User:Tohline/PGE/FirstLawOfThermodynamics

First Law of Thermodynamics

Standard Presentation

Following the detailed discussion of the laws of thermodynamics that can be found, for example, in Chapter I of [C67] we know that, for an infinitesimal quasi-statical change of state, the change <math>~dQ</math> in the total heat content <math>~Q</math> of a fluid element is given by the,

Fundamental Law of Thermodynamics

<math>~dQ</math>

<math>~=</math>

<math>~ d\epsilon + PdV \, , </math>

[C67], Chapter II, Eq. (2)
[H87], §1.2, Eq. (1.2)
[KW94], §4.1, Eq. (4.1)
[HK94], §1.2, Eq. (1.10)
[BLRY07], §1.6.5, Eq. (1.124)

where, <math>~\epsilon</math> is the specific internal energy, <math>~P</math> is the pressure, and <math>~V</math><math>~= 1/</math><math>~\rho</math> is the specific volume of the fluid element. Generally, the change in the total heat content can be rewritten in terms of the gas temperature, <math>~T</math>, and the specific entropy of the fluid, <math>~s</math>, via the expression,

<math>~dQ</math>

<math>~=</math>

<math>~T ds \, .</math>

[C67], Chapter I, Eq. (76) & Chapter II, Eq. (44)
[H87], §1.4, p. 16
[HK94], §1.2, Eq. (1.10)

If, in addition, it is understood that the specified changes are occurring over an interval of time <math>~dt</math>, then from this pair of expressions we derive what will henceforth be referred to as the,

Standard Form
of the First Law of Thermodyamics

<math>T \frac{ds}{dt} = \frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr)</math>

[T78], §3.4, Eq. (64)
[Shu92], Chapter 4, Eq. (4.27)
[HK94], §7.3.3, Eq. (7.162)

If the state changes occur in such a way that no heat seeps into or leaks out of the fluid element, then <math>~ds/dt = 0</math> and the changes are said to have been made adiabatically. For an adiabatically evolving system, therefore, the First Law assumes what henceforth will be referred to as the,

Adiabatic Form
of the First Law of Thermodyamics

<math>~\frac{d\epsilon}{dt} + P \frac{d}{dt} \biggl(\frac{1}{\rho}\biggr) = 0</math>

[C67], Chapter II, Eq. (13)
[T78], §3.4, Eq. (70)

Clearly this form of the First Law also may be viewed as a statement of specific entropy conservation.

Entropy Tracer

Initial Recognition

Multiplying the Adiabatic Form of the First Law of Thermodynamics through by <math>~\rho</math> and rearranging terms, we find that,

<math>~0</math>	<math>~=</math>	<math>~ \rho\frac{d\epsilon}{dt} + \rho P \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr) </math>
	<math>~=</math>	<math>~ \frac{d(\rho\epsilon)}{dt} - \epsilon \frac{d\rho}{dt} - \frac{P}{\rho} \frac{d\rho}{dt} </math>
	<math>~=</math>	<math>~ \frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon) \frac{1}{\rho}\frac{d\rho}{dt} </math>
	<math>~=</math>	<math>~ \frac{d(\rho\epsilon)}{dt} - (P + \rho\epsilon)\frac{d\ln\rho}{dt} \, , </math>

is an equally valid statement of the conservation of specific entropy in an adiabatic flow. In combination, first, with

Form B
of the Ideal Gas Equation

<math>~P = (\gamma_\mathrm{g} - 1)\epsilon \rho </math>

and, second, with the

Lagrangian Form
of the Continuity Equation

<math>\frac{d\rho}{dt} + \rho \nabla \cdot \vec{v} = 0</math>

we may furthermore rewrite this expression as,

<math>~\frac{d(\rho\epsilon)}{dt}</math>	<math>~=</math>	<math>~ \gamma_g (\rho\epsilon)\frac{d\ln\rho}{dt} </math>
<math>~\Rightarrow ~~~ \frac{1}{\gamma_g} \frac{d\ln(\rho\epsilon)}{dt}</math>	<math>~=</math>	<math>~ \frac{d\ln\rho}{dt} </math>
<math>~\Rightarrow ~~~ \frac{d\ln(\rho\epsilon)^{1/\gamma_g}}{dt}</math>	<math>~=</math>	<math>~ - \nabla\cdot\vec{v} \, . </math>

This relation has the classic form of a conservation law. It certifies that, within the context of adiabatic flows, the entropy tracer,

<math>~\tau \equiv (\rho\epsilon)^{1/\gamma_g} = \biggl[ \frac{P}{(\gamma_g - 1)} \biggr]^{1/\gamma_g} \, ,</math>

is the volume density of a conserved quantity. In this case, that conserved quantity is the entropy of each fluid element.

