Supplementary material
Structures of the interconnection matrix M and the perturbation block Δ
The two key blocks in μ-analysis problem ( Δ M ) shown in equation (5) are the
interconnection matrix M and the perturbation block Δ. The interconnection matrix M is defined
as follows:
0  yˆ p M13 
 0


M    yˆ p I
0
0 
M
0 
 31 M 32
(A.1)
where  ŷ p is a real scalar parameter that represents the unknown element in equation (5) and I is
an identity matrix of size N  1 N  1 . The three additional matrices shown in (A.1) are
defined as follows:

M13  d1K,, d nd K

T

M31  h0,1,, hN ,1,, h0,q ,, hN ,q ,, h0,nd ,, hN ,nd

(A.2)

M 32  h0,1 ,, hN ,1 ,, h0,q ,, hN ,q ,, h0,nd ,, hN ,nd
(A.3)

(A.4)
where K is a row vector of length N+1. As shown in (A.2)-(A.4), the key entries to build M are
the coefficients of the each of the uncertain impulse response models that describe the response
in output p due to changes in the process disturbances, i.e., ŷ p , and the maximal change of each
process disturbance considered in the analysis, i.e., δd (see definition (2)). The perturbation
block Δ included in the mixed structured singular value problem presented in (5) is defined as
follows:
Δ  diag Δ d , Δ h ,  p 
(A.5)


(A.6)


(A.7)
Δ d   d  d  1 ; Δ d  nd  N 1
Δ h   h  h  1 ; Δ h   nd  N 1
1
 p  1;  p  C
(A.8)
where Δ d is norm-bounded block that accounts for the uncertainty in the process disturbances at
specific sampling instants. Likewise, the norm-bounded block Δ h represents the uncertainty in
the impulse response coefficients at a particular sampling interval. Each of the elements in Δ d
and Δ h are assumed to be independent real scalars. Moreover, δp is a complex scalar that
measures the system’s dynamic performance. Upper and lower bounds for the Δ M problem
can be estimated using numerical subroutines available in standard engineering software, e.g.,
the mussv built-in function MATLAB®.
Remarks on the worst-case performance tool
The use of uncertain impulse response models as an approximation to the actual nonlinear
process model may add conservatism in the computation of  *yˆ , d*p and ywc,p since each of the
p
model coefficients in (3) is bounded by lower and upper limits. Thus, the optimal process and
control design obtained by the present method may be conservative. However, tight bounds are
expected for each model coefficient included the uncertain models because the identification test
is performed in closed-loop. Accordingly, it is expected that the feedback loops and the
controller actions will reduce the process nonlinearities in the presence of fluctuations in the
process disturbances, which will subsequently reduce the conservatism in the computation of
 * , d*p and ywc,p, respectively. Moreover, the bound  * can be used as a metric of the worstyˆ p
yˆ p
case performance of the output p. That is,  *yˆ can be used to evaluate the dynamic performance
p
and the process constraints considered in the optimal process and control design formulation.
Although the use of  *yˆ reduces the computational costs, it also adds conservatism in the
p
solution because of the use of a bound (  *yˆ ) for the evaluation of the complete optimization
p
function. However, a preliminary process design analysis using only  *yˆ may return educated
p
starting points in a short period of time. Accordingly, the solution obtained from that preliminary
analysis can be used to initialize the computation of an optimal design using the actual worst2
case performance of the process outputs, which is a more intense calculation since it requires
simulations of the closed-loop process model using the critical realizations in the disturbances.
Due to use of norm-bounded calculations, the critical trajectory obtained for d*p can produce the
worst-case performance in the positive direction (worst-case increase) or in the negative
direction (worst-case decrease). As shown in (6), Δ*d is a vector that describes the critical timetrajectory in the disturbances (in a normalized form) in either the positive or the negative
