Integral governing equations

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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. A mathematical relation that allows one to mathematically and physically link the conservation laws for a control volume with that of a fixed-mass system will be derived.
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==Transformation Formula==
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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.  
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Figure 2.1 shows the flow field under consideration. At time <math>t</math>, the control volume shown by the solid line coincides with a single-phase fixed-mass system depicted by the dashed line. At time <math>t+dt</math>, a portion of the fixed-mass system moves outside of the boundaries of the control volume. It can be seen from Fig. 2.1 that region I is occupied by the system at time <math>t</math> only, region II is common to the system at both <math>t</math> and <math>t+dt</math>, and region III is occupied by the system at <math>t+dt</math> only. For a system with a fixed-mass, the change of the general property <math>\Phi</math>, which has a specific value (per unit mass) <math>\phi</math>, can be written as [[#References|(Welty, 1978)]]:
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<center><math>{\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} }</math></center>
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<center><math>    \qquad \qquad( ) </math></center>(2.1)
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Considering that the control mass occupies regions I and II at time <math>t</math>, one obtains <math></math> At time <math>t+dt</math>, the fixed-mass system occupies both regions II and III, i.e., <math></math> Therefore, eq. (2.1) can be rewritten as
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''See Main Article'' [[Transformation formula]]
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==Continuity Equation==
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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
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<center><math>\frac{\partial }{{\partial t}}\int_V {\rho dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} }  = 0    </math></center>
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''See Main Article'' [[Integral continuity equation]]
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==Momentum==
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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
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<center><math>     \qquad \qquad( ) </math></center>(2.2)
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<center><math>\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}  </math></center>
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<center>
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[[Image:Gov_(1).jpg|400 px|alt=Relation between a fixed mass system and a control volume.|Figure 1: Relation between a fixed mass system and a control volume.]]</center>
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The first term on the right-hand side of eq. (2.2) may be written as 
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<center><math></math></center>
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''See Main Article'' [[Integral momentum equation]]
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and represents the rate of change of property <math>\Phi</math> within the control volume, because region II becomes coincident with the control volume as <math></math>. For the general case of variable <math>\Phi</math> within a control volume, it is appropriate to write the time derivative of <math>\Phi</math> as
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<center><math></math></center>
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==Energy==
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where <math>V</math> is the volume of the control volume.
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The second and third terms on the right-hand side of eq. (2.2), respectively, represent the property <math>\Phi</math> leaving and entering the control volume due to mass flow across its boundary. If the absolute velocity is <math>V</math>, and the reference frame moves with a constant velocity <math>{V_{ref}}</math>, the relative velocity is <math></math>. For the control volume’s entire surface area, <math>A</math>, the rate of movement of property <math>\Phi</math> due to mass flow may be written as (see Fig. 2.2)
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<center><math></math></center>
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The first law of thermodynamics can be applied to a control volume with one phase to obtain the following integral energy equation:
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<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>
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where <math>n</math> is the normal direction of the control volume.
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<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}  </math></center>
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Therefore, eq. (2.2) can be rewritten as
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<center><math>    \qquad \qquad( )</math></center>(2.3)
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''See Main Article'' [[Integral energy equation]]
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<center>[[Image:Gov_(2).jpg|400 px|alt=Control volume in a flow field|Figure 1: Control volume in a flow field.]]</center>
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==Entropy==
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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:
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which is the final form of the transformation formula that relates the change of property for a fixed-mass system to that of the control volume. It states that the rate of change of a property <math>\Phi</math> for a fixed-mass system is equal to the rate of change of <math>\Phi</math> in the control volume (the first term on the right-hand side) plus the net rate of efflux of <math>\Phi</math> by mass flow into or out of the control volume (the second term on the right-hand side).
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<center><math>\begin{array}{l}
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\frac{d}{{dt}}\int_V {\rho sdV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})sdA} }  \\
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  + \int_A {\frac{{{\mathbf{q''}} \cdot {\mathbf{n}}}}{T}dA}  - \int_V {\frac{{q'''}}{T}} dV = \int_V {{{\dot s'''}_{gen}}} dV \ge 0 \\
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\end{array}
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    </math></center>
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It should be pointed out that the control volume moves with the reference frame, which moves with a constant velocity, <math>{V_{ref}}</math>. The coordinate system is attached to and moves with the reference frame. In other words, the coordinate system is stationary relative to the reference frame. The reference frame is inertial, so Newton’s second law is valid in the coordinate system that moves with the reference frame. Equation (2.3) will be used to obtain the macroscopic (integral) formulation of the basic laws for a control volume.
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''See Main Article'' [[Integral entropy equation]]
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==Conservation of Mass Species==
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For a system containing one phase but more than one component, the total mass of the system is composed of different species. If the concentrations of each of these species are not uniform, mass transfer occurs in a way that makes the concentrations more uniform. Therefore, it is necessary to track the individual components by applying the principle of conservation of species mass. The conservation of species mass that contains only one phase is:
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<center><math>\frac{\partial }{{\partial t}}\int_V {{\rho _i}dV + \int_A {{\rho _i}({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} }  =  - \int_A {{{\mathbf{J}}_i} \cdot {\mathbf{n}}dA}  + \int_V {{{\dot m'''}_i}dV}  </math></center>
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''See Main Article'' [[Integral conservation of mass species equation]]
==References==
==References==
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Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Burlington, MA.
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Faghri, A., Zhang, Y., and Howell, J. R., 2010, ''Advanced Heat and Mass Transfer'', Global Digital Press, Columbia, MO.
==Further Reading==
==Further Reading==
==External Links==
==External Links==

Current revision as of 13:40, 5 August 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.

{\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} }

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

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}

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}

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}

See Main Article Integral entropy equation

Conservation of Mass Species

For a system containing one phase but more than one component, the total mass of the system is composed of different species. If the concentrations of each of these species are not uniform, mass transfer occurs in a way that makes the concentrations more uniform. Therefore, it is necessary to track the individual components by applying the principle of conservation of species mass. The conservation of species mass that contains only one phase is:

\frac{\partial }{{\partial t}}\int_V {{\rho _i}dV + \int_A {{\rho _i}({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} }  =  - \int_A {{{\mathbf{J}}_i} \cdot {\mathbf{n}}dA}  + \int_V {{{\dot m'''}_i}dV}

See Main Article Integral conservation of mass species equation

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.

Further Reading

External Links