Steady-state Conservation Equations Research Articles

Theoretical modeling of the boiling annular-mist flow in vertical channels

Accurate prediction of the boiling annular-mist flow as well as the occurrence of the dryout is of particular importance for the safe operation of heat systems. Therefore, this paper focuses on the analysis of the boiling annular-mist flow and aims to propose a mechanistic model to describe this boiling process. Firstly, the physical model is established which divides the flow field into the bubble layer region and the core region in the radial direction. The bubble layer serves as a pseudo boundary for generating bubbles, whose thickness is adopted as the bubble departure diameter. In addition, the annular-mist flow is divided into the core region evaporation flow section and the bubble layer evaporation flow section in the axial direction, based on the relationship between the thickness of the bubble layer and the film thickness. Then, a set of two-dimensional steady-state conservation equations is proposed, which considers the energy, mass and momentum exchanges in both the bubble layer region and the core region. The accuracy of this model is verified with the experimental data and simulation results of void fraction and CHF. Based on this model, the variations of two-phase flow parameters in the bubble layer region and the core region can be obtained separately, such as the velocity, void fraction and pressure. The dryout point can be predicted based on the variation of the film thickness. These findings promote the fundamental understanding of the thermal–hydraulic characteristics of the annular-mist flow.

Quasi-Steady Simulation of Glaze Ice Accretion and Heat Transfer in the Supercooled Large Droplet Regime in Atmospheric Turbulent Air Flow

Abstract To avoid the use of a computationally intensive full unsteady simulation while providing an accurate solution, a quasi-steady simulation has been performed to study glaze and mixed ice accretion in the supercooled large droplet (SLD) regime in turbulent flow of atmospheric air. We have attempted to find the minimum time-step necessary to adequately simulate the icing process on an airfoil surface. Based on node displacement, a mesh morphing scheme has been adopted in the computations to account for the moving boundaries that are caused by the continuous ice buildup. We have modeled ice accretion on an airfoil surface for a time period of 232 s using several time steps. At each time-step we solved the steady-state conservation equations for the air and droplet phases and then used the results as initial conditions for the next time position. The magnitude of the time-step ranged from a least accurate two-shot simulation (where the time-step is 116 s) to a most accurate 2320-shot simulation (where the time-step is 0.1 s). In-between these two extreme time steps, we have performed a three-shot simulation (where the time-step is 77.33 s), a four-shot simulation (where the time-step is 58 s), a six-shot simulation (where the time-step is 38.67 s), a 46-shot simulation (where the time-step is 5 s), and a 232-shot simulation (where the time-step is 1 s). We have done so to find out the degree of accuracy (or inaccuracy) of the multishot simulation approach and to find out the appropriate time-step needed for a successful and valid quasi-steady simulation. A valid quasi-steady simulation needs to use a time-step that is small enough to reproduce the full time-dependent solution within a very small error band. We have found that both the 1 s and the 0.1 s time steps produce virtually identical results. This is the primary litmus test that proves the validity of the quasi-steady-state assumption. The results adopted in this paper are thus all based on the more conservative 0.1 s time-step. In the process of performing the simulations, remeshing was required in order to maintain the grid density in zones of high curvature to be able to capture the full physics in those zones. After successfully modeling glaze ice accretion over the airfoil surface using the 0.1 s quasi-steady simulation approach, the effects of supercooled large droplets (SLDs) impacting the surface have been examined and presented in terms of the variation of the local collection efficiency, the water film thickness, and the heat transfer rate. Examination of the variation of the angle that the ice horn makes with the airfoil chord line demonstrated a 20% improvement in angle prediction when the time-step is reduced from 116 s to 0.1 s. The analysis also reveals a 12% or 8.5% increase in the maximum collection efficiency, βmax, depending on whether the value of the liquid water content (LWC) has been doubled from 0.5 g/m3 to 1 g/m3, or the value of the freestream velocity has been doubled from 75 m/s to 150 m/s, respectively. Because of the need to monitor the local collection efficiency and convective heat fluxes at each shot (regardless of the number of shots employed), the approach adopted here was found to be effective in successively and successfully reproducing the curvature of the glaze ice horn.

