Constraint effects on fracture toughness of impact-loaded, precracked Charpy specimens

Impact-loaded, precracked Charpy specimens often play a crucial role in irradiation surveillance programs for nuclear power plants. However, the small specimen size B = W = 10 mm limits the maximum value of cleavage fracture toughness J_c that can be measured under elastic—plastic conditions without loss of crack tip constraint. In this investigation, plane strain impact analyses provide detailed resolution of crack tip fields for impact-loaded specimens. Crack tip stress fields are characterized in terms of J—Q trajectories and the toughness-scaling model which is applicable for a cleavage fracture mechanism. Results of the analyses suggest deformation limits at fracture in the form of b > MJ_c/σ₀, where M approaches 25–30 for a strongly rate-sensitive material at impact velocities of 3–6 m s⁻¹. Based on direct comparison of the static and dynamic J values computed using a domain integral formulation, a new proposal emerges for the transition time, the time after impact at which interial effects diminish sufficiently for simple evaluation of J using the plastic η factor approach.     

Part-through cracks in pipes under cyclic bending

The propagation of a circumferential external surface crack in a metallic round pipe under cyclic bending loading is examined through a two-parameter theoretical model. A finite element analysis is carried out to determine the stress-intensity factor distribution along the front of the flaw, which is assumed to present an elliptical-arc shape with aspect ratio α=a_el/b_el (a_el, b_el=ellipse semi-axes). The relative depth ξ of the deepest point on the front is equal to the ratio between the maximum crack depth, a, and the pipe wall thickness, t. The parameter R/t, with R=internal radius of the pipe, is made to vary from 1 to 10. The fatigue growth of the surface flaw occurs according to preferred paths in the diagram of α against ξ.     

Hydrogen Concentration Dependence of Tritium Tracer Diffusion Coefficient in Alpha Phase of Niobium

In order to examine influences of coexistent hydrogen isotopes on diffusion behavior of tritium in niobium, tracer diffusion coefficients D_t of tritium in alpha phase of hydrogenated and deuterized niobium (α-NbH_xT_y and α-NbD_xT_y (x<0,8,y<<x)) have been measured at 473 K, 493 K and 553 K. The data on D_t show typical hydrogen concentration dependence: D_t of tritium for both α-NbH_xT_y, and α-NbD_xT_y, decreases with hydrogen concentration under all experimental conditions. The obtained concentration dependence of D_t of tritium differs from that of D_t of protium in α-NbH_x or of deuterium in α-NbH_xT_y. On the other hand, no appreciable differences in the concentration dependence of D_t of tritium between α-NbH_xT_y, and α-NbD_xT_y, are observed: there are no definite isotope effects due to the coexisting hydrogen isotopes. This result suggests that D_t of tritium for a tritiated niobium (α-NbT_x) is not very different from that for α-NbH_xT_y and α-NbD_xT_y. The chemical diffusion coefficient D* of tritium is also evaluated on the basis of the obtained D_t of tritium and of a literature value of a thermodynamic factor F for Nb-H and Nb-D systems.     

Break up time of fragmentating-solidifying UO2 spheres when quenched in sodium

When molten UO₂ is quenched in sodium, a sand-like debris results containing about 80% of fractured particles and 20% of smooth particles and spheres. The production of the fractured particles is normally explained by the thermal stress fragmentation model. Previously brittle fracture mechanics was applied to the complete solid shell of a freezing UO₂ drop, i.e. where 954°C < T < 2850°C; a calculation of fragmentation time was not possible. In this contribution the solid shell is continuously subdivided in a plastic or ductile layer for 1300°C < T < 2850°C and a brittle one for 945°C < T < 1300°C. Cracking occurs in the brittle layer only. In the present model a layer of a predescribed depth is assumed to ablate instantaneously, when the temperature reaches the transition point of elastic of ductile behavior (T = 1300°C) at its inner boundary. A new layer is formed within a time step, governed by the heat conduction equation. The discontinuous ablation process is thus related to the continuous progression of the solidification front. A calculation of the fragmentation time is possible: in principle it comprehends the summation of a large number of time steps for the formation of brittle layers. The thickness of the cracked brittle layer is parametrized to 20, 10, 5 and 1 μm. The concept of instantaneous ablation was suggested by the experience that the violent boiling forces of sodium are very effective on the UO₂ surface. The introduction of these minor changes makes the thermal stress model more realistic, because it can explain now, why UO₂ does not fragment in argon and water. The fragmentation time assessed for a UO₂ drop of 7.2 mm diameter in sodium, brittle layer 10 μm, is 250 ms.     

