Mass and energy analysis of control volumes undergoing unsteady processes

Studying unsteady systems: conservation of mass

Integrating both sides

( ) (0)CV CVinlets outlets

m t m m m

CV

inlets outlets

mdm

mdt

V+

-

Studying unsteady systems with the conservation of mass and energy equation

• General energy equation with the assumption of uniform flow at inlets/outlets

2 2( / 2 ) ( / 2 )CV

inlets ouCV

tlets

m h V gz mdE

Wz Qt

hd

V g

2 2( ) (0) ( / 2 ) ( / 2 )CV CVinle

CVts outlets

t E m h V gz dt m h V gz dt QE W

Time integrated form

2 2( ) (0) ( / 2 ) ( / 2 )CV CVinlets outlet

CVs

t E m h V gz m h V gz QE W

Assume further that states at the inlets and outlets are constant with time

Calculating energy change

0

( ) (0)CV CV

CV CVt

tE E edV edV

0t

udV udV Assuming KE and PE effects are negligible.

( ) ( ) (0) (0)CV CVm t u t m u Assuming propertiesare uniform with position within the CV at final andinitial states (e.g. when the control volume is atthermodynamic equilibrium atThe begin and end states).

Conservation of mass and energy for an unsteady system: final usable forms

2 2

( ) ( ) (0) (0)

( / 2 ) ( / 2 )

CV CV

inlets outletsCV

t u t m u

m h V gz m h V

m

gz Q W

( ) (0)CV CVinlets outlets

m t m m m V

+

-

Assuming • properties are uniform with position

within the CV at final and initial states • states at the inlets and outlets are constant

with time• KE and PE change of the CV can be

neglected

Example problem: heat transfer during the filling of an evacuated bottle (also in Tutorials)

Consider a rigid and evacuated container (bottle) of volume V that is surrounded by the atmosphere (T0, P0). At some point in time, the neck valve of the bottle opens, and atmospheric air flows in. The wall of thebottle is thin and conductive enough so that the trapped air and the atmosphere eventually reach thermal equilibrium. In the end, the trapped air and the atmosphere are also in mechanical equilibrium, because the neck valve remains open. Determine the net heat interaction that takes place through the wall of the bottle during the entire filling process.

Solve this problem by (a) a closed system approach (b) by an open system approach