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International Journal of Environmental Research
University of Tehran
ISSN: 1735-6865 EISSN: 2008-2304
Vol. 4, Num. 1, 2010, pp. 11-26

International Journal of Environmental Research, Vol. 4, No. 1, 2010, pp. 11-26

Simultaneous Energy and Water Optimization in Multiple - Contaminant Systems with Flowrate Changes Consideration

Ataei, A. and Yoo, C. K.^*

Center for Environmental Studies / Green Energy Center, Dept. of Environmental Science and Engineering, College of Engineering, Kyung Hee University, Seocheon-dong 1, Giheung-gu, Yongin-Si, Gyeonggi-Do, 446-701, Korea
*Corresponding author E-mail: ckyoo@khu.ac.kr

Received 7 Nov. 2008; Revised 16 Aug. 2009; Accepted 25 Aug. 2009

Code Number: er10002

ABSTRACT

In this paper, a new systematic design methodology has been developed for the simultaneous energy and water minimization in multiple-contaminant systems that also feature maximum re-use of water. In addition to allowing re-use of water in multiple-contaminant systems, issues about heat losses and flowrate changes inside unit operations have also been incorporated in this new design method. To implement such a design, two new design aspects are introduced; new method for “Non-isothermal Mixing” point identification and new “Separate System” generation. The first aspect involves “non-isothermal mixing”, which enables direct heat recovery between water streams, and therefore allows the reduction of the number of heat transfer units. The other aspect is the generation of “separate system “in heat exchanger network design. The flexibility of mixing and splitting of water streams allows separate systems to be created as a cost-effective series of heat-exchanger units between freshwater and wastewater streams. The new design aspects have been illustrated with two examples.

Key word: Heat loss, Non-isothermal mixing, Separate system, Heat recovery, Heat-exchanger network

INTRODUCTION

Water is one of the most widely used raw materials in chemical and petroleum industries. Significant amounts of water are required in washing, stripping, and manufacturing processes. As water resources face scarcities, everincreasing prices, and more stringent environmental regulations, much attention has been paid to reduce freshwater consumption and wastewater generation (Kim & Smith, 2002), (Ataei et al., 2009d) There are conceptual and automated approaches as two traditional methods to design water networks with re-use of water (Panjeshahi and ataei, 2008). The former analysis exploits graphical tools to explore the possibilities of water reuse, whilst the latter employs mathematical optimization models to obtain a costeffective solution (Alva-Argaez, 1999), (Ataei et al., 2009c). The analysis of water management generally involves water distribution among water using operations with the criteria of contaminant concentration levels (Mann & Liu, 1999).

In some cases such as sterilization and process-washing, temperature of water becomes as important as the quality of water (Bagajewicz et al., 2002). The water system is now subject to not only the constraints of contaminant concentration levels, but also those of the temperature levels. Water streams need to be heated up or cooled down to satisfy the temperature requirements of the operations and energy consumption become necessary for these heating and cooling tasks (Smith, 2005). Under these circumstances, energy and water management needs to be considered simultaneously. Therefore, the problem has become a combined analysis of water and energy systems (see Fig. 1).

The simultaneous energy and water minimization was first addressed by Savulescu (1999). In this methodology, several assumptions are made for problem simplification but these assumptions make the design inaccurate. Some of these assumptions are:

Each water-using operation has a fixed temperature and runs isothermally.
The water flowrate through an operation does not change.
Only single contaminant operations are considered.