Temperature distribution in nuclear fuel rod and variation of the neutronic performance parameters in (D–T) driven hybrid reactor system

Temperature distribution in nuclear fuel rod and variation of the neutronic performance parameters are investigated for different coolants under various first wall loads (P_w=2, 5, 7, 8, 9, and 10 MW m⁻²) in (D, T) (deuterium and tritium) driven and fueled with UO₂ hybrid reactors. Plasma chamber dimension, DR, with a line fusion neutron source is 300 cm. The fissile fuel zone is considered to be cooled with four different coolants with various volume fractions, the volumetric ratio of coolant-to-fuel [(Vm/V_f) = 1:2, 1:1, and 2:1], gas (He, CO₂), flibe (Li₂BeF₄), natural lithium (Li), and eutectic lithium (Li₁₇Pb₈₃). Calculation in the fuel rods and the behavior of the fissile fuel have been observed during 4 years for discrete time intervals of Δt=15 days and by a plant factor (PF) of 75%. As a result of the calculation, cumulative fissile fuel enrichment (CFFE) value indicating rejuvenation performance has increased by increasing P_w for all coolants and . Although CFFE and neutronic performance parameter values increase to the higher values by increasing P_w, the maximum temperature in the centerline of the fuel roads has exceeded the melting point (T_m>2830°C) of the fuel material during the operation periods. However, the best CFFE (11.154%) is obtained in gas coolant blanket for =1:2 (29.462% coolant, 58.924% fuel, 11.614% clad), under 10 MW m⁻² first wall load, followed by flibe with CFFE=11.081% for =2:1 (62.557% coolant, 31.278% fuel, 6.165% clad), under 7 MW m⁻², and flibe with CFFE=9.995% for =1:1 (45.515% coolant, 45.515% fuel, 8.971% clad), under 7 MW m⁻² during operation period without reaching the melting point of the fuel material. While maximum CFFE value has been obtained in fuel rod row#10 in gas, natural lithium, and eutectic lithium coolant blankets, it has been obtained in fuel rod row#1 in flibe coolant blanket for all and P_w. At the same condition, the best neutronic performance parameter values, tritium breeding ratio (TBR)= 1.4454, energy multiplication factor (M)= 9.2018, and neutron leakage (L)= 0.0872, have been obtained in eutectic lithium coolant blankets for the =1:2, followed by gas, natural lithium, and flibe coolant blankets. The isotopic percentage of ²⁴⁰Pu is higher than 5% in all blankets for P_w 7 MW m⁻², so that plutonium component in all blankets can be never reach a nuclear weapon grade quality during the operation period.     

A faster procedure for the calculation of the J(ξ, β) function

One of the biggest difficulties in obtaining an analytical expression for the J(ξ, β) function is its explicit dependence on the Doppler broadening function ψ(x,ξ). The objective of this paper is to present a method for the fast and accurate calculation for the J(ξ, β) function based on the recent advances in the calculation of the Doppler broadening function and on a systematic analysis of its integrand. The methodology proposed uses an analytical formulation for the calculation of ψ(x, ξ) and a representation in series for error functions with complex argument. The results were satisfactory from the accuracy and processing time standpoint and are an option to other calculation methods found in the literature.     

An experimental investigation on thermal striping: Mixing phenomena of a vertical non-buoyant jet with two adjacent buoyant jets as measured by ultrasound Doppler velocimetry

