EKC338: REACTOR DESIGN & ANALYSISCore Course for
B.Eng.(Chemical Engineering)Semester II (2014/2015)
Mohamad Hekarl Uzir([email protected])
School of Chemical EngineeringEngineering Campus, Universiti Sains Malaysia
Seri Ampangan, 14300 Nibong Tebal, Seberang Perai Selatan, PenangEKC338-SCE p. 1/164
Syllabus
1. External Diffusion:External diffusion effectsMass Transfer CoefficientDiffusion with chemical reaction
2. Internal Diffusion:Internal diffusion effectsEffective diffusivityDiffusion and chemical reaction in a cylindrical poreThiele Modulus, and effectiveness factor, Falsified kinetics
EKC338-SCE p. 2/164
Syllabus
3. Bioreactor Analysis and Operation:Mixing and transfer of masses: Oxygen transfer andKla
Bioreactor kinetics: substrate consumption,biomass production, product formation and kineticsmodelsDesign of bioreactorsRole of transport processes in bioreactor design
EKC338-SCE p. 3/164
Syllabus
4. Design of Multiple-Phase ReactorsGas-liquid-solid reactionTrickle-bed reactorSlurry reactorThree-phase fluidised-bed reactors
5. Projects on COMPUTER APPLICATIONS (MATLABr)in REACTOR DESIGN
EKC338-SCE p. 4/164
External & Internal Diffusion
1. Diffusion FundamentalsConsider a tubular-typed reactor, where the molarflow rate of reaction mixture in the z-direction isgiven by;
FAz = AcWAz
where WAz is the flux and Ac is the cross-sectionalarea.Diffusionspontaneous mixing of atoms ormolecules by random thermal motion which givesrise to the motion of the species relative to themotion of the mixture.
EKC338-SCE p. 5/164
External & Internal Diffusion
CA,b
CA,s
CA(r)
External
diffusion
Internal
diffusion
Porous catalyst
pellet
External
surface
EKC338-SCE p. 6/164
External & Internal Diffusion
1. Diffusion FundamentalsMolecules of a given species within a single phasewill diffuse from regions of higher concentrations toregions of lower concentrations (this gives aconcentration gradient per unit area between the 2regions).External mass transfer:
(a) Consider a non-porous particle where the entiresurface is uniformly accessible.
(b) The average flux of reactant, CA to the fluid-solidinterface can be written as;
NA = kA(CA,b CA)
EKC338-SCE p. 7/164
External & Internal Diffusion
1. Diffusion FundamentalsExternal mass transfer:
(b) where CA,b is the bulk concentration of reactant Aand CA is the concentration at the solid-liquidinterface and kA is the mass-transfer coefficient.(c) let the reaction rate, rA follows first order reaction;
rA = kCA
where k is the first order rate constant. Therefore,at steady-state;
kCA = kA(CA,b CA)
EKC338-SCE p. 8/164
External & Internal Diffusion
1. Diffusion FundamentalsExternal mass transfer:
(d) defining the dimensionless parameters;
x =CACA,b
Da =k
kA
thus;Da =
1 xx
(e) where Da is defined as the ratio of reaction ratewith the convective/diffusive mass transfer rate.
EKC338-SCE p. 9/164
Heterogeneous Reaction
Introduction to Heterogeneous and Multiphase ReactionsFor pseudo-homogeneous assumption:
Mass and heat transfer resistances betweendifferent phases are neglectedthe reactor contentscan be treated as a single phase.Useful for preliminary designtruly homogeneoussystem.
For heterogeneous modelused when temperatureand concentration need to be distinguished betweenthe phases.
EKC338-SCE p. 10/164
Heterogeneous Reaction
Introduction to Heterogeneous and Multiphase ReactionsFor real reactor: (multiphasesMulti-Phase Reactors)
Should be heterogeneous typeNormally used for systems involving fluid-fluidinteractions [liquid-liquid or gas-liquid]
EKC338-SCE p. 11/164
Heterogeneous Reaction
Introduction to Heterogeneous and Multiphase ReactionsFor solid state:
solid as porous catalyst pellet:1. not being consumed during reaction BUT
changes in physical & chemical states2. pore blocking due to deposits of carbonaceous
by-products [coking]3. metal particles [active catalyst]coalesce at high
temperaturetherefore reduce surface area forreaction hence reducing rate constant [sintering]
EKC338-SCE p. 12/164
Heterogeneous Reaction
Introduction to Heterogeneous and Multiphase ReactionsFor solid state:
solid as non-catalyst:1. dissolution of solid through reaction with fluid2. burning off coke in catalyst pellet for its
regeneration3. most common utilisation of solid catalyst in
fixed-bed catalytic reactor -FBCR4. could also be used in turbular reactor packed with
catalyst through which the fluid species flow
EKC338-SCE p. 13/164
Heterogeneous Reaction
Introduction to Heterogeneous and Multiphase ReactionsFor solid state:
Advantages of FBCR:1. no solids handling2. little solids attribution3. high surface area through use of porous catalyst4. plug flow operation can be achieved5. no separation of catalyst from reaction products
needed
EKC338-SCE p. 14/164
Heterogeneous Reaction
Introduction to Heterogeneous and Multiphase ReactionsFor solid state:
Disadvantages of FBCR:1. pressure drop2. complex arrangement (e.g. multitubular) for
reactions requiring high heat-exchange duties3. large down-time for catalyst which deactivate
rapidly
EKC338-SCE p. 15/164
Heterogeneous Reaction
Interfacial gradient effects: Reaction at catalyst surface
CA
CsAs
CAs
Boundary layer Active centres
Concentration within the catalyst
Concentration at thecatalyst surface
Transfer flux
Bulk concentration
NA
z
FLUID SOLID0 EKC338-SCE p. 16/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsFor first order reaction:
reaction rate at the catalyst surface:
rsAs = ksCsAs (1)
where ks is the rate constant at the catalyst surfaceand CsAs is the concentration at the active surface atz = 0
at steady-state:
rsAs = NA = rA (2)
whereNA = kmc(CA CsAs) (3)
EKC338-SCE p. 17/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsFor first order reaction:
the mass-transfer coefficient can also be expressedin terms of mole fraction & pressure:
kmy =NA
(yA ysAs)and
kmp =NA
(pA psAs)and kmc = kmp = kmy
EKC338-SCE p. 18/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsFor first order reaction:
substitute (3) into (1):NA = ksC
sAs
ksCsAs = kmc(CA CsAs)
CsAs =kmcCAks + kmc
(4)
EKC338-SCE p. 19/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsFor first order reaction:
substitute into (1) and upon rearrangement gives;1
ko=
1
kmc+
1
ks(5)
where ko is the overall rate constant.Limiting cases:
1. kmc >> ks [rapid mass transfer]ko ks
andCsAs CA
EKC338-SCE p. 20/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsFor first order reaction:
Limiting cases:2. ks >> kmc [rapid reaction]
ko kmcand
CsAs 0
