Energy Integral Equations Research Articles

Instability analysis for MHD boundary layer flow of nanofluid over a rotating disk with anisotropic and isotropic roughness

The study focuses on the instability of local linear convective flow in an incompressible boundary layer caused by a rough rotating disk in a steady MHD flow of viscous nanofluid. Miklavčič and Wang's (Miklavčič and Wang, 2004) [9] MW roughness model are utilized in the presence of MHD of Cu-water nanofluid with enforcement of axial flows. This study will investigate the instability characteristics with the MHD boundary layer flow of nanofluid over a rotating disk and incorporate the effects of axial flow with anisotropic and isotropic surface roughness. The resulting ordinary differential equations (ODEs) are obtained by using von Kàrmàn (Kármán, 1921) [3] similarity transformation on partial differential equations (PDEs). Subsequently, numerical solutions are obtained using the shooting method, specifically the Runge-Kutta technique. Steady-flow profiles for MHD and volume fractions of nanoparticles are analyzed by the partial-slip conditions with surface roughness. Convective instability for stationary modes and neutral stability curves are also obtained and investigated by the formulation of linear stability equations with the MHD of nanofluid. Linear convective growth rates are utilized to analyze the stability of magnetic fields and nanoparticles and to confirm the outcomes of this analysis. Stationary disturbances are also considered in the energy analysis. The investigation indicates the correlation between instability modes Type I and Type II, in the presence of MHD, nanoparticles, and the growth rates of the critical Reynolds number. An integral energy equation enhances comprehension of the fundamental physical mechanisms. The factors contributing to convective instability in the system are clarified using this approach.

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.

Instantaneous acceleration-based modeling of pumping systems response under transient events

Water hammer can cause pipeline eruptions after uncontrolled pump trip. In order to protect the pipeline from water hammer effects and monitor the transient flow, it is imperative to accurately predict pressure oscillation and surge energy dissipation for longer simulation periods. This study investigates the effectiveness of the water hammer control strategy by adjusting the rotational inertia of a pumping system alongside valve closing operations. In addition, it also explores the complex nature of the transient energy dissipation by using the one-dimensional instantaneous acceleration-based (IAB) unsteady friction models. To this end, the IAB unsteady friction (UF) models were improved to calculate the water hammer pressure by investigating the elastic pipe's response under transient events that are caused by pump trip and valve closure. Various boundary equations were derived and combined with water hammer governing equations to form the final system of equations to predict the transient pressure variation with time. After that, the numerical results of two IAB friction models (i.e., one and two-decay coefficient models, respectively) were compared with previous experimental data to verify the proposed transient solver. This solver was based on Method of Characteristics (MOC) and UF models. The results indicate that one-coefficient and two-coefficients UF models yield almost similar results that match with the experimental outcomes; however, the latter model is comparatively more complex in terms of formulation. Furthermore, a parametric study shows that increasing the moment of inertia of the pump and integrating it with the optimal operation of the valve helps in significantly controlling the transient pressure. This helps in reducing the check-valve slam and controls the water hammer with no need for additional protection devices. Results of the integral energy equation describe that the role of UF on the total dissipation of transient energy diminishes with an increase in the valve's closing time and inertia of the pumping system. Finally, the IAB-UF model acts as a valuable numerical tool that provides comprehensive information about the physical parameters required to protect high-head water supply systems from transient incidents.

Application of Cold Radiant Surfaces in the Buildings Development in Summer Conditions

This paper presents a numerical study of sustainable energy application in the buildings construction using cold radiant surfaces in summer conditions. The study is made in a University building and the proposed technology is used to improve the internal thermal comfort conditions in summer conditions. This numerical study uses a software that simulates the whole Building Dynamic Response with complex topology in transient conditions. The numerical model is founded on the balance of energy and mass integral equations. The energy balance integral equations are taken for the indoor air of the compartments, the transparent (windows) bodies of the building, the interior and surroundings bodies of the building and the opaque (walls and doors) bodies of the building. The mass balance integral equations, taken into consideration for the water vapor and the air contaminants, are developed for the spaces of the buildings and the solid matrix (opaque and interior bodies). The software is used to assess the human thermal comfort and indoor air quality levels, the cold radiant system, indoor thermal variables, and other parameters. The building analyzed in this work has 107 compartments, of which 33 are for classes. The radiant system is based on the use of subterraneous cold water. Three situations were analyzed: without cold radiant surfaces, with horizontal cold radiant surfaces and with all compartment cold radiant surfaces. The thermal comfort level, using the Predicted Mean Vote (PMV) index, and the indoor air quality, using the carbon dioxide concentration, are evaluated. The results demonstrate that the implemented Heated Ventilation and Air Conditioned system, working in Ventilation and Radiant methodology, with the use of all surfaces equipped with a cold radiant system, allows to guarantee, in the morning, acceptable, and, in the afternoon, near acceptable levels of thermal comfort by PMV index values according to category C of the standard ISO 7730. During occupancy, the indoor air quality levels obtained in the compartments are near the acceptable limit provided by the ASHRAE 62.1 standard.