direction. However, another critical disturbance vector is obtained by multiplying this vector by
minus one, i.e.,  Δ*d . Although both vectors are bounded by  *yˆ , Δ*d produces the worst-case
p
performance in one direction while the other critical vector (  Δ*d ) generates the worst-case
output performance in the opposite direction.
The systems identification method introduced by the present approach estimates the coefficients
of an uncertain model from a single input-single output (SISO) identification test. Hence, the
family of uncertain models Y, identified in the present analysis to describe the closed-loop
process dynamics, do not consider the simultaneous effect of multiple disturbances on the
process outputs. Accordingly, the worst-case performance obtained for each process output does
not account for that condition. Although a more intensive multiple input-multiple output
(MIMO) identification of an uncertain model with nd inputs and ny outputs can be implemented
in the present method, the worst-case performance in that MIMO approach will be obtained
under the assumption that the critical realizations in all the process disturbances will affect the
system simultaneously, which leads to the specification of a conservative process design. For
that reason, the present approach relaxes that condition and estimates the worst-case performance
following the procedure described above and schematically shown in Figure 1. That is, the
present analysis assumes that only one worst-case disturbance affects the process at a time while
the rest of the disturbances remain at their nominal operating point. Although one of the authors
had recently proposed an integration of design and control methodology that accounts for the
evaluation of the worst-case performance in the presence of multiple process disturbances,32 that
3
methodology did not include structural changes in the process flowsheet and the control structure
because of the computational costs of adding integer decisions within the methodology’s
optimization framework.
Derivation of the quadratic Lyapunov stability test
The stability test is based on the concept of Lyapunov Theroy51-53 and can be stated as follows:
given a positive definitive quadratic Lyapunov function V(x):
V( x)  xT Qx
Q  QT , Q  0
(B.1)
where x represents the system states in closed-loop given by the state space model shown in (12)
whereas Q is a positive definitive matrix that is independent of the system states x. Matrix Q is
also referred to as the Lyapunov matrix51-53. The system, specified by the states x, is ensured to
be asymptotically stable if and only if51-53:
 ( x)  0  xT Qx  x T Qx  0
V
(B.2)
Since the stability of a system is determined by the elements in the A state space matrix, the
robust stability condition (B.2) can be reformulated as follows:
Aθ  Q  QA θ   0
T
(B.3)
where the symbol θ, formally defined below, is used in (B.3) to denote that A is an uncertain
matrix as it is described in (13). Condition (B.3) represents a set of Linear Matrix Inequalities
(LMI’s); each LMI represents a specific combination between the elements in the A matrix.
However, matrix A cannot be directly evaluated since each of its elements can take an infinite
number of values limited by the uncertainty description provided for the elements of A and
shown in (13). To address this limitation, the present method relax the specification for the
elements of A in (13) and considers that each of its elements can only take two extreme values,
i.e.,
ai , j  ai , j  ai , j , ai , j  ai , j 
(B.4)
4
This approximation reduces the rigorous robust stability condition (B.3) into a finite set of LMI’s
as it is shown in (14). The formal derivation of the stability test using a quadratic Lyapunov
matrix can be found elsewhere51-53.
Implementation of the simultaneous process synthesis and control design methodology to
the case study: A system of CSTRs in series.
Cost Function
The annualized cost function specified for this case study is as follows:
ΦCSTR  CCCSTR  DPCCSTR

