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Analyzing hydriding performance in full-scale depleted uranium beds
Energy 193 (2020) 116742
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Analyzing hydriding performance in full-scale depleted uranium beds Seongjin Yun, Geonhui Gwak, Masoomeh Ghasemi, Jaeyoo Choi, Hyunchul Ju* Department of Mechanical Engineering, Inha University, 100 Inha-ro Michuhol-Gu, Incheon, 22212, Republic of Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 April 2019 Received in revised form 14 November 2019 Accepted 10 December 2019 Available online 13 December 2019
In this study, the hydrogen absorption characteristics of depleted uranium (DU) are analyzed using a three-dimensional transient DU hydride model. The DU model is first validated using experimental data taken from a copper foam-based DU hydride bed, where it is found that the simulation results agree well with the experimental DU temperature evolution and hydriding time. Using the experimentally validated model, we then compare two DU bed designs: one designed to contain 1.86 kg of DU for a tritium capacity of 70 g and loaded with the copper foam to enhance the internal heat transfer during the exothermic hydrogen absorption process, and the other based on copper fins and 5.26 kg of DU to increase a hydrogen isotope (tritium and deuterium) capacity. The simulation results for full-scale DU beds reveal differences in the DU hydriding behavior of the two DU designs, information that is useful in the development of optimal bed design strategies to achieve optimal hydrogen charging performance. © 2019 Published by Elsevier Ltd.
Keywords: Tritium storage Depleted uranium Metal hydride model Numerical simulation
1. Introduction Depleted uranium (DU) has been recognized as one of the most suitable getter materials for the storage of heavier hydrogen isotopes such as tritium [1]. Compared to the wide variety of other materials available, DU requires a relatively low pressure for the hydriding reaction in the temperature range of interest and exhibits a wide plateau region in pressure-composition isotherm curves. In addition to these favorable thermodynamic features, DU exhibits faster reaction kinetics for the hydriding and decomposition of hydrides and higher thermal diffusivity than do other metal hydride materials. Therefore, DU has been employed in large-scale hydrogen isotope (tritium and/or deuterium) storage systems, including the storage and delivery system (SDS) in the International Thermonuclear Experimental Reactor (ITER). While the selection of an appropriate metal hydride material is critical for the rapid delivery and recovery rates of hydrogen isotopes, the optimal design and analysis of the metal hydride bed for the selected material also requires attention, particularly for largecapacity metal hydride bed designs such as the DU bed in the ITER. The modeling and simulation of metal hydride beds are thus essential to gaining a comprehensive understanding of various reaction and transport processes and to optimizing bed design and
* Corresponding author. School of Mechanical Engineering, Inha University, 253 Yonghyun-Dong, Nam-Gu, Incheon, 402-751, Republic of Korea. E-mail address: [email protected] (H. Ju). https://doi.org/10.1016/j.energy.2019.116742 0360-5442/© 2019 Published by Elsevier Ltd.
operating conditions. A number of metal hydride models of varying sophistication have been developed and simulated for different metal hydride materials and bed designs [2e10]. These range from simple empirical or physics-based mathematical models [2e4] to multi-physics and multidimensional models [5e10]. Although significant advances have been made in terms of metal hydride modeling, only a few models have been constructed to represent the DU hydriding and dehydriding process [4,11e14]. The DU bed modeling process begins by establishing experimental DU hydriding and dehydriding rates as a function of temperature and pressure. Various kinetic models for hydrogen-DU reactions have been reported in the literature. For example, Bloch and Mintz [15] derived the reaction kinetics of DU and hydrogen over a wide range of pressures and temperatures. They conducted mathematical analysis to obtain a rate law that considers both hydriding and dehydriding in a single equation. Kirkpatrick and coworkers [16] developed a DU hydride model (referred to as the CeK model) that considers several key transport parameters such as hydrogen diffusivity and solubility in the DU hydride phase, as well as hydriding reaction kinetics. The model calculated the velocity of the spall front in a uranium piece and the constant reaction rate from an equation that was expressed in a closed form except for one numerical integration. Powell et al. [17,18] further improved the CeK model [16], approximating the DU hydriding rate as a function of the half order pressure. While several DU hydriding and dehydriding rate models have been reported in the literature, few attempts have been made to
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model and simulate DU hydriding and dehydriding processes and resultant heat and mass transfer phenomena. Bhattacharyya et al. [11] conducted a simplified mathematical analysis of the DU hydriding process in which a shrinking core model was used in the spherical DU particle domain to account for the DU and DU hydride phases and the movement of the phase boundary for the DU particles. They reported that overall DU hydriding kinetics are governed by the diffusion of uranium hydride. Salloum and Gharagozloo [4] developed an empirical model to predict the decomposition time of a lumped mass of uranium hydride and physics-based mathematical model to evaluate the kinetics by simulating the rate of hydrogen diffusion in a spherical hydride particle. Measured decomposition times were found to fall within the range of the uncertainty predicted by their model, while quantified uncertainty accounted for the scattering found in the experiments at operating temperatures. Ju and coworkers compared the performance of DU and ZrCo hydride beds for hydrogen charging [12] and discharging [13] using threedimensional (3-D) hydrogen absorption and desorption models. The simulation results clearly indicated that the DU bed outperformed the ZrCo bed due to the combination of faster hydrogen absorption/desorption kinetics and higher thermal diffusivity. However, in their comparison, the effect of the change of volume in DU was neglected, even though the change in volume due to DU hydride formation is considerable at around 75%. Gwak et al. [14] later attempted to consider the effects of the volumetric expansion of DU in hydrogen absorption simulations using multiple computational domains. Their simulation results showed that the consideration of volume changes in DU hydride resulted in a faster rise in temperature and slower hydrogen absorption. In this study, a fully 3-D transient DU hydride model is applied to real-scale experimental bed geometries. A parallel computational methodology is adopted to considerably reduce the computational iteration time and to manage the system memory. The DU hydride model is first validated against experimental measurements of changes in DU hydride temperature and hydrogen reservoir vessel pressure during the DU hydriding process. The experimental bed is designed to house 1.86 kg of DU for tritium storage of 70 g and is loaded with copper foam to improve the internal heat transfer capability between the DU particles and the bed wall surface [19]. The DU model is also applied to another DU bed design that contains 5.26 kg of DU and that is loaded with copper fins to improve internal heat transfer [20]. Figs. 1 and 2 present the computational domains for the two DU bed geometries loaded with copper foam or copper fins, respectively. The simulation results from the two DU bed designs are analyzed in order to accurately estimate the scale-up performance of the DU bed for a DU load of up to 9.3 kg. 2. Numerical model 2.1. Model assumptions The following assumptions were applied to the proposed 3-D DU hydride model: (1) the gas phase (hydrogen) obeys the ideal gas law; (2) the powdery DU hydride is an isotropic and homogeneous porous medium characterized by uniform porosity, permeability, and tortuosity; (3) there is a local thermal equilibrium between the solid metal and hydrogen gas; and (4) the porous properties of the DU hydride, such as the porosity and permeability, remain constant during the hydriding process. 2.2. Conservation equations and source terms Based on the assumptions listed above, the DU hydride model is
governed by the conservation equations for mass, momentum, and energy. 2.2.1. Mass conservation Hydrogen:
vεrg ! þ V , ðrg u Þ ¼ Sm vt
(1)
In Eq. (1), under the assumption that hydrogen obeys the ideal gas law, the hydrogen density rg can be expressed as follows:
rg ¼
P g Mg RT
(2)
DU hydride:
ð1 εÞ
vrs ¼ Sm vt
(3)
In Eqs. (1) and (3), ε denotes the porosity of the DU hydride, and Sm represents the volumetric mass source term, which is the local hydriding rate per unit volume. Sm is expressed as follows:
Ea P Ln Sm ¼ Ca exp Drð1 XÞn Peq RT
(4)
where Ca and Ea are the rate constant and activation energy for the hydriding process, respectively. Dr denotes the fictitious change in DU density during the hydriding process, i.e. the difference between the density of DU hydride at saturation and the hydrogen free-density of the DU powder (Dr ¼ rssat rsemp ). X is the fraction of absorbed hydrogen and is defined as the molar ratio of absorbed hydrogen to the maximum amount of hydrogen that can be absorbed by uranium, i.e. X ¼ ðH=UÞ=ðH=UÞsat . The index of Eq. (4) n decreases linearly from 2.3 to 1.0 with variation in the H/U ratio; it was chosen somewhat arbitrarily to fit the experimental data. The assumption may be reasonable due to the porous structure of the DU hydride region, especially given that the tortuosity of the hydrogen transport continuously changes during the hydrogen absorption process. 2.2.2. Momentum conservation
! 1 vrg u 1 !! ! þ V , ðrg u u Þ ¼ VP þ V , t þ rg g þ Su ε ε vt
(5)
In Eq. (5), Su denotes the momentum loss in the porous DU hydride region. Therefore, Su was set to zero for the nonporous region, such as the expansion volume. On the other hand, the porous region, such as DU hydride or copper foam, can be represented by Darcy’s law as a function of permeability K and dynamic viscosity m as follows:
Su ¼
m K
! u
(6)
2.2.3. Energy conservation According to assumption (3) above, there is a local thermal equilibrium between the solid DU hydride and hydrogen gas and thus the energy equation can be represented using the single temperature variable T as follows:
g! vrCp T þ V , rg cp u T ¼ V , keff VT þ ST vt
(7)
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Fig. 1. (a) Copper-foam-based DU bed geometry designed for a DU load of 1.86 kg (representing a tritium capacity of 70 g) [19], (b) schematic diagram of the computational domains with boundary conditions, and (c) mesh configuration.
