User:Marc Goossens/First & Second Law

The first law as a special case of a balance law

Many of the above traditional formulations of the first law of thermodynamics, are special cases of the law of conservation of energy in continuum mechanics, extended to include heat. In turn, this constitutes a specific application—to energy—of the general form of a balance law for some physical quantity, say Ψ. Looking at this general form first, helps to attain a clearer understanding of the special case in its different (integral and local) appearances afterwards.

General formulation of a balance law

Abstract form and purpose

The first law of thermodynamics expresses the conservation of energy, including the transfer and transformation of thermal energy or heat. For some, indeed any^[1] (and possibly time dependent) chosen control volume V(t), a generic balance law in classical continuum mechanics for some physical quantity Ψ (e.g. mass, momentum, energy, ...) takes the following general form^[2]:

[A = rate of change of Ψ inside V] = — [B = convective outflow of Ψ through the surface of V] — [ C = outward flux of Ψ through the surface of V] + [D = rate of production of Ψ within V] + [E = bulk supply or "capture" of Ψ within V].

The quantity Ψ is a time dependent field Ψ(x,t); it (and correspondingly its density ψ) may be a scalar, vector or tensor valued function, depending on the quantity being considered.

The equation expresses a physically motivated condition on the field Ψ: it is a field equation for this quantity. As such, it helps to determine (or calculate) Ψ by "solving the field equation". One condition like this may not suffice to determine the solution Ψ completely: further field equations, possibly supplemented with model assumptions for specific materials ("constitutive equations") may be needed.

This abstract formulation also settles the sign convention for the "surface flux" terms [B] and [C] in particular.

Integral form

Further assuming that Ψ posesses a specific density with respect to mass, ψ. With ρ being the mass density of the body with respect to volume, this means that the bulk terms [A], [D] and [E] in the abstract equation may be expressed as volume integrals over dV, weighted by ρ. (See also balance laws in continuum mechanics.)

${\cfrac {d}{dt}}\left[\int _{V(t)}\rho \psi ~{\text{dV}}\right]=\int _{\partial V(t)}\rho \psi [w_{n}-\mathbf {v} \cdot \mathbf {n} ]~{\text{dA}}+\int _{\partial V(t)}g~{\text{dA}}+\int _{V(t)}\rho h~{\text{dV}}~$ (balance law for Ψ with specific density ψ, integral form).

where

$\rho =\rho (\mathbf {x} ,t)$, the mass density,

$\psi =\psi (\mathbf {x} ,t)$, the specific density with respect to mass of the quantity Ψ , and

$g=g(\mathbf {x} ,t)$, the flux of Ψ at x on ∂V

$h=h(\mathbf {x} ,t)$, the specific supply of Ψ at x within V;

and

$\mathbf {v} =\mathbf {v} (\mathbf {x} ,t)$, is the velocity vector field or flow of the material; this means that the particle trajectories are integral curves of v, with

$\mathbf {n} =\mathbf {n} (\mathbf {x} ,t)$, the outward unit normal at x to the surface ∂V(t) of the control volume;

$w_{n}=w_{n}(\mathbf {x} ,t)$, the speed at which ∂V is moving at x in the direction of n.

Dealing with finite (as opposed to "infinitesimal") volumes, the integral formulation still allows for a clear intuitive understanding of what it expresses. However, as an integro-differential equation, it is often difficult to solve, so that one seeks to replace it by a mere differential equation. Though less directly accessible for physical interpretation, it is expected to be easier to handle and solve mathematically.

Local or differential form

Applying the divergence theorem, the surface integral terms may be transformed into volume integrals of the divergence of the integrand. Then, as the balance law is postulated to hold for all volumes, however small, in regular points (i.e. where the fields and densities involved are smooth) it may be reduced to the equivalent (but mathematically easier to handle) differential equation by dropping the integrals and keeping the integrands only.

This gives the equation: ... (local or differential form)

Generalizations and simplified special cases

Depending on the job at hand, the statement may be further generalized or simplified by regarding special cases.

• Generalizations — The balance law may be stated in a "weak" or distributional sense, or be made to include "jump conditions" at non-regular points),

• Closed systems — One simplification concerns the convective flow [B] at a point x of the surface ∂V, which corresponds to the component of the bulk flow ρψ[vector] that is orthogonal to the bounding surface ∂V at x. If the volume V(t) is co-moving (i.e. "transported along" with the flow of the material), or indeed in the static case (no mechanical flow of the material) this term vanishes: the system considered is a closed one.

Thermo-mechanical balance of energy

Next, we apply the above to the balance of energy.

In the context of the first law, thermodynamics complements a purely mechanical description of the motion and the dynamics of continua by postulating the existence of internal energy in addition to the kinetic energy of the overall motion of such a body. For the surface terms, this adds a corresponding flow of internal energy, the heating flux, in addition to the purely mechanical contact stresses on the surface of the control volume V(t). Finally, the supply term indicates that internal energy may be absorbed in the bulk of the material from external sources, say through radiation. We further omit any possible internal production of internal energy.

This leads to the following equations:

[a] (abstract form, also giving the physical interpretation)

${\cfrac {d}{dt}}\left[\int _{V(t)}f(\mathbf {x} ,t)~{\text{dV}}\right]=\int _{\partial V(t)}f(\mathbf {x} ,t)[u_{n}(\mathbf {x} ,t)-\mathbf {v} (\mathbf {x} ,t)\cdot \mathbf {n} (\mathbf {x} ,t)]~{\text{dA}}+\int _{\partial V(t)}g(\mathbf {x} ,t)~{\text{dA}}+\int _{V(t)}h(\mathbf {x} ,t)~{\text{dV}}~.$

(integral form), and

[a] (local form at regular points),

relating the functions of time and spatial position

for a given body (material) which undergoes a thermo-mechanical process of displacement and deformation (flow) and heating. Implicitly therefore, the mathematical equation also posits the existence of these functions (and for its local form their smoothness).

As a "law of nature", this equation is put forward as a mathematical axiom. Physically, it claims that any real physical process a continuous body may experience, is such that the equation is fulfilled at all times. $At least for the materials and processes within the scope of classical continuum mechanics extended with thermodynamics.$

Notes

1. ^a At least any sufficiently regular region V(t).
2. ^b The formulation given here is "abstract" in the sense that it avoids the use of the machinery from analysis (such as integrals, derivatives, ...) as one traditionally expects from "a formula". Precisely in this way though, this form remains a small step closer to direct physical interpretation, and has the advantage of readability. Using measure therory, it may also be rendered mathematically rigorous^[2].
3. ^b See for instance (Gurtin, Williams 1967).

References

• ^b Gurtin, Morton E. (1967). "An Axiomatic Foundation of Continuum Thermodynamics". Archive for Rational Mechanics and Analysis. 26 (2). Springer: 83–117. doi:10.1007/BF00285676. {{cite journal}}: Unknown parameter |coauthors= ignored (|author= suggested) (help)