A summary of influence of temperature on the water flux.
\\n\\n
Released this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\\n\\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
\\n"}]',published:!0,mainMedia:null},components:[{type:"htmlEditorComponent",content:'IntechOpen is proud to announce that 191 of our authors have made the Clarivate™ Highly Cited Researchers List for 2020, ranking them among the top 1% most-cited.
\n\nThroughout the years, the list has named a total of 261 IntechOpen authors as Highly Cited. Of those researchers, 69 have been featured on the list multiple times.
\n\n\n\nReleased this past November, the list is based on data collected from the Web of Science and highlights some of the world’s most influential scientific minds by naming the researchers whose publications over the previous decade have included a high number of Highly Cited Papers placing them among the top 1% most-cited.
\n\nWe wish to congratulate all of the researchers named and especially our authors on this amazing accomplishment! We are happy and proud to share in their success!
Note: Edited in March 2021
\n'}],latestNews:[{slug:"intechopen-partners-with-ehs-for-digital-advertising-representation-20210416",title:"IntechOpen Partners with EHS for Digital Advertising Representation"},{slug:"intechopen-signs-new-contract-with-cepiec-china-for-distribution-of-open-access-books-20210319",title:"IntechOpen Signs New Contract with CEPIEC, China for Distribution of Open Access Books"},{slug:"150-million-downloads-and-counting-20210316",title:"150 Million Downloads and Counting"},{slug:"intechopen-secures-indefinite-content-preservation-with-clockss-20210309",title:"IntechOpen Secures Indefinite Content Preservation with CLOCKSS"},{slug:"intechopen-expands-to-all-global-amazon-channels-with-full-catalog-of-books-20210308",title:"IntechOpen Expands to All Global Amazon Channels with Full Catalog of Books"},{slug:"stanford-university-identifies-top-2-scientists-over-1-000-are-intechopen-authors-and-editors-20210122",title:"Stanford University Identifies Top 2% Scientists, Over 1,000 are IntechOpen Authors and Editors"},{slug:"intechopen-authors-included-in-the-highly-cited-researchers-list-for-2020-20210121",title:"IntechOpen Authors Included in the Highly Cited Researchers List for 2020"},{slug:"intechopen-maintains-position-as-the-world-s-largest-oa-book-publisher-20201218",title:"IntechOpen Maintains Position as the World’s Largest OA Book Publisher"}]},book:{item:{type:"book",id:"2261",leadTitle:null,fullTitle:"Entrepreneurship - Born, Made and Educated",title:"Entrepreneurship",subtitle:"Born, Made and Educated",reviewType:"peer-reviewed",abstract:"Entrepreneurship has a tremendous impact on the economic development of a country. 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This lifelong infection suggests that herpesviruses have critical features to evade immunosurveillance. The knowledge about herpesvirus molecular genetics has been critical to design new therapies based on Herpes simplex vectors.
\r\n\r\n\tThis book summarizes the main aspects of Herpesviruses infection, the key molecular mechanisms associated to latency and reactivation, mechanism related to immune evasion, immunosuppression and cellular stress, the contribution of herpesvirus (Herpes simplex type 1, Human herpevirus 6) to the neurodegenerative disease and autoimmunity, development of new drugs and vaccines against Herpesviruses, and development of gene therapy against cancer based on herpes simplex vectors.
\r\n\t
Osmosis, one of the most fundamental transport processes responsible for homeostasis in living organisms, has a rich history of applications—ranging from food preservation to water treatment and drug delivery. Osmosis occurs when a solute concentration difference is established across a semipermeable membrane. Due to the chemical potential imbalance, the water molecules will spontaneously migrate across the membrane toward the higher solute concentration side. Such a process has been regarded as one of the most central mechanisms that dictates the membrane-based water treatment technologies. The most widely utilized process, in our opinion, is reverse osmosis (RO) for solute removal, which requires an external hydraulic pressure to overcome the osmotic pressure difference across the membrane. In contrast to RO, the process that exploits the spontaneous transport of solvent molecules driven by the osmotic phenomenon is referred to as forward osmosis (FO) or direct osmosis (DO), which is, in principle, the same as the original osmosis.
\nFO was first conceptualized by Batchelder as a means for water treatment since the 1960s [1]. Since then, there has been a growing interest in applying FO to wastewater treatment technologies either as a stand-alone or in combination with other technologies such as membrane distillation, thermal distillation, or reverses osmosis [2]. Particularly, FO has been utilized in space stations for wastewater reclamation due to its excellent long-term stability and low energy consumption [3, 4]. Not limited to wastewater treatment, FO has also been explored extensively for many useful applications such as seawater desalination [5, 6, 7], portable hydration bags [8], food processing [9, 10, 11], pharmaceutical systems [12, 13, 14], and energy conversion [15, 16].
\nUnlike RO, FO is purely an osmosis-driven process, which is thermodynamically spontaneous. The osmotic pressure difference \n
for weakly interacting molecules, where \n
In the FO process, the solvent (water) transport is driven solely by osmotic pressure difference without the need of any external hydrostatic pressure, allowing for lower energy consumption compared to RO. To extract water from the feed solution, the osmotic pressure at the opposite side of the membrane must be higher, which requires a highly concentrated solution; this concentrated solution is typically referred to as the draw solution. Draw solutes need to be inert and easily removable. A semipermeable membrane separates the feed solution and the concentrated draw solution where the chemical potential difference allows the water to flow through the membrane while leaving behind the solutes in the feed stream. Regions of high and low solute concentrations refer to those of low and high solvent chemical potentials, respectively. As the semipermeable membrane restricts the solute transport and maintains chemical potential differences of both solute and solvent, water migrates from its high solvent chemical-potential region (i.e., of low solute concentration) to low solvent chemical-potential region (i.e., of high solute concentration). Such a water transport leads to dilution of the draw solution where the diluted draw solution can be further recycled such that the initial solute concentration is recovered. Particularly for desalination applications, the solutes in the draw solution (osmotic agent or draw solutes) are chosen to be inert, nontoxic, and easily removed to obtain the desalinated water with ease. One example includes \n
The water flux across the membrane results in concentration of the feed solution and dilution of the draw solution since the membrane mainly allows passage of water molecules. This phenomenon, referred to as concentration polarization (CP), has an adverse impact on the efficacy of the FO process since such an effect reduces the effective osmotic pressure difference across the membrane, thus hindering water transport.
