ChE 367: Lecture 7
Combest
2/3/2012
1
Agenda
System vs.
Control Volume
The Reynolds
Transport
Theorem
Announcements
• Intensive vs. Extensive
Property
• Integral
• Continuity Equation
2
System vs. Control Volume
3
System vs. Control Volume
System
Defined as a collection of matter of fixed identity
4
System vs. Control Volume
System
Defined as a collection of matter of fixed identity
[1]
Chemical
species in a
reactor
5
System vs. Control Volume
System
Defined as a collection of matter of fixed identity
[2]
[1]
Chemical
species in a
reactor
Air in your lungs
6
System vs. Control Volume
System
Defined as a collection of matter of fixed identity
[2]
[1]
Chemical
species in a
reactor
Air in your lungs
[3]
Cereal in a bowl
7
System vs. Control Volume
System
Defined as a collection of matter of fixed identity
Control Volume
Geometric entity through which a fluid flows that is
bound by a control surface.
8
System vs. Control Volume
System
Defined as a collection of matter of fixed identity
Control Volume
Geometric entity through which a fluid flows that is
bound by a control surface.
[4]
Engine Block
9
System vs. Control Volume
System
Defined as a collection of matter of fixed identity
Control Volume
Geometric entity through which a fluid flows that is
bound by a control surface.
[4]
Engine Block
Square Box
10
System vs. Control Volume
11
System vs. Control Volume
Control volumes can be…
(a) Fixed
(b) Fixed or moving
(c ) Deforming
12
Reynolds Transport Theorem
13
Reynolds Transport Theorem
Intensive vs. Extensive Property
14
Reynolds Transport Theorem
Intensive vs. Extensive Property
Intensive Property
A property that is independent of the amount of material.
15
Reynolds Transport Theorem
Intensive vs. Extensive Property
Intensive Property
A property that is independent of the amount of material.
Examples
Density
Temperature
Viscosity
Heat Capacity
Velocity
Elasticity
16
Reynolds Transport Theorem
Intensive vs. Extensive Property
Intensive Property
A property that is independent of the amount of material.
Examples
Density
Temperature
Viscosity
Heat Capacity
Velocity
Elasticity
Extensive Property
A property that is dependent on the amount of material.
17
Reynolds Transport Theorem
Intensive vs. Extensive Property
Intensive Property
A property that is independent of the amount of material.
Examples
Density
Temperature
Viscosity
Heat Capacity
Velocity
Elasticity
Extensive Property
A property that is dependent on the amount of material.
Examples
Momentum
Volume
Total Energy
Moles Present
Mass
Entropy
18
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
19
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
Extensive Property
Intensive Property
20
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
Extensive Property
If B equals…
Intensive Property
Then b equals…
21
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
Extensive Property
If B equals…
Intensive Property
Then b equals…
22
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
Extensive Property
If B equals…
Intensive Property
Then b equals…
K.E. per
unit mass
23
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
Extensive Property
If B equals…
Intensive Property
Then b equals…
K.E. per
unit mass
24
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
Extensive Property
If B equals…
Intensive Property
Then b equals…
K.E. per
unit mass
Momentum
per unit mass
25
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
26
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
27
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
28
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
Change of the property in
the system
Change of the property in
the control volume
29
Reynolds Transport Theorem
Intensive vs. Extensive Property
In terms of RTT
Change of the extensive
property in the system
Change of the extensive
property in the control
volume
RTT relates the time rate of change of the
extensive property in the system and control
volume
30
Reynolds Transport Theorem
Intensive vs. Extensive Property
t=0
Control Volume
t>0
System
31
Reynolds Transport Theorem
Intensive vs. Extensive Property
At t > 0
32
Reynolds Transport Theorem
Intensive vs. Extensive Property
At t > 0
33
Reynolds Transport Theorem
Intensive vs. Extensive Property
At t > 0
34
Reynolds Transport Theorem
The Integral Form
35
Reynolds Transport Theorem
The Integral Form
Remembering the divergence theorem
36
Reynolds Transport Theorem
The Integral Form
Remembering the divergence theorem
37
Reynolds Transport Theorem
The Integral Form
38
Reynolds Transport Theorem
The Integral Form
39
Reynolds Transport Theorem
Continuity Equation
40
Reynolds Transport Theorem
Continuity Equation
If B equals…
Then b equals…
41
Reynolds Transport Theorem
Continuity Equation
If B equals…
Then b equals…
mass
42
Reynolds Transport Theorem
Continuity Equation
If B equals…
Then b equals…
mass
43
Reynolds Transport Theorem
Continuity Equation
If B equals…
Then b equals…
mass
44
Reynolds Transport Theorem
Continuity Equation
45
Reynolds Transport Theorem
Continuity Equation
If the fluid is incompressible and constant density
46
Reynolds Transport Theorem
Continuity Equation
If the fluid is incompressible and constant density
47
Reynolds Transport Theorem
Continuity Equation
If the fluid is incompressible and constant density
Great, but wait a minute….
48
Reynolds Transport Theorem
Continuity Equation
49
Reynolds Transport Theorem
Continuity Equation
Using Divergence Theorem
50
Reynolds Transport Theorem
Continuity Equation
Using Divergence Theorem
51
Reynolds Transport Theorem
Continuity Equation
What does this
term really
mean?
52
Reynolds Transport Theorem
Continuity Equation
53
Reynolds Transport Theorem
Continuity Equation
54
Reynolds Transport Theorem
Continuity Equation
• Velocity vector points into
the volume
• Surface normal points
outward
55
Reynolds Transport Theorem
Continuity Equation
• Velocity vector points into
the volume
• Surface normal points
outward
56
Reynolds Transport Theorem
Continuity Equation
• Velocity vector points into
the volume
• Surface normal points
outward
• Velocity vector points
parallel to surface
• Surface normal points
outward
• Vectors perpendicular
(orthogonal)
57
Reynolds Transport Theorem
Continuity Equation
• Velocity vector points into
the volume
• Surface normal points
outward
• Velocity vector points
parallel to surface
• Surface normal points
outward
• Vectors perpendicular
(orthogonal)
58
Reynolds Transport Theorem
Continuity Equation
59
Reynolds Transport Theorem
Continuity Equation
Captures the net Outflow-Inflow
60
Reynolds Transport Theorem
Continuity Equation
Captures the net Outflow-Inflow
61
Reynolds Transport Theorem
Continuity Equation
Captures the net Outflow-Inflow
Accumulation + Outflow-Inflow= 0
62
Reynolds Transport Theorem
Closure
• RTT is the integral form of the material derivative
• Both of which are the basis for the transport equations for
heat, mass, and momentum
• Go through the book and glance over the derivation in
detail
63
Reynolds Transport Theorem
Problems
64
Some Announcements
• Test 1 week from today @ 9AM sharp
• Let me know ASAP if you are taking it early
• HW 3 Due Wednesday (make a photocopy if
you want in in the test)
• Answer Key 1 and 2 will be up tonight
• Have a good weekend!
65
Sources
1.
2.
3.
4.
http://pinnacleequipments.tradeindia.com/Exporters_Suppliers/Exporte
r19109.308646/Chemical-Reactor.html
http://www.saburchill.com/chapters/chap0019.html
http://blog.echolife.com.au/2010/08/12/unhealthy-breakfastcereals/cereal/
http://www.ansys.com/Products/Workflow+Technology/ANSYS+Workbe
nch+Platform/ANSYS+Meshing/Features
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