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In Eq. (7), rCp and keff denote the overall heat capacity and effective thermal conductivity, respectively. Because the hydrogen gas and the solid mixture of DU and DU hydride are combined, they can be expressed as porosity-weighted functions as follows:
rCp ¼ ð1 εÞrs C sp þ εrg C gp
(8)
H kUHsat keff ¼ εkg þ ð1 εÞ U H H H kU þ U sat U U sat
(9)
In Eq. (9), the thermal conductivity of the DU hydride phase is assumed to be linearly related to the H/M ratio. The volumetric energy source term ST can be represented in terms of the enthalpy change during the DU hydriding process (DH) and the hydriding rate (Sm) as follows:
h i ST ¼ Sm DH T C gp C sp
(10)
Because ST accounts for the release of heat during the DU hydriding process, it is positive in the porous metal sub-region during the exothermic DU hydriding reaction.
2.3. Thermal properties combined with copper foam The copper foam installed in the DU bed to enhance the internal heat transfer alters the effective thermal mass and thermal conductivity, thus Eqs. (8) and (9) require further modification. In particular, estimating the effective thermal conductivity for the combination of DU hydride and copper foam is key to accurately simulating copper-foam-based DU beds. In this study, the thermal conductivity correlation proposed by Bhattacharya et al. [21] was adopted: eff keff ¼ Akeff par þ ð1 AÞkser mf
(11)
eff þ 1 εcf kcf keff par ¼ εcf k
(12)
" keff ser
¼
εcf
keff
1 εcf þ kcf
#1 (13)
where εcf and kcf denote the porosity and thermal conductivity of the copper foam, respectively. keff is the effective thermal conductivity described in Eq. (9). In Eqs. (12) and (13), keff par represents the upper limit, with the combined regions of copper foam, DU powder, and hydrogen gas arranged parallel to the direction of the heat flow
Fig. 2. (a) Copper-fin-based DU bed geometry designed for a DU load of 5.26 kg [20], (b) schematic diagram of the computational domains with boundary conditions, and (c) mesh configuration.
path, whereas keff ser is the lower limit, with the combined regions arranged in series. A is the correlation factor between the parallel and series arrangements. According to Bhattacharya et al. [21], A ¼ 0.35 is the best fit for the experimental thermal conductivity data for various air-metal foam systems. The overall specific heat and density for the combined region of copper foam, DU hydride power, and hydrogen gas can be averaged by weight and volume, respectively, as follows:
C eff p ¼
mðMHþH2 Þ C eff p þ mmf Cp;mf mðMHþH2 Þ þ mmf
(14)
S. Yun et al. / Energy 193 (2020) 116742
reff ¼ εmf rðMHþH2 Þ þ 1 εmf rmf
(15)
5
which the DU hydride code is massively parallelized to be capable of running on multiprocessor computers. 3. Results and discussion
2.4. Initial and boundary conditions and numerical implementation As shown in Figs. 1 and 2, the two DU beds considered in this study differ in terms of size and the strategy for enhancing the internal heat transfer. One bed contains 1.86 kg of DU for a tritium capacity of 70 g and is loaded with copper foam to improve the internal heat transfer (Fig. 1a). The other bed is larger, with 5.26 kg of DU for a tritium capacity of 200 g and the installation of copper fins. These DU beds are assumed to initially exist in a state of thermodynamic equilibrium. The initial conditions for the DU hydriding simulations are defined as
T ¼ T0 ; P ¼ P0 ; rs ¼ rs0
(16)
In Eq. (16), rs0 denotes the initial density of the hydrogen-free DU because hydrogen gas was assumed to be absent from the DU hydride bed at the beginning of the hydriding reaction. In addition, the initial velocity of hydrogen gas should be equal to zero, as follows:
! u ¼0
(17)
To set the boundary conditions for the DU beds, no-slip velocity and symmetrical conditions were applied to the external surfaces except for the hydrogen inlet regions. As illustrated in Fig. 1, the convection boundary conditions for the temperature calculations are applied to the outer faces as follows: eff
k
vT ! ¼ hðT Tamb Þ vn
(18)
! where n is the unit normal vector out of the DU bed wall and Tamb is the ambient temperature around the DU hydride bed. The hydrogen inlet pressure Pin of the DU bed is assumed to be equal to the pressure of the hydrogen storage reservoir Pres during the DU hydriding process. Pres decreases with time depending on the amount of hydrogen absorbed within the DU bed. Therefore, the time dependence of Pin can be expressed as [22]:
Pin ¼ Pres ¼ P0:res
1 R TH2 ;res H rs VU 2 VH2 ;res U MU
3.1. Pressure-composition isotherm for a DU-hydrogen system The reaction of DU with hydrogen gas is expressed as
x U þ H2 /UHx ; 0 x 3 2
(20)
where the maximum value of x for saturated DU hydride is set at 3, thus (H/U)sat in Eq. (9) equals 3. Unlike other metal hydride materials, DU has a wider plateau region in its pressure-composition isotherm curve, meaning the equilibrium pressure for the DU hydriding process (Peq) can be approximated as a function of temperature only based on the van ‘t Hoff equation as follows [23]:
Peq ¼ P0 10
6:264410 T
(21)
where P0 represents the reference pressure, i.e. 1 bar. Fig. 3 plots Eq. (21) at various temperatures along with the experimental data measured by Libowitz and Gibb Jr [23]. In addition, Table 1 summarizes the key input parameters for the DU hydriding simulations, including the thermo-physical properties of the DU bed and the operating conditions. 3.2. Experimental validation of the DU hydride model The DU hydride model presented in the previous chapter was applied to a full-scale experimental DU bed equipped with copper foam and designed to contain 1.86 kg of DU for a tritium capacity of 70 g (Fig. 1). Full 3-D simulations were carried out to validate the 3D DU hydride model during the DU hydriding process. The experimental values used in this study represent the most recently available figures from the literature, and details of the experiment can be found in Kang et al. [19]. Fig. 4 compares the simulation results with the experimental data in terms of DU hydride temperature and H/U atomic ratio evolution curves during the DU
(19)
where P0:res denotes the initial hydrogen pressure within the hydrogen storage vessel. The proposed 3-D DU hydride model is numerically implemented in ANSYS Fluent V. 16.1, the commercial computational fluid dynamics (CFD) package, using user-defined functions (UDFs). Therefore, all source terms of the governing equations, i.e. Sm ; Su ; and ST in Eqs. (4), (6) and (10), respectively and all correlations for physical and transport properties are specified in the user codes. The convergence criteria are set to 108 for the equation residuals. The grid size used for DU hydriding simulations were determined based on the previous grid independent study. As shown in Figs. 1c and 2c, the number of grid points needed to sufficiently resolve spatial gradients in the computational domain is 116,857 for the smaller DU bed with the copper foam in Fig. 1c (i.e., approximately 1820 and 64 grid points for the area of base and height, respectively) and 121,335 for the larger DU bed in Fig. 2c (i.e., approximately 1751 and 69 grid points for the area of base and height, respectively). A large-scale simulation that is capable of resolving all three dimensions is made possible by parallel computing, for
Fig. 3. Equilibrium hydrogen pressure as a function of temperature for the DUhydrogen system. The symbols represent the experimental data measured by Libowitz and Gibb Jr [22]. and the solid lines are calculated from Eq. (21).
