Additive Property of Separative Power

A separative power of a separating element, whose heads and tails separation factors are α and β is expressed by φ _b (α, β)= [α(β-1)1n α-(alpha;-1) in β]/(α β-1) for the unit flow of the desired material and φ _a (α, β) (≡ φ _b (β, α) for that of undesired material. The additive properties of the functions φ _b and (φ _a were demonstrated by calculations of various types of ideal cascades, but the origin of the property is not obvious. The present study has furnished the mathematical basis of the additivity based on the special functional equation. First, for symmetric processes (α, β), the functional equation which describes the function representing the quality of separation f(α, α) concerning the desired material was obtained and solved to give the functional form of f(α, α,). The result was extented to the function f(α, β) representing the quality of asymmetric separation (α, ≠ β). The derived function f(α, β) was demonstrated to be equal to φb(α, β)and it was verified that functions φb(α, β) and φ _a (α, β) have the additive property in themselves.     

Sodium Vapor Deposition,on Simulated Model of LMFBR Rotating Shield Plug Annulus, (I)

Experimental study was made to investigate the controlling factors on the vapor deposition rate on reactor operational shield plug annulus, which is exposed to the vapor entrained cover gas during reactor operation. Two simulated test assemblies having annuli were made for this purpose and were installed into a small test vessel. In the experiment, the average deposition rates on the annular walls of the test assemblies were measured for various pool temperatures, and their dependents upon such parameters as pool temperature, T_s (or the saturated vapor pressure P_s at pool surface), cover gas pressure P_g , and temperature drop ΔT_a across cover gas, were studied.

The results revealed that the dominant controlling factor was the vapor pressure P_s at pool surface. Dependent of the average deposition rate φbar_d. upon the above parameters was simply expressed by: φbar_d=Bσp_sD_sΔT_G , where, σ_s is the saturated vapor concentration at pool surface, D_s , the vapor diffusion coefficient, and B the proportional constant.

To these experimental results, the previously published evaporation rate data and the theoretical evaporation rate equation based on Epstein & Rosner's theory were reviewed. Then correlation between the deposition and the evaporation rates was discussed.     

Applications of Artificial Neural Network for the Prediction of Flow Boiling Curves

An artificial neural network (ANN) was applied successfully to predict flow boiling curves. The databases used in the analysis are from the 1960's, including 1,305 data points which cover these parameter ranges: pressure P=100–1,000 kPa, mass flow rate G=40–500 kg/m²-s, inlet subcooling ΔT_sub =0–35°C, wall superheat ΔT_w = 10–300°C and heat flux Q=20–8,000kW/m². The proposed methodology allows us to achieve accurate results, thus it is suitable for the processing of the boiling curve data. The effects of the main parameters on flow boiling curves were analyzed using the ANN. The heat flux increases with increasing inlet subcooling for all heat transfer modes. Mass flow rate has no significant effects on nucleate boiling curves. The transition boiling and film boiling heat fluxes will increase with an increase in the mass flow rate. Pressure plays a predominant role and improves heat transfer in all boiling regions except the film boiling region. There are slight differences between the steady and the transient boiling curves in all boiling regions except the nucleate region. The transient boiling curve lies below the corresponding steady boiling curve.     

Analysis of Two Phase Mass Transfer Kinetics by Logarithmic Driving Force Based on Chemical Thermodynamics

Interphase transfer kinetics of neodymium and nitric acid was studied using a single drop column with recycling organic phase: via an external mixing vessel in H₂O-HNO₃/NaNO₃-Nd(NO_3-) tri-n-butylphosphate system. Experimental data have been analyzed by two new concepts for driving forces for transport: synthesized linear and logarithmic forms. The former is defined as geometrical-mean driving force, and the latter is the logarithm of the product of reciprocals of concentration ratios: x/x^e and: y/y^e against equilibrium states in each phase,: i.e. In {(x^e-y^e)/(x-y)}.

