One-dimensional Momentum Equations Research Articles

Numerical investigation on transpiration cooling with phase change based on multiphysics coupled finite element method

Transpiration cooling with phase change is an efficient active thermal protection method in aerospace. However, conducting numerical simulation of transpiration cooling with phase change is still a challenge, because porous flow, heat transfer and phase change are coupled with each other, which lead to numerical difficulties. In this article, the previous model is modified and the Zero Thickness Area Model (ZTAM) is proposed, which improves the efficiency and convergence significantly. In addition, the local thermal nonequilibrium effect and latent heat can be considered. To verify this ZTAM model, one-dimensional control equations of mass, momentum, and energy transpiration are solved using finite element method. Furthermore, the position of phase change is determined by fixed point iteration numerical algorithm, which shows good convergence. Finally, the influence on cooling effect by heat flux, coolant mass flow rate, porosity and pore diameter is investigated numerically.

A NUMERICAL STUDY ON THE REDUCTION OF GREENHOUSE GASEOUS COMPONENT (CO2) DUE TO THE ADDITION OF H2 IN THE FUEL STREAM OF THE COUNTERFLOW CH4/AIR DIFFUSION FLAME

In this study, a series of 1-D and steady-state numerical simulations have been performed for the prediction of the effect of the addition of H2 on the characteristics of a non-sooting counterflow CH4/Air diffusion flame using detailed chemical reaction model, which is composed of 325 elementary chemical reactions and 53 chemical species. Under the steady-state assumption, a set of one-dimensional transport equations of mass, momentum, species, and energy along with the equation of state has been solved numerically at the atmospheric conditions over the counterflow configuration by exploiting an efficient numerical code, OPPDIF (a Fortran Program for Computing Opposed-Flow Diffusion Flames). The grid adaption technique has been used to achieve better convergence as well as to ensure the maximum accuracy of the simulated results. It is found that the flame temperature is increased due to the addition of H2 with CH4, which is injected into the fuel stream. The elevation in the temperature is caused by the augmentation of the integrated heat release rate of the elementary reactions supported by the active radicals (H, O, and OH), which are generated by the higher reactivity of H2. Besides, it is found that the mole fractions of H2O are increased as the percentage of H2 in the loading fuel (CH4) is increased and also, it is identified that the chain propagating reaction, OH + H2 => H2O + H is dominating one which produces highest amount of H2O. Furthermore, it is noticed that the indirect greenhouse gas or precursor, CO is reduced when H2 is added to CH4. Consequently, the mole fraction of the principle greenhouse gas, CO2 is decreased significantly when the fuel, CH4 percentage is modified by the higher percentage of H2. The sensitivity analysis of elementary reactions reveals the fact that the chemical reaction: OH + CO => H + CO2 is a dominating reaction in producing a lower amount of CO2 when the volume fraction of H2 is increased in the fuel (CH4) stream. In the presence of 75 % H2 in CH4, the pressure-dependent reaction, O + CO (+M) => CO2 (+M) appears as another chemical route that also generates greenhouse gas, CO2 but its contribution is negligibly small.

Simulation of decay heat removal by active means following emergency shutdown in SFRs

Decay heat removal (DHR) is an important safety function in sodium-cooled fast reactors (SFRs). Dedicated systems are provided in the plant for performing DHR functionalities. Transients following reactor shutdown need to be studied to understand the dynamics of DHR systems, design the systems, and establish reactor safety. DHR systems can operate by active as well as passive means, thanks to the phenomenon of natural circulation. In the present work, a DHR system riding on the Balance of Plant (BoP) of a typical medium-sized SFR has been modeled and incorporated into the in-house developed plant dynamics code DYANA-P. DYANA-P already has models of primary and secondary circuits of SFR. With this code enhancement, the simulation of DHR in an integrated manner became possible. In particular, thermal models of the Moisture Separator Tank (MST), Decay Heat Removal Condenser (DHRC), and hydraulic model of the DHRC flow circuit were developed. Two-phase one-dimensional mass, momentum, and energy conservation equations with homogeneous flow assumptions were used to create the models. Simulation of SCRAM in a typical SFR has been carried out to demonstrate the applications of the modified code. Two different DHR deployment schemes were studied and compared – 1. Using DHR systems designed for high pressure of ∼170 bar, 2. Using DHR systems designed for low pressure of ∼15 bar. In the first scheme, reactor cooling was achieved at the desired rate of ∼20 K/h. In the second scheme, the reactor cooling rate will be higher at ∼60 K/h.

