Fluides et systèmes énergétiques (FISE)
http://hdl.handle.net/10985/174
2017-11-30T13:33:29ZNumerical Simulation of Yield Stress Fluid Flow in Capillary Bundles: Influence of the Form and the Axial Variation in the Cross Section
http://hdl.handle.net/10985/12237
MALVAULT, Guillaume; AHMADI, Azita; OMARI, Abdelaziz
Transport in porous media
In this paper, we investigate possible improvements that can be made to the bundle
of capillaries model in order to better represent the flow of yield stress fluids through porous media. This was examined by performing extensive and progressive numerical simulations and by introducing the non-circularity of channels’ cross section and/or its variability along the channels’ axis. It is shown that if only the non-circularity of channels’ cross section is taken into account, a moderate influence is observed on both critical pressure gradient for the flow onset and the flow rate/pressure gradient Q(∇P) relationship. However, the axial variation in capillaries’ cross section has proved to be more impacting the computed flow rate/pressure gradient data. We show hence that when available pore throat and pore body size distributions are used to construct the bundle of axially varying capillaries, the obtained Q(∇P) data do fit well experimental results corresponding to the flow of a Bingham-like fluid through a bed of randomly packed mono-sized spheres.
2017-01-01T00:00:00ZMALVAULT, GuillaumeAHMADI, AzitaOMARI, AbdelazizIn this paper, we investigate possible improvements that can be made to the bundle
of capillaries model in order to better represent the flow of yield stress fluids through porous media. This was examined by performing extensive and progressive numerical simulations and by introducing the non-circularity of channels’ cross section and/or its variability along the channels’ axis. It is shown that if only the non-circularity of channels’ cross section is taken into account, a moderate influence is observed on both critical pressure gradient for the flow onset and the flow rate/pressure gradient Q(∇P) relationship. However, the axial variation in capillaries’ cross section has proved to be more impacting the computed flow rate/pressure gradient data. We show hence that when available pore throat and pore body size distributions are used to construct the bundle of axially varying capillaries, the obtained Q(∇P) data do fit well experimental results corresponding to the flow of a Bingham-like fluid through a bed of randomly packed mono-sized spheres.Colloidal Particle Deposition in Porous Media Under Flow: A Numerical Approach
http://hdl.handle.net/10985/12235
LI, Yajie; SARISHVILI, Otar; OMARI, Aziz; AHMADI, Azita; PU, Hongting
International Journal of Modeling an optimization
The objective of this study is to simulate the transport and deposition of colloidal particles at the pore scale by means of computational fluid dynamics simulations (CFD). This consists in the three-dimensional numerical modeling of the process of transport and deposition of colloidal particles in a porous medium idealized as a bundle of capillaries of circular cross section. The velocity field obtained by solving the Stokes and continuity equations is superimposed to particles diffusion and particles are let to adsorb when they closely approach the solid wall. Once a particle is adsorbed the flow velocity field is updated before a new particle is injected. Our results show that both adsorption probability and surface coverage are decreasing functions of the particle’s Péclet number. At low Péclet number values when diffusion is dominant the surface coverage is shown to approach the Random Sequential Adsorption value while it drops noticeably for high Péclet number values. Obtained data were also used to calculate the loss of porosity and permeability.
2017-01-01T00:00:00ZLI, YajieSARISHVILI, OtarOMARI, AzizAHMADI, AzitaPU, HongtingThe objective of this study is to simulate the transport and deposition of colloidal particles at the pore scale by means of computational fluid dynamics simulations (CFD). This consists in the three-dimensional numerical modeling of the process of transport and deposition of colloidal particles in a porous medium idealized as a bundle of capillaries of circular cross section. The velocity field obtained by solving the Stokes and continuity equations is superimposed to particles diffusion and particles are let to adsorb when they closely approach the solid wall. Once a particle is adsorbed the flow velocity field is updated before a new particle is injected. Our results show that both adsorption probability and surface coverage are decreasing functions of the particle’s Péclet number. At low Péclet number values when diffusion is dominant the surface coverage is shown to approach the Random Sequential Adsorption value while it drops noticeably for high Péclet number values. Obtained data were also used to calculate the loss of porosity and permeability.Flow of yied stress fluids through porous media : simulations, experiments and applications
http://hdl.handle.net/10985/12156
AHMADI, Azita; RODRIGUEZ DE CASTRO, Antonio; MALVAULT, Guillaume; OMARI, Abdelaziz
A Yield Stress fluids injection porosimetry Method (YSM) has recently been developed as a simple potential alternative to the extensively used Mercury Intrusion Porosimetry (MIP). Its main advantage is the use of a nontoxic fluid instead of mercury used in MIP. Using this method, the Pore Size Distribution (PSD) of a porous medium is obtained by measuring the flow rate / pressure gradient relationship obtained by injecting a yield stress fluid in the porous medium. The principle of the method and some experimental results obtained using this technique will be presented and will be compared to those obtained by Mercury Intrusion Porosimetry (MIP).
