Assumption Of Thermal Equilibrium Research Articles

Numerical investigation of porous stack for a solar-powered thermoacoustic refrigerator

This article proposes a numerical analysis for performance improvement of the stack, which represents a crucial element on the solar-powered thermoacoustic refrigerator. The stack is considered as a saturated parallelepiped homogeneous porous media. Numerical simulation of the flow and the heat transfer in the thermoacoustic refrigerator is carried out. The physical flow is governed by the modified Darcy–Brinkmann–Forchheimer model. The governing equations are solved numerically using the lattice Boltzmann method. Furthermore, the local thermal equilibrium assumption is applied to examine heat transfer. Particular attention is paid a new form of the lattice Boltzmann equation system describing the flow and the heat transfer in porous media and fluid regions. The effects of several parameters characterizing the thermal behaviour in the porous medium are studied. The parametric results lead to the optimization of the porous media form. The presence of the viscous dissipation term in the heat transfer formulation within the thermoacoustic system is particularly highlighted, due to its significant effects introduced by the Eckert number.

Effect of heat source on forced convection in a partially-filled porous channel under LTNE condition

An analysis of forced convection is presented within a channel partially filled by porous medium inserted in the core, with the inclusion of internal heat source in fluid and solid phases. By employing Brinkman extended Darcy model and local thermal non-equilibrium (LTNE) model in porous region, exact solutions are obtained including fluid velocity, solid and fluid temperatures and Nusselt number, and pertinent parameters' effects on heat transfer enhancement are performed. In addition, the solutions are also compared to that by applying the local thermal equilibrium (LTE) assumption and the LTE validity is discussed. The results find that for a partially-filled porous channel, existed heat source terms can intensify the occurrence of the singularity in Nusselt number, especially with the variation of porous filling ratio more than one singularity points can be observed under both LTE and LTNE conditions. In addition, even if the singularity exists the Nusselt number can reach to LTE condition at a high Biot number (Bi) or at a very high solid to fluid thermal conductivity ratio (K), and at a small K the LTE also holds with the fluid heat source only. Overall, partially-filled porous systems with internal heat source have more rich thermal characteristics.

Numerical Analysis of Thermochemical Nonequilibrium Flows in a Model Scramjet Engine

This numerical study was conducted to investigate the flow properties in a model scramjet configuration of the experiment in the T4 shock tunnel. In most numerical simulations of flows in shock tunnels, the inflow conditions in the test section are determined by assuming the thermal equilibrium of the gas. To define the inflow conditions in the test section, the numerical simulation of the nozzle flow with the given nozzle reservoir conditions from the experiment is conducted by a thermochemical nonequilibrium computational fluid dynamics (CFD) solver. Both two-dimensional (2D) and three-dimensional (3D) numerical simulations of the flow in a model scramjet were conducted without fuel injection. Simulations were performed for two types of inflow conditions: one for thermochemical nonequilibrium states obtained from the present nozzle simulation and the other for the data available using the thermal equilibrium and chemical nonequilibrium assumptions. The four results demonstrate the significance of the modelling approach for choosing between 2D or 3D, and thermal equilibrium or nonequilibrium.

Observation of persistent species temperature separation in inertial confinement fusion mixtures

The injection and mixing of contaminant mass into the fuel in inertial confinement fusion (ICF) implosions is a primary factor preventing ignition. ICF experiments have recently achieved an alpha-heating regime, in which fusion self-heating is the dominant source of yield, by reducing the susceptibility of implosions to instabilities that inject this mass. We report the results of unique separated reactants implosion experiments studying pre-mixed contaminant as well as detailed high-resolution three-dimensional simulations that are in good agreement with experiments. At conditions relevant to mixing regions in high-yield implosions, we observe persistent chunks of contaminant that do not achieve thermal equilibrium with the fuel throughout the burn phase. The assumption of thermal equilibrium is made in nearly all computational ICF modeling and methods used to infer levels of contaminant from experiments. We estimate that these methods may underestimate the amount of contaminant by a factor of two or more.

