# Systems and Methods for Transformation and Degradation Analysis

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**transformation–phenomenological entropy generation (TPEG)**theorem with practical use in the design, health/performance monitoring, diagnostics and optimization of all macroscopic systems. We establish a direct relationship between the instantaneous transformation (and degradation) of systems undergoing irreversible processes—real processes including multiple concurrent, high-rate, anisothermal processes—and the accompanying phenomenological entropy generation. Example applications for several multi-disciplinary engineering systems are demonstrated.

## 2. Definitions

- Observable: a metric such as a physical property or performance indicator is observable if it can be sensed and measured directly.
- Phenomenological: characterized by observable phenomena, such as volume expansion.
- Transformation: a change in state quantified by the difference between the instantaneously observable time varying value of a
**non-monotonic**transformation measure (or performance indicator) and its initial/reference value. - Phenomenological transformation/degradation ${w}_{phen}$: the instantaneous transformation/degradation of a system or material via a
**non-monotonic**transformation/degradation measure (or performance indicator). - Phenomenological entropy generation ${\mathbb{S}}_{phen}$: the instantaneous entropy generation along the transformation path through state space Z, observable through the state variables that characterize the active interactions, is always the sum of all active work and compositional change entropy generations and MST entropy [27]. Unlike entropy generation S′, which is always non-negative, ${\mathbb{S}}_{phen}$ is positive for energy addition and negative for energy extraction, in accordance with IUPAC sign convention.
- Reversible transformation ${w}_{rev}$: the idealized, quasi-static transformation of a system or material. ${w}_{rev}$ can be estimated as the healthiest state transformation.

## 3. A Review of the Degradation–Entropy Generation Theorem

#### Statement

## 4. A Review of the Phenomenological Entropy Generation Theorem

#### 4.1. Statement

**quantifies energy dissipation/degradation while**${\mathit{\delta}\mathbb{S}}_{\mathit{p}\mathit{h}\mathit{e}\mathit{n}}$

**characterizes total energy transformation including conversion and dissipation.**Details of the PEG theorem, including statements and derivation, can be found in reference [27].

#### 4.2. Corollary

#### 4.2.1. Heat and Work Lines—Instantaneous Orthogonality

#### 4.2.2. Dissipation Factor J and Entropic Efficiency ${\eta}_{S\prime}$

## 5. Transformation–Phenomenological Entropy Generation Theorem

#### 5.1. Reversible/Quasi-Static Transformation ${w}_{rev}$

#### 5.2. Phenomenological Transformation ${w}_{phen}$

#### 5.3. Phenomenological Transformation–Phenomenological Entropy Generation Theorem

**Theorem**

**1.**

**Z**and time-varying phenomenological variables ${\zeta}_{r}={\zeta}_{r}\left(t\right)$ such that ${p}_{r}={p}_{r}[Z,{\zeta}_{r}(t\left)\right],r=1,2,...,n$, the phenomenological transformation

**Proof.**

**While degradation**$\mathit{w}$

**involves energy dissipation, transformation**${\mathit{w}}_{\mathit{p}\mathit{h}\mathit{e}\mathit{n}}$

**includes energy conversion and dissipation**. From Equation (13), over a time period beginning with zero initial states, the accumulated phenomenological transformation is

## 6. Transformation/Degradation Analysis via TPEG Methods

#### Generalized Transformation Analysis Procedure

- (i)
- Identify a measurable transformation parameter, w, that is observable to the transformation characteristics;
- (ii)
- Measure or estimate the transformation ${w}_{meas}$, and evaluate the concurrent phenomenological entropy generation terms ${S}_{W,r}^{\prime}$ and ${S\prime}_{\mu T}$ due to the active processes during the interactions;
- (iii)
- Obtain the coefficients ${B}_{W,r}$ and ${B}_{\mu T}$ by correlating transformation ${w}_{meas}$ increments, accumulations or rates to phenomenological entropy generation increments, accumulation or rates (model calibration step);
- (iv)
- Re-combine the now-evaluated (or calibrated) coefficients ${B}_{W,r}\mathrm{a}\mathrm{n}\mathrm{d}{B}_{\mu T}$ with entropies ${S}_{W,r}^{\prime}$ and ${S\prime}_{\mu T}$ via Equation (15), to obtain instantaneous transformations in ${w}_{phen}$ which were hitherto unobservable.