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Table 1 Thermal-physical properties and operating conditions. Description
Design Copper foam
Initial/ambient temperature, T0/Tamb, C Initial hydrogen reservoir pressure, P0,res, kPa Reference pressure, P0, kPa Wall temperature for cooling, Tw, C Porosity of the DU, ε Porosity/pore diameter of copper foam, εmf/dmf Permeability of the metal, K, m2 Specific heat of hydrogen gas, Cgp, kJ(kg$K)1 Thermal conductivity of hydrogen, kg, W(m$K)1 Hydriding rate constant, Ca, s1 Activation energy, Ea, kJ$mol1 Reaction enthalpy, DH, kcal$mol1 Correlation factor, A Specific heat of the DU, Csp, kJ(kg$K)1 Thermal conductivity of the DU, kM, W(m$K)1 Thermal conductivity of the fully hydrided DU, kMHsat, W(m$K)1 Density of the DU, rM, kg$m3 Density of the fully hydride DU, rMHsat, kg$m3 Thermal conductivity of stainless steel, ksus, W(m$K)1 Thermal conductivity of copper, kcu, W(m$K)1 Specific heat of stainless steel, Cp,sus, kJ(kg$K)1 Specific heat of copper, Cp,cu, kJ(kg$K)1 Heat transfer coefficient, h, W(m2$K)1
hydriding process. The location of the thermocouple installed to measure the DU hydride temperature is indicated in Fig. 4a, and the simulation also used this location for an accurate comparison. The change in the H/U atomic ratio represents the average amount of hydrogen absorbed into DU hydride, which is obtained by monitoring the pressure of the hydrogen storage reservoir. In general, the simulation results closely agree with the experimental data, indicating that the proposed DU hydride model reasonably captures key experimental trends during the DU hydriding process. In Fig. 4a, both the measured and calculated DU temperature rapidly rises in the early stages of DU hydriding, which means that the rate of the DU hydriding reaction is initially rapid due to the availability of sufficient DU surface area for hydrogen absorption. The temperature peak predicted by the model is around 240.05 C after 2.8 min, which is close to the measured values (229.60 C after 2.8 min). Following these peaks, both the calculated and measured DU temperature gradually decrease due mainly to the reduction in the active DU surface area and the resultant lower DU hydriding rate. The comparison of the H/U ratio and inlet pressure curves presented in Fig. 4b is generally satisfactory except for the early DU hydriding stages (i.e., within the first 10 min), during which the DU hydriding rate is underpredicted by the simulation. The discrepancy may be attributed to the use of the adjustable parameter, n in Eq. (4) to fit the experimental data, implying that additional physics related to fast DU hydriding rate for the early stages should be accounted for in the present DU hydride model. In the latter stages, however, the model successfully reflects the experimentally measured DU hydriding performance. In the experiment, hydrogen absorption levels of 90% (H/U ¼ 2.7) and 95% (H/U ¼ 2.85) are achieved at 20.0 and 28.7 min, respectively, which are close to the model predictions (20.5 and 27.5 min, respectively). Fig. 5 shows the DU hydride temperature and H/U atomic ratio contours inside the DU bed taken at five points in time. As shown in Fig. 4, the rise in DU hydride temperature and DU hydriding rate is vigorous in the early DU hydriding stages (within 10 min). Therefore, the DU hydride temperature and H/U distributions are relatively non-uniform at 50 s, 100 s, and 500 s. It is observed that the DU hydride temperature and H/U gradients mainly form in the
25 150 101.325 25 0.895 0.97/4.0 mm 10e10 14.890 0.167 0.51 25.2 16 0.35 0.104 4.6 1.0 19,050 10,920 16.2 400.38 0.383 0.468 10
Ref. Copper fin
0.812 or 0.895 e
e
[22] e e [22] e e [13] [10] [10] [18] [18] [15] [21] [24] [24] [24] [24] [24] [25] [26] [25] [27] [28]
radial direction, with an almost uniform distribution in the axial direction. The non-uniform temperature distribution is mainly due to differences in the cooling rates between the bed wall and core regions, while the non-uniform H/U distribution along the radial direction is related to hydrogen gas transport through the DU hydride particles. In the latter stages (1000 s and 2000 s), the DU hydride is almost saturated, resulting in almost uniform DU hydride temperature and H/U distributions. 3.3. Comparison of copper-foam- and copper-fin-based DU bed designs Because DU ingot and DU hydride powder densities are reported to be rU;ingot ¼ 19:05 g=cm3 and rU;powder ¼ 2:0 g=cm3 , respectively [24], a high degree of swelling is expected of the DU hydride during hydrogen absorption. The copper-foam-based DU bed geometry (Fig. 1) used in the model validation study has sufficient expansion volume, allowing the free expansion of DU hydride during hydrogen absorption. Therefore, the porosity of the freely expanded DU hydride powder can be estimated to be 0.895 (1 rU;powder . On the other hand, the copper-fin-based DU bed in Fig. 2 is r U;ingot
designed to contain 5.26 kg of DU for a larger hydrogen isotope (tritium þ deuterium) capacity [20], thus it compacts the DU hydride particles and provides insufficient room for the particles to expand into during hydrogen absorption. In other words, the swelling of the DU hydride particles in the copper-fin bed is inevitably restricted by the bed components during the DU hydriding process. As a result, the DU hydride porosity is reduced to 0.812 for a DU loading of 5.26 kg. To compare the performance of the copperfoam- and copper-fin-based DU bed designs, two DU hydride porosity levels are defined for the copper-fin bed simulations: 0.812 for the intended DU loading of 5.26 kg and 0.895 for a reduced DU loading of 2.94 kg to produce the same porosity as the copper-foam bed. Fig. 6 displays a comparison of the DU hydride temperature and H/U atomic ratio evolution curves for the three cases. It can clearly be seen that the copper-fin design leads to a higher temperature peak during the DU hydriding process than does the copper-foam
S. Yun et al. / Energy 193 (2020) 116742
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Fig. 4. Comparison of simulated and measured (a) DU hydride temperature and (b) average H/M atomic ratio and inlet pressure evolution profiles during the DU hydriding process. The experimental values were obtained from Kang et al. [19].
design. The lower porosity copper-fin design (ε ¼ 0.812, DU ¼ 5.26 kg) produces a higher rise in temperature because the denser DU hydride powder results in the greater release of volumetric heat during the exothermic hydrogen absorption process. As presented in Fig. 6b, more rapid DU hydriding is achieved using the lower porosity copper-fin design (ε ¼ 0:812), indicating that the
greater rise in temperature at ε ¼ 0:812 is beneficial for DU hydriding performance. When the two cases with the same porosity (ε ¼ 0:895) are compared, both the copper-fin and copperfoam designs exhibited a similar DU hydriding performance, although the DU hydride temperature profiles in Fig. 6a differ significantly. This suggests that there is a tradeoff between the DU
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Fig. 6. Comparison of (a) DU hydride temperature and (b) H/U ratio evolution profiles for the copper-foam design and the two copper-fin designs with different porosities (0.895 and 0.812).
Fig. 5. Three-dimensional contours of (a) DU hydride temperature and (b) H/U atomic ratio during DU hydriding process at 50 s, 100 s, 500 s, 1000 s, and 2000 s.
hydride kinetics and the equilibrium pressure as the DU hydride temperature increases. The rise in temperature increases both DU hydride kinetics and equilibrium pressure, with the former being beneficial to the DU hydriding reaction and the latter suppressing it. To analyze the effects of the rise in temperature on the DU hydriding performance in more detail, the temperature and pressure dependent terms for the DU hydriding rate equation Sm in Eq. (4) are plotted in Fig. 7a. The temperature dependent term is the Arrhenius expression of the DU hydriding kinetics, expð Ea =RTÞ; while the pressure dependent term is related to the equilibrium pressure, i.e. LnðP =PeqÞ. These individual terms are normalized to those of copper foam as follows:
S. Yun et al. / Energy 193 (2020) 116742
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negative effect of increasing the equilibrium pressure. As a result, Fig. 7b shows that larger source terms (Sm) in the two copper-fin designs are predicted due to their greater temperature rise. In contrast, when the DU hydride temperature in the copper-foam design becomes higher than that of the copper-fin design (after 4 min with ε ¼ 0:895 and after the 8 min with ε ¼ 0:812), the opposite trend is clearly observed (Fig. 7). After analyzing Figs. 6 and 7, it can be seen that the rise in DU temperature up to 350 C promotes the overall DU hydriding reaction despite the increase in the equilibrium pressure. Figs. 8 and 9 display detailed 3-D contours of the DU hydride temperature and H/U atomic ratio, respectively, for the two copperfin designs at five different elapsed times during hydrogen absorption. In the early stages when the DU hydriding rate is rapid, a noticeable temperature gradient between the copper fin and the interior is observed (Fig. 8). When Figs. 8 and 9 are compared, the hotter interior DU hydride region leads to slower DU hydriding at 50 s and 100 s, indicating that more effective cooling near the copper fin enhances the DU hydriding rate. In the latter stages, however, the DU hydriding rate near the copper fin slows down due to a reduction in the free DU surface for hydrogen absorption and thus the DU hydride temperature there cools due to less generation of exothermic reaction heat. Therefore, the local DU hydride temperature and H/U gradients between the copper fin and interior DU regions are lower at 1000 s and 2000 s. Detailed analysis of the 3-D contours in Figs. 5, 8 and 9 indicate that the copper foam and copper fin DU bed designs exhibit different behaviors in terms of local DU temperature and H/U distributions. Unlike the copperfoam design in Fig. 5, the hotter DU hydride region in the copperfin-based DU beds corresponds to a lower H/U region.