Prediction of flow and heat transfer through a microtube filled with bidisperse porous medium under local thermal nonequilibrium condition

AbstractIn this paper, the thermal and hydrodynamic solutions of a microtube filled with bidisperse porous medium (BDPM) under the local thermal nonequilibrium (LTNE) condition are presented. Considering the LTNE condition, the energy equations have been numerically solved. The rarefaction effects are considered for Knudsen numbers ranging from 0 to 0.1; therefore, first‐order boundary condition is applied on the wall. The temperature distribution of each phase is examined with respect to the involved parameters in the BDPM system. For the first time, the Nusselt number ratio (NRDP) is introduced to study the influence of Darcy number on the Nusselt number more precisely. Also, the effect of different thermophysical parameters on the Nusselt number is studied. The advantage of BDPM system over monodisperse porous medium (MDPM) structure is examined through the heat transfer performance parameter. The findings exhibit a good agreement with the literature. Also, the LTNE condition produces more realistic results in comparison to local thermal equilibrium assumption. On the whole, although implementing the BDPM enhances the heat transfer rate compared with the MDPM, it does not improve the thermal hydrodynamic performance significantly.

Testing of a new non-equilibrium heat and moisture transfer model in porous building materials

In this paper the new non-equilibrium model of heat and moisture transfer in heterogenous building materials is presented and tested. The new hygro-thermal model differs from the other approaches which are based on the classical assumption of instantaneous local mechanical, thermal and hygric equilibrium between vapour and water in the pores in building materials. Instead of this assumption the model uses the finite rate of transition of moisture from the liquid to the vapour state and vice versa while still keeping mechanical and thermal equilibrium between components of the medium. The linear kinetics of this transition is applied. The assessment of the model correctness is also performed in the paper. In the first step of the testing the model predictions were successfully verified with the reference data obtained numerically. Then the model was validated using reference data obtained experimentally. Finally, the influence of volumetric mass transfer coefficient between vapour and water as well as water and vapour in pores is investigated, i.e., kinetics of the vapour-water/water-vapour transition is analysed. During the model testing traditional building material were considered (i.e., ceramic brick). However, the model may be used for investigation of hygro-thermal behaviour of bio-based materials.

Study of Rayleigh–Bénard Convection of a Newtonian Nanoliquid in a High Porosity Medium Using Local Thermal Non-equilibrium Model

In the paper we make linear and non-linear stability analyses of Rayleigh–Benard convection in a Newtonian, nanoliquid-saturated porous medium using local thermal non-equilibrium model (LTNE). The LTNE assumption results in advanced onset of convection and increase in heat transport when compared to that of local thermal equilibrium assumption. Free–free and rigid–rigid, isothermal boundaries are considered for investigation. The Galerkin method is used to obtain the critical eigen value. The influence of inter-phase heat transfer coefficient, ratio of thermal conductivities, Brinkman number, porous parameter on the onset of convection as well as on heat transport has been presented graphically and discussed in detail. The effect of increasing the value of porosity modified thermal conductivities advances the onset of convection and enhances the amount of heat transport whereas the remaining parameters have an opposing influence on both onset of convection as well as heat transport.

Real-fluid injection modeling and LES simulation of the ECN Spray A injector using a fully compressible two-phase flow approach

A fully compressible two-phase flow model consisting of four balance equations including two mass, one momentum, and one internal energy equation, formulated with the mechanical and thermal equilibrium assumptions is developed in this article. This model is closed with a real fluid equation of state (EoS) and has been applied to the simulation of different 1D academic cases, in addition to the 3D Large-Eddy Simulation (LES) of the Engine Combustion Network (ECN) Spray A injector including the needle to target part with and without the phase change (i.e. frozen) assumptions. The obtained numerical results from the model with phase change have proven to be able to accurately predict the liquid, vapor penetrations and rate of injection compared to experimental data. However, the frozen model has presented some uncertainties and deviations in predicting the penetration length as with different measure criterions, even though an excellent agreement can be achieved in the estimation of rate of injection, near-nozzle mass and velocity distribution. Several conclusions are drawn from the simulations: (1) the initial in-nozzle flow has a strong effect on the early jet development; (2) considering phase change is still essential in the high temperature, high pressure (HTHP) injection modeling since it strongly affects the temperature distribution, turbulence intensity and thereby the jet development; (3) significant variations of liquid compressibility factor and density, as well as the cooling effect through the nozzle are highlighted. Overall, the detailed analysis of the numerical results reported in this article may complement the Engine Combustion Network (ECN) experimental database.

