Thermodynamics of grease degradation

Tribology International 137 (2019) 433–445

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Thermodynamics of grease degradation

T

∗

Jude A. Osara , Michael D. Bryant Mechanical Engineering Department, The University of Texas at Austin, Austin, TX, 78712, USA

A R T I C LE I N FO

A

bstr ac t

Keywords: Greases Lubrication Degradation Shear stress

A review of grease degradation mechanisms and existing grease models is presented, followed by a detailed thermodynamic analysis of grease degradation. Fundamental formulations are derived from thermodynamic laws and applied to instantaneous nonlinear response of grease to loading. The resulting model of grease degradation relates grease shear stress response to mechanical work, temperatures and environmental effects via the irreversible entropy generated by these dissipative interactions. The model is experimentally verified, and a universally consistent grease characterization methodology is described. A near 100% agreement between the grease model and nonlinear data from uncontrolled experimental measurements is achieved. The model also gives rise to new material and process parameters to characterize grease for performance and hence, system optimization.

1. Introduction

1.1. Thixotropy

Grease mixes and disperses lubricating oils into a thickener to form a gelatinous product [1–3] that lubricates surfaces in contact. High load applications such as rolling contact bearings and some gears are greased. Because the base oil, suspended in the high shear strength thickener, does not flow out of the clump, grease is a semi-permanent lubricant. However, grease lubricant properties degrade over time, which can result in catastrophic failure of equipment. Manufacturers’ tests and studies characterize greases based on application. Methods of ASTM and NLGI classify grease and predict life [1,4–6]. Needed are improved grease degradation models for failure prediction. Grease base oils are mineral oils e.g. Naphthenic oils [1], and synthetic oils. Since thickeners determine overall properties, grease is classified based on thickener [2,3]. Desired properties also vary with operating conditions and environment. High-temperature thickeners withstand heat, food-processing machines need non-toxic thickeners, and water applications require water-resistant thickeners. Most thickeners are soap or non-soap based. Common soap-based thickeners contain soap made from fats, oils (e.g. animal fat) and alkali such as caustic soda NaOH. Non-soap greases contain inorganic thickeners such as silica clays or organic thickeners such as amides. Additives that improve grease properties range from anti-oxidants, anti-wear and corrosion inhibitors, among others. Fillers such as graphite and metal oxides also improve performance [1–3]. A typical general-purpose grease has about 85% base oil, 10% thickener and 5% additives/fillers [3].

Grease as a non-Newtonian fluid deforms under applied forces, which changes rheological properties and impacts load-bearing and lubrication performance. Mechanical shearing changes grease microstructure over a time determined by thickener type, content and additives. After load removal, grease slowly tends back to original state [7–9]. Thixotropy generally applies to isothermal viscoelastic changes in grease microstructure, observed in the particle distribution and bond density (e.g. intermolecular hydrogen bonds) as grease breaks down under shear and rebuilds during relaxation [7,9,10]. Another explanation suggests recovery due Brownian motion [8]. The sheared sample is unable to spontaneously return to its original state in finite time during relaxation, thereby causing permanent structural breakdown over time. This increases with shear rate [9,11,12] until recovery is significantly diminished. Desired is a parameter that describes breakdown of microstructure. Attempts to relate macroscopic properties to thixotropy include thixotropic index, viscosity, consistency, shear stress, modulus, and interparticle bonds. Shear stress, viscosity and ASTM consistency [6] are common. Choice of parameter/model depends on convenience and measurement methods.

∗

1.2. Degradation measures and mechanisms Resistance to shear is grease's most significant property, typically determined by strain response to stress or stress response to strain. Shear stress in grease is time-dependent, indicating thixotropy. At a

Corresponding author. E-mail addresses: [email protected] (J.A. Osara), [email protected] (M.D. Bryant).

https://doi.org/10.1016/j.triboint.2019.05.020 Received 4 March 2019; Received in revised form 7 May 2019; Accepted 14 May 2019 Available online 16 May 2019 0301-679X/ Published by Elsevier Ltd.

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J.A. Osara and M.D. Bryant

given shear rate, shear stress τ (or strength) reduces to the yield stress τy wherein grease completely breaks down and flows [13–15]. Other commonly used grease performance and degradation measures include apparent viscosity [10,12], consistency [4–6,8,14,16,17] and drop point. a few of which have been related to shear stress [4,11,13,18]. Degradation occurs mechanically, thermally and chemically. Mechanical and thermal mechanisms reduce grease consistency and break down thickener [1,5,5,10,13,16,19]. Chemical degradation can oxidize base oil and thickener, separate/evaporate oil from thickener and/or breakdown the oil-thickener mixture [13,20]. Conditions determine which simultaneous mechanisms dominate [1]. Solid boundaries (bearing housings) make shearing the most common degradation mechanism, typically accompanied by thermal degradation at high temperatures. Grease oxidation is slow but common during long-term storage or high temperatures [3]. Degradation proceeds irreversibly at a rate dependent on the active dissipative processes. Even with hightemperature greases, heat can induce oxidation and evaporation of base oil. A review of common grease degradation measures is presented in Appendix A.

models [4,8,10,13,22–24], mostly empirical, inconsistently and inadequately characterize observed trends. 1.4. Existing energy/entropy models and limitations Energy-based formulations which relate microstructure stability of grease to viscous energy density are more consistent and less restricted. Kuhn's [25] rheological energy density, quotient of rheological input work to grease volume

d¯ erh = ηγ˙ ⎛ e ⎞ h ⎝ 0⎠ ⎜

ϕ=

(1)

tf

1 2 3 4 5

τt MPa-s

AW MJ/m3

S′W MJ/m3/K

S′ μT

MJ/m3

1.02 0.67 0.04 2.79 0.66

29.7 18.3 1.5 74.0 17.9

−857.0 −526.9 −41.9 −2136.6 −517.1

−505.0 −303.8 −11.3 −761.1 −306.0

−2.78 −1.72 −0.13 −6.81 −1.68

−1.63 −0.98 −0.04 −2.44 −0.99

AμT

energy were τ (t ) = τlim

MJ/m3/K

1 2

and e (t ) = τlim

dS = S˙ irr (t ) + S˙ Q (t ) + m˙ in s˙in − m˙ out s˙out dt

erh = Tf (ρout sout ) − Tf (m˙ in s˙in − SQ )/ Vout

2.80 2.36 1.01 2.34

70.7 60.4 25.6 59.7

−2038.6 −1741.7 −737.8 −1723.9

−187.9 −109.8 −100.6 −93.0

−6.52 −5.59 −2.40 −5.46

−0.60 −0.35 −0.33 −0.29

Duration hours

τt MPa-s

AW MJ/m3

AμT

S′W MJ/m3/K

S′ μT

MJ/m3

2.26 1.46

23.3 19.5

−672.2 −563.2

5.1 23.5

−2.24 −1.88

0.02 0.07

(

1 −n + 1

−n + 1 t , tlim

)( )

where n

(4)

Kuhn evaluated an apparent rheological energy density from the combined first and second laws of thermodynamics at steady state, defined entropy generation in terms of frictional energy, and estimated frictional energy from entropy transfer by mass and heat as (5)

Kuhn [29] measured two different greases under similar conditions and plotted normalized degradation versus normalized entropy flow as done in Ref. [33], but excluded oxidation. Via entropy transfer, Kuhn [31] applied the Degradation-Entropy Generation theorem [34] to structural grease degradation, obtaining a formulation for degradation coefficient via entropy versus change in number of particles np, from which a curvilinear relationship between np and γ˙ was obtained for three different greases. In Ref. [32], Kuhn showed self-reorganizing behavior in greases via Nosonovsky's [35] thermodynamic stability criterion, based on Prigogine's [36–38] minimum entropy generation theorem and dissipative structures. In Ref. [39] accumulated energy density depended on grease composition and shear rate. Frictional and

(a) N

−n t tlim

( )

is structural degradation intensity. Kuhn had difficulty isolating shear degradation from thermal and pressure effects, and effect of interface wear in boundary and mixed friction measurements. More recent works by Kuhn [29–32] replaced friction energy with irreversible entropy generation. Using the open system entropy balance [29,30].

