Arbitrary Temperature Distribution Research Articles

Thermoacoustic velocity and pressure oscillations in one-dimensional combustors with spatially varying heat sources and thermal distributions

In this work, an asymptotically based framework using a naturally occurring perturbation parameter is further developed in the context of a one-dimensional tube with an open–open end point configuration without and with a spatially variable heat source. This approach is shown to produce accurate predictions of not only pressure waveforms, as previously demonstrated, but also of velocity mode shapes and frequencies for arbitrary temperature distributions that mimic a wide variety of flow heating arrangements. These include those associated with a Rijke tube as well as other thermoacoustic sound-generation devices. The underlying formulation consists of two linearly coupled partial differential equations that can be solved simultaneously while using a Green's function to capture the thermoacoustically induced velocity. The strategy leading to accurate predictions of the thermoacoustic velocity is described and then applied to several representative cases. Results pertaining to the acoustic velocity and pressure are systematically discussed and compared to other models in the literature. Overall, we find the axially dependent thermal gain to have the most significant impact on the mode shape structure, frequencies, and nodal locations. For example, higher thermal gains lead to elevated wave propagation speeds in the downstream segment of the tube. As such, they cause the velocity nodes to gradually shift upstream with successive increases in the thermal gain. Moreover, due in part to a non-homogeneous pressure–temperature pumping term that appears exclusively in the acoustic velocity formulation, the main difference between the acoustic pressure and velocity oscillation frequencies can be directly attributed to specific derivatives of the spatially varying thermal profiles.

Heat transfer analysis of compressible laminar flow in a parallel-plates channel via integral transforms

The present work analyzes a coupled nonlinear mathematical model for heat transfer in compressible laminar flow within a parallel plates channel. The governing partial differential equations are obtained considering the conservation of mass, momentum and energy, based on a two-dimensional steady laminar flow of an ideal gas with constant physical properties and taking into account viscous dissipation. Two kinds of boundary conditions at the wall were considered, namely, prescribed arbitrary temperature or heat flux longitudinal distributions. The mathematical model was solved using the hybrid numerical-analytical method known as the Generalized Integral Transform Technique (GITT). A convergence analysis of the proposed eigenfunction expansions for velocity and temperature fields was undertaken, indicating that fairly low truncation orders are required for the accurate representation of the potentials. The proposed hybrid solution was also critically compared with the Mathematica NDSolve finite difference-based routine and additional theoretical and experimental results in the literature, with good agreement in all cases considered.

BTE-Barna: An extension of almaBTE for thermal simulation of devices based on 2D materials

We present BTE-Barna (Boltzmann Transport Equation - Beyond the RTA for NAnosystems), a software package that extends the Monte Carlo (MC) module of the almaBTE solver of the Peierls-Boltzmann transport equation for phonons (PBTE) to work with nanosystems based on 2D materials with complex geometries. To properly capture how the phonon occupations evolve in momentum space as a result of scattering, we have supplemented the relaxation-time approximation with an implementation of the propagator for the full linearized version of the PBTE. The code can now find solutions for finite and extended devices under the effect of a thermal gradient, with isothermal reservoirs or with an arbitrary initial temperature distribution in space and time, writing out the temperature and heat flux distributions as well as their spectral decompositions. Besides the full deviational MC solver, a number of useful approximations for highly symmetric devices are also included. Program summaryProgram Title::BTE-BarnaCPC Library link to program files:https://doi.org/10.17632/t58z289tpc.1Developer's repository link:https://github.com/sousaw/BTE-BarnaCode Ocean capsule:https://codeocean.com/capsule/4171806Licensing provisions: Apache-2.0Programming language: C++Nature of problem: Calculation of temperature profiles, thermal flux and/or effective thermal conductivities for nanosystems based on 2D materials.Solution method: For highly symmetric systems confined along some direction(s), the linearized phonon Boltzmann transport equation is solved iteratively by partially suppressing the phonon lifetimes due to boundaries. For more complex geometries, an energy-based deviational Monte Carlo method including off-diagonal terms in the linearized scattering operator arising from the phonon-phonon interaction is used.Additional comments including restrictions and unusual features: Depends on the following external libraries: boost, Eigen, HDF5, MessagePack, MPI, oneTBB, spglib, and almaBTE.