4. Conclusion
Fig. 7. Comparison of the calculated ratio of the copper-fin designs with different porosities (0.985 and 0.812) to the copper-foam design: (a) the Arrhenius equation (exp(-Ea/RT)) and pressure term (Ln(P/Peq)) and (b) the mass source.
Ea exp RT fin RatioT ¼ Ea exp RT
(22)
foam
ln PPeq fin RatioP ¼ ln PPeq
(23)
foam
As seen in Fig. 6a, the copper-fin designs with ε ¼ 0:895 and 0.812 have a higher rise in DU temperature than does the copperfoam design for the first 4 min and for the first 8 min, respectively. During the initial DU hydriding time, the increase in RatioT is greater than the decrease in RatioP for both porosity levels, which indicates that, as the DU hydride temperature increases, the positive effect of improving the DU hydriding kinetics dominates the
In this study, large-scale transient DU hydriding simulations of full-size DU beds were successfully carried out to investigate key DU hydriding characteristics, including a detailed analysis of the local DU hydride temperature and H/U atomic ratio distributions within the DU bed geometries. Major conclusions can be summarized as follows: 1. The DU hydride model was successfully validated against experimental DU hydriding data measured from a copper foambased DU bed loaded with 1.86 kg of DU. In general, good agreement between the simulation results and experimental data was achieved in terms of DU hydride temperature and H/U atomic ratio evolution curves. However, the agreement became weaker at the early DU hydriding stages when the DU hydriding rate was fast, which indicates that the present model should be improved to more accurately simulate fast DU hydriding process. 2. As the DU hydride temperature increased during hydrogen absorption, there was a tradeoff between the absorption kinetics and the equilibrium pressure: the kinetics improved the DU hydriding rate whereas the rise in equilibrium pressure with temperature limited it. The detailed simulation results indicate that the rise in DU temperature up to 350 C favors DU hydriding, demonstrating that the effect of increasing kinetics dominates the effect of increasing equilibrium pressure. It is worth noting that the above trend only occurs in the copperfoam-based DU bed design. In the copper-fin-based DU bed designs, the higher rise in DU hydride temperature resulted in the slower DU hydride rate. Therefore, DU hydriding characteristics and performance are highly influenced by the effective
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Fig. 8. Three-dimensional DU hydride temperature contours for the copper-fin designs with different porosities (0.895 and 0.812) during the DU hydriding process at 50 s, 100 s, 500 s, 1000 s, and 2000 s.
S. Yun et al. / Energy 193 (2020) 116742
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Fig. 9. Contours for the change in the three-dimensional H/U atomic ratio for the copper-fin designs with different porosities (0.895 and 0.812) during the DU hydriding process at 50 s, 100 s, 500 s, 1000 s, 2000 s.
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thermal conductivity of the DU hydride region along with the local DU hydride temperature and equilibrium pressure. 3. The higher DU loading and resultant lower DU hydride porosity induced the faster hydrogen absorption performance in the copper-fin-based DU bed, which is mainly owing to the higher effective thermal conductivity of the DU hydride region. Therefore, the high degree of compaction of DU powders is found to be significant design parameter for copper-fin-based DU bed design. Acknowledgements This study was supported by the Technology Innovation Program of Korea Evaluation Institute of Industrial Technology (KEIT) funded by the Ministry of Trade, Industry & Energy (MOTIE) (10052823) as well as by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government (MOTIE) (20183030032010). We also thank Taesung S&E, Inc. Korea for lending technical support for the ANSYS Fluent software. Nomenclature A Ca Cp DU Ea EXV H/U DH h K k M P R S SUS T t ! u V X
correlation factor hydriding rate constant, s1 specific heat, kJ∙kg1∙K1 depleted uranium activation energy, kJ∙kmol1 expansion volume hydrogen to metal atomic ratio reaction heat of formation, J∙kg1 convection heat transfer coefficient, W∙m2∙K1 permeability, m2 thermal conductivity, W∙m1∙K1 molecular weight, kg∙kmol1 pressure, bar universal gas constant, 8.314 J∙mol1 K1 source term stainless steel temperature, K time, s velocity vector, m∙s1 volume, m3 reaction fraction
Greek symbols ε porosity m dynamic viscosity, kg∙m1∙s1 r density, kg∙m3 t stress tensor Superscripts eff effective value emp empty eq equilibrium g gas phase s solid phase sat saturation value Subscripts c cu Eq H2
cooling copper equilibrium hydrogen
in U UH m cf par ref res sat ser T u 0
inlet depleted uranium metal depleted uranium metal hydride mass equation copper foam parallel reference value reservoir saturation value series energy equation momentum equation initial value
References [1] Penzhorn RD, devillers M, Sirch M. Evaluation of ZrCo and other getters for tritium handling and storage. J Nucl Mater 1990;170(3):217e31. [2] Gadre SA, Ebner AD, Al-Muhtaseb SA, Ritter JA. Practical modeling of metal hydride hydrogen storage systems. Ind Eng Chem Res 2003;42(8):1713e22. [3] Førde T, Maehlen JP, Yartys VA, Lototsky MV, Uchida H. Influence of intrinsic hydrogenation/dehydrogenation kinetics on the dynamic behaviour of metal hydrides: a semi-empirical model and its verification. Int J Hydrogen Energy 2007;32(8):1041e9. [4] Salloum M, Gharagozloo PE. Empirical and physics-based mathematical models of uranium hydride decomposition kinetics with quantified uncertainty. Chem Eng Sci 2014;116:452e64. [5] Hardy BJ, Anton DL. Hierarchical methodology for modeling hydrogen storage systems. Part I: scoping models. Int J Hydrogen Energy 2009;34. 2269e227. [6] Hardy BJ, Anton DL. Hierarchical methodology for modeling hydrogen storage systems. Part II: detailed models. Int J Hydrogen Energy 2009;34(7):2992e3004. [7] Valizadeh M, Aghajani Delavar M, Farhadi M. Numerical simulation of heat and mass transfer during hydrogen desorption in metal hydride storage tank by Lattice Boltzmann method. Int J Hydrogen Energy 2016;41(1):413e24. [8] Førde T, Næss E, Yartys VA. Modelling and experimental results of heat transfer in a metal hydride store during hydrogen charge and discharge. Int J Hydrogen Energy 2009;34(12):5121e30. [9] Wang Y, Adroher XC, Chen J, Yang XG, Miller T. Three-dimensional modeling of hydrogen sorption in metal hydride hydrogen storage beds. J Power Sources 2009;194(2):997e1006. [10] Nam J, Ko J, Ju H. Three-dimensional modeling and simulation of hydrogen absorption in metal hydride hydrogen storage vessels. Appl Energy Jan. 2012;89(1):164e75. [11] Bhattacharyya R, Bandyopadhyay D, Bhanja K, Mohan S. Mathematical analysis of the hydrogen-uranium reaction using the shrinking core model for hydrogen storage application. Int J Hydrogen Energy 2015;40(29):8917e25. [12] Yoo H, Kim W, Ju H. A numerical comparison of hydrogen absorption behaviors of uranium and zirconium cobalt-based metal hydride beds. Solid State Ion 2014;262:241e7. [13] Kyoung S, Yoo H, Ju H. Numerical comparison of hydrogen desorption behaviors of metal hydride beds based on uranium and on zirconium-cobalt. Fusion Sci Technol 2015;67(2):394e9. [14] Gwak G, Kim M, Oh K, Kyoung S, Ferekh S, Ju H. Analyzing effects of volumetric expansion of uranium during hydrogen absorption. Int J Hydrogen Energy 2017;42(6):3723e30. [15] Bloch J, Mintz MH. Kinetics and mechanism of the U-H reaction. J Less Common Met 1981;81(2):301e20. [16] Kirkpatrick JR, Condon JB. The linear solution for hydriding of uranium. J Less Common Met 1991;172e174:124e35. no. PART A. [17] Powell GL, Harper WL, Kirkpatrick JR. The kinetics of the hydriding of uranium metal. J Less Common Met 1991;172e174:116e23. no. PART A. [18] Powell GL, Harper WL, Ceo RN, Kirkpatrick JR. The kinetics of the hydriding of uranium metal II. Zeitschrift fur Phys. Chemie 1993;181:275e82. [19] Kang H-G, Chung D, Chang MH, Yun S-H. Development of depleted uranium bed for tritium fuel cycle and basic absorption/desorption experiments. Fusion Eng Des 2018;132:86e9. [20] W. K. Hollis and D. Dogrual, “Self-assaying tritium accountability and containment unit for ITER (STACI) review and experimental results from hydriding and dehydriding,” [Los Alamos]. [21] Bhattacharya A, Calmidi VV, Mahajan RL. Thermophysical properties of high porosity metal foams. Int J Heat Mass Transf 2002;45(5):1017e31. [22] Koo D-S, et al. Fabrication and test of thin double-layered annulus metal hydride bed. Fusion Eng Des 2010;86(9e11):2196e9. [23] Libowitz GG, Gibb TRP. High pressure dissociation studies of the uranium hydrogen system. J Phys Chem 1957;61(6):793e5. [24] Kolasinski RD, Shugard AD, Tewell CR, Cowgill DF. Uranium for hydrogen storage applications: a material science perspective. Sandia Natl. Labs 2010;37(August). [25] Yoo H, Ko J, Ju H. A numerical investigation of hydrogen absorption phenomena in thin double-layered annulus ZrCo beds. Int J Hydrogen Energy