Substantiation

To further substantiate this claim, we note that,

<math>~\frac{\tau}{\rho}</math>	<math>~=</math>	<math>~ \epsilon^{1/\gamma_g} \cdot \rho^{1/\gamma_g - 1} </math>
<math>~\Rightarrow ~~~ \ln\biggl(\frac{\tau}{\rho}\biggr)</math>	<math>~=</math>	<math>~ \frac{1}{\gamma_g} \biggl[ \ln\epsilon - ( \gamma_g-1)\ln\rho \biggr] \, . </math>

Now, from the first law, we can write,

<math>~ds</math>	<math>~=</math>	<math>~\frac{1}{T} \biggl[ d\epsilon - \frac{P}{\rho} {d\ln\rho} \biggr] </math>
	<math>~=</math>	<math>~ c_V~ d\ln\epsilon - \frac{\Re}{\mu} ~{d\ln\rho} </math>
<math>~ \Rightarrow ~~~ \frac{ds}{c_P} </math>	<math>~=</math>	<math>~ \frac{c_V}{c_P}~ d\ln\epsilon - \frac{\Re/\mu}{c_P} ~{d\ln\rho} </math>
	<math>~=</math>	<math>~ \frac{1}{\gamma_g} \biggl[ d\ln\epsilon - (\gamma_g-1){d\ln\rho} \biggr] \, , </math>

which, upon integration, gives,

<math>~\frac{s}{c_P}</math>

<math>~=</math>

<math>~ \frac{1}{\gamma_g} \biggl[ \ln\epsilon - (\gamma_g-1)\ln\rho \biggr] + \mathrm{constant} \, . </math>

To within an additive constant, this is precisely the expression for the logarithm of the entropy tracer, as provided immediately above. Hence, we see that,

<math>~s = c_P \ln\biggl( \frac{\tau}{\rho} \biggr) + \mathrm{constant} \, ,</math>

that is, we see that the variable, <math>~\tau</math>, traces the fluid entropy just as <math>~\rho</math> traces the fluid mass.

We have found one other instance in the literature — although there are undoubtedly others — where the role of this entropy tracer previously has been identified. In chapter IX of [LL75] we find that, "apart from an unimportant additive constant," the specific entropy is,

<math>~s</math>

<math>~=</math>

<math>~c_P \ln \biggl(\frac{P^{1/\gamma_g}}{\rho} \biggr) \, .</math>

[LL75], §80, Eq. (80.12)

Given that <math>~\tau \propto P^{1/\gamma_g}</math>, this is clearly the same expression as we have derived for the specific entropy of the fluid.

Incorporation Into the First Law

Multiplying the Standard Form of the First Law of Thermodynamics through by <math>~\rho</math>, we can now write,

<math>~\rho T ~\frac{ds}{dt}</math>	<math>~=</math>	<math>~ \frac{d(\rho\epsilon)}{dt} - \gamma_g (\rho\epsilon) ~\frac{d\ln\rho}{dt} </math>
<math>~\Rightarrow~~~ \frac{\rho T}{\gamma_g(\rho\epsilon)} ~\frac{ds}{dt}</math>	<math>~=</math>	<math>~ \frac{d\ln(\rho\epsilon)^{1/\gamma_g}}{dt} - \frac{d\ln\rho}{dt} </math>
<math>~\Rightarrow~~~ \frac{1}{c_P} ~\frac{ds}{dt} </math>	<math>~=</math>	<math>~ \frac{d\ln(\tau/\rho)}{dt} </math>
<math>~\Rightarrow~~~ \frac{1}{c_P}\biggl( \frac{\tau}{\rho} \biggr) ~\frac{ds}{dt} </math>	<math>~=</math>	<math>~ \frac{d(\tau/\rho)}{dt} </math>
	<math>~=</math>	<math>~ \frac{1}{\rho} \biggl[ \frac{d\tau}{dt} - \frac{\tau}{\rho}\frac{d\rho}{dt}\biggr] </math>
	<math>~=</math>	<math>~ \frac{1}{\rho} \biggl[ \frac{d\tau}{dt} + \tau \nabla\cdot\vec{v} \biggr] </math>
<math>~\Rightarrow~~~ \biggl( \frac{\tau}{c_P} \biggr) ~\frac{ds}{dt} </math>	<math>~=</math>	<math>~ \frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) </math>