Nonlinear Flutter Analysis of Labyrinth Seals

Abstract A simple nonlinear model to describe labyrinth seal flutter has been developed to assess the aeromechanic stability of straight-through labyrinth seals subjected to large gap variations. The model solves the one-dimensional integral mass, momentum, and energy equations of the seal for a prescribed motion numerically until a periodic state is reached. The model accounts for the effect, previously neglected, of high clearance variations on the stability. The results show that when the vibration amplitudes are small, the work-per-cycle coincides with the prediction of the Corral and Vega model (2018, “Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models. Part I: Theoretical Background,” ASME J. Turbomach., 140(10), p. 121006) and Corral et al. (2021, “Higher-Order Conceptual Model for Seal Flutter,” ASME J. Turbomach., 143(7), p. 071006), but for large vibration amplitudes nonlinearities alter the stability limit. In realistic cases, when the discharge time of the seal is much longer than the vibration period, the nonlinear effects are significant and tend to increase the unstable range of operating conditions. Furthermore, seals supported either on the high-pressure or low-pressure sides, stable for small vibration amplitudes, can destabilize when the vibration amplitude increases. The linear stability, though close in many situations to the nonlinear threshold, is not conservative, and attention must be paid to nonlinear effects.

Partition method of wall friction and interfacial drag force model for horizontal two-phase flows

The improvement of thermal-hydraulic analysis techniques is essential to ensure the safety and reliability of nuclear power plants. The one-dimensional two-fluid model has been adopted in state-of-the-art thermal-hydraulic system codes. Current constitutive equations used in the system codes reach a mature level. Some exceptions are the partition method of wall friction in the momentum equation of the two-fluid model and the interfacial drag force model for a horizontal two-phase flow. This study is focused on deriving the partition method of wall friction in the momentum equation of the two-fluid model and modeling the interfacial drag force model for a horizontal bubbly flow. The one-dimensional momentum equation in the two-fluid model is derived from the local momentum equation. The derived one-dimensional momentum equation demonstrates that total wall friction should be apportioned to gas and liquid phases based on the phasic volume fraction, which is the same as that used in the SPACE code. The constitutive equations for the interfacial drag force are also identified. Based on the assessments, the Rassame-Hibiki correlation, Hibiki-Ishii correlation, Ishii-Zuber correlation, and Rassame-Hibiki correlation are recommended for computing the distribution parameter, interfacial area concentration, drag coefficient, and relative velocity covariance of a horizontal bubbly flow, respectively.

Development of dynamic motion models of SPACE code for ocean nuclear reactor analysis

Lately, ocean nuclear power plants have attracted attention as one of diverse uses of nuclear power plants. Because ocean nuclear power plants are movable or transportable, it is necessary to analyze the thermal hydraulics in a moving frame of reference, and computer codes have been developed to predict thermal hydraulics in large moving systems. The purpose of this study is to incorporate a three dimensional dynamic motion model into the SPACE code (Safety and Performance Analysis CodE) so that the code is able to analyze thermal hydraulics in an ocean nuclear power plant. A rotation system that describes three-dimensional rotations about an arbitrary axis was implemented, and modifications were made to the one-dimensional momentum equations to reflect the rectilinear and rotational acceleration effects. To demonstrate the code's ability to solve a problem utilizing a rotational frame of reference, code calculations were conducted on various conceptual problems in the two-dimensional and three-dimensional pipeline loops. In particular, the code results for the three-dimensional pipeline loop with a tilted rotation axis agreed well with the multi-dimensional CFD results.

