4.3.2 Thermal flux

The thermal flux consists of conductive and convective transports, proportional to ∇T and ∇P, respectively. Neglecting the solute diffusion in Eq. (101), the coupled equations of heat and fluid flows are simplified as

$$
\begin{bmatrix} J\_q \\ J\_v \end{bmatrix} = \begin{bmatrix} a & \beta \\ \beta & \gamma \end{bmatrix} \begin{bmatrix} X\_q \\ X\_v \end{bmatrix} \tag{113}
$$

using Onsager's reciprocal relationship in that the off-diagonal coefficients are symmetrical. The driving forces are

$$
\begin{bmatrix} X\_q \\ X\_\nu \end{bmatrix} = \begin{bmatrix} \nabla \mathbf{1}/T \\ \nabla P/T \end{bmatrix} = \frac{-\mathbf{1}}{T^2} \begin{bmatrix} \mathbf{1} & \mathbf{0} \\ P & -T \end{bmatrix} \begin{bmatrix} \nabla T \\ \nabla P \end{bmatrix} \tag{114}
$$

The substitution of Eq. (113) into (114) gives

$$\begin{aligned} \begin{bmatrix} -T^2 \\ J\_\nu \end{bmatrix} &= \begin{bmatrix} a & \beta \\ \beta & \gamma \end{bmatrix} \begin{bmatrix} 1 & 0 \\ P & -T \end{bmatrix} \begin{bmatrix} \nabla T \\ \nabla P \end{bmatrix} \\ &= \begin{bmatrix} a + P\beta & -T\beta \\ \beta + P\gamma & -T\gamma \end{bmatrix} \begin{bmatrix} \nabla T \\ \nabla P \end{bmatrix} \end{aligned} \tag{115}$$

or

$$-T^{2}\begin{bmatrix}J\_{q}/\beta\\J\_{v}/\gamma\end{bmatrix} = \begin{bmatrix}a/\beta + P & -T\\\beta/\gamma + P & -T\end{bmatrix} \begin{bmatrix}\nabla T\\\nabla P\end{bmatrix} \tag{116}$$

Subtracting the first row by the second row of Eq. (116) provides

Fundamentals of Irreversible Thermodynamics for Coupled Transport DOI: http://dx.doi.org/10.5772/intechopen.86607

$$\left(\frac{I\_q}{\beta} - \frac{J\_v}{\gamma}\right) = -\frac{1}{T^2} \left(\frac{a}{\beta} - \frac{\beta}{\gamma}\right) \nabla T \tag{117}$$

$$J\_q = \frac{\beta}{\chi} J\_v - \frac{1}{T^2} \left( a - \beta \frac{\beta}{\chi} \right) \nabla T \tag{118}$$

Through physical interpretation, one can conclude that

$$\frac{\beta}{\gamma} = \tilde{h} \tag{119}$$

where ~ h represents the system enthalpy as a function of temperature. Finally, the coupled heat transfer equation is

$$J\_q = -\kappa\_q \nabla T + \tilde{h} J\_v \tag{120}$$

where

<sup>T</sup> <sup>∂</sup>μ<sup>s</sup> ∂c � ��<sup>1</sup>

which leads to

Non-Equilibrium Particle Dynamics

4.3.2 Thermal flux

or

96

where NA is the Avogadro constant.

mole number per unit volume is straightforward.

symmetrical. The driving forces are

Xq Xv � �

The substitution of Eq. (113) into (114) gives

�T<sup>2</sup> Jq Jv

�T<sup>2</sup> Jq=<sup>β</sup> Jv=γ � �

" #

coupled equations of heat and fluid flows are simplified as

Jq Jv � �

<sup>¼</sup> <sup>∇</sup>1=<sup>T</sup> ∇P=T � �

> <sup>¼</sup> α β β γ

Subtracting the first row by the second row of Eq. (116) provides

T

Lss <sup>¼</sup> kBT 3πηdp

<sup>¼</sup> <sup>T</sup> <sup>1</sup> RT=c

> c <sup>R</sup> <sup>¼</sup> NATc 3πηdp

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

2 <sup>c</sup> <sup>¼</sup> <sup>R</sup> D J 2 ss c

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

The thermal flux consists of conductive and convective transports, proportional

to ∇T and ∇P, respectively. Neglecting the solute diffusion in Eq. (101), the

<sup>¼</sup> α β β γ � � Xq

using Onsager's reciprocal relationship in that the off-diagonal coefficients are

<sup>¼</sup> �<sup>1</sup> T2

" # 1 0

<sup>¼</sup> <sup>α</sup>=<sup>β</sup> <sup>þ</sup> <sup>P</sup> �<sup>T</sup> β=γ þ P �T � � ∇T

<sup>¼</sup> <sup>α</sup> <sup>þ</sup> <sup>P</sup><sup>β</sup> �T<sup>β</sup> β þ Pγ �Tγ

P �T

" # ∇T

Xv � �

1 0 P �T

" # ∇T

� � ∇T

∇P � �

∇P

∇P � �

" # (115)

∇P

" #

<sup>σ</sup><sup>s</sup> <sup>¼</sup> DR ð Þ <sup>∇</sup><sup>c</sup>

¼ c

<sup>R</sup> (110)

(111)

(112)

(113)

(114)

(116)

$$\kappa\_q = a - \beta \frac{\tilde{h}}{T^2} \tag{121}$$

is the thermal conductivity.
