EGM 6812, F11, University of Florida, M. Sheplak
1/24
Section 4, Basic Laws
4 Basic Laws
4.1
Review of
Reynold’s Transport Theorem
(RTT)
Adopted from and figures from Potter and Foss Chapter 2
4.1.1 NonDeformable RTT
Starting with the basic equation for the relation between an arbitrary extensive property, N, and
its corresponding intensi
ve property, η:
N
d
From here we take the time rate of change of the system, N
sys,
as:
0
(
)
( )
lim
sys
sys
sys
sys
t
DN
N
t
t
N
t
D
d
Dt
Dt
t
First we apply this formulation to a fixed, nondeformable control volume, as depicted in the
following figure:
Note: the shape of the control volume is intentionally arbitrary to indicate that it can be formed
as the application requires. Proper selection of control volume is an essential component to
solving such problems.
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EGM 6812, F11, University of Florida, M. Sheplak
2/24
Section 4, Basic Laws
At time,
t
, both the system and control volume share the same space. As we progress in time, to
t+∆t
, the control volume occupies its original space, but the system has moved to a new position.
Using the figure notation the terms from the previous equation are recast as:
3
2
(
)
(
)
(
)
sys
N
t
t
N
t
t
N
t
t
2
1
( )
( )
( )
sys
N
t
N
t
N t
Thus, we rewrite the equations as:
3
2
2
1
0
(
)
(
)
( )
( )
lim
sys
t
DN
N
t
t
N
t
t
N
t
N t
Dt
t
2
1
2
1
3
1
0
(
)
(
)
( )
( )
(
)
(
)
lim
t
N
t
t
N
t
t
N
t
N
t
N
t
t
N
t
t
t
Where in the second equation
1
(
)
N t
t
has been added and subtracted to achieve the desired
form. Once more referring to the figure above, the equation is manipulated to become:
. .
. .
3
1
0
0
(
)
( )
(
)
(
)
lim
lim
sys
c v
c v
t
t
DN
N
t
t
N
t
N
t
t
N
t
t
Dt
t
t
By definition, the first term on the right hand side is:
. .
. .
. .
0
(
)
( )
lim
c v
c v
c v
t
N
t
t
N
t
d
d
t
dt
The second term is analyzed by realizing that
1
(
)
N t
t
and
3
(
)
N
t
t
are the total quantity of
the extensive property contained in their respective regions at time,
t+∆t.
Thus:
3
1
3
1
3
1
. .
ˆ
ˆ
(
)
(
)
(
)
ˆ
ˆ
A
A
A
A
c s
N
t
t
N
t
t
V ndA t
V n dA t
V ndA t
V ndA t
This leads us to a form of the system to control volume transformation of
. .
. .
ˆ
sys
c v
c s
DN
d
d
V ndA
Dt
dt
,
But the control volume is not deforming, so the
t
.
Therefore, the derivative with respect
to time has been brought into the integral in accordance with Leibniz' rule, as the control volume
does not change with respect to time. This equation represents the Reynold's Transport Theorem.
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 Fall '09
 RENWEIMEI
 Fluid Dynamics, Energy, Force, Kinetic Energy, Flux, M. Sheplak

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