Diffusion 12 and Reaction

Diffusion
and Reaction
12
Research is to see what everybody else sees, and
to think what nobody else has thought.
Albert Szent-Gyergyi
concentntion in
the internal surface
of the pellet is less
than lhatOf
external surface
Overview This chapter presents the principles of diffusion and reaction.
Wile the fmus is primarily on catalyst pellets. examples iUustrating these
principles are also drawn from biomaterials engineering and mimlecttonics.
in our discussion of cataIytic reactions in Chapter 10, we assumed each
point an the interior of catalyst surface was accessible to the same concentration. However, we b o w there are many, many situations where this equal
accessibility will not be h e . For exmple, whan the reacrants must diffuse
inside the catalyst pellet in order to react, we krlow the crmcenlration at the
pore mouth must be higher than that inside the pime. C o wquently, the entire
catalytic surface is not accessible to the same concentration; therefore, the
rate of reaction throughout the pellet will vary. To account for variations in
reaction rate throughout the pellet, we introduce a parameter known as the
effectiveness factor, which is the ratio of the overall reaction rate in the pellet to the reaction rate at the external surface of the pellet. In this chapter we
wilI develop mdeIs for diffusion and readan in two-phase systems, which
include catalyst pellets, tissue generation, and chemical vapor deposition
(CVD). The types of reactors discussed in, this chapter will include packed
beds, bubbling fluidized beds, slurry reactors, trickle bed reactors,and C W
boat reactors. After studying this chapter, you wilt be able to describe diffusion and reaction in two- and three-phase,syskms,determine when internal
diffusion Iimits the overall rate of reaction, describe how to go aboi
nating this limitation, and develop models for s:ysterns in which bo
sion and reaction pIay a role (e-g,. tissue growth1, m 1 .
814
Dittusion and Reaction
Chap. 12
In a heterogeneous reaction sequence, mass transfer of reactants first sakes
place from the bulk fluid to the external surface of the pellet. The reactants
then diffuse from the external surface into and through the pores within the
pellet, with reaction taking place only on the catalytic surface of the pores. A
schematic representation of this two-step diffusion process is shown in Figures
10-6and 12-1.
surface
Figure 12-1 Mass transfer and reaction steps for a catalyst pellet.
12.1 Diffusion and Reaction
in Spherical Catalyst Pellets
In this section we will develop the internal effectiveness factor for spherical
cataIyst pellets. The development of models that treat individual pores and pellets of different shapes is undertaken in the problems at the end of this chapter.
We will first look at the internal mass transfer resistance to either the products
or reactants that occurs between the external pelIet surface and the interior of
the pellet. TQ iJIusmate the salient principles of this model, we consider the
irreversible j sorneriza~ion
A-B
that occurs on the surface of the pore walls within the spherical pellet of mdius R.
12.1-1 Effective Diffusivity
The pores in .the pellet are not straight and cyljndsical; rather, they are a series
of tortuous, interconnecting paths of pore bodies and pore throats with varying
cross-sectional areas. It would not be fruitful to describe diffusion within each
and every one of the tortuous pathways individually; consequently, we chaIl
define an effective diffusion cwfficient so as to describe the average diffusion
taking place at any position r in tHe pellet. We shall consider only radial variations in the concentration; the radial flux W,, will be based on the total area
(voids and soIid) normal to diffusion transport c.e., 4i~r2)rather than void area
alone. This basis for W,, is made possible by proper definition of the effective
diffusivity 5),
The effective diffusivity accounts for the fact that:
.
1. Not all of the area normal to the direction of the flux is available (i.e..
the area occupied by solids) for the molecules to diffuse.
Sw. 12.1
825
Diffusion and Reaction in Spherical Catalyst Pellets
2. The paths are tortuous.
3. The pores are of varying cross-sectional areas.
An equation that relates D, to either the bulk or the Knudsen djffusivit~is
The effective
diffusivity
where
? = tortuosityl =
Actual distance a moIecule travels between two points
Shortest distance between those two points
$, = pellet porosity =
Volume of void space
Total volume (voids and solids)
u, = Constriction factor
The constriction factor, u,, accounts for the variation in the
cross-sectional area that is normal to diffusion.* It is a function of the ratio of
maximum to minimum pore areas (Figure 12-2(a)). When the two areas, A ,
and A * , are equal, the consrrjction factor is udty, and when P = 10, the constriction factor is approximately 0.5.
p = area A;,
area A,
(a1
Figure 1L2
Example 12-1
(b)
(a) Pore consrriction; (b) pore tonuosity.
Findittg the TorluesiQ
Calculate the tonuosity for the hypotheticar pore of length. L (Figure 12-2(b)). from
the definition of ?.
1
?
Some investigators lump consuicfion and tortuosity into one factor. called the tortuosity factor, and set it equal to 3nC.C. N. Safterfield. Muss Trun<fer in Heterngeneour
Caraljsis (Cambridge, Mass.: MIT Press, 1970). pp. 33-47. has an excellent discussion on this point.
See E. E. Petersen. Chemical Reacrion A11a1ysi.r(tUpper Saddle River, N.J.: Prenticc
Hall. 19651, Chap. 3: C. h'. Satterfield and T. K . Shemood. The Role ({Drflusion irl
Cnral~.rr.r(Reading. Mass.: Addlson-Wesley. 1963). Chap. 1 .
816
Dtffus~onand React~on
Chap.
7 = Actual distance n~oleculetravels from A to B
Shortest distance between I-I and B
The %honestdistance between potnts A and B is
ecule travels from A to B is 2L.
,E . The actual distance the mc
Although this value ih rcasonablc for i.
values for ? = 6 to IO are not unknow~
Typical values of the constriction fac~or,the tortuosity, and the pellet porosity an
respxtively, crc = 0.8, i= 3.0. and 0, = 9.10.
12.1.2 Derivation of the Differential Equation
Describing Diffusion and Reaction
we will derive
the cuncentration
profile oFre,ctant
4 In the pellet.
We now perform a steady-state mole balance on species A as it enters. leaves
and reacts in a spherical sheIl of inner radius r and outer radius r + Ar of tb
pellet (Figure 12-3). Note that even though A is diffusing inward toward th
center of the pel jet, the convention of our shell balance dictates that the ff ux bl
in the direction of increasing r. We choose the flux of A to be positive in thl
direction of increasing r (i.e.. the outward direction). Because A is actuall]
diffusing inward. the Aux of A will have some negative value, such a:
- 10 mol/m2.s. indicating that the flux i s acttlaIly in the direction of decreas
ing r.
-
Figure 13-3 Shell balance on a catniyst pellet.
We now proceed to perform our shell balance on A. The area that appears
in the balance equation is the total area (voids and solids) normal to the ditec-
tion of the molar Rux:
Rate of A in at r = W,,;Area = W,, X 4ar21,
IrfAr
Rate of A out at ( r + Ar) = WAr.Area = WArX 4 ~ r Z
( 12-2)
( 1 2-3)
Sec. 12.1
817
Diffusion and React~on~nSpher~calCatalyst Pellets
1
Rate of
generation
Rate of reaerion]
A withina
= [Mass of cnaiyrr
sheIl thickness
of"
1
+
reaction jnsldc the
cat:~Iy+lpellet
btole balnnce
Volume
x [volume of sheli]
1
-
Mole balance for
diffusion and
[~ars
.
PC
x
x
4~lri~r
where r, is some mean radius between r and r + br that is used to approximate the volume A Y of the shelt and p, is the density of the pellet.
The mole balance over the shell thickness L\r is
- (Out at r + b ) + (Generation within b)= 0 ( 1 2-5)
- ( W A , x 4 ~ r 2 ] , + b+) ( x
A )
=o
After dividing by ( - 4 r A r ) and taking the limit as Ar -+ 0,we obtain
(In at r)
( WArx4nr2Ir)
the following differential equation:
Because 1 mol of A reacts under conditions of constant temperature and
pressure to form 1 moE of 3,we have Equal Molar Counter Diffusion (EMCD)
at constant total concentration (Section 11.2.1A), and. therefore.
The
B t ~ xequation
where C , is the number of moles of A per dm3 of open pore volume (i-e.,volto (rnol/vol of gas and solids).-In systems where we
do not have EMCD in catalyst pores. it may still be possible to use Equation
( 12-7) if the reactant gases are present in dilute concentrations.
After substituting Equation ( t 2-7) into Equation / 12-6$, we arrive at the
following differential equation describing diffusion with reaction in a catalyst
ume of gas) as opposed
pellet:
We now need to incorporate the rate law. In the past we have based the
rate of reaction in terms of either per unit volume,
-
-
or per unit mass of cataIyst,
-r,[=](moVg cat. s )
818
Dtffuskm and Reaction
Chap. 12
we study reactions on the internal surface area of catalysts, the rate of
reaction and rate law ax often based on per unit surface area.
When
As a result, the surface area af the catalyst per unit mass of catalyst,
is an important property of the catalyst. The rate of reaction per unit mass of
catrtlyst, -rl , and the rate of reaction per unit surface area of catalyst are
related through the equation
gm,
catalyst may cover
as much surface area
as a football field
The rate law
-6 -rzsB
5
A typical value of S, might be 150 m2/gof catalyst.
As mentioned previously, at high temperatures, the denominator of the
catalytic rate law approaches 1 . Consequently, for the moment, it is reasonable
to assume that the surface reaction is of nth-order in the gas-phase concentration of A within the pellet.
where
'
L
-
Similarly,
per unit Surgace Area:
per unit Mass of Catalyst: k; = k
per unit Volume: k
p ~=[m
,
( k g , s )1
=~~S,~,[S-]]
kn
Differential
equation and
=pcsakRpckn151
(gr
Substituting the rate law equation (12-9) into Equation (12-8) gives
boundary condition^
dercrib~ngdiffusion
and reaction in a
catalys~pellet
d [r2(-De
dC,/dr)]
dr
+r2 ~'LS,P,'C~
=o
(12-101
By differentiating the first term and dividing through by -r2D,, Equation
( 1 2- 3 0) becomes
See. 12.1
-
Diffusion arid Reaction in Spherical Catalyst Pellets
The boundap conditions are:
1. The concentration remains finite at the center of the pellet:
C, is finite
at r = 0
2. At the extemaI surface of the catalyst pellet. the concentration is CAs:
12.1.3 Writing the Equation in Dimensionless Form
We now introduce dimensionless variables and A so that we may arrive at a
parameter that is frequently discussed in cataIytic reactions, the Thiele modulus. Let
With the transformation of variables, the boundary condition
CA = CAa
at r = R
and the boundary condition
C, is finite
at r = 0
becomes
yf is finite
at h = 0
We now rewrite the differential equation for the molar flux in terms of our
dimensionless variables. Starting with
CA
w*,= -n, ddr
we use the chain rule to write
820
Diffusion and Reaction
Ch:
Then differentiate Equation (12-12) with respect to \y and Equation ( 1 2
with respect to r, md substitute the resulting expressions.
