Reactions and Separations
Modeling
Eugeny Kenig
Univ. of Paderborn
Panos Seferlis
Aristotle Univ. of Thessaloniki
R
Effective modeling of reactive absorption
enhances system design, improves
experiment planning for parameter
estimation, and facilitates process
operation and control decisions.
eactive absorption — i.e., the absorption of gases in
liquid solutions accompanied by chemical reactions
— is an important industrial operation for the production of basic chemicals (e.g., sulfuric or nitric acid) and for the
removal of harmful substances (e.g., H2S) from gas streams.
In recent decades, this process has become especially important for the purification of gases to high purities. Unlike physical absorption (without reactions), reactive absorption is able
to provide high throughput at moderate partial pressures and
without requiring large amounts of solvent.
The advantages of combining chemical reactions with
absorption are realized only in the region of low gas-phase
concentrations due to the liquid-load limitations imposed
by the reaction stoichiometry or equilibrium. Other factors
that may limit the efficiency of reactive absorption are the
heat liberated by exothermal reactions and the difficulty of
solvent regeneration. Most reactive absorption processes
are steady-state operations involving reactions in the liquid
phase, although some applications involve both liquidphase and gas-phase reactions.
Reactive absorption is a complex rate-controlled process
that occurs far from thermodynamic equilibrium. Therefore,
the equilibrium concept is often insufficient to describe it,
and instead, accurate and reliable models involving the
process kinetics (rate-based models) are required. The effectiveness of online model-based applications, such as process
control and optimization, depends strongly on the quality of
the available model predictions.
Equilibrium stage model
Modeling and design of reactive absorption processes
are usually based on the equilibrium stage model, which
assumes that each gas stream leaving a tray or a packing
segment (stage) is in thermodynamic equilibrium with the
corresponding liquid stream leaving the same tray or segment. For reactive absorption, the chemical reaction must
also be taken into account.
With very fast reactions, the reactive separation
process can be satisfactorily described assuming reaction
equilibrium. A proper modeling approach is based on the
nonreactive equilibrium stage model, which is extended
by simultaneously considering the chemical equilibrium
relationship and the tray or stage efficiency. If the reaction rate is slower than the mass-transfer rate, the influence of the reaction kinetics increases and becomes a
dominating factor. This tendency is taken into account by
integrating the reaction kinetics into the mass and energy
balances. This approach is widely used today.
In real reactive absorption processes, thermodynamic
equilibrium is seldom reached. Therefore, correlation
parameters such as tray efficiencies or height equivalent to
a theoretical stage (HETS) values are introduced to adjust
the equilibrium-based theoretical description to real column
conditions. However, reactive absorption occurs in multicomponent mixtures, for which this simplified concept
often fails (1).
The acceleration of mass transfer due to chemical reacCEP
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65
Reactions and Separations
Two-Film Model
Stage 1
A similar equation can be written
for the gas phase. Thus, the gas-liquid mass transfer is modeled as a
yiI
yiB
combination of the film model
Stage S
I
B
presentation and the Maxwellxi
xi
Stefan diffusion description. In this
Liquid Phase
Gas Phase
stage model, equilibrium is
δy
δx
Stage N
assumed at the interface only.
The film thickness represents a
Packed
Tray
Column
Column
model parameter that can be estimated using the mass-transfer
■ Figure 1. The column discretization and the two-film model for the stage description.
coefficient correlations governing
the mass transport dependence on
physical
properties
and
process
hydrodynamics. These
tions in the interfacial region is often accounted for via socorrelations
are
usually
obtained
experimentally and are
called enhancement factors (2, 3). These are either obtained
available
from
the
literature.
Another
important parameter
by fitting experimental results or derived theoretically
of
the
film
model
is
the
specific
contact
(interfacial) area,
based on simplified model assumptions. However, it is not
which
is
also
estimated
from
experimental
data.
possible to derive the enhancement factors properly from
Balance
equations.
The
component
mass-balance
equadata on binary experiments, and a theoretical description of
tions
of
the
rate-based
models
are
written
separately
for
reversible, parallel or consecutive reactions is based on
each phase, and, due to the presence of chemical reactions,
rough simplifications.
include the reaction source terms (6). Considering the
process dynamics, these equations become:
Rate-based stage model
B
A more physically consistent way to describe a column
∂mLi
∂
B I
=−
LxiB + N Li
a + rLiBφ L Ac i = 1 … n
(2)
stage is the rate-based approach (4). This approach directly
∂t
∂l
considers actual rates of multicomponent mass and heat
B
transfer and chemical reactions.