Investigation of water-air flows in nozzles

The undoubted importance of these problems allows us to conclude that the research aimed at creating and using a simple model of the flow of a two-phase medium is relevant and of interest not only from a scientific, but also from a practical point of view. Two approaches to their description can be distinguished: the study of flows of two-phase media taking into account relaxation processes between phases with a microscopic description of the interaction between phases, or the study of flows of two-phase media with a macroscopic description of the medium in the form of a one-speed one-temperature continuum. However, sometimes, when calculating, it is possible to ignore the structural two-phase medium and consider the medium as a one-speed one-temperature continuum. This proposal allows us to calculate the averaged flow parameters of a two-phase medium, which is required for engineering calculations. In this paper, the calculation of the flow of the gas-drop flow in the Laval nozzle is given. The method is described, which is based on integral energy equations for two-phase dispersed currents. In the calculations, the two-phase flow is considered as a single-speed, single-temperature continuum. When modeling in the ANSYS Fluent software package, a package of Euler equations is used to compare with analytical results obtained from integral energy equations.

Theoretical analysis of unsteady convective heat transfer from a flat plate with time-varying and spatially-varying temperature distribution

Convective heat transfer due to laminar, incompressible flow past a flat plate has been extensively researched due to applications in a variety of engineering systems. In several cases, such as thermal management of Li-ion cells and phase change based energy storage, the flat plate temperature may vary as a function of both time and space, and the resulting plate heat flux is of interest. This paper presents an integral method based analytical model for predicting the heat flux distribution from a flat plate with a time-varying and spatially-varying temperature. Third-order Karman-Pohlhausen polynomial forms of velocity and temperature distributions are used in the integral energy equation to derive an ordinary differential equation that predicts the plate heat flux distribution in response to an arbitrary time- and space-dependent plate temperature distribution. The generalized results derived here are shown to reduce to previously reported results for special cases of only time-dependent or only spatially varying or constant plate temperature. Results are also shown to be in good agreement with finite-element simulations. The analytical model developed here is used to predict the plate heat flux in response to realistic plate temperature distributions that may be encountered in practical applications. These results improve the fundamental understanding of an important convective heat transfer process, and may help in the design and optimization of a broad variety of engineering systems involving convective heat transfer.

Higher Order Conceptual Model for Labyrinth Seal Flutter

Abstract A simple nondimensional model to describe the flutter onset of two-fin straight labyrinth seals (Corral, R., and Vega, A., 2018, “Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models. Part I: Theoretical Background,” ASME J. Turbomach., 140(10), p. 121006) is extended to account for nonisentropic flow perturbations. The isentropic relationship is replaced by the more general integral energy equation of the inter-fin cavity. A new expression for the Corral and Vega stability criterion is derived, which is very consistent with the previous model in the whole design space of the seal but for torsion centers located in the high-pressure side close to the seal. The new model formally depends on more dimensionless parameters since the existing parameter grouping of the previous model does not hold anymore, but this dependency is weak in relative terms. The model blends the limit where the discharge time of the inter-fin cavity is much longer than the vibration period, and the flow is nearly isentropic, and the opposite limit, where the perturbations are isothermic, gracefully. A few numerical examples obtained using a three-dimensional linearized frequency domain solver are included to support the model and show that the trends are correct, but the body of the numerical work will be presented in a separated article. The matching between the work-per-cycle obtained with the model and frequency domain solver is good. It is shown that some weird trends obtained using linearized unsteady simulations are qualitatively consistent with the current model but not with the previous one (Corral, R., and Vega, A., 2018, “Conceptual Flutter Analysis of Labyrinth Seals Using Analytical Models. Part I: Theoretical Background,” ASME J. Turbomach., 140(10), p. 121006). The largest differences between the new and the previous model are seen when the seal is supported at the high-pressure side.

