ARTICLE pubs.acs.org/IECR
Optimization of Direct Recycle Networks with the Simultaneous Consideration of Property, Mass, and Thermal Effects Houssein Kheireddine,† Younas Dadmohammadi,† Chun Deng,‡ Xiao Feng,‡ and Mahmoud El-Halwagi*,† † ‡
Chemical Engineering Department, Texas A&M University, College Station, Texas 77843, United States Department of Chemical Engineering, Xi’an Jiaotong University, Xi’an, Shaanxi, 710049, People’s Republic of China ABSTRACT: There is a growing need to develop systematic and cost-effective design strategies for direct recycle strategies that lead to the reduction in the consumption of fresh materials and in the discharge of waste streams. Traditionally, most of the previous research efforts in the area of designing direct-recycle networks have considered the chemical composition as the basis for process constraints. However, there are many design problems that are not component-based; instead, they are property-based (e.g., pH, density, viscosity, chemical oxygen demand (COD), basic oxygen demand (BOD), toxicity). Additionally, thermal constraints (e.g., stream temperature) may be required to identify acceptable recycles. In this work, we introduce a novel approach to the design of recycle networks that allows the simultaneous consideration of mass, thermal, and property constraints. Furthermore, the devised approach also accounts for the heat of mixing and for the interdependence of properties. An optimization formulation is developed to embed all potential configurations of interest and to model the mass, thermal, and property characteristics of the targeted streams and units. Solution strategies are developed to identify stream allocation and targets for minimum fresh usage and waste discharge. A case study is solved to illustrate the concept of the proposed approach and its computational aspects.
’ INTRODUCTION The efficient use of natural resources is a key challenge to industrial facilities seeking to operate in a sustainable manner. One of the promising means to accomplish the sustainability objectives is material recovery and effective allocation of resources. Over the past two decades, significant progress has been made in developing systematic process integration techniques for conservation of mass. This effort in the field of mass integration has emerged as an effective technique to identify performance targets for the maximum extent of material recovery within individual processes.1-4 Direct recycle is recognized as an effective saving tool in reducing the consumption of raw materials, generation of industrial wastes, and cost. Much research has been performed to design cost-effective material (e.g., water, hydrogen, solvent) recycle networks. Recent surveys can be found in the literature.4-7 Three general approaches have been developed: graphical,8-13 algebraic,14-17 and mathematical programming.18-24 Early mass integration methodologies were based on stream compositions. Nonetheless, there are many wastewater streams that are characterized by properties in addition to concentrations. These problems can be effectively addressed by the propertyintegration framework, which is defined as “a functionality-based holistic approach for the allocation and manipulation of streams and processing units, which is based on functionality tracking, adjustment and assignment throughout the process”.25 Using the algebraicproperty-based approach, several methodologies have been developed for the design of recycle/reuse networks. These include graphical,25-27 algebraic,28,29 and optimization30-35 techniques. This paper expands the scope of recycle/reuse network by introducing, for the first time, a systematic approach that accounts for the simultaneous consideration of mass, property, and operating temperature constraints to satisfy a set of process r 2011 American Chemical Society
and environmental regulations. The paper also addresses the dependence of properties on composition and temperature. The problem is formulated as a nonlinear programming (NLP) problem that minimizes the total annualized cost (TAC) of the system while satisfying the process and environmental constraints.