AFTER 4 MONTHS OF REST \/ 6 7 8 9

t → t lim

t0

isothermal process with steady state dissipation, specific energy e∗ = erh/ ϕ measured the unsteady dissipation of the grease's available friction energy, with a limiting value of 1. In the experiment, the friction energy depended on the shear motion of the solid boundaries. Friction energy and wear intensity parameter e* were then unified using the steady state value of the dissipation function. Kuhn further investigated thixotropic behavior [28] using rheometric measurements of NGLI 2 grease and identified elastic and plastic regions in shear. Defining a maximum degradation condition as degradation rate greater than or equal to −0.005, time dependence of shear stress and friction

Table 1 Datasets of Helmholtz energy/entropy and DEG parameters for calcium greases (a) Aeroshell 14 NLGI 4 Aircraft calcium grease and (b) Valvoline NLGI 2 general multi-purpose lithium grease showing different iteration durations (shear rate = 28.8 s−1). Each iteration N was a different experiment on the same grease sample. Duration hours

(3)

energy density erh = γ˙ ∫ τ (t ) dt with limiting elim = lim erh (t ) . For an

Equation (1) and other grease models reviewed in Table A1 of Appendix A, typically involve one degradation mechanism, are inadequate to consistently model degradation over time [4,8,10,13,22–24], and limit the range of shear rates and grease types. Pre-shearing makes difficult establishing a consistent initial condition for shearing. Other issues include wall slip for low shear rate tests, equipment inertia, and dependence on soap composition. Physical shearing models are isothermal. Without temperature control, data must be normalized to an approximate constant temperature. Overall

N

τ 2VG ∗ ∗ Derh h0

and established a maximum degradation point based on viscosity loss. References [25,26] are linked via the apparent rheological frictional energy density erh = Wrh/Vrh. Different energy levels for different greases make these formulations grease sample- and process-specific. Kuhn et al. [27] evaluated erh in equation (2) using mechanical dissipation with measured values of shear stress and shear rate. Frictional

Models of grease behavior are mostly experimental. Thixotropic models should include time dependence to shearing. Czarny [21] proposed an experimentally verified grease shear stress 1

(2)

a function of grease properties and operating conditions, was related to e a friction coefficient ηf = prh irh where η is dynamic viscosity (Pa s), D r −1 shear rate (s ), de middle diameter of micro contact (mm) and ho* central film thickness (mm). Kuhn [26] defined structural degradation rate

1.3. Physical models and limitations

τ (t ) = [k1 t (m − 1)τ01 − m + (τ0 − τr )1 − m]1 − m + τr

⎟

MJ/m3/K

(b)

434

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J.A. Osara and M.D. Bryant

2.1. Statement

rheological tests on specially manufactured grease samples with different soap concentrations verified Leider-Bird's time-dependent shear stress

⎡ ⎛ ⎞ ˙ − 1⎟ τ (t ) = kγ˙ n ⎢1 + ⎜bγt ⎝ ⎠ ⎣

i

⎤

∑ wi ei (−t / λn⎥ 1

Given an irreversible material transformation caused by i = 1, 2, …, n underlying dissipative processes and characterized by an energy, work, or heat pi . Assume effects of the mechanism can be described by variable w that measures the effects of the degradation transformation, i.e. w = w ( pi ) = w ( p1 , p2 , … , pn ), i = 1,2, …, n and is monotonic in the degradation and each pi . Then the rate of degradation of the material

⎦

(6)

and showed the “yielding” energy density e1 depended on shear rate, soap concentration and base oil viscosity whereas yield stress τy depended only on soap concentration and base oil viscosity. Khonsari et al.'s [40] wear rate w˙ av = ψw Pd depended on frictional power dissipation (evaluated from oscillation speed and normal torque) with wear energy dissipation coefficient ψw . Temperature was included via a linear relationship between power dissipation and temperature rise Pd = ψT ΔT with coefficient ψT . Via finite element thermal analysis, they predicted ΔT for wear rate. Rezasoltani and Khonsari [41], using internal energy, formulated entropy generation in terms of shear stress τ , shear rate γ˙ and steady temperature T as tf

Sg = γ˙ (t )

∫ t0

τ (t ) dt T

w˙ =

is a linear combination of the rates of irreversible entropies S˙ ′i generated by the dissipative processes pi , where the degradation/ transformation process coefficients

Bi =

Pen = 4.162Sg + 0.071

(9)

w=

tf

t0

(14)

2.2. Generalized degradation analysis procedure Bryant et al.'s [34] structured DEG theorem-based degradation analysis embeds the physics of the dissipative processes into the energies pi = pi (ζij ), j = 1,2, …, m ; derives entropy generation S˙ i ′ = S′i ( pi ) as a function of the pi and expresses the rate of degradation w˙ as a linear combination of all entropy generation terms, equation (12). Here the pi can be energy dissipated, work lost, heat transferred, change in thermodynamic energy (internal energy, enthalpy, Helmholtz or Gibbs free energy), or some other functional form of energy and the ζij are time-dependent phenomenological variables (loads, kinematic variables, material variables, etc.) associated with the dissipative processes pi . The degradation coefficients Bi must be measured using equation (13). The approach.

(10)

had variable temperature, defined and measured shear stress, and was linear in generated entropy. From this, they developed an isothermal, constant-shear rate, time-dependent shear stress formulation

∝γ˙ t⎞ τ (t ) = τ∞ + (τ0 − τ∞)exp ⎛− ⎝ T ⎠

∑ Bi S′i

which is a linear combination of the accumulated entropies, S′i .

˙

∫ τ (Tt )(γt )(t ) dt

(13)

pi

i

Experiments to determine Sg verified the formulations. Rezasoltani's and Khonsari's [42].

Sg =

∂w ∂S′i

are slopes of degradation w with respect to the irreversible entropy generation S′i = S′i ( pi ), and the |pi notation refers to the process pi being active. The theorem was stated and proved in Ref. [34]. Integrating equation (12) over time, composed of cycles wherein the Bi are constant, yields the total accumulated degradation

for mechanical degradation of grease. Citing the Degradation-Entropy Generation theorem [34], they obtained a linear relationship between net penetration Pen as degradation measure, and viscous energy accumulated ε /entropy produced Sg (8)

(12)

i

(7)

Pen = 0.014ε + 0.069

∑ Bi S˙ ′i

(11)

experimentally verified with different combinations of grease type, shear rate and temperature. Although this illustrated the entropy versus consistency relation in Ref. [41], the result is not applicable when other degradation modes are significant. Based on [41], Zhou et al. [43] investigated grease degradation via mechanical shearing under isothermal ambient conditions. Using controlled experiments, the authors verified entropic approach for grease characterization, and observed a two-phase aging mechanism. Reviewed was grease degradation from physical and manufactureradjusted empirical models to thermodynamics-based models. Most models struggle to combine different mechanisms (mechanical, chemical and thermal). Using thermodynamics and the DegradationEntropy Generation DEG theorem, a universal and consistent model will be formulated to account for simultaneous grease degradation modes.