Development of a Double Skin Facade System Applied in a Virtual Occupied Chamber

In this study a system constituted by seven double skin facades (DSF), three equipped with venetian blinds and four not equipped with venetian blinds, applied in a virtual chamber, is developed. The project will be carried out in winter conditions, using a numerical model, in transient conditions, and based on energy and mass balance linear integral equations. The energy balance linear integral equations are used to calculate the air temperature inside the DSF and the virtual chamber, the temperature on the venetian blind, the temperature on the inner and outer glass, and the temperature distribution in the surrounding structure of the DSF and virtual chamber. These equations consider the convection, conduction, and radiation phenomena. The heat transfer by convection is calculated by natural, forced, and mixed convection, with dimensionless coefficients. In the radiative exchanges, the incident solar radiation, the absorbed solar radiation, and the transmitted solar radiation are considered. The mass balance linear integral equations are used to calculate the water mass concentration and the contaminants mass concentration. These equations consider the convection and the diffusion phenomena. In this numerical work seven cases studies and three occupation levels are simulated. In each case the influence of the ventilation airflow and the occupation level is analyzed. The total number of thermal and indoor air quality uncomfortable hours are used to evaluate the DSF performance. In accordance with the obtained results, in general, the indoor air quality is acceptable; however, when the number of occupants in the virtual chamber increases, the Predicted Mean Vote index value increases. When the airflow rate increases the total of Uncomfortable Hours decreases and, after a certain value of the airflow rate, it increases. The airflow rate associated with the minimum value of total Uncomfortable Hours increases when the number of occupants increases. The energy production decreases when the airflow increases and the production of energy is higher in DSF with venetian blinds system than in DSF without venetian blinds system.

The effect of κ-distributed trapped electrons on fully nonlinear electrostatic solitary waves in an electron–positron-relativistic ion plasma

Based on the hydrodynamic model, the existence and propagation features of fully nonlinear electrostatic solitary waves in an unmagnetized, collisionless, homogenous three-component plasma have been investigated. The plasma containing cold relativistic ions, Boltzmann positrons, and trapped electrons modelled by κ-trapped distribution function. Employing the pseudo-potential method, the Sagdeev pseudo-potential and the first integral energy equation for the system as a function of the electrostatic potential (disturbance) have been derived. The influence of the relevant plasma configurations including the propagation pulse velocity, the superthermality index, the characteristic trapping parameter, the relativistic strength parameter, the positron density ratio, and the positron temperature ratio, on the properties of electrostatic solitary pulse profile has been determined. The results of our study may be helpful in better interpretation of the existence of localized structures in astrophysical and space plasmas as well as in laboratory plasmas, where the positron-ion plasmas with nonthermal trapped electrons can exist.

A convective analytical model in turbulent boundary layer on a flat plate based on the unifying heat flux formula

The two-dimensional (2D) convective heat fluxes are formulated for quasi-steady incompressible turbulent thermal boundary layer flow over a flat plate based on the unifying heat-flux formula developed previously. It is indicated that the total turbulent convection heat flux is the combination of three heat transport mechanisms: conduction due to random molecular motion, laminar advection due to time-averaged motion of fluid bulk and turbulent advection due to fluctuations of velocity and temperature. The interdependent relationship between two heat flux components in different directions is obtained for the quasi-steady turbulent boundary layer flow, which is proved to be equivalent to the turbulent differential and integral energy equations with constant physical properties, respectively. The turbulent thermal boundary layer is divided into laminar sublayer and turbulent layer, in which the linear and one-seventh-power temperature (velocity) profiles are adopted, respectively. Considering the different thicknesses of turbulent thermal and velocity boundary layers, the explicit analytical expression of convective turbulent heat transfer coefficient, as well as the simplified analytical expressions of boundary layer thickness ratio, is proposed based on the total turbulent convection heat transfer structure. The suggested analytical turbulent convective heat transfer coefficients (or Stanton numbers) for air and water are validated by comparing with those of previous theories and experimental measurements. Relevant key characteristics and physical insights of the turbulent structure on the flat plates are revealed in virtue of 2D analytical convective heat fluxes proposed.

Application of passengers’ thermoregulation integral model used in the evaluation of thermal comfort in vehicles compartment equipped with internal curtains

This work evaluates the passengers thermal comfort level inside a vehicles compartment. The numerical study, made in winter conditions, consider a bus indoor environment equipped with internal curtains, internal seats, lateral panels, ceiling, floor and occupied by 52 passengers. The numerical model considers the passengers and vehicle grid generation, passengers body and clothing thermal response and passengers thermal comfort level. The grid generation is used to evaluate the view factors and Mean Radiant Temperature that the passengers are subjected. In this calculus the passengers and the vehicles surfaces shading devices are considered. The thermal response numerical models consider the energy and mass balance integral equations. The thermal comfort evaluation considers the heat produced inside the body and the heat exchange between the body and the environment. The human body numerical model considers also the thermoregulatory system to control the temperature. The numerical model is used to evaluate the thermal comfort level that seated passengers are subjected in a bus equipped with asymmetrical warm curtains. Three Cases studies were developed: the first one considers the temperature curtains equal to the indoor bus surfaces, while the other two consider higher temperatures values. All Cases are thermally comfortable according to the standards.