In the Yield Stress fluid injection porosimetry method, the main assumption is that the porous medium is described as a bundle of straight capillaries of circular cross-section following a given pore size distribution. This simple model is revisited by introducing both non-circular and axially varying cross-sections. Two key points are tackled using numerical simulations: the flow onset at minimal pressure drop and the variation of the flow rate vs the pressure gradient. These results are finally used to show that the flow rate / pressure gradient relationship of a yield stress fluid through a porous medium can be more closely predicted using a bundle of capillaries of irregular cross-sections rather than using the classical bundle of straight circular capillaries.
2016-01-01T00:00:00ZAHMADI, AzitaRODRIGUEZ DE CASTRO, AntonioMALVAULT, GuillaumeOMARI, AbdelazizA Yield Stress fluids injection porosimetry Method (YSM) has recently been developed as a simple potential alternative to the extensively used Mercury Intrusion Porosimetry (MIP). Its main advantage is the use of a nontoxic fluid instead of mercury used in MIP. Using this method, the Pore Size Distribution (PSD) of a porous medium is obtained by measuring the flow rate / pressure gradient relationship obtained by injecting a yield stress fluid in the porous medium. The principle of the method and some experimental results obtained using this technique will be presented and will be compared to those obtained by Mercury Intrusion Porosimetry (MIP).
In the Yield Stress fluid injection porosimetry method, the main assumption is that the porous medium is described as a bundle of straight capillaries of circular cross-section following a given pore size distribution. This simple model is revisited by introducing both non-circular and axially varying cross-sections. Two key points are tackled using numerical simulations: the flow onset at minimal pressure drop and the variation of the flow rate vs the pressure gradient. These results are finally used to show that the flow rate / pressure gradient relationship of a yield stress fluid through a porous medium can be more closely predicted using a bundle of capillaries of irregular cross-sections rather than using the classical bundle of straight circular capillaries.Colloidal Particle Deposition in Porous Media Under Flow: A Numerical Approach
http://hdl.handle.net/10985/12155
LI, Yajie; SARISHVILI, Otar; OMARI, A.; AHMADI, Azita; PU, Hongting
The objective of this study is to simulate the transport and deposition of colloidal particles at the pore scale by means of computational fluid dynamics simulations (CFD). This consists in the three-dimensional numerical modeling of the process of transport and deposition of colloidal particles in a porous medium idealized as a bundle of capillaries of circular cross section. The velocity field obtained by solving the Stokes and continuity equations is superimposed to particles diffusion and particles are let to adsorb when they closely approach the solid wall. Once a particle is adsorbed the flow velocity field is updated before a new particle is injected. Our results show that both adsorption probability and surface coverage are decreasing functions of the particle’s Péclet number. At low Péclet number values when diffusion is dominant the surface coverage is shown to approach the Random Sequential Adsorption value while it drops noticeably for high Péclet number values. Obtained data were also used to calculate the loss of porosity and permeability.
2017-01-01T00:00:00ZLI, YajieSARISHVILI, OtarOMARI, A.AHMADI, AzitaPU, HongtingThe objective of this study is to simulate the transport and deposition of colloidal particles at the pore scale by means of computational fluid dynamics simulations (CFD). This consists in the three-dimensional numerical modeling of the process of transport and deposition of colloidal particles in a porous medium idealized as a bundle of capillaries of circular cross section. The velocity field obtained by solving the Stokes and continuity equations is superimposed to particles diffusion and particles are let to adsorb when they closely approach the solid wall. Once a particle is adsorbed the flow velocity field is updated before a new particle is injected. Our results show that both adsorption probability and surface coverage are decreasing functions of the particle’s Péclet number. At low Péclet number values when diffusion is dominant the surface coverage is shown to approach the Random Sequential Adsorption value while it drops noticeably for high Péclet number values. Obtained data were also used to calculate the loss of porosity and permeability.Origin of the Inertial Deviation from Darcy's Law: an 2 Investigation from a Microscopic Flow Analysis on 2D Model
http://hdl.handle.net/10985/12154
AGNAOU, M.; LASSEUX, D.; AHMADI, A.
