Nonlinear Dynamics of Heat-Exchanger Tubes Under Crossflow: A Time-Delay Approach

Fluid-conveying heat-exchanger tubes in nuclear power plants are subjected to a secondary cross-flow to facilitate heat exchange. Beyond a critical value of the secondary flow velocity, the tube loses stability and vibrates with large amplitude. The equation governing the dynamics of a heat-exchanger tube is a delay differential equation (DDE). In all the earlier studies, only the stability boundaries in the parametric space of mass-damping parameter and reduced flow-velocity were reported. In this work using Galerkin approximations, the spectrum (characteristic roots) of the DDE is also obtained. The rightmost characteristic root, whose real part represents the damping in the heat-exchanger tube is included in the stability chart for the first time. The highest damping is found to be present in localized areas of the stability charts, which are close to the stability boundaries. These charts can be used to determine the optimal cross-flow velocities for operating the system for achieving maximum damping. Next, the interaction between the tube and the surrounding cladding at the baffle-plate makes it vital to determine the optimal design parameters for the baffle plates. The linear stability of a heat-exchanger tube modeled as a single-span Euler-Bernoulli cantilever beam subjected to cross-flow is studied with two parameters: (i) varying stiffness of the baffle-cladding at the free end and (ii) varying flow velocity. The partial delay differential equation governing the dynamics of the continuous system is discretized to a set of finite, nonlinear DDEs through a Galerkin method in which a single mode is considered. Unstable regions in the parametric space of cladding stiffness and flow velocity are identified, along with the magnitude of damping in the stable region. This information can be used to determine the design cladding stiffness to achieve maximum damping at a known operational flow velocity. Moreover, the system is found to lose stability by Hopf bifurcation and the method of multiple scales is used to analyze its post-instability behavior. Stable and unstable limit cycles are observed for different values of the linear component of the dimensionless cladding stiffness. An optimal range for the linear cladding stiffness is recommended where tube vibrations would either diminish to zero or assume a relatively low amplitude associated with a stable limit cycle. Furthermore, heat-exchanger tubes undergo thermal expansion, and are consequently subject to thermal loads acting along the axial direction, apart from design-induced external tensile loads. Nonlinear vibrations of a heat-exchanger tube modeled as a simply-supported EulerBernoulli beam under axial load and cross-flow have been studied. The fixed points (zero and buckled equilibria) of the nonlinear DDE are found, and their linear stability is analyzed. The stability of the DDE is investigated in the parametric space of fluid velocity and axial load. The method of multiple scales is used to study the post-instability behavior for both zero and buckled equilibria. Multiple limit-cycles coexist in the parametric space, which has implications on the fatigue life calculations of the heat-exchanger tubes

Missile Materials

The development of missile materials is the key to indigenous materials and their components. The selection of materials includes some the properties such as high strength to weight ratio, easy fabrication, good corrosion resistance, reliable quality, and high fracture toughness. The materials used for the airframe and propulsion system are alloys of aluminum, titanium, magnesium, and maraging steel. Non-metallic materials are also used such as carbon-carbon composites and polymer materials. High purity materials like phosphorous, and silicon are also important for material advancements. The paper outlines the needs and challenges of research and its solutions

A note on damping in heat-exchanger tubes subjected to cross-flow

The equation governing the dynamics of a heat-exchanger tube is a delay differential equation (DDE). In all the earlier studies, only the stability boundaries in the parametric space of mass-damping parameter and reduced flow-velocity were reported. The contour plots showing the damping in different regions of the stability chart has never been reported, due to the complexity in solving the infinite-dimensional nonlinear eigenvalue problem associated with characteristic roots of the governing DDE. In this work using Galerkin approximations, the spectrum (characteristic roots) of the DDE is obtained. The rightmost characteristic root, whose real part represents the damping in the heat-exchanger tube is included in the stability chart. Interestingly, it is found that the highest damping is present in localized areas of the stability charts, which are close to the stability boundaries. These stability charts can be used to determine the optimal cross-flow velocities for operating the heat-exchanger tube for achieving maximum damping. Explicit evaluation of the characteristic roots allows us to show that the roots cross the stability boundary with a non-zero velocity, clearly indicating the existence of Hopf bifurcation at the stability boundary

