This presentation explores a non dimensional analysis of various types of heat exchangers

1. Non-dimensional Analysis of Heat Exchangers
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
The culmination of Innovation …..

2. 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.

3. 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  .

4. 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.

5. 0
0.2
0.4
0.6
0.8
0 10 20 30
Pressure ratio
1872, Dr Franz Stikze’s Paradox
cycle

nd
net
W ,

6. 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.

7.  
c
h
tot
h
tot
h
p
hot
c
tot
c
p
cold
T
T
A
dA
dT
UA
c
m
dT
UA
c
m





,
,


Capacity of An infinitesimal Size HX
 
c
h
tot
h
p
hot
tot
h
c
p
cold
tot
c
T
T
A
dA
c
m
UA
dT
c
m
UA
dT



,
,


 
c
h
tot
h
tot
h
p
hot
c
tot
c
p
cold
T
T
A
dA
dT
UA
c
m
dT
UA
c
m



,
,



8. Define a Non Dimensional number N
 
c
h
tot
hot
h
cold
c
T
T
A
dA
N
dT
N
dT



Maximum Possible Heat Transfer ?!?!?!?
Let hot
cold N
N 
Then h
c dT
dT 
Cold fluid would experience a large temperature change.

9. For an infinitely long counter flow HX.
i
h
e
c T
T ,
, 
 
i
c
i
h
c
p
c T
T
c
m
Q ,
,
,
max 

 

Counter Flow HX
   
i
c
i
h
p T
T
c
m
Q ,
,
min
max 

 

 min
min
max
p
cap
c
m
UA
N
NTU


 
Maximum Number of Transfer Units

10. Number of transfer units for hot fluid:
h
p
hot
hot
c
m
UA
NTU
,


Number of transfer units for cold fluid:
c
p
cold
cold
c
m
UA
NTU
,



11. For an infinitely long Co flow HX. e
h
e
c T
T ,
, 
Let
Then
h
c dT
dT 
 
i
c
e
h
c
p
c
p T
T
c
m
Q ,
,
,
max, 

 

hot
cold N
N 
Co Flow HX

12. 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.
 
i
c
i
h
c
p
c
per
high T
T
c
m
Q ,
,
, 

 


 
i
c
o
h
c
p
c
per
low T
T
c
m
Q ,
,
, 

 


If hot
cold N
N 
First law for Heat Exchangers !!!!

13. 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.
 
i
c
i
h
h
p
h
per
high T
T
c
m
Q ,
,
, 

 


 
o
c
i
h
h
p
h
per
low T
T
c
m
Q ,
,
, 

 


If cold
hot N
N 
First law for Heat Exchangers !!!!

14. Second Law for HXs
•It is impossible to construct an
infinitely long counter flow HX.
•What is the maximum possible?

15. Effectiveness of A HX
• Ratio of the actual heat transfer rate to maximum available
heat transfer rate.
max
Q
Qact




• Maximum available temperature difference of minimum
thermal capacity fluid.
i
c
i
h
fluid T
T
T ,
,
max, 


• Actual heat transfer rate:
LMTD
act T
UA
Q 



16.    
i
c
i
h
p
LMTD
T
T
c
m
T
UA
,
,
min





 
i
c
i
h
LMTD
T
T
T
NTU
,
,
max




 
 
i
c
i
h
comm
comm
comm
comm
T
T
T
T
T
T
NTU
,
,
1
,
2
,
1
,
2
,
max
ln















17.     





















max
min
1
,
2
, 1
1
exp
p
p
comm
comm
c
m
c
m
UA
T
T


 
 
  





















max
min
min
1
,
2
,
1
exp
p
p
p
comm
comm
c
m
c
m
c
m
UA
T
T



 
  





















max
min
max
1
,
2
,
1
exp
p
p
comm
comm
c
m
c
m
NTU
T
T



18.  
  





















max
min
max
1
,
2
,
1
exp
p
p
comm
comm
c
m
c
m
NTU
T
T


 
 
i
c
i
h
comm
comm
comm
comm
T
T
T
T
T
T
NTU
,
,
1
,
2
,
1
,
2
,
max
ln















19. Counter Flow Heat Ex















 1
exp
max
min
,
,
C
C
NTU
T
T in
comm
out
comm
Tci
Tce
Thi
The
 

















 1
exp
max
min
C
C
NTU
T
T
T
T ci
he
ce
hi