S. Yun et al. / Energy 193 (2020) 116742 2013;38(18):7697e703. [26] Chung CY, Lee MT, Tsai MY, Chu CH, Lin SJ. High thermal conductive diamond/ Cu-Ti composites fabricated by pressureless sintering technique. Appl Therm Eng 2014;69(1e2):208e13. [27] Brooks CR, Norem WE, Hendrix DE, Wright JW, Northcutt WG. The specific
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heat of copper from 40 to 920 C*. J Phys Chem Solids 1968;29(4):565e74. [28] Wang Y, Adroher XC, Chen J, Yang XY, Miller T. Three-dimensional modeling of hydrogen sorption in metal hydride hydrogen storage beds. J Power Sources 2009;194:997e1006.
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Analyzing hydriding performance in full-scale depleted uranium beds Seongjin Yun, Geonhui Gwak, Masoomeh Ghasemi, Jaeyoo Choi, Hyunchul Ju* Department of Mechanical Engineering, Inha University, 100 Inha-ro Michuhol-Gu, Incheon, 22212, Republic of Korea
a r t i c l e i n f o
a b s t r a c t
Article history: Received 12 April 2019 Received in revised form 14 November 2019 Accepted 10 December 2019 Available online 13 December 2019
In this study, the hydrogen absorption characteristics of depleted uranium (DU) are analyzed using a three-dimensional transient DU hydride model. The DU model is first validated using experimental data taken from a copper foam-based DU hydride bed, where it is found that the simulation results agree well with the experimental DU temperature evolution and hydriding time. Using the experimentally validated model, we then compare two DU bed designs: one designed to contain 1.86 kg of DU for a tritium capacity of 70 g and loaded with the copper foam to enhance the internal heat transfer during the exothermic hydrogen absorption process, and the other based on copper fins and 5.26 kg of DU to increase a hydrogen isotope (tritium and deuterium) capacity. The simulation results for full-scale DU beds reveal differences in the DU hydriding behavior of the two DU designs, information that is useful in the development of optimal bed design strategies to achieve optimal hydrogen charging performance. © 2019 Published by Elsevier Ltd.
Keywords: Tritium storage Depleted uranium Metal hydride model Numerical simulation
1. Introduction Depleted uranium (DU) has been recognized as one of the most suitable getter materials for the storage of heavier hydrogen isotopes such as tritium [1]. Compared to the wide variety of other materials available, DU requires a relatively low pressure for the hydriding reaction in the temperature range of interest and exhibits a wide plateau region in pressure-composition isotherm curves. In addition to these favorable thermodynamic features, DU exhibits faster reaction kinetics for the hydriding and decomposition of hydrides and higher thermal diffusivity than do other metal hydride materials. Therefore, DU has been employed in large-scale hydrogen isotope (tritium and/or deuterium) storage systems, including the storage and delivery system (SDS) in the International Thermonuclear Experimental Reactor (ITER). While the selection of an appropriate metal hydride material is critical for the rapid delivery and recovery rates of hydrogen isotopes, the optimal design and analysis of the metal hydride bed for the selected material also requires attention, particularly for largecapacity metal hydride bed designs such as the DU bed in the ITER. The modeling and simulation of metal hydride beds are thus essential to gaining a comprehensive understanding of various reaction and transport processes and to optimizing bed design and
* Corresponding author. School of Mechanical Engineering, Inha University, 253 Yonghyun-Dong, Nam-Gu, Incheon, 402-751, Republic of Korea. E-mail address: [email protected] (H. Ju). https://doi.org/10.1016/j.energy.2019.116742 0360-5442/© 2019 Published by Elsevier Ltd.
operating conditions. A number of metal hydride models of varying sophistication have been developed and simulated for different metal hydride materials and bed designs [2e10]. These range from simple empirical or physics-based mathematical models [2e4] to multi-physics and multidimensional models [5e10]. Although significant advances have been made in terms of metal hydride modeling, only a few models have been constructed to represent the DU hydriding and dehydriding process [4,11e14]. The DU bed modeling process begins by establishing experimental DU hydriding and dehydriding rates as a function of temperature and pressure. Various kinetic models for hydrogen-DU reactions have been reported in the literature. For example, Bloch and Mintz [15] derived the reaction kinetics of DU and hydrogen over a wide range of pressures and temperatures. They conducted mathematical analysis to obtain a rate law that considers both hydriding and dehydriding in a single equation. Kirkpatrick and coworkers [16] developed a DU hydride model (referred to as the CeK model) that considers several key transport parameters such as hydrogen diffusivity and solubility in the DU hydride phase, as well as hydriding reaction kinetics. The model calculated the velocity of the spall front in a uranium piece and the constant reaction rate from an equation that was expressed in a closed form except for one numerical integration. Powell et al. [17,18] further improved the CeK model [16], approximating the DU hydriding rate as a function of the half order pressure. While several DU hydriding and dehydriding rate models have been reported in the literature, few attempts have been made to
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model and simulate DU hydriding and dehydriding processes and resultant heat and mass transfer phenomena. Bhattacharyya et al. [11] conducted a simplified mathematical analysis of the DU hydriding process in which a shrinking core model was used in the spherical DU particle domain to account for the DU and DU hydride phases and the movement of the phase boundary for the DU particles. They reported that overall DU hydriding kinetics are governed by the diffusion of uranium hydride. Salloum and Gharagozloo [4] developed an empirical model to predict the decomposition time of a lumped mass of uranium hydride and physics-based mathematical model to evaluate the kinetics by simulating the rate of hydrogen diffusion in a spherical hydride particle. Measured decomposition times were found to fall within the range of the uncertainty predicted by their model, while quantified uncertainty accounted for the scattering found in the experiments at operating temperatures. Ju and coworkers compared the performance of DU and ZrCo hydride beds for hydrogen charging [12] and discharging [13] using threedimensional (3-D) hydrogen absorption and desorption models. The simulation results clearly indicated that the DU bed outperformed the ZrCo bed due to the combination of faster hydrogen absorption/desorption kinetics and higher thermal diffusivity. However, in their comparison, the effect of the change of volume in DU was neglected, even though the change in volume due to DU hydride formation is considerable at around 75%. Gwak et al. [14] later attempted to consider the effects of the volumetric expansion of DU in hydrogen absorption simulations using multiple computational domains. Their simulation results showed that the consideration of volume changes in DU hydride resulted in a faster rise in temperature and slower hydrogen absorption. In this study, a fully 3-D transient DU hydride model is applied to real-scale experimental bed geometries. A parallel computational methodology is adopted to considerably reduce the computational iteration time and to manage the system memory. The DU hydride model is first validated against experimental measurements of changes in DU hydride temperature and hydrogen reservoir vessel pressure during the DU hydriding process. The experimental bed is designed to house 1.86 kg of DU for tritium storage of 70 g and is loaded with copper foam to improve the internal heat transfer capability between the DU particles and the bed wall surface [19]. The DU model is also applied to another DU bed design that contains 5.26 kg of DU and that is loaded with copper fins to improve internal heat transfer [20]. Figs. 1 and 2 present the computational domains for the two DU bed geometries loaded with copper foam or copper fins, respectively. The simulation results from the two DU bed designs are analyzed in order to accurately estimate the scale-up performance of the DU bed for a DU load of up to 9.3 kg. 2. Numerical model 2.1. Model assumptions The following assumptions were applied to the proposed 3-D DU hydride model: (1) the gas phase (hydrogen) obeys the ideal gas law; (2) the powdery DU hydride is an isotropic and homogeneous porous medium characterized by uniform porosity, permeability, and tortuosity; (3) there is a local thermal equilibrium between the solid metal and hydrogen gas; and (4) the porous properties of the DU hydride, such as the porosity and permeability, remain constant during the hydriding process. 2.2. Conservation equations and source terms Based on the assumptions listed above, the DU hydride model is