∂mGi
∂
B I
=
GyiB − N Gi
a − rGiB φG Ac i = 1 … n
(3)
Mass transfer at the gas-liquid interface can be described
∂t
∂l
using different theoretical concepts (1, 5). Usually, the twoEquations 2 and 3 are valid for continuous systems
film model or the penetration/surface-renewal model are
(packed
columns). For discrete systems (tray columns),
used, and the model parameters are estimated via empirical
the
differential
terms on the right-hand side become finite
correlations. The advantage of the two-film model is that
differences
and
the balances are reduced to ordinary difthere is a broad spectrum of correlations available in the litferential
equations
(5).
erature for all types of column internals.
If
chemical
reactions
take place in the liquid phase only
In the two-film model (Figure 1), it is assumed that the
(which
is
true
for
most
reactive
absorption processes), the
resistance to mass transfer is concentrated entirely in thin
reaction
term
in
Eq.
3
is
omitted.
films adjacent to the phase interface, and that mass transfer
Equations 2 and 3 are supplemented by the summation
occurs within these films by steady-state molecular diffuequation
for the liquid and gas bulk mole fractions:
sion alone. Outside the films, in the fluid bulk phases, the
n
n
level of mixing is assumed to be sufficiently high so that
B
x
=
1
yiB = 1
(4)
∑
∑
there is no composition gradient — i.e., one-dimensional
i
i
i
=
1
=
1
diffusion transport normal to the interface takes place.
The volumetric liquid hold-up, φL, depends on the gas
Multicomponent diffusion in the films can be described
by the Maxwell-Stefan equations that relate components’
and liquid flowrates in the column and is calculated by
diffusion fluxes to their chemical potential gradients. In a
empirical correlations. To determine axial temperature progeneralized form, the Maxwell-Stefan equations can be
files, differential energy balances that include the product
used for the description of real gases and liquids (1). For
of the liquid molar hold-up and the specific enthalpy as
the liquid phase:
energy capacity are formulated. The energy balances for
n x N −x N
continuous systems are:
xi d μi
i Lj
j Li
i =1 … n
di =
=∑
(1)
∂U LB
∂
RT dz
cLt Dij
j =1
=−
LhLB + QLB a I Ac - qLloss
(5)
∂t
∂l
Film
Interface
Film
( ) (
)
( ) (
)
( ) (
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January 2009
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)
∂UGB ∂
=
GhGB − QGB a I Ac - qGloss
∂t
∂l
(
) (
)
(6)
Mass transfer and reaction coupling in the fluid film.
The component fluxes, NiB, in Eqs. 2 and 3 are determined
based on the mass transport in the film region. The description of the film phenomena is usually reduced to a steadystate problem (6). The key assumptions of the film model
result in one-dimensional mass transport normal to the
interface, and the differential component-balance equations
including simultaneous mass transfer and reaction in the
film are:
dN Li
− rLi = 0
(7)
i =1 … n
dz
Equation 7 is generally valid for both liquid and gas
phases if reactions take place in both phases. It represents
the differential mass balance for the film region including
the source term due to the reaction. The component fluxes
are expressed in terms of concentrations using Eq. 1,
whereas the source terms result from the reaction kinetics
description and usually represent nonlinear dependencies
on the mixture composition and temperature of the corresponding phase.
The boundary conditions for Eq. 7 are those typical for
the film model and specify the values of the mixture composition at both film boundaries. They are applicable to
both phases; for the liquid phase they are:
xi ( z = 0 ) = xiI
xi ( z = δ L ) = xiB
i =1 … n
(8)
Combining Eqs. 7 and 8 results in a set of vector-type
boundary value problems that permits the component concentration profiles to be obtained as functions of the film
coordinate. These concentration profiles allow the component fluxes to be determined. Thus, the boundary value
problems describing the film phenomena have to be solved
in conjunction with all other model equations.