\nCP is highly influenced by the morphology of the membranes. The membranes used in the FO process consist of a thin, dense layer that rejects the solutes (active layer) followed by a coarse, thick porous layer (support layer or porous substrate) to reinforce the mechanical stability against fluid pressure and shear. This configuration makes the membrane asymmetric in which the orientation of the membrane with respect to the direction of the water flux (i.e., from low to high osmotic pressure) leads to significantly different transport dynamics [20].
\nTypically in the FO process, the active layer is placed against the feed stream in order to minimize fouling since the support layer is more susceptible to colloidal fouling due to the large pores. This configuration is called FO mode, as shown in Figure 1(a). However, the downside of placing it in this way is that there is a significant dilutive internal concentration polarization (ICP) in the thick porous substrate. This is because the support layer is in contact with the concentrated draw solution hindering the solute diffusion, which significantly reduces the water flux (Figure 1(a)).
\nInfluence of CP on the osmotic pressure distribution in the FO process. The membrane is configured in (a) FO mode and (b) PRO mode.
In contrast, when the active layer is placed against the draw stream, one can expect a higher water flux since this configuration can avoid the dilutive ICP at the expense of accelerated membrane fouling. This configuration is called the pressure-retarded osmosis (PRO) mode, as shown in Figure 1(b), typically realized in standard PRO systems. To avoid any confusion, we will refer to the membrane configuration in which the active layer is placed against the feed solution as the
The first quantitative experiments on temperature-dependent osmosis go back almost a century ago [21]. Traxler demonstrated the osmosis of pyridine by using a thin rubber sheet as a semipermeable membrane within a uniform system temperature, ranging from 5 to 85°C (Figure 2(a)). He showed that as the temperature is increased, the transport of pyridine across the membrane is also increased (Figure 2(b)). In this chapter, such a uniform temperature will be refered to as ‘system temperature’ indicating the absence of local or transmembrane temperature gradient.
\nThe first quantitative experiments reported on the effect of temperature on the osmosis phenomenon. (a) A schematic of the experimental setup that allows temperature control via a thermostat. (b) Transport of pyridine across a rubber membrane under various temperature conditions. Reprinted with permission from Ref. [
From the van’t Hoff equation, the osmotic pressure is directly proportional to the system temperature, which is an indispensable factor for the FO process. However, temperature not only influences the osmotic pressure but also impacts many other key properties that are important to the transport process such as viscosity, diffusivity, solubility, density, and so forth. Such a change in the properties not only influences the water flux but also alters the solute rejection/diffusion and membrane fouling. In this section, we provide a summary of how the system temperature influences the water transport, solute rejection, and membrane fouling. We note that the experimental studies that will be covered in the following sections employ a circulating crossflow type setup (in contrast to a dead-end type as seen in Traxler’s experiments in Figure 2).
\nThe most direct consequence of raising the system temperature is the increased water flux across the membrane due to lowered water viscosity and increased water diffusivity, which effectively increases the water permeability across the membrane. Since the transport of water through the active layer of the membrane follows the solution-diffusion mechanism [22], it is commonly believed (and also observed) that the diffusivity \n
On the basis of our survey, the available literature related to the temperature-dependent FO reported increased water flux with temperature. Table 1 provides a summary of experimental conditions and resulting water flux from the available literature [23, 24, 25, 26, 27, 28, 29, 30, 31]. Here, we define a new quantity to indicate how much solvent flux increases with respect to the system temperature, as indicated in the last column of Table 1:
\nReference | \nFeed solution (concentration) | \nDraw solution (concentration) | \nMembrane1 | \nMode2 | \nTemperature (°C) | \n\n\n | \n\n\n | \n
---|---|---|---|---|---|---|---|
[25] | \nTomato juice (0.13 M) | \nNaCl (3.9 M) | \nPA | \n\n | 26–58 | \n1.5 | \n0.030 | \n
[26] | \nNaCl (0–86 mM) | \nKCl (0.5–3 M) | \nCT | \nFO | \n25–45 | \n19 | \n0.43 | \n
[27] | \nDeionized water | \nNaCl (0.5 M) | \nCT | \n\n | 20–40 | \n5.5 | \n0.14 | \n
PA | \n\n | \n | 17 | \n0.49 | \n|||
[23] | \nSucrose (0–1.65 M) | \nNaCl (2–4 M) | \nCT | \n\n | 20–30 | \n24 | \n0.91 | \n
Sucrose (0–0.7 M) | \nNaCl (4 M) | \nPA | \n\n | \n | 2.5 | \n0.15 | \n|
[28] | \nNaCl (0.1 M) | \nNaCl (1 M) | \nCT | \nPRO | \n20–40 | \n11 | \n0.89 | \n
\n | FO | \n\n | 9.4 | \n0.59 | \n|||
[29] | \nNaCl (0–1 M) | \nNaCl (1.5 M) | \nCT | \nPRO | \n20–40 | \n43 | \n1.4 | \n
\n | \n | \n | \n | FO | \n\n | 18 | \n0.63 | \n
[24] | \nNaCl (60 mM) | \n\n\n | \nCT | \n\n | 25–45 | \n15 | \n0.35 | \n
[30] | \nNaCl (0.2–0.5 M) | \n\n\n | \nCE | \nPRO | \n30–50 | \n5.4 | \n0.10 | \n
[31] | \nDeionized water | \nNaCl (1.2 M) | \nCT | \nFO | \n20–30 | \n14 | \n0.61 | \n
A summary of influence of temperature on the water flux.
PA: polyamide; CT: cellulose triacetate; CE: cellulose ester
FO mode: active layer placed against feed solution; PRO mode: active layer placed against draw solution
where \n
McCutcheon and Elimelech were the first to study the influence of temperature on the CP phenomena [29]. Raising the temperature increases the water flux because of the decreased water viscosity in solutions (and/or solubility) and increased water solubility and diffusivity within the membrane. At the same time, however, the higher flux also increases both the ICP and ECP, which essentially limit the water flux as a feedback hindrance. Therefore, such a self-limiting behavior driven by two counteracting effects leads to the fact that the temperature has a “modest” effect on the water flux at high water flux conditions [29]. This self-hindering effect of the solvent flux is unavoidable in most membrane separation processes. It is similar to the fact that, in RO, applying high pressure initiates increasing permeate flux, which will eventually bring more solutes from the bulk phase to the membrane surface, enhancing the CP. Therefore, additional gain of the RO permeate flux is not as much as anticipated when the pressure is increased.
