A Method For Producing Ultrapure Water

Abstract

The present invention relates to a method for producing purified water comprising a step (a) of passing water through a mixed bed ion exchanger comprising beads having a diameter between 0.2 and 0.7 mm and a step (b) of passing water through a fibrous ion-exchange material. The invention further relates to a module comprising the mixed bed ion exchange resin and the fibrous material and to a water treatment system for producing ultrapure water comprising the mixed bed ion exchange resin and the fibrous material.

Claims

1. A method for producing purified water comprising a step (a) of passing water through a mixed bed ion exchanger comprising beads having a diameter between 0.2 and 0.7 mm and a step (b) of passing water through a fibrous ion-exchange material.

2. A method according to claim 1, wherein the purified water is ultrapure water.

3. A method according to claim 1, wherein step (a) is performed before step (b).

4. A method according to claim 1, wherein the mixed bed ion exchanger consists of a mixture of anion exchange particles and cation exchange particles.

5. A method according to claim 1, wherein the mixed bed ion exchanger is based on styrene divinylbenzene co-polymer.

6. A method according to claim 1, wherein the fibrous ion-exchange material is a non-woven fibrous ion-exchange material.

7. A method according to claim 1, wherein the fibrous ion-exchange material comprises layers of fibrous cation-exchange material comprising a substrate into which cation-exchange groups have been introduced by radiation-induced graft polymerization and layers of fibrous anion-exchange material comprising a substrate into which anion-exchange groups have been introduced by radiation-induced graft polymerization.

8. A method according to claim 1, wherein the fibrous ion-exchange material is based on polypropylene substrate into which sulfonic groups or quaternary ammonium groups have been introduced by radiation-induced graft polymerization.

9. A method according to claim 1, wherein the ratio of the volume of the mixed bed ion exchanger to the volume of the fibrous ion-exchange material is between 10:1 and 1:5.

10. A method according to claim 1, wherein the method comprises a further step (c) of passing water through an activated carbon bed.

11. A method according to claim 1, wherein the method comprises a further step (d) of treating water by reverse osmosis and/or a further step (e) of treating water by electrodeionization, wherein step (d) and step (e) are performed prior to steps (a) and (b).

12. A module comprising a mixed bed ion exchanger comprising beads having a diameter between 0.2 and 0.7 mm and a fibrous ion-exchange material.

13. A module according to claim 12, wherein the mixed bed ion exchanger is based on styrene divinylbenzene co-polymer.

14. A module according to claim 12, wherein the fibrous ion-exchange material is a non-woven ion-exchange material.

15. A module according to claim 12, wherein it further comprises an activated carbon bed.

16. A water treatment system for producing ultrapure water comprising a mixed bed ion exchanger comprising beads having a diameter between 0.2 and 0.7 mm and a fibrous ion-exchange material.

17. A water treatment system according to claim 16, wherein the mixed bed ion exchanger and the fibrous ion-exchange material are provided in a single module comprising a mixed bed ion exchanger comprising beads having a diameter between 0.2 and 0.7 mm and a fibrous ion-exchange material.

18. A water treatment system according to claim 16, wherein the mixed bed ion exchanger and the fibrous ion-exchange material are provided in at least two modules.

19. A water treatment system according to claim 16, further comprising an activated carbon bed.

20. The water treatment system according to claim 19, wherein said activated carbon bed is mixed with said mixed bed ion exchanger.

Description

FIGURES

[0105] FIG. 1: Three zones of an ion exchange cartridge.

[0106] FIG. 2: The flow schematics of the experimental set-up for Example 1.

[0107] FIG. 3: Configurations of cationic and anionic exchange material layers as tested in Example 1.

[0108] FIG. 4: Comparison of the deionization efficiency of conventional mixed bed ion exchange resin and fibrous material as described in Example 1.

[0109] FIG. 5: Dynamic capacity test for configuration 5 according to Example 3.

[0110] FIG. 6: Dynamic capacity test for configuration 7 according to Example 4.

[0111] FIG. 7: Dynamic capacity test for extended configuration 5 according to Example 5, using 11 cm of standard resin (FIG. 7A), 5.5 cm of standard resin and 5.5 cm of Organex resin (FIG. 7B), and 3 cm of Organex resin and 8 cm of standard resin (FIG. 7C). FIG. 7D compares all dynamic capacity curves including original configuration according to Example 2.