1. Identifies the degradation measure w, dissipative process energies pi and phenomenological variables ζij , 2. Finds entropy generation S′ = S′i ( pi ) caused by the pi , 3. Evaluates coefficients Bi by measuring increments/accumulation or rates of degradation versus increments/accumulation or rates of entropy generation, with process pi active. This approach can solve problems consisting of one or many variegated dissipative processes. Previous applications of the DEG theorem analyzed friction and wear [33,34,44], metal fatigue [45–48], and battery degradation [49]. 3. Thermodynamic formulations This section reviews the first and second laws of thermodynamics [36,38,50-56].

2. Degradation-Entropy Generation Theorem [34] review The Degradation-Entropy Generation theorem [34] relates material/system degradation w to the irreversible entropy S′i produced by the active dissipative physical processes pi that drive the degradation. Entropy measures disorganization in materials. Since degradation is advanced and permanent disorganization, entropy generation is fundamental to degradation. Here i indexes the dissipative processes.

3.1. First law – energy conservation The first law

dU = δQ − δW + 435

∑k dNk

(15)

Tribology International 137 (2019) 433–445

J.A. Osara and M.D. Bryant

for a stationary thermodynamic system, neglecting gravity, balances the change dU in internal energy, δQ the heat exchange across the system boundary, δW the energy transfer across the system boundary by work, and ∑k dNk the internal energy changes due to chemical reactions, mass transport and diffusion, where k are chemical, flow and diffusion potentials, Nk = N r k + N e k + N d k are numbers of moles of material species k with N r k , N e k and N d k the reactive, transferred and diffusive species respectively. Inexact differential δ indicates path-dependent variables. For chemical reactions governed by a stoichiometric equation [36,38,57], ∑k dNk = Adξ ,where A is reaction affinity and dξ is reaction extent.

dA = dArev is the free energy change in the system via the reversible (rev) path, maximum for energy transfer out of the system and minimum for energy transfer into the system. According to the thermodynamic State Principle, changes in system energy and entropy are path-independent: can be determined via reversible (linear) and irreversible (nonlinear) paths between system states. Equality of equations (17) and (18) applies this principle. Eliminating δQ from equation (15) via equation (18) gives, for compression work PdV [33,36,38,44,53], dU = dUirr = TdSe − TδS′ − PdV +

Known as the Clausius inequality, the change in closed system entropy according to the second law of thermodynamics is

δQ T

dA = dAirr = −SdT − PdV +

(16)

δQrev T

(17)

which approximates a quasi-static (very slow) process, in which total entropy change occurs via reversible heat transfer δQrev . Expressed as the equality dS = dSe + δS′ [36,38], the second law equates the change in entropy dS to the measured entropy flow dSe across the system boundaries from heat transfer and mass transfer (for open systems), plus any entropy δS′ produced within the system boundaries by dissipative processes. Entropy generation δS′ measures the permanent changes in the system when the process constraint is removed or reversed [36,51], allowing the system to evolve. For a closed system (no mass transfer across system boundaries), the second law becomes [44,59].

dS = dSirr =

δQ + δS′ T

δAphen = −SdT − PdV +

(18)

δS′ = −

For a system undergoing quasi-static heat transfer and compression work, equation (15), replacing δQ with δQrev = TdSrev from equation (17) gives the combined form of the first and second laws (19)

−

and

reversible

(ideal)

Helmholtz

entropy

as (26)

where for energy extraction, (dSrev ≤ δSphen ) < 0 , and for energy addition, 0 < (dSrev ≤ δSphen ) . A comparison of equations (17) and (18), and (21) and (23), applying the State Principle, showed that changes in Helmholtz entropy and energy between two states are path-independent, whether the process path are reversible or irreversible, i.e.,

(20)

∑k dNk

(25)

δS′ = δSphen − dSrev ≥ 0

results in an alternate form of the first law which can measure maximum boundary work obtainable from a thermodynamic system. Differentiating equation (20) and substituting equation (19) for dU into the result give the Helmholtz fundamental relation

dA = dArev = −Srev dT − PdV +

(24)

∑ dNk δAphen SdT PdV dArev dArev − + k − = − ≥0 T T T T T T

∑ dNk PdV + kT T dArev dSrev = T , restated SdT T

commonly known as the TdS equation [55,58]. Here P is pressure, V volume, T temperature and S entropy. Replacing entropy with temperature as the independent variable and performing a Legendre transform with Helmholtz free energy

A = U –TS

∑k dNk

which satisfies the second law. During energy extraction, dT ≥ 0, dV ≥ 0, dNk ≤ 0 and dArev ≤ 0 , rendering δS′ ≥ 0 . For energy addition or product formation, dT ≤ 0, dV ≤ 0, dNk ≥ 0 and dArev ≥ 0 , reversing the signs of the middle terms in equation (25) to preserve accordance with the second law δS′ ≥ 0 [36]. Equation (25) defines entropy production as the difference between δAphen δSphen = T = − phenomenological Helmholtz entropy

3.3. Combining first and second laws with Helmholtz potential

∑k dNk

(23)

due only to changes in physically observable and measurable intensive and extensive properties of a real system undergoing a real (irreversible) process. Note terms dT , dV and dNk are readily measured (observed). With a known constant dArev and equation (24), equations (21) and (23) can be combined to give the fundamental Helmholtz-based entropy production relation

where dSirr is entropy change along the irreversible (real) path, δQ/T is entropy flow by heat transfer which may be positive or negative, and T is the temperature of the boundary where the energy/entropy transfer takes place. The second law also asserts that entropy generated δS′ ≥ 0 .

dU = dUrev = TdSrev − PdV +

∑k dNk − TδS′ ≤ 0

for a spontaneous process where dA = dAirr is the free energy change in the system via the irreversible (irr) path, also maximum for energy transfer out of the system and minimum for energy transfer into the system. Both equations (21) and (23) assess total change in Helmholtz free energy for all systems and show that dA can be evaluated via an ideal change dArev (wherein δS′ = 0 and S = Srev which reverts equation (23)–(21)), or a real change (wherein entropy production δS′ > 0 ) obtained by adding the energy lost due to entropy production TδS′ to the physically observable Helmholtz energy loss via active processes, e.g., from a dynamically loaded component. From equation (23), define the phenomenological Helmholtz energy change

indicating that the entropy change is equal to or greater than the measured entropy transfer across system boundary. For open systems, the right side of equation (16) includes a mass transfer term. For a reversible process,

dS = dSrev =

(22)

the irreversible combined first and second laws, or the TδS′ equation, where reversible entropy change dSrev was replaced by entropy flow dSe and entropy generation δS′ . Differentiating Helmholtz energy equation (20) and substituting equation (22) for dU into the result give the irreversible or TδS′ form of the Helmholtz fundamental relation

3.2. Second law – entropy balance

dS ≥

∑k dNk

dS = dSrev = δSirr = dSphen − δS′δA = dArev = δAirr = δAphen − TδS′ (27) Entropy and energy changes along the irreversible path are a sum of instantaneously observed/measured phenomenological entropy and energy changes and the internally permanent entropy generation and energy loss respectively. Helmholtz energy change dA = dArev and

(21)

the quasi-static change in Helmholtz energy of the system between two states according to the first law, valid for all systems, where 436

