Abstract

The term “Molten salt reactors” refers to a broad class of nuclear reactors that use a molten alkali-halide salt as the primary coolant fluid. This paper pertains to thermal spectrum liquid fuel molten fluoride salt reactors with graphite moderator (MSRs), where the molten salt also dissolves the actinide fuel. Xenon isotope 135,135Xe, is a fission product that is produced during nuclear fission energy production, and it acts as a neutron poison. Due to the circulating nature of the fuel salt in MSRs, there is a qualitative difference in the behavior of 135Xe in an MSR compared to a solid fueled reactor. Some of the 135Xe produced in fission may end up in the pore space of the porous graphite moderator used in an MSR. This paper examines the transfer and storage of 135Xe in MSR graphite. Prior publications are reviewed, the porosity of the MSR graphite is examined, governing equations are detailed, film layer production and destruction are discussed, the graphite/salt interface is explored, transport pathways are considered, transfer processes are exposited, the effect of charged species is examined, the solubility of noble gases in molten fluoride salts is examined, the mass diffusion coefficient in molten salts is explored, and the calculation of mass transfer coefficients is described.

1 Introduction

This paper discusses the behavior of 135Xe in liquid fueled molten salt reactors with graphite moderator, or molten salt reactors (MSRs) for short. MSRs are a type of liquid-fueled nuclear reactor, where an alkali-halide salt melt dissolves fuel actinides. The liquid molten salt fuel circulates around the primary loop. Fission heat is generated in the graphite core region and removed in the primary heat exchanger. The isotope 135Xe is a gaseous by-product of fission and is produced in all nuclear reactors. Xenon is a noble gas and chemically inert under normal circumstances.135Xe is primarily produced through the subsequent beta decays of 135I and 135Te, which are also fission products. In solid-fueled nuclear reactors, the 135Xe is immobilized by the solid fuel matrix. Conversely, in MSRs, the 135Xe is free to migrate. This migrational behavior creates unique challenges to the prediction and description of 135Xe behavior in MSRs. The authors hope this paper serves as an aid to some of the more recent research efforts in MSR graphite development, such as the 2020 paper by Lee et al. [1], which surveyed technologies to protect MSR graphite from gas permeation.

There are numerous locations to which 135Xe can migrate in an MSR. Examine Fig. 1, which shows a schematic illustration of a general MSR. A mixture of molten metal halide salts and molten actinide halide salts, known as the fuel salt, enters into the system through the reactor inlet pipe, 1. The fuel salt is distributed radially through the flow distributor, 2, before entering the flow distribution orifices, 3. Fuel salt then moves downwards through the downcomer, 4, and collects in the lower head, 5. Fuel salt then flows upwards through the fuel channels, 6, which have been cut into the graphite moderator, 7. While in the fuel channels, the actinides in the fuel salt undergo fission. Finally, the fuel salt collects in the upper head, 8, before exiting the reactor through the reactor outlet piping, 9. During this entire circulation process, gases, including 135Xe, are circulating with the fuel.

Fig. 1
Illustrative cutaway drawing of a molten salt reactor
Fig. 1
Illustrative cutaway drawing of a molten salt reactor
Close modal

The migration of 135Xe in an MSR is further complicated by engineered systems that strip the 135Xe from the fuel salt. Such a device is called a xenon stripper, and is illustrated in Fig. 2. In this figure, a side stream of the fuel salt enter the device through the inlet, 1, before being distributed to a torus, 2. Fuel salt exits the torus through the spray holes, 3, where it forms a stream that impacts on the surface of the foaming chamber, 4. The impact of the fuel salt stream on the surface creates a large surface area that allows gases, including 135Xe, entrained in the fuel salt to escape through the outlet, 5. Gas pressure in the xenon stripper is controlled through the gas inlet, 6, and gas outlet, 7, lines.

Fig. 2
Xenon stripper cutaway drawing
Fig. 2
Xenon stripper cutaway drawing
Close modal

In prior MSR designs, such as the Molten Salt Reactor Experiment (MSRE).1 The moderator was comprised of porous graphite, which has a gaseous pore space (or void space) and a solid graphite matrix. The pore space is further subdivided into the open pore space, which can be accessed by gases at the periphery of the graphite, and the closed pore space, which is inaccessible to outside gases. 135Xe can migrate from the fuel salt into the open graphite pore space. The graphite used in the MSRE was “grade CGB”2 graphite. The porous structure of the graphite is caused by its manufacturing method, which, as described ORNL-P-150, [4], and ORNL-TM-1854, [3], is through binding extruded petroleum coke with coal tar pitch. ORNL-4148, [5], shows some microscopies of the MSRE graphite, and one of these microscopies has been recreated and is shown in Fig. 3 to illustrate the distinction between the pore space and the graphite matrix. The authors believe the dark-colored regions in Fig. 3 show the gaseous pore space, and the light-colored regions the solid graphite. Please see the endnote, at the end of this paper, for details on this claim about microscopy coloring. Additionally, see the 2020 paper by Lee et al., [1], for more recent microscopies of graphite and its pore space.

Fig. 3
Recreation of a microscopy of MSRE graphite from ORNL-4148, [32]
Fig. 3
Recreation of a microscopy of MSRE graphite from ORNL-4148, [32]
Close modal

Although a reactor system may have a xenon stripper installed, during reactor operation, some of the 135Xe that is produced necessarily remains in the fuel salt. This is because (1) xenon is continuously produced during fission, and (2) the xenon stripper can never be a perfectly efficient at removing 135Xe from the fuel salt. A proportion of the 135Xe produced that remains will migrate to the graphite pore space through mass transfer processes. This paper explores in detail the relevant structure and behavior of 135Xe in MSR graphite. In particular, this paper focuses on how 135Xe behaved in the MSRE graphite, and although the focus is only on this one reactor, it suggests that the behavior of 135Xe in MSRE graphite could be used as a guide for how 135Xe would behave in a future MSR under design that has a porous graphite moderator.

This paper examines some aspects of 135Xe behavior in MSR graphite. This paper includes a review of prior work; a discussion of the porosity of MSR graphite; a description of equations that have been used in prior work to describe the behavior of MSR 135Xe; a discussion of the physics of the film layer behavior with a lumped parameter assumption; a discussion of the molten salt/graphite interface; a discussion of transport pathways; a discussion of the transfer pathways between the fuel salt and 135Xe; a discussion of the solubility of noble gases in molten fluoride salts; a discussion of the mass diffusion coefficient; and, a brief discussion of heat and mass transfer correlations.

An understanding of mass transfer is a perquisite to understanding the physics of 135Xe migration. Mass transfer, in the experience of authors, is not covered in a typical nuclear engineering education. This paper therefore compiles the following list of relevant resources. At the level of an undergraduate text or an illustrative introduction is Chapter 14 the 2015 book by Cengel and Ghajar, [6]. A more in depth treatment is given by Cussler’s 2007 book, [7]. Mass transport in molten salts in particular is discussed in the 1971 book chapter by Moynihan [8]. Finally, Vrentas and Vrentas wrote a 2013 book [9], that can be used as either a graduate or reference text. Additionally, for readers unfamiliar with MSRs, reference material has been compiled in the 2017 by book Dolan [10]. For readers lacking a background on 135Xe physics in solid-fueled reactors, more information can be found in any introductory nuclear engineering texts such as Lamarsh’s 2001 book [11].

A note should be made here, before the paper begins, about the metastable form of Xe-135, Xe-135m. This isotope has a cross section many times that of Xe-135, and therefore, despite its relatively low abundance in reactor gas content, it does make a significant contribution to overall system Xe-135 behavior. We posit that Xe-135 and Xe-135m are affected by the same mass transfer processes equally, since the species are identical electronically.

Although our previous papers (see Ref. [12]), for example, have touched on xenon behavior in graphite, or examined the reactivity effects of xenon in graphite (see [13]). This paper attempts to provide a deep dive into how xenon migrates to and within the graphite of MSRs.