By applying thermodynamic logarithmic form of driving force along reaction coordinate, the net transfer fluxes of neodymium and nitric acid have been represented by chemical affinity under high ionic strengths over a wide range of solvent loading as flux=flux/(1—exp(—A/RT)).     

Some problems for bundle CHF prediction based on CHF measurements in simple flow geometries

A comparison of critical heat flux (CHF) fuel bundles data with CHF data obtained in simple flow geometries was made. The base for the comparison was primary experimental data obtained in annular, circular, rectangular, triangular, and dumb-bell shaped channels cooled with water and R-134a. The investigated range of flow parameters (pressure, mass flux, and critical quality) in R-134a was chosen to be equivalent to modern nuclear reactor water flow conditions (p=7 and 10 MPa, G=350–5000 kg (m² s)⁻¹, x_cr=−0.1–1). The proper scaling laws were applied to convert the data from water to R-134a equivalent conditions and vise versa. The effects of flow parameters (p, G, x_cr) and the effects of geometric parameters (D, L) were evaluated during comparison. The comparison showed that no one simple flow geometry can be used for accurate and reliable bundle CHF prediction in wide range of flow parameters based on local (critical) conditions approach. The comparison also showed that the limiting critical quality phenomenon is unique characteristic for each flow geometry which depends on many factors: flow conditions (pressure and mass flux), geometrical parameters (diameter or surface curvature, gap size, etc.), flow obstructions (spacers, appendages, turbulizers, etc.) and others.     

Combined Forced and Free Convective Heat Transfer Characteristics in a Narrow Vertical Rectangular Channel with 2.5 mm in Gap Heated from Both Sides

Heat transfer characteristics of mainly combined forced and free convective flow in a vertical rectangular flow channel with a gap of 2 5 mm, which was quite narrow compared with those investigated in previous experiments, were studied experimentally for water. As a result, the following heat transfer characteristics were made clear, using a non-dimensional parameter Gr _x /Re ²¹ _x /⁸ Pr ^1/2 similarly to the case for the 18 mm gap which was already reported by the authors.

(1) When the Gr _x /Re ²¹ _x /⁸ Pr ^1/2 is less than 10^?4, both upward flow and downward flow show the nature of forced convective heat transfer.

(2) When the Gr _x /Re ²¹ _x /⁸ Pr ^1/2 is between 10^?4 and 10^?2, heat transfer coefficients for both upward flow and downward flow are higher than any of those predicted by the previous correlations for turbulent forced convection along a flat plate and turbulent free convection along a vertical flat plate. This is, differently from the case of 18 mm gap, due to the effect of the acceleration of main flow induced by the development of the boundary layer along the channel.

(3) When the Gr _x /Re ²¹ _x /⁸ Pr ^1/2 is larger than 10^?2, the upward flow shows the nature of free convective flow even with the gap as narrow as 2.5 mm in the vertical rectangular flow channel. Heat transfer correlations which have been developed for the 18 mm gap channel, are also available for the described-above regions of 2.5 mm gap channel.     

Selectivity of Alkali Metal Ion and Lithium Isotopes on Ion Exchangers in H Form Prepared from LiTi x Zr2-x (PO4)3 (x=0, 1)

Inorganic metal (IV) phosphate-based ion exchanges in the hydrogen form, HTi _x Zr_2-x (PO₄)₃ (x =0, 1), have been prepared by leaching lithium ions from the precursors, LTi _x Zr_2-x (PO₄)₃ (x=0, 1), through ion exchange with protons. The degrees of leaching of lithium ion were more than 99%. Both ion exchangers showed high selectivity toward lithium and sodium ions and little affinity to rubidium and cesium ions among alkali metal ions. The lithium ion uptakes from aqueous solutions were monotonously increasing functions of pH. Isotopically, both ion exchangers were ⁶Li-specific like other inorganic ion exchangers so far examined. The ⁷Li-to-⁶Li isotopic single-stage separation factor, S, of HTiZr(PO₄)₃ was larger than that of HZr₂(PO₄)₃ at a given pH-value. The relatively large S-values of 1.022 and 1.042 were found for HZr₂ (PO₄)₃ for HTiZr(PO₄)₃, respectively, at 25°C when the pH of the solution phase was around 5.     