’ PROBLEM STATEMENT The problem can be expressed as follows. Given is a set of sinks with the constraints for the inlet flow rates and allowable compositions, properties, and temperatures. Also given is a set of fresh and process sources, which can be recycle/reused in sinks. Each source has a known flow rate, composition, property, and temperature. The fresh sources must be purchased to supplement the use of process sources in sinks. In addition, the discharged waste must meet the environmental regulations. The objective is to find an optimal direct recycle/reuse network while simultaneously considering property, mass, and thermal effects and minimizing the cost the overall system. Furthermore, the devised approach should also account for the heat of mixing and the interdependence of properties. ’ MODEL FORMULATION A source-sink mapping diagram (Figure 1) is used to represent the superstructure of the problem embedding potential configurations of interest. Each source is split into fractions that are Special Issue: Water Network Synthesis Received: June 5, 2010 Accepted: February 1, 2011 Revised: February 1, 2011 Published: February 11, 2011 3754
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Component material balance for component c in sink j:
mixed with fractions of other streams to form the feeds to the process sinks that must meet the process constraints expressed as bounds on concentrations, temperature, and properties. Mass balance for the ith source: X Fi, j þ Fi, waste i ∈ NSOURCE ð1Þ Fi ¼ j ∈ NSINK
A similar mass balance can be applied for fresh source r without assigning any fresh waste: X Fr, j r ∈ NFRESH ð2Þ Fr ¼ j ∈ NSINK
Fj zin j, c ¼
X
Fr , j þ
r ∈ NFRESH
X
r ∈ NFRESH
Fr , j z r , c þ
i ∈ NSOURCE
Fi, j zi, c ð4Þ
Note that the component material balances should be limited to the key components upon which constraints are imposed or to those that highly impact the heat of mixing. If the heat effect of mixing is involved, the heat balance for sink j is rewritten as X
Fj Cpj ðTjin - T0 Þ¼
r ∈ NFRESH
Fi, j j ∈ NSINK
X
c ∈ NCOMP, j ∈ NSINK
Mass balance for sink j: Fj ¼
X
ð3Þ
þ
i ∈ NSOURCE
Fr, j Cpr ðTr - T0 Þ
X
i ∈ NSOURCE
Fi, j Cpi ðTi - T0 Þ
þ Fj ΔHjmix j ∈ NSINK
ð5Þ
where Cp can be calculated as Cp ¼
X
xc Cpc c ∈ NCOMP
ð6Þ
c
where xc denotes the mole fraction of component c and Cp for each component can be calculated using a temperature-dependent expression. For example, the following linearized equation may be used: Cpc ¼ ac þ bc T c ∈ NCOMP
Figure 1. Source/sink allocation with direct reuse/recycle.
ð7Þ
Figure 2. Process flowsheet of the production of phenol from cumene.
Table 1. Data for the Sources and Fresh Water (Scenario 1) impurity concentration, source
flow rate (kg/h)
zi (mass fraction)
temperature, T (C)
vapor pressure (kPa)
Washer101 Decanter101
3661 1766
0.016 0.024
75 65
38 25
Washer102
1485
cost ($/tonne)
0.220
40
7
Freshwater1
0.000
25
3
1.32
Freshwater2
0.012
35
6
0.88
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Table 2. Sink Data and Constraints (Scenario 1) water flow
maximum inlet impurity concentration,
minimum
maximum
minimum vapor
maximum vapor
rate (kg/h)
zmax (mass fraction) j
temperature (C)
temperature (C)
pressure (kPa)
pressure (kPa)
Wash101
2718
0.013
60
80
20
47
Wash102
1993
0.013
30
75
4
38
Neutralizer R104
1127
0.1
25
65
3
25
sink
Table 3. Data for the Sources and Fresh Water (Scenario 2) impurity concentration, cost ( 10-3 $/kg)
source
flow rate (kg/h)
zi (mass fraction)
temperature (C)
vapor pressure (kPa)
pH
Washer101
3661
0.016
75
38
5.4
Decanter101
1766
0.024
65
25
5.1
Washer102
1485
0.220
40
7
4.8
Freshwater1
0.000