[0112] FIG. 8: Dynamic capacity at different flow rates according to Example 6.

[0113] FIG. 9: Resistivity of combination cartridges determined for a full scale model on a Milli-Q system according to Example 7.

[0114] FIG. 10: TOC values of combination cartridges determined for a full scale model on a Milli-Q system according to Example 7.

[0115] FIG. 11: Resistivity (FIG. 11A) and TOC values (FIG. 11B) of combination cartridges determined for a full scale model on a Milli-Q system according to Example 8.

[0116] FIG. 12: Comparison of the resistivity of standard resin and fibrous material over time using feed water containing weak acids according to Example 9.

[0117] FIG. 13: Dynamic capacity test for different media according to Example 10.

EXAMPLES

Example 1

Ion-Exchange Efficacy of Fibrous Ion-Exchange Media

[0118] In the following experiments fibrous anion exchange material and cation exchange material obtained from Ebara Clean Environment are used. These materials are based on a polypropylene microfiber non-woven substrate (fiber nominal diameter 50 m), subjected to radiation graft polymerization for introducing a copolymer (chloromethylstyrene and glycidylmethacrylate) and functionalized to quaternary amine and sulfonate. Material being approximately 1 mm thick possesses 1.4 meq/g dry anionic or 2.7 meq/g dry cationic exchange capacity, according to ASTM D2187-2009 total capacity equivalent method.

[0119] Non-regenerated fibrous ion exchange material which are not treated for ultrapure water production have to be regenerated and purified before use according to the present invention.

[0120] Materials are first die-cut into disks with an adequate diameter of 35 mm or 69 mm. Non-regenerated cation exchangers are exposed to 2N HCl solution (prepared from 25% HCl (EMSURE, Merck KGaA)) exceeding more than 10 times the ion exchange capacity of the cation exchangers for more than 12 hours at room temperature on an orbital mixing table. The materials are then rinsed three times with Milli-Q water. Finally the ion exchanger disks are filled into the column and rinsed with Milli-Q water at 0.89 cm/sec linear velocity for 15 minutes.

[0121] For the anion exchanger, 2N NaOH (prepared from 50% NaOH (EMSURE, Merck KGaA)) is used instead of HCl.

[0122] Layered ion exchange material is put in a 35 mm diameter flow through column to test the exchange kinetics.

[0123] NaCl is added into the water in order to adjust the feed ionic conductivity. Injected salt is mixed in recirculation and salt in feed and effluent is determined using a conductivity meter to calculate the deionization efficiency.

[0124] For simulating typical feed water conditions in laboratory, NaCl (Merck EMSURE) is spiked to a conductivity of 25 S/cm into ultrapure water prepared by Elix 100 system (Merck KGaA, Darmstadt, Germany), SDS 200 (Merck KGaA, Darmstadt, Germany) and Mill-Q Reference A+ (Merck KGaA, Darmstadt, Germany).

[0125] In the test bench, ultrapure water stored in 10 L PE tank recirculates through a make-up polisher (Quantum TEX polishing cartridge, Merck KGaA, Darmstadt, Germany) and a test column containing ion exchange resin samples. Upstream of the test tube, a salt injection point is located where a precise injection pump (ISMATEC MCP-CPF process pump+PMOCKC pump head) spikes concentrated salt solution prepared at 30 g/L to target conductivity of 25 S/cm. Resistivity sensors (Thornton 770MAX, Mettler Toledo) mesure water resistivity at the inlet and outlet of the test column.

[0126] The diameter of the test column is 35 mm as scale model. The flow rate of water recirculation is adjusted to a linear velocity of 0.69 cm/sec, i.e. 0.4 L/min.

[0127] The flow schematics of this experimental set-up is shown in FIG. 2 (dynamic capacity test bench).