It should be noted that for particular operations, temperature of water changes and hence isothermally running assumption for practical water-using operations cannot be correct. Furthermore, in industrial practices, many waterusing operations have fixed flowrate requirements, such as in many vessel-cleaning operations (Young et al., 2006). Also, there may be a fixed flowrate of water loss (e.g., cooling-tower evaporation) or gain (e.g., dewatering filter). Accordingly, the flowrate changes should be considered in design of water-using networks with minimum water and energy consumption. Savulescu design considers only the networks with single contaminant and the non-isothermal mixing point identification is based on water-pinch analysis and synthesis. Therefore, extension of this method for multiple contaminant problems may be tedious. Accordingly, An NLP model should be formulated to identify feasible non-isothermal mixing points, which satisfy minimum freshwater and utility requirements in both of single and multiple contaminant problems. Two main stages are suggested for simultaneous water and energy minimization through Savulescu methodology: Stage 1: Two dimensional grid diagram for designing a water network.

Stage 2: Separate system approach for designing a heat exchanger network.

This method is a sequential approach that follows a set of design rules in the first stage, to provide a water network with less heat exchanger units required. These rules, however, do not always guarantee minimum utility requirement. In other words, the actual utility requirement of the design is higher than the utility target and the design with small number of heat exchangers could be obtained but with utility penalty. Furthermore, in the presented design method, temperature of some water streams in the network may increase to above the normal boiling temperature. This temperature increasing can cause many operational problems for the process; however, increasing of the process pressure, which suggested in this method, cannot be a no-cost and easy solution for these problems.In the second stage, the idea of generating separate systems to simplify a heat exchanger network design was introduced. Nevertheless, the generation of separate systems has not been fully explored from the recognition that a smaller number of heat exchanger units could be acquired. Moreover, the optimum heat transfer area in each separate system should be explored by introducing a tradeoff between the capital cost of heat exchanger and the power losses because of the pressure drops of each fluid to achieve minimum total annual cost. Accordingly, a new methodology should be developed to construct a water structure without the utility penalty and the increasing of water streams temperature to above the normal boiling point, and provide a heat exchanger network with minimum number of units and optimum heat transfer area.This paper addresses the simultaneous management of energy and water as an approach for multiple-contaminant systems with maximum re-use of water. In addition, the heat loss and water flowrate changes through operations have been considered in this new methodology. In other words, In addition to overcome the aforementioned limitations of Savulecsu design method, the simplifier assumptions of it have been relaxed in this new simultaneous water and energy minimization approach.The new simultaneous water and energy minimization technique has been tested through two illustrative examples. Related coding in GAMS optimization package was used for illustrative examples to get optimal values in the proposed design method computations.

MATERIALS & METHODS

The new systematic design methodology has been developed for the simultaneous management of energy and water in multiple-contaminant systems that also feature maximum re-use of water. In addition to allowing re-use of water, issues about heat losses and water flowrate changes inside unit operations have also been incorporated in this design method. The general features of the problem involve a set of waterusing operations with specifications of flowrates, temperature and contaminant concentration levels, a selection of water sources with different qualities, and a number of heat transfer units. It is desired to determine water and energy targets and specify the distribution of water among the water-using operations as well as the allocation of heat exchangers between these water streams in order to complete the overall network configuration.The new design method comprises two new design aspects; new method for “Non-isothermal Mixing” point identification to design a water network with the minimum freshwater and energy requirements and new “Separate System” generation for designing a heat exchanger network with minimum number of heat exchanger units and optimum heat transfer area. Moreover, in the proposed method, the optimum detail design of the heat exchanger related to each separate system can be achieved. . Fig 2 shows an example of the non-isothermal mixing area and separate systems in the cold and hot composite curves.

New Method for “Non-Isothermal Mixing”Point Identification

Non-isothermal mixing enables direct heat recovery between water streams, and therefore allows the reduction of the number of heat transfer units. However, non-isothermal mixing can cause the degradation of temperature driving forces, and also reduces the number of possibilities of indirect heat transfer matching between hot and cold streams (Baldyga et al., 1998). Thus, in the introduction of non-isothermal mixing, a water network without utility penalty should be considered.