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Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsFor Second order reaction:
the rate of reaction is expressed by;
rsAs = ksCsAs
2 (6)
at steady-state;
ksCsAs
2 = kmc(CA CAs)2ksC
sAs
2 + 2kmcCACsAs kmcCsAs2 = kmcC2A
EKC338-SCE p. 22/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsFor Second order reaction:
Limiting cases:1. kmc >> ks:
rA ksC2A[second order dependent] overall is reactionrate controlled
2. ks >> kmc:rA kmcCA
[first order dependent] overall is diffusioncontrolled regime
EKC338-SCE p. 23/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsFor Complex reactions (analytical SOLUTION notusually possible):
mass-transfer can lead to difficulties inexperimentally determining rate coefficient & orderscan work under conditions:
1. reaction controlled:
kmc >> ks
[reduce TEMPERATURE (lower rate), increasefluid turbulence]
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Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsFor Complex reactions (analytical SOLUTION notusually possible):
can work under conditions:2. diffusion controlled:
ks >> kmc
[increase temperature]
EKC338-SCE p. 25/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsDetermining the km value:
usually defined as the mass-transfer coefficient ofequimolar counter diffusion, kmrelationship between km and km
1. Equimolar counter diffusion:
NA = NBthe total mass flux of component A:
NA = NTyA + CDABdyAdz
(7)
EKC338-SCE p. 26/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsDetermining the km value:
relationship between km and km1. since
NT = NA +NB = 0
thusNA = CDAB
dyAdz
(8)
EKC338-SCE p. 27/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsDetermining the km value:
relationship between km and km1. upon integration of this leads to;
NA =CDAB
l(yA ysAs) (9)
sincekmy =
CDABl
and for equimolar counter diffusion;
kmy = kmyEKC338-SCE p. 28/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsDetermining the km value:
relationship between kmy and kmy1. which then gives;
kmc =kmyC
=DAB
l(10)
2. For reaction in which total moles are notconserved
aA bB
EKC338-SCE p. 29/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsDetermining the km value:
relationship between kmy and kmy2. which gives;
NB = baNA (11)
substitute into Equation (7) leads to;
NAl = CDABa
bln
yAysAs
(12)
EKC338-SCE p. 30/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsDetermining the km value:
relationship between kmy and kmy2. for NA = kmy(yA ysAs) where
kmy =kmyyfA
andyfA =
(1 + AyA) (1 + AysAs)ln(
1+AyA1+Ay
sAs
)where A = (ba)a
EKC338-SCE p. 31/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsDetermining the km value:
relationship between kmy and kmy2. for general equation of the form;
aA + bB + . . . qQ + rR + . . .
therefore;
A =(q + r + . . .) (a+ b+ . . .)
a
forA 0, yfA 1
EKC338-SCE p. 32/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsDetermining the km value:
relationship between kmy and kmy2. thus; kmy = kmy
the j-factor:1. jD-factor:
defined as;jD =
kmMmG
Sc23
km can be taken as kmy/kmp, as long as;
km = kmyyfA = kmpPyfA = kmpPfAEKC338-SCE p. 33/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsthe j-factor:1. for a flow in a packed-bed with spherical particles
and b = 0.37;
jD = 1.66Re0.51, for Re < 190
jD = 0.983Re0.41, for Re > 190
2. jH-factor:defined as;
jH =hfCpG
Pr23
EKC338-SCE p. 34/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsConcentration partial pressure differences acrossexternal film:1. if CA/PA 0 that is (yA 0) where the mass
transfer is very fast, therefore, rA can be expressedas function of bulk CA or PA
rA = rsAs = ksCA
since CA CsAs2. using differential definition of rA, thus;
rA
(mol
kgcat s)
EKC338-SCE p. 35/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsConcentration partial pressure differences acrossexternal film:2. with the correction factor for area, am given by;
rA = kmcam(CA CsAs) (13)but in terms of concentration (mole fraction);
rA = amkmy(yA)
and upon rearrangement gives;
kmy =kmyfA
EKC338-SCE p. 36/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsTemperature differences across the external film:1. taking energy balance at steady-state;
rA(Hr) = hfam(T ss T ) (14)but it is known that, rA = kmyamyA uponsubstitution gives;
T = Hr(jDjH
)(Pr
Sc
) 23(yAyfA
)(1
MmCp
)(15)
T increases with the increase of yA. whenmass-transfer resistances is HIGH.
EKC338-SCE p. 37/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsTemperature differences across the external film:1. for gaseous flow in a packed-beds;
T 0.7[ HrMmcp
]yAyfA
(16)
for maximum T T |max occurs when ysAs = 0(for irreversible reaction)and for reversible reaction,
ysAs = yAequilibrium and yfA =AyA
ln (1 + AyA)
EKC338-SCE p. 38/164
Transport Processes inHeterogeneous Catalysis
Interfacial gradient effectsTemperature differences across the external film:1. for maximum temperature difference, substitute the
above terms into Equation (17) then, T |max gives;
T |max = 0.7[ HrMmcp
]ln (1 + AyA)
A(17)
EKC338-SCE p. 39/164
Transport Processes inHeterogeneous Catalysis
Mass Transfer on Metallic Surfaces:for a packed bed, concentration gradient, C variationis SMALLusually negligiblemass transfer may be significant when catalyst is aMETALLIC SURFACE1. catalyst monolith/honeycomb[e.g. catalytic
converter]2. wire gauze[oxidation of NH3]advantages of this unit:1. LOW P (due to porous structure)2. particulate in feed (NO clog-up bed)
EKC338-SCE p. 40/164
Transport Processes inHeterogeneous Catalysis
Intra-Particle Gradient Effects:Catalyst internal structure:
reaction rate catalyst surface areaarea range: 10 200 m2/gactivated carbon: 800 m2/gsand: 0.01 m2/g
EKC338-SCE p. 41/164
Transport Processes inHeterogeneous Catalysis
Intra-Particle Gradient Effects:Catalyst internal structure:
high areas through highly porous structure give highsurface area to volume ratiopore sizes are not uniformpore sizes distributionexistspore size classifications:
1. Micropores: dpore < 0.3nm2. Mesopores: 0.3nm < dpore < 20nm3. Macropores: dpore > 20nmIN CALCULATION use MEAN PORE SIZE!!some catalystshave bimodal distribution of poresizes ZEOLITE CATALYST
EKC338-SCE p. 42/164
Transport Processes inHeterogeneous Catalysis
Intra-Particle Gradient Effects:Catalyst internal structure:
non-ZEOLITE catalystsactive metal dispersedand supported within a macroporous support matrixsuch as SILICA and ALUMINAFURTHER COMPLICATION: DIFFUSION RATEAND MECHANISMS VARY WITH PORE SIZE!