A BUBBLE-LAYER-BASED MECHANISTIC MODEL FOR THE SLIGHTLY SUBCOOLED FLOW BOILING IN VERTICAL TUBES

Bubble behaviors and interactions are quite complex in the slightly subcooled flow boiling including sliding, lift-off and condensation. Currently, the predicting of this process mainly relies on numerical simulations combined with bubble-behavior-related sub-models, or a combination of one-dimensional empirical correlations without considering bubble behaviors. In contrast, the theoretical analysis is still lacking. Therefore, the present paper aims to develop a mechanistic model for this process which divides the flow field into the bubble layer region and the core region based on a bubble layer adjacent to the wall. Bubble behaviors in each region, mass and energy exchanges at the interfaces of different regions, and variations of parameters along the channel direction are considered and analyzed through a new set of two-dimensional steady-state conservation equations of mass and energy combined with bubble-behavior-related sub-models. Based on this model, void fraction and liquid temperature distributions are obtained which are verified with experimental results with fairly good accuracy. In addition, distributions of velocity and mass flow rate in each region in the axial direction, and mass flow rate between different regions in the radial direction are also obtained and analyzed. In comparison, the present model can reveal more detailed information than one-dimensional correlations and provides a simple, fast and stable method for the predictions of subcooled flow boiling especially for industrial applications.

Pipeline design with flow assurance constraints in offshore production lines

Multiphase flow in petroleum pipelines became more challenging due to the emergence of deeper wells at extreme environmental conditions. Flow assurance strategies ensure the production of hydrocarbons uninterruptedly. In this work we propose a simple and straightforward framework for design strategies for flow assurance constraints, implementing the most widely used models and correlations available. One-dimensional steady-state conservation equations were used for the modeling through a system of differential-algebraic equations, extending the previous work of Teixeira et al. (Rev IFP Energies Nouvelles 70(3):497–510, 2015) to incorporate flow assurance principles. Design strategies used for flowline internal diameter and insulation layer sizing were aggregated to the process modeling. The proposed modeling approach provided results with great agreement with the standard software used in oil industry (OLGA®) at industrial full-scale flow conditions, therefore providing a simple, robust and accurate tool for flow assurance calculations. Also comparing to OLGA®, the proposed framework presented a much shorter simulation time when more refined pipeline discretization is required. The strategy adopted here proved to be efficient in making initial estimates of production field design, in a faster way.

Fire Smoke Transport and Opacity Reduced-Order Model (Fire-STORM): A New Computer Model for High-Rise Fire Smoke Simulations

The problem of smoke spread through elevator shafts in high rise buildings is analyzed theoretically and numerically in this paper. While experiments and computational fluid dynamics (CFD) models have been used for such exercises, there is a need for fast reduced-order models for such scenarios. Towards this goal, a transient network model called High-rise fire smoke transport and opacity reduced-order model (Fire-STORM) was developed to investigate heat and mass transfer through the elevator shaft during fires. The model numerically solves the coupled set of differential equations of the fire floor in conjunction with the steady state conservation equations of the elevator shaft. The model is validated in two stages. First, the stack effect in a non-fire scenario is analyzed. Pressure differences through exterior doors and elevator doors are compared with experimental data available in the literature and results of a computational fluid dynamics tool. Then, a first-floor fire scenario is considered for the same high-rise building in four different cases which are combinations of different building tightness and ambient temperatures. The results are compared with CFD simulations. For the four different building envelope and ambient thermal conditions, the soot mass fractions and optical visibilities were calculated and compared to CFD predictions. Overall, Fire-STORM is a simple and fast tool to model the evolution of heat and mass transfer in a high-rise building affected by fire. While Fire-STORM is excellent in predicting transient smoke transport for buildings with loose envelopes, it should be used with caution for buildings with tight envelopes since the errors for these cases are relatively high. Despite this, the relative computational speed difference between Fire-STORM and the CFD model highlights the utility of a reduced-order model for firefighter decision making and building control system design.

