What is Entropy Generation Rate (EGR) and why is it important? When energy is transferred or used for an objective, new entropy is always created. This new entropy is called the entropy generation. The rate at which it is produced is called the entropy generation rate. Unlike energy, entropy is not a conserved quantity, so this newly produced entropy adds to the overall entropy of the universe. This entropy generation invariably results in a loss of efficiency in any process. The distribution of the new entropy depends on the process. In the picture below new entropy is produced because of the energy (heat) that flows down a temperature gradient.
The entropy generation rate is given the symbol.
It can only be zero or positive.
Entropy generation S gen rate for heat transfer between two temperatures connected by a metal rod. The direct power extracted, P max, is zero in this configuration.
Why is the 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 predicts how complex systems will behave (resilience) or self-organize in nature. The maximum entropy generation rate predicts S-shaped transformations over time and pattern formation under steady-state conditions.
Applications: Tropospheric weather, solidification (crystallization), transformation, chemical reactions, metal deformation, sintering, patterns in nature, bird organization, self-organization, heat pumps, energy efficiency calculations, or flow 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.
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 means z≥-1. For an exponential distribution (λ=1) (μ=1 and σ=1 and z=(x-1/λ)/(1/λ)) are 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 (something that also looks like a S-curve) is also shown below. The total (cumulative) entropy produced (generated) by a skewed distribution process is lower than from a normal distribution behaved process.
