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