Different perspectives on predicting the thermoacoustic energy conversion response in a Rijke tube

In this work, six different perspectives on characterizing the thermoacoustic field in an open-ended Rijke tube are considered and discussed. These begin with a three-pronged approach consisting of theoretical, experimental, and numerical investigations of the Rijke tube's time-dependent field. It is followed by a discussion of effective techniques that rely on either Green's function or differential equation models. Finally, a perturbation expansion is introduced that leverages a naturally occurring small parameter in the open tube configuration. This approach is shown to produce accurate predictions of pressure modal shapes and frequencies for an arbitrary specified temperature distribution. It also leads to a set of linear partial differential equations that can be solved in conjunction with a Green's function expression for the thermoacoustic pressure, velocity, and heat oscillations. In this study, the underlying framework is presented and evaluated for the pressure disturbance only. Another fundamental result includes a similarity parameter, coined the Rijke number, which plays an essential role in driving thermoacoustic oscillations, namely, by relating heat-flux fluctuations to the acoustic velocity and pressure. In this context, we find that the peak value of the energy-flux vector modulus, which stands for the modular product of acoustic velocity and pressure, does indeed occur at the heat source location and increases with the heat power input.

An analytical approach to the cooling of a flat plate

The present work deals with the cooling of a flat plate having an initial non-homogeneous temperature distribution, which undergoes a sudden temperature drop at its surfaces. Hence, starting from an arbitrary steady-state temperature distribution, the two surfaces of the slab are suddenly cooled down to the value θ0, resulting in a strong perturbation of the initial temperature distribution. A general analytical solution is provided by adopting the variable separation technique to the Fourier equation in transient conditions, thus obtaining a general solution in terms of a very fast converging series, which includes a definite integral of the initial steady-state distribution. Common engineering applications are investigated and the related results, obtained thanks to an extremely low computational effort, are presented and discussed.

A model of micro lubrication between two walls with unequal temperature distribution based on kinetic theory

Micro lubrication of a gas between two walls with arbitrary and independent temperature distributions is studied on the basis of the Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation. The BGKW equation is studied analytically using the slowly varying approximation. Following the author's previous study [T. Doi, “A model of micro lubrication between two walls with an arbitrary temperature difference based on kinetic theory,” Phys. Fluids 32, 052005 (2020)], the leading-order approximation, which ought to be the solution of the nonlinear heat transfer problem, is replaced by its free molecular solution. A lubrication model of the Reynolds-type equation is derived in closed form. A direct numerical analysis of the lubrication flow subject to localized heating or cooling of the walls is conducted for an assessment of the lubrication model. The lubrication lift calculated using the model agrees with that of the direct numerical solution within an error of 5% when the Knudsen number based on the gap size lies between 0.1 and 10. The result of the lubrication model agrees also with that of the Boltzmann equation for a variable hard sphere gas. A sharp peak arises in the pressure distribution for large Knudsen numbers owing to the effect of thermal creep flows induced by localized heating.

On formation of dry spots in heated liquid films

Here, the phenomenon of food sticking when frying in a frying pan is experimentally explained. Thermocapillary convection causes a dry spot formation in the center of the frying pan upon heating of the sunflower oil film. It is shown that the speed of formation of a dry spot is similar to the speed of receding motion of the edge of a droplet upon impact and spreading on a solid surface. This allows theoretical determination of the speed of dewetting. For the thin liquid film flowing vertically over a solid surface, the critical volumetric flow rate qcr partitions two regimes: metastable or subcritical, when small perturbation of the film free surface results in the film rupture (q < qcr) and stable or supercritical at q > qcr. For the falling thin liquid film, the critical volumetric flow rate qcr partitions two regimes: metastable or subcritical (q < qcr) and stable or supercritical at q > qcr. At q < qcr, small deformations of the film free surface result in the film rupture. For the case of the temperature distribution in the form of a unit step function, the fundamental solution G1(x) describing the deformation of the film free surface has been derived by the perturbation technique. This solution is important by itself since it describes the most “dangerous” film surface profile at a prescribed value of the temperature drop. For an arbitrary surface temperature distribution θ (ξ), the convolution of G1(ξ) and θ ′(ξ) yields the film thickness profile.