\nThe change in the temperature influences the CP phenomena in different ways depending on the orientation of the membrane. This is because the formation of the ICP, which is the most critical factor that limits the driving force, is dependent on the membrane configurations. In the PRO mode, the concentrative ICP is developed in the feed side (see Figure 1(b)). By reducing the ICP using deionized water as the feed, the water flux was shown to be highly dependent on the temperature, confirming the impact of ICP on the FO process [29].
\nIn the presence of solutes in the feed side so that the ICP is present, however, the water flux was shown to be almost insensitive to the temperature, at least in the operating temperature range (20–40°C). This behavior is attributed to the coupled interaction between ICP and ECP. Although the increased solute diffusion at higher temperature mitigates the concentrative ICP in the support layer so that the water flux can be increased, such an increased water flux carries more solutes from the feed bulk phase to the vicinity of the support layer surface and enhances the dilutive ECP, thereby reducing the osmotic driving force. Therefore, the two opposing effects on the water transport effectively limit the enhancement of the water flux such that the temperature has a marginal effect on the overall water flux. If both water and solute diffusivities increase in a similar behavior, the net diffusive transport must be more or less the same.
\nIn the FO mode, however, the water flux was shown to be significantly influenced by the temperature. Overall, the water flux was observed to be lower than the PRO mode due to the presence of the dilutive ICP. This was proven mathematically using the method of proof by contradiction [32]. Such a low water flux effectively suppresses the extent of concentrative ECP in the feed side. Also, the influence of concentrative ECP on the water flux is less important than the dilutive ECP in the draw solution side because the initial solute concentration in the bulk phase is much lower at the feed solution than the draw solution. This implies that the ECP has a minor effect on the driving force in the FO mode. Therefore, when the membrane is placed in the FO mode, the water flux is significantly influenced by the temperature since the ICP is the only major factor that determines the driving force.
\nOne assumption McCutcheon and Elimelech had made while analyzing their data were the insignificant solute diffusion across the membrane [29], which otherwise leads to further ICP. Obviously, commercially available membranes are known to permit diffusion of the solutes, which can impact the formation of the CP effect. Since the solute diffusion is also sensitive to the temperature, the transmembrane solute flux should also lead to a change in the water flux. We discuss the effect of temperature on the solute diffusion and rejection in the following section.
\nIt is of general consensus that the transmembrane solute diffusion increases with temperature. A number of groups have recently investigated experimentally the temperature effect on the transmembrane solute diffusion and the solute rejection [26, 27, 28].
\nXie et al. recognized that the effective size of the solute molecules was the most important parameter for the transmembrane solute diffusion [27], which was predicted theoretically using the integral equation theory [33]. Hydration of charged organic solutes results in an increase in the effective solute size, which directly influences the solute diffusion and rejection rate, as it was well understood that the rejection of the charged organic solutes would be much higher than the neutral organic solutes. In this regard, neutral solutes were more likely to diffuse across the pores than the charged solutes in both the cellulose triacetate membranes and polyamide membranes. This implies that increasing the temperature leads to higher solute diffusion due to the increased solute diffusivity. Moreover, increasing the temperature leads to faster dissolution of the solutes into the membrane such that even hydrophobic neutral solutes absorb into the membrane at an order of magnitude higher rate at elevated temperatures.
\nNotably, the ratio between the water flux \n
Meanwhile, You et al. showed that the transmembrane solute diffusion was also shown to be dependent on the membrane orientation regardless of the temperature in which the PRO mode was shown to exhibit higher solute flux across the membrane than the FO mode, which is similar to the behavior of the water flux [28].
\nMembrane scaling occurs when the solute concentration is high enough to initiate precipitation. This is directly related to solute rejection and the CP phenomena, implying that membrane scaling should also be temperature-dependent.
\nZhao and Zou studied how the temperature influences the membrane scaling over time, which is important in long-term operations [24]. Due to the fast water flux at elevated temperature, increase in the final concentration of the feed solution (i.e., concentration after 28 hours of running) was accelerated by more than 100% when the temperature was raised from 25 to 45°C, which led to faster membrane scaling. Concentrative polarization is also enhanced when the water flux is increased, which results in an accelerated membrane scaling. This was confirmed by directly visualizing the fouled membrane by using a scanning electron microscope (Figure 3(a)–(d)) and also by measuring the decrease in flow rate over time (Figure 3(e)), showing faster decline of water flux over time at elevated temperatures due to the scaling. In addition to higher solute concentration near the membrane surface driven by the temperature-enhanced solvent flux, the changes in solubility limits for inorganic species may contribute to the accelerated fouling behavior.
\nTemperature-dependent membrane fouling and associated water flux decline. (a–d) Scanning electron microscope images of the (a) virgin and (b–d) fouled membranes at various temperatures; (b)
One step further, we can also consider a case where the temperature is unevenly distributed across the membrane. In such a case, the temperature gradient may allow independent control of transport on either side of the membrane. In practice, temperature gradients can occur frequently; temperature of the feed solution can increase due to the heat released from the hydraulic pumping or when the solution is pretreated. Likewise, the temperature of the draw solution may change due to the post-treatment process for recovery and recycling of draw solutes such as thermal and membrane distillation. Since heating only on one side of the solution requires lower energy than heating up the entire system, imposing a temperature gradient across the membrane may offer an energy-efficient control over the osmotic phenomena.
\nIn the presence of a temperature gradient, van’t Hoff’s law (of Eq. (1)) cannot be used directly to calculate the osmotic pressure difference since it relies on the assumption of the constant system temperature. A full theory accounting for the temperature gradient in osmosis may result in highly nonlinear effects on the FO performance. Furthermore, the temperature gradient may provide an additional complexity to the coupled mass and heat transfer phenomena within the membrane. In this section, we provide a summary of how the temperature difference between the feed and the draw solution influences the FO performance, including the water transport and solute diffusion/rejection.
