# Methods to Calculate Entropy Generation

^{1}

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^{*}

## Abstract

**:**

## 1. Introduction

**open and closed systems**as the difference

#### 1.1. Local Equilibrium

#### 1.2. States, Paths and Path (Line) Integrals

- Thermodynamics often involves changes in variables between two states. Variables include a set of the independent thermodynamic state variables
**Z**chosen and measured for a particular system, and state dependent system properties which are functions of the states of**Z**. The**Z**characterize a system’s thermodynamic state and can include temperature T, pressure P, and number of moles N, among others. Changes in system properties such as energy E, entropy S, temperature T and ${\mathbb{S}}_{rev}$ use the exact differential d; are path independent, wherein changes in properties over an irreversible path (irr) are identical to changes in properties over a reversible path (rev), e.g., dE = dE_{irr}= dE_{rev}; and the line integral ${\int}_{o}^{f}dE=\u2206E={E}_{f}-{E}_{o}$ depends only on the property values at the beginning and end states (o and f). This is the thermodynamic state principle. - Path-dependent variables such as work W, heat transfer $Q,$ entropy generation ${S}^{\prime}$ and ${\mathbb{S}}_{phen}$ depend on what occurs along the path between states o and f, and use the inexact differential $\delta $ such that ${\int}_{o}^{f}\delta W=W$ must be accumulated over all instants of time t along the (assumed known) transformation path between times ${t}_{o}$ and ${t}_{f}$. The path-dependent parameters will depend on a set of variables $\mathcal{Z}$= {
**Z**$,{\zeta}_{k}$} assumed to be time dependent, observable, and measurable. The**Z**characterize the thermodynamic state, whereas the ${\zeta}_{k}$ characterize any active irreversible dissipative processes. Via a suitable numerical integration such as the trapezoid rule, with the $\mathcal{Z}\left(t\right)$ = {**Z**(t), ${\zeta}_{k}\left(t\right)$} measured as points $\mathcal{Z}\left({t}_{j}\right)$= {**Z**$({t}_{j})$, ${\zeta}_{k}({t}_{j})\}$ suitably spaced at time instants ${t}_{o}<$ ${t}_{j}$ < ${t}_{f}$ in accord with the sampling theorem [14], the increments $\delta W$ can be accumulated into the line integral ${\int}_{o}^{f}\delta W=W$. - Exact differentials $dE$, state functions of the independent state variables
**Z**(t), if intractable, can be numerically integrated over the (ideal) reversible path per methods of the prior paragraph. The reversible path must transit states o to f in quasi-equilibrium and be continuous and maximally smooth over time, which can be approximated by linear functions with slope determined by the end states, for example, if $dE\left(t\right)=dE(\mathit{Z}(t))$, components of**Z**(t) with slope $\frac{{Z(t}_{f})-{Z(t}_{o})}{{t}_{f}-{t}_{o}}$ where ${Z(t}_{f})$ and ${Z(t}_{o})$ must be measured or known at the beginning and final times ${t}_{o}$ and ${t}_{f}$. With this, the line integral ${\int}_{o}^{f}dE={\int}_{{t}_{o}}^{{t}_{f}}\frac{dE}{dt}dt={\int}_{{t}_{o}}^{{t}_{f}}\sum _{Z}\frac{dE}{dZ}\frac{dZ}{dt}dt=\sum _{Z}\frac{{Z(t}_{f})-{Z(t}_{o})}{{t}_{f}-{t}_{o}}{\int}_{{t}_{o}}^{{t}_{f}}\frac{dE}{dZ}dt$, where sum index Z denotes a sum over all the components of**Z**, and dE/dZ must be evaluated at each time instant ${t}_{o}<$ ${t}_{j}$ < ${t}_{f}$ along the reversible path. - For reversible processes ${d\mathbb{S}}_{rev}$ with initial and final states ${{d\mathbb{S}}_{rev}}_{o}$ and ${{d\mathbb{S}}_{rev}}_{f}$, ${d\mathbb{S}}_{rev}(t)=\frac{{{d\mathbb{S}}_{rev}}_{f}-{{d\mathbb{S}}_{rev}}_{o}}{{t}_{f}-{t}_{o}}(t-{t}_{o})+{{d\mathbb{S}}_{rev}}_{o}$ satisfies the reversible path approximations of item 3.
- A phenomenological path (phen) through the thermodynamic state space enclosing $\mathcal{Z}$= {
**Z**$,{\zeta}_{k}$} defined in item 2 includes nonzero ${\zeta}_{k}\left(t\right)$. A reversible path (rev) in the reversible subspace {**Z**} of $\mathcal{Z}$ [15] involves only the thermodynamic states**Z**, not the ${\zeta}_{k}$. The projection of the set of points $\mathcal{Z}\left({t}_{j}\right)$= {_{rev}**Z**$\left({t}_{j}\right)$, ${\zeta}_{k}({t}_{j})\}$ that comprise an irreversible phenomenological path onto the reversible subspace is the set of points {${\mathit{Z}}_{\mathit{r}\mathit{e}\mathit{v}}\left({t}_{j}\right)\}$ [15]. The system inputs, active mechanisms and dynamics are determined by the phenomenological path.