Pore diffusion:for a gas diffusion through a single cylindrical pore ratio of dpore to mean free path, the ratio determines whether OR not pore wallaffects the diffusion behaviour
EKC338-SCE p. 43/164
Transport Processes inHeterogeneous Catalysis
Intra-Particle Gradient Effects:
dpore
where is the distance between the two molecules of gasfor collision.
for dpore >> :1. molecular diffusion dominatesFickian Diffusion2. for example; gases at HIGH pressure or liquids
EKC338-SCE p. 44/164
Transport Processes inHeterogeneous Catalysis
Intra-Particle Gradient Effects:for dpore
Transport Processes inHeterogeneous Catalysis
Intra-Particle Gradient Effects:for dpore
Transport Processes inHeterogeneous Catalysis
Intra-Particle Gradient Effects:
dpore
when dpore
Transport Processes inHeterogeneous Catalysis
Correlations for Diffusion Coefficient:For binary molecular diffusion; (for gases)
Dmi,k T
32
P
Diffusion coefficient for the key component through amixture of the other components, Dmi,m
Ni = yi
Nck=1
Nk CDmi,mdyidz
EKC338-SCE p. 48/164
Transport Processes inHeterogeneous Catalysis
Correlations for Diffusion Coefficient:With the Stefan-Maxwell equation for diffusion, Dmi,mcan be calculated from the actual binary diffusion datausing;
1
Dmi,m
=
Nck=1
1Dmi,k
(yk yi vkvi )1 yi
Nck=1
vkvi
where v is the stoichiometric coefficient.The Knudsen diffusion coefficient, DK can becalculated using;
Dki
(T
Mmi
) 12
dporeEKC338-SCE p. 49/164
Transport Processes inHeterogeneous Catalysis
Correlations for Diffusion Coefficient:And
Dki 6= f(P )when P : transport regime can switch fromKnudsen to molecular diffusion.Micropore diffusion coefficient difficult to predict and always relies on experimental measurementFor NON-zeolite catalysts molecular & Knudsendiffusion dominate and the pore diffusion coefficient,Dp is a function of Dm and Dk
EKC338-SCE p. 50/164
Transport Processes inHeterogeneous Catalysis
Correlations for Diffusion Coefficient:Where Dp the pore diffusion coefficient for a singlepore
dpore
> 20
(molecular diffusion controlling) thus,Dp = Dm
dpore
< 0.2
(Knudsen diffusion controlling) thus,Dp = Dk
EKC338-SCE p. 51/164
Transport Processes inHeterogeneous Catalysis
Correlations for Diffusion Coefficient:For intermediate values, both diffusion types areimportant.Use the Bosanquet Equation to estimate Dp where;
1
Dp=
1
Dk+
1
Dm
EKC338-SCE p. 52/164
Transport Processes inHeterogeneous Catalysis
Correlations for Diffusion Coefficient:If given Dp, the approximation of Deff is given by;
Deff =Dpp
where Deff is the effective diffusion coefficient, p is theintraparticle void fraction and p is the tortuosity factor.
EKC338-SCE p. 53/164
Transport Processes inHeterogeneous Catalysis
Correlations for Diffusion Coefficient:Comparing diffusion in a single pore, (a) & diffusion ina porous pellet, (b):
ANA = -Dp dCA/dz
CA,1 z CA,2
tortuous path
(a) (b)
EKC338-SCE p. 54/164
Transport Processes inHeterogeneous Catalysis
Correlations for Diffusion Coefficient:The cross-sectional area available for diffusion = Ap,thus, lower NA.Tortuous molecules path and changing porecross-sectional area due to constrictions, thus dCA
dzis
reduced.Therefore;
NA = Dpp
dCAdz
For zeolite;p = 3 10
EKC338-SCE p. 55/164
Transport Processes inHeterogeneous Catalysis
Correlations for Diffusion Coefficient:NOTE:
p =tortuosity
constriction factor
where;
tortuosity =actual diffusion path length
shortest radial pellet length
If Deff is given, then the combined diffusion & reactionwithin a catalyst pellet can be considered.Reaction at the surfacediffusion & reaction take placesimultaneously rather than consecutively.
EKC338-SCE p. 56/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Concentration profile for porous catalyst pellet:
Concentration
Position
significant external mass
transfer
negligible external mass
transfer
central axis of pellet
C
A
T
A
L
Y
S
T
external
film
CsA,sCA
bulk concentration
concentration
on the surface
CA,s concentration
within the catalyst
0rpr
EKC338-SCE p. 57/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:The rate of reaction is measured under conditionswhere external and internal mass-transfer resistancesare negligible; rA [use small particle!]When mass-transfer is important;
CA > CAs
1. CANNOT use bulk concentration to calculate theactual (observed) reaction rate.
2. NEED to relate rA to rA using the EffectivenessFactor:
=rArA
EKC338-SCE p. 58/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet: < 1 for ISOTHERMAL or ENDOTHERMIC reaction. is useful for DESIGN CALCULATIONFor rigorous calculations, particularly for COMPLEXREACTION KINETICS and NON-ISOTHERMALoperation, BETTER to solve the simultaneousequations governing diffusion and reaction.
EKC338-SCE p. 59/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:For packed-bedexternal film mass-transferresistances SMALL
ASSUME: situation depicted by the solid line inprevious graphrA is the reaction rate measured if all of the pelletsgive concentration of CsAs, thus;
rA = rAs[CsAs] = r
sAs
and =
rArsAs
EKC338-SCE p. 60/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Pseudo-First Order Reaction: [A Product]
consider material balance through the incrementalsection of a catalyst SLAB of area, a;
r = 0
r
r + r
rp
r r
Incremental
section
NA
EKC338-SCE p. 61/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Pseudo-First Order Reaction: [A Product]
INOUT = CONSUMPTION(NA a)|r+r (NA a)|r = rAsar
dividing by ar and let limr0 gives;
dNAdr
= rAs = kvCAs
EKC338-SCE p. 62/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Pseudo-First Order Reaction: [A Product]
For no convective flow in pellet, Ficks Law isobeyed;
NA = DeAdCAsdr
upon substitution gives;
DeA
d2CAsdr2
= kvCAs (18)
for constant DeA with respect to radius, r.
