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Conservation of Energy

Chapter One Section 1.3



Alternative Formulations

• Alternative Formulations

Time Basis

At an instant

or

Over a time interval

Type of System

Control volume

Control surface

• An important tool in heat transfer analysis! often

provi"ing the #asis for "etermining the temperature

of a system.

CO$SE%&AT'O$ OF E$E%()

*F'%ST +A, OF T-E%O/)$A'CS0



• At an 'nstant of Time

$ote representation of system #y a

control surface *"ashe" line0 at the #oun"aries.

Surface henomena

! energy transfer across the control

su

rate of thermal an"2or mechanical

"ue to heat transfer! flui" flo an"2or or4rface interactions.

in out E E g g

&olumetric henomena

rate of "ue to conversion from another enegy form

*e.g.! electrical! nuclear! or chemical05 energy conversion proces

thermal en

s occurs

ergy

ithi

generatio

n the m

n

syste .

g E g

energy storage in the system rate of change .of st E

g

Conservation of Energyin g out st

dE st

dt E E E E + − = ≡g g g g (1.11a)

Each term has units of 62s or ,.

A+'CAT'O$ TO A CO$T%O+ &O+7E

• Over a Time 'nterval

Each term has units of 6.

in g out st E E E E + − = ∆ (1.11b)

C& at an 'nstant an" over a Time 'nterval



At an instant

dU q W

dt

− =g

(1.11d)

• Special Cases *+in4ages to Thermo"ynamics0

*i0 Transient rocess for a Close" System of ass * M) Assuming Heat Transfer

to the System (Inflow) and Work one !y the System ("utflow)#

Over a time interval

$ U W − = ∆

(1.11c)

Close" System



E8ample 1.3 Application to thermal response of a con"uctor ith Ohmic

heating *generation0

• 'nvolves change in thermal energy an" for an incompressi#le su#stance

t dU dU dT

M%dt dt dt = =

• -eat transfer is from the con"uctor *negative )q

• (eneration may #e viee" as electrical or4 "one on the system *negative 0W g

E8ample 1.3



E8ample 1.9 Application to isothermal soli":li;ui" phase change in a container

+atent -eat

of Fusion

1at sf U U M ∆ = ∆ = h

E8ample 1.9



*ii0 Stea"y State for Flo through an Open System ithout hase Change or

(eneration

( ) flo o r4 &' →•

( ) enthalpy u &' i+ ≡ →•

( )

i"eal gas constant specific heat• For an ith

i o & i oi i % T T − = −

( )

( ) ( )

incompressi#le li;ui"• For an

<

i o i o

i o

u u % T T

&' &'

− = −− ≈

( ) ( )

= =

For systems ith significant heat transfer

<= =

<

i

i

o

g g o

− ≈ ÷ ÷

− ≈

•

At an 'nstant of

Time=

<= o

m u &' g W

− + + + − = ÷

• •=

= i

m u &' g q + + + + ÷

•

(1.11e)

Open System



Surface Energy Balance

A special case for hich no volume or mass is encompasse" #y the control surface.

Conservation Energy *'nstant in Time0

<out in

E E − =g g

(1.12)

• Applies for stea"y:state an" transient con"itions

Consi"er surface of all ith heat transfer #y con"uction! convection an" ra"iation.

<%ond %on' rad q q q′′ ′′ ′′− − =

( ) ( )9 91 =

= = = < sur

T T k T T T T

*ε σ

∞

−− − − − =h

• ,ith no mass an" volume! energy storage an" generation are not pertinent to the energy

#alance! even if they occur in the me"ium #oun"e" #y the surface.

T-E S7%FACE E$E%() BA+A$CE



etho"ology

• On a schematic of the system! represent the control surface #y

"ashe" line*s0.

• Choose the appropriate time #asis.

• '"entify relevant energy transport! generation an"2or storage terms

#y la#ele" arros on the schematic.

• ,rite the governing form of the Conservation of Energy re;uirement.

• Su#stitute appropriate e8pressions for terms of the energy e;uation.

• Solve for the un4non ;uantity.

ET-O/O+O() OF F'%ST +A, A$A+)S'S



ro#lem 1.93 Thermal processing of silicon afers in a to:>one furnace.

/etermine *a0 the initial rate of change of the afer

temperature an" *#0 the stea"y:state temperature.

ro#lem Silicon ,afer

SCHEMATIC



ro#lem Silicon ,afer *cont.0

in out st E E E − =& & &

or! per unit surface area

! ! ! !w

rad h rad % %' u %' l d T

q q q q %d

dt

ρ ′′ ′′ ′′ ′′+ − − =

( ) ( ) ( ) ( )9 9 9 9!!

ww sur % w u w l w sur h

d T T T T T h T T h T T %d

dt εσ εσ ρ ∞ ∞− + − − − − − =

( ) ( )? = 9 9 9 ? = 9 9 9 99<.@ .@ 1< , 2 m 1<< 3<< <.@ .@ 1< , 2 m 33< 3<<

− −× × × − + × × × −

( ) ( )= =?, 2 m 3<< << 9 , 2 m 3<< << − × − − × − =

3=<< 4g 2 m ?6 2 4g × × ( ) i

<.<<<? m " T 2 "t×

( ) i"T 2 "t 1<9 2 s=



ro#lem Silicon ,afer *cont.0

( ) ( )9 9 9 9 9 9!ss !ss<.@A 1A<< T C <.@A 33< T C σ σ − + −

( ) ( )= =3!ss 3!ss?, 2 m T << 9 , 2 m T << <− × − − × − =

3!ssT 1=A1 C =

ro#lem Cooling of Spherical Canister




ro#lem 1.9? Cooling of spherical canister use" to store reacting chemicals.

/etermine *a0 the initial rate of change of the canister temperature!

*#0 the stea"y:state temperature! an" *c0 the effect of convection

on the stea"y:state temperature.





SCHEMATIC:





100 400 800 2000 6000 10000

Convection coefficient, h(W/m^2.K)

300

400

500

600

700

800

900

1000

T e m

e ! " t # ! e ,

T ( K )

01b- chapter 1-sec 1.3 black text

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