Optimal Design of Multistage Membrane Distillation Systems for Water

ARTICLE pubs.acs.org/IECR

Optimal Design of Multistage Membrane Distillation Systems for Water Purification Yanyue Lu and Junghui Chen* R&D Center for Membrane Technology, Department of Chemical Engineering, Chung-Yuan Christian University, Chung-Li, Taiwan 320, Republic of China ABSTRACT: In this paper, a novel optimal design strategy is proposed to produce pure water in a multistage membrane distillation (MD) system. It is a systematic approach to modeling and analyzing multistage MD processes. The proposed method can generate the optimal system structure and operation conditions and avoid poor integration and suboptimal configuration of multistage MD systems. In this approach, an innovative module model is first established, and then the cost function relating the capital and the operation cost to the design variables and the structural variables of the designed system is introduced in the objective function. A superstructure flow sheet is presented to describe all the possible flow routes for the cold stream. Then optimal design problem can be formulated as a mixed-integer nonlinear programming (MINLP) problem to minimize the total annualized cost. Based on our simulation studies, the mode of the commonly used countercurrent operation is the optimal flow route for the MD system. The design results would also determine the configuration of the optimal separation stage, the optimal operating mode, and the corresponding operating conditions.

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, and use of the industrial waste heat and solar energy as a heat source. Capital and utility costs for MD are less expensive than conventional separation processes, such as distillation and reverse osmosis.14 Therefore, MD is suitable for applications of water desalination, food processing, wastewater treatments, and concentration of solutions. Although the MD technology is widely applied in practice, there are still many engineering problems to be solved, e.g., the optimization design and process integration of the MD system in the operating plant. In the process design, model-based process design is a good way to perform the system optimization design. The proper transfer models for the descriptions of the heat- and mass-transfer mechanisms of MD processes are necessary. In 1998, Gryta et al. investigated the MD process using a model of laminar stream flows, and the applicability of the model to describing the heat transfer in MD processes was presented and verified experimentally.5 In 2005, Guijt et al. presented a predictive model for air-gap MD in a countercurrent flow configuration using fiber membranes; in their study, the water vapor transport across the membrane was described by the dusty-gas model that presented simultaneous Knudsen diffusion, molecular diffusion, and viscous flow.6,7 Based on these generally accepted transfer models, several techniques using the numerical simulation analysis have been applied to the study of MD processes. The sensitivity of the permeate flux to the main parameters has been investigated, including the temperature, concentration, velocity, etc., for direct contact membrane distillation (DCMD) and air gap membrane distillation (AGMD).8 The comprehensive models of the heat and mass transfer associated with the hollow fiber AGMD and the hollow fiber DCMD have also been developed separately.9,10 Although these simulation studies of rigorous mass, r 2011 American Chemical Society

momentum, and energy balances provide comprehensive explanations on the nature of the process, their models include too many immeasurable parameters that complicate the relationship between measurable parameters. It is not convenient to design a large system in practice. In 1998, Liu et al. simplified the complex vapor transfer model into some correlations for AGMD processes.11 The correlations represent just part of the entire membrane module and they cannot be used for the optimization design of the MD system. Based on the study of the mass transfer in the MD process, the optimization design of the MD system that improves the system efficiency has been studied in the past. In 2007, Gilron et al. presented a short-cut design method to determine the number of modules and to extract the maximum heat recovery for a cascade structure of crossflow DCMD.12 This method deduced an operating line relationship between the brine temperature drop across a stage and the temperature difference across the membrane. Then, the number of stages and the exit temperature of brine can be estimated using the similar McCabeThiele diagram. The results of Gilron’s study showed that if the waste heat from other processes was applied to the DCMD process, the energy costs may be even lower. However, this study is specifically aimed at DCMD modules. With the integration of DCMD and traditional circulating cooling water processes, the waste heat for the production cost of optimal pure water was also studied,13 but how to utilize the waste heat carried by multiple streams has not been studied systematically. MD is the most potential method for water purification. It is worthy to explore the design methodology of MD processes, but the design work has rarely been tackled from a system engineering Received: August 9, 2010 Accepted: April 19, 2011 Revised: March 28, 2011 Published: April 19, 2011 7345

dx.doi.org/10.1021/ie1016877 | Ind. Eng. Chem. Res. 2011, 50, 7345–7354

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perspective. In this paper, an optimal design strategy of a multistage MD process is presented for water production. Multiple steps are often utilized in the most separation systems because multiple steps can perform the same operation to purify materials and to meet output purity goals, but the high capital costs and long payback times associated with multistage systems call for efficient management over time. Therefore, a complex multistage configuration, which is a so-called “superstructure”, must be considered to describe the possible processes of MD. This structure provides most possible operating modes for the improvement of the system efficiency, especially the distributed manner of the coolant streams. To determine the optimal separation stage number, the number of modules in each stage, the optimal operating mode, and the corresponding operating conditions with the minimum production cost, a new superstructure-based mixed-integer nonlinear programming (MINLP) formulation is presented in this paper. Finally, the feasibility of the proposed optimal design strategy is demonstrated through a case study.