Physical Review E
Inertial flow in porous media occurs in many situations of practical relevance among which one can cite flows in column reactors, in filters, in aquifers, or near wells for hydrocarbon recovery. It is characterized by a deviation from Darcy’s law that leads to a nonlinear relationship between the pressure drop and the filtration velocity. In this work, this deviation, also known as the nonlinear, inertial, correction to Darcy’s law, which is subject to controversy upon its origin and dependence on the filtration velocity, is studied through numerical simulations. First, the microscopic flow problem was solved computationally for a wide range of Reynolds numbers up to the limit of steady flow within ordered and disordered porous structures. In a second step, the macroscopic
characteristics of the porous medium and flow (permeability and inertial correction tensors) that appear in the macroscale model were computed. From these results, different flow regimes were identified: (1) the weak inertia regime where the inertial correction has a cubic dependence on the filtration velocity and (2) the strong inertia
(Forchheimer) regime where the inertial correction depends on the square of the filtration velocity. However, the existence and origin of those regimes, which depend also on the microstructure and flow orientation, are still not well understood in terms of their physical interpretations, as many causes have been conjectured in the literature. In the present study, we provide an in-depth analysis of the flow structure to identify the origin of the deviation from Darcy’s law. For accuracy and clarity purposes, this is carried out on two-dimensional structures. Unlike the previous studies reported in the literature, where the origin of inertial effects is often identified on a heuristic basis, a theoretical ustification is presented in this work. Indeed, a decomposition of the convective
inertial term into two components is carried out formally allowing the identification of a correlation between the flow structure and the different inertial regimes. These components correspond to the curvature of the flow streamlines weighted by the local fluid kinetic energy on the one hand and the distribution of the kinetic energy
along these lines on the other hand. In addition, the role of the recirculation zones in the occurrence and in the form of the deviation from Darcy’s law was thoroughly analyzed. For the porous structures under consideration, it is shown that (1) the kinetic energy lost in the vortices is insignificant even at high filtration velocities and (2) the shape of the flow streamlines induced by the recirculation zones plays an important role in the variation of the flow structure, which is correlated itself to the different flow regimes.
2017-01-01T00:00:00ZAGNAOU, M.LASSEUX, D.AHMADI, A.Inertial flow in porous media occurs in many situations of practical relevance among which one can cite flows in column reactors, in filters, in aquifers, or near wells for hydrocarbon recovery. It is characterized by a deviation from Darcy’s law that leads to a nonlinear relationship between the pressure drop and the filtration velocity. In this work, this deviation, also known as the nonlinear, inertial, correction to Darcy’s law, which is subject to controversy upon its origin and dependence on the filtration velocity, is studied through numerical simulations. First, the microscopic flow problem was solved computationally for a wide range of Reynolds numbers up to the limit of steady flow within ordered and disordered porous structures. In a second step, the macroscopic
characteristics of the porous medium and flow (permeability and inertial correction tensors) that appear in the macroscale model were computed. From these results, different flow regimes were identified: (1) the weak inertia regime where the inertial correction has a cubic dependence on the filtration velocity and (2) the strong inertia
(Forchheimer) regime where the inertial correction depends on the square of the filtration velocity. However, the existence and origin of those regimes, which depend also on the microstructure and flow orientation, are still not well understood in terms of their physical interpretations, as many causes have been conjectured in the literature. In the present study, we provide an in-depth analysis of the flow structure to identify the origin of the deviation from Darcy’s law. For accuracy and clarity purposes, this is carried out on two-dimensional structures. Unlike the previous studies reported in the literature, where the origin of inertial effects is often identified on a heuristic basis, a theoretical ustification is presented in this work. Indeed, a decomposition of the convective
inertial term into two components is carried out formally allowing the identification of a correlation between the flow structure and the different inertial regimes. These components correspond to the curvature of the flow streamlines weighted by the local fluid kinetic energy on the one hand and the distribution of the kinetic energy
along these lines on the other hand. In addition, the role of the recirculation zones in the occurrence and in the form of the deviation from Darcy’s law was thoroughly analyzed. For the porous structures under consideration, it is shown that (1) the kinetic energy lost in the vortices is insignificant even at high filtration velocities and (2) the shape of the flow streamlines induced by the recirculation zones plays an important role in the variation of the flow structure, which is correlated itself to the different flow regimes.What can be learnt about dispersivity from transport experiments in unsaturated double-porosity soils?
