2.1.1 Example 2.1: Miner’s Rule Derivation

One basic example of how the important result of Equation (2.2) can be used is to help explain what is termed Miner’s fatigue rule, which is a key area in physics of failure.

Consider a subsystem that is made up of, say, metal with defect sites such as a cell lattice shown in Figure 2.2. Just as the entropy of each subsystem is the sum of the entropies, so too is the entropy damage for each cell C which adds up for a subsystem that is under some sort of stress such as cyclic fatigue, as shown in Figure 2.2. Suppose we have n damaged cells so the total entropy damage is

(2.5)

where n is the total number of cells with entropy damage S_{damage-cell,i.} We assume equal entropy damage per cell. Consider that we have M total cells of potential damage to failure that would lead to a maximum cumulative damage S_max. Then the cumulative entropy damage can be normalized:

(2.6)

where s_n is the entropy damage of the nth cell, and m represents a damaged cell such that Σm is the number of damaged cells observed.

If all the cells were damaged the entropy would be at a maximum, then the damage is equal to 1 and this should be equal to counting the number of damaged cells and dividing by the total number of cells that could cause failure. This is one way to measure damage. Here damage is assessed as a value between 0 and 1. Let’s assume that there is a linear relationship between the number of cells damaged and the number of fatigue cycles. Fatigue is also dependent on the stress of the cycle, for example, if we are bending metal back and forth, the stress level would be equal to the size of the bend. So for any ith stress level the number of cells damaged for n cycles is some stress fraction f_i of the number of cycles:

(2.7)

Note that N is the number of cycles to failure and n is the number of cycles performed (N > n). Then we can write for the ith stress level:

(2.8)

When Damage = 1, failure occurs. This equation is called Miner’s rule [1]. Its accuracy then depends on the assumption of Equation (2.7). Furthermore, it follows that we can also invent here a related term called

(2.9)

When Damage = 1, the cumulative strength vanishes. Equation (2.7) turns out to be a good approximation, but it is of course not perfectly accurate. We do know that Miner’s rule is widely used and has useful validity. Miner’s rule is not just applicable to metal fatigue; it can be used for most systems where cumulative damage occurs. Later we have an example in Chapter 5 for secondary batteries. We have now derived it here using the cumulative entropy principle (Equation (2.2)). In Chapter 4 we provide a more accurate way to assess fatigue damage through the thermodynamic work principle.

2.1.2 Example 2.2: Miner’s Rule Example

Aluminum alloy has the following fatigue characteristics:

Stress 1, N = 45 cycles,

Stress 2, N = 310 cycles, and

Stress 3, N = 12 400 cycles.

How many times can the following sequence be repeated?

n ₁ = 5 cycles at stress 1,

n ₂ = 60 cycles at stress 2, and

n ₃ = 495 cycles at stress 3

Solution

The fractions of life exhausted in each cycle type are:

The fraction of life exhausted in a complete sequence yields an approximate cumulative damage of 0.345 (with the cumulative strength decreasing down to 0.655). The life is entirely exhausted when damage D = 1 and has been produced 3 times when D = 1.035. The sequence can then be repeated about 2.9 times.

One interesting thing to note in fatigue damage is that even though damage accumulates with stress cycles, the degradation may not be easily detectable as the metal system is capable of performing its intended function as long as the system’s cumulative strength (Equation (2.9)) is greater than the stress that it is under. We typically find catastrophic failure occurs in a relatively short cycle time compared with the system’s cycle life. How then can we detect impending failure at such a mesoscopic level (mesoscopic is defined in Section 2.5)? One possible solution may be in what we term noise detection. This is discussed later in this chapter (see Sections 2.5–2.9).

2.1.3 Non-Cyclic Applications of Cumulative Damage

The thermodynamic second law requirement that

leads one to realize that even non-cyclic processes must cumulate damage. That is, we can write similar to Miner’s Rule, cumulative damage for non-cyclic time-dependent processes

(2.10)

where, by analogy with Miner’s cyclic rule, t_i is the time of exposure of the ith stress level and τ_i is the total time to cause failure at the ith stress level. Not all processes age linearly with time as is found in Chapter 4, so the exponent P has been added for the generalized case where P is different from unity (see Chapter 4 for examples and Table 4.3). Equation (2.10) could be used for processes such as creep (0 < P < 1) and wear (P = 1) or even semiconductor degradation, any process where it might make sense to estimate the cumulative damage and/or the cumulative strength left (i.e., 1–Cum Damage). Note the time to failure does not have to be defined as a catastrophic failure. It may be prudent to apply this to parametric failure where τ_i then represents a parametric threshold time for failure. For example, τ_i could be the time when a certain strength, such as voltage output of a battery, degrades by 15%.

The main difference between a cyclic and non-cyclic process is that for a closed system undergoing a quasistatic cyclic process, the internal energy ΔU is often treated theoretically as unchanged in a thermodynamic cycle (see Section 3.3). This is a reversible assumption.

2.4.1 Measuring System Entropy Damage with Temperature

The most popular continuous intensive thermodynamic variable at the system level is temperature. In both mechanical and electrical systems, system internal temperature can be a key signature of disorder and increasing entropy. Simply put, if entropy damage increases, typically this is the result of some resistive or frictional heating process. Some examples will aid in our understanding of how such measurements can be accomplished with the aid of temperature observations.