## 7. Transformation and Degradation Analysis Examples

#### 7.1. Friction Sliding of Copper against Steel at Steady Speed—Steady State

^{−1}[23]. The steady-state friction force $F$ and temperature T were measured during sliding (row 2) to render $dT=0$; hence, ${S\prime}_{\mu T}=0$, and frictional entropy ${S}_{W}^{\prime}=\underset{{t}_{o}}{\overset{{t}_{f}}{\int}}\frac{F\dot{x}}{T}dt$ (row 3). Via step (iii) of Section 6, the normalized steady-state wear—the degradation measure—was correlated (curve-fitted) with frictional entropy to obtain the DEG coefficient (row 4). A near-linear correlation is evident between measured wear and single-variable entropy generation, per the DEG theorem, Equation (3).

#### 7.2. Lubricants—Grease

_{k}= 0) [13,34,35]. Table 3 demonstrates the proposed degradation analysis procedure described in Section 6, for lubricant grease shearing. Two different-composition greases—multipurpose lithium grease, NLGI 2, and aircraft lithium grease, NLGI 4—were sheared in a cup using an impeller at constant strain rates. In Table 3, the measured shear stresses and temperatures (row 2) and strain rates and grease material properties for both greases were substituted into the terms in Equation (5) to obtain the shear entropy and MST entropy densities (row 3) which were, in turn, substituted into Equation (7) to obtain the degradation model (row 4). Via step (iii) of Section 6, the DEG coefficients were obtained by simultaneously correlating (i.e., curve fitting) both time-based entropies with concurrently accumulated shear stress, the transformation measure.

#### 7.3. Energy Storage Systems—Li-ion, Ni-MH, Pb-Acid Batteries, Supercapacitors and Fuel Cells

#### 7.4. General Fatigue—Cyclic Bending and Torsion of Metal Rods

#### 7.5. Pump Flow—Pressure and Flow Rate (Internal Energy)

## 8. Elements of the TPEG Methodology

#### 8.1. PEG Terms: Work (Including Flow and Reaction) Entropy and MST/ECT Entropy

#### 8.2. Degradation, A Geometric Problem: TPEG Trajectories, Hypersurfaces, and Domains

**transformation–phenomenological entropy generation (TPEG) trajectory**—lie on tilted

**TPEG hypersurface(s).**The orthogonal multi-dimensional space occupied by TPEG trajectories and surfaces is the component’s material-dependent

**TPEG domain/space**. Processes with one significant primary interaction, e.g., the unsteady (anisothermal) mechanical shearing of grease, have a three-dimensional TPEG space. Processes with two significant interactions, e.g., the unsteady mechanical shearing of oxidizing grease, have a four-dimensional TPEG domain. Transformation mechanisms with more active processes will require yet higher dimensional TPEG spaces, which are possible mathematically. TPEG trajectories characterize loading conditions (battery discharge/charge rate, grease shear/oxidation rate, metal torsion/bending stress/strain amplitudes, flow rates, etc.). TPEG hypersurfaces characterize system/material composition and dissipative process rates. The TPEG domain defines the operating/aging/failure region, fully specifying the component’s life, transformation and degradation path for all loading conditions and active mechanisms. Proper formulation of the phenomenological entropy generation of the active processes is required to accurately determine contributions to overall entropy accumulation and degradation during system transformation, loading or operation.

#### 8.2.1. TPEG Coefficients

#### 8.2.2. Entropy Generation Subspace and Reversible Transformation Subspace: Dissipation Factor J and Entropic Efficiency ${\eta}_{S\prime}$

## 9. Instability and Critical Phenomena

#### 9.1. MST/ECT Entropy and Critical Failure Entropy ${S\prime}_{CF}$

#### 9.2. Thermal Runaway in Batteries

## 10. Discussion

## 11. Summary and Conclusions

^{2}$\approx $ 1 indicating near-100% accuracy (near-perfect correlation between theory and uncontrolled experiments). Application to system instability and critical phenomena was presented. The flexibility of degradation/transformation parameter selection was demonstrated. New parametric and geometric features—dissipation factor and entropic efficiency—were discussed. The TPEG (DEG + PEG) methodology can directly compare designs and materials for manufacturing and loading applications, in addition to in situ diagnostic performance/health monitoring and optimization. This article successfully verified theory with non-intrusive measurements of temperature and active process parameters, with consistent results. Without being system- or material-dependent, the TPEG methodology is universal, system- and material-characteristic, consistent and readily adaptable to all systems undergoing real-world dynamic loading, energy addition and/or system formation.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Nomenclature | Name | Unit |

