Integration Design of Heat Exchanger Networks into Membrane

Article pubs.acs.org/IECR

Integration Design of Heat Exchanger Networks into Membrane Distillation Systems to Save Energy Yanyue Lu†,‡ and Junghui Chen†,* †

R&D Center for Membrane Technology and Department of Chemical Engineering, Chung-Yuan Christian University, Chung-Li 32023, Taiwan, ROC ‡ Key Laboratory of Chemical and Biological Transforming Process, College of Chemistry and Chemical Engineering, Guangxi University for Nationalities, Nanning, Guangxi ABSTRACT: Membrane distillation requires a lower operation temperature than other separation processes. It is also able to use low-grade thermal energy. In this study, an integrated design method that utilizes a membrane distillation (MD) system to produce pure water is presented. This method is the integration of the heat exchanger network (HEN) into the MD system, and it uses the waste heat from HEN as the energy source of the MD process so as to greatly reduce the water cost. In this proposed method, a multistage MD system is designed to generate the optimal system structure and operating conditions. Then the output and input MD streams are merged into the existing HEN so that the energy cost of pure water can be determined by redesigning the HEN with the trans-shipment models. The efficiency of the proposed method is demonstrated through an industrial case. sequential synthesis and simultaneous synthesis methods.8 Owing to the complication of the HENS problems, sequential synthesis methods, which were mainly based on pinch analysis, were applied earliest to obtain a network design. Three primary objectives are separately used by sequential synthesis methods to design a HEN with minimum investment and operating costs, including minimum utility consumption, the number of units, and the heat transfer area. The pinch design method utilizes the minimum utility consumption to synthesize a HEN since the utility consumption is the most important design objective for an energy efficient network. Papoulias and Grossmann developed a series of trans-shipment models.9 They used the minimum utility consumption and the number of units to synthesize a HEN. The trans-shipment model formed the subproblems of the HENs, which were also a series of mathematical programming problems. This way, the minimum utilities, the minimum number of matches, and the minimum capital cost would be determined in sequence. Sequential synthesis methods are easy to compute and they are suitable for large HENs with many streams, but they can just generate suboptimal networks due to the stepwise nature. On the other hand, simultaneous synthesis methods can find the optimal network without decomposition of the problem. A complete MINLP simultaneous synthesis formulation was first presented by Yee and Grossmann.10 Their model was based on the stagewise superstructure representation which assumes that the split streams were mixed at equal temperatures. Subsequently, lots of improvement work has been done based on Yee’s model in order to reduce the computational effort.11 It is noted that the MINLP simultaneous synthesis approach is rigorous in describing heat exchange processes

1. INTRODUCTION Membrane distillation (MD) is a promising membrane separation technology. It has many significant advantages, such as less sensitivity of fluxes to salinity, compact system construction, low operating temperature and pressure. Moreover, the ability of MD to utilize the low-grade heat in a form of waste heat or renewable energy sources has boosted the interest in research in order to find suitable applications. However, MD has not been implemented yet in industry for water purification. Although MD is a conceptually simple process, the total mass transfer is asymptotically limited when the membrane area increases by adding modules in series.1 Thus, a proper design decision for MD with respect to selection of operating conditions and module configurations is significantly needed. Most of the studies in MD are focused on the heat and mass transfer mechanism of MD processes and the MD system operation.2−5 The problem of integration design of the MD system received little attention. Gilron et al. presented a shortcut design method to determine the number of modules and to extract the maximum heat recovery for a cascade structure of crossflow MD.6 The results of Gilron’s study showed that if the waste heat from other processes was applied to the MD process, the energy costs may be even lower. With the integration of MD and traditional circulating cooling water processes, Wang et al. used the waste heat for pure water production.7 This process technique could effectively reduce the pure water production cost, but how to utilize waste heat from multiple streams has not been systematically studied. Because of the energy exchange occurring in the MD system, it is a feasible way to merge the streams of MD into the heat exchanger networks (HENs) of the existing operation processes for sufficient utilization of the waste heat. Through heat exchanger network synthesis (HENS), the waste heat can be recycled so that the energy consumption would be cut down. Considerable research has been done about HENS over the past 40 years, and most of the research can be classified as © 2012 American Chemical Society

Received: Revised: Accepted: Published: 6798

October 22, 2011 February 22, 2012 April 5, 2012 April 10, 2012 dx.doi.org/10.1021/ie2024245 | Ind. Eng. Chem. Res. 2012, 51, 6798−6810

Industrial & Engineering Chemistry Research

Article

Figure 1. Schematic diagram of the integrated system of AGMD and HEN.

using mathematical models, but it does not always guarantee optimality for industrial scale problems even with advanced optimization solvers. Sequential synthesis problems can be solved with very little or modest computational effort. The approaches to these problems have been widely applied in industry.9 In most cases, they can be used to obtain optimal or nearly optimal solutions;9 therefore, this sequential method will be used in this paper. For sufficient utilization of the waste heat and production of low-cost pure water, an integrated design method which combines the MD system and the HEN is presented in this paper. The MD system is first designed to generate an optimal system structure in the target operating conditions of the maximum heat efficiency and the minimum capital cost. Then the output and input MD streams would be merged into the existing HEN for redesigning HEN to minimize the energy cost of pure water.

afterward, these two outlet streams from the MD would go through the HEN and enter the MD system again. The design results would determine the optimal structure and operating conditions of the MD system, and the optimal network of heat exchange. The objective function of minimizing the water cost Φ ($/m3) can be represented by ⎡ CMD + C HEN ⎤ Φ = min⎢ ⎣ tmd × CF × WT ⎥⎦