Combined scalar tracking, computational fluid dynamics, and multi-objective genetic algorithm method to accelerate optimization of film cooling systems

This paper presents a method to significantly accelerate optimization of film cooling systems. The method combines high-fidelity computational fluid dynamics with scalar tracking implemented, a proxy model (linear superposition model) initialized with the computational fluid dynamics solution, and a multi-objective evolutionary algorithm approach. The proposed method is structured as follows: the computational fluid dynamics solution is used to predict the (generally complex) flow domain for the film cooling system; the scalar tracking method identifies the contributions of individual holes to an overall cooling effectiveness distribution by associating a unique passive scalar variable to the flow associated with each hole, and solving an additional advection–diffusion (scalar transport) equation; the proxy model is a (generally linear) superposition model implemented – for example – in Matlab, which inherits the scalar values from the computational fluid dynamics solution, and allows extrapolation of solutions to new design points as part of an optimization process; the optimization process is handled with a multi-objective evolutionary algorithm approach which iterates the proxy model to optimize for a defined objective function. The process works with inner and outer convergence loops. The inner convergence loop is the multi-objective evolutionary algorithm interfacing with the proxy model, which achieves convergence against a design target. At the end of each inner loop cycle, a high-fidelity computational fluid dynamics simulation is run, and this is used to recalibrate the proxy model. Convergence for a given objective function is typically achieved with six outer-loop iterations (high-fidelity computational fluid dynamics runs) and 10,000 inner-loop iterations per outer-loop iteration. The significant advantage of the proposed method is that for certain optimization problems, the computational cost can be reduced by several orders of magnitude, replacing thousands of high-fidelity computational fluid dynamics runs with approximately six computational fluid dynamics runs. The process is demonstrated by applying the optimization method to the film cooling of a flat plate. In our example we have an objective function which maximizes the component life (related to the difference from an arbitrary target temperature distribution) and minimizes the mixing loss introduced by the films. The flow environment was moderately compressible. The optimization converged after six computational fluid dynamics runs. A 30% reduction in mixing loss, a 11% increase in component life, and a 30% reduction in cooling mass flow rate were achieved. The advantages and limitations of the proposed method are also discussed in detail.

Hydrostatic equilibrium preservation in MHD numerical simulation with stratified atmospheres

Context. Many astrophysical processes involving plasma flows are produced in the context of a gravitationally stratified atmosphere in hydrostatic equilibrium, in which strong gradients can exist with gas properties that vary in small regions by several orders of magnitude. The standard Godunov-type schemes with polynomial reconstruction used to numerically solve these problems fail to preserve the hydrostatic equilibrium owing to the appearance of spurious fluxes generated by the numerical unbalance between gravitational forces and pressure gradients. Aims. The aim of this work is to present local hydrostatic reconstruction techniques that can be implemented in existing codes with Godunov-type methods to obtain well-balanced schemes that numerically satisfy the hydrostatic equilibrium for various conditions. Methods. The proposed numerical scheme is based on the Godunov method with second order MUSCL-type reconstruction, as is extensively used in astrophysical applications. The difference between the scheme and the standard formulations is only given by calculating the pressure and density Riemann states on each intercell face and by computing the gravitational source term on each cell. Results. The local hydrostatic reconstruction scheme is implemented in the FLASH code to verify the well-balanced property for hydrostatic equilibrium with constant or linearly variable temperature and constant or variable gravity. In addition, the behavior of the scheme for hydrostatic equilibrium with arbitrary temperature distributions is also analyzed together with the ability to propagate low-amplitude waves and to capture shock waves. Conclusions. The scheme is demonstrated to be robust and relatively simple to implement in existing codes. This approach produces good results in hydrostatic equilibrium preservation, satisfying the well-balanced property for the preset conditions and strongly reducing the spurious fluxes for extreme configurations.

Thermal-Mechanical Fracture Analysis Considering Heat Flux Singularity

Abstract Precise modeling of thermoelastic cracks remains challenging due to the fact that both heat flux and stress fields have singularity issue. In the previous studies, the first author proposed different types of symplectic analytical singular element (SASE) for thermal conduction and stress analysis of cracks. It has been demonstrated that these crack-tip elements of which the interior fields are defined by analytical solutions are highly accurate and efficient. However, the thermal mechanical coupling problem of crack cannot be treated with the existing SASEs. The main difficulty is that the analytical solution of the crack problem considering arbitrary temperature distribution is not available. Approximate solution may lead to significant numerical instabilities. Moreover, the construction of a crack-tip singular element for both thermal conduction and stress analysis is complicated and requires more efforts. In this study, the governing symplectic dual equation of thermoelastic crack is restudied. The analytical solution considering arbitrary temperature distribution is obtained in close form which, to the best of the authors' knowledge, has not been found before. Then, the finite element formulation of a new SASE for thermal-mechanical fracture analysis is derived analytically through a variational approach. A two-step analysis procedure is proposed to calculate the mixed mode thermal stress intensity factors (TSIFs)) and the analysis can be done on a fixed finite element mesh. Mesh refinement around the crack tip is unnecessary, and the mixed-mode TSIFs can be solved accurately without any postprocessing.