New Phase Transitions in Non-Stoichiometric U3O8-x at High Temperatures, (II)

Electrical conductivity and X-ray diffraction studies on non-stoichiometric U₃O_8-x phase were carried out simultaneously in the range 765°≦T≦995°C and 10^?4≦Po₂≦1 atm. The plot of logσ vs. logPo₂ showed many refractions which corresponded with the phase transitions determined by thermogravimetry reported in the preceding paper. Based on the data of both electrical conductivity and thermogravimetry, the non-stoichiometric defect structures of various U₃O_8-x phases are interpreted as consisting of singly charged oxygen interstitials (O_l′) and doubly charged oxygen vacancies (Vo^.)? Some of the X-ray diffraction lines were found to undergo splitting with decreasing oxygen partial pressure. These splits are qualitatively discussed in reference to the out-of-step structure model. The mechanism of electrical conduction in the high temperature hexagonal U₃O_8-x phases is surmised to be the hopping of small polarons.     

Subcritical multiplication factor and source efficiency in accelerator-driven system

The subcritical multiplication factor k_s   and the external neutron source efficiency φ^∗${\phi }^{\ast }$ are important parameters in the accelerator-driven system (ADS) performance assessment. The theoretical relation between k_s and the effective multiplication factor k_eff in a subcritical system is discussed in different cases of subcritical system. On the basis of the theoretical background, the dependence of k_s   and φ^∗${\phi }^{\ast }$ on subcriticality, source position, and energy is numerically investigated using a simple thermal subcritical model. For the sake of experimental evaluation of k_s   and φ^∗${\phi }^{\ast }$, the ADS experiments have been carried out in the subcritical systems combined with 14 MeV pulsed neutrons of the Kyoto University Critical Assembly (KUCA). The k_s   and φ^∗${\phi }^{\ast }$ parameters are successfully measured by utilizing the reaction rate distribution obtained by the optical fiber detectors in the subcritical system, within a relative difference of less than 7% and 12% for k_s   and φ^∗${\phi }^{\ast }$, respectively, between measured and calculated values for most studied cases.     

Thermodynamic properties of ThxU1−xO2 (0 < x < 1) based on quantum-mechanical calculations and Monte-Carlo simulations

Th_xU_1−xO_2+y binary compositions occur in nature, uranothorianite, and as a mixed oxide nuclear fuel. As a nuclear fuel, important properties, such as the melting point, thermal conductivity, and the thermal expansion coefficient change as a function of composition. Additionally, for direct disposal of Th_xU_1−xO₂, the chemical durability changes as a function of composition, with the dissolution rate decreasing with increasing thoria content. UO₂ and ThO₂ have the same isometric structure, and the ionic radii of 8-fold coordinated U⁴⁺ and Th⁴⁺ are similar (1.14 nm and 1.19 nm, respectively). Thus, this binary is expected to form a complete solid solution. However, atomic-scale measurements or simulations of cation ordering and the associated thermodynamic properties of the Th_xU_1−xO₂ system have yet to be determined. A combination of density-functional theory, Monte-Carlo methods, and thermodynamic integration are used to calculate thermodynamic properties of the Th_xU_1−xO₂ binary (ΔH_mix, ΔG_mix, ΔS_mix, phase diagram). The Gibbs free energy of mixing (ΔG_mix) shows a miscibility gap at equilibration temperatures below 1000 K (e.g., E_exsoln = 0.13 kJ/(mol cations) at 750 K). Such a miscibility gap may indicate possible exsolution (i.e., phase separation upon cooling). A unique approach to evaluate the likelihood and kinetics of forming interfaces between U-rich and Th-rich has been chosen that compares the energy gain of forming separate phases with estimated energy losses of forming necessary interfaces. The result of such an approach is that the thermodynamic gain of phase separation does not overcome the increase in interface energy between exsolution lamellae for thin exsolution lamellae (10 Å). Lamella formation becomes energetically favorable with a reduction of the interface area and, thus, an increase in lamella thickness to >45 Å. However, this increase in lamellae thickness may be diffusion limited. Monte-Carlo simulations converge to an exsolved structure [lamellae || ] only for very low equilibration temperatures (below room temperature). In addition to the weak tendency to exsolve, there is an ordered arrangement of Th and U in the solid solution [alternating U and Th layers || {1 0 0}] that is energetically favored for the homogeneously mixed 50% Th configurations. Still, this tendency to order is so weak that ordering is seldom reached due to kinetic hindrances. The configurational entropy of mixing (ΔS_mix) is approximately equal to the point entropy at all temperatures, indicating that the system is not ordered.     