governed by the conservation equations for mass, momentum, and energy. 2.2.1. Mass conservation Hydrogen:
vεrg ! þ V , ðrg u Þ ¼ Sm vt
(1)
In Eq. (1), under the assumption that hydrogen obeys the ideal gas law, the hydrogen density rg can be expressed as follows:
rg ¼
P g Mg RT
(2)
DU hydride:
ð1 εÞ
vrs ¼ Sm vt
(3)
In Eqs. (1) and (3), ε denotes the porosity of the DU hydride, and Sm represents the volumetric mass source term, which is the local hydriding rate per unit volume. Sm is expressed as follows:
Ea P Ln Sm ¼ Ca exp Drð1 XÞn Peq RT
(4)
where Ca and Ea are the rate constant and activation energy for the hydriding process, respectively. Dr denotes the fictitious change in DU density during the hydriding process, i.e. the difference between the density of DU hydride at saturation and the hydrogen free-density of the DU powder (Dr ¼ rssat rsemp ). X is the fraction of absorbed hydrogen and is defined as the molar ratio of absorbed hydrogen to the maximum amount of hydrogen that can be absorbed by uranium, i.e. X ¼ ðH=UÞ=ðH=UÞsat . The index of Eq. (4) n decreases linearly from 2.3 to 1.0 with variation in the H/U ratio; it was chosen somewhat arbitrarily to fit the experimental data. The assumption may be reasonable due to the porous structure of the DU hydride region, especially given that the tortuosity of the hydrogen transport continuously changes during the hydrogen absorption process. 2.2.2. Momentum conservation
! 1 vrg u 1 !! ! þ V , ðrg u u Þ ¼ VP þ V , t þ rg g þ Su ε ε vt
(5)
In Eq. (5), Su denotes the momentum loss in the porous DU hydride region. Therefore, Su was set to zero for the nonporous region, such as the expansion volume. On the other hand, the porous region, such as DU hydride or copper foam, can be represented by Darcy’s law as a function of permeability K and dynamic viscosity m as follows:
Su ¼
m K
! u
(6)
2.2.3. Energy conservation According to assumption (3) above, there is a local thermal equilibrium between the solid DU hydride and hydrogen gas and thus the energy equation can be represented using the single temperature variable T as follows:
g! vrCp T þ V , rg cp u T ¼ V , keff VT þ ST vt
(7)
S. Yun et al. / Energy 193 (2020) 116742
3
Fig. 1. (a) Copper-foam-based DU bed geometry designed for a DU load of 1.86 kg (representing a tritium capacity of 70 g) [19], (b) schematic diagram of the computational domains with boundary conditions, and (c) mesh configuration.
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In Eq. (7), rCp and keff denote the overall heat capacity and effective thermal conductivity, respectively. Because the hydrogen gas and the solid mixture of DU and DU hydride are combined, they can be expressed as porosity-weighted functions as follows:
rCp ¼ ð1 εÞrs C sp þ εrg C gp
(8)
H kUHsat keff ¼ εkg þ ð1 εÞ U H H H kU þ U sat U U sat
(9)
In Eq. (9), the thermal conductivity of the DU hydride phase is assumed to be linearly related to the H/M ratio. The volumetric energy source term ST can be represented in terms of the enthalpy change during the DU hydriding process (DH) and the hydriding rate (Sm) as follows:
h i ST ¼ Sm DH T C gp C sp
(10)
Because ST accounts for the release of heat during the DU hydriding process, it is positive in the porous metal sub-region during the exothermic DU hydriding reaction.
2.3. Thermal properties combined with copper foam The copper foam installed in the DU bed to enhance the internal heat transfer alters the effective thermal mass and thermal conductivity, thus Eqs. (8) and (9) require further modification. In particular, estimating the effective thermal conductivity for the combination of DU hydride and copper foam is key to accurately simulating copper-foam-based DU beds. In this study, the thermal conductivity correlation proposed by Bhattacharya et al. [21] was adopted: eff keff ¼ Akeff par þ ð1 AÞkser mf
(11)
eff þ 1 εcf kcf keff par ¼ εcf k
(12)
" keff ser
¼
εcf
keff
1 εcf þ kcf
#1 (13)
where εcf and kcf denote the porosity and thermal conductivity of the copper foam, respectively. keff is the effective thermal conductivity described in Eq. (9). In Eqs. (12) and (13), keff par represents the upper limit, with the combined regions of copper foam, DU powder, and hydrogen gas arranged parallel to the direction of the heat flow
Fig. 2. (a) Copper-fin-based DU bed geometry designed for a DU load of 5.26 kg [20], (b) schematic diagram of the computational domains with boundary conditions, and (c) mesh configuration.
path, whereas keff ser is the lower limit, with the combined regions arranged in series. A is the correlation factor between the parallel and series arrangements. According to Bhattacharya et al. [21], A ¼ 0.35 is the best fit for the experimental thermal conductivity data for various air-metal foam systems. The overall specific heat and density for the combined region of copper foam, DU hydride power, and hydrogen gas can be averaged by weight and volume, respectively, as follows:
C eff p ¼
mðMHþH2 Þ C eff p þ mmf Cp;mf mðMHþH2 Þ þ mmf
(14)
S. Yun et al. / Energy 193 (2020) 116742
reff ¼ εmf rðMHþH2 Þ þ 1 εmf rmf
(15)
5
which the DU hydride code is massively parallelized to be capable of running on multiprocessor computers. 3. Results and discussion
2.4. Initial and boundary conditions and numerical implementation As shown in Figs. 1 and 2, the two DU beds considered in this study differ in terms of size and the strategy for enhancing the internal heat transfer. One bed contains 1.86 kg of DU for a tritium capacity of 70 g and is loaded with copper foam to improve the internal heat transfer (Fig. 1a). The other bed is larger, with 5.26 kg of DU for a tritium capacity of 200 g and the installation of copper fins. These DU beds are assumed to initially exist in a state of thermodynamic equilibrium. The initial conditions for the DU hydriding simulations are defined as
T ¼ T0 ; P ¼ P0 ; rs ¼ rs0
(16)
In Eq. (16), rs0 denotes the initial density of the hydrogen-free DU because hydrogen gas was assumed to be absent from the DU hydride bed at the beginning of the hydriding reaction. In addition, the initial velocity of hydrogen gas should be equal to zero, as follows:
! u ¼0
(17)
To set the boundary conditions for the DU beds, no-slip velocity and symmetrical conditions were applied to the external surfaces except for the hydrogen inlet regions. As illustrated in Fig. 1, the convection boundary conditions for the temperature calculations are applied to the outer faces as follows: eff
k
vT ! ¼ hðT Tamb Þ vn
(18)
! where n is the unit normal vector out of the DU bed wall and Tamb is the ambient temperature around the DU hydride bed. The hydrogen inlet pressure Pin of the DU bed is assumed to be equal to the pressure of the hydrogen storage reservoir Pres during the DU hydriding process. Pres decreases with time depending on the amount of hydrogen absorbed within the DU bed. Therefore, the time dependence of Pin can be expressed as [22]:
Pin ¼ Pres ¼ P0:res
1 R TH2 ;res H rs VU 2 VH2 ;res U MU
3.1. Pressure-composition isotherm for a DU-hydrogen system The reaction of DU with hydrogen gas is expressed as
x U þ H2 /UHx ; 0 x 3 2
(20)
where the maximum value of x for saturated DU hydride is set at 3, thus (H/U)sat in Eq. (9) equals 3. Unlike other metal hydride materials, DU has a wider plateau region in its pressure-composition isotherm curve, meaning the equilibrium pressure for the DU hydriding process (Peq) can be approximated as a function of temperature only based on the van ‘t Hoff equation as follows [23]:
Peq ¼ P0 10
6:264410 T
(21)
where P0 represents the reference pressure, i.e. 1 bar. Fig. 3 plots Eq. (21) at various temperatures along with the experimental data measured by Libowitz and Gibb Jr [23]. In addition, Table 1 summarizes the key input parameters for the DU hydriding simulations, including the thermo-physical properties of the DU bed and the operating conditions. 3.2. Experimental validation of the DU hydride model The DU hydride model presented in the previous chapter was applied to a full-scale experimental DU bed equipped with copper foam and designed to contain 1.86 kg of DU for a tritium capacity of 70 g (Fig. 1). Full 3-D simulations were carried out to validate the 3D DU hydride model during the DU hydriding process. The experimental values used in this study represent the most recently available figures from the literature, and details of the experiment can be found in Kang et al. [19]. Fig. 4 compares the simulation results with the experimental data in terms of DU hydride temperature and H/U atomic ratio evolution curves during the DU