## 2. Irreversible Thermodynamics and Entropy Generation

#### 2.1. Combining Internal Energy and Entropy Balances

**closed system**as $dS\ge \frac{\delta Q}{T}$ where δQ/T is entropy flow by heat transfer, which can be positive or negative depending on the transfer direction, and T is the temperature of the boundary where the energy/entropy transfer takes place. Via the thermodynamic state principle—item 1 of Section 1.2—and the entropy balance [5,6,12], entropy change accompanying an

**open system**process can be evaluated along a real and often nonlinear

**irreversible path (irr) as**

**reversible rev (ideal and linear) path**with $\delta {S}^{\prime}=0$,

_{l}are intensive variables such as pressure P, strength/stress $\sigma $, voltage $\nu $, etc.; ${X}_{l}$ are the system’s extensive variables such as volume V, strain $\epsilon $, charge $q$; and $T{\delta \mathbb{S}}_{U}\left(S,{X}_{l},{N}_{k}\right)$, a path-dependent inexact differential, is defined by and equal to the middle expression of Equation (5).

**The script notation distinguishes**${\mathbb{S}}_{\mathit{U}}$

**as an entropy related function of the independent variables in parenthesis**. Subscript $phen$ indicates evaluation along the phenomenological path, the observable path where the independent states and dissipative process variables are available and measured at each instant. Since the product of temperature and entropy change ($TdS$) in Equation (5) subsumes the heat and flow transfer terms in Equations (2) and (3), Equation (5) is valid for all systems, open and closed [7,8,9,10,12]. Equation (5), which governs the system along an irreversible path, has a pair of unknowns $\delta {S}^{\prime}>0$ and $dU$; all other terms are observable and can be measured, as shown later.

#### 2.2. Entropy Generation of Various System Classes via Thermodynamic Potentials

**all systems, open or closed, steady or unsteady.**The thermodynamic potentials (with PV work replaced by generalized YX work)––enthalpy $H=U+YX$, Helmholtz free energy $A=U\u2013TS$ and Gibbs free energy $G=U+YX\u2013TS$––are differentiated and solved for $dU$, the result of which is then inserted into Equation (5) to get:

- ${\delta \mathbb{S}}_{phen}$: evaluated via the expressions of the middle equalities of Equations (5) and (8) at points over the irreversible phenomenological path defined in items 2 and 5 of Section 1.2, where salient quantities can be measured.
- ${d\mathbb{S}}_{rev}$: evaluated as an exact differential over a reversible path between the beginning and final values of the irreversible path for ${\delta \mathbb{S}}_{phen}$, as discussed in list items 3, 4 and 5 of Section 1.2. Here, the independent states in the parentheses of Equations (5) and (8) are linear in time, as per items 3 and 4 of Section 1.2.