An experimental investigation on the thermal mixing phenomena of three quasi-planar vertical jets, with the central jet at a lower relative temperature than the two adjacent jets, was conducted. The central jet was unheated (‘cold’), while the two adjacent jets were heated (‘hot’). The temperature difference and velocity ratio between the heated (h) and unheated (c) jets were, ΔT_hc=5°C, 10°C and r=V_cold,exit/V_hot,exit=1.0 (isovelocity), 0.7, 0.5 (non-isovelocity) respectively. The typical Reynolds number was Re_D=1.8×10⁴, where D is the hydraulic diameter of the exit nozzle. Velocity measurement of a reference single-jet and triple-jet arrangement were taken by ultrasound Doppler velocimetry (UDV) while temperature data were taken by a vertically traversed thermocouple array. Our UDV data revealed that, beyond the exit region, our single-jet data behaved in the classic manner. In contrast, the triple-jet exhibited, for example, up to 20 times the root-mean-square velocity values of the single-jet, especially in the regions in-between the cold and hot jets. In particular, for the isovelocity case (V_exit=0.5 m/s) with ΔT_hc=5°C, we found that the convective mixing predominantly takes place at axial distances, z/D=2.0–4.5, over a spanwise width, x/D|2.25|, centered about the cold jet. An estimate of the turbulent heat flux distribution semi-quantitatively substantiated our results. As for the non-isovelocity case, temperature data showed a localized asymmetry that subsequently delayed the onset of mixing. Convective mixing however, did occur and yielded higher post-mixing temperatures in comparison to the isovelocity case.     

Elastic–plastic analysis of an elbow with a through-wall axial crack at crown under in-plane bending

Piping elbows under bending moment are vulnerable to cracking at crown. The structural integrity assessment requires knowledge of the J-integral. The J-integral values for axially through-wall cracked thick elbows under in-plane bending moment are not available in open literature. This paper presents the J-integral results for 90°, long radius elbows subjected to in-plane opening bending moment based on a large number of finite element analyses covering a wide range of standard geometries. The non-linear elastic–plastic finite element analyses were performed using WARP3D software. Both geometric and material non-linearity were considered in the study. The geometry considered were for R_m/t = 5, 6, 7.5, 9, 12, 15, 20 and 25 with crack angles of 9°, 18°, 27° and 36° and strain hardening exponent, ‘n’ varied for 2, 3, 5, 7 and 10.     

Transformation and fragmentation behavior of molten metal drop in sodium pool

In order to clarify the fragmentation mechanism of a metallic alloy (U–Pu–Zr) fuel on liquid phase formed by metallurgical reactions (liquefaction temperature = 650 °C), which is important in evaluating the sequence of core disruptive accidents for metallic fuel fast reactors, a series of experiments was carried out using molten aluminum (melting point = 660 °C) and sodium mainly under the condition that the boiling of sodium does not occur. When the instantaneous contact interface temperature (T_i) between molten aluminum drop and sodium is lower than the boiling point of sodium (T_c,bp), the molten aluminum drop can be fragmented and the mass median diameter (D_m) of aluminum fragments becomes small with increasing T_i. When T_i is roughly equivalent to or higher than T_c,bp, the fragmentation of aluminum drop is promoted by thermal interaction caused by the boiling of sodium on the surface of the drop. Furthermore, even under the condition that the boiling of sodium does not occur and the solid crust is formed on the surface of the drop, it is confirmed from an analytical evaluation that the thermal fragmentation of molten aluminum drop with solid crust has a potential to be caused by the transient pressurization within the melt confined by the crust. These results indicate the possibility that the metallic alloy fuel on liquid phase formed by the metallurgical reactions can be fragmented without occurring the boiling of sodium on the surface of the melt.     

A performance comparison of lithium,helium, and flibe cooled tube/header fusion blankets

