Integral governing equations
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''See Main Article'' [[Integral momentum equation]] | ''See Main Article'' [[Integral momentum equation]] | ||
| - | + | ==Energy== | |
| - | ''See Main Article'' [[Integral energy equation | + | |
| - | + | The first law of thermodynamics can be applied to a control volume with one phase to obtain the following integral energy equation: | |
| + | <center><math>\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} } </math> <br> | ||
| + | <math> = - \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)</math></center> | ||
| + | |||
| + | ''See Main Article'' [[Integral energy equation]] | ||
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===Entropy=== | ===Entropy=== | ||
''See Main Article'' [[Integral entropy equation|Entropy]] | ''See Main Article'' [[Integral entropy equation|Entropy]] | ||
Revision as of 03:47, 27 June 2010
Contents |
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
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)](/encyclopedia/images/math/f/c/6/fc6c5d89a3e034693e01320bec685419.png)
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:
![= - \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)](/encyclopedia/images/math/4/f/9/4f95c23c0d673b0e876eb2a675b49c06.png)
See Main Article Integral energy equation
Entropy
See Main Article Entropy
Conservation of Mass Species
See Main Article Conservation of Mass Species