The pressure surge in pipes due to change in operating conditions exerts an axial load on elbows proportional to the change in momentum of fluid and unbalanced pressure forces. The response of piping structure to such load needs the full time history analysis in three dimensional spaces which is cumbersome process due to high computing memory requirements and long simulation time. In present work it has been shown that using Rayleigh energy balance for each elbow-load configuration, the system can be reduced to equivalent 1D spring mass system and the response can be estimated by solving 1D equation of motion. Then it has been recommended to simulate the response of each elbow which gives good approximation of dynamic amplification of displacement also called as Dynamic Load Factor (DLF). These dynamic load factors for each elbow can be used for the interaction of forces using static equivalent response in 3D space. This approach is pseudo static equivalent analysis where the load amplifications factors DLF are estimated from the dynamic force profile and system response in one-dimensional space. An algorithm is developed for the above explained process. Most of the engineers are using the DLF = 2 for the load estimation due to absence of method to estimate the dynamic load factor. The approach was proposed by Goodling in 1989 and still widely followed in the industry. The present paper discusses uncertainty and inaccuracy involved in performing approximate analysis and shows the significance and need of performing full force time history analysis. The proposed method shows very good agreement with the time consuming 3D full force time history results. There are also limitations for the proposed method. As the spring mass system is simulated with dimensional reduction to single frequency domain, the pipe supports and guides should be properly placed before applying the present approach. It has been shown that with proper support configuration, this simplified approach yields very good approximation of surge load on pipes with reduced time.

This paper addresses an instability problem of a plate in a uniform incompressible and irrotational flow. In particular, this study is aimed at developing a stability analysis method with low computational cost. An analytical dynamics framework for constraint systems is applied to derivation of the fluid force acting on the plate surface. This formulation gives an algebraic relation between the plate displacement and the generalized momentum regarding the velocity potential as a constraint, that is, this relation plays the role of the fluid-structure interaction in the present formulation. Then, the algebraic equation for calculating the fluid force is obtained by a second order time derivative of the constraint. Specifically, the fluid force is given as the Lagrange multiplier, which is introduced for imposing the constraint regarding the fluid-structure interaction mentioned above. The resulting equations are comprised of differential equations regarding the plate displacement and the algebraic equation giving the fluid force. This kind of equations are referred to as the differential algebraic equations. Moreover, derived equations for the fluid-structure interaction are reduced to the eigenvalue problem by employing the generalized Fourier series expansion technique. Then, the stability of this system is evaluated by the eigenvalue analysis. After that, the results are verified by comparing with the results by the previous method.

Flow-induced vibration of heat-exchangers tubes is particularly studied in the nuclear industry for safety and cost reasons. It implies to have, among others, relevant characterizations of the random buffeting forces the cross-flow applies to the tube bundle. Work is still needed in this domain, particularly for two-phase flow, to improve the available data as the ones for PWR steam generator, currently very envelope. In parallel to get new experimental data, using “real” or substitutional mixtures (e.g. air-water instead of steam-water for PWR), it is essential to understand the basic excitation mechanisms which induce the vibrations under two-phase flow, as e.g. the influence of flow regimes. In this general framework, what can be learnt from deliberately simple models may be a contributive help. As a first attempt on this issue, the paper deals with the elementary case of a single rigid tube under air-water cross flow. This case is part of experiments carried out at CEA-Saclay with bundles where both tube support reactions and flow characteristics are measured, with respectively piezo-electrical sensors and bi-optical probes (BOP). The information provided by the BOP (mean interface velocity, statistical distribution, etc.) feeds a primitive model of water “droplet” impulses on the tube, based on a lot of crude assumptions about impact velocity, momentum conservation, impulse shape, statistical independence, etc., and which uses analytical results of random processes constructed from the superposition of random pulses. The “equivalent” excitation force, obtained in terms of dimensional PSD, is compared to the one measured in the drag and lift direction with an acceptable agreement, at least in order of magnitude. Comments and lessons are drawn from this first attempt, and some paths are advanced to improve this kind of primitive models, especially for treating rigid square bundles under air-water cross flow.