Indirect noise prediction in compound, multi-stream nozzle flows

An analytical framework to compute the acoustic response of one-dimensional planar waves in a compound-compressible nozzle is presented. Using the steady-state conservation equations, a set of algebraic relations governing compound flow is solved numerically. In each stream, the linearized Euler equations are recast using the invariant formulation and solved with a Magnus expansion following Duran & Moreau (Journal of Fluid Mechanics 723 (2013) 190–231). The modeled acoustic response is compared to the direct solution of the non-linear Euler equations with a maximum error of 5%. A parametric investigation of the acoustic response of a two-stream compound nozzle is presented. The effects of the bypass inflow Mach number and bypass ratio on the magnitude and phase of the transmitted and reflected acoustic waves are studied. It is shown that indirect noise generation is maximum in the peripheral bypass stream, not the core jet and that the initial Mach number of the bypass greatly impacts the phase shift of the resulting acoustic waves. The work provides an insight for acoustic response mitigation of high-speed confined jets.

Variable viscosity effects on third-grade liquid flow in post-treatment analysis of wire coating in the presence of nanoparticles

PurposeThe features of coated wire product are measured by the flow and heat transport occurring in the interior of dies. Therefore, an understanding of characteristics of polymers momentum, heat mass transfer and wall shear stress is of great interest. Enhancement of heat transfer rate is fundamental need of wire coating process. Therefore, this study aims to investigate the effect of suspended nanoparticles in heat and mass transport phenomena of third-grade liquid in post-treatment of wire coating process. Buongiorno model for nanofluid is adopted. Two cases of temperature dependent viscosity are considered.Design/methodology/approachThe governing equations are modelled with the help of steady-state conservation equations of mass, momentum, energy and nanoparticle concentration. Some appropriate dimensionless variables are introduced. Numerical solutions for the nonlinear problem are developed through Runge–Kutta–Fehlberg technique. The outcome of sundry variables for dimensionless flow, thermal and nanoparticle volume fraction fields are scrutinised through graphical illustrations.FindingsThe study’s numerical results disclose that the force on the total wire surface and shear stress at the surface in case of Reynolds Model dominate Vogel’s Model case. Impact of nanoparticles is constructive for force on the total wire surface and shear stress at the surface. The velocity of the coating material can be enhanced by the non-Newtonian property.Practical implicationsThis study may provide useful information to improve the wire coating technology.Originality/valueEffect of nanoparticles in wire coating analysis by using Brownian motion and thermophoresis slip mechanisms is investigated for the first time. Two different models for variable viscosity are used.

Investigating the effect of Brownian motion and viscous dissipation on the nanofluid heat transfer in a trapezoidal microchannel heat sink

In the present study, laminar forced convection of copper-oxide nanofluid in a trapezoidal microchannel-heat-sink (MCHS) is studied using the Eulerian–Eulerian two-phase approach. The incompressible, three dimensional and steady state conservation equations are solved using finite volume method. The substrate material is assumed to be silicon, mean diameter of spherical nanoparticles are considered 100nm and in the 1–4% volume concentration range. The effects of viscous dissipation, Brownian motion and geometry change on thermal performance of MCHS are evaluated. It is observed that the Brownian motion depends on three parameters, namely, nanofluid inlet temperature, nanoparticles diameter and volume fraction. Also, it is found that with the increase in aspect factor at constant Re, average Nusselt number and pressure drop reduce, while at a specified length of MCHS, average Nusselt number decreases. Finally, it is shown that considering the viscous dissipation, pressure drop varies very slowly, while with the increase in nanoparticles volume fraction and therefore increase in viscous dissipation, heat transfer reduces non-linearly.

Modeling of a two-phase CO2-filled vertical borehole for geothermal heat pump applications

A numerical model has been developed to study the complex thermal behavior of a CO2-filled vertical geothermal borehole consisting of a long copper U-tube embedded in a solid material (grout). The model accounts for combinations of single-phase and two-phase fluid flow which may occur along the U-tube length. An explicit solution for fully coupled steady-state conservation equations of mass, momentum and energy is applied. The borehole is divided into small volume elements due to relatively significant changes in the thermophysical properties of the fluid. Borehole wall temperature variations as well as thermal interaction between tube sections are taken into account and the model predicts the fluid temperature, pressure and two-phase quality profiles.A case in which the borehole is assumed to be an evaporator of a direct-expansion ground coupled heat pump is studied. The model is used to assess the thermal performance of the borehole as well as pressure, temperature, and quality variations of CO2 along the U-tube. According to the results, the pressure variation of the fluid in the two-phase region has unavoidable consequences on the temperature variation. Results indicate that the proposed borehole can extract relatively large amounts of energy due to the superior two-phase heat transfer characteristics of CO2. Furthermore, the heat exchange rate between the ground and the upward flow leg in the two-phase region is greater than that between the ground and the downward flow, which is the opposite of conventional boreholes with a performance drop from the downward flow to the upward flow.