2.4.2 Example 2.3: Resistor Aging

Resistor aging is a fundamental example, since resistance generates entropy. A resistor with value R ohms is subjected to environmental stress over time at temperature T₁ while a current I₁ passes through it for 1 month. Determine a measurement process to find the entropy at two thermodynamic non-aging states before the stress is applied and after it has been applied at times t_initial and t_final, respectively. Determine if the resistor has aged from the measurement process and the final value for the resistor.

Before and after aging at temperature T₁, we establish a quasistatic measurement process f to determine the entropy change of the resistor at an initial time t_i and final time t_f. A simple method would be to thermally insulate the resistor and pass a current through it at room temperature T₂ for a small time t, and monitor the temperature rise T₃. The internal energy of the resistor with work done on it is:

(2.15)

This yields

(2.16)

and

(2.17)

The entropy for this quasistatic measurement process at initial time t₁ over the time period Δt is considered reversible so that the entropy can be written over the integral:

(2.18)

It is important to note that the entropy change is totally a function of the measurement process f, often termed path dependent in thermodynamics, which is why the notation δQ is used in the integral. The integral for this process (d(Volume) = 0) is:

(2.19)

where T₃ is the temperature rise of the resistor observed after time period Δt.

A month later we repeat this exact quasistatic measurement process at time t_f and find that aging occurred as the temperature observed is now T₄, where T₄ > T₃ and the entropy damage change observed over the final measurement time is

(2.20)

and

(2.21)

Therefore the entropy damage change related to resistor degradation is

(2.22)

or, in terms of an aging ratio,

(2.23)

The resistance change is

(2.24)

We note

(2.25)

An analysis of the problem indicates that the material properties and the dimensional aspects need to be optimized to reduce the observed aging/damage that occurred.

One might now ask, so what! We can easily measure the resistance change of the aging process with an ohm meter. On the other hand, we note that our aging measurement was independent of the resistor itself. If we are only looking at aging ratios, we do not even have to know any of its properties. In some cases, we are unable to make a direct measurement; we can detect aging using thermodynamic principles. As well, the next example may be helpful to understand some advantages to this approach for complex systems.

2.4.3 Example 2.4: Complex Resistor Bank

In the above example we dealt with a simple resistor, but what if we had a complex resistor bank such as a resistance bridge or some complex arrangement? The system’s aging can still be detected even though we may not easily be able to make direct component measurements to see which resistor or resistors have aged over environmental stress conditions. In fact, components are often sealed, so they are not even accessible for direct measurement. In this case, according to our thermodynamic damage theory, if we isolate the system and its environment the total entropy change is, from Equation (2.1):

(2.26)

If we are able to use the exact same measurement process f as we did for Section 2.4.2 then for a complex bank the aging ratio is still:

(2.27)

2.4.4 System Entropy Damage with Temperature Observations

In thermodynamic terms, if a part is incompressible (constant volume), and if a part is heating up over time t₂ compared to its initial time at t₁ due to some degrading process, then its entropy damage change is given by (e.g., see Equation (2.19)):

(2.28)

where C_avg is the specific heat (C_p = C_v = C) and for simplicity we are using an average value.

Then for a system made up of similar type parts, in accordance with Equation (2.28) the total entropy damage is

(2.29)

This is an interesting result. It helps us to generalize how the state variable of temperature can be used to measure generated entropy damage for a system that is undergoing some sort of damage related to thermal issues. Note that we do not need to know the temperature of each part; temperature is an intensive thermodynamic variable. It is a roughly uniform indicator for the system. Thus, monitoring temperature change of a large system is one key variable that is a major degradation concern for any thermal-type system.

2.4.5 Example 2.5: Temperature Aging of an Operating System

Prior to a system being subjected to a harsh environment, we make an initial measurement M₁⁺ at ambient temperature T₁ and find its operation temperature is T₂. Then the system is subjected to an unknown harsh environment. We then return the system to the lab and make a measurement M₂⁺ in the exact same way that M₁ was made, where we note it is found to have a different operational temperature T₃. Find the aging ratio where

• ,
• 

The system has aged by a factor of 1.58. We next need to determine what aging ratio is critical.

2.4.6 Comment on High-Temperature Aging for Operating and Non-Operating Systems

Reliability engineers age products all the time using heat. If this is the type of failure mechanism of concern, then they will often test products by exposing their system to high temperature in an oven. This of course is just one type of reliability test. The product may be operational or non-operational. Often the operational product will add more heat, such as a semiconductor test having a junction temperature rise. Then the true temperature of the semiconductor junction is the junction rise plus the oven temperature. You might ask is there a threshold temperature for aging. The answer is, not really. Temperature aging mechanisms are thermally activated. In a sense, defects are created by surmounting a free energy barrier. Thus, there is a certain probability for a defect to be activated at any given temperature. The most common type of thermally activated aging law is called the Arrhenius Law. The barrier height model is discussed in Chapter 6. However, the Arrhenius Law is found with numerous examples in Chapters 4–10.

2.5.1 Example 2.6: Gaussian Noise Vibration Damage

As an example, consider Gaussian white noise which is one common case of randomness and often reflects many real world situations (note that not all white noise is Gaussian). When we find that a system has Gaussian white noise, the probability density function (PDF) function f(x) is

(2.34)

When this function is inserted into the differential entropy equation, the result is given by [4]

(2.35)

This is a very important finding for system noise degradation analysis, especially for complex systems, as we will see. Such measurements can be argued to be more on the macroscopic level. However, their origins are likely due to collective microstates which we might argue are on the mesoscopic level.