2 Prior Publications of Interest to 135Xe Behavior Within Graphite

One of the earliest reports on the subject of MSR 135Xe transport in graphite was the 1962 report ORNL-TM-262, [14]. This report provides insight into the early concerns related to the deployment of unclad graphite in direct contact with the fuel salt of the MSRE. The report states there are four contingencies due to the introduction of unclad graphite: (1) the potential for solid UO2 to be deposited on graphite; (2) the potential for fuel salt to ingress into the open pore space of the graphite; (3) the variation of core reactivity balance due to ingress of 135Xe into the graphite stringers; and (4) the potential for there to be a high nominal 135Xe reactivity load due to the 135Xe content of the graphite stringers. It is interesting to note that the MSRE began construction in 1962 (see Ref. [15]), so it can therefore be deduced that the answer to the four aforementioned concerns were unknown when the MSRE began construction.

ORNL-TM-262 [14] describes calculations to determine the maximum 135Xe poison fraction, examines variation of the 135Xe poison fraction with 135Xe removal rate, and provides a simplified expression to determine the 135Xe poison fraction in the MSRE as it was envisioned at the time. Another aspect of interest in this report is the difference between the graphite parameters described in the report compared to the graphite parameters of the operating reactor. These parameters are compared in Table 1, in which the left most column indicates the name of the parameter being compared, the second column indicates the (pre-operational and preconstruction) value of the parameter as indicated in ORNL-TM-262 (see Ref. [14]), the third column indicates the value of the parameter during the operational phase of the MSRE (during operation and after construction), and the final, right most, column indicates the source from which the operational value of the parameter was taken. The authors were unable to find the operational value of the “exposed area per stringer” in the literature. Thus, the number presented in the table was derived via the following calculation: 1520 ft2 exposed surface area ([16]) ÷ 1140 fuel channels ([2]).

Table 1

Comparison of the pre-operational graphite parameters presented in ORNL-TM-262, [8], and the actual values of the parameters for the operating MSRE as presented elsewhere in the literature

ParameterORNL-TM-262OperatingSource
Total graphite volume2.18 m31.95 m3[25]
Number of graphite stringers565513[47]
Exposed area per stringers0.351 m20.123 m2Derived
ParameterORNL-TM-262OperatingSource
Total graphite volume2.18 m31.95 m3[25]
Number of graphite stringers565513[47]
Exposed area per stringers0.351 m20.123 m2Derived

In 1967, the first of a two-part report that described gas transport in MSRE graphite was released, ORNL-4148 [17]. The successor to this report was released in 1969, ORNL-4380 [18]. These provided a review of the theory of porous media gas transport and reported on several experiments, which investigated the gas flow properties of MSRE graphite. Of particular interest to understanding the evolution of gas behavior in MSRE graphite is the claim in ORNL-4148 [17], that the graphite specimen was expected to exhibit considerable nonuniformity near the surface regions of the graphite. That is, properties of the graphite, such as the porosity, degree of connectivity of the pores, and the mass diffusion coefficient of the graphite varied with respect to spatial position within the graphite. This claim of nonuniformity contrasts with the uniform models of 135Xe distribution in MSRE graphite, which were deployed in ORNL-4069 [16] and ORNL-TM-3464, [19]. The point that the graphite is nonhomogeneous is reinforced in the conclusion of the report [17], which states the impregnation treatment, which the graphite is subject to, imparts nonhomogeneity in the structure of the graphite. Thus, it can be expected that there would be an effective graphite volume, which contains the pore space accessible to the 135Xe, in contrast to the total graphite volume, which would contain the closed pore space as well.

Another 1967 report, ORNL-TM-1810 [20], documents a model that was developed at ORNL that computed the migration of noble gases (speci 140Xe and 141Xe) into MSRE graphite. The model uses a reaction diffusion equation in three dimensional Cartesian coordinates to model gas transport within the graphite. The report also claims that “In the analysis of Xe-135 poisoning in the MSRE [a reference to ORNL-4049, [16], is placed here], it was seen that the 135Xe concentration in salt at the interface was very small compared with the concentration in the bulk salt,” [20]. This statement is surprising as it is not clear where in ORNL-4069 this claims is derived from; the best the authors can conjecture is that this claim was derived from Annex-B of ORNL-4069. The report further examines the predicted daughter radionuclide concentrations and compares them to measurements of daughter radionuclide concentrations. The report concludes with the claim that “the model predicts very short lived noble gas and daughter products in graphite fairly well,” [20]. The claim can therefore be made that since the model in ORNL-TM-1810 uses the reaction diffusion equation to model short lived gaseous radionuclide transport, and since the model in ORNL-TM-1810 can be reasonably said to be validated, it therefore follows that it is likely true that the transport of radionuclides with a relatively longer half-life (such as 135Xe) can also be predicted with the reaction diffusion equation. In conclusion, the results of ORNL-TM-1810 add credibility to the use of the reaction diffusion equation in 135Xe transport modeling.

In 1968, Greenstreet published ORNL-4327 [21], a survey of the mechanical properties of artificial graphite. Information in the report relevant to gas transport processes include details on the graphite impregnation process, differences between artificial graphite density and the density of a graphite crystal, and details related to the thermally induced expansion of graphite as well as creep based variation of graphite volume.3

The report ORNL-TM-2136 [22], written in 1969, describes the penetration of gases into Molten Salt Breeder Reactor (MSBR) graphite and claims, (1) the mass diffusion coefficient and the void fraction of the graphite control the quantity of 135Xe residing within the graphite; (2) there is a numerical equivalence between the graphite porosity and the void coefficient (presumably this means that the same results can be obtained with different values of mass diffusion coefficient and porosity); and (3) the diffusion behavior of gases within MSBR graphite is typified by Knudsen flow conditions.

Another 1969 report, ORNL-4389 [18], provides details related to the impregnation and attempts thereof of MSRE graphite, a discussion of the theoretical aspects of gas transport in MSRE graphite, and also provides a discussion of 135Xe behavior in MSRE graphite as it relates to previous theoretical gas transport considerations.

The final document of interest published in 1969 was a journal article, [23], by Scott and Eatherly on aspects of MSBR design that involve graphite and 135Xe behavior. The article, after discussing aspects related to the lifetime of graphite, as well as internal stresses within the graphite, discusses the behavior of 135Xe and how it relates to the graphite of the reactor. The paper further investigates reducing 135Xe content in the reactor through side stream removal of 135I, notes that in the MSBR design considered in the paper the majority of neutron interaction with 135Xe occurs within the graphite, discusses means to control the 135Xe transport into the graphite, and finishes with a discussion of 135Xe interaction with circulating bubbles.

We conclude this section by outlining the problems, which we encountered in the literature, with xenon transfer to MSR graphite:

  • Solid UO2 may deposit on the graphite surface and affect heat transfer, mass transfer, local fuel chemistry, and system reactivity balance.

  • Salt may ingress into the open pore space of the graphite and alter the fuel-to-moderator ratio of the reactor.

  • 135Xe or other gaseous neutron poisons may ingress into the pore space of the graphite stringers.

  • The graphite stringers may become the predominate hold up of 135Xe in the reactor system, reducing the breeding capability of the reactor.

  • Graphite may be nonuniform, especially near its surface. This may present challenges for reduced order modeling efforts as it changes the distribution of 135Xe in the graphite stringers.

  • The totality of the pore space may be noncontiguous. If this is the case, then there are regions of the graphite stringers to which 135Xe cannot migrate and therefore cause spatial reactivity imbalances.

  • Since, if modeled with the reaction-diffusion equation, the rate ass transfer into the graphite is dependent on the term Dϵ, where D is the graphite mass diffusion coefficient and ϵ is the porosity of the graphite, there are certain ranges of values of D and ϵ that are indistinguishable from each other—The same result will be produced by D and ϵ as 2D and 2ϵ.

  • Graphite, irradiated by neutrons, is subject to shrinkage follow by expansion. This changes the pore space volume of the graphite with respect to operational time.