An analytical solution of this boundary value problem in
a closed matrix form can be obtained (5) if some further
assumptions concerning the linearization of the diffusion and
reaction terms are made. On the other hand, the boundary
values need to be determined from the total system of equations describing the process. The bulk values in both phases
are found from the balance relationships, Eqs. 2 and 3. The
interfacial liquid-phase concentrations are related to the relevant gas-phase concentrations, yiI, by the thermodynamic
equilibrium relationships and by the continuity condition for
the molar fluxes at the interface (1, 5).
Due to the chemical conversion in the liquid film, the
molar fluxes at the interface and at the boundary between
the film and the bulk of the phase differ. The system of
Nomenclature
aI
= specific contact area, m2/m3
Ac
= cross-sectional column area, m2
c
= molar concentration, kmol/m3
d
= mass-transport driving force, m-1
D
= Maxwell-Stefan diffusivity, m2/s
F
= Faraday’s constant = 9.65×104 C/mol
G
= gas-phase stream molar flowrate, kmol/s
h
= molar enthalpy, kJ/kmol
l
= axial coordinate, m
L
= liquid-phase stream molar flowrate, kmol/s
m
= length-specific molar hold-up, kmol/m
n
= number of components, dimensionless
N
= molar flux, kmol/(m2-s)
P
= pressure, Pa
= length-specific heat loss, kJ/(m-s)
qloss
Q
= heat flux, kW/m2
r
= reaction rate, kmol/(m3-s)
R
= universal gas constant = 8.3144 kJ/(kmol-K)
t
= time, s
T
= temperature, K
U
= length-specific energy hold-up, kJ/m
x
= mole fraction of component in liquid phase
y
= mole fraction of component in gas phase
z
= normal coordinate, m
zi
= ionic charge, dimensionless
Greek Letters
δ
= film thickness, m
φ
= volumetric hold-up, m3/m3
ϕ
= electrical potential, V
η
= film coordinate, dimensionless
μ
= chemical potential, kJ/kmol
Subscripts
G
= gas phase
i, j
= component or reaction index
L
= liquid phase
t
= mixture property
Superscripts
B
= bulk phase
I
= interface
equations is completed with the continuity equations for
the mass and energy fluxes at the phase interface.
Handling electrolyte systems
Reactive absorption processes occur mostly in aqueous systems, with both molecular and electrolyte species.
These systems demonstrate substantially nonideal behavior. Two basic models are employed for the thermodynamic description of electrolyte-containing mixtures, the
Electrolyte NRTL model and the Pitzer model. The
Electrolyte NRTL model is able to estimate the activity
coefficients for both ionic and molecular species in aqueous and mixed-solvent electrolyte systems based on the
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67
Reactions and Separations
Rate-Based/Staged Model
Rate-Based/OCFE Model
binary pair parameters (7). The Pitzer model (8, 9) can
be used for aqueous electrolyte systems up to 6 mol/kg
ionic strength. The models require different parameters,
such as pure-component dielectric constants of nonaqueous solvents, Born radii of ionic species, and binary and
(for the Pitzer model) ternary interaction parameters.
The electrolyte solution chemistry involves a variety
of chemical reactions in the liquid phase. These reactions
occur very rapidly, so chemical equilibrium conditions
are often assumed. Therefore, chemical equilibrium calculations are of special importance for electrolyte systems. Concentration or activity-based reaction equilibrium constants as functions of temperature can be found in
the literature.
The presence of electrolyte species makes calculation
of relevant diffusion coefficients crucial. The effective
diffusion coefficients for electrolyte components can be
obtained from the Nernst-Hartley equation for dilute solutions and from the Gordon equation for higher electrolyte
concentrations (10). The driving force due to electrical
potential difference also needs to be taken into account
(1), which is done by introducing the electrical potential
gradient into the generalized driving force di:
x 1 d μi
F 1 dϕ
di = i
+ xi zi
i =1 … n
(9)
RT δ L dη
RT δ L dη
In dilute electrolyte systems, the diffusional interactions
can usually be neglected, and the generalized MaxwellStefan equations are reduced to the Nernst-Planck equations (1). In all cases, the electroneutrality condition must
be met at each point of the liquid phase:
n
∑ xi zi = 0
(10)
i =1
Reducing the size of the model
The models described in the previous sections, when
applied to a tray or packed absorption column (Figure 1),
will eventually result in a large set of complex and highly
coupled nonlinear equations whose solution may become
quite tedious. Several online applications, such as operator
training, and process control and optimization, require the
solution within a reliably defined timeframe. This can be
achieved by formulating the rate-based model in a compact
form, for example by using the orthogonal collocation on
finite elements (OCFE) model. The OCFE formulation
reduces the overall size of the reactive absorption process
model (in terms of the total number of equations) while
preserving the model’s structure and accuracy.