(19)
where P0:res denotes the initial hydrogen pressure within the hydrogen storage vessel. The proposed 3-D DU hydride model is numerically implemented in ANSYS Fluent V. 16.1, the commercial computational fluid dynamics (CFD) package, using user-defined functions (UDFs). Therefore, all source terms of the governing equations, i.e. Sm ; Su ; and ST in Eqs. (4), (6) and (10), respectively and all correlations for physical and transport properties are specified in the user codes. The convergence criteria are set to 108 for the equation residuals. The grid size used for DU hydriding simulations were determined based on the previous grid independent study. As shown in Figs. 1c and 2c, the number of grid points needed to sufficiently resolve spatial gradients in the computational domain is 116,857 for the smaller DU bed with the copper foam in Fig. 1c (i.e., approximately 1820 and 64 grid points for the area of base and height, respectively) and 121,335 for the larger DU bed in Fig. 2c (i.e., approximately 1751 and 69 grid points for the area of base and height, respectively). A large-scale simulation that is capable of resolving all three dimensions is made possible by parallel computing, for
Fig. 3. Equilibrium hydrogen pressure as a function of temperature for the DUhydrogen system. The symbols represent the experimental data measured by Libowitz and Gibb Jr [22]. and the solid lines are calculated from Eq. (21).
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Table 1 Thermal-physical properties and operating conditions. Description
Design Copper foam
Initial/ambient temperature, T0/Tamb, C Initial hydrogen reservoir pressure, P0,res, kPa Reference pressure, P0, kPa Wall temperature for cooling, Tw, C Porosity of the DU, ε Porosity/pore diameter of copper foam, εmf/dmf Permeability of the metal, K, m2 Specific heat of hydrogen gas, Cgp, kJ(kg$K)1 Thermal conductivity of hydrogen, kg, W(m$K)1 Hydriding rate constant, Ca, s1 Activation energy, Ea, kJ$mol1 Reaction enthalpy, DH, kcal$mol1 Correlation factor, A Specific heat of the DU, Csp, kJ(kg$K)1 Thermal conductivity of the DU, kM, W(m$K)1 Thermal conductivity of the fully hydrided DU, kMHsat, W(m$K)1 Density of the DU, rM, kg$m3 Density of the fully hydride DU, rMHsat, kg$m3 Thermal conductivity of stainless steel, ksus, W(m$K)1 Thermal conductivity of copper, kcu, W(m$K)1 Specific heat of stainless steel, Cp,sus, kJ(kg$K)1 Specific heat of copper, Cp,cu, kJ(kg$K)1 Heat transfer coefficient, h, W(m2$K)1
hydriding process. The location of the thermocouple installed to measure the DU hydride temperature is indicated in Fig. 4a, and the simulation also used this location for an accurate comparison. The change in the H/U atomic ratio represents the average amount of hydrogen absorbed into DU hydride, which is obtained by monitoring the pressure of the hydrogen storage reservoir. In general, the simulation results closely agree with the experimental data, indicating that the proposed DU hydride model reasonably captures key experimental trends during the DU hydriding process. In Fig. 4a, both the measured and calculated DU temperature rapidly rises in the early stages of DU hydriding, which means that the rate of the DU hydriding reaction is initially rapid due to the availability of sufficient DU surface area for hydrogen absorption. The temperature peak predicted by the model is around 240.05 C after 2.8 min, which is close to the measured values (229.60 C after 2.8 min). Following these peaks, both the calculated and measured DU temperature gradually decrease due mainly to the reduction in the active DU surface area and the resultant lower DU hydriding rate. The comparison of the H/U ratio and inlet pressure curves presented in Fig. 4b is generally satisfactory except for the early DU hydriding stages (i.e., within the first 10 min), during which the DU hydriding rate is underpredicted by the simulation. The discrepancy may be attributed to the use of the adjustable parameter, n in Eq. (4) to fit the experimental data, implying that additional physics related to fast DU hydriding rate for the early stages should be accounted for in the present DU hydride model. In the latter stages, however, the model successfully reflects the experimentally measured DU hydriding performance. In the experiment, hydrogen absorption levels of 90% (H/U ¼ 2.7) and 95% (H/U ¼ 2.85) are achieved at 20.0 and 28.7 min, respectively, which are close to the model predictions (20.5 and 27.5 min, respectively). Fig. 5 shows the DU hydride temperature and H/U atomic ratio contours inside the DU bed taken at five points in time. As shown in Fig. 4, the rise in DU hydride temperature and DU hydriding rate is vigorous in the early DU hydriding stages (within 10 min). Therefore, the DU hydride temperature and H/U distributions are relatively non-uniform at 50 s, 100 s, and 500 s. It is observed that the DU hydride temperature and H/U gradients mainly form in the
25 150 101.325 25 0.895 0.97/4.0 mm 10e10 14.890 0.167 0.51 25.2 16 0.35 0.104 4.6 1.0 19,050 10,920 16.2 400.38 0.383 0.468 10
Ref. Copper fin
0.812 or 0.895 e
e
[22] e e [22] e e [13] [10] [10] [18] [18] [15] [21] [24] [24] [24] [24] [24] [25] [26] [25] [27] [28]
radial direction, with an almost uniform distribution in the axial direction. The non-uniform temperature distribution is mainly due to differences in the cooling rates between the bed wall and core regions, while the non-uniform H/U distribution along the radial direction is related to hydrogen gas transport through the DU hydride particles. In the latter stages (1000 s and 2000 s), the DU hydride is almost saturated, resulting in almost uniform DU hydride temperature and H/U distributions. 3.3. Comparison of copper-foam- and copper-fin-based DU bed designs Because DU ingot and DU hydride powder densities are reported to be rU;ingot ¼ 19:05 g=cm3 and rU;powder ¼ 2:0 g=cm3 , respectively [24], a high degree of swelling is expected of the DU hydride during hydrogen absorption. The copper-foam-based DU bed geometry (Fig. 1) used in the model validation study has sufficient expansion volume, allowing the free expansion of DU hydride during hydrogen absorption. Therefore, the porosity of the freely expanded DU hydride powder can be estimated to be 0.895 (1 rU;powder . On the other hand, the copper-fin-based DU bed in Fig. 2 is r U;ingot
designed to contain 5.26 kg of DU for a larger hydrogen isotope (tritium þ deuterium) capacity [20], thus it compacts the DU hydride particles and provides insufficient room for the particles to expand into during hydrogen absorption. In other words, the swelling of the DU hydride particles in the copper-fin bed is inevitably restricted by the bed components during the DU hydriding process. As a result, the DU hydride porosity is reduced to 0.812 for a DU loading of 5.26 kg. To compare the performance of the copperfoam- and copper-fin-based DU bed designs, two DU hydride porosity levels are defined for the copper-fin bed simulations: 0.812 for the intended DU loading of 5.26 kg and 0.895 for a reduced DU loading of 2.94 kg to produce the same porosity as the copper-foam bed. Fig. 6 displays a comparison of the DU hydride temperature and H/U atomic ratio evolution curves for the three cases. It can clearly be seen that the copper-fin design leads to a higher temperature peak during the DU hydriding process than does the copper-foam
S. Yun et al. / Energy 193 (2020) 116742
7
Fig. 4. Comparison of simulated and measured (a) DU hydride temperature and (b) average H/M atomic ratio and inlet pressure evolution profiles during the DU hydriding process. The experimental values were obtained from Kang et al. [19].
design. The lower porosity copper-fin design (ε ¼ 0.812, DU ¼ 5.26 kg) produces a higher rise in temperature because the denser DU hydride powder results in the greater release of volumetric heat during the exothermic hydrogen absorption process. As presented in Fig. 6b, more rapid DU hydriding is achieved using the lower porosity copper-fin design (ε ¼ 0:812), indicating that the
greater rise in temperature at ε ¼ 0:812 is beneficial for DU hydriding performance. When the two cases with the same porosity (ε ¼ 0:895) are compared, both the copper-fin and copperfoam designs exhibited a similar DU hydriding performance, although the DU hydride temperature profiles in Fig. 6a differ significantly. This suggests that there is a tradeoff between the DU
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S. Yun et al. / Energy 193 (2020) 116742
Fig. 6. Comparison of (a) DU hydride temperature and (b) H/U ratio evolution profiles for the copper-foam design and the two copper-fin designs with different porosities (0.895 and 0.812).
