01b- chapter 1-sec 1.3 black text
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Conservation of Energy
Chapter One Section 1.3
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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
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• 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
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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
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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
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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
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*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
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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
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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
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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
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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=
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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
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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.
ro#lem Cooling of Spherical Canister
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ro#lem Cooling of Spherical Canister
SCHEMATIC:
ro#lem Cooling of Spherical Canister
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ro#lem Cooling of Spherical Canister
ro#lem Cooling of Spherical Canister