Improvements for mass-exchange networks design

Chemical Engineering Science 54 (1999) 1649—1665

Improvements for mass-exchange networks design P. Castro, H. Matos*, M.C. Fernandes, C. Pedro Nunes Department of Chemical Engineering, Instituto Superior Te& cnico, 1096 Lisboa Codex, Portugal Received 19 March 1998; accepted 16 October 1998

Abstract This paper addresses minimum utility cost mass-exchange network design by considering the special case of water minimisation. Two different situations are considered, re-use and regeneration re-use for single contaminants. For re-use, three different methods of targeting are presented, one of them being simultaneously a design method. This novel design method has the advantage of considering flowrate constraints only in the final stage of design. The concept of multiple pinches is introduced to prevent designing networks that do not lead to minimum cost distributed effluent treatment systems. For regeneration re-use, this paper presents the first known algorithm for targeting minimum water consumption in all possible situations. The targeted flowrate is then used to design the mass-exchange network, that almost always features splitting of operations.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Wastewater minimisation; Mass exchange-networks; Re-use; Regeneration; Multiple pinch analysis

1. Introduction The awareness of the danger to the environment resulting from the over extraction, and stricter discharge regulations, has caused the price of freshwater and waste treatment facilities to rise. Therefore, there is a growing interest in the process industries to reduce both freshwater consumption and wastewater production. Processes generate wastewater as by-products of reaction, or when water comes into contact with process materials in mass transfer and washing operations, vessel cleaning and hosing operations, steam ejectors, etc. Also, wastewater is generated by utility systems from boiler feedwater treatment processes, boiler and cooling tower blowdown and condensate loss. Wastewater is characterised by the volume and contaminant load carried. If we exclude the possibility of making fundamental changes to processes to reduce their inherent demand for water, for example, using air coolers instead of cooling towers, there are three possibilities for reducing wastewater (Rossiter, 1995): E

Re-use. Wastewater can be re-used directly in other operations. This might require wastewater being *Corresponding author.

E

E

blended with wastewater from other operations and/or freshwater. Regeneration re-use. Wastewater can be regenerated by partial treatment to remove the contaminants, which would otherwise prevent its re-use, and then re-used in other operations (water does not re-enter processes in which it has previously been used). Regeneration recycling. In this case water can re-enter processes in which it has previously been used.

The problem of optimal water allocation in a petroleum refinery has been addressed by Takama et al. (1980). Their approach first generated a superstructure of all possible re-use and regeneration opportunities. This superstructure was then optimised and uneconomic features of the design removed. El-Halwagi and Manousiouthakis (1989), addressed the more general problem of synthesising cost-effective mass-exchange networks, in which, a key process contaminant, present in a set of rich process streams, is transferred to a set of lean process streams. Their approach was adapted from the methodology developed for heat exchanger networks by Linnhoff and Hindmarsh (1983). Later, they automated the approach and included regeneration (El-Halwagi and Manousiouthakis, 1990a, b). In the first stage of their automated approach,

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thermodynamic constraints were used to formulate a linear programming (LP) problem whose solution determined the minimum cost of lean streams (mass separating agents), also referred to as utilities, and the thermodynamic bottlenecks (pinch points) which restrict the exchange of mass between the two different set of streams. Their formulation could account for special constraints in the design, such as forbidden matches (due to safety problems, control considerations, transportation problems, etc.), or compulsory ones. In the second stage, a mixed integer linear program (MILP) transhipment problem was solved, to identify the minimum number of mass exchange units consistent with the minimum utility cost. More recently, Wang and Smith (1994a), considered the case of a single contaminant and a single lean stream, presented a conceptually based approach, and applied it to wastewater minimisation problems. In the first stage of their approach a graphical construction was used to represent the limiting data of the problem. Then, they identified the pinch point, and targeted the minimum water flowrate to be used in the overall process, for the three different scenarios mentioned. Two different design methods were shown to achieve the target. In the end, they presented a general algorithm to extend the approach to multiple contaminants. Later, new design rules were given, based on local recycling and splitting of operations, to handle processes with fixed flowrate requirements (Wang and Smith, 1995). These include existing water networks and operations such as vessel cleaning, hosing, hydraulic transport, etc. Water losses and multiple sources of freshwater, were also considered in that paper. In order to solve the more general problem of several input streams, at different purities, in a single operation, and or several output streams, a newer approach which involved a combination of new graphical and mathematical techniques, trademarked WaterPinch, was presented (Dhole et al., 1996). The graphical procedure involves plotting the required aqueous streams input purities of all relevant operations, into a combined ‘‘demand composite’’ form, on a graph having purity as its vertical axis and aqueous-stream flowrate as its horizontal axis, and constructing the ‘‘source composite’’ with the output water streams of all operations. The resulting diagram indicates the potentialities for water re-use, helping the engineer at the same time to identify the design modifications needed. Research has been directed to improve existing water networks, and so water re-use with flowrate constraints has been the most studied wastewater reducing possibility. This, because those network operations are unlikely to use intermediate regeneration processes (its use seeks to reduce water requirements in the system, which conflicts with flowrate constraints). The regeneration re-use possibility becomes more important if a new process is to

be implemented, since all design options should be considered and evaluated. The targeting methodology developed so far for regeneration re-use may lead to an insufficient flowrate before or after the regeneration process. This paper presents an algorithm that efficiently eliminates this problems, while introducing the multiple pinch points concept as well as a novel re-use network targeting/design methodology for a single contaminant.

