Influence of thermal boundary conditions on the dynamic behaviour of a rectangular single-phase natural circulation loop

Natural circulation of distilled water and FC43 has been experimentally investigated in a rectangular loop characterized by internal diameter of 30 mm and total length of 4.1 m. The aim of the present study is to analyse the influence of thermal boundary conditions on the flow regimes inside the pipes and on the stability of the system. The new aspect of the present research is the possibility of tuning the heat sink temperature in a range between −20 °C and +30 °C by means of a cryostat. This kind of analysis could be useful for the design of systems characterized by a wide range of environment temperatures, as for example for aerospatial applications. The other parameters investigated were the heat flux transferred to the fluid, which varied between 0.1 kW and 2.5 kW, and the thermo-physical properties of the working fluid. The system showed both stable and unstable behaviour. In particular, in the case of FC43 the loop was more unstable and it was characterized by higher velocities and frequencies compared to the case of distilled water.It was found that the stability threshold could be crossed by varying only the heat sink temperature, demonstrating the importance of this boundary condition on the dynamics of the system. Different flow regimes and fluid velocities were observed. In the case of steady-state flow, Vijayan’s correlation (Vijayan et al., 2000) was tested and found to give good agreement with experimental data. Linear stability analysis was made following the Vijayan’s model. In particular, the effect of heat sink temperature was considered in the dimensionless Stanton number based on the overall heat transfer coefficient at the heat sink. Finally, Ultrasound Pulsed Doppler Velocimeter (UPDV) was used on a natural circulation loop for the first time, and gave a preliminary validation of the traditional fluid velocity measurement method based on the frequency analysis.     

Steady state and stability characteristics of single-phase natural circulation in a rectangular loop with different heater and cooler orientations

Effect of the heater and cooler orientations on the single-phase natural circulation was studied in a rectangular loop. From steady state considerations, the maximum flow for a specified operating condition is achievable for the orientation with both the heater and the cooler horizontal. However, this orientation is found to be least stable and the orientation with both heater and cooler vertical is found to be most stable. Three oscillatory modes were observed in the experiments, two of which are observable only for certain heat addition paths. A hysteresis (conditionally stable) regime wherein stable or unstable flow can prevail depending on the operating procedure is also observed. Generalized steady state equation valid for all the orientations is possible whereas generalized stability equation is possible for a given orientation and flow direction. Steady state flow rate can be predicted with reasonable accuracy using the generalized equation. The stability map predicted by the generalized equation is found to be highly conservative in that its unstable zone is much larger than that observed in the experiments. Both the linear and the nonlinear stability analyses gave the same stability map for identical initial conditions. With nonlinear analysis, it was possible to reproduce the observed unstable oscillatory modes albeit at significantly different power levels. Also, the predicted limit cycles were significantly different in shape which is attributed to the multidimensional nature of the flow especially during the low flow part of the oscillation cycle.     

Experiments in a single-phase natural circulation mini-loop

This study reports an experimental investigation related to a rectangular single-phase natural circulation mini-loop, which consists of two horizontal copper tubes (heat transfer sections) and two vertical tubes (legs) made of copper, connected by means of four glass 90° bends. The loop inner diameter is 4 mm. The lower heating section consists of an electrical heating wire made of nicromel on the outside of the copper tube; the upper cooling system consists of a coaxial cylindrical heat exchanger with a water–glycol mixture, set at controlled temperature and flowing through the annulus. The loop has an imposed heat flux in the lower heating section and an imposed temperature in the cooler. The mini-loop was placed onto a table which can assume different inclinations. The parameters investigated during the experiments were: power transferred to the fluid and inclination of the loop. The preliminary results show a stable behaviour with a steady temperature difference across the heat sinks. It has been confirmed that the fluid velocity is very small (order of millimetres per second).     

An analysis of the relaxation oscillations of a nonlinear thermosyphon

The oscillatory behavior of an asymmetrically forced thermosyphon constituted by two connected vessels has been subjected to an asymptotically valid analysis using the vessel-volume ratio as expansion parameter. Due to the structure of the governing equations, the problem could not be dealt with using standard techniques; instead a phase-plane analysis was conducted. The analytically determined corrections to the previously established lowest-order discontinuous results proved to be useful even for comparatively large values of the expansion parameter. The relationship between these asymptotically valid corrections and the physics underlying the relaxation oscillation as well as the behavior of the system for strong thermal forcing is discussed. The study is concluded by an overview of some specific inconsistencies associated with the discontinuous lowest-order analysis and how these were alleviated by the asymptotically valid corrections.     

