h3 conduction 2
DESCRIPTION
"Heat can I vent" Cole WorldTRANSCRIPT
-
(3) Tube wall with the BC of the 1st kind
( ) 112121 ln
ln)( sss Tr
rrrTTrT +
= ( ) 2221
21 lnln
)( sss Trr
rrTTrT +
=
01 =
drdTr
drd
r
=
=
22
11
)()(
s
s
TrTTrT
BCs: (1st kind)
Heat Equation:
or
-
01 =
drdTr
drd
r
Heat Equation:
BCs: (1st kind)
=
=
22
11
)()(
s
s
TrTTrT
Solution:
01 =
drdTr
drd
rC
drdTr constant,=
rC
drdT
= DrCT += ln
Using the two BCs, ( )21
21ln rr
TTC ss = ( )( )21121
1 lnln
rrrTTTD sss
=
Thus,
( ) 112121 ln
ln)( sss Tr
rrrTTrT +
= ( ) 2221
21 lnln
)( sss Trr
rrTTrT +
=or
-
Heat flux & heat transfer rate:
( )( ) T
krr
TTLkq ssr
=
area)(~ln
2
12
21
( )( ) rrrr
TTkrCk
drdTkq ssr
1~ln 21
21 ===
constant~22
==
rkCrLqrLq rr
Standard form for qr
( )( ) ( ) ( )211212
12ln
2ssr TTrr
krr
Lrrq
=
Lr2area, where ( )1212
ln rrrrr =
logarithmic mean radius LrLA
-
- What does the form of the heat equation tell us about the variation of the heat flux with r in the wall?
01 =
drdTr
drd
r
=
=
22
11
)()(
s
s
TrTTrT
BCs: (1st kind)
Heat Equation:
- Is the foregoing conclusion consistent with the energy conservation requirement?
- How does the transfer rate qr vary with r ?
rq
-
Heat flux depends on r, but the heat flow rate is independent of r.
- What does the form of the heat equation tell us about the variation of with r in the wall?
01 =
drdTr
drd
rCconstant ==
drdTr
rkC
drdTkqr ~= constant~2~2
=
rkCrLqrLq rr
- Is the foregoing conclusion consistent with the energy conservation requirement?
- How does the heat transfer rate qr vary with r ?
rq
-
Heat Flux :
Temperature profile:
( ) 112121 ln
ln)( sss Tr
rrrTTrT +
=
( ) 222121 ln
ln)( sss Tr
rrrTTrT +
=
( ) ( )21122 ssLrr TTrr
kAqrLq
==
( ) ( )2112ln ssrTT
rrrk
drdTkq ==
Heat Transfer Rate:
where
* Summary:
( )1212
ln rrrrr =LrA LL 2= and LMR
-
(4) Tube wall with the BC of the 3rd kind
01 =
drdTr
drd
r
BCs: (3rd kind)
Heat Equation:
( )( )
==
==
22222
11111
;At ;At
TThAqrxTThAqrr
sr
sr
A1=2r1L (inner surface area) A2=2r2L (outer surface area)
T1 and T2 are prescribed. But Ts1 and Ts2 are not known a priori.
-
Solution: - The temperature profile will be same as the previous case. i.e.,
01 =
drdTr
drd
r
Heat Equation: BCs: (3rd kind)
( )( )
==
==
22222
11111
;At ;At
TThAqrxTThAqrr
sr
sr
( ) 112121 ln
ln)( sss Tr
rrrTTrT +
=
( ) ( )11111111 2 ssr TTLhrTThAq ==
( ) ( ) ( ) ( )211221122 ssLssLr TTrr
kLrTTrr
kAq
=
=
( ) ( )22222222 2 == TTLhrTThAq ssr
Ts1 and Ts2 are not known a priori and to be determined
- Recall that the heat transfer rate is constant. i.e.,
------------ (1)
-------- (2)
------------ (3)
-
- Ts1 and Ts2 can be determined from equations (1)~(3) although complicated.
- Temperature profile itself, however, is less interesting than the heat transfer rate, qr
( )2121 =
= TTAUR
TTqtot
r expression for Rtot, U, A? (A1 or A2?)
( )221122
12
11
112
12
ln2
1AUAULhrLk
rrLhr
Rtot ==
++=
( )
++==
22
1121
11
1
ln11hrr
krrr
hRA
U tot
( )
++==
2
122
11
22
2
1ln1hk
rrrhrrRA
U tot
Too complicated to remember! Any easier way?
-
( )221122
12
11
1111UAUAhAkA
rrhA
RL
tot ==
+
+=
totRAU 111
=
21
111AhAk
LAhAU
Rtot ++==
Recall the slab problem.
( )2121 =
= TTAUR
TTqtot
r
where
Then, for the present problem in cylindrical geometry
1T1h
2T2h
( ) LrrrrLrA LL
==
12
12ln
22 totRAU 221
=where
U1 : Overall heat transfer coefficient based on the inner surface U2 : Overall heat transfer coefficient based on the outer surface
-
(5) Composite Wall with Negligible Contact Resistance
( )11112 sr TTLhrq =
( )( )32
23B
r TTrrlnLk2q =
( )( )s43
34
Cr TTrrln
Lk2q =
( )( )21s
12A
r TTrrlnLk2q =
( )44442 = TTLhrq sr
( ) ( )4122411141 ==
= TTAUTTAUR
TTqtot
r
-
( ) ( ) ( )
( ) ( ) ( )
ooii
ooCBAii
ooCCLBBLAALii
oCBAitot
AUAU
LhrLkrr
Lkrr
Lkrr
Lhr
hAkArr
kArr
kArr
hA
RRRRRR
11
21
2ln
2ln
2ln
21
11
342312
,
34
,
23
,
12
==
++++=
+
+
+
+=
++++=
But, U is tied to specification of an interface. Ui : overall heat transfer coefficient based on the inner surface Uo : overall heat transfer coefficient based on the outer surface
Note that Rtot is a constant that is independent of position r.
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