-
~ C =ACAs
dyr
and
&
- = -1
dr
R
into the equation for the concentration gradient to obtain
The flux of A in terns of the dimensionless variables,
and X, is
The total rate of
consumption of A
~nsidethe pellet.
IM, (mollsl
At steady state, the net flow of species A that eaters into the @let at
external pellet surface reacts completely within the pellet. The overall rate
reaction is therefore equal to the total molar flow of A into the catalyst pel
The over311 rate of reaction, MA,can be obtained by rnuItiplying the molar B
into the pellet at the outer surface by the external surface area of the pel1
47r R2:
All the reactant
that d~ffusesinto
the pellet is
consumed
(a black hole)
MA=-4icR'W,,l
r=8
=4xRDeC,&1
=+4nR2Desl
dr
r=R
(12-1
k=I
Consequently, to determine the overall rate of reaction, which is given 1
~ ~ u a t i o( 12n 13), we first solve Equation (12-I 1) for CA,
wi
respect to r, and then substitute the resulting expression into Equation (12-1;
Differentiating the concentration gradient, Equation (12-13, yields
differentiate^^
After dividing by CAsIR2.the dimensionless form of Equation f 12-11 ) is wri
ten as
Then
Dimensionless form
of equations
describing diffusion
and reaction
Sec, 12.1
821
Diffusion and React~onin Spher~calCatalyst Pellets
( 12-20)
Th~eIemodulus
The square root of the coefficient of v*. (i.e., h,) is called the Thiele
moduius. The Thiele modulus, 4,. will always contain a subscript (e.g., n),
which will distinguish this symbol from the symbol for porosity, 4, defined in
Chapter 4. which has no subscript. The quantity &! i s a measure of the M I ~ O
of "a" surface reaction rate to "a" rate of diffusion through the catalyst pellet:
-
$; =
k,R2C?;' knRCXq - "a" surface reaction rate
D,
D,[{CAC,,
-O)/R 1
"a" diffusion rate
When the Thiele rnodrrlus is large, inwrnnl diffzuion ~ ~ s t t a litnits
l l ~ rl~e
overall rate of reaefion; when +,, is stnnfl, [he ~sttface reacrion is t t ~ t ~ n l i y
mte-limiri~zg.If for the reaction
the surface reaction were rate-limiting with respect to the adsorption of A and
the desorption of 13, and if species A and B are weakly adsorbed (i.e., low coverage) and present in very dilute concentrations, we can write the apparent
first-order rate law
The units of K; are rn3/rn's (= mJs).
For a first-order reaction. Equation {I 2- 19) becomes
where
822
Diilusion and Reaction
Chap. 12
The boundary conditions are
32-1.4 Solution to the Differential Equation
for a First-Order Reaction
Differential equation (12-22) is readjIy solved with the aid of the transfonnation y = yA:
With these transformations, Equation (1 2-22) reduces to
This differentia1 equation has the following solution (Appendix A.3):
= A , cash+, )l + B , sinhb, h
The arbitrary constants A* and El can easily be evaluated with the aid of the
b o u n d a ~conditions. At h = 0;cosh4, h
I, ( I l k ) + =. and sinhblA + 0.
Because the second boundary condition requires to be finite at the center
(i.e.. h = 0). therefore A , must be zero.
The constant B, is evaluated from B.C. 1 (i.e., iy = I. h = 1) and the
dimensionless concentration profile is
Cancentralion
profile
Figure 12-4 shows the concentration profile for three different values of
the Thiele modulus, b , . Small values of the Thiele modulus indicate surface
reaction controls and a significant amount of the reactant diffuses well in to the
pellet interior without reacting. Large values of the Thiele modulus indicate
that the surface reaction is sapid and thal the reactant is consumed very close
to the external pellet surface and very little penetrates into the interior of the
Sec. 12.1
I
Dfttusionand Reaction ~n Spherical Catalyst Pelbets
823
pellet. Consequently, if the porous pellet is to be plated with a precious metal:
catalyst (e-g., Pt). it should only be plated in the imrnedia~evicinity of the
external surface when large values of &,, characterize the diffusion and xaction. That is, it would be a waste of the precious metal to plate the entire pellet
when internal diffusion is limiting because the reacting gases are consumed
near the outer surface. Consequently, the reacting gases would never contact
the center ponion of the pellet.
For large values of
the Thiele modulus,
~nternaldiffusion
lirnln the rate of
reaction.
Figure 12-4
Example 12-2
I
Concentration profile in a spherical cataly.rt pcllc~
application.^ of DifSusion arrd Reactio~;ia ESSIIP
Engirreen'ng
The equations describing diffuusion and reaction In pornus cataly\t!, also can he uqed
rates of tiswe growth. One important area of tissue growth I S in cartilage
tissue in joints such as the knee. Over 200.NK)patients per year receive knee joint
replacements. Alternative stratep~es include rhe growth af cartilage to repair the
damaged knee.?
One approach cumntly being researched by Professor Krifti Anqeth at the
University of Colorado is to deliver cartilage forming cells in a hydrogel to the damaged area such as the one shown in Figure E12-7.1.
to denve
I
Figure Ell-2.1 Damaged c;in~lugr, Flpvrc c o u n r i y ol
\'i,i~
iil
r ~ iScplrmhur
.
3. 1IXl I . i
824
Biffuslon and React~on
Chap.
Here the patient's own cells are obtained from a biopsy and embedded if
hydro gel. wh~ch1s a cross-l~nkedpolymer network that is swollen in water. In on
for the celEs to riurvive and grow new tissue, many propenies of the gel must
tuned to allow diffusion of important specles in and out le.g., nutrients in a
cell-secreted extracellular molecules our. such as collagen). Because there is
blood flow through the cartilage, oxygen transport to the canilage cells is prirnar
by diffusion. Consequently, the design rnuqt be that the gel can maintain the necr
aary nre'; of diffusion of nutrients le.g., O?) into the hydrogel. These rates
exchange in the gel depend on the geometry and the thickness of the gel. To ill1
trate the application of chemical reaction engineering principles to tissue engine,
ing, we wiH examine the diffusion and consumption of one of the nutrients, oxygr
Our examination of diffusion and reaction in catalyst pellets showed that
many cases the reactant concentration near the center of the particle was virtual
zero. Ifthis condition were to occur in a hydrogel. the cells at the center would dl
Consequently, the gel thickness needs to be designed to allow rapid transport
oxygen.
Let's consider rhe simple gel geometry shown in Figure El?-2.2,
02
Figure El2-2.2 Schematic of cnrtilage cell system.
We want to find the gel thickness at which the minimum oxygen consumption rat
is 10-l3 moticell/h. The cell density in the gel is loL0celisidrn3, and the bulk con
centration of oxygen = 0)is 2 x 1W rnoYdm3, and the diffusivity is IPScm2/s.
Solldlion
A mole balance on oxygen, A. in the volume A V = A&
F*I=-F*I:+,+r*A,~ = 0
Dividing by Az and taking the limit as d* + 0 gives
is
Sec. 12.1
D~ffuSlOnand
Reactron rn Spherical Catalyst Pellets
For dilute concentrations we neglect
tE 12-2.3) to obtain
ffc,
825
and combine Equatims (E12-2.2) and
If we assume the 0:consumption rate is zero order. then
I
Purring our equation in dimcnrionlcu form using
= C,.,fCAfjand :/L7 vc obtain
Recognizing the second term is just the ratio of a reaction rate to a diffusion rate for
a zero order reaction, we call this mrio the Thiele modulus, @,., We divide and multiply by two to facilitate the integration:
The boundary conditions are
At
h=U
v=I
At
L=l
ko
A.
CA= CAI)
(E 12-2.9)
Symmetry condition (E12-2.10)
Recall that at the midplane (=
: L, h = 1) we have symmetry so that there is no diffusion across the midplane so the gradient is zero at A = I .
tntegrating Equation (El2-2.8) once yields
Using the symmetry condition that there is no gradient across the midplane, Equation (E12-2.101, gives k; = - 3$0:
integrating a second time gives
yr = $ , x ~ - ~ ~ , x + K ~
826
Diffusion and Reaction
Using the boundary condition
concentration profile is
\y
Chap. 12
= I at h = 0, we find K2 = I . The dimensionless
Nore: The dimensionless concentration profile given by Equation (E12-2.13) is only
valid for values of the Thiele modulus less than or equal to 1 . This restriction can
he easily seen if we set & = 10 and then calculate yr at Ir = 0.1 to find y~ = -4.9,
which 1s a negative concentrarion!! This condition i s explored further in Problem
PI?-10,.
Evaluating the zem-order rate constant. k. yields
-13
k
l ~=' ~ c e. 10
t ~ s mole 01= 10-~moie,dm3,
dm3
cell . h
,,
and then the ratio
The Thiele module is
(a)
Consider the gel to be completely effective such tha! the concentration of
oxygen I S reduced to zero hy the time it reaches the center of the pet. That is,
~f y = O at 7L = I. we solve Equation (El 2-2.13) to find that bo = 1
Solving for the gel half thickness L yields
(hl
Lel's critiqi~ethis answer. We said the oxygen concentration was zero st the
center. and the cells can't furvive w~tboutoxygen. Consequently, we need to
redcsign <u C>, is not zero at the center.
Now confider ;he case whet;: the minimum oxygen concentration for the cells
to survrve is 0.1 mmolldm', ~ h i c hi s one half that at the surface (i.e., y = 0.5
at = l .O). Then Equnt~on[El 2-2. I31 give<
cm'
Solving Eqi~arion(EI2-2.171 for L gives
Sec. 12.2
(c)
Internal Effectiveness Factor
827
Consequently, we see that the maximum thickness of the cartilage gel (2L) is
the order of 1 mm. and engineering a t h i c k tissue is challenging.
One can consider orher perturbations to the preceding analysis by considering
the reaction kinetics to foliow a first-order rate law, -r, = k,C,, or Monod
kinetics,
The author notes the similarities to this problem wirh his research on wax build-up
in subsea pipeline gels! Here as the parafin diffuses into the gel to form and grow
wax particles. these particles cause paraffin molecules to take a longer diffusion
path, and as a consequence the diffusivity is reduced. An analogous diffusion pathway for oxygen in the hydrogel containing collagen is shown for in Figure E12-2.3.