2. Mass transfer operations Consider a number of rich process streams, that need to reduce the content of a certain species (contaminant) by a mass separating agent, in mass exchangers. Without loss of generality, we will assume that this mass separating agent is water, used in many operations throughout the process industries, such as washing, absorption, stream stripping, etc. Each rich stream has a mass flow rate G and has to be brought from a supply composition G y  to a target composition y. Accordingly, each water G G stream will have a mass flowrate ¸ , an initial composiG tion x  and a final composition x. G G Consider the mass exchanger in Fig. 1. The mass to be transferred in this unit, Dm , can be determined by estabG lishing a mass balance on the solute: Dm "G (y !y)"¸ (x!x  ). (1) G G G G G G G It was assumed that the mass flow rate of each stream remains essentially unchanged as it passes through the network. This assumption is reasonable, when relatively small variations in compositions are required, or when some counterdiffusion takes place from the lean streams to the rich streams. When the changes in flow rates are considerable, it is necessary to use mass flowrates of the non-transferrable (inert) components instead of those of the whole streams, and mass ratios (mass of transferrable component/mass of inert components) instead of mass fractions. Eq. (1) tell us that for a certain x  , the higher the outlet G concentration the lower the water flowrate for the same mass exchanged, with minimum water flowrate being achieved when water outlet composition is at its maximum, x . Maximising the outlet composition is G always the best option. When using freshwater, x "0, G this will lead both to minimum water consumption and wastewater generation. When using wastewater, x  highG er than the environmental limit for discharge, this will

Fig. 1. A mass exchanger.

P. Castro et al. /Chemical Engineering Science 54 (1999) 1649—1665

lead to the distributed effluent treatment system with lower flowrate, and consequently with lower investment and operating costs (Wang and Smith, 1990b). Thus, outlet composition will always be fixed to its maximum value: x"x . (2) G G For a certain x, minimum flowrate is achieved by G using freshwater, x "0. Although freshwater will lead G to minimum wastewater generation for a single operation, the same is not necessarily true when there are more than one. If it is possible to re-use part of the wastewater in other operations, less water needs to be used and the global process will generate less wastewater. Wastewater re-utilisation possibilities will be determined by maximum inlet compositions, x  . G Let us assume that in the range of compositions involved, the equilibrium relation governing the distribution of the solute between the reach stream and the water stream is linear (if the relation is not linear, one can always linearise it over small composition ranges with both coefficients being varied from one range to the other), y"mx#b,

(3)

where both m and b are assumed to be constants. When the value of x  is specified, maximum theoretical outlet G composition of the water stream, in equilibrium with y , G can only be achieved in an infinitely large exchanger. The same applies when the outlet composition, x, is speciG fied. Therefore it is necessary to assign a minimum value for the difference between the operating and the equilibrium compositions of the water stream, e, to prevent designing an excessively large unit. Two conditions need to be satisfied: y!b x )x  " G !e G  m and y !b !e . x)x " G G  m

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Trade off between capital and operating costs can be achieved using the targeting methods developed below for the operating costs, and the ‘‘Supertargeting’’ procedure of Hallale and Fraser (1998) for the capital cost, both of which applied ahead of any design. At the optimum value of e, design methods that use minimum water flowrate and minimum number of units lead to a network that has a capital cost close enough to the targeted cost, as shown by those authors for an example problem. Thus, in this paper only operating costs will be considered to design the mass exchange network. In order to maintain the same notation as Wang and Smith, we will use concentrations instead of compositions, thus substituting x   by C and x  by G  G G C . When C "C , the water flowrate ¸ is  G G  G G a limiting one, and will be represented as f . G Having stated the mass-exchange problem we now focus on the different ways of targeting minimum water flowrate. To illustrate our approach an example problem will be used.

3. Example problem Consider a process in which water is used to reduce the content of a certain contaminant. It consists of four operations, with limiting data shown in Table 1. Determine a network that produces minimum wastewater by re-use and regeneration re-use. In this last situation, assume an outlet concentration from the regeneration process of 80 ppm.