2. PROBLEM DESCRIPTION In the field of pure water production, the comparison of several desalination technology 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 of the MD process will become more competitive than that of RO.14 The goal of this study is to develop optimal design of the multistage MD system to reduce the cost of pure water production. A schematic diagram of this multistage system is shown in Figure 1. With the lowest heat loss, AGMD is the most versatile MD configuration, which can be applied to almost any application and is selected in this paper. In this figure, the hot feed stream and the coolant stream enter the AGMD system under the countercurrent operation mode. The pure water is produced by the membrane separation process with temperature driving force. Through the process of membrane distillation, the temperatures of the hot feed stream and the coolant stream are decreased and increased, respectively. The design results would determine the optimal structure and operation conditions of the MD system. In MD processes, the pure water output is proportional to the water vapor pressure difference across the membrane and the membrane area. However, in comparison with the membrane area, the transmembrane temperature difference has more significant effects on the water production. Besides, the thermal efficiency of AGMD processes, η (the ratio of vapor enthalpy to the total heat flux which transfers across the membrane), is another factor that influences energy consumption, because it is affected by the system structure and operation conditions. In MD design, the requirement is to maximize thermal efficiency and to minimize the membrane module investment. The multiobjective optimization problem is represented by min Φ1 ¼

CAGMD md 3:6 24 330 max η

system, and the conversion factor for the unit is 3.6. Here, it is assumed that the AGMD system is operated 24 h a day and 330 days a year. In this design problem, the temperature and the flow rate of the hot feed stream entering the MD system are given. In the next section, the unit model and the system model of the AGMD process are first established; then, the multistage MD system is designed to yield the maximum pure water output. Finally, the feasibility of the proposed optimal design strategy is discussed through a case study.

3. AGMD UNIT MODEL A simple mass-transfer model for AGMD processes, shown in Figure 2, in which the heat and mass transfer with temperature polarization are taken into consideration, is used here.11 J ¼

ð2Þ

ΔT RMD

ð3Þ

2:1 RMD ¼ DTMD þβ

ð4Þ

ΔT ¼ Th Tc

ð5Þ

Th þ Tc 2

ð6Þ

TMD ¼

ð1Þ

where Φ1 ($/m ) is the production cost of pure water. CAGMD ($/yr) is the annual cost of the AGMD system, including the annualized capital cost of the membrane module and the energy cost of pumps. md (kg/s) is the pure water output of the AGMD 3

Figure 1. Schematic diagram of multistage AGMD systems.

where J (kg/s 3 m2) is the mass flux, and ΔT (°C) is the transmembrane temperature difference. Th and Tc (°C) are the temperatures of the hot side and the cold side of MD, respectively. RMD denotes the membrane distillation resistance, which can be calculated using the simplified relationship described by eq 3. TMD is a parameter defined by eq 6, while ∂ and β are coefficients that are mainly dependent on the geometric 7346

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equal to the inlet temperature of the coolant stream when the flow rate of the coolant stream is larger than that of the hot feed stream. Then the AGMD effectiveness, ε, which is defined as the ratio of the actual heat transfer in a given AGMD module to the maximum possible amount of heat transfer, is represented by ε¼

Cw mc ðTco Tci Þ Cw mbi ðTbi Tci Þ

ð10Þ

By combining eqs 9 and 10, the outlet temperature Tbo can be solved by Tbo ¼ Tbi εðTbi Tci Þ

ð11Þ

In eq 11, it is necessary to first determine the AGMD effectiveness (ε). A relationship similar to the heat exchanger effectiveness is employed here: mbi 1 exp NTU 1 mc ð12Þ ε¼ mbi mbi 1 exp NTU 1 mc mc NTU ¼

Figure 2. Diagram of the AGMD module structure.

characteristics of the AGMD module and the property of the feed solution. They can be determined by regressing with experimental data for different aqueous solutions. In order to use the previously mentioned mass-transfer model to calculate the permeate flux of an AGMD module, some simplified hypotheses are brought up as follows. For the AGMD module shown in Figure 2, the temperature difference of the hot feed stream between the inlet temperature (Tbi) and the outlet temperature (Tbo) is small, because of the existence of the air gap; therefore, Th can be regarded as the average temperature of Tbi and Tbo. Similarly, the cold side temperature (Tc), which also can be regarded as the average value of the entering temperature (Tci) and the leaving temperature (Tco). Tbi þ Tbo Th ¼ 2

ð7Þ

Tci þ Tco 2

ð8Þ

Tc ¼

Thus, the average value of the permeate flux of the module with the average temperatures of the hot stream and the cold stream can be calculated using eqs 36. In this paper, it is assumed that the outlet temperature of the permeate water is equal to the outlet temperature of the hot feed stream. Thus, the AGMD module can be considered as a heat exchanger. The heat balance for hot and cold stream is Q ¼ Cw mbi ðT bi Tbo Þ ¼ Cw mc ðTco Tci Þ

