Exergy as Energy Quality

Kinetic energy, electrical energy, chemical and nuclear energy can all be converted fully into heat energy, while heat energy can only be partially converted back again. This is captured in the 2nd Law of Thermodynamics. We can thus say that heat energy is of lower quality compared with the other forms. More generally, the term exergy is used as a measure of quality of energy of fundamental importance for all forms of life and society as ability to do work.

We can make this more precise by recalling that the quality of heat energy comes to expression in radiative and conductive heat transfer from a body B1 of temperature $T_1$ to a neighbouring body B2 of lower  temperature $T_2<T_1$ in basic cases according to Stefan-Boltzmann's Law or Fourier's Law:

• $Q = (T_1^4-T_2^4)$            (SB)
• $Q = (T_1-T_2)$                    (F)

with $Q$ heat energy per time unit. Heat energy of higher temperature thus can be considered to have higher quality than heat energy of lower temperature, which of course also plays role in conversion of heat energy to other forms of energy. The maximal efficiency of a heat engine operating between $T_1$ and $T_2$ and transforming heat energy to mechanical work, is equal to $\frac{T_1-T_2}{T_1}$ displaying the higher quality of $T_1$.

Heat energy at high temperature is the major source for useful mechanical work supporting human civilisation, while heat energy at lower temperatures appears as a useless loss e g in the cooling of a gasoline engine.

But what is the real physics behind (SB) and (F)? This question was addressed in a previous post viewing (F) to be a special case of (SB) with the physics behind (SB) displayed in the analysis of Computational Blackbody Radiation. 

The essence of this analysis is a high-frequency cut-off $\frac{T}{h}$ allowing a body of temperature $T$ to only emit frequencies $\nu <\frac{T}{h}$, where $h$ is a constant. This allows a body B1 of temperature $T_1$ to transfer heat energy to a body B2 of lower temperature $T_2$ via frequencies $\frac{T_2}{h}<\nu <\frac{T_1}{h}$, which cannot be balanced by emission from B2.  

High frequency cut-off increasing linearly with temperature represents Wien's displacement law (W), giving improved exergy with increasing temperature.

The high-frequency cut-off can be seen as an expression of finite precision limiting the frequency being carried and emitted by an oscillating atomic lattice in coordinated motion, with frequencies above cut-off being carried internally as heat energy as uncoordinated motion. 

Higher temperature thus connects to higher quality heat energy or better exergy. The standard explanation of this basic fact is based on statistical mechanics, which is not physical mechanics. 

PS Radiative heat transfer without high-frequency cut-off would boil down to (F), while (SB) is what is observed, which gives support to (W).

Thermodynamics of War and Peace

Opposing ordered armies at the moment before turbulent destruction. 

The recent posts on 2nd law of thermodynamics describe a process where increasing spatial gradients eventually reach a level (from convection and opposing flow) where further increase is no longer possible because it would bring the process to brutal stop, and so some form of equilibration of spatial differences must set in where 

• each particle tends to take on the mean-value of neighbouring particles.   (M)

This is the process in turbulent fluid flow transforming ordered large scale kinetic energy into small scale disordered kinetic energy taking the form internal heat energy in a turbulent cascade of turbulent dissipation. Here (M) is necessary to avoid break-down into a stop. The flow or show must go on.

(M) is also the essence of the diffusion process of heat conduction seeking to decrease gradients, even if not absolutely necessary as in turbulent fluid flow.

It is natural to connect turbulence to the violent break-down of large scale ordred structures into rubble in a war necessarily resulting from escalation of opposing military forces in direct confrontation which at some level cannot be further escalated and so have to be dissipated in a war. 

It is then natural to connect the equilibration (M) in heat conduction to a geopolitical/parliamentary process in peace time, where each country/party takes on the mean value of neighbouring countries/parties keeping gradients small. 

While (M) is necessary in turbulence to let the flow go on, one may ask what the physics of (M) in the case of heat conduction, and find answer in this post. 

The mathematics is elaborated in: 

• Computational Turbulent Incompressible Flow
• Computational Thermodynamics
• Computational Blackbody Radiation

The geopolitical/parliamentary situation today evolves towards sharpened gradients, while politicians refuse to follow (M) and so there is a steady march towards break-down... 