(1)

where CMD ($/yr) is the annual costs of the MD system, including the annualized capital cost of the membrane module and the energy cost of the pumps. CHEN ($/yr) is the annual costs of the HEN resulting from the MD streams, including the additional energy cost and the capital cost of the heat exchanger produced by merging MD streams into HEN. tmd (kg/s) is the pure water production of the MD system. CF is the conversion factor for the flow rate unit from kg/s to m3/h; it is 3.6 here. WT denotes the working time in a year for this integrated system. Here it is assumed that the integrated system is operated 24 h a day and 330 days a year. As a HEN consists of multiple hot streams and cold streams in industry processes, it is computationally difficult to integrate the MD and HEN with the simultaneous synthesis approach; therefore, a sequential synthesis approach for the design problem is considered here. Equation 1 is solved by two phases. The outline of the design process is presented as Figure 2. In this figure, the objective function Φ in eq 1 is decomposed into two subobjective functions: one for the MD system operation design and the other for the HEN energy cost design. The MD design problem is conducted first since the stream data must be known before the HEN design is handled. In the MD system design, with a mathematical programming method, the cost of the MD system (Φ1, to be defined later) is minimized. As a result, the MD stream data can be determined. Since the cost of heat energy consumed by the MD process is not considered in the MD system design phase, the cost (Φ2, to be defined later) in the HEN design phase is computed through the cost difference between HEN with and without merging MD streams. The details will be described in sections 3 and 4. The costs of the MD (Φ1) and the HEN (Φ2) should be defined first. As shown in Figure 2, the MD design system phase, the optimal MD system structure, and the operation condition are determined. Also, the thermal energy consumption should be considered as the pure water output is proportional to the transmembrane temperature difference. The thermal efficiency

2. PROBLEM DESCRIPTION Membrane distillation is a thermal-driven process. The transmembrane vapor pressure difference, which is generated because of the temperature difference between the hot feed stream and the coolant stream, is the driving force of vapor through the microporous membrane. For this reason, the high thermal energy consumption becomes one of the main barriers of MD to realize common commercial application. In the field of pure water production, the comparison of several desalination technologies shows that the cost of pure water produced by reverse osmosis (RO) is the least; however, if the low grade energy sources, such as the waste heat and the solar energy, are used in the MD process, the expected cost for the MD process will become more competitive than RO.12 The goal of this study is to develop an optimal design method integrating the MD system with a HEN to reduce the cost of pure water production. The schematic diagram of this integrated system is shown in Figure 1. In the upper part of this figure, the hot feed stream (in solid line) and the coolant feed stream (in dashed line) enter the MD system under the countercurrent operation mode. The pure water in the dashed dotted line of Figure 1 is produced by the MD process with the driving force of temperature difference. Through the process of MD, the temperatures of the hot feed stream and the coolant stream are respectively decreased and increased. Then, in the lower part of this figure, these streams are merged into a HEN. Through the HEN, the heat transfers between streams from the MD and the streams from the operating plant are involved; 6799

dx.doi.org/10.1021/ie2024245 | Ind. Eng. Chem. Res. 2012, 51, 6798−6810


Article

TACHEN = C huQ hu + CcuQ cu +

∑ ∑ (Cf + C1Si,jC ) 2

i∈H j∈C

(6)

The annual cost of HEN includes the operating cost (utility consumption) and the capital cost (i.e., the heat exchanger investment). Where Chu and Ccu denote the costs of hot utility and cold utility per unit, respectively. Qhu and Qcu denote the amount consumed of hot utility and cold utility, respectively. Cf denotes the fixed charge for exchangers. C1 and C2 denote the area cost coefficients.18,19 Si,j denotes the heat transfer area for the exchanger with two streams i and j. H and C denote the sets of hot and cold streams, respectively. The last item at the right side of this equation denotes the annualized capital cost of heat exchanger. To solve eq 4, the successive decomposition method is employed here. Since the annual costs of a HEN consist of the cost of utility consumption and the capital cost resulting from the heat transfer area, and the number of heat exchangers, the design of a HEN is generally decomposed into several steps. As shown in Figure 2, in this paper, the process can be divided into three steps: (1) minimizing utility consumption; (2) minimizing the number of heat exchanger units; and (3) synthesizing an optimal HEN structure. The detailed models of the HEN design will be presented in section 4. As a result, the water cost, Φ, is the sum of the relevant MD cost, Φ1, and the relevant HEN cost, Φ2. A nearly optimal integration MD system producing pure water with relatively low cost would be designed using this sequential synthesis method. Simultaneously, the HEN is also redesigned to sufficiently recover the heat energy. In the next section, the MD system is designed to yield pure water at a minimum cost. In section 4, the design method of the HEN is presented to calculate the minimum annual cost and to determine the heat exchanger network structure. In section 5, a case study is discussed to illustrate this integrated design method.

Figure 2. The outline of synthesis design strategy for the integrated system of AGMD and HEN.

of the MD processes (ηm) is the ratio of vapor enthalpy to the total heat flux which transfers across the membrane. It is considered as an important interaction parameter which relates to the thermal energy consumption as well as MD structure. In designing the MD system, however, its energy is provided by HEN; therefore, the objective function of the MD design should be set to minimize the cost of the MD system (Φ1) and to maximize thermal efficiency (ηm). The multiobjective optimization problem is represented by ⎤ ⎡ CMD Φ1 = min⎢ ⎣ tmd × CF × WT ⎥⎦

(2)

max ηm

(3)

3. THE MD SYSTEM DESIGN According to the nature of the membrane permeate side, MD systems can be classified into four configurations: direct contact membrane distillation (DCMD), air gap membrane distillation (AGMD), sweeping gas membrane distillation, and vacuum membrane distillation. Because of the lowest heat loss, AGMD has the most versatile MD configuration that can be applied to almost all applications.12−16 In this paper, the design method for AGMD systems would be developed. First, the AGMD model is set up based on a great amount of experimental data and theoretical derivation; the detailed AGMD models have been described in our previous paper.17 In AGMD, as the air gap between the membrane and the cold surface considerably reduces the energy loss by heat conduction, the temperature of the hot feed stream in an AGMD module still maintains at a high level. To effectively utilize the energy contained in the hot feed stream and increase the product recovery, a multistage structure of the membrane modules shown in Figure 3 is proposed. This multistage AGMD system consists of the same membrane modules with a certain combination. The number of stages and the number of modules in each stage are set to be design variables, and the optimal system structure could be determined by solving these variables by the mathematic programming method. Through the optimal combination of the