Tribology International 137 (2019) 433–445

J.A. Osara and M.D. Bryant

grease that flows out of the tribo-control volume [61] is continuously replenished. After shearing and heating cease, the grease cools to the ambient temperature and the microstructure rebuilds. The resulting Helmholtz energy regained during relaxation is

entropy change dS = dSrev from equation (27) are widely used: only end state measurements of system variables (before and after active process interactions) are required for evaluation, unlike dA = δAirr = δAphen − TδS′ and dS = δSirr = δSphen − δS′ which require instantaneous account of all active processes. However, a system's energy change dA and entropy change dS during a process can be negative or positive, depending on energy flow TdSe or entropy flow dSe across system boundaries, hence neither dA nor dS measures the permanent changes in the system. This limits success of energy and entropy formulations in characterizing measurable permanent system changes. On the other hand, entropy generation, equation (25) or (26), evolves monotonically as stipulated by the second law. In subsequent sections, entropy generation is simplified for experimental measurements and applied to grease degradation via the DEG theorem. With δS′ = 0 indicating an ideal (reversible) system-process interaction, equation (26) also indicates that a portion of any real system's energy is always unavailable for external work, δS′ > 0 . Equation (26), which gives the entropy generated by the system's internal irreversibilities alone, is in accordance with experience, similar to the Gouy-Stodola theorem of availability (exergy) analysis [54,56,59,60]. The foregoing equations are in accord with the IUPAC convention of positive energy into a system. Note that reversible and irreversible imply thermodynamic reversibility: a real system undergoing a spontaneous—natural and uncontrolled—process cannot “revert” back to its original state without work from an external source, hence is thermodynamically irreversible.

where dT ≤ 0 gives dAr ≥ 0 during recovery which can be estimated from models like Maxwell's shear stress relaxation. After relaxation is recovery equilibrium where dA|T = 0 , determined experimentally as dT → 0. At recovery equilibrium, from start of one cycle to start of next, total Helmholtz energy change

where breakdown b is much faster than spontaneous recovery r [10] as in most applications, see section 1.3.2,

rendering dAr negligible. Entropy generation, via equation (25) with generalized boundary loading and active chemical reaction, is

δS′ = −

Vτdγ SdT dArev − − ≥0 T T T

(35)

which suggests S ’ = S ’(T , γ , m) . Shaft rotational work MT dθ (input work across grease boundaries into grease) with torque MT and angular displacement θ can replace shear work. Vτdγ. 4.1.1. Rates and accumulations Rewriting equations (29) and (34) in rate form,

μ Vτγ˙ ST˙ m˙ A˙ ˙ ˙′ = − mS A˙ = −ST˙ − Vτγ˙ + − + − rev Mm T T Mm T T

(37)

With no independent chemical interaction, m˙ ≈ 0. Integrating over time obtains the total change in Helmholtz energy and entropy generation from time t0 to t as

(28)

where Mm is the grease (or reactive component – oil or thickener) molecular mass. Combining gives the grease Helmholtz energy loss

t

ΔA = −

t

˙ − ∫ Vτγdt ˙ ∫ STdt t0

(29)

t

S′ = −

where dT ≥ 0 , dγ ≥ 0 and dm ≤ 0 . Equation (29) follows the IUPAC sign convention. The system has three independent state variables when all three modes occur simultaneously and independently, which suggests

A = A (T , γ , m)

(34)

Equation (34) accumulates the entropy generated by three simultaneous active processes. Mechanical and thermal loss of shear strength is the most significant degradation mechanism for grease in service and oxidation does not begin until significant heating, hence for grease degradation below the drop point dominated by mechanical and thermal loading (no significant material removal or chemical transformation dm ≃ 0 ),

Equation (21) gives the loss of Helmholtz energy in a compressible system. To represent all forms of dynamic loading, generalized thermodynamic boundary work YdX1 will replace compression work PdV . Here Y is a generalized load and X is a generalized displacement. For small-deformation shear stress τ and shear strain γ [27], YdX = Vτdγ . Term ∑k dN ′k (= dN for a closed system with one reactive species) gives energy loss for independent chemical processes such as corrosion, where [13] dN ′ = dN ′react + dN ′evap . Also,

μ dm ≤ 0 Mm

Vτdγ SdT dm dArev − + − ≥0 T T Mm T T

δS′ = −

4.1. Helmholtz energy and entropy

dA = −SdT − τdγ +

(33)

dAb ≫ dAr

Grease is typically sheared between solid boundaries. After excitations cease, grease (the thermodynamic system) spontaneously settles to a new equilibrium state. The system is closed (no mass transfer) with heat transfer to the surroundings. The system is at equilibrium before and after operation. Established will be thermodynamic analyses that include mechanical, chemical and thermal interactions.

dm Mm

(32)

dA = dAb + dAr

4. Grease degradation analysis

dN ′ =

(31)

dAr = −SdT,

(38)

t0

˙

t

t

˙

˙

Vτγ A dt − ∫ rev dt dt − ∫ ∫ ST T T T t0

t0

(39)

t0

Note A˙ rev < 0 for grease in operation. At thermal equilibrium, Vτγ˙ A˙ S˙ ′ΔT = 0 = − T − Trev . Cyclic Continuous Shearing: Grease is repeatedly sheared and relaxed, hence a cyclic analysis using datasets from several iterations can be performed via equations (38) and (39). Accumulated entropy production after N load iteration cycles

(30)

Equation (29) also applies to an open-system that leaks, or where 1

Derivations involving pressure-volume work in equation (19) and subsequent equations such as (41) originated from the general work term δW in the first law equation (15). Reformulating with generalized force-displacement work YdX instead of pressure-volume work PdV allows replacement of pressure-volume terms in these formulations, without loss of generality.

S′total =

∑ ⎛∫ΔtN ⎜

N

⎝

ST˙ dt + T

∫Δt

N

Vτγ˙ dt − T

∫Δt

A˙ rev ⎞ dt T ⎠ ⎟

N

where ΔtN is the time duration of the Nth iteration. 437

(40)

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4.2. Entropy content S and internal free energy dissipation –SdT

maximum strength coefficient τ ′0 (assumes an ideal/reversible strength equal to the grease maximum strength, sometimes called yield stress). Similarly, from equation (46), MST energy density

The Helmholtz fundamental relation, equation (21) or (29), introduced –SdT, free energy dissipated and accumulated internally by grease during operation. This includes effects of shearing and heat from an external source. Temperature change dT is driven by grease entropy content S. For shear power Vτγ˙ , equation (29) indicates Helmholtzbased entropy depends on grease temperature T, volumetric shear Vγ and mass m. Replacing m with number of moles N for convenience, entropy change via partial derivatives

∂S ∂S ∂S ⎞ dS = ⎛ ⎞ dT + ⎜⎛ ⎟⎞ Vdγ + ⎛ dN ∂ T ∂ γ ∂ ⎝ ⎠γ , N ⎝ N ⎠T , γ ⎝ ⎠T , N

t t0

where

(for

heat

capacity

solids Cγ ≈ CP ≈ CV = C ), α = pansion and κT = −

( )

∂γ ∂τ T , N

at

( )

∂γ ∂T τ , N

constant

shear

S′ phen = −

˙

(48)

t0

t

˙

˙

∫ − ⎛ρc ln T + καT γ ⎞ TT dt + BW ∫ − τγT dt t0

BμT =

α Vγ κT

⎜

⎟

⎝

⎠

(49)

t0

wtotal =

⎟

⎛ρc ln T + α γ ⎞ Tdt ˙ − κT ⎠ ⎝ ⎜

⎟

t

˙

⎜

⎟

⎠

⎝

⎟

∫Δt

N

Vτγ˙ ⎞ dt ⎟ T ⎠

(51)

t

(46)

BμT =

(52)

∂τ ∂τ ; BW = ∂S′ μT ∂S′W

(53)

2/ G

˙

5. Grease experiments

(47)

In equation (47), the first term is the microstructurothermal MST

(

⎝

α ⎞ T˙ γ dt + BWN κT ⎠ T

t0

entropy density S′ μT = − ∫ ρc ln T + t0

⎜

where the shear stress-entropy coefficients

t0

t

⎜

∫ τdt = BμT S′μT + BW S′W

˙ τγdt

t0

t

(50)

For shear rate-controlled loading with cumulative shear stress as degradation parameter, equation (49) becomes

t

∫

∑ ⎛BμTN ∫ΔtN −⎛ρc ln T + N

(45)