\nAlthough the temperature dependence on the water flux shows an agreeable consensus as shown in Table 1, the anisotropic temperature effect is shown to differ largely across various studies. When the temperature on either side of the solutions is increased, the water flux becomes higher than that at the base temperature, but lower than when the temperatures of both sides of the solutions are increased. It is, however, left unclear which side of the solution has more influence on the FO process when heated as this does not have an agreeable consensus. Table 2 provides a summary of the effect of temperature difference on the FO process under various experimental parameters. For simplicity, we define
\nReference | \nFeed solution | \nDraw solution | \nMembrane | \nMode | \nTemperature (°C) | \n\n\n | \n\n\n | \n\n\n | \n
---|---|---|---|---|---|---|---|---|
[10] | \nPineapple juice (0.37 M) | \nSucrose (40 wt%)+NaCl(12 wt%) | \nCT | \n\n | 25–45 | \n1.2 | \n0.045 | \n\n |
[26] | \nNaCl (0–86 mM) | \nKCl (0.5–3 M) | \nCT | \nFO | \n25–45 | \n19 | \n0.048 | \n0.12 | \n
[27] | \nDeionized water | \nNaCl (0.5 M) | \nCT | \n\n | 20–40 | \n5.5 | \n0.045 | \n0.065 | \n
\n | \n | \n | PA | \n\n | \n | 17 | \n0.125 | \n0.175 | \n
[35] | \nNaCl (0–0.5 M) | \nNH4HCO3 (1–4 M) | \nCT | \nPRO | \n25–45 | \n2.5 | \n\n | 0.028 | \n
\n | \n | \n | \n | FO | \n\n | 1.9 | \n\n | 0.018 | \n
[28] | \nNaCl (0.1 M) | \nNaCl (1 M) | \nCT | \nPRO | \n20–40 | \n11 | \n0.54 | \n0.19 | \n
\n | \n | \n | \n | FO | \n\n | 9.4 | \n0.41 | \n0.18 | \n
[36] | \nAnthocyanin (24 \n | \nNaCl (6 M) | \nCT | \nPRO | \n25–40 | \n4.9 | \n0.013 | \n\n |
\n | \n | \n | \n | FO | \n\n | 13 | \n0.53 | \n\n |
[31] | \nDeionized water | \nNaCl (1.2 M) | \nCT | \nFO | \n20–30 | \n14 | \n0.22 | \n0.54 | \n
A summary of influence of temperature difference on the water flux.
and
\nas included on the right-hand side of Table 2. Eqs. (3) and (4) refer to the water flux increase per temperature change when the
In general, regardless of either the feed or draw, raising the temperature on either side leads to increase in both the water flux and the solute flux. Xie et al. stated that raising the feed solution temperature leads to enhanced diffusivity of the water molecules, whereas raising the draw solution temperature leads to decreased draw solution viscosity and increased draw solute diffusivity, both of which lead to increased water flux and reverse solute flux [27]. However, the degree to which the water flux and solute flux are increased varies across the literature [10, 26, 27, 28, 31, 35, 36] (see Table 2).
\nPhuntsho et al. showed that increasing the
You et al. showed, however, that regardless of the membrane orientation, the water flux increased more when the
Interestingly, in Nayak and Rastogi’s study [36], the water flux in the FO mode was shown to be higher than the water flux in the PRO mode particularly when the molecular size of the feed solute is large enough such that the external concentration polarization cannot be ignored. They also showed that this is indeed true for concentrating anthocyanin, which is a large sugar molecule. In their work, the water flux in the FO mode was measured to be 260% higher than that in the PRO mode.
\nAs mentioned in the preceding section, Xie et al. showed that the neutral solutes are more likely to diffuse through the membrane than the charged ones due to their smaller hydrodynamic size [27]. In this sense, transmembrane temperature differences barely influenced the solute rejection rate for the
To the best of our knowledge, effects of temperature and its gradient on the osmosis phenomena and FO processes have been investigated only phenomenologically without fundamental understanding. The theoretical research is currently in a burgeoning state in explaining the transmembrane temperature gradient effect on the FO performance. In this section, we first briefly review the conventional FO theories [37, 38] based on the solution-diffusion model and van’t Hoff’s law. Then, we revisit statistical mechanics to identify the baseline of the osmosis-diffusion theories, where the isothermal condition was first applied. We then develop a new, general theoretical framework on which FO processes can be better understood under the influence of the system temperature, temperature gradient, and chemical potentials.
\nThe solution-diffusion model is widely used to describe the FO process, which was originally developed by Lonsdale et al. to explain the RO phenomena using isothermal-isobaric ensemble [39]. In the model, the chemical potential of water is represented as a function of temperature, pressure, and solute concentration, i.e. \n
where the integration is over the membrane region. From the basic thermodynamic relationship,
\nis used where \n
and hence we derive \n
which becomes
\nwhere \n
where \n
Figure 4(a) shows a schematic representing the PRO and FO modes altogether. Concentrations in the PRO and FO modes are denoted as \n
A schematic representation of (a) concentration polarization across a skinned membrane during FO process in the PRO and FO modes, represented using the solid and dashed lines, respectively and (b) arbitrary temperature profile increasing from the active layer to the porous substrate.
In the PRO mode, the solvent flux (in magnitude) is
\nwhere \n
and that in the porous substrate:
\nIn a steady state, \n
and
\nrespectively, where \n
is interpreted as the characteristic mass transfer resistance, proposed by Lee et al. [37]. Following the convention of standard mass transfer theory, \n
where
\nis an integer to toggle between the two modes. Any theoretical development can be initiated from Eq. (17) to consider universally both the FO and PRO modes, and then a proper value of \n
In the theory, there are several key assumptions during derivations of Eqs. (14) and (15). These assumptions are summarized in the following for the PRO mode for simplicity, but conceptually are identical to those in the FO mode.
\n1. Mass transfer phenomena are described using the solution-diffusion model in which the solvent and solute transport are proportional to the transmembrane differences in the osmotic pressures and solute concentrations, respectively [39]. If one sees these combined phenomena as diffusion, the solvent transport can be treated as semibarometric diffusion. In other words, under the influence of pressure, the solute transport can be treated as Fickian diffusion, driven by the concentration gradient. In a universal view, the net driving forces of the solvent and solutes are their chemical potential differences.
\n2. In the flux equations, \n
3. The osmotic pressure is presumed to be linear with the solute concentration \n
using \n
4. Rigorously saying, mass transport phenomena are assumed to be in a steady state and equilibrium thermodynamics are used to explain the filtration phenomena. Although the FO phenomenon occurs in an open system, transient behavior is barely described in the literature.
