Entrained entropy. Show that the rate at which entropy...

Entrained entropy. Show that the rate at which entropy enters region 1 in Figure 28.5 is $$ \frac{\mathrm{d} S_1}{\mathrm{~d} t}=\frac{1}{T}\left[\left(L_{U U}-\frac{L_{U N} L_{N U}}{L_{N N}}\right) f_U+\left(\frac{L_{U N}}{L_{N N}}-\mu\right) \dot{N}_1\right] . $$ Interpret the first term, and hence show that the second term is the entrained entropy, that is, the extra entropy carried along by the particle current. Note, this is not the same as (and has little relation to) the equilibrium entropy per particle in the fluid.

Entrained entropy. Show that the rate at which entropy enters region 1 in Figure 28.5 is
$$
\frac{\mathrm{d} S_1}{\mathrm{~d} t}=\frac{1}{T}\left[\left(L_{U U}-\frac{L_{U N} L_{N U}}{L_{N N}}\right) f_U+\left(\frac{L_{U N}}{L_{N N}}-\mu\right) \dot{N}_1\right] .
$$
Interpret the first term, and hence show that the second term is the entrained entropy, that is, the extra entropy carried along by the particle current. Note, this is not the same as (and has little relation to) the equilibrium entropy per particle in the fluid.

Key Concepts

Entrained Entropy

Entrained entropy refers to the additional entropy that is carried along with a particle (or mass) flux during transport processes. This concept distinguishes the entropy transferred by the mere movement of particles from the intrinsic equilibrium entropy per particle, highlighting an extra contribution that arises when particles carry along disorder as they move, a key consideration in coupled transport phenomena.

Phenomenological Coefficients

These coefficients, often denoted by symbols like L_UU, L_UN, etc., characterize the proportionality between thermodynamic forces and fluxes in the linear regime near equilibrium. They encapsulate the material's transport properties and determine how a system responds to nonequilibrium driving forces.

Onsager Reciprocal Relations

These are symmetry relations obeyed by the phenomenological coefficients linking fluxes and forces in near-equilibrium systems. They imply that the cross-couplings between two different types of fluxes are equal under appropriate conditions, which is essential for simplifying and interpreting the transport equations and understanding how changes in one variable affect the other.

Thermodynamic Fluxes and Forces

In non-equilibrium thermodynamics, fluxes (such as energy or particle currents) are linked to thermodynamic forces (such as temperature gradients or differences in chemical potential). This correspondence allows one to quantify the rate at which properties are transferred through a system and to understand the coupling between different types of transfer processes.

Non-equilibrium Thermodynamics

This concept deals with systems that are not in thermodynamic equilibrium and describes how small deviations from equilibrium result in fluxes of energy, mass, or other quantities in response to gradients (in temperature, chemical potential, etc.). In this framework, one constructs linear relationships between driving forces (gradients) and the resultant fluxes, enabling the use of phenomenological coefficients to predict how systems evolve over time.

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