3 Porosity of Molten Salt Reactor Experiment Graphite

The ratio of a porous solid’s pore space volume to its total volume is its porosity. In prior MSR xenon modeling efforts, the porosity has affected the rate of mass transfer, the spatial distribution of the xenon and the equilibrium concentration of xenon in the graphite.

The manner by which the porosity affects both the spatial distribution of xenon and steady-state concentration can be seen by inspection of Eq. (5) in Sec. 8. The effect of the porosity on the equilibrium concentration of xenon can be seen by noting that variation in pore-space volume, which is computed by multiply the graphite volume by the porosity, affects the amount of space the xenon is able to diffuse into. Therefore, in order to accurately model MSR xenon behavior, an accurate model of graphite porosity is necessary.

The grade CGB graphite that was used in the MSRE had numerous measurements of its porosity reported in the ORNL literature. The two major 135Xe modeling efforts for the MSRE, ORNL-4069 [16], and ORNL-TM-3464, [19], both used a 10% figure to describe the porosity. Converse to this, the experimental data from which the 10% figure was derived is not as unanimous. This section describes the numerous quantities reported that describe the porosity of the MSRE graphite. This section includes both the reports describing the 135Xe modeling efforts, as well as several reports that described or referenced experimental porosity data. Finally, commentary is given on how the experimental porosity data should be used in modeling efforts First, however, examination of two experimental methods is presented, the gas expansion method and the kerosene method, which were both mentioned in the ORNL literature as techniques by which the porosity of the graphite can be measured.

The first experimental method is the gas expansion method which was mentioned in several ORNL reports such as ORNL-3789 [24]. The gas expansion method was also claimed to have been used to determine a sample’s porosity in the report ORNL-4389 [25]. Although the none of the ORNL reports examined describe the gas expansion method, the authors of ORNL-4389 did cite a 1953 Bureau of Mines report by Rall, Hamontre, and Taliferro. No literature was found that was identical to this reference in ORNL-4389; however, a Bureau of Mines report was found [26], with identical information save for a 1954 publication date and this report describes a gas expansion method. For the sake of clarity, a qualitative description of the gas expansion method is provided here.

Figure 4 shows two containers, 1, and 2. Initially, the container 1 is gas filled and the container 2 is evacuated. Container 1 and container 2 are connected through a pipe with a valve installed in it, 3. The valve is initially set to closed to prevent the flow of gas between the containers. The volumes of both containers and the pipe are known beforehand, and instrumentation is connected to the containers to measure the pressure of the system when the valve is opened. The sample whose porosity is to be measured, 4, has its volume measured through some method such as a water displacement method. The sample, 4, is then placed inside of the container 1 and the container 1 is sealed with the gas and sample inside of it. The valve, 3, is then opened and gas is allowed to flow from container 1 to container 2 until equilibrium is reached. The pressure of the system is then measured, and gas laws are used to calculate the volume of gas that would have needed to be trapped within the pore space of the sample to produce the observed pressure. A more in depth description of the gas expansion method is found in the 2012 book by Dullien, [27] or the original 1922 paper [28], by Washburn and Bunting.

Fig. 4
Illustration of the gas expansion method that was used in several measurements of MSRE graphite porosity
Fig. 4
Illustration of the gas expansion method that was used in several measurements of MSRE graphite porosity
Close modal

The second experimental method is the porosity available to kerosene. Both ORNL-TM-0728 [16] and ORNL-4069 [2] describe a porosity that is “available to kerosene.” This refers to a standard method of kerosene immersion porosimetry used in geochemistry. A detailed description is available in a 1979 article by Nuhfer [29]; to summarize, the kerosene method involves drawing the air out of the sample using vacuum and allowing kerosene to impregnate the now empty pore space, the weight differential is measured and the void fraction computed therefrom.

With these two experimental methods described, ORNL literature that lists figures representing the MSRE graphite porosity is examined. A detailed MSRE design report was published in 1965, ORNL-TM-728 [2]. This report states [2], the volume of graphite that was “interconnected and accessible from the surface” was 7%. There is also a more sophisticated discussion of MSRE graphite porosity that claims [2], the porosity accessible to kerosene was 7.9% and the inaccessible porosity was 9.8%. Apparently, the authors of the report drew the inference that the porosity measurements made through kerosene-based experiments were representative of the porosity of the graphite that was interconnected to surface accessible pores or directly surface accessible. This inference may have underestimated the porosity, which was surface accessible and interconnected since, as we will see, the porosity as measured through the gas expansion method exceeded the porosity as measured through kerosene experiments. This underestimation is somewhat intuitive, since kerosene is a liquid that has a surface tension which prevents it from accessing the smallest pores, whereas gases do not have any sort of surface tension properties.

Another 1964 report, ORNL-3708, [30], reported the porosity of the MSRE graphite had an “accessible” porosity of 4.0% and an “inaccessible” porosity of 13.7%. The kerosene accessible porosity of 7%, mentioned in ORNL-TM-0728 [2], falls between the values of the accessible porosity and the inaccessible porosity which are mentioned in ORNL-3708, [30]. Engel stated in his section in ORNL-3708, [30], that there would be a “reactivity effect associated with permeation of 7% of the graphite volume fuel salt.” Engel apparently estimated this 7.0% permeation figure by experiments that measured the porosity of the graphite with kerosene. Contrary to Engel’s claim, later in ORNL-3708, in a section on MSRE graphite that was written by Cook, [30], explicitly states “The molten fluoride salts do not wet the graphite, and it has been calculated that a differential pressure of >300 psia would be required to force the salt into these voids [of the graphite]. This is supported in the impregnation tests.” Therefore, it appears that the knowledge that the fuel salt would not wet the graphite nor permeate into the pore space was not uniformly understood by everyone working on the MSRE in 1964. Engel’s claim also suggests some of the scientists and engineers involved in the MSRE believed kerosene was an appropriate analog for fuel salt in predicting its absorptive behavior in graphite.

In 1965, ORNL-3789 [24], reports the porosity of grade CGB and CEY graphite were determined by a gas expansion method to be 9.0% and 11.1%, respectively. As was previously mentioned, this estimate of the MSRE graphite porosity through the gas expansion method exceeds measurements made through kerosene-based experiments, which were reported in both ORNL-TM-0728 [2], and ORNL-3708, [30].

Four years later, in 1969, a report that described the first major MSRE 135Xe model, ORNL-4069, [16], listed three quantities related to porosity: a porosity accessible to kerosene, a theoretical porosity, and a porosity available to xenon and krypton. These quantities were 4.0, 17.7, and 10%, respectively. These quantities are examined in detail in the following paragraphs.

The origin of the claim in ORNL-4069 that stated the MSRE graphite had 4.0% porosity that available to kerosene is unclear. One possibility is that the 4.0% figure originated from ORNL-3708 [30]. There is a 4.0% porosity figure mentioned in ORNL-3708 [30]; however, the report does not mention how the 4.0% porosity figure was measured. There is also a measurement of porosity through a kerosene method mentioned in ORNL-3708 [30]; however, the kerosene method based measurement was listed as 7% rather than the 4.0% figure mentioned in ORNL-4069 [16].