The motivation behind this study is to simulate high pressure gas flow through the clearance between a valve seat and disc when in a closed position using a representative model. This leakage phenomenon is common in metal-to-metal seal pressure relief valves. As a pressure relief valve reaches the set pressure, it is known for the leakage to increase. The representative model that we studied is of an ideal-gas flow through a 2D microchannel in the slip flow regime. We used a laminar continuum flow solver which solved the mass, momentum and energy equations. In addition, we applied low pressure slip boundary conditions at the wall boundaries which considered Maxwells model for slip. The channel height was varied from 1μm to 5μm while the length remained at 1.25 mm, this means the length to height ratio varied from 1250 to 250. Inlet pressure was varied from a low pressure (0.05 MPa) to a high pressure (18.6 MPa), while the outlet remained constant at atmospheric. The calculated mass flow rate is compared to an analytical solution giving very good agreement for low pressure ratios and high length to height ratios.

This paper addresses a formulation and an aeroelastic instability analysis of a plate in a uniform incompressible and irrotational flow based on the classical variational principle framework. Because of an intrinsic algebraic relation between the plate displacement and the velocity potential, this system has to be formulated as the constrained system. In this study, we tried to apply the Hamiltonian mechanics to the formulation of the fluid-structure interaction problem with mixed boundary condition. As a result, we obtain the canonical equations, that consist of the evolution equations for the plate displacement, the velocity potential, the Lagrange multiplier and canonically conjugate momenta for those physical quantities. In particular, it was found that the Lagrange multiplier was just the pressure. In other words, the equations of time evolution could be derived for not only the plate displacement and the velocity potential but also the pressure (the Lagrange multiplier). The stability of this system was analyzed by the eigenvalue analysis. Then, flutter modes, their frequencies and growth rates were discussed. The proposed technique has the advantage that it can reduce iteration procedures in the stability analysis. As a consequence, it can be expected that the stability of this system can be evaluated efficiently. This paper introduces a formulation of the only two dimensional problem, and the stability analysis of a clamped-free plate is implemented as an numerical example. Howerver, this formulation can be applied to three dimensional problems without intrinsic difficulties.

In most cases, the method of characteristics is used to calculate the propagation of water hammer in hydraulic systems due to the size of those pipings, although three-dimensional effects are known to occur. In order to investigate and quantify these effects, a three-dimensional computational fluid dynamics simulation of water hammer through a bend geometry was performed. For the resolution of the developing high spatial and temporal gradients an adequate mesh and suitable physical model was generated using a commercial code. The applicability of unsteady Reynolds-averaged Navier-Stokes simulation was evaluated considering the turbulent properties of the flow using results from the literature. Furthermore velocity, pressure, wall shear stress and vorticity distributions are presented. The effect of the 90° bend as three-dimensional element was identified and the impact on the flow field is presented. In the end, the annular effect is discussed. Due to the high forces of inertia in the boundary layer and the dominating viscous forces close to the wall, high velocity gradients are developing resulting in high wall shear stresses. It is shown that the viscous and turbulent transport of momentum in the radial direction reduces these velocity gradients and limits the maximum occurring wall shear stress.

Thermal fatigue may occur downstream from mixing tee junctions where hot and cold water are turbulently mixed. Depending on the momentum of the incoming flows, different characteristic eddies are formed which create a local hot spot on the inner surface of the pipe. The hot spot have an unstable discrete boundary that spatially oscillates and this creates an oscillating multidimensional thermal load that induces high cycle thermal fatigue. To predict the thermal fatigue lifetime of a pipe subject to this kind of thermal load, transient FEM simulations with an ideal hot spot in two and three dimensions were conducted. The crack lengths induced by the frequency of the oscillating boundary was analysed and it was found that the most harmful frequency for a pipe is in the nondimensional frequency of 0.01 to 1.0. Additionally, the simulation method was validated against data publications of FAT3D experiments. The simulations have been compared to the current proposed Japanese design code for mixing tee junctions. The study shows that this multidimensional effect induces larger crack lengths than the current design code. The reason of this was analysed.