ð9Þ

The heat released from the hot feed stream is carried away by the coolant stream. Cw is the specific heat of water, mbi denotes the flow rate of the hot feed stream entering the module, and mc is the flow rate of the coolant stream. In the AGMD module, the air gap is very minute, in comparison with the membrane area and the cold plate area; thus, the maximum possible amount of heat transfer is available. In such a case, the outlet temperature of the hot feed stream is

UA Cw mbi

ð13Þ

where NTU is defined as the number of transfer units. U denotes the overall heat-transfer coefficient. A denotes the area, and it is approximately equal to the membrane area for the flat sheet membrane module or the tubular membrane module. In AGMD processes, the total heat flux QT (kJ/m2) consists of two parts, the latent heat by water evaporation QV and the sensible heat by heat conduction QC, which is expressed by QT ¼ QV þ QC ¼ kv ΔT þ ks ΔT ¼ UΔT

ð14Þ

where kv denotes the heat-transfer coefficient of the vapor heat and ks denotes the heat-transfer coefficient of the conductive heat. Both kv and ks comprise the overall heat-transfer coefficient U. The vapor heat is also expressed by QV ¼ Jλ

ð15Þ

where λ is the latent heat of evaporation. With the combination of eqs 3, 4, 14, and 15, kv can be determined by kv ¼

λ 2:1 þ β DTMD

ð16Þ

For the heat conduction, the heat-transfer resistance between the hot feed stream and the coolant stream includes the hot solution, the membrane, the air gap, the condensate film, the cooling plate, and the cold solution. The relevant heat-transfer resistance in the heat conduction through the membrane is shown in Figure 3. Thereby, the overall heat-transfer coefficient of the conductive heat ks is represented by !1 1 δm δa δcp 1 1 ks ¼ þ þ þ þ þ ð17Þ h1 km ka kcp hf h2 where h1 and h2 are the heat-transfer coefficients of the hot solution and the cold solution, respectively; hf is the heat-transfer coefficient of the condensate film; and km, ka, and kcp represent the thermal conductivity of the membrane, air, and cooling plate, respectively. Here, the effect of the temperature on the thermal conductivity is neglected. δm, δa, and δcp are the thickness of the 7347



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mbi, y ¼ mbo, y þ Jy Sy

ð22Þ

Tco, y ¼ Tci, y1

ð23Þ

where Tbo,y1 denotes the temperature of the hot feed stream leaving the (y 1)th stage. It is equal to the temperature of the hot feed stream entering the yth stage, Tbi,y. ΔTstage,y is the temperature difference of the hot feed stream at the yth stage. A similar convention is used for the mass flow rate of the hot feed stream and the temperature of the coolant stream. Jy is the pure water flux of the membrane module at the yth stage and can be calculated by eq 3. Sy denotes the total membrane area of the yth stage. The flow rate of the coolant stream (mc) is a given constant. For the multistage AGMD system design, the number of stages is set to be the design variable. The constraints on checking whether a stage exists or not should be included: Zy Ls e Sy e Zy Us

ð24Þ

Figure 3. Relevant heat-transfer resistance in the AGMD process.

membrane, the air gap, and the cooling plate, respectively. Among these heat-transfer resistances, the air gap contributes most significantly to the resistance; hence, the overall heattransfer coefficient of the conductive heat ks is mainly dependent on the separation distance of the air gap and the thermal conductivity of the air. For a unit of the module, the mass balance of the hot feed stream is represented by mbi ¼ mbo þ JSm

ð18Þ

where mbo denotes the flow rate of the hot feed stream leaving the module and Sm is the membrane module area.

4. THE MULTISTAGE AGMD SYSTEM MODEL In AGMD, since 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 exiting an AGMD module remains high. In order to effectively utilize the energy contained in the hot feed stream and increase the product recovery, a multistage structure of membrane modules, such as that shown as Figure 1, is proposed. This multistage AGMD system is a combination of identical membrane modules. The number of stages and the number of modules in each stage are the design variables, and the optimal system structure could be determined by solving these variables with the mathematic programming method. Through the optimal combination of the membrane modules, the pure water yield would increase as the thermal efficiency of the system increases. Here, the stage number 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. Thus, the energy balance and the mass balance between the membrane modules are Tbo, y 1 ¼ Tbi, y

