Integral governing equations

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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


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


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


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:

 \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 \\ 

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


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

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