Now,

<math>~c_P</math>	<math>~=</math>	<math>~c_V \gamma_g</math>
	<math>~=</math>	<math>~\biggl(\frac{c_V}{\rho\epsilon}\biggr) \gamma_g \tau^{\gamma_g}</math>
	<math>~=</math>	<math>~\biggl(\frac{1}{\rho T}\biggr) \gamma_g \tau^{\gamma_g} \, .</math>

Hence,

<math>~ \frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) </math>

<math>~=</math>

<math>~ \biggl( \frac{1}{\gamma_g \tau^{\gamma_g-1}} \biggr) ~\rho T~\frac{ds}{dt} \, . </math>

Marcello & Tohline (2012), §2.2, Eq. (31)

Notice, as well, that we can write,

<math>~ \frac{ds}{dt} </math>	<math>~=</math>	<math>~c_P~ \frac{d\ln(\tau/\rho)}{dt} </math>
	<math>~=</math>	<math>~c_V \biggl[ \frac{d\ln(\tau/\rho)^\gamma}{dt} \biggr] </math>
	<math>~=</math>	<math>~c_V \frac{d}{dt}\biggl[ \ln\biggl( \frac{\rho\epsilon}{\rho^{\gamma_g}}\biggr) \biggr] </math>
<math>~ \Rightarrow ~~~ \rho T ~\frac{ds}{dt} </math>	<math>~=</math>	<math>~\rho\epsilon ~\frac{d}{dt}\biggl[ \ln\biggl( \frac{\rho\epsilon}{\rho^{\gamma_g}}\biggr) \biggr] </math>
<math>~ \Rightarrow ~~~ \frac{P}{(\gamma_g - 1)} ~\frac{d}{dt}\biggl[ \ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr) \biggr] </math>	<math>~=</math>	<math>~ \rho T ~\frac{ds}{dt} \, . </math>

Specifically for the case, <math>~\gamma_g = \tfrac{5}{3}</math>, this gives,

<math>~ \frac{3}{2} ~P ~\frac{d}{dt}\biggl[ \ln\biggl( \frac{P}{\rho^{5/3}}\biggr) \biggr] </math>

<math>~=</math>

<math>~ \rho T ~\frac{ds}{dt} \, . </math>

[Shu92], Chapter 9, Eq. (9.26)

It is fair to say, therefore, that in this specific case [Shu92] also recognized the relevance of and the conservative nature of, what we have referred to as, the entropy tracer.

Nonadiabatic Environments

Left-Hand Side (LHS)

There are several potentially useful expressions for the time-rate of change of fluid entropy.

<math>~\rho T \frac{ds_\mathrm{fluid}}{dt}</math>

<math>~=</math>

<math>~ \rho\frac{d\epsilon}{dt} + \rho P \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr) </math>

<math>~=</math>

<math>~ \rho\frac{d\epsilon}{dt} + P\nabla\cdot \vec{v} </math>

<math>~=</math>

<math>~ \gamma_g \tau^{\gamma_g-1}\biggl[ \frac{\partial \tau}{\partial t} + \nabla\cdot (\tau \vec{v}) \biggr] </math>

<math>~=</math>

<math>~ \rho\epsilon \biggl[ \frac{d\ln(\tau/\rho)^{\gamma_g}}{dt} \biggr] </math>

<math>~=</math>

<math>~ \frac{P}{(\gamma_g - 1)} ~\frac{d}{dt}\biggl[ \ln\biggl( \frac{P}{\rho^{\gamma_g}}\biggr) \biggr] \, . </math>

Clearly to within an additive constant, an expression for the fluid entropy, itself, is

<math>~s</math>

<math>~=</math>

<math>~ c_P \ln\biggl( \frac{\tau}{\rho} \biggr)</math>

<math>~=</math>

<math>~c_V \ln \biggl(\frac{P}{\rho^{\gamma_g}} \biggr) \, .</math>

In optically thick environments where the radiation field is intermixed and in equilibrium with the fluid (gas), the time-rate-of-change in the entropy of the radiation field is characterized by the expression,

<math>~\rho T \frac{ds_\mathrm{rad}}{dt}</math>

<math>~=</math>

<math>~ \rho\frac{d}{dt}\biggl(\frac{E_\mathrm{rad}}{\rho}\biggr) + \rho P_\mathrm{rad} \frac{d}{dt}\biggl(\frac{1}{\rho} \biggr) </math>

<math>~=</math>

<math>~ \frac{4a_\mathrm{rad} \rho T}{3} \frac{d}{dt} \biggl( \frac{T^3}{\rho}\biggr) </math>

Hence, to within an additive constant, an expression for the entropy of the radiation field is,

