Enthalpy Transport Equation Research Articles

A pressure-based solver for low-Mach number flow using a discontinuous Galerkin method

Over the past two decades, there has been much development in discontinuous Galerkin methods for incompressible flows and for compressible flows with a positive Mach number, but almost no attention has been paid to variable-density flows at low speeds. This paper presents a pressure-based discontinuous Galerkin method for flow in the low-Mach number limit. We use a variable-density pressure correction method, which is simplified by solving for the mass flux instead of the velocity. The fluid properties do not depend significantly on the pressure, but may vary strongly in space and time as a function of the temperature.We pay particular attention to the temporal discretization of the enthalpy equation, and show that the specific enthalpy needs to be ‘offset’ with a constant in order for the temporal finite difference method to be stable. We also show how one can solve for the specific enthalpy from the conservative enthalpy transport equation without needing a predictor step for the density. These findings do not depend on the spatial discretization.A series of manufactured solutions with variable fluid properties demonstrate full second-order temporal accuracy, without iterating the transport equations within a time step. We also simulate a Von Kármán vortex street in the wake of a heated circular cylinder, and show good agreement between our numerical results and experimental data.

Effect of Pressure on Solidification Process and Mechanical Properties During Semi-Solid Casting by Computational Fluid Dynamics (CFD)

This paper focused on the development of Computational Fluid Dynamics (CFD) by a mathematical model based on the control volume method, which would enable efficient simulation of the semi-solid processing of complex industrial casting parts. Theoretical basis of the numerical simulation was briefly introduced. The latent heat was incorporated using the effective specific heat. The mass, momentum and enthalpy transport equation for each phase were solved. The application of the model allowed determining the temperature fields in the metal and the mold at 1 sec and the liquid fraction at time step 0.15 s with three different pressure regimes. The calculated pressure distribution and the evolution of liquid fraction through the material were examined and used to clarify their influence further investigated with tensile and hardness testing.

Modelling heat loss effects in high temperature oxy-fuel flames with an efficient and robust non-premixed flamelet approach

The non-premixed steady flamelet model is extended by two simple, robust and effective heat loss modelling approaches. The heat release damping (HRD) approach decreases the chemical source term in the energy equation by a constant factor, while the artificial radiation (AR) approach introduces an augmented temperature dependent radiative source term. The models are tested in a simulation of the 0.78 MWth IFRF (International Flame Research Foundation) pilot scale, non-premixed, natural gas/oxygen flame. Both approaches are applied in steady laminar flamelet calculations with detailed chemistry, tabulating the thermo-chemical state as a function of mixture fraction and normalised heat loss. The turbulence-chemistry interaction is modelled using the β-PDF approach. An enthalpy transport equation is solved to keep track of the heat loss, while radiative heat transfer is calculated by the P-1 model. We observe that the major species, temperature, velocities and velocity fluctuations show a good agreement with the available experimental data. The heat loss modelling yields a significant improvement over the adiabatic model. Interestingly, both heat loss models (HRD and AR) show negligible differences in the simulations of the turbulent flame and permit to apply the steady laminar flamelet model to oxy-fuel processes in a simple, robust and user friendly manner.

Modelling of soot formation in laminar diffusion flames using a comprehensive CFD-PBE model with detailed gas-phase chemistry

A discretised population balance equation (PBE) is coupled with an in-house computational fluid dynamics (CFD) code in order to model soot formation in laminar diffusion flames. The unsteady Navier–Stokes, species and enthalpy transport equations and the spatially-distributed discretised PBE for the soot particles are solved in a coupled manner, together with comprehensive gas-phase chemistry and an optically thin radiation model, thus yielding the complete particle size distribution of the soot particles. Nucleation, surface growth and oxidation are incorporated into the PBE using an acetylene-based soot model. The potential of the proposed methodology is investigated by comparing with experimental results from the Santoro jet burner [Santoro, Semerjian and Dobbins, Soot particle measurements in diffusion flames, Combustion and Flame, Vol. 51 (1983), pp. 203–218; Santoro, Yeh, Horvath and Semerjian, The transport and growth of soot particles in laminar diffusion flames, Combustion Science and Technology, Vol. 53 (1987), pp. 89–115] for three laminar axisymmetric non-premixed ethylene flames: a non-smoking, an incipient smoking and a smoking flame. Overall, good agreement is observed between the numerical and the experimental results.

