DEVELOPMENT OF AN EQUATION-ORIENTED STEADY-STATE EVAPORATION PLANT SIMULATOR

Abstract

An equation-oriented process simulator was developed for simulating evaporation plants. The simulator graphical user interface was written in Python 2.7, and its engine, in C++. The simulator orders and partitions the system of equations that describe the evaporator system under study and solves the partitions sequentially using the Newton-Raphson method. If no good initial estimates can be provided by the user, the simulator solves a simplified problem to generate the estimates, which greatly facilitates convergence. Calculations are based on steam table correlations and on black liquor enthalpy correlations described in the literature. Two multiple-effect counter-current evaporator scenarios were extracted from the literature and used to validate the simulator: a simple 3-effect system and a realistic 6-effect system. The simulator converged to the solutions with relative ease, provided that the model equations were written as described, that equations were ordered and partitioned, and the simplified model was used to generate good initial estimates. Good agreement was found between the calculated values and those reported in the literature, indicating that the proposed simulation framework could be extended to accommodate more complex systems. Deviations from the reported values can be explained by the different choices of physical properties correlations.

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DEVELOPMENT OF AN
EQUATION-ORIENTED STEADY-STATE
EVAPORATION PLANT SIMULATOR
Authors: Márcio R. Vianna Neto 1, Marcelo Cardoso 1, Esa K. Vakkilainen 2, Éder D. Oliveira 1
1 Universidade Federal de Minas Gerais. Brazil
2
Lappeenranta University of Technology. Finland
ABSTRACT
An equation-oriented process simulator was developed for
simulating evaporation plants. e simulator graphical user
interface was written in Python 2.7, and its engine, in C++.
e simulator orders and partitions the system of equations
that describe the evaporator system under study and solves
the partitions sequentially using the Newton-Raphson
method. If no good initial estimates can be provided by the
user, the simulator solves a simplied problem to generate the
estimates, which greatly facilitates convergence. Calculations
are based on steam table correlations and on black liquor
enthalpy correlations described in the literature. Two multiple-
eect counter-current evaporator scenarios were extracted
from the literature and used to validate the simulator: a simple
3-eect system and a realistic 6-eect system. e simulator
converged to the solutions with relative ease, provided that
the model equations were written as described, that equations
were ordered and partitioned, and the simplied model was
used to generate good initial estimates. Good agreement was
found between the calculated values and those reported in the
literature, indicating that the proposed simulation framework
could be extended to accommodate more complex systems.
Deviations from the reported values can be explained by the
dierent choices of physical properties correlations.
Keywords: simulation, multiple-eect, evaporation, equation-
oriented, partitioning
1. INTRODUCTION
It is desirable to increase the energetic eciency of pulping
plants to make the process more competitive. Moreover, by
increasing its energetic eciency, less energy resources need to
be consumed, lowering fossil-red carbon dioxide generation,
which is key to achieving sustainable development [1,2]. For
these reasons, the energetic optimization of chemical pulping
plants has received attention in the literature [3,4].
Black liquor is a residue produced during the chemical
pulping process. In the chemical recovery cycle, black liquor is
burned in the recovery boiler to produce energy and to recover
chemicals that are consumed during the cooking process.
is step is key to making the pulping process economically
feasible [4, 5–7]. Before the black liquor can be burned in the
recovery boiler, it is necessary to reduce its water fraction
to increase combustion eciency. Evaporation and drying
are the most energy intensive steps in the chemical pulping
process. Evaporation is typically carried out in a Multiple-
Eect Evaporator (MEE) train, usually constituted of 5 to 7+
evaporator bodies. MEE trains concentrate black liquors from a
dry solids mass fraction of approximately 15% to about 80-85%
[8]. e evaporation step makes up 24-30% of the total energy
used by a pulp mill, which justies its optimization [9].
e simulation and optimization of MEE systems has
received considerable attention in the literature. Cardoso et
al. (2009) simulated and optimized a pulp mill for minimum
energy consumption [6]. Saturnino (2012) modeled the
chemical balance of a Kra mill. e evaporator train was
simulated using two commercial process simulators [7]. Ji
et al. (2012) constructed a simplied evaporator model and
conducted process integration through a mixed-integer linear
programming (MILP) approach [4]. Mesfun and Toolo (2013)
applied a process integration methodology to optimize the
evaporator train and the CHP system of a Kra pulp and paper
mill [3]. Several other deterministic and non-deterministic
approaches to simulate MEE trains and optimize their energy
consumption have also been tried [10–22]. Verma et al. (2019)
recently published a review of methods used evaporator train
simulation and optimization [19].
However, these approaches rely on the user having to
hard code each individual system to be optimized, which
is tedious and error-prone for large systems. In this paper
we present an equation-oriented steady-state simulator
constructed specifically to facilitate the optimization
Corresponding author: Márcio R. Vianna Neto. Universidade Federal de Minas Gerais. Av. Pres. Antônio Carlos, 6627 – Pampulha.
Belo Horizonte – MG. 31270-901. Brazil. Phone: +55-31-34091735. marciorvneto@ufmg.br

