Chemical Reaction Engineering

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Chemical Reaction Engineering
Lecture 14
Lecturer ???
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This course focuses on models for nonideal
reactors.

• Three concepts used to describe nonideal
  reactors
• the distribution of residence times in the system
  (two bounds (i.e., the segregation and maximum
  mixedness) were determined)
• the quality of mixing
• the model used to describe the system

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Models for nonideal reactors

• One-parameter model
• tanks-in-series model
• dispersion model
• Two-parameter model
• reactor with bypassing and dead volume
• Combination and/or modification of ideal reactors
• The RTD is used to evaluate the parameter(s) in
  the model.

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Models for real reactors

• mathematically tractable
• realistic describe the characteristics of the
  reactor
• no more than 2 adjustable parameters
• the less the better
• Give me four adjustable parameters and I can fit
  an elephant give me five and I can include his
  tail!

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One-parameter model

• One-parameter?
• the reactor dead volume, VD
• the fraction of fluid bypassing the reactor, f
• the number of tanks for the tanks-in-series
  model, n
• the dispersion coefficient for the dispersion
  model, Da
• Modeling of a tubular reactor
• real the velocity profile is not flat there is
  axial mixing
• tanks-in-series model the dispersion model

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Tanks-in-series model
If there are three tanks
Pulse injection
For a single CSTR, a material balance (V V1
V2 V3 ? ? 1 ?2 ? 3)
on the first reactor gives
on the second reactor gives
on the third reactor gives
The fraction of material leaving the system of
three reactors that has been in the system
between time t and t ?t is
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n tanks-in-series
n 10
E(?)
n 4
n 2
dimensionless
?
How many tanks are needed for modeling?
P 875
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The number of tanks in series is determined by
the tracer experiment
1
1
The number of tanks necessary to model the real
reactor as n ideal tanks in series.
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Dispersion model
It is used to describe nonideal tubular reactors.
In addition to transport by bulk flow, UCAc,
every component is transported through any cross
section of the reactor at a rate,
, resulting from molecular and convective
diffusion.
The molar flow rate of tracer is
Mole balance on the inert tracer
This compensates not only axial mixing but also
radial mixing and other nonflat velocity profiles
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Dispersion in a tubular reactor with laminar flow
For a laminar flow reactor, the axial velocity
varies in the radial direction
the Hagen-Poiseuille equation
For a laminar flow reactor, the RTD function is
IF there is axial and radial dispersion, what
will the RTD be?
The P.D.E describing the concentration at a
particular r, x, and t
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Similarly, the P.D.E describing the variation of
the average concentration with t and x is
The value of Da for turbulent flow is shown in
Fig. 14-6 and for PBR is shown in Fig. 14-7.
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Determination of Da from the RTD experimental data
dimensionless
We want to solve this P.D.E. with certain B.C.s
characteristic length
Per the reactor Peclet number
U/? the average interstitial velocity
Pef the fluid Peclet number
(in a packed bed)
(in an empty tube)
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Boundary conditions
(1) closed-closed vessel B.C.
at x 0
the Danckwerts boundary conditions
in dimensionless form
at ? 0
at x L
at ? 1
at t 0
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B.C.
at ? 0
at ? 1
at t 0
A pulse injection
We can theoretically solve the P.D.E..
How to obtain Per for the equation?
The mean residence time tm ?
Bischoff and Levenspiel, 1963
From experiments
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(1) open-open vessel B.C.
at x 0
at x L
at t 0
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together with the B.C.s in the previous slide, we
have
the dimensionless effluent tracer concentration
The mean residence time
We obtain Per and then Da.
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When there are dispersion and reaction in a
tubular reactor
U
Ac
z L
z?z
z
z 0
At steady-state, the mass balance on A over the
volume element ?VAc?z is
rate in - rate out rate of generation
rate of accumulation
dimensionless
where Da is the Damköhler number for convection
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B.C.
(use the Danckwerts boundary conditions for
close-close system)
at ? 0
at ? 1
The O.D.E. is solved
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The comparison between the dispersion and
tanks-in-series models
The first-order reaction is
carried out in a 10-cm-diameter tubular reactor
6.36 m in length. The specific reaction rate is
0.25 min-1. The results of a tracer test carried
out on this reactor are (the effluent tracer
concentration as a function of time)
Calculate conversion using (a) the closed vessel
dispersion model (b) PFR (c) the
tanks-in-series model (d) a single CSTR
?2
C(t)
E(t)
tm
tm ?
close-close vessel dispersion model
cf
open-open vessel dispersion model
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(a) the closed vessel dispersion model
We have Per from
Per 7.5
Da 1.29
We need Da
We have X 0.68
(b) ideal PFR
We have X 0.725
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(c) the tanks-in-series model
The number of tanks necessary to model
The conversion for n tanks-in-series is
We have X 0.677
(d) a single CSTR
n 1
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Two-parameter models

• A real reactor might be modeled by one of two
  different combinations of ideal reactors.
• A tracer experiment is used to evaluate the model
  parameters.

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A real CSTR is modeled as an ideal CSTR with a
bypassing and a dead space
CA0
vs (1-?)v0
v0
Bypass
Vd (1-?)V
vb ?v0
Vs ?V
deadzone
CA0
CAs
CA
v0
A first-order reaction
Mole balance on species A
Mole balance on species A in the reactor
Assume
We want to determine the values of these two
parameters (using RTD).
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For an inject of tracer, T, as a positive-step
input, the unsteady-state mass balance on T is
Positive injection (B.C.)
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Example CSTR with dead space and bypassing
The elementary reaction
is carried out in a CSTR with bypassing and
a stagnant region in the reactor. The measured
reactor volume is 1 m3 and the flow rate is 0.1
m3/min to the reactor. The reaction rate constant
is 0.28 m3/kmol.min. The feed is equimolar in A
and B with an entering concentration of A equal
to 2.0 kmol/m3. The tracer output for the reactor
is in the table. Calculate the conversion
expected in this reactor.
CT0 2000
? 0.7 ? 0.2
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Calculation of conversion use this 2-parameter
model
Mole balance on reactor volume
Mole balance on species A
Combining
From
, we obtain the values of vs, ?s
cf, if an ideal CSTR, X 0.66
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A real CSTR modeled as two-ideal-CSTR-interchange
v0
v0
v0
v1
CSTR 2
CSTR 1
CA2
CA1
v1
V
V2
V1
CA1
v0
Mole balance on reactor 1
Mole balance on reactor 2
For a first-order reaction
These equations are solved together
We want to determine the values of ? and ?
(using RTD)
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A tracer pulse injection at t 0 is used to
determine the value of the two parameters
Mole balance on reactor 1
Mole balance on reactor 2
We also have
They are used to determine the values of ? and
?! Numerical method might be used (page 9023)
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Other models of nonideal reactors using CSTRs
and PFRs

• CSTR
• almost all agitated tank reactors have a
  well-mixed zone in the vicinity of the agitator
  and can be represented by a CSTR
• Dead volume
• Bypassing
• PFRs
• channeling
• A number of combinations can be obtained.