<math>~s_\mathrm{rad}</math>

<math>~=</math>

<math>~ \frac{4 a_\mathrm{rad} T^3}{3\rho} \, .</math>

Right-Hand Side (RHS)

Example A

One physically reasonable pair of sources/sinks of entropy in the fluid arise in the context of what [LL75] identify as the general equation of heat transfer, namely,

<math>~\rho T \frac{ds_\mathrm{fluid}}{dt}</math>

<math>~=</math>

<math>~ - \nabla\cdot \vec{F}_\mathrm{cond} + \Psi \, . </math>

[LL75], §49, p. 185, Eq. (49.4)
[Shu92], Vol. II, §3, p. 30, Eq. (3.26)
[P00], Vol. I, §8.4, p. 369, Eq. (8.35)

In this expression,

<math>~\vec{F}_\mathrm{cond}</math>

<math>~=</math>

<math>~-\mathcal{K}_\mathrm{cond} \nabla T \, ,</math>

[Shu92], Vol. II, §3, p. 28, Eq. (3.19)

where, <math>~\mathcal{K}_\mathrm{cond}</math> is the coefficient of thermal conductivity; and the rate of viscous dissipation,

<math>~\Psi</math>

<math>~\equiv</math>

<math>~ \pi_{ik} \frac{\partial v_i}{\partial x_k} \, , </math>

[Shu92], Vol. II, §3, p. 29, following Eq. (3.25)

where <math>~\pi_{ik}</math> is the "viscous stress tensor," as defined, for example: by equation (15.3) on p. 48 of [LL75]; by equation (44) on p. 52 of [T78]; and by equation (8.34) on p. 369 of [P00]. Note that when [Shu92] defines <math>~\pi_{ik}</math> — see his equations (3.19) and (3.20) on p. 28 — he implicitly zeroes out the coefficient of bulk viscosity component, keeping only the shear viscosity component because it is the piece that is usually of interest in astrophysical discussions. [Shu92] goes on to explain — see on p. 23 immediately following his equation (2.36) — together, the pair of terms on the right-hand-side express the "time rate of adding heat (through heat conduction and the viscous conversion of ordered energy in differential fluid motions to disordered energy in random particle motions)."

Example B

In addition to the pair of source/sink terms that arise from the general equation of heat transfer, [T78] includes another pair of terms that often arise in discussions of stellar structure and evolution. Specifically, on p. 56, his equation (65) states,

<math>~\rho T \frac{ds_\mathrm{tot}}{dt} = \rho T \frac{d}{dt}\biggl( s_\mathrm{fluid} + s_\mathrm{rad} \biggr)</math>

<math>~=</math>

<math>~ \Psi - \nabla\cdot \vec{F}_\mathrm{cond} + \rho \epsilon_\mathrm{nuc} - \nabla \cdot \vec{F}_\mathrm{rad} \, . </math>

[T78], §3.4, p. 56, Eq. (65)
[Shu92], Vol. II, §4, p. 53, Eq. (4.40)

(Note, that [T78] uses the variable notation <math>~\Phi_v</math> in place of <math>~\Psi</math>.) In this expression, <math>~\epsilon_\mathrm{nuc}(\rho,T)</math> expresses the rate at which (specific) energy is released via thermonuclear reactions, and

<math>~\vec{F}_\mathrm{rad}</math>	<math>~=</math>	<math>~- \frac{c}{3\rho\kappa_R} \nabla (a_\mathrm{rad}T^4) </math>	<math>~=</math>	<math>~-\chi_\mathrm{rad} \nabla T \, ,</math>
[Shu92], Vol. I, §2, p. 17, Eq. (2.17)			and    [T78], §3.4, p. 57, Eq. (67)

where [T78] refers to

<math>~\chi_\mathrm{rad}</math>

<math>~\equiv</math>

<math>~ \frac{4c a_\mathrm{rad} T^3}{3\kappa \rho} \, . </math>

[T78], §3.4, p. 57, Eq. (68)

as the coefficient of radiative conductivity. The expression for the radiation flux, <math>~\vec{F}_\mathrm{rad}</math>, presented by [T78] is identical in form to the expression presented above for the flux due to heat conduction, <math>~\vec{F}_\mathrm{cond}</math>. This highlights the similarities between the manner in which nature handles transport processes ("Fourier's law") — whether by heat conduction (electrons) or radiative diffusion (photons).

