# Integral entropy equation

The second law of thermodynamics requires that the entropy generation in a closed system (fixed-mass) must be greater than or equal to zero. The entropy change for a system with fixed-mass and only one phase can be obtained by setting Φ = S,φ = s in eq. ${\left. {\frac{{d\Phi }}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho \phi dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\phi dA} }$
from transformation formula i.e., ${\left. {\frac{{dS}}{{dt}}} \right|_{system}} = \frac{d}{{dt}}\int_V {\rho sdV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA} } \qquad \qquad(1)$

The change of entropy in a closed system results from heat transfer and/or entropy generation: ${\left. {\frac{{dS}}{{dt}}} \right|_{system}} = \int_A {\frac{{ - {\mathbf{q''}} \cdot {\mathbf{n}}}}{T}dA} + \int_V {\frac{{q'''}}{T}} dV + \int_V {{{\dot s'''}_{gen}}} dV \qquad \qquad(2)$

where, on the right-hand side of eq. (2), the first term represents the change of entropy due to heat transfer across the boundary of the control volume, and the second term represents the change of entropy due to internal heat generation in the control volume. The last term represents entropy generation, which should always be greater than or equal to zero, i.e., $\int_V {{{\dot s'''}_{gen}}} dV \ge 0 \qquad \qquad(3)$

Combining eqs. (1) and (2) and applying eq. (3), one obtains the integral form of the second law of thermodynamics for single phase systems: $\begin{array}{l} \frac{d}{{dt}}\int_V {\rho sdV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA} } \\ + \int_A {\frac{{{\mathbf{q''}} \cdot {\mathbf{n}}}}{T}dA} - \int_V {\frac{{q'''}}{T}} dV = \int_V {{{\dot s'''}_{gen}}} dV \ge 0 \\ \end{array} \qquad \qquad(4)$

If the control volume includes Π phases, the second law of thermodynamics must be obtained by integrating over the two phases separately (Faghri and Zhang, 2006), $\sum\limits_{k = 1}^\Pi {\left[ {\frac{d}{{dt}}\int_{{V_k}(t)} {{\rho _k}{s_k}dV} + \int_{{A_k}(t)} {{\rho _k}({{\mathbf{V}}_{k,rel}} \cdot {{\mathbf{n}}_k}){s_k}dA} + \int_{{A_k}(t)} {\frac{{{{{\mathbf{q''}}}_k} \cdot {{\mathbf{n}}_k}}}{{{T_k}}}dA} - \int_{{V_k}(t)} {\frac{{{{q'''}_k}}}{{{T_k}}}} dV} \right]}$ $= \sum\limits_{k = 1}^\Pi {\int_{{V_k}(t)} {{{\dot s'''}_{gen,k}}} dV} + \int_{{A_I}(t)} {{{\dot s''}_{gen,I}}} dA \ge 0 \qquad \qquad(5)$

The entropy generation for a control volume including Π phases consists of entropy generation in each phase, plus that in the interfaces. The second law of thermodynamics requires that each of these entropy generations be greater than or equal to zero.

## References

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

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