(b) Conservation of Energy applied to a Control Volume

Let = total mass inside the CV at time

total mass inside the CV at time

specific energy of matter inside the control volume at time t

= specific energy of matter inside the control volume at time

and = Pressure at the inlet and exit ports respectively

and = Flow velocity at the inlet and exit ports respectively

and = specific volumes at the inlet and exit ports respectively

and = specific energy of the material at the inlet and exit ports respectively

= Rate energy flow as heat into the control volume

= Rate of shaft work done by the control volume

The mass contained in the region A which enters the control volume during the time interval

The mass contained in the region B which leaves the control volume during the time interval

From the mass balance, we can write

(12.6)

where is the mass entering the control volume during the differential time . To accommodate this, the mass inside the control volume has to be compressed such that its volume decreases by the amount . This is accomplished by the pressure acting on the material entering the control volume. Therefore, the work done =

Since the mass has to leave the control volume at the exit port, the work done = ,

Energy of the system at time

Energy of the system at time

During the time interval We may account for the following

Energy transferred as heat to the system =

Shaft work done by the system

Energy flow as heat into the control volume and the shaft work delivered by the control volume are taken as positive. By applying the first law of thermodynamics, we get

(12.7)

or,

(12.8)

or,

(12.9)

In the limiting case,

(12.10)

Where,

and

elevation of the exit and inlet ports above the datum level. The above expression can now be rearranged as

(12.11)

Rate of energy accumulation = Rate of energy inflow - Rate of energy outflow