 M & S  max
max
CCCSTR  60.22761r 
 V1,wc  DV2,wc
280


5
DPCCSTR  5.25 x10  B w1 1  D   Dw2 
   


 3 

*
*
min
w1  max w1max
, wc  wB , wB  w1, wc 
(C.1)
*
*
min
w2  max w2max
, wc  wB , wB  w2 , wc 
where CCCSTR and DPCCSTR represents the capital cost and the dynamic performance cost of this
process. As shown in (C.1), the capital cost specified for this process is a direct function of the
sizes (capacities) of the reactors included in the process flowsheet superstructure (see Figure 4).
The parameters r and M&S represent the rate of return (r=20%) and the Marshall and Swift cost
index for this process. The present case study assumes that the plant was built in 2011
(M&S=1597.7). 55 In the present analysis, the CSTRs are assumed to be vertical cylindrical
vessels made of carbon steel that operate at atmospheric pressure. The parameters α (1.066) and
β (0.802) in (C.1) are constant factors specified for this type of vessels.56 The output variables
max
V1,max
wc and V2, wc represent the largest deviation (worst-case performance) expected for the liquid
max
level on each reactor due to changes in the process disturbances, d. That is, V1,max
wc and V2, wc are
included in the set ywc shown in (9) and are obtained from simulations of the complete closedloop process model, which is specified by equations (16)-(17). The inputs to simulate the closedloop process model are the critical time-trajectories in the disturbances that produces the worstcase performance for the liquid level in first and in the second reactor, respectively, i.e.,
5
d*V1 and d*V2 . These critical realizations are estimated from the roots problem presented in (5) and
max
expression (6). Both V1,max
wc and V2, wc are estimated around a nominal operating point specified by
the set ξ, which includes the decision variables considered in the optimal process and control
max
design formulation. Therefore, V1,max
wc and V2, wc need to be estimated for every new operating
condition specified by the simultaneous design and control optimization algorithm.
The dynamic performance cost function (DPCCSTR) in (C.1) is an index that aims to measure the
dynamic performance of the CSTR system in economic terms. The key idea is to assign a cost to
the largest variability in the process outputs that are used to measure the system’s performance.
Therefore, DPCCSTR is problem specific and needs to be defined based on the variables used to
measure the design and control goals. As described above, a key goal for the present case study
is to obtain an outlet stream that maintains product B’s flowrate at its target value despite sudden
time-varying fluctuations in the process disturbances. Therefore, the dynamic performance of
this process can be measured based on the deviations of B’s molar flowrate with respect to its
target value. While a large increase of B’s molar flowrate with respect to its target value does not
affect the productivity of this process, it represents a loss in profit since an outlet (product)
stream with purity in B that is higher than that required is being produced. On the other hand, a
significant decrease in B’s molar flowrate with respect to its target value in the presence of
disturbances represents an economic loss since the plant is not meeting a design requirement.
Accordingly, the DPCCSTR function proposed for this case study assigns a cost to the difference
between the largest deviation expected in B’s molar flowrate and the design (target) flowrate of
B at the outlet stream ( wB*  198 mol / min ). That is, the function DPCCSTR penalizes the
deviations in B’s molar flowrate at the outlet stream with respect to its target value ( wB* ).
Therefore, it is expected that the resulting process design and control configuration obtained by
the present analysis reduces the variability in this variable thus improving the dynamic
performance of the system in the presence of critical realizations in the inlet feedstreams’
flowrate and temperature. As shown in (C.1), w1 and w2 represent the worst-case performance
expected for B’s molar flowrate at the outlet stream of the first and the second reactor,
6
min
respectively. Both w1 and w2 are functions of wimax
,wc and wi ,wc , which represent the worst-case
performance of B’s molar flowrate at the outlet stream of the ith CSTR in the positive (worst-case
increase) and in the negative (worst-case decrease) direction, respectively. Estimates for wimax
,wc
and wimin
,wc are obtained from simulations of the closed-loop model using the critical realizations in
the disturbances that produce the worst-case performance of B’s molar flowrate, i.e., d*wi . The
min
values for wimax
,wc and wi ,wc are obtained from simulations of the closed-loop process model using
two critical realizations for d*wi , one obtained from Δ*d and another obtained from  Δ*d (see
Remarks in supplementary material, section B). This is done to ensure that the worst-case
increase and the worst-case decrease in B’s molar flowrate are properly accounted for in the
evaluation of DPCCSTR. Figure C.1 shows a graphical representation of the computation of wi .
Once wi have been estimated, a cost  B (0.001$/mol of product B) is assigned to variability of
product B at the product stream. The constant terms that appear in CCCSTR and DPCCSTR are unit
conversion factors. Following (C.1), structural changes in the process design have been explicitly
considered in ΦCSTR since CCCSTR and DPCCSTR are defined in terms of the binary decision
variable D.
Operational constraints
The design and control goals specified for the present case study impose limitations on the
dynamic operability of the process in the presence of critical realizations in the inlet feedstreams’
flowrate and temperature. These operability constraints need to be included in the simultaneous
design and control formulation to ensure process dynamic feasibility. As shown in problem (9),
the set ρ includes the process feasibility constraints considered in the analysis. Thus, ρ can be
reformulated for the present case study as follows:
7


ρ   g  g  0 ; g  1,,5
1  T1,max
wc  500
 2  T1,min
wc  400
(C.2)
3  DT2max
, wc  500 
 4  DT2min
, wc  400 

 

