System and Control Volume Approaches

Let V(x, y, z, t) be the velocity vector field given in a stationary space coordinate system x, y, z and time coordinate t. Shown in Fig. 2.4 is the closed system composed of air coalescing with a control volume at time t. The control volume remains the same at time t + At the system, however, as the collection of same particles, moves and deforms with the flow as shown in Fig. 2.4.

Let N be the total thermodynamical property in our system. Because of the flow field, there will be a change with time in the property N as DN/Dt. Let g be the specific and local value of property N, which is distributed throughout the control volume. The total value of this property can be represented as an integral as follows: N = Jgp dV. Here, dV shows the infinitesimal volume element in the control volume. Now, we can relate the time rate of change of property g in the control volume in terms of its flux through the control surface as the control volume coincides with the system as At approaches zero. Under this condition, the flux of g from the control surface will be fflgp(V ■ dA), (Fox and McDonald 1992). If we consider the limiting case as the system coinciding with the control volume, the total derivative of the property N in the system can be related to the control volume as follows

DNN=I# gp# gp(VdV) (2.27)

where V = ~. Now, we can apply the conservation laws of mechanics to Eq. 2.27 and obtain the strong forms of the governing equations.