Pipes with a corrugated inner surface, as used in flexible pipes for gas production and transport, can be subject to Flow-Induced Pulsations when the flow velocities are higher than a certain onset velocity. The onset velocity for classical corrugated pipes can be predicted on basis of the geometry of the corrugations, the operational conditions and the geometry of the topside and subsea piping. A newly developed inner carcass design for flexible pipes features smaller corrugation cavities. The effect of narrow cavities on the whistling of corrugated cavities is evaluated. In this paper, small-scale tests performed on corrugated tubes are reported. The tested geometries include both “classical” profiles, similar to the inner profile of agraff flexible risers, and profiles with less typical variations, such as deeper and narrower cavities. These tests were performed in order to evaluate the validity of a prediction model for the onset of pulsations, for corrugated pipes with these kinds of atypical variations. The experimental results show that the validity of the model remains reasonable, except when the cavities are very narrow. In this case, the model becomes overly conservative. The deviation is attributed to the momentum thickness of the boundary layer, which is too large compared to the cavity width. In this case, any instability of the shear layer is destroyed, which prevents whistling. Furthermore, the shift towards higher frequencies of the acoustic source term due to narrower cavities, and the possible coupling with higher acoustic modes, are considered.

An innovative time integration method that incorporates spurious high-frequency dissipation capability into the so called “high precision direct integration algorithm” is presented, and its numerical stability and accuracy is discussed. The integration algorithm is named “high precision” to emphasize its numerical capability in reaching computer hardware precision. The proposed procedure employs the well-known state space approach to solve the simultaneous ordinary differential equations, the exact solution of which contains an exponential matrix to be efficiently computed using the truncated Taylor series expansion together with the power-of-two algorithm. The proposed method, belonging to the category of explicit methods, is found to provide better accuracy than many other existing time integration methods, and the integration scheme remains numerically stable over a wide range of frequencies of engineering interest. This paper is also devoted to study numerical accuracy of the Precise Integration Method in solving forced vibration problems, particularly near resonance conditions. The numerically obtained transfer functions are then compared with the analytical exact solution to detect spurious resonance. Finally, numerical examples are used to illustrate its high performance in numerical stability and accuracy. The proposed method carries the merit that can be directly applied to solve momentum equations of motion with exactly the same procedure.

We report on the derivation of a simplified but accurate model to describe gas flow through a bore-piston ring contact. This is achieved by making use of a scale analysis on the classical mass, momentum and energy equations assuming that the gas obeys ideal gas law. The main regime of interest for practical application in reciprocating machines, corresponding to the compressible flow with inertia is identified and is shown to be free of unsteady terms in the simplified flow equation. For this regime, a quasi analytical solution is further provided that allows the estimation of the axial gas flow rate through the contact. This predictive model is successfully compared to direct numerical simulations of the complete initial set of balance equations performed on a model aperture field of sinusoidal shape. This simplified quasi analytical solution is of particular practical interest since it allows an accurate gas flow rate estimate through a real contact using the aperture field as the only input datum which would not permit a tractable direct numerical simulation otherwise.

The effects of a distributed radial load on an elbow or bend within a piping system, caused by large changes in momentum due to fluid flow, are often represented by a single force. The method presented here will result in more accurate results. In this paper, equations are derived to predict deformations of a pipe bend or curved beam cantilever at its free end due to a uniformly distributed radial force. Assuming isotropic, linearly elastic materials of uniform cross section, uniform bend radius, and small deformations, equations are derived using Castigliano’s theorem. Deformation due to shear and axial stresses is also considered. Derived equations are validated through a case example which compares them to a model consisting of a number of straight beam elements assembled to model a curved beam. The example demonstrates that free end deformations can be integrated into a piping analysis program by using the direct stiffness method in order to obtain the resulting displacements, forces, and moments that result from restraint of the bend due to the stiffness of the attached pipe.