Since all the design variables of the MD system have been determined by solving the above optimization problem, in the next phase, HEN would be designed to minimize the cost data related to HEN, Φ2. The streams coming from the MD system accomplish heat exchange via merging into the existing HEN. To measure the effectiveness of HEN, the additional HEN cost, CHEN, is defined as the cost difference between HENs with and without merging AGMD streams. Then the subobjective function of the relevant HEN cost is expressed by ⎤ ⎡ C HEN Φ2 = min⎢ ⎣ tmd × CF × WT ⎥⎦

(4)

C HEN = TACHEN2 − TACHEN1

(5)

where TACHEN2 and TACHEN1 denote the total minimum annual costs of HEN with the MD streams and without the MD streams, respectively. Each TACHEN can be determined by optimizing the design of respective HEN, which can be calculated by the following equation: 6800



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where Cm denotes the unit cost of the membrane area. CR is the capital recovery factor. It is assumed that the lifetime of the membrane module is 5 years, so CR is 0.2. Ce denotes the power price. Phot and Pcold are the pump power of the hot feed stream and the coolant stream, respectively. γw is specific weight of water. Hhot and Hcold are the pumping heads of the hot stream pump and the cold stream pump, respectively. ηp denotes the efficiency of the pump. Here it is assumed that the AGMD system is operated 24 h a day and 330 days a year. For the AGMD process, the thermal efficiency ηm is defined by ηm =

membrane modules, the pure water yield would increase with high thermal efficiency of the system. For the multistage AGMD system design, the number of stages is set to be the design variable. Here the number of stages of the system structure increases along the flow direction of the hot feed stream. The hot feed stream and the coolant stream are in the countercurrent operation. The constraints on checking whether a stage exists or not should be included:

Ny =

ηm ≥ ηt

Sy (8)

where Zy is a binary variable. Its value is 1 when the yth stage exists and 0 if otherwise. Sy denotes the total membrane area of the yth stage. Ls and Us are the low-bound and the up-bound of the membrane area respectively. Ls is usually set to be the area of one membrane module Sm. Ny denotes the number of membrane modules at the yth stage. To reduce the computation complexity, Ny is considered as a continuous variable, and the approximate result would be obtained by rounding the variable. To establish the above multistage AGMD system, the costs that mainly include the annualized capital cost of the membrane module and the energy cost of the pumps shall be minimized. The costs are expressed as follows: Y

CMD = CR × Cm ∑ Sy + Ce(Phot + Pcold) × WT y=1

Phot =

Pcold =

(10)

γwHcoldmci , Y ηp

(13)

4. THE HEN DESIGN The stream data of the hot feed stream and the coolant stream of the AGMD system could be determined after the AGMD system design is accomplished. Then the two streams would be merged into the existing heat exchanger networks to heat the feed stream and to cool the coolant stream. Many design methods about the HEN have been developed. Papoulias and Grossmann developed the trans-shipment models targeting the

(9)

γwHhotmbi ,1 ηp

(12)

In this way, the optimization problem could be formulated into a single-objective optimization problem, and its objective function is represented by eq 2. Depending on the mathematical model describing AGMD processes, the design problem of the multistage AGMD system can be solved by the optimization method. In this optimization problem, the temperatures of the hot feed stream and the coolant stream entering the AGMD system are given. The designed variables include the flow rates of the hot feed stream and the coolant stream entering the AGMD system, the number of stages, and the number of modules in each stage. This procedure is carried out by introducing an excessive number of stages as an initial guess while at the optimum certain design variables, the total membrane area of the yth stage, Sy, is either set to zero or to a value, indicating the absence or the presence of the specific stage, respectively. The detailed design is referred to our previous paper.17 Finally, the optimal system structure and the operating variables of the AGMD system can be determined.

(7)

Sm

QT

where QT (kJ/m2) is the total heat flux in AGMD processes, which consists of two parts: the latent heat of water evaporation QV and the sensible heat of heat conduction. QT is affected by the flow rate of the hot stream and the cold stream, the temperatures at both sides, and the number of stages. The detailed model is described in our previous paper.17 From the AGMD model, it is found that the thermal efficiency is affected by the temperatures of the hot side and the cold side of AGMD. With a decrease of the temperatures at both sides, the thermal efficiency also decreases. Thereby for the multistage AGMD system, the number of stages would affect the process thermal efficiency. Since the low investment cost and the high system thermal efficiency are desired, the optimization design problem of the multistage AGMD system is a multiobjective optimization problem (eqs 2−3). If an acceptable value of thermal efficiency (ηt) is predefined, the objective function of maximizing thermal efficiency could be converted to a constraint,

Figure 3. The AGMD system of the multistage structure.

ZyLs ≤ Sy ≤ ZyUs

QV

(11) 6801



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minimum utility and the minimum number of the heat exchanger to synthesize a HEN.9 Since this method is easy to carry out and suitable for a large HEN with many streams, it is employed in this paper. The trans-shipment models formulate the minimum utility cost and the minimum number of units of HEN into linear programming (LP) and mixed-integer linear programming (MILP) in turn. At first, with a given value for the minimum approach temperature, ΔTmin, the whole temperature range is partitioned into K temperature intervals based on the inlet and the highest and lowest temperatures of the streams. The flow of the heat content between the temperature intervals is represented in Figure 4. In this heat cascade diagram, heat is

Rk − Rk−1 −

Wk

∑ Q uh,s ,k + ∑ Q uc, w , k s=1

Hk

=

w=1 Ck

∑ Q i,k − ∑ Q j,k i=1

(15)

j=1

Q i , k = F cpi ΔTki

(16)

Q j , k = F cpj ΔTkj

(17)

R 0 = RK = 0

(18)

Rk ≥ 0

(19)

Q uh, s , k ≥ 0,

Q uc, w , k ≥ 0

(20)

where Rk and Rk−1 are the heat residual at interval k and k − 1, respectively. Fcpi and Fcpj are the heat capacity flow rates. They are the product of the flow rate and the specific heat capacity for a stream. ΔTik and ΔTjk are the temperature differences of the hot stream i and the cold stream j at interval k, respectively. Qi,k and Qj,k are the heat contents of the hot stream i and the cold stream j at interval k, and they can be calculated by eq 16 and 17. Hk and Ck denote respectively the sets of the hot stream and the cold stream that exist at interval k. Equation 18 indicates that the heat residuals entering the first interval and leaving the last interval are set to zero. In this LP model (eqs 14−20), Rk, Quh,s,k, and Quc,w,k are the non-negative variables. They can be solved by the mathematical programming approach. The solution of this model provides the loads of the hot and the cold utilities and the location of pinch points, since the temperature interval k corresponds to a pinch point if Rk = 0 at the optimal solution. With the information of the location of pinch points from the LP model, the temperature intervals are partitioned into subnetworks above the pinch points, below the pinch points, or between the pinch points. The known hot and cold utilities from the LP model are treated as additional hot and cold streams. Then the problem with the fewest number of heat exchangers can be formulated as an MILP trans-shipment model for each subnetwork. Its objective function is the sum of the binary variables representing all possible matches.