The basic concept behind this efficient approximating
technique (mathematical details can be found in Refs.
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Rate-Based
Interface
Gas
Bulk
Gas
Film
Liquid Liquid
Film
Bulk
Materials Streams
Energy Streams
■ Figure 2. Schematic of the OCFE and the full column model
formulation for a configuration with multiple feed and recycle
streams (blue lines) and heat-transfer streams (orange lines).
11–15) is that the stagewise domain within a column section,
defined as the part of the column between two consecutive
streams entering or leaving the column, is considered to be a
continuous analog. Subsequently, molar and enthalpy
flowrate profiles within such a column section are treated as
continuous functions of the longitudinal coordinate in the
column. The approximating power of the OCFE model lies
in the fact that the behavior within a specific column section
can be accurately approximated by fewer balance equations
(applied only at selected points — namely the collocation
points) than a traditional stage-by-stage formulation. The
greatest advantage is that the form of the balance equations
at the selected collocation points is entirely preserved; hence,
the rate-based model formulation as previously described is
fully maintained.
An important feature of the OCFE approximation model
shown schematically in Figure 2 is that stages at which
feed or side-draw material streams are attached are treated
as discrete stages, thus isolating abrupt and sharp changes
in the concentration and temperature profiles in the column
approximating scheme. The extent to which the model
order is reduced is dictated by the shape of the approximated concentration and temperature profiles along a given
column section. Consequently, column sections with steep
temperature and concentration profiles generally require a
denser pattern of collocation points than column regions
with relatively flat profiles.
In design optimization, the decision variables involve
the overall column configuration (e.g., the number of
stages in each column section, the locations of material
and energy feed and side-draw streams, etc.) and the column operating conditions (e.g., solvent flowrate, stage
cooling/heating, recycle scheme, etc.). The OCFE model
formulation transforms the otherwise discrete decision
problem (e.g., calculation of the optimal number of discrete stages in the column) to a continuous design optimization problem, where the size of the column sections
becomes a continuous degree of freedom, thus facilitating
solution of the problem. Rounding up to the next-highest
integer value leads to the optimal number of stages. This
is of particular interest in synthesis problems where the
optimal interconnectivity of multiple sequential separations is sought (16) as the overall number of required integer decision variables is significantly reduced.
Enhancing the predictive power
of a reactive absorption model
Liquid- and gas-side film thicknesses depend strongly
on the flow pattern in the column, the type of column
internals, and gas and liquid physical properties, such as
surface tension, diffusivity and viscosity. Similarly, specific contact area, which, in general, differs from the
geometric surface area of the column internals, strongly
depends on the flow pattern and physical properties of
the fluids. In simple packed beds (e.g., packed with
rings or saddles), it is possible that, at moderate liquid
loads, the packing surface is not completely covered, so
the mass transfer occurs through a smaller area than the
geometric surface area. The opposite situation, in which
the phase interface is larger than the geometric surface
area, is also possible.
Values of film thicknesses, specific contact area, and
mass transfer coefficients are most often determined by
empirical correlations, which allow scale-up to different
operating states. The liquid- and gas-phase mass-transfer
coefficients are usually related to Sherwood number (Sh);
the latter is represented as a function of Reynolds number
(Re), Schmidt number (Sc) and other dimensionless
process characteristics (17, 18). It is important that the
correlations are applied to conditions within their range
of validity.
Additional parameters related to the empirical correlations are the liquid hold-up on the relevant column internals and the pressure drop caused by the flow resistance in
the column. The liquid hold-up is necessary both for the
liquid-phase reaction description and for the estimation of
the gas-phase hold-up in the case of gas-phase reactions.
The pressure drop can influence primarily the phase equilibrium and hold-up. These parameters also depend on the
operating conditions, column internals type, and physical
properties. In some cases, hold-up and pressure drop are
coupled and cannot be calculated explicitly, so they are
determined iteratively (18). The range of application for
each correlation depends on the actual column loads, as
both hold-up and pressure drop strongly depend on hydrodynamic interactions.