In this study, an NLP model is formulated to identify feasible non-isothermal mixing points, which satisfy not only the inlet requirements (temperature and contaminant concentration levels) of the operations but also achieve the minimum freshwater and utility requirements and create an overall water network with fewer number of heat exchanger units. By using this mathematical model, the water network design with small number of heat exchangers and minimum operating cost can be obtained without utility penalty. . Fig 3 depicts a general water-using operation i. Here, we define the operation with a fixed mass load of contaminant j(j=1,2,3^,...,ⁿcontaminants ), to be transferred, ∆m_{i, j,tot}, and with maximum allowable concentrations of contaminant j(j=1,2,3,...,n_contaminants) at the inlet, C_{i, j,in}^max and outlet, C_{i, j,out^max}of the operation. We include inlet streams from the freshwater source at temperature T₀ and heated to T_fi with a flowrate, f_i(i=1,2,3,...,n_{operati
ons}), as well as streams reused from other operations, k(k=1,2,3,...,n_{operati
ons}), at a flowrate, Xi,k, temperature of and a contaminant (j=1,2,3,...,n_contaminants) concentration, C_k,j,out. Likewise, we consider an outlet stream to wastewater treatment at a flowrate, W_i, temperature of T_i,outand a contaminant (j=1,2,3,...,n_contaminants) concentration, C_i,j,out ,and outlet streams for reuse in other operations, k(k=1,2,3,...,n_{operati ons}) at flowrates, X_k,i^,temperature of T_i,out and concentration of contaminant (j=1,2,3,...,n_contaminants),C^{_i,j,out^.}

The total operating cost, as the objective function, is expressed in Eq. (1) (Ataei et al., 2009a);

We formulate the constraints governing water reuse from the maximum inlet and outlet concentrations as well as the fixed mass load of contaminants transferred in each operation. We calculate the average inlet concentration of contaminant j , C_{i, j,in}, by the flowrate-weighted average of the concentrations provided by the fresh water source and reused from other operations;

We relate the outlet concentration of contaminant j from operation i ,_{Ci, j,out}, to C_i,j,in and the change in concentration due to the fixed mass load of contaminant j transferred, ∆mi, j,tot, as follows:

Substituting for Ci,in from Eq. (2) into Eq. (3) gives;

By re-arranging Eqs. (3) and (4), a set of more linear constraints can be formed as follows;

In addition, we include a mass balance on water around each operation i as follows;

We specify that all concentrations and flowrates be positive. The temperature of inlet water stream to the operation i , Ti,inand the temperature of outlet water stream from the operationi ,T_{_i_,_out} , are fixed and known parameters. The constraint related to the fixed and known amount of inlet water temperature can be expressed as Eq. (8);

The energy requirement for heating of the inlet freshwater to the operation i from temperature T₀ to T_fiis given by Eq.(9);

Q_i = Kf_iC_p (T_fi - T_o ) (9)

The nonlinear program to optimize the water-using network, without water flowrate changes consideration, is to minimize the total operating cost, OPCOST expressed in Eq. (1), subject to Eqs. (5), (6), (7), (8) and (9).As we develop constraints for mathematical optimization, we have a greater freedom to tailor our model for the type of waterusing operations involved. For non-isothermal mixing point identification in water networks with water flowrate changes, (water gain, water loss and fixed water flowrate), the presented NLP model should be revised.For a water gain, we can formulate the constraints governing water reuse from the maximum inlet and outlet concentrations as well as the fixed mass load of contaminant j transferred in each operation just as in Eqs. (5) and (6). However, we include an increase in the flowrate through operation i , fi,gain . The mass balance on water around each operationi with only a water gain becomes;

When Eq. (10) is included in place of Eq. (7) as a mass balance on water, the optimization procedure will make an additional flowrate of fi,gain available for reuse from operation i. Equations (5) and (6) remain valid as the constraints on the limiting inlet and outlet concentrations for operation i , respectively. In this condition, the constraints governing inlet and outlet water temperatures as well as fresh water heating from temperature T₀to T_fi just as in Eqs. (11) and (12). Therefore, Eqs. (11) and (12) should be included in place of Eqs.(8) and (9).