4. Water re-use For the simplest situation, using a single water source, three different methods can be applied to target the minimum water flowrate. These will be presented in order of increasing complexity.

(5)

The minimum allowable composition difference e is an optimizable parameter (El-Halwagi and Manousiouthakis, 1989). When it is closed to zero, infinitely large separators will be required and, consequently, the capital cost of the mass exchanger network will be infinite. However, utility costs will be at a minimum. When the e is increased, the operating cost will increase whereas the fixed cost will decrease. In general, the annualised cost will pass through a minimum, which corresponds to the optimal value of the minimum allowable composition difference. A lower value can be specified if other problems need to be considered, such as excessive equipment fouling, corrosion limitations, possible material precipitation from solution, possible settling of solid material, etc.

5. Mass problem table El-Halwagi and Manousiouthakis (1989) introduced the composition interval table (CIT) for mass-exchange networks, to calculate the minimum requirement of

Table 1 Limiting operation data for the example problem Operation

f (t/h) G

C (ppm)  

C (ppm)  

1 2 3 4

25 60 30 70

0 50 150 300

250 300 400 500

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external mass separating agents to supplement the separation of a contaminant from the reach streams to the lean streams. Since in the problem formulation used in this paper, the data of the rich streams is incorporated into the lean streams, a different table needs to be used, which will be called mass problem table (MPT). We use mass instead of composition, in order to have a better correspondence between the mass exchanger network (MEN) problems and the heat exchanger network (HEN) problems. This because the composition intervals of the CIT are constructed in the same way as the enthalpy intervals, and not as the temperature intervals. The division process in both enthalpy and temperature intervals can be found in Floudas (1995). In the MPT an arrow represents any process operation. The tail of each arrow corresponds to the limiting inlet concentration of the water stream, C , while  G the head represents the limiting outlet concentration, C . We begin by establishing a series of mass inter G vals in such a way, that along each interval there are always the same operations present. The interval extremes correspond to the heads and tails of these arrows. The number of concentration intervals, N , can be  related to the number of operations, N , through the M following expression: N )2N !1 (6)  M with the equality applying in cases where no two heads or tails coincide. Intervals numbering, concentrations and operations representation constitutes the first three columns of the MPT. For the remaining columns, simple calculations are required. The fourth column indicates the sum of the limiting water flowrates, R f , of the operations present in G each interval, while the fifth tells us the amount of mass transferred in each interval. For interval j, this is calculated by




Dm " f ;(C !C ) H G H H\ H G with

(7)

C "min +C , H  G G Cumulative mass transferred, given by

(8)

Dm "Dm #2#Dm #Dm    H  H\ H H " Dm (9) I I is presented in the next column. The MPT can only be applied for a single water source. For it to be capable of solving the entire massexchange duty, the concentration in the contaminant, C , must verify the following condition: UQ C )C . (10) UQ H

Minimum water flowrate that can be used, f , while UQ respecting the constraints of the problem, is determined from the values in the seventh column, by





Dm    H . f "max (11) UQ C !C H UQ H The upper limit of the concentration interval that leads to the maximum, indicates the first pinch point of the problem, C . Each pinch represents a point where   the driving force for mass transfer goes to a minimum. In cases where more than one interval leads to the same f value, the first pinch point will be the one with lower UQ concentration. This will have important implications when we consider the regeneration re-use possibility. The first pinch point is the only pinch point considered by Wang and Smith (1994a). After this point, sufficient water exists to remove the remaining contaminant mass, but in order to achieve an optimum distributed effluent treatment system (Wang and Smith, 1994b), it is important to use as less water as possible. Therefore, we treat the remaining problem as a new one, and calculate by Eq. (13) the minimum amount of wastewater, f at ?N C , required. This in cases where   C (max+C ,, (12)    G Dm !Dm    J      . f "max (13) ?N C !C J   J In Eq. (13), l represents a certain mass interval with higher concentration than C . The point where the   maximum occurs, determines a second pinch point. More pinch points can arise, until a maximum equivalent to the number of concentration intervals. The concentration intervals are formed by considering only the outlet concentration from the operations (as opposed to considering both the inlet and outlet concentrations, in mass intervals), which means that there are a maximum of N pinch points. Note that it is easier to determine the M *m for a mass interval than for a concentration interval H (Eq. (7) cannot be applied because the inlet concentrations can be different than the lower limit of the concentration interval), even though more values are required. Flowrate requirements after a certain pinch point, is lower than the flowrate required until then. The difference between these values gives the flowrate rejected by the network to the treatment system





f "f !f , (14)   K ?NK\ ?NK where f "f . ?N UQ The only water source able to solve the example problem, with a concentration of 0 ppm, is known as freshwater. The mass problem table is presented in Table 2. One can see that there are two pinch points, having concentrations of 400 and 500 ppm. Minimum freshwater flowrate is equal to 89.375 t/h, and the water

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streams rejected at the pinch points have flowrates equal to 19.375, and 70 t/h, respectively.