\n5. In the porous substrate, the bulk porosity is assumed to be uniform, which implies isotropic pore spaces. Moreover, the interfacial porosity between the active and porous layers is assumed to be equal to the bulk porosity. An in-depth discussion on the interfacial porosity can be found elsewhere [40]. In the same vein, the tortuosity is a characteristic geometric constant of the substrate, which is hard to measure independently. More importantly, tortuosity is included in the definition of the structural parameter \n
6. The solute diffusivity \n
7. Finally, temperatures of the draw and the feed streams are assumed equal although hydraulic and thermal conditions of these two streams can be independently controlled. As a consequence, heat transfer across the membrane is barely discussed in the literature.
\nIn practice, solvent and solute permeability \n
This chapter aims to explain how the temperature across the FO membrane, which consists of the active and porous layers, may affect the performance of the mass transfer at the level of statistical physics. The transmembrane temperature gradient prevents from using the abovementioned assumptions and approximations, which are widely used in the FO analysis. First, the SD model is purely based on isothermal-isobaric equilibrium in a closed system. Second, the external concentration polarizations in the draw and feed sides cannot be neglected at the same level because the temperature gradient causes a viscosity difference across the membrane. Third, the weighting factor connecting \n
Figure 4(b) shows an arbitrary temperature profile across the FO membrane, increasing from the active layer side to the porous layer side. In bulk phases of the active and porous sides, temperatures are maintained at \n
where subscripts BA and BP indicate bulk phases in the active and porous layer sides, respectively, and AL and PL mean the active layer and porous layer, respectively. The net temperature difference across the membrane is \n
Note that Eq. (24) assumes that the heat transfer is solely based on thermal conduction without thermal convection, that is, transfer rate of heat by solvent flux. In the FO process with the transmembrane thermal gradient, Eqs. (21) and (22) should be revised as
\nwhere \n
where
\nThis heat balance analysis is very similar to that of membrane distillation [43, 44], but the FO process does not have any solvent phase transition so that the latent heat is not considered.
\nIn statistical mechanics, Gibbs energy is the master function of the isothermal-isobaric ensemble. Consider a box in which two regions are separated by a semipermeable membrane. In equilibrium, the maximum entropy condition requires that the chemical potential divided by the temperature should be constant, i.e.
\nwhich converts to the constant chemical potential for the isothermal environment, i.e. \n
which can perhaps be extended intuitively to \n
where \n
Fick’s law is a phenomenological equation based on experimental observations. The equation states that the diffusive flux \n
In the dilute limit, the diffusivity is independent of concentration \n
where \n
In the presence of the spatial variation of \n
Thus, substitution of the Stokes-Einstein diffusivity into Eq. (36) gives
\nwhich is valid if the solvent viscosity \n
where one can write the chemical potential gradient as
\nSubstitution of Eq. (39) into (38) gives
\nwhich defines the thermal diffusion coefficient \n
In the conventional isothermal theory of FO, one can write a conceptual relationship between the water flux and the transmembrane osmotic pressure difference as [33]
\nwhich clearly indicates that \n
If we write intuitively the anisothermal osmotic pressure as
\nacross the membrane with \n
As both \n
so that both the solvent and solute fluxes increase with the mean temperature \n
On the basis of our investigation, temperature effects on the osmotic phenomena are not as simple as expected from the linear van’t Hoff equation, but highly correlated through the temperature-dependent material constants of solvent \n
This chapter provides a comprehensive review on the effect of temperature on the FO process. Although the motivation for studying the temperature effect comes from the fact that osmosis is a thermodynamically spontaneous process, changing the system temperature either locally or globally can offer more effective ways of engineering the FO process with lower energy consumption. However, as evidenced by the scattered data across the literature and a lack of theoretical descriptions, more robust and systematic studies are warranted for deeper understanding of the phenomena. For example, most of the temperature-dependent FO studies relate the changes in the water and the solute flux to the change in the physical properties of the bulk solution only, neglecting any changes in the membrane properties such as water permeability \n
The mathematical models representing the nuclear reactor physics are based mainly on two theoretical areas: neutron transport theory and neutron diffusion theory, where it is necessary to remark that neutron diffusion theory is really a simplification of the neutron transport theory.
\nNumerical methods are used to solve the partial differential equations representing the nuclear reactor physics, and these methods are derived from discretization techniques. For numerical solutions in any scientific area, computational tools have been developed including software and hardware. In the past, the former computer processing was the sequential execution of computer commands, meaning to say that program tasks are carried out one after one. Modern computational tools have been developed for parallel processing, executing several tasks concurrently.
\nThe computing branch dealing with the system architecture and appropriate software related to the simultaneous execution of computer instructions and applications is known as parallel computing science. Former developments in parallel computing were made in the late 1950s, following the construction of supercomputers throughout the 1960s and 1970s. Nowadays, clusters are the workhorse of scientific computing and are the dominant architecture in data centers.
\nSince the late 1950s, the performance of safety analyses was essential in the nuclear industry, in research reactors, but mainly safety analyses of nuclear power plants for commercial purposes. Scientific computing calculations were vital to these safety analyses, but with important limitations in computer/computing capabilities. At the beginning, the objective was to give a solution to partial differential equation models based on neutron diffusion or neutron transport with technology and methods available in those years. Numerical techniques were used first with finite differences and finite element approaches, and gradually up to now, with nodal finite element methods (NFEMs). Despite the numerical method employed, the computer code user faces the problem of solving extremely large algebraic systems challenging hardware/software capabilities. Generation of results for any reactor simulation in considerable short times is a desirable achievement for computer code users [1].
\nRecent developments of high-performance computer equipment and software have made the use of supercomputing in many scientific areas possible. The appropriate selection of parallel computing software, like newly developed linear algebra libraries, to be used in a specific project may result in a suitable platform to simulate nuclear reactor states with relatively prompt results.
\nThroughout the world, several research projects in the last decade have been developed with the main objective of making full tridimensional (3D) coupling simulations of nuclear reactor cores, leaving aside the obsolescence of the point kinetics theory. Most of the modern nuclear reactor simulators are based on neutron transport theory, or on neutron diffusion theory, to obtain detailed 3D results. As light water is used for cooling/moderating light water reactors (LWRs), a comprehensive analysis of the reactor core physics must include thermal-hydraulic phenomena, so that modern simulations perform reactor calculations with thermal-hydraulic feedback coupled with neutron kinetics calculations.