Q_i =K(f _i + f_{i ,gain})C_p (T_fi -T_o ) (12)

However, if we wish to include a water loss at the limiting inlet concentration of contaminant j , we must modify both constraints on the limiting inlet and outlet concentrations as well as the water balance. The average inlet concentration of contaminant j,C_{_i_,_j_,_in} , is given by the flowrateweighted average of the concentrations from all other operations and the freshwater. Note that the numerator contains a term, f_{i
,lossi}C _{j ,in} ^max, to account , for a water loss at the limiting inlet concentration of contaminant j,to operation i;

The outlet concentration is the sum of the average inlet concentration of contaminant j , C_{i, j,in}, and the change in concentration of contaminant j due to the fixed mass load of contaminant j transferred,∆m^fixed_{_i_,_j} by only the flowrate of water that passes completely through the operation;

Substituting for C from Eq. (13) into Eq. (14) gives; (15)

We rearrange Eqs.(14) and (15) to form another linear constraint as follows;

In addition, we form a mass balance on water around each operation i ;

Equations (8) and (9) remain valid as the constraints on the inlet water temperature to operation i and heating of the inlet fresh water to operation i from temperature T₀ to T_fi^,respectively.We may choose to model an operation i with a fixed flowrate ( f_i^fixed) while maintaining maximum inlet and outlet concentrations. A water balance across the operation gives;

Equations (5) and (6) remain applicable to the constraints on the inlet and outlet concentrations, respectively. Also Eqs. (8) and (9) remain valid as the constraints on the inlet water temperature to operation i and heating of the inlet fresh water to operation from temperature T₀ to T_fi^,respectively.The presented NLP model can be a useful tool to determine water and energy targets and specify the distribution of water among the water-using operations with and without flowrate changes consideration.After the connections between operations are established by using the above mentioned model, heat exchanger network design is considered to complete the overall network configuration. In the next section, a new separate system approach will be introduced to design the heat exchanger network.

New Method for 'Separate System'Generation

Once the non-isothermal mixing for the water re-use streams is completed, the remaining design is to identify the matching of water streams by generating separate systems and appropriate location of separate systems. The remaining problem of heat recovery involves only fresh water streams as cold streams and wastewater streams as hot streams, which enables a simple heat exchanger network design with fewer heat transfer units (Kim et al., 2001). To design a costeffective heat exchanger network for the water system, new separate system generation has been developed. As each separate system represents a heat transfer unit between hot and cold streams, the number of separate systems should be minimized in order to achieve the minimum number of heat exchanger units. Besides, the temperature driving forces in each separate system should be maximized to reduce heat transfer area (Savulescu et al., 2002). Moreover, the optimum heat transfer area in each separate system should be explored by introducing a trade-off between the capital cost of heat exchanger and the cost related to compensation of pressure drops in tube and shell sides, for achieving the minimum total annual cost. Therefore, the concept of new separate system approach intends to create minimum number of separate systems and optimum heat transfer area in each separate system. The procedure of the new separate system approach is based on the five steps as follows:

Step 1; Construct the energy composite curves

The initial energy composite curves are generated based on individual thermal stream data extracted from the water network. As shown in Fig. 2 the minimum demand for fresh water can be targeted by the slope of the fresh water supply line from the cold composite curve. The energy target obtained from the analysis of these composite curves is the same as the value of energy consumption estimated in the stage of nonisothermal mixing point identification.

Step 2; Minimize the number of separate systems

In order to achieve the minimum number of separate systems and consequently fewer heat transfer units, separate systems should be generated following kink points on the composite curve with fewer kink points. Then, the boundaries of separate systems can be defined at kink points from the selected curve.

Step 3; Maximize temperature driving force in each separate system.