25
3
7
1.32
Freshwater2
0.012
35
6
6.8
0.88
Table 4. Sink Data and Constraints (Scenario 2) maximum inlet impurity water flow
concentration, zmax j
minimum
maximum
rate (kg/h)
(Mass Fraction)
temperature (C)
temperature (C)
pressure (kPa)
pressure (kPa)
Wash101 Wash102
2718 1993
0.013 0.013
60 30
80 75
20 4
47 38
4.5 4
7 8
Neutralizer
1127
0.1
25
65
3
25
4.5
7
5
9
sink
minimum vapor maximum vapor minimum pH maximum pH
R104 Waste
0.15
The heat of mixing can be calculated as 2 !3 GE 6 7 6D RT 7 7 mix 26 7 ΔH ¼ - RT 6 6 DT 7 5 4
ð8Þ
P, x
where GE is the excess Gibbs free energy and T is the absolute temperature. The excess Gibbs free energy is the difference between the actual value of Gibbs free energy for the real solution and the value for the ideal mixture at the same composition, temperature, and pressure. In this paper, the Wilson equation36,37 is selected for the estimation of the excess Gibbs free energy. For the case of binary systems, it contains only two binary parameters, referenced as Λ1,2 and Λ2,1 and is expressed as GE ð9aÞ ¼ - x1 lnðx1 þ x2 Λ12 Þ - x2 lnðx2 þ x1 Λ21 Þ RT with ln Λ12 ¼ a12 þ ln Λ21 ¼ a21 þ
b12 T
value of Gibbs free energy for the real solution and the value for the ideal mixture at the same composition, temperature, and pressure. The activity coefficients are expressed in terms of the binary parameters and mole fractions: Λ12 Λ21 ln γ1 ¼ - lnðx1 þ x2 Λ12 Þ - x2 x1 þ x2 Λ12 x2 þ x1 Λ21 ð9bÞ ln γ2 ¼ - lnðx2 þ x1 Λ21 Þ - x1
ð9cÞ A key advantage of the Wilson equation is its ability to describe multicomponent systems using the binary parameters. For multicomponent systems, the Wilson equation36,37 is given by 0 1 E X X G ¼ ð9dÞ xi ln@ xj Λij A RT i j and
0 ln γi ¼ 1 - ln@
b12 T
where x is the mole fraction and T is the absolute temperature. The excess Gibbs free energy is the difference between the actual
Λ12 Λ21 x1 þ x2 Λ12 x2 þ x1 Λ21
X
1 xj Λij A -
j
X xk Λki P xj Λkj k
ð9eÞ
j
where Λij takes a value of 1 when i = j. 3756
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Hence, for the binary sytems, substituting eq 9 into eq 8, we get Λ12 b12 Λ21 b21 mix ΔH ¼ - Rx1 x2 þ ð9Þ x1 þ x2 Λ12 x2 þ x1 Λ21
Here, the property is dependent on the temperature and other properties p0 . The operator can be considered as a function of temperature and other properties: jp ðpÞ ¼ f ðT, p0 Þ
Alternatively, when the mixing of entropy is assumed to be ideal, then the excess Gibbs free energy is equal to the excess heat of mixing and eq 9a can be used directly to estimate the change in enthalpy resulting from nonideal mixing. Property balance for property p in sink j: X Fr, j jp ðpr, p Þ Fj jp ðpin j, p Þ ¼ þ
i ∈ NSOURCE
in max zmin j, c e zj, c e zj, c
c ∈ NCOMP, j ∈ NSINK
ð12Þ
Sink temperature constraints: Tjmin e Tjin e Tjmax
Fi, j jp ðpi, p Þ
j ∈ NSINK
ð13Þ
Sink property constraints:
p ∈ NPROP, j ∈ NSINK
Piping Cost ($ h/(kg yr)) Process Source, i
in max pmin j, p e pj, p e pj, p
ð10Þ
Table 5. Piping Costs for the Case Studya
a
ð11Þ
Sink composition constraints:
r ∈ NFRESH
X
" p0 6¼ p
Fresh Source, r
sink, j
1
2
3
1
2
1
11.0231
4.4092
6.6138
9.9208
5.5115
2
7.7161
2.2046
11.0231
6.6138
2.2046
3
4.4092
8.8184
4.4092
7.7161
3.3069
p ∈ NPROP, j ∈ NSINK
ð14Þ
It is important to point out that one of the sinks is the environmental discharge system with eqs 13-15 correspond to the environmental regulations. The mass balance for the waste is given as X Fi, waste ð15Þ Fwaste ¼ i ∈ NSOURCE
The component c load in the waste stream can be obtained using the following component mass balance: X ¼ Fi, waste zi, c c ∈ NCOMP ð16Þ Fwaste zwaste c i ∈ NSOURCE
Data taken from ref 32.
