Temperature as Quality Measure of Energy.

In ideal gas dynamics temperature appears as an intensive variable $T$ connected to internal energy $e$ and density $\rho$ by 

• $T=\frac{e}{\rho}$                          

with a corresponding pressure law 

• $p=\gamma e$

where $\gamma$ is a gas constant. Internal energy is viewed as small scale kinetic energy from small scale molecular motion. Internal energy can transformed into mechanical work in expansion, which without external forcing (or gravitation) is an irreversible process.  

For a solid body viewed as a vibrating atomic lattice temperature scales with total internal energy as the sum of small scale kinetic energy and potential energy, which can be transferred by radiation and conduction to a body of lower temperature.   

In both cases temperature appears as a quality measure of internal energy as an intensive variable. 

The maximal efficiency of a Carnot heat engine transforming heat energy into work operating between two temperatures $T_{hot}>T_{cold}$ is equal to $1-\frac{T_{cold}}{T_{hot}}$. 

Radiative heat transfer form a hot body of temperature $T_{hot}$ to a cold body of temperature $T_{cold}$, scales with $(T_{hot}^4-T_{cold}^4)$ according to Stephan-Boltzmann-Planck. 

Conductive heat transfer scales with $(T_{hot}-T_{cold})$ according to Fourier.

In both cases the heat transfer from hot to cold can be seen as transformation from high quality energy into low quality energy in an irreversible process in conformity with the 2nd Law of Thermodynamics. 

The Nobel Prize in Physics in 2008 was awarded to experimental detection of Cosmic Microwave Background CMB radiation with perfect Planck spectrum as an after-glow of a Bing Bang with temperature of  2.725 K and corresponding very low quality energy.  

With radiation scaling with $T^4$ the difference between 300 K as global temperature and 3 K as deep space CMB comes out with a factor of $10^{-8}$. The contribution to global warming from CMB thus appears to be very small. 

We see from $e=\rho T$ that low density and low temperature both connect to low energy quality making both wind and solar energy inefficient compared to fossil and nuclear energy.    

How to Generate Heat Energy

We recall from a previous post:

• Heat energy can be generated from large scale kinetic energy by compression. 
• Kinetic energy can be generated from heat energy by expansion.

More precisely, we saw in the previous saw that heat energy at high temperature can generate useful mechanical work. Heat energy at high temperature can be created by nuclear/chemical reactions. 

Heat energy typically at lower temperatures also appears as losses from electrical currents subject to resistance, fluid motion subject to turbulent/laminar viscosity and friction between solid bodies. These losses appear as substantial, unavoidable and irreversible as expressions of a 2nd Law.

We have seen that heat energy $\sim T\nu^2$ of frequency $\nu$ carried by an atomic lattice of temperature $T$ subject to high-frequency cut-off $\nu <\frac{T}{h}$ expressing ordered synchronised atomic oscillation or kinetic motion, can be radiated. Here $h$ is a constant. 

We can view turbulent dissipation in fluid flow as a form of high-frequency forcing above present cut-off which cannot be reradiated and so is absorbed as internal energy in the form of unordered small scale kinetic energy. We can similarly view viscosity and friction as forms of high-frequency forcing supplying internal energy. 

The contribution to internal energy increases the temperature and so allows unordered small scale motion to be synchronised to higher frequency and then radiated.  

The key is thus that turbulent, viscous and frictional dissipation all represent high-frequency forcing above   present cut-off, which cannot be represented and reradiated and so shows up as internal energy as small scale kinetic energy. 

Rubbing hands is one way to transform large scale kinetic motion into small scale kinetic motion as heat energy. The brakes on your car work the same way. 

Finite Precision Computation/Physics and Heat Energy

Euler CFD as a parameter free Theory of Everything ToE for slightly viscous incompressible fluid flow is a prime example of the idea of finite precision digital computation capable of simulating physics as a form of finite precision analog computation. Euler CFD captures turbulent flow from a principle of best possible digital solution of Euler's equations in a situation where exact (laminar) solutions are all unstable without permanence over time and so are unphysical and cannot be observed. 

The essence of turbulent flow captured by Euler CFD is the production of heat energy in turbulent dissipation from residual stabilisation in a situation where Euler residuals can be made small only in a weak mean value sense, but blow up in a strong pointwise sense. The dissipative mechanism thus expresses an impossibility to computationally resolve the flow because of finite precision, which in physical terms means production of heat energy as small scale unordered motion. 