\nAll the discussions included in this chapter are centered in a simulator for light water reactors. The computer code AZtlan KInetics in Neutron Diffusion (AZKIND) is part of the neutronic codes selected for their implementation in the AZTLAN Platform1 project in which neutron transport and neutron diffusion codes are being developed in Mexico. A (TH) model has been implemented recently and coupled with the neutronic (NK) model, and both models are based on HPC implementations.
\nAlthough there has been growing interest in the transport-based core neutronics analysis methods for a more accurate calculation with high-performance computers, it is yet impractical to apply them in the real core design activities because their performance is not so practical on ordinary desktop or server computing machines. For this reason, most of the neutronics codes for reactor core calculations are still subject to the two-step calculation procedure, which consists of (1) homogenized group neutron parameters generation and (2) neutron diffusion core calculation.
\nIn the core calculation steps that are the main concern of this work, nodal codes based on the diffusion theory have been used to determine the neutron multiplication factor and the corresponding core neutron flux (or power) distribution. Practically, almost all nuclear reactor simulation codes employ the two-group approach involving only fast and thermal neutron energy groups for the applications to light water reactors (LWR). However, numerical calculations with the two-group structure are not appropriate in the analysis of cores loaded with mixed oxide fuels or analysis of fast breeder reactors, since the neutron spectrum is influenced more by the core environment, requiring much more energy groups than only two groups.
\nAs settled in Ref. [2], even using a high-performance computer, a direct core calculation with several tens of thousands of fuel pins is difficult to perform in its heterogeneous geometry model form, using fine groups of a prepared reactor cross-section library. The Monte Carlo method can handle such a core calculation (see also the Serpent code), but it is not easy to obtain enough accuracy for a local calculation or small reactivity because of accompanying statistical errors, besides the large calculation times. Instead of using neutron transport computer codes, the nuclear design calculation is performed in two steps: (1) lattice calculation in a two-dimensional infinite arrangement of fuel rods or assemblies for the generation of homogenized lattices jointly with their corresponding homogenized cross-sections and (2) core calculation in a three-dimensional whole core, with a neutron diffusion code using the information of the previous step.
\nAs shown in Figure 1 [2], the lattice calculation prepares few-group homogenized cross sections which maintain the energy dependence (neutron spectrum) of nuclear reactions, and these reduce the core calculation cost in terms of time and memory. The final core design parameters are not concerned with continuous energy dependence, but spatial dependence, such as power distribution, is important to avoid high local neutron fluxes or high absorbing materials causing significant neutron flux gradients, mainly when safety analyses are performed upon the final proposed core designs.
\nTypical lattice calculation process flow for light water reactors [
In the core calculations with space-dependent data (cross sections and neutron flux), the effective cross sections are processed, with a little degradation in the accuracy as possible, by using the results from the multi-group lattice calculation. Lattice code calculation and codes are not discussed here.
\nThere are two processes followed for lattice calculation. One is the homogenization to lessen the space-dependent information and the other is group-collapsing to reduce the energy-dependent information as shown in Figure 2. The fundamental idea of both methods is to preserve neutron reaction rate. The next step is to consider the conservation of reaction rate in the energy group
Homogenization and group collapsing of cross sections [
The number of few groups depends on reactor type and computation code. Two or three groups are adopted for the NK- and TH-coupled core calculation of LWRs and much more groups (18, 33, etc.) are used for the core calculation of LMFRs (Liquid Metal Fast Reactors). Currently, revised methods exist for the improvement of cross-sections generation using computer codes dedicated to lattice calculation for few-groups approach, like in Ref. [3], where three topics are involved: (1) improved treatment of neutron-multiplying scattering reactions; (2) group constant generation in reflectors and other non-fissile regions, leading to the use of discontinuity factors in neutron diffusion codes; and (3) homogenization in leakage-corrected criticality spectrum, in which several leakage corrections are used to attain criticality, accounting for the non-physical infinite-lattice approximation. Another improvement was done in Monte Carlo codes [4], implementing reliable multi-group cross-sections calculations for collapsed flux spectrum. Ref. [4] focuses on calculating scattering cross sections, including the group-to-group scattering.
\nThe following sections contain, as a matter of example, summarized explanations of the AZKIND nuclear reactor simulator in which the reactor physics is based on neutron diffusion theory.
\nFor
In addition to boundary conditions for neutron fluxes, initial conditions must be satisfied by neutron fluxes and neutron precursor functions. Parameters involved in the above equations are described in [5].
\nThe spatial discretization of Eqs. (1) and (2) is strongly connected with the discretization of a nuclear reactor core of volume Ω. Representing the neutron flux and the precursor concentrations in terms of base functions defined over Ω, it is possible to write
\nwhere
where \n
Matrix | \nType | \nDimension | \nElements | \n
---|---|---|---|
\n\n | \nMass | \n\n\n | \n|
\n\n | \nMass | \n\n\n | \n|
\n\n | \nStiffness | \n\n\n | \n|
\n\n | \nMass | \n\n\n | \n|
\n\n | \nMass | \n\n\n | \n|
\n\n | \nMass | \n\n\n | \n|
\n\n | \nMass | \n\n\n | \n
Matrix elements from the spatial discretization.
As fully explained in [6] and summarized in [1], a simple NFE element is characterized by the fact that for each node, the function unknowns to be determined are the (00) Legendre moment (average) of the unknown function over each face of the node and the (000) Legendre moment over the node volume. Figure 3(a) shows a physical domain Ω graphically represented after generating an
Discretization of reactor volume Ω and a local node Ωe. (a) Domain Ω. (b) Physical local node Ωe.
In the NFE method RTN-0, the normalized zero-order Legendre polynomials defined over the unit cell Ω
The matrix elements are quantified introducing the following nodal basis functions [7]:
\nwhere \n
An extensive discussion on nodal diffusion methods can be found in Ref. [7] for space discretization using simplification approaches for calculating the moments over a node.