Selecting the proper correlation is mostly a question of
column operating regime and user experience. Masstransfer correlations must be compared and validated with
experimental data generated through experiments in the
reactive absorption column. During the experimental runs
(planned using statistical design of experiments techniques (19)), one or more input process variables to the
reactive absorption column are changed deliberately in
order to record the effect these changes have on the output process variables (responses).
The accuracy of model parameter estimates is
improved by centering the joint confidence region (JCR)
of the model parameter estimates on their true values, and
their precision is subsequently increased by reducing the
volume of the multi-dimensional JCR. Statistical correlation of model parameter estimates provides a measure of
the degree to which two or more parameters co-vary
under certain experimental conditions, and is mainly characterized by the orientation of the JCR. In general, a
small (with respect to volume) and spherical JCR centered at the true values of the model parameters is the
most desirable situation.
To satisfy (all or a subset of) the aforementioned objectives, the inputs to the process system should be selected in
such a way that the system response becomes sensitive to
its major model parameters. Reactive absorption process
models can then be utilized in the calculation of optimal
experimental runs. Several techniques are available (19)
based on a variety of objective functions that are intended
to improve the quality of parameter estimates. Key decisions in designing new steady-state and dynamic experiments involve the choice and characteristics of the input
variables to the system (usually these are the manipulated
variables of the control system), the selection of a suitable
set of measured variables (e.g., stages with temperature
sensors or composition analyzers), and the sampling rate.
The duration of an experimental run depends on the system
dynamics and the time it takes to reach the final steady
state (which is proportional to the dominant system time
constant). Combining all these elements within a unified
optimal experimental-design framework that utilizes the
rate-based model provides a well-defined context for
improving the predictive power of the model for reactive
absorption columns.
NOx absorption example
Absorption of nitrous gases is an important operation in
the chemical process industries, used mainly in the production of nitric acid and in the purification of exhaust gas
streams. It is a highly complex process due to the interaction of several components and chemical reactions in both
the liquid and gas phases. This example demonstrates the
ability of the rate-based modeling approach to accurately
predict the steady-state and dynamic column behavior of
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69
Reactions and Separations
Table 1. Nitric acid is produced
through a series of reactions.
Table 2. Design parameters for
the industrial sieve tray column.
Gas-Phase
RR1
2 NO + O2 → 2 NO2
RR2
2 NO2 ↔ N2O4
RR3
3 NO2 + H2O ↔ 2 HNO3 + NO
RR4
NO + NO2 ↔ N2O3
RR5
NO + NO2 + H2O ↔ 2 HNO2
Column Diameter
Number of Trays
Plate Spacing
Weir Height
Weir Length
Flow Path
Number of Holes
Hole Diameter
Distance between
Holes (Pitch)
Hole Diameter
industrial and experimental columns.
Chemical reactions are important in NOx absorption
because they enhance the absorption of components that
are otherwise insoluble in water (e.g., NO) through their
chemical transformation to more soluble components (e.g.,
NO2). Nitric acid is produced through the complex reaction
mechanism detailed in Table 1 (20–22). The oxidation of
NO to NO2 (RR1) is the slowest reaction, and thus is the
limiting step (21). Diffusion coefficients in the gas phase
were estimated using the Chapman-Enskog-Wilke-Lee
model (7) and in the liquid phase using the method proposed in Ref. 23. The liquid-phase activity coefficients
were calculated using the NRTL activity model, and all
other necessary thermodynamic calculations were based on
the Soave-Redlich-Kwong equation of state.
Validation for a packed column. The model was implemented using the commercial simulator Aspen Custom
Modeler (www.aspentech.com), which utilizes Aspen
Properties to calculate the required physical properties.
Validation was performed by comparing simulation
results with experimental data for three pilot-scale
columns connected countercurrently (22). Simulations for
two of these columns are presented here.