Figure EX2-2.3 Diffusion of Ol amund collagen.
where a and F, are predetermined parameters that account for diffusion around particles. Specifically, for cotlagen. a is the aspect ratio of the collagen panicle and F ,
is weight fraction of 'kolid" coliagen obstructing the diffusion.' A similar rnodification could be made for cartilage growth. These situations are left as an exercise in
the end-of-the-chapter problems. e.g., PI 2-ZIb).
12.2 Internal Effectiveness Factor
The magnitude of the effectiveness factor (ranging from 0 to 1 ) indicates the
relative importance of diffusion and reaction limitations. The internal effectiveness factor is defined as
-a
P. Singh. R. Venkatesan, N. Nagarajan, and H,S, Fogler, AIChE J., 46, 1054 (2000).
Diffusion and Reaction
828
7 1s a measure of
huw hr the
reactant
diffuws
into the pellet
before reacttne
Cbap
ActuaI overail rate of reaction
= Rate of reaction that would result if entire interior surface were
exposed to tlie external peIlet surface conditions C,,
'
The overall rate, -r;, is also referred to as the observed rate of react]
[-r,(ubs)]. ln terms of symbols. the effectiveness factor is
To derive the effectiveness factor for a first-order reaction, It is easiest to wc
in reaction rates o f (moles per unit time). M,. rather than in moles per UI
time per volume of catalyst (i.e.. -r, )
- -r,xVolume of catalyst particle =>11.1
q=-- -r,,
- I . , , x Volume of catalyst particle
MM4,
First we shall consider the denominator, MA,. If the entire surface we
exposed to the concentration at the external surface of the pellet, CAs.the ra
for n fitst-order reactinn wcluld be
MA,=
Rate at external surfacexYolume of carslyrl
Volume
The subscript s indicates that the rate -rAs is evaluated at the condition
present at the external surface of the pellet (i.e., h = 1).
The actual rate of reaction is the rate at which the reactant diffuses int
the peltet at the outer surface. We recall Equation (12-17) for the actual rate c
reaction,
The actual rate
of reactton
Differentiating Equation (12-77) and then evaluating the result at X = 1 yield:
-
Substituting Equation (1 2-30) into (12- 17) gives us
MA = 45iRDeCA,(mlcothib1 - I )
We now substitute Equations (12-293 and (12-31) into Equation
obtain an expression for the effectiveness factor:
(12-31;
(12-28) to
Sec. 12.2
829
Internal Effectiveness Factor
Internal
effectiveness factor
for a first-order
reaction in a
spherical catalyst
( 12-32)
gellet
A plot of the effectiveness factor as a function of the Thiele modulus is shown
in Figure 12-5. Figure 12-5(a) shows q as a function of the Thiele modulus 4,
for a spherical catalyst pellet for reactions of zero, first, and second order. Figure 12-5(b) corresponds to a first-order reaction uccurring in three differently
shaped pellets of volume V, and external surface area A,, and the Thiele modulus for a first-order reaction, &,, is defined differently for each shape. When
volume change accompanies a reaction (i.e., & + b ) the corrections shown in
Figure 12-6 apply to the effectiveness factor for a first-order reaction.
We observe that as the particle diameter kcomes very small, b, decreases,
If $ , 7 2
s o that the effectiveness factor approaches 1 md the reaction is surface-reacthen ~ ( = - [ + ~ - l ] tion-limited, On the other hand, when the Tbiele modulus 4, is large (-301, the
internal effectiveness factor q is small ( i t . , 7 4 1). and the reaction is diffusion-limited within the pellet. Consequently, factors inf uencing the rate of exterthen ?-nal mass urnsport will have a negligible effect on the overall reaction rate. For
@I
large values of the Thiele modulus,the effectiveness factor can be written as
To express the overall rate of reaction in terms of the Thiefe modulus, we aearrange Equation (12-28) and use the rate law for a first-order reaction in Equation (12-29)
-rA =
1
rate
reaction rate at C,,)
Reaction rate at C,
=7(~ICAJ
(12-34)
Combining Equations (12-33) and (12-341, the overall rate of reaction for a
first-order, internal-diffusion-limitedreaction is
Diffusion and Reaction
Chap. 12
Thiele modulus,
-4
Zero order
= ~jw~
+,o=~
+ , = ~ J k i : =
Second order
1
4
~
Reaction
Internal
effectiveness factor
for different
reaction orders and
catalyst shapes
0.1
I
1
I
I
0.4 0.6
I
1
I
I
I
1
2
4
6
I
10
m
Sphere
$1
Cylinder
$1
Stab
( n ~ ~ h z= $lm
i x
= ( ~ 2 ) J =m
$hz
=
+ , = L J ~ =
Figure 12-5 (a) Effectiveness factor plot for nth-order kinetics spherical catalysl
particles (from Mass T~ansfer m Heterogeneous Caralysrs, by C. N. Salkrtield.
1970; repnnt editron: Roben E. Krieger Publikhing Co., 1981; reprinted by
permiss~onof the au~hor).Ib) First-order reactlon in differem pellet peometrics
(from R. Aris. Inrmducr~on!o rhe Analwir of Cltemicrrl Rtacrors. 1965, p 13 I :
reprtnted by permi.rf~onof Pren~ice-Hall.Englewood Cliffs. N.J.).
Sec. 12.2
Internal Effecttveness Factor
Comction for
volume change with
reaction (ire., c+O )
T'
--
?
- Factor in the presence of volume chmnqe
Foctor in obsence of volume change
Figure 12-6 Effectiveness factor satins for first-order kine~icson spherical
catalyst pellets for various values of the Thiele modulus of a sphere. 6,.as a
function of volume change. [From V. W.Weekman and R. t.Gor~ng,J. Carat. 4,
260 (1965j.j
can the rate
of reaction be
increased?
HOW
Therefore, to increase the overall rate of reaction, -ri: ( 1 ) decrease the radius
R (makepellets smaller); (2) increase the tempemure; (3) increase the concentration; and (4) increase the internal surface area. For reactions of order n, we
have, from Equation (12-20),
For large values of the Thiele modulus, the effectiveness factor i s
Consequently, for reaction orders greater than 1 , the effectiveness factor
decreases with increasing concentration at the external pellet surface.
The preceding discussion of effectiveness factors is valid only far isothermal conditions. When a reaction is exothermic and nonisothemal, the
effectiveness factor can be significantly greater than 1 as shown in Figure 1 2-7.
Values of q greater than 1 occur because the external surface temperature of
the pellet is less than the temperature inside the pellet where the exothermic
reaction is taking place. Therefore, the rate of reaction inride the pellet is
greater than the rate at the surfzce. Thus, because the effectiveness hcror is the
Oitfusion and Reaction
Chap,
Can you find regions
where multiple wlulions (MSS) exist?
4'1
Figure 12-7 Nonirothermal effectiveness Factor.
ratio of the actual reaction rase to the rate at surface conditions, the effecdvt
ness factor can be greater than 1, depending on the magnitude of the paramt
ters p and y . The parameter y is sometimes referred to as the Arrhenit
number, and the parameter p represents the maximum temperature differenc
that could exist in the pellet relative to the surface temperature T,.
-
y = Arrhenius number = t
RT,
(See Problem P12-138 for the derivation of P.) The Thiele modulus for
first-order reaction,
is evaluated at the externaI surface temperature. Typicr
values of y for industrial processes range from a value of y = 6.5 (P = 0.022
b, = 0.22) for the synthesis of vinyl chloride from HC1 and acetone to a valu
of y = 29.4 (P = 6 X
= 1.2) for the synthesis of amrnonian5Th
lower the ?henna1 conductivity k, and the higher the heat of reaction, the greate
the temperature difference (see Problems P 12- 13, and P12-14,). We observ
+,,
TypicaI parameter
values
+,
H. V. Hlavacek,
N. Kubicek, and M. Marek, J, C~ataL,15, 17 ( 1969).
Ssc. 12.3
I
Ctiterton for no
I\lSSs in thc pellet
Falsified Kinetics
from Figure 12-7 that multiple stead!: states can exist for values of the Thjele
modulus less than I and when P is greater than approximately 0.2. There will
be no multiple steady stares when the criterion developed by Lussh is fulfiIled.
/ 12-36]
12.3 Falsified Kinetics
YOU may not be
measuring what
you th~nkyc~uare.
There are circumstances under which the measured reaction order and activation energy are not the true values. Consider the case in which we obtain reaction rate data in a differenrial reactor. where precautions are taken to virtually
eIiminate external mass transfer resistance (i.e., C, = CAb}.From these data
we construct a log-log plot of the measured rate of reaction -r, as a function
of the gas-phase concentration, CAs(Figure 12-81, The slope of this plot is the
apparent reaction order n' and the rate law takes the form
Measured rate with
apparent reaction
order n'
,c
log
Figure 12-8 Determining the apparent reaction order I-r, = p, I-r;)).
We will now proceed to relate this measured reaction order n' to the true reaction order n. Using the definition of the effectiveness factor. note that the
actual rate -r,, is the product of r( and the rate of reaction evaluated at the
external surface, k , C l , , i.e.,
For large values of the Tbiele modulus d,, we can use Equation ( 17-35) to substitute into Equation (12-38) to obtain
D. LUSS.
Ckem. Eng. Sci.,23, 1 249 ( 1968).
Diffusion and Reactinn
Chap 72
We equate the true reaction rate, Equation (12-39), to the measured reaction
rate, Equation (12-371, to get
We now compare Eqations (12-39) and (12-40). Because the overall exponent
of the concenlration, CAq,must be the same for both the analytical and measured rates of reaction, the apparent reaction order n' is related to the true reaction order n by
The true and the
apparent reaction
order
In addition to an apparent reaction order, there is also an apparent activation energy, EApp.
This value is the activation energy we would calculate using
the experimental data, from the slope of a plot of In (-r,) as a function of 1IT
at a fixed concentration of A. Substituting for the measured and true specific
reaction rates in terms of the activation energy gives
measured
into Equation (12-401, we find that
true
Taking the natural log of both sides gives us
where ET is the true activation energy.
Comparing the temperature-dependent terms on the right- and left-hand
sides of Equation (12-42), we see that the true activation energy is equal to
twice the apparent activation energy.