Figure 4. Heat cascade for the trans-shipment model.

considered as a commodity transferred from the hot streams and the hot utilities to the cold streams and the cold utilities in the corresponding intervals, while the heat residuals in one interval can cascade down to the lower interval. By applying heat balances to each temperature interval, as shown in Figure

Figure 5. Heat flows in interval k.

min J2 =

∑ ∑

Zi , j (21)

i∈H ,S j∈C ,W

where J2 is the target of the minimum number of heat exchangers. Zi,j is the binary variable. When the value of Zi,j is 1, it indicates there is match between the hot stream i and the cold stream j. Then a heat exchange unit is needed in this case. When the value of Zi,j is 0, the hot stream i and the cold stream j do not match. H and S denote the sets of hot streams and hot utilities, respectively. C and W are the sets of cold streams and cold utilities, respectively. The energy balance and the constraints of the heat exchanger for the hot stream i and the cold stream j are shown as follows:

5, the minimum utility cost problem can be formulated as an LP model: K

min J1 =

Sk

Wk

∑ (∑ Cuh,sQ uh,s ,k + ∑ Cuc, wQ uc, w , k) k=1

s=1

w=1

(14)

where J1 is the target of the minimum utility cost. Chu,s denotes the unit cost of the sth hot utility, and Ccu,w is the unit cost of the wth cold utility. Quh,s,k and Quc,w,k denote respectively the heat loads of the hot utility s and the cold utility w at interval k. Sk and Wk denote respectively the sets of the hot utility and the cold utility that exist at interval k. The energy balance around each interval can be written as

R i,k − R i,k−1 +

∑

Q i ,j,k = Q i ,k

j ∈ Ck , Wk

∑ i ∈ Hk′ , Sk′

6802

Q i ,j,k = Q j,k

i ∈ Hk′ , Sk′ (22)

j ∈ Ck , Wk (23)



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temperatures of the hot stream i. ti,jCin and ti,jCout denote respectively the inlet and the outlet temperatures of the cold stream j for the heat exchanger unit (i,j). Therefore, the two-level hierarchical MD design architecture shown in Figure 2 is proposed to get the minimum water cost. The design procedure can be summarized as follows. MD Design Phase. Step 1. Calculate the MD streams by minimization of the cost function of eq 2, subject to constraints in eqs 7−13. Get the MD cost, Φ1. HEN Design Phase. (1). HEN with Merging MD Streams. Step 2. With all the given stream data, minimize utility consumption in eq 14, which is subject to constraints in eqs 15−20. Step 3. Based on the previously calculated minimum utility consumption, minimize the number of heat exchanger units using the objective function in eq 21, which is subject to constraints in eqs 22−28. Step 4. Synthesis an optimal HEN structure and get TACHEN2 using eq 6. (2). HEN without Merging MD Streams. Step 5. Repeat the same procedures from Step 2 to Step 4 but the MD streams are not mergered into the HEN system. Get TACHEN1 using eq 6. Total Design Cost. Step 6. With the calculated TACHEN1 and TACHEN2, the HEN cost (Φ2) can be determined using eq 4. Thus, the total design cost (Φ = Φ1 + Φ2) can be computed.

Kq

∑ Q i ,j ,k − Ui ,jZi ,j ≤ 0

i ∈ H, S

j ∈ C, W (24)

k=1 Kq

Kq

Ui , j = min{∑ Q i , k , k=1

∑ Q j , k}

i ∈ H, S

j ∈ C, W

k=1

(25)

R i ,0 = R i , Kq = 0

R i,k ≥ 0 Q i ,j,k ≥ 0

i ∈ Hk′ , Sk′

i ∈ Hk′ , Sk′ i ∈ Hk′ , Sk′

(26) (27)

j ∈ Ck , Wk

(28)

where Ri,k and Ri,k−1 are the heat residuals which correspond to the hot stream i at temperature intervals k and k − 1, respectively. Qi,j,k is the heat exchanged between the hot stream i and the cold stream j at temperature interval k. Qi,k is the heat content of the hot stream or the hot utility i at temperature interval k. H′k and S′k denote respectively the sets of the hot stream and the hot utility that exist at or above the temperature interval k. Qj,k is the heat content of the cold stream or the cold utility j at temperature interval k. Equation 22 represents the energy balance calculation of the hot stream or the hot utility i at interval k, while eq 23 represents the energy balance calculation of the cold stream or the cold utility j at interval k. In eq 24, if the binary variable Zi,j is equal to zero, the match (i,j) does not exist and the heat exchanged for this match must be zero. When Zi,j is equal to 1, the match (i,j) does exist. Here Ui,j denotes the upper bound of the exchanged heat for the match (i,j). It can be determined by eq 25. Kq is the last temperature interval in the subnetwork q. The summation of the heat exchange Qi,j,k in the subnetwork q must be less than the minimum heat content between the hot stream i and the cold stream j. In this MILP model (eqs 21−28), the variables include the integer variable Zi,j and the non-negative variables, Ri,k and Qi,j,k. For each subnetwork, the problem of the minimum number of units could be solved independently using the objective function in eq 21, which is subject to constraints at eqs 22−28, the results would determine the number of matches and the heat exchanged at each of these matches; then the final network structure can be determined by the pinch analysis to rectify the HEN until it is optimal.8 Finally the integral HEN configuration is given by joining the structure of each subnetwork. The sequential synthesis approach presented above first synthesizes the optimal network structure, and then the total annual cost of the HEN which is expressed as eq 6 can be calculated using the corresponding stream data. Meanwhile, the heat transfer area of the heat exchanger can be given by Si , j =