The next porosity figure listed in ORNL-4069 [16], was the theoretical porosity of 17.7%. This 17.7% figure also appears in ORNL-4069 and the claim is made that “slightly over half [of the pore space] is accessible to gas such as xenon.” The report ORNL-3789 [24] is cited as reference number two in ORNL-4069 as a reference for the 17.7% claim; however, no instances of a statement that would indicate graphite in the MSRE had a 17.7% porosity were found in ORNL-3789. Indeed, the only figure for grade CGB graphite porosity given in ORNL-3789 was 11.1% [24]. A total porosity of 17.7% was found to be listed in ORNL-3708 [30], reference three of ORNL-4069. However, the subsequent claim of ORNL-4069, that half of the pore space was accessible to xenon, was not found to be substantiated in ORNL-3789. Rather, ORNL-3789 [24], claimed the accessible porosity was 4.0%, between a fifth and a quarter of the total pore space. This 4.0% figure in ORNL-3789 [24], appears to mirror the 4.0% porosity that was available to kerosene mentioned in ORNL-4069 [16]. ORNL-3789 does not, however, explicitly say that the 4.0% accessible porosity figure was accessible to xenon. Therefore, assuming that ORNL-4069 [16], derived its 17.7% total porosity figure from ORNL-3708 [30], rather than ORNL-3789 [24], it may not necessarily follow that a second mistake was made that ascribed one half rather than one quarter or one fifth to the proportion of the total porosity accessible to xenon. Indeed, the fact that the 17.7% porosity figure was found elsewhere in the literature referenced in ORNL-4069, but the claim that half of the 17.7% was accessible to xenon was not found, and the fact that the document in which the 17.7% figure was found in does not even mention xenon or krypton accessibility in its discussion of porosity (see ORNL-3708, [30]), these two facts together suggest that there is another source (document, discussion, or otherwise) somewhere that explored the total porosity of MSRE graphite and how much of it would be accessible to xenon gas.

The last figure mentioned in ORNL-4069 [16], that is discussed in this paper is the 10% quantity, which represented the porosity available to krypton and xenon. ORNL-4069 cited ORNL-3789, [24], for this 10% figure; however, a page number was not provided for this reference. Our investigation was unable to find 10% porosity figure ORNL-3789; however as previously mentioned, an 11.1% porosity was reported in ORNL-3789 [24]. This 11.1% porosity figure was found through the gas expansion method for grade CGB graphite, and this is near the 10% figure. Therefore, it seems that the authors of ORNL-4069 believed that the porosity measurements that were achieved through the gas expansion method would approximate the porosity accessible to krypton and xenon gas, and thus approximated the 11.1% porosity measurement reported in ORNL-3789 with 10% in their calculations. The assumption that the porosity found through the gas expansion method approximates the porosity accessible to xenon seems intuitively justified since the method used to measure the quantity of interest, the gas expansion method, and the quantity of interest to be predicted, xenon and krypton gas behavior, both involve gases.

Another 1969 report, ORNL-4389 [25], provides a more sophisticated investigation and description of the porosity of the MSRE graphite. The report describes the porosity for the base stock graphite as well as an impregnated one. The 1964 semi-annual progress report ORNL-3708 [30], states the low porosity of MSRE graphite is obtained through a series of impregnations; therefore, the in situ graphite used in the MSRE would have been impregnated. Numerous measurements of porosity are listed, including a “connected porosity” as well as porosity measurements associated with peaks of pore size distribution. This report substantiates the use of a 10% porosity figure in ORNL-4069 [16], since the “open porosity” of a graphite specimen described in ORNL-4389 [25], ranges from 10.1% to 11.1%. Furthermore, the numerous types of porosity measurements listed in ORNL-4389 indicate that although the MSRE graphite porosity can be characterized by a single number, there are actually many nuanced measurements that can lead to this single figure and the porosity can be described at a substantially more sophisticated level of detail than a single number representation. This insight is a relevant detail to modeling efforts since it is apparent that although modeling efforts typically seek a single number to describe the graphite porosity, that single number can represent numerous different measurements of porosity, each of which would, in a complete description of the system, need its own physics considerations.

In 1970, a report on fluid dynamic studies performed on the MSRE, ORNL-TM-3229 [31], claims the kerosene accessible porosity of MSRE graphite was 6.5%. This 6.5% figure is similar to the early claim in ORNL-TM-0728 that the volume of graphite accessible to kerosene was 7%.

Finally, in 1971, a report was published that detailed the development of a model that attempted to describe the MSRE 135Xe behavior, ORNL-TM-3464 [19]. This report states [19], that the gas volumes used in the model described in the report were “based on 10% accessible void volume in graphite.” First, note the change in terminology: the term porosity and a percent void volume effectively describe the same thing. Second, the 10% porosity figure is identical to the porosity value used in the development of the ORNL-4069 135Xe model, [16].

Table 2 summarizes the data presented in this section. This table has three columns. The first column describes where the data are from, the second column lists the porosity of the graphite as measured, and the third column briefly describes the measurement as the data is presented in the report. Figure 5 plots these data as a bar chart. The data show that there is substantially more inaccessible void space. Furthermore, although there is only one measurement that is explicitly from the gas expansion method, it appears the gas expansion method measures a larger void space than the void space as measured by the kerosene method.

Fig. 5
Bar chart summarizing the MSRE porosity data from ORNL reports detailed in Sec. 3
Fig. 5
Bar chart summarizing the MSRE porosity data from ORNL reports detailed in Sec. 3
Close modal
Table 2

Summary of MSRE porosity data from ORNL reports detailed in Sec. 3

ReportPorosity (%)Description
ORNL-TM-7287Interconnected and accessible
7.9Kerosene accessible
9.8Kerosene inaccessible
ORNL-37084Accessible
13.7Inaccessible
ORNL-37899Gas expansion method
ORNL-40694Kerosene accessible
17.7Theoretical
10Available to krypton and xenon
ORNL-438910.1Open porosity lower measurement
11.1Open porosity upper measurement
ORNL-TM-32296.5Kerosene accessible
ORNL-TM-346410Accessible void space
ReportPorosity (%)Description
ORNL-TM-7287Interconnected and accessible
7.9Kerosene accessible
9.8Kerosene inaccessible
ORNL-37084Accessible
13.7Inaccessible
ORNL-37899Gas expansion method
ORNL-40694Kerosene accessible
17.7Theoretical
10Available to krypton and xenon
ORNL-438910.1Open porosity lower measurement
11.1Open porosity upper measurement
ORNL-TM-32296.5Kerosene accessible
ORNL-TM-346410Accessible void space

The documents reviewed in this section show that porosity used in MSR xenon modeling is not necessarily as simple as what is suggested by using a single number to represent it. There are numerous measurements of porosity, and it is not necessarily true that the porosity as measured is representative of the porosity that is accessible to xenon. Indeed, it can be said that the porosity as defined by the ratio of the volume of pore space to the total volume of a sample almost certainly is not the same as the porosity that is accessible to xenon. One unexplored avenue is how the porosity of MSR graphite evolves with the reactor life. Presumably, the nuclear evolution of gases in the graphite, either by nuclear reactions or decay that transmute the gases into solids, the migration through diffusion of material from the fuel salt into graphite, and the structural changes in graphite due to mechanical, radiation, and thermal effects would change the pore structure of the graphite in an operating MSR, especially during a multidecade long operational life. The one postirradiation study of the MSRE graphite available, ORNL-TM-4174, [33], did not include a postirradiation examination of the porosity.

Regardless, in order to produce accurate models of MSR xenon behavior, accurate models of mass transfer of xenon into graphite are needed, and these models need accurate porosity data. The measurement of stringer’s porosity directly affects the predicted volume into which the xenon can migrate. This, in turn, affects the concentration of xenon in the stringer—a lower volume means the same amount of xenon will produce a larger concentration. Since the concentration differential is the driving for mass transfer, this affects rate equations used to determine xenon transient calculations. Furthermore, since porosity affects concentration, this also produces a change in the steady-state xenon poisoning of a system.

4 Governing Equations for 135Xe Transport Processes Within and to the Graphite Stringers

No evidence was found that indicated that advective transport mechanisms were present in the graphite stringers. It therefore seems reasonable to conclude that mass transport in the graphite stringers is governed by diffusive mechanisms, which can be described, in one dimension, by Fick’s second law for porous media
(1)

where C=C(x,t) is the concentration of 135Xe, D is the effective diffusion coefficient for the porous media, ϵ is the porosity of the diffusing media, and x and t are the spatial and time coordinates, respectively. Diffusion equations of this form are ubiquitous in the literature. For example, Dye and Dallvale state in their 1958 paper [34], that the diffusion of gases in porous media can be described by an equation of the form of Eq. (1).