6. Limiting composite curve El-Halwagi and Manousiouthakis (1989), first introduced the idea of a concentration composite curve. Later, Wang and Smith (1994a, 1995) used the concentration composite curve in a different context. Using the limiting water profiles of the operations, and combining them within mass intervals, they attained the limiting composite curve, and used it to determine the flowrate targets for re-use, regeneration re-use and regeneration recycle both for a single water source, and for multiple water sources and losses. Although the composition of operations within mass intervals can be quite simple for small problems, it can be

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quite complicated as the problem size increases. A simpler way can be used after the mass problem table has been made. One needs only to join together the N #1 points, with co-ordinates (*m ; C ), in     H H increasing concentration. Naturally, *m "0.     The limiting composite curve for the example problem is shown in Fig. 2. The graphical representation of the limiting composite curve is a very important tool in determining all possible mass-exchange alternatives. As indicated by Wang and Smith (1994a), all possible alternatives have composite water supply lines underneath the limiting composite curve, with optimised solutions having composite water supply lines touching the limiting composite curve in at least one point, the pinch. Thus, to determine the target for a single water source, and beginning at point (0; C ), UQ one needs only to increase the water supply line slope (as higher the slope the lower the flowrate) until it touches

Table 2 Mass problem table for the example problem Interval

1 2 3 4 5 6

C H (ppm)

Operations

0 50 150 250 300 400 500

1 2 3 4

Rf G (t/h)

Dm H (kg/h)

Dm    H (kg/h)

f UQ (t/h)

f ?N (t/h)

25 85 115 90 100 70

1.25 8.5 11.5 4.5 10 7

1.25 9.75 21.25 25.75 35.75 42.75

25 65 85 85.83 89.375 85.5

70

Fig. 2. Limiting composite curve and water supply line for the example problem. Re-use possibility.

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the limiting composite curve. At the pinch, we continue the supply line construction by proceeding in a similar manner, leading to the second pinch, and so on. The water supply line for the example problem, having a second part superimposed on the limiting composite curve, is represented in Fig. 2.

7. Water sources diagram The design method developed here is based on the learning from one of UMIST’s course (1996) and leads to the same results as the so-called second network design method presented by Wang and Smith (1994a). It has a different graphical aspect, responsible for less confusing representation of operations and water streams, the water sources diagram (WSD), and several more advantages, namely: E

E

E

It can be used as a targeting method for re-use with single or multiple water sources. Fewer intervals need to be considered, since we use concentration intervals instead of mass intervals. Flowrate constraints need only to be considered in the final stage of design.

To establish the water sources diagram several steps are required. The methodology will be described in detail, by applying it to the example problem. Step 1. Identify every possible water source to the process: E

N external water sources. CUQ N internal water sources. M Only water sources with different concentrations are represented, by a rectangle, in the diagram. Therefore the

E

f

ent the inlet concentration beneath the box. The limiting water flowrates, f , should be written down on the left side G of the diagram (Fig. 4). Step 3. After the representation of the necessary process data we can start to build the mass-exchange network. We proceed from the least contaminated concentration interval to the higher contaminated ones. In this way, it is ensured that internal water sources flowrates are generated prior to being used. All operations with inlet concentrations lower than the one of the first internal water source are required to use the available external water sources (demineralised, potable and borehole water). One should use, if possible, the lowest quality water since it is generally cheaper. To attain minimum flowrate in the process, two simple rules must be satisfied: Rule 1. Use external water sources only when internal water sources are not available, in either quality or quantity. Rule 2. For a certain operation, the water source(s) used in a certain concentration interval, must transfer exactly the amount of mass that needs to be transferred (this ensures that the maximum inlet concentrations are not exceeded). In concentration interval p, the mass transferred in operation i can be calculated by Dm "f ;(C !max+C ,C ,). (16) GN G UQN> UQN  G Flowrate requirements of operation i, from water source k, in interval p, can be determined from the following equations. If k is an external water source one must use Eq. (17). If k is an internal water source, use Eq. (18). Dm ! N [f ;(C !C )] ?I> UQ?GN UQN> UQ? , f " GN UQIGN C !C UQN> UQI (17)

Dm !f ;(C !C )! N\ [ f ;(C !C )] UQNGN UQN> UQN ?I> UQ?GN UQN> UQ? . " GN UQIGN C !C UQN> UQI

total number considered, N , satisfies the following conUQ dition: N )N #N . (15) UQ M  On top of the rectangle there is an indication of the concentration level of the water source, C , where UQI k"1,2, N (Fig. 3). UQ Step 2. Represent the mass-exchange operations in the diagram by arrows, where the tail, connected to a box that holds the identification of the operation, corresponds to the maximum inlet concentration, and the head to the maximum outlet concentration. When the inlet concentration of a certain operation is different from the concentration of the water wells, it is necessary to repres-