To clarify, for a Gaussian noise system the deferential entropy is only a function of its variance σ² (independent from the mean μ). This is logical in the sense that, according to the definition of the continuous entropy, it is equal to the expectation value of the log (PDF). For white noise we note that the expectation value of the mean is zero as it fluctuates around zero (see Figure 2.3), but the variance is non-vanishing (see Figure 2.3). Note that the result is also very similar if the noise is lognormally distributed. For a system that is degrading by becoming noisier over time, the entropy damage can be measured in a number of ways where the change in the entropy at two different times t₂ and t₁ is then [2]

(2.36)

2.5.2 Example 2.7: System Vibration Damage Observed with Noise Analysis

Prior to an engine system being subjected to a harsh environment, we make an initial measurement M₁⁺ of the engine vibration (fluctuation) profile which appears to be generating a white noise spectrum in the time domain as is shown in Figure 2.3.

Then the system is subjected to an unknown harsh environment. We again return the system to the lab and make a second measurement M₂⁺ in the exact same way that M₁ was made. It is observed that the engine is again generating white noise. The noise measurement spectrum results are

M ₁: Engine exhibits a constant power spectral density (PSD) characteristic of 3G_rms content in the bandwidth from 10 to 500 Hz.

M ₂: Engine exhibits a constant PSD characteristic of 5G_rms content in the bandwidth from 10 to 500 Hz.

(Note that standard deviation = G_rms for white noise with a mean of zero.) The system noise damage ratio is then:

(2.37)

Note this is the damage aging ratio (see Example 2.5) compared to the damage entropy change given by Equation (2.36). This level of entropy damage ratio indicates a likely issue. It would of course be prudent to have more measurements of other similar engines to assess what the statistical true outliers are.

Interestingly enough, noise engineers are quite used to measuring noise with the variance statistic. That is, one of the most common measurements of noise is called the Allan Variance [5]. This is a very popular way of measuring noise and is in fact very similar to the Gaussian Variance, given by:

(2.38)

where N ≥ 100 generally and τ = sample time. By comparison, the true variance is

(2.39)

We note the Allan Variance commonly used to measure noise is a continuous pair measurement of the population of noise values, while the True Variance is non pair measurement over the entire population. The Allan Variance is often used because it is a general measure of noise and is not necessarily restricted to Gaussian-type noise.

The key results here are that entropy of aging for system noise is dependent on the variance which is also a historical way of measuring noise and is likely a good indicator of the entropy damage of many simple and complex systems.

There are a number of historical options of how noise can be best measured. The author has recently [6, 7] proposed an emerging technology for reliability detection using noise analysis for an operating system. Concepts will be discussed in Section 2.9.

2.6.1 Stationary versus Non-Stationary Entropy Process

Measured data can often be what is termed non-stationary, which indicates that the process varies in time, say with its mean and/or variances that change over time. Although a non-stationary behavior can have trends, these are harder to predict than a stationary process which does not vary in time.

Therefore, non-stationary data are typically unpredictable and hard to model for autocorrelated analysis. In order to receive consistent results, the non-stationary data need to be transformed into stationary data. In contrast to the non-stationary process that has a variable variance and a mean that does not remain near, or return to, a long-run mean over time, the stationary process reverts around a constant long-term mean and has a constant variance independent of time.

Although aging processes are non-stationary, they may vary slowly enough so that in the typical measurement time frame they are stationary. Furthermore, disorder is cumulative which means there is a memory in the process. This means there is a likely trend. The issue is: this is a new way of looking at aging. There are, to this author’s knowledge, few studies on the subject. We describe below one good study on aging of the human heart to help the reader understand the potential importance of noise autocorrelation measurements.

We summarize below what we would anticipate from such a study.

1. Entropy randomness mesoscopic measurements.
2. If a system has a thermodynamic process that generates energy (battery, engine, electric motor,…) or has energy passing through the system (transistor, resistor, capacitor, radio, amplifier,…), the subtle irreversibilities occurring in the system are likely generating noise in the operating system.
3. Non-damage flow exchange that is part of the exchange process related to entropy damage exhibits randomness that we term noise.
4. For such subtle system-level damage, we employ the method that we term here as mesoscopic entropy noise measurement analysis to detect subtle changes in the system due to entropy damage. Such damage is not easily detectable by other means.
5. This randomness, if measured by one or more thermodynamic state variables, is likely observable from the entropy exchange process with the aid of the autocorrelation function of the state variable.
6. Over time, as the system ages, we suspect that the increase in disorder in the system will tend to reduce the autocorrelation result so that the normalized value of R_S,S(τ) will approach 0 when the system is in a state of maximum entropy (disorder).
7. The aging occurring is likely on a longer time scale than the measurement observation time so that the measurement process can be treated as stationary.

2.8.1 Noise Measurements Rules of Thumb for the PSD and R

Since people are not familiar with noise measurements, we provide some simple rules of thumb to remember.

• As the absolute value of the autocorrelation function decreases to zero, |R_norm| → 0, noise levels are increasing in the system.
• As the PSD( f ) levels increase, noise is increasing in the system.
• The slope of the autocorrelation function is related to the slope of the PSD spectrum. As the autocorrelation slope increases, the PSD slope decreases. The slope is an indication of the type of system-level noise characteristic detailed in the next section.