\nOnce the spatial discretization is done, the
where \n
For time integration, parameters
Once the formulation to be used for time integration is established, the
For a known vector \n
Once the computer model to solve the reactor kinetics Eqs. (1) and (2) is able to provide the neutron flux profile, the next objective is to know the power distribution in the reactor configuration. It is necessary to be aware that the neutron flux is by itself the shape of the power distribution in multiplicative materials. The numerical methods presented in previous sections to solve Eq. (9) produce an algorithm capable to obtain the neutron flux profile for a reactor steady state. The calculated neutron flux has the following property over the domain Ω: \n
Theoretically, it would be best to determine the flux level which resulted in a critical reactor \n
where \n
where \n
In a more general way, for a reactor volume
Therefore, using the reference total thermal power specified by the code user, the flux normalization factor can be written as
\nwhere the factors “kappa-fission” are \n
In summary, once the NK model is used to generate the neutron flux distribution in the reactor core, expression (12) can be used to calculate the thermal power being generated along all the nodes in a thermal-hydraulic channel of area \n
The description contained in this section is based on a work published by Ceceñas in Ref. [9] about a TH model developed for boiling water reactors. The TH model was modified from a point kinetics approach with an extension of the NK model to 3D and implemented in the development of AZKIND.
\nThe treatment of neutron kinetics in [9] has been improved by coupling a 3D solution of the neutron diffusion equations with an arrangement of TH channels in parallel. Each channel independently contemplates three regions: (1) one phase, (2) subcooled boiling, and (3) bulk boiling. The objective was to implement a detailed model of a nuclear reactor core, which is somehow perturbed to simulate NK-TH coupling. These perturbations are obtained when the power generated in a group of channels changes and thus affecting the TH state of each channel.
\nThe original [9] TH model is based on a generic channel, which is adapted by transferring to it the operational data as flow area, generated power, axial power profile, and subcooling, among other parameters. Each channel is associated with a number of nuclear fuel assemblies and an axial power profile. Although the neutron model is a two-dimensional model for the radial power profile in each
For the implementation in AZKIND of the TH model of Ceceñas, the grouping of fuel assemblies was maintained for generating a reduced number of TH channels; operational data are also used. The main difference is that the NK model recursively computes the axial power profile for each channel, and this thermal power is the updated source of power for TH model. Therefore, a “new” thermal-hydraulic condition is generated, and it is used by the NEMTAB model to update the nuclear data to generate new thermal power profiles with the NK model. The process is iterative, and it stops when the convergence is met. Convergence is achieved when updated conditions do not change in both NK and TH models.
\nThe NK-TH coupling in AZKIND performs core calculations as described above to obtain a steady-state reactor core condition. For transient conditions in a time interval
The TH model comprises the solution of the mass, momentum, and energy conservation equations in the three regions contemplated by the channel: (1) one phase, (2) subcooled boiling, and (3) bulk boiling. The system receives heat through a non-uniform source whose profile is axially defined plane by plane. This axial use of the power profile allows the inclusion of a wide range of axial profiles, from relatively flat to profiles with their peak value at some axial point in each channel in the core.
\nIn the following subsections, there are several expressions for which the corresponding parameters are defined in Refs. [10, 11].
\nThe heat transfer and temperature distribution in the fuel and cladding can be calculated by a simple model where the heat diffusion equation is solved in one dimension (radial) for a fuel rod, since the conduction in axial direction is small compared to the radial one, it can be neglected. An energy balance per unit length yields
\nwhere \n
The conservation equations of mass, energy, and momentum are applied in this case to a flow of water along a vertical channel, where the dynamics of the fluid heated by the wall of the fuel is modeled. Conservation equations can be expressed as [10]
\nIn this work, the conservation equations are solved by the Integral Moment method [11], according to which it is assumed that the refrigerant is incompressible but thermally expandable, and the density is a function of enthalpy at a constant pressure
\nNeglecting terms related to pressure changes and wall friction forces, the energy equation is simplified as
\nwhere the axial flow variation can be obtained by
\nThis equation provides the flow variations with respect to an average value imposed as a boundary value or provided by the dynamics of the coolant recirculation system. Three regions are defined by which the coolant circulates as it ascends into the channel: a one-phase region, a subcooled boiling region, and a bulk boiling region. The first region begins at the bottom of the channel, where the coolant enters with known enthalpy and ends at the point of separation of the bubbles
where, in this case, the parameters ks and
ks = 0.71 + 1.2865 × 10−3
The total pressure drop in the channel is made up of the contributions of each region. Every term in each region includes the contribution by acceleration, gravity, and friction. For the channel arrangement, the steady state is obtained by iterating the coolant flow rate of each channel to obtain the same pressure drop for all of them. This iteration consists of a correction to the flow defined by the deviation of the pressure drop of the channel with respect to the average of all the channels:
\nwhere \n
It is observed that even though the pressures are equaled, the value of the pressure drop in the core is not imposed as a boundary condition. Convergence is achieved when the following relationship is met: \n
Although reference [12] has important issues to be considered in the development of an NK-TH-coupled model, those issues are not repeated here, but taken into account. The most direct way of coupling NK module and TH module, as implemented in AZKIND, consists simply in that axially both NK mesh and TH mesh have the same partition, making possible to assign an NK node at position
As it can be seen in Figure 4, before initiating the NK-TH feedback process, the initial nuclear parameters and kinetics parameter (XS) are loaded from files constructed in NEMTAB format, previously generated by means of a lattice code. Then, following the reading of the nuclear reactor burn-up state and thermo-physics initial conditions, the XS parameters are obtained from the Nemtab multi-dimensional tables by means of interpolation calculations.
\nThe NK-TH feedback process in AZKIND.
The process continues as follows. The corresponding neutron flux is calculated in the NK module with the
The important variables sent to the NK module are the fuel temperature (
HPC was implemented in AZKIND with the support of the linear algebra solvers library PARALUTION [13]. This open-source library is optimized for parallel computing process using graphics processing units (GPUs). For the numerical solution of an algebraic system \n
To demonstrate the HPC implementation in AZKIND, as described in Ref. [1], very large matrices were constructed for fine spatial discretization of arrangements of nuclear fuel assemblies of an LWR. Fine discretization means that each fuel assembly was subdivided in a mesh of size 10 × 10. As an example, an arrangement of 6 × 6 fuel assemblies consists of a square with 36 fine-discretized fuel assemblies. The corresponding algebraic system for each fuel arrangement was solved with parallel processing performed by the
Assemblies array: Matrix dimension ( | \n1 × 1 126,200 1,332,400 | \n2 × 2 492,800 5,305,600 | \n4 × 4 1,947,200 21,174,400 | \n6 × 6 4,363,200 47,606,400 | \n10 × 10 12,080,000 132,160,000 | \n
---|---|---|---|---|---|
Serial | \n24 | \n124 | \n372 | \n994 | \n2471 | \n
GTX 860 M | \n2.1 | \n7.9 | \n31.3 | \nNo memory | \nNo memory | \n
Tesla K20c | \n1.3 | \n4.0 | \n16.6 | \n40.1 | \nNo memory | \n
GTX TITAN X | \n1.0 | \n2.6 | \n10.4 | \n26.7 | \n95.4 | \n
Parallel processing time (seconds) in different architectures [1].