NUMERICAL ANALYSIS OF SOLAR TOWER SYSTEM UTILIZED WITH FLAT PLATE AND POROUS ABSORBER

The performance of solar updraft tower investigates numerically by comparing between two solar collectors with and without porous absorber flat plate. In this study, copper metal foam (10 and 40) PPI at the same porosity (0.9) are used as an absorber plate. The effect of the absorbing porous medium is studied to increase the air flow towards the updraft tower by tilting of the porous absorber at an angle (2° and 6°) from the horizontal line for (10 and 40) PPI and compared with the horizontal absorber flat plate. To simulate the physical quantities inside the porous medium, at steady state, symmetry, three dimensional, Darcy model and energy numerical model with local thermal equilibrium (LTE) assumption are adopted and numerical models is approximated by a RNG (Re-Normalization Group) k- ϵ turbulent model and discrete ordinates (DO) radiation model equations. The numerical study is analyzed by Fluent software package (version 18.2) to solve the governing equations. The results showed that the tilting of a porous absorber plate at an angle (2° and 6°) from the horizontal line lead to increase in the mass flow rate inside the solar updraft tower and the maximum performance is found by using 40 PPI at tilt angle 2°.The performance of solar updraft tower investigates numerically by comparing between two solar collectors with and without porous absorber flat plate. In this study, copper metal foam (10 and 40) PPI at the same porosity (0.9) are used as an absorber plate. The effect of the absorbing porous medium is studied to increase the air flow towards the updraft tower by tilting of the porous absorber at an angle (2° and 6°) from the horizontal line for (10 and 40) PPI and compared with the horizontal absorber flat plate. To simulate the physical quantities inside the porous medium, at steady state, symmetry, three dimensional, Darcy model and energy numerical model with local thermal equilibrium (LTE) assumption are adopted and numerical models is approximated by a RNG (Re-Normalization Group) k- ϵ turbulent model and discrete ordinates (DO) radiation model equations. The numerical study is analyzed by Fluent software package (version 18.2) to solve the governing equations. The results showed that the tilting of a porous absorber plate at an angle (2° and 6°) from the horizontal line lead to increase in the mass flow rate inside the solar updraft tower and the maximum performance is found by using 40 PPI at tilt angle 2°.

A simple finite element model to study the effect of plasma plume expansion on the nanosecond pulsed laser ablation of aluminum

In this paper, a simple model was proposed using finite element analysis (FEA) with a commercial FEA software ABAQUS to simulate the two-dimensional (2-D) laser heat transfer in an aluminum material. Without relying on the conventional hydrodynamic model, the proposed model not only predicts the evolutions of the temperature field and ablation profiles in the target material, but also provides an estimation on the evolutions of electron density, plasma temperature, and plasma absorption coefficient. The assumptions used in the model include the local thermal equilibrium and additional assumptions regarding the average plasma temperature and vapor density. The assumptions allowed the laser heat transfer equation to be solved together with the Saha–Eggert equation and conservation equations of matter and charge. When compared to the existing hydrodynamic models, the proposed model solves a less number of nonlinear equations and hence is computationally more efficient. The proposed FE model was employed to study the plasma-shielding effect on PLA produced by a 193 nm Excimer laser and a 266 nm Nd:YAG laser. The predictions of ablation depths, electron density, and plasma temperature agree well with the experimental data. Moreover, effects of the laser intensity and the average plasma temperature on the efficiency of the plasma shielding during PLA were also investigated and discussed in this study.

Transient thermoporoelastic model under local thermal non-equilibrium

Wellbore stability is a key challenge for oil and gas industry since it adds a great deal of additional cost to the industry. Traditional wellbore stability models such as elastic and poroelastic models may not produce reliable stress field around a wellbore since they consider isothermal wellbore condition. During the drilling phase, the drilling fluid temperature is different from the formation temperature due to geothermal gradient and circulation of the fluid inside the wellbore. Therefore, the assumption of isothermal condition will not predict the correct wellbore stability condition, especially for high pressure and high temperature wells. Hence, consideration of a thermoporoelastic model in stress estimation is more appropriate. Although there are several thermoporoelastic models in the literature, they employ the assumption of local thermal equilibrium (LTE) in their modeling. However, thermal properties of phases in a porous medium are rarely similar. This dissimilarity evokes the validation of LTE assumption. The local thermal non-equilibrium assumption ignores additional pore and thermal stresses in the porous medium caused by temperature variations of the fluid and solid phases. Under pure heat conduction, LTE is valid if thermal boundary conditions in a medium is constant or thermal conductivities of phases are similar according to Vadasz (2005) 1. Hence, using a local thermal non-equilibrium (LTNE) approach is required to establish each phase temperature, and update thermoporoelastic model accordingly. In this paper, the effects of induced thermal stresses on wellbore stress resolution and LTNE on thermoporoelastic model around a wellbore during drilling phase are investigated in low and high permeable formations. The LTNE thermoporoelastic model developed herein is a transient model considering conductive and convective heat transfers.