An integral constitutive law for viscoelastic fluids based on the concept of evolving natural configurations: stability analysis

A general conceptual framework has recently been developed within the context of large deformations for the study of materials (solids and fluids) that either undergo microstructural changes or for which stress states are related to the presence of several relaxation mechanisms. The corner stone of this new approach is that all materials exhibit an infinity of stress-free natural configurations that are evolving in a thermodynamically admissible process. In this paper, that is exploratory in nature, we study the behavior of a non-linear integral viscoelastic constitutive equation (CE) for a fluid assumed to posses an infinity of evolving natural configurations. The CE stability pattern with respect to small perturbations about the rest state is also addressed.     

An experimental investigation of single-phase natural circulation behavior in a rectangular loop with Al2O3 nanofluids

In this paper, the natural circulation behavior in a rectangular loop was investigated experimentally with water and different concentration of Al₂O₃ nanofluids (0.3–2% by wt. and particle size 40–80 nm). It was demonstrated that, not only the flow instabilities are suppressed but also the natural circulation flow rates are enhanced with nanofluids. The enhancement in natural circulation flow rate and suppression of instabilities were found to be dependent on the concentration of nanoparticles in water.     

Heat transfer from a vertical tube bundle under natural circulation conditions

A rectangular loop (thermosyphon) was used to measure the average heat transfer coefficients for water at atmospheric pressure under natural circulation conditions. A twenty-one tube bundle with tubes 1.65 m long and 9.55 mm in diameter, and a pitch-to-diameter ratio of 1.33, was used as a test heat exchanger in one of the vertical legs of the loop. A natural circulation flow in the loop developed due to buoyancy differences of the fluid in its two vertical legs. Flow visualization experiments were performed to determine the flow regimes associated with natural circulation flow longitudinal to a tube bundle. Empirical correlations for the average Nusselt number have been developed and are reported. Grid spacers arranged on tube bundles were shown to enhance heat transfer, especially for laminar flow, without any noticeable increase in pressure drop.     

Inlet throttling effect on the boiling two-phase flow stability in a natural circulation loop with a chimney

 Experiments have been conducted to investigate an effect of inlet restriction on the thermal-hydraulic stability. A Test facility used in this study was designed and constructed to have non-dimensional values that are nearly equal to those of natural circulation BWR. Experimental results showed that driving force of the natural circulation at the stability boundary was described as a function of heat flux and inlet subcooling independent of inlet restriction. In order to extend experimental database regarding thermal-hydraulic stability to different inlet restriction, numerical analysis was carried out based on the homogeneous flow model. Stability maps in reference to the core inlet subcooling and heat flux were presented for various inlet restrictions using the above-mentioned function. Instability region during the inlet subcooling shifted to the higher inlet subcooling with increasing inlet restriction and became larger with increasing heat flux. Received on 17 January 2000     

The influence of the wall thermal capacity and axial conduction over a single-phase natural circulation loop: 2-D numerical study

 In this paper the natural circulation phenomenon of a differentially heated single-phase natural circulation loop has been numerically investigated. The conservation equations of mass, momentum and energy are solved using a two-dimensional finite difference method. The preliminary results will be presented. The transient and steady-state behavior of the circuit has been investigated. The dependency of the temperature and velocity field on the power input and the wall thermal capacity has been analyzed, as well as the relationship between the most important non-dimensional numbers (Nusselt, Reynolds and the friction factor versus the modified Grashof number). A comparison of the numerical results is made with conventional one-dimensional analysis and with the experimental data. Generally, a good agreement is achieved in the stable region. Received on 17 January 2000     

Investigation of the stability of a natural circulation two-pass boiler

In this paper, the results of a theoretical analysis of the static flow instability, namely the reverse flow, for a two-pass steam generator (see Fig. 1) will be presented. The aim of the work was the development of design criteria to avoid the reversal of flow. To determine the critical heat absorption ratio between the two riser systems of a two-pass boiler several dynamic simulations of a warm start-up of the two-pass boiler with different heat absorption ratios, different heat profiles along the lower heated riser system 1 and modified geometry of the boiler (influence of the flow resistance) are done. On the basis of the results of this investigations it can be shown that for every natural circulation system with unequally heated risers and a common downcomer a special critical heat absorption ratio exists. The investigations clearly indicate that such types of steam generators should be designed only in that way that the operation below the critical heat absorption ratio is possible. The modification of the flow resistance in the riser systems can help to extend the range of stable operation.     

