Under this conditions the Sherwood number is 2, i.e.,
Sh 
k cdp
2
D AB
(E72)
Then
kc  2
D AB
dp
(E73)
And
rc 
d 2 p
  2d2 p
12D AB
(E74)
Consequently, if external diffusion is controlling and there isno shear between the
particles and the fluid, the slope of a plot of rcr versus dp should be 2. Since the
particle moves with the fluid, increasing the stirring would have no effect in
increasing the overall rate of reaction.
Case 2: Shear between Particles and the Fluid. If the particles are sheared by the
fluid motion, one can neglect the 2 in the Frossling correlation between the
Sherwood number and the Reynolds number and
1
1
Sh  2  0.6 Re 2 Sc 3
(E75)
Becomes
1
Sc Re 2
(E76)
then
k cdp
D AB
(
dpU

1
)2
(E77)
or
1
k c
U2
1
d2
(E78)
p
And
k ca c
1
2
U
d 1.5 p
(E79)
32
rc   3 d p1.5
(E80)
Another correlation for mass transfer to spheres in a liquid moving at low velocity
gives
Sh2  4.0  1.21(Re Sc)
2
3
(E81)
From which one obtains, upon neglecting the first term on the right hand side,
rc   4 d 1.7 p
(E82)
Figure 18. Effect of Particle Size on Controlling Resistance
If it is found that if the combined resistance varies with dp from 1.5-1.7power, the
external resistance is controlling and mixing (stirring speed) is important. Figure
18. shows the plot of the combined resistance rcr as a function of particle diameter
dp on log-log paper for the various rate-limiting steps.
Given a set of reaction rate data, we can carry out the following procedure to
determine which reaction step is limiting:
•
Construct a series of plots of C1/R as function of 1/m.
•
Determine the combined resistance from the slopes of these plots for each
corresponding particle in diameter.
•
Plot rcr as a function of dp on log-log paper. From the slope of this plot
determine which step is controlling. The slope should be 0,1,1.5,1.7 or 2
•
If the slope is in between any of these values say 0.5, this suggests that more
than one resistance is limiting.
33
4. Limiting conditions
The variables that influence the reactor under each of the limiting conditions are
discussed in the table below
Table 1.Variables Affecting the Observed Rate
Variables with:
Controlling step
Major influence
Minor influence

transport
Influence

Stirring rate

Reactor design
of liquid phase
(impeller, gas
reactant
Temperature


distributor,
Gas-liquid mass
Insignificant
baffling, etc.)

Concentration
Amount of
catalyst

Concentration
of reactant in a
Catalyst
particle size

gas phase
Concentration
of active
component(s)
on catalyst

Liquid-solid mass


Temperature
catalyst

Stirring rate
of liquid-phase
Catalyst

Reactor
reactant
particle size
transport(gaseous
reactant)
Amount of

design
Liquid-solid mass

reactant)

Concentration

Viscosity
of active
of reactant in

Relative
component(s)
densities
on catalyst
Amount of

Temperature
catalyst

Stirring rate
of gas-phase
Catalyst

Reactor
reactant
particle size
transport(liquid

Concentration
Concentration
gas phase


design


Concentration
Concentration
Concentration

Viscosity
of active
of reactant in

Relative
component(s)
densities
on catalyst
liquid phase
34

Temperature

Stirring rate

Amount of

Reactor design
catalyst

Catalyst
Chemical
Reaction

(insignificant
pore diffusion
Reactant
particle size
concentrations

resistance)
Concentration
of active
component(s)
on catalyst

Amount of
catalyst

Chemical
Reaction(significa
nt pore diffusion
resistance)

Pore structure

Stirring rate

Reactor design
Reactant
concentrations

Temperature

Catalyst
particle size

Concentration
of active
component(s)
on catalyst
B. Trickle Bed Reactors
A trickle bed reactor utilizes a fixed bed over which liquid flows without filling the
void spaces between particles. The liquid usually flows downward under the influence
of gravity, while the gas flows upward or downward through the void spaces amid the
catalyst pellets and the liquid holdup.
Industrial trickle beds are typically 3 to 6 m deep and up to 3 m in diameter and are
filled with catalyst particles ranging from to In. in diameter. The pores of the catalyst
are filled with liquid. In petroleum refining, pressures of 34 to 100atm and
temperatures or 350 to 425'C are not uncommon. A pilot-plant trickle bed reactor
might be about 1m deep and 4 cm in diameter. Trickle beds are used in such processes
35
as the hydrodesulphurization of heavy oil stocks, the hydro treating of lubricating oils,
and reactions such as the production of butynediol from acetylene and aqueous
formaldehyde over a copper acetylide catalyst. It is on this latter type of reaction,
A( g ,l )  BL  Cl
(R5)
1. Fundamentals
The basic reaction and transport steps in trickle bed reactors are similar to slurry
reactors. The main differences are the correlations used to determine the masstransfer coefficients. In addition, if there is more than one component in the gas
phase(e.g., liquid has a high vapor pressure or one of the entering gases is inert),
there is one additional transport step in the gas phase. Figure 19 shows the various
transport steps in trickle bed reactors. Following our analysis for slurry reactors we
develop the equations for the rate of transport of each step.
Transport from the bulk gas phase to the gas-liquid interface:
The rate of transport per mass of catalyst is
 rA  k g a i