#### 2.3. Phenomenological Entropy Generation Theorem

**Phenomenological Entropy Generation (PEG) Theorem:**

#### 2.4. Phenomenological Entropy Generation Functions for Various System Classes

#### 2.4.1. Thermal Systems: Internal Energy and Enthalpy

#### 2.4.2. Boundary-Loaded (Work-Capable) Systems: Helmholtz Potential

_{rev}is the maximum/minimum (theoretical) work possible. During work output, energy extraction or system loading, $dT\ge 0,d{X}_{l}\ge 0,d{N}_{k}\le 0$ and ${dA}_{rev}\le 0$, rendering $\delta {S}^{\prime}\ge 0$. For work input, energy addition or product formation, $dT\le 0,d{X}_{l}\le 0,d{N}_{k}\ge 0$ and ${dA}_{rev}\ge 0$, reversing the respective signs of terms in Equation (9b) to accord with the second law $\delta {S}^{\prime}\ge 0$ [5,8,9,19]. A comparison of Equations (6) and (9b) shows that the Helmholtz relation conveniently absorbs $d{U}_{rev}$ and $dS$ into $d{A}_{rev}$ and $-{S}_{A}dT$, removing the need to measure heat or mass transfer across the system boundary and the need to determine $d{U}_{rev}$, which is ambiguous for non-thermal systems. Most work-capable systems have a standardized maximum work $d{A}_{rev}$ obtainable, e.g., the elastic energy function for deformable solids. With a specified $d{A}_{rev}$, $\delta {S}^{\prime}$ in Equation (9b) measures the irreversible entropy generation pertaining to dissipation of useful energy via work across the thermodynamic boundary, which requires the instantaneous evaluation of the Helmholtz phenomenological entropy generation [8,9] terms in Equation (8b),

_{l}and dN

_{k}. For a non-reactive non-diffusive (${{dN}^{r}}_{k}={{dN}^{d}}_{k}=0$) system, such as a lubricated mechanical interface, a fatigue-loaded component, and others, the last term of Equation (10c) can be neglected. Note that the very slow and/or (typically laboratory-controlled) isothermal case gives minimum entropy generation ${\delta S}_{min}=\frac{-\sum {Y}_{l}d{X}_{l}}{T}$. Various forms of work $\sum {Y}_{l}d{X}_{l}$ include frictional ${F}_{f}dx$, electrical $vdq$, shear $V\tau d\gamma $, compression $PdV$, and magnetic BdM, among others.

#### 2.4.3. Internally Reactive Systems and Energy Storage Systems: Gibbs Potential

_{k}. For constant-pressure reactive system-process interactions such as cycling of electrochemical energy systems [7,10], the term XdY/T can be neglected. For non-reactive (${dN}_{k}=0$) energy systems such as hydraulic/pressure accumulators, the last term in Equation (10d) can be dropped.

#### 2.5. Generalized Material Properties, Entropy Content S, Internal Free Energy Dissipation “–SdT”

**entropy**changes

**load modulus**E’ also derived from the second partial isothermal Helmholtz derivative (Appendix A). The equation $\frac{\alpha}{{\kappa}_{T}}=\alpha {E}^{\prime}=\beta >0$ is the thermal coefficient of generalized force (pressure, stress, voltage, etc.); ${\lambda}_{X}>0$ and ${\lambda}_{Y}>0$ are the coefficients of thermal chemico-transport decay (for solids ${\lambda}_{X}\approx {\lambda}_{Y}=\lambda $) including the combined effects of internal reaction and mass flow on entropy content.

**loadability**, a measure of a material/system’s “cold” response to boundary loading:

**loadability**is

**compressibility**(inverse of bulk modulus) for a compressible system [16],

**bendability**for a beam under bending [9],

**shearability**(inverse of shear modulus) for shearing or torsional loading [8], and

**conductance**(inverse of resistance) for electrical work. Derivations and detailed discussions of the properties in Equations (12) are presented in Appendix A and in reference [22] specifically for a shear-loaded system.