[0128] The following configurations (Table 3) are tested (C stands for cation exchange material; A for anion exchange material) (see also FIG. 3):

TABLE-US-00003 TABLE 3 Number of Number of H, Ionic Configuration discs C discs A mm removal Sandwich 4 (C + 3A) 4 12 21 54% Sandwich 7 (C + 2A) + C 8 14 30 92% Sandwich 9 (C + 2A) + C 10 18 40 95% Sandwich 9 (C + 3A) + C 10 27 50 95% Sandwich 13 (C + 2A) + C 14 26 55 99% Group (3C + 6A) 3 6 11 27% Group (10C + 18A) 10 18 40 67%

[0129] Results:

[0130] Three configurations with different media thicknesses are evaluated. The sandwich configuration with a C/A ratio of 1/2 (C+2A) or 1/3 (C+3A) showed good results. The deionization efficiency is improved when the stack thickness is increased. The comparison of 9(C+2A) and 9(C+3A) shows that the 1/3 ratio configuration does not improve deionization despite the increased anionic capacity. In contrast, the separated media group configuration demonstrates poor results.

[0131] It should be noted that the 1/2 ratio is ionically equilibrated considering the media capacities (2.7 meq/g and 1.4 meq/g for cation and anion exchange media, respectively).

[0132] In a further experiment the deionization efficiency is compared between conventional mixed bed ion exchange resin (Jetpore, used in Milli-Q consumable cartridge, Merck KGaA) and the fibrous material in the 1/2 ratio (C+2A) as described above.

[0133] The results are shown in FIG. 4: The fibrous ion-exchange material shows considerably faster kinetics. For achieving the same deionization, the standard ion exchange resin requires a column height more than three times higher than for the fibrous media.

Example 2

Combination of Ion Exchange Resin and Fibrous Ion Exchanger

[0134] The combination of standard resin and fibrous ion exchange media is evaluated with different feed water qualities. As fibrous media the same media as in Example 1 is used. Jetpore mixed bed resin (Merck KGaA) is used as standard resin. The flow schematic of the dynamic capacity tests is the same as described in Example 1 and FIG. 2. The test cartridges containing standard resin and fibrous media in combination are prepared in 35 mm diameter columns. For the fibrous media the configuration 7(C+2A)+C is used. The flow rate is adjusted at LV 0.89 cm/sec or 0.5 L/min at a 35 mm test column, being equivalent to 2 L/min for a 69 mm diameter cartridge.

[0135] The initial resistivity value is read at early period of NaCI injection at target conductivity (maximum point or plateau). The NaCI dynamic capacity is calculated by retained NaCI quantity to resistivity drop down to 10 M.Math.cm. The total capacity is obtained from volumes of ion exchangers and their capacity values obtained by ASTM D2187-2009 total capacity method. The table below (Table 4) shows the test configurations and test results.

TABLE-US-00004 TABLE 4 Feed NaCl NaCl Flow conduc- Initial dynamic Total rate, tivity, resistivity, capacity, capacity, Configuration L/min S/cm M .Math. cm g/cartridge g/cartridge 1) Standard resin, 0.5 5 13 1.4 3.3 8 cm 2) Standard resin, 0.5 5 >18 3.8 8.6 21 cm 3) Standard resin 0.5 1 18 1.7 3.4 8 cm + fibrous media 1 cm 4) Standard resin 0.5 1 17 1.5 3.4 8 cm + fibrous media 2 cm 5) Standard resin 0.5 1 >18 1.9 3.5 8 cm + fibrous media 3 cm 6) Standard resin 0.5 5 >18 1.8 3.5 8 cm + fibrous media 3 cm 7) Standard resin 0.5 5 >18 5.7 8.8 21 cm + fibrous media 3 cm 8) Standard resin 0.5 50 >18 5.8 8.8 21 cm + fibrous media 3 cm

Example 3

Combination of Ion Exchange Resin and Fibrous Ion Exchanger: Configuration 5

[0136] Using standard resin alone at 8 cm height does not allow to achieve an ultrapure water quality >18 M.Math.cm. The combination with fibrous material requires a layer of at least 3 cm. Such combination achieves ultrapure water polishing at the outlet of the cartridge. The fast kinetics of fibrous media allow for the trapping of ionic leakage from the standard resin. FIG. 5 shows the result of the dynamic capacity test for configuration 5 according to Example 2 above. In this configuration the standard resin compartment allows for deionization from 1 S/cm to >10 M.Math.cm (<0.1 S/cm). In terms of ionic load this means that almost 1/20 of ions are to be retained by the fibrous media, while the total capacity gap between the two media is around 1/20. Ideally, for an equal media saturation and simultaneous exhaustion at the end of the cartridge lifetime, the ionic load gap should be close to the capacity gap of the media. Based on the feed water quality and the capacity gap determined from the standard resin and fibrous media specification, the target ionic leak and media quantity ratio can be calculated. For typical feed water quality of 0.5-1 M.Math.cm (typical for RO to EDI water stored in a tank) the combination of 8 cm standard resin and 3 cm fibrous resin material is optimum.