http://hdl.handle.net/10985/12137
TRAN NGOC, T.D.; AHMAD, Azita; BERTIN, Henri
Dispersivity is assumed to be an intrinsic property which characterizes the heterogeneity scale of porous media. When the medium is unsaturated by two fluid phases (water and air), dispersivity depends strongly on the saturation. “Double-porosity” medium concept can be attributed to a class of heterogeneous soils and rocks in which a strong contrast in local pore size characteristics is observed. In this work, we revisited the intrinsicity of the dispersivity of a double-porosity soil with different saturations, by performing a series of one-dimensional experiments of chloride tracer dispersion under different boundary conditions. The physical double-porosity model was composed of solidified clayey balls, distributed periodically in a more permeable sandy matrix. The dependence of the dispersivity on the saturation in the double-porosity soil was established and compared with the trends obtained for the single-porosity soils in previous studies.
RÉSUMÉ
Il est généralement admis que la dispersivité est une propriété intrinsèque qui caractérise l'échelle d'hétérogénéité des milieux poreux. Lorsque le milieu est saturé par deux phases fluides (eau et air), elle dépend fortement de la saturation. Le concept du milieu à « double-porosité » peut être attribué à une classe de sols et de roches hétérogènes dans lesquels on observe un fort contraste de tailles caractéristiques locales de pores. Dans ce travail, nous avons revisité la nature intrinsèque de dispersivité d'un sol à double porosité avec des saturations différentes, en effectuant une série d'expériences unidimensionnelles de la dispersion d’un traceur, pour différentes conditions aux limites. Le modèle physique à double porosité a été composé de sphères argileuses solidifiées, réparties périodiquement dans une matrice de sable plus perméable. La dépendance de la dispersivité de la saturation dans le sol à double porosité a été établie et comparée avec les tendances obtenues pour les sols à simple-porosité dans des études antérieures.
2017-01-01T00:00:00ZTRAN NGOC, T.D.AHMAD, AzitaBERTIN, HenriDispersivity is assumed to be an intrinsic property which characterizes the heterogeneity scale of porous media. When the medium is unsaturated by two fluid phases (water and air), dispersivity depends strongly on the saturation. “Double-porosity” medium concept can be attributed to a class of heterogeneous soils and rocks in which a strong contrast in local pore size characteristics is observed. In this work, we revisited the intrinsicity of the dispersivity of a double-porosity soil with different saturations, by performing a series of one-dimensional experiments of chloride tracer dispersion under different boundary conditions. The physical double-porosity model was composed of solidified clayey balls, distributed periodically in a more permeable sandy matrix. The dependence of the dispersivity on the saturation in the double-porosity soil was established and compared with the trends obtained for the single-porosity soils in previous studies.
RÉSUMÉ
Il est généralement admis que la dispersivité est une propriété intrinsèque qui caractérise l'échelle d'hétérogénéité des milieux poreux. Lorsque le milieu est saturé par deux phases fluides (eau et air), elle dépend fortement de la saturation. Le concept du milieu à « double-porosité » peut être attribué à une classe de sols et de roches hétérogènes dans lesquels on observe un fort contraste de tailles caractéristiques locales de pores. Dans ce travail, nous avons revisité la nature intrinsèque de dispersivité d'un sol à double porosité avec des saturations différentes, en effectuant une série d'expériences unidimensionnelles de la dispersion d’un traceur, pour différentes conditions aux limites. Le modèle physique à double porosité a été composé de sphères argileuses solidifiées, réparties périodiquement dans une matrice de sable plus perméable. La dépendance de la dispersivité de la saturation dans le sol à double porosité a été établie et comparée avec les tendances obtenues pour les sols à simple-porosité dans des études antérieures.Thermographic and tomographic methods for tridimensional characterization of thermal transfer in silica/phenolic composites
http://hdl.handle.net/10985/12134
BESSON, Gerard; AHMADI-SENICHAULT, Azita; CATY, Olivier; AYVAZYAN, Vahan; GREGORI, M.L; COSTA, S.F.; VIGNOLES, G.L.