Heat capacity and electrical conductivity of non-stoichiometric U4O9-y from 300 to 1200 K

The heat capacity and the electrical conductivity of non-stoichiometric U₄O_9-y with various compositions were measured simultaneously by direct heating pulse calorimetry from 300 to 1200 K. As well as the heat capacity anomaly due to the α-β transition around 350 K, two small heat capacity anomalies due to the β-γ transition were observed around 1000 and 1100 K, which are superimposed on a monotonie increase in the heat capacity above 800 K, presumably due to the onset of the γ-U₄O_9-y-UO_2+x transition. The change in the slope of the electrical conductivity curve as also observed at the phase transitions. The excess entropy due to the overall transition from α-U₄O₉to UO_2+x was evaluated to be 5.95 J K^-1mol^-1, which is in agreement with the value calculated on the assumption that the excess entropy consists of the contribution of the electronic disordering of U⁴⁺ and U⁵⁺ ions and that of the atomic disordering of oxygen atoms.     

Analysis of Core Physics Experiments on Fresh and Irradiated PWR UO2 Fuels in the REBUS Program

Critical experiments of two cores each loaded with fresh 5 × 5 test PWR-type fuel rods of ²³⁵U enrichment of 3.8 wt% or irradiated 5 × 5 test rods of rod average burnup of 55 GWd/t in the REBUS program were analyzed using diffusion, transport, and continuous-energy Monte Carlo calculation codes coupled with nuclear data libraries based on JENDL-3.2 and JENDL-3.3. Biases in effective multiplication factors k _eff's of the critical cores were about ?1:2%Δk for the diffusion calculations (JENDL-3.2), ?0:5%Δk for the transport calculations (JENDL-3.3), and ?0:5 and 0.1%Δk for the Monte Carlo calculations (JENDL-3.3 and JENDL-3.2, respectively). The measured core fission rate and Sc- or Co-activation rate distributions were generally well reproduced using the three types of calculation. The burnup reactivity determined using the measured water level reactivity coefficients was ?2:35 ± 0:07Δk/kk′. The calculated result of the Monte Carlo calculations agreed with it; however, the diffusion and transport calculations overestimated the absolute value by about 7%, which would be mainly attributed to the errors in the calculation of the reactivity caused by changing the fuel compositions from fresh fuel to irradiated fuel.     

Residual stress behavior in fast neutron irradiated SA AISI 304l stainless steel cylindrical tubing

Residual stress measurements were made on solution-annealed (SA) AISI 304L stainless steel (SS) irradiated in EBR-II over a temperature range from 402 to 524°C by axially slitting short sections of tubing. The data were analyzed by using SA AISI 304 SS physical properties and SA AISI 304L SS swelling and irradiation creep empirical equations to calculate the slit width change (δ) versus fluence (φt) curve. At temperatures equal to and above 445°C, δ versus φt calculations indicate that the stress effect on swelling is sufficiently large to reduce the swelling rate temperature gradient, and consequently the on-power stress gradient, to zero. This behavior is confirmed by void volume gradient measurements. At lower temperatures, δ versus φt calculations indicate that stress affected swelling is smaller and does not relax the swelling rate temperature gradient. Void volume gradient measurements confirm the presence of a swelling gradient. Calculations of the δ versus φt curve were made with four different empirical swelling equation fluence dependencies, and the best agreement with the δ versus φt data was obtained with a power form type swelling equation. The equations fit the immersion density data ($\text{ΔV}\text{V}{}_0\text{versus}\text{φt}$) within experimental scatter, but predict significantly different δ versus φt behavior. These results show that the slit tube results are very sensitive to the empirical swelling equation form.     