A | chemical affinity | J/mol |

B | DEG coefficient | Ah K/Wh |

C | charge, charge transfer or capacity | Ah |

$\Delta C$ | charge fade or capacity fade | Ah |

F | Faraday’s constant | C/mol |

G | Gibbs energy | Wh |

I | discharge/charge current or rate | A |

k_{B} | Boltzmann constant | J/K |

m | mass | kg |

N | cycle number | |

N, N_{k} | number of moles of substance | mol |

p | dissipative process energy | J |

P | pressure | Pa |

q | charge | Ah |

Q | heat | J |

R | gas constant | J/mol·K |

S | entropy or entropy content | Wh/K |

S’ | entropy generation or production | Wh/K |

t | time | s, min, h |

T | temperature | degC or K |

v | voltage | V |

V | volume | m^{3} |

w | degradation measure | |

W | work | J |

Symbols | ||

μ | chemical potential | |

ζ | phenomenological variable | |

Subscripts and acronyms | ||

Ω | Ohmic | |

0 | initial | |

c, ch | charge | |

d, disch | discharge | |

f | final | |

vT, ECT | ElectroChemicoThermal | |

μT, MST | MicroStructuroThermal | |

rev | reversible | |

irr | irreversible | |

phen | phenomenological | |

DEG | degradation–entropy generation | |

PEG | phenomenological entropy generation | |

NLGI | National Lubricating Grease Institute |

## References

- Osara, J.A. Thermodynamics of Manufacturing Processes—The Workpiece and the Machinery. Inventions
**2019**, 4, 28. [Google Scholar] [CrossRef] - Bryant, M.D. Chapter 3: Thermodynamics of Ageing and Degradation in Engineering Devices and Machines. In The Physics of Degradation in Engineered Materials and Devices: Fundamentals and Principles; EBL-Schweitzer; Momentum Press: New York, NY, USA, 2014; ISBN 978-1-60650-468-0. [Google Scholar]
- Lemaitre, J.; Chaboche, J.L. Mechanics of Solid Materials; Cambridge University Press: New York, NY, USA, 1994; ISBN 978-0-521-47758-1. [Google Scholar]
- Bejan, A.; Tsatsaronis, G.; Moran, M.J. Thermal Design and Optimization; Wiley: New York, NY, USA, 1996. [Google Scholar]
- Bejan, A. Entropy Generation Minimization: The Method of Thermodynamic Optimization of Finite-Size Systems and Finite-Time Processes; Mechanical and Aerospace Engineering Series; CRC Press: Boca Raton, FL, USA, 2013; ISBN 978-1-4822-3917-1. [Google Scholar]
- Kuhn, E. An Energy Interpretation of Thixotropic Effects. Wear
**1991**, 142, 203–205. [Google Scholar] [CrossRef] - Kuhn, E. Description of the Energy Level of Tribologically Stressed Greases. Wear
**1995**, 188, 138–141. [Google Scholar] [CrossRef] - Naderi, M.; Khonsari, M.M. An Experimental Approach to Low-Cycle Fatigue Damage Based on Thermodynamic Entropy. Int. J. Solids Struct.
**2010**, 47, 875–880. [Google Scholar] [CrossRef] - Amiri, M.; Naderi, M.; Khonsari, M.M. An Experimental Approach to Evaluate the Critical Damage. Int. J. Damage Mech.
**2011**, 20, 89–112. [Google Scholar] [CrossRef] - Naderi, M.; Khonsari, M.M. A Comprehensive Fatigue Failure Criterion Based on Thermodynamic Approach. J. Compos. Mater.
**2012**, 46, 437–447. [Google Scholar] [CrossRef] - Naderi, M.; Khonsari, M.M. Thermodynamic Analysis of Fatigue Failure in a Composite Laminate. Mech. Mater.
**2012**, 46, 113–122. [Google Scholar] [CrossRef] - Rezasoltani, A.; Khonsari, M.M. On the Correlation between Mechanical Degradation of Lubricating Grease and Entropy. Tribol. Lett.
**2014**, 56, 197–204. [Google Scholar] [CrossRef] - Rezasoltani, A.; Khonsari, M.M. An Engineering Model to Estimate Consistency Reduction of Lubricating Grease Subjected to Mechanical Degradation under Shear. Tribol. Int.
**2016**, 103, 465–474. [Google Scholar] [CrossRef] - Aghdam, A.B.; Khonsari, M.M. Prediction of Wear in Grease-Lubricated Oscillatory Journal Bearings via Energy-Based Approach. Wear
**2014**, 318, 188–201. [Google Scholar] [CrossRef] - Sosnovskiy, L.; Sherbakov, S. Surprises of Tribo-Fatigue; Magic Book: Minsk, Belarus, 2009. [Google Scholar]
- Sosnovskiy, L.A.; Sherbakov, S.S. Mechanothermodynamic Entropy and Analysis of Damage State of Complex Systems. Entropy
