Deduction of Fundamental Laws for Heat Exchangers

P M V SubbaraoProfessor

Mechanical Engineering Department

I I T Delhi

Modification of Basic Laws for Design of General Templates for HXs?!?!?!

Thermodynamic Vision of Science & Engineering

Primitiveconcepts

AcceptTheory

Secondaryconcepts

Propose Theory

TestTheory

Impact on Society

IndustrialProcesses

EngineeringRelations

ParticularRelations

IND

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ION

Increase or Decrease of Temperature : Fluid in A Container

• Heating of a control mass:

Constant Volume Heating 1Q2 = U2 – U1

pdVdUQ

• Consider a homogeneous phase of a substance with constant composition.

• Define Specific Heat: The amount of heat required per unit mass/mole to raise the temperature by one degree.

• No change in other forms of energy, except internal energy.

Increase or Decrease of Temperature : Flowing Fluid

Thermodynamic Perspective of HX.

• The rate of enthalpy gained by a cold fluid

)( cfcoldcfhotfluidcoldabsorbed hhmQ

)( hfcoldhfhotfluidhotdonated hhmQ

donatedabsorbed QQ

• The rate of enthalpy lost by hot fluid

• Thermal Energy Balance:

Heat Transfer Perspective of HX.

• Estimation and Creation of primary driving force.

• To the hot fluid loose thermal energy?

• To help cold fluid gain thermal energy?

• Provision of thermal infrastructure to satisfy law of conservation of energy.

donatedabsorbed QQ

• How to model this mutual interaction using principles of Heat Transfer ?

Understanding of precise role of thermodynamic Parameters…

Incompleteness in Basic Laws of Heat Teat Transfer

• For Heat communication between cold and hot.

• A Simple adiabatic Heat Exchanger model.

phihhot cTm ,, ,

pciccold cTm ,, ,

phehhot cTm ,, ,

pceccold cTm ,, ,

Fourier’s law of heat conduction

L

TkA

L

TTkAq coldhot

x

This is called as Fourier Law of Conduction

A Constitutive Relation

Global heat transfer rate:

dx

dTkAqx

Use of Fourier Law of Conduction for HXs

phihhot cTm ,, ,

pciccold cTm ,, ,

phehhot cTm ,, ,

pceccold cTm ,, ,

Local Heat flux in a slab:

dx

dTkqx ''

Mathematical Description

• Temperature is a scalar quantity.

• Heat flux is defined with direction and Magnitude : A Vector.

• Mathematically it is possible to have:

kqjqiqq zyxˆ''ˆ''ˆ'' ''

Using the principles of vector calculus:

Tkq ''

kz

Tj

y

Ti

x

Tkq ˆˆˆ''

Further Physical Description

• Will k be constant from one end of HX to the other end?

• Will k be same in all directions?• Why k cannot be different each direction?• Why k cannot be a vector variable?

kzyxkjzyxkizyxkzyxk zyxˆ,,ˆ,,ˆ,, ,,

Will mathematics approve this ?

What is the most general acceptable behavior of k, approved by both physics and mathematics?

Most General form of Fourier Law of Conduction

dx

dTkqx ''

Tkq ''

kkjkikk zyxˆˆˆ

We are at cross roads !!!!!

Local Heat flux in a slab along x-direction :

Local Heat flux vector :

Tzyxkzyxq ,, ,,''

Physical-mathematical Feasible Model

• Taking both physics and mathematics into consideration, the most feasible model for Fourier’s Law of conduction is:

Tkq .~~

''

Thermal conductivity of a general material is a tensor.

z

T

y

Tx

T

kkk

kkk

kkk

q

q

q

zzzyzx

yzyyyx

xzxyxx

z

y

x

''

''

''

Surprising Results !!!

z

Tk

y

Tk

x

Tkq xzxyxxx ''

z

Tk

y

Tk

x

Tkq yzyyyxy ''

z

Tk

y

Tk

x

Tkq zzzyzxy ''

Newton’s Law of Convection Cooling

• Convection involves the transfer of heat between a surface at a given temperature (Ts) and fluid at a bulk temperature (Tb).

• Newton’s law of cooling suggests a basic relationship for heat transfer by convection:

bs TThAQ

h is called as Convection Heat Transfer Coefficient, W/m2K

donatedabsorbed QQ

coldbcoldscoldcoldabsorbed TTAhQ ,.

hotbhotshothotdonated TTAhQ ,.

Realization of Newton’s Law Cooling

• A general heat transfer surface may not be isothermal !?!

• Fluid temperature will vary from inlet to exit !?!?!

• The local velocity of flow will also vary from inlet to exit ?!?!

• How to use Newton’s Law in a Real life?

Local Convection Heat Transfer

Consider convection heat transfer as a fluid passes over a surface of arbitrary shape:

Apply Newton’s law cooling to a local differential element with length dx.

