Entropy equation

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$\begin{array}{l} \frac{d}{{dt}}\int_V {\rho sdV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA} } \\ + \int_A {\frac{{{\mathbf{q''}} \cdot {\mathbf{n}}}}{T}dA} - \int_V {\frac{{q'''}}{T}} dV = \int_V {{{\dot s'''}_{gen}}} dV \ge 0 \\ \end{array}$

To obtain the differential form of the second law of thermodynamics, the surface integrals in the above equation can be rewritten as volume integrals:

$\int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA} = \int_V {\nabla \cdot (\rho {{\mathbf{V}}_{rel}}s)dV} \qquad \qquad(1)$ $\int_A {\frac{{{\mathbf{q''}}}}{T} \cdot {\mathbf{n}}dA} = \int_V {\nabla \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right)dV} \qquad \qquad(2)$

Substituting eqs. (1) and (2) into the integral entropy equation and considering $\frac{d}{{dt}}\int_V {\rho \phi } dV = \int_V {\frac{{\partial (\rho \phi )}}{{\partial t}}} dV$ yields

$\int_V {\left[ {\frac{\partial }{{\partial t}}\left( {\rho s} \right) + \nabla \cdot (\rho {{\mathbf{V}}_{rel}}s) + \nabla \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T}} \right]dV} = \int_V {{{\dot s'''}_{gen}}dV} \ge 0 \qquad \qquad(3)$

In order for eq. (3) to be true for any arbitrary control volume, the integrand in eq. (3) should always be positive, i.e.,

$\frac{\partial }{{\partial t}}\left( {\rho s} \right) + \nabla \cdot (\rho {{\mathbf{V}}_{rel}}s) + \nabla \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0 \qquad \qquad(4)$

Equation (4) can be rewritten as

$s\left[ {\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot (\rho {{\mathbf{V}}_{rel}})} \right] + \rho \left[ {\frac{{\partial s}}{{\partial t}} + {{\mathbf{V}}_{rel}} \cdot \nabla s} \right] + \nabla \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0 \qquad \qquad(5)$

Considering the continuity equation, $\frac{{\partial \rho }}{{\partial t}} + \nabla \cdot \rho {{\mathbf{V}}_{rel}} = 0$ , and definition of the substantial derivative, $\frac{D}{{Dt}} \equiv \frac{\partial }{{\partial t}} + {{\mathbf{V}}_{rel}} \cdot \nabla$ , eq. (5) can be reduced to

$\rho \frac{{Ds}}{{Dt}} + \nabla \cdot \left( {\frac{{{\mathbf{q''}}}}{T}} \right) - \frac{{q'''}}{T} = {\dot s'''_{gen}} \ge 0 \qquad \qquad(6)$

where the three terms on the left-hand side represent rate of change of entropy per unit volume, rate of change of entropy per unit volume by heat transfer and internal heat generation, respectively. Equation (6) means that the entropy generation per unit volume must not be negative at any time or location.

For a multicomponent system without internal heat generation ( $q''' = 0$ ), Curtiss and Bird (1999; 2001) obtained the entropy flux vector and the entropy generation as

${\mathbf{s''}} = \frac{1}{T}\left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right) \qquad \qquad(7)$ $T{\dot s'''_{gen}} = - \left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right) \cdot \nabla \ln T - \sum\limits_{i = 1}^N {\left( {{{\mathbf{J}}_i} \cdot \frac{{c{R_u}T}}{{{\rho _i}}}{{\mathbf{d}}_i}} \right)} - {\mathbf{\tau }}:\nabla {\mathbf{V}} - \sum\limits_{i = 1}^N {\frac{{{{\bar g}_i}}}{{{M_i}}}{{\dot m'''}_i}} \qquad \qquad(8)$

where $d i$ is the diffusional driving force [see eq. (2) – (6)], ${\bar g_i}$ is partial molar Gibbs free energy, ${\mathbf{q''}}$ is total heat flux, obtained by (see Introduction to Heat Transfer)

${\mathbf{q''}} = - k\nabla T + \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} + c{R_u}T\sum\limits_{i = 1}^N {\frac{{{x_i}{x_j}}}{{{\rho _i}}}\frac{{D_i^T}}{{{{D}_{ij}}}}} \left( {\frac{{{{\mathbf{J}}_i}}}{{{\rho _i}}} - \frac{{{{\mathbf{J}}_j}}}{{{\rho _j}}}} \right)$

which includes conduction, interdiffusional convection, and Dufour effect.

References

Curtiss, C.F., and Bird, R.B., 1999, “Multicomponent Diffusion,” Industrial and Engineering Chemistry Research, Vol. 38, pp. 2115-2522.

Curtiss, C.F., and Bird, R.B., 2001, “Errata,” Industrial and Engineering Chemistry Research, Vol. 40, p. 1791.

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

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Entropy equation

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