A Novel Analytical Method for Evaluating the Characteristics of Hydraulic Jump at a Positive Step

We present a new method to evaluate the hydraulic jump characteristics in a horizontal rectangular channel with a positive step. We considered the flow curvature effect and the free surface’s small rise at the A-type hydraulic jump’s end. First, we present a novel method to give jump length estimation based on the similarity of the jump and the turbulent wall-jet, considering the pressure gradient. Then, considering the jump as a curvilinear flow and using a one-dimensional momentum equation, we present an accurate expression for the conjugate flow depth regarding the initial Froude number and step height. Finally, we compute the jump’s energy dissipation rate. Compared to the theoretical models for conjugate flow depth in a hydraulic jump, the proposed equation in this study fit the experimental data better, even for high steps and large initial Froude numbers. However, for low Froude numbers (F1 < 5), the equation was less accurate in estimating the jump length. Regarding the jump’s energy dissipation rate, the results agreed well with the experimental data from previous investigations. However, it is noted that the increased energy dissipation rate dwindled in larger Froude numbers.

One-dimensional numerical simulation of cavitation surge in pumping system considering cavity response delay

Cavitation instabilities such as rotating cavitation and cavitation surge often occur in high speed turbopumps. In the present study, numerical simulation of cavitation surge, which numerically solves the one-dimensional momentum and continuity equations with modelled dynamic cavitation characteristics, is conducted. The phase lag of cavity response against the inlet pressure and the suction flow rate variations is modeled in the form of the first-order lag system considering that the frequency of cavitation surge is small. Differently from linear stability analysis made in our previous study, the equations are solved in time domain with retaining some of non-linear terms. The method is validated through the comparison with the stability analysis. Then the effect of phase lags in dynamic cavitation characteristics is examined.

Study on interfacial friction in the inverted annular film boiling regime

The interfacial friction factor is an important closure model in the two-fluid model for the inverted annular film boiling (IAFB) regime. Almost all the interfacial friction factor correlations used in IAFB modeling were obtained based on experimental data from annular two-phase flows. This paper first reviews the interfacial friction factor models/correlations used in IAFB modeling. A laminar vapor film model is then derived for the smooth IAFB region based on a laminar flow assumption. Experimental data obtained from subcooled and low-quality film boiling experiments in the literature are reduced to calculate the interfacial friction factor by solving one-dimensional two-fluid momentum equations for the liquid and vapor phases. The trend of the interfacial friction factor indicates that the transition from the smooth to wavy IAFB regions occurs at a void fraction range of 0.2 to 0.4. Comparisons between the laminar vapor film model and corresponding experimental data and nine correlations in the literature indicate that the laminar vapor film model appears to provide the lower bound of the interfacial friction factor in the smooth IAFB region. Two interfacial friction factor correlations from the literature are recommended for the smooth and wavy IAFB regions based on the analysis. It can be concluded that the interfacial friction factor in the smooth IAFB region is dependent on the gas Reynolds number and vapor film thickness, while in the wavy IAFB region, it is primarily dependent on the amplitude of the interfacial waves.

Hydraulic Characteristics of Submerged Flow In Non-Prismatic Stilling Basins.(Dept.C)

Recently, the non-prismatic stilling basins attracted many researchers due to their effective dissipation of excess energy. Radial stilling basins are characterizing with symmetric flow patterns that in turn minimize the expected damages of the erodible bed downstream the stilling basins. In this research, the submerged flow in radial stilling basin was provided with negative step and an end sill was analyzed both theoretically and experimentally. The effects of various parameters such as submergence, relative height of sill, relative height of step and inlet Froude number were presented and discussed. The experimental study is conducted on 30 cm wide, 45 cm deep and 12 m long open channel. The theoretical study utilized the one-dimensional momentum and continuity equations. The developed equations are verified using the obtained experimental data. Also, prediction models to estimate the length of jump are developed using the dimensional analysis and statistical methods. The models are calibrated and verified using the experimental data.