Production of thermal energy in University building greenhouses in cold climate conditions

The present work focuses on the production of thermal energy in University building greenhouses in cold climate conditions. The building model uses a system of energy and mass balance integral equations, which are solved by the Runge–Kutta–Felberg method with error control. This numerical study is about the thermal behaviour of a university building with complex topology, in winter and transient conditions. The thermal comfort of the occupants, using the Predicted Mean Vote index, and the indoor air quality, using the carbon dioxide concentration, are evaluated. This building has 319 compartments distributed by four floors and it is equipped with one internal greenhouse in the third floor. This greenhouse is located on the south facing facade and the heated air in this space will be transported to compartments located on the north facing façade. The spaces subject to the influence of the heated air coming from the greenhouse improve the level of thermal comfort of its occupants. The level of indoor air quality in occupied spaces is acceptable according to international standards.

A Unified Treatment for Local Thermal Nonequilibrium Free, Forced, and Mixed Convective Boundary Layer Flows Over an Arbitrarily Shaped Nonisothermal Body in a Fluid Saturated Porous Medium

Abstract A unified integral solution procedure has been proposed to analyze all possible Darcian local thermal nonequilibrium (LTNE) free, forced, and mixed convective boundary layer flows, commonly encountered in porous media engineering applications. The heated body may be arbitrarily shaped, and its temperature may vary over the surface. The integral energy equation for the solid phase yields an algebraic equation between the dimensionless fluid thermal boundary layer thickness and its ratio to the solid-phase counterpart, while the integral energy equation for the fluid phase reduces to a first order ordinary differential equation in terms of the dimensionless fluid thermal boundary layer thickness. This set of the equations for determining the local Nusselt number of our primary interest proved to be valid for all possible Darcian cases of LTNE free, forced, and mixed convective boundary layer flows over an arbitrarily shaped nonisothermal body in a fluid saturated porous medium. Asymptotic expressions for the cases of arbitrary shapes were also obtained analytically for both leading edge and far downstream regions. The results are found to agree well with available direct numerical integration results. Furthermore, the regime map has been constructed to show the boundary layer transition point from the LTNE to equilibrium. The proposed unified method is found quite useful when designing thermal engineering systems associated with fluid saturated porous media.

Numerical Study of Mach Number on Solutions in a Warm Negative Ion Plasma With Flat-Topped Electron Distribution

The influence of the Mach number on the localized ion acoustic solitons is investigated in a collisionless one-dimensional unmagnetized plasma system. The plasma model used here is a negative ion plasma having electrons complying flat-topped distribution. Using the basic fluid equations and applying proper boundary conditions we derive the energy integral equations containing the Sagdeev potential. The Sagdeev potential obtained, depends on various parameters out of which the effect of Mach number on the highly nonlinear structures is studied. It is found that depth of the Sagdeev potential, amplitude of the solitary structures and bipolar electric field pulses rises with rise in the values of M . Also narrower solitons starts appearing with increase in the solitary speed.

Thermocapillary migration of a planar droplet at small and large Marangoni numbers: effects of interfacial rheology

In this paper, roles of interfacial rheology on thermocapillary migration of a planar droplet at small and large Marangoni numbers are analyzed. Under quasi-steady-state assumption, the time-independent momentum and energy equations of thermocapillary droplet migration with boundary conditions are determined. An exact solution of the steady thermocapillary migration of the deformed droplet at small Marangoni numbers is obtained. It is found that the deformed droplet has an oblate shape. The deviation from the circular section depends on the Weber number and the migration speed. The surface shear viscosity, the dilatational viscosity and the surface internal energy parameter affect the deformation of the droplet through reducing the migration speed. The validity of the steady thermocapillary droplet migration at small Marangoni numbers is confirmed by determining the conservative overall integral energy equations. At large Marangoni numbers, the non-conservative overall integral energy equations imply that thermocapillary droplet migration is always an unsteady process.