The true activation
energy
Sec 12.4
Overall Effectiveness Factor
835
This measurement of the apparent reaction order and activation energy results
primarily when internal diffusion limitations are present and is referred to as
disguised or falsified kinetics. Serious consequences could occur if h e laboratory data were taken in the disguised regime and the reactor were operated in
Imponant industrial a different regime. For example, what if the particle size were reduced so that
consquence
Of
jnternaI diffusion limitations became negligible? The higher activation energy,
falsified kinetic
ET,would cause the reaction to be much more temperature-sensitive, and there
is the possibility for runoway reaciion conditions to occur.
12.4 Overall Effectiveness Factor
For firstorder reactions we can use an overall effectiveness factor to help us
analyze diffusion, flow, and reaction in packed beds. We now consider a situation where external and internal resistance to mass transfer to and within the
pellet are of the same order of magnitude (Figure I 2-9). At steady state, the
rransport of the reactant.(s)from the bulk Ruid to the external surface of the catalyst is equal to the net rate of reaction of the reactant within and an the pellet.
Here, borh internaI
and external
diffusion are
important.
Figurn 12-9
Mass transfer and reaction steps.
The molar rate of mass transfer from the bulk fluid to the external surface is
Molar rate = (Molar flux) - {External surface area)
MA = W,, (Surface area/Volume) (Reactor volume)
= WA,-a,dV
112-44)
where oa,is the external surface area per unit reactor volume (cf. Chapter 11)
and AV is the voIume.
This molar rate of mass transfer to the surface. MA, is equal ro the net
(total) rate of reaction on and n-irhin the pellet:
Diffusion and Reaction
836
(
External area =
iM, = -ri (External area + Internal area j
Mass of catalyst
= P C 11 - 4 )
= Purosity
Mass of catalyst
Vol~~rne
of catalyst
) MA=+;[a,
See nomenclature note
1
area x Reaclor v o l u m ~
Reactor volume
area = Internal area
$I
Chap
volume of catalystxReactor valum,
Reactor volume
AV+S,pb A V ]
I
( 12-4
Combining Equations ( 1 2-44) and (12-45) and canceling the voIurne P
one obtains
For most catalysts the internal surface area is much greater than the extern
surface area (i.e., Sapb 4 a,). in which case we have
where -r," is the overall rate af reaction within and on the pellet per unit su
face area. The relationship for the rate of mass transport is
where k, is the external mass transfer coefficient ( d s ) . Because internal diffi
sion resistance is also significant, not all of the interior surface of the pellet
accessible to the concentration at the external surface of the pellet, C,,. M
have already learned that the effectiveness factor is a measure of this surfar
accessibility [see Equation (12-3811:
Assuming that the surface reaction is first order with respect to A, we can ut
lize the internal effectiveness factor ta write
Sec. 12.4
Overalk Effect~venessFactor
837
We need to eliminate the surface concentration from any equation involving the rate of reaction or rate of mass transfer, because CAscannot be measured by standard techniques. To accomplish this elimination. first substitute
Equation (12-48) into Equation ( 12-46):
Then substitute for WA,a, using Equation (12-47)
I
1
Solving for CAs,we obtain
Concentration at
the pellet surface as
a function of bulk
gas concentndon
Substituting for CAsin Equation (12-48) gives
In discussing the surface accessibility, we defined the internal effectiveness factor q with respect to the concenwation at the external surface of the
pellet, CAs:
'=
Two different
effectivenes~factors
Actual overall rate of reaction
Rate of reaction that would result if entire interior surface were
exposed to the external pellet surface conditions, C,,, T,
We now define an overall effectiveness factor that is based on the bulk
concen~ratjon:
Rate that would result if the entire surface were
exposed to the bulk conditions, CAb,Tb
Dividing the numerator and denominator of Equation (12-51) by kc@,, we
obtain the net rate of reaction (total molar flow of A to the surface in t m s of
the bulk fluid concentration), which is a measurable quantity:
838
Diffusion and Reaction
Chap. ? 2
Consequently, the overall rate of reaction in terms of the bulk concentration
CAbis
where
Overall effectiveness
factor for a
first-order reac~ion
The rates of reaction based on surface and bulk concentrations are related by
where
The actual rate of reaction is related to the reaction rate evaluated at the bulk
concentrations. The actual rate can be expressed in terms of the rate per unit
volume, -r,, the rate per unit mass, -ri, and the rate per unit surface area,
-r,", which are relazed by the equation
In terms of the overall effectiveness factor for a first-order reaction and the
reactant concentration in the bulk
-r,=rA,1;2=r~,p,Q=-r~,S,pb~=k;CAbSap,fl
(12-57)
where again
Overall
effectiveness
factor
Recall that k;' is given in terms of the catalyst surface area (rn3/m2-s).
12.5 Estimation of Diffusion- and Reaction-Limited
Regimes
In many instances it is of interest to obtain "quick and dirty" estimates to learn
which is the rate-limiting srep in a heterogeneous reaction.
Set. 12.5
Estimation of DMusion- and Reaction-Limited Regimes
12.5.1 Weisz-Prater Criterion for Internal Diffusion
The Weisz-Prater criterion uses measured values of the rate of reaction,
-r; (obs), to determine if internal diffusion is limiting the reaction. This criterion can be developed intuitively by first rearranging Equation (12-32) in the
fom
Showing where
the Welsz-Prater
comes from
The left-hand side is the Weisz-Prater parameter:
cWP
=q~b:
- Observed (actual) reaction rate:,
Reaction rate evaluated at C,,
Reaction rate evaluated at CA,
A diffusion rate
- Actual reaction rate
A d~ffusionrate
Substituting for
q=- -rA(obs)
and
-
3, P, R2 - 4,
R2
D,C*,
D,C*,
$:=
-.as
PC
in Equation (12-59) we have
AR there any
internal diffusion
Iirnitations
All the terms in Equation (12-61) are either measured or known. Consequently,
we can calculate Cw Howwer, if
indicated from the
0
C,,
Weisz-Prater
criterion?
c<
1
there are no diffusion limitations and consequently no concentration gradient
exists within the pellet. However, if
internal diffusion limits the reaction severely. Ouch!
Example 12-3
Estimating Thiele Modulus and Eflecctiveness Factor
The first-order reaction
840
1
Diffusion and Rpac!jon
was carried out over two different-sized pellets. The- pellets were contained j r
spinning basket reactor that was operated at sufficteatly high rotation speeds tl
external mass transfer resistance was negligible. The results of two experimen
nlns made under identical conditions are as given in Table E13-3. I . (a) Estimate t
Thiele modulus and effectiveness factor for each peliet. (b) How small should t
pellers be made to virtually eliminate all internal diffusion resistance?
TABLE
E12-3.1 DATA
mow
A
SP~NN~NGBASKET
RWCTIO?~'
Menstwed Rare (obsl
cat . s ) X I@
The.% two
experiments yield an
enomlous amount of
information,
Chap.
[mollg
Run 1
Run 2
Allet Raditt.~
(m)
3.0
15.0
0.01
0.00 t
See Figure 5- II(c),
(a) Combining Equations (11-58) and ( 1 2-6 t ), we obtain
Letting the subscripts 1 and 2 refer to runs 1 and 2, we apply Equation (E12-3.1) t~
runs 1 and 2 and then take the ratio 50 obtain
The terms p, D,,and C
,, cancel because the runs were carried out unda
identicai conditions. The Thiclc modulus is
Taking the ratio of the Thielc moduli for runs 1 and 2, we obtain
Substituting for h , , in Equation (E12-3.2) and evaluating -ri and R for nrns 1 and
2 gives us
Sec. 12.5
1
Estimation of D ~ k s i o n and
Reaction-Limited Regimes
We now hove one equation and one unknvwn. Solving Equation i E 1 2-3.7) we find
that
61I= rod,?= 16.5
The corresponding effectiveness factors
mentill polnts. one
can predict the
particle cize where
internal macs
transfer does not
limit the rate of
reaction.
For R,:
for R l = 0.01
n
rn
q2=
ForR,: q, = 3(16.5 coth 16.5-1)
(b) Next we calcuIate the panicle radius needed to virtually eliminate internal diffusion control (say. a = 0.95):
Solution to Equation (E12-2.&)yields
= 0.9:
(i.e.. q = 0.95).
12.5.2 Mears' Criterion lor External Diffusion
The Mears7 criterion, like the Weisz-PMter criterion, uses the measured rate of
reacrian, - r i . (kmoilkg catas) to learn if mass transfer from the bulk gas
phase to the catalyst surface can be neglected. Mears proposed that when
I s external
diffusion limiting?
external mass transfer effects can be neg2ected.
'D. E. Mem, Ind
Eng. Chem Pmcess Des Der. 10. 541 (1971). Other interphase
transport-limitingcriteria can be found in AlChE S ~ m p Sez
. 143 (S. W. Weller, ed.).
70 ( 1974).
842
where
Diffusion and Readion
Chap. 12
n = reaction order
R = catalyst particle radius, m
ph = bulk density of catalyst bed, kg/rn3
= (1 +)p, (4 = porosity)
p, = bulk density of catalyst bed, kg/m3
CAb= solid catalyst density, kg/m3
kc = mass transfer coefficient, m/s
-
The mass transfer coefficient can be calcuIated from the appropriate correlation, such as that of Thoenes-Garners, for the flow conditions through the
bed. When Equation (12-62) is satisfied, no concentration gradients exist
between the bulk gas and external surface of the catalyst pellet.
Mears also proposed that the bulk fluid temperature, T, will be virtually
the same as the temperature at the external surface of the pellet when
Is there a
temperature
gradient?
where
h = heat transfer coefficient between gas and pellet, kJ/m2.s-K
R, = gas constant, Wmol. K
AHR, = heat of reaction, !d/mol
E = activation energy, kYkmol
and the other symbols are as in Equation (12-62).
12.6 Mass Transfer and Reaction
in a Packed Bed
We now consider the same isomerization taking place in a packed bed of catalyst pellets rather than on one single pellet (see Figure 17-10). The concentration CAis the bulk pas-phase concentration of A at any point along the length
of the bed.
z=o
z
Z*AZ
r=L
Rgure 12-10 Packed-bed reacior.
We shall perform a balance on species A over the volume element 11'
neglecting any radial variations in concentration and assuming that the bed is
Sec. 12.6
Mass Transfer and Reaction in
B
Packed Bed
843
operated at steady state. The following symbols will be used in developing our
model:
A, = cross-sectional area of the tube, dm1
CAb= bulk gas concentration of A, molldm3
pb = bulk density of the catalyst bed, g/dm3
v, = volumetric flow rate, dmVs
U = superficial velocity = v, /A,, dmls
Mole Balance
A mole balance on the volume element (A,&) yields
[Rate in]
-
Ac'Lnlr
- AcWkl:+a;
[Rate out]
$.