Figure 3. Optimal water network (Scenario 1). 3757
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Figure 4. Optimal water network (Scenario 2).
Figure 5. Retrofitted process flowsheet based on the optimized results.
Considering the heat effects of the mixing, the temperature for the waste can be calculated as X in Fwaste Cpwaste ðTwaste - T0 Þ ¼ Fi, waste Cpi ðTi - T0 Þ i ∈ NSOURCE
The indent of the objective function is to minimize the total annualized cost (TAC), which involves the cost for the fresh sources, cost for the waste discharge, and cost for the pipeline. TAC ¼
mix ð17Þ þ Fwaste ΔHwaste The property p load in the waste stream is expressed through the following property mixing rule: X in Fwaste jp ðpwaste, p Þ ¼ Fi, waste jp ðpi, p Þ p ∈ NPROP ð18Þ
X
Costr Fr HY þ Costwaste Fwaste HY
r ∈ NFRESH
þ
X
r ∈ NFRESH j ∈ NSINK
i ∈ NSOURCE
3758
Pipr, j Fr, j þ
X
Pipi, j Fi, j ð19Þ
i ∈ NSOURCE j ∈ NSINK
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The cost of the fresh is calculated as the product of the unit cost of the fresh resource multiplied by the hourly flow rate multiplied by the number of operating hours per year. The cost of treating the waste is obtained by multiplying the unit cost of treatment by the flow rate of the waste by the number of operating hours per year. The cost of piping from fresh source r to sink j or from process source i to sink j is obtained by multiplying the annualized fixed cost of the pipe per unit flow rate of the pumped stream by the flow rate of the stream.
Table 6. Comparison of the Optimal Results without and with the pH Constraints
’ CASE STUDY Figure 2 shows a schematic representation of the phenol production process from cumene hydroperoxide (CHP). More details on the process can be found in the literature.32 Cumene is fed into the reactor, along with air and Na2CO3 (which works as a buffer solution). In the reactor, cumene is oxidized to CHP. The mixture of CHP and cumene is then sent to a washing operation to remove the excess of the buffer solution and water-soluble materials. Fresh water is used for washing. A stream of wastewater is discharged out of the process to the wastewater treatment facility. Next, the stream leaving the washer is sent to a concentration unit, to increase the low concentration of CHP to 80 wt % or higher. After that, the concentrated CHP stream is fed to the cleavage units, where the CHP is decomposed to form phenol and acetone in the presence of sulfuric acid. The resulting cleavage stream is neutralized with a small amount of sodium hydroxide. Next, decantation is used to separate the stream into an organic phase and an aqueous phase. The aqueous phase is discharged as wastewater and is fed to the treatment facility. The organic phase (which is mainly a mixture of phenol, acetone, and cumene) is washed with water to remove the excess alkali and is finally sent to distillation columns, where it is fractionated into the pure products phenol and acetone. The excess water is discharged as a wastewater stream. Data Extraction with Concentration, Thermal, and VaporPressure Constraints but Not Properties (Scenario 1). Phe-
nol is chosen as the key pollutant, because of its environmental hazards and carcinogenic effects. The property studied in this case study is the vapor pressure, because of its significant contribution volatility, which affects both safety and environmental impacts. The lower and upper bound constraints on vapor pressure guarantee compliance with the operational conditions as well as the environmental regulations. The following mixing rules are used for the pH and the vapor pressure (p): X xi 10pHi ð21Þ 10pH ¼ i
and p¼
X
xi pi
ð22Þ
i
where xi is the fractional contribution of stream i. Below is the list of sources, sinks, and available fresh water sources: • Process sinks: (1) Waterwash Cumene peroxidation section (Wash101) (2) Neutralizer (R104) (3) Waterwash cleavage section (Wash102) • Process sources: (1) Stream 8 from Wash101 (2) Stream 22 from Decanter (D101)