Experimental and numerical investigations on flashing-induced instabilities in a single channel

During the start-up phase, natural circulation BWRs (NC-BWRs) need to be operated at low pressure conditions. Such conditions favor flashing-induced instabilities due to the large hydrostatic pressure drop induced by the tall chimney. Moreover, in novel NC-BWR designs the steam separation is performed in the steam separators which create large pressure drops at the chimney outlet, which effect on stability has not been investigated yet.In this work, flashing-induced oscillations occurring in a tall, bottom heated channel are numerically investigated by using a simple linear model with three regions and an accurate implementation for estimating the water properties. The model is used to investigate flashing-induced instabilities in a channel for different values of the core inlet friction value. The results are compared with experiments obtained by using the CIRCUS facility at the same conditions, showing a good agreement. In addition, the experiments on flashing-induced instabilities are presented in a novel manner allowing visualizing new details of the phenomenon numerical stability investigations on the effect of the friction distribution are also done. It is found that by increasing the total restriction in the channel the system is destabilized. In addition, the chimney outlet restriction has a stronger destabilizing effect than the core inlet restriction. A stable two-phase region is observed prior to the instabilities in the experiments and the numerical simulations which may help to pressurize the vessel of NC-BWRs and thus reducing the effects of flashing instabilities during start-up.     

Two dimensional analysis of transient flow and heat transfer in a natural circulation loop

The transient heat transfer, fluid flow and pressure in a natural circulation loop have been studied under laminar flow conditions. Most studies of these systems have utilized a onedimensional approach which requires a priori specifications of the friction and the heat-transfer coefficients. In the present work the variation of the friction and heat-transfer coefficients are determined. Detailed pressure, temperature and velocity distributions are presented.     

Characterization of instabilities during two-phase natural circulation in typical PWR conditions

Strong oscillations in fluid velocities and densities were measured in the LOBI test facility during natural circulation experiments. A sort of “siphon condensation” occurs in the U-tubes of steam generators when primary-side mass inventory reaches roughly 64% of the initial value.

The present paper deals with the characterization of the phenomenon considering flooding and condensation dynamics of U-tubes; RELAP5/MOD2 calculations made it possible to select system parameters affecting the oscillation characteristics. An attempt was made to evaluate the possibility of instability occurring in real plant situations.     

Asymptotic solutions of the axisymmetric moist Hadley circulation in a model with two vertical modes

A simplified model of the moist axisymmetric Hadley circulation is examined in the asymptotic limit in which surface drag is strong and the meridional wind is weak compared to the zonal wind. Our model consists of the quasi-equilibrium tropical circulation model (QTCM) equations on an axisymmetric aquaplanet equatorial beta-plane. This model includes two vertical momentum modes, one baroclinic and one barotropic. Prior studies use either continuous stratification, or a shallow water system best viewed as representing the upper troposphere. The analysis here focuses on the interaction of the baroclinic and barotropic modes, and the way in which this interaction allows the constraints on the circulation known from the fully stratified case to be satisfied in an approximate way. The dry equations, with temperature forced by Newtonian relaxation towards a prescribed radiative equilibrium, are solved first. To leading order, the resulting circulation has a zonal wind profile corresponding to uniform angular momentum at a level near the tropopause, and zero zonal surface wind, owing to the cancelation of the barotropic and baroclinic modes there. The weak surface winds are calculated from the first-order corrections. The broad features of these solutions are similar to those obtained in previous studies of the dry Hadley circulation. The moist equations are solved next, with a fixed sea surface temperature at the lower boundary and simple parameterizations of surface fluxes, deep convection, and radiative transfer. The solutions yield the structure of the barotropic and baroclinic winds, as well as the temperature and moisture fields. In addition, we derive expressions for the width and strength of the equatorial precipitating region (ITCZ) and the width of the entire Hadley circulation. The ITCZ width is on the order of a few degrees in the absence of any horizontal diffusion and is relatively insensitive to parameter variations.     

Experimental investigation on the flow instability behavior of a multi-channel boiling natural circulation loop at low-pressures

Natural circulation as a mode of heat removal is being considered as a prominent passive feature in the innovative nuclear reactor designs, particularly in boiling-water-reactors, due to its simplicity and economy. However, boiling natural circulation system poses many challenges to designer due to occurrence of various kinds of instabilities such as excursive instability, density wave oscillations, flow pattern transition instability, geysering and metastable states in parallel channels. This problem assumes greater significance particularly at low-pressures i.e. during startup, where there is great difference in the properties of two phases. In light of this, a parallel channel loop has been designed and installed that has a geometrical resemblance to the pressure-tube-type boiling-water-reactor, to investigate into the behavior of boiling natural circulation. The loop comprises of four identical parallel channels connected between two common plenums i.e. steam drum and header. The recirculation path is provided by a single downcomer connected between steam drum and header. Experiments have been conducted over a wide range of power and pressures (1–10 bar). Two distinct unstable zones are observed with respect to power i.e. corresponding to low power (Type-I) and high power (Type-II) with a stable zone at intermediate powers. The nature of oscillations in terms of their amplitude and frequency and their evolution for Type-I and Type-II instabilities are studied with respect to the effect of heater power and pressure. This paper discusses the evolution of unstable and stable behavior along with the nature of flow oscillation in the channels and the effect of pressure on it.     