TTTThq ss ''

h is called as Local Convection Heat Transfer Coefficient, W/m2K

The total energy emitted by a real system, regardless of the wavelengths, is given by:

4syssurfacesyssysemitted TAQ

• where εsys is the emissivity of the system,• Asys-surface is the surface area, • Tsys is the temperature, and • σ is the Stefan-Boltzmann constant, equal to 5.67×10-8 W/m2K4. • Emissivity is a material property, ranging from 0 to 1, which

measures how much energy a surface can emit with respect to an ideal emitter (ε = 1) at the same temperature

Radiation from a Thermodynamic System

Radiative Heat Transfer between System and Surroundings Consider the heat transfer between system surface with surroundings, as shown in Figure. What is the rate of heat transfer into system surface ?

4, sursursuremittedsur TAQ

This radiation is emitted in all directions, and only a fraction of it will actually strike system surface. This fraction is called the shape factor, F.

To find this, we will first look at the emission from surroundings to system. Surrounding Surface emits radiation as described in

The amount of radiation striking system surface is therefore:

4, sursursursyssurinceidentsys TAFQ

The only portion of the incident radiation contributing to heating the system surface is the absorbed portion, given by the absorptivity αB:

4, sursursursyssursysabsorbedsys TAFQ

Above equation is the amount of radiation gained by System from Surroundings. To find the net heat transfer rate for system, we must now subtract the amount of radiation emitted by system:

4, syssyssysemittedsys TAQ

The net radiative heat transfer (gain) rate at system surface is

emittedsysabsorbedsyssys QQQ ,,

44syssyssyssursursursyssursyssys TATAFQ

Similarly, the net radiative heat transfer (loss) rate at surroundings surface is

44sursursursyssyssyssursyssursur TATAFQ

What is the relation between Qsys and Qsur ?

Wall Surfaces with Convection

2112

2

0 CxCTCdx

dT

dx

TdA

Boundary conditions:

110

)0(

TThdx

dTk

x

22 )(

TLThdx

dTk

Lx

Rconv,1 Rcond Rconv,2

T1 T2

dido Di

TubularFlow

Annular Flow

Overall heat transfer coefficient of a used HX, based on outside area:

ofowallfi

i

out

ii

outoutsideOverall

hRRR

AA

hAA

U11

1,

A Simple Heat Exchanger

owall

ii

outoutsideOverall

hR

hAA

U11

1,

i

owall d

d

kR ln

2

1

Overall heat transfer coefficient of a new/cleaned HX, based on outside area:

Thermal resistance of any annular solid structure:

MeanoutDPHX TUAQ

Mean Temperature Difference

Simple Counter Flow Heat Exchangers: C >1

cpcold

hphot

cm

cmC

,

,

Simple Counter Flow Heat Exchangers: C < 1

Simple Parallel Flow Heat Exchangers

Thermodynamics of An Infinite HX

• All properties of thermal structure remain unchanged in all directions.

• All properties of thermal structure are independent of temperature .

• An unique surface area of heat communication is well defined.

hhphotloss dTcmQ ,

,hph

lossh cm

QdT

ccpcoldgain dTcmQ ,

cpc

gainc cm

QdT

,

Thermal Resistance of infinitesimal Heat Exchanger

commcommth

comm

effcold

gain

effhot

loss TdUR

Td

A

Q

A

Q

,,,

• Thermal resistance of an Infinitesimal adiabatic Heat Exchanger

Th

Tc

Variation of Local temperature difference for heat communication:

chchcomm dTdTTTdTd

cpc

gain

hph

losscomm cm

Q

cm

QTd

,,

Heat Transfer in an infinitesimal HX

dATTUQQ chgainloss

Synergism between HT & TD:

cpchphchcomm cmcm

dATTUTd,,

11

cpchphcommcomm cmcm

dATUTd,,

11

cpchphcomm

comm

cmcmUdA

T

Td

,,

11

A

cpchph

T

T comm

comm dAcmcm

UT

Tdcomm

comm 0,,

112,

1,

For A finite HX:

Acmcm

UT

T

cpchphcomm

comm

,,1,

2, 11ln

For A finite HX:

Q

TT

cmQ

TT

cm

TTcmQTTcm

outhinh

hphot

incoutc

cpcold

outhinhhphotincoutccpcold

,,

,

,,

,

,,,,,,

1

1

ATTTTQ

U

T

Tincoutcouthinh

comm

comm,,,,

1,

2,ln

ATTQ

U

T

Tcommcomm

comm

comm2,1,

1,

2,ln

1,

2,

1,2,

lncomm

comm

commcomm

T

T

TT

UA

Q

A representative temperature difference for heat communication:

1,

2,

1,2,

lncomm

comm

commcommLM

T

T

TTT

Discussion on LMTD

• LMTD can be easily calculated, when the fluid inlet temperatures are know and the outlet temperatures are specified.

• Lower the value of LMTD, higher the value of overall value of UA.

• For given end conditions, counter flow gives higher value of LMTD when compared to co flow.

• Counter flow generates more temperature driving force with same entropy generation.

• This nearly equal to mean of many local values of T.