2.8.2 Literature Review of Traditional Noise Measurement

There are countless noise measurements in the literature [11]. Here we would like to review well-known noise measurements as we feel it will be helpful in eventually working with degradation problems.

We believe that noise analysis can be crucial in prognostics as we suspect that noise analysis will be a highly important contributing tool. The problem is unfortunately that, at the time of writing of this book, there seems to be little work in the area of noise degradation analysis similar to the human heart study that we have illustrated (i.e., looking at noise as a system ages).

It is clear from our example on the human heart and as we have theoretically shown in Section 2.5 that the noise variance increases with system damage as it is a dependent variable of entropy. This is an indication that the autocorrelation function for system noise is becoming less and less correlated over time for non-stationary processes. As such, we anticipate that the frequency spectrum might have a tendency to higher levels of noise values.

Put another way, as the autocorrelation function becomes less correlated, the PSD spectrum increases. The slope of the PSD curve is an indication of the type of system-level noise observed. Note the slopes of the autocorrelation function and the PSD curve are related. We can clearly see that as the autocorrelation slope decreases (from Figures 2.8b, 2.9b, and 2.10b), the slope of the PSD curve increases (from Figures 2.8c, 2.9c, and 2.10c). Note in Figure 2.8b the slope is hard to see as it decreases quickly near zero time lag. It is possible that the slope may also change in a non-stationary process; we at least expect the noise level to increase. The problem is, as we have mentioned, there is little work in this area to date for non-stationary noise processes.

Noise is sometimes characterized with color [11]; the three main types in this category are white, pink, and brown noise. Each has a specific type of broadband PSD characteristic. We will first overview these then we will describe some reliability-related type of noise measurements. Each type has an interesting sound that is audible in the human audio frequency range (~20 Hz to 20 kHz) of the spectrum.

Figure 2.8 is a well-known time series that we previously discussed for white noise; see Section 2.5.1 and Figure 2.3. In Figure 2.8b we see the autocorrelation function that starts at 1 (τ = 0) and quickly diminishes, becoming uncorrelated. The PSD spectrum appears to be flat (constant) over the frequency region. However, the variance is non-vanishing. Much work has been done in the area of thermal noise, also known as Johnson noise or Johnson–Nyquist noise, which has a white noise spectrum (see Section 2.9.1).

Figure 2.9 is a well-known phenomenon called 1/f noise, also referred to as pink noise or flicker noise. While many measurements have been carried out and many theories have been proposed, no one theory uniquely describes the physics behind 1/f noise. It is generally not well understood, but is considered by many to be a non-stationary process [11, 12]. From the time series, the autocorrelation function has more correlation over the lag time τ compared to white noise. The fact that it is called 1/f has to do with the slope of the PSD line as shown in Figure 2.9c. In the PSD plot, we see that there is more “power” at low frequencies and appears to diverge at 0.

If we compare figures again, we see that a rapid variation in the time series such as White noise in Figure 2.8 is harder to correlate than a slower variation such as Figure 2.10. In terms of the PSD spectrum, we carefully plotted 1/f and 1/f² in Figures 2.9 and 2.10c so one can see the relative slope change on the same scale. This is also compared in Figure 2.11. The slope yielding 1/f has been measured over longer periods of lag of just 1 s. There are reports of lags say over 3 weeks with a frequency measured down to 10^−6.3 Hz, and the slope continuous to diverge as 1/f with no change in shape [13]. Using geological techniques, the slope of 1/f has been computed down to 10⁻¹⁰ Hz or 1 cycle in 300 years [14].

The author anticipates that, if the measurement was taken on semiconductors that had been subjected to parametric aging, the following would hold.

A system undergoing parametric degradation will have its noise level increase. Studies suggest (see Section 2.8.3) for example that

where t is time, subscript V is a system-level state parameter (such as voltage; see Table 1.1), and γ is a power related to the aging mechanism. The system noise would become more uncorrelated yielding a decrease in R for the autocorrelation function and a higher noise level in the PSD spectrum (see Section 2.8.3).

This of course is a statement that the time series changes in aging and, at maximum entropy where failure occurs, we would observe the maximum noise characteristic. There are a number of reasons for this. One key reason is that the material degrades and in effect its geometry is changing due to increasing disorder. For example, a resistor will effectively have less area for current to pass through as the material degrades. In effect the resistor is becoming smaller. We will see in Section 2.8.3 when we look at resistor 1/f type noise characteristics published in the literature that resistors with less area display larger noise levels. Resistors with smaller area handling the same current will have current crowding, which increases noise.

The interesting issue with 1/f noise [11, 12] is that it has been observed in many systems:

• the voltage of currents of vacuum tubes; Zener diodes; and transistors;
• the resistance of carbon resistors; thick-film resistors; carbon microphones; semiconductors; metallic thin-films; and aqueous ionic solutions;
• the frequency of quartz crystal oscillators; average seasonal temperature;
• annual amount of rainfall;
• rate of traffic flow;
• rain drops;
• blood flow;
• and many other phenomena.