The creation of separate systems involves non-isothermal stream mixing in order to achieve the temperatures required by the water-using operations. Through non-isothermal mixing of hot wastewater streams, the hot composite curve should be modified to maintain maximum driving force in each separate system for reducing the heat transfer area.

Step 4; Determine water distribution between separate systems and operations

Since some modifications have been made to the composite curves, water distribution between the separate systems and the operations should be determined. The water distribution involving non-isothermal mixing of wastewater streams can be carried out by solving a simple series of mass and heat balance equations.

Step 5; Optimize heat transfer area in each separate system

After determination of cold and hot streams in each separate system in step 4, the optimum heat transfer area in each separate system should be explored by introducing a trade-off between the capital cost of heat exchanger and the cost related to compensation of pressure drops in the tube side and shell side, for achieving the minimum total annual cost.

Here we examine a procedure for optimizing the heat transfer area in each separate system. We assume the heat exchanger, which represented by each separate system, is a baffled shell-and-tube, single-pass, counter flow heat exchanger ( Fig. 4). in which the tube fluid is in turbulent flow but no change of phase of fluids takes place in the shell or tubes. It should be noted that the inlet and outlet flowrates and temperatures to and from the tube side and shell side of the heat exchanger in each separate system are known in this stage (Edgar et al., 2001). Also, the tube spacing and tube inside and outside diameters should be specified a priori by the designer (Nordman & Berntsson, 2001). Note that the presented optimization procedure is specified for a general separate system j. Thus, this procedure should be carried out for each of separate systems individually.The total cost of the heat exchanger in the separate system j, as the objective function in dollars per year, is formulated as follows (Ataei et al., 2009c);

MinTC _j= A_oj(C_Aj+C_ijE_ij+ C_ojE_oj) (19)

The rate of indirect heat transfer in the separate system j is given in Eq.(20) (Polley et al., 1990);

F_tj is unity for a single-pass exchanger for the separate system j (Glavic, 2001). U _oj is given by the values of h_oj, h_ij, and the fouling coefficient h_tj in the separate system j, as follows (Polley & Panjeshahi, 1991);

h_tj is a combined coefficient for tube wall and dirt films, based on tube outside area. This parameter is expressed in Eq.(22) (Jarzebski et al., 1977), (Ramalho &Alabastro, 1966);

Cichelli and Brinn (1956) showed that the annual pumping cost terms in Eq. (19) could be related to h_ij and h_oj by using friction factors for tube flow and shell flow;

E_ij = φ_ij h_ij^3.5(23)

E_oj = φ_oj h_oj^4.75(24)

The coefficients φij and φoj depend on fluid specific heat, thermal conductivity, density and viscosity as well as the tube diameters in the separate system j . φoj is based on either in-line or staggered tube arrangements.If we substitute for Eij , Eoj in Eq. (19), the resulting objective function can be expressed as Eq.(25) (Woods et al., 1976);

Min TC_j= C_AjA_oj+ C_ijφ_ij h_ij^3.5A_oj + C_oj φ_oj h_oj^4.75A_oj(25)

To accommodate the constraint on the fixed and known indirect heat transfer rate in the separate system j , a Lagrangian function L_j is formed by augmenting TC _j with Eq. (26), using a Lagrange multiplier ω _j as follows (Ataei et al., 2009a);

Eq. (26) can be differentiated with respect to four variables (h_ij, h_oj, ∆t_{2 j}and A_oj) . After some rearrangement, a relationship between the optimum h_oj and h_ijcan be obtained as follows (McAdams, 1954);

The value of h_ij in the separate system j can be obtained by solving the following equation;

Accordingly, the following algorithm can be used to obtain the optimal values of heat transfer coefficients, power loss inside and outside tubes because of pressure drops and heat transfer area in the separate system j without the explicit

calculation of ω j ;

I.Solve for h_ij from Eq. (28).

II.Obtain h_oj from Eq. (27).

III.Calculate U oj from Eq. (21).