A numerical study on the ability to predict the heat release rate using CH* chemiluminescence in non-sooting counterflow diffusion flames

Numerical studies on 1-D, non-sooting counterflow diffusion flames were performed to determine the precision with which the total heat release rate can be calculated using light emission, namely, chemiluminescence, from the reaction zone. A detailed reaction mechanism, incorporating sub-reaction models for excited state radicals (CH* and OH*, where * denotes the excited state), was employed in this study. A set of 1-D, steady state conservation equations was solved under standard atmospheric conditions over a counterflow configuration utilizing the CHEMKIN-PRO package. A variety of fuels (CH4 and C3H8), velocities (0.1m/s − near the extinction condition), diluents (N2, H2O, CO2, and Ar), detailed reaction mechanisms for C1–C3 hydrocarbons (GRI-Mech 3.0, Hai Wang’s-, and San Diego-mechanism), different sub-reaction models for excited radicals, and excited radical transport properties were examined for the current purpose. It was found that a one-to-one correlation between total chemiluminescence from CH* and the total heat release rate cannot be sustained when the flame experiences a relatively high stretch and dilution, even though the condition is still far away from extinction. This trend is consistent with the different types of fuels, and it is understood that the reduction of the ethynyl radical (C2H), a potential precursor of CH*, is the main cause of the one-to-one correlation not being sustained. To this end, it was concluded that the observable light emission can only be used to predict the total heat release rate when non-sooting diffusion flames exist under velocity conditions from 0.1m/s to 1.5m/s. In other words, the chemiluminescence intensity does not always correlate with the total heat release rate of highly stretched flames found in practical combustors.

Computationally-efficient hybrid strategy for mechanistic modeling of fuel cell stacks

In general, detailed mechanistic models for fuel cell stacks that seek to capture the local transport phenomena are computationally expensive. In this context, we propose a hybrid modeling strategy, in which the steady-state conservation equations are solved iteratively in two separate groups: The first comprises asymptotically-reduced governing equations for momentum, mass, and species, which are solved as a transient-like propagation problem; the second comprises the full set of equations for energy and charge, which are solved as an elliptic stationary problem. Physically, the segregation is justified by the nature of the dependent variables; in essence, the first group covers local variables on the cell level and the second involves global variables on the stack level. We demonstrate the methodology for a steady-state detailed mechanistic model of a proton exchange membrane fuel cell stack comprising 2 to 350 cells subjected to non-uniform operating conditions across the cells; e.g., a stack comprising 350 cells takes less than an hour to solve. The proposed methodology is generic and can also be employed for other systems where transport phenomena occur on different length scales and involve some form of slenderness.

Sedimentary radioactive tracers and diffusive models

This paper examines the underlying assumptions and consequences of applying a steady-state equation to sediment profiles of radioactive tracers in order to deconvolute sedimentation from bioturbation processes modelled as a diffusive type process. Several factors follow immediately from this investigation: (i) if the observed radioactive concentration increases with depth over any finite depth range then the proposed steady-state, constant flux equation is not applicable. Any increase in radioactive concentration with depth implies a negative mixing coefficient which is a physical impossibility; (ii) when the radioactive concentration systematically decreases with increasing sedimentary depth then solutions to the steady-state conservation equation exist only when either the constant solid state flux to the sediment surface is small enough so that a positive mixing coefficient results or when the mixing coefficient is small enough so that a positive flux results. If the radioactive concentration, porosity and/or density of the solid phase are such that the proposed equation is inappropriate (because no physically acceptable solution exists) then one must abandon the proposed steady-state equation. Further: if the flux of solid sediment to the sediment surface varies with time then, of course, a steady-state conservation equation is also inappropriate. Simple examples illustrate that the assumption of steady-state restricts the applicability of this modelling approach to a relatively small sub-set of expected situations in the real world.