Composites Part B: Engineering
Silica/phenolic composite materials are often used in thermal protection systems (TPS) for atmospheric re-entry. The present work aims to compare two different approaches to assess heat transfer properties of these materials: i) using standard and specific xperimental methods, and ii) with the development of 3D thermal transfer multiscale model using 2D (microscopy) and 3D (tomography) images. The latter procedure, based on computations on images, is a two-step change of scale from microscopic scale to
mesoscopic scale and then to the macroscopic one. Two silica/phenolic composites with different spatial organizations are studied and their thermal properties are compared. Several experimental methods have been used, including space-resolved diffusivity determinations. Numerical results are compared to experimental ones in terms of transverse and longitudinal thermal conductivities of the composites, and
were found to be in good agreement. A discussion is made on the different possible sources of uncertainty for both methods.
2016-11-01T00:00:00ZBESSON, GerardAHMADI-SENICHAULT, AzitaCATY, OlivierAYVAZYAN, VahanGREGORI, M.LCOSTA, S.F.VIGNOLES, G.L.Silica/phenolic composite materials are often used in thermal protection systems (TPS) for atmospheric re-entry. The present work aims to compare two different approaches to assess heat transfer properties of these materials: i) using standard and specific xperimental methods, and ii) with the development of 3D thermal transfer multiscale model using 2D (microscopy) and 3D (tomography) images. The latter procedure, based on computations on images, is a two-step change of scale from microscopic scale to
mesoscopic scale and then to the macroscopic one. Two silica/phenolic composites with different spatial organizations are studied and their thermal properties are compared. Several experimental methods have been used, including space-resolved diffusivity determinations. Numerical results are compared to experimental ones in terms of transverse and longitudinal thermal conductivities of the composites, and
were found to be in good agreement. A discussion is made on the different possible sources of uncertainty for both methods.Experimental Investigation of Inverter Open-Circuit Fault Diagnosis for Bi-Harmonic Five-Phase Permanent Magnet Drive
http://hdl.handle.net/10985/11934
TRABELSI, Mohamed; SEMAIL, Eric; NGUYEN, Ngac Ky
IEEE Journal of Emerging and Selected Topics in Power Electronic
This paper proposes a procedure that is suitable for experimental investigation of real-time open-switch and open-phase faults diagnosis of a five-leg Voltage Source Inverter (VSI) feeding a five-phase bi-harmonic Permanent Magnet Synchronous Machine (5-Φ B-PMSM). The algorithm is based on the specific characteristics of multiphase machines, which allows inverter fault detection with sufficient robustness of the algorithm in the presence of fundamental and third harmonic components. Firstly, the inverter fault effects analysis is achieved in the characteristic subspaces of the five-phase PMSM. Specificities that are interesting for the elaboration of a real-time Fault Detection and Identification (FDI) process are highlighted. Original and particular algorithms are used for an accurate two-dimensional normalized fault vector extraction in a defined fault reference frame. This frame is dedicated only for fault detection and identification. To ensure the high immunity of the FDI process against transient states, a particular normalization procedure is applied. The normalized diagnostic signals are formulated from the defined frame and others variables derived from the reference and measured currents. Simulation and experimental results of open-switch and open-phase faults are provided to validate the proposed algorithm.