The simulated units have a simple configuration, with
one liquid inlet stream at the top and one gas inlet
stream at the bottom. Both columns have a diameter of
0.254 m and are filled with a random packing (16-mm
steel Pall rings) to a packing height of 6 m. In Column
Packing Height, m
5
Column 1
NOx, Simulation
NOx, Experiment
4
3
Column 2
NOx, Simulation
NOx, Experiment
2
13 mm
2.2 mm
1, a gas stream containing 20 mol% nitrogen oxides is
absorbed by an aqueous solution containing 5 mol%
nitric acid. The inlet gas stream of Column 2 contains
10.2 mol% nitrogen oxides and is treated by an aqueous
solution containing 2.65 mol% nitric acid.
Figure 3 compares the simulated axial profiles of the
total NOx concentration in the gas phase for Columns 1
and 2 with reported experimental measurements (shown
with 5% error bars). For Column 1, the calculated total
NOx concentration at the top of the column shows good
agreement with the experiments, as the maximum deviation
is less than 5%. Figure 4 presents the simulated axial profiles of the liquid-phase temperature and measured values
for both columns. The liquid temperature profiles reveal a
maximum in the lower section of each column, which is
typical for NOx absorption processes (24). The absolute
deviation between the simulated and measured liquid temperatures is 4.5°C for Column 1 and 3.8°C for Column 2,
which can be attributed to heat losses through the column
wall because the experimental column was not insulated.
Validation for a tray column. The design details of a
tray column are given in Table 2. A gas stream containing
0.77 mol% nitrogen oxides is treated with an aqueous solution containing 0.68 mol% nitric acid. To maintain a fairly
constant temperature profile, seven trays in the lower part
5
Packing Height, m
Liquid-Phase
RR6
N2O4 + H2O → HNO2 + HNO3
RR7
3 HNO2 → HNO3 + H2O +2 NO
RR8
N2O3 + H2O → 2 HNO2
RR9
2 NO2 + H2O → HNO2 + HNO3
3.8 m
20
0.9 m
0.25 m
3.015 m
2.3 m
54,000
2.2 mm
Column 1
Simulation
Experiment
4
Column 2
Simulation
Experiment
3
2
1
1
0
0
0.04
0.06
0.08
0.10
0.12
Molar Fraction
0.14
0.18
0.22
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January 2009
25
30
35
40
45
50
Temperature, ºC
■ Figure 3. An axial profile of the total gas-phase NOx
concentration of Columns 1 and 2 and their measured values.
70
20
CEP
■ Figure 4. An axial profile of the liquid temperature of
Columns 1 and 2 and their measured values.
55
18
Column Height, m
Table 3. Stream data for the
NOx absorption column shown in Figure 6.
NOx, Simulation
HNO3, Simulation
NOx, Experiment
HNO3, Experiment
16
14
12
10
8
6
4
2
0
0.00
0.01
0.1
1
Gas Inlet Stream (Bottom)
Liquid Inlet Stream (Top)
NO
NO2
N2O4
O2
N2
T
P
H2O
T
21.83 mol/s
58.08 mol/s
20.11 mol/s
82.74 mol/s
1016.44 mol/s
332 K
5.6 bar
Molar Fraction
■ Figure 5. An axial profile of the total NOx concentration in
the gas phase, the nitric acid concentration in the liquid phase,
and the measured values.
of the column are equipped with cooling coils fed with
152 m3/h water at an average temperature of 23.6°C.
The resulting rate-based model was solved using
gPROMS, an integrated process modeling environment
(www.psenterprise.com). Figure 5 shows the calculated
axial profiles of the total NOx concentration in the gas
phase and the nitric acid concentration in the liquid phase,
as well as the measured inlet and outlet values (with a
measurement error of 15%). The experimental and simulated values of the total NOx reveal a maximum deviation of
8.5%, and the deviation of the simulated HNO3 concentration from the experimental value is within 10%. Thus, the
agreement is good, as all deviations lie within the measurement error margins. The simulated outlet temperature of
the cooling water is 24.8°C. This agrees very well with the
measured value of 24.9°C (deviation <1%). From the
results of both pilot plant and industrial applications, it can
be concluded that the suggested model demonstrates good
accuracy for the highly complex NOx absorption process.
Improving column operation through design optimization and control. A similar example involves an industrial
column that comprises 44 trays with an internal diameter of
3.6 m. Tray spacing is 0.9 m, except at the bottom of the
column. The oxidation reaction (RR1) takes place mainly
over the bottom six trays, where larger spacing is used for
higher gas-phase hold-up to enhance the NO oxidation.