∫ ⎛ρc ln T + καT γ ⎞ TT dt − ∫ τγT dt − τ0T ⎝

∂w ∂w ; BW = ∂S′ μT ∂S′W

which pertain to microstructurothermal (MST) entropy S’ μT and boundary work entropy S’W respectively, and can be evaluated from measurements of slopes of w versus phenomenological entropy production components S′i for dissipative processes pi . In iterative applications, since entropy accumulates, degradation during the Nth iteration relates to entropy production through equation (49). Total accumulated degradation sums over N iterations, giving

(44)

and Helmholtz entropy generation density

t0

⎠

t

w = BμT

hereby named the MicroStructuroThermal MST energy dissipation. Substituting the rate form of equation (45) into (38) and (39) and dividing through by volume V gives change in grease Helmholtz energy density during shearing

S′ = −

⎝

Via equation (13), the DEG coefficients

⎜

t0

⎟

Applying the DEG theorem to phenomenological entropy generation estimates the concomitant grease degradation as

(43)

α − SdT = −⎛C ln TdT + VγdT ⎞ κT ⎠ ⎝

∫

⎜

4.3. Degradation-Entropy Generation (DEG) analysis

is isothermal shearability, obtained via ma-

Note initial entropy S0 , the constant of integration in equation (44), was assumed zero, as in new/unsheared grease sample. The first righthand side term is the entropy from temperature change (thermal energy storage) while the second term emanates from internal changes in microstructure and configuration. From equation (44),

t

t

˙

∫ ⎛ρc ln T + καT γ ⎞ TT dt − ∫ τγT dt

If unknown, dArev and consequently, dSrev , may be neglected for analysis convenience and equation (48) applied.

Integrating gives entropy content

ΔA = −

t0

t

t0

is the coefficient of thermal ex-

C α dT + Vdγ. T κT

S = C ln T +

˙ . = ∫ τγdt

t0

nipulation of mixed partial second derivatives, similar to isothermal compressibility for compression work [51]. While C and α measure response to heat/temperature changes, κT measures the grease's “cold” response to shearing. For a constant-composition system (no independent chemical transformations or phase changes), dN = 0 to give, from equations (41) and (42)

dS =

W

t

(42)

t

˙ and A ) Tdt

manufacture as the maximum energy in the grease's microstructure, between leaving the factory and until full degradation; or locally (cyclically) just before onset of loading (start of cycle) as the maximum energy change before and after loading. Hence ΔArev is relatively inactive in the characteristic path-dependent evolution of entropy generation in grease. Neglecting the constant (between 2 states) reversible term in equation (47) as in Prigogine et al.'s irreversible entropy formulations for active process/work interactions [36,53], the phenomenological entropy production in grease is

(41)

Cγ ⎛ ∂S ⎞ μ ∂τ α ∂S ⎞ ⎛ ∂S ⎞ = ; ⎜ ⎟ =⎛ ⎞ = ; ⎛ = −⎛ ⎞ T ⎝ ∂γ ⎠T , N ⎝ ∂T ⎠γ , N κT ⎝ ∂N ⎠T , γ ⎝ T ⎠γ , N ⎝ ∂T ⎠γ , N is

α γ κT

∫ dArev dt = ΔArev is constant and defined globally at

In grease,

Via Callen's derivative reduction technique and Maxwell's relations [51], equation (41) can be re-stated using established and measurable thermodynamic system parameters [38,51,52].

Cγ

(

AμT = − ∫ ρc ln T +

α γ κT

)

T˙ dt T

A rotational grease shearing test was performed to verify formulations and evaluate degradation coefficients. Via measured speeds and torques, energy rate due to applied shear work was estimated.

characterizing internal

dissipation in grease, the second is the boundary work term t

S ’W = ∫ t0

τγ˙ dt T

5.1. The mechanical shearer

characterizing energy dissipation via useful work output,

and the third is the reversible/elastic shear entropy S′rev =

(

tained from Maxwell's equation τ˙ = G γ˙ −

τ η

),

τ0 2 / G T

A motorized stirrer sheared grease in a cup continuously, and the resulting temperature rise determined energy dissipated as heat in the grease. The total grease rheological energy density was determined

ob-

defined using the 438

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Fig. 1. Monitored parameters during (a) 3-hr grease shearing of Aeroshell 14 NLGI 4 aircraft calcium grease (higher consistency) and (b) Valvoline NLGI 2 general multi-purpose lithium grease, at a constant shear rate of 28.8 s−1. Temperatures are on the right axes and instantaneous shear stresses on the left.

indicated by a drop in steady state shear stress below required values.

from the stirrer's power (torque times rotational speed). A paint mixer, with impeller diameter 63.5 mm and a 9.5 mm shaft which extended through a hole in the cup cover to the motor sheared the grease. An overhead stirrer driven by a brushless DC motor that kept the set angular speed to within 1% despite grease viscosity changes, powered the system and established a constant shear rate. The change in viscosity/ shear stress was derived from motor torque. A current probe with a voltage output estimated the current. Instantaneous power by the shearer gives the frictional energy

W˙ = 2πω (MT − MT0)

6. Results and data analysis Using equations for energy loss in grease and entropy production via work and thermal energy changes, Table 1 was generated. Integrals were evaluated using the trapezoidal rule on data over time increment Δt . Sampling at 0.1 Hz rendered the time interval between data points Δt = 10s for all data. Results are for Aeroshell 14 NLGI 4 aircraft calcium grease (subsequently referred to as #4 grease) and Valvoline NLGI 2 general multi-purpose lithium grease (subsequently referred to as #2 grease). Constants include grease sample mass m = 0.247 kg (#4) and 0.202 kg (#2); specific heat capacity Cgrease = 2300 J/kg K; drop point = 140 °C (#4) and 130 °C (#2) which indicates tendency of oxidation. Table 1 (#4 in (a) and #2 in (b)) was populated using equations (46) and (47). Both grease samples were sheared at irregular intervals for different durations to show the robustness of the DEG approach. Randomly selected iterations 4 (#4 grease) and 1 (#2) are used to break down observed trends. Each iteration N (column 1) was a separate test on the grease sample. In accordance with formulations, dissipative energy changes and entropy production during shearing are are on negative axes. Fig. 1 plots shear stress as a function of torque MT (via equation (56)), grease temperature T and ambient temperature T∞, showing a maximum temperature of about 45 °C for the higher-consistency—hence more resistance to shearing and higher viscous heat dissipation—grease, (Fig. 1(a)), suggesting no considerable oxidation. Fig. 1 shows 2.8 h of continuously shearing Aeroshell 14 NLGI 4 aircraft calcium grease and 2.3 h of shearing Valvoline NLGI 2 general multi-purpose lithium grease, both at shear rate 28.8 s−1. Beyond static inertia, the shear stress curves show an initial transient region in which microstructure quickly breaks down. After the first half hour, the instantaneous shear stress continues to diminish, albeit at a reduced rate, due to grease thixotropy. An initial steep rise in grease temperatures during the first hour eventually levels off exponentially towards steady state, attributed to the initially high viscosity dissipating more heat (hysteresis). #4 grease with higher consistency shows much higher shear stress and shear stress fluctuations than #2 grease. Cumulative shear stress, column 3,

(54)

where ω is frequency or angular speed, MT is instantaneous torque and MT0 is initial torque. In terms of measured current I and voltage V ,

W˙ = V (I − I0) φ

(55)

where φ is the motor constant. Subscript 0 values measured with the impeller rotating in air indicate torque dissipated by the driving actuator. Hence equations (54) and (55) pertain to power dissipated in the grease. From instantaneous shear stress [62].