1
C
C
1   B  p A g  Ai g 
mol 

 gcat
 s 


(E83)
Where
a i  interphacial area per volume of bed,
k g  gas phase transfer coefficient,
ρ p  density of catalyst pellet,
m2
,
m3
m
s
kg
m3
1   B   volume of solids per volume of bed (voids + solids)
C A g   bulk gas - phase concentration of A,
C Ai g   concentration of A at interphase,
kmol
m3
kmol
m3
Equilibrium at gas-liquid interface:
C Ai 
C Ai g 
H
(E84)
36
Where:
C Ai  concentration of A in liquid at the interphase,
kmol
m3
H  Henry' s constant
Transport from interface to bulk liquid:
 rA  k l a i
Where:


1
C Ai  C Ab ,  mol 
1   B  p
 g cat   s 
(E85)
k l  liquid phase mass - transfer coefficien t
C Ai  concentrat ion of A in liquid interphase
C Ab  bulk liquid concentration of A,
m
s
Transport from bulk gas to gas-liquid interface to bulk liquid to solid-liquid
interface Diffusion and reaction in catalyst pellet
Figure 19. Tricle Bed Reactor
Figure 20. Reactant Concentration Profile
37
Transport from bulk liquid to external catalyst surface:
 mol 

 rA  k c a c C Ab  C As , 
 g cat   s 
area
ac 
mass
Where:
k c  liquid - solid mass - transfer coefficient,
(E86)
m
s
C As  concentrat ion of A at solid - liquid interphase
2. Diffusion and reaction in the pellet
If we assume first-order reaction in dissolved gas A and in liquid B, we have
 mol 

 rA  kC As C Bs , 
 g cat   s 
(E87)
η  effectiven ess
Where:
C As  concentrat ion at the external surface of the pellet
k  specific reaction rate
Combining (E83) through (E87) and rearranging an identical manner to that
leading to the development of reactions for slurry reactors, we have
1
 mol 
H

(E88)
 rA 
C A g  , 
1   b  p 1   b  p
g cat   s 
1
1




Hk g a r
k lar
k c a c kC Bs
k vg
 rA  k vg C A g 
(E89)
Where:
k vg  overall transfer coefficient for the gas into the pellet,
m3
g cat  s
Mole balance on species A gives
dFA
 rA  k vg C A g 
dW
(E90)
We next consider the transport and reaction of species B, which does not leave the
liquid phase.
38
(a) Transport of B from bulk liquid to solid catalyst interface:
 mol 

 rA  k c a c (C B  C Bs )
 g cat   s 
(E91)
Where
C B and CBs are the concentration of B in the bulk fluid and at the solid
interphase, respectively.
(b) Diffusion and the reaction of B inside the catalyst pellet
rB  kC As C Bs
(E92)
Combining equation (E91) and (E92)and rearranging, we have
 rB 
1
1
1

k c a c kC As
k vl
 mol 

C B 
 g cat   s 
 r   k vl C B
(E93)
(E94)
A mole balance on species B gives
dFa
dCB
 v1
 rB  k vl C B
dW
dW
(E95)
One notes that the surface concentrations of A and B, CAs and CBs, appear in the
denominator of the overall transport coefficients. Consequently(E88), (E90),
(E93), and (E93) must be solved simultaneously. In some cases analytical
solutions are available, but for complex rate laws, one resorts to numerical
solutions. However, we shall consider some limiting situations.
3. Limiting situations
Mass Transfer of the Gaseous Reactant Limiting For this situation we assume that
either the first three terms in the denominator of (E88) are dominant, or that the
liquid-phase concentration of species B does not vary significantly through the
trickle bed. For these conditions Kvg constant, and we can integrate the mole
balance. For negligible volume change, then
W
vg
kg
ln
C A in 
C A out 

vg
k vg
ln
1
1XA
(E96)
Mass Transfer and Reaction of Liquid Species Limiting Here we assume that the
liquid phase is entirely saturated with gas throughout the column. As a result, CAs
39
is a constant and therefore is kvl. Consequently, we can integrate the combined
mole balance and rate law to give
W
CBin 
vl
v
1
ln
 1 ln
k vl CB out  k vl 1  X A
(E97)
4. Evaluating the transport coefficient
The mass-transfer coefficients, kg, kl, and kc depends on a number of variables,
such as type of packing, flow rates, wetting of particle, and geometry of the
column, and as a result the correlations vary significantly from system to system.
The plug-flow design equation may be applied successfully provided the ratio of
reactor length L to particle diameter dp satisfies the criteria (Satterfield,1975)
L 20
1

n ln
d p Pe
1X
Where:
Pe  Peclet number 
(E98)
dpU t
D Ax
D ax  axial dispersion coefficien t
n  reaction order
The CSTR design equations apply to the trickle bed when
L
4

d p Pe
(E99)
40
VII.
References

Levenspiel, Octave., Chemical Reactions Engineering, 3rd Ed., John Wiley
and Sons, Inc., 1999

Felder, Richard M. and Rousseou, Ronald W, Elementary Principles of
Chemical Processes, 3rd Ed., John Wiley and Sons, Inc., 2005

Fogler, Scott H., Elements of Chemical Reaction Engineering, 3rd Ed.,
Prentice-Hall Inc.,
41
42
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