Generalized Synthetic Kernel Approximation for Elastic Moderation of Fast Neutrons

A method of synthetic kernel approximation is examined in some detail with a view to simplifying the treatment of the elastic moderation of fast neutrons. A sequence of unified kernel {f _N } is introduced, which is then divided into two subsequences {W _n } and {G_n} according to whether N is odd (W _n =f _2n-1, n=1, 2,…) or even (G _n =f _2n , n=0,1, …). The W ₁ and G ₁ kernels correspond to the usual Wigner and GG kernels, respectively, and the W _n and G _n kernels for n ≧ represent generalizations thereof. It is shown that the W _n kernel solution with a relatively small n (≧2) is superior on the whole to the G_n kernel solution for the same index n, while both converge to the exact values with increasing n. To evaluate the collision density numerically and rapidly, a simple recurrence formula is derived. In the asymptotic region (except near resonances), this recurrence formula allows calculation with a relatively coarse mesh width whenever h_a ≧0.05 at least. For calculations in the transient lethargy region, a mesh width of order ?/10 is small enough to evaluate the approximate collision density φ _N with an accuracy comparable to that obtained analytically. It is shown that, with the present method, an order of approximation of about n=7 should yield a practically correct solution diviating not more than 1% in collision density.     

Effects of fluid properties on CCFL characteristics at a vertical pipe lower end

The purpose of this study is to derive a counter-current flow limitation (CCFL) correlation and evaluate its uncertainty for steam generator (SG) U-tubes in a pressurized water reactor (PWR). Experiments were conducted to evaluate effects of the liquid viscosity on CCFL characteristics using air–40 wt% or air–60 wt% glycerol water solution and saturated steam–water at atmospheric pressure with vertical pipes simulating the lower part of the SG U-tubes. The steam–water experiments confirmed that CCFL characteristics could be expressed in terms of the Wallis parameters (J_G* and J_L*) for the pipe diameters of D = 14, 20, and 27 mm. A CCFL correlation was derived using the ratio μ_G/μ_L of the viscosities of the gas and liquid phases, μ_G and μ_L, as a correction term representing effects of fluid properties, where J_G^*1/2(μ_G/μ_L)^?0.07 was expressed by a cubic function of J_L^*1/2(μ_G/μ_L)^0.1. In the correlation, the constant C indicating the value of J_G^*1/2(μ_G/μ_L)^?0.07 at J_L* = 0 was (1.04 ± 0.05), and this uncertainty of ±0.05 would cover most of the previous experimental data including the ROSA-IV/LSTF data at 1, 3, and 7 MPa.     

Effect of Resonance Scattering of Sodium on Resonance Absorption of U-238

The interference effects of the resonance scattering of Na on the resonance absorption of ²³⁸U were investigated. The results for self-shielding factor obtained from exact treatment were compared with those from the usual conventional method, as well as with results based on the assumption that φ(u)σ_t=const. (φ(u): flux, σ_t: total cross section).

In the exact treatment, the neutron spectrum used is obtained with accurate treatment of the slowing down by the elastic scattering of light and medium elements. In the φ(u)σ_t = const, assumption, the energy dependence of the Na resonance scattering cross section is taken into account as in the exact treatment, whereas only the averaged value is used in the conventional method.

Self-shielding factors and their temperature dependence were calculated for several compositions and energy regions. It was found that the conventional method is not satisfactory, while the assumption that φ (u)σ_t= const, compares well with the exact treatment.     