τ=

(MT − MT0) 2πr 2l

time-dependent viscosity η =

(56) τ . γ˙

With a known volume V , shear rate

W˙ . Vτ

γ˙ = Process parameters for the grease shearer test were continuously recorded. Aeroshell 14 NLGI 4 aircraft calcium grease, used in aircraft applications, and Valvoline NLGI 2 general multi-purpose lithium grease, used in automotive applications, were tested. Note that other empirical formulations based on more accurate rheometric measurements are available like the Metzner-Otto formulation for grease shearing [63-65] and Nguyen et al.'s [62] shear rate equation. 5.2. Procedure The area was ventilated and surroundings temperatures uncontrolled. 1. Thermocouples measured temperatures: in the cup through the hole in the grease cup cover. at the exterior of the cup. about 80 mm from the cup, with sensing electrode in the air. 2. With shearer attached to motor, stirred at constant speeds to determine the no-load condition. 3. Started data logger and recorded initial state of system for 3 min. 4. Shearing: Inserted impeller in grease and stirred at 1 Hz for 10 min, to establish the grease's pre-shear history (initial condition). Sheared the grease at 3 Hz continuously for about an hour. 5. With data still logging, grease cooled to surrounding temperature. 6. Stopped data logger. 7. Repeated steps continuously until the grease sample degraded,

• • •

τt =

t

n

t0

1

∫ τ (t ) dt ≈ ∑ ⎡⎣ τn +2τn−1 ⎤⎦ Δt

(57)

increased from t0 to t where index 1, 2, 3, …, n corresponds to times t1 t2 t3 , …, tn and Δt = tn − tn − 1. The unit MPa-s which measures stress accumulated over time is not viscosity, the ratio of shear stress to shear rate.

• •

6.1. Helmholtz energy and entropy densities Accumulated viscous or shear toughness AW is the primary 439

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Fig. 2. Components of phenomenological Helmholtz entropy generation versus accumulated shear stress for (a) Aeroshell 14 NLGI 4 aircraft calcium grease and (b) Valvoline NLGI 2 general multi-purpose lithium grease.

Fig. 3. Phenomenological Helmholtz entropy generation rates over time during shearing of (a) #4 grease and (b) #2 grease.

a profile significantly influenced by the measured temperature profile but less steep due to the microstructural recovery effect (second righthand side term in equation (59), and fluctuations in Fig. 2(a); note differences in orders of magnitude of axes values). Accurate determination of MST entropy should include effects of instantaneous temperature, particularly for anisothermal conditions. Except for iterations 5 and 7 (#4 grease) and iteration 2 (#2 grease), shear work AW and shear entropy S’W are more significant than MST energy AμT and MST entropy S’ μT changes during grease shearing, as anticipated, with values that directly depend on shear rate. At every instant, shear (viscous work) entropy S’W and an accompanying MST (viscous heat) entropy S’ μT are produced, both at the instantaneous boundary temperature. The need to keep grease below its drop point for continuous shearing underscores the influence of MST entropy on grease degradation. Both AμT and S’ μT are higher for #4 grease than #2 grease. Fig. 3 plots rates of phenomenological Helmholtz entropy generation components – shear and MST – versus time during grease shearing. Shear entropy rate (blue curves, left axes labels) is initially high and quickly steadies as quasi-steady temperature is approached, similar to shear stress. Fluctuations in #4 grease shear entropy rate (Fig. 3(a), blue curve) emanate from fluctuations in shear stress (see Fig. 3(a)), perhaps indicating grease viscoelastic response to shearing. MST entropy rate (red curves, right axes labels) shows more significant fluctuations with a few large spikes. In both greases, it is observed that the amplitudes of the MST entropy rate fluctuations increase as shear entropy rate steadies, possibly verifying the pseudo-stabilizing effect of the opposing fluctuations (about a mean of zero), with #4 grease showing the higher MST entropy rate fluctuation amplitude.

component of the Helmholtz free energy density, the boundary work t

obtained 1

from

equation

AW = − ∫ τγ˙

(54),

0

t

n

{

Δt

dt = − V ∫ 2πω (M − M0) dt ≈ − ∑1 [πω (MTn + MTn − 1 − MT 0)] V t0

}

and

tabulated in column 4 of Table 1. Microstructurothermal (MST) energy dissipation, first right-hand side term of equation (46), n

(

α

)

AμT ≈ − ∑1 ρc ln Tn + κ γn ΔTn , column 5, is the loss of available T grease Helmholtz toughness due to thermal and microstructural changes during loading. Fig. 2 plots components of phenomenological Helmholtz entropy density, equation (48), viscous or shear entropy density S’W (column 6) and viscous heat or MST entropy density (column 7) S’ μT versus accumulated stress, where

S′W = −

t

tf

t0 n

t0

M − M0) dt ∫ τγT˙ dt = − ∫ 2πω (V T

Δt ⎫ ≈ − ∑ ⎧ [πω (Mn + Mn − 1 − M0)] ⎨ Tn ⎬ V ⎭ 1 ⎩ t

S′ μT = −

˙

(58)

n

∫ ⎛ρc ln T + καT γ ⎞ TT dt ≈ − ∑ ⎛ρc ln Tn + καT γn ⎞ ΔTTnn t0

⎜

⎟

⎝

⎠

1

⎜

⎟

⎝

⎠

(59)

A linear relationship is observed between shear entropy density (Fig. 2, blue plot, left axes labels) and cumulative shear stress. Microstructurothermal MST entropy density (Fig. 2, red plot, right axes labels) progresses similar to grease temperature change. MST energy and entropy changes depend directly on grease composition (viz. heat capacity, coefficient of thermal expansion and isothermal shearability), ongoing process (viz. shear rate) and temperature evolution during loading, showing for #4 grease (Fig. 2(a)) a temperature-driven rise that eventually steadies, while for #2 grease (Fig. 2(b)), the effect of the minimal temperature rise appears overcome by instantaneous elastic recovery (the fluctuations) making MST energy and entropy mostly positive during shearing, and hence the final values in Table 1(b). Kuhn [35] reported a duration of “self optimization” in grease during shearing. #4 grease, having total temperature rise of about 18 °C, shows

6.2. DEG analysis – cumulative stress versus cumulative entropy generation By associating data from various time instants, data (S’W , S’ μT , τt ) of accumulated entropies S’W and S’ μT versus accumulated stress τt (equation (57)) are 3-dimensionally represented in Fig. 4 for iterations 4 of #4 grease and 1 of #2 grease. Both greases’ entropy trajectories can be seen to lie in planes whose orientations define BW and BμT via equation (53). Fig. 4 shows coincidence of measured data points with 440

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coincident with a plane – its DEG surface. The DEG surface suggests a linear dependence of shear stress accumulation on both the shear (actual output work/boundary loading) and microstructurothermal MST (internal dissipation) entropies at every instant. The orthogonal 3-D space, the grease DEG domain enclosing the DEG surface appears to characterize the allowable regime in which the particular grease can be sheared. The DEG domain, spanned by shear stress, shear entropy and MST entropy can define consistent parameters for identifying desired characteristics of any grease for any application. The dimensions of the horizontal pair of axes are determined by the accumulation of the entropy generation components. In Fig. 4(a) and (b), both greases are shown in the same DEG domain for visual comparison, with the orange trajectory and plane representing #4 grease and the much smaller blue trajectory and plane representing #2 grease. For a closer look, Fig. 4(c) and (d) present DEG domain for #2 grease only. In (b) and (d), end views of the DEG domains verify the coincidence of the DEG trajectories with the DEG planes, hence the coefficient of determination R2 = 1, rare for most experiments under uncontrolled conditions, especially dissipation measurements obtained from severely nonlinear materials such as grease. Note that #2 grease MST dimension (Fig. 4(c)) extends in both positive and negative directions as a result of the fluctuations during shearing. 6.2.1. Degradation coefficients Bi Degradation coefficients BW and BμT , partial derivatives of cumulative shear stress with respect to shear and MST entropies respectively via the DEG theorem, were estimated from the orientations of the surfaces in Fig. 4. For #4 grease, BW = −10.36 Pa-s K/J, and BμT = −0.504 Pa-s K/J, and for #2 grease, BW = −10.38 Pa-s K/J, and BμT = −0.031 Pa-s K/J. It is observed that both greases have similar BW as a result of the same shear rate and the use of cumulative shear stress as transformation measure, making BW a process characterization constant (note the identical slopes of both DEG planes in Fig. 4(b)). BμT measures the nonlinear impact of the instantaneous fluctuations introduced via the MST entropy. Grease manufacturers specify temperature ranges outside of which catastrophic degradation can occur (e.g., oxidation at induction temperatures, base oil separating from thickener at drop point, etc.). Many grease studies involve low-rate shearing to minimize MST entropy and employ an isothermal assumption. In Table 1, low BμT minimized MST-related degradation from iteration to iteration. Note that thermocouple movements during shearing may have introduced fluctuations to temperature measurements. A temperature measurement setup having no physical interference with the rotating stirrer will enhance evaluation of BμT .

