Energy Equation fora Control Volume
Transcript of Energy Equation fora Control Volume
Energy Equation for a Control Volume
(Lecture 5)
2021년 1학기열역학 (M2794.001100.002)
송한호(*) Some texts and figures are borrowed from Sonntag & Borgnakke unless noted otherwise.
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Energy Equation for a Control Volume
4.1 Conservation of Mass and the Control Volume
è Control volume: space of interest enclosed by control surface
• Size and shape are arbitrary
• Surface may move
• Mass, heat, work transfers
across the boundary
• Accumulation of mass
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Energy Equation for a Control Volume
è Continuity equation = Conservation of mass
è For several different states of the mass,
è In case of known velocity and area normal to the control surface,
åå -=
-+=
eiVC mm
dtdm
!!..
outin change of rate
avgavgavgavglocallocal
avglocal
vAAdAm
AdAV
VVV
VV
===
==
òò
rr!
!
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Energy Equation for a Control Volume
4.2 The Energy Equation for a Control Volume
è Energy equation for a control mass:
è For a control volume, the additional consideration is needed for energy transfer associated with the mass flow across the control surface.
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Energy Equation for a Control Volume
è The energy per unit mass associated with the fluid (or the substance):
è There is additional mode of energy transfer associated with the fluid flow:
è Finally, the total energy per unit mass associated with the flow of mass:
(Flow work)
PvwmPvW flowflow == or dd
è Then, the energy equation for a control volume,
è For multiple entering and leaving mass flow rates,
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( ) ( )
÷øö
çèæ ++-÷
øö
çèæ +++-=
+-++-=
-+-+-=
eeeeiiiiVCVC
eeeeiiiiVCVC
outflowinfloweeiiVCVCVC
gZhmgZhmWQ
vPemvPemWQ
WWememWQdtdE
22....
....
......
V21V
21
!!!!
!!!!
!!!!!!
Energy Equation for a Control Volume
etoteitotiVCVCVC
eeeeiiiiVCVCVC
hmhmWQdtdE
gZhmgZhmWQdtdE
,,......
22....
..
or
V21V
21
åå
åå
-+-=
÷øö
çèæ ++-÷
øö
çèæ +++-=
!!!!
!!!!
gZhhtot ++= 2V21 where (total enthalpy)
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Energy Equation for a Control Volume
4.3 The Steady-State Process
è Steady-state process considers the long-term steady operation of
devices.
è The necessary assumptions are:
1. CV does not accelerate relative to the coordinate frame.
2. The state of the mass at each point in CV does not vary with time.
3. The mass flux and the state of the mass across the control surface
do not vary with time. The rates of heat and work transfers remain
constant.
These assumptions lead to 0,0 .. ==dtdE
dtdm vcvc
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0,0 .. ==dtdE
dtdm vcvc
Energy Equation for a Control Volume
åååå =®-= eieiVC mmmm
dtdm
!!!! :equation Continuity ..
..22
..
22....
..
V21V
21
V21V
21
:equationEnergy
VCeeeeiiiiVC
eeeeiiiiVCVCVC
WgZhmgZhmQ
gZhmgZhmWQdtdE
!!!!
!!!!
+÷øö
çèæ ++=÷
øö
çèæ +++®
÷øö
çèæ ++-÷
øö
çèæ +++-=
åå
åå
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Energy Equation for a Control Volume
For single in- and out-flow systems:
Rearranging this equation (divide by mass flow),
mmm ei !!! == :equation Continuity
..22
.. V21V
21 :lawFirst VCeeeiiiVC WgZhmgZhmQ !!!! +÷
øö
çèæ ++=÷
øö
çèæ +++
)/( ),/( where
V21V
21
....
22
kgkJmWwkgkJ
mQq
wgZhgZhq
VCVC
eeeiii
!
!
!
!==
+++=+++
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Energy Equation for a Control Volume
4.4 Examples of Steady State Processes
è Energy equation for multiple inlet-outlet systems at steady state:
or for single inlet-outlet systems at steady state:
..22
.. V21V
21
VCeeeeiiiiVC WgZhmgZhmQ !!!! +÷øö
çèæ ++=÷
øö
çèæ +++ åå
)/( ),/( where
V21V
21
....
22
kgkJmWwkgkJ
mQq
wgZhgZhq
VCVC
eeeiii
!
!
!
!==
+++=+++
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Energy Equation for a Control Volume
Heat exchanger
è A simple fluid flow through a pipe or system of pipes, where heat is transferred to or from the fluid.
è Only a small pressure drop due to friction in a real device.è No work transfer. è Changes in KE and PE negligible.
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Energy Equation for a Control Volume
Nozzle (↔ Diffuser)
è A steady-state device whose purpose is to create a high-velocity fluid stream at the expense of the fluid’s pressure.
è No work transfer. è Change in PE negligible.è Typically, no heat transfer.è Inlet velocity usually small, unless otherwise noted.
Pressure: highVelocity: low
Pressure: lowVelocity: high
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Energy Equation for a Control Volume
Throttle
è Throttling occurs when a fluid flowing in a line encounters a sudden restriction in the flow passage (e.g. plate with a small hole, partially closed valve, capillary tube, expansion valve, etc.)
è Abrupt pressure drop. è Change in KE and PE negligible.è Typically, little heat transfer and no work transfer.è Constant enthalpy process.
)/( ),/( where
V21V
21
....
22
kgkJmWwkgkJ
mQq
wgZhgZhq
VCVC
eeeiii
!
!
!
!==
+++=+++
ei hh =
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Energy Equation for a Control Volume
Turbine
è A rotary steady-state machine whose purpose is to produce shaft work at the expense of the pressure of the working fluid.
è Change in PE and inlet and exit KE’s negligible.è Little or no heat transferè Work output corresponds to the decrease
in enthalpy.
(source: pixabay.com)
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Energy Equation for a Control Volume
Compressor (gas) or Pump (liquid)
è A rotary steady-state machine whose purpose is to increase the pressure of a fluid by putting in shaft work.
è Change in PE and inlet and exit KE’s negligible.è Little or no heat transferè Work input corresponds to the increase in enthalpy.
è Working principle is in opposite to the turbine.
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Energy Equation for a Control Volume
Combination of Components
Simple steam power plant Refrigerator
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Energy Equation for a Control Volume
4.6 The Transient Process
è Now consider the system with unsteady flows.
è e.g. filling closed tanks with a gas or liquid, or discharging from
closed vessels.
è Simplifying assumptions can be made:
è State is uniform throughout the entire C.V.
è State of mass crossing each of the areas of flow on the C.S. is
constant with time (Mass flow rates may vary with time).
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Energy Equation for a Control Volume
è Continuity equation (mass conservation):
è 1st law of Thermodynamics:
(Rate basis)
(From state 1à2)
÷øö
çèæ ++= gZumE vc
2.. V
21 where (Rate basis)
(From state 1à2)with NO state
change on C.S.
( )
)2
()2
(
2222
..
.
1
21
112
22
22.12
ee
eeii
iivcvc
vcvc
gZVhmgZVhmWQ
gZVumgZVumEE
åå ++-+++-=
úû
ùêë
é÷÷ø
öççè
æ++-÷÷
ø
öççè
æ++=-