5. CASE STUDY In this section, an example is presented to illustrate the proposed design methodology. On the basis of the proposed sequential synthesis approach as shown in Figure 2, first the AGMD system would be designed. The known conditions are as follows: the inlet temperatures of the hot feed stream and the coolant stream are respectively fixed at 80 °C and 20 °C, the system heat efficiency should be over 90%, and the total required yield of pure water is no less than 0.2 kg/s. The design results are listed in Table 1, and the detail can be found in our previous work.17 From the design results, it is Table 1. The Design Results of the AGMD System number of stages, y inlet flow rate of the hot feed stream (kg/s), mbi,1 inlet flow rate of the coolant stream (kg/s), mci,3 system heat efficiency, η annualized capital cost of the membrane ($/y) energy cost of the pump ($/y) total yield of pure water (m3/y) unit cost of pure water in the AGMD system ($/m3) outlet temperature of the coolant stream at the first stage (°C), Tco,1 outlet temperature of the hot feed stream at the third stage (°C), Tbo,3

Q i,j Ψi , jLMTDi , j

(29)

found that the AGMD system consists of three stages with 21, 17, and 16 membrane modules at each stage. The total yield of pure water is 5808 m3/yr. The unit cost of the product is 0.14$/m3 for this AGMD system. It does not include the cost of heat energy. The cost data correspond with the symbol Φ1 in Figure 2. In the meantime, the design results present the explicit data of the streams coming from the AGMD system; the flow rates of both the hot feed stream and the coolant stream are 5 kg/s. The inlet and the outlet temperatures of the hot feed stream are 80 and 56 °C, respectively, while those of the coolant stream are 20 and 44 °C, respectively. The system

LMTDi , j = ⎡⎣((tiH, j in − tiC, jout)(tiH, j out − tiC, jin)(tiH, j in − tiC, jout + tiH, j out 1/3 − tiC, jin))/2⎤⎦

3 5 5 0.92 170.6 633.6 5808 0.14 44 56

(30)

where Ψi,j is the overall heat transfer coefficient for the match (i,j). LMTDi,j is the log mean temperature difference for the match (i,j). Here the Chen approximation, eq 30, is used to compute LMTDi,j in order to avoid numerical difficulties. tHi,j in and ti,jHout denote respectively the inlet and the outlet 6803



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Table 2. The Stream Data for Case 1a process stream

AGMD stream

hot 1 hot 2 cold 1 cold 2 US UW hot feed stream coolant stream

Tin (°C)

Tout (°C)

FCp (kw °C−1)

h (kw m−2 k−1)

150 90 20 25 180 10 56 44

60 60 125 100 180 15 80 20

20 80 25 30

2 2 2 2 2 2 2 2

21 21

HRAT = 20 °C; hot utility cost =80 ($ kw−1 yr−1); cold utility cost =20 ($ kw−1 yr−1); annualized capital cost for exchanger = 1000 + 50×A ($ yr−1), (A = m2).

a

with the multistage structure has been improved more than the single module in terms of the total yield. This illustrates the efficiency of the proposed design method. The streams leaving the AGMD system would be merged into the HEN to achieve the heat exchange. It is assumed that the hot feed stream is recycled and made up in time to recover the inlet flow rate and the makeup flow rate is very small. Thus, the change of the outlet temperature of the hot feed stream is negligible. Then the synthesis of HEN would be performed to decide the part cost related to HEN. Since different types of HENs would cause different changes in the total annual cost of HEN when they are merged with the AGMD streams, three cases based on different types of HENs would be presented to illustrate the practicality of the proposed design method. 5.1. Case 1: HEN with Extra Hot and Cold Contents. In the first case, the process streams taken from ref 18 are assumed to be the existing process streams which can provide the heat exchange for the AGMD streams. These process streams consist of two hot streams, two cold streams, one hot and one cold utility. The stream data are presented in Table 2, where Tin and Tout denote the inlet and outlet temperatures of the stream. FCp is the heat capacity flow rate. US and UW denote the hot utility and the cold utility, respectively. Here, the heat transfer coefficient of each stream, h, is assumed to be 2 kw m−2 k−1. The specified heat recovery approach temperature (HRAT) is 20 °C. It is assumed that the lifetime of the heat exchanger is set to be 10 years and its annualized capital charge rate is 10%. The annualized capital cost function for the heat exchanger is also presented in Table 2. The AGMD streams of Table 1 would be merged into these process streams to exchange heat. Without the AGMD streams, an HEN is first synthesized based on those existing process streams using the proposed trans-shipment models, which consists of LP (including the objective function eq 14 and constraints eqs 15−20) and MILP (including the objective function eq 21 and constraints eqs 22−28). LP is used to calculate the utility consumption, and MILP can calculate the number of heat exchanger units. Thereafter, the total annual cost of HEN can be calculated by eqs 29−30 and eq 6. It corresponds with the symbol TACHEN1 in Figure 2. In the procedure of HEN synthesis, the temperature range is partitioned into temperature intervals as shown in Figure 6, based on the inlet, the highest and lowest temperatures of the streams. The problem of the minimum utility consumption is solved by the LP trans-shipment model. The results indicate that the hot utility consumption is 1075 kw, the cold utility consumption is 400 kw, and the pinch is at 70 °C−90 °C. Since there is only one pinch point, the synthesis problem is decomposed into two subnetworks, including the network

Figure 6. The temperature intervals of the HEN without AGMD streams for Case 1.

above the pinch point and the network below the pinch point. The amount of consumed utility given by the LP model would be set to be parameters in the next MILP trans-shipment model; then the MILP problem is solved for each subnetwork to determine the minimum number of units. The results of the matches between streams and the heat exchanger are shown in Table 3. On the basis of the information given in Table 3, an Table 3. The Design Results of HEN without the AGMD Streams in Case 1 LP Model hot utility consumption cold utility consumption the pinch