**These formulations can be used to define new system- and process-specific material properties for assessing system/material performance and behavior**.

**MicroStructuroThermal**(

**MST)**energy dissipation [8,9] to suggest the dissipated energies as the source of heat. For electrochemical energy systems such as batteries and capacitors—where the last terms in Equation (16) and the second of Equations (15) are expressed via Faraday’s electrolysis law in terms of cell charge capacity $q$ and potential v—these equations are more specifically named

**ElectroChemicoThermal**(

**ECT)**energy dissipation [7,10]. An application is presented in Section 4.3.

#### 2.6. Stress vs. Strength Sign Conventions

**useful force, strength or potential**in line with the definitions of the free energies as maximum

**useful work**obtainable from a system (Gibbs: internal work or compositional change; Helmholtz: external or boundary work). All the energy and entropy balances here accord with the IUPAC convention of representing energy leaving the system via work (and heat) as negative. In such a loaded system, dY is the

**decrease in strength,**and hence is negative and dX is the increase in displacement. This accords with an expanding gas, for which pressure drops with increasing volume. However, in mechanics and other science/engineering fields that deal with solid materials, it is common to observe and use

**increase in stress**as the system is loaded. In such cases, dY is positive. The derivations in this article and appendix consider dY < 0 the

**decrease in strength**in a loaded system.

#### 2.7. Helmholtz-Gibbs Coupling

#### 2.8. Rates

#### 2.9. Open Systems: Pumps, Compressors, Fuel Cells, etc.

## 3. Evaluating Total Entropy Generation and Path (Line) Integrals

## 4. Sample (Phenomenological) Entropy Generation Calculations

#### 4.1. Friction Sliding of Copper against Steel at Steady Speed—(Steady State)

^{−1}under carefully maintained thermal and lubricated boundary conditions [23]. Measured were friction force $F$ and temperatures at three locations in the copper, to estimate friction heat generation $F\dot{x}$, heat flow and surface temperature $T$, all of which were steady during sliding to render $\dot{T}=0$, which when substituted into Equation (18c) yields ${S}_{phen}^{\prime}=-{\int}_{{t}_{o}}^{{t}_{f}}\frac{F\dot{x}}{T}dt$. Using measured values, the integral was evaluated over the test time interval to obtain the entropy generation plot in Figure 1.

#### 4.2. Mechanical Shearing of Grease—Shear Stress and Shear Strain (Helmholtz Potential)

_{T}ω (the product of torque M

_{T}and rotational speed ω) and temperatures (via thermocouples) of grease T and ambient, which are plotted versus time in Figure 2a. In terms of native grease internal variables, ${M}_{T}\omega =V\tau \dot{\gamma}$ gives the shear power as the product of volume $V$, shear stress $\tau $ and shear strain rate $\dot{\gamma}$. Helmholtz entropy content density S

_{A}—the first of Equations (14) divided by V—was substituted into Equation (18c) with ${\dot{N}}_{k}=0,Y=\tau $, $\dot{X}=V\dot{\gamma}$, to yield phenomenological entropy density

#### 4.3. Discharge of Lithium-Ion Battery—Voltage and Charge (Helmholtz-Gibbs Coupling)

#### 4.4. Fatigue of Metals—Stress and Strain (Helmholtz Potential)

_{A}—the first of Equations (14) divided by volume V—was substituted into (18c) with ${\dot{N}}_{k}=0$ to obtain

_{W}(blue plot) are the first and second integrals of Equation (24), respectively. For low-cycle fatigue, with significant plastic deformation, the modulus defined in Equations (12) and (13), substituted into entropy content density S

_{A}in Equation (24), is the hardness modulus.