EKC338-SCE p. 63/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Pseudo-First Order Reaction: [A Product]
integrating Equation (18) using the followingboundary conditions;
r = rp : CAs = CsAs
r = 0 :dCAsdr
gives;
CAsCsAs
=cosh
(r
kvDeA
)cosh
(rp
kv
DeA
) (19)EKC338-SCE p. 64/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Pseudo-First Order Reaction: [A Product]
where Thiele Modulus can be defined as;
slab = rp
kv
DeA
EKC338-SCE p. 65/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Pseudo-First Order Reaction: [A Product]
1.0
1.0 0.0
C
A
s
/
C
s
A
s
r/rp
slab = 0
slab = rp(kv/DeA)1/2
As slab increases - the rate
constant becomes SMALLER
INCREASING
EKC338-SCE p. 66/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Pseudo-First Order Reaction: [A Product]
for spherical pellet, asphere = 4pir2applying the same method as for SLAB; the finalequation leads to;
CAsCsAs
=rpr
sinh(r
kvDeA
)sinh
(rp
kvDeA
) (20)
EKC338-SCE p. 67/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Pseudo-First Order Reaction: [A Product]
for cylindrical-shaped pellet, acylinder = 2pir(L+ r)applying the same method as for SLAB; the ratiogives;
CAsCsAs
=I1I0
r
kvDeA
rp
kvDeA
(21)
where I is the Bassel function given by;
In(r) = rn
m=0
(1)mr2m22m+nm!(n+m)!
EKC338-SCE p. 68/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Pseudo-First Order Reaction: [A Product]
GENERALLY;
1
rm
d
dr(rmNA) = rAs (22)
where;1. for SLAB; m = 02. for CYLINDER; m = 13. for SPHERE; m = 2
EKC338-SCE p. 69/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
It is given by;
e =observed reaction rate
reaction rate at pellet surface conditions
e = rArAs
(23)
EKC338-SCE p. 70/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
Isothermal and Endothermic reactions; rsAs gives amaximum reaction ratesince;
CsAs > CAs
AND[rsAs = kvC
sAs] [rA = kvCAs]
AND therefore;e 1
EKC338-SCE p. 71/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
For a very HIGH diffusional resistances withincatalyst, NEGLIGIBLE penetration of reactant intopellet;
CAs = 0, rAs = 0, e = 0
thus, the range of e;
0 e 1
EKC338-SCE p. 72/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
With the value of e, rA can be determined using;
rA = e rsAs rA = e(kvCsAs) rA = e(kvCA)
NOTE: This is only for NEGLIGIBLE external filmmass transfer resistances!
EKC338-SCE p. 73/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
FOR SLAB:The rate of reaction is given as;
rAs = kvCAs
substitute into the average rate of reaction gives rAwhich can be used to obtain eFinal solution for SLAB-type catalyst;
e =tanhslabslab
(24)
EKC338-SCE p. 74/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
FOR SLAB:NOTE:
slab 0, e 1
slab , e 1slab
EKC338-SCE p. 75/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
FOR SPHERE:By applying Equation (20), the Effectiveness factorfor spherical shape is given by;
e =3
sphere
{1
tanhsphere 1sphere
}(25)
NOTE:sphere 0, e 1
sphere , e 3sphere
EKC338-SCE p. 76/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
FOR CYLINDER:
e =I1(2cylinder)
I0(2cylinder)
1
cylinder(26)
NOTE:cylinder 0, e 1
cylinder , e 2cylinder
For a very SMALL , e will always converge toUNITY (1)!
EKC338-SCE p. 77/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:The Effectiveness Factor for First Order Reaction:
10 20 30
1.0
cylinder
slab
sphere
EKC338-SCE p. 78/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
The equations (e and ) for sphere and cylinderare rather complexFrom the previous plot, the trend is similar only theline shift in the x-axisThiele Modulus can be redefined for any pelletgeometry such that e and curve coincide
EKC338-SCE p. 79/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (for First-order reaction)
Curve for sphere and cylinder coincide with slabcurve such that a relatively simple expressionreduces into;
e =tanh
where is generally given by;
=VpAp
kv
DeA(27)
EKC338-SCE p. 80/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (General Order Reactions)
For general order & reversible reactions;
=VpAp
rsAs2
{ CsAsC
As
DeArAsdCAs
} 12
(28)
where CAs is the equimolar concentration of thelimiting reactant (= 0 for an irreversible reaction)The above equation accounts for DeA varies withCAs
EKC338-SCE p. 81/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Effectiveness Factor, e (General Order Reactions)
It also assumes HIGH differential resistances suchthat within the region of e 1ELSE, CAs in the above equation needs to becalculated using;
rp =
CsAsC
As
DeAdCAs[2 C
A
CAs
DeArAsdC
A
] (29)
EKC338-SCE p. 82/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Criteria for Intraparticle Diffusional Limitations:
For known reaction kinetics e can be calculated(e < 1 indicates diffusional limitation)The Weisz-Prater Criteria:Using;
=VpAp
kv
DeA
EKC338-SCE p. 83/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Criteria for Intraparticle Diffusional Limitations:
upon rearrangement gives;
2(ApVp
)2DeA = kv
for First-order reaction;
rA = ersAs = kvC
sAs
EKC338-SCE p. 84/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Criteria for Intraparticle Diffusional Limitations:
eliminating kv gives;
=rA
DeACsAs
(VpAp
)2= e
2 (30)
is the Weisz-Prater ParameterCsAs CA under typical conditions.
EKC338-SCE p. 85/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Criteria for Intraparticle Diffusional Limitations:
The RHS of Equation (30) is measurable, then;1. NEGLIGIBLE diffusional limitations; when;
1, e 1therefore;
1
EKC338-SCE p. 86/164
Transport Processes inHeterogeneous Catalysis
Diffusion and Reaction within a Catalyst Pellet:Criteria for Intraparticle Diffusional Limitations:
The RHS of Equation (30) is measurable, then;2. CONSIDERABLE diffusional limitations; when;
1, e 1
therefore; 1
The above method can be generalised to anyreaction scheme where appropriate for the ThieleModulus.