ð19Þ

Tbi, y ¼ Tbo, y þ ΔTstage, y

ð20Þ

mbo, y1 ¼ mbi, y

ð21Þ

Ny ¼

Sy Sm

ð25Þ

where Zy is a binary variable. Zy = 1 when the yth stage exists and vice versa. Ls and Us are the lower and upper bounds of the membrane area, respectively. Ls is usually set to be the area of one module (Sm). Ny denotes the number of membrane modules at the yth stage. In order to reduce the computation complexity, Ny is considered to be a continuous variable, and the approximate result would be obtained by rounding the variable. As shown in Figure 1, the membrane modules are arranged parallel in each separation stage. In the yth separation stage, the calculated variable Sy would be a positive value; the number of module Ny in this stage can be directly calculated by eq 25; otherwise, the values of Sy and Ny are zero. Thus, eq 24 is used to judge whether there is a separation stage. At the same stage, the stream outlet temperature and the mass and energy balance can also be calculated using eqs 318, but the membrane area of a unit of the module (Sm) is substituted by the total membrane area of the yth stage (Sy). Since identical membrane modules are employed in the AGMD system, each module at the same stage is considered to have uniform operating conditions. If the yth separation stage exists, the flow rate entering each module, ms,bi,y, can be calculated by dividing the overall flow rate, mbi,y, entering every module at each stage. According to the rule of the operating module, the flow rate entering a module should be kept within bounds. If the yth separation stage does not exist, it is not necessary to calculate the flow rate entering each module; however, the overall flow rate (mbi,y) still exists, so it continues to flow to the next stage. Thus, eq 26 can be used to judge whether the flow rate entering the module would be calculated, while eq 27 is a constraint for the scope of the flow rate entering a module. Ny ms, bi, y mbi, max ð1 Zy Þ e mbi, y e Ny ms, bi, y þ mbi, max ð1 Zy Þ

ð26Þ

Lm e ms, bi, y e Um

ð27Þ

where mbi,max is a large positive number and it is usually equal to the maximum flow rate of the hot feed stream entering the AGMD system. Lm and Um are, respectively, the lower and upper bounds of the flow rate of the hot feed stream entering each module. Equation 26 indicates that, when the yth stage exists, the 7348



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flow rate entering each module (ms,bi,y) can be calculated by combining eqs 21, 24, and 25. If the yth stage does not exist, the flow rate of the hot feed stream entering the yth stage (mbi,y) is just equal to the flow rate of the hot feed stream leaving the (y 1)th stage (mbo,y1). The water output (md,y) in the yth separation stage can be calculated if the stage exists and the above constrains are met. Thus, the total water output is the summation of all the stage outputs and is expressed as md ¼

Y

Y

∑ md, y ¼ y∑¼ 1 JySy y¼1 md g mmin d

ð28Þ ð29Þ

but it should meet the requirement of the minimum yield, mmin d . Jy can be computed using eqs 36. To establish the above multistage AGMD system, the cost that includes the annualized capital cost of the membrane module and the energy cost of pumps shall be minimized. The cost is expressed as follows: CAGMD ¼ Cm

Y

∑ Sy þ Ce ðPhot þ Pcold Þ 24 330 y¼1

ð30Þ

where Phot ¼

γw Hhot mbi, 1 ηp

ð31Þ

Pcold ¼

γw Hcold mci, Y ηp

ð32Þ

and

Here, Cm denotes the unit cost of the membrane area, and Ce denotes the power price. Phot and Pcold are the pump power of the hot feed stream and the coolant stream, respectively. γw is the 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 ¼

QV QT

ð33Þ

From eqs 14 and 16, it is found that the thermal efficiency is affected by the temperatures of the hot side and the cold side of AGMD. As the temperatures at both side decrease, the heattransfer coefficient of the vapor heat kv decreases; as a result, the thermal efficiency also decreases. Therefore, for the multistage AGMD system, the number of stages would affect the process thermal efficiency. Since low investment cost and high system thermal efficiency are desired, the optimization design problem of the multistage AGMD system is a multiobjective optimization problem (see eqs 1 and 2). If an acceptable value of thermal efficiency (ηt) is predefined, the objective function of maximizing thermal efficiency could be converted to a constraint: η m g ηt

ð34Þ

In this way, the optimization problem could be formulated into a single-objective optimization problem, and its objective function is represented by eq 1. In this study, assume that the

waste heat as the energy source of the MD process is free. It is possible to produce more pure water by consuming more heat energy; therefore, the efficiency is set above a certain value (eq 34) to avoid the MD modules with less efficiency at the lower temperature. With the above mathematical model describing AGMD processes (i.e., eqs 1934), the design problem of the multistage AGMD system can be solved using the optimization method. During this progression, the permeate flux and the heat flux can be calculated based on the AGMD unit model (i.e. eqs 318). 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 rate 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 is a MINLP problem. The optimal system structure and the operating variables of the AGMD system can be determined.

5. THE MULTISTAGE AGMD SYSTEM WITH DISTRIBUTION STRUCTURE OF THE COOLANT STREAM In Section 4, the design method assumes that the countercurrent operation mode is employed. In this operation, the hot feed stream flows along the first stage to the last stage, while the coolant stream flows in an opposite direction from the last stage to the first stage. Although the mode of the countercurrent operation is commonly used in practice, this operation mode is chosen based on our experience; it cannot be proved that, in theory, the configuration of Figure 1 is the optimal flow route for the AGMD system. For this reason, the flow routes of the cold streams would be changed; they can enter and leave the AGMD system via different routes, which constitute the superstructure flow sheet shown in Figure 4, to describe all the possible flow routes for the cold streams. These flow routes of the superstructure mode actually encompass those of the first operating mode shown in Figure 1. Since the permeate flux is proportional to the transmembrane temperature difference, one feasible way to increase the flux is by changing the route of the coolant stream entering and leaving the AGMD system. Figure 4 shows the route map of the coolant stream. In this figure, the coolant stream from the cooling tower or other operation systems can enter the system from different stages, and this, together with the stream leaving the previous module, forms the inlet coolant stream. Similarly, the stream leaving a module can either enter the next module or exit the system. Under this operation mode, the mixture and separation of the coolant stream would be designed to increase the permeate flux. Therefore, the optimal distribution structure mode, including the flow routes of the cold stream, the configuration of the AGMD system, and the operation condition, should be considered. In order to illustrate all the possible appearance of the coolant stream mixture or separation, the state-space approach is used to describe the route of the coolant stream, which is shown in Figure 5. In this figure, the nodes e and h indicate the entrance and the exit of the AGMD system, respectively, for the coolant stream. The sets of nodes py and qy indicate the entrance and the exit of the membrane module at the yth stage, respectively, for the coolant stream. The coolant stream coming from the node e can be distributed to each entrance node py; then, the relationship of the mass balance of the coolant stream at the set of nodes py is expressed as mtc, y ¼ mtc, in ay 7349