(18)

In the preceding equations, water quality considerations imply that k)p. In the water sources diagram, dashed lines are used to connect C to C (when different) or to UQI  G C (when pOk). On top of the lines, the flowrates UQN figures are represented, as can be seen in Fig. 5 for the first interval of the example problem. From top to bottom, these correspond to f , f and f . UQ UQ UQ Since *m "0, f "0.  UQ In the first concentration interval there is only one water source available, and so there are no other constructing alternatives. In the following concentration intervals, if caused by external water sources, the same applies, since the kind of water to use is

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Fig. 3. First step of the water sources diagram construction for the example problem.

Fig. 4. Second step of the water sources diagram construction for the example problem.

Fig. 5. Third step of the construction process, applied to the first concentration interval of the example problem. Re-use possibility.

chosen from economic considerations, as described. The alternatives arise when we enter concentration intervals located to the right of the first internal water source. In these, there are both internal and external

water sources available, having the first priority, as stated by rule 1. Consider the example problem. In the second concentration interval, there are 25 t/h available from operation 1,

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which as been completed, 48 t/h from operation 2 and 12 t/h from operation 3. In order to attain minimum number of mass transfer units, operation 2 and 3 must use at least the same flowrate as before. This can be stated generally, as I\ f * f . UQIGN UQ?GN\ ?

(19)

If we apply the equality in Eq. (19) and make f "48 and f "12, we can determine the UQ UQ additional requirements of water source 2 in interval 2, by Eq. (18). These are equal to 12 t/h in operation 2, and 18 t/h in operation 3. Each one can be fully satisfied with the wastewater from operation 1, but not both. Therefore a design option must be made. Different options will correspond to different networks, each with its own capital cost. We choose using 18 t/h in operation 3, and 7 t/h in operation 2. Adding this values to those already determined, results in f "55 t/h and f "30 t/h. UQ UQ In interval p, after exhausting the C water source, UQN one should use the preceding water sources. This movement from the most contaminated water sources to the least contaminated ones is assumed in Eq. (18). If no more internal water sources are available, we proceed to an external water source. For the example problem there remains only freshwater to satisfy the remaining of operation 2, in interval 2. Thus, f "0.833 t/h. The result can be seen in UQ Fig. 6. Applying the same procedure to the remaining two intervals of the example problem, the following flowrates result: f "30 t/h, f "55.833 t/h, f " UQ UQ UQ 3.542 t/h, f "70 t/h. UQ The pinch points of the problem are found whenever the rejected flowrates, at C (k"N ,2, N ), are UQI CUQ UQ positive values. These can be calculated by Eq. (20), and are indicated beneath the corresponding water sources in the WSD. To determine total external water sources

flowrate, Eq. (21) can be used (k"1,2, N ). As exCUQ pected, f "89.375 t/h. UQ






I\ ,UQ\ F " f , (20) ! f  I UQ?GI\ UQIG@ G ? @I f "!F . (21) UQI  I The final stage of this design is to eliminate all loops, in order to achieve the minimum number of mass-exchange units. Each extra arrow represents a loop. For the example problem, as can be seen in Fig. 7, there are five loops (0#2#1#2). Loop elimination is quite simple, since the construction process ensures that the mixing of the water streams creates a stream (see Appendix A for proof ): (i) that removes the same mass of contaminant; (ii) with a lower or equal concentration than the maximum inlet concentration of the respective operation. For the example problem, this fourth step is shown in Fig. 8. If there are flowrate constraints an extra step needs to be added. Wang and Smith (1995) have proven that local recycling around individual or several operations, can satisfy minimum flowrate constraints. This can be seen in Fig. 9, for operation 2 of the example problem, if its limiting flowrate had to be satisfied (4.167"60!55.833). The mass-exchange network for the example problem, without flowrate constraints, is presented in Fig. 10.

8. Regeneration re-use Water consumption can be further reduced if we allow intermediate regeneration. Many different types of process can be used to regenerate wastewater, for example

Fig. 6. Third step of the construction process, applied to the second concentration interval of the example problem.

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Fig. 7. Third step of the construction process, applied to the third and fourth concentration intervals of the example problem.

Fig. 8. Fourth step of the water sources diagram construction for the example problem.

