reactor design and analysis (mhu)

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


CA,b

CA,s

CA(r)

External

diffusion

Internal

diffusion

Porous catalyst

pellet

External

surface

EKC338-SCE p. 6/164


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


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


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


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


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



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



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



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


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



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



substitute (3) into (1):NA = ksC

sAs

ksCsAs = kmc(CA CsAs)

CsAs =kmcCAks + kmc

(4)

EKC338-SCE p. 19/164



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



Limiting cases:2. ks >> kmc [rapid reaction]

ko kmcand

CsAs 0

EKC338-SCE p. 21/164


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


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


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]

EKC338-SCE p. 24/164


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


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



relationship between km and km1. since

NT = NA +NB = 0

thusNA = CDAB

dyAdz

(8)

EKC338-SCE p. 27/164



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



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



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



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



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



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


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


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


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


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


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


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


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


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



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



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


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


Intra-Particle Gradient Effects:for dpore


Intra-Particle Gradient Effects:

dpore

when dpore


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


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


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


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


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


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


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


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


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


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


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


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


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


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



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



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



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



where Thiele Modulus can be defined as;

slab = rp

kv

DeA

EKC338-SCE p. 65/164



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



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



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



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


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



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



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



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



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



FOR SLAB:NOTE:

slab 0, e 1

slab , e 1slab

EKC338-SCE p. 75/164



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



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


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



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



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


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


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


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



upon rearrangement gives;

2(ApVp

)2DeA = kv

for First-order reaction;

rA = ersAs = kvC

sAs

EKC338-SCE p. 84/164



eliminating kv gives;

=rA

DeACsAs

(VpAp

)2= e

2 (30)

is the Weisz-Prater ParameterCsAs CA under typical conditions.

EKC338-SCE p. 85/164



The RHS of Equation (30) is measurable, then;1. NEGLIGIBLE diffusional limitations; when;

1, e 1therefore;

1

EKC338-SCE p. 86/164



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


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


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


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


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


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


Temperature Gradient Within Catalyst Pellet:and both and is defined as;

=r2pA0e

DeA

and =

E

RT ss

EKC338-SCE p. 93/164


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


Temperature Gradient Within Catalyst Pellet:

< 0:

= 0:

> 0: Exothermic

Isothermal

Endothermic

1.0

0.0010.1

EKC338-SCE p. 95/164


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



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



Therefore, the Global Effectiveness Factor can bedefined as;

G =rate observed

rate at bulk fluid concentration

G =rA

rAsCA

EKC338-SCE p. 98/164



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



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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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


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: 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: 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


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



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



also;uoCA = fbubCAb + feueCAe (49)

and the boundary conditions are;for bubble-phase:

z = 0 : CAb = CAo

EKC338-SCE p. 134/164



for emulsion-phase:

z = 0 : DzedCAedz

= ue(CAo CAe)

z = L :dCAedz

= 0

EKC338-SCE p. 135/164


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


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



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



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



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



5. Dze : diffusion coefficient of emulsion phase:Using;

Dze = f(uo, dt)

6. ue: emulsion velocity:Using

ue =umfmf

EKC338-SCE p. 141/164



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


Modelling of fluidised-bed reactors:Three-phase model:

ub

ue

emulsion

cloud

bubble

EKC338-SCE p. 143/164



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



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


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


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



using equations for emulsion and could phasesand substitute into the bubble phase equationgives;

ubdCAbdz

= kCAb (51)

EKC338-SCE p. 148/164



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



Integration of Equation (49) with boundaryconditions;

z = 0; CAb = CAo

leads to;CAbCAo

=CACAo

= eKb (52)

where b = Lfub

EKC338-SCE p. 150/164


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


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



The diagram is given;

solidgasms (kg/s)uo (m/s)

A

EKC338-SCE p. 153/164



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



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



The function can be of the form;

= 1 tOR

= et

Therefore Equation (53) becomes;

uodCAdz

= rAA(1 )p (57)

EKC338-SCE p. 156/164



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



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



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



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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