EKC338-SCE p. 87/164
Transport Processes inHeterogeneous Catalysis
Temperature Gradient Within Catalyst Pellet:Temperature gradient, T can be calculated byconsidering simultaneously the intraparticle mass andenergy balances.For spherical pellet; the mass balance is given by;
1
r2DeA
d
dr
(r2dCAsdr
)= rAs
similarly for energy balance;
1
r2e
d
dr
(r2dTsdr
)= rAs Hr (31)
EKC338-SCE p. 88/164
Transport Processes inHeterogeneous Catalysis
Temperature Gradient Within Catalyst Pellet:Equation (31) is known as Fouriers Law where e isthe effective thermal conductivity of the pellet.By eliminating rAs and integrating twice leads to;
Ts = (Ts T ss ) =HrDeA
e(CAs CsAs) (32)
For irreversible reaction, Ts is maximum whenCAs = 0 (OR CAs for an equimolar reversible reaction)thus;
Ts|max = HrDeAe
CsAs (33)
EKC338-SCE p. 89/164
Transport Processes inHeterogeneous Catalysis
Temperature Gradient Within Catalyst Pellet:Equation (33) is applicable to all pellet catalystgeometries.For many industrial applications;
Ts|maxT ss
< 0.1
that is for small Ts, T (external film) can be large.EXCEPT for HIGHLY exothermic reactions such assome oxidation and hydrogenation reactions.The effect of Ts on e is complex since, it willinfluence DeA as well as kv.
EKC338-SCE p. 90/164
Transport Processes inHeterogeneous Catalysis
Temperature Gradient Within Catalyst Pellet:Consider the First-order non-isothermal reaction on apellet; the mass balance is given by;
1
r2DeA
d
dr
(r2dCAsdr
)= rAs
andrAs = kvCAs
wherekv = A0e
(
ERT0
)
EKC338-SCE p. 91/164
Transport Processes inHeterogeneous Catalysis
Temperature Gradient Within Catalyst Pellet:Upon substitution gives;
1
r2DeA
d
dr
(r2dCAsdr
)= A0e
(
ERT0
)CAs
putting into dimensionless form leads to;
d2C
dr2= Ce(1T )
whereC =
CAsCsAs
T =TsT ss
r =r
rpEKC338-SCE p. 92/164
Transport Processes inHeterogeneous Catalysis
Temperature Gradient Within Catalyst Pellet:and both and is defined as;
=r2pA0e
DeA
and =
E
RT ss
EKC338-SCE p. 93/164
Transport Processes inHeterogeneous Catalysis
Temperature Gradient Within Catalyst Pellet:Similarly, for energy balance;
d2T
dr2= 2Ce(1T )
where =
(Ts)maxT ss
=HrDeACsAs
eT ss
EKC338-SCE p. 94/164
Transport Processes inHeterogeneous Catalysis
Temperature Gradient Within Catalyst Pellet:
< 0:
= 0:
> 0: Exothermic
Isothermal
Endothermic
1.0
0.0010.1
EKC338-SCE p. 95/164
Transport Processes inHeterogeneous Catalysis
Combined Interfacial [External] and Intraparticle [Internal]Resistances:
In the solution of intraparticle diffusional equation, CsAswas assumed known;
CsAs = CA
and it remains constant.When the external-film resistances are important, theBOUNDARY CONDITIONS for the solution of theintraparticle diffusion equation become;
r = rp : kmc(CA CsAs) = DeAdCAsdr
rp
EKC338-SCE p. 96/164
Transport Processes inHeterogeneous Catalysis
Combined Interfacial [External] and Intraparticle [Internal]Resistances:
and;
r = 0 :
dCAsdr0
= 0
For slab pellet with a First-order reaction, the solutionwith the above boundary conditions gives;
CAs =CA cosh
r
rp
cosh+DeA
rpkmcsinh
EKC338-SCE p. 97/164
Transport Processes inHeterogeneous Catalysis
Combined Interfacial [External] and Intraparticle [Internal]Resistances:
Therefore, the Global Effectiveness Factor can bedefined as;
G =rate observed
rate at bulk fluid concentration
G =rA
rAsCA
EKC338-SCE p. 98/164
Transport Processes inHeterogeneous Catalysis
Combined Interfacial [External] and Intraparticle [Internal]Resistances:
Which then gives;
1
G=
1
+
2
Bim(34)
where Bim is Biot number for mass-transfer given by;
Bim =kmcrpDeA
For Bim 1.0, G = e.
EKC338-SCE p. 99/164
Transport Processes inHeterogeneous Catalysis
Combined Interfacial [External] and Intraparticle [Internal]Resistances:
For the region of strong intraparticle diffusionallimitations, where;
and
e =1
thus,1
G= +
2
Bim(35)
EKC338-SCE p. 100/164
Fixed-Bed Catalytic Reactor Design
Describing the homogeneous models and modelsaccounting for interfacial and intrafacial gradientsusing;1. Effectiveness factor2. Actual pellet phase mass and energy balancesPLUG-FLOW REACTOR (PFR) model:
the simplest PFR model is given by;
dnidV
= ri = rib =vi|vA|r
Ab (36)
EKC338-SCE p. 101/164
Fixed-Bed Catalytic Reactor Design
PLUG-FLOW REACTOR (PFR) model:when ni = uaCi and dV = adz, thus, the equationreduces into;
d
dz(uCi) = rib =
vi|vA|r
Ab (37)
since u 6= constant, therefore momentum equationis required.
EKC338-SCE p. 102/164
Fixed-Bed Catalytic Reactor Design
PLUG-FLOW REACTOR (PFR) model:Using the Ergun equation of the form;
dp
dz= E1u E2u2 (38)
to find the pressure along the bed, where;
E1 =180(1 b)2
d2p3b
andE2 =
1.8(1 b)gMmdp3b
EKC338-SCE p. 103/164
Fixed-Bed Catalytic Reactor Design
PLUG-FLOW REACTOR (PFR) model:If the flow is highly TURBULENT, E1 can beneglected.If the flow is LAMINAR, E2 can be omitted.While for a perfect gas;
i
Ci =P
RT= g
For non-isothermal operation, energy balance isrequired to describe Tz variation
EKC338-SCE p. 104/164
Fixed-Bed Catalytic Reactor Design
PLUG-FLOW REACTOR (PFR) model:Energy balance across a fix-bed reactor is given as;
dT
dV= (i
nicpi) + br
Ar Qav = 0 (39)
whereQ = U(Tc T ) (J/m2s)
and av is the surface area per unit reactor volume,(m1), therefore;
dT
dz= (U
i
nicpi) + br
Ar Qav = 0 (40)
EKC338-SCE p. 105/164
Fixed-Bed Catalytic Reactor Design
PLUG-FLOW REACTOR (PFR) model:where; U is the overall heat transfer coefficient,(J/m2s.K)and Tc is the temperature of cooling fluid (K)For no-separation of reactor species due to differentrates of axial dispersion OR intra-particle diffusion,Ci can be related to CA using the reactionstoichiometry;
(nAo nA) mol A reactedthus;
ni = nio +i|nA|(nAo nA)
EKC338-SCE p. 106/164
Fluidised-Bed Reactors
These involve catalyst beds which are not packed inrigid but either suspended in fluid (for fluidised-bedreactor) or flowing with the fluid (transport reactor)Fluidisation Principles (Overview):
Downward flow in packed bedno relativemovement between particles
1. P u for LAMINAR flow2. P u2 for TURBULENT flow
EKC338-SCE p. 107/164
Fluidised-Bed Reactors
Fluidisation Principles (Overview):Upward flow through bed P is the same asdownward flow at LOW flow rate:
when frictional drag on particles become equal totheir apparent weight (actual weight LESSbuoyancy)particle rearrange and offer LESSresistance to flowresults in bed EXPANSION.as u increases, process continues until bedassumes its loosest stable form of packing.