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of MEE systems. The equation-oriented architecture is
advantageous for optimization because it exposes the
equations that constitute the system under study, allowing
their mathematical properties to be fully exploited. The
simulator is general enough to enable systems of arbitrary
complexity to be constructed, simulated and optimized. In
this paper, we will focus on the architecture and validation
of the results calculated by the simulator engine.
METHODS
Simulator architecture
e simulator is equation-oriented, which means that
the unit processes involved in the simulation are abstracted
as a system of equations, which is then solved. A Graphical
User Interface (GUI) is exposed, through which it is possible
to assemble the block diagram corresponding to the MEE
system under study and to input process parameters. In these
diagrams, unit processes are represented as blocks, which are
interconnected through streams.
Figure1 summarizes the calculation process carried out by
the simulator. Initially, the user inputs a block diagram and its
respective process parameters through the GUI. Blocks and
streams, collectively referred in Figure1 as elements, supply
equations to the simulation engine, which assembles the Global
System of Equations (GSE). e equations are then ordered to
facilitate convergence.
e ordering process begins by constructing the bipartite
graph corresponding to the GSE, as shown in [24]. In this graph,
two sets of vertices exist: equation vertices and variable vertices.
An edge connects an equation vertex i to a variable vertex j
if variable j takes part in equation i. e maximum bipartite
matching M for this graph is then determined using the Ford-
Fulkerson algorithm. e bipartite graph is then converted into
a directed graph through the following process: for every edge
e connecting the equation vertex i to the variable vertex j, if e
belongs to M, replace it by a directed edge connecting i to j. If e
is not in M, replace it by a directed edge connecting j to i. e
topological ordering of this directed graph gives the ordering of
the equations.
Once the equations ordering has been determined, the
simulator proceeds to determine the clusters of equations that
need to be solved simultaneously. is step is referred to as
partitioning the GSE.
Having completed these steps, the simulator solves each
partition following the determined equation ordering using
the Newton-Raphson (NR) method. Initial estimates for the
variables may either be supplied by the user or asked to be
calculated by the simulator. In the latter case, the simulator is
run using simplied models for each unit process to facilitate
convergence, as will be detailed in the next section. e values
found by solving the simplied models are then used as initial
estimates for the original system. e stopping criterion for NR
is the Euclidean distance between two successive iterations,
which must be smaller than some small error tolerance input
by the user. Once the simulator has concluded its calculations,
the results can be either displayed to the user via the GUI, or be
fed to an objective function, if optimization is to be performed.
e GUI was developed using Python 2.7, whereas the
graph algorithms, as well as the numerical routines were
implemented in C++.
Figure 1. Simulator architecture
Simulator
Block diagram (GUI) Simulation engine
Global system of eqs.
Element 1
Equation 1
Equation 1
Equation 2
Equation 3
...
Equation n
Equation n-p
System of equations
partitioning
Equation ordering
Newton-Raphson
Error tolerance
Initial estimates Results
Equation m
Equation е
...
...
Element n