**2016**, 18, 268. [Google Scholar] [CrossRef] - Basaran, C.; Nie, S. An Irreversible Thermodynamics Theory for Damage Mechanics of Solids. Int. J. Damage Mech.
**2004**, 13, 205–223. [Google Scholar] [CrossRef] - Gomez, J.; Basaran, C. A Thermodynamics Based Damage Mechanics Constitutive Model for Low Cycle Fatigue Analysis of Microelectronics Solder Joints Incorporating Size Effects. Int. J. Solids Struct.
**2005**, 42, 3744–3772. [Google Scholar] [CrossRef] - Basaran, C. Introduction to Unified Mechanics Theory with Applications; Springer International Publishing: Berlin/Heidelberg, Germany, 2023; ISBN 978-3-031-18621-9. [Google Scholar]
- Osara, J.; Bryant, M. A Thermodynamic Model for Lithium-Ion Battery Degradation: Application of the Degradation-Entropy Generation Theorem. Inventions
**2019**, 4, 23. [Google Scholar] [CrossRef] - Osara, J.A.; Bryant, M.D. Performance and Degradation Characterization of Electrochemical Power Sources Using Thermodynamics. Electrochim. Acta
**2021**, 365, 137337. [Google Scholar] [CrossRef] - Osara, J.A. Thermodynamics of Degradation; The University of Texas at Austin: Austin, TX, USA, 2017. [Google Scholar]
- Osara, J.A.; Bryant, M.D. Thermodynamics of Grease Degradation. Tribol. Int.
**2019**, 137, 433–445. [Google Scholar] [CrossRef] - Osara, J.A.; Bryant, M.D. Thermodynamics of Fatigue: Degradation-Entropy Generation Methodology for System and Process Characterization and Failure Analysis. Entropy
**2019**, 21, 685. [Google Scholar] [CrossRef] [PubMed] - Osara, J.A.; Bryant, M.D. A Temperature-Only System Degradation Analysis Based on Thermal Entropy and the Degradation-Entropy Generation Methodology. Int. J. Heat Mass Transf.
**2020**, 158, 120051. [Google Scholar] [CrossRef] - Osara, J.A.; Bryant, M.D. Thermodynamics of Lead-Acid Battery Degradation: Application of the Degradation-Entropy Generation Methodology. J. Electrochem. Soc.
**2019**, 166, A4188. [Google Scholar] [CrossRef] - Osara, J.A.; Bryant, M.D. Methods to Calculate Entropy Generation. Entropy
**2024**, 26, 237. [Google Scholar] [CrossRef] - Bryant, M.D.; Khonsari, M.M.; Ling, F.F. On the Thermodynamics of Degradation. Proc. R. Soc. Math. Phys. Eng. Sci.
**2008**, 464, 2001–2014. [Google Scholar] [CrossRef] - Moran, M.J.; Shapiro, H.N. Fundamentals of Engineering Thermodynamics, 5th ed.; Wiley: Hoboken NJ, USA, 2004. [Google Scholar]
- Kondepudi, D.; Prigogine, I. Modern Thermodynamics from Heat Engines to Dissipative Structures; John Wiley & Sons, Inc.: New York, NY, USA, 1998. [Google Scholar]
- Osara, J.A.; Ezekoye, O.A.; Marr, K.C.; Bryant, M.D. A Methodology for Analyzing Aging and Performance of Lithium-Ion Batteries: Consistent Cycling Application. J. Energy Storage
**2021**, 42, 103119. [Google Scholar] [CrossRef] - Burghardt, M.D.; Harbach, J.A. Engineering Thermodynamics, 4th ed.; HarperCollins College Publishers: New York, NY, USA, 1993. [Google Scholar]
- Çengel, Y.A. Thermodynamics: An Engineering Approach, 6th ed.; McGraw-Hill Higher Education: Boston, MA, USA, 2008. [Google Scholar]
- Kuhn, E. Correlation between System Entropy and Structural Changes in Lubricating Grease. Lubricants
**2015**, 3, 332–345. [Google Scholar] [CrossRef] - Lugt, P.M. Grease Lubrication in Rolling Bearings; John Wiley & Sons Ltd.: Hoboken, NJ, USA, 2013; ISBN 0824772040. [Google Scholar]
- Rincon, J.S.; Osara, J.; Bryant, M.D.; Fernández, B.R. Shannon’s Machine Capacity & Degradation-Degradation-Entropy Generation Methodologies for Failure Detection: Practical Applications to Motor-Pump Systems. In Proceedings of the 37th International Pump Users Symposium; Turbomachinery Laboratory, Texas A&M Engineering Experiment Station, Houston, TX, USA, 13 December 2021. [Google Scholar]
- Bryant, M.D.; Osara, J.A. On Degradation-Entropy Generation Theorems and Vector Spaces for Irreversible Thermodynamics. Appl. Mech.
**2024**. in submission. [Google Scholar] - Feng, X.; Ouyang, M.; Liu, X.; Lu, L.; Xia, Y.; He, X. Thermal Runaway Mechanism of Lithium Ion Battery for Electric Vehicles: A Review. Energy Storage Mater.
**2018**, 10, 246–267. [Google Scholar] [CrossRef] - Ouyang, M.; Ren, D.; Lu, L.; Li, J.; Feng, X.; Han, X.; Liu, G. Overcharge-Induced Capacity Fading Analysis for Large Format Lithium-Ion Batteries with LiyNi
_{1}/_{3}Co_{1}/_{3}Mn_{1}/_{3}O_{2}+ LiyMn_{2}O_{4}Composite Cathode. J. Power Sources**2015**, 279, 626–635. [Google Scholar] [CrossRef] - Rohatgi, A. WebPlotDigitizer. Version 4.8. Available online: https://apps.automeris.io/wpd4/ (accessed on 25 March 2024).
- Bejan, A.; Kestin, J. Entropy Generation through Heat and Fluid Flow. J. Appl. Mech.
**1983**, 50, 475. [Google Scholar] [CrossRef]