Dynamic flight stability of a model dronefly in vertical flight

The dynamic flight stability of a model dronefly in hovering and upward flight is studied.The method of computational fluid dynamics is used to compute the stability derivatives and the techniques of eigenvalue and eigenvector used to solve the equations of motion.The major finding is as following.Hovering flight of the model dronefly is unstable because of the existence of an unstable longitudinal and an unstable lateral natural mode of motion.Upward flight of the insect is also unstable,and the instability increases as the upward flight speed increases.Inertial force generated by the upward flight velocity coupled with the disturbance in pitching angular velocity is responsible for the enhancement of the instability.     

A solution for the vertical variation of stress,rather than velocity,in a three-dimensional circulation model

A simple technique is presented that allows a numerical solution to be sought for the vertical variation of shear stress as a substitute for the vertical variation of velocity in a three-dimensional hydrodynamic model. In its most general form the direct stress solution (DSS) method depends only upon the validity of an eddy viscosity relation between the shear stress and the vertical gradient of velocity. The rationale for preferring a numerical solution for shear stress to one for velocity is that shear stress tends to vary more slowly over the vertical than velocity, particularly near boundaries. Consequently, a numerical solution can be obtained much more efficiently for shear stress than for velocity. When needed, the velocity profile can be recovered from the stress profile by solving a one-dimensional integral equation over the vertical. For most practical problems this equation can be solved in closed form. Comparisons are presented between the DSS technique, the standard velocity solution technique and analytical solutions for wind-driven circulation in an unstratified, closed, rectangular channel governed by the linear equations of motion. In no case was the computational effort required by the velocity solution competitive with the DSS when a physically realistic boundary layer was included. The DSS technique should be particularly beneficial in numerical models of relatively shallow water bodies in which the bottom and surface boundary layers occupy a significant portion of the water column.     

Variational formulation and stability analysis of a three dimensional superelastic model for shape memory alloys

This paper presents a variational framework for the three-dimensional macroscopic modelling of superelastic shape memory alloys in an isothermal setting. Phase transformation is accounted through a unique second order tensorial internal variable, acting as the transformation strain. Postulating the total strain energy density as the sum of a free energy and a dissipated energy, the model depends on two material scalar functions of the norm of the transformation strain and a material scalar constant. Appropriate calibration of these material functions allows to render a wide range of constitutive behaviours including stress-softening and stress-hardening. The quasi-static evolution problem of a domain is formulated in terms of two physical principles based on the total energy of the system: a stability criterion, which selects the local minima of the total energy, and an energy balance condition, which ensures the consistency of the evolution of the total energy with respect to the external loadings. The local phase transformation laws in terms of Kuhn–Tucker relations are deduced from the first-order stability condition and the energy balance condition.The response of the model is illustrated with a numerical traction–torsion test performed on a thin-walled cylinder. Evolutions of homogeneous states are given for proportional and non-proportional loadings. Influence of the stress-hardening/softening properties on the evolution of the transformation domain is emphasized. Finally, in view of an identification process, the issue of stability of homogeneous states in a multi-dimensional setting is answered based on the study of second-order derivative of the total energy. Explicit necessary and sufficient conditions of stability are provided.     

An investigation of the stability in the large for an autonomous second-order two-degree-of-freedom system

An extension to an algorithm due to Simpson has been developed for the analysis of a second-order two-degree-of-freedom autonomous system. The form of equations considered arises from the study of mechanical systems with a single concentrated non-linearity and the method assumes a solution made up of harmonic terms whose amplitudes vary slowly in time. For a system possessing a stable equilibrium point and an unstable limit cycle arising from a subcritical Hopf bifurcation, the method has been applied to the problem of predicting the basin of attraction of the equilibrium point. The method reduces the problem from a search in four-dimensional phase space to a search for a boundary in a plane defined by amplitudes a₁ and a₂ in the assumed form of the solution. The method was applied to four weakly non-linear systems in which the non-linearity was due to either a linear spring with a small amount of cubic hardening or a linear spring with freeplay. Agreement was shown to be good in the cases considered. However, it would be expected that the method would not give such accurate results if the non-linear effect was more significant. This was illustrated for the case of the cubic hardening non-linearity.