Analysis of Herchel-Bulkley fluid flows in planar and circular microducts with constant heat flux and slip boundary condition

The hydrodynamically and thermally fully developed Herchel-Bulkley fluids flows in circular and planar microducts with constant heat flux at the wall was investigated by applying second law analysis while imposing slip boundary condition and including viscous dissipation. To determine tendencies of effects of Brinkman number, slip boundary condition, flow behavior index, cross sectional ratio and heat flux on the Nusselt number, entropy generation rate and thus, irreversibility distribution ratio were investigated. One-dimensional momentum and energy equations with viscous dissipation term were analytically solved to obtain equations for the velocity, temperature distribution, Nusselt number, entropy generation and irreversibility distribution ratio. It was observed that the slip coefficient and the Brinkman number and thus viscous dissipation affecting the Nusselt number, entropy generation and thus, irreversibility distribution ratio cannot be ignored in the analysis. This research illustrates that the Nu increases with increasing the slip coefficient and decreases with increasing the Br and n. The entropy generation rate and thus, the irreversibility distribution ratio increases with increasing the n and Br and decreases with increasing the β. The obtained results from this study are compared with those obtained from the related experimental and theoretical studies and reasonable agreement is achieved.

Analysis of Optimum Performance of Air Vessels Used in Damping Water Hammer Pressure Wave

Transient flow or water hammer is the sudden change of flow conditions due to an unexpected event such as a sudden closing of a valve installed on the pipeline system. Water hammer can pose serious threats to the pipeline system, as the flow's pressure can be changed above or below the limits which the pipeline network can withstand. Controlling the water hammer pressure becomes indispensable, so a transient surge analysis is required to investigate the points of severe changes along with the pipeline system. Compressed air vessel (CAV) is a pressure control device that is used to control the positive and negative pressure changes. There are two parameters affect the sizing of the air vessel; the initial trapped air volume and throttling the aperture of the air vessel. A computer model based on the unsteady one-dimensional flow equations of momentum and continuity is established to examine the proper size of the two parameters. The equations are solved by the method of characteristics while the vessel is modeled as a quasi-one-dimensional flow system. An experimental test rig provided with a rapid closing magnetic valve and pressure sensors are used to validate the model results. Both the experimental and model results show the high capability of the air vessel to dampen water hammer pressure. In addition, the results show that the introduced throttling action has a wide effect on the size of the air vessel and that the diameter of the throttling shall not be less than 0.3 the diameter of the main pipe. Besides, the initial volume of the trapped air should be in the range of 53% to 78% of the total volume of the air vessel, so that the air vessel works effectively with optimum value of 60%.

Numerical and experimental investigation of the steady-state performance characteristics of loop heat pipes

A numerical model to analyze the steady-state operational characteristics of loop heat pipes (LHPs) is presented. The mathematical model uses a new approach and is developed in a way that it can be easily extended to a transient model. The LHP components are discretized into control volumes. The one-dimensional mass, momentum and energy conservation equations are solved by using a staggered grid. The model uses both single-phase and two-phase correlations to determine the heat transfer coefficient and friction factor for each control volume. All the fluid interfaces in the LHP is tracked by using a phase-check algorithm. Four main experimentally-determined accommodation parameters are considered in the model. Later, it is shown that two of these parameters can be ignored without a significant effect on the results. The model results are validated by comparing them with experimental values obtained from the two different LHPs and the predictive capability of the model is demonstrated. Finally, a detailed sensitivity analysis is presented to study the effects of various essential variables including the accommodation parameters on the model results.