Supercritical and Subcritical Hopf Bifurcations in a Delay Differential Equation Model of a Heat Exchanger Tube Under Cross-Flow

Nonlinear vibrations of a heat exchanger tube modeled as a simply-supported Euler-Bernoulli beam under axial load and cross-flow have been studied. The compressive axial loads are a consequence of thermal expansion and tensile axial loads can be induced by design. The fluid forces are represented using an added mass, damping, and a time-delayed displacement term. Due to the presence of the time-delayed term, the equation governing the dynamics of the tube becomes a partial delay differential equation (PDDE). Using the modal-expansion procedure, the PDDE is converted into a nonlinear delay differential equation (DDE). The fixed points (zero and buckled equilibria) of the nonlinear DDE are found, and their linear stability is analyzed. It is found that stability can be lost either via supercritical or subcritical Hopf bifurcation. Using Galerkin approximations, the characteristic roots of the DDE are found and reported in the parametric space of fluid velocity and axial load. Furthermore, the stability chart obtained from the Galerkin approximations is compared with the critical curves obtained from analytical calculations. Next, the method of multiple scales (MMS) is used to derive the normal-form equations near the supercritical and subcritical Hopf bifurcation points for both zero and buckled equilibrium configurations. The steady-state amplitude response equation, obtained from the MMS, at Hopf bifurcation points is compared with the numerical solution. The coexistence of multiple limit-cycles in the parametric space is found and has implications in the fatigue life calculations of the heat exchanger tubes

Effect of nonlinear cladding stiffness on the stability and Hopf bifurcation of a heat-exchanger tube subject to cross-flow

The linear stability of a heat-exchanger tube modeled as a single-span cantilever beam subjected to cross-flow has been studied with two parameters: (1) varying stiffness of the baffle-cladding at the free end and (2) varying flow velocity. A mathematical model incorporating the motion-dependent fluid forces acting on the beam is developed using the Euler–Bernoulli beam theory, under the inextensible condition. The partial delay differential equation governing the dynamics of the continuous system is discretized to a set of finite, nonlinear delay differential equations through a Galerkin method in which a single mode is considered. Unstable regions in the parametric space of dimensionless cladding stiffness and flow velocity are identified, along with the magnitude of damping in the stable region. This information can be used to determine the cladding stiffness at which the system should be operated to achieve maximum damping at a known operational flow velocity. Furthermore, the system is found to lose stability by Hopf bifurcation and the method of multiple scales is used to analyze its post-instability behavior. Stable and unstable limit cycles are observed for different values of the linear component of the dimensionless cladding stiffness. A global bifurcation analysis indicates that the number of limit cycles decreases with increasing linear cladding stiffness. An optimal range for the linear cladding stiffness is recommended where tube vibrations would either diminish to zero or assume a relatively low amplitude associated with a stable limit cycle

SARS-CoV-2 vaccination modelling for safe surgery to save lives: data from an international prospective cohort study

Background: Preoperative SARS-CoV-2 vaccination could support safer elective surgery. Vaccine numbers are limited so this study aimed to inform their prioritization by modelling. Methods: The primary outcome was the number needed to vaccinate (NNV) to prevent one COVID-19-related death in 1 year. NNVs were based on postoperative SARS-CoV-2 rates and mortality in an international cohort study (surgical patients), and community SARS-CoV-2 incidence and case fatality data (general population). NNV estimates were stratified by age (18-49, 50-69, 70 or more years) and type of surgery. Best- and worst-case scenarios were used to describe uncertainty. Results: NNVs were more favourable in surgical patients than the general population. The most favourable NNVs were in patients aged 70 years or more needing cancer surgery (351; best case 196, worst case 816) or non-cancer surgery (733; best case 407, worst case 1664). Both exceeded the NNV in the general population (1840; best case 1196, worst case 3066). NNVs for surgical patients remained favourable at a range of SARS-CoV-2 incidence rates in sensitivity analysis modelling. Globally, prioritizing preoperative vaccination of patients needing elective surgery ahead of the general population could prevent an additional 58 687 (best case 115 007, worst case 20 177) COVID-19-related deaths in 1 year. Conclusion: As global roll out of SARS-CoV-2 vaccination proceeds, patients needing elective surgery should be prioritized ahead of the general population