5  1  D  C Amax  D C Amax  0.03
1, wc
2 , wc
min
max
where Ti ,wc
and Ti ,wc
represent the worst-case increase and the worst-case decrease in the
temperature of the ith CSTR, respectively. Likewise, C Amax
represents the largest variability (in
i ,wc
the positive direction) of the concentration of reactant A at the outlet stream of the ith CSTR.
Estimates of the worst-case performance of these outputs are obtained following the procedure
min
described in Section 2. For example, to compute T2max
, wc and T2 , wc , uncertain impulse response
models like (3) must be identified from the each process disturbance, i.e., qF1,TF1, qF 2 and TF 2 , to
the temperature of the second reactor, T2. This results in 4 uncertain models, each describing the
dynamic behaviour between a single process disturbance and T2. The nominal and the uncertain
impulse response coefficients that describe each uncertain model, i.e., hi ,q and hi , q (see
definition (3)), together with the uncertainty descriptions specified for the process disturbances
in (19), are used to build the interconnection matrix M and the perturbation block Δ presented in
(A.1)-(A.8). These structures are the key inputs that need to be specified to solve the roots
problem shown in (5), which returns a bound on the worst-case performance for T2 (  T*ˆ2 ), and a
norm-bounded vector that includes the critical realizations in the disturbances that produces the
worst-case performance for T2 ( Δ*d ). Expression (6) is used to obtain the actual critical
realizations in the process disturbances for this output, i.e., d*T2 . Two critical disturbance profiles
are obtained from (6), one using Δ*d and another profile using  Δ*d .Then, the complete
mechanistic process model, given by equations (16)-(17), is simulated using the two critical
profiles in d*T2 as the model inputs. The procedure to simulate the closed-loop process model
using d*T2 as inputs is schematically shown in Figure 1, i.e., one critical realization in the
disturbance set is simulated at a time while the rest of the disturbances are set to their nominal
8
operating condition. The results from these simulations are then used to seek for the worst-case
increase and the worst-case decrease of the temperature inside the second CSTR, i.e., T2max
, wc and
min
max
T2min
, wc , respectively. Both T2, wc and T2 , wc are used in (C.2) to evaluate the compliance of the
temperature constraints for the second reactor in the presence of the most critical time-varying
min
fluctuations expected for qF1,TF1, qF 2 and TF 2 , respectively. Note that T2max
, wc and T2 , wc have been
obtained around the nominal operating state specified by ξ (see Figure 1). Thus, the procedure
described above needs to be repeated for every new nominal operating state (ξ). The worst-case
performance for the rest of the process output variables shown in (C.2) are estimated in a similar
fashion and is not discussed here for brevity.
Optimal design and control
The cost function ΦCSTR shown in (C.1) and the process feasibility constraints presented in (C.2)
are the key expressions needed to build the simultaneous design and control problem for the
present case study. Thus, problem (9) can be reformulated for this case study using the
descriptions shown in (C.1)-(C.2), respectively. The optimization variables specified for this case
study are the set of binary decisions that determine the process flowsheet, ψ  D, R , the binary
set that specifies the control structure, ω , the nominal (optimal) operating temperature for each
reactor, T1 and T2 , the nominal (optimal) liquid level on each reactor, V1 and V2 ,and the
controllers’ tuning parameters included in the control superstructure, K C and τ I . Hence, the
resulting optimization problem is challenging since it includes 14 binary decision variables and
28 continuous decision variables. The set of decision variables included in the optimization
problem are used to specify the nominal (optimal) operating state for this process, i.e., ξ. The rest
of the process variables that are needed to fully specify the operating state, e.g., unm , CA1 , C A2 ,
