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A synthesis of some variational theorems for the extended irreversible thermodynamics of nonstationary heat and mass transfer

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Abstract

The purpose of the paper is to present and asses several advanced variational theorems for the extended thermodynamics of energy and mass transfer described by wave equations.

The classical functionals of Hamilton's type are presented and the physically important relation between the thermodynamic irreversibility and time reversal in these functions is underlined. The role of the entropy of mass and energy currents as well as kinetic energy of diffusion is the variational formulations is shown for Eulerian (field) and Lagrangian representations of diffusive motion. The overdamped nature of the physical regime described by wave equations and action functionals is pointed out.

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Abbreviations

a :

heat diffusivity

a j :

Lagrangian coordinate of j-th component

C, C :

capacity and capacity matrix, respectively

C P :

heat capacity at the constant pressure

c, c 0 :

light speed and propagation speed, respectively

D, D :

diffusivity and diffusivity matrix, respectively

E, E :

thermodynamic disturbance and vector of disturbances, respectively

\(\frac{{ - \partial \phi }}{{\partial x}}j\) :

external force per unit mass of component j

H :

col(H 1, H 2 ... H n−1, H q ) column matrix of Biots vectors

h, h n :

specific enthalphy and partial enthalpy of n-th component, resp.

I, \(I\frac{{C^{ - 1} }}{{\rho C_0^2 }}\) :

inertial relaxation coefficient and inertial matrix, respectively

J :

col(J 1, J 2 ... J n−1, J n =J q ) column matrix of all independent fluxes

J i , J q :

mass diffusion flux of i-th component and energy flux, respectively

L :

Onsager's matrix

m :

thermodynamic system mass

P i :

partial pressure of i-th component

P :

matrix of flux transformation

q :

pure heat flux

R, R=L −1 :

resistance, and resistance matrix, respectively

r :

radius vector

\(\tilde S\) :

action functional for irreversible process

s :

specific entropy

T :

absolute temperature

t :

time

u :

\( = col(\frac{{\mu _n - \mu _1 }}{T}... \frac{{\mu _n - \mu _{n - 1} }}{T},\frac{1}{T})\), column matrix of transfer potentials

V :

total volume

v :

velocity

W, W e :

mean moisture content in solid and equilibrium moisture content, respectively

x :

radius vector composed of coordinates, x, y, z

y j :

mass fraction of j-th component

z :

=col(y 1, y 2 ... y n−1, h) column matrix of thermodynamic state at mechanical equilibrium

▽2 :

Laplace operator

γ :

relativistic correction factor, eq. (14)

σ :

entropy source

δ :

variation

ζ, ζ j :

mass density and density of j-th component, respectively

τ, τ :

relaxation time and relaxation matrix, respectively

μ i :

chemical potential of i-th component

ω, ω 0 :

frequency and current maximizing frequency, respectively

φ :

gravitational potential per mass unit

Λ, \(\tilde \Lambda\) :

Lagrangian of reversible and irreversible process, respectively

Λ, \(\tilde \Lambda\) :

Lagrangian tensor, or reversible and irreversible process, respectively

λ :

thermal conductivity

m:

maximum

−1:

reverse matrix

*:

transformed quantity

T:

transpose matrix

′:

resting frame

-:

averaged or constant quantity

⋁:

modified quantity

⋀:

extended quantity

e:

equilibrium

f:

final quantity

h:

heat

m:

mass diffusion

q:

energy

s:

solid

T:

thermal diffusion

v:

volumetric quantity

0:

initial state (t=0)

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  1. Institute of Chemical Engineering, Warsaw Technical University Warynskiego 1, 00-645, Warsaw, Poland

    Stanisław Sieniutycz

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Sieniutycz, S. A synthesis of some variational theorems for the extended irreversible thermodynamics of nonstationary heat and mass transfer. Appl. Sci. Res. 42, 211–228 (1985). https://doi.org/10.1007/BF00539341

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