Example 4

Combination of Ion Exchange Resin and Fibrous Ion Exchanger: Configuration 7

[0137] FIG. 6 shows the result of the dynamic capacity test for configuration 7 according to Example 2 above, wherein 21 cm of standard resin are combined with 3 cm of fibrous non-woven material. The cartridge capacity is determined at the 10 M.Math.cm resistivity set point: The standard resin alone only results in a capacity equivalent of 15.2 g NaCl. This resin should have 34 g NaCl total capacity with this bed volume. The addition of 3 cm of fibrous media results in a recovery of the capacity to 22.8 g NaCl. Surprisingly, this capacity increase is not directly linked to the additional capacity of the fibrousmedia (which is estimated to be only about 0.2 g NaCl), but the media allows for accessing the residual capacity of the standard resin and thereby extending the apparent resin capacity.

Example 5

Combination of Ion Exchange Resin, Activated Carbon and Fibrous Ion Exchanger: Expansion of Configuration 5

[0138] FIG. 7 shows the results of the dynamic capacity tests for extended configuration 5 and its derivatives. In comparison to configuration 5 according to Example 2 above, wherein 8 cm of standard resin are combined with 3 cm of fibrous non-woven material, the extended model of configuration 5 comprises 11 cm of standard resin (FIG. 7A). This additional height of standard resin promises more capacity and lifetime of the cartride.

[0139] Extended volume for standard resin can be addressed to other purification media such as activated carbon which is able to remove dissolved organics. Two further configurations in which 50% (5.5 cm+5.5 cm) and 27% (3 cm+8 cm) of 11 cm height standard ion exchange resins are replaced by Organex resin are tested. In these configurations a homogeneous mixture of standard mixed bed resin and spherical synthetic activated carbon in same average diameter is used (FIG. 7B and FIG. 7C).

[0140] FIG. 7D compares all dynamic capacity curves in the same graph including the original configuration 5 according to Example 2.

Example 6

Impact of the Flow Rate

[0141] The flow rate is investigated by comparing a cartridge filled with standard resin only (24 cm mixed bed resin as described above) and a cartridge filled with a combination of standard resin with fibrous non-woven material (21 cm standard resin with 3 cm fibrous non-woven material, according to configuration 7 as described above). The dynamic capacity is determined as described in Example 1. For the cartridge containing standard resin a flow rate of 0.5 L/min (LV 0.22 cm/min) is used. For the combination cartridge the flow rate is 2.0 L/min (LV 0.89 cm/min). The feed water is RO water produced from municipal water in Guyancourt, France by RiOs 200 system with a conductivity range of 17-22 S/cm and dissolved CO.sub.2 around 10-15 ppm.

[0142] The result is shown in FIG. 8: The ion exchange cartridge (Quantum IX, QTUM000IX, Merck KGaA) containing only mixed bed standard ion exchange resin (Jetpore) and supposed to produce ultrapure water can operate at a flow rate of around 0.5 L/min. At higher flow rates the resin cannot achieve >18 M.Math.cm water quality because of the lack of ion exchange kinetic. In contrast, the combination of 21 cm standard resin with 3 cm fibrous non-woven material is able to increase the resistivity to >18 M.Math.cm. Both cartridges have the same length (24 cm), but the cartridge combining standard resin with fibrous non-woven material can be operated at a much higher flow rate (4 times) without experiencing a capacity loss.

Example 7

Full Scale Experiment

[0143] The concept is also demonstrated in a full scale model with full scale cartridges on a Milli-Q system (commercially available from Merck KGaA, Darmstadt, Germany). In this experiment the cartridge diameter is 69 mm (instead of 35 mm as in some examples above). The flow schematic is as follows: An Elix system comprising means for RO and EDI produces pure water which is filled in a 60 L polyethylene tank. Two identically modified Milli-Q systems with 8 cm standard resin and 3 cm non-woven fibrous material produce ultrapure water periodically. The TOC (total organic carbon) measurement is accomplished by an Anatel A-1000 TOC analyzer and conductivity cells. The flow rate is adjusted up to 2 L/min.

