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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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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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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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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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

!!!!

!!!!

!!!!!!


etoteitotiVCVCVC

eeeeiiiiVCVCVC

hmhmWQdtdE

gZhmgZhmWQdtdE

,,......

22....

..

or

V21V

21

åå

åå

-+-=

÷øö

çèæ ++-÷

øö

çèæ +++-=

!!!!

!!!!

gZhhtot ++= 2V21 where (total enthalpy)

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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


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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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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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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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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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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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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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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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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Combination of Components

Simple steam power plant Refrigerator

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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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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++=-

Energy Equation fora Control Volume

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