1075 kw 400 kw at 70−90 °C MILP Model match S−C1 match H1−C1 match H1−C2 match H1−C2 match H1−W match H2−C1 match H2−C2

subnetwork 1

subnetwork 2

utility cost investment cost total annual cost

QS,C1 = 1075 kw QH1,C1 = 300 kw QH1,C2 = 900 kw QH1,C2 = 200 kw QH1,W = 400 kw QH2,C1 = 1250 kw QH2,C2 = 1150 kw $94,000 yr−1 $14,725 yr−1 $108,725

integrated HEN targeting the minimum utility and the minimum number of units can be derived by some heuristic rules of pinch analysis. The configuration of the HEN shown in Figure 7 consists of five heat exchangers, one heater and one condenser. After the configuration of HEN is determined, the required areas of each exchanger can be calculated using eqs 29−30. This HEN features the total annual cost of $108,725, 6804



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Figure 7. The configuration of the original HEN for Case 1.

which includes the utility cost of $94,000 yr−1 and the investment cost of $14,725 yr−1. For the AGMD streams merged into this existing HEN to exchange heat, the repartitioned temperature intervals shown in Figure 8 should be conducted. This new HEN would be

Table 4. The Design Results of HEN with the AGMD Streams in Case 1 LP Model hot utility consumption cold utility consumption the pinch MILP Model match S−C1 match H1−C1 match H1−C2 match H1−C3 match H1−C1 match H1−W match H2−C1 match H2−C2 match H2−C3 match H3−W

subnetwork 1

subnetwork 2


Figure 8. The temperature intervals of HEN with AGMD streams for Case 1.

1285 kw 610 kw at 70−90 °C QS,C1 = 1285 kw QH1,C1 = 90 kw QH1,C2 = 900 kw QH1,C3 = 210 kw QH1,C1 = 494 kw QH1,W = 106 kw QH2,C1 = 756 kw QH2,C2 = 1350 kw QH2,C3 = 294 kw QH3,W = 504 kw $115,000 yr−1 $20,470 yr−1 $135,470

including AGMD streams are similar to the ones in Figure 6 and Figure 8 of Case 1. In the design procedure, the original HEN is first synthesized with the series of trans-shipment models. The design results are presented in Table 6. The configuration of the HEN is shown in Figure 10. After the AGMD streams are merged, the design results and the configuration of HEN are respectively presented in Table 7 and Figure 11. In Tables 6 and 7, it is found that both HENs have no pinch point and they do not need the hot utility. There is still surplus heat to be cooled by the cold utility. When the AGMD streams are merged, the high grade surplus heat of the process streams can be used to heat the hot feed stream of AGMD and the low grade heat of the coolant stream of AGMD needs to be cooled; as a result, the utility consumption of HEN does not have any change. Figures 10 and 11 show that the original HEN consists of three exchangers and one condenser while the HEN with the AGMD streams consist of five exchangers and two condensers. Since the unit number of exchangers has increased to be three for the heat exchange of the AGMD streams, it would cause an increasing capital cost of $6,476 when the AGMD streams are merged into the existing HEN. When the increasing cost is regarded as the partial cost of the AGMD process, the unit cost of $1.25/m3 is obtained by solving eq 1 based on the design results in Table 1. In this case, there is no additional energy to be consumed for the AGMD process, and the capital cost of the heat exchanger is the main expense. Thus, the AGMD process is more competitive than other methods in the pure water production because it can utilize the waste heat of the process

redesigned by the same method and the procedure presented above to determine its total annual cost which just corresponds with the symbol TACHEN2 in Figure 2. The design results are listed in Table 4, and the derived configuration of HEN is shown in Figure 9. In Table 4, it is found that the utility cost is $115,000 yr−1, which increases $21,000 relative to $94,000 of the original HEN. The investment cost is $20,470, which increases $5,745 relative to the original HEN. As a result, there would be an increasing total annual cost of $26,745 when the AGMD streams are merged into the existing HEN. The cost data correspond with the symbol CHEN in Figure 2. It is regarded as the partial cost of pure water produced by the AGMD system. The unit cost of $4.74/m3 for the pure water product would be derived by computing eq 1. The price of this case is higher than that of the other production processes (MED, RO, etc.) because the energy cost related to AGMD is still high. Therefore, in this case, the integration of the AGMD process and the HEN does not yield an obvious effect of energy saving although this HEN brought a large amount of energy saving. 5.2. Case 2: HEN with Extra Net Heat Content. Similar to Case 1, the process streams in the second case are used to synthesize the original HEN. The HEN can supply net heat content after full heat recovery. The stream data are listed in Table 5, in which Tin and FCp of Hot 1 are different from those in Table 2. Since the AGMD streams are the same in both Case 1 and Case 2, the partitioned temperature intervals excluding or 6805



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Figure 9. The configuration of HEN with the AGMD streams for Case 1.

Table 5. The Stream Data for Case 2a

process stream

AGMD stream

hot 1 hot 2 cold 1 cold 2 US UW hot feed stream coolant stream

Tin (°C)

Tout (°C)

FCp (kw °C−1)

h (kw m−2 k−1)

180 90 20 25 180 10 56

60 60 125 100 180 15 80

30 80 25 30

21

2 2 2 2 2 2 2

44

20

21

2

Table 7. The Design Results of HEN with the AGMD Streams in Case 2 LP Model hot utility consumption cold utility consumption the pinch

0 kw 1125 kw none MILP Model match H1−C1 match H1−C2 match H1−C3 match H2−C1 match H2−C2 match H2−W match H3−W

HRAT = 20°C; hot utility cost = 80 ($ kw−1 yr−1); cold utility cost =20 ($ kw‑1 yr−1); annualized capital cost for exchanger =1000 + 50 × A ($ yr−1), (A = m2).

a


Table 6. The Design Results of Original HEN in Case 2

QH1,C1 = 1375 kw QH1,C2 = 1721 kw QH1,C3 = 504 kw QH2,C1 = 1250 kw QH2,C2 = 529 kw QH2,W = 621 kw QH3,W = 504 kw $22,500 yr−1 $18,309 yr−1 $40,809

