non-dimensional analysis of heat exchangers p m v subbarao professor mechanical engineering...
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Non-dimensional Analysis of Heat Exchangers
P M V SubbaraoProfessor
Mechanical Engineering Department
I I T Delhi
The culmination of Innovation …..
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The Concept of Space in Mathematics
• Global Mathematical Space: The infinite extension of the three-dimensional region in which all concepts (matter) exists.
• Particular Mathematical Space: A set of elements or points satisfying specified geometric postulates.
• Euclidian Space: The basic vector space of real numbers.
• A Hilbert space is an abstract vector space possessing the structure of an inner product that allows length and angle to be measured.
• A Sobolev space is a space of functions with sufficiently many derivatives for some application domain.
• Development of a geometrical model for Hx in A compact Sobolev space helps in creating new and valid ideas.
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The Compact Sobolev Space for HXs
• A model for hx is developed in terms of positive real parameters.
• High population of these parameters lie in 0 p .
• The most compact space is the one where all the parameters defining the model lie in 0 p .
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History of Gas Turbines
• 1791: A patent was given to John Barber, an Englishman, for the first true gas turbine.
• His invention had most of the elements present in the modern day gas turbines.
• The turbine was designed to power a horseless carriage. • 1872: The first true gas turbine engine was designed by Dr
Franz Stikze, but the engine never ran under its own power.
• 1903: A Norwegian, Ægidius Elling, was able to build the first gas turbine that was able to produce more power than needed to run its own components, which was considered an achievement in a time when knowledge about aerodynamics was limited.
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0
0.2
0.4
0.6
0.8
0 10 20 30Pressure ratio
1872, Dr Franz Stikze’s Paradox
cycle
ndnetW ,
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First turbojet-powered aircraft – Ohain’s engine on He 178
The world’s first aircraft to fly purely on turbojet power, the Heinkel He 178.
Its first true flight was on 27 August, 1939.
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chtot
htot
hphotc
tot
cpcold TTA
dAdT
UA
cmdT
UA
cm ,,
Capacity of An infinitesimal Size HX
chtot
hphot
tot
h
cpcold
tot
c TTA
dA
cm
UAdT
cm
UAdT
,,
chtot
htot
hphotc
tot
cpcold TTA
dAdT
UA
cmdT
UA
cm ,,
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Define a Non Dimensional number N
chtothot
h
cold
c TTA
dA
N
dT
N
dT
Maximum Possible Heat Transfer ?!?!?!?
Let hotcold NN
Then hc dTdT
Cold fluid would experience a large temperature change.
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For an infinitely long counter flow HX.
ihec TT ,, icihcpc TTcmQ ,,,max
Counter Flow HX
icihp TTcmQ ,,minmax
min
minmaxp
cap cm
UANNTU
Maximum Number of Transfer Units
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Number of transfer units for hot fluid:
hphothot cm
UANTU
,
Number of transfer units for cold fluid:cpcold
cold cm
UANTU
,
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For an infinitely long Co flow HX. ehec TT ,,
Let
Thenhc dTdT
icehcpcp TTcmQ ,,,max,
hotcold NN
Co Flow HX
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For A given combination of fluids, there exist two ideal extreme designs of heat exchangers.
High Performance HX: Infinitely long counter flow HX.
Low Performance HX: Infinitely long co flow HX.
icihcpcperhigh TTcmQ ,,,
icohcpcperlow TTcmQ ,,,
If hotcold NN
First law for Heat Exchangers !!!!
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For A given combination of fluids, there exist two ideal extreme designs of heat exchangers.
High Performance HX: Infinitely long counter flow HX.
Low Performance HX: Infinitely long co flow HX.
icihhphperhigh TTcmQ ,,,
ocihhphperlow TTcmQ ,,,
If coldhot NN
First law for Heat Exchangers !!!!
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Second Law for HXs
•It is impossible to construct an infinitely long counter flow HX.
•What is the maximum possible?
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Effectiveness of A HX
• Ratio of the actual heat transfer rate to maximum available heat transfer rate.
maxQ
Qact
• Maximum available temperature difference of minimum thermal capacity fluid.
icihfluid TTT ,,max, • Actual heat transfer rate:
LMTDact TUAQ
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icihp
LMTD
TTcm
TUA
,,min
icih
LMTD
TT
TNTU
,,max
icih
comm
comm
commcomm
TT
T
T
TT
NTU,,
1,
2,
1,2,
max
ln
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maxmin1,
2, 11exp
ppcomm
comm
cmcmUA
T
T
max
min
min1,
2, 1expp
p
pcomm
comm
cm
cm
cm
UA
T
T
max
minmax
1,
2, 1expp
p
comm
comm
cm
cmNTU
T
T
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max
minmax
1,
2, 1expp
p
comm
comm
cm
cmNTU
T
T
icih
comm
comm
commcomm
TT
T
T
TT
NTU,,
1,
2,
1,2,
max
ln
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Counter Flow Heat Ex
1exp
max
min,, C
CNTUTT incommoutcomm
Tci
Tce
Thi
The
1exp
max
min
C
CNTUTTTT cihecehi