[Rate of formation of A] = o
+
Dividing by A, hz and taking the limit as bz
6 pbAc
=0
+ 0 yields
Assuming that the total concentration c is constant, Equation (11-14] can
be expressed as
Also. writing the bulk flow term in the form
B A=
~Y A ~ ( W+
A ~W B ~ =
) ?IAN
= UCA~
Equation (12-64) can be written in the form
Now we will see
how to use .
Iand
to calculate
conversion in a
nacked hed.
The term DAR(d2CAbJd~')
is used to represent either diffusion andtor dispersion in the axial direction. Consequently. we shall use the symbol D,for the
dispersion coefficient to represent either or both of these cases, We wiIl come
back to this form of the diffusion equation when we discuss dispersion in
Chapter 14. The overall reaction sate within the pellet. -I-;.
is the overalI rate
of reaction within and on the catalyst per unit mass of catalyst. It is a function
of the reactant concentration within the cataIyst. This overall rate can be
related ro the rate of reaction of A that would exist if the entire surface were
exposed to the bulk concentration CAhthrough the overail effectiveness factor fl:
For the first-order reaction considered here.
-rAb = -rib Sa= k3'SaCAh
844
Diffus~onand React~on
Chap
Substituting Equation ( 1 2-66) into Equation ( 12-57). we obtain the overall
of reaction per unit mass of catalyst in terms of the bulk concentration C,,
Substituting this equation for -ri into Equarion ( 12-65). we form the c
ferential equation describing diffusion with a first-order reaction in a cat$
bed:
Flow and firstorder reaction in a
packed bed
I
As an example. we shall solve this equation for the case in which the Row ra
through the bed is very large and the axial diffusion can be neglecte
Finlaysons has shown that axial dispersion can be neglected when
Criterion For
neglecting axial
dispersion{diffusion
where W,, is the superficial velocity, d, the panicle diameter. and W, is tt
effective axial dispersion coefficient. In Chapter 14 we will consider solutior
to the complete form of Equation (12-67).
Neglecting axial dispersion with respect to forced aria1 convection,
Equation (12-67) can be arranged in the form
With the aid of the boundary condition at the entrance of the reactor.
C,, = C,4wl
at t = 0
Equation (12-693 can be integrated to give
-(~Ksaibvc!
C A=~C A ~e O
The conversion at the reactor's exit. z = L. is
Conversion in a
pack&-kd -tor
L. C.Young and B. A. Finlayson, Ind Eng. Chem, Fund. 12, 412 (1973).
Sec. 12.6
Mass Transfer and Reaction ~n a Packed Bed
845
Example 12-4 Reducing Niiroc~sOxides in a Phnl Efluent
I n Section 7.1.4 we saw the role that nitric oxide plays in smog formation and the
incentive we would have for reducing its concentration in the atmosphere. It is proposed to reduce the concentration of NO in an effluentstream from a plant by passing it through a packed bed o f spherical porous carbonaceous solid pellets. A 2%
NO-98% air mixture flows at a rate of 1 X 1W6m3/s (0.001 drn3/s) through a
2-in.-ID tube packed wi!h porous solid at a temperature of F I73 K and a pressure of
101.3 kFa. The reaction
is first order in NO,that is.
-.j, = VSC
,,
and occurs primarily in the pores inside the pellet, where
Green
S,= Internal surface area = 530 m21g
chemical reaction
engineering
Calculate the weight of porous solid necessary to reduce the NO concentration to a
level of 0.004%, which is below the Environmental Protection Agency limit.
Additional information:
At I173 K. the fluid properties are
v = Kinematic viscosity = 1.53 X
lWsm2/s
DAB
= Gas-phase diffusivity = 2.0 X IOV8m2/s
D, = Effective diffusivity
The properties of the catalyst and bed
Also X e Web site
ww.mwan.edu/
= 1.82 X
m2ts
are
p, = Density of catalyst particre = 2.8 g/cm3 = 2.8 x
pmengincering
106 g/m'
b = Bed porosity = 0.5
- 4) =
pb = Bulk density of bed = p,(t
R = Pellet radius = 3 X
1.4
x 106 glm'
m
y = Sphericity = 1.0
Solulion
It is desired to reduce the NO concentntion from 2.0% to 0.0048. Neglecting my
volume change at these low concentrations gives us
1
where A represents NO.
846
Diffusion and Reaction
Chap. 12
The variation of NO down the length of the reactor is given by Equation
(12-69)
Multiplying she numerator and denominator on the right-hand side of Equation
(12-69) by the cross-sectional ma, A,, and realizing that the weight of solids up to
a p i n t z i n the bed is
(Mole baIancej
+
+
W = pbArz
(Rate law)
the variation of NO concentration with solids is
(Overall
effectiveness
factor)
Because NO i s present in dilute concentrations. we shall take E 6 I and set
o = v , . We integrate Equation (EF2-4.1) using the boundary condition that when
W = 0,then CAC= CAM:
(El 2-4.2)
where
1
Rearranging. we have
1. Calculating fhc internal effectiveness facror for spherical pellets in which a
first-order reaction is wcurring. we obtained
As a first approximation, we shall negIect any changes in the pellet size
resulting from the reactions of NO with the porous carbon. The Thiele modulus for t h ~ ssystem isg
.
where
R = pellet radius = 3 x 10-3 m
D, = effective diffusivity = 1.82 x
'L. K
m21s
p, = 2.8 glcm3 = 2.8 X IF pJm'
.ha", A. F. Sarohm, and J.
M. Beer, Comht~sr,Flame, 52, 37 (1983).
Set. 12.6
I
Mass Transfer and Reaction ~ne Packed Bed
847
k = specific reaction rate = 4.42 x 10-lo rn3tm2-s
4 , = I8
Because. L, is large,
2. To calculate the external mass fransfer cneficient. the Thwnes-Kramers correlation i s used. From Chapter 1 1 we recall
Sh' = ( ~ ~ ' ) ! / 2 $ ~ 1 / 3
( 1 1-65)
For a 2-in.-ID pipe, A, = 2.03 X lo-' m2. The supefidal velocity is
Re' =
Prdum
Calculate
Re '
Sc
Ud,
( 1 -0)v
-- (4.93x 1 OL4 m / S ) ( 6 ~10-3 m)= 386.7
( 1 -0.5)(1.53x
m2/s)
Nomenclature note: 4 with subscript 1. 4, = Thiele moduFus
& without subscript. d~ = porosity
Then
I"? I
Sh'
Sh'
= (386.T)117(0.765)v3
= (19.7)(0.915)= 18.0
3. Calcubtin~the extcrnal area per unit reactor volume, we obtain
I
= soo m2/s3
4. Evaluating the m~emlleff~ctivencssfactnr.Substituting into Equation (12-55).
w e have
Dlffus~onand React~on
848
Chap.
In this example we see that both the external and internal resistances to mn
transfer are significant.
5. Calruluti~lgthe weighr of . d i d necessnty to nchievc 99.8CTc cmtr~ersion.Su
stituting into Equntion (E 12-4.3). we obtain
I
6. The reactor length is
12.7 Determination of Limiting
Situations from Reaction Data
For externa1 mass transfer-limited reactions in packed beds. the rate of reactio
at a point in the bed is
Variation of
reaction
with
system v:iriahleo
for the mass transfer coefficient. Equation 11 1-66), shows th
kc is directly proportional to the square root of the veracity and inversely pro
portional to the square root of the particle diameter:
The correlation
We recaIl from Equation (EI2-4.5), a, = 6(1 - @)Ed,,that the variation of exter
nal surface area with catalyst particle size is
Consequently, for external mass transfer-limited reactions, the rate is inversel:
proportional to the particle diameter to the three-halves power:
Sec, 12.8
Many
heterogeneou'
reuction\ are
diffilsion limited,
From Equation ( 1 1.-72) we see that for gas-phase externat mass trinsfer-limited
reactions, the rate increases approximately linearly with temperature.
When internal diffusion limits the rate of reaction, we observe from
Equation (12-39) that the rate of reaction varies inversely with partick diameter, is independent of veiocity, and exhibits an exponential temperature dependence which is not as strong as that for surface-reaction-controlling reactions.
For surface-reaction-limited reactions the rate is independent of particle size
and is a strong function of temperature (exponential). Table f2-I summarizes
the dependence of the rate of reaction on the velocity through the bed, particle
diameter, and temperature for the three types of limitations that we have been
discussing.
1
Very Important
Table
Mulliphase Reactors
,
I
Variation of Rcnction Rate w i h :
Type of
Limitation
Velocip
Pnrticle Size
Tmpemture ,
1
External diffusion
Internal diffusion
Surface reaction
~ l n
Independent
Independent
(dP)-312
(dp)-'
Independent
=Lineat
Exponential
Exponential
i
,
The exponential temperature dependence for internal diffusion lirnitatians is usually not as suong a function of temperature as is the dependence for
surface reaction limitations. IF we would calculate an activation e n e s y
between 8 and 24 WJmol, chances are that the reaction is strongly diffusion-limited. An activation energy a€ 200 kT/moE, however, suggests that the
reaction is reaction-rate-limited.
12.8 Multiphase Reactors
Multiphase reactors are reactors in which two or more phases are necessary to
carry out the reaction. The majority of multiphase reactors involve gas and liquid phases that contact a sotid. In the case of the sturry and trickle bed reac-
'Reference Shelf
tors, the reaction between the gas and the liquid takes place on a solid catalyst
surface (see Table 12-21. However, in some reactors the liquid phase is an inert
medium for the gas to contact the solid catalyst. The latter situation arises
when a l q e heat sink is required for highly exothermic reactions. In many
cases the cataIy st life is extended by these milder operating conditions.
The multiphase reactors discussed in this edition of the book are the
durry reactor, fluidized bed. and the trickle bed reactor. The trickle bed reactor,
which has reaction and transport steps sirniIar to the slurry reactor, is discussed
in the first edition of this book and on the CD-ROM along with the bubbIin8
850
Diffusion and Asaction
TABLE
12-2.