Fig. 5. Three-dimensional contours of (a) DU hydride temperature and (b) H/U atomic ratio during DU hydriding process at 50 s, 100 s, 500 s, 1000 s, and 2000 s.
hydride kinetics and the equilibrium pressure as the DU hydride temperature increases. The rise in temperature increases both DU hydride kinetics and equilibrium pressure, with the former being beneficial to the DU hydriding reaction and the latter suppressing it. To analyze the effects of the rise in temperature on the DU hydriding performance in more detail, the temperature and pressure dependent terms for the DU hydriding rate equation Sm in Eq. (4) are plotted in Fig. 7a. The temperature dependent term is the Arrhenius expression of the DU hydriding kinetics, expð Ea =RTÞ; while the pressure dependent term is related to the equilibrium pressure, i.e. LnðP =PeqÞ. These individual terms are normalized to those of copper foam as follows:
S. Yun et al. / Energy 193 (2020) 116742
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negative effect of increasing the equilibrium pressure. As a result, Fig. 7b shows that larger source terms (Sm) in the two copper-fin designs are predicted due to their greater temperature rise. In contrast, when the DU hydride temperature in the copper-foam design becomes higher than that of the copper-fin design (after 4 min with ε ¼ 0:895 and after the 8 min with ε ¼ 0:812), the opposite trend is clearly observed (Fig. 7). After analyzing Figs. 6 and 7, it can be seen that the rise in DU temperature up to 350 C promotes the overall DU hydriding reaction despite the increase in the equilibrium pressure. Figs. 8 and 9 display detailed 3-D contours of the DU hydride temperature and H/U atomic ratio, respectively, for the two copperfin designs at five different elapsed times during hydrogen absorption. In the early stages when the DU hydriding rate is rapid, a noticeable temperature gradient between the copper fin and the interior is observed (Fig. 8). When Figs. 8 and 9 are compared, the hotter interior DU hydride region leads to slower DU hydriding at 50 s and 100 s, indicating that more effective cooling near the copper fin enhances the DU hydriding rate. In the latter stages, however, the DU hydriding rate near the copper fin slows down due to a reduction in the free DU surface for hydrogen absorption and thus the DU hydride temperature there cools due to less generation of exothermic reaction heat. Therefore, the local DU hydride temperature and H/U gradients between the copper fin and interior DU regions are lower at 1000 s and 2000 s. Detailed analysis of the 3-D contours in Figs. 5, 8 and 9 indicate that the copper foam and copper fin DU bed designs exhibit different behaviors in terms of local DU temperature and H/U distributions. Unlike the copperfoam design in Fig. 5, the hotter DU hydride region in the copperfin-based DU beds corresponds to a lower H/U region.
4. Conclusion
Fig. 7. Comparison of the calculated ratio of the copper-fin designs with different porosities (0.985 and 0.812) to the copper-foam design: (a) the Arrhenius equation (exp(-Ea/RT)) and pressure term (Ln(P/Peq)) and (b) the mass source.
Ea exp RT fin RatioT ¼ Ea exp RT
(22)
foam
ln PPeq fin RatioP ¼ ln PPeq
(23)
foam
As seen in Fig. 6a, the copper-fin designs with ε ¼ 0:895 and 0.812 have a higher rise in DU temperature than does the copperfoam design for the first 4 min and for the first 8 min, respectively. During the initial DU hydriding time, the increase in RatioT is greater than the decrease in RatioP for both porosity levels, which indicates that, as the DU hydride temperature increases, the positive effect of improving the DU hydriding kinetics dominates the
In this study, large-scale transient DU hydriding simulations of full-size DU beds were successfully carried out to investigate key DU hydriding characteristics, including a detailed analysis of the local DU hydride temperature and H/U atomic ratio distributions within the DU bed geometries. Major conclusions can be summarized as follows: 1. The DU hydride model was successfully validated against experimental DU hydriding data measured from a copper foambased DU bed loaded with 1.86 kg of DU. In general, good agreement between the simulation results and experimental data was achieved in terms of DU hydride temperature and H/U atomic ratio evolution curves. However, the agreement became weaker at the early DU hydriding stages when the DU hydriding rate was fast, which indicates that the present model should be improved to more accurately simulate fast DU hydriding process. 2. As the DU hydride temperature increased during hydrogen absorption, there was a tradeoff between the absorption kinetics and the equilibrium pressure: the kinetics improved the DU hydriding rate whereas the rise in equilibrium pressure with temperature limited it. The detailed simulation results indicate that the rise in DU temperature up to 350 C favors DU hydriding, demonstrating that the effect of increasing kinetics dominates the effect of increasing equilibrium pressure. It is worth noting that the above trend only occurs in the copperfoam-based DU bed design. In the copper-fin-based DU bed designs, the higher rise in DU hydride temperature resulted in the slower DU hydride rate. Therefore, DU hydriding characteristics and performance are highly influenced by the effective
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Fig. 8. Three-dimensional DU hydride temperature contours for the copper-fin designs with different porosities (0.895 and 0.812) during the DU hydriding process at 50 s, 100 s, 500 s, 1000 s, and 2000 s.
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Fig. 9. Contours for the change in the three-dimensional H/U atomic ratio for the copper-fin designs with different porosities (0.895 and 0.812) during the DU hydriding process at 50 s, 100 s, 500 s, 1000 s, 2000 s.
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thermal conductivity of the DU hydride region along with the local DU hydride temperature and equilibrium pressure. 3. The higher DU loading and resultant lower DU hydride porosity induced the faster hydrogen absorption performance in the copper-fin-based DU bed, which is mainly owing to the higher effective thermal conductivity of the DU hydride region. Therefore, the high degree of compaction of DU powders is found to be significant design parameter for copper-fin-based DU bed design. Acknowledgements This study was supported by the Technology Innovation Program of Korea Evaluation Institute of Industrial Technology (KEIT) funded by the Ministry of Trade, Industry & Energy (MOTIE) (10052823) as well as by the Korea Institute of Energy Technology Evaluation and Planning (KETEP) grant funded by the Korea government (MOTIE) (20183030032010). We also thank Taesung S&E, Inc. Korea for lending technical support for the ANSYS Fluent software. Nomenclature A Ca Cp DU Ea EXV H/U DH h K k M P R S SUS T t ! u V X
correlation factor hydriding rate constant, s1 specific heat, kJ∙kg1∙K1 depleted uranium activation energy, kJ∙kmol1 expansion volume hydrogen to metal atomic ratio reaction heat of formation, J∙kg1 convection heat transfer coefficient, W∙m2∙K1 permeability, m2 thermal conductivity, W∙m1∙K1 molecular weight, kg∙kmol1 pressure, bar universal gas constant, 8.314 J∙mol1 K1 source term stainless steel temperature, K time, s velocity vector, m∙s1 volume, m3 reaction fraction
Greek symbols ε porosity m dynamic viscosity, kg∙m1∙s1 r density, kg∙m3 t stress tensor Superscripts eff effective value emp empty eq equilibrium g gas phase s solid phase sat saturation value Subscripts c cu Eq H2
cooling copper equilibrium hydrogen
in U UH m cf par ref res sat ser T u 0
inlet depleted uranium metal depleted uranium metal hydride mass equation copper foam parallel reference value reservoir saturation value series energy equation momentum equation initial value
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