are obtained from the solution of the process model equations at steady-state using as inputs the
values assigned to the decision variables, the production target specified for this process (
wB*  198 mol / min ), the nominal values in the process disturbances, see description (18), and the
process model parameters that remain constant during the analysis, e.g., k0, ΔH0, Cp.
9
The MINLP optimization problem specified for this case study was implemented in MATLAB®.
A numerical integrator solver (ode23s), which implements a modified version of the Rosenbrock
method,57 was embedded within the MINLP formulation to integrate the process model
equations (16) and the controller equations (17) specified for this case study. Although finite
element methods or finite differences can also be used to integrate (16)-(17), these methods
return estimates in the process variables that are often less accurate than those provided by
numerical integrators. The MINLP problem for the present case study was solved using the
glcdirect built-in function available in TOMLAB, which implements a modified version of the
Direct Algorithm.58 For each new set of decision variables specified by the optimization
algorithm, the worst-case performance of the process variables included in the cost function and
the process constraints need to be obtained. Hence, dynamic simulations of (16)-(17) need to be
performed for each of these variables using as inputs excitations signals for the process
disturbances. The data obtained from simulations is then used to identify uncertain models that
are aimed to capture the transient characteristics between each disturbance and each process
output considered in the analysis. The nominal and the uncertain model parameters obtained
from the identification step, together with the uncertainty description provided for the process
disturbances, are the key inputs that need to be specified for the roots problem presented in
problem (5). The solution to problem (5) returns a bound on the worst-case performance and a
norm-bounded vector that contains the individual critical realizations in the disturbances that
produces the worst-case performance for a specific process output. The actual worst-case
disturbance vector, obtained from (6), is used to simulate (16)-(17) following the procedure
described in Section 2.1 and Figure 1. The simulation results are used to search for the worstcase increase or the worst-case decrease in that process output. This result, the outputs’ worstcase performance, is then used to evaluate the cost function (C.1) and the process constraints
(C.2) included in the MINLP formulation. This process is repeated until an optimization criterion
is satisfied. Following Figure 3, the solution obtained from the case study’s MINLP problem was
tested for robust stability using condition (14).
10
Figure C.1 Schematic representation of the evaluation of DPCCSTR: System of CSTRs in
series.
Ternary separation system: process model superstructure, control superstructure and
objective function
The case study presented in Section 4 assumes that the dynamic behaviour of the ternary
distillation system is described using a mechanistic tray-by-tray process model, adapted from
Mohideen and co-workers,12 which includes mass component, total mass and energy balances for
each tray considered in the design. The physical properties needed to evaluate the tray-by-tray
process model include equilibrium relationships and empirical correlations. In the present model,
the number of stages is denoted by N whereas the feedstream stage is denoted as Nfeed. Similarly,
stage 1 represents the condenser/reflux drum whereas stage N represents the reboiler and the
column base.
Component mass balances
The composition of component i on each tray is given by the following expressions:
Stage 1:
1
dxi ,1
 V2 Ki , 2 xi , 2  xi ,1 L1  D1 
dt
(D.1)
11
Stages 2 to N-1:
r
dxi , r
 Vr 1Ki , r 1xi , r 1  Vr Ki , r xi , r  Lr xi , r  Lr 1xi , r 1
dt
(D.2)
Feedstream stage (r=Nfeed):
r
dxi , r
 Vr 1Ki , r 1xi , r 1  Vr Ki , r xi , r  Lr xi , r  Lr 1xi , r 1  Fr xi , feed
dt
(D.3)
Stage N:
N
dxi , N
 LN 1 xi , N 1  xi , N VN Ki , N  BN 
dt