Chap. 72
A m i c ~ n o ~OFsTHREE-PHASE
REACTORS
I. S b r reactor
~
A. Hydrogenation
1. of fatty ackds over a suppwted nickel catalyst
2. of 2-butyne-1,4-dl01 wer a Pd-CeC03 catalyst
3. of glucose over a Raney nickel catalyst
B. Oxidation
I . of CzH4in an inen liquid over a PdC1:-ca&n catalyst
2. of SO2 in inen water over an activated carbon catalyst
C. Hydmformation
of CO w~thhigh-molecular-weight olefins on either a cubah or ruthenium complex bound
to polymers
D. Ethynylation
React~onof acetylene with formaldehyde over a CaC12-supported catalyst
11. Trickle hed reartr8rs
A. Hydnndesulfurization
Removal of sulfur compounds from crude oil by reaction with hydrogen on Co-Mo on
alumina
R. Hydrogenation
1. of anikine over a Ni-ctay catalyst
2. of 2-htyne-1.4-dm1 over a supponed Cu-Ni catalyst
3. of benzene, a-CH3 styrene. and crotonaldehyde
4. of aromatics in napthenic lube oil distillate
C. Hydrodeni~ropenalion
1 . of lube oil dist~llate
2. of cracked Iight furnace oil
D Oxidation
1. of cumene wer activated carbon
2. of SO, over carbon
Sourre: C.N. Satterfield. AIChE J., 21, 209 (1975): F.A. Ramachandmn and R. V. Chaudhari.
Chcm. Ens.,87c241. 74 (19801: R. '4. Chaudhari and P. A. Rarnachandran. AEChE I . , 16, 177
1 1980).
Reference
fluidized bed. Tn slurry reactors, the catalyst is suspended in the liquid, and gas
is bubbled through the liquid. A slurry reactor may be operated in either a
semibatch or continuous mode.
($-%;~z
12.B.f
Slurry Reactors
A complete description of the slurry reactor and the transport and reaction
steps are given on the CD-ROM,along with the design equations and a number of examples. Methods to det~rminewhich of the transport and reaction
steps are rate limiting are included. See Professional Reference Shelf R12.1.
12.8.2 Trickie Bed Reactors
The CD-ROM includes all the material on trickle bed reactors from the first
edition of this book. A comprehensive example problem for trickle bed reactor
design is included. See Professional Reference Shelf R 1 2.2.
Chap. 12
851
Summary
12.9 Fluidized Bed Reactors
The Kunii-kvenspiel model for fluidization is given on the CD-ROM along
with a comprehensive example problem. The rate limiting transport steps are
also discussed. See Professional Reference Shelf R 1 2.3.
12.10 Chemical Vapor Deposition (CVD)
,
VU
,
'1 Chemical Vapor Deposition in boat reactors is discussed and modeled. The
equations and parameters which affect wafer thickness and shape are derived
and analyzed. This material i s taken directly from the second edition of this
book. See Professional Reference Shelf R12.4.
Closure A h r completing this chapter, the reader should be able to derive
differential equations describing diffusion and reaction, discuss the meaning
of the effectiveness factor and its relationship to the Thiele modulus, and
identify the regions of mass transfer control and mction rate control. The
reader should be able to apply the Weisz-Prater and Mears criteria to identify gradients and diffusion limitations. These principles should be able to
be applied to catalyst particles as well as bjomaterid tissue engineering.
The reader should be able to apply the overall effectiveness factor to a
packed bed reactor to calculate the conversion at the exit of the reactor. The
reader should be able to describe the reaction and transport steps in slurry
reactors, trickle bed reactors, fluidized-bed reactors, and CVD boat reactors
and to make calculations for each reactor.
SUMMARY
1. The concentration profile for a first-order reaction occumng in a spherical
cataJyst pellet i s
where bl is the Thiele modulus. For a first-order reaction
2. The effectiveness factors are
Internal
effectiveness = 11 =
factor
Actual rate o f reaction
Reaction rate if entire interior
sirrfacc I S exposed to concentration
at the external pellet surface
"l
ool
b,
,,
.a
Diffusion and React~on
Chap. 1
Overall
Actual sate of reaction
effectiveness = !2 =
Reaction rate if entire surface area
factor
is exposed to bulk concentration
3. For large values of the Thzele modulus for an nh order reaction,
4.
For internal diffusion control, the m e reaction order i s related to the measured reaction order by
Ttte me and apparent activation energies are related by
Errue
= ZEapp
(S12-51
5. A. The Weisz-Prater Parameter
Cw,=
9:rl =
-6 (observed) p, RZ
D~CA~
The Weisz-Prater criterion dictates that
If CwpQ 1
no internal diffusion limitations present
If C, B 1
internat diffusion limitations present
B. Mears Criteria for Neglecting External Diffusion and Heat Transfer
and
CD-ROM MATERIAL
* Learning Resaucees
1. Summary Notes
Summary Notcr
Chap. f 2
CD-ROM Material
Professional Reference Shelf
R 12. I .S l u r y Rencrors
Trancpon Step%and Re~istanoes
mmh
mmIICI1
Reference Shclf
A. Description of Use of Slurry Reactors
Example R 12- I Industrial Sluny Reactor
B. Reaction and Trampon Steps in a Step in Slurry Reactor
C. Determining the Rate-Limiting Step
1. Effect of Loading. Particle Size and Gas Adsorption
2. Effect of Shear
Example RI?-2 Detennin~ngthe Controlling Resistance
D. Slurry Reactor Design
Example R12-3 S l u q Reactor Design
R 12.2. Trickle Bed
Renctors
A. Fundamentals
B. Limiting Situations
C. Evaluating the Transport Coefficients
h -.,
Dlftusion and Reacfbn
Chao. 12
R12.3. Fluidized-Bed Reactors
A. Descriptive Behavior of the Kunii-LRvenspitl Bubbling Bed Modd
B. Mechanics of Fluidized Beds
Example R12-4 Maximum Solids Hold-Up
C. Mass Transfer in Fluidized Beds
D. Reaction in a Fluidized Bed
E. Solution to the Balance Equations for a First Order Reaction
Example R12-5 Catalytic Oxidation of Ammonia
F+
limiting Situations
Example R 17-6 Calculation of the Resistances
Example R12-7 Effect of Particle Size on Catalyst Weight for
a Slow Reaction
Example R 12-8 Effect of Catalyst Weight for a Rapld Reaction
R11.4.Chemical Vapor D ~ p o s ~ i i oRroctor.r
n
Refereace Shelf
a*-
C*LW"?I
6YS11u
-
>
:
u
=/
\-,-A
Chap. 12
Questions and Problems
A. Chemical Reaction Engineering in Microelectronic b e s s i n g
B. Fundamentals of CVD
C, Effectiveness Factors for Boar Reactors
Example R12-9 Diffusion Between Wafers
Example R 12-10 CVD Boat Reactor
~efcrcnceShelf
QUESTIONS AND PROBLEMS
The subscript to each of the problem numbers indicates the level of difficulty: A. least
dificult: 1). most difficult.
Mall of Fame
Fl2-lc Make up an original problem using the concepts presented in Section
(your instructor will specify the section). Extra credit will be gtven if you obtain
and use real data from the literature. (See Prohrem P4-1 for thc guidelines.)
P12-2R (a) Example 12-1. Efferril?~Drffusivin: Make a sketch of a diffus~onpath
for n h ~ c hthe tortuosity is 5. How ~ ~ o u your
l d effective gas-phase diffuslv~ty
change if the ahsolute pressure were tnpled and the temperature were
increased by SO%?
How would your answers change if
(b) Example 12-2. 7issr1eEtlginrerir~~.
the reaction kinetics were ( I ) first order In O? concentration with
k , = lW2 h-I? ( 2 ) Monod kinetics with p,,,, = 1.33 x
h-I and
K, = 0.3 rnolldm'. (3) zero-order kinet~cscarry out a quasi steady state
analysis uslnf Equations (E12-2.19) along with the overall balance
to predict the 0?Aux and collagen build-up as a function of time. ,V01r:
V = A,L. Assume o: = 10 and the ~toichiorne~ric
coefficient for oxygen In
collagen. v,. is 0.05 mass liaction of cclllmol 0:.
(c) ExampFe 15-3. ( I ) What is the percent of the total reri<t;~ncefor internal
diffusion and for reaction rale for each of [Ire thrcc panicks qtudied. ( 2 )
Apply the Weisz+kater criteria to a panicre O.OO7 rn tn dlarneter
(d) Example 12-4. OlberallEf~rrirerir.s\ h r ~ o r (: 1 ) Cnlcula~ethe percent of
the total resistance for external diffuclon. inrernal diffi~sion.and surface
reachon. Qualrta~ivelyhow would each of your percentage< change ( 2 ) i f
the temperature were incrcawd rlgnilicanll>? (3I ~f the Fac vclnc~tywere
D~Rus~on
ano Rsact~on
Chap.
tripled? (4) if the particle v i ~ ewere decreased by a factor of 27
would the reactor length change in each case? (5)What length would
required to achieve 99.99% conversion of the pollutant NO?
What it..
(e) you app11c-dthe Mean and Weisz-Prater criteria to Examples E 1-4 a
12-4'? What would you find? What would you learn if AHR, = kcal/moI. h = 100 Btulh-ft:."F and E = 20 k callmol?
(f) we let y = 30, = 0.4, and d~ = 0.4 in Figure 12-7? What wot~ldcau
you to go from the upper steady state to the lower steady state and vi
versa?
(g) your internal surface area decrrnsed with time because of sintering. Hr
would your effectiveness factor change and the rate of reaction chan
with time if k, = 0.01 h-1 and q = 0.01 at r = O? Explain.
[h) someone had used the False kinetics (it..wrong E. wrong n)'? Wou
their catalyst weight be overdesigned or underdesigned? What are 0th
positive or negative etficts that occur?
(i) you were asked to compare the conditions (e.g.. catalyst charge, convt
sions) and s~zesof the reactors in CDROM Example R l 2 . l . What diffc
ences would you find? Are there any fundamental discrepancies betwe1
the two? If so, what are they, and what are some reasons for them?
(j) you were to assume the ressstance to gas abwrption in CDROM Examp
Rl2.1 were the same as in Example RZZ.3 and that the liquid phase rea
tor volume in Example R12.3 was 50% o f the total, could you estima
the controlling resistance? If so, what is it? What other things could yc
calcuIat~in Example R12.1 (e.g., selectivity. conversion, molar flow rat,
in and out)? Hint: Some of the other reactions that occur include
a
Green engineering
Web site
zrrvw row~-lrrr.ed~r/
.qrt-enengineerirrg
(k) the temperature in CUROM Example R12.2 were increased? How woul
the relative resistances in the slurry reactor change?
(I) you were asked for alI the things that could go wrong in the operation (
a slurry reactor, what would you say?
P12-3R The catalytic reaction
takes place within a fixed bed containing spherical porous catalyst X22. Fil
ure P12-3 shows the overalf rates of reaction at a p i n t in the reactor as
function of temperature For various entering total molar flow rates. Fm.