A question may be asked as to why the right-hand side of Eq. (1) was divided by the porosity factor, ϵ. A hint as to why the diffusion coefficient is divided by the porosity may be given in Liu’s 2018 book, [35] which states,

“This [tortuous diffusion] effect can be considered using the effective porosity defined in terms of areas, which can be approximated using the porosity of the porous material defined in terms of volumes.”

There is a large body of literature related to the theory of transport in porous media. A review of the development of the porous media diffusion equation has been presented by LaBolle, Quastel, and Fogg in Sec. 2 of their 1998 paper, [36]. An elementary explanation is achieved by multiplying both sides of Eq. (1) with ϵ and noting that the change of concentration, Ct, only occurs in the pore space.

This section has reviewed the literature related to the porosity of the graphite used in the MSRE. Methods used to measure the porosity of the graphite of the MSRE were reviewed, as well as some claims about the porosity of the graphite and the porosity figures used in various MSRE modeling efforts. Since the MSRE is the only MSR that has experimental data available, Xe-135 modeling efforts will likely require this data until new data from new reactors is available.

5 Negligible Film Layer Production and Destruction

Consider a mass Flux of 135Xe, J, transferring at a mass velocity (mass transfer coefficient) km, between a liquid phase fuel salt with dissolved 135Xe concentration, CL, and a gas space with a concentration of 135Xe, CG. This transfer occurs at temperature T, and the 135Xe in fuel salt has a Henry’s constant of H. The rate of mass transfer of 135Xe from the fuel salt to the graphite pore space can be modeled using an expression of the form
(2)

When this equation is used, an assumption is made:4 That assumption is, there are negligible rates of 135Xe production and destruction within the boundary.

The reason why this assumption is implicit in Eq. (2) is nuanced. To see why, examine Fig. 6. This figure illustrates the so-called two film model of mass transfer, which is described in detail on Reibile’s 1998 book [37], or the 2006 book by Baehr and Stephan [38]. Figure 6 shows, on the left side, a detailed view of a liquid/gas interface, and on the right-hand side shows a plot of gas concentration versus a positional coordinate that can be said to represent the distance of penetration. This figure is bounded on the top by a bulk gas phase, 1, and on the bottom by a bulk liquid, 7. There is a gas side film layer, 3, and a liquid side film layer, 5. The gas side film layer starts at the gas side film layer extent, 2, and the liquid side film layer starts at the liquid side gas layer extent, 6. The gas and liquid phases are separated from each other through the gas/liquid interface, 4. The bulk gas phase concentration (which can be related to the partial pressure through a gas law, such as the ideal gas law) is assumed to be constant throughout the bulk gas phase and is denoted CBG. Likewise, the bulk liquid phase gas concentration (dissolved gas) is assumed to be constant and is denoted CBL. The concentration of gas in the gas side film layer, 3, is assumed to vary linearly from the gas side film layer extent, 2, to the gas/liquid interface, 4. Similarly, the concentration of gas in the liquid side film layer, 5, is assumed to vary linearly from the liquid side film layer extent, 6, and the gas/liquid interface, 4.

Fig. 6
Diagram of the two film model that is used in mass transfer calculations
Fig. 6
Diagram of the two film model that is used in mass transfer calculations
Close modal

When Eq. (1) is derived, as shown in our 2019 paper, [12], the gas concentration at the liquid side of the liquid/gas interface, the bottom of 4, is related to the gas concentration in the bulk gas phase, 1, through (1), assuming the diffusion velocity in the gas film layer, 3, is much higher than the diffusion velocity in the liquid film layer, 5, and, (2), relating the gas concentration on the liquid side of the liquid/gas interface, the bottom of 4, to the gas concentration on the gas side of the liquid/gas interface, the top side of 4. Nowhere in the two film model nor the two assumptions described in the last sentence is the production or destruction of the dissolved gas species considered. Therefore, Eq. (2) cannot have considered the production or destruction of gas species, and it is exactly this sort of production or destruction, which is caused by nuclear processes such as fission or transmutation, that one would expect to see in an MSR.

6 The Molten Salt/Graphite Interface

The amount of surface area available for mass transfer of 135Xe to the graphite is not necessarily trivial. Indeed, a complete theory of the molten salt/graphite interface should be able to predict the available area for mass transfer from the molten salt to the graphite. Since the graphite is porous, some of the salt facing surface area is pore space and the remainder is the solid graphite matrix.

One explanation is that the mass transfer area is equal to the surface area. This appears to be the assumption made in ORNL-4069 [16], since the graphite surface area used for their 135Xe model. Specifically, according to data from ORNL-0728, the graphite fuel channels had an approximate surface area of [2] (2×1.2+2×0.4)×68×1140=1723ft2 compared to the 1520 ft2 used in the ORNL-4069 model.

There is some evidence that supports the one to one theory for mass transfer surface area. ORNL-4069 [16], modeled fuel salt/graphite interface as a system in which the bulk fuel salt was separated from the pore space by a boundary layer and a pure diffusion region.

A diagram of the model used in ORNL-4069’s analysis is shown in Fig. 7, and we will now expand upon it—note, the nomenclature has been changed from that used in ORNL-4069 for clarity. The figure depicts a fuel salt/graphite interface with the bulk fuel salt on the left and the graphite on the right. The names of the regions in the ORNL-4069 fuel salt/graphite interface were only given as sub- and superscripts, so we take some liberty here with naming the regions. Furthermore, for the purposes of description, we deviate from the original nomenclature in the ORNL-4069 fuel salt/graphite interface model. With that caveat made explicit, the bulk fuel salt, 1, has a mean bulk dissolved gas concentration, CBlk. Moving left to right, the boundary layer, 2, is characterized by a lack of eddy current motion and has a different dissolved gas concentration, CBL. The pure diffusion region, 3, is between the boundary layer, and the graphite pore space, 4. The interior of the pore space, 4, is dry and gaseous, and gas concentration of the dissolved gas species, CPS. The pore space sits within a solid graphite matrix, 5. The interface between the pure diffusion region, 3 and the pore space/graphite interface, 4/5, is called the film interface, 7, and has a dissolved gas concentration denoted, CBLI. On the other side of the pure diffusion region, 3, is the film surface, 6, and has a dissolved gas concentration denoted CBLF. Ultimately, by modeling the film interface as a hemispherical bubble, the ORNL-4069 analysis showed that the gas concentration at the surface of the boundary layer was approximately equal to the gas concentration at the interface of the boundary layer, CBLFCBLI.

Fig. 7
Simplified diagram of the ORNL-4069 salt/graphite interface model
Fig. 7
Simplified diagram of the ORNL-4069 salt/graphite interface model
Close modal

The difference in dissolved krypton concentration between the pure diffusion region boundaries was found to be sufficiently negligible so as to claim the dissolved krypton concentration at the salt film is approximately equal to the dissolved krypton concentration at the salt/pore gas interface.

Another uninvestigated aspect of the graphite/molten salt interface is the presence of surface clinging microbubbles. This aspect was first postulated in the 1965 report ORNL-TM-1060 [39], (Sec. 6.6.2). The total surface energy of a single large bubble is less than that of two smaller bubbles. Therefore, if there are surface clinging microbubbles, they will be driven to coalescence with circulating bubbles through surface tension energetic.

One aspect that appears uninvestigated is the effect fission and radiation interaction has on mass transfer processes. As described in Sec. 5, the mass transfer occurs over a film layer, or two film layers in the two film model. It is foreseeable that a certain fraction of nuclear reactions occur within this film layer of the fuel salt. If fissions or decays occur within the film layer, then they impart energy into the film layer and affect the energetics of any mass transfer processes occurring. It is therefore an open research question, how does the occurrence of fission or interaction of radiation within a film layer affect mass transfer processes?

