V 1. Conservation of Mass dz dy dx A continuous fluid can be thought of as being made up of a large group of elemental fluid particles. These particles jostle along shoulder to shoulder as the fluid moves.

V 2. Fluid Particle dz dy dx Each particle moves at a velocity (V) and has a volume (dx, dy, dz) and density (). Each has an individual mass ( m= .dx.dy.dz).

3. Conservation of Particle Mass When moving with the flow, particles may be distorted (stretched, compressed, sheared or rotated) due to the pressure and temperature fields that they pass through. However their individual masses remain unchanged.

4. Control Volume  For the control volume (), the mass contained within will be the sum of the masses of all the fluid particles.

5. Flow through Control Volume If the fluid is moving, then all fluid particles can be tracked over a small period of time (t). Particle mass will be conserved. The change in fluid mass in the volume will equal the difference in mass of particles entering compared to mass of particles leaving.

Outflow V (t) C Inflow B 6. Inflow and Outflow This can be written as Mass Change in Volume () = Inflow (B) –Outflow mass(C) Outflow V (t) C Inflow B

In limit as t 0 7. Equations of Mass Flow Mass Change in Volume () = Inflow (B) –Outflow mass(C) In limit as t 0

-Vn dl V dA B  n 8. Inflow Rate The amount of mass entering the control volume through an elemental surface, area dA, can be calculated using the component of velocity normal to the surface -Vn dl V Thus incoming flow rate is dA B  n

Vn  V dl dA C 9. Outflow Rate The amount of mass exiting the control volume through surface, dA, can also be calculated using the component of velocity normal to this surface Vn  V dl Thus the flow rate through the exit surface is dA C

10. Integral Equation Form Substituting for the incoming and outgoing flow rates in the mass conservation balance for volume () leads to the result, Integral Form