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Centre for Catalysis Research
Department of Chemical EngineeringUniversity of Cape Town • Rondebosch 7701
South Africa
Residence time distribution in real reactors
Linda H. CallananEric van Steen
1. Recap on RTD and conversion in real reactors
2. Dispersion model3. Tank-in-series model4. Compartment model
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Residence Time Distribution (RTD)
Determined by introducing a non-reactive, non-adsorbing tracer into the feed and measuring the concentration of the tracer in the exit line.
Pulse input
Step input
( ) ( )
( )∫∞=
=
⋅
= t
0tdttC
tCtE
( ) ( )( )dt
tFdtE =( ) ( )( )0tCtCtF
inlet >=
Cex
it(t)
t
E(t)
t
Cex
it(t)
t
F(t)
t
E(t)
t
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Evaluation of Residence Time Distributions (RTD)
Consistence checkPulse input
Step input
Mean Residence timeThe mean or average residence time:The mean residence time should be equal to the space time:
VarianceSpread in the residence time distribution is commonly measured by the variance:
vVt
?m =τ=
∫∞=
=
⋅⋅=t
0tm dtEtt
fluid
injectedt
0t vn
dtC =⋅∫∞=
=
( ) inletCtC =∞=
( )∫∞
⋅⋅−=σ 02
m2 dtEtt
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Conversion in real reactors -Segregation Model
The fluid consisting of non-interacting elements
Each exit stream is thought to consist of elements having spent various times in the reactor.
The exit concentration of the reactant is given by:
Assumptions:Valid for linear processes: Each fluid element does not react with any other Mixing occurs as late as possible (at the reactor exit)
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
+
⋅
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
+
=⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛
∑
dttandtbetween
ageofstreamexitoffraction
dttandtbetweenageof
elementinleftttanreacof.conc
streamexitinionconcentrat
mean
elementsexitAll
[H. Scott Fogler, “Elements of Chemical reaction Engineering” 4th
Ed., Prentice Hall, 2006]
∫∞=
=
⋅⋅=t
0telement,AA dtECC
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Determination of conversion from E-curve
The residence time distribution in a reactor has been determinedand the E(t) data have been determined as shown below:
The reactor is to be used for a liquid phase decomposition. The rate of reaction is 1st order with respect to the reactant (k = 0.1 min-1). Determine:
Time t, min 0 5 10 15 20 25 30 35E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0
1. The mean residence time2. The conversion, which would have been obtained if the reactor
is an ideal PFR/CSTR3. The conversion obtained in this reactor according the
segregation model
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Determination of mean residence time from E-curve
∫∞=
=
⋅⋅=t
0tm dtEttMean residence time:
t, min 0 5 10 15 20 25 30 35
E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0
t.E 0 0.15 0.50 0.75 0.80 0.50 0.30 0
Integral 0.375 1.625 3.125 3.875 3.25 2.00 0.75
( ) ( )( ) ( )1212 tttEtE21
−⋅⋅+⋅⋅
Mean residence time: 15 min
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Determination of conversion in ideal reactors
PFR:
CSTR
0,A
A
0,A
ACvr
Fr
dVdX
⋅−
=−
=0,A
AC
rddX −
=τ
1st order reaction: ( )X1kC
CkC
rddX
0,A
A
0,A
A −⋅=⋅
=−
=τ
( ) τ⋅=−− kX1ln τ⋅−−= ke1XConversion in ideal PFR: 77.8 %
0,A
A
0,A
ACvr
Fr
VX
⋅−
=−
=0,A
AC
rX −=
τ
1st order reaction: ( )X1kC
CkC
rX0,A
A
0,A
A −⋅=⋅
=−
=τ τ⋅+
τ⋅=
k1kX
Conversion in ideal CSTR: 60 %
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Conversion using segregation model
Segregation model
The concentration of the reactant in each element depends on thetime it spent in the reactor. Each element can be seen as a batch-type reactor.