(a) Is the reaction limited by external diffusion?
(b) If your answer to part (a3 was "yes." under what conditions [nf thos
shown (i.e.. T, F,)] 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 show
(i-e., T, Fm)] is the reaction limited by the rate of the surface reactions'
(e) Is the reaction limited by internal diffusion?
(fl If your answer to part (e) was "yes;' under what conditions [of thos
shown (i.e., 7; FTu)]is the reaction limited by the rate of internal diffusion
(g) For a flow ratate of ID g molth. deterrmne (if possible) the overall effec
tiveness factor, R, at 360 K.
Ih) Estimate (if possible) the internal effectiveness factor, r(. at 367 K.
Chap. 12
Quest~onsand Problems
I
0 350
1
I
I
I
I
360
370
380
390
400
T(k)
Figure P12-3 Reaction ntes in a catalyst bed.
i
If the concentration at the external catalyst surface is 0.01 mol/dm3, calculate (if possible) the concentration at r = R12 inside the porous catalyst at 337 K. (Assume a first-order reaction.)
Addirionnl information:
Bed properties:
Gas properties:
Biffusivity: 0.I cmzls
Density: 0.001 g/cm3
Viscosity: 0.0001 g l c m - s
P12-.la The reaction
Tortuosity of pellet: 1.415
Bed permeability: 1 millidarcy
Porosity = 0.3
A - B
is carried out in a differential packed-bed reactor at different temperatuses, flow
rates, and particle sizes, The results shown in Figure P12-4 were obtained.
Mall c f Fame
t
Figure P I 2 4 Reaction rates in a camlyst hzd.
8 58
P12-5,
Diffusion and Readion
Chap. 12
ta) What regions (i.e.. conditions d,, 7, F,) are external mass transfer-limited?
(b) What regions are reaction-rate-limited?
lc) What region is internal-diffusion-controlled?
(d) What i s the internal effectiveness factor at T = 400 and d, = 0.8 cm?
Curves A, B. and C in Figure P12-5show the variations in reaction rate for
three different reactions catalyzed by solid catalyst pellets. What can you say
ahout each reactlon?
!
.
I
'
Figure P12-5 Temperature dependence of three reactions
PI2-6,
A first-order heterogeneous irreversible reaction is ttking place within a
spherical catalyst pellet which is plated with platlnum throughout the pellet
(see Figure 12-3). The reactant concentration halfway between the external
surface and the center of the pellet f i x . . r = RI?) is equal to one-tenth the
concentratlon of the pellet's external surface. The concentratlon at the external
surface is 0 001 g molfdrn3.the diameter ( 2 R ) is 2 X 30-hcm. and the dtffuslon c r ~ f f i c i c nis~ 0.1 cm=ls.
crn in
(a) What is the concentration of reactant at a distance of 3 X
from the external pellet surface? {Ans.: C, = 2.36 X 10-%ol/dm4.)
(bk To wwh atdiameter should the pellet be reduced if the effectiveness factor
. = 6.8 X I F 4 cm. Critique this answer!)
is lo be 0.8? I A I I ~ .dp
(c) Ifthe catalyst suppon were not yet plated with platinum. how would you
suggest that the catalyst support be plated afrer it had been reduced by
Applioaflon
Pending
rut Pmblem
Hall of
grinding?
PIZ-7,, The SMimming rate of a small organism fJ. Theorpi. Biol., 26, 1 1 (I970)] is
related to the energy released by the hydrolysis of adenosine triphosphate
(ATPI to adenosine diphosphate (ADP). The rate of hydroly~isis equal to the
rate of diffuvon of ATP from the rnidp~eceto the tail (see Figure P 12-71. The
diffusion coefficient of ATP in the midpiece and tail is 3.6 X
cmZls.AQP
is converted to ATP ~n the midsection. where its concentrafion is 4.36 X loc-'
mollcm'. The cross-rcctional area of the tail is 3 X IO-IU cm2.
Figure P12-7 Swirnming of an organicrn.
(a1 Derive an equation for diffusion and reactlon in the tail.
Ihl Derirc an equation Tor the effectiveness Factor in the tail
Chap, t 2
Questions and Problems
859
(c) Taking the reaction in rfie tail lo be of zero order. calcufate the lmgth of
the tail. The rate of reaction in the tail is 23 X 10- I R molts.
(d) Compare your answer with the average tail length of 41 pm. What
possible sources of error?
P12.BB A first-order, heterogeneous, irreversible reaction is taking place within a catalyst pore which i s plated with platinum entirely along the length of the pore
(Figure P12-8). The reactant concentration at the plane of symmetry (i.e.,
equal distance from the pore mouths) of the pore is equal to one-tenth the
concentmtion of the pore mouth. n e concenmlion at the pore mouth is 0.001
g moIldm3, the pore length (2L) i s 2 x
cm. and the diffusion coefficient
is 0.1 cmYs.
Figum P12-8 Single catalyst pore.
(a) Derive an equation for the effectiveness factor.
(b) What is the concentration of reactant at LIZ?
(c) To what length should the pore length be reduced if the effectiveness factor is to be 0.8?
(d) If the catafyst suppon were not yet plated with platinum, bow would you
suggest the catalyst support be plated after the pore length, L, had been
reduced by grinding?
P12-9, A first-order reaction is taking place inside a porous catalyst. Assume dilute
concentrations and neglect any variations in the axial (x) direction.
(a) Derive an equation for both the internal and overall effectiveness factors
for the rectangular porous slab shown in Figure PI 2-9,
(b) Repear part (a) for a cylindrical catalyst pellet where the reactants diffuse
inward in the radial direct~on.
Figure P12-9 Flow over porous catalyst slab.
Pf2-10, The irreversible reaction
is taking place in the porous catalyst disk shown in Figure P12-9. The reaction is zero order in A.
(a) Show that the concentration profile using the symmetry B.C.is
Hall of Fame
Diffusion and Reaction
Char
where
(b) For a Thiele modulus of 1.0, at what point in the disk is the ioncenml
zero? For
= 4?
(c) What i s the concentration you calculate at z = 0.1 L and = 10 us
Equation (P12-10.I)'? What do you conclude about using this equatio,
(d) Plot the dimensionless concentration profile y = CAiCAIas a function
h = z/L for Q,, = 0.5. 1, 5 , and 10. H i ~ r there
:
are regions where the c~
centration i s zero. Show that h, = 1 - I/@, is the start of this reg
where the gradient and coocedtntion are both zero. [L. K. Jang, R.
York. J. Ctln. and L. R. Hile, Inst. Chem. Engr., 34, 319 (2003).]Sh
that v=O; X'-2@,,($,)- 1 ) A+(&I ) = f o r & 5 h c 1.
(e) The effectiveness factor can be written ns
@,,
where z , (&)is the point where both the concentration gradients and flux ,
zero and A, is the crors-sectional area of the disk. Show for a rem-ord
to
reaction that
for ib,, 5 1.0
q = { i - x c = ~ t-orhzl
$0
(f) Make a sketch for versus dosimiEar to the one shown in Figure 12-5
(g) Repeat parts (a) to ( f ) for a spherical catalyst pelIet.
(h) Repeat parts {a) to (0 for a cylindrical catalyst petlet.
ti) What do you believe to be the point of thls probiem?
P12.11c The second-order decomposition reaction
is carried out in a tubular reactor packed with catalyst pellets 0.4 cm in diarnt
ter. The reaction is inkmaldifft~sion-limited.
Pure A enters the reactor at
superficial velocity of 3 mls, n temperature of 25O"C, and a pressure of 500 kPi
Experiments carried out on smaller gellets where surfilca reaction is lirmtin,
yielded a specific reaction rate of 0.05 mblrnoE g cat -s. Calculate the length o
bed necessary to achieve 80% conversion. Critique the numerical answer.
-
Effective diffusivity: 2.66 X LO-8 m2ls
lneffectlve diffusivity: 0.00 m2/s
Bed porosity: 0.4
Pellet density: 2 X 1P g/m3
Internal surface area: 400 m2/g
Chap. 72
861
Questions and Promems
P12-lZc Derive the concentration profile and effectiveness factor for cylindrical pellets
0,2 crn in diameter and 1.5 crn in length. Neglect diffusion through the ends
of the pellet.
(a) Assume that the reaction is a first-order isomerization. (Hint: Look for a
Bessel function.)
(b? Rework Problem 812- I1 for these pAlets.
P12-13c Reconsider diffusion and reaction in a sphericaIcatalyst pellet forthe case where
the reaction i s dot isothermal. Show rhat the energy balance can be written as
where k, is the effective thermal conductivity, calls - cm - K of the pellet with
dTldr = 0 at r = 0 and T = T, at s = R.
(a) Evaluate Equation (12-11) for a first-order reaction and combine with
Equation (P12-15.1) to arrive at an equation giving the maximum temperature in the peIIet.
w e . ' At Tm,, c, = 0.
(b) Choose representative values of the parameters and use a software package to solve Equations (12-11) and (P12-13.1) simultaneously for T(r)
and CA(r)when the reaction is carried out adiabatically. Show that the
resuiting solution agrees qualitatively with Figure 12-7.
P12-14cDetermine the effecrrveness factor for a nonisothermal spherical catalyst pellet
in which a tirst-order isomerisation is taking place.
Additional infomtion:
A, = 100 m2/m3
hH, = -800,000 Jlmol
D, = 8.0 X
m2/s
CAs= 0.01 kmol/m3
External surface temperature of pellet, T, = 400 K
E = 120.000 Jlrnol
ThermaE conductivity of pellet = 0.004 JErn .s
.K
d, = 0.005 m
Specific reaction rate = 10-I m/s at 400 K
Density of calf's liver = 1.1 gJdm3
How would your answer change if the pelIets were I t 2 , 1 P , and l P 5 m in
diameter? What are typical temperature gradients in catalyst pellets?
P12-15, Extension of Problem P12-8. The elementary isomerization reaction
is taking place on the walls of a cylindrical catalyst pore. (See Figure P12-8.)
In one run a catalyst poison P entered the reactor together with the reactant A.
To estimate the effect of poisoning, we assume that the poison renders the catdyst pore walls near the pore mouth ineffective up to a distance z,, so that no
reaction takes place on the walls in this entry region.
862
DifIuslon and Reaction
C h a ~ 12
.
(a) Show that before poisoning of the pore occurred, the effectiveness factor
was given by
where
with
k = reaction rate constant (lengthltirne)
r = pore radius (length)
I), = effective molecular diffusivity larealtime)
(b) Derive an expression for the concentration profile and also for the molar
flux of A in the ~neffecriveregion 0 < x < 2,. in iems of:,. DAD,,.