(ii) a removal ratio RR,

Fig. 9. Fifth and final step of the water sources diagram construction applied to operation 2 of the problem.

gravity settling, filtration, membranes, activated carbon, biological treatment, etc. A regeneration process must perform to either of the following: (i) a minimum outlet concentration of C , M C )C (22)  M C , being the outlet concentration at given process  conditions;

f ;C !f ;C    . RR"  (23) f ;C   Throughout this work it will be assumed that the flowrate before, f , and after regeneration, f , is   unchanged. Regeneration flowrate is represented by f .  9. Water and regenerated water targeting In the network, the regeneration process must be located appropriately. Wang and Smith (1994a) have shown, using the limiting composite curve, that minimum water flowrate for regeneration re-use without recycling, is achieved for regeneration at the first pinch concentration. Thus, C "C .   

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Mass removal in the regeneration process allows for water re-use in operations with C (C . In  G   other words, the same water can be used twice in the interval [C , C ]. This increases the complexity of M   the problem, making targeting more difficult. Wang and Smith (1994a) calculated minimum water flowrate, f , by considering the mass balance before the UQ pinch: Dm "f ;(C !C )      UQ   UQ #f ;(C !C ) (25)    M with f "f . Equation (25) worked fine for the example  UQ considered, but it can lead to targets impossible to achieve. One of these situations occurs when f is insuffiUQ cient to satisfy flowrate requirements until C (composite M water supply line crosses the limiting composite curve in the region ]0; C [), and was identified by those authors. M

To solve this situation, they equalled f to the inverse of UQ slope of the limiting composite curve, and determined f by Eq. (25).  This work now presents an algorithm that leads to the optimum feasible target. Besides the situation mentioned, it is capable of avoiding limiting composite curve crossing in the other two regions, ]C ; C [ and M   ]C ; max+C ,[. This is what happens in the    G example problem, if Eq. (25) is applied with f "f , as  UQ can be seen in Fig. 11. Crossing in ]C ; C [ can be avoided by decreasing M   the slope of the composite water supply line. This is done by increasing freshwater flowrate until the slope of the water supply line is equal to the maximum slope of the limiting composite curve in this region. Regenerated water flowrate is then calculated by Eq. (25). The result is presented in Fig. 12.

Fig. 10. Mass exchange network featuring reutilization, for the example problem.

Fig. 11. Target from Eq. (25). The composite water supply line crosses the limiting composite curve in two regions.

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Fig. 12. Increasing freshwater flowrate and decreasing regenerated water flowrate avoids crossing in the region ]C ; C [. There is still crossing in M NGLAF the last region.

After the first pinch, the available flowrate, f , may be UQ lower than the flowrate needed, f , leading to the cross ?N of the limiting composite curve. Again this is avoided by decreasing the slope of the composite water supply line. In doing so, the first pinch point (the reutilization pinch) will not represent a bottleneck in regeneration re-use design and the regeneration pinch point will be different. A further analysis is then necessary to determine regeneration pinch, and consequently water and regenerated water flowrate. Not all-limiting composite curve points located above the first pinch, can be the regeneration pinch. As will now be shown, the only possible candidates are the other reutilisation pinch points. Let h be one non-pinch point of the limiting composite curve, verifying the condition C 'C . If the comF   posite water supply line touches h, at C it removed   an amount of mass equal to Dm plus a pos     itive amount d. Thus Dm !(Dm #d)      f "    F UQ C !C F   Dm !Dm     "f . (    F (26) F C !C F   By definition of the second pinch point, it is known that: f )f , (27) F ?N and so the targeted flowrate leads to an infeasible network. Regeneration pinch is calculated by trial and error. We force the composite water profile to touch the limiting composite curve at the second pinch point, determine the

necessary flowrate of freshwater, and see if it is enough to satisfy the flowrate needed after that pinch. If the answer is yes then the regeneration pinch point is found. If not, we proceed to the next pinch and so on. The algorithm used to determine the amount of freshwater and regenerated water needed is shown in Fig. 13. It requires information about the water source used, the performance of the regeneration process and some of the parameters of the MPT, mainly the coordinates (Dm , C ) of mass intervals and reutilisation   H H pinch points. All reutilisation pinch points after the regeneration first pinch point, are also regeneration pinch points. For the example problem, the regeneration re-use algorithm targets a consumption of 60.208 t/h of freshwater and 39.518 t/h of regenerated water. There is only one regeneration pinch point, with a concentration of 500 ppm. Therefore, the process rejects all the flowrate at this concentration. Graphical representation of the solution is given in Fig. 14.

10. Network design If for the re-use possibility the WSD design method developed is able to achieve the network without previous knowledge of water requirements, the same is not true for regeneration re-use. This, because there is now an extra internal water source, regenerated water, with unknown flowrate requirements. Since we can only consider one external water source, Eq. (15) becomes N )N #2 UQ M

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Fig. 13. Regeneration re-use algorithm.

