# Differential formulation of governing equations

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- | + | The microscopic (differential) formulations to be presented here include conservation equations and jump conditions. The former apply within a particular phase, and the latter are valid at the interface that separates two phases. The phase equations for a particular phase should be the same as those for a single-phase system. Most textbooks (e.g., [[#References|White, 1991; Incropera and DeWitt, 2001; Bejan, 2004; Kays ''et al''., 2004)]] obtain the governing equations for a single-phase system by performing mass, momentum, and energy balances for a microscopic control volume. We will obtain the conservation equations by using the integral equations for a finite control volume that includes only one phase. Jump conditions at the interface will be obtained by applying the conservation laws at the interfaces. | |

- | + | ==Continuity Equation== | |

- | ''See Main Article'' [[Continuity equation | + | <center><math>\frac{{D\rho }}{{Dt}} + \rho \nabla \cdot {{\mathbf{V}}_{rel}} = 0 </math></center> |

+ | ''See Main Article'' [[Continuity equation]] | ||

- | + | ==Momentum Equation == | |

- | + | ||

- | == | + | <center><math>\rho \frac{{D{{\mathbf{V}}_{rel}}}}{{Dt}} = \sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} - \nabla p + \nabla \cdot (\mu \nabla {{\mathbf{V}}_{rel}})</math></center> |

- | + | ||

- | ===Entropy=== | + | ''See Main Article'' [[Momentum equation]] |

- | ''See Main Article'' [[Entropy equation | + | |

+ | ==Energy Equation == | ||

+ | <center><math>\rho {c_p}\frac{{DT}}{{Dt}} = \nabla \cdot (k\nabla T) + T\beta \frac{{Dp}}{{Dt}} + q''' + \nabla {{\mathbf{V}}_{rel}}:{{\mathbf{\tau }}_{rel}} </math></center> | ||

+ | |||

+ | ''See Main Article'' [[Energy equation]] | ||

+ | |||

+ | ==Entropy Equation == | ||

+ | |||

+ | For a multicomponent system without internal heat generation (<math>q''' = 0</math>), the entropy flux vector and the entropy generation are | ||

+ | |||

+ | <center><math>{\mathbf{s''}} = \frac{1}{T}\left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right) </math></center> | ||

+ | |||

+ | <center><math>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}} </math></center> | ||

+ | |||

+ | ''See Main Article'' [[Entropy equation]] | ||

+ | |||

+ | ==Conservation of mass species equation== | ||

+ | |||

+ | <center><math>\rho \frac{{D{\omega _i}}}{{Dt}} = - \nabla \cdot {{\mathbf{J}}_i} + {\dot m'''_i} </math></center> | ||

- | |||

''See Main Article'' [[Conservation of mass species equation|Conservation of Mass Species]] | ''See Main Article'' [[Conservation of mass species equation|Conservation of Mass Species]] | ||

+ | |||

+ | ==References== | ||

+ | |||

+ | Bejan, A., 2004, ''Convection Heat Transfer'', 3<sup>rd</sup> ed., John Wiley & Sons, New York. | ||

+ | |||

+ | 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. | ||

+ | |||

+ | Incropera, F.P., and DeWitt, D.P., 2001, ''Fundamentals of Heat and Mass Transfer'', 5<sup>th</sup> ed., John Wiley & Sons, New York. | ||

+ | |||

+ | Kays, W.M., Crawford, M.E., and Weigand, B., 2004, ''Convective Heat Transfer'', 4<sup>th</sup> ed., McGraw-Hill, New York, NY. | ||

+ | |||

+ | White, F.M., 1991, ''Viscous Fluid Flow'', 2<sup>nd</sup> ed., McGraw-Hill, New York. | ||

+ | |||

+ | ==Further Reading== | ||

+ | |||

+ | ==External Links== |

## Current revision as of 13:41, 5 August 2010

The microscopic (differential) formulations to be presented here include conservation equations and jump conditions. The former apply within a particular phase, and the latter are valid at the interface that separates two phases. The phase equations for a particular phase should be the same as those for a single-phase system. Most textbooks (e.g., White, 1991; Incropera and DeWitt, 2001; Bejan, 2004; Kays *et al*., 2004) obtain the governing equations for a single-phase system by performing mass, momentum, and energy balances for a microscopic control volume. We will obtain the conservation equations by using the integral equations for a finite control volume that includes only one phase. Jump conditions at the interface will be obtained by applying the conservation laws at the interfaces.

## Contents |

## Continuity Equation

*See Main Article* Continuity equation

## Momentum Equation

*See Main Article* Momentum equation

## Energy Equation

*See Main Article* Energy equation

## Entropy Equation

For a multicomponent system without internal heat generation (*q*''' = 0), the entropy flux vector and the entropy generation are

*See Main Article* Entropy equation

## Conservation of mass species equation

*See Main Article* Conservation of Mass Species

## References

Bejan, A., 2004, *Convection Heat Transfer*, 3^{rd} ed., John Wiley & Sons, New York.

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.

Incropera, F.P., and DeWitt, D.P., 2001, *Fundamentals of Heat and Mass Transfer*, 5^{th} ed., John Wiley & Sons, New York.

Kays, W.M., Crawford, M.E., and Weigand, B., 2004, *Convective Heat Transfer*, 4^{th} ed., McGraw-Hill, New York, NY.

White, F.M., 1991, *Viscous Fluid Flow*, 2^{nd} ed., McGraw-Hill, New York.