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

## 5. Discussion

**Here, reversible implies thermodynamic reversibility: any real system undergoing a spontaneous process cannot “revert” back to its original state without work from an external source, hence is thermodynamically irreversible.**The reversible forms of Equations (9) (wherein $\delta {S}^{\prime}=0$) derived directly from Legendre transforms of entropy by Francois Massieu are called the Massieu functions [16,27]. Here, by using the irreversible form of internal energy change (Equation (5)), Equations (9) are termed

**the irreversible Massieu functions.**

#### 5.1. Phenomenology

#### 5.2. Steady vs. Unsteady Systems: Minimum Entropy Generation (MEG) vs. MicroStructuroThermal (MST) Entropy

#### 5.3. Dissipation Factor and Entropic Efficiency

#### 5.4. Thermal vs. Non-Thermal Systems—Internal Energy and Enthalpy vs. the Free Energies

**free entropies**” particularly convenient for non-thermal systems and interactions where heat transfer is not readily measurable and for which thermal mechanisms primarily emanate from energy dissipation. The free entropies are more consistent for assessing system transformations from observable and measurable system variables, irrespective of surrounding conditions.

## 6. Summary and Conclusions

**Phenomenological Entropy Generation (PEG)**theorem was derived and proposed, expounding the significances of the newly introduced MicroStructuroThermal MST entropy (or ElectroChemicoThermal ECT entropy for electrochemical power systems) and previously neglected reversible entropy to characteristic entropy generation. Extending and generalizing Gibbs theory of thermodynamic stability, the Clausius inequality, Rayleigh’s energy dissipation principle, Onsager’s reciprocity, and Prigogine’s entropy balance—the hallmarks of classical and modern irreversible thermodynamics [4,5,6,12,13,33]—this article, using results from recently published experimental works [7,8,9,10], demonstrated that:

- a combination of the thermodynamic potentials and the
**irreversible**form of the TdS equation yields the**irreversible Massieu functions**; - steady-state systems generate entropy at a minimum rate which, for a boundary-loaded, internally reactive open system, is the sum of boundary work/load entropy ${{S}^{\prime}}_{W}$ and compositional change entropy ${{S}^{\prime}}_{N}$ rates;
- for unsteady non-thermal systems, a microstructurothermal MST entropy is included to characterize the accompanying instantaneous transients during system transformation;
- phenomenological entropy generation ${{S}^{\prime}}_{phen}$ is the sum of boundary work/load entropy ${{S}^{\prime}}_{W}$, compositional change entropy ${{S}^{\prime}}_{N}$ and microstructurothermal MST entropy ${{S}^{\prime}}_{\mu T}$ (ElectroChemicoThermal ECT entropy ${{S}^{\prime}}_{VT}$ for electrochemical systems);
- entropy generation is the difference between phenomenological ${{S}^{\prime}}_{phen}$ and reversible ${S}_{rev}$ entropies at every instant, named the
**Phenomenological Entropy Generation (PEG) Theorem**; - along the phenomenological path, thermodynamic and other states germane to evaluating ${{S}^{\prime}}_{phen}$ and ${S}_{rev}$ are observable (measurable), permitting evaluation of entropy generation $\delta {S}^{\prime}$;
- entropy generation is always non-negative in accordance with the second law, while its constituent terms ${{S}^{\prime}}_{phen}$ and ${S}_{rev}$ are directional, negative for a loaded system and positive for an energized system. This implies $\left|{{S}^{\prime}}_{phen}\right|\le \left|{S}_{rev}\right|$ during load application/work output or active species decomposition, and $\left|{{S}^{\prime}}_{phen}\right|\ge \left|{S}_{rev}\right|$ during energization/work input or active species formation, in accordance with experience and thermodynamic laws. In plain words, measurable energy obtained from a real system is always less than the theoretical maximum/reversible energy, and measurable energy added to a real system is always more than the minimum/reversible energy. (Modulus signs indicate magnitudes only).