**Figure 1.**Illustrations of the phenomenological entropy generation theorem for (

**a**) energy addition/system formation $0<{d\mathbb{S}}_{rev}\le {\delta \mathbb{S}}_{phen}$ and (

**b**) energy extraction/system loading ${d\mathbb{S}}_{rev}\le {\delta \mathbb{S}}_{phen}<0$. Reproduced from [27].

**Figure 2.**Cyclic capacity fade measured during consistent cycling of Samsung single-cell 3.6 V, 2.5 Ah lithium-ion polymer battery [31]. (

**a**) Discharge steps; (

**b**) recharge steps.

**Figure 3.**TPEG domain/space and subspace representations of the (

**a**) $\mathit{d}\mathit{i}\mathit{s}\mathit{s}\mathit{i}\mathit{p}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}\mathit{f}\mathit{a}\mathit{c}\mathit{t}\mathit{o}\mathit{r}=\frac{\mathit{M}\mathit{S}\mathit{T}/\mathit{E}\mathit{C}\mathit{T}\mathit{e}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}}{\mathit{W}\mathit{o}\mathit{r}\mathit{k}\mathit{e}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}}=\frac{\mathit{D}\mathit{e}\mathit{p}\mathit{t}\mathit{h}}{\mathit{L}\mathit{e}\mathit{n}\mathit{g}\mathit{t}\mathit{h}}$ in the entropy generation subspace EGS and (

**b**) $\mathit{e}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{i}\mathit{c}\mathit{e}\mathit{f}\mathit{f}\mathit{i}\mathit{c}\mathit{i}\mathit{e}\mathit{n}\mathit{c}\mathit{y}=\frac{\mathit{W}\mathit{o}\mathit{r}\mathit{k}\mathit{e}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}}{\mathit{R}\mathit{e}\mathit{v}\mathit{e}\mathit{r}\mathit{s}\mathit{i}\mathit{b}\mathit{l}\mathit{e}\mathit{e}\mathit{n}\mathit{t}\mathit{r}\mathit{o}\mathit{p}\mathit{y}}=\frac{\mathit{L}\mathit{e}\mathit{n}\mathit{g}\mathit{t}\mathit{h}}{\mathit{I}\mathit{d}\mathit{e}\mathit{a}\mathit{l}\mathit{l}\mathit{e}\mathit{n}\mathit{g}\mathit{t}\mathit{h}}$ in the reversible transformation subspace. The green and pink curves are the transformation trajectories. Axes are not to scale.