MINIMUM fluidisation velocity, umfis the velocity ata point where fluidisation occurs!
EKC338-SCE p. 108/164
Fluidised-Bed Reactors
Fluidisation Principles (Overview):When superficial velocity > umf ;
1. LIQUID fluidisation;bed continues to EXPAND with uit maintains a uniform characterand AGITATION of particleincreasesparticulate fluidisation
EKC338-SCE p. 109/164
Fluidised-Bed Reactors
Fluidisation Principles (Overview):When superficial velocity > umf ;
2. GAS fluidisation;gas bubble formation within a continuousphase consisting of fluidised solids.continuous phase refers to as thedense/emulsion phaseaggregation fluidisationat HIGH inlet flow rate: flow in emulsion phaseto particulate remains approx. constant butbubbles may be more rigorous.at HIGH inlet flow rate and a deepbedbubbles coalesce forming slugs of gasthat occupy the entire cross-section of the bed.
EKC338-SCE p. 110/164
Fluidised-Bed Reactors
Fluidisation Principles (Overview):An increase of bubbles within the bed gives V andthis lowers the transfer area.HIGH volume of bubbles also gives high residencetime.It behaves like fluidhydrostatic forces aretransmitted and solid objects FLOAT when;densities of objects < density of bed
Intimate mixing and rapid heat transfer easy tocontrol the TEMPERATURE (even for highlyEXOTHERMIC reaction)Type of fluidisation depends on [i] the particle sizeand [ii] relative density of the particles (s g)
EKC338-SCE p. 111/164
Fluidised-Bed Reactors
WHY Fluidisation?Can achieve a GOOD control of TEMPERATURECan work with VERY FINE particles for which
e 1As catalyst improvesthe rates of reactionINCREASE resulted form higher kv BUT;
=rp3
kv
DeA
when fv , the ONLY way to keep SMALL and eclose to 1 is to decrease rp
EKC338-SCE p. 112/164
Fluidised-Bed Reactors
WHY Fluidisation?NOTE: an increase of kv will increase , therefore itwill be MASS TRANSFER controlling and NOTkinetics (reaction) the possible way is to REDUCErp
EKC338-SCE p. 113/164
Fluidised-Bed Reactors
P versus uo for fluidised bed:
hysterisis due to
pressure differentblown out particles
(initiation of
particle entrainment)
log P
log uo
umf
EKC338-SCE p. 114/164
Fluidised-Bed Reactors
P versus uo for fluidised bed:NOTE:
1. LAMINAR FLOW:P
L= E1uo
log (P ) = C + log uo2. TURBULENT FLOW:
P
L= E2u2o
log (P ) = C + 2 log uoEKC338-SCE p. 115/164
Fluidised-Bed Reactors
P versus uo for fluidised bed:Calculation of P across fluidised bed: Consider adiagram below;
A
L
P1
P2
F1
F2
uo
uo = superficial velocity
at bed inlet
ut = terminal velocity
when pellet are
blown out of the
bed
EKC338-SCE p. 116/164
Fluidised-Bed Reactors
P versus uo for fluidised bed:Resolving forces on the bed;
F1 = F2P1A = P2A+ (s g)(1 )ALg
(P1 P2) = (s g)(1 )LgP = (s g)(1 )Lg (41)
As P1 , P also , and therefore, as the bedexpendsOR resistance as the gas by-pass throughbubbling and P remains the same.
EKC338-SCE p. 117/164
Fluidised-Bed Reactors
Calculation of the minimum fluidisation velocity, umf ;For LAMINAR flow;Using the previously defined Ergun equation[Equation (38)];
PmfLmf
= E1umf
umf = (1 mf)(s g)gE1
(42)
whereE1 =
180(1 mf)2d2p 3mf
EKC338-SCE p. 118/164
Fluidised-Bed Reactors
Calculation of the minimum fluidisation velocity, umf ;For LAMINAR flow;Substitute into Equation (40) and simplify gives;
umf =1
180
3mf d2p(1 mf)
(s g)g
(43)
For mf 0.4 the bed is packed with isometricparticles.
EKC338-SCE p. 119/164
Fluidised-Bed Reactors
Calculation of the minimum fluidisation velocity, umf ;For TURBULENT flow [usually for coarse particles];Similarly, applying the Ergun equation;
PmfLmf
= E1umf E2u2mf = (1 mf)(s g)g
and solving for umf explicitly gives;
Ga = 180(1 mf)
3mfRemf +
1.75
3mfRe2mf (44)
EKC338-SCE p. 120/164
Fluidised-Bed Reactors
Calculation of the minimum fluidisation velocity, umf ;For TURBULENT flow [usually for coarse particles];where
Ga =g(s g)gd3p
2
is the Galileos Number and
Remf =gumfdp
is the Reynolds Number for minimum fluidisation.in reality, expect Darcys Law and Ergun equationto overestimate Pmf .
EKC338-SCE p. 121/164
Fluidised-Bed Reactors
Calculation of the minimum fluidisation velocity, umf ;For LAMINAR flow, many investigations haveshown that it is more accurate to use a value of 120rather than 180 in Equation (41).Equation (42) for TURBULENT flow DOES NOTaccount for;
1. Channeling of fluid2. Electrostatic forces between particles3. Agglomeration of particles4. Friction between fluid and vessel walls.
EKC338-SCE p. 122/164
Fluidised-Bed Reactors
Calculation of terminal velocity, ut;
Force exerted by flowing gas
mg
when the drag force exerted on a spherical particleby the upflowing gas, the gravitational force (basedon the apparent density) on the particle, then theparticle will be BLOWN OUT of the bed!