An iterative fission matrix scheme for calculating steady-state power and critical control rod position in a TRIGA reactor

Although Monte Carlo simulations are the most accurate representation of nuclear reactor cores, they can be computationally expensive. This computational expense can prohibit the use of Monte Carlo in situations that require sequences of neutronics calculations, such as accounting for temperature feedback. In the present work, we demonstrate a fission matrix methodology to calculate the fission source in the Penn State Breazeale Reactor while accounting for control rod movement and fuel temperature feedback. A fission matrix A, represents the rate of fission neutron production in a reactor core, where the aij entry of the matrix is equal to the neutrons produced in cell i per neutron born from cell j. The fission source distribution and multiplication factor keff are the principle eigenvector and principle eigenvalue, respectively, of the fission matrix. Given a varied core temperature distribution, a new fission matrix can be obtained by interpolating from a database of pre-calculated fission matrices. This approach shows promise in accurately and quickly calculating the reactivity and fission source distributions for a system with an arbitrary temperature distribution. In the present work, the largest resulting relative error of the calculated power distribution per pin was 1.52% compared to a Serpent criticality simulation, and all calculated eigenvalues were within 50 pcm, which corresponds to 0.1–0.4 cm of rod insertion depending on whether rods are inserted in the center or edge of the core. In contrast, the critical rod position varies by 13.49 cm for the cases considered (0–650 kW). The present database is composed of data from 78 Monte Carlo calculations, but enables subsequent neutronics calculations to be done in a fraction of a second. Future work would be to implement correction factors for neighboring temperature differences in three-dimensional models, incorporate xenon feedback, and investigate ways to decrease database size.

A general model for the evolution of non-equilibrium systems

The paper addresses the problem of the evolution of systems that are initially in a state of non-equilibrium. The model we propose leads to an equation of motion that starts from a rephrasing of the classical non-equilibrium Ginzburg-Landau equation by reinterpreting in the sense of an exergy evolution paradigm. This paper may be considered as a logical corollary to -and at the same time as an e conceptual extension of-the solution to the problem of the existence and quantification of a non-equilibrium exergy presented in previous articles by the present Authors. In previous papers it was shown that, if both energy and exergy are considered a priori concepts, the evolution of the exergy of a solid body subject to a sufficiently smooth relaxation process can be calculated for arbitrary initial temperature or concentration distributions with an accuracy that depends only on the information about the initial distribution of the system properties at the initial time and on the availability of proper material relations. It was shown that the non-equilibrium exergy, i.e., the extra ideal work that can be extracted from the body, relaxes to zero as the system tends to its equilibrium state, so that the total exergy content (given by the sum of non-equilibrium and equilibrium exergy) attains the value given by its classical definition. The evolution history depends of course on the imposed b.c. and on the “gradient” that drives the relaxation. In this paper, we formalize the dependence of the non-equilibrium exergy on its possible drivers (pressure, temperature or concentration gradients) and derive a general “equation of motion” that links the former to the latter. The solution is analytical, and therefore there is no need to postulate local equilibrium, as long as we are dealing with a continuum (scales sufficiently removed from the atomic ones). A few applications to ideal and real processes are presented and discussed, while the application of the method to more complex and industrially relevant cases is left for later studies. The paradigm is theoretically simple and the resulting model of relatively easy implementation: we therefore hope that applications of the proposed framework may be systematically developed in the fields of engineering and natural science, to gain a better insight into real non-equilibrium processes.

Single- and two-phase water jet impingement heat transfer on a hot moving surface

In this study, the laminar flow and heat transfer of water jet impingement on a hot moving plate is investigated. A similarity solution is applied to momentum and energy equations formulating the single-phase forced convection in order to determine the flow velocity and heat transfer. The heat flux in flow boiling regime is predicted by a superposition approach which is based on the combination of the single-phase and nucleate pool boiling components. The effects of surface motion and arbitrary surface temperature distribution on important forced convection and nucleate boiling heat transfer parameters for both stationary and moving plates are examined in the stagnation line and its nearby region. The results show that surface motion does not affect the rate of heat transfer in stagnation region when surface temperature is constant, while this motion is found to decrease heat transfer for a non-uniform surface temperature distribution state. However, it is observed that in fully developed nucleate boiling regime, the parameters including the surface velocity, the surface temperature gradient and the local distance from the stagnation line have negligible effect on the rate of heat transfer from the surface.