Secret of Conductive and Radiative Heat Transfer

This is a continuation of the previous post on Heat Conduction in Solids as Radiative Heat Transfer with  clarifying analysis from Mathematical Physics of Blackbody Radiation and Computational Blackbody Radiation.

The key aspect of both conductive and radiative heat transfer is interaction in a coupled system of weakly damped oscillators of different frequencies tending to an equilibrium with all oscillators having the same temperature as the system temperature. The damping can be frictional (1st order time derivative) or radiative (3rd order time derivative) 

There are two main questions: (i) Why do different systems take on the same temperature? (ii) Why do oscillators with different frequencies in a system take on the same temperature?  

The answer is hidden in the interaction between incoming radiation, oscillator and outgoing radiation in a weakly radiatively damped oscillator analysed in detail in the above texts. The essence is that under near resonance between incoming frequency and oscillator frequency,  

• incoming radiation is balanced by outgoing radiation plus internal heating. 

This is a non-trivial basic fact reflecting that the forcing and oscillator are out-of-phase with a shift of half a period as a consequence of small radiative damping and near resonance. 

Two coupled oscillators thus interact with outgoing from one oscillator acting as incoming for the other and vice versa and so are led to take on the same temperature, which is then spread over the oscillators of a system and also over systems. 

The essential components in this equilibration process are thus

• weakly damped oscillators generating outgoing radiation and internal heating 
• out-of-phase balance between forcing and damping from near resonance
• high-frequency cut-off increasing with temperature from finite precision computation.  

This analysis connects to Planck's derivation of his law of radiation with statistics replaced by finite precision thus replacing non-physics by physics. 

Heat Conduction in Solids as Radiative Heat Transfer

What is the physics of heat conduction in a solid like a metal? The trivial story is that "heat flows from warm to cold" or "there is a flux of heat from warm to cold" which scales with the temperature difference or gradient. 

But heat is not a substance like water in a river flowing from high-altitude/warm to low-altitude/cold, which connects to the caloric theory and also to phlogiston theory presenting fire as form of substance, both debunked at the end of the 18th century. 

In any case there is a law of physics named Fourier's Law:

• $q =- \nabla u$          (F)

which combined with a law of conservation 

• $\nabla\cdot q = f$

leads to the following heat equation (here in stationary state for simplicity) in the form of Poisson's equation

• $-\Delta u = f$.                    (H)

where $u(x)$ is temperature and $f(x)$ heat source depending on a space variable $x$, and $q(x)$ is named "heat flux" although it has no physical meaning; heat is not any substance which flows or is in a state of flux. 

Let us now seek the physics of (F) in the case of metallic body as a lattice of atoms, and so seek an explanation of the observation that the temperature distribution $u(x)$ of the body tends to an equilibrium state with $u(x)=U$ with $U$ a constant (assuming no interaction with the surrounding and no internal heating for simplicity). 

We thus ask: 

• What is the physics of the process towards equilibrium with constant temperature?
• How is heat transferred from warm to cold?
• Why is (F) valid?

We then recall our analysis of radiative transfer of energy at distance in a system of bodies/parts separated in space which (without external forcing) leads to an equilibrium state with all bodies having the same temperature, based on the following physical model:

• Each body is a vibrating lattice of atoms described by a wave equation with small radiative damping. 
• The bodies interact by electromagnetic waves through resonance. 
• There is a high-frequency cut-off increasing linearly with temperature with the effect that heat transfer mediated by electromagnetic waves between two bodies, is one-way from high temperature to low temperature. 

The key is here the high-frequency cut-off increasing with temperature, which makes heat transfer one-way. The cut-off can be seen as a form of finite precision threshold allowing coordinated lattice vibration only below cut-off, thus allowing a high temperature lattice to carry higher frequencies. It is like a warmed-up opera soprano being able to reach higher frequencies.

We can view metallic body as a system composed of parts/atoms interacting by 

electromagnetic waves at small distance. 

Heat conduction will then come out as a special case of electromagnetic heat transfer 

between atoms of different temperature with high-frequency cut-off guaranteeing one-way transport as expressed by (F) and exposed above. 