Modelling of Soot Formation in a Laminar Coflow Non-premixed Flame with a Detailed CFD-Population Balance Model

In this work, a discretised Population Balance Equation (PBE) model is coupled with a detailed in-house Computational Fluid Dynamics (CFD) code, to investigate soot formation in a laminar co-flow non-premixed ethylene flame. The unsteady Navier-Stokes, species conservation and enthalpy transport equations are solved in a segregated manner, combined with comprehensive gas-phase chemistry and an optically thin radiation model to predict the structure of the flame. The objective of this numerical methodology is to compute the complete Particle-Size-Distribution (PSD) of the soot particles, using particle volume as the sole internal coordinate. The following mechanisms are implemented into the discretised PBE: nucleation, surface growth, oxidation and coagulation, using acetylene based soot model. Validation will be undertaken between the numerical and the experimental results of Santoro's jet burner [R. J. Santoro, H. G. Semerjian, and R. A. Dobbins, Combust. Flame 51:203-218, 1983]. The importance of comprehensive fluid dynamics, reactions and PBE modelling is explored for the investigation of problems where all of these phenomena are coupled.

A novel 1D/2D model for simulating conjugate heat transfer applied to flow boiling in tubes with external fins

This study presents a novel, simplified model for the time-efficient simulation of transient conjugate heat transfer in round tubes. The flow domain and the tube wall are modeled in 1D and 2D, respectively and empirical correlations are used to model the flow domain in 1D. The model is particularly useful when dealing with complex physics, such as flow boiling, which is the main focus of this study. The tube wall is assumed to have external fins. The flow is vertical upwards. Note that straightforward computational fluid dynamics (CFD) analysis of conjugate heat transfer in a system of tubes, leads to 3D modeling of fluid and solid domains. Because correlation is used and dimensionality reduced, the model is numerically more stable and computationally more time-efficient compared to the CFD approach. The benefit of the proposed approach is that it can be applied to large systems of tubes as encountered in many practical applications. The modeled equations are discretized in space using the finite volume method, with central differencing for the heat conduction equation in the solid domain, and upwind differencing of the convective term of the enthalpy transport equation in the flow domain. An explicit time discretization with forward differencing was applied to the enthalpy transport equation in the fluid domain. The conduction equation in the solid domain was time discretized using the Crank–Nicholson scheme. The model is applied in different boundary conditions and the predicted boiling patterns and temperature fields are discussed.

A velocity divergence constraint for large-eddy simulation of low-Mach flows

The velocity divergence (rate of fluid volumetric expansion) is a flow field quantity of fundamental importance in low-Mach flows. It directly affects the local mass density and therefore the local temperature through the equation of state. In this paper, starting from the conservative form of the sensible enthalpy transport equation, we derive a discrete divergence constraint for use in large-eddy simulation (LES) of low-Mach flows. The result accounts for numerical transport of mass and energy, which is difficult to eliminate in relatively coarse, engineering LES calculations when total variation diminishing (TVD) scalar transport schemes are employed. Without the correction terms derived here, unresolved (numerical) mixing of gas species with different heat capacities or molecular weights may lead to erroneous mixture temperatures and ultimately to an imbalance in the energy budget. The new formulation is implemented in a publicly available LES code called the Fire Dynamics Simulator (FDS). Accuracy of the flow solver for transport is demonstrated using the method of manufactured solutions. The conservation properties of the present scheme are demonstrated on two simple energy budget test cases, one involving a small fire in a compartment with natural ventilation and another involving mixing of two gases with different thermal properties.

Numerical Analysis of Compressible Flow in Labyrinth Seal

Numerical simulation code for compressible flow based on OpenFOAM-1.6 was developed to compute a high Mach number flow. Total enthalpy transport equation was solved to calculate temperature and a flow around a rotor was solved in whirling rotating frame to predict a hydraulic force on an eccentric whirling rotor. Straight and high-low labyrinth seals were simulated by the developed code and numerical results regarding hydraulic force were compared with experimental ones to validate the prediction accuracy. The non-diagonal component of spring constant was predicted well, but the diagonal component of damping constant was largely under-predicted. The maximum and minimum pressures were observed at the circumferential positions of 270° and 90° respectively in inlet region of the straight labyrinth seal. Therefore large hydraulic force in the whirling direction was generated. On the other hand, the maximum and minimum pressures were observed at 180° and 0° respectively in the high-low labyrinth seal. Therefore the hydraulic force in the whirling direction was lower than that of the straight labyrinth seal. Pressure distribution in the circumferential direction was uniform in a large clearance region surrounded by two fins in the straight labyrinth seal, where the hydraulic force in whirling direction was largely decreased.

