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 …..
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.
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 .
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.
0
0.2
0.4
0.6
0.8
0 10 20 30Pressure ratio
1872, Dr Franz Stikze’s Paradox
cycle
ndnetW ,
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.
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 ,,
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.
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
Number of transfer units for hot fluid:
hphothot cm
UANTU
,
Number of transfer units for cold fluid:cpcold
cold cm
UANTU
,
For an infinitely long Co flow HX. ehec TT ,,
Let
Thenhc dTdT
icehcpcp TTcmQ ,,,max,
hotcold NN
Co Flow HX
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 !!!!
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 !!!!
Second Law for HXs
•It is impossible to construct an infinitely long counter flow HX.
•What is the maximum possible?
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
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
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
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
Counter Flow Heat Ex
1exp
max
min,, C
CNTUTT incommoutcomm
Tci
Tce
Thi
The
1exp
max
min
C
CNTUTTTT cihecehi