During the start of this research, the area available for the transfer of 135Xe from the fuel salt to the graphite pore space was considered as equal to the surface area of the fuel channels. There are, however, several considerations that cast uncertainty on this assumption. Since the graphite is porous, the definition of the graphite surface is ambiguous in the same way that a length of a coastline is ambiguous. With increasing resolution, the porosity increases the effective surface. There is additionally the potential for clogging of the graphite pores due to fission or corrosion product deposition. If clogging occurs, it would reduce the area available for mass transfer of 135Xe into the graphite stringers. Therefore, the area available for mass transfer of 135Xe into the graphite stringers is not a straight forward matter, and it may be preferable to consider an effective area that is a function of the graphite porosity.

7 Transport Pathways

There are several potential pathways for 135Xe to transfer into the graphite stringers. A model of 135Xe mass transfer pathways is illustrated in Fig. 8. In this model, 135Xe can either be in the circulating gas phase in the reactor, 1, dissolved in the fuel salt, 2, contained with the microbubbles (described in Sec. 6 attached to the surface of the graphite stringers, 3, or within the pore space itself, 4. Potentially, there may be mass transfer directly between contacting stringers (not illustrated). The following discussion in this section is made with reference to Fig. 8; however, explicit references to the figure are omitted for the sake of concision.

Fig. 8
Diagram of potential pathways for 135Xe mass transfer in a MSR
Fig. 8
Diagram of potential pathways for 135Xe mass transfer in a MSR
Close modal

The mass transfer process between the bubbles,51, and the 135Xe dissolved in the fuel salt, 2, to the best of our knowledge is not disputed. The transfer of mass between the fuel salt and bubbles is explored in detail in our 2020 paper [40].

The pathway between 135Xe in attached surface microbubbles, 3, and the circulating bubbles, 1, has not been previously explored in the literature. The surface energy of a bubble is given by
(3)

where γ is the surface tension between two immiscible substances, ES is the surface energy, and AS is the surface area. The fact that a single large bubble of a given volume has less surface energy than two smaller bubbles that encompass the same volume indicates that there would be a tendency for the surface attached bubbles, 3, to coalesce with the circulating bubble, 1, and since the resultant bubble would have a greater buoyant force the direction of the coalescence would be in the direction of the circulating bubbles, 1. More information on surface energy is provided in the 2013 book by Butt, Graf, and Kappl, [42] (Sec. 3.4).

There are two arguments for and against the transfer of 135Xe from the circulating bubbles, 1, to the graphite pore space, 4. The arguments against are:

  • The graphite is cooled by the fuel salt, and therefore hotter than the fuel.6 As the fuel salt is colder than the graphite, any bubbles circulating in the fuel salt would also be colder than the graphite. It seems unlikely that matter could transfer from a colder medium to a hotter one.

  • In our 2020 paper that details modeling of 135Xe behavior in the MSRE, [44], the model is able to fit both the startup and shut down 135Xe transients in the MSRE without the transfer of 135Xe from the bubbles to the graphite.

The arguments for are:

  • The temperature may be just one driving force of many involved in the determination of the direction of mass transfer. Indeed, heat, pressure, chemical potential, or any other potential involved in the Gibbs energy would likely be involved in determining the direction of mass transfer.

  • ORNL-TM-3464 [19], did allow for direct transfer of 135Xe from the bubbles to the graphite in some of their parametric fitting studies, [19]. The authors did report, [19], that the inclusion of the bubble to graphite mass transport mechanism shown explanatory power.

Ultimately, the question if 135Xe transfer from the circulating bubbles to the graphite is a question that can be answered definitely positive if the phenomenon can be observed experimentally.

Finally, the transfer of 135Xe from fuel salt, 2, to the graphite pore space, 4, is covered in detail in Sec. 8.

As a caveat to this discussion, note, the pathways presented in Fig. 8 do not include any transfer of 135Te or 135I, which are radioactive precursors to 135Xe. The in-solution assumption, which is described in detail in Sec. 2.1 of our 2019 paper [12], states that tellurium and iodine remain suspended within the fuel salt melt on the time scales relevant to 135Xe production. If the in-solution assumption is false, then the pathways presented in Fig. 8 would not be limited only to species of 135Xe but also include 135Te and 135I. As also mentioned in the introduction of this paper 135mXe is expected to follow the same mass transfer pathways that 135Xe follows.

8 Transfer of 135Xe From the Fuel Salt to the Graphite Pore Space

In the modeling of 135Xe transfer from the fuel salt to the graphite pore space, two aspects have remained constant in all known research efforts:

  1. a reaction diffusion equation or an approximation thereof has been used to model mass transfer rates;

  2. the concentration of 135Xe dissolved in the fuel salt has been related to the partial pressure of 135Xe in the graphite pore space through Henry’s law, discussed in Sec. 10.

Miller’s 1961 report, ORNL-CF-61-5-62 [45], provides a derivation of the steady-state reaction diffusion7 equation
(4)

where D is the mass diffusion coefficient of the graphite, CXe is the 135Xe concentration as a function of positional coordinate, x, σXe is the 135Xe xenon neutron absorption cross section, ϕ is the neutron flux, and λXe is the xenon decay constant. Further, Miller [45], provides details regarding the use of Henry’s law for the coupling conditions for the graphite/salt interface.

The 1969 report, ORNL-4069 [16], uses a reaction diffusion equation in cylindrical coordinates
(5)

where the same nomenclature is used as in Eq. (4), with the addition of ϵ is the porosity and r, the radial coordinate. Henry’s law was used in the boundary conditions developed for Eq. (5), see [16].

The explicit inclusion of a porosity term, ϵ, in Eq. (5), indicates that between 1961 and 1969, the importance of porosity in xenon behavior within graphite was recognized. During this eight year period, in 1967, a report was published, ORNL-4148 [5], that provides information on the theory of counterdiffusion as well as descriptions of several experiments on the diffusive properties of MSRE graphite. ORNL-4148 [5] uses the concept of porosity in its text. Therefore, to the best of our knowledge, this indicates at least six years elapsed between the first publication of an equation, which would undergo a major change and the publication of the first report that mentioned the core science upon which the major change to the equation would be based. If the goal is to deploy MSRs this decade, the authors believe that it would be wise to keep in mind that, in prior MSR development efforts, a period of time greater than half a decade elapsed between the introduction of an equation believed to be relevant to the physics of MSRs and a major change to that equation.

In 1971, ORNL-TM-3464 [19], does mention the distribution of 135Xe in the MSRE graphite can be modeled as the solution to a cylindrical reaction diffusion equation. That said, the actual equation that was implemented in the computational model presented in ORNL-TM-3464 [19], was a lumped parameter model. The graphite was discretized into four adjacent regions8 and an equation of the form
(6)

was used to model 135Xe migration into graphite.9 Equation (6) can be seen as a form of a reaction diffusion equation in that there are terms which include reaction and diffusion, but the assumptions and manipulations necessary to transform a partial differential equation such as Eq. (5) or Eq. (4) into an equation of the form of Eq. (6) were not detailed. As a final note, these equations do not take into account any cracking nor dimensional changes in the graphite due to irradiation. A more advanced 135Xe model could examine the effects of pore evolution and cracking on 135Xe mass transfer behavior.

9 Charged Interface and Ionized Species

The Helmholtz layer is another region of the fuel salt/graphite interface that may be affected by radiation or fission. We are aware of no theories that describe the Helmholtz layer of a molten salt/graphite interface, h; however, there is some literature related to a molten salt/metal interface.

According to Coway [46], The Question of “Structure” of the Double-Layer in Molten Salts, the double layer in a molten salt/metal interface is qualitatively different from the double layer in a solution/metal interface and, due to the lack of a solvent, there are inherit difficulties in visualizing a double layer for a molten salt/metal interface. Nevertheless, there is a charged electric interface between metals and molten salts, and it stands to reason that this interface would be affected by irradiation. Although Coway [46] does review literature pertaining to the molten salt/metal interface, the authors are unaware of any information available, theoretical or experimental, that describes how radiation interacts with the Helmholtz layer.