The average concentration of the reactant in the exit stream is given by:
tk0,Aelement,A eCC ⋅−⋅=
∫∞=
=
⋅⋅=t
0telement,AA dtECC
∫∞=
=
⋅− ⋅⋅⋅=t
0t
tk0,AA dtEeCC
∫∞=
=
⋅− ⋅⋅−=−=t
0t
tk
0,A
A dtEe1CC1X
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Conversion using segregation model
t, min 0 5 10 15 20 25 30 35
E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0
e-k.t.E 0 0.018 0.018 0.011 0.005 0.002 0.000 0
Integral 0.045 0.091 0.074 0.041 0.018 0.005 0.001
( ) ( )( ) ( )121tk
2tk ttEeEe
21 12 −⋅⋅+⋅⋅ ⋅−⋅−
X = 1-Σintegral
Comparison of conversions:PFR 77.8%CSTR 60.0%Segregation model 72.4%
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Dispersion model
convectionTc
dispersion
TcT CAu
dzdCADF ⋅⋅+⋅⋅−=
Pulse input
t
C(t)
t
C(t)
moleculardiffusionfluid flow
t
C(t)
t
Flux of inert tracer:
Mole balance on inert tracer:
FT: molar flux of tracerD: diffusion coefficient for tracerAc: cross sectional area of tubez: lengthu: linear velocityt
CAz
F Tc
T∂∂⋅=
∂∂
−
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Development of dispersion model
tC
zCu
zCD TT2T
2
∂∂
=∂∂⋅−
∂
∂⋅Mole balance on inert tracer:
θ⋅= duLdtL
utt ⋅=τ=θIntroducing a dimensionless time:
a dimensionless length: Lz=λ λ⋅= dLdz
2222 dLdL2dzz2dz λ⋅=λ⋅λ⋅⋅=⋅⋅=
Substituting dimensionless numbers in mole balance on inert tracer:
θ∂∂
=λ∂
∂−
λ∂
∂⋅
⋅TT
2T
2 CCCLu
D
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Development of dispersion model
DLu ⋅
=Bo
LD
θ∂∂
=λ∂
∂−
λ∂
∂⋅
⋅TT
2T
2 CCCLu
D
Bodenstein number Bo (also called Per)
Ratio of rate of convective transport relative to rate of transport by diffusion
Convective transport large (Bo ∞) PFR-behaviourTransport by diffusion ((Bo 0) mixing by diffusion CSTR-behaviour
For constant conditions (temperature/pressure, etc.), Bo increases with increasing L (length of reactor)
Long reactors, B0 ∞ approaching plug flow behaviour!
moleculardiffusionfluid flow
Characteristic rate of diffusion:
Characteristic rate of convective flow: u
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Development of dispersion model
θ∂∂
=λ∂
∂−
λ∂
∂⋅ TT
2T
2 CCC1Bo
(discontinuity in Bo at λ=0 and λ=1)Closed system
λ=0 λ=1
(discontinuity in Bo at λ=0)Closed-open system
Open system
τ=t ( )⎟⎠⎞
⎜⎝⎛ −⋅−⋅=σ −Bo
2BoBoe122t22
( )Bo+⋅τ= 1t ⎟⎠⎞
⎜⎝⎛ +⋅=σ 3BoBo
32t22
τ=t ⎟⎠⎞
⎜⎝⎛ +⋅=σ 2BoBo
82t22
( )θ⋅⋅θ−−
⋅θ⋅π
⋅= 41 2
e21E
BoBo
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Dispersion modelcomparison open and closed systems
0
0.2
0.4
0.6
0.8
1
0.0001 0.01 1 100 10000
Bodenstein number
Rel
ativ
e va
rianc
e,
σ2 cl
osed
/ σ2 op
en
DLu ⋅
=Bo
Insignificant change forBo >50
For Bo>50 all systems can beconsidered to be open systems(i.e. solvable!)
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Dispersion modelopen systems
0
2
4
6
8
0 0.5 1 1.5 2
Relative time, θ=t.u/L
E(θ)
Bo = 0.01 ~ 0(mixed flow; enlargement *10)
Bo = 5
Bo = 50
Bo = 500
Bo = 50000 ~ ∞(plug flow)
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Determination of Bo from E-curve
⎟⎠⎞
⎜⎝⎛ +⋅=σ 2BoBo
82t22
( ) 20
2m
2 min5.47dtEtt ⋅=⋅⋅−=σ ∫∞
The residence time distribution in a reactor has been determinedand the E(t) data have been determined as shown below:
Time t, min 0 5 10 15 20 25 30 35E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0
Assuming an open system: Bo = 12.5
Average residence time: 15 min
Variance:
Assuming a closed system: Bo = 8.3( )⎟⎠⎞
⎜⎝⎛ −⋅−⋅=σ −Bo
2BoBoe122t22
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Conversion and dispersion model (1st order rxn – Closed system)
Damköhler number:
1st order reactions:
Solving the mole balance of species A in the reactor (closed system) (second order differential equation) :
τ⋅⋅= −1n0ACkDa
( ) ( ) 2q
22q
2
2
0A
L,A
eq1eq1
eq4CC
X1 ⋅⋅
⋅−−⋅+
⋅⋅==− Bo-Bo
Bo
BoDa41q ⋅
+=
τ⋅= kDa