Figure 2.10 is another well-known phenomenon called 1/f ² noise, also referred to as brown (sometimes red) noise, and is a kind of signal-to-noise issue related to Brownian motion. This is sometimes referred to as random walk which is displayed in Figure 2.10a. Here again the slope of the PSD spectrum goes as 1/f ² (hence the name).

Figure 2.11 compares noise-type slopes on the same graph scale so one can see the relative slopes for a PSD plot.

2.8.3 Literature Review for Resistor Noise

We have mentioned in Section 1.7 that entropy can be produced from resistance. Resistance is a type of friction which increase temperature, creates heat, and often entropy damage. In the absence of friction, the change in entropy is zero. Therefore, it makes sense in our discussion on noise as an entropy indicator that we look at noise in electrical resistors. One excellent study is that of Barry and Errede [15]. These authors found for 1/f noise in carbon resistors that noise measurements on 0.5, 1, and 2 W resistors indicated that the 0.5 W resistors had the largest noise and the 2 W resistors had the least noise. Figure 2.12 illustrates the PSD noise level for carbon resistors as well as the 1/f noise type slopes for these resistors.

We see that resistors with reduced volume or cross-sectional area produce more noise. Other authors have found that 1/f noise intensity for thick-film resistors varies directly with resistance [16, 17]. This is a similar observation as resistor wattage goes as its volume. We can interpret this as an increase in resistance produces higher levels of entropy, increasing disorder, and increasing system-level noise. From an electrical point of view, smaller resistors can produce higher current densities increasing noise. The fact that the noise characteristics are 1/f-type (pink noise) is related to the type of noise phenomenon in the system.

2.9.1 System Noise Temperature

It is not surprising that noise and temperature are related. For example, thermal noise also known as Johnson–Nyquist noise has a white noise spectrum. The noise spectral density in resistors goes directly with temperature and resistance according to the Nyquist theorem, defined as

where R is resistance, T is the temperature, and k_B is Boltzmann’s constant [11]. Simply put, as temperature and/or resistance increases, so too will the noise level observed in resistors. Thermal noise is flat in frequency as there is no frequency dependence in the noise spectral density; therefore it is a true example of white noise.

2.9.2 Environmental Noise Due to Pollution

We have asserted that temperature, noise, and failure rate are some good key system thermodynamic state variables. However, depending on how you define the system and the environment, they can also be state variables for the environment. An internal combustion (IC) engine is a system which interacts with the environment causing entropy damage to the environment. This pollution damage and that of other systems, on a global scale, are unfortunately measurable as meteorologists track global climate change. The temperature (global warming) effect is the most widely tracked parameter. However, it might also make sense from what we determined above to look at environmental noise degradation. As our environment ages, we might expect from our above finding that its variance will increase. Some indications of environmental variance change are larger swings in wind, rain, and temperature, causing more frequent and intense violent storms. It might be prudent to focus on such environmental noise issues and not just track global temperature rise [2].

2.9.3 Measuring System Entropy Damage using Failure Rate

So far we have two state variables for measuring aging. Another possible state variable, somewhat atypical but often used by engineers, is the reliability metric called the failure rate λ. We do not think of the failure rate as a thermodynamic state variable. In fact, to any reliability engineer, it is second nature that the failure rate is indeed a key degradation unit of measure that helps to characterize a system in terms of its potential for failure over time. For consistency, we would like to put it in the context of entropy damage to see what we find. For complex systems, the most common distribution used by reliability engineers comes from the exponential probability density function:

(2.56)

The differential entropy can be found easily (see Equation (2.33)) from the negative of the expectation of the natural logarithm of f(x), defined [9]:

(2.57)

where for the exponential distribution the mean E[x] = 1/λ. A way to view this is that the exponential distribution maximizes the differential entropy over all distributions with a given mean and supported on the positive half line (0,∞). This is not a very exciting result, but does provide some physical insight from an entropy point of view on why it is one of the more popular distribution used in reliability for complex systems. So in general, we turned the problem around to the fact that any PDF will maximize the entropy subjected to certain constraints, in this case E[x] = 1/λ. For a normal distribution it is subjected to a mean and sigma over the interval (−∞,∞). In the case of white noise, the results proved helpful in our noise analysis (see Section 2.5).

For the exponential distribution, we note that if the environmental stress changes we can measure the entropy change. In this case, the entropy damage change would be for this system under two different environments:

(2.58)

where the subscripts E1 and E2 are for environments 1 and 2.

2.10.1 Example 2.9: Thermal Equilibrium

Consider a system that can exchange energy with its environment subject to the constraint of overall energy conservation. The total conserved energy is written

(2.63)

where we have dropped the “system” subscript. Under a quasistatic exchange of energy we have

Then, from Equation (2.60) we have

(2.64)

Let’s apply a physical situation to this equation; we can use the conjugate pressure P and volume V thermodynamic work function PdV for YdX. Figure 2.14 displays this example. In the figure we imagine a perfect insulating cylinder so that heat does not escape. It is divided into two sections by a piston which can move without friction. The piston is also a heat conductor. We take the left side to represent the environment and the right side that of the system.

