# Differential formulation of governing equations

(Difference between revisions)
 Revision as of 20:52, 9 November 2009 (view source) (Created page with '
Differential Formulation
===Continuity=== ''See Main Article'' Continuity ===Momentum=== ''See Main Article'' [[Momentum equation|Momen…')← Older edit Current revision as of 13:41, 5 August 2010 (view source) (10 intermediate revisions not shown) Line 1: Line 1: -
Differential Formulation
+ 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=== + ==Continuity Equation== - ''See Main Article'' [[Continuity equation|Continuity]] +
$\frac{{D\rho }}{{Dt}} + \rho \nabla \cdot {{\mathbf{V}}_{rel}} = 0$
+ ''See Main Article'' [[Continuity equation]] - ===Momentum=== + ==Momentum Equation == - ''See Main Article'' [[Momentum equation|Momentum]] + - ===Energy=== +
$\rho \frac{{D{{\mathbf{V}}_{rel}}}}{{Dt}} = \sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} - \nabla p + \nabla \cdot (\mu \nabla {{\mathbf{V}}_{rel}})$
- ''See Main Article'' [[Energy equation|Energy]] + - ===Entropy=== + ''See Main Article'' [[Momentum equation]] - ''See Main Article'' [[Entropy equation|Entropy]] + + ==Energy Equation == +
$\rho {c_p}\frac{{DT}}{{Dt}} = \nabla \cdot (k\nabla T) + T\beta \frac{{Dp}}{{Dt}} + q''' + \nabla {{\mathbf{V}}_{rel}}:{{\mathbf{\tau }}_{rel}}$
+ + ''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 + +
${\mathbf{s''}} = \frac{1}{T}\left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right)$
+ +
$T{\dot s'''_{gen}} = - \left( {{\mathbf{q''}} - \sum\limits_{i = 1}^N {{h_i}{{\mathbf{J}}_i}} } \right) \cdot \nabla \ln T - \sum\limits_{i = 1}^N {\left( {{{\mathbf{J}}_i} \cdot \frac{{c{R_u}T}}{{{\rho _i}}}{{\mathbf{d}}_i}} \right)} - {\mathbf{\tau }}:\nabla {\mathbf{V}} - \sum\limits_{i = 1}^N {\frac{{{{\bar g}_i}}}{{{M_i}}}{{\dot m'''}_i}}$
+ + ''See Main Article'' [[Entropy equation]] + + ==Conservation of mass species equation== + +
$\rho \frac{{D{\omega _i}}}{{Dt}} = - \nabla \cdot {{\mathbf{J}}_i} + {\dot m'''_i}$
- ===Conservation of Mass Species=== ''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'', 3rd 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'', 5th ed., John Wiley & Sons, New York. + + Kays, W.M., Crawford, M.E., and Weigand, B., 2004, ''Convective Heat Transfer'', 4th ed., McGraw-Hill, New York, NY. + + White, F.M., 1991, ''Viscous Fluid Flow'', 2nd 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.

## Continuity Equation

$\frac{{D\rho }}{{Dt}} + \rho \nabla \cdot {{\mathbf{V}}_{rel}} = 0$

See Main Article Continuity equation

## Momentum Equation

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

See Main Article Momentum equation

## Energy Equation

$\rho {c_p}\frac{{DT}}{{Dt}} = \nabla \cdot (k\nabla T) + T\beta \frac{{Dp}}{{Dt}} + q''' + \nabla {{\mathbf{V}}_{rel}}:{{\mathbf{\tau }}_{rel}}$

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

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

See Main Article Entropy equation

## Conservation of mass species equation

$\rho \frac{{D{\omega _i}}}{{Dt}} = - \nabla \cdot {{\mathbf{J}}_i} + {\dot m'''_i}$

See Main Article Conservation of Mass Species

## References

Bejan, A., 2004, Convection Heat Transfer, 3rd 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, 5th ed., John Wiley & Sons, New York.

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

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