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Unit processes, blocks and streams
e unit processes currently supported by the simulator are:
evaporation, black liquor and condensate ashing, black liquor
mixing, vapor and condensate mixing. Streams represent mass
ows, and are divided into black liquor, vapor and condensate
streams. Table 1 lists the currently supported process blocks
and the equations corresponding to each of them. Table 2
lists the variables that describe each type of stream. In these
equations, variable subscripts denote the streams to which they
correspond, except the subscript sat, which denotes saturation.
Mass ows are indicated by
m
, enthalpies by H , temperatures
and pressures by T and P respectively, the dissolved solids mass
fraction by
xD
and the boiling point rise of black liquor, by BPR.
As mentioned before, if no initial estimates are supplied by
the user, the simulator solves simplied models to obtain those
estimates. e simplied model equations are the same as those
listed in Table1, for all blocks, except the Evaporator. Its energy
balance is replaced by the equation
mS=mV
and the heat of
steam condensation is approximated by
Q=2200
⋅
mS
.
Table 1. Unit processes supported by the simulator and their respective equations.
Process block Description Equations
Evaporator
e black liquor stream F and the inlet
vapor stream S enter the Evaporator
block. e outlet black liquor stream
L, the condensate stream C and the
outlet vapor stream V exit the block.
Evaporator blocks take as parameters the
heat transfer coecient U and the heat
transfer area A.
mS
=
mC
mF
=
mL+mV
mFxD,F
=
mLxD,L
PS
=
PC
TC
=
Tsat(PS)
TV
=
Tsat PV+BPR(PV,xD,L)
TV
=
TL
Q
=
mS(HS-HC)
Q
=
UA(TS-TL)
Q+ mFHF
=
mLHL+mVHV
Flash tank
e inlet black liquor or condensate
stream F enters the Flash block. e
outlet vapor stream V and the outlet
black liquor or condensate stream L
exit the block. e ash tank pressure is
determined by the outlet vapor stream
pressure, PV.
mF
=
mL+mV
TV
=
TL
PV
=
Psat(TV)
PL
=
Psat(TV)
mFxD,F
=
mLxD,L
Black liquor mixer
An arbitrary number of black liquor
streams Li enter the mixer block, and a
single combined black liquor stream Lout
exits it.
mi
i
=
mout
mi
i
Hi
=
moutHout
mi
i
xD,i
=
moutxD,out
Vapor mixer
An arbitrary number of vapor or
condensate streams Vi enter the mixer
block, and a single combined vapor or
condensate stream Vout exits it. In this
work, the vapor mixer is assumed to
cause negligible pressure drop and to
impose that all pressures be equal.
mi
i
=
mout
mi
i
=
mout
P1=P2=…=Pn=Pout