A method of estimating aerosol particle removal rates using one-dimensional two-fluid equations for venturi scrubbers in filtered containment venting

One-dimensional two-fluid mass, momentum, and energy conservation equations for gas and liquid droplet and the particle removal rate are formulated to estimate aerosol particle removal rates of venturi scrubbers used in filtered containment venting systems. Compared with existing one-dimensional models, present method enables describing actual two-phase interactions by calculating the local gas and droplet velocities hence local droplet diameters as well as the other physical properties. The results show that the particles are mostly removed at the entrance region of the throat where droplets are introduced and thus the two-phase velocity difference is the largest. This coincides well with previous experimental studies. The present model predicts the existing experimental data considered here better than the previous one- or two-dimensional models. Especially, in the conditions of low liquid-to-gas flow ratio, the predictions of the present model are very close to the experimental data.

Study on the similarity of wet gas pressure drop in long-throat Venturi

This study investigates the effect of pipe diameter on pressure drop with the same diameter ratio, similar pressure-sampling position and throat length of long-throat Venturi. Considering the factors including the void fraction, the friction between the two phases and the entrainment in the gas core, the one-dimensional momentum equation for gas has been solved in the axial direction of long-throat Venturi. A novel void fraction model is established, by considering the effects of dryness and gas-liquid density ratio, then predicting the distribution of wet gas static pressure between the two pressure tapings of the long-throat Venturi. The comparison between the values predicted by the model and those measured experimentally reveals that all the relative deviations of the predicted points by the modified model were within ±15%. In the same entrance conditions, the effect of pipe diameter on pressure drop in long-throat Venturi is similar.

Study of intake manifolds of an internal combustion engine: A new geometry based on experimental results and numerical simulations

During the intake period in internal combustion engines, it has been noted that the cylinder volume is not completely occupied due to the variation of specific volume and the pressure drop along the supply system. Consequently, the volumetric efficiency in the cylinder is lowered, negatively affecting the engine power output. In this paper, numerical and experimental studies were performed to investigate different geometries of intake manifolds of internal combustion engines, with the aim of improving its efficiency. The numerical solutions were obtained through a 1D commercial code (GT-Power®) which solves the one-dimensional equations of continuity, momentum and energy, using a finite volume scheme. An experimental set-up was built and mass flow rate and pressure measurements of the intake air were carried out. The set-up is composed by an internal combustion engine, driven by an electric motor, and the experiments were performed without fuel and combustion. Numerical and experimental results were compared and a good agreement was found, thus confirming that the commercial code used is capable of simulating the kind of phenomena studied. Based on the Helmholtz resonator effect theory, a new intake manifold geometry was proposed. Measurements showed the novel geometry to improve the volumetric efficiency of the engine by 6% at 3500 rpm, the more common speed for this kind of engine, consequently increasing the indicated power by 3.68 kW (4.93 hp). The observed increase is worthy of note, considering the inherent difficulty to improve the volumetric efficiency near its maximum value for this type of engine, occurring around 3000 rpm.

The comprehensive analysis of coolability limits of passive external reactor vessel cooling under in-vessel retention

The induced two-phase natural circulation flow and the critical heat flux (CHF) on the outer surface of lower head are crucial factors for the assessment of the external reactor vessel cooling (ERVC) under in-vessel retention which is considered as an effective management strategy to mitigate the core degradation accident. In this work combined with the two-phase drift flux model, an analysis program for the induced two-phase circulation flow has been established based on one-dimensional steady-state mass, momentum and energy conservation equations. The calculation results are compared with the simulation results using RELAP5 within the same geometric parameters to verify the rationality and correctness of the analysis program. Combining the analysis program with the SULTAN CHF correlation, the characteristics of induced two-phase flow and the CHF on the outer surface of lower head are studied comprehensively to analyze the coolability limits of ERVC under in-vessel retention. The effects of concerned parameters including subcooling of water, decay power of molten corium, flooding levels, height of riser channel, loss coefficient at the inlet and gap clearance between the reactor pressure vessel and thermal insulation are studied.