Engine-Scalable Rotor Casing Convective Heat Flux Evaluation Using Inverse Heat Transfer Methods

In this paper, we propose a two-step methodology to evaluate the convective heat flux along the rotor casing using an engine-scalable approach based on discrete Green's functions . The first step consists in the use of an inverse heat transfer technique to retrieve the heat flux distribution on the shroud inner wall by measuring the temperature of the outside wall; the second step is the calculation of the convective heat flux at engine conditions, using the experimental heat flux and the Green functions engine-scalable technique. Inverse methodologies allow the determination of boundary conditions; in this case, the inner casing surface heat flux, based on measurements from outside of the system, which prevents aerothermal distortion caused by routing the instrumentation into the test article. The heat flux, retrieved from the inverse heat transfer methodology, is related to the rotor tip gap. Therefore, for a given geometry and tip gap, the pressure and temperature can also be retrieved. In this work, the digital filter method is applied in order to take advantage of the response of the temperature to heat flux pulses. The discrete Green's function approach employs a matrix to relate an arbitrary temperature distribution to a series of pulses of heat flux. In this procedure, the terms of the Green's function matrix are evaluated with the output of the inverse heat transfer method. Given that key dimensionless numbers are conserved, the Green's functions matrix can be extrapolated to engine-like conditions. A validation of the methodology is performed by imposing different arbitrary heat flux distributions, to finally demonstrate the scalability of the Green's function method to engine conditions. A detailed uncertainty analysis of the two-step routine is included based on the value of the pulse of heat flux, the temperature measurement uncertainty, the thermal properties of the material, and the physical properties of the rotor casing.

Thermo-structural Analysis of Solid Rocket Scarfed Nozzle with Composite Ablative Liners for Crew Escape Solid Motor

Crew escape system (CES) is one of the most critical sub-systems in a human-rated launch vehicle. CES has four different types of solid motors. One is the high-altitude escape motor (HEM). This motor has the scarfed nozzle region at the divergent aft-end side which is different from conventional nozzle to accommodate the nozzle inside the envelope of the crew module shroud. The composite ablative phenolic liners are bonded to the nozzle metallic hardware by means of adhesive. The structural integrity of this scarfed nozzle region due to temperature and pressure plays an important role for the success of the motor. The nozzle hardware of this motor comprises of three subassemblies connected together by flanged joint. Composite ablative liners are primarily designed to satisfy the thermal and internal ballistics constraint requirements. Though metallic nozzle hardware is designed to bear the complete internal pressure loads, liners share a substantial part during operation due to its stiffness which is comparable with that of the metallic backup. In addition, the high thermal gradient also results in stresses near the inner surface of the liner. These stresses cannot be estimated by closed-form solutions considering the complexity in geometry, direction-dependent material property, and arbitrary temperature distribution arising during operation. The temperature of the liner at its inner surface is the highest due to its direct contact with hot gases. The temperature within the liner decreases across thickness. It is required to be ensured that the temperature at the liner–hardware interface does not exceed the safe-operating temperature. Thermo-structural analysis of composite ablative liners is essential to estimate the complete stress state in liners and to arrive at the minimum available structural margins. The temperature and scarfed geometry make the analysis all the more complicated. Challenges in modelling liners with the 3D contact surface elements are highlighted. Varying pressure load along the inner surface of the liners is simulated in an exclusive load step, and in addition, temperature data estimated from transient thermal analysis are applied in another load step. Temperature-dependent thermo-physical and structural properties are used for the analysis. The temperature distribution across the liner thickness and stresses in the liner and metallic nozzle are reported at different cross sections and their criticality is analysed. This paper highlights the integrated 3D finite-element modelling and analysis of composite liners of solid rocket scarfed nozzle and also covers the comparison of the mechanical strain and thermal parameters with post-static test values.

Experimental Validation of a Boundary Layer Convective Heat Flux Measurement Technique

If a steady thermal boundary layer is sufficiently thick, wall heat fluxes and associated convective heat transfer coefficients can be directly calculated from measured temperature distributions taken within it using a traversing thermocouple probe. The boundary layer can be laminar, turbulent, or transitional and on a surface of arbitrary surface temperature distribution and geometry. Herein, this technique is presented and validated in a steady, turbulent, two-dimensional boundary layer on a flat, uniform-heat-flux wall. Care is taken to properly account for radiation from the wall and conduction within the thermocouple wire. In the same setting, heat flux measurements are made for verification purposes using an energy balance on a segment of the test wall carefully designed to minimize and include radiation and conduction effects. Heat flux values measured by the boundary layer measurement technique and by the energy balance measurement agree to within 4.4% and the difference between the two lie completely within their respective measurement uncertainties of 5.74% and 0.6%.

