# Integral governing equations

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 Φ = m and φ = 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

$\frac{\partial }{{\partial t}}\int_V {\rho dV + \int_A {\rho ({{\mathbf{V}}_{rel}} \cdot {\mathbf{n}})dA} } = 0 \qquad \qquad(2)$

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.

For a control volume containing multiple phases separated by interfaces, the conservation of mass can be similarly obtained (Faghri and Zhang, 2006):

$\sum\limits_{k = 1}^\Pi {\left[ {\frac{\partial }{{\partial t}}\int_{{V_k}(t)} {{\rho _k}dV} + \int_{{A_k}(t)} {{\rho _k}({{\mathbf{V}}_{k,rel}} \cdot {{\mathbf{n}}_k})dA} } \right]} = 0 \qquad \qquad(3)$

where k denotes the kth phase in the multiphase system, and Π is the number of phases.

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Burlington, MA.