The first two steps of the construction method remain the same. The result for the example problem is presented in Fig. 15. As can be seen, the regenerated process is represented in an opposite direction compared to the others operations, since its outlet concentration is lower than its inlet concentration. Therefore, regenerated water can only be used, when a f flowrate of freshwater, after  passing by some of the operations, reaches C . As   this flowrate is already known, it is written down below the N limiting flowrates. The rejected flowrates can also M be represented right away. With regeneration, step three of the construction method is not as straightforward, even though the same rules are to be satisfied. This, because our external water resource is limited to f and we need to generate f at UQ 

the C concentration. To achieve these targets step 3   will be divided in two parts. In the first, a flowrate of freshwater equal to f will be used to generate the water  stream that will be regenerated, and in the second, the remaining flowrate of freshwater and the regenerated water, will be used. In the first concentration interval, only freshwater can be used. Flowrate requirements are determined by Eq. (17). For the example problem this results in f "25, f "48 and f "12 t/h. Since UQ UQ UQ there are available only 39.518, 25 t/h will be used in operation 1 and the other operations divided in such a way, that the remaining 14.518 t/h remove the fraction of contaminant before the regeneration (until the next internal water source available, in this case 250 ppm). If

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Fig. 14. Composite water supply line of the regeneration re-use possibility for the example problem.

Fig. 15. First two steps of the construction process for the example problem. Regeneration re-use possibility.

a represents the fraction of the operation before regenG eration this means that: 250!50 250!150 60;a ; #30;a ; "14.518   250 250 and 80!50 60;(1!a ); "20.69,  80 where 20.69 t/h is the remaining freshwater, to use after the regeneration. The solution to this system is a "0.08044 and a "0.88806 and so both operations   must be divided. The limiting flowrate of the first part of operation 3 is equal to 26.642 t/h, which is less than the amount that

needs to be regenerated. Since this is the only operation ending at C the remaining flowrate can only be   achieved by mixing water stream(s) at higher concentration(s) with stream(s) at lower concentration(s). Since operation 4 is the only one ending after C , it will   have to be used. To determine the exact amount, one can use the freshwater mass balance: f ;(C !C )" a ;Dm .    UQ G G G For the example problem this becomes

(29)

39.518;(400!0)"1;6250#0.08044;15000 #0.88806;7500#a ;14 000(")a "0.12072.   The WSD can now be constructed. As determined, f #f "3.861 and f "10.657 t/h. For UQ UQ UQ

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the next concentration intervals we would get f "4.826, f "26.642, f "4.826, UQ UQ UQ f "2.416 and f "0.483 t/h. Total flowrate UQ UQ to regeneration ( f ) is obtained after mixing the effluents  from streams 1, 3 and 4. After mixing, the concentration of contaminant in the water stream is equal to 400 ppm (C ). The result is given in Fig. 16.   The third step can easily be completed by using the regenerated water and the remaining freshwater, as can be seen in Fig. 17 for the example problem. In Fig. 17 it can be seen that operations 2—4 are divided in two. These loops cannot be eliminated, since the regen-

eration re-use network must satisfy one important property: E

The same water cannot pass by the same operation twice, since this would correspond to a recycling situation.

For the other kind of loops we proceed as described before, resulting in the flowsheet of Fig. 18. The regeneration re-use target usually implies splitting of at least one of the operations, meaning that the minimum number of mass transfer units cannot be achieved in design. This increases the capital cost of the network

Fig. 16. First part of the third step of the construction process, for regeneration re-use.

Fig. 17. Conclusion of the third step of the construction process, for regeneration re-use (first part of these third step in grey).

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Fig. 18. Mass-exchange network featuring regeneration, for the example problem.

Fig. 19. A minimum number of units mass-exchange network for the example problem, featuring regeneration re-use.

and may not be desired. For a small penalty in freshwater consumption it is possible to eliminate operation(s) splitting. To achieve a minimum number of units network with wastewater production close to its minimum, it is recommended that operations with a higher than 0.5 should not use regenerated water. So, with this heuristic rule, the problem can easily be solved. The effluents from this network are then regenerated, and the remaining network achieved by using the regenerated water and, if necessary, freshwater. For the example problem the network without splitting is obtained with a penalty of 5.5% of freshwater (Fig. 19). One other difference between this network and the previous, is that the inlet concentration in the regeneration process is now 371.6 ppm. To maintain the performance of the regeneration process in terms of removal ratio, the outlet concentration would now be 74.3 ppm.