(D.4)
where the subscripts r and i are used to denote the stage number and the component,
respectively. Hence, the subscript r can take any integer value between 1 and N. Likewise, the
present system considers a ternary mixture of heptane (hep), n-hexane (hex) and toluene (tol);
thus, i={hex, hep, tol}. Accordingly, xhex,r, xhep,r and xtol,r represent the mole fractions of nhexane, heptane and toluene at stage r, respectively. Following (D.1)-(D.4), r represents the
molar holdup at stage r whereas Vr and Lr are the vapour and liquid flow rates leaving stage r,
respectively. Moreover, D1 represents the distillate product flowrate whereas BN is the total
product flowrate at the bottoms. Likewise, Ki,r represents the equilibrium constant of component
i at stage r; Fr represents the feed flowrate that enters at the feedstream stage (r=Nfeed) whereas
xi , feed represents the mole fraction of component i in the feedstream. The ratio between the
distillate product flowrate ( D1 ) and liquid reflux at the top of the column ( L1 ) specifies the
reflux ratio (RR), i.e.,
RR 
L1
D1
(D.5)
In the present analysis, (RR) is assumed to be a process variable that is available for control, i.e.,
RR is a manipulated variable.
Mass balances
12
The present model assumes constant molar holdup. Accordingly, the corresponding total mass
balance on each tray is evaluated from the mass conservation equations shown in (D.6)-(D.9).
These equations are used to estimate the liquid and vapour flowrates leaving each stage r.
Stage 1:
V2  D1  L1  0
(D.6)
Stages 2 to N-1:
Vr 1  Lr 1  Vr  Lr  0
(D.7)
Feedstream stage (r=Nfeed):
Vr 1  Lr 1  Vr  Lr  Fr  0
(D.8)
Stage N:
LN 1  BN  VN  0
(D.9)
Energy balances
The energy conservation balances on each tray are shown in (D.10)-(D.13). These equations
estimate the variations in the temperature profile along the column due to changes in the heat
duties of the condenser and the reboiler units, respectively. The terms hL,r and hV,r represent the
specific enthalpies of the liquid and vapour respectively leaving tray r. Similarly, Qcond and Qreb
are the heat duties in the condenser and the reboiler, respectively. The condenser heat duty ( Qcond
) is a function of the pressure in the condenser (Pcond) and the amount of cooling water required
to condensate the overhead vapour stream leaving stage 2 ( V2 ). Moreover, the reboiler heat duty
( Qreb ) is considered as a process variable that can be adjusted to control the operation of the
distillation unit, i.e., Qreb is a manipulated variable. Furthermore, hL , feed represents the specific
enthalpy of the feedstream, which is assumed to be liquid.
Stage 1:
13
1
dhL,1
 V2hV , 2  hL,1 L1  D1   Qcond
dt
(D.10)
Stages 2 to N-1:
r
dhL, r
 Vr 1hV , r 1  Vr hV , r  Lr hL, r  Lr 1hL, r 1
dt
(D.11)
Feedstream stage (r=Nfeed):
r
dhL, r
 Vr 1hV , r 1  Vr hV , r  Lr hL, r  Lr 1hL, r 1  Fr hL, feed
dt
(D.12)
Stage N:
N
dhL, N ˆ
 LN 1hL, N 1  VˆN hV , N  Bˆ N hL, N  Qreb
dt
(D.13)
Physical properties
A few of the model parameters included in the previous equations require the evaluation of the
physical properties of the mixture at certain operation conditions. In the present model, the
equilibrium constant of component i at stage r (Ki,r) is obtained from Raoult’s law,61 i.e.,
Ki,r 
Pi ,vap
r
Pr
(D.14)
where Pr represents the pressure at stage r whereas Pi ,vap
is the vapour pressure of component i at
r
stage r and is calculated using Antoine’s equation,61 i.e.,
log 10 Pi ,vap
r  i 
i
Tr   i
(D.15)
where Tr is the temperature at stage r whereas αi, βi and γi are component-specific Antoine’s
parameters that can be obtained from the literature.62 As shown in (D.1)-(D.4), the composition
in the vapour phase of component i at stage r is obtained from the equilibrium constants and the
composition in the liquid phase of component i, i.e.,
14
yi , r  K i , r xi , r
(D.16)
Moreover, the energy balance equations shown in (D.10)-(D.13) calculates the molar specific
enthalpy of the liquid stream leaving stage r, i.e., hL,r. The corresponding molar specific
enthalpies for the vapour phase and stage temperatures are calculated from the following
expressions:62