2017-01-01T00:00:00ZTRABELSI, MohamedSEMAIL, EricNGUYEN, Ngac KyThis paper proposes a procedure that is suitable for experimental investigation of real-time open-switch and open-phase faults diagnosis of a five-leg Voltage Source Inverter (VSI) feeding a five-phase bi-harmonic Permanent Magnet Synchronous Machine (5-Φ B-PMSM). The algorithm is based on the specific characteristics of multiphase machines, which allows inverter fault detection with sufficient robustness of the algorithm in the presence of fundamental and third harmonic components. Firstly, the inverter fault effects analysis is achieved in the characteristic subspaces of the five-phase PMSM. Specificities that are interesting for the elaboration of a real-time Fault Detection and Identification (FDI) process are highlighted. Original and particular algorithms are used for an accurate two-dimensional normalized fault vector extraction in a defined fault reference frame. This frame is dedicated only for fault detection and identification. To ensure the high immunity of the FDI process against transient states, a particular normalization procedure is applied. The normalized diagnostic signals are formulated from the defined frame and others variables derived from the reference and measured currents. Simulation and experimental results of open-switch and open-phase faults are provided to validate the proposed algorithm.Approach for numerical modeling of airfoil dynamic stall
http://hdl.handle.net/10985/11888
VELKOVA, Cvetelina; DOBREV, Ivan; TODOROV, Michael; MASSOUH, Fawaz
The aim of the computational study is to present different approach for numerical modeling of airfoil dynamic stall as the airfoil is pitched at a constant rate from zero incidences to a high angle of attack. An application of the Detached-Eddy Simulation model on a NACA 0012 airfoil is presented. The DES model is a method for predicting turbulence in CFD computations, which combines a Reynolds Averaged Navier-Stokes (RANS) method in the boundary layer with a Large Eddy Simulation (LES) in the free shear flow. (DES) turbulence model gives a good accuracy of the flow field because its solves an additional equation for turbulent Reynolds number in a shear stress transport version (SST), which solves a first equation for the turbulent energy K and a second equation for the specific turbulent dissipation rate w. The approach using DES turbulence model is effective because it gives better visualization of flow field, the unsteady separation flow and vortex shedding. Consequently the suggested approach is suitable and it can be used in prediction of dynamic stall phenomenon in the stage of helicopter rotors, wind turbine rotors and aircraft wings design purposes.
http://e-university.tu-sofia.bg/e-publ/files/882_BulTrans_12_Velkova.pdf
2012-09-01T00:00:00ZVELKOVA, CvetelinaDOBREV, IvanTODOROV, MichaelMASSOUH, FawazThe aim of the computational study is to present different approach for numerical modeling of airfoil dynamic stall as the airfoil is pitched at a constant rate from zero incidences to a high angle of attack. An application of the Detached-Eddy Simulation model on a NACA 0012 airfoil is presented. The DES model is a method for predicting turbulence in CFD computations, which combines a Reynolds Averaged Navier-Stokes (RANS) method in the boundary layer with a Large Eddy Simulation (LES) in the free shear flow. (DES) turbulence model gives a good accuracy of the flow field because its solves an additional equation for turbulent Reynolds number in a shear stress transport version (SST), which solves a first equation for the turbulent energy K and a second equation for the specific turbulent dissipation rate w. The approach using DES turbulence model is effective because it gives better visualization of flow field, the unsteady separation flow and vortex shedding. Consequently the suggested approach is suitable and it can be used in prediction of dynamic stall phenomenon in the stage of helicopter rotors, wind turbine rotors and aircraft wings design purposes.Influence of the Manufacturing Process of a Claw-Pole Alternator on its Stator Shape and Acoustic Noise
http://hdl.handle.net/10985/11835
TAN-KIN, Antoine; HAGEN, Nicolas; LANFRANCHI, Vincent; CLENET, Stephane; COOREVITS, Thierry; MIPO, Jean Claude; LEGRANGER, Jerome; PALLESCHI, frederic
Transcations on Industry Applications
This paper shows the influence of the manufacturing process of a claw-pole alternator on its acoustic noise. First, the stator welds and the assembly of the stator in the
brackets are linked to deformations of the inner diameter of the stator. Then, the influences of these deformations on the magnetic forces and the subsequent acoustic noise are investigated. Results show that the deformations caused by the
manufacturing process significantly increase the sound power level of particular orders.
2017-05-25T00:00:00ZTAN-KIN, AntoineHAGEN, NicolasLANFRANCHI, VincentCLENET, StephaneCOOREVITS, ThierryMIPO, Jean ClaudeLEGRANGER, JeromePALLESCHI, fredericThis paper shows the influence of the manufacturing process of a claw-pole alternator on its acoustic noise. First, the stator welds and the assembly of the stator in the
brackets are linked to deformations of the inner diameter of the stator. Then, the influences of these deformations on the magnetic forces and the subsequent acoustic noise are investigated. Results show that the deformations caused by the
manufacturing process significantly increase the sound power level of particular orders.