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Entropy of flux relaxation and variational theory of simultaneous energy and mass transport governed by non-Onsager phenomenological equations

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Abstract

Using the relaxation phenomenological equations describing the non-stationary coupled heat and mass transport in the locally non-equilibrium fluid, an entropy of the diffusive fluxes of mass and energy is obtained. It is called the relaxation entropy as it is associated with the tendency of every element of the continuum to recover the thermodynamic equilibrium during vanishing of the fluxes. By exploiting this relaxation entropy a function of thermodynamical action is introduced, having the dimension of the product of entropy and time, whose stationarity conditions are the phenomenological and the conservation equations (constituting the hyperbolic set). The engineering significance of the variational principle given consists in finding approximate fields of transfer potentials and fluxes using the direct variational methods.

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Abbreviations

a :

thermal diffusivity

C p :

specific heat

C=[c ik]:

capacity matrix, equation (5)

c 0 :

propagation speed of second sound wave (assumed as a constant quantity)

D :

generalized matrix of diffusivities, equation (4)

H i :

Biot's vector connected with flux J i (J i =H i )

G=[g ik]:

inertial matrix, equation (2)

H=col (H 1, H 2 ... H n−1, H q ):

column matrix of Biot's vectors

h :

specific enthalpy

J q :

density of diffusive energy flux

J 1, J 2 ...J n−1 :

densities of diffusive mass fluxes

J=col (J 1, J 2 ...J n−1, J q ):

column matrix containing all independent fluxes

L=[L ik]:

Onsager matrix

M, M :

molar mass, matrix exponential function, respectively

r :

radius vector

s, s′ :

specific static entropy and total entropy, respectively

S, ST :

action functional and its surface term, respectively

T :

temperature

t :

time

\(u = {\text{col}}\left( {\frac{{\mu _n - \mu _1 }}{T} \ldots ,\frac{{\mu _n - \mu _{n - 1} }}{T},\frac{1}{T}} \right)\) :

column matrix of transfer potentials

V :

volume

\(\begin{gathered} X = {\text{col}}\left( {{\text{grad}}\frac{{\mu _n - \mu _1 }}{T} \ldots ,} \right. \hfill \\ \left. { {\text{grad}}\frac{{\mu _n - \mu _{n - 1} }}{T},{\text{grad}}\frac{1}{T}} \right) \hfill \\ \end{gathered} \) :

column matrix of classical thermodynamic forces

x, y, z :

cartesian coordinates

y=col (y 1, y 2 ...y n−1):

column matrix of independent mass fractions

z=col (y 1, y 2 ...y n−1, h):

column matrix of thermodynamic state

∇2 :

Laplace operator

Δs r :

relaxation entropy of unit mass

σ, σ′ :

classical and non-classical entropy source, respectively

δ :

variational symbol

ρ :

mass density

τ :

matrix of relaxation coefficients

μ i :

chemical potential of component i

Λ:

Lagrangian

q′ :

heat

q :

energy in coupled process

r :

relaxation

T :

transpose matrix

−1:

reverse matrix

∼:

functional for irreversible process

□:

specified distribution of potentials, u=u □, or specified normal component of vector H i (H □ i )

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Authors and Affiliations

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

    Stanisław Sieniutycz

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Sieniutycz, S. Entropy of flux relaxation and variational theory of simultaneous energy and mass transport governed by non-Onsager phenomenological equations. Appl. Sci. Res. 39, 87–103 (1982). https://doi.org/10.1007/BF00457012

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