Integral governing equations

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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, conservation of mass can be obtained by setting the general and specific property forms to <math>\Phi = m</math> and <math>\phi  = 1</math> in eq. (2.3), i.e.,
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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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<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>{\left. {\frac{{dm}}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} }      \qquad \qquad(1)</math></center>
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''See Main Article'' [[Transformation formula]]
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where the first term on the right hand side represents the time rate of change of the mass of the contents of the control volume, and the second term on the right hand side represents the net rate of mass flow through the control surface. The term <math>({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA</math> in the mass flow integral represents the product of the velocity component perpendicular to the control surface and differential area. This term is the volume flowrate through <math>dA</math>, and becomes the mass flowrate when multiplied by density, ''ρ''.
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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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Since the mass of a fixed-mass system is constant by definition, and the fixed-mass system contains only one phase, the resulting formulation of conservation of mass 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    \qquad \qquad(2)</math></center>
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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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which shows that the time rate of change of the mass of the contents of the control volume plus the net rate of mass flow through the control surface must equal zero. In other words, the sum of the mass flow rate into and out of the control volume must be equal to the accumulation and depletion of the mass within the control volume.
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''See Main Article'' [[Integral continuity equation]]
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For a control volume containing multiple phases separated by interfaces, the conservation of mass can be similarly obtained [[#References|(Faghri and Zhang, 2006)]]:
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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>\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><math>\sum\limits_{k = 1}^\Pi  {\left[ {\frac{\partial }{{\partial t}}\int_{{V_k}(t)} {{\rho _k}dV}  + \int_{{A_k}(t)} {{\rho _k}({{\mathbf{V}}_{k,rel}} \cdot {{\mathbf{n}}_k})dA} } \right]}  = 0    \qquad \qquad(3)</math></center>
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''See Main Article'' [[Integral momentum equation]]
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where <math>k</math> denotes the <math>k</math><sup>th</sup> phase in the multiphase system, and <math>\Pi </math> is the number of phases.
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==Energy==
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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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<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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''See Main Article'' [[Integral energy equation]]
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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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<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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''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==
Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Burlington, MA.
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