Three independent water-cooling systems control the column temperature. In general, low column temperatures
favor both the NO oxidation reaction and the absorption of
NO2 in water. Empirical correlations were used for pressure drop, liquid-phase hold-up, film thickness, and stage
interfacial area for sieve plate calculations (15).
In the column configuration shown in Figure 6, a gas
stream with a high concentration of NOx enters the bottom
of the reactive absorption column, a liquid water stream
enters at the top, and a weak solution of nitric acid enters
as a side feed stream. The bottom liquid stream, an aqueous nitric acid solution, is partially recycled in the column
4.55 mol/s
293 K
Side Feed Stream
H2O
HNO3
NO2
T
34.62 mol/s
6.02 mol/s
0.41 mol/s
306 K
Recycle Stream Flowrate 4.55 mol/s
for better control of the HNO3 concentration in the product
stream, which is subject to quality constraints. Similarly,
the concentration of NOx components in the gas stream at
the top of the column is subject to composition constraints
due to environmental regulations. Inlet stream data for the
column are provided in Table 3.
Applying the OCFE model formulation, the absorption
column is partitioned into three sections with boundaries
defined by the location of the side feed and draw streams
attached to the column (15). Each column section is further
partitioned into a number of finite elements. However, the
six oxidation stages at the bottom of the column are treated
as discrete stages. The resulting model is only one-third the
size of the respective tray-by-tray model.
Control of the total NOx composition in the fluegas
stream is achieved by adjusting the temperature profile in
the column. Three PID controllers use temperature measurements at selected stages, which act as inferential variables of NOx
composition at
H2 O
the effluent gas
AC
Fluegas
stream, to
NOx Specifications
manipulate the
cooling water
flowrate in the
Cooling System
stage coils.
Another PID
H2O, HNO3
TC
controller
directly controls
TC
the NOx composition at the
fluegas stream
TC
through the
manipulation of
LC
the solvent
(water) flowrate
Air, NOx
at the top of the
column.
Dynamic simu- ■ Figure 6. A schematic representation
of a NO absorption column with its
lation runs using control xsystem.
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71
Reactions and Separations
NO2 Composition, Molar Fraction
0.015
Table 4. Column design configuration
at different operating conditions.
0.0145
< 0.0125
> 0.32
< 0.0125
> 0.32
19
18
10
27
Liquid Feed Stream Flowrates
Solvent (Top), mol/s
Side Stream, mol/s
25
44.93
25
42.67
Cooling Water Flowrates
Side Stream Feed Stage, mol/s
Middle Section, mol/s
Bottom Section, mol/s
10
247.47
100
10
247.09
100
Outlet Stream Concentrations
NOx in Gas Outlet*
HNO3 in Liquid Product*
Total Cost, $/yr
0.0117
0.3222
72,634
0.0115
0.3216
69,074
0.0135
0.013
Column Specifications
NOx in Gas Outlet*
HNO3 in Liquid Product*
0.012
0.0125
0
0.5
1
1.5
Time, h
2.5
2
3
■ Figure 7. Fluegas composition closed-loop response to a step
change in the NOx composition of the inlet gas stream.
gPROMS, shown in Figure 7, demonstrate the ability of
the control system to successfully compensate for the
effects of disturbances in the NOx concentration of the
inlet gas stream.
A viable alternative for improving the operation of the
NOx absorption column (25) involves determining a suitable column configuration at different operating modes.
Absorption columns are often equipped with piping to supply the feed streams at multiple locations. Design optimization determines the best location and distribution of the side
feed and recycle streams at different operating conditions
(e.g., variable feed gas composition and/or variable fluegas
and product stream specifications). Table 4 shows that a
4.9% improvement in operating costs can be achieved
through a suitable column configuration (calculation of the
optimal position of the side feed streams to the column),
solvent flowrates, and cooling policy in the entire column.
Experimental design for improved parameter estimation.
An experimental design was performed for the NOx
absorption column described to estimate the kinetic parameters in reactions RR1 and RR3 (Table 1). Since the rates
of reactions RR1 and RR3 are strongly affected by temperature, the cooling water flowrates at the column oxidation
1.1
Kinetic Parameter for RR3
Case A
(Operation
Optimization)
Case B
(Operation
and Column
Configuration
Optimization)
0.014
Arbitrary Design
Optimal Design
1.05
1
0.95
0.9
0.85
0.986 0.988 0.99 0.992 0.994 0.996 0.998
1
■ Figure 8. Joint confidence region for the kinetic parameter
estimates for RR1 and RR3.