Heat and fluid flow analysis of metal foam embedded in a double-layered sinusoidal heat sink under local thermal non-equilibrium condition using nanofluid

The present study aims to enhance the hydrothermal performance of a porous sinusoidal double-layered heat sink using nanofluid. The optimum thickness of metal foam (nickel) for different Reynolds numbers ranging from 10 to 100 for the laminar regime and Darcy numbers ranging from 10−4 to 10−2 is obtained. At the optimum porous thicknesses, nanofluid (silver–water) with three volume fractions of nanoparticles equal to 2, 3, and 4% is employed to enhance the heat sink thermal performance. Darcy–Brinkman–Forchheimer model and the local thermal non-equilibrium model or two equations method are employed to model the momentum equation and energy equations in the porous region, respectively. It was found that in the cases of Darcy numbers 10−4, 10−3, and 10−2 the dimensionless optimum porous thicknesses are 0.8, 0.8, and 0.2, respectively. It was also obtained that the maximum PEC number is 2.12 and it corresponds to the case with Darcy number 10−2, Reynolds number 40, and volume fraction of nanoparticles 0.04. The validity of local thermal equilibrium (LTE) assumption was investigated, and it was found that increasing the Darcy number which results in an enhancement in porous particle diameter leads to some errors in results, under LTE condition.

Generalized Hydrodynamics on an Atom Chip.

The emergence of a special type of fluidlike behavior at large scales in one-dimensional (1D) quantum integrable systems, theoretically predicted in O. A. Castro-Alvaredo etal., Emergent Hydrodynamics in Integrable Quantum Systems Out of Equilibrium, Phys. Rev. X 6, 041065 (2016)10.1103/PhysRevX.6.041065 and B. Bertini etal., Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents, Phys. Rev. Lett. 117, 207201 (2016)10.1103/PhysRevLett.117.207201, is established experimentally, by monitoring the time evolution of the insitu density profile of a single 1D cloud of ^{87}Rb atoms trapped on an atom chip after a quench of the longitudinal trapping potential. The theory can be viewed as a dynamical extension of the thermodynamics of Yang and Yang, and applies to the whole range of repulsive interaction strength and temperature of the gas. The measurements, performed on weakly interacting atomic clouds that lie at the crossover between the quasicondensate and the ideal Bose gas regimes, are in very good agreement with the theory. This contrasts with the previously existing "conventional" hydrodynamic approach-that relies on the assumption of local thermal equilibrium-which is unable to reproduce the experimental data.

Kinetic mass-transfer calculation of water isotope fractionation due to cloud microphysics in a regional meteorological model

Abstract. In conventional atmospheric models, isotope exchange between liquid, gas, and solid phases is usually assumed to be in equilibrium, and the highly kinetic phase transformation processes inferred in clouds are yet to be fully investigated. In this study, a two-moment microphysical scheme in the National Center for Atmospheric Research (NCAR) Weather Research and Forecasting (WRF) model was modified to allow kinetic calculation of isotope fractionation due to various cloud microphysical phase-change processes. A case of a moving cold front is selected for quantifying the effect of different factors controlling isotopic composition, including water vapor sources, atmospheric transport, phase transition pathways of water in clouds, and kinetic-versus-equilibrium mass transfer. A base-run simulation was able to reproduce the ∼ 50 ‰ decrease in δD that was observed during the frontal passage. Sensitivity tests suggest that all the above factors contributed significantly to the variations in isotope composition. The thermal equilibrium assumption commonly used in earlier studies may cause an overestimate of mean vapor-phase δD by 11 ‰, and the maximum difference can be more than 20 ‰. Using initial vertical distribution and lower boundary conditions of water stable isotopes from satellite data is critical to obtain successful isotope simulations, without which the δD in water vapor can be off by about 34 ‰ and 28 ‰, respectively. Without microphysical fractionation, the δD in water vapor can be off by about 25 ‰.