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Validation scenarios
Two MEE plants were taken as case studies to validate the
results given by the simulator. e rst scenario was a simple
counter-current 3-eect MEE system, depicted in Figure2. In
this scenario, condensate leaving the rst and second eects are
ashed and fed to the second and third eects, respectively. is
system was adapted from [5] and, due to its relative simplicity,
was intended to aid in the testing of the unit process models
and the simulator code. Black streams denote black liquor, red
streams denote vapor and blue streams, condensate.
e input values for this system are listed in Table3. All heat
exchange areas were assumed to be equal.
Table 2. Stream types and their describing variables.
Stream type Variables
Black liquor
m
,
T
,
xD
Vap or
m
,
T
,
P
Condensate
m
,
T
,
P
Physical properties
Energy balances require black liquor, water steam enthalpies
to be known. Water and steam enthalpies were calculated from
steam table correlations, implemented in C++ as described in
the 2007 revised release on the International Association for the
Properties of Water and Steam IAPWS Industrial Formulation
of 1997 standard [25].
Black liquor enthalpies were calculated from the correlation
described by Zaman and Fricke (1996), which expresses the
enthalpy of black liquor at 80 °C, H80, as shown in Equation(1)
[26]. In this equation, Hw denotes the water enthalpy at 80 °C,
xD is the black liquor dissolved solids fraction, and the constants
and depend on the type of black liquor being considered. In
this work, it was assumed that kJ/kg.K and .
H80=Hw+b -1+exp
xD
c
(1)
In order to account for black liquor enthalpies at other
temperatures, H80 is corrected using the black liquor specic
heat correlation given by Equation(2), where t stands for the
temperature measured in °C [5].
c
p
=4.216 1-x
D
+1.675+
3.31t
1000.0
x
D
+4.87+
20t
1000.0
1-x
D
x
D
3
(2)
e black liquor Boiling Point Rise (BPR) also needs to be
considered in evaporator calculations. BPR is accounted for
using Equations(3) and(4), where TP is the boiling temperature
of water at pressure P.
BPR(P,xD)=BPRatm(xD)[1+0.6(TP-373.16)/100]
(3)
BPRatm xD=6.173xD-7.48xD
1.5+32.747xD
2
(4)
Figure 2. The simple 3-effect MEE system, adapted from [5]
Table 3. Input values for the 3-effect MEE system.
Variabl e Value Units
Live steam temperature 120 °C
Black liquor inlet mass ow 50 kg/s
Black liquor inlet
temperature 70 °C
Black liquor inlet
dissolved solids 20 %
Vapor temperature
from E3 60 °C
Heat transfer coecient
of E1 (U1) 1.2 kW/m²K
Heat transfer coecient
of E2 (U2) 1.6 kW/m²K
Heat transfer coecient
of E2 (U3) 2.0 kW/m²K
Outlet black liquor
dissolved solids 50 %

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A more realistic 6-effect system was adapted from [9]
and is shown in Figure3. It should be noted that black
liquor exiting the first effect is flashed and its vapor is
fed to the fourth effect. In their original paper, Jyoti and
Khanam (2014) used polynomial fits to estimate steam table
properties and correlations to calculate the evaporators
heat transfer coefficients. The input values for this system
are listed in Table4. All heat transfer areas were assumed
to be equal.
Figure 2: Realistic 6-eect MEE system, adapted from [9].
e two systems were input to the simulator without
supplying any initial estimates for the variables, which means
that they were calculated by rst solving the simplied models.
e NR error tolerance was set to 10-6.
Figure 3. Realistic 6-effect MEE system, adapted from [9].
Table 4. Input values for the 6-effect MEE system
Variabl e Value Units
Live steam temperature to E1 140 °C
Live steam temperature to E2 147 °C
Vapor temperature from E7 52 °C
Black liquor inlet mass ow 15.6 kg/s
Black liquor inlet temperature 64.7 °C
Black liquor inlet dissolved solids 11.8 %
Black liquor outlet dissolved solids 31 %
Heat transfer coecient of E1 (U1) 0.296 kW/m²K
Heat transfer coecient of E2 (U2) 0.4303 kW/m²K
Heat transfer coecient of E2 (U3) 0.2584 kW/m²K
Heat transfer coecient of E2 (U4) 0.6955 kW/m²K
Heat transfer coecient of E2 (U5) 0.839 kW/m²K
Heat transfer coecient of E2 (U6) 0.9698 kW/m²K
Heat transfer coecient of E2 (U7) 1.224 kW/m²K

