Thermodynamics is something of a curious subject, for in many ways it deals directly with only the equilibrium states of macroscopic systems, rather than the dynamical processes which interpolate between them. Recall that the equilibrium states of a system are the states in which the system is spatially homogeneous, and the pressure, density and temperature of the system can be characterised by single values. Here we need to recall also the distinction between microstates and macrostates:

A microstate is the complete, detailed description of a system, which specifies the components of position and the components of momentum, for all the particles in the system. A system consisting of

*n*particles has a 6

*n*-dimensional state space, called the

*phase space*, and each microstate corresponds to a point in this 6

*n*-dimensional space.

In contrast, a macrostate is a collection of empirically indistinguishable microstates. Now, a macrostate can be either an equilibrium macrostate, or a non-equilibrium macrostate. An equilibrium macrostate is uniquely specified by the homogeneous temperature, pressure and density of the system. In contrast, a non-equilibrium macrostate must be specified by something like the Boltzmann distribution function

*f(q,v)*, where

*q*denotes position and

*v*denotes velocity. This distribution specifies the velocity profile of the system (the distribution in

*v*), and the distribution of the particles in space. The idea is that

*q*and

*v*should each be partitioned into macroscopically small, but microscopically large cells, and a macrostate of the

*n*-particle system can then be specified by the number of particles in each cell. The equilibrium macrostates then correspond to the special class of Boltzmann distribution functions in which the particles are uniformly distributed in space, and the velocity profile has a Maxwell-Boltzmann distribution. Systems with different equilibrium temperatures simply have Maxwell-Boltzmann velocity distributions with different mean values.

The entire 6

*n*-dimensional phase space of a system can be exhaustively partitioned into non-overlapping macrostates, some of which are non-equilibrium macrostates, and some of which are equilibrium macrostates. The system can also be stratified into a 1-parameter family of 6

*n*-1 dimensional hypersurfaces, each consisting of states with the same internal energy

*E*. Assuming the system is unconstrained in space, then within each constant energy hypersurface there is a unique equilibrium macrostate, and this macrostate will have a volume overwhelmingly larger than any of the other macrostates on that hypersurface. The

*entropy*of a macrostate is proportional to the volume of the macrostate, hence each equilibrium macrostate is the highest entropy macrostate on its respective constant energy hypersurface. Non-equilibrium macrostates tend to evolve towards equilibrium macrostates, in accordance with the Second Law of thermodynamics.

If we loosen the assumption that the system is unconstrained in space, then it is no longer true to say that there is a unique equilibrium macrostate for each value of the system's internal energy

*E*. Instead, for each fixed volume within which the system is bounded, and for each internal energy of the system bounded within that volume, there is a unique equilibrium macrostate.

Now, to return to our main expository objective, the Equation of State

*P(V,T)*, is an expression which links the pressure, volume (or density) and temperature of a material's equilibrium macrostates. Each different substance has its own Equation of State. In graphical terms, the Equation of State can be represented by a two-dimensional surface which specifies the pressure corresponding to each possible combination of volume (or density) and temperature. The temperature

*T*determines the (internal) energy

*E*hypersurface of phase space, and the curve across the Equation of State surface at a fixed volume

*V*, specifies the pressure of the volume-V system in an equilibrium macrostate, at any possible temperature.

Here, however, we should add that the Equation of State is not necessarily restricted to the

*global*equilibrium states of a system. Even in a spatially inhomogeneous substance, if a macroscopically small but microscopically large region around every point of the substance has equilibrated, then the system is said to be in

*local thermodynamic equilibrium*(LTE). The thermodynamical relations which apply to global equilibrium states, can also apply to the fields which represent pressure p(x), density ρ(x), and temperature T(x), in a substance out of global equilibrium.

So that's the Equation of State. Now, the release of chemical energy in a combustible gas, can be idealised by representing the region in which combustion occurs to be infinitely thin, and the combustion to be instantaneous. This, however, introduces a discontinuity into the flow, and the discontinuity is called the reaction front, or combustion front. The presence of the discontinuity entails that the equations of conservation of mass, momentum and energy, normally used in gas dynamics, have to be replaced by the Rankine-Hugoniot jump relations.

By combining the jump relations with the Equation of State of the gas in question, we obtain a curve, centred on the initial pressure and volume of the gas, called the Hugoniot curve. This curve doesn't specify the sequence of states of the gas during combustion; rather, each point specifies a possible post-combustion equilibrium macrostate of the gas.

Strictly speaking, the curve exists in a five-dimensional space in which the coordinates are pressure, volume (or density), internal energy, speed of the discontinuity

*U*, and the particle speed after the passage of the discontinuity

_{s}*u*. Typically, however, the curve will be projected onto a plane, such as the

_{p}*P-V*plane.

The Hugoniot curve can be divided into two branches, the upper and the lower. The upper branch specifies those states reached through detonation, in which the discontinuity becomes a supersonic shock compression, whilst the lower branch specifies those states reached through a so-called deflagration, in which the discontinuity moves with subsonic speed, via thermal conductivity. It is only the latter process which engine designers consider to be a useful form of combustion.

In fact, a detailed analysis of the combustion fronts in deflagration reveal not an infinitely thin layer, but a two-zone structure, with a leading pre-heat zone in which the temperature is raised by conduction, and a trailing reaction zone. The portion of the flame ahead of the reaction zone is typically five times as thick as the reaction zone itself.

All of which constitutes enough for now.

## 2 comments:

Just as a side note, in France the Otto cycle is known -as you probably know- as Beau de Rochas cycle... I am always amazed to discover how often -even in sciences- some nations prefer to ignore historical facts and data because it suits them better ;-)

Not pointing my finger at any country in particular, just stating a fact.

... and thanks for your facinating blog, sometimes not easy to read, but worth the effort anyway!

Thanks for the compliments Ago.

I once knew a man with a name similar to yours; it was, however, Long Ago.

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