EKC338-SCE p. 123/164
Fluidised-Bed Reactors
Calculation of terminal velocity, ut;this can be shown by;
Fdrag = Vp(s g)gbut (FROM FLUID FLOW NOTES);
Fdrag =1
2gu
2tCD Ap
where CD is the drag coefficient. with Ap = pid2p
4thus;
Fdrag =pid2p8 gu2t CD
EKC338-SCE p. 124/164
Fluidised-Bed Reactors
Calculation of terminal velocity, ut;upon rearrangement gives;
ut =
4dp(s g)g
3CDg(45)
for spherical particles and Re < 0.4 where
Re =gutdp
EKC338-SCE p. 125/164
Fluidised-Bed Reactors
Calculation of terminal velocity, ut;and the Drag coefficient is given by;
CD =24
Re
and Equation (43) reduces into Stokes Law of theform;
ut =(s g)gd2p
18(46)
EKC338-SCE p. 126/164
Fluidised-Bed Reactors
Calculation of terminal velocity, ut;for 1 < Re < 103;the Drag coefficient is given by;
lnCD = 5.50 + 69.43lnRe + 7.99
and for Re > 103;the Drag coefficient CD = 0.43, which gives;
ut =
3.1dp(s g)g
g
EKC338-SCE p. 127/164
Fluidised-Bed Reactors
Fluidisation regimes:For COARSE PARTICLES:
bubbles appear as soon as umf is exceeded.in TURBULENT regimesbubbles life time isSHORT due to bubbles burst. Bed is quiteuniformshort circuiting of gas through bubbles isless likely.umf and particle blow-out coincide.in FAST fluidisation regimethere is the netentrainment of solids.in TRANSPORT regimethere is solid flow in thedirection of gas flow.carry-over (entrainment) separates particles bysize.
EKC338-SCE p. 128/164
Fluidised-Bed Reactors
Fluidisation regimes:For FINE PARTICLES:
bubbles DO NOT appear as soon as minimumfluidisation is reachedinstead, there is a uniformexpansion of bed.bed is more coherent rather than particlesbehaving independently.TURBULENT regime sets in well after uo exceedsut of an individual particle, thus, operate at higheruo.carry-over DOES NOT separate particles bysizea more cohesive bed.
EKC338-SCE p. 129/164
Fluidised-Bed Reactors
Fluidised-Bed Reactors: The ApplicationsIt is useful for highly EXOTHERMIC systemsAND/OR systems requiring close temperaturecontrol such as oxidation reactions.In a classical fluidised-bed operation, catalystparticles are retained in bedlittle catalystentrainment.Some of the systems of reactions that usefluidised-bed include:
1. Oxidation of napthalene into phtalic anhydride.2. Ammoxidation of propylene to acrylonitrile.3. Oxychlorination of ethylene to ethylene dichloride.4. Coal combustion (injection of limestone for the
in-situ capture of SO2).EKC338-SCE p. 130/164
Fluidised-Bed Reactors
Fluidised-Bed Reactors: The ApplicationsSome of the systems of reactions that usefluidised-bed include:
5. Roasting of oresEven with classical fluidised-bed, region above thesurface of bed contains some solid concentration.This concentration becomes constant as it is movedaway from the surface.
EKC338-SCE p. 131/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Two-phase model:
the model is based on the interchange betweenthe two phases;
Bubble
phase
Emulsion
phase
uo, CAo
CAb|out CAe|out
CA
ub ue
CAb CAe
EKC338-SCE p. 132/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Two-phase model:
for ISOTHERMAL fluidised-bed in emulsionphase, the material balance is given by;for bubble-phase:
fbubdCAbdz
+ kI(CAb CAe) + fbgbrA = 0 (47)
for emulsion-phase:
feuedCAedz
feDzed2CAedz2
kI(CAbCAe)+(1fb)gerA = 0(48)
EKC338-SCE p. 133/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Two-phase model:
also;uoCA = fbubCAb + feueCAe (49)
and the boundary conditions are;for bubble-phase:
z = 0 : CAb = CAo
EKC338-SCE p. 134/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Two-phase model:
for emulsion-phase:
z = 0 : DzedCAedz
= ue(CAo CAe)
z = L :dCAedz
= 0
EKC338-SCE p. 135/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Model simplification:
If ub ue, that is when ub umf , then theemulsion-phaseclosed (relatively negligible inletOR outlet flow). Thus Equation (46) reduces into;
kI(CAb CAe) = (1 fb)gerA (50)also neglecting the DISPERSION.The above equation assumes a stagnantemulsion phase BUT, CAe varies with bed lengthz.
EKC338-SCE p. 136/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Estimation of parameters appearing in thetwo-phase model:
1. ub: bubble velocity:this is given by;
ub = (uo umf) + ubrwhere ubr is the bubble rise velocity when there isa SWARM of bubbles. This is separately given by;
ubr = dbg
EKC338-SCE p. 137/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Estimation of parameters appearing in thetwo-phase model:
1. ub: bubble velocity:where = 0.64 for dt < 0.1m OR = 1.6d0.4t for0.1m < dt < 1.0m OR = 1.6 for dt > 1.0m
2. fb: bubble friction:this is given by;
fb =uo umf
ub
EKC338-SCE p. 138/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Estimation of parameters appearing in thetwo-phase model:
2. fb: bubble friction:BUT for ub umf
fb uoub
3. fe: emulsion friction:This is given by
fe + fb = f
where f is the VOIDAGE of a fluidised-bed.EKC338-SCE p. 139/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Estimation of parameters appearing in thetwo-phase model:
4. Lf and f : length of bed and bed voidage:Given that the volume of solids constant, where;
Lf(1 f) = Lmf(1 mf) = L(1 b)
1 f1 mf =
LmfLf
= 1 fb
given that fb and mf 0.4, then Lf and f can becalculated.
EKC338-SCE p. 140/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Estimation of parameters appearing in thetwo-phase model:
5. Dze : diffusion coefficient of emulsion phase:Using;
Dze = f(uo, dt)
6. ue: emulsion velocity:Using
ue =umfmf
EKC338-SCE p. 141/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Estimation of parameters appearing in thetwo-phase model:
7. gb and ge: mass of solid in bubble andemulsion phases respectively:Using;
fbgb + (1 fb)ge = mA Lf
8. kI : gas interchange coefficient:For two-phase modelskI often used as a fittingparameter such that model agrees with plantdata.