IV.Determine Eij and Eoj from h_ij and h_oj using Eqs. (23) and (24).

V.Calculate A_ojfrom Eq. (20).

Note that steps I to V require that several nonlinear equations be solved one at a time.

Optimal Detail Design of the Heat Exchanger Related to Each Separate System

Once the optimal four variables (h_ij, h_oj, ∆t_2i^,and A_oj) were calculated in the previous stage, the physical dimensions of the heat exchanger in each separate system can be determined. Accordingly, the following algorithm can be used to obtain the optimal detail design of the heat exchanger related to each separate system;

I. Determine the optimal v_ij and v_oj from h_ij and h_ojusing the appropriate heat transfer correlations (McAdams, 1954); recall that the inside and outside tube diameters are specified a priori.

II. The number of tubes N_tj can be found from a mass balance as follows;

III. The length of the tube can be found from Eq. (30);

A_oj= N_tjπD_ojL_tj (30)

IV. The number of clearances can be found from N_tj based on either square pitch or equilateral pitch. The flow area S_oj is obtained from ν _oj(flow normal to a tube bundle). Finally, baffle spacing (or the number of baffles) is computed from S_oj, A_oj, N_tj and N_cj.

RESULTS & DISCUSSION

The application of the new simultaneous water and energy minimization technique presented in this paper is demonstrated on two different examples. The design specifications for both of examples have been given in Table 1. As presented in Table 1. the temperature of the fresh water supply in these examples is assumed to be fixed (20 ^oC) and the effluent discharge temperature is assumed to be 30^oC. Therefore, heat can be recovered from the effluent until ∆T_min (10 ^oC) is achieved.

Example 1

The first example is a multiple-contaminant problem without water flowrate changes but with heat loss inside unit operations. The limiting waterusing operations data of example 1 are given in Table 2. Applying the new NLP model to illustrative example 1, through the commercial mathematical optimization software package GAMS, an optimum water network, which can achieve both minimum freshwater (70 t/h) and hot utility (1983.3 kW) consumption, is identified in Fig. 5.

As shown in Fig. 5, the network includes two nonisothermal mixing points (direct heat transfer). One is the mixing of a freshwater stream and a reuse stream at the inlet of Operation 2. The other is the mixing of a freshwater stream and a reuse stream at the inlet of Operation 3. These mixings can reduce the number of heat exchanger units required in the design without non-isothermal mixing. The targeting results for example 1 are given in Table 3. After the connections between operations are created, design of heat exchanger network through the new separate system approach is considered to complete the optimum overall network configuration. The thermal data of streams referred to the optimum water network ( Fig. 5) are given in Table 4.

The initial energy composite curves based on the thermal stream data and a minimum temperature approach (10 °C) which indicates the minimum water and energy requirements in the new water network (example 1) are shown in Fig. 6. As represented in Fig. 6, these composite curves assure that the energy requirements in the new water network achieve the utility target to 1983.3 kW hot utility and 0 kW cold utility.To achieve the minimum number of separate systems in example 1, separate systems are created following kink points on the cold composite curve. Then, the boundaries of separate systems can be defined at kink points from the cold composite curve as shown in Fig. 6. In addition, the hot composite curve is modified to maintain maximum driving force in each separate system. Heat loads exchanged between wastewater and freshwater streams in the separate systems are vertically transferred, and the shaded areas between the original and the modified hot composite curves represent the non-isothermal mixing points of hot wastewater streams from operations.

According to Fig. 6. by applying the new separate system generation method to example 1, only two heat exchangers represented by two separate systems can be enough to complete overall network configuration.The optimum heat transfer area and detail design for each heat exchanger related to the represented separate systems are found by the introduced trade-off between the capital cost of heat exchanger and the cost related to compensation of pressure drops in the tube side and shell side. Fig. 7 illustrates the effect of the heat transfer area on the total annual cost of heat exchangers 1 and 2 related to the represented separate systems in example 1. The optimum heat transfer area achieves the minimum total annual cost. The optimum design of heat exchangers 1 and 2 has been given in Table 5. In example 1, the total number of heat transfer units is three, as there are two heat exchangers (separate systems) plus one heater. The new and conventional network configurations for example 1 is presented in Fig. 8.