If 135Te, 135I, or 135Xe is ionized by irradiation and the molten salt/metal interface has a distribution of electrical charge, then there is a potential for the behavior of 135Te, 135I, or 135Xe to be dependent on the properties of the molten salt/metal interface. If it is true that irradiation affects the behavior of the Helmholtz layer of a molten salt/metal interface, then it is also reasonable to consider there would be effects in a molten salt/graphite Helmholtz layer induced by irradiation.

10 The Solubility of Noble Gases in Molten Fluorides

If mass transfer of 135Xe occurs from the liquid fuel salt, the fractionation of 135Xe between the fuel salt and the graphite governed by Henry’s constant.

The earliest study of noble gases in molten fluoride salts was the 1958 study by Grimes, Smith, and Watson, [47], that investigated He, Ar, Ne, and Xe in melts of NaF–ZrF4 and NaF–ZrF4–UF4. Critical commentary on the study by Grimes, Smith, and Watson [47] is provided by Gerrard on Sec. 16.9 of his 1976 book [48].

The following year, in 1959, Blander, Grimes, Smith, and Watson wrote a paper [32] that presented their experimental investigation of the noble gases, He, Ne, Ar, and Xe in a LiF–NaF–KF eutectic mixture.

In 1962, Watson, Evans III, Grimes, and Smith published a paper [49] that presented details on experiments that measured the solubility and Henry’s constant of He, Ne, Ar, and Xe in LiF–BeF2.

Both the 1959 paper by Blander et al. [32] and the 1962 paper by Watson et al. [49] compare their experimental measurements of Henry’s constant to results predicted by the equation
(7)

The nomenclature for Eq. (7) is as follows: K is Henry’s constant as expressed by a dimensionless ratio of gas concentration in solution to gas concentration in the gas phase; r is the radius of the dissolving atom, in Angstroms, γ is the surface tension, in (ergs cm−3); and T is the absolute temperature, in (K). Details on the derivation of Eq. (7) can be found in Braunstien’s 1971 book [Braunstien1971] or the 1959 paper by Blander, Grimes, Smith, and Watson [32]. A critical discussion of the hole theory, upon which Eq. (7) is based, is given by Gerrard on Sec. 17.10 of his 1980 book, [50].

11 Mass Diffusion Coefficient in Molten Salts

One of the most important parameters to estimate correctly is the mass diffusion coefficient. It is used in the to calculate dimensionless numbers that are used in mass transfer analogies. The 1967 report, ORNL-4069, [16], briefly mentions the Einstein Stokes equation, and the Wilke Change equation are two methods by which the mass transfer coefficient may be estimated. This section describes these methods and provides reference to where more information may be found.

The first method, the Einstein Stokes equation, that states
(8)
where kb is Boltzmann’s Constant (1.38×1023mkg2s1K2), T is the temperature of the fuel salt (K), μ is the viscosity of the fuel salt, and rXe is the radius of 135Xe. The 1998 book by Brockis and Reddy [51] provides detailed derivation of the Einstein Stokes equation in Sec. 4.4.8, and discusses its validity in ionic liquids (molten salts) in Sec. 5.6.4.
The other method, the Wilke Chang equation, described in Brodkey and Hershey’s 1988 book [52] states
(9)

in which ϕ is a dimensionless, substance deponent, association parameter, T is the absolute temperature in (K), M is the molecular weight of the solvent in gmol1, μ is the viscosity of the solvent in (cP), and V is the molar volume of the solute at its normal boiling point in cm3mol1.

A challenge to application of the Wilke Chang equation is the magnitude of the association parameter, ϕ. The authors are unaware of any measurements or predictions of the association parameter, ϕ, for molten salts; however this may be immaterial. In 1963, Sitaraman, Ibrahim, and Kuloor published a paper [53] that presented a correlation, based on the Wilk Chang equation, that provides the mass diffusion coefficient of a substance as
(10)

where Ms is the molecular weight of the solvent,10Ls is the latent heat of vaporization of the solvent at the normal boiling point in (cal. g1), T is the temperature,11μ is the viscosity of the solvent (the molten salt without any dissolved species) in (cP), Vm is the molecular volume in (cm3g1mol). An uncertain aspect is that the term LS appears in both the numerator and denominator of Eq. (10). This may be in error since the nomenclature of [53] lists both LS, the latent heat of vaporization of the solvent, and L, the latent heat of vaporization of the solute, in its nomenclature. Although the generation of Eq. (10) did not involve any data from molten salts, if it is applicable to molten salts, then it provides a means by which the mass diffusion coefficient of noble gases in molten salts may be calculated using only macroscopic variables that can be measured by a physical experiment.

Since the ORNL-4069 [16] was published in 1969, numerous other predictive theories of diffusion have been developed. Chapter 6 of the 1984 book by Tyrrell and Harris presents a review and discussion of such theories.

12 Heat Transfer Correlations to Mass Transfer

This first step in deriving a mass transfer coefficient for use in computing the rate of 135Xe mass transfer between the fuel salt and the graphite is to select an appropriate heat transfer correlation. A heat transfer correlation at its highest level is a mathematical function that describes the heat transfer coefficient of a particular geometry as a function of various system parameters that are constrained to a certain range. Commonly, this takes the form of a correlation between several dimensionless numbers such as the Nusselt Number, Nu, the Reynolds number, Re, and the Peclet number, Pe. These correlations are available from numerous sources. Two example sources of heat transfer correlation sources are (1) the 2003 handbook compiled and edited by Bejan and Kraus, [54], or (2) the 1998 handbook edited and compiled by Rohsenow and Cho, [55]. Note, these dimensionless numbers require use of the diffusion coefficient—this should be the diffusion coefficient of the fuel salt, and not the graphite, as the two numbers are not the same.

It is important that the geometry, flow, and material conditions of the situation to which the heat transfer correlation is applied are as close as possible to the geometry, flow, and material conditions of the situation from which the heat transfer correlation is derived. This close correspondence between applied, and experimental situations is not always possible in practice. For example, as shown by the diagram in ORNL-4069, [16], the relevant part of which is recreated in Fig. 9, the fuel channels of the MSRE are circtangular in shape whereas the heat transfer correlation that is used to derive mass transfer coefficients used in 135Xe analysis of the MSRE, the Dittus Boetler correlation (which was introduced by McAdams, see the 1998 paper by Winterton, [56]), is derived from experiments using a pipe with a circular cross section [56].

Fig. 9
The arrangement of four graphite stringers. The reactor core was arranged using this as the unit cell.
Fig. 9
The arrangement of four graphite stringers. The reactor core was arranged using this as the unit cell.
Close modal
There are several heat transfer correlations that require information about both the bulk fluid temperature, Tbulk, and the temperature of the wall, Twall, and these two temperatures are not necessarily the same in an operating MSR. Consider the Sieder Tate correlation for laminar flow
(11)

in which Nu is the Nusselt number, Re is the Reynolds number, Pr is the Prandtl number, and μ(Tbulk) and μ(Twall) are the viscosities of the fuel salt at the bulk and wall temperature, respectively.

The Sieder Tate correlation is valid for 13 < Re <2030. Although no information was found that described the temperature of the fuel salt in the graphite boundary layer, information has been found that describes the temperature of the graphite and fuel salt separately. Figure 10 recreates a plot from ORNL-TM-0378 that shows the midplane temperatures of the fuel salt and graphite of the MSRE during steady-state operations as a function of radial position. At all radial positions, the graphite temperature is higher than the fuel salt temperature. The graphite is hotter than the fuel salt (and is therefore cooled by the fuel salt) because the graphite is subject to heating due to reactions with neutrons, gamma, and other radiation sources generated by the fission process.

Fig. 10
The temperature versus position for the fuel salt and the graphite in the MSRE
Fig. 10
The temperature versus position for the fuel salt and the graphite in the MSRE
Close modal

The fuel salt, conversely, is able to constantly have its heat removed by the primary to secondary side heat exchanger. A potential method to estimate the fuel salt boundary layer temperature is to assume the boundary layer temperature is the same as the graphite temperature. Alternatively, since the factor (μ(Tbulk)μ(Twall))0.14 is raised to the exponent 0.14, its impact on the final answer is smaller than any of the other factors and therefore one may assume the ratio of viscosities to be sufficiently close to unity so as to replace the entire factor by 1.