Fig. 4. 3D plots and surface fits of cumulative shear stress vs shear entropy and MST entropy for #4 grease iteration 4 (orange points and orange plane, (a) and (b)) and #2 grease iteration 1 (blue points and blue plane, (c) and (d)), showing linear dependence on 2 active processes. (b) and (d) show all points coincident with linear surfaces for both greases (R2 = 1 goodness of fit). Axes are not to scale.

Fig. 5. Measured and phenomenological shear stresses versus shearing time for (a) #4 grease and (b) #2 grease.

6.2.2. Prediction After a few shearing iterations—listed datasets in the first half of Table 1(a)—the #4 grease sample was stored for 4 months. At the end of this period, the stored sample was re-sheared at the same shear rate and new coefficient values obtained from the new data sets as BW = −10.93 Pa-s K/J, and BμT = 0.565 Pa-s K/J, indicating a slight permanent change in the grease since initial characterization from iteration 1 of Table 1(a), i.e. higher BW magnitude indicates higher stress accumulation for the same entropy generation, with the more sensitive BμT indicating the short-term changes in microstructure as well as new test conditions. This is expected, given the grease was unused during the 4-month period, hence more significant microstructure recovery.

Figure 6. (a) Multiple DEG trajectories from iterations 1 to 5 (Table 1a) plotted on the same DEG surface for #4 grease. Different trajectory lengths indicate different durations. Trajectories start from lower right corner. The long trajectory belongs to iteration 4 with the longest shearing duration, see Table 1. (b) End view shows visual of goodness of fit. Trajectories overlap and axes are not to scale.

7. Discussion Similar to Prigogine's extension of reversible thermodynamics to irreversible and non-equilibrium processes and states [36-38,53], this study derived and verified a universally consistent system- and utilitybased, time-dependent entropy generation. Based on energy conservation and entropy balance, this article demonstrated that.

planar surfaces (a goodness of fit R2 = 1 was obtained in all cases, datasets in Table 1). Fig. 4 shows each grease path or DegradationEntropy Generation DEG trajectory measured during shearing 441

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phenomenological entropy generation S′ is the sum of boundary • work/shear entropy S’ and microstructurothermal MST entropy

Amiri and Khonsari [46,47,67,68] have in independent studies verified different versions of a material-dependent fatigue fracture entropy FFE for metals and composite laminate. DEG's critical failure entropy is inherent in the second law as given in equations (26) and (47). With the required entropy generation evolution criterion for process continuity Srev ≤ Sphen < 0 , an abrupt spike in Sphen due to sudden instability results in the second law-prohibited negative entropy generation, indicating discontinuity of the process.

phen

W

S′MST ;

• entropy generation is the difference between phenomenological and reversible S′ Helmholtz entropies at every instant; S′ • entropy generation is always non-negative per the second law, phen

rev

whereas components S′ phen and S′rev are directional. This implies S′ phen < |S′rev | during load application/energy extraction (e.g. grease shearing) in accordance with experience and thermodynamic laws. (Modulus sign indicates magnitude only).

7.2. Features of the DEG theorem and coefficients

7.1. Phenomenological shear stress

The DEG theorem provides a structured approach to system/component degradation modeling, removing the need for multitudinous measurements, numerous curve fits, and multiple analysis tools. DEG coefficients determined from a reference iteration, e.g. iteration 1, can accurately anticipate the transformation/degradation paths in subsequent iterations. This suggests that degradation coefficients can be determined at any point in grease life using simple measurements, without prior history. These coefficients show grease's true response to prevalent processes and conditions by quantifying the processes' individual dissipative effect on the grease. Boundary interaction DEG coefficients such as shear entropy coefficient BW are negative for positive evolution of fatigue measure: shear entropy is negative during loading. Microstructurothermal MST coefficient BμT has varying sign. To understand BμT sign changes, rearrange equation (61) as

All systems undergoing real processes/interactions degrade continuously towards failure. As a corollary constraint of the second law, the Carnot limitation which governs the availability of a system's energy for work, was expressed by Burghardt [56] for a heat source as

Energy added = Available energy + Unavailable energy

(60)

τphen a phenomenological (actual and physically observable) shear stress which can be decomposed into τmeas a measured component and Δτphen representing the fluctuations or deviation from the measured transformation τmeas , can be correlated directly with phenomenological entropy terms (right-hand side terms) in equation (48) as τphen = τmeas + Δτphen = BμT S′ μT + BW S′W

(61)

where Δτphen , a portion of τphen not available for boundary (useful) work during operation due to instantaneous dissipation from heating and configurational changes, is the difference between actual observable transformation τphen and measured transformation τmeas . Hence, equation (52) describes τ = τphen , the grease phenomenological shear stress and DEG model for evaluation of instantaneous transformation/degradation. While τmeas can be any existing transformation/response measure, e.g., shear stress of this study, actual entropy generation-induced transformation τphen capable of detecting instabilities is not directly measurable, making equations (49), (52) and (61)—all equivalent forms—which require only measurements of load conjugate variables τ and γ˙ , and temperature T convenient for all practical applications. Substituting BW = −10.36 Pa-s K/J, BμT = −0.50 Pa-s K/J into equation (61) gives τphen = −0.50S′ μT − 10.36S′W which is plotted in Fig. 5 (purple plots), showing high-frequency nonlinear effects of temperature and microstructure changes not observed in measured shear stress τmeas (blue plots). For experimentally determined system response e.g. shear stress as done in this study, isothermal control will give for t > t0, τmeas = τphen , verifying the often observed and soughtafter accuracy in isothermal low-shear rate characterization of grease, achievable via expensive equipment under controlled laboratory conditions. This is more easily observed in Fig. 5(b) for #2 grease (lower consistency, hence less resistance to shearing and heat dissipation), which shows only slight fluctuations in phenomenological shear stress (purple plot), relative to measured shear stress (blue plot) due to minimal MST entropy during shearing. With the exception of energy storage systems, all systems after manufacture are irreversibly, and often dynamically, loaded towards failure. For general system evolution, “transformation” and “degradation” have been used interchangeably in this article.