Radiative heat transfer also involves an aspect of finite precision in the sense that a body viewed as a set of oscillators is capable of radiating only frequencies below a certain cut-off frequency scaling with temperature because synchronisation of the oscillators necessary for radiation is in finite precision impossible above cut-off. 

There is connection between turbulent flow and radiative heat transfer in that the heat energy generated in turbulent dissipation ultimately is released in radiation, and so gives a meaning to heat energy as unordered motion as unsynchronised oscillatory motion.  

Recall from the blog post 2nd Coming of the 2nd Law that finite precision computation/physics explains why certain processes are irreversible as processes where large scale kinetic energy/ordered motion is transformed into small scale kinetic energy/disordered motion, which cannot be reversed because the precision required to restore large scale order from small scale disorder is not there. This is like restoring all your manuscripts after a tornado has swept them into little pieces, or your hard disk has collapsed. 

Finite precision computation open an approach to the 2nd Law which is different from the standard based on statistics. Small scale disorder is the result of turbulent dissipation as a finite precision resolution of increasingly complex flow arising from flow instability, a resolution which cannot be reversed in finite precision.  It is like necessary (because storage is limited) chopping digits/details, which cannot be retrieved.

Heat energy as internal energy as small scale disordered motion is low quality energy in the sense that transformation to other forms of energy such as large scale motion comes with severe losses.  This puts limits to the efficiency of steam and combustion engines transforming heat energy into piston motion. On the other hand electric energy is high quality energy typically generated from large scale motion in generators allowing efficient electrical motors returning large scale motion. Heating by electricity is thus involves a form of quality degradation, which can be expensive, while heating by burning fossil fuels is efficient and cheap.    

What Is Heat Energy?

Heat energy is a central element in both thermodynamics and radiative heat transfer. But what is in fact heat energy?

The 1st Law of Thermodynamics states that the total energy as kinetic energy plus (internal) heat energy remains constant in a system (without chemistry/fission/fusion) with no energy exchange with its surrounding. The 2nd Law of Thermodynamics states that transformation of kinetic energy into heat energy is irreversible. 

The Planck-Stefan-Boltzman Law (PSB Law) expresses that the transfer of heat energy by electromagnetic radiation from a warmer body of temperature $T_w$ to a colder body of temperature $T_c<T_w$ scales with $T_w^4 -T_c^4$. 

Computational Thermodynamics and Computational BlackBody Radiation present a new approach to uncover the mysteries of both the 2nd Law of Thermodynamics and the PSB Law based on a principle of finite precision computation/physics. In  this setting heat energy takes the form of small scale unordered kinetic motion.

In thermodynamics kinetic energy thus takes the form of large scale ordered motion and small scale unordered motion which is the result of turbulent dissipation into heat energy. The 2nd Law expresses that the process of turbulent dissipation is irreversible because in finite precision unordered small scale motion cannot be coordinated into large scale ordered motion. Heat energy here appears as "internal energy" with limits set by the 2nd Law as concerns transformation to "external energy" as large scale kinetic motion.

In radiative heat transfer the temperature of a body determines a cut-off frequency scaling with temperature with heat energy as atomic vibrations with only frequencies below cut-off appearing in synchronized ordered form capable to generating outgoing radiation. Here the finite precision limit thus scales with the inverse of the temperature and the heat transfer from a warm to a cold body consist only of the frequencies above cut-off for the colder and below cut-off for the warmer. 

In both cases heat energy is a result of an impossibility arising from finite precision computation. In thermodynamics heat energy is unresolvable unordered small scale kinetic motion. In thermodynamics a body absorbs heat energy as unordered kinetic motion for frequencies above cut-off. 

In short, heat energy emerges as a rest product of finite precision computation/physics meeting unresolvable scales of motion. 

Even if now heat energy is a form of rest product, it does not mean that it cannot be recycled into useful energy to some extent. In thermodynamics a gas expanding into a larger volume creates turbulence which is turned into heat energy, which can be used to do work when expanding into an even bigger volume. In radiative heat transfer a colder body when heating up by absorbing heat in unordered form from a warmer body, increases its cut-off and so can radiate higher frequencies in synchronised ordered form.