LP Model hot utility consumption cold utility consumption the pinch

0 kw 1125 kw none MILP Model match H1−C1 match H2−W match H1−C2 match H2−C2


QH1,C1 = 2625 kw QH1,C2 = 975 kw QH2,C2 = 1275 kw QH2,W = 1125 kw $22,500 yr−1 $11,833 yr−1 $34,333


condition, the synthesis of the AGMD system and HEN is desirable. 5.3. Case 3: HEN with Extra Net Cold Content. In the third case, the process streams taken and modified from ref 19 are assumed to be the existing process streams which can provide heat exchange for the AGMD streams. These process streams consist of three hot streams, two cold streams, and one hot and one cold utility. The specified temperature of the heat recovery approach is 10 °C. It is assumed that the AGMD streams coming from Table 1 are employed to exchange heat with other process streams in this example. The stream data are presented in Table 8. The two temperature intervals excluding or including the AGMD streams are partitioned as shown in Figure 12 and Figure 13. In these two figures, it is found that the main difference from the previous cases is the existence of

streams. This type of original HEN used in this case features a great amount of surplus heat which needs to be cooled; simultaneously, the surplus heat must be at or above the temperature interval of the hot feed stream of AGMD. In this

Figure 10. The configuration of original HEN for Case 2. 6806



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Table 8. The Stream Data for Case 3a

process stream

AGMD stream

hot 1 hot 2 hot 3 cold 1 cold 2 US UW hot feed stream coolant stream

Table 9. The design results of original HEN in Case 3

Tin (°C)

Tout (°C)

FCp (kw °C−1)

h (kw m−2 k−1)

155 80 200 10 10 180 10 56

30 40 40 160 100 180 15 80

8 15 15 20 20

21

2 2 2 2 2 2 2 2

44

20

21

2

LP Model hot utility consumption cold utility consumption the pinch

800 kw 0 kw none MILP Model match S−C2 match H1−C2 match H2−C1 match H3−C1


QS,C2 = 800 kw QH1,C2 = 1000 kw QH2,C1 = 600 kw QH3,C1 = 2400 kw $64,000 yr−1 $13,931 yr−1 $77,931

HRAT = 10 °C; hot utility cost = 80 ($ kw−1 yr−1), cold utility cost = 20 ($ kw−1 yr−1); exchanger cost = 1000 + 50 × A ($ yr−1), (A = m2). a

Figure 14. The configuration of original HEN for Case 3.

Table 10. The Design Results of HEN with the AGMD streams in Case 3 LP Model hot utility consumption cold utility consumption the pinch

Figure 12. The temperature intervals of HEN without AGMD streams for Case 3.

800 kw 0 kw none MILP Model match S−C1 match S−C3 match H1−C2 match H2−C2 match H3−C1 match H4−C1 match H4−C2


QS,C1 = 296 kw QS,C3 = 504 kw QH1,C2 = 1000 kw QH2,C2 = 600 kw QH3,C1 = 2400 kw QH4,C1 = 304 kw QH4,C1 = 200 kw $64,000 yr−1 $20,518 yr−1 $84,518

Figure 13. The temperature intervals of HEN with AGMD streams for Case 3.

the cold streams with low temperature and the cold streams can utilize the heat exhaust of the AGMD coolant stream. On the basis of the design procedure proposed above, the original HEN is first synthesized with the series of trans-shipment models. The design results are presented in Table 9 and the configuration of the HEN is shown in Figure 14. After the AGMD streams are merged, the design results and the configuration of the HEN are respectively presented in Table 10 and Figure 15.


The design results show that both HENs have no surplus heat to be cooled with the cold utility while they all need the hot utility to heat the streams. There are no pinch points in both HENs. In the original HEN, the hot utility consumption is 800 kw. After the AGMD streams are merged, the heat exhaust 6807



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combination of the AGMD streams with HEN can achieve the heat exchange. Therefore, the integrated process of AGMD and the HEN is a feasible way to produce pure water with low costs although the integrated effectiveness is affected by the type of HEN. The comparison of the three cases in the integrated system indicates that Case 2 and Case 3 have better integrated effectiveness. In these two cases, heat can be mutually transferred among the process streams and the AGMD streams. As a result, there is no additional utility to be consumed after the AGMD streams are merged into HEN. That is, in the HEN design stage, there is no energy cost for the AGMD process except the investment cost. Thereby a relatively low unit cost can be obtained. In these two cases, HEN has only the hot utility consumption or the cold utility consumption. This situation only occurs above the pinch or below the pinch of HEN. Therefore, if all the AGMD streams can be merged into the section above the first pinch or below the last pinch of this HEN, like Case 2 and Case 3, the effectiveness of saving energy can be achieved. On the contrary, in Case 1, HEN needs the hot utility as well as the cold utility. When the AGMD streams are respectively merged into HEN above the pinch and below the pinch, the heat of the AGMD streams cannot be mutually transferred through the process streams. Thereby it would cause additional utility consumption and generate a unit cost of pure water higher than that in Case 2 and 3 although the integrated system saves more energy than the individual AGMD process. In Case 2 and Case 3, the investment cost of the heat exchanger is the main expense. If the price of the heat exchanger can be reduced, the unit cost of pure water would be even lower.

of the AGMD coolant stream can be utilized to heat the cold streams with low temperature while the hot streams are used to increase the temperature of the process streams or of the hot feed stream of AGMD. As a result, there is no change about the utility consumption of the HEN. Figure 14 shows that the match of H3−C1 requires two exchangers to achieve the heat exchange for the limit in the minimum HRAT. Thereby compared with the original HEN, two heat exchangers are added when the AGMD streams are merged. This situation is similar to Case 2 except the investment cost of the exchanger for the heat exchange of the AGMD streams, and there are no more costs. The investment cost of the exchanger would cause the increase of the total annual cost by $6,587 relative to the original HEN. Thereby the unit cost of $1.27/m3 is obtained by solving eq 1 based on the design results in Table 1. For the AGMD process, the integration with this type of HEN is effective since there is no additional energy to be consumed in this process. It finally generates an acceptable product cost. This type of original HEN used in this case has only the hot utility consumption. Simultaneously, cold streams with low temperature can utilize the heat exhaust of the AGMD coolant stream. In this situation, the synthesis of the AGMD system and HEN is desirable. Through HEN designed with or without the AGMD streams, the partial cost related to HEN can be determined. The partial cost related to the AGMD system has been determined in Table 1; the unit cost of pure water resulting from the AGMD process is determined by solving eq 1. The cost comparison between different cases presented above is listed in Table 11. In Table 11. The Cost Comparison between Different Cases integrated system of AGMD and HEN