Since and, when the volume changes incrementally, we will also have , we can write

(2.65)

and

(2.66)

In order to ensure that the total entropy goes to a maximum, we have positive increments under the exchange of energy in which the system energy change is dU. If the environmental temperature is more than the system temperature, that is, , then this dictates that and energy flows out of the environment into the system. If the environmental temperature is less than the system temperature, that is, , then this dictates that . The energy then flows out of the system into the environment from the higher-temperature region to the lower-temperature region. A similar argument can be made for the pressure; if P > P_env then dV > 0 and the piston is pushed to the left. If the entropy is at the maximum value, then the first order differential vanishes to yield

(2.67)

At equilibrium the energy exchange will come to a halt when the environmental temperature is equal to the system temperature, that is, T = T_env, and P = P_env. This is in accordance with the maximum entropy principle, which occurs when the system and the environment are in equilibrium. This of course is an irreversible process; in thermodynamic terms there is a tendency to maximize the entropy and the pressure and temperature will not seek to go back to the non-equilibrium states. In terms of aging, we might have a transistor that is being cooled by a cold reservoir. Pressure may be related to some sort of mechanical stress. If the transistor fails due to the mechanical stress, the transistor comes to equilibrium with the environment as the pressure is released and the pressure and temperature of the system and environment are equal.

2.10.2 Example 2.10: Equilibrium with Charge Exchange

Shown in Figure 2.15 is a system (capacitor) in contact with an environment (battery).

A (possibly) non-linear capacitor is connected to a battery. The capacitor and the battery can exchange charge and energy subject to conservation of total energy and total charge. Then with these constraints we wish to maximize

(2.68)

with the total energy and charge

(2.69)

and

(2.70)

obeying

(2.71)

and

(2.72)

The entropy for the capacitor and battery requires

(2.73)

Then we can write from the combined first and second law equation

(2.74)

or, equivalently,

(2.75)

and

(2.76)

As before when T_env > T, then dU > 0 energy flows out of the environment into the capacitor system. For E > V, dq > 0 and charge is flowing out of E (battery) into V (capacitor system). For equilibrium when the entropy is maximum, T = T_env and V = E, so that dS_total = 0. The battery has in a sense degraded its energy in charging the capacitor so that, in the absence of capacitor leakage, no more useful work will occur. Again, the entropy is maximized in the irreversible process. There is no tendency for the charge to flow back into the battery.

2.10.3 Example 2.11: Diffusion Equilibrium

In Chapter 8 for spontaneous diffusion processes we make an analogy to Example 2.10. We find for equilibrium of diffusion into an environment that when the entropy is maximum T = T_env and μ = μ_env. Here T, T_env, μ, and μ_env are system and environmental temperatures and chemical potentials, respectively, prior to diffusion. As well the pressure (P = nRT/V) is the same in system and environmental regions at equilibrium where the system and environment are shown in Figure 8.1. Therefore, atoms or molecules migrate so as to remove differences in chemical potential. Diffusion ceases at equilibrium, when μ = μ_env. Diffusion occurs from an area of high chemical potential to low chemical potential. The reader may be interested to work the analogy and compare results to that found in Section 8.1.

2.10.4 Example 2.12: Available Work

Consider a system in a state with initial energy U and entropy S which is not in equilibrium with its neighboring environment. We wish to find the maximum amount of useful work that can be obtained from the system if it can react in any possible way with the environment [4]. Let’s say the system wants to expand to come to equilibrium with the environment and we have a shaft on it that is doing useful work (see Figure 2.16).

From the first law we have

(2.77)

The entropy production is

(2.78)

where dS_damage is the increase in entropy damage and δQ/T₀ is the entropy outflow. Combining these two equations then the useful shaft work is

(2.79)

Integrating from the initial to final state we have for the actual work

(2.80)

The lost work or the irreversible work associated with the process is

(2.81)

From the second law S_gen > 0 so that the useful work

(2.82)

S _gen is positive from Equation (2.80) or zero, so that

(2.83)

is the available work (free energy). Thermodynamics sometimes refers to terms such as exergy or anergy (for an isothermal process) or maximum available work when the final state of the system is reduced to full equilibrium with its environment [4]. In reliability terms this can be a catastrophic failed state.

Consider the situation where W₁₂ has as state 2 the lowest possible energy state so that it is in equilibrium with the environment. This is the maximum availability that gives the maximum useful work. We then take the derivative of this availability A, so with

(2.84)

which gives

(2.85)

Solving, we get the thermodynamic definitions of temperature and pressure:

(2.86)

We see that these conditions are satisfied when the system has the same temperature and pressure as the environment at the final state. Thus the largest maximum work is possible when the system ends up in equilibrium with the environment. It is also the maximum possible entropy damage S_D when the final state of the system has catastrophically failed. Then it is impossible to extract further energy as useful work.

In general, one system can provide more work than another similar system. We can assess the irreversible and actual work as

(2.87)

The efficiency (η) and inefficiency (1–η) of work-producing components can be assessed as

(2.88)

2.11.1 The Helmholtz Free Energy

The Helmholtz free energy is the capacity to do mechanical work (useful work). The maximum work is for a reversible process. This is similar to potential energy having the capacity to do work. To develop this potential, we use the chain rule for d(TS) and for PdV work we have:

(2.92)

We insert this expression into the internal energy and arrange the equation as

(2.93)

where we identify the function F = U − TS, known as the Helmholtz free energy. This type of transform is called a Legendre transform; it is a method to change the natural variables. The full expression with particle exchange is

(2.94)

We see the Helmholtz free energy is a function of the independent variables T,V, denoted .