A multicomponent real-fluid fully compressible four-equation model for two-phase flow with phase change

A fully compressible four-equation model for multicomponent two-phase flow coupled with a real-fluid phase equilibrium-solver is suggested. It is composed of two mass, one momentum, and one energy balance equations under the mechanical and thermal equilibrium assumptions. The multicomponent characteristics in both liquid and gas phases are considered. The thermodynamic properties are computed using a composite equation of state (EoS), in which each phase follows its own Peng-Robinson (PR) EoS in its range of convexity, and the two-phase mixtures are connected with a set of algebraic equilibrium constraints. The drawback of complex speed of the sound region for the two-phase mixture is avoided using this composite EoS. The phase change is computed using a phase equilibrium-solver, in which the phase stability is examined by the Tangent Plane Distance approach; an isoenergetic-isochoric flash including an isothermal-isobaric flash is applied to determine the phase change. This four-equation model has been implemented into an in-house IFP-C3D software. Extensive comparisons between the four-equation model predictions, experimental measurements in flash boiling cases, and available numerical results were carried out, and good agreements have been obtained. The results demonstrated that this four-equation model can simulate the phase change and capture most real-fluid behaviors for multicomponent two-phase flows. Finally, this validated model was applied to investigate the behaviors of n-dodecane/nitrogen mixtures in one-dimensional shock and double-expansion tubes. The complex wave patterns were unraveled, and the effects of dissolved nitrogen and the volume translation in PR EoS on the wave evolutions were revealed. A three-dimensional transcritical fuel injection is finally simulated to highlight the performance of the proposed four-equation model for multidimensional flows.

Azimuthal angle dependence of the longitudinal spin polarization in relativistic heavy ion collisions

The azimuthal angle dependence of quark spin polarization in the longitudinal beam direction of non-central relativistic heavy ion collisions is studied in the chiral kinetic approach. Contrary to the prediction from models based on the assumption of thermal equilibrium of spin degrees of freedom that the quark spin polarization always points along the direction of local vorticity field, we find the two can have opposite directions due to the effect from the transversal component of vorticity field, which can lead to a redistribution of axial charges in the produced matter. Our finding is consistent with the azimuthal angle dependence of the longitudinal spin polarization of $\Lambda$ hyperons, which is mainly determined by that of the strange quark, recently measured in the experiments by the STAR Collaboration at the Relativistic Heavy Ion Collider (RHIC).

Critical review of the local thermal equilibrium assumption in heterogeneous porous media: Dependence on permeability and porosity contrasts

Increasing demand for green energy sparked a renewed interest in extracting geothermal energy resources. An accurate description of the temperature field and related heat extraction potential remains a challenge and is critical for successful geothermal energy exploitation. Typically, two main approaches are used when modeling heat exchange and transport in geothermal reservoirs. The standard approach assumes immediate local thermal equilibrium between solid and fluid phases (LTE), while another approach considers heat transport separately for fluid and solid (LTNE). Both are then coupled through a heat exchange term. In these simple and common models, the host system is described as a heterogeneous porous media in which regions of higher fracture density are represented implicitly by areas of higher permeability within the porous media.In this work, we investigate both, LTE and LTNE, approaches and how they are affected by fluid and host rock characteristic properties. We especially focus on the influence of hydraulic heterogeneity in the porous material on the temperature difference between both approaches. We compare LTE and LTNE heat transfer models in a simple, two dimensional model with a layer of different hydraulic parameters embedded in a host rock, and study in detail the role of heterogeneous distributions of permeability and porosity. This simple geometrical model allows us to gain insights into the complex interaction between the different heat transfer models and fluid flow. Porosity and permeability contrast determine separate regimes in which the LTE assumption is lower or upper bound of outflow temperature. Using these results, we compare LTE and LTNE models in a synthetic field case with a complex heterogeneous porous media with large permeability contrast. We show that both models can diverge up to 7% in local fluid temperature, while the LTNE model is the lower bound if high permeability regions are also assumed to have a higher porosity.