Thermal performance of a cubic cavity with a solar control coating deposited to a vertical semitransparent wall

We present a theoretical and experimental study of combined heat transfer in a cubic cavity containing non-participating air. The cubic cavity has a vertical semitransparent wall (glazing) with a solar control coating (SCC); an opaque black isothermal wall forms its opposite side. The top, bottom and side walls are opaque, gray and adiabatic. In the theoretical study, the 3-D steady state conservation equations for the mass, momentum and energy, along with the coupled radiation and conduction equations, were solved numerically by the finite volume method. The conduction for the semitransparent wall and the radiative energy flux were coupled through their boundary conditions at the convection model. Also, the semitransparent wall with SCC exchanges heat by convection and radiation to the exterior of the cavity. In the experimental study, the solar absorptance of the SCC was simulated experimentally using a thin film electrical resistance located on the glazing surface. Infrared imaging thermography was used to measure the temperature of the exterior surface temperature of the glazing. The interior air temperatures of the cavity were measured using thermocouples. The measured exterior surface temperatures of the glazing were introduced into the theoretical model as a boundary condition and the temperatures of the air at the interior of the cavity were compared with the theoretical ones predicted from the computational code for Ra = 2.3 × 10 6. Their average difference was 1.86%. Through these results, detailed descriptions of the air flow and temperature profiles in the cubic cavity are presented. The influence of radiative process on the overall heat transfer in the cavity is given particular attention, thus distinguishing the convective and radiative heat transfer in the cavity was shown separately. A parametric study was carried out for SCC absorptances of 0.08, 0.50 and 0.64 and exterior temperatures of 15 °C, 25 °C and 30 °C. It was found that for an exterior temperature of 25 °C, the radiative heat flux increases as the absorptance of the SCC increases from 0.08 to 0.64, but the solar heat gain coefficient (SHGC) decreases from 0.94 to 0.52. A new correlation for the Nusselt number as a function of the SCC absorptance is introduced as Nu = 0.9525 α + 10.985 for an ambient temperature of 25 °C.

Transient linear stability of a Simons-process gas–liquid electrochemical flow reactor using numerical simulations

A transient mathematical model and a modified linear perturbation analysis are developed to examine the stability of a gas–liquid, parallel-plate electrochemical flow reactor. Dynamic terms are added to our one-dimensional steady-state conservation equations of momentum, continuity, and thermal energy (Drake et al., Ind. Eng. Chem. Res. 40 (2001a), 3117–3116). The stability analysis is applied to a previously unconsidered, relatively general form for dynamic equations. The perturbation analysis method of linearizing a transient mathematical system and solving the resultant eigenvalue differential equations for stability is automated using a numerical algorithm. At the inlet temperature, exit pressure, and liquid feed flow rate considered, our linear perturbation analysis and nonlinear transient simulations demonstrate the stability of some states and instability of others. Steady-state operation below 60 mA/ cm 2 average current density proves stable to infinitesimal disturbances. Beyond this production rate, however, a gas–liquid vertical flow regime, called heading flow, occurs in part of the reactor, causing unstable steady states. This flow regime permits a particular coupled flow disturbance to magnify. Namely, a liquid-flow-rate increase with position leads to local depletion of the liquid phase, as required by mass conservation. Then, the depletion of liquid creates a lighter two-phase mixture, further accelerating the fluid. Beyond the short times accessible by nonlinear transient simulation, we speculate that the unstable disturbance results in an oscillatory state or a new high-potential state. Operating conditions for avoiding instability are outlined.

Prediction of the steady flame spread rate by the principle of minimal entropy production

The propagation of a two-dimensional diffusive flame over a combustible material is studied by solving steady-state conservation equations written in the coordinate system fixed on the flame front. The analysis of a two-dimensional problem featuring flame spread rate as an eigenvalue has shown that there is no unique solution relative to the flame spread rate unless some additional condition is involved. A novel approach to the prediction of flame spread rate is proposed using the principles of irreversible thermodynamics. The steady flame propagation is considered as a stationary non-equilibrium thermodynamic state, which can be characterized, according to the formulation of Prigogine, by minimal entropy production. A numerical algorithm for the prediction of flame spread rate has been developed and tested on the investigation of downward flame spread over thin sheets of paper. The adequate physical background of the proposed approach and satisfactory agreement with experimental data have been shown.