ð35Þ



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Figure 5. Route map of the coolant stream via the state-space approach.

Figure 4. Route map of the coolant stream entering and leaving the AGMD system.

Y

∑ ay ¼ 1 y¼1

ð36Þ

where m tc,in indicates the total flow rate of the coolant stream and m tc,y is the subflow rate of the total coolant stream entering the yth stage. These values can be determined by the stream split ratio (a y). Thus, the flow rate entering the yth stage (m ci,y), which includes the coolant stream leaving the (y þ 1)th stage and the stream coming from the node e, can be computed by mci, y ¼ mtc, in ay þ mco, y þ 1 by

ð37Þ

mci, y ¼ mco, y

ð38Þ

mci, Y ¼ mtc, in aY

ð39Þ

where mco,yþ1 is the flow rate coming from the (y þ 1)th stage, and by is the stream split ratio of mco,yþ1, which determines the flow rate entering the yth stage. The other part of mco,yþ1 would exit the AGMD system. Equation 39 indicates that the inlet flow rate of the last stage only comes from the total coolant stream. The relationships of energy balance of the coolant stream at the set of node py are as follows: Cw Tci, y mci, y ¼ Cw Ttc, in mtc, in ay þ Cw Tco, y þ 1 mco, y þ 1 by ð40Þ Tci, Y ¼ Ttc, in

ð41Þ

where Tci,y is the temperature of the coolant stream entering the yth stage. Tco,yþ1 is the temperature of the coolant stream leaving the (y þ 1)th stage. It can be calculated by eqs 9 and 10. Tci,Y denotes the temperature of the coolant stream entering the last stage. Ttc,in denotes the temperature of the total coolant stream. In this design method, the value of Ttc,in would be assigned beforehand. For system exit node h, the following mass and energy balances are given: mtc, out ¼

Y1

∑ mco, yþ1ð1 byÞ þ mco, 1

y¼1

Cw Ttc, out mtc, out ¼ Cw

ð42Þ

Y1

∑ Tco, y þ 1mco, y þ 1ð1 by Þ

y¼1

þ Cw Tco, 1 mco, 1

ð43Þ

where mtc,out and Ttc,out are the flow rate and the temperature of the total coolant stream exiting the AGMD system, respectively. mco,1 and Tco,1 are the flow rate and the temperature of the coolant stream leaving the first stage, respectively. With the combination of eqs 3543, the AGMD unit model, and the system model, the optimization design of the AGMD system would be solved and its objective function is eq 1. The design result would yield the optimal system structure, the operating variables, and the distribution of the coolant stream.

6. CASE STUDY In this section, an example is presented to illustrate the proposed design methodology. With the inlet temperature of both feed streams, the AGMD system would be designed. Before implementing the system design, the sensitivity of the process thermal efficiency to the operating temperature is analyzed to determine the range of operating temperatures that maintains the high thermal efficiency. The unit model presented earlier is used 7350



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Table 1. The Model Parameters and Module Features parameter

value

coefficients in eq 4 ∂

3.2 107

β

6 103 2

area of membrane module, Sm (m )

1.6

range of flow rate for each membrane module, ms,bi,y (kg/s)

0.240.3

heat-transfer coefficients hot solution, h1 (W/m2 3 K)

cold solution, h2 (W/m2 3 K) condensate film, hf (W/m2 3 K) thickness

500 800 192

membrane, δm (mm)

0.4

air gap, δa (mm)

2

cooling plate, δcp (mm)

1.5

thermal conductivity membrane, km (W/m 3 K)

0.2

air, ka (W/m 3 K) cooling plate, kcp (W/m 3 K)

0.03 60

latent heat of evaporation, λ (kJ/kg)

2257.2

specific heat of water, Cw (kJ/kg 3 °C)

4.2

range of operating temperatures hot feed stream, Tbi (°C)

4080

coolant stream, Tci (°C)

2050

unit cost of membrane area, Cm ($/m2)

10

power price, Ce ($/kW 3 h) pumping heads

0.06

hot stream pump, Hhot (m)

12

cold stream pump, Hcold (m)