Assemblies | \n1 × 1 | \n2 × 2 | \n4 × 4 | \n6 × 6 | \n10 × 10 | \n
---|---|---|---|---|---|
GTX 860 M | \n11 | \n16 | \n12 | \n– | \n– | \n
Tesla K20c | \n18 | \n31 | \n22 | \n24 | \n– | \n
GTX TITAN X | \n24 | \n48 | \n36 | \n37 | \n26 | \n
Speedup comparison (S) [1].
Figure 5 [1] shows the distribution of nuclear fuel assemblies in the core of a boiling water reactor. Excepting the blue-shaded zone, colors are for different types of fuel assemblies. In the plane
A map of fuel assemblies in an LWR [
As described in [1], a reactor power transient was simulated as the capability to remove neutrons was highly increased in the perturbed assembly shown in Figure 5. An increase as step function in the neutrons removal capability during 3 s is implemented in the perturbed assembly, after that the perturbation finishes and the transient lasts for two more seconds, giving a reactor power reduction. The time step used in this simulation was 0.1 s. Figure 6 shows the power behavior over time, departing from a normalized value of 1.0 and reducing the power reactor to almost 80% of its original value. This reactor power transient was simulated with the AZKIND code, running on the three different GPUs listed in Tables 2 and 3. The right side of Figure 6 shows the time spent by AZKIND in a logarithmic scale, running in a sequential mode (Serial bar) and the times spent by each GPU card.
\nSimulation of a reactor power transient—serial and parallel processing.
A simple example was prepared to show the capability of the AZKIND code running with NK-TH coupling, and the thermal-hydraulic effect on power distribution is compared to the power distribution resulted from the NK model running standalone.
\nThis example was prepared for a two energy group, that is, fast neutrons and thermal neutrons. In LWR, the nuclear fissions of the fuel atoms are mainly coming from the thermal neutrons present in the reactor core. The effect observed in Figure 7 is that the TH feedback induces an increase in the thermal neutrons population and so increasing power. As the coolant/moderator enters the reactor core through the bottom part of the reactor and the core is beginning the production cycle, the core design allows more power generation in the first third of the core active fuel. Also, as it was expected, in the map of fuel assemblies of the reactor core, the location of the fuel assembly with the highest generation of thermal power remained unchanged with the insertion of TH feedback.
\nAxial power peaking profile location.
In the last two decades, there have been significant advances in the development of nuclear reactor codes for 3D simulation with coupling NK-TH, supported with new modeling techniques and modern computing capabilities in software and hardware. Some examples of these advances are listed subsequently:
DYNSUB: Pin-based coupling of the simplified transport (SP3) version of DYN3D with the sub-channel code SUBCHANFLOW. See [16, 17]. The new coupled code system allows for a more realistic description of the core behavior under steady state and transient conditions. DYNSUB has successfully been applied to analyze the behavior of one eight of a PWR core during an REA transient by a pin-by-pin simulation consisting of a huge number of nodes. Some insights are pointed out on the convergence process with a detailed coupling solution modeling neighbor sub-channels and modeling adjacent assembly channels.
DYN3D: The code comprises various 3D neutron kinetics solvers, a thermal-hydraulics reactor core model, and a thermo-mechanical fuel rod model, see [18]. The following topics are delineated in the reference: the latest developments of models and methods, a status of verification and validation; code applications for selected safety analyses; multi-physics code couplings to thermal-hydraulic system codes, CFD, and sub-channel codes as well as to the fuel performance code TRANSURANUS.
TRACE/PARCS: See [19]. The study of the coupling capability of the TRACE and PARCS codes by analyzing the “Main Steam Line Break (MSLB) benchmark problem,” consisting of a double-ended MSLB accident assumed to occur in the Babcock and Wilcox Three Mile Island Unit 1. The model TRACE/PARCS generated data showing that these codes have the capability to predict expected phenomena typical of this transient and the related NK-TH feedback.
COBAYA3: See [20]. This reference describes a multi-physics system of codes including the 3D multi-group neutron diffusion codes, ANDES and COBAYA3-PBP, coupled with the sub-channel thermal-hydraulic codes COBRA-TF, COBRA-IIIc, and SUBCHANFLOW, for the simulation of LWR core transients. Implementation of the PARALUTION library to solve sparse systems of linear equations was done. It features several types of iterative solvers and preconditioners which can run on both multi-core CPUs and GPU devices without any modification from the interface point of view. By exploring this technology, namely the implementation of the PARALUTION library in COBAYA3, the code can decrease the solution time of the sparse linear systems by a factor of 5.15 on GPU and 2.56 on a multi-core CPU using standard hardware.
CNFR: See [21]. This reference summarizes three methods, implemented for multi-core CPU and GPU, to evaluate fuel burn-up in a pressurized light water nuclear reactor (PWR) using the solutions of a large system of coupled ordinary differential equations. The reactor physics simulation of a PWR with burn-up calculations spends long execution times, so that performance improvement using GPU can imply in a better core design and thus extended fuel life cycle. The results with parallel computing exhibit speed improvement exceeding 200 times over the sequential solver, within 1% accuracy.
The state of the art in the topic of nuclear reactor simulations shows significant advances in the development of computer codes. A wide range of applications focusing, besides on improving nuclear safety, on more efficient analyses to improve fuel cycles/depletion have been found in a recent study. A considerable “saving time” factor in obtaining nuclear reactor analyses has been observed.
\nOne important part of a nuclear reactor simulator is the benchmarking process to demonstrate reliability and repeatability in the simulation of real cases, for which data from reactor operation or comprehensive data from experiments are well documented. In this sense, extensive documentation is necessary for theoretical basis, numerical techniques and tools, and validation of both codes and simulation models.
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