The objective of this study is to provide a comparison of thermal-hydraulic and structural performance of lithium, helium, and flibe cooled fusion blankets based on a tube/header geometry in a liquid lithium breeder. Type 316 stainless steel and TZM are considered as representative near-term and long-term, high temperature blanket structural materials, respectively, to show the potentials of each coolant. The flibe-TZM system has the best characteristics, while lithium-316SS, helium-316SS, and helium-TZM are comparable but definitely more limited in operating conditions. These results suggest that molten salt-refractory metal systems deserve more attention.Nomenclature a radial direction half-width of region cooled by single tube (m) - A A=_st/_cD - A _w first wall area (m²) - b azimuthal half-width of region cooled by single tube (m) - B magnetic field strength (T) - C _p specific heat of coolant (J/kg-°C) - C ₁ pumping power ratio - D _h ,D _t header and cooling tube diameter (m) - E _F energy deposited in the blanket region per fusion neutron, determined from neutronic calculations; 15.2 MeV used in this study - F _c allowance factor in pressure loss calculations for lithium system - h heat transfer coefficient (W/m²-°C) - Ha Hartmann number,Ha=BD _c /gm - J the ratio of percent change of first wall loading to percent change of a design parameter - K _c ,K _Li,K _s thermal conductivity of coolant, lithium, and structure (W/m-°C) - L major on-axis circumference of reactor (m) - M blanket energy multiplication factor,M=E _F /14.1 - n number of coolant tubes per header - N number of blanket modules (or headers) azimuthally - N _t total number of coolant tubes - Nu Nusselt number,Nu = hD_t/K_c - P coolant pressure (Pa) - P header and total pressure loss (Pa) - P _r Prandtl number - q _w first wall neutron energy loading (W/m²) - q average volumetric heat generation rate in the blanket (W/m³) - q(r) volumetric heat generation rate in blanket (W/m³) - r radial distance from first wall (m) - r _e radial position of the tube close to the hottest spot in the lithium pool - R gas constant - R _w first wall radius (m) - S defined by Eq. (25) - t _t ,t _h coolant tube and header tube thickness (m) - ¯T average coolant temperature (°C) - T _in inlet temperature (°C) - T _Li,max maximum lithium pool temperature (°C) - T _w,max maximum tube temperature (°C) - T _c coolant temperature rise across blanket (°C) - T _F film temperature rise (°C) - T _m temperature rise between coolant tube and maximum in pool (°C) - T _w wall temperature rise (°C) - U _h coolant velocity at header inlet for lithium system (m/s) - U _t coolant velocity in coolant tubes (m/s) - U _h ,max maximum inlet velocity for the lithium system, given by Eq. (13) - W _s surface heat flux in coolant tube (W/m²) - V _m voltage drop across the tube in flibe system (V) - V _t total blanket volume (m³) - X axial length of coolant tubes (m) - X _e entry and exit tube length in flibe system (m) - Z radial thickness of blanket (m) - _c , _s fraction of blanket volume occupied by coolant and structural material (exclusive of header region) - ratio of the minimum value ofq(r) to q, 0.4 - coolant viscosity (kg/m-s) - fiction coefficient - coolant density (kg/m³) - _t tube density (m^–3) - _c , _s electrical conductivity of coolant and structure (1/-m) - _h hoop stress (Pa) - _y structural material design yield stress limit (Pa)     

In-Pile Eddy Current Test on PWR Fuel Rod Failed by Pellet-Cladding Interaction

Zirconium oxide nodules formed on BWR fuel rods were characterized quantitatively and correlated statistically with the rod operational parameters. Cladding specimens were obtained from fuel rods irradiated in a commercial BWR. Their burnup and fast neutron fluence ranged 17~38 GWd/t and 4×10²⁵~8×10²⁵ n/m², respectively. Characteristic variables of the nodules such as maximum thickness T ^max (μm) were measured on metallographs of the cladding cross sections. These variables were correlated by multiple regression analyses with the operational parameters, such as irradiation time t (d), linear heat rate p (kW/m) and fast neutron flux ø (n/m²-s). For example, the maximum thickness depended on linear heat rate and showed a saturating tendency with burnup B (GWd/t) (T^max ∝ t0.8+0.5 p2.3±0.9 or T ^max ∝B0.8+0.4p1.5±0.5). This decrease of growth rate with irradiation time was interpreted in terms of a microstructure change of Zircaloy-2 during neutron irradiation. Results of transmission microscopy and energy dispersive X-ray spectroscopy indicated that the alloying elements such as Fe, Cr and Ni dissolved from intermetallic precipitates into the base metal during neutron irradiation. Dissolution of the alloying elements might be effective in decreasing the growth rate of nodules.     

Existence theory and simulations for one-dimensional radiative flows

We consider the coupling of the radiative heat transfer equations and the energy equation for the temperature T of a compressible fluid within the finite segment [0, L]. Using the technique of upper and lower sequences associated to integro-parabolic equations, we establish the existence and uniqueness of a classical solution T, 0 ? Λ₋ ? T(x, t) ? Λ₊ < ∞ with corresponding radiative intensity I(x, Ω, ν, t). The boundary is considered to be semi-reflexive with reflection coefficient ρ, 0 ? ρ(μ) ? 1. The existence of the solution for the coupled system does not depend on any additional hypotheses besides that the total absorption coefficient is bounded and that the ratio between the coefficients of scattering and total absorption is uniformly bounded. As well we present numerical results for the coupled evolutive problem. Using the operational representation encountered in the course of establishing the existence theory, we derive vector Green’s functions for the transport equation which allow us to solve numerically the coupled system.     