[0144] Resistivity:

[0145] FIG. 9 shows the resistivity determined in the experiment. The system is able to produce water of >18 M.Math.cm when it is fed with Elix grade water (>1 M.Math.cm). The combination of the solid line (outlet of combination cartridge (combo pack)) and broken line (intermediate resin outlet) show the resistivity on each step (standard resin, fibrous resin) and each system (L=left system, R=right system, F=feed water). If a conventional ultrapure water system is used a column of at least 24 cm is necessary in order to achieve the same water quality.

[0146] TOC values:

[0147] FIG. 10 shows the TOC values determined in the experiment (R=right system, L=left system, F=feed water. At the start-up TOC rinsing the initial TOC value is around 5 ppb and decreases by the rinsing effect to <5 ppb. The TOC value remains at a low level over 10000 L capacity.

Example 8

Full Scale Experiment: Comparison of Configuration 5, Extended Configuration 5 and Prior Art

[0148] Combination of standard resin and fibrous media according to configuration 5 of Example 3 (8 cm standard resin+3 cm fibrous media), extended configuration 5 of Example 5 (8 cm standard resin+3 cm fibrous media) and prior art standard ion exchange cartridge (25 cm standard resin alone) are compared in a full scale test setting according to Example 7. Test cartridges are prepared in diameter 69 mm columns for combination of standard resin and fibrous media, Quantum TIX (QTUM 0TIX1, Merck KGaA) is used as prior art representative of 25 cm height standard resin.

[0149] In addition to TOC measurement, the water quality is evaluated by ion chromatography (Dionex ICS-3000) and graphite furnace atomic absorption spectrometry (Perkin Elmer AAnalyst 600) for element analysis. For microbiology related quality, microbial count is done by Milliflex system with membrane funnel (MXHAWG124) and R2A agar culture media cassette (MXSMCRA48) after 72 hours incubation at 22 C., endotoxin is determined by Limulus Amebocyle Lysate Kinetic-QCL (Lonza).

[0150] Resistivity

[0151] FIG. 11A shows the resistivity determined in the experiment. The systems with combinations of standard resin and fibrous media are able to produce water of >18 M.Math.cm when it is fed with Elix grade water (>1 M.Math.cm) as well as prior art 25 cm standard resin alone cartridge.

[0152] TOC Values:

[0153] FIG. 11 B shows the TOC values determined in the experiment. There is no significant difference on TOC values between the three configurations including the prior art cartridge.

[0154] Ion Chromatography Analysis:

[0155] The trace ion analysis is done by ion chromatography. The results of three major ions abundant in pure and ultrapure water are shown in the tables below (Tables 5, 6, 7).

TABLE-US-00005 TABLE 5 Chloride analysis by ion chromatography, all units in g/L (ppb) 1 Invention 2 Invention 8 cm 11 cm 3 Prior art standard standard 25 cm resin + 3 cm resin + 3 cm standard fibrous media fibrous media resin Feed water 550 L <0.2 <0.2 <0.2 0.6 1250 L <0.2 <0.2 <0.2 <0.2 2100 L <0.2 <0.2 <0.2 <0.2 3000 L <0.2 <0.2 <0.2 <0.2 46000 L <0.2 <0.2 <0.2 0.4

TABLE-US-00006 TABLE 6 Sodium analysis by ion chromatography, all units in g/L (ppb) 1 Invention 2 Invention 8 cm 11 cm 3 Prior art standard standard 25 cm resin + 3 cm resin + 3 cm standard fibrous media fibrous media resin Feed water 550 L <0.2 <0.2 <0.2 2.0 1250 L <0.2 <0.2 <0.2 0.8 2100 L <0.2 <0.2 <0.2 1.0 3000 L <0.2 <0.2 <0.2 0.9 46000 L <0.2 <0.2 <0.2 0.5

TABLE-US-00007 TABLE 7 Pottasium analysis by ion chromatography, all units in g/L (ppb) 1 Invention 2 Invention 8 cm 11 cm 3 Prior art standard standard 25 cm resin + 3 cm resin + 3 cm standard fibrous media fibrous media resin Feed water 550 L <0.03 <0.03 <0.03 0.60 1250 L <0.03 <0.03 <0.03 0.09 2100 L <0.03 <0.03 <0.03 0.28 3000 L <0.03 <0.03 <0.03 0.13 46000 L <0.03 <0.03 <0.03 0.08

