1 lec 12: closed system, open system. 2 for next time: –read: § 5.4 outline: –conservation of...
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![Page 1: 1 Lec 12: Closed system, open system. 2 For next time: –Read: § 5.4 Outline: –Conservation of energy equations –Relationships for open systems –Example](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649d555503460f94a333dc/html5/thumbnails/1.jpg)
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Lec 12: Closed system, open system
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• For next time:– Read: § 5.4
• Outline:– Conservation of energy equations– Relationships for open systems– Example problem
• Important points:– Memorize the general conservation of mass
and energy equations– Know how to translate the problem statement
into simplifications in the mass and energy conservation equations.
– Remember the conversion factor between m2/s2 and J/kg.
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Remember the difference between closed and open systems
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Property variations
• Closed system--properties at any location in the system are the same (though in a transient problem they may change with time).
• Open system--properties vary with location in a control volume--for example between the entrance to an air compressor and the exit.
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Steady-flow assumption
Extensive and intensive properties within the control volume don’t change with time, though they may vary with location.
Thus mCV, ECV, and VCV are constant.
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• With VCV constant and VCV=0, there is no boundary work.
Steady-flow assumption
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Steady-flow assumption
• With mCV and ECV constant,
0dt
dEcv 0dt
dmcv
• This allows the properties to vary from point-to-point but not with time.
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Steady-flow assumption
• However, material can still flow in and out of the control volume.
• The flow rate terms are not zero.m
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Consider a simple two-port system (one inlet/one outlet)
Control Volume
m cv, Ecv
Q W
m 2
m 1
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Assumptions
• No generation of mass or energy in the control volume
• No creation of mass or energy in the control volume
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Conservation of mass
c.v1 2
dMm m
dt
C.V.
INTO RATE
FLOW MASS
C.V.
OF OUT RATE
FLOW MASS
C.V.
THE IN MASS OF
CHANGE OF RATE
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Conservation of mass
• If we assume steady-flow,
dt
dmmm c.v
21 0
21 mm
outin mm
• and
• or
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• This can be extended to multiple inlets and outlets:
Conservation of mass
outin mm
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Apply energy conservation equation
C.V.
INTO ENERGY
OFRATE
C.V.
THE IN ENERGY OF
CHANGE OF RATE
C.V.OF
OUT ENERGY
OFRATE
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How does energy enter the control volume?
Heat Energy
Work Energy
Movement of fluid
inQ
inW
ininθm
[Could be if there is more than one source]
inQ
[Could be if there is more than one source]
inW
(this also could be a summed term)
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Conservation of energy
• Remember
kepehθ
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Thus, energy input is:
C.V.
INTO ENERGY
OFRATE
inQ inW ininθm
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We can develop a similar expression for rate of energy leaving the control volume
C.V.OF
OUT ENERGY
OFRATE
outQ outW outoutθm
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We’ve seen the transient term before in the closed system
analysis
C.V.
THE IN ENERGY OF
CHANGE OF RATE
dt
dEcv
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Let’s look at the heat transfer terms first:
We want to combine them into a single term to give us the net heat transfer
outQnetQ inQ
For simplicity, we’ll drop the “net” subscript
Q netQ
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We’ll do the same thing with work
Work becomes
netWW outin WW
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Energy equation
dt
dEmθmθ)W-W(Q-Q c.v
outoutininoutinoutin
Q Wdt
dEmθmθ c.v
outoutinin
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We’ll now write the energy equation as:
dt
dEmθmθW-Q c.v
outoutinin
dt
dEmθmθW-Q c.v
eeii
dt
dEmθmθW-Q c.v
2211
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Comment on work
• Work includes, in its most general case, shaft work, such as that done by moving turbine blades or a pump impeller; the work due to movement of the CV surface (usually the surface does not move and this is zero); the work due to magnetic fields, surface tension, etc., if we wished to include them (usually we do not); and the work to move material in and out of the CV. However, we have already included this last pv term in the enthalpy.
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e
2e
eei
2i
iiCV gz
2
Vhmgz
2
VhmWQ
dt
dE
and we finally have a useful expression for conservation of energy for an open system:
eeiiCV θmθmWQ
dt
dE
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More on the work term
The work term does not include boundary work (=0 because the control volume does not change size) and it does not include flow work.
Shaftcv WWW
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For multiple inlets and outlets, the first law will look like:
2
1 2
ki
Shaft i i ii
Q W h gz mV
2.
1 2
lj c v
j j jj
dEh gz m
dt
V
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Two port devices with steady state steady flow (SSSF) assumption
Conservation of mass:
1 2m m m
Conservation of energy:
21
1 12ShaftQ W h gz mV
2
22 2 0
2h gz m
V
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The energy equation can be simplified even more…..
Divide through by the mass flow:
m
Heat transfer per unit mass
m
Ww shaft
shaft
Shaft work per unit mass
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We get the following for the energy equation
2 2
2 12 1 2 1( - )
2 2Shaftq w h h g z zV V
Or in short hand notation:
pekehwq shaft
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TEAMPLAYTEAMPLAY
On a per unit mass basis, the conservation of energy for a closed system is
pekehwq shaft
pekeu wq Conservation of energy for an open system was just derived
Explain the difference of the meaning of each term between the open and closed system expressions.
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Let’s Review - for two port system
Conservation of Mass
dt
dmmm c.v
21
Conservation of energy
2
1 2
ki
Shaft i i ii
Q W h gz mV
2.
1 2
lj c v
j j jj
dEh gz m
dt
V
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Example ProblemExample Problem
Steam enters a two-port device at 1000 psia and 1000F with a velocity of 21.0 ft/s and leaves as a dry saturated vapor at 2 psia. The inlet area is 1 ft2 and the outlet area is 140 ft2.
A) What is the mass flow (lb/hr)?
B) What is the exit velocity (ft/s)?
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Draw DiagramDraw Diagram
1
2
Turbine
P1 = 1000 psia
T1 = 1000F
V1 = 21.0 ft/s
A1=1 ft2
P2 = 2 psia
x2 = 1.0
A1=140 ft2
STATE 1 STATE 2
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State assumptionsState assumptions
• Steady state (dm/dt = 0)• One inlet/one outlet• Uniform properties at inlet and outlet
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Basic EquationsBasic Equations
Conservation of mass for 2 port steady state:
21 )A
()A
(VV
m
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Get property data from steam tables:
31 m0.831ft / lb
32 m173.75 ft / lb
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Calculate mass flow:Calculate mass flow:
1)A
(V
m
)3600(831.0
)1(0.21
3
2
hr
s
hrft
ftsft
m
hr/lb975,90m m
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Exit VelocityExit Velocity
2)A
(V
m
A
mV
2
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Exit Velocity - page 2Exit Velocity - page 2
)s3600
hr(
ft140
)hr
lb975,90(
lb
ft75.173
V2
m
m
3
2
sftV /4.312
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TEAMPLAYTEAMPLAY
Water at 80 ºC and 7 MPa enters a boiler tube of constant inside diameter of 2.0 cm at a mass flow rate of 0.76 kg/s. The water leaves the boiler tube at 350 ºC with a velocity of 102.15 m/s. Determine (a) the velocity at the tube inlet (m/s) and (b) the pressure of the water at the tube exit (MPa).