hL, r  i 1 xi , r hV0 ,i , r  Hivap
3
hV , r  i 1 hV0 ,i , r yi , r
3
hV0 ,i , r   z 1
5

(D.17)
(D.18)
 i , z Tr  Tr0 R
z
(D.19)
where  i, r represents a component-specific fitting coefficient that is available in the literature for
different species.62 Similarly, H ivap represents the heat of vaporization of component i, R is the
ideal gas constant, Tr0 represents the reference temperature at which the fitting coefficients were
obtained ( Tr0  298.15K ), and hV0 ,i , r is the specific molar enthalpy of component i at stage r.
Furthermore, the constraints shown (D.20)-(D.21) are included in the model to ensure that the
sum of mole fractions in the liquid phase and the vapour phase at each stage r are satisfied at all
time.

x 1
(D.20)

yi , r  1
(D.21)
3
i 1 i , r
3
i 1
Control superstructure
The present analysis assumes that the reboiler heat duty (Qreb) and the reflux ratio (RR) are
available for control (manipulated variables) whereas the composition of toluene in the distillates
(xtol,1) and the composition of heptane at the bottoms (xhep,N) are variables that can be measured
on-line (controlled variables). Therefore, a 2x2 multi-loop control structure composed of PI
15
feedback controllers is proposed here to maintain the operation of the distillation unit within
specifications (see Table 2). The control superstructure proposed for this process in presented in
(D.22). A set of binary decision variables ( ω  0,1
2x2
) is included in the control superstructure
to specify the pairings between the manipulated and the controlled variables in the feedback
control loop. The subscripts n and m in (D.22) denote a manipulated variable and a controlled
variable, respectively. Logical constraints on the binary variables ωnm are included to ensure the
*
feasibility of the control structure. The parameter z nm
represents the set point for the control loop
*
specified by the n-m pairing; hence, z nm
can be either the composition of heptane in the bottoms
*
*
( xhep
, N ) or the composition of toluene in the distillates ( xtol,1 ). The PI tuning parameters,
K C   2 x 2 and τ I  2 x 2 , the nominal (steady-state) value in the manipulated variables, un , and
*
*
the PI controllers’ set points ( xhep
,N and xtol,1 ) have been included as continuous optimization
variables in the simultaneous process synthesis and control design formulation.
e 
cnm  nm  nm 
  I nm 
un  un  m1 unm
2
unm  nm K C nm enm  cnm 
*
enm  znm
 znm


2
m 1
2
n 1
(D.22)
nm  1 n  1,2
nm  1 m  1,2
nm  0,1
Objective function
The cost function proposed in this work was adapted from Bansal and co-workers13 and is
defined as the addition of the capital and operating costs for this system, i.e.,
Φdist  CCdist  OPdist
(D.23)
The annualized capital cost for the distillation unit ( CCdist ) represents the costs associated with
each major process unit included in this system, i.e.,
16
CCdist  Cshell  Ctrays  Creb  Ccond  Ccool
(D.24)
where Cshell, Ctrays, Creb, Ccond and Ccool represent the capital costs associated with the distillation
column’s shell, the theoretical trays, the partial reboiler, the total condenser and the cooler unit,
respectively. The individual capital costs of these units are calculated as follows:13
1.066
Cshell
 M & S  100 Dcol 
 151.22


 280  30.48 
t
N  15
0.802
spa
(D.25)
1.55
 M & S  100 Dcol 
Ctrays  2.193


 280  30.48 
Creb
4
 M & S  10 Areb 

 89.9544

 280  929.03 
t spa N
(D.26)
0.65
(D.27)
Ccond
4
 M & S  10 Acond 


 66.6554

 280  929.03 
Ccool
4
 M & S  10 Acool 

 66.6554

 280  929.03 
0.65
(D.28)
0.65
(D.29)
where the Areb, Acond and Acool represent the heat transfer areas for the partial reboiler, the total
condenser and the cooler unit, respectively. Each of these design variables are estimated from the
following empirical relations:13
Areb 
Qreb
U o, reb Tsteam  TN 
Acond 
Qcond
U o, cond


T T

ln  2 water , out 




 T1  Twater ,in 
 T  T
  T1  Twater,in 
 2 water , out


(D.30)
(D.31)
17
Acool 
Qcool
U o, cool


 T  Twater , out 

ln  N





 Tb, out  Twater ,in 
 T  T

water , out   Tb , out  Twater , in 
 N



(D.32)
where Qreb , Qcond and Qcool represent the heat duties of each unit at a nominal (optimal)
operating state. The nominal operating state for the reboiler heat duty ( Qreb ) is estimated from
the steady-state solution of process model equations using an inputs the decision variables
specified in the simultaneous design and control formulation (see Section 4). Likewise, Qcond is
calculated based on Pcond and the total amount of cooling water required to totally condensate the
vapours produced at V2 whereas Qcool is estimated based on the amount of water required to cool
the bottoms at a given (desired) temperature (60 ºC). The rest of the parameters shown in (D.30)(D.32) are described in Table D.1.
Following (D.25)-(D.26), the capital costs of the distillation column’s shell and the trays inside
the column are a direction function of the number of theoretical trays (N), the tray spacing (tspa)
and the distillation column’s diameter (Dcol). In the present analysis, the distillation column’s
diameter is specified such that it can accommodate the highest vapour flowrate that is expected
to occur within the column due to critical changes in the operation of the process. Accordingly,
Dcol is estimated as follows:
Dcol 
VF 
4  VF 


