# Integral governing equations

(Difference between revisions)
 Revision as of 00:59, 6 November 2009 (view source)← Older edit Revision as of 01:04, 6 November 2009 (view source) (Blanked the page)Newer edit → Line 1: Line 1: - 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 $\Phi = m$ and $\phi = 1$ in eq. (2.3), i.e., -
${\left. {\frac{{dm}}{{dt}}} \right|_{system}} = \frac{\partial }{{\partial t}}\int_V {\rho dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} } \qquad \qquad(1)$
- - 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 $({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA$ 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 $dA$, and becomes the mass flowrate when multiplied by density, ''ρ''. - - 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 - -
$\frac{\partial }{{\partial t}}\int_V {\rho dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} } = 0 \qquad \qquad(2)$
$\sum\limits_{k = 1}^\Pi {\left[ {\frac{\partial }{{\partial t}}\int_{{V_k}(t)} {{\rho _k}dV} + \int_{{A_k}(t)} {{\rho _k}({{\mathbf{V}}_{k,rel}} \cdot {{\mathbf{n}}_k})dA} } \right]} = 0 \qquad \qquad(3)$
- - where $k$ denotes the $k$th phase in the multiphase system, and $\Pi$ is the number of phases. - - ==References== - - Faghri, A., and Zhang, Y., 2006, ''Transport Phenomena in Multiphase Systems'', Burlington, MA. - - ==Further Reading== - - ==External Links==