12

efficiency of the pump, ηp

0.9

to simulate the AGMD process. Its model parameters and module features are listed in Table 1, with reference to refs 8 and 11. The model parameters ∂ and β in eq 4 are first regressed with the experimental data. The model predicted results are validated by our membrane module experimental values. Figure 6 shows that the model predicted values and experiments are in very good agreement. Based on the established AGMD unit model, the process simulation has been performed under a series of inlet temperatures of the hot feed stream (Tbi) and the coolant stream (Tci), and the sensitivity of the process thermal efficiency to the operating temperature is analyzed. The simulation result is shown in Figure 7. In Figure 7a, the value of thermal efficiency (ηm) increases as Tbi increases, because the permeate flux increases with the enlarging transmembrane temperature difference. In Figure 7b, reducing Tci increases the permeate flux but decreases ηm, because of the decrease of the contribution of the mass-transfer resistance of the cold stream to the total masstransfer resistance.15 In this figure, the value of ηm can range from 0.85 to 0.95 when Tbi and Tci vary within the operating scope. Hence, in this case, the desired high thermal efficiency (ηt) in eq 34 can be assumed to be 0.9, since it is possible for the membrane module employed to maintain ηm at this relatively high value by regulating Tbi and Tci. The inlet temperatures of the hot feed stream and the coolant stream are fixed at 80 and 20 °C, respectively. The total required productivity of pure water is no less than 0.2 kg/s. If only the

Figure 6. (a) Effect of the inlet temperature of the hot feed stream on the permeate flux, in comparison with experiments when Tci = 25° C. (b) Effect of the inlet temperature of the cold feed stream on the permeate flux, in comparison with experiments when Tbi = 50 °C.

single-stage structure is applied, the design results shown in Table 2 are obtained. Under the same conditions, the multistage structures based on Figures 1 and 4 are separately conducted, and two types of operating modes of the coolant stream are presented. The first operating mode shown in Figure 1 is that the hot feed stream flows along the first stage to the last stage, while the coolant stream flows in an opposite direction from the last stage to the first stage. From the design results shown in Table 3, it is found that the AGMD system consist of three stages with 21, 17, and 16 membrane modules at each stage. The total productivity of pure water is 5808 m3/yr and the unit cost of the product is $0.25/m3 for this AGMD system. Although the total productivity of the system with the multistage structure is similar to that of the single module, the inlet flow rate of the hot feed stream in the single-stage AGMD system is almost double that of the multistage system. This means that the single-stage system has a lower productive rate and consumes more energy; therefore, it results 7351



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Table 3. Design Results of the AGMD System for the First Operating Mode parameter

value

number of stages, y

3

inlet flow rate of the hot feed stream, mbi,1 (kg/s) inlet flow rate of the coolant stream, mci,3 (kg/s)

5 5

system heat efficiency, η

0.92

capital cost of the membrane ($/yr)

853

energy cost of the pump ($/yr)

633.6

total yield of pure water (m3/yr)

5808

unit cost of pure water in the AGMD system ($/m3)

0.25

For the First Stage rounded number of modules, n

21

permeate flux at the first stage, J (kg/s 3 m2) average flow rate of the hot feed stream entering each

0.003 0.24

module at the first stage, ms,bi,1 (kg/s) inlet temperature of the coolant stream at the first

33

stage, Tci,1 (°C) outlet temperature of the coolant stream at the

44

first stage, Tco,1 (°C) temperature drop of the hot feed stream at the first stage (°C)

11

For the Second Stage rounded number of modules, n permeate flux at the second stage, J (kg/s 3 m2)

flow rate of the hot feed stream entering the second

17 0.002 4.91

stage, mbi,2 (kg/s) average flow rate of the hot feed stream entering each

0.3

module at the second stage, ms,bi,2 (kg/s)

Figure 7. (a) Effect of the hot feed stream (Tbi) on the permeate flux (J) and the thermal efficiency (ηm) when Tci = 20 °C. (b) Effect of the cold feed stream (Tci) on the permeate flux (J) and the thermal efficiency (ηm) when Thi = 70° C.

inlet temperature of the hot feed stream at the second stage, Tbi,2 (°C)

69

inlet temperature of the coolant stream at the second

25.8

stage, Tci,2 (°C) temperature drop of the hot feed stream at the

7.2

second stage (°C) For the Third Stage

Table 2. Design Results of the Single-Stage AGMD System parameter

value

rounded number of module, n

16

permeate flux at the third stage, J (kg/s 3 m2) flow rate of the hot feed stream entering the third

0.002 4.85

stage, mbi,3 (kg/s) flow rate of the hot feed stream leaving the third

inlet flow rates hot feed stream, mbi (kg/s)

9.6

coolant stream, mci (kg/s) hot feed stream temperatures

9.6

4.8

stage, mbo,3 (kg/s) average flow rate of the hot feed stream entering each

0.3

module at the third stage, ms,bi,3 (kg/s)

inlet temperature, Tbi (°C)

80

inlet temperature of the hot feed stream at the third

62

outlet temperature, Tbo (°C)