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RESULTS AND DISCUSSION
Table5 displays some of the key variables found for the
3-eect MEE system, and their respective values as reported
from the original reference. Notice that there is a strong
agreement between calculated and reference values. ere is,
however, the simulator underestimated the heat transfer area
and the black liquor outlet temperature, as compared to the
reference values. is discrepancy is justied by the fact that
the black liquor enthalpies in this work were calculated from
dierent correlations than those of the original reference.
Table6 displays some of the key variables found for the
6-effect MEE system, and their respective values as reported
from the original reference. As before, there is a strong
agreement between calculated and reference values, and the
discrepancy are likely due to the different correlations used.
Notice that the heat transfer areas are not shown in Table6,
as they were not reported in the original publication.
It should be noted that the steps of ordering the GSE,
partitioning it and obtaining initial estimates by solving the
simplified model were critical to ensure that the simulator
would converge. In both scenarios, the combination of
these steps allowed the simulator to converge with relative
ease. This, however, was not the case when any of the steps
were skipped. Solving all the equations simultaneously
via NR proved to be a particularly poor approach, as poor
initial estimates often led to singular Jacobians during the
NR iterations.
It must also be emphasized that the simulator determines the
ordering of the GSE based on the form taken by each equation
Table 5. Comparison between calculated values and reference values for the 3-effect system
Variable Calculated value Reference value Units
Heat transfer areas (all eects) 810.0 1040.0 m²
Vapor temperature from E1 93.7 91.6 °C
Vapor temperature from E2 74.2 73.3 °C
Vapor temperature from E3 60.0 60.0 °C
Black liquor dissolved solids from E1 50.0 50.0 %
Black liquor dissolved solids from E2 33.3 33 %
Black liquor dissolved solids from E3 25.2 25 %
Outlet black liquor temperature 93.7 99.8 °C
Live steam mass ow 11.6 11.3 kg/s
Table 6. Comparison between calculated values and reference values for the 6-effect system
Vari a b le Calculated value Reference value Units
Vapor temperature from E1 129,0 106,3 °C
Vapor temperature from E2 128,5 126,8 °C
Vapor temperature from E3 93,6 89,5 °C
Vapor temperature from E4 78,3 77,2 °C
Vapor temperature from E5 67,1 67,2 °C
Vapor temperature from E6 58,3 58,7 °C
Vapor temperature from E7 52,0 52 °C
Black liquor dissolved solids from E1 29.17 31.65 %
Black liquor dissolved solids from E2 26.59 26.29 %
Black liquor dissolved solids from E3 23.08 24.42 %
Black liquor dissolved solids from E4 19.59 20.32 %
Black liquor dissolved solids from E5 16.66 17.29 %
Black liquor dissolved solids from E6 14.67 15.12 %
Black liquor dissolved solids from E7 13.24 13.45 %
Total live steam mass ow 2.2 1.9 kg/s

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i.e. the set of variables present in each equation. In other words,
the way the equations are written has a direct impact on the
performance of the simulator. e equations described in Table1
are written in the way that yielded the most satisfactory results.
CONCLUSIONS
e simulator was capable of calculating realistic MEE
systems. To facilitate convergence, it is desirable to make
sure that the equations take a suitable form and that they
are ordered and partitioned. If no good initial estimates can
be given, it is also desirable to solve a simplied model to
generate reasonable estimates.
e simulator currently assumes that the heat transfer
coecient is given by the user, as opposed to being calculated
by the simulator. In the future, this issue may be addressed by
introducing heat transfer correlations. In future work, more unit
processes can be modeled, such as condensate preheaters and
recovery boilers, which would allow for more realistic scenarios
to be simulated. More complex systems should also be tested to
verify if the current solution strategy remains robust.
ACKNOWLEDGEMENTS
e authors are indebted to the Conselho Nacional de
Desenvolvimento Cientíco e Tecnológico (CNPq/BRAZIL) and
Fundação de Amparo à Pesquisa do Estado de Minas Gerais
(Fapemig/BRAZIL) for supporting this study. n
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