Note that the reference text Conduction of Heat in Solids by Carslaw and Jaeger presents (F) as an ad hoc physical law without physics.

Recall that the standard explanation of radiative heat transfer from warm to cold is based on statistics without physics, which if used to explain heat conduction would again invoke statistics without physics thus not very convincing. 

Also note that the standard explanation of heat transfer in a gas involves collisions of molecules of different kinetic energy, which is not applicable to a metal with atoms in a lattice.

PS1 Fluid flow in a river from higher to lower altitude is driven by pressure. "Heat flow" from warm to cold is not driven by pressure and so the physics is different. 

PS2 Also compare with one-way osmotic transport of material driven by pressure. 

Unphysical Schwarzschild Equation for Heat Transfer by Conduction

Schwarzschild's equation for heat transfer by radiation through the atmosphere is a two-stream upwelling-downwelling heat transfer model with the net heat transfer expressed as the difference between upwelling and downwelling streams. It is the basic model for radiative heat transfer used in climate models.

To get perspective on this two-stream model, let us see what a two-stream model for heat transfer by conduction instead of radiation would look like. We shall then compare with the standard one-stream model for vertical heat conduction through a horisontal layer, which is Fourier's Law:

• $q(x) = -\gamma\frac{T(x+dx)-T(x)}{dx}$   or $q(x) = -\gamma\frac{dT}{dx}$,    (1)

where $q(x)$ is net heat transfer/flow, $\gamma$ is a heat conductivity coefficient, $T(x)$ is temperature, $x$ is a vertical coordinate and $dx$ a small increment.

A two-way version of Fourier's Law takes the form

• $q(x) =\gamma\frac{T(x)}{dx} - \gamma\frac{T(x+dx)}{dx}$   (2) 

where the net heat transfer $q(x)$ is expressed as the difference of two gross streams  $\gamma\frac{T(x)}{dx}$ and $\gamma\frac{T(x+dx)}{dx}$ in opposite directions. But this two-way version is unstable because you are dividing temperatures $T(x)$ and $T(x+dx)$ by the small quantity $dx$, and so it is both uncomputable (by dividing by small $dx$) and unphysical since an unstable system does not have permanence over time. Note that the derivative $\frac{dT}{dx}$ does not suffer from the same instability since $dT$ is small.

We conclude that a Schwarzschild two-stream model for heat transfer by conduction is unstable, uncomputable and unphysical. Can we expect that Schwarzschild's two-stream model for heat transfer by radiation does not suffer from the same deficiency?  

It remains to formulate a stable one-stream model for heat transfer by radiation through an absorbing/emitting gas. An interesting such model is presented here, to which I will return.  

We can take the argument one step further by expressing conservation of heat energy in the form 

• $\frac{q(x)-q(x-dx)}{dx} = f(x)$  (3)

where $f(x)$ is a heat source. Combined with (2) this gives the model (with $\gamma =1$)

• $\frac{T(x)}{dx^2} - \frac{T(x+dx)}{dx^2} - \frac{T(x-dx)}{dx^2} +\frac{T(x)}{dx^2}=f(x)$  (4) 

which in the spirit of Schwarzschild splitting the source $f(x)$ into a contribution to "upwelling heat flux" $q_{up}$ and "downwelling heat flux" $q_{down}$, we can write as  

• $q_{up}(x)\equiv\frac{2T(x)}{dx^2} =\frac{f(x)}{2}$                                                   
• $q_{down}(x)\equiv\frac{T(x+dx)}{dx^2} + \frac{T(x-dx)}{dx^2}=-\frac{f(x)}{2}$ 

with net flow $q(x)=q_{up}(x)-q_{down}(x)$ balancing $f(x)$ according to (4). These are unstable unphysical equations that do not make sense. We compare with combining (3) with (1) into the standard heat equation 

•  $-\frac{d^2T}{dx^2} \approx -\frac{T(x+dx)-2T(x)+T(x-dx)}{dx^2}=f(x)$

which makes perfect sense as a differential equation allowing stable solution. We understand that Schwarzschild's two-stream model for heat transfer by conduction is no good. Is it then no good also for heat transfer by radiation?