EKC338-SCE p. 142/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Three-phase model:
ub
ue
emulsion
cloud
bubble
EKC338-SCE p. 143/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Three-phase model:
there is an interchange of gas from bubble tocloud, then from cloud to emulsion in sequentialstepthis can be depicted in the diagram below;
kI,b
kI,e
CA,b CA,b CA,e
bubble cloud emulsion
EKC338-SCE p. 144/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Three-phase model:
different mixing regimes in different phases canbe assumed.Kunnii-Levenspiel Model (k-L) assumesemulsion phase with no net gas flow.this is usually achieved for
uoumf
> 6
EKC338-SCE p. 145/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Example: k-L Model for First-order reaction
Consider the material balances:Bubble phase:
fbubdCAbdz
+ kIb(CAb CAc) + fbgbkCAb = 0
Emulsion phase:
kIe(CAc CAe) = (1 fb f c)gekCAeCloud phase:
kIb(CAb CAc) = kIe(CAc CAe) + f cgckCAcEKC338-SCE p. 146/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Example: k-L Model for First-order reactionfc is with the units of m
3cloud
m3bed
gc is in the form of kgm3cloud which is approx. equal to
ge =b
1 fband f c is normally given by;
f c fb =1.17
1.17 +ubue
EKC338-SCE p. 147/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Example: k-L Model for First-order reaction
using equations for emulsion and could phasesand substitute into the bubble phase equationgives;
ubdCAbdz
= kCAb (51)
EKC338-SCE p. 148/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Example: k-L Model for First-order reaction
and K is given by;
K = k
gb +
1kfbkIb
+ 1gcf c+
1kfbkIe
+ 1ge(1fbf
c)
fb
which is the effective rate constant for athree-phase fluidised-bed model k-L rateconstant.
EKC338-SCE p. 149/164
Fluidised-Bed Reactors
Modelling of fluidised-bed reactors:Example: k-L Model for First-order reaction
Integration of Equation (49) with boundaryconditions;
z = 0; CAb = CAo
leads to;CAbCAo
=CACAo
= eKb (52)
where b = Lfub
EKC338-SCE p. 150/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Example: Fluid Catalytic Crackingfast reactions(small required) and rapid catalyst deactivation.Velocity of SOLIDS velocity of GAS. That is, NOSLIP VELOCITYUsually employed FINE SOLIDS such that e 1For NO catalyst DEACTIVATION, riser is very muchlike pseudo-homogeneous Plug-Flow reactor (PFR)but
> b
EKC338-SCE p. 151/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Calculation of :
Given that;
(m3gm3b
)=
Auo
Auo +msp
(53)
where p is the pellet density with units of kgm3pelletUpon simplification of Equation (51) gives;
(m3gm3b
)=
1
1 + msAuop
(54)
EKC338-SCE p. 152/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Calculation of :
The diagram is given;
solidgasms (kg/s)uo (m/s)
A
EKC338-SCE p. 153/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Calculation of :
From Equation (52);ms uo : 1ms uo : 0
for Packed-Bed reactor; b 0.4For NO catalyst deactivation:
uodCAdz
= rA(1 )p (55)
EKC338-SCE p. 154/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Calculation of :
Catalyst deactivation in Fluid-Catalytic Crackingis believed to arise from:
1. coke deposition2. adsorption of certain species present in the
feedThus will give a reduction in the reaction rate(s)and therefore with time, with DeactivationFunction given by;
A =rA(t)
rA(0)= f(t) (56)
EKC338-SCE p. 155/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Calculation of :
The function can be of the form;
= 1 tOR
= et
Therefore Equation (53) becomes;
uodCAdz
= rAA(1 )p (57)
EKC338-SCE p. 156/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Calculation of :
Where t = zuo
(NO SLIP) and it represents thetime for a particular catalyst to have spent in theriser.Sometimes, is given as a function of the cokeconcentration on the catalyst pellets. It is practicalto express the concentration in the form of;
Cc
(kgcoke
kgcatalyst
)
EKC338-SCE p. 157/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Calculation of :
And the rate of formation of coke is given by;
rc
(kgcoke
kgcatalyst s)
where rc can itself be deactivated as the coke isbeing produced!The balances for coke deposition is given by;
msA dCcdz
= rccp(1 ) (58)
EKC338-SCE p. 158/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Calculation of :
The energy balances for the ADIABATIC riser canbe written as;
mgcpg + mscpsA
dT
dz= [rAA(HA) + rcc(Hc)] p(1 ) (59)
where cpg and cps are the specific heat capacitiesof gas and solid respectively in kJ
kgKand mg is the
mass flow rate of gas in kgs
EKC338-SCE p. 159/164
Fluidised-Bed Reactors
Modelling of Transport Reactor (Riser):Calculation of :
And mg is given by;
mg =AuopoRTo
Mg
EKC338-SCE p. 160/164
Multiphase Reactors
Involved GAS and LIQUID phases in contact with aSOLID.The SOLID may be of the form of;1. catalyst particles dispersed in the liquid phase (Eg.
SLURRY REACTOR)2. packing for liquid distribution (Eg. PACKED-BED
ABSORBER)3. packing for liquid distribution and catalyst support
(Eg. TRICKLED-BED REACTOR and PACKEDBUBBLE REACTOR)
4. plates for liquid-gas contact (Eg. DISTILLATIONCOLUMN)
EKC338-SCE p. 161/164
Multiphase Reactors
Reactors can also be classified in terms of whichphase is continuous and which is dispersed.
Referring to the diagram below:LIQUID: continuous
GAS: disperse
LIQUID: disperse
GAS: continuous
LIQUID: continuous
GAS: continuous
GAS GAS GAS
LIQUID
LIQUID
LIQUID
Bubble reactor
Slurry reactor
Fermentation vessel
Spray tower
Trickle-bed reactor
Packed-bed reactor
Wetted-wall reactor
(falling film)
EKC338-SCE p. 162/164
Multiphase Reactors
If mass-transfer resistance located in the liquid-film,use DISPERSEgas phase and CONTINUOUSliquidphase.If mass-transfer resistance located in the gas-film,use CONTINUOUSgas phase and DISPERSEliquidphase.Residence time, of reactant and heat transferconsideration will also dictate the type of reactor;1. plate columns can achieve long contact times
between gas and liquid, BUT poor TEMPERATUREcontrol
EKC338-SCE p. 163/164
Multiphase Reactors
Residence time, of reactant and heat transferconsideration will also dictate the type of reactor;2. stirred-tank (BUBBLE and SLURRY), will have large
LIQUID:GAS ratio, BUT yet, cope with HIGH GASflow rates and therefore GOOD TEMPERATUREcontrol.
Reactors can have co- OR counter- current flow ofGAS and LIQUID to utilise driving force for MASS andHEAT transfers.Where reactors are employed for GAS purification,then it is referred to as ABSORBERS.
EKC338-SCE p. 164/164
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