**Figure 4.**(

**a**) Ohmic and ECT entropies during the over-discharge of a lead–acid battery. The sudden drops in ECT entropy coincided with a sudden 2-step transition to over-discharge, following full discharge. (

**b**) Phenomenological entropy (the sum of Ohmic and ECT entropies) and reversible entropy showing the 2-step drops violating the second law momentarily when ${\mathbb{S}}_{\mathit{r}\mathit{e}\mathit{v}}>{\mathbb{S}}_{\mathit{p}\mathit{h}\mathit{e}\mathit{n}}$. (

**c**) Phenomenological charge and reversible charge showing the 2-step drops violating the DEG theorem momentarily when ${w}_{rev}>{w}_{phen}$. Reproduced from [26].

**Figure 5.**TPEG analysis/monitoring of overcharge-induced thermal runaway in a 20 Ah commercial lithium-ion battery, as a function of the state of charge SOC. (

**a**) Voltage and temperature data extracted from [39]. (

**b**) Phenomenological entropy components: Ohmic and ECT. (

**c**) Instantaneous capacity fade.

**Table 1.**Single-variable entropy generation $\mathit{\delta}{\mathit{S}\prime}_{\mathit{W},\mathit{r}}$ for various systems.

Work | $\mathit{\delta}{\mathit{S}\prime}_{\mathit{W},\mathit{r}}$ |
---|---|

Frictional | $\frac{{F}_{f}dx}{T}$ |

Magnetic | $\frac{BdM}{T}$ |

Shear | $\frac{V\tau d\gamma}{T}$ |

Electrical | $\frac{vdq}{T}$ |

Rotational shaft | $\frac{{M}_{T}\omega}{T}$ |

Chemical | $\frac{\sum {\mu}_{k}d{N}_{k}}{T}$ |

Flow | $\frac{\sum \left[{\left({hdN}_{e}\right)}_{exit}-{\left({hdN}_{e}\right)}_{inlet}\right]}{T}$ |

**Table 2.**Pseudo-steady-state frictional wear analysis via the degradation–entropy generation methodology.

# | Characterization Step | Model and Graphical Representation |
---|---|---|

(i) | Measured or input data | |

(ii) | Phenomenological Entropy GenerationFrictional entropy ${{S}^{\prime}}_{W}=-{\int}_{{t}_{0}}^{t}\frac{F\dot{x}}{T}dt$ MST entropy ${{S}^{\prime}}_{\mu T}=0$ | ${\mathit{S}\prime}_{\mathit{p}\mathit{h}\mathit{e}\mathit{n}}={{\mathit{S}}^{\prime}}_{\mathit{W}}$ |

(iii) | Degradation–Entropy Generation Material wear is the degradation measure. Slope yielded ${B}_{W}=-0.0366$ K/J | Degradation model: $\mathit{w}={\mathit{B}}_{\mathit{W}}{{\mathit{S}}^{\prime}}_{\mathit{W}}$ |

**Table 3.**Grease analysis via the transformation–phenomenological entropy generation methodology. $\rho $ is grease density, c

_{γ}is specific heat capacity at constant shear, τ is shear stress, γ is shear strain, $\beta ={\left(\frac{\partial \tau}{\partial T}\right)}_{\gamma ,N}=\alpha G\prime $ is the thermal stress coefficient, where $\alpha ={\left(\frac{\partial \gamma}{\partial T}\right)}_{\tau ,N}$ is the thermal strain coefficient, and $G\prime ={\left(\frac{\partial \tau}{\partial \gamma}\right)}_{T,N}$ is the storage modulus [23,27]. MST: MicroStructuroThermal. Excluding row 4 (step (iii)) where both greases are overlaid on the same set of axes, the plots on the left are for NLGI 4 grease and on the right are for NLGI 2 grease. Reproduced from [23].

# | Characterization Step | Model and Graphical Representation |
---|---|---|

(i) | Measured or input data | |

(ii) | Phenomenological Entropy GenerationMST entropy density ${{S}^{\prime}}_{\mu T}=-{\int}_{{t}_{0}}^{t}\left(\rho {c}_{\gamma}lnT+\beta \gamma \right)\frac{\dot{T}}{T}dt$ Shear entropy density ${{S}^{\prime}}_{W}=-{\int}_{{t}_{0}}^{t}\frac{\tau \dot{\gamma}}{T}dt$ | ${\mathit{S}\prime}_{\mathit{p}\mathit{h}\mathit{e}\mathit{n}}={{\mathit{S}}^{\prime}}_{\mathit{\mu}\mathit{T}}+{{\mathit{S}}^{\prime}}_{\mathit{W}}$ |