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January 2009
CEP
*molar fraction
stages (the six stages from column bottom) were selected
as input variables in order to obtain the necessary information for parameter estimation. Temperature and the composition of the liquid and gas bulk phases at the bottom oxidation stages were selected as feasible measured variables.
The optimal steady-state experimental conditions (as
dictated by the cooling water flowrates) were calculated by
minimizing the volume of the joint confidence region for
the kinetic parameter estimates (19). The experimental
design capabilities of gPROMS were used to solve the
resulting optimization problem using the rate-based/OCFE
model of the absorption column.
Figure 8 shows the surface reduction of the JCR for the
two kinetic parameters. The dashed contour corresponds to
the JCR of the kinetic parameter estimates for two arbitrarily selected steady-state experiments. The solid contour
represents the JCR for the kinetic parameters achieved
with the addition of an optimally designed experiment. In
the optimally designed experimental run, the bottom three
stages are operated at higher temperatures and stages four
through six at lower temperatures. In both cases, model
parameters have been scaled to unity at the nominal operating point to enable good numerical conditioning.
1.002 1.004
Kinetic Parameter for RR1
72
Number of Stages
Top Section
Middle Section
Final thoughts
Detailed and accurate models are essential for numerous
online and offline reactive absorption column applications,
including control performance evaluation, operator training, design optimization, and steady-state operation optimization. Rate-based models using the two-film theory and
adapted to the particular features of reactive absorption
units provide a reliable modeling framework. OCFE techniques allow for significant reduction of model size (in
terms of the number of modeling equations) without compromising the accuracy of the model predictions. Hence,
the predictive power of rate-based models has been efficiently combined with the systematic and adaptable
approximating properties of the OCFE technique.
Examples given in this article demonstrate the accuracy
of the models and the successful implementation in design
and control applications.
CEP
Acknowledgements
The experimental data for industrial-scale one-pass sieve tray column in the
NOx absorption example were provided in the context of research funded
by the European Commission, Project OPT-ABSO (G1RD-CT-2001-00649).
EUGENY KENIG, PhD, is professor and chair of the Fluid Process Engineering in
the Mechanical Engineering Dept. at the Univ. of Paderborn (Pohlweg 55,
33098 Paderborn, Germany; Phone: +49 5251 60 2408; E-mail:
eugeny.kenig@upb.de). He worked for the Russian Academy of Sciences, and
later became a postdoctoral fellow at the Univ. of Dortmund. He also worked
for BASF AG in Ludwigshafen, Germany, before returning to Dortmund as a
professor. His main expertise is in the mathematical modeling and simulation
of complex process systems (reactive and hybrid separations,
microseparations, design and optimization of column internals). He has
authored more than 200 publications and has participated in the research
and coordination activities of several large European and joint national
projects. A member of DECHEMA, Kenig holds an MS in applied mathematics
from the Moscow Univ. of Oil and Gas, a PhD in chemical engineering from
the Russian Academy of Sciences, and a DSci and a Venia Legendi from the
Univ. of Dortmund.
PANOS SEFERLIS, PhD, is an assistant professor in the Dept. of Mechanical
Engineering at Aristotle Univ. of Thessaloniki (AUTh) and a collaborating
researcher in the Chemical Process Engineering Research Institute (CPERI) at
the Centre for Research and Technology – Hellas (CERTH) (P.O. Box 484, 54124,
Thessaloniki, Greece; Phone: +30 2310 99 4229; E-mail: seferlis@auth.gr). His
interests are in the areas of automatic control of process and mechanical
systems, integrated process design and control, and optimization. Prior to
coming to AUTh in 2006, he worked for CPERI for six years, he was a
postdoctoral fellow at Delft Univ. of Technology, the Netherlands, and he
worked for Honeywell Hi-Spec Solutions in Canada. He has authored more
than 50 publications and is a member of AIChE and the Society for Industrial
and Applied Mathematics (SIAM). He has a BS in chemical engineering from
AUTh and a PhD in chemical engineering from McMaster Univ. in Canada.
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