Analytical solutions of fluid flow and heat transfer in a partial porous channel with stress jump and continuity interface conditions using LTNE model

Forced convection heat transfer is analytically performed in a channel partially filled with porous media located at two inner walls under local thermal non-equilibrium (LTNE) condition. A constant heat flux is imposed at the channel walls. The Brinkman extended Darcy model is applied in the porous region and the stress jump and continuity conditions are employed at the interface. Exact solutions are obtained for velocity, pressure drop, the fluid and solid temperatures and Nusselt number. The effects of pertinent parameters on the fluid flow and heat transfer are conducted. Furthermore, the solution for the Nusselt number is compared to that by applying the local thermal equilibrium (LTE) assumption and the validity of the LTE is examined. It is shown that by applying LTNE model for different solid to fluid effective thermal conductivity ratios (K) and Biot numbers (Bi), the variations of Nusselt number with hollow ratio include three types of curves, which are: a maximized Nusselt number occurs at a small optimum hollow ratio, Nusselt number monotonically decreases by increasing hollow ratio and a minimized Nusselt number occurs at a small hollow ratio, respectively. For high K, a small critical value of S at which the Nusselt number reaches to LTE Nusselt number occurs and it lowers with the increase of Bi number and the decrease of Darcy number; while for low K, the LTNE Nu number versus hollow ratio is almost the same with LTE Nu number and therefore the LTE is valid. The stress jump at the interface is found to have negligible effect on the Nusselt number and the pressure drop, except in a high Darcy number with a low stress jump coefficient where the calculation of pressure drop need to account for the stress jump effect at the interface and the Nusselt numbers for both LTE and LTNE models slightly differs from the case of stress continuity interface condition.

Numerical simulation of ballistic electron dynamics and heat transport in metallic targets exposed to ultrashort laser pulse

The role of ballistic electrons generated during ultrashort pulse laser (USPL) absorption in metallic targets was investigated in a wide range of laser intensities using our developed simulation package FEMTO-2D. The simulation package is based on the numerical solution of the two-temperature model with the assumption of local thermal equilibrium for electron and lattice subsystems within the simulation cell at any time step. Electron thermodynamic parameters were calculated through the processes of material transition from the cold solid state into the dense plasma state during and after the pulse based on the collision theory. The appropriate model for temperature dependent thermodynamic parameters allows defining the heat transport during an early stage of the USPL-matter interaction directly, without relying on the effective absorption depth model. The study investigated, for the first time, using integrated computer simulation the role of ballistic electrons in energy transfer and heat conduction during USPL deposition. The simulation predictions of the electron heat transport dynamics during and shortly after the laser pulse were benchmarked for the gold target against available experimental data and were able to confirm the dominant role of the ballistic electrons in the initial heat propagation within 100–120 nm of the target at laser intensities below 1013 W/cm2.

An evaluation of equilibrium conditions and temperature-dependent speciation in a laser-produced air plasma

A laser-produced air plasma is a dynamical system with fundamental physical parameters that change significantly during its lifespan. The evolution of the spectral features from an air plasma shows broadband radiation at early times followed by discrete electronic transitions from ions, atoms, and rotational and vibration bands from molecules. The molecular band emission from an air plasma typically appears at times ⪞5 μs and persists for 100's of μs. In this article, an evaluation is made on the temperature evolution and speciation in a laser-produced air plasma. The air plasmas are generated by focusing 1064 nm, 6 ns pulses from an Nd:YAG laser. High-resolution emission spectra of various molecules such as N2, N2+, CN, OH, NH, and NO are acquired and fit using spectral models. Fitting of atomic and molecular emission features permits tracking of the air spark temperature evolution from 1 to 200 μs. Though the excitation and molecular temperatures show a good overlap at times ⪝10 μs, the molecular temperatures obtained from different species show a discontinuity at times ∼30 μs and this is related to shock collapse and subsequent changes in hydrodynamics and chemistry of the plume. The fitting of multiple species in broadband spectra has permitted calculation of the relative concentrations of various molecules as a function of temperature that provides insight into the air spark speciation. The measured relative mole fractions from laser-induced air plasma are within factors of 2–6 of the values estimated by speciation model under the assumption of thermal and chemical equilibrium across the temperature ranges studied. Probable reasons for the observed deviation in the relative fractions are discussed.