Closed-end Tube Research Articles

Anisotropic creep behaviour of zirconium alloys in a fast neutron flux

During fabrication, zirconium alloys develop strong crystallographic textures, which give the materials anisotropic properties. To derive the anisotropic factors which apply to creep deformation at relatively low stresses, temperatures of 260° to 300°C and fast neutron flux from 0.6 × 1013 to 3.5 × 1013 n/cm2s (>1 MeV), test results are analyzed using the effective stress and effective strain relations. The analysis is an extension of the conventional method representing the multiaxial stress state based on the generalized Von Mises yield function proposed by Hill.For the tubes evaluated the ratio of transverse creep in a closed-end tube to a uniaxial creep rate in the axial direction at the same stress, flux and temperature is about 0.3. The ratio of uniaxial creep rates in the transverse and axial directions is about 0.5. Tentative values for the anisotropic factors are determined for pressure tube materials which allow one to calculate strain behaviour under any stress state. The factors are broadly related to the inverse of the resolved fraction of prism planes oriented for slip. Résumé La texture prononcée qui se forme au cours de la fabrication d'alliages de zirconium engendre une anisotropie des propriétés. Afin de determiner les facteurs d'anisotropie qui influencent la déformation due au fluage sous contraintes relativement faibles, à des températures variant de 260° à 300°C. et sous un flux de neutrons rapides variant de 0.6 × 1013 à 3.5 × 1013 n/cm2 s (>1 MeV), les résultats d'essais ont été analysés à l'aide des équations pour la contrainte et la déformation effectives. L'analyse est un développement de la méthode conventionnelle de représentation de l'état de contraintes multiaxiales, basé sur la généralisation du critère d'écoulement de von Misès proposé par Hill.Pour les tubes étudiés, on a obtenu environ 0.3 pour le rapport du fluage transversal dans un tube à fermés, divisé par le taux de fluage uniaxial dans la direction axiale, à la même contrainte, à la même température, et sous le même flux. Le rapport des taux de fluage uniaxiaux dans les directions transversale et axiale est d'environ 0.5. En utilisant des valeurs approachées pour les facteurs d'anisotropie de matériaux pour tubes de force, on peut calculer le comportement en déformation pour tout état de contrainte. En première approximation on peut dire que les facteurs sont reliés à l'inverse de la fraction réduite des plans orientés favorablement pour le glissement.

Resonant Oscillations: A Regular Perturbation Approach

The resonant oscillations of a gas in a closed-end tube are treated. The time-periodic motion is given by a regular perturbation method in which the expansion parameter is the Mach number of the resulting flow. The solution at the first order exhibits a shock traveling in the tube, but, to this order, the waves do not interact. The interaction of the waves is calculated at the second order. The perturbation procedure proceeds systematically by an application of the Fredholm alternative at each stage.

The in-reactor creep of cold-worked Zircaloy-2 and Zirconium-2.5 wt % niobium pressure tubes