(iii) | Transformation–Phenomenological Entropy GenerationShear stress accumulation is the trans-formation measure. Orthogonal slopes yielded ${B}_{W}=-10.4$$\mathrm{Pa}-\mathrm{s}\mathrm{K}/\mathrm{J}\mathrm{and}{B}_{\mu T}=-0.03$$\mathrm{Pa}-\mathrm{s}\mathrm{K}/\mathrm{J}\mathrm{for}\mathrm{NLGI}2\mathrm{grease}\mathrm{and}{B}_{W}=-10.4$$\mathrm{Pa}-\mathrm{s}\mathrm{K}/\mathrm{J}\mathrm{and}{B}_{\mu T}=-0.50$ Pa-s K/J for NLGI 4 grease. | Transformation model: ${\int}_{{\mathit{t}}_{0}}^{\mathit{t}}\mathit{\tau}\mathit{d}\mathit{t}={\mathit{B}}_{\mathit{\mu}\mathit{T}}{{\mathit{S}}^{\prime}}_{\mathit{\mu}\mathit{T}}+{\mathit{B}}_{\mathit{W}}{{\mathit{S}}^{\prime}}_{\mathit{W}}$ |

(iv) | Change in shear strength/stress is the degradation measure. ${J}_{NLGI2}=0.021$ ${J}_{NLGI4}=$ 0.396 |

**Table 4.**Battery degradation analysis via the transformation–phenomenological entropy generation methodology. $v$ is battery voltage, T is temperature, $q$ is charge content, I is current and $\Delta q$ is capacity fade. vT, ECT: ElectroChemicoThermal. The arrows indicate process direction, i.e., charge C follows discharge D. Reproduced from [20].

# | Characterization Step | Model and Graphical Representations |
---|---|---|

(i) | Measured or input data. | |

(ii) | Phenomenological Entropy GenerationECT entropy $S{\u2019}_{vT}=\int \frac{qv}{T}dt$ Ohmic entropy $S{\u2019}_{\Omega}=\int \frac{vI}{T}dt$ | ${\mathit{S}\prime}_{\mathit{p}\mathit{h}\mathit{e}\mathit{n}}={{\mathit{S}}^{\prime}}_{\mathit{v}\mathit{T}}+\mathit{S}{\mathit{\u2019}}_{\mathit{\Omega}}$ |

(iii) | Transformation–Phenomenological Entropy GenerationCharge content is the transformation measure. TPEG coefficients: ${B}_{\Omega}=76.6$ Ah K/Wh and ${B}_{VT}=113$ Ah K/Wh for discharge. For charge, ${B}_{\Omega}=75.5$ Ah K/Wh and ${B}_{VT}=28.3$ Ah K/Wh. | Transformation model: $\mathit{q}={\mathit{B}}_{\mathit{v}\mathit{T}}\mathit{S}{\mathit{\u2019}}_{\mathit{v}\mathit{T}}+{\mathit{B}}_{\mathit{\Omega}}\mathit{S}{\mathit{\u2019}}_{\mathit{\Omega}}$ |

(iv) | Charge capacity fade is the degradation measure. For this cycle, $\Delta {q}_{disch}=1.3$ Ah $\Delta {q}_{ch}=0.3$ Ah ${J}_{disch}=0.063$ ${J}_{ch}=$0.033 ${\eta}_{S\prime disch}=$0.72 ${\eta}_{S\prime ch}=$ 0.93 | Degradation model: $\mathit{\Delta}\mathit{q}={\mathit{B}}_{\mathit{v}\mathit{T}}\mathit{S}{\mathit{\u2019}}_{\mathit{v}\mathit{T}}+{\mathit{B}}_{\mathit{\Omega}}\mathit{S}{\mathit{\u2019}}_{\mathit{\Omega}}-{\mathit{q}}_{\mathit{r}\mathit{e}\mathit{v}}$ |

**Table 5.**A summary of the transformation–phenomenological entropy generation methodology for bending and torsion fatigue. $\rho $ is density, ${c}_{\epsilon}$ is specific heat capacity, $\sigma $ is normal stress and $\epsilon $ is normal strain for bending. $\beta ={\left(\frac{\partial \sigma}{\partial T}\right)}_{\epsilon ,N}=\frac{\alpha}{{\kappa}_{T}}$ is the thermal stress coefficient, where $\alpha ={\left(\frac{\partial \epsilon}{\partial T}\right)}_{\sigma ,N}$ is the thermal strain coefficient, and ${\kappa}_{T}=-{\left(\frac{\partial \epsilon}{\partial \sigma}\right)}_{T,N}$ is isothermal loadability; ${{S}^{\prime}}_{\mu T}$ is MST (MicroStructuroThermal) entropy, and ${{S}^{\prime}}_{W}$ is work/load entropy. For torsion, shear stress $\tau $ and shear strain $\gamma $ are used. Excluding row 4 (step (iii)) where both load types are overlaid, plots on the left are for bending, and plots on the right are for torsion. Reproduced from [24].