without pH
with pH
Min Cost ($/yr)
93 825
94 016
Fresh1 (kg/h)
1159.86
1175.46
Fresh2 (kg/h)
0.00
0.00
Fresh1,1 (kg/h)
575.84
591.45
Fresh1,2 (kg/h)
584.01
584.01
Fresh1,3 (kg/h)
0.00
0.00
Fresh2,1 (kg/h)
0.00
0.00
Fresh2,2 (kg/h)
0.00
0.00
Fresh2,3 (kg/h)
0.00
0.00
F1,1 (kg/h)
2009.73
1962.90
F1,2 (kg/h)
988.33
988.33
F1,3 (kg/h)
662.94
662.94
F2,1 (kg/h)
132.43
163.65
F2,2 (kg/h)
420.65
420.65
F2,3 (kg/h)
0.00
0.00
F3,1 (kg/h) F3,2 (kg/h)
0.00 0.00
0.00 0.00
F3,3 (kg/h)
464.06
464.06
w1,waste (kg/h)
0.00
46.83
w2,waste (kg/h)
1212.91
1181.70
w3,waste (kg/h)
1020.94
1020.94
z1 z2
0.013 0.013
0.013 0.013
z3
0.1
0.1
zwaste
0.11
0.11
T(sink1) (C)
64.88
64.49
T(sink2) (C)
59.30
59.30
T(sink3) (C)
61.22
61.22
T(waste) (C)
53.40
53.72
P(sink1) (kPa)
29.95
29.60
P(sink2) (kPa)
25.00
25.00
P(sink3) (kPa)
25.24
25.24
P(waste) (kPa)
16.77
17.10
pH(sink1)
6.37
pH(sink2) pH(sink3)
6.49 5.24
pH(waste)
5.00
ΔHmixing (sink1) (kJ/(K h))
3131.93
ΔHmixing (sink2) (kJ/(K h))
2114.71
2114.71
ΔHmixing (sink3) (kJ/(K h))
-263.33
-263.33
3115.31
(3) Stream 25 from Wash102 • Fresh water sources: (1) Freshwater1: 0.000 impurity concentration (2) Freshwater2: 0.012 impurity concentration (mass fraction) 3759
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Industrial & Engineering Chemistry Research Next, the relevant data are gathered from a developed ASPEN Plus simulation. The data are tabulated in Tables 1 and 2 for the sources and the sinks. Note that Scenario 1 will be calculated with and without heat of mixing considerations. Data Extraction with Concentration, Thermal, VaporPressure, and pH Constraints (Scenario 2). This scenario is an extension of Scenario 1, with the consideration of pH in addition to the vapor pressure, chemical components, and thermal effects. The lower and upper bound constraints guarantee compliance with the operational conditions as well as the environmental regulations. The data are tabulated in Tables 3 and 4 for the sources and the sinks of the second scenario. The piping costs are given in Table 5. Solution and Results. Next, the proposed methodology is applied. The optimization software LINGO 11.0 is used to solve the developed NLP model, using the embedded Global Solver. The value of the objective function, which is the cost of fresh water, the cost of piping, and the cost of waste treatment, is evaluated for each case. The amount of fresh water needed prior to direct recycle strategy is 5838 kg/h. These optimal results are illustrated in Figures 3-5. Note that the distribution of sources to sinks is not impacted by accounting for the nonideal heat of mixing. This is primarily attributed to the relatively low concentrations of the impurities. Hence, there were no significant changes in the overall optimal source sink allocation as a result of including the nonideal heat of mixing. This may not be the case in other case studies, especially when relatively high concentrations of solutes are involved. For Scenario 1, direct recycle results in a reduction of the fresh water usage, from 5838 kg/h to 1160 kg/h. This is substantial, especially given the fact that no new pieces of equipment were added. When the pH constraints are added, the solution of Scenario 2 gives a fresh water usage of 1175 kg/h. The TAC values of Scenarios 1 and 2 are $93 825 and $94 016/yr, respectively. As expected, the incorporation of property constraints leads to an increase in cost. Although, in this case, the effect of including pH constraints is not significant, it can be more profound in other cases. Finally, note that the distribution of the flows from sources to sinks is different in both scenarios, because of the need to satisfy all the constraints at minimum cost. A comparison of the results with and without pH constraints is given in Table 6.