  vel 
  RT
(D.34)
P
vel  0.107
(D.33)
 L ,   V , 
 L, 
(D.35)
18
where the subscript  denotes the stage with the highest vapour molar flowrate. Accordingly, VF
and vel represent corresponding volumetric flowrate and the velocity at stage  , which are
obtained using the Souders-Brown equation.63 Likewise,   represents the vapour molar
flowrate at stage  whereas V ,  and  L ,  represent the densities of the vapour and the liquid
phases at stage  . Furthermore, P and T are the corresponding pressure and temperature at
stage  . The parameters M&S and tspac shown in (D.25)-(D.29) are specified in Table D.1. The
capital costs of each process unit included in (D.24) have been estimated assuming a payback
period of 3 years and that the units operate continuously for 8,150 hours each year.
The annualized operating costs for the present system ( OPdist ) are directly related to the steam
and water consumption costs in the system, i.e.,
OPdist  Cwater Fwater , cond  Fwater , cool   CsteamFsteam, reb
(D.36)
where Cwater and Csteam are the cooling water and steam costs, respectively. Likewise, Fsteam, reb ,
Fwater , cond and Fwater , cool represent the utility flowrates in the reboiler, condenser and cooler,
respectively. These flowrates are calculated as follows:13
Fwater , cond  29340
Qcond
c p , water Twater , out  Twater ,in 
(D.37)
Fwater , cool  29340
Qcool
c p , water Twater , out  Twater ,in 
(D.38)
Qreb
vap
H water
(D.39)
Fsteam, reb  528120
The constant values shown in (D.37)-(D.39) are unit conversion factors. The values used for the
vap
specific heat capacity of water ( c p , water ) and the heat of vaporization of water ( H water
) are listed
in Table D.1.
19
The number of trays N and the location of the feedstream Nfeed were implemented in the dynamic
flexibility and feasibility formulation using 0-1 binary variables as it is presented in12.
Accordingly, the binary variable Nr is introduced in the formulation to determine the existence of
the rth theoretical tray in the column, i.e., Nr=1 denotes the existence of tray r in the column.
Likewise, the binary variable Nfeed,r is incorporated in the formulation to denote the existence of a
feedstream entering the column at the rth tray, i.e., Nfeed,r=1 indicates that the feedstream enters
the column at the rth tray. The complete set of logical constraints that involve Nr and Nfeed,r are
presented in12. For example, the following logical constraints are included in the dynamic
feasibility and flexibility formulation to ensure that the feedstream (Ffeed) only enters at one tray
in the column:
Fr  F feed N feed ,r  0 r  1,..., R
R
F
r 1
r
 F feed
R
N
r 1
feed , r
1
N feed ,r  N r
(D.40)
r  1,..., R
N r  [0,1]R
N feed ,r  [0,1]R
where R is the total number of trays considered for the column including the reboiler and the
condenser (see Figure 6). The last constraint shown in (D.40) ensures that the feed tray has to be
assigned to an existing tray in the column. The following constraint ensures that the (r+1)th tray
exist in the column only if the rth tray exists:
Nr 1  Nr
r  1,..., R  1
(D.41)
The total number of trays (N) in the column is calculated as follows:
R
N   r N r  N r 1 
r 1
(D.42)
20
Table D.1 – Model parameters: Ternary distillation system. Adapted from Bansal and coworkers.13
Parameter
M&S
tspa
Definition
Marshall and Swift index for 2008
Tray spacing
Cp,water
Heat capacity of cooling water
Twater,in
Temperature of available cooling water
Twater,out
Maximum outlet temperature of cooling
water
Estimate
1449.3
0.6096m (2
ft)
4.187 J/g.K
25°C
45°C
Tb,out
Desired bottoms product temperature
60°C
Td,out
Desired distillate product temperature
60°C
Tsteam
Temperature of available steam
146.85°C
ΔHvapwater
Enthalpy of condensing steam @ 146.85°C
38,253 J/mol
Uo,cool
Overall heat transfer coefficient for cooler
1000 W/m2K
Uo,reb
Overall heat transfer coefficient for reboiler
2000 W/m2K
Uo,cond
Overall heat transfer coefficient for
condenser
1000 W/m2K
Csteam
Cost of steam
0.01 $/kg
Cwater
Cost of cooling water
10-5 $/kg
21