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Simultaneous Synthesis of Property-Based Water
Reuse/Recycle and Interceptions Networks for Batch
Processes
Denny K. S. NG(1), Arwa RABIE(2) ,
Mahmoud M. EL-HALWAGI(2), Dominic C. Y. FOO(1)
(1)
School of Chemical & Environment Engineering
University of Nottingham Malaysia, Broga Road, 43500 Semenyih, Selangor, Malaysia.
(2)
Chemical Engineering Department
Texas A&M University, College Station, TX 77843, USA
As industry endeavors to prevent pollution and conserve resources, there is an
increasing emphasis given to the optimal management of water usage and wastewater
discharge. In this context, recycle/reuse strategies play an instrumental role. Water may
be directly reused/recycled or purified then recycled to reduce both fresh water and
wastewater. The objective of this paper is to introduce a systematic technique for the
synthesis of cost-effective batch water networks. Using the new concept of property
integration, a property-based approach is adopted. The water streams (sources) are
characterized by a number of key properties. Additionally, constraints on the feed to
water-using units (sinks) are given in terms of bounds on properties. The procedure will
also consider a number of interception devices that can be used to modify the properties
of the streams. A source-interception-sink representation is developed. Storage tanks are
used throughout the network to enable mixing and scheduling. The procedure is
supported by an optimization formulation whose solution identifies optimal allocation
of sources to tanks, interception devices, and sinks. The solution also determines an
optimal policy for scheduling and operating the water network. A case study is solved to
illustrate the important aspects of the devised procedure.
1. Introduction
The area of synthesizing water recycle/reuse networks has received considerable
research interest. Examples of such research efforts can be found in literature (e.g.,
Wang and Smith, 1994; Almató, 1997, 1999; Savelski and Bagajewicz, 2000; Hallale,
2002; El-Halwagi et al., 2003; Manan et al., 2004; Foo et al., 2005; Majozi, 2005; ElHalwagi, 2006).
Most of the recycle/reuse work used composition-based
characterization of the streams and constraints for the sinks. However, in addition to
composition of pollutants, there are many applications that require the consideration of
properties. For instance, the use and discharge of water streams may be dependent on
various characteristics such as pH, conductivity, turbidity, toxicity, theoretical oxygen
demand, and color. In order to address design problems that are governed by
functionalities and properties, the framework of property integration has been recently
introduced, which may be defined as “a functionality-based, holistic approach to the
allocation and manipulation of streams and processing units, which is based on the
tracking, adjustment, assignment, and matching of functionalities throughout the
process” (El-Halwagi et al., 2004). Shelley and El-Halwagi (2000) developed the
concept of property-based clusters to enable the conserved tracking of properties.
Subsequently, graphical, algebraic, and optimization techniques have been developed
for property-based recycle/reuse (Kazantzi and El-Halwagi, 2005; Foo et al., 2006; Qin
et al., 2004). It is worth noting that these research efforts have been limited to steadystate problems. With numerous batch processes using and discharging water, it is
necessary to develop systematic techniques for the synthesis of batch water networks
with property-based constraints. This is the objective of this paper. A structural
representation of the problem is introduced to embed potential configurations of
interest. An optimization formulation is developed and its solution provides optimal
allocation of sources, storage, interception, network configuration, and scheduling.
1.1 Problem Statement
The problem definition of a property-based batch water network are stated as follows:
Given is a batch process with a cycle time (τ). It is characterized by the following:
• A set of water sources: SOURCES = {i | i = 1,2, …, Nsources} composed of process
water streams that may be recycled or discharged. Each source has a flowrate, fi (t)
and is characterized by a set of properties: PROPERTIES = {p | p = 1,2, …, Np}.
The properties of the sources are designated by pi,p (t), and t is the time from the
beginning of the cycle (0 ≤ t ≤ τ).
• A set of process sinks (units): SINKS = {j | j = 1,2, …, Nsinks}. Sinks are process
units that can accept the sources. Each sink requires a flowrate, gj (t) and property
values, pj,p (t), that satisfy the following constraints:
p min
≤ p j , p ≤ p max
j,p
j,p
j ∈ SINKS, p ∈ PROPERTIES
(1)
max
where p min
j , p and p j , p are lower and upper bounds on acceptable properties to unit j.