Finite element fracture mechanics analysis of creep rupture of fuel element cladding

A finite element fracture mechanics technique is applied for simulating the elevated temperature creep rupture behavior of initially defected austenitic stainless steel fuel element cladding. The basic analytical approach consists of determining total instantaneous strain energy release rates G_T, and the corresponding values of the stress intensity factor K_T from sequential linear elastic finite element solutions and relating these to either an effective creep fracture toughness parameter G_ec (or K_ec) or to creep crack growth rates , obtained from test results.An initial application of this approach has been made to simulate the creep rupture behavior of initially defected type 316 austenitic stainless steel fuel element cladding in the 20% cold worked condition, tested at 650°C. This application has provided a relationship in the simple familiar form: , where σ is the nominal loop stress, a is the initial depth of a longitudinal crack, h is the cladding thickness, t_r is the time to rupture, and q is a structure sensitive parameter which accounts for the influence of the environment. is a function, obtained from finite element solutions, which accounts for the geometric differences between the present structure and the classical Griffith plate. The function ) is obtained from creep rupture tests of cladding with varying initial flaw depths and times to rupture under corrosive as well as inert environments.Performing time-dependent analyses, a preliminary relationship is obtained between the instantaneous values G_T and K_T, and crack growth rates under corrosive and non-corrosive environments. The analytical predictions of critical combinations of cladding flaw configurations, stresses, times to rupture and crack growth rates are in good agreement with the limited test data available for comparison. Current applications are aimed at the long-term cyclic creep fracture behavior of fast reactor fuel elements, using a nonlinear finite element code. In addition, multiple intergranular fracture configurations are being investigated.     

Cross sections for the photoionization of with the formation of , and vibrational overlaps and R-centroids for the associated vibrational transitions

Cross sections for the photoionization of , initially in vibrational levels v_i = 0–14, with the production of in vibrational levels v_f = 0–18 are tabulated for the full vibrational array at 24 photon wavelengths ranging from 912 Å to 450 Å. The associated vibrational overlap integrals v_ifv_f and R-centroids, v_i|Rⁿ|v_f/v_i|v_f, N = 1 and 2 are also presented together with accurate curve fits of the bound-free (H₂-H₂⁺ + e) electronic matrix elements.     

Results on research of templates from Kozloduy-1 reactor pressure vessel

The studies on the specimens manufactured from the templates cut out from the weld 4 of Kozloduy NPP Unit 1 reactor vessel have been conducted. The data on chemical composition of the weld metal have been obtained. Neutron fluence, mechanical properties, ductile to brittle transition temperature (DBTT) using mini Charpy samples have been determined. The phosphorus and copper content averaged over all templates is 0.046 and 0.1 wt.%, respectively. The fluence amounted up to 5×10¹⁸ n cm⁻² within 15–18 fuel cycles, and about 5×10¹⁹ n cm⁻² for the whole period of operation. These values agree well with calculated data. DBTT was determined after irradiation (T_k) to evaluate the vessel metal state at the present moment, then after heat treatment at the temperature of 475°C to simulate the vessel metal state after thermal annealing (T_an), and after heat treatment at 560°C to simulate the metal state in the initial state (T_k0). As a result of the tests the following values were obtained: T_k, +91.5°C; T_an, +63°C; and T_k0, 54°C. The values of T_k and T_an obtained by measurements were found to be considerably lower than those predicted in accordance with the conservative method accepted in Russia (177°C for T_k and 100°C for T_an). Thus, the obtained results allowed to make a conclusion that it is not necessary to anneal Kozloduy NPP Unit 1 reactor vessel for the second time. The fractographic and electron-microscopic research allowed to draw some conclusions on the embrittlement mechanism.     