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

Nomenclature | Name | Unit |

A | Helmholtz free energy | J |

B | magnetic field | T |

C | Heat capacity | J/K |

F | Faraday’s constant | C/mol |

G | Gibbs free energy | J |

I | discharge/charge current or rate | A |

m | mass | kg |

n’ | number of charge species | |

M | magnetic dipole moment | J/T |

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

P | pressure | Pa |

q | charge | Ah |

Q | heat | J |

R | gas constant | J/mol·K |

S | entropy or entropy content | J/K (Wh/K) |

$\mathbb{S}$ | entropy generation function | J/K (Wh/K) |

S’ | entropy generation or production | J/K (Wh/K) |

t | time | sec |

T | temperature | °C or K |

U | internal energy | J |

v | voltage | V |

V | volume | m^{3} |

W | work | J |

Z | thermodynamic state variable | |

Symbols | ||

μ | chemical potential | |

ζ | phenomenological variable | |

ρ | density | |

Subscripts & acronyms | ||

0 | initial | |

c | specific heat capacity | |

d | diffusion | |

e | flow | |

f | final | |

ECT, VT | Electro-Chemico-Thermal | |

MST, μT | Micro-Structuro-Thermal | |

rev | reversible | |

irr | irreversible | |

phen | phenomenological | |

DEG | Degradation-Entropy Generation | |

PEG | Phenomenological Entropy Generation | |

NLGI | National Lubricating Grease Institute |

## Appendix A. Generalized Material Properties from First Principles

^{‘}≈ 0 in Equations (8c) and (8b) for one active/reactive species (k = 1) yields, respectively,

^{’}> 0. Equations (A1) and (A2) indicate that the Gibbs free energy G = G(T,Y,N) is a function of temperature T, generalized force Y and amount of active species N, while the Helmholtz free energy A = A(T,X,N) is a function of temperature T, generalized position X and amount of active species N. Here, active species include open systems through which active matter flows and chemically reactive systems in which quantity of active matter changes.

_{G}and S

_{A}for evaluating the MicroStructuroThermal (MST) energy change ”–SdT” (Section 2.5).

#### Appendix A.1. Heat Capacities ${C}_{X}$ and ${C}_{Y}$

_{G}or S

_{A}must increase in response to increase in temperature for the system to remain stable. When the entropy increase is the result of heat transfer $\delta Q$ into the system only, approximated by slowly heating the stationary system, Equations (A8) and (A10) are combined and rewritten for practical purposes as

#### Appendix A.2. Isothermal Loadability κ_{T} and Load Modulus E’

_{T}of a material/system. Substituting the second equality of Equation (A4) into the second partial Gibbs derivative gives

^{’}of a material. Substituting the second equality of Equation (A6) into the second partial Helmholtz derivative yields

#### Appendix A.3. Thermal Displacement Coefficient α and Thermal Force (Strength) Coefficient β

_{G}must increase in response to decrease in generalized force Y for process continuity and system stability. It also establishes that the position/displacement X in the system spontaneously increases (e.g., expansion, strain, etc.) with increase in temperature for stability and process continuity.

#### Appendix A.4. Isothermal Chemical Resistances ${R}_{{N}_{Y}}$ and ${R}_{{N}_{X}}$

#### Appendix A.5. Thermal Chemico-Transport Decay Coefficients ${\lambda}_{Y}$ and ${\lambda}_{X}$

#### Appendix A.6. Gibbs Properties vs. Helmholtz Properties

**Table A1.**The Gibbs and Helmholtz potential-based material properties derived in Appendix A.

Category | Gibbs-Based | Helmholtz-Based |
---|---|---|

Thermal | Constant-Force Heat Capacity ${C}_{Y}={\left.\frac{\delta Q}{\partial T}\right|}_{Y,N}$ | Constant-Displacement Heat Capacity ${C}_{X}={\left.\frac{\delta Q}{\partial T}\right|}_{X,N}$ |

Thermo-mechanical | Thermal Displacement Coefficient | Thermal Force Coefficient |