Temperature-explicit formulation of energy equation for thermal–hydraulic analyses

A temperature equation which is derived from an enthalpy transport equation by using an assumption of a constant specific heat is very attractive for analyses of heat and fluid flows. It can be used for an analysis of a solid–fluid conjugate heat transfer, and it does not need a numerical method to obtain temperature from a temperature–enthalpy relation. But its application is limited because of the assumption. A new method is derived in this study, which is a temperature-explicit formulation of the energy equation. The enthalpy form of the energy equation is used in the method. But the final discrete form of the equation is expressed with temperature. The discretized equation from the temperature-explicit formulation can be used for a heat transfer analysis in a solid–fluid coupled region without any special treatment at the solid–fluid interface. And it can be applied for multiphase flows with a real gas effect. It is found by numerical tests in this study that the proposed method is very efficient and as accurate as the standard enthalpy formulation.

Modeling of radiation and nitric oxide formation in turbulent nonpremixed flames using a flamelet/progress variable formulation

A model for the prediction of the nitric oxide (NO) formation in turbulent nonpremixed flames is proposed. Since the NO formation has a strong temperature sensitivity, the accurate prediction of the flame temperature under the consideration of radiative heat losses is required. The first part of the paper addresses the extension of a flamelet-based combustion model to account for radiative heat loss effects by introducing enthalpy as an additional parameter. A transport equation for enthalpy is solved, and the radiative sink term in this equation is obtained from unsteady flamelet solutions. The model is applied to a large-eddy simulation (LES) of Sandia flame D, and the importance of the interaction between turbulence and radiation on temperature and mixture fraction is investigated. Based on the radiative flamelet formulation, a consistent model for the prediction of NO formation is developed in the second part of the paper. In this model, an additional transport equation for the NO mass fraction is solved, and the chemical source term is obtained from a flamelet library. Since the consumption rate is dependent on the NO mass fraction, this term requires modeling, which is discussed in this paper. By employing a scale similarity argument, a closure model for application in LES is presented. After the analysis of the proposed model for the thermal, nitrous oxide, and prompt pathways for NO formation, the NO model is integrated into the extended flamelet/progress variable model and applied in LES of Sandia flame D and a Pratt & Whitney aircraft engine combustor configuration.

Large eddy simulations of a turbulent thermal plume

Large eddy simulations of a three-dimensional turbulent thermal plume in an open environment have been carried out using a self-developed parallel computational fluid dynamics code SMAFS (smoke movement and flame spread) to study the thermal plume’s dynamics including its puffing, self-preserving and air entrainment. In the simulation, the sub-grid stress was modeled using both the standard Smagorinsky and the buoyancy modified Smagorinsky models, which were compared. The sub-grid scale (SGS) scalar flux in the filtered enthalpy transport equation was modeled based on a simple gradient transport hypothesis with constant SGS Prandtl number. The effect of the Smagorinsky model constant and the SGS Prandtl number were examined. The computation results were compared with experimental measurements, thermal plume theory and empirical correlations, showing good agreement. It is found that both the buoyancy modification and the SGS turbulent Prandtl number have little influence on simulation. However, the SGS model constant Cs has a significant effect on the prediction of plume spreading, although it does not affect much the prediction of puffing.

Flow and Mixing Fields for Transported Scalar PDF Simulations of a Piloted Jet Diffusion Flame (‘Delft Flame III’)

Numerical simulation results are presented for ‘Delft Flame III’, a piloted jet diffusion flame with strong turbulence–chemistry interaction. While pilot flames emerge from 12 separate holes in the experiments, the simulations are performed on a rectangular grid, under the assumption of axisymmetry. In the first part of the paper, flow and mixing field results are presented with a non-linear first order k–e model, with the transport equation for e based on a modeled enstrophy transport equation, for cold and reactive flows. For the latter, the turbulence model is applied in combination with pre-assumed β-PDF modeling for the turbulence–chemistry interaction. The mixture fraction serves as conserved scalar. Two chemistry models are considered: chemical equilibrium and a steady laminar flamelet model. The importance of the turbulence model is highlighted. The influence of the chemistry model is noticeable too. A procedure is described to construct appropriate inlet boundary conditions. Still, the generation of accurate inlet boundary conditions is shown to be far less important, their effect being local, close to the nozzle exit. In the second part of the paper, results are presented with the transported scalar PDF approach as turbulence–chemistry interaction model. A C1 skeletal scheme serves as chemistry model, while the EMST method is applied as micro-mixing model. For the transported PDF simulations, the model for the pilot flames, as an energy source term in the mean enthalpy transport equation, is important with respect to the accuracy of the flow field predictions. It is explained that the strong influence on the flow and mixing field is through the turbulent shear stress force in the region, close to the nozzle exit.