Example 2

The second example is a multiple-contaminant problem with heat loss inside unit operations and with water loss in operations 1 & 5, water gain in operation 3 and water fixed flowrate in operations 2 & 4. The limiting water-using operations data of example 2 are given in Table 6.Formulating and solving the presented NLP model to illustrative example 2, through the GAMS optimization software, an optimum water network, which can achieve both minimum freshwater (93.02 t/h) and hot utility (4923.1 kW) consumption, is shown in Fig. 9. The targeting results for example 2 are given in Table 7.

After the connections between operations are created, optimum design of heat exchanger network can be achieved similar to example 1. In other words, only formulation of the presented NLP model for non-isothermal mixing point identification is different between example 1 and example 2, but the method for separate systems generation and optimum design of heat exchangers are the same in both of examples. The new and conventional network configurations for example 2 are presented in Fig. 10.As shown in Fig. 10 the total number of heat transfer units for example 2 is three, as there are two heaters plus one heat exchanger (separate system). A comparison of designs from the conventional and new approaches for examples 1 and 2 is made in Table 8. As presented in Table 8. the new approach provides a better design with less utility usage, fewer heat transfer units and smaller total annual cost for both of examples.

CONCULSION

Process integration has been highlighted in this paper to provide a new systematic design methodology for the problem of simultaneous energy and water minimization in multiplecontaminant systems with consideration of flowrate changes and heat losses inside unit operations.

The method relies on two sequential design aspects to achieve the water and energy targets; new method for non-isothermal mixing points identification and new separate system generation. In the new method for non-isothermal mixing point’s identification, reuse options of water within the water-using systems with multiple-contaminant are exploited not only from the point of view of contaminant concentration, but also considering energy. An NLP model is proposed to identify feasible non-isothermal mixing points, which create an overall water network with minimum freshwater and utility consumption. Then, new separate system generation is developed to design a simplified heat exchanger network. The new approach provides a heat exchanger network with fewer heat transfer units and optimal heat transfer area.

ACKNOWLEDGEMENTS

This work is supported by Brain Korea 21 project (environmental informatics program), the Korea Science and Engineering Foundation (KOSEF) grant funded by the Korea government (MEST) (KRF-2009-0076129) and funded by Seoul R&BD Program (CS070160).