Fluid transients have been a major topic of research due to its possible fatal consequence. For example, the sudden closure of valves or operation of pumps can create pressure waves. High pressure pulses propagate through pipelines and generate huge unbalanced forces, known as waterhammer. A lack of consideration of fluid transient forces can even lead to pipe ruptures. The increasing interest in dynamic safety demands the necessity of deep considerations for fluid transients. An assessment process is required that includes the internal unbalanced forces and the dynamic system response in the pipework. This paper presents the comparison between a simplified method and time history analysis, which are analytical methods for transient forces. A simplified method is manual calculation based on the conservation of mass and momentum equations, and the impact loading theory. Time history analysis is computer simulation for transient analysis and piping structural analysis. The paper illustrates the deviations of result which contain waterhammer force, displacements and resultant stresses for the piping system by applying both analytical methods.

The paper presents a one-dimensional transient mathematical model of compressible thermal two-phase flows of multi-component gas mixture and water in pipes. The set of mass, momentum and enthalpy conservation equations is solved for the gas and water phases. Thermo-physical properties of multi-component gas mixture are calculated by solving the Equation of State (EOS) model. The Soave-Redlich-Kwong (SRK-EOS) model is chosen. Gas mixture viscosity is calculated on the basis of the Lee-Gonzales-Eakin (LGE) correlation. The proposed mathematical model is successfully validated on the experimental measurements of rapid decompression in conventional dry natural gas mixtures at low temperature and shows very good agreement with the experimental data at high and low pressure. The influence of temperature and water on rapid gas decompression process is investigated numerically. The HLP crack velocity model is used together with the proposed mathematical model in order to determine the minimum fracture arrest toughness in piping systems containing a natural gas mixture.

This paper describes hydrogen and nitrous oxide detonation experiments that were performed using an approximately 200-ft long 2-inch schedule 40 piping system. The objective of these experiments was to develop an understanding of the loads and forces imposed by internal detonation on piping combinations representative of a typical industrial piping system. The apparatus contained numerous straight pipe lengths with 90° and 45° bends, 90° elbows, and a tee along with rigid foundation supports that were connected to the pipe using typical u-bolt fasteners. As a detonation wave propagates through a gas-filled piping system, the pipe begins to respond globally once a detonation encounters a change in flow direction, such as a bend, causing a pressure imbalance due to both the internal detonation pressure and change in momentum of the reaction products. The resultant force imparts both axial forces and moments on the pipe exciting both extensional and bending modes. The test data was used to validate two finite element (FE) models developed using the ANSYS finite-element program: a hybrid model that made use of both shell and beam elements, to determine the interaction between shell and beam modes, and an all beam element model. An additional beam element model was developed using the Bechtel National Inc. software ME101 that was also found to be in agreement with the measured and ANSYS calculated frequencies and support loads. In addition to the detonation testing, the finite-element models were validated against experimental modal analysis data of the piping system that identified the primary modal frequencies and vectors. These data were compared to the modes extracted from finite-element models of the piping system.