Numerical modeling of inert gas-condensing vapor thermoacoustic engines

Recent theoretical work by Slaton and Raspet et al. describe the acoustic propagation equation [J. Acoust. Soc. Am. 114, 1414–1422] and the second-order enthalpy and mass transport equations [J. Acoust. Soc. Am. 114, 1423–1430] for an inert gas-condensing vapor mixture in a porous medium with an imposed temperature gradient. The acoustic propagation and enthalpy transport equations show that the vapor diffusion effects in the mixture are analogous to the heat diffusion effects in the thermoacoustics of inert gases, and that these effects occur in parallel with the heat diffusion effects in the wet system for proper choice of inert gas and vapor. Writing the acoustic propagation equation as two coupled first-order differential equations in terms of the volumetric velocity and acoustic pressure amplitude and utilizing the conservation of enthalpy in the stack allows the system of equations to be solved numerically by interfacing with the well-established thermoacoustic modeling code, DeltaE. Modeling of various thermoacoustic engines utilizing an inert gas-condensing vapor working fluid will be presented. It will be shown how the COP relative to Carnot and the heat pumping power for thermoacoustic refrigerators can be increased significantly by proper choice of gas mixture. [Work supported by ONR.]

MHD free convection in a liquid-metal filled cubic enclosure. I. Differential heating

The buoyancy-driven magnetohydrodynamic flow in a liquid-metal filled cubic enclosure was investigated by three-dimensional numerical simulation. The enclosure was differentially heated at two opposite vertical walls, all other walls being adiabatic, and a uniform magnetic field was applied orthogonal to the temperature gradient and to the gravity vector. The Rayleigh number was 10 5 and the Prandtl number was 0.0321 (characteristic of Pb–17Li at 573 K). The Hartmann number was made to vary between 10 2 and 10 3 and the electrical conductance of the walls between 0 and ∞. The continuity, momentum and enthalpy transport equations, in conjunction with a Poisson equation for the electric potential, were solved by a finite volume method using the general-purpose CFX-4 package with some necessary adaptations. Steady-state conditions were assumed. With respect to the case of parallel flow in an infinitely tall enclosure, studied in previous work, the suppression of convective motions due to magnetohydrodynamic interactions was stronger in the core, and a complex three-dimensional flow (with secondary motions) and current pattern was established in the fluid domain. Increasing the Hartmann number suppressed convective motions and exalted the square-shape of the circulation cells. Increasing the wall conductance ratio from perfectly insulating to perfectly conducting walls also resulted in an increasing suppression of convection. The related case of an internally heated enclosure is discussed in a companion paper.

Modelling Differential Diffusion in Nonpremixed Reacting Turbulent Flow: Application to Turbulent Jet Flames

The modelling of differential diffusion effects in a well documented turbulent hydrogen-air jet diffusion flame is considered- The species and enthalpy transport equations are modelled by the conditional moment closure (CMC) method. In the presence of differential diffusion the terms involving the fluctuations around the conditional means, the eyterms, must not be neglected and need modelling. They have been modelled by an expression which is described in detail in a companion paper. Using these non-unity Lewis number modelling strategies, super-equilibrium temperatures on the lean side of stoichiomelric and increased nitric oxide concentrations are predicted giving improved agreement with experiment. The results show that the eymodel which has been derived with the aid of DNS is applicable to real flame geometries. However, it is also demonstrated that differential diffusion is not the only cause of substantial underpredictions of WOclose to the jet exit. This is in contrast to all previous studies on these hydrogen flames.

A comparative study of radiative heat transfer modelling in gas-fired furnaces using the simple grey gas and the weighted-sum-of-grey-gases models

Radiative heat transfer in, (a) two model enclosures containing a high-temperature CO 2–H 2O mixture, and (b) a real multiburner natural-gas-fired furnace, was studied using the simple grey gas model and the more realistic weighted-sum-of-grey-gases model of gas radiative properties. The radiative transfer equation was solved by the discrete-ordinates method. When the radiative transfer equation and the enthalpy transport equation are solved simultaneously, the simple grey gas model yields wall heat flux distributions in reasonably good agreement with those of the weighted-sum-of-grey-gases model but underpredicts gas temperature levels. When the temperature field is specified and the radiative transfer equation is then solved, however, the wall heat flux distributions predicted using the simple grey gas model are in serious error compared to those from the weighted-sum-of-grey-gases model. The weighted-sum-of-grey-gases model predicts more accurate heat flux and gas temperature distributions than the simple grey gas model in the full modelling of the gas-fired furnace.