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Conversion and dispersion model (1st order rxn – Closed system)
0%
20%
40%
60%
80%
100%
0.001 0.01 0.1 1 10 100 1000
Bodenstein number, Bo
Con
vers
ion
for 1
st o
rder
re
actio
n
Da = 10Da = 5
Da = 2
Da = 1
Da = 0.5
Da = 0.2Da = 0.1
The reactor (average residence time: 15 min) is to be used for aliquid phase decomposition. The rate of reaction is 1st order with respect to the reactant (k = 0.1 min-1). Da = 1.5 (Boclosed system = 8.3)
Comparison of conversions:PFR 77.8%CSTR 60.0%Segregation model 72.4%Dispersion model 73.2%
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Conversion and Dispersion model (1st order rxn)
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Tank-in-series model
C0v0
C1v1
C2v2
C3v3
V1 V2 V3
Reactor is assumed to contain n equally sized CSTRs in series (PFR can be viewed as an infinite number of CSTRs in series)
( )( )
i/t
in
1ne
!1nttE τ−−
⋅τ−
=n
22 τ=σ
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Determination of number of CSTRs in series from E-curve
( ) 20
2m
2 min5.47dtEtt ⋅=⋅⋅−=σ ∫∞
The residence time distribution in a reactor has been determinedand the E(t) data have been determined as shown below:
Time t, min 0 5 10 15 20 25 30 35E, min-1 0 0.03 0.05 0.05 0.04 0.02 0.01 0
Assuming tank in series model:n = 4.7
Average residence time: 15 min
Variance:
n
22 τ=σ
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Conversion from tank-in-series model
i
i1ii Ck
CC⋅−
=τ −
( )i
1ii0i r
XXC−−⋅
=τ −
i1i
ik11
CC
τ⋅+=
−
Performance equation CSTR:
For 1st order reaction:
Assuming tank in series model (n = 4.7; τi = 15/4.7 = 3.16 min; k.τi = 0.3) X= 72.8%
( ) nni0
i
nk1
1k11
CCX1
⎟⎠⎞
⎜⎝⎛ τ
⋅+
=τ⋅+
==−
Comparison of conversions:PFR 77.8%CSTR 60.0%Segregation model 72.4%Dispersion model 73.2%Tank-in-series model 72.8%
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Compartment models
Considering the actual reactor as a set of ideal set-ups:1. Ideal reactor(s) with dead volume2. Ideal reactor(s) with by-pass
4. Ideal reactors in series (tank-in-series model) 5. Combinations of dead volume, bypassing, ideal reactors in
series/parallel
3. Ideal reactor(s) in parallel
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Compartment modelsIdeal reactors with dead volume
Vplug
Vdead
Vreactor = Vplug + Vdead
VCSTR
Vdead Vreactor = VCSTR + Vdead
v v
vVt reactorectedexp =
( )t
Vv
CSTR
CSTReV
vtE⋅−
⋅=
E(t)
E(t)
vV
t plug=
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Compartment modelsIdeal reactors with bypassing
Vplug
v = v1 + v2
VCSTR
v
v
v1
v2v2
v1
E(t)
vvArea 1=
vvArea 2=
( )t
Vv
CSTR
1 CSTR1
eV
vtE⋅−
⋅=
1
plugv
Vt =
vvArea 1=E
(t) vvArea 2=
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Compartment modelsideal PFRs in parallel
Vplug,1v v1
v2Vplug,2
v = v1 + v2
1
1,plug
vV
t =
vvArea 1=E
(t)
vvArea 2=
2
2,plugv
Vt =
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Compartment modelsideal CSTRs in parallel (1)
VCSTR,1E
(t)
v1
( )t
Vv
2,CSTR
2t
Vv
1,CSTR
1 2,CSTR2
1,CSTR1
eV
veV
vtE⋅−⋅−
⋅+⋅=
VCSTR,2
v2
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Compartment modelsideal CSTRs in parallel (2)
0
0.2
0.4
0.6
0.8
1
0 1 2 3 4 5time
E(t)
V1 = 0.75 * Vtotalv2 = 0.1 * v
VCSTR,1
v
VCSTR,2
v2
0.01
0.1
1
0 1 2 3 4 5time
E(t)
V1 = 0.75 * Vtotalv2 = 0.1 * v
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Compartment modelscombination of models
Vd
Vp
Eθ
(or C
θ)
Vm
υ2
υ1υ
VVP
1υυ
υυ1area =
υυ2area =
Vd
Vp
Eθ
(or C
θ)
Vm
υ2
υ1υ
VVP
1υυ
υυ1area =
υυ2area =
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What is the “best” model
Dispersion model:Gives insight in the design phase to anticipate non-ideal behaviour and the consequences
Compartment model“Visualizes” the origin of possible non-ideal behaviour:
by-passingdead volumemixing zones