To assist in the design of the pressure tube as a core component, the creep behaviour of cold-worked Zircaloy-2, cold-worked zirconium-2.5 wt % niobium, and heat-treated zirconium-2.5 wt % niobium materials is being studied by measuring the inside diameter of reactor pressure tubes during reactor shutdown. The results show a significant increase in creep rate which can be directly related to the fast neutron flux. This paper covers briefly the equipment and procedures used to measure diametral creep, and the results gained from several experimental pressure tubes in the NRU and NPD reactors. The diametral creep rate of a cold-worked Zircaloy-2 pressure tube operating at 270 °C, 3 × 10 13 n/ cm 2 · s (> 1 MeV) and 14000 psi under a transverse to longitudinal stress ratio of 2: 1 (closed end tube) is about 1.8 × 10 -7/ h, and is about ten times the creep rate out-of-reactor. The effects of stress and temperature are not as well defined as the effect of flux. Besults indicate that for tubes operating between 10 000 to 14 000 psi stress, the exponent n in the relation ε = βσ n ( ε = strain rate, σ = stress) is closer to 1 than to the value of about 3 found in the in-reactor uniaxial teats. Results also indicate that the in-reactor creep rate under biaxial stressing is less temperature dependent than under uniaxial stressing. As a first approximation, the equation, creep rate ε t = 4 × 10 -27 σ tφ( T− 160 ° C)/h could be used for cold-worked Zircaloy-2 in the temperature ( T) range from 250 to 300 °C, hoop stress (σ t) up to 20 000 psi, and with fast neutron flux (φ) up to 3.5 × 10 13 n/ cm 2 · s (> 1 MeV). At present both uniaxially stressed specimens and tube results show the effect of fast flux but there is no direct correlation between them. The creep of cold-worked zirconium-2.5 wt % niobium is accelerated by the fast neutron flux but the rates are about one-third those of cold-worked Zircaloy-2. Besults have been obtained at stresses from 15 500 to 23 000 psi at 270 to 285 °C and up to 3.1 × 10 13 n/ cm 2 · s fast neutron flux (> 1 MeV).

Fluid Drag on a Sphere Rolling in a Tube

The incompressible fluid drag force on a sphere rolling at constant speed in a closed-end tube has been analyzed for the case where the tube inside diameter is only slightly larger than the sphere diameter. One drag law is found for Reynolds numbers much less than 75π/4λ1/2, where λ is a parameter defined by the sphere and tube diameters. A second drag law is found for Reynolds numbers much larger than 75π/4λ1/2. Experimental results show good agreement with these drag laws. The first law is almost identical with the results of Christopherson and Dowson, and of McNown, et al., for a sphere falling in a vertical tube.

Tensile Instability in Thin-Walled Tubes

An analytical solution giving conditions at instability of a thin-walled tube subjected to internal pressure and independent axial load is presented. An attempt is made to explain the physical reasons for the occurrence of an instability. The analysis shows that a given material has greatest ductility when the ratio of the hoop stress to axial stress has the value one half. The material has least ductility when it is in the form of a closed-ended tube subjected to internal pressure. The terms ‘tensile strength’ and ‘percentage elongation’ are discussed within the framework of the present investigation.

Plastic Stress-Strain Relationships—Further Experiments on the Effect of Loading History

Further tests have been carried out on thin closed-ended tubes of alpha brass subjected to various combinations of torque and internal pressure. The effect of loading, unloading, and reloading along different paths has been investigated. The loading paths were based on a yield function which has previously been found to correlate initial radial loadings for this material, which possesses one degree of anisotropy. However, the results obtained from the second loadings suggest a cross effect which is greater than would be obtained from a nested set of yield surfaces of the foregoing form. There appears to be no evidence to support the presence of a corner in the yield surface.

Plastic Stress-Strain Relationships—Some Experiments on the Effect of Loading Path and Loading History

Abstract Tests have been carried out on thin closed-ended tubes of alpha brass subjected to various combinations of torque and internal pressure. The effect of loading, unloading, and reloading along different loading paths has been investigated, including the effect on the shape of the yield surface, and the form of the curve of representative stress and representative strain. The behavior of the material for initial loading suggests that the material is isotropic in the plane of the surface of the tube but is anisotropic in a radial direction. A form of yield criterion and representative stress which correlates these results has been deduced, but the results of the tests for the second loadings of the specimens cannot be correlated in terms of a nested set of similar yield surfaces. The evidence for or against the existence of corners or pointed vertices on the yield surface is inconclusive.

On the Stagnation of Natural-Convection Flows in Closed-End Tubes

Abstract An analysis of the laminar natural-convection flow and heat transfer in a closed-end tube with a linear wall temperature and large but finite length-radius ratio is presented. It is found that for a given relation between the two physical parameters of the problem, the flow will fill the entire tube length. Representative velocity and temperature profiles are presented to show the effects of the parameters on the flow and heat transfer.