Alternatively,† "… recognizing <math>~aT^4</math> as the energy density of blackbody radiation, we see that [the expression for <math>~\vec{F}_\mathrm{rad}</math> that appears as equation (2.17) in Volume I of Shu92] has the general form for diffusive fluxes (Fick's law): diffusive flux <math>~=</math> <math>~- \mathcal{D} \nabla</math>(density of quantity being diffused), where <math>~\mathcal{D}</math> is the diffusivity. Indeed, this comparison allows us to identify the radiative diffusivity as having the characteristic formula, <math>~\mathcal{D}_\mathrm{rad} = \frac{1}{3} c \ell \, ,</math> where <math>~\ell \equiv 1/\rho\kappa_R</math> is the (Rosseland) mean-free path of the diffusing particles (photons). A 'random walk' slows down the free-flight speed <math>~c</math> by a typical factor of <math>~\ell/R_\odot</math>, so that the time <math>~R_\odot^2/\mathcal{D}_\mathrm{rad}</math> for photons to diffuse to the surfacce of the Sun is roughly <math>~3R_\odot/\ell</math> times longer than the free-flight time <math>~R_\odot/c</math> of 2 s. This process prevents the Sun from releasing its considerable internal reservoir of photons in one powerful blast, but instead regulates it to the stately observed luminosity of <math>~L_\odot = 3.86 \times 10^{33}</math> erg s-1."

†Text in a green font has been taken directly from Volume I, §2, p. 17 of [Shu92].

Example C

In astrophysical discussions of the time-rate-of-change of the fluid entropy, it is not unusual to include a scalar function, <math>~\Gamma</math>, that accounts in a generic manner for volumetric gains of energy due to local sources, and another scalar function, <math>~\Lambda</math>, that accounts in a generic manner for volumetric loses of energy due to local sinks. In place of the above "Example A" right-hand-side expression, then, we would expect to see,

<math>~\rho T \frac{ds_\mathrm{fluid}}{dt}</math>

<math>~=</math>

<math>~ - \nabla\cdot \vec{F}_\mathrm{cond} + \Psi + \Gamma - \Lambda \, . </math>

When, for example, the fluid (gas) is exposed to photon radiation, heating of the fluid by the radiation is handled by setting,

<math>~\Gamma = c\kappa_E E_\mathrm{rad} \, ,</math>

and the fluid cools — returning energy to the radiation field — according to the reciprocating expression,

<math>~\Lambda = 4\pi \kappa_p B_p = 4 \kappa_p \sigma T^4 \, ,</math>

where, <math>~\sigma \equiv \tfrac{1}{4}c a_\mathrm{rad}</math> is the Stefan-Boltzmann constant. In such a case, the right-hand-side of the equation describing the corresponding time-rate-of-change of the entropy of the radiation field, <math>~s_\mathrm{rad}</math>, would necessarily contain the same two terms, but in both cases with opposite signs. That is, the entropy of the radiation field sees <math>~\Lambda</math> as a source while it sees <math>~\Gamma</math> as a "sink."

[In addition to <math>~\Gamma</math> and <math>~\Lambda</math>, other terms involving spatial variations in the velocity field and in the radiation energy density also appear on the right-hand-side of the expression for <math>~ds_\mathrm{rad}/dt</math>. For simplicity, and because these other terms are not relevant to the principal point we are making, we have opted not to detail the entire expression for <math>~ds_\mathrm{rad}/dt</math> here. The additional terms and details can be found in, for example, J. C. Hayes et al. (2006) or D. C. Marcello & J. E. Tohline (2012).]

First Elaboration

When the expressions for <math>~ds_\mathrm{fluid}/dt</math> and <math>~ds_\mathrm{rad}/dt</math> are added together to obtain a prescription for the time-rate-of-change of <math>~s_\mathrm{tot}</math> — see, for example, "Example B" above — neither of the functions, <math>~\Gamma</math> or <math>~\Lambda</math>, will appear explicitly because they have opposite signs in the two separate expressions. This will be the case whether the environment is optically thin or optically thick.

Second Elaboration

In an optically thick environment where local thermodynamic equilibrium has been achieved, <math>~E_\mathrm{rad} = a_\mathrm{rad}T^4</math>, so,

<math>~\Gamma = c\kappa_E a_\mathrm{rad}T^4 = \biggl( \frac{\kappa_E}{\kappa_p} \biggr) \Lambda \, .</math>

In such an environment, we also expect <math>~\kappa_E \leftrightarrow \kappa_p</math>, so the heating and cooling terms will cancel out each other. As a result, the quantity <math>~(\Lambda - \Gamma) </math> will disappear from the separate expressions for <math>~ds_\mathrm{fluid}/dt</math> and <math>~ds_\mathrm{rad}/dt</math>.

 

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