Thermo-hydraulic characteristics and second-law analysis of a single-phase natural circulation loop with end heat exchangers

In this work, a three-dimensional transient numerical simulation of heat transfer characteristics in a single-phase natural circulation loop (SPNCL) with water as the working fluid in the loop and end heat exchangers is performed. A 3-D computational model of a SPNCL with end heat exchangers is developed. The second-law analysis for the SPNCL is presented and total entropy generation is given. Meanwhile, the transient response is obtained based on the finite difference method solving the one-dimensional momentum and energy equations. Firstly, a mesh dependence test and a model verification with the laminar correlation presented by Vijayan are conducted. Then the effects of driving temperature difference and heating or cooling fluid velocity on the thermos-hydraulic performance and dimensionless entropy generation number are investigated by fixing the cooling fluids temperature of 273K. Results show that a stable flow can be reached for all the cases. The formation of the stable natural circulation is the consequence of coupling between heating and cooling fluid. The time of quiescent state is relatively short due to the thermal asymmetry of the SPNCL. The mass flow rate and heat transfer rate at steady state increase with the increase of driving temperature difference and heating or cooling fluid velocity. It is obviously shown the internal heat transfer coefficient is mainly dependent on the driving temperature difference and increases with the increase of driving temperature difference. The thermal effectiveness and dimensionless entropy generation number also increase with the increase of driving temperature difference and heat transfer irreversibility is dominated in total irreversibility. Moreover, the time evolution of dimensionless mass flow rate shows a damped oscillation behavior.

Development of one-dimensional two-fluid model with consideration of void fraction covariance effect

ABSTRACT Accurate evaluation of gas-liquid two-phase flow behavior within rod bundle geometry is crucial for the safety assessment of the nuclear power plants. In safety assessment codes, two-phase flow in rod bundle geometry has been treated as a one-dimensional flow. In order to obtain the reliable one-dimensional two-fluid model, it is essential to utilize proper area-averaged models for governing equations and constitutive relations. The area-averaged interfacial drag term utilized to evaluate two-phase interfacial drag force is typically given by the drift-flux parameters which consider the velocity profile in two-phase flow fields. However, in a rigorous sense, the covariance due to void fraction profile is ignored in traditional formulations. In this paper, the rigorous formulation of one-dimensional momentum equation was derived by taking consideration of void fraction covariance, and a new set of one-dimensional momentum equation and constitutive relations for interfacial drag was proposed. The newly obtained set of formulations was embedded into TRAC-BF1 code and numerical simulation was performed to compare against the traditional model without covariance. It was found that effect of covariance was almost negligible for steady-state adiabatic conditions, but for high void fraction condition with added perturbation, the traditional model underpredicted the damping ratio at around 8%.

Novel online simulation-ready models of conjugate heat transfer in combustion chamber waterwall tubes of supercritical power boilers

This paper presents two models of fluid heating in waterwall tubes of supercritical steam boilers. The models are named 1D/2D and 1D/3D. The models are formulated and discussed in detail with an industrial application example. For the 1D/2D model, a two-dimensional (2D) transient heat conduction equation is solved for the tube wall with the fin domain, while one-dimensional (1D) mass, momentum, and energy equations are solved for the fluid domain. The 1D/3D model considers a three-dimensional tube wall domain (3D) and one-dimensional fluid domain. At the fluid-solid interface, a conjugate heat transfer model is applied. The model is based on convective flux between the fluid and solid domains. Nonlinear governing balance equations of mass, momentum, and energy for fluid are solved using the forward time backward space (FTBS) scheme. The proposed models allow incorporation of the effect of heat flux nonuniformities along the waterwall tube and on its outer circumference. Transient simulations are carried out to determine the temperature histories for both the fluid and the tube wall in the selected cross sections. The computations are performed using the finite volume method formulation. The results obtained from the 1D/2D model are nearly the same as those from the 1D/3D model; however, the computation time is more than five times shorter. Therefore, an efficient 1D/2D model can be used in power unit simulators.