A certain nuclear power plant uses a siphon to bring cooling water into the plant during emergency operations. This siphon was “hardened” by adding a nuclear grade vacuum system to the siphon high points. A float valve, sized for the required air flow rate, was used to prevent water carryover into the vacuum system. In order to pass the required flow rate, the valve’s orifice size would be the largest produced by any manufacturer to date. Initial system testing revealed that the valve was not fully opening due to the high vacuum and the design of the valve opening mechanism. Analysis and testing developed a solution to the opening problem. However, a new problem was introduced — excessive liquid carryover. The liquid carryover rate was estimated at 6 gallons per minute. The liquid carryover was postulated to be due to droplet entrainment as high velocity air bubbles entered the valve body, ruptured the air-water interface, and carried some of the resultant droplets through the valve outlet as the valve opened due to falling water level. Additionally, valve operating mechanism changes resulted in it responding slower. Using information learned from modifying the original valve, a new valve design was conceived that would be based on the following 3 principles: 1) liquid momentum must cause it to separate from the air upon entrance into the valve body; 2) enough distance and space must exist inside the body to allow gravity to act upon any liquid droplets to allow them to fall back into the liquid pool instead of being carried out of the valve toward the vacuum system; 3) valve operating mechanism must respond quickly enough to close the valve upon rapid liquid influx. The valve manufacturer reviewed the conceptual design and produced a prototype. The prototype valve was successfully tested with zero water carryover plus better flow versus pressure loss performance than the original valve. The valve also functioned acceptably during shaker table testing to simulate earthquake conditions.

NuScale Power, Inc. is commercializing a 45 Megawatt electric light water nuclear reactor NuScale Power Module (NPM). Each NPM includes a containment vessel, a reactor vessel, a nuclear reactor core, an integral steam generator, and an integral pressurizer. The NuScale Power Module is cooled by natural circulation. The primary coolant in the Reactor Pressure Vessel is heated in the nuclear core, it rises through a central riser, it spills over and encounters the helical coil steam generator, it is cooled as steam is generated inside the steam generator, and it is again heated in the nuclear core. The Steam Generator also must be designed to provide adequate heat transfer, to allow adequate primary reactor coolant flow, and to provide adequate steam flow to produce the required power output. This paper presents the CFD results that describe the transport phenomena on the heat transfer and fluid flow dynamics in helical coil steam generator tubes. The ultimate goal of the CFD modeling is to predict the steam outlet conditions associated with the chosen helical coil tube geometries, solving the primary and secondary flow region together coupled with the helical coil tube. However, current studies are focused on the primary side with the heat flux boundary condition assigned on the outer surface of the helical coil steam generator. In this study, the ANSYS CFX v. 12.1 [1] was used to solve the three-dimensional mass, momentum and energy equations. The helical coil steam generator has complex geometry and modeling entire geometry requires the enormous memory that is beyond our hardware capability and is not practical. Therefore, geometry was limited to 1 degree of the wedge and 5% of the total length in the middle. Only external flow, single phase flow around the helical coils, is simulated using the standard k-ε model and shear stress transport model. From the results of the numerical simulation, the pressure drop and temperature profiles were determined. It is important to understand thermal hydraulic phenomena for the design and performance prediction of the reactor internal.

This study discusses on the implementation of an upwind method for a one-dimensional two-fluid model including the surface tension effect in the momentum equations. This model consists of a complete set of six equations including two-mass, two-momentum, and two-internal energy conservation equations having all real eigenvalues. Based on this equation system with upwind numerical method, the present authors first make a pilot code and then solve some benchmark problems to verify whether this model and numerical method is able to properly solve some fundamental one-dimensional two-phase flow problems or not.

This paper reports the results of experiments, analytical models, and finite element simulations on the structural response of piping systems to internal detonation loading. Of particular interest are the interaction of detonations with 90° bends and the creation of forces that lead to axial and bending structural response of the piping system. The piping systems were fabricated using 304 stainless steel, 2-in. (50 mm) diameter schedule 40 commercial pipe with a nominal wall thickness of 0.154-in. (3.8 mm) and welded construction to ASME B31.3 standards. The piping was supported using custom brackets or cantilever beams fastened to steel plates that were bolted to the laboratory walls. Nearly-ideal detonations were used in a 30/70 H 2 -N 2 O mixture at 1 atm initial pressure and 300 K. The detonation speeds were close (within 1%) to the Chapman-Jouguet velocity and detonation cell sizes much smaller than the tube diameter. Pressure, displacement, acceleration and hoop, longitudinal, and support strains were measured using a high-speed (1 MHz) digital data acquisition system and calibrated signal conditioners. Detonation propagation through a bend generates a longitudinal stress wave in the piping that can be observed on the strain gauges and is predicted by both analytical models and finite element simulations. The peak magnitude of the bend force is approximately twice that due to the pressure alone since the peak momentum flux of the flow behind the detonation front is comparable to the pressure in the front. With relatively simple models, quantitative predictions of the bend forces can be made for the purposes of design or safety analysis of piping systems with internal detonations.

