PPT: Conservation Equations & Analysis of Finite Control Volume | Fluid Mechanics for Mechanical Engineering PDF Download

 Page 1


Finite Control V olume Analysis
CVEN 311
Application of Reynolds Transport Theorem
Page 2


Finite Control V olume Analysis
CVEN 311
Application of Reynolds Transport Theorem
Moving from a System to a Finite 
Control V olume
?Mass
?Linear Momentum
?Moment of Momentum
?Energy
?Putting it all together!
Page 3


Finite Control V olume Analysis
CVEN 311
Application of Reynolds Transport Theorem
Moving from a System to a Finite 
Control V olume
?Mass
?Linear Momentum
?Moment of Momentum
?Energy
?Putting it all together!
Conservation of Mass
B = Total amount of ____ in the system
b = ____ per unit mass = __ 
ˆ
sys
cv cs
DM
dV dA
Dt t
rr
¶
= + ×
¶
òò
Vn
ˆ
cs cv
dA dV
t
rr
¶
× = -
¶
òò
Vn
mass
1
mass
But DM
sys
/Dt = 0!
cv equation
mass leaving - mass entering = - rate of increase of mass in cv
ˆ
sys
cv cs
DB
bdV b dA
Dt t
rr
¶
= + ×
¶
òò
Vn
Continuity Equation
Page 4


Finite Control V olume Analysis
CVEN 311
Application of Reynolds Transport Theorem
Moving from a System to a Finite 
Control V olume
?Mass
?Linear Momentum
?Moment of Momentum
?Energy
?Putting it all together!
Conservation of Mass
B = Total amount of ____ in the system
b = ____ per unit mass = __ 
ˆ
sys
cv cs
DM
dV dA
Dt t
rr
¶
= + ×
¶
òò
Vn
ˆ
cs cv
dA dV
t
rr
¶
× = -
¶
òò
Vn
mass
1
mass
But DM
sys
/Dt = 0!
cv equation
mass leaving - mass entering = - rate of increase of mass in cv
ˆ
sys
cv cs
DB
bdV b dA
Dt t
rr
¶
= + ×
¶
òò
Vn
Continuity Equation
Conservation of Mass
12
1 1 1 2 2 2
ˆ ˆ 0
cs cs
dA dA rr × + × =
òò
V n V n
1
2
V
1
A
1
If mass in cv 
is constant
Unit vector     is ______ 
to surface and pointed 
____ of cv
ˆ
cs cv
dA dV
t
rr
¶
× = -
¶
òò
Vn
ˆ n
normal
out
ˆ
cs
dA r ×=
ò
Vn
m
ˆ
cs
dA
V
A
×
=
ò
Vn
VA r =
ˆ n
We assumed uniform ___ on 
the control surface
r
is the spatially averaged 
velocity normal to the cs
V
[M/T]
Page 5


Finite Control V olume Analysis
CVEN 311
Application of Reynolds Transport Theorem
Moving from a System to a Finite 
Control V olume
?Mass
?Linear Momentum
?Moment of Momentum
?Energy
?Putting it all together!
Conservation of Mass
B = Total amount of ____ in the system
b = ____ per unit mass = __ 
ˆ
sys
cv cs
DM
dV dA
Dt t
rr
¶
= + ×
¶
òò
Vn
ˆ
cs cv
dA dV
t
rr
¶
× = -
¶
òò
Vn
mass
1
mass
But DM
sys
/Dt = 0!
cv equation
mass leaving - mass entering = - rate of increase of mass in cv
ˆ
sys
cv cs
DB
bdV b dA
Dt t
rr
¶
= + ×
¶
òò
Vn
Continuity Equation
Conservation of Mass
12
1 1 1 2 2 2
ˆ ˆ 0
cs cs
dA dA rr × + × =
òò
V n V n
1
2
V
1
A
1
If mass in cv 
is constant
Unit vector     is ______ 
to surface and pointed 
____ of cv
ˆ
cs cv
dA dV
t
rr
¶
× = -
¶
òò
Vn
ˆ n
normal
out
ˆ
cs
dA r ×=
ò
Vn
m
ˆ
cs
dA
V
A
×
=
ò
Vn
VA r =
ˆ n
We assumed uniform ___ on 
the control surface
r
is the spatially averaged 
velocity normal to the cs
V
[M/T]
Continuity Equation for Constant 
Density and Uniform Velocity
12
1 1 2 2
0 V A V A rr - + =
12
12
V A V A Q ==
12
1 1 1 2 2 2
ˆ ˆ 0
cs cs
dA dA rr × + × =
òò
V n V n
Density is constant across cs
Density is the same at cs
1
and cs
2
[L
3
/T]
Simple version of the continuity equation 
for conditions of constant density. It is 
understood that the velocities are either 
________ or _______ ________.
1 1 2 2
V A V A Q ==
uniform spatially averaged