• A set of interception units: INTERCEPTORS = {k | k =1,2, …, NInt}. Interception
devices are units that may be added to the process to change the source properties.
The interceptors are operated dynamically with the following performance:
pkint,p = f ( piin,p , z k , rk , t )
(2)
where pkint,p (t) is the value of the intercepted property leaving interceptor k, zk and rk
are design and operating variables of interceptor k, and t is time of operation.
A general mixing rule is needed to define all possible mixing patterns among these
individual properties. One such form for mixing is the following equations (e.g. Shelley
and El-Halwagi, 2000):
ψ ( p ) = ∑ xiψ ( pi )
i
(3)
where ψ(pi) and ψ ( pi ) are operators on property pi and mixture property pi
respectively; xi is the fractional contribution of source i of the total mixture flowrate, i.e.
f
xi = i
(4)
fi
∑
i
Figure 1 Source -interception-tank-sink representation
The problem can be schematically represented by a source-interception-tank-sink
allocation, as shown in Figure 1. According to this representation, each source i, is fed
to an interception unit k. The intercepted sources are then sent to tank v where they are
stored and finally dispatch to the appropriate sink j.
1.2 Approach
The following approach will be used to synthesize an optimal batch water network of
minimum total annualized cost:
1. Each source is assigned to an interception unit.
2. For a given size and model of an interception unit, the outlet intercepted source
property piint,p (t) will vary with time according to the following model.
3.
4.
5.
6.
Reformulation of intercepted sources and sinks into discrete events.
Determine the minimum total annualized cost of the network.
Synthesize a direct-recycle water network that’s meets the min TAC using storage
and dispatch tanks.
Schedule an optimum operating scheme.
The main objective of this work is to synthesize and schedule an optimal batch network
which meets the minimum total annualized cost and meets all process constraints.
Hence the objective function can be expressed as:
N Fresh N sin k
Minimize total cost =
∑∑ C
r =1
j =1
r
fr, j +
N Int
∑C I
k
k =1
k
+2
N Tanks
∑C I
v
v
(5)
l =1
where Cr is the cost coefficient of the fresh resources, fr,j is the fresh resources that feds
into the jth sink (mass per batch cycle, τ), Ck is the cost coefficient associated with
interception device k. Ck is a function of flowrate, inlet and outlet composition, system
design as well as operating variable. Cv is the cost coefficient of the storage and
dispatch tanks. Ik and Iv are binary integers that take the value of 0 or 1 designating the
absence or presence of an interception device and tanks respectively. From Equation 5,
it is noted that the number of tanks is double than the tanks required. This is because
the role of each set of the tanks will alternate after each cycle. One set of tanks will first
be used for collection of water sources while the other set dispatches the stored water.
In the subsequent cycle, their roles are reversed. The objective function is formulated to
minimise the total annualized cost that is subject to the following constraints:
Splitting of sources to respective interception device and waste treatment:
Fi =
∑ (w
N Sources
i ,k
+ wi ,waste )
i = 1, 2,…, NSources
(6)
i =1
Property operator of interception devices ψ kint,p are driven by interception device models:
ψ kint,p = f (ψ u , p , z k , rk , t )
k = 1, 2,…, NInt ; p = 1, 2,…, NP
(7)
Splitting of sources from interception devices k to all pairs of tanks v for storage:
N Int
wk = ∑ wk ,v
k = 1, 2,…, NInt
(8)
v = 1, 2,…, NTank v
(9)
k =1
Splitting of dispatch sources from tanks v:
wv = ∑ (g v , j + g v ,waste )
NTanks
v =1
Property operator of the storage tanks v is given by:
N Int
ψ v, p =
∑w
ψ k,p
k ,v
k =1
N Int
k = 1, 2,…, NInt ; p = 1, 2,…, NP
∑ wk ,v
(10)
k =1
Overall material balance around the mixing point of the feed to the sink:
N Fresh
NTnaksv
r =1
v =1
G j = ∑ f r , j + ∑ g v, j
r = 1, 2,…, NFresh ; v = 1, 2,…, NTanks v
(11)
Material property operator constraints around the mixing point of feed to the sink j:
NTanksv
G j ×ψ inj , p = f r , j ×ψ r , p + ∑ g v , j ×ψ vout, p
j = 1, 2,…, NSinks; p = 1, 2,…, NP
(12)
v =1
Sink constraints:
ψ min
≤ ψ j , p ≤ ψ max
j,p
j,p
j = 1, 2,…, NSinks; p = 1, 2,…, NP
(13)
All the unused sources flow for reuse/recycle to the process sinks are fed to the waste
treatment system before discharge to the environment:
waste =
NTanksv
∑g
v ,waste
v = 1, 2,…, NTanks v
(14)
v =1
From the formulation described above, it is noted that the mathematic formulation is a
mixed integer nonlinear program (MINLP) which can be solved to determine the
minimum total annualised cost for the water network with interception units.
