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Tra
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Transport processes – Part 3a
Ron ZevenhovenÅbo Akademi University
Thermal and Flow Engineering / Värme- och strömningstekniktel. 3223 ; [email protected]
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more general: T=T1
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more general: θ = (T –T1)/(T0 –T1)
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Orthogonality
)cos()sin()(cos1
0
2kkkk d
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Note: if the source gives transport in only one direction, then take 2x in that direction and 0x in opposite direction
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),(),()()(
)( )(1
txc
QtxTx
c
QxT
DiracdxxdxQ
TcTdxcQ
pp
pp
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b is a position !
But: note the Second Law of Thermodynamics!!!
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t
dtdt
dTttxG
0
''
)',( p
w
pw
cdt
dT
dt
dTc
dt
dQ
"
"
'
''
"
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)'(4
2
tta
rq
~ (3.22)
~ (3.23)
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112
44
1")()(
24
4
1"
44
4
1"
4
1":)39.3(
44
4444
22
2222
atat
ec
Qerferfat
ate
c
Q
at
ydeat
ate
c
Qdyee
atc
Q
at
x
p
at
x
p
at
y
at
x
p
at
y
at
x
p
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at
T
atT
dx
dT:x@
ateT
dx
dT atx
11
41
2
120
2
122
)(
2)( 2
dx
xderfce
dx
xderf x
Penetrationdepth
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dt
dc
dx
dD
dx
dc
dt
dD
dt
d "m
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~ (2.87)
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Example contact temperature
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Contact temperature – example /1
• A (human) tongue may freeze to a bridge-rail depending on the material.
• Consider the contact temperature for two types of bridge-rail material:
steel: λb = 40 W/(m.K), ab = 10-5 m2/s
wood: λb = 0.2 W/(m.K), ab = 1.2·10-7 m2/s
• Tongue data: Ta0 = 35°C, λa = 0.55 W/(m.K), aa = 1.7·10-7 m2/s
• Bridge temperature: Tb0 < 0°C
• Calculate Tb0 giving contact temperature 0°C
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Contact temperature – example /2
• Use (equation 3.59):
• This gives
for steel: Tb0 = - 3.7°C
for wood: Tb0 = - 81°C
• This shows that a wooden bridge need not be a problem.
6
60
1071550
1071550350
..
..
bb
bbb
a
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x = L
x = bx = 0
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Sources used(besides course book Hanjalić et al.)
• Beek, W.J., Muttzall, K.M.K., van Heuven, J.W. ”Transport phenomena” Wiley, 2nd edition (1999)
• R.B. Bird, W.E. Stewart, E.N. Lightfoot ”Transport phenomena” Wiley, New York (1960)
• * C.J. Hoogendoorn ”Fysische Transportverschijnselen II”, TU Delft / D.U.M., the Netherlands 2nd. ed. (1985)
• * C.J. Hoogendoorn, T.H. van der Meer ”Fysische Transport-verschijnselen II”, TU Delft /VSSD, the Netherlands 3nd. ed. (1991)
• D. Kaminski, M. Jensen ”Introduction to Thermal and Fluids Engineering”, Wiley (2005)
• S.R. Turns ”Thermal – Fluid Sciences”, Cambridge Univ. Press (2006)
* Earlier versions of Hanjalić et al. book but in Dutch
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