4.3.1 Solute diffusion

The primary driving force for the solute transport is Xs ¼ �∇ð Þ μs=T , if temperature and pressure gradients are not significant in solute transport. We consider the diffusive flux of solute only in an isothermal and isobaric process and neglect terms of Lqs and Lvs:

$$J\_s = -L\_{sr} \nabla \frac{\mu\_s}{T} = -\frac{L\_{sr}}{T} \nabla \mu\_s \tag{102}$$

which is equivalent to Fick's law of

$$J\_s = -D\nabla \mathfrak{c} \tag{103}$$

where D [m<sup>2</sup> /s] is a solute diffusion coefficient. If Eq. (102) is expressed in terms of concentration gradient, we have

$$J\_{sr} = -\frac{L\_{sr}}{T} \left(\frac{\partial \mu\_s}{\partial c}\right) \nabla c \tag{104}$$

By Eqs. (103) and (104), one can find

$$\frac{L\_{\rm sr}}{T} \left( \frac{\partial \mu\_{\rm s}}{\partial c} \right)\_{T} = D \tag{105}$$

Then, the entropy-changing rate based on the solute transport is calculated as

$$
\sigma\_{\rm s} = f\_{\rm s} X\_{\rm s} = \frac{L\_{\rm sr}}{T^2} (\nabla \mu\_{\rm s})^2 = \frac{D/T}{(\partial \mu\_{\rm s}/\partial \mathfrak{c})\_T} (\nabla \mu\_{\rm s})^2 \tag{106}
$$

Next, we consider the Stokes-Einstein diffusivity:

$$D\_{\rm SE} = \frac{k\_B T}{3\pi \eta d\_p} \tag{107}$$

where kB is Boltzmann constant, η is the solvent viscosity, and dp is the diameter of a particle diffusing within the solvent medium. The phenomenological coefficient Lqq is represented as

$$L\_{\rm ss} = \frac{D\_{\rm SE} \, T}{(\partial \mu\_s / \partial c)\_T} = \frac{k\_B T}{3 \pi \eta d\_p} \cdot T \left(\frac{\partial \mu\_s}{\partial c}\right)\_T^{-1} \tag{108}$$

For weakly interacting solutes, the solute chemical potential is

$$
\mu = \mu\_0 + \mathcal{R}T \ln a \tag{109}
$$

where μ<sup>0</sup> is generally a function of T and P, which are constant in this equation, R is the gas constant, and a is the solute activity. For a dilute solution, the activity represented by a is often approximated as the concentration c (i.e., a≃c). The proportionality between Lss and DSE is

$$T\left(\frac{\partial\mu\_s}{\partial c}\right)\_T^{-1} = T\frac{\mathbf{1}}{\mathcal{R}T/c} = \frac{c}{\mathcal{R}}\tag{110}$$

Jq <sup>β</sup> � Jv γ 

Fundamentals of Irreversible Thermodynamics for Coupled Transport

Jq <sup>¼</sup> <sup>β</sup> γ Jv � <sup>1</sup>

Through physical interpretation, one can conclude that

where ~

where

97

the coupled heat transfer equation is

DOI: http://dx.doi.org/10.5772/intechopen.86607

is the thermal conductivity.

5. Concluding remarks

¼ � <sup>1</sup> T2

> β <sup>γ</sup> <sup>¼</sup> <sup>~</sup>

Jq ¼ �κq∇<sup>T</sup> <sup>þ</sup> <sup>~</sup>

κ<sup>q</sup> ¼ α � β

In this chapter, we investigated diffusion phenomenon as an irreversible process. By thermodynamic laws, entropy always increases as a system of interest evolves in a non-equilibrium state. The entropy-increasing rate per unit volume is a measure of how fast the system changes from the current to a more disordered state. Entropy concept is explained from the basic mathematics using several examples. Diffusion phenomenon is explained using (phenomenological) Fick's law, and more fundamental theories were summarized, which theoretically derive the diffusion coefficient and the convection-diffusion equation. Finally, the dissipation rate, i.e., entropy-changing rate per volume, is revisited and obtained in detail. The coupled, irreversible transport equation in steady state is applied to solute diffusion in an isothermal-isobaric process and heat transfer that is consisting of the conductive and convective transport due to the temperature gradient and fluid flow, respectively. As engineering processes are mostly open in the steady state, the theoretical approaches discussed in this chapter may be a starting point of the future

development in irreversible thermodynamics and statistical mechanics.

α β � β γ

γ

<sup>T</sup><sup>2</sup> <sup>α</sup> � <sup>β</sup> <sup>β</sup>

h represents the system enthalpy as a function of temperature. Finally,

~ h

<sup>∇</sup><sup>T</sup> (117)

<sup>∇</sup><sup>T</sup> (118)

h (119)

hJv (120)

<sup>T</sup><sup>2</sup> (121)

which leads to

$$L\_{ss} = \frac{k\_B T}{3\pi\eta d\_p} \frac{c}{\mathcal{R}} = \frac{N\_A T c}{3\pi\eta d\_p} \tag{111}$$

where NA is the Avogadro constant.

For a dilute isothermal solution, we represent the entropy-changing rate as

$$
\sigma\_{\rm s} = DR \frac{\left(\nabla c\right)^{2}}{c} = \frac{R}{D} \frac{J\_{\rm s}^{2}}{c} \tag{112}
$$

for an isothermal and isobaric process. Assuming that D is not a strong function of c, Eq. (112) indicates that the diffusive entropy rate σ<sup>s</sup> is unconditionally positive (as expected), increases with the diffusive flux, and decreases with the concentration c. Within this analysis, c is defined as molar or number fraction of solute molecules to the solvent. For a dilute solution, conversion of c to a solute mass or mole number per unit volume is straightforward.