1.3 Case Study
A semi-batch chemical process operates with an eight-hour cycle time. The process
produces two recyclable water sources and has two process sinks that require water. The
feed to process sinks is constrained by criteria on flowrate, composition of a sulfur
pollutant, and pH. Tables 1 and 2 provide a summary of the pertinent data for the
sources and sinks.
Table 1 Data for the process sources of the case study (t is start time of the cycle, hr)
Function for flowrate
(kg/s)
6.0
2.0
Source
1
2
Function for pollutant
composition (ppm)
20 × t + 10
30
Function for
pH
2 × t + 8.0
7.7
Start time (hr) End time (hr)
0
3
1
4
Table 2 Data for the process sinks of the case study
Minimum Maximum Minimum allowable Maximum allowable
Minimum Maximum Start End
flowrate flowrate
pollutant
pollutant
pH entering pH entering time time
Sink
demand demand composition entering composition entering
the sink
the sink
(hr) (hr)
(kg/s)
(kg/s)
the sink (ppm)
the sink (ppm)
1
8.0
8.5
0
17
6.0
7.2
5
6
2
3.0
3.2
0
20
7.0
7.5
7
8
Acid
0.03 kg/s
pH = 5.0
Tank
Interception
(from t =0 to 1 hr)
Source 1
Source 2
2.0 kg/s
(from t = 3 to 4 hr)
1.0 kg/s
Fresh
(from t = 3 to 4 hr)
(storing from
t=0 to 1 hr
and dispatching
from t= 5 to 6 hr)
Sink 1
Tank
(storing from
t=3 to 4 hr
and dispatching
from t= 7 to 8 hr)
Sink 2
Figure 2 Optimal Solution of Case Study
To adjust the pH, neutralization units using an acid (pH = 5, cost = $0.2/kg) or an alkali
(pH = 11, cost = $0.1/kg) may be used as interceptors. For pollutant removal, activated
carbon adsorption may be used for interception (removal efficiency = 90%, cost =
$0.05/kg intercepted water). Fresh water (zero content of the pollutant and a pH of 7.0,
cost = $0.01/kg) may be used as needed. The total annualized cost of a tank (including
pumping and piping) is $20,000 per year. The mixing rule for pH is given by:
10 − pH =
∑ (x ×10 )
− pH i
i
(15)
i
The objective is to synthesize a cost-effective water-recycle network. Following the
proposed approach, the identified solution is shown by Fig. 2. The minimum total
annualized cost of the system is $181,912/yr.
2. Conclusion
A systematic procedure has been developed for the synthesis of batch water networks.
Property-based constraints are used to determine acceptable feed to the process sinks.
Also, property-based characterization of sources is used to describe the various
recyclable water streams in the process. A source-interceptor-tank-sink representation
was used to incorporate potential configurations of interest, which then solved by a
mathematical-optimization approach. The solution of optimization determines the duty
and location of each interception device, the assignment of sources to interceptors, tanks
and sinks. A case study was solved to illustrate the usefulness of the approach.
3. Acknowledgement
Funding from the Texas Hazardous Waster Research Center and the Texas Water
Resources Institute is deeply appreciated. Besides, the financial support from
University of Nottingham Research Committee through New Researcher Fund (NRF
3822/A2RBR9) and Research Studentship are gratefully acknowledged.
4. References
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Use in the Batch Process Industry. Comp. Chem. Eng. 21: S971-976.
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