$\alpha =\frac{1}{X}\frac{\partial X}{\partial T}$ | $\beta =\alpha $E’ | |

Mechanical | Isothermal Loadability | Load Modulus |

${\kappa}_{T}=-\frac{1}{X}{\left.\frac{\partial X}{\partial Y}\right|}_{T,N}$ | ${E}^{\prime}=-X{\left.\frac{\partial Y}{\partial X}\right|}_{T,N}$ | |

Chemical | Isothermal Constant-Force Chemical Resistance ${R}_{{N}_{Y}}={\left.\frac{\partial \mu}{\partial N}\right|}_{T,Y}$ | Isothermal Constant-Displacement Chemical Resistance ${R}_{{N}_{X}}={\left.\frac{\partial \mu}{\partial N}\right|}_{T,X}$ |

Thermo-chemical | Constant- Force Thermal Decay Coefficient | Constant-Displacement Thermal Decay Coefficient |

${\left.{\lambda}_{Y}=\frac{\partial \mu}{\partial T}\right|}_{Y,N}$ | ${\left.{\lambda}_{X}=\frac{\partial \mu}{\partial T}\right|}_{X,N}$ |

Property | Physical Interpretation |
---|---|

Heat Capacity | Material’s ability to withstand heat without an increase in temperature. |

Thermal Displacement/Force Coefficient | Impact of temperature on displacement/force. Physical response to temperature change. |

Load Modulus | Physical response to non-thermal load. |

Chemical Resistance | Resistance to chemical conversion of reactive species. |

Thermal Decay Coefficient | Chemical response to temperature change. |

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**Figure 2.**(

**a**) Monitored parameters during grease shearing and (

**b**) phenomenological entropy density terms: shear entropy density and MST entropy density over time [8].

**Figure 3.**(

**a**) Monitored parameters during Li-ion battery cycling showing 1.5 h discharge starting at ~5 A, followed by 1.5 h charge at 3 A. (

**b**) phenomenological entropy terms: Ohmic entropy and ECT entropy over time [7].

**Figure 4.**(

**a**) Monitored parameters during steel rod fatigue loading and (

**b**) phenomenological entropy terms: load entropy density and MST entropy density over time [9].

**Figure 5.**(

**a**) Monitored inlet/exit pressures and flow rate during a centrifugal motor pump operation, and (

**b**) phenomenological entropy terms: flow entropy and load entropy over time.

**Figure 6.**Illustrations of the Phenomenological Entropy Generation theorem, showing reversible (green) and phenomenological (purple) paths; the vertical difference between the paths (see black arrows) defines entropy generation or energy dissipation (orange). (

**a**) Rates $d{\mathbb{S}}_{rev},\delta {\mathbb{S}}_{phen},\delta {S}^{\prime}$ and (

**b**) Accumulations ${\mathbb{S}}_{rev},{\mathbb{S}}_{phen},{S}^{\prime}$.

**Figure 7.**Illustrations of the Minimum Entropy Generation theorem, showing reversible (green) and minimum (purple) paths; the difference between the paths defines minimum entropy generation (orange). (

**a**) Rates $d{\mathbb{S}}_{rev},\delta {\mathbb{S}}_{min},\delta {S}^{\prime}$ and (

**b**) Accumulations ${\mathbb{S}}_{rev},{\mathbb{S}}_{min},{S}^{\prime}$.

**Table 1.**Summary of entropy generation formulations for various system categories, with examples. Here, subscript e, and superscripts r, p and d represent flow, reactants, products and diffusion, respectively. For thermal systems, $\mu $ is the molar enthalpy $\frac{\partial H}{\partial N}$. For other (chemical, etc.) reactive/energy systems, $\mu $ is the molar Gibbs energy or chemical potential $\frac{\partial G}{\partial N}$. More information on $c$, $\beta $ and $\lambda $ is in Section 2.5 and Appendix A.