BμT =

1 (τphen − BW S′W ) S′ μT

(62)

where phenomenological shear stress τphen fluctuates about the loadbased BW S′W (Fig. 5), making the parenthesis expression in equation (62) fluctuate about zero during operation. Also observe from Fig. 2 and Table 1 that MST entropy S’ μT can be negative (as for #4 grease in Fig. 2(a) and Table 1(a)) or positive (as for #2 grease in Fig. 2(b) and Table 1(b)). Kuhn [35], using formulations based on Prigogine's universal evolution criterion [36], experimentally verified structural selfreorgarnization in grease under mechanical shear. Equation (48), the DEG theorem and equation (49) fully define the actual transformation paths for all grease. 7.2.1. Characteristic DEG elements – DEG trajectories, surfaces and domains Fig. 6 which plots DEG trajectories from iterations 1–5 of the #4 grease, suggests a characteristic DEG surface containing all DEG trajectories the grease can “draw” at a given shear rate. The orientation of the DEG surface determines the characteristic coefficients. Similar to multi-dimensional orthogonal spaces that have been consistently used in thermodynamics to describe thermodynamic states of reversible processes [36,41,47], this study introduces the DEG domain, a multi-dimensional space that characterizes a system's phenomenological transformation path. DEG trajectories appear characteristic of grease response to loading, and can overlap under consistent operating conditions; DEG surfaces appear characteristic of shear rates and grease composition; and the DEG domain characterizes the grease (all iterations, operational conditions and all shear rates). Grease having a domain with large cumulative shear stress dimension and small MST entropy dimension is able to carry more load more efficiently and operate in service longer. Fig. 4 shows that #4 grease has significantly higher shear entropy and MST entropy dimensions than #2 grease for similar-duration and same-shear rate shearing. In addition to the observed instantaneous “recovery” fluctuations in #2 grease, this is in accordance with experience as #4 grease has higher consistency, i.e. resistance to shear, hence higher shear entropy rate, and with heat as the direct effect of viscous processes like grease shearing, the higher MST entropy is also expected for #4 grease. This is additionally verified by the typical heavy-duty applications, with

7.1.1. Critical failure entropy S′CF and instability in operational grease For sudden changes in operational grease like yielding or thermal instability where base oil separates from thickener leading to catastrophic failure of a lubricated interface, a parameter that consistently describes critical phenomenon is often desired. A corollary of the DEG theorem states that “if a critical value of degradation measure exists at which failure occurs, there must also exist critical values of accumulated irreversible entropies” [34]. Sosnovskiy and Sherbakov [66], and Naderi, 442

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adequate cooling, for which NLGI 4 grease (also known as helicopter grease) is primarily suited. Chemical degradation mechanisms such as oxidation of base oil or thickener would appear as additional entropy generation terms in equations (13), (14), (49) and (50). These terms, formulated via the kinetics and stoichiometry from the known chemistry, would append more orthogonal entropy generation axes to the DEG trajectory plots, which would extend the 3D plots of Fig. 6 to four or more dimensions. More intense thermal degradation at higher temperatures would increase the microstructurothermal MST entropy generation (via equation (45)) and possibly increase the value of BμT relative to other B coefficients in equations (49) and (50). For grease-lubricated contacts characterized by a tribo-control volume [37], the DEG methodology, via the phenomenological entropies appropriately defined to include effects of interacting interfaces (work across boundaries and temperatures at the boundaries), can be used for grease (and grease material) selection, and predict life/degradation of both grease and the lubricated contacts.

significant transients, verified mechanical shearing. The contribution of microstructurothermal MST entropy to total grease entropy generation and degradation was demonstrated. 100% accuracy was observed in experimental results with coefficients of determination—of trajectories and surface fits—R2 = 1 for all iterations, indicating consistent robustness of the DEG methodology. DEG coefficients and elements (trajectories, surface and domain) appear to fully and consistently characterize grease operational life. An appropriate combination of thermodynamic analysis and the DEG theorem, being universally instantaneous, could allow manufacturers and researchers to directly compare thickener and base oil compositions during grease manufacture and/or study, without the need for special equipment or isothermal conditions. Measurements and appropriate data analyses via DEG give industrial users a tool to compare various grease types, determine suitability for an intended application (high temperatures, high shear rates, etc.), as well as real-time monitoring of grease-lubricated interfaces to prevent catastrophic failures.

8. Summary and conclusion

Acknowledgement

A mathematical combination of the first and second laws of thermodynamics with the Helmholtz potential employed the instantaneous applicability of local equilibrium to model grease degradation from multiple simultaneous mechanisms. Experimental data including

The authors thank Mr. Mark Philips—for his assistance with the grease shearer experiments, and Professor Ronald Matthews—for the laboratory space used, both of the University of Texas at Austin's Mechanical Engineering Department.

Appendix A Listed in Table A1 are physical (half empirical, half theoretical – Newtonian, Hookean, Arrhenian) models that are commonly used to characterize grease. Table A1. Physical grease models based on steady response to steady impulse – combined theoretical and experimental. Here k is consistency factor and n is flow index.

Degradation Mechanism

Degradation Measure

Published Model

Notes

Mechanical

Shear stress (Yield stress)

Power-law: τ = kγ˙ n Herschel-Bulkley: τ = τy + Kγ˙ n

Earliest widely adopted model, limited to a narrow range of medium shear rates [13]. Currently the most widely used. Good correlation with data at shear rates between 0.001 and 1000 s−1 [16,69]. n ≈ 0.5 for greases.

Sisko [23]: τ = Kγ˙ n + ηb γ˙ Maxwell:

Typically applied to high shear rates (> 1000 s−1) [23].

(

τ η

τ˙ = G γ˙ −

Describes viscoelasticity via a series spring and viscous damper. Gives accurate time-based shear stress response to constant shear rate, but not accurate to constant shear stress [70].

)

Gecim and Winer [71]:

γ˙ = Shear strain

+

τL tanh−1 μ

=

( ) Connects above elements in parallel and accounts for constant shear stress time-dependent strain response. Does not accurately predict relaxation [70].

τ − γG η

Mewis [8]: dη dt

Gives rate of change of viscosity at constant shear rate. n

= k [ηe (η˙ 1) − η]

Gives rate of bond breakdown in grease in terms of number of linkages N, which can be further related to viscosity.

Cross [72]: dN dt

Thermal

Shear stress

Yield stress

= k2 P − (k 0 + k1 γ˙1m) N

Arrhenius:

Uses Arrhenius formulation to describe response to temperature changes [70].

τ = γ0 G ⎡exp ⎣ Lugt [13]: τy

Viscosity

= exp ⎡ τy0 ⎣ Arrhenius:

(

η = η0 ⎡exp ⎣ Chemical

( ) ⎤⎦ Ea RT

Extends Arrhenius formulation to yield stress.

T0 − T b

) ln2⎤⎦ Arrhenius formulation – viscosity.

( ) Ea RT

⎤ ⎦

Shear stress

dτ dt

= −τ0 kexp (−kt )

Viscosity

dη dt

= −η0 kexp (−kt )

Mass

Lugt [20]: dm dt

Adds nonlinearity to Newtonian component in Maxwell model using limiting shear stress concept [71].

τ τL

Kevin-Voigt: dγ dt

Viscosity

τ˙ G∞

Based on Rhee's [73] % degradation = e-kt Extends Rhee's [73] % degradation = e-kt to viscosity. Describes oxidation in grease via mass change.

= −m0 k exp(−kt )

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Abbreviations Nomenclature Name Unit A B C M N N, Nk P P Q S S′ t T U V w W Symbols

Helmholtz free energy J DEG coefficient Pa-s K/J Heat capacity J/K torque Nm cycle number number of moles of substance mol dissipative process energy J pressure Pa heat J entropy or entropy content J/K entropy generation or production J/K time sec temperature degC or K internal energy J volume m3 degradation measure work J

α thermal expansion coefficient/K kT isothermal shearability μ chemical potential density ρ τ shear stress MPa γ˙ shear rate s-1 apparent viscosity η ζ phenomenological variable Subscripts & acronyms 0 initial MST,μT Micro-Structuro-Thermal rev reversible irr irreversible phen phenomenological DEG Degradation-Entropy Generation

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