C,,,
and CAs.Without solving any funher differential equations. obtain the
new effectiveness factor T ' far the poisoned pare.
P12-16B Fals$ed Kit~erics.The irreversible gas-phase dirnerization
Is carried out at 8.2 atm in a stirred contained-solids reactor to which only
pure A is fed. There is 40 g of catalyst In each of the four spinning baskets.
The following runs were carried out at 227°C:
Toral Molar Feed Rote,
F , (g rnollmin)
1
2
4
6
11
20
The following experiment was carried out at 237°C:
(a) What are the apparenl reaction order and the apparent activation energy?
(b) Determine the true reaction order, specific reaction rate. and activation
energy.
(c) Calculate the Thiele modulus and effectiveness factor.
(d) What diameter of pellets should be used to make the catalyst more effective?
(el Calculate the rate of reaction on a rotating disk made of the catalytic matenal when the gas-phase reactant concentration 1s 0.01 g mallL and the
temperature is 527°C. The disk i s flat, nonporous, and 5 crn in diameter.
Effective diffusivity: 0.23 cm7s
Surface area of porous catalyst: 49 m31g cat
Density of caraly\t pellets: 1.3 g/crn3
Radius of catalyst pellet$: 1 cm
Color of pellets: blushrng peach
Chap. 12
863
Journal Article Problems
P12-17, Derive Equation (12-35). Hint: Multiply both sides of Quation (12-25) for
nth order reaction, that is.
by 2&I&
rearrange to get
and solve using h e boundary conditions d p l d = O at h = 0.
JOURNAL ARTICLE
PROBLEMS
P12J-I The article in Tmns. ini. Chcrn. Eng.. 60, 131 (1982) may be advantageous in
answering the fotlowing questions.
(a) Describe the various types of gas-liquid-solid reactors.
(b) Sketch the concentration profiles for gas absorption with:
(3) An instantaneous reaction
(2) A very slow reaction
(3) An intermediate reaction rate
P12J-2 After reading the journal review by Y.T. Shah et al. [AIClrE J., 28. 353
(1982)j. design the following bubbie column reactor. One percent carbon
dioxidd in aiT is to be removed by bubbling through a solution of sodlum
hydroxide. The reaction i s mass-uansfer-limited. Calculate the reactor size
(length and diameter) necessary to remove 99.9% of the COz. Also specify a
type of sparger. The reactor is to operate in the bubbly flow regime and still
process 0.5 m"ls of gas. The liquid flow rate through the column is lo-' rn3/s.
JOURNAL CRITIQUE PROBLEMS
P12C-1 Use the Weisz-Praler criterion to determine if the reaction discussed in AIChE
J., 10. 568 (1964) is diffusiun-rate-limited.
P12C-2 Use the references given in Ind. Eng. Chem. Pmd. Res. Dei:, 14. 226 (1975)
to define the iodine value, saponification number. acid number, and experimental setup. Use the slurry reactor analysis to evaluate the effects of mass
uansfer and determine If there are any mass transfer I~rnitations.
Additional Homework Pmblems
CDPlZ-AB Determine the catalyst size that gives the highest conversion in a
packed bed reactor.
CDP12-BB Determine importance of concentration and temperature gradients in a
packed bed reactor.
CDP12-C, Determine concentration profile and effecttveness factor for the first
order gas phax reaction
864
Diffusion and Reaction
Cha~.
Slurry Reactors
CDPIZ-D,
Hydrogenation of methyl linoleate-comparing cataiyst. 13rd 1
PI2-IYJ
CDPlZ-E, Hydrogenation of methyl linolente. Find the rate-limiting step.
13rd Ed. PI2-201
CDP12-FR Hydrogenation of 2-butyne-1.4-diol to butenediol. Calculate perc,
resistance of total of each for each step and the conversion. [3rd E
PI?-211
%
CVD Boat Reactors
CDPIZ-GI, Derennine the temperature profile to ach~eve\>nuniform thickne
\>
CDPXt-H,
CDPl2-IR
CDP12-.Ic
CDPIZ-Kc
P
Mcrcbcr
Mall of Fame
Green Engineering
,
[Znd Ed. P1 1-18)
Explain how varying a number of the parameters in the CVD bc
rerrctor will affect the wafer shape. [Znd Ed. P11-141j
Determine the wafer shape in a CVD boat reactor for a series of opt
ating condttions. [Znd Ed. P 1 3 -201
Model the build-up of a silicon wafer on parallel sheets. [2nd E
PI I-21J
Rework CVD boat reactor accounting for the reaction
SiH4
SiH2 + HI
[Znd ed. Pll-221
Trickle Bed Reactors
CDP12-LB Hydrogenation of an unsaturated organic is carried out in a trick.klebt
reactor. [2nd Ed. PI2-71
CDPlZ-M, The oxidation of ethanol is camed out in a trickle bed recrcror. [2r
Ed. P12-91
CDPlt-Nc Hydrogenation of aromatics in a srickle brd wauror ['nd Ed. P12-R:
Fluidized Red Reactors
CDPIZ-0, Open-ended Ruidizrttion problem that requires critical thinking 1
compare the two-phase fluid models with the three-phase bubblin
bed model.
CDP12-P, Calculate reaction ntes at the top and the bottom of the bed fc
Example R12.3-3.
CDP12-Q, Calculate the conversion for A + I3 in a bubbling fluidized bed.
CDP12-Rk Calculate the effect of operating parameters on conversion for th
reaction limited and transport limited operation.
CDP12-Se Excellent Problem Calculate all the parameters in Example R 12-3.
for a different reaction and different bed.
CDPf2-Ta Plot conversion and concentration as a function of bed height in
bubbling fluidized bed.
CDP12-UB Use RTD studies to compare bubblingkd with a fluidized bed.
CDP12-VB New problems on web and CD-ROM.
CDPIZ-WB Green Engineering, www.rowan.edu/greenengineering.
Ghap. 12
Supptemefltary Reading
865
SUPPLEMENTARY READING
I. There are a number of hooks that d ~ ~ internat
c u ~ diffusion In catalyst peliets:
however. one of the first b ~ k that
s should be consulted on this and other topics
on heterogeneous catnlyhis is
L ~ P m u s L..
, and N.R. AMU~DSOY.
Chernicnl ReachtorTheor?.:A Review Upper
Saddle River, N.J.:Prentice Hall. I977.
In addition, see
ARIT.R.. E f e m e t t ~ nChemicnl Rrrrcror Ann(v3i.r. Upper Saddle River. N.1.:
Prentice Hall. 1969. Chap. 6. One should f n d the references listed at the
end of this reading particularly useful.
Lvss. D., "Diffusion-Reaction Interactions in Catalyst Pellets," p. 239 in ChernicnI Rerrcrion and Rerrcror Err,yineeritrg. New York: Marcel Dekker. 1987.
The effects o f mass transfer on reactor performance are also discussed in
DENBIGH,
K.. and J. C. R. TURNER,
Chemical Reactor Theov. 3rd ed. Cambridge: Cambridge Univer~ityPress. 1984. Chap. 7.
SATERFIELD,
C . N . , Heremgcneous Catnivsis in Industrinl Pmcrice, 2nd ed.
New York: McGraw-Hill, 199 1 .
2. Diffusion with homogeneous reaction is discussed in
ASTARITA,
G.,and R. OCONE,Speci~rlTopics in Transport Phenomena. New York:
Elsevier, 2002.
DANCKWERTS.
P.V.. Ens-Liqrtid Rencrions. New York: McGraw-Hill. 1970.
Gas-liquid reactor design is also discussed in
CHARPG~E
J. RC.,
, review article. Trans. lnst. Chem. Eng,60, 131 (L982).
S H A HY
. .T., Gas-Liquid-Solid Rencror Design. New York: McGmw-Hill. 1979.
3. Modeling of CVD reactors is discussed in
HESS, D.W.. K. F. JENSEA,and T. I. ANDERSON,"Chemical Vapor Deposition:
A Chemicat Engineering Perspective." Rev. Chrm. Eng.. 3, 97, 1985.
JENSEN, K. E, "Modeling of Chemical Vapor Deposition Reactors for the Fabrication of Microelectronic Devices," Clternicul and Cntal,vric Rencror
Modeling. ACS Syrnp. Ser. 237. M. F. Dudokavic, P.L. Mills. eds., Washington, D.C.: American Chemical Society. 1984. p. 197.
LEE, Id. H.. fundamm~nis of ,Wicroelectronics Pmcessing . New York:
McGnw-Hill. 1990.
4. Multiphase reactors are discussed in
R~UACHANDRAN,
P.A.. and R. V. CHAUDHARI.Three-Plrase Caraiyfic Reactors.
New York: Gordon and Breach, 1983.
RODRIG~~ES,
A. E..J. M. COLO,and N. H. SWEED.
eds.. Mulriphnse R~actorx,
Vol. I : F~ndDme??~~ljs.
Alphen aan den Rijn, The Netherlands: Sitjhoff and
Noordhoff, 198 1.
RODRIGUES.
A. E., J. M. COLO.and N. H . SWEED, eds.. Muftiphase Renctors,
Val. 1: Design Metitoris. Alphen aan den Rijn. The Netherlands: Sitjhoff
and Nmrdhoff. 198 1.
866
Diffusion and Reaction
Chap. 12
SHAH. Y. T.. 8 . G. KELKAR, S. P. GODBOLE,and W. D. DECKWEK"Design
Parameters Estimations for Bubble Column Reactors" (journal review),
AIChE J.. 28, 353 (1982).
TARHAN.
M.O.,Catalytic Reactor Design. New York: McGraw-Hill, 1983.
YATES,J. G.,Fundamenrals of Fluidized-Bed Chemical Processes, 3rd ed. London: Butternorth, 1983.
The following Advances in Chemistry Series volume discusses a number of mul-
tiphase reactors:
F'OGLERH, S., ed., Chernical Reactors. ACS Syrnp. Ser. 168. Washington.
D.C.: Arnencan Chemical Society, 198 1, pp. 3-255.
5. Fluidization
In addition to Kunii and Levenspiel's book, many comlations can be found in
DAV~DSON,
J. F., R. CLIFF,and D. HARRISON,
Fluidization, 2nd ed. Orlando:
Academic Press, 1985.
A discussion of the different models can be found in
YATES.I. G.. Fluidizad Bed Chemical Processes. London: Butterworth-Heinemann, 1983. Also qee GELDART,
D. ed. Gas Fluidi:ation Technolop.
Chichester: Wiley-Interscience, 1986.