Structured thermal surface for radiative camouflage

Thermal camouflage has been successful in the conductive regime, where thermal metamaterials embedded in a conductive system can manipulate heat conduction inside the bulk. Most reported approaches are background-dependent and not applicable to radiative heat emitted from the surface of the system. A coating with engineered emissivity is one option for radiative camouflage, but only when the background has uniform temperature. Here, we propose a strategy for radiative camouflage of external objects on a given background using a structured thermal surface. The device is non-invasive and restores arbitrary background temperature distributions on its top. For many practical candidates of the background material with similar emissivity as the device, the object can thereby be radiatively concealed without a priori knowledge of the host conductivity and temperature. We expect this strategy to meet the demands of anti-detection and thermal radiation manipulation in complex unknown environments and to inspire developments in phononic and photonic thermotronics.

An Analytical Solution for Transient Heat Conduction in a Composite Slab with Time-Dependent Heat Transfer Coefficient

An analytical solution is derived for one-dimensional transient heat conduction in a composite slab consisting of n layers, whose heat transfer coefficient on an external boundary is an arbitrary function of time. The composite slab, which has thermal contact resistance at n-1 interfaces, as well as an arbitrary initial temperature distribution and internal heat generation, convectively exchanges heat at the external boundaries with two different time-varying surroundings. To obtain the analytical solution, the shifting function method is first used, which yields new partial differential equations under conventional types of external boundary conditions. The solution for the derived differential equations is then obtained by means of an orthogonal expansion technique. Numerical calculations are performed for two composite slabs, whose heat transfer coefficient on the heated surface is either an exponential or a trigonometric function of time. The numerical results demonstrate the effects of temporal variations in the heat transfer coefficient on the transient temperature field of composite slabs.

Effect of temperature and spring-mass systems on modal properties of Timoshenko concrete beam

An exact solution for the title problem was obtained in closed-form fashion considering general boundary conditions. The expressions of moment, shear and shear coefficient (or shear factor) of cross section under the effect of arbitrary temperature distribution were first derived. In view of these relationships, the differential equations of Timoshenko beam under the effect of temperature were obtained and solved. Second, the characteristic equations of Timoshenko beam carrying several spring-mass systems under the effect of temperature were derived based on the continuity and force equilibrium conditions at attaching points. Then, the correctness of proposed method was demonstrated by a Timoshenko laboratory beam and several finite element models. Finally, the influence law of different temperature distribution modes and parameters of spring-mass system on the modal characteristics of Timoshenko beam had been studied, respectively.

A Novel Derivation of the Time Evolution of the Entropy for Macroscopic Systems in Thermal Non-Equilibrium

The paper discusses how the two thermodynamic properties, energy (U) and exergy (E), can be used to solve the problem of quantifying the entropy of non-equilibrium systems. Both energy and exergy are a priori concepts, and their formal dependence on thermodynamic state variables at equilibrium is known. Exploiting the results of a previous study, we first calculate the non-equilibrium exergy En-eq can be calculated for an arbitrary temperature distributions across a macroscopic body with an accuracy that depends only on the available information about the initial distribution: the analytical results confirm that En-eq exponentially relaxes to its equilibrium value. Using the Gyftopoulos-Beretta formalism, a non-equilibrium entropy Sn-eq(x,t) is then derived from En-eq(x,t) and U(x,t). It is finally shown that the non-equilibrium entropy generation between two states is always larger than its equilibrium (herein referred to as “classical”) counterpart. We conclude that every iso-energetic non-equilibrium state corresponds to an infinite set of non-equivalent states that can be ranked in terms of increasing entropy. Therefore, each point of the Gibbs plane corresponds therefore to a set of possible initial distributions: the non-equilibrium entropy is a multi-valued function that depends on the initial mass and energy distribution within the body. Though the concept cannot be directly extended to microscopic systems, it is argued that the present formulation is compatible with a possible reinterpretation of the existing non-equilibrium formulations, namely those of Tsallis and Grmela, and answers at least in part one of the objections set forth by Lieb and Yngvason. A systematic application of this paradigm is very convenient from a theoretical point of view and may be beneficial for meaningful future applications in the fields of nano-engineering and biological sciences.