AGM

HEN

annualized capital cost of the membrane, $/yr energy cost of pump, $/y cost of extra heat energy, $/yr capital cost of heat exchangers, $/yr energy cost related to AGMD, $/yr capital cost of heat exchangers related to AGMD, $/yr output of pure water, m3/yr unit cost of pure water, $/m3

AGMD system only

Case 1

Case 2

Case 3

170.6

170.6

170.6

170.6

633.6 50 400

633.6 0

633.6 0

633.6 0

3641

0

0

0

21,000

0

0

5745

6476

6587

5808

5808

5808

5808

9.44

4.74

1.25

1.27

6. CONCLUSION MD is regarded as a promising method of producing pure water. It takes the advantage of the low operating temperature. However, the high thermal energy consumption is still one of the main barriers for MD to realize common commercial applications. Utilizing waste heat is a feasible way of reducing energy consumption. To sufficiently utilize waste heat and produce pure water with low cost, an integrated design method which combines the MD system and HEN is presented in this paper. The main conclusions are summarized as follows: (1) The sequential optimization design method for integrating MD and HEN system is presented. It consists of two steps. In the first step, the stream data is calculated for the MD design; in the second step, HEN design is done. This sequential optimization design can simplify the calculation effort and can be easily used in industry. (2) Based on the system model, the MD system is designed by the mathematical programming method. The design results present an optimal multistage MD system which can improve the productive rate of the system and keep high efficiency of energy utilization; simultaneously, the optimal operating conditions and explicit stream data are also given. (3) Based on the existing process streams, the trans-shipment models are employed to synthesize HEN. The cost difference between HENs with and without the MD streams is considered as the partial cost of pure water resulting from the MD process. The trans-shipment model is a sequential synthesis method to synthesize HEN in minimum utility and the minimum number of units, respectively. This method can conveniently determine the energy consumption and the investment cost of the MD process in the HEN aspect.

this table, only the AGMD system is also included to compare with other cases. The system is not integrated with the HEN. Its hot feed stream is directly heated by the extra hot utility and the coolant stream is cooled by the extra cold utility. In Table 11, the AGMD system only has the largest unit cost because of higher energy costs required to heat or cool the streams. This indicates that the AGMD process is not practical for the pure water production unless the low cost energy or waste heat is employed. Since the temperature grade of the hot feed stream to be heated is higher than that of the coolant stream which needs to discharge the surplus heat for the AGMD streams, both streams cannot directly exchange heat. However, the 6808



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Sk = sets of the hot utility that exist at interval k Sm = area of one membrane module Sy = total membrane area of the yth stage tmd = pure water production of the MD system, (kg/s) tHi,j in, tHi,j out = inlet and the outlet temperatures of the hot stream i tCi,jin, tCi,jout = inlet and the outlet temperatures of the cold stream j Tbi, Tbo = inlet temperature and outlet temperature of hot feed stream Tci, Tco = inlet temperature and the outlet temperature of the coolant stream TACHEN1 = annual costs of HEN without the MD streams TACHEN2 = annual costs of HEN with the MD streams Wk = sets of the cold utility that exist at interval k WT = working time in a year for this integrated system y = number of stage in MD systme Z = binary variable Us = up-bound of the membrane area

Since the low cost water can be obtained by the above method, this proposed integrated method is efficient and practical for the MD process to produce pure water. However, due to the decomposing strategy, the design result can just give a near optimal solution. To pursue the global optimal solution of this integration problem, a simultaneous design method needs to be investigated in a further study.

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AUTHOR INFORMATION

Corresponding Author

*Tel.: +886-3-2654107. Fax: +886-3-2654199. E-mail jason@ wavenet.cycu.edu.tw. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The authors wish to express their sincere gratitude to the Ministry of Economic Affairs for financially supporting this project.

■

Greek Letters

NOMENCLATURE C = sets of cold streams C1, C2 = coefficients in annualized cost function for heat exchanger Ce = power price Cm = unit cost of the membrane area CMD = annual cost related to the MD system, ($/yr) CHEN = annual cost related to HEN, ($/yr) Chu, Ccu = costs of hot utility and cold utility per unit Cf = fixed charge for exchangers CF = conversion factor for the flow rate unit CR = annualized capital charge rate FCp = heat capacity flow rates H = sets of hot streams Hhot = pumping heads of the hot stream pump Hcold = pumping heads of the cold stream pump J1 = objective function of the minimum utility cost J2 = objective function of the minimum number of heat exchangers k = temperature intervals Ls = low-bound of the membrane area LMTDi,j = log mean temperature difference for the match (i,j) Ny = number of membrane modules at the yth stage mbi, mbo = flow rate of the hot feed stream entering and leaving the module mci, mco = flow rate of the coolant stream entering and leaving the module Phot, Pcold = pump power of the hot feed stream and the coolant stream Qhu, Qcu = amount consumed of hot utility and cold utility Qhu,s,k = heat loads of the hot utility s at interval k Qcu,w,k = heat loads of the cold utility w at interval k Qi,k = heat contents of the hot stream i at interval k Qj,k = heat contents of the cold stream j at interval k Qi,j,k = heat exchanged between the hot stream i and the cold stream j at temperature interval k QT = total heat flux in AGMD processes, (kJ/m2) QV = latent heat of water evaporation Rk = heat residual at interval k Si,j = heat transfer area for the exchanger with two streams i and j

■

Φ = objective function, unit cost of pure water, ($/m3) Φ1 = sub objective function related to MD system Φ2 = sub objective function related to HEN ηm = thermal efficiency of MD processes ηp = efficiency of the pump γw = specific weight of water Ψi,j = overall heat transfer coefficient for the stream i,j ΔTmin = minimum approach temperature ΔTik = temperature differences of the hot stream i at interval k ΔTjk = temperature differences of the cold stream j at interval k

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