What is Entropy? Entropy is a scientific thermodynamic measure (of the universe, system, or control volume) that can increase with time because it is mostly a consequence of rearranging energy and distributing it more uniformly. While it is most commonly described as a “measure of disorder or randomness,” engineers more precisely define it as a measure of how many different ways a system can rearrange itself without changing its overall energy or state. Entropy is given the symbol S, with units of joules per kelvin (J/K). Or the lowercase “s,” which is the entropy per unit volume or per mole.
What is Entropy Generation Rate (EGR) and why is it important? When energy is transferred, rearranged, or used for an objective, new entropy is always created. This new entropy production is called entropy generation. It arises from gradients of temperature, pressure, chemical potential, electrical charge, or surfaces. Whenever there is a gradient, new entropy is generated. The rate at which new entropy is generated (produced) is called the entropy generation rate (EGR). Unlike energy, entropy is not a conserved quantity, so this newly produced entropy adds to the overall entropy of the universe. More importantly, while dissipating the gradient, the entropy generation can produce beautiful patterns. This is the basis of observable patterns in a self-organizing system. Patterns have repeatable spacings and so often influence engineering properties, such as resilience. For example, grain patterns influence the measured engineering hardness in crystallized metallic solids. A lower grain diameter commonly indicates a higher hardness. A higher rate of entropy production yields a finer pattern.
This entropy generation invariably results in a loss of traditional thermodynamic efficiency (in extracting work, e.g., electrical energy). Still, it can be used in several creative ways for energy storage and other useful purposes (such as resilience to cracks and dislocations due to pattern spacing). Pattern formation is linked to the rate of entropy generation, especially in steady-state open systems. Weather selection and cloud-type formation are related to the rate of tropospheric entropy generation. This steady-state entropy generation rate can also be applied to bird f
light formations. The distributions of newly produced entropy and the overall energy depend on the process. In the picture below, new entropy is produced by the energy (heat) that flows down a temperature gradient in a metal rod held between two reservoirs at temperatures T1 and T0. This is a simple example that shows the entropy generation rate is maximized for several processes- see the article. Source (link).
Are you wondering what patterns are related to solid-state heat transfer? In metals, or at least when solids are near their melting points, heat transfer can cause noticeable atomic diffusion. Atoms migrate from hotter (higher-stress) regions to cooler regions along the boundaries between crystalline grains, which themselves respond to thermal conditions. This bulk movement of atoms is a microscopic analog of fluid convection. In some solids, electron motion and related patterns can be driven by thermal gradients, and vice versa.
The entropy generation rate is given the symbol.
which can only be zero or positive. The units are joules per kelvin per second (J/K · s).
Entropy generation rate (Sgen rate) by heat transfer between two temperatures connected by a metal rod of length L. The direct power (rate of work) extracted, Pmax (J/s), is zero in this configuration.
Why is the concept of Maximum Entropy Generation Rate (MEPR) important? The Maximum Entropy Generation Rate (MEPR) principle – also sometimes referred to as the Maximum Entropy Production Principle (MEPP) – is important because it likely predicts how complex systems will behave (resilience) or self-organize in nature. The maximum entropy generation rate appears to predict S-shaped transformations over time. At equilibrium, gradients are dissipated.
Applications: Tropospheric weather development, solidification patterns (crystallization), transformation kinetics, chemical reactions, metal deformation, sintering, surface patterns in nature, bird organization, self-organization of chemical and sociological groups, heat pumps, energy efficiency calculations, or flow-flow resistance of a fluid in a pipe, i.e., wherever it takes a finite amount of time to complete a process. There is a quantum mechanical analogy to entropy generation that is not discussed here.
Keeping the entropy generation rate maximized seems to be a natural outcome of spontaneous processes. Most high-efficiency applications preserve energy quality.
A Statistical Basis. Although an S-curve based on a symmetrical (normal) distribution of rate can be used to represent a self-organizing process, one should be aware that skewness in a distribution can also yield an S-curve-like transformation (albeit with a tail), as shown.
By definition, a normal distribution has skewness 0. A skewness of 2 corresponds to an exponential distribution. Normalized for the standard normal distribution with mean μ, std. dev. σ, and z=(x-μ)/σ) are shown below.
PDF(z) (Normal Distribution) = e^(-z^2/2)
CDF(z) (Normal Distribution) = 1/2 [1+erf (z/√2)]. This is a classic S-Curve
For an Exponential (λ=1) (so μ=1and σ=1). The domain is x ≥ 0, which implies z ≥ -1. For an exponential distribution (λ=1) (μ=1 and σ=1 and z=(x-1/λ)/(1/λ)) as shown below.
PDF(x) (Exponential Distribution) = λe^(-λx), x≥0
CDF(x) (Exponential Distribution) = 1-e^(-λx), x≥0
(λe^(-λx))/[(1-e^(-λx))^2] = (λe^(-(z+1)))/(1-e^(-(z+1)) )^2 for z≥-1
The PDF (e.g., rate of change of temperature vs. time) and CDF (S-shaped curves, e.g., for temperature) for each distribution are shown for the normal and exponential distributions. The PDF/(CDF)^2 approximates the entropy generation rate. The cumulative integral (which also looks like an S-curve) is shown below. The total (cumulative) entropy produced (generated) by a skewed distribution process is lower than from a normal distribution-behaved process.
One of our AI friends, after reading the source article above and the key references, opined that the real question is: What kind of order best dissipates the available gradients produced by entropy generation? The real mystery may not be why entropy increases. The deeper question may be why entropy generation so often produces beautiful, persistent, and resilient structures (patterns) and connectivity along the way.