NOMENCLATURE

A_ij Inside tube surface area in separate system j, m²A_lmjLog mean of inside and outside tube surface area s in separa te system j, m²A_ojOutside tube surface area in separate system j, m²C Cooler
C_Aj Annual cost of heat exchanger per unit outside tube surface area in separate system j, $/m₂yr
C_e Annual cost of energy, $/kWyr
C_{i, j ,in} Ave rage contaminant j concentration of inlet str eam to oper ation i, ppm
C_ij Cost of supplying 1 kW ele ctr icity to pump tube side fluid in separate system j, $/kWyr
C_{i, j,out} Ave rage contaminant j concentration of outlet stream from operation i, ppm
C_{k , j,out}Ave rage contaminant j concentration of outlet stream from operation k, ppm
C_ojCost of supplying 1 kW ele ctr icity to pump shell side fluid in separate system j, $/kWyr
Cw Annual cost of fresh water, $.h/t.yr
C^max_{i, j,in}Maximum a llowable concentrations of contaminant j at the inlet of operation i, ppm
D_ij Tube inside diameter in separate system j, m
D_oj Tube outside diameter in separate system j, m
E_ijPower loss inside tubes per unit outside tube area in separate system j, kW/m²
E_oj Power loss outside tubes per unit outside tube area in separate system j, kW/m2
f _Aj A_ij/A_oj
f_iInle t fresh water flowrate to operation i, t/h
f_i^fixed fixed flowrate of operation i, t/h
f _i,gainincrease in the flowrate through operation i, t/h
f_{i,lo ss}water loss through operation i, t/h
C p Spec ific he at capac ity, k J/k goC
C_i,^max_j,out ma ximum a llowa ble conce ntrations of contamina nt j at the outlet of operation i, ppm
hf _ij Fouling coef fic ie nt of inside tubes in separate system j, W/m^2oC
hf _oj Fouling coe ffic ie nt of outside tube s in separate system j, W/m^2oC
h_ijCoef fic ie nt of heat transfer inside tube s in se pa rate system j, W/m^2oC
h_oj Coef fic ie nt o f heat tra nsfer outside tube s in se pa rate system j, W/m^2oC
h_tj Combined c oe fficie nt for tube wa ll and dir t films in separ ate system j, W /m^2oC
k Unit c onversion factor, 0.2778
k_wjTherma l conductiv ity of tube wall in separate system j, W/m^oC
L_jLagrangia n function f or se pa rate system j, $/yr
L_tj Length of tubes in separate system j, m
l' _j Thic kness of tube wall in separa te system j, m
M_ij Flowrate o f fluid inside tubes in separa te system j, t/h
Min Minimization
N_cjNumber of clear ances for flow be twee n tubes ac ross shell axis in separa te system j
NLP Non- linear pr ogramming
n_{operation s}Number of ope rations
N_tj Number of tubes in the exchange r in separate system j
OPCOST Tota l anuual cost o f opera ting, $/y
OP_{1,2, 3,4} Water-using oper ations
Q_i Energy re quirement for heating of inlet fre shwater stream to operation i, kW
F_tj Multipass exchanger factor in se parate system j
H Heater
Q_{Re covery}Heat r ecovery, kW
q _j Ra te of indirec t heat transfer in separate system j, kW
S _ojMinimum c ross-sectional ar ea for flow ac ross tubes in separa te system j, m2
T Tempe rature , ° C
T₀ Tempe rature o f freshwa ter sour ce, °C
t_{1 j} Shell side inlet temperature in separate system j, °C
t_{2 j} Shell side outlet tempera ture in separate system j, °C
T_{1 j}Tube side outlet tempera ture in separate system j, °C
T_{2 j} Tube side inle t temperature in separate system j, °C
TC_jTo ta l a nn ual c ost of the h ea t exchanger in se pa rate system j, $/yr
T _fi Tempe rature of inle t fresh water str eam to oper ation i, °C
T_i,inAverage temperature of inle t str eam to oper ation i, °C
T _i,outAverage temper ature of outlet stream from ope ration i, °C
T _j,out Average temper ature of outle t str eam from ope ration j, °C
U _ojOverall coe ffic ie nt of hea t transfer based on outside tube ar ea in separate system j, W /m² °C
W_iFlowrate of steam from opera tion i to wastewa te r treatme nt, t/h
X_i,kFlowrate o f stream f rom opera tion k to oper ation i, t/h
X_k,i Flowrate o f stream f rom opera tion i to operation k , t/h

Greek Letters

Δm_{i, j,tot} Total mass transfer load o f contaminant j in ope ration i, kg/h
Δm^fixed_{i , j} change in concentration o f contam inant j due to the fixe d ma ss load o f contamina nt j tra nsfer red through operation i, kg /h
ν_ij Average velocity of fluid inside tube s in separate system j,m/s
ν _ojAverage ve locity of fluid outside tube s at shell axis in separate system j,m/s
ω_j Lagrange multiplier for separa te system j, $W/yr°C
φ_ij Fa ctor relating fr iction loss to hij
φ_oj Fa ctor relating fr iction loss to h_oj

REFERENCES

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