[0156] GFAAS Analysis:

[0157] The following table shows the GFAAS analysis for some specific elements (Table 8):

TABLE-US-00008 TABLE 8 Trace elements ana; ysis by graphite furnace atomic absorption spectrometry, sampled at 150 L, all units in g/L (ppb) 1 Invention 2 Invention 8 cm 11 cm 3 Prior art standard standard 25 cm resin + 3 cm resin + 3 cm standard fibrous media fibrous media resin Beryllium <0.02 <0.02 <0.02 Aluminium 0.3 0.2 0.2 Silicon <1 <1 13 Chromium <0.1 <0.1 <0.1 Copper <0.1 <0.1 <0.1 Zinc 0.3 0.2 0.1 Arsenic <0.3 <0.3 <0.3 Selenium <0.2 <0.2 <0.2 Silver <0.02 <0.02 <0.02 Cadmium <0.006 <0.006 <0.006 Thalium <0.4 <0.4 <0.4 Lead <0.2 <0.2 <0.2

[0158] The table below shows the result of microorganism and endotoxin levels (Table 9).

TABLE-US-00009 TABLE 9 Microbial contamination, sampled at 700 L 1 Invention 2 Invention 8 cm 11 cm 3 Prior art standard standard 25 cm resin + 3 cm resin + 3 cm standard fibrous media fibrous media resin Feed water Bacteria 3 4 2 31 count, cfu/ml Endotoxin, <0.005 <0.005 <0.005 0.03 EU/ml

[0159] The combination cartridges comprising standard ion exchange resin and fibrous media can achieve equivalent water quality in resistivity, TOC, ions, trace elements and biological contamination index to conventional ion exchange cartridge in Milli-Q ultrapure water system condition. Without any compromise in purity of ultrapure water, the systems with the cartridges th.sub.rough this invention realizes compactness of cartridges.

Example 9

Weak Acid Retention of Fibrous Fibrous Material

[0160] A similar experiment as described in Example 1 is performed, wherein the feed water contains 10 ppb formate and acetate (column diameter 35 mm, standard resin bed depth 3 cm or fibrous media bed depth 3 cm (7(C+2A)+C), flow rate 400 ml/min, 1 S/cm equivalent, LV =0.69 cm/sec). The result is shown in FIG. 12: Weakly dissociated organic acids as formate and acetate show faster exchange kinetics with fibrous material than with standard resin. The fibrous material retains better the acetate and formate under challenging conditions. Such components are difficult to remove from ultrapure water by conventional mixed bed resins. Plastic components of the water system may however potentially release such organics. Therefore the retention ability of the fibrous material at the polishing step may be of great value.

Example 10

Combination of Ion Exchange Resin and Fibrous Ion Exchanger

[0161] Examples 1, 2 and 3 are repeated using different fibrous ion exchange material (Institute of Physical Organic Chemistry of the National Academy of Science of Belarus. This material, similar to the material used in Example 1, is processed by irradiation initiated graft polymerization to introduce ion exchange fuctional groups on polypropylene fibers. Fibrous substrate is a needle punched non-woven fabric. The thickness is 3 to 12 mm (instead of 1 to 2 mm). The capacity density (meq/g) is similar. The ion exchange capacities for dried anion exchanger and cation exchanger are 2.5 meq/g and 2.6 meq/g, respectively. Since characteristic of this media is different from the one in previous examples, media arrangement is optimized by capacity equilibrium rule and anion/cation sandwich configuration. Media are treated by the same procedure as described in Example 1 for disc preparation, regeneration and purification. Configuration 5 in Example 2 (8 cm standard resin+non-woven media combination) and 3 are reproduced with this media in a scale model (Table 10).

TABLE-US-00010 TABLE 10 Number of Number of H, Configuration discs C discs A mm Invention 1 Sandwich 8 14 30 Example 1 7 (C + 2A) + C Invention 2-1 Sandwich 6 5 30 5 (C + A)) + C Invention 2-2 Sandwich 7 6 38 6 (C + A)) + C Invention 2-3 Sandwich 8 7 45 7 (C + A)) + C

[0162] The reproduction of the performance as shown in Example 2 is possible if the capacity equilibrium rule is respected, as can be seen in FIG. 13.