67.4

stage, Tbi,3 (°C) outlet temperature of the hot feed stream at the third

56

coolant stream temperatures inlet temperature, Tci (°C)

20

outlet temperature, Tco (°C)

32.6

rounded number of module, n

40

capital cost of the membrane ($/yr) energy cost of the pump ($/yr)

640 1216.5

total yield of the pure water (m3/yr)

5909.7


0.32

stage, Tbo,3 (°C) temperature drop of the hot feed stream at the third stage (°C)

6

in a higher unit cost of the product than the multistage one. This illustrates the efficiency of the proposed design method. The sensitivity analysis is conducted when the AGMD system is designed based on the first operating mode. The results 7352



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Table 4. Design Results of the AGMD System for the Second Operating Mode parameter

parameter

value

number of stages, y

3

inlet flow rate of the hot feed stream, mbi,1 (kg/s) inlet flow rate of the coolant stream, mci,3 (kg/s)

5.2 15

system heat efficiency, η

0.91

capital cost of the membrane ($/yr)

1002

energy cost of the pump ($/yr)

1277

3

Table 4. Continued

total yield of pure water (m /yr)

6743


0.34

value

inlet temperature of the hot feed stream at the third stage, Tbi,3 (°C)

58.7

outlet temperature of the hot feed stream at the third

52.3

stage, Tbo,3 (°C) outlet temperature of the coolant stream at the third

26.4

stage, Tco,3 (°C) temperature drop of the hot feed stream at the third

6.4

stage (°C) split ratio of the total coolant stream entering the third stage, a3

0.33

For the First Stage rounded number of modules, n

22

permeate flux at the first stage, J (kg/s 3 m2) average flow rate of the hot feed stream entering each

0.004 0.24

module at the first stage, ms,bi,1 (kg/s) outlet temperature of the hot feed stream at the first

67.4

stage, Tbo,1 (°C) outlet temperature of the coolant stream at the first

32.6

stage, Tco,1 (°C) temperature drop of the hot feed stream at the first

12.6

stage (°C) split ratio of the total coolant stream entering the first

0.34

stage, a1 split ratio of the coolant stream coming from the second

0

stage, b2 For the Second Stage rounded number of modules, n permeate flux at the second stage, J (kg/s 3 m2)

21 0.003

flow rate of the hot feed stream entering the second

5.0

stage, mbi,2 (kg/s) flow rate of the hot feed stream leaving the second

4.9

stage, mbo,2 (kg/s) average flow rate of the hot feed stream entering each

0.24

module at the second stage, ms,bi,2 (kg/s) inlet temperature of the hot feed stream at the

67.4

second stage, Tbi,2 (°C) outlet temperature of the hot feed stream at the

58.7

second stage, Tbo,2 (°C) outlet temperature of the coolant stream at the

28.7

second stage, Tco,2 (°C) temperature drop of hot feed stream at the second stage (°C)

8.7

split ratio of the total coolant stream entering the

0.33

second stage, a2 split ratio of the coolant stream coming from the

0

third stage, b3 For the Third Stage rounded number of module, n permeate flux at the third stage, J (kg/s 3 m2)

flow rate of the hot feed stream entering the third stage,

21 0.003 4.9

mbi,3 (kg/s) flow rate of the hot feed stream leaving the third stage,

4.8

mbo,3 (kg/s) average flow rate of the hot feed stream entering each module at the third stage, ms,bi,3 (kg/s)

0.24

indicate that the objective function is most sensitive to the heattransfer coefficient of the vapor heat (kv). It will decrease 10 units of the objective function when kv increases by one unit, since kv is dependent on the module configuration and the temperature of the hot stream and the cold stream. Thus, designing a module based on the above factors may be the prior choice to improve the system performance. Second increasing the membrane area can also reduce the objective function. When it increases one unit, the objective function would decrease 1.3 units. On the other hand, the flow rate of the hot stream and the cold stream have a little effect on the objective function. Based on the proper description of the distribution structure, the system design would be performed under the same design conditions as the first operating mode. The calculation results can be obtained by solving the mathematical programming model presented in Section 5. The results indicate that the optimal solution is the same as the first case shown in Table 3. That is, the flow route specified in the first operating mode is the optimal way. In order to compare the design results under different operation conditions, it is assumed that the inlet temperatures of the coolant stream at each stage are all set to be 20 °C. The second operating mode of the multistage AGMD system then is designed under the distribution structure. The design results of the main variables are presented in Table 4. It indicates that this multistage AGMD system, which is employing the second operating mode, consists of three stages with 22, 21, and 21 membrane modules at each stage. The split ratios of the total coolant stream entering each stage are 0.34, 0.33 and 0.33, respectively. In this way, the inlet temperatures of the coolant stream for each stage are lower than those in the first operation mode; then ,it could result in higher permeate flux, because of the increase in the transmembrane temperature difference. However, as the total flow rate of the coolant stream increases to supplement the coolant stream at each stage and decrease the stream temperature, the energy consumption of the cold stream pump must be increased. As a result, the additional energy cost leads to a higher unit cost of pure water ($0.34/m3) in this second operating mode, although it has higher productivity under the previous operation conditions. In Tables 3 and 4, it is found that these two operating modes have high heat efficiency (over 90%); however, the heat efficiency of the second operating mode (in Table 4) has a slight declining trend, because their coolant streams have a low temperature, leading to more heat loss by way of the conduction. Also, the design results show that the second operating mode has more capital cost and energy cost than the first one, because more membrane modules are needed 7353


Industrial & Engineering Chemistry Research to produce sufficient pure water in the second mode. As a result, through the evaluation of the heat efficiency, the capital cost, and the energy cost, the first operating mode has better performance.