Supersonic flow of two-phase gas- droplet flows in nozzles

For practical applications, the description of processes occurring during the flow of two-phase gas-liquid mixtures requires a simple physical and mathematical model that describes the behavior of a two-phase medium in the entire range of phase concentrations changes and in a wide range of pressure changes. Problems of this kind arise in various branches of industry and technology. In the space industry, one often has to deal with the movement of various gases in rocket nozzles, consider the combustion, condensation of various vapors on the nozzle walls and their further impact on the velocity sublayer at the nozzle wall. The large acoustic effect arising from the engines affects the gas-liquid mixture in the nozzles of rocket engines. In the metal industry, metal cooling occurs with the help of nozzles in which the emulsion mixture is supplied under high overpressure. But this is only a short list of applied issues in which one has to deal with a problem of this type. The paper presents the results and directions of study of the problems of two-phase dispersed gas-droplet flows in the nozzles. The main methods of investigation of two- phase heterogeneous flows are described. The main characteristics of heterogeneous two-phase flows in the nozzles, which were confirmed by experimental results, are presented. The calculation of the air-droplet flow in the Laval nozzle is given. The technique, which is based on integral energy equations for two-phase dispersed flows, is described. The main problems and questions concerning the further description and studying of two-component flows are stated.

The instability of non-Newtonian boundary-layer flows over rough rotating disks

We are concerned with the local linear convective instability of the incompressible boundary-layer flows over rough rotating disks for non-Newtonian fluids. Using the Carreau model for a range of shear-thinning and shear-thickening fluids, we determine, for the first time, steady-flow profiles under the partial-slip model for surface roughness. The subsequent linear stability analyses of these flows (to disturbances stationary relative to the disk) indicate that isotropic and azimuthally-anisotropic (radial grooves) surface roughness leads to the stabilisation of both shear-thinning and -thickening fluids. This is evident in the behaviour of the critical Reynolds number and growth rates of both Type I (inviscid cross flow) and Type II (viscous streamline curvature) modes of instability. The underlying physical mechanisms are clarified using an integral energy equation.

Theory and Validation of a 2D Finite-Volume Integral Boundary-Layer Method for Icing Applications

A two-dimensional integral boundary-layer method is developed to enable fast and economical computations of boundary-layer flows. The ultimate goal is to provide some experience for the extension of this method in three dimensions. In this study, the unsteady momentum and kinetic energy integral equations are solved numerically, together with a set of closure relations based on assumed velocity profiles for laminar and turbulent flows. The robustness of the method is ensured by a finite-volume formulation based on an upwind scheme and a semi-implicit time discretization. Moreover a control method has been developed in order to avoid the Goldstein singularity. The accuracy of the numerical method in the vicinity of the stagnation point is strongly improved by introducing a consistent corrective source term in the right-hand side of the equation system. The chosen closure relations are validated with test cases of self-similar flows. Numerical results are also compared with those of a full Prandtl equations code for NACA0012, GLC305, and MS317 airfoils test cases to demonstrate the capabilities of the method. Finally, preliminary results are shown proving the ability of the method to deal with iced airfoils even for complex glaze ice shapes.

Application of HVAC Systems with Control Based on PMV Index in University Buildings with Complex Topology

In this work the control of HVAC systems based on PMV index, in university buildings with complex topology, is numerically developed. A numerical software that simulates the building thermal behavior and the occupant thermal comfort, in transient thermal conditions, is applied. This numerical software is based in energy and mass balance integral equations. The numerical simulation is made in a university building, equipped with 344 compartments and with 2072 occupants, in summer and winter conditions. The number of compartments equipped with HVAC system, control physiology system based on PMV index, are 171. The PMV index is calculated as a function of the air temperature, air velocity, air relative humidity, mean radiant temperature, clothing level and activity level. The numerical results show that the application of this type of control allows to guarantee, in an efficient way, good levels of thermal comfort, both in winter and summer conditions, in most of the rooms of the studied building.

Stability of the shear-thinning boundary-layer flow over a flat inclined plate

In this study, we consider the boundary-layer flow of an inelastic non-Newtonian fluid over an inclined flat plate. Using two popular generalized Newtonian models, we determine base flow profiles and associated linear stability results for a range shear-thinning fluids. In addition to neutral stability curves, we also present results concerning the linear growth of the Tollmien–Schlichting waves as they propagate downstream. Furthermore, to gain an insight into the underlying physical mechanisms affecting the destabilization of the disturbances, an integral energy equation is derived and energy calculations are presented. Results from all three analyses suggest that the effect of shear-thinning will act to stabilize the boundary-layer flow. Consequently, it can be argued that the addition of shear-thinning agents could act as a passive control mechanism for flows of this nature.