The change in F (ΔF) is also useful in determining the direction of spontaneous change and evaluating the maximum work that can be obtained in a thermodynamic process. To see this we note that for the first law the total work done by a system with heat added to it also includes a decrease in the internal energy dU = δQ – δW and since δQ < TdS (with the inequality for reversible process) then

(2.95)

or

(2.96)

Therefore, when we have

(2.97)

For an isothermal process (dT = 0), the Helmholtz free energy bounds the maximum useful work that can be performed by a system. As the system degrades, the change in the work capability or lost potential work is due to irreversibilities building up in the system; this change in irreversible work resulting from system degradation causes entropy damage that builds up (or is “stored”) in the system. We note that W can be any kind of work; it can include electrical ore mechanical (pressure–volume) work including stress strain, that is, not just gases.

The equal sign occurs for a quasistatic reversible process, in which case:

(2.98)

Spontaneous degradation: if we had δW = PdV type of work and then decided to hold the volume constant, so that δW = PdV, Equation (2.68) becomes

(2.99)

where

We note that when the system spontaneously degrades (losing its ability to perform useful work), the process requires . Note this only requires that the change in the Helmholtz free energy has the same temperature and volume in the initial and final states; along the way the temperature and/or volume may go up or down. This does not necessarily account for the total entropy damage S_damage of the system.

We can replace the inequality in Equation (2.97) by adding an irreversible term. We find that when we talk about irreversible processes in Section 2.11.5, for dT = 0 the actual work change is

(2.100)

This is an important result as it connects the concept of work, free energy, and entropy damage.

2.11.2 The Enthalpy Energy State

Enthalpy is the capacity to do non-mechanical work plus the capacity to release heat. It is often used in chemical processes. To develop enthalpy, we write PdV using the chain rule

(2.101)

Then we substitute this into the internal energy and manipulate it as

(2.102)

H is called the enthalpy and we can write it as . The full expression for the enthalpy with particle exchange is

(2.103)

We see that enthalpy is a function of the natural variables (S, P), that is, . At constant pressure (isobaric) and without particle exchange ; then . This says the enthalpy tells one how much heat is absorbed or released by a system. If for example we have an exothermic reaction where heat flows out of the system to the surroundings, the entropy of the surroundings increases so . It is common knowledge that a chemical reaction will be spontaneous when the Gibbs free energy (described in the following section) is negative. We mention this because the Gibbs free energy G can be given in terms of the enthalpy which helps explain ΔH. So from this relation, we note that if ΔH is negative and the entropy is increasing than and the reaction will proceed.

2.11.3 The Gibbs Free Energy

The Gibbs free energy is the capacity to do non-mechanical work. The maximum work is for a reversible process. This is similar to potential energy having the capacity to do work. To develop the Gibbs free energy we again use the chain rule for d(PV). This time we start by inserting it into the Helmholtz free energy for convenience, obtaining

(2.104)

which can be arranged as

(2.105)

Here we identify the function , which is called the Gibbs free energy. We see the Gibbs free energy is a function of the independent variables . Some other relations of interest include or writing it in terms of the enthalpy .

We have that , writing , where other work refers to non P–V mechanical work such as electrical work. Then at constant temperature and pressure

(2.106)

But by definition, dH = dU + PdV so that

(2.107)

For an isothermal process at constant pressure, the Gibbs free energy bounds the maximum amount of useful work that can be performed by a system, that is:

(2.108)

As the system degrades, the change in the work capability or lost potential work is due to irreversibilities building up in the system; this change in irreversible work resulting from system degradation causes entropy damage that is built-up (or “stored”) in the system.

In a similar manner to Equation (2.100), we can replace the inequality above by adding the irreversible term

(2.109)

If a system is in contact with a heat reservoir meaning constant temperature, and if the pressure is constant, then the change in the Gibbs free energy is

(2.110)

The change in the entropy of the system and the reservoir is

(2.111)

where Q = −Q_R. Since ΔS must increase in any process then the Gibbs free energy change must decrease having a negative change in value, ΔG ≤ 0, when the temperature and pressure are constant in a degradation process.

The Gibbs free energy is a popular function to use when there is particle exchange with the surroundings. In this case, the internal energy is also a function of N so U = U(S,V,N).

From Table 1.1, inserting chemical work, we can write

(2.112)

From the above Legendre transform, following the same methods (Equation (2.92)),

(2.113)

Most commonly one considers reactions at constant P and T, so the Gibbs free energy is the most useful potential in studies of chemical reactions as it simplifies to

(2.114)

2.11.4 Summary of Common Thermodynamic State Energies

From these definitions we can say that ΔU is the energy added to the system, ΔF is the total work done on it, ΔG is the non-mechanical work done on it, and ΔH is the sum of non-mechanical work done on the system and the heat given to it. When we are concerned with the conjugate variables (S, T) and (P, V), we note there are only four possible pairs of independent variables. The only time that we would need to change variables is when we use other types of thermodynamic work besides (P, V).

Table 2.2 summarizes the four types of energies.

Name	Formula	Properties	Derivatives
Internal energy U(S, V)		First law: ΔU = Q – W
Helmholtz free energy F(T, V)		ΔF ≤ 0; T, V constant
Enthalpy H(S, P)		Isobaric process: ΔH = Q
Gibbs free energy G(T, P)	Also of interest:	ΔG ≤ 0; T, P constant when G = G(T, P, N) and

2.11.5 Example 2.13: Work, Entropy Damage, and Free Energy Change

Let’s consider an irreversible process to illustrate some of the ideas in this section with respect to system entropy damage, free energy, and internal energy.