Effect of thermal radiation on rewetting of top spray emergency coolant

A general physical model for top spray rewetting during an emrgency core cooling (ECC) transient is proposed which takes into account thermal radiation in the dry region. The model is employed to study the effect of thermal radiation on rewetting a single rod and a 3 × 3 rod bundle up to 2100°F. The results show that rewetting in a bundle is slower than for an isolated rod, due to reduced thermal radiation heat transfer in the dry region. Also, there is a definite correlation between the decreased radiation heat flux Δq_R and the corresponding decrease in rewetting velocity Δu. Values of Δu are not significant unless Δq_R is larger than 6000 Btu/hr ft², where Δq_R cannot exceed a value of 6000 Btu/hr ft² below a temperature of 1100°F, even in the most adverse conditions. Hence, it is concluded that radiant heat transfer does not significantly affect rewetting velocities up to an initial rod temperature of 1100°F. Beyond this temperature, the rewetting velocities change by more than 1.5% and hence radiation must be included in the model for top spray rewetting.     

Socket weld integrity in nuclear piping under fatigue loading condition

The purpose of this paper is to evaluate the integrity of socket weld in nuclear piping under the fatigue loading. The integrity of socket weld is regarded as a safety concern in nuclear power plants because many failures have been world-widely reported in the socket weld. Recently, socket weld failures in the chemical and volume control system (CVCS) and the primary sampling system (PSS) were reported in Korean nuclear power plants. The root causes of the socket weld failures were known as the fatigue due to the pressure and/or temperature loading transients and the vibration during the plant operation. The ASME boiler and pressure vessel (B & PV) Code Sec. III requires 1/16 in. gap between the pipe and fitting in the socket weld with the weld leg size of 1.09 × t₁, where t₁ is the pipe wall thickness. Many failure cases, however, showed that the gap requirement was not satisfied. In addition, industry has demanded the reduction of weld leg size from 1.09 × t₁ to 0.75 × t₁. In this paper, the socket weld integrity under the fatigue loading was evaluated using three-dimensional finite element analysis considering the requirements in the ASME Code. Three types of loading conditions such as the deflection due to vibration, the pressure transient ranging from P = 0 to 15.51 MPa, and the thermal transient ranging from T = 25 to 288 °C were considered. The results are as follows; (1) the socket weld is susceptible to the vibration where the vibration levels exceed the requirement in the ASME operation and maintenance (OM) code. (2) The effect of pressure or temperature transient load on socket weld in CVCS and PSS is not significant owing to the low frequency of transient during plant operation. (3) ‘No gap’ is very risky to the socket weld integrity for the systems having the vibration condition to exceed the requirement specified in the ASME OM Code and/or the transient loading condition from P = 0 and T = 25 °C to P = 15.51 MPa and T = 288 °C. (4) The reduction of the weld leg size from 1.09 × t₁ to 0.75 × t₁ may induce detrimental effect on the socket weld integrity.     

Stresses at the intersection of two cylindrical shells

The stress analysis based on the theory of a thin shell is carried out for two normally intersecting cylindrical shells with a large diameter ratio. Instead of the Donnell shallow shell equation, the modified Morley equation, which is applicable to ₀(R/T)^1/2 1, is used for the analysis of the shell with cut-out. The solution in terms of displacement function for the nozzle with a non-planar end is based on the Love equation. The boundary forces and displacements at the interaction are all transformed from Gaussian coordinates (α, β) on the shell, or Gaussian coordinates (ξ, θ) on the nozzle into three-dimensional cylindrical coordinates (π, θ, z). Their expressions on the intersecting curve are periodic functions of θ and expanded in Fourier series. Every harmonics of Fourier coefficients of boundary forces and displacements are obtained by numerical quadrature.The results obtained are in agreement with those from the finite element method and experiments for d/D 0.8.     

Novel porous media formulation for multiphase flow conservation equations

Multiphase flows consist of interacting phases that are dispersed randomly in space and in time. An additional complication arises from the fact that the flow region of interest often contains irregularly shaped structures. While, in principle, the intraphase conservation equations for mass, momentum, and energy, and their initial and boundary conditions can be written, the cost of detailed fluid flow and heat transfer analysis with explicit treatment of these internal structures with complex geometry and irregular shape often is prohibitive, if not impossible. In most engineering applications, all that is required is to capture the essential features of the system and to express the flow and temperature field in terms of local volume-averaged quantities while sacrificing some of the details. The present study is an attempt to achieve this goal by applying time averaging after local volume averaging.Local volume averaging of conservation equations of mass, momentum, and energy for a multiphase system yields equations in terms of local volume-averaged products of density, velocity, energy, stresses, and field forces, together with interface transfer integrals. These averaging relations are subject to the following length scale restrictions:

dℓL,