’ CONCLUSIONS This paper has introduced a systematic procedure that addresses, for the first time, the simultaneous handling of concentrations, temperature, and properties to characterize the process streams and constraints. This has been done, taking into account the interdependency of properties and their dependency on concentrations and temperature. An optimization formulation has been developed to identify optimal allocation of sources to sinks that will minimize the network cost while satisfying all process and environmental constraints. Finally, a case study on water recycle in a phenol production plant is solved. ’ AUTHOR INFORMATION Corresponding Author
*Tel.: þ1 979 845 3361. Fax: þ1 979 845 6446. E-mail address:
[email protected].
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’ ACKNOWLEDGMENT Financial support provided by the National Natural Science Foundation of China (under Grant No. 50876079) is deeply appreciated. ’ NOTATION (i) Indices
c = index for the components i = index for the internal sources or index for component i in the Wilson equation j = index for the sinks or index for component j in the Wilson equation k = index for multiple components in the Wilson equation p = index for the properties r = index for the fresh sources waste = index for waste (ii) Sets
NCOMP = {c|c is one of the components} NFRESH = {r|r is a fresh source} NPROP = {p|p is one of the properties} NSINK = {j|j is an internal sink} NSOURCE = {i|i is an internal source} (iii) Parameters
ac = parameter in linearized temperature-dependent expression for heat capacity of the pure component; ac = 1.3724 J/(g K) for water, 0.4685 J/(g K) for phenol bc = parameter in linearized temperature-dependent expression for heat capacity of the pure component; bc = 0.0083 J/(g K) for water, 0.0044 J/(g K) for phenol a12 = binary parameter in the Wilson equation for phenol and water solution, a12 = 2.4395 a21 = binary parameter in the Wilson equation for phenol and water solution, a21 = -3.2239 b12 = binary parameter in the Wilson equation for phenol and water solution, b12 = -2229.9297 K b21 = binary parameter in the Wilson equation for phenol and water solution; b21 = 1046.1246 K Cpc = heat capacity of the pure component Cpi = heat capacity, as a function of temperature of process source i Cpr = heat capacity, as a function of temperature of fresh source r Costr = unit cost of fresh source r Costwaste = unit cost of waste Fi = total mass flow rate from process source i Fj = total mass flow rate inlet process sink j Hy = number of operating hours per year pr,p = property p of fresh source r pi,p = property p of process source i pmin j,p = minimum property for property p of process sink j pmax j,p = maximum property for property p of process sink j T0 = reference temperature, assumed to be 0 C Tr = temperature of fresh source r Ti = temperature of process source i = minimum temperature of process sink j Tmin j = maximum temperature of process sink j Tmax j R = ideal gas constant; R = 8.314 J/(K mol) zr,c = composition for component c of fresh source r zi,c = composition for component c of process source i zmin j,c = minimum composition for cth component of process sink j 3760
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Industrial & Engineering Chemistry Research zmax = maximum composition for cth component of process j,c sink j (iv) Variables
Cpj = heat capacity, as a function of the temperature of process sink j Cpwaste = heat capacity, as a function of the temperature of the waste Fi,j = segregated mass flow rate from process source i to sink j Fi,waste = segregated mass flow rate from process source i to the waste stream Fr = total flow rate consumed from fresh source r Fr,j = segregated mass flow rate from fresh source r to sink j Fwaste = total mass flow rate of the waste GE = excess Gibbs free energy, J/(K mol) pin j,p = inlet property for property p of process sink j pin waste,p = inlet property for property p of the process waste Pipi,j = cost of piping from source i to sink j expressed as annualized fixed cost per flow rate transported from source i to sink j Pipr,j = cost of piping from fresh source r to sink j expressed as annualized fixed cost per flow rate transported from fresh source r to sink j Tin j = inlet temperature of process sink j Tin waste = inlet temperature of the waste zwc = composition for component c of the waste w zin j,c = inlet composition for component c of process sink j (v) Greek Symbols
γ = activity coefficient ΔHmix waste = enthalpy change in the mixing node before the waste = enthalpy change in the mixing node before process sink j ΔHmix j jp(p) = property operator of property p Λ12, Λ21 = binary variables in the Wilson equation
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