We wish to see how much work can be extracted from a system even during an irreversible process. One way to think of this is to take in heat and see how much work the system can provide. This is illustrated in Figure 2.17. By conservation of energy from the first law for Figure 2.17 we have

(2.115)

Associated with irreversible process will be damage that is occurring in the system (or device). Some of the heat goes into damage (say due to friction) and some into work. Keeping tabs on the system entropy we have

(2.116)

Solving for δQ we have

(2.117)

and by rearranging terms in Equation (2.115) and inserting the above we obtain

(2.118)

For the moment let’s consider a reversible process so that dS_damage = 0. This gives us the maximum work for a reversible process

(2.119)

If dT = 0 for an isothermal process we can write this as

(2.120)

Therefore, the change in the Helmholtz free energy is defined as the maximum amount of useful work that can be obtained from a system. The work found from the integral going from a system state 1 to 2 in a reversible way is

(2.121)

Now what about the irreversible process? We need to somehow account for what happens to the system’s free energy. When in state 1 we know that its free energy is F₁. In that state it had a certain capacity to do work which was bounded by the free energy F₁, so the work capability was .

The path-dependent work causes resistance or friction and the actual work in going to state 2 was really

(2.122)

So now state 2 has a different free energy then state 1 due to the entropy damage. Note that we have dropped the minus sign for the work as it is understood we seek the useful work that a system does.

Now if we continued to do work to a final equilibrium state F_N in an incremental way we would get to the minimum value of the free energy F_min, but over the same path we would have

(2.123)

We can visualize this incremental process with the aid of the Figure 2.18 for a system that has undergone an irreversible process and incurred entropy damage between state 1 and state 2. There is no way to reverse the entropy damage. According to the second law, dS > 0 for the irreversible process and entropy increase means the system disorder has increased. If state 2 happens to be its lowest energy state then no further work can occur; the system has likely failed catastrophically so entropy damage stops, , and the change in the free energy goes to zero .

When we talk about dT = 0 it is more appropriate to say that only the initial and final states need to have the same temperature. In between they can fluctuate. We can write the incremental change in the work as

(2.124)

or

(2.125)

This is consistent with Equation (2.87). The efficiency of the work process is then

(2.126)

At this point we can start modeling the actual work and the irreversible work. For example we might write

(2.127)

Since the reversible work (the maximum work) can be treated as fixed, the rate of actual work is some function of the rate of the irreversible work over the work path

(2.128)

These last equations seem harmless enough but they are important results. These work functions are independent of whether we talk about a chemical process where we are using the Gibbs free energy, an electric, or a mechanical process using the Helmholtz free energy. If you are a pragmatic person, the last few equations give you a great bottom line for thermodynamics because no matter how complicated thermodynamics is, a big part of it is undeniably all about work.

For this problem the general expression for power is:

(2.129)

Now the work left that is available in state 2 can be more time related than power loss related. That is, the system has lost some strength from the irreversible process U₁₂; however, it may very well be able to keep on performing its intended function until its free energy is all used up. We provided an example in Section 2.1 on Miner’s rule (see also Chapter 4) that is commonly used to interpret the work left in terms of cycles of remaining (time) and not power or strength loss.

We also note that the entropy damage is path dependent on how the work was performed. As we mentioned in Chapter 1 we are staying away from using too many confusing symbols but, in reality, we should be mindful about the path dependence related to the work path for which we should be writing dS_damage and dS_non-damage as δS_damage and δS_non-damage.

2.11.6 Example 2.14: System in Contact with a Reservoir

Consider a special case of a system that is in contact with a heat reservoir, as shown in Figure 2.19. If there is no mechanical work PdV = 0 with constant volume ΔU = ΔQ. Then

(2.130)

Then the total entropy change is

(2.131)

where .

Note that the total entropy so we can also write

(2.132)

The Helmholtz free energy never increases in a spontaneous process with fixed temperature and volume in a system in contact with a heat reservoir. We see this causes the entropy to increase since . To emphasize this, as the system’s entropy increases, its free energy will have a tendency to decrease if it can. Figure 2.20 shows the possible spontaneous decay of a system’s free energy over time to its final descent to its equilibrium state and the corresponding total entropy increase.

We note that we have sketched the free energy which decays in a non-linear way. This is not atypical of systems.

2.11.6.1 Full Expression for Potential with Entropy Damage

General work-related processes and exchanges with the system’s internal energy require different relationships. For example, consider the full assessment without any conditions on the Helmholtz free energy and temperature where particles can flow in or out of the system; then the general change in the work is

(2.133)

where N_i is the number of particles or moles in the system of species i. Note in Section 2.11.1 we used conditions dT = dN = 0. One needs to account for sign changes depending on the particular conditions as well.

If we have a situation of constant temperature and pressure (dT = dP = 0) we will obtain the similar results in terms of the Gibbs free energy

(2.134)

The general assessment without any conditions on the Gibbs free energy is

(2.135)

If we have a situation of constant pressure and entropy flow (dT = dS = 0) we will obtain similar results in terms of the enthalpy:

(2.136)

The general assessment without any conditions on the enthalpy is

(2.137)

Lastly, the internal energy is

(2.138)