Optimum Cut of Separating Element as Function of Total Separation Factor and Mole Fraction of Component to be Separated

For a simplified model of separating elements where a total separation factor αβ is independent of values of cut ? (0≤?≤1), an optimum cut ?_opt in the sense that the cut makes a separative power δU maximum, was derived in terms of αβ, and a mole fraction x_F (0≤x _F≤1) of the component to be separated. When values of x^F is nearly equal to zero, the optimum cut ?^xF _opt?0 decreases and approaches to near 0, as the total separation factor becomes larger. On the contrary, when x_F is nearly equal to 1, the optimum cut ?^xF _opt?1 was found from calculation to be 1-?^xF _opt?0, and increases and approaches to near 1, as the total separation factor becomes larger. Moreover, in the case of x_F =0.5, the optimum cut is 0.5 regardless of αβ. Generally, the optimum cut ?_opt(x_F ) was solved to be in the form of a linear interpolation of the boundary values, ?x^F _opt?0 and ?x^F _opt?1.     

Natural convection heat transfer from horizontal rod bundles in liquid sodium. Part 2: Correlations for horizontal rod bundles based on theoretical results

Natural convection heat transfer from horizontal rod bundles in N_xm × N_ym arrays (N_xm, N_ym = 5–9) in liquid sodium was numerically analyzed for three types of the bundle geometry (in-line rows, staggered rows I and II). The unsteady laminar two-dimensional basic equations for natural convection heat transfer caused by a step heat flux were numerically solved until the solution reaches a steady state. The PHOENICS code was used for the calculation considering the temperature dependence of thermophysical properties concerned. The surface heat fluxes for each cylinder were equally given for a modified Rayleigh number, R_f, ranging from 0.0637 to 63.1 (q = 1×10⁴ to 7×10⁶ W/m²). S_x/D and S_y/D for the rod bundle, which are the ratios of the distance between center axes on the abscissa and the ordinate to the rod diameter, respectively, were ranged from 1.6 to 2.5 on each bundle geometry. The spatial distribution of Nusselt numbers, Nu, on horizontal rods of a bundle was clarified. The average value of Nusselt number, Nu_av, for three types of bundle geometry with various values of S_x/D and S_y/D were calculated to examine the effect of the array size, S/D and R_f on heat transfer. The bundle geometry for the higher Nu_av value under the condition of S_x/D×S_y/D = 4 was examined by changing the ratio of S_x/S_y. A correlation for Nu_av for the three types of bundle geometry above mentioned including the effects of S_x/D and S_y/D was developed. The correlation can describe the theoretical values of Nu_av for the three types of bundle geometry in N_xm × N_ym arrays (N_xm, N_ym = 5–9) for S_x/D and S_y/D ranging from 1.6 to 2.5 within 10% difference.     

Condensation experiments for counter-current flow limitation in an inverted U-tube

In this study, we measured counter-current flow limitation (CCFL) characteristics in an inverted U-tube (18.4 mm diameter and 1.0 m straight-part length) simulating steam generator (SG) U-tubes under conditions of steam condensation at pressures of 0.1–0.14 MPa. Differential pressure ΔP between the top of the inverted U-tube and the lower tank was measured, and the flow patterns wave estimated by comparing the waveforms of ΔP with those in air–water experiments. As a result, we classified the flow patterns under CCFL conditions into CCFL-P, CCFL-L and CCFL-T. The falling water flow rate under CCFL conditions slightly increased as the pressure increased and the cooling water temperature decreased (subcooling of cooling water increased). In the case of CCFL-L, CCFL characteristics in the inverted U-tube were between those in air–water and saturated steam–water experiments at 0.1 MPa. Furthermore, we derived a Wallis type CCFL correlation and its uncertainty from CCFL data, including previously measured data, i.e., J*^1/2_G + 0.88J_L*^1/2 = 0.76 ± 0.05.