# Integral governing equations

(Difference between revisions)
 Revision as of 03:47, 27 June 2010 (view source) (→Energy)← Older edit Revision as of 04:29, 27 June 2010 (view source) (→Entropy)Newer edit → Line 26: Line 26: ''See Main Article'' [[Integral energy equation]] ''See Main Article'' [[Integral energy equation]] - ===Entropy=== + ==Entropy== - ''See Main Article'' [[Integral entropy equation|Entropy]] + 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 integral form of the second law of thermodynamics for single phase systems: - + +
$\begin{array}{l} + \frac{d}{{dt}}\int_V {\rho sdV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA} } \\ + + \int_A {\frac{{{\mathbf{q''}} \cdot {\mathbf{n}}}}{T}dA} - \int_V {\frac{{q'''}}{T}} dV = \int_V {{{\dot s'''}_{gen}}} dV \ge 0 \\ + \end{array} + \qquad \qquad(4)$
+ + ''See Main Article'' [[Integral entropy equation]] + ===Conservation of Mass Species=== ===Conservation of Mass Species=== ''See Main Article'' [[Integral conservation of mass species equation|Conservation of Mass Species]] ''See Main Article'' [[Integral conservation of mass species equation|Conservation of Mass Species]]

## Transformation Formula

A fixed-mass system describes an amount of matter that can move, flow and interact with the surroundings, but the control volume approach depicts a region or volume of interest in a flow field, which is not unique and depends on the user. So conservation laws for a fixed-mass system need to be transformed to apply to a control volume. The transformation Formula allows one to mathematically and physically link the conservation laws for a control volume with that of a fixed-mass system.

See Main Article Transformation formula

## Continuity Equation

The law of the conservation of mass dictates that mass may be neither created nor destroyed. For a control volume that contains only one phase, the integral continuity equation is

$\frac{\partial }{{\partial t}}\int_V {\rho dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} } = 0 \qquad \qquad(1)$

See Main Article Integral continuity equation

## Momentum

Newton’s second law states that, in an inertial reference frame, the time rate of momentum change of a fixed mass system is equal to the net force acting on the system, and it takes place in the direction of the net force. The momentum equation for the control volume of a single-phase system is

$\begin{array}{l} \int_V {\left[ {\sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} } \right]} dV + \int_A {{{{\mathbf{\tau '}}}_{rel}} \cdot {\mathbf{n}}dA} \\ = \frac{\partial }{{\partial t}}\int_V {\rho {{\mathbf{V}}_{rel}}dV} + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}}){{\mathbf{V}}_{rel}}dA} \\ \end{array} \qquad \qquad(2)$

See Main Article Integral momentum equation

## Energy

The first law of thermodynamics can be applied to a control volume with one phase to obtain the following integral energy equation:

$\frac{\partial }{{\partial t}}\int_V {\rho \left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2}} \right)dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})\left( {e + \frac{{{\mathbf{V}}_{rel}^2}}{2}} \right)dA} }$
$= - \int_A {{\mathbf{q''}} \cdot {\mathbf{n}}dA + \int_V {q'''dV} } + \int_A {({\mathbf{n}} \cdot {{{\mathbf{\tau '}}}_{rel}}) \cdot {{\mathbf{V}}_{rel}}dA} + \int_V {\left[ {\sum\limits_{i = 1}^N {{\rho _i}{{\mathbf{X}}_i}} } \right] \cdot {{\mathbf{V}}_{rel}}dV} \qquad \qquad(3)$

See Main Article Integral energy equation

## Entropy

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 integral form of the second law of thermodynamics for single phase systems:

$\begin{array}{l} \frac{d}{{dt}}\int_V {\rho sdV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA} } \\ + \int_A {\frac{{{\mathbf{q''}} \cdot {\mathbf{n}}}}{T}dA} - \int_V {\frac{{q'''}}{T}} dV = \int_V {{{\dot s'''}_{gen}}} dV \ge 0 \\ \end{array} \qquad \qquad(4)$

See Main Article Integral entropy equation

### Conservation of Mass Species

See Main Article Conservation of Mass Species