Category | Phenomenological Entropy Generation ${\mathit{\delta}\mathbb{S}}_{\mathit{p}\mathit{h}\mathit{e}\mathit{n}}$ | Example |
---|---|---|

OpenInternal Energy | ${\delta \mathbb{S}}_{U,phen}=$ $\frac{{C}_{X}dT}{T}+\frac{\left(u+Pv\right){dN}_{e}}{T}-\frac{\sum {Y}_{l}d{X}_{l}}{T}$ Equation (10a) (for a non-reacting system) | Compressor (with temperature rise and boundary work): |

${\dot{\mathbb{S}}}_{U,phen}=\frac{{C}_{V}\dot{T}}{T}+\frac{\sum \left[{\left({\dot{N}}_{e}h\right)}_{exit}-{\left({\dot{N}}_{e}h\right)}_{inlet}\right]}{T}-\frac{Y\dot{X}}{T}$ (Rate form in Equation (19)) | ||

ThermalEnthalpy | ${\delta \mathbb{S}}_{H,phen}=$ $\frac{{C}_{Y}dT}{T}+\frac{\sum {X}_{l}d{Y}_{l}}{T}+\frac{\sum {\mu}_{k}d{N}_{k}}{T}$ Equation (10b) (for a closed system) | Combustion systems without boundary work: |

${\delta \mathbb{S}}_{H,phen}=\frac{{C}_{p}dT}{T}+\frac{\sum {\mu}_{k}{{dN}^{r}}_{k}}{T}-\frac{\sum {\mu}_{k}{{dN}^{p}}_{k}}{T}$ | ||

Boundary-LoadedHelmholtzpotential | ${\delta \mathbb{S}}_{A,phen}=$ $\frac{-{S}_{A}dT}{T}-\frac{\sum {Y}_{l}d{X}_{l}}{T}+\frac{\sum {\mu}_{k}d{N}_{k}}{T}$ Equation (10c) | Mechanical loading, e.g., shearing without oxidation: |

${\delta \mathbb{S}}_{A,phen}=-\left(\rho {c}_{\gamma}lnT+\beta \gamma \right)\frac{dT}{T}-\frac{\tau d\gamma}{T}$ | ||

Reactive/EnergyGibbspotential | ${\delta \mathbb{S}}_{G,phen}=$ $\frac{-{S}_{G}dT}{T}+\frac{\sum {X}_{l}d{Y}_{l}}{T}+\frac{\sum {\mu}_{k}{\dot{N}}_{k}}{T}$ Equation (10d) | Electrochemical energy systems, e.g., batteries: |

${\delta \mathbb{S}}_{G,phen}=-\left(ClnT-{\lambda}_{Y}q\right)\frac{dT}{T}+\frac{\sum {\mu}_{k}{{dN}^{r}}_{k}}{T}-\frac{\sum {\mu}_{k}{{dN}^{p}}_{k}}{T}+\frac{\sum {\mu}_{k}{{dN}^{d}}_{k}}{T}$ | ||

Via Gibbs-Helmholtz coupling: | ||

${\delta \mathbb{S}}_{phen}=-\left(ClnT-{\lambda}_{X}q\right)\frac{dT}{T}+\frac{vdq}{T}$ | ||

Via Gibbs-Duhem formulation: | ||

${\delta \mathbb{S}}_{phen}=\frac{-qdv}{T}+\frac{vdq}{T}$ |

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**MDPI and ACS Style**

Osara, J.A.; Bryant, M.D.
Methods to Calculate Entropy Generation. *Entropy* **2024**, *26*, 237.
https://doi.org/10.3390/e26030237

**AMA Style**

Osara JA, Bryant MD.
Methods to Calculate Entropy Generation. *Entropy*. 2024; 26(3):237.
https://doi.org/10.3390/e26030237

**Chicago/Turabian Style**

Osara, Jude A., and Michael D. Bryant.
2024. "Methods to Calculate Entropy Generation" *Entropy* 26, no. 3: 237.
https://doi.org/10.3390/e26030237