When the value of the flow conditions or fluid properties exceed the range of flow conditions or fluid properties that a particular heat transfer correlation can predict, another heat transfer correlation can be used to predict the heat transfer coefficient at the flow condition or fluid property point of interest. For example, although the Sieder Tate correlation has an upper bound of Re=2030, Nusselt numbers can be predicted for Reynolds numbers between 2300 <Re<105 with the Whitaker correlation. Using the same notation that was used for the Sieder Tate correlation, the Whitaker correlation is written
(12)

More information can be found about the Sieder Tate and Whitaker correlations in the 2007 book by Tosun [57].

This section has examined heat transfer correlations and their peculiarities in application to MSRs.

13 Conclusions

The purpose of this review paper was to expose that the modeling of 135Xe mass transfer between the fuel salt and the graphite of an MSR with a porous moderator is not as trivial as one might have initially thought. Indeed, our initial analyses of 135Xe behavior in an MSR did not include any of the nuances mentioned in this paper and, as far as the authors are aware, no MSR 135Xe analysis to date has included considerations on many of the aspects detailed herein. Three practical considerations to emphasize are as follows:

  • The uncertainties in effective surface area for mass transfer area are a significant source of potential error in MSR 135Xe analysis, and uncertainties in effective mass transfer area likely present an equal if not greater source of error than uncertainties in the mass transfer coefficient. This is because measurements of mass transfer coefficients is not a new art, whereas there are no measurements of effective mass transfer area to graphite in an operating MSR. An error in mass transfer area would become apparent in a transient analysis, but not in a steady-state analysis.

  • The uncertainties in porosity will have an impact on both steady-state and transient calculations. The evolution of the pore space with reactor operation due to irradiation is one aspect in particular into which there has been minimal investigation. Furthermore, the large surface area for gases to adsorb onto the internal surface of the graphite stringer pore space presents a potentially very large source or sink of 135Xe in an operating molten salt reactor.

  • The potential for surface attached micro scale bubbles to cling to the graphite stringers, or other reactor surfaces, represents an uninvestigated phenomenological aspect of MSR 135Xe behavior, and therefore, it would be reasonable to doubt that the current models of MSR 135Xe behavior are phenomenologically complete.

14 Endnote on Microscopy Coloring

The claim that light-colored regions are solid, whereas dark-colored regions are gaseous is contrary to the claim in ORNL-4148 [5], which states light colored regions are pores.

The area covered by light-colored regions is substantially greater than the covered by the dark-colored regions. ORNL-4069 [16], claims the porosity of the MSRE graphite was 17.7%. Since there is more solid graphite than pore space, and the micrograph in Fig. 3 must have more solid than gas.

The micrograph of Fig. 3 has more light-colored region than dark colored region. Since there is more solid matrix than pore space, the light colored region must correspond to the solid graphite matrix and the dark colored region must correspond to the gaseous pore space. This conclusion is somewhat intuitive, since a solid region would reflect microscopy light and become brightly lit.

Acknowledgment

We acknowledge the funding from the Mitacs accelerate grant that made this research possible, as well as the funding from Dr. George Bereznai that saw this project through to completion.

Funding Data

  • University of Texas Cockrell School of Engineering (No. UTA20-000500).

Data Availability Statement

The authors attest that all data for this study are included in the paper.

Nomenclature

Acronyms
ARE =

aircraft reactor experiment

AS =

surface area

CGB =

nothing was found that would elucidate what the acronym CGB refers to

CBL =

concentration in boundary layer

CBlk =

concentration in bulk liquid

CG =

gas concentration

CG =

graphite concentration

CL =

liquid concentration

CPS =

concentration in the pore space

CGAdj. =

graphite concentration in the adjacent region

CDRF =

concentration on the film side of the diffusion region

CDRI =

concentration on the interface side of the diffusion region

CXe =

xenon concentration

D =

mass diffusion coefficient

ES =

surface energy

H =

Henry’s constant

J =

mass flux

KB =

Boltzmann constant

km =

mass transfer coefficient

L =

latent heat

M =

molecular weight

MS =

molecular weight of solvent

MSBR =

molten salt breeder reactor

MSR =

molten salt reactor

MSRE =

molten salt reactor experiment

ORNL =

Oak Ridge National Laboratory

R =

ideal gas constant

r =

radial coordinate

T =

temperature

TBulk =

bulk temperature

TWall =

wall temperature

V =

specific volume

VG =

volume of graphite

Vm =

molecular volume

x =

spatial coordinate

γ =

surface tension

ϵ =

porosity

λXe =

Xe-135 decay constant

σXe =

Xe-135 neutron absorption cross section

ϕ =

neutron flux/association parameter

Footnotes

1

MSRE was a 7.4 MWth MSR that operated at Oak Ridge National Laboratory (ORNL), 1964–1969. For more information on the MSRE, see the 1965 MSRE Design and Operations Report, [2].

2

No information was found that explained what the CGB acronym referred to. A graphite grade with a similar name is grade CEY graphite. Grade CEY graphite grade is also referred to in many locations in the ORNL literature such as ORNL-TM-1854, [3], but no indication of what its acronym stood for was found either.

3

A complete analysis would show how 135Xe behavior changes with graphite volume during operation of the reactor. This effect is likely negligible, but a proof based on physical arguments would be useful.

4

The astute observer will note that this Equation is not parameterized by the porosity, ϵ. That parameter is hidden in the equation, since the concentration of the xenon in the graphite, CG, can be computed by NXeϵVG, where NXe is the molar quantity of xenon in the graphite, and VG is the total volume of the graphite.

5

As an aside, it appears that some researchers are under the impression that an MSR can be designed to operate bubble free. For example, the model reported in the 2017 paper by Wu et al., [41], does not consider any effects of bubbles. Note, the same assumption was made at the start of the MSRE program and was later found to be incorrect. Consider that there are numerous gas sources in an MSR besides direct ingress of cover gas or xenon, e.g., hydrogen and helium production from ternary fission and neutron absorption on light elements such as lithium.

6

ORNL-TM-378, [43] reports the graphite in the MSRE was hotter than the fuel.

7

Miller calls this equation a diffusion equation and omits the reaction terminology. It is nonetheless a reaction diffusion equation rather than just a diffusion equation.

8

Figure 10 of ORNL-TM-3464 shows five boxes but note X15 is repeated twice.

9

Note, we’ve adopted a different nomenclature here than in the report to generalize and clarify this Equation. We are trying to describe a set of equations without specifying a particular Equation. Nomenclature: J is the mass flux; A is the area available for mass transfer. VG is the graphite volume being transferred into; ϵ is the porosity of the graphite (note, this term is not included in the original Equations, but added here for clarity); CL is the 135Xe concentration in the salt; CG is the 135Xe concentration in the graphite pore space; H is Henry’s constant for the fuel salt; R is the ideal gas constant; λXe is the decay constant of the xenon. ϕ is the neutron flux σXe is the microscopic absorption cross section; D is the mass diffusion coefficient for the graphite; CGAdj. is the concentration of 135Xe in the adjacent graphite stringers.

10

The unit for this quantity is not given in the paper’s nomenclature, but considering the units that are presented are in CGS, it seems reasonable to posit this number is to be in (gmol1).

11

The units for temperature were not found to be explicitly stated in the paper. The temperature in both Eqs. (1) and (2) of [53] use the same symbol. Equation (1) in Ref. [53] is the Wilke Chang equation. The Wilke Chang equation takes its temperature in kelvin. Given these considerations, it is reasonable to conclude that the temperature of Eq. (2) in Ref. [53] is in Kelvin, just as the Eq. (1) in Ref. [53] appears to be.

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