7. CONCLUSION The research on membrane distillation (MD) over the past 25 years has shown that MD is regarded as a promising method of producing pure water. It takes advantage of the low operating temperature. In many areas of chemical process design and operation, the process synthesis and optimization have shown significant benefits. The design problem of the membrane distillation process has rarely been tackled from a system engineering perspective. In this research, the multistage air gap membrane distillation (AGMD) system is designed by the mathematical programming method, based on the system model. The design results present an optimal multistage AGMD system which can improve the water production rate of the system and maintain a high efficiency of energy utilization; simultaneously, the optimal operation conditions and explicit stream data are also given. MD has been found to be most suitable when the required energy is supplied by waste heat, because MD is energy-intensive. In the future, a large part of the membrane module, which uses saline water as cooling water, will be developed in a manner similar to that of the classic multistage flash distillation and will be compared with the proposed MD structure. It could be worthwhile to see if the design cost would be reduced. Also, the integration of the heat-exchanger networks and MD systems will be investigated. It utilizes the waste heat from the heat-exchanger networks as the energy source of the MD process, to improve the efficiency of the process and greatly reduce the water cost.

ARTICLE

(8) Alklaibi, A. M.; Lior, N. Heat and mass transfer resistance analysis of membrane distillation. J. Membr. Sci. 2006, 282, 362–369. (9) Cheng, L.-H.; Wu, P.-C.; Chen, J. Modeling and optimization of hollow fiber DCMD module for desalination. J. Membr. Sci. 2008, 318, 154–166. (10) Cheng, L.-H.; Wu, P.-C.; Chen, J. Numerical simulation and optimal design of AGMD-based hollow fiber modules for desalination. Ind. Eng. Chem. Res. 2009, 48, 4948–4959. (11) Liu, G. L.; Zhu, C.; Cheung, C. S.; Leung, C. W. Theoretical and experimental studies on air gap membrane distillation. Heat Mass Transfer 1998, 34, 329–335. (12) Gilron, J.; Song, L.; Sirkar, K. K. Design for cascade of crossflow direct contact membrane distillation. Ind. Eng. Chem. Res. 2007, 46, 2324–2334. (13) Wang, J.; Fan, B.; Luan, Z.; Qu, D.; Peng, X.; Hou, D. Integration of direct contact membrane distillation and recirculating cooling water system for pure water production. J. Cleaner Prod. 2008, 16, 1847–1855. (14) Meindersma, G. W.; Guijt, C. M.; Haan, A. B. Desalination and water recycling by air gap membrane distillation. Desalination 2006, 187, 291–301. (15) Alklaibi, A. M.; Lior, N. Comparative study of direct-contact and air-gap membrane distillation. Ind. Eng. Chem. Res. 2007, 46, 584–590.

’ AUTHOR INFORMATION Corresponding Author

*Tel.: þ886-3-2654107. Fax: þ886-3-2654199. E-mail address: [email protected].

’ ACKNOWLEDGMENT This research work has been funded by grants from the Center-of-Excellence Program on Membrane Technology, the Ministry of Education, Taiwan, ROC. ’ REFERENCES (1) Lawson, K. W.; Lloyd, D. R. Membrane distillation. J. Membr. Sci. 1997, 124, 1–25. (2) Alklaibi, A. M.; Lior, N. Membrane-distillation desalination: Status and potential. Desalination 2004, 171, 111–131. (3) Charcosset, C. A review of membrane processes and renewable energies for desalination. Desalination 2009, 245, 214–231. (4) Hanemaaijer, J. H.; Van Heuven, J. W. Methods for the purification of a liquid by membrane distillation, in particular for the production of desalinated water from seawater or brackish water or process water. U.S. Patent No. 6,716,355, 2004. (5) Gryta, M.; Tomaszewska, M. Heat transport in the membrane distillation process. J. Membr. Sci. 1998, 144, 211–222. (6) Guijt, C. M.; Meindersma, G. W.; Reith, T.; de Haan, A. B. Air gap membrane distillation. 1. Modelling and mass transport properties for hollow fibre membranes. Sep. Purif. Technol. 2005, 43, 233–244. (7) Guijt, C. M.; Meindersma, G. W.; Reith, T.; de Haan, A. B. Air gap membrane distillation. 2. Model validation and hollow fibre module performance analysis. Sep. Purif. Technol. 2005, 43, 245–255. 7354


Optimal Design of Multistage Membrane Distillation Systems for Water

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