11. Conclusions This paper presents a simultaneous targetingdesign, easy to apply method (for single contaminants), that minimises external water sources requirements by maximising water re-use. Utilities minimisation causes the appearance of the first pinch point. Other pinch points result if internal water sources usage is minimised, in order to achieve the minimum cost distributed effluent treatment system. Multiple pinches is introduced as an entirely new concept in water minimisation. An intermediate regeneration process can be used to decrease freshwater requirements. With the knowledge of the reutilisation pinch points from one of the three targeting methods presented (MPT, Limiting Composite Curve and WSD), minimum freshwater and regenerated water consumption can be targeted with the regeneration

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re-use algorithm. This is the first known algorithm presented for these problems. During targeting with the regeneration re-use algorithm, the regeneration pinch points are determined. Regeneration first pinch point can be different from the re-utilisation first pinch point, in fact, it can be any one of the pinch points. The WSD presented as a targeting-design tool for re-use, can be used with freshwater and regenerated water targets from the regeneration re-use algorithm to achieve a regeneration re-use network. The construction procedure is divided into two parts, in order to avoid water being used in the same operation twice. First, freshwater must be used in order to achieve the targeted flowrate of regenerated water, and then, as a second step, the regenerated water and the remaining freshwater (if f 'f ). As a result the mass exchange network is not, UQ  in most cases, a minimum number of units network. A minimum number of units network, with a small penalty in freshwater consumption, can be achieved by following a simple heuristic rule.

f  f   K F  I

Notation

x  G

b C  C G C  G C H C J

C M C  C  G C  K C UQ C UQI f ?NK f G f  f 

intercept of equilibrium line for contaminant inlet concentration to regeneration process, ppm inlet concentration of contaminant in water for operation i, ppm maximum inlet concentration of contaminant in water for operation i, ppm upper limit concentration of mass interval j, in ppm upper limit concentration of composition interval l, l having an upper limit concentration higher than the first pinch concentration, ppm minimum outlet concentration from regeneration process, ppm outlet concentration from regeneration process at given process conditions, ppm maximum outlet concentration of contaminant in water from operation i, ppm mth pinch point concentration, ppm concentration of contaminant in water source capable of solving the mass-exchange problem, ppm concentration level of the kth water source, ppm water flowrate needed after the mth pinch, t/h limiting flowrate of operation i, t/h flowrate to regeneration process, t/h flowrate from regeneration process, t/h

f UQ f UQIG f UQIGN G G ¸ G m N  N  N M N UQ RR x

x   G x G x  G y  G y G

regeneration process flowrate, t/h flowrate rejected at the mth pinch, t/h flowrate rejected at internal water source k, t/h minimum flowrate of water source that can be used to solve the problem, t/h flowrate requirements of operation i, from water source k, t/h flowrate requirements of operation i, from water source k, in interval p, t/h mass flowrate of rich process stream i, kg/h mass flowrate of water stream i, kg/h slope of equilibrium line for contaminant number of external water sources number of mass intervals in the mass problem table number of mass exchange operations, number of internal water sources total number of water sources removal ratio in the regeneration process mass fraction of contaminant in a water stream supply mass fraction of contaminant in water stream i maximum inlet mass fraction of contaminant in water stream i target mass fraction of contaminant in water stream i maximum outlet mass fraction of contaminant in water stream iy - mass fraction of contaminant in a rich stream supply mass fraction of contaminant in rich stream i target mass fraction of contaminant in rich stream i

Greek letters a G

fraction of the operation that uses freshwater d positive amount of mass, kg/h Dm cumulative mass exchanged until the end    H of mass interval j, kg/h Dm cumulative mass exchanged until the end    J of mass interval l, l having an upper limit concentration higher than the first pinch concentration, kg/h Dm cumulative mass exchanged until the     K mth pinch, kg/h Dm mass of contaminant exchanged in operaG tion i, kg/h Dm mass exchanged in operation i, in concenGN tration interval p, kg/h Dm mass of contaminant exchanged in mass H interval j, kg/h

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e 

minimum allowable inlet composition difference for contaminant minimum allowable outlet composition difference for contaminant

e

Appendix A: Proof of the two WDS properties that allow for easy loop elimination We begin to prove that the mixing of water streams creates a stream that removes the same mass of contaminant, and then, that its concentration is not higher than the maximum inlet concentration of the corresponding operation. Let us consider that the total mass of operation i, is removed by k different water sources, each having a certain f flowrate and a C concentration. Then UQIG UQI I Dm "m #2#m " [ f ;(C !C )] G  I UQ?  G UQ? ? I I "C ; f ! (f ;C )  G UQ?G UQ?G UQ? ? ? I (f ;C ) I UQ? ! ? UQ?G " f ; C  G UQ?G I f ? UQ?G ? I " f ;(C !C ). (A.1) UQ?G  G G ? The last step in the demonstration resulting from the definition of C . G To relate this concentration to the maximum inlet concentration, we recall that the maximum flowrate that can be used to get the water stream to the maximum outlet concentration in a mass-exchange operation, is the limiting water flowrate. Then




 

I I f )f (") f UQ?G G UQ?G ? ? Dm I G ) (") f UQ?G C !C  G  G ?



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I (f );(C !C )  G G ) ? UQ?G C !C  G  G (") C !C !C !C  G  G  G G (") C )C (A.2) G  G as stated.

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