The structure of mass‐loading shocks: 3. Magnetohydrodynamics

The dissipative, mass‐loading, multifluid model, developed by Zank et al. [1993, 1994] to describe the structurally complex bow shocks observed at comets, has been extended to include the effects of a magnetic field. After describing the model, the shock structure problem associated with the one‐dimensional, steady state conservation equations is solved. The solutions are compared to those obtained from previous cold and warm gasdynamic models. The characteristic speeds of the system are calculated, and it is found that the physically relevant solutions correspond to fast magnetosonic shocks. The extended multifluid model is used to explore a new type of wave, called an “MHD draping shock,” which was proposed by Neubauer et al. [1990] to explain the unusual behavior of the magnetic field across the outbound Halley bow shock. It is found that mass loading induces switch‐on behavior at quasi‐parallel shocks, even in a high plasma beta environment. Across quasi‐parallel shocks we find switch‐on rotations as large as 30° in the cold solar wind case. This value decreases with increasing solar wind plasma beta, to a value of 16° at a value of βsw = 1. We do not find rotations large enough to explain the 69° twist in the field observed by Giotto, nor do we find any cases where the perpendicular component of the field changes sign.

A vertical buoyant jet with high momentum in a long ventilated tunnel

In this paper the flow resulting from the release of buoyant material within a long tunnel is investigated. The source fluid is discharged through a nozzle of small radius with sufficiently high flow rate to ensure that the axial lengthscale of the buoyant jet (subsequently called the ‘jet-length’) is several times the depth of the tunnel, d. The ends of the tunnel may be either open or closed and a number of ventilation points may exist along it. Consideration of a source with high momentum is an important development in confined jet flow models, as most previous models have assumed that the source has little or no initial momentum. It is found that circulation cells are driven near to the source and that the concentration within them increases to a steady-state maximum. At a distance of about 2.5d from the source the buoyancy forces are then sufficiently strong to drive a two-layered stratified counterflow. The steady-state conservation equations are analysed in order to calculate the mean flow variables. The flow past a ventilation point and the characteristics of the secondary outflow are derived, enabling the calculation of the total number of vents needed to vent the buoyant layer. The time dependence of the mean concentration in the circulation cell near to the source is also deduced. This could be used to calculate time-dependent solutions for the other mean flow variables. All of the theoretical results are compared with experimental measurements.

Physical and numerical modelling of a solar chimney-based ventilation system for buildings

This paper describes an experimental and numerical study to analyse the thermal performance of a bio-climatic building prototype in Nigeria. The roof performs as a solar chimney, generating an air flow through the living space of the building to provide cooling. Experimental tests on a 1:12 small-scale model of the prototype are outlined, and the results, bith qualitative and quatitative, are used to validate a two-dimensional flow simulation model, in which the steady state conservation equations of mass, momentum and thermal energy are solved using a finite volume formulation. The experimental and numerical results, expressed in terms of temperature and velocity fields, for two different window geometries are critically evaluated and compared with good agreement.

Vapor condensation on liquid surface due to laminar jet-induced mixing

The effects of system parameters on the interface condensation rate in a laminar jet-induced mixing tank are numerically studied. The physical system consists of a partially filled cylindrical tank with a subcooled jet discharge from the center of the tank bottom toward the liquid-vapor interface which is at a saturation temperature corresponding to the constant tank pressure. Liquid is also withdrawn from the outer part of the tank bottom to maintain the constant liquid level. The steady-state conservation equations in nondimensiohal form are solved by a finite-difference method for various system parameters including liquid height to tank diameter ratio (////>), tank to jet diameter ratio (Did}, liquid outflow to jet area ratio (Aout/A7), and a heat leak parameter (Nh) which characterizes the uniform wall heat-flux. It is found that the average Condensation Stanton number (Stc) is decreasing with increasing H/D, Did, and Nh. For small Nh9 Stc is nearly independent of Aout/A,. For R€J ^ 600, Stc is essentially equal to (d/D)2 if (Did} > 20 and Nh « 1.

Application of computational fluid dynamics in room ventilation

This paper presents the results of a computer program developed for solving 2- and 3-D ventilation problems. The program solves, in finite difference form, the steady-state conservation equations of mass, momentum and thermal energy. Presentation of the fluctuating velocity components is made using the k-ϵ turbulence model. Predicted results of air velocity and temperature distribution in a room are corroborated by experimental measurements. The numerical solution is extended to other room ventilation problems of practical interest.