Temperature fluctuations generated by the mixing of hot and cold streams at T-junctions can cause high cycle thermal fatigue failure. One of the important parameters for determining the degree of thermal fatigue damage is related to the flow patterns downstream of the mixing junction. Many papers have been published identifying the characteristic equations for classifying the flow patterns at the T-junctions, which have been shown to be related to the momentum ratios between the main pipe flow and the branch pipe flow. The flow patterns and governing equations studied in the past are only applicable to 90-degree tee junctions (T-junctions). The intention of this paper is to extend the work of others to include angled tee junctions other than 90 degrees (Y-junctions). Y-junctions especially with 45-degree angle are also used in the chemical and refinery plants of our concern for lowering the pressure drop in the mixing zone and weakening the impingement of the branch pipe stream against the main pipe. This paper presents the proposal for a set of extended characteristic equations for classifying the flow patterns in mixing tees which can be applied to all angles of tee junctions. The modified characteristic equations have been verified using computational fluid dynamics (CFD) simulations. Models with branch angles other than 90 degrees have been used to confirm the robustness and validity of the modified equations. The effect of the pressure drop caused by the varying opening sizes has been shown to play only a minor role in determining the overall flow patterns, with the key variables being the velocity ratio and the interaction area between the flow in the main pipe and the jet exiting from the branch pipe. The results have also highlighted that the angle of the branch pipe has significant impact on increasing the velocity ratio range for the less damaging deflecting jet flow pattern, which is an important finding that could be used to extend the current design options for piping systems where high cycle thermal fatigue may be a concern.

The analysis of transient flow in a piping system containing one or more trapped gas volumes is considerably simplified when the liquid can be treated by bulk flow theory without propagation effects, complicated by moving boundaries. Many cases that have occurred in the power industry, which involve pump operation, valve opening and closure, and even some pipe ruptures, have time intervals and/or piping segments that can be shown to be dominated by bulk flow with negligible propagation effects. Often, propagation effects become important only for short time intervals at the beginning or toward the end of a given transient. This observation can greatly simplify the solution of such problems for maximum pressures and pipe loadings to a relatively simple procedure. Most models which have received attention or undergone development are based on the mass conservation and momentum principles, with gas state equations, and this is the correct approach for tracking liquid motion and gas volume compression. The energy conservation principle is not required to solve for the pressure and velocity of the liquid in the pipe, and to obtain the resulting forces. However, energy considerations are extremely useful in gaining physical insight to the overall problem. The major insight offered by the energy principle is based on the fact that when liquid flow is initiated in a pipe system containing gas pockets, the total mechanical energy added to fluid in the pipe must be distributed between the liquid and the gas in mechanical forms, namely kinetic, gravitational potential, and compressive or “spring” type energy. If one examines the gas and liquid when it has come to rest and neglects the kinetic and gravitational potential energies, the only stored energy forms are compressive. If the liquid is incompressible, or “rigid,” all the energy is stored in the gas and the resulting pressure is P R . If the liquid is elastic, or compressible, the energy will be shared so that both the gas and liquid will be at the same pressure, P . By comparing pressure P with P R , it is possible to determine if the compressible liquid is capable of storing enough energy to reduce the pressure significantly from that which would result if the liquid were incompressible and all the energy were stored in the gas. A procedure and several examples are given for starting with a given system, predicting the pressure with a rigid or incompressible liquid model, and then determining if a modified analysis is justified, incorporating liquid compressibility and propagation effects.