# | Characterization Step | Model and Graphical Representation |
---|---|---|

(i) | Measured or input data | |

(ii) | Phenomenological Entropy Generation${{S}^{\prime}}_{\mu T}=-{\int}_{{t}_{0}}^{t}\left(\rho {c}_{\epsilon}lnT+\beta \epsilon \right)\frac{\dot{T}}{T}dt$${{S}^{\prime}}_{W}=-{\int}_{{t}_{0}}^{t}\frac{{N}_{dt}{\sigma}_{N}}{T}:\left[{\dot{\epsilon}}_{eN}+\left(\frac{1-n\prime}{1+n\prime}\right){\dot{\epsilon}}_{pN}\right]dt$ | ${\mathit{S}\prime}_{\mathit{p}\mathit{h}\mathit{e}\mathit{n}}={{\mathit{S}}^{\prime}}_{\mathit{\mu}\mathit{T}}+{{\mathit{S}}^{\prime}}_{\mathit{W}}$ |

(iii) | Transformation–Phenomenological Entropy GenerationStrain is the transformation measure. Orthogonal slopes yielded ${B}_{W}=-0.92$ %m ^{3}K/MJ and ${B}_{\mu T}=0.22$ %m^{3}K/MJ (bending), and ${B}_{W}=-1.96$ %m^{3}K/MJ and ${B}_{\mu T}=0.42$ %m^{3}K/MJ (torsion), prior to failure onset. | Transformation model: $\mathit{\epsilon}={\int}_{{\mathit{t}}_{0}}^{\mathit{t}}\dot{\mathit{\epsilon}}\mathit{d}\mathit{t}={\mathit{B}}_{\mathit{\mu}\mathit{T}}{{\mathit{S}}^{\prime}}_{\mathit{\mu}\mathit{T}}+{\mathit{B}}_{\mathit{W}}{{\mathit{S}}^{\prime}}_{\mathit{W}}$ |

(iv) | Change in strain is the degradation measure. At onset of failure, ${J}_{bending}\ge 0.04$ ${J}_{torsion}\ge $0.10 |

**Table 6.**Characterizing a water pump using the transformation–phenomenological entropy generation methodology.

# | Characterization Step | Model and Graphical Representation |
---|---|---|

(i) | Measured or input data | |

(ii) | Phenomenological Entropy GenerationFlow entropy ${{S}^{\prime}}_{N}=\underset{{t}_{o}}{\overset{{t}_{f}}{\int}}\frac{{\sum}_{}^{}\left[{\left(\dot{m}h\right)}_{exit}-{\left(\dot{m}h\right)}_{inlet}\right]}{T}dt$ Work entropy ${{S}^{\prime}}_{W}=\underset{{t}_{o}}{\overset{{t}_{f}}{\int}}\frac{{M}_{T}\omega}{T}dt$ | ${\mathit{S}\prime}_{\mathit{p}\mathit{h}\mathit{e}\mathit{n}}={\mathit{S}}_{\mathit{N}}^{\prime}+{{\mathit{S}}^{\prime}}_{\mathit{W}}$ |

(iii) | Transformation–Phenomenological Entropy GenerationPressure drop is the transformation measure. Orthogonal slopes give ${B}_{N}=-0.057$ MPa-h K/kJ and ${B}_{W}=0.018$ MPa-h K/kJ | Transformation model: $\u2206\mathit{P}={\mathit{B}}_{\mathit{N}}{{\mathit{S}}^{\prime}}_{\mathit{N}}+{\mathit{B}}_{\mathit{W}}{{\mathit{S}}^{\prime}}_{\mathit{W}}$ |

(iv) | Change in pressure drop measures degradation. |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Osara, J.A.; Bryant, M.D.
Systems and Methods for Transformation and Degradation Analysis. *Entropy* **2024**, *26*, 454.
https://doi.org/10.3390/e26060454

**AMA Style**

Osara JA, Bryant MD.
Systems and Methods for Transformation and Degradation Analysis. *Entropy*. 2024; 26(6):454.
https://doi.org/10.3390/e26060454

**Chicago/Turabian Style**

Osara, Jude A., and Michael D. Bryant.
2024. "Systems and Methods for Transformation and Degradation Analysis" *Entropy* 26, no. 6: 454.
https://doi.org/10.3390/e26060454