Chapter 4: Heat Transfer from Extended Surfaces

4.1 …………. Introduction and Types of Fins

4.2 …………. Governing Equation of Fin

4.3 …………. Heat Dissipation through Rectangular Fin

4.4 …………. Heat Dissipation from Infinitely Long Fin

4.5 …………. Heat Dissipation from a Fin Insulated at the Tip

4.6 …………. Heat Dissipation from a Fin Losing Heat at a Tip

4.7 …………. Fin Efficiency and Effectiveness

4.8 …………. Estimation of error in temperature measurement in a thermometer well

Convection: Heat transfer between a solid surface and a moving fluid is governed by the Newton’s cooling law: q = hA(Ts-T). Therefore, to increase the convective heat transfer, one can

Increase the temperature difference (Ts-T) between the surface and the fluid.

Increase the convection coefficient h. This can be accomplished by increasing the fluid flow over the surface since h is a function of the flow velocity and the higher the velocity, the higher the h. Example: a cooling fan.

Increase the contact surface area A. Example: a heat sink with fins.

An extended surface (also know as a combined conduction-convection system or a fin) is a solid within which heat transfer by conduction is assumed to be one dimensional, while heat is also transferred by convection (and/or radiation) from the surface in a direction transverse to that of conduction.

4.1 … Introduction and Types of Fins

Introduction and Types of Fins (continue….)

The term extended surface (Fin) is commonly used in reference to a solid that experiences energy transfer by conduction within its boundaries, as well as energy transfer by convection (and/or radiation) between its boundaries and the surroundings.

Extended surfaces may exist in many situations but are commonly used as fins to enhance heat transfer by increasing the surface area available for convection (and/or radiation).

Note: They are particularly beneficial when h is small, as for a gas and natural convection. T2

T 1

T 1 T2T (x)

0

L

qcon v FluidToo ,h

qx,1

qx,2

T 1 T2 Too

APPLICATIONS: the arrangement for cooling engine heads on motorcycles and lawn-mowers or for cooling electric power transformers the tubes with attached fins used to promote heat exchange between air and the working fluid of an air conditioner.

Introduction and Types of Fins (continue….)

Straight fin of uniform cross section

Straight fin of non uniform cross section

Annular fin Pin fin

Types of Fins

Tb

cond,in cond,out conv,outq q q

Ac(x)

cond,inq cond,outq

conv,outq

dAs(x)x

T∞, h

dx

dx

4.2 … Governing Equation of Fin

Assumptions

• Heat transfer is assumed to be in only one dimensional i.e., in the longitudinal (x) direction, even though conduction within the fin is actually two dimensional.

• The rate at which the energy is convected to the fluid from any point on the fin surface must be balanced by the rate at which the energy reaches that point due to conduction in the transverse ( y,z ) direction. However, in practice the fin is thin and temperature changes in the longitudinal direction are much larger than those in the transverse direction.

• Steady state conditions are assumed. • Thermal conductivity is assumed to be constant .

• Radiation from the surface is assumed to be negligible .

• Convection heat transfer coefficient is assumed to be uniform over the surface.

Governing Equation of Fin (continue….)

dx

Ac(x)dAs(x)

cond,inq cond,outq

conv,outq

cond,in cond,out conv,outq q q

x

( ) ( )c cdT d dTkA x kA x dxdx dx dx

( ) ( )shdA x T x T

cond,inq ( )x cdTq kA xdx

cond,outq x dxq

conv,outq

T∞, h

T(x)

convdq

When k = constant,

( ) ( ) ( ) 0c sd dTkA x dx hdA x T x Tdx dx

( ) ( ) 0sc

dAd dTkA x h T x Tdx dx dx

( ) ( ) 0sc

dAd dT hA x T x Tdx dx k dx

Fins of Uniform Cross-Sectional Area

Ac(x) = constant, and dAs = Pdx

P: fin perimeter

x

Ac

Tb

L

dAsdx

,T h

( ) ( ) 0sc

dAd dT hA x T x Tdx dx k dx

2

2 ( ) 0c

d T hP T x Tdx kA

P

boundary conditions

at x = 0:

excess temperature : q(x) = T(x) - T∞

2

2 ( ) 0c

d T hP T x Tdx kA

22

2 0d mdxq q

where2

c

hPmkA

dxx

Tb, T h

(0) bT T (0) (0)T Tq bT T bq

L

T(x)

1 2( ) mx mxx C e C eq 1 2sinh( ) cosh( )D mx D mx

at x = L: ---- 3 cases

1) very long fin (L → ∞):

2) convection tip:

3) negligible heat loss: adiabatic tip

dxx

Tb, T h

L( ) ( )x T x Tq

( )T L T ( ) 0T Tq

( )L

dTk h T L Tdx ( ) 0

L

d h Ldx kq q

0L

dTkdx

0L

ddxq

T(x)

Temperature distribution

2) convection tip:

1) long fin:

3) adiabatic tip:

( ) mx

b

x eqq

cosh ( ) / sinh ( )( )cosh / sinhb

m L x h mk m L xxmL h mk mL

qq

( ) cosh ( )coshb

x m L xmL

qq

Total heat loss by the fin

1) long fin:

3) adiabatic tip:

2) convection tip:

,0

f c bx

dTq kAdx

or ( )f

f Aq h T x T dA

x

AcdAs

Tb

L

dx

P, T h

f c bq hPkA Mq

sinh / coshcosh / sinhf c b

mL h mk mLq hPkA

mL h mk mLq

tanh tanhf c bq hPkA mL M mLq

Temperature distribution for fins of different configurations

Case Tip Condition Temp. Distribution Fin heat transfer A Convection heat

transfer: hq(L)=-k(dq/dx)x=L mLmk

hmL

xLmmkhxLm

sinh)(cosh

)(sinh)()(cosh

MmLmk

hmL

mLmkhmL

sinh)(cosh

cosh)(sinh

B Adiabatic (dq/dx)x=L=0 mL

xLmcosh

)(cosh mLM tanh

C Given temperature: q(L)=qL

mL

xLmxLmb

L

sinh

)(sinh)(sinh)( qq

mL

mLM b

L

sinh

)(cosh qq

D Infinitely long fin q(L)=0

mxe M

bCbb

C

hPkAMTT

kAhPmTT

qqq

q

,)0(

, 2

Note: This table is adopted from Introduction to Heat Transfer by Frank Incropera and David DeWitt

How effective a fin can enhance heat transfer is characterized by the fin effectiveness f: Ratio of fin heat transfer and the heat transfer without the fin. For an adiabatic fin:

ChPkA tanh( )tanh( )

( )If the fin is long enough, mL>2, tanh(mL) 1, it can be considered an infinite fin (case D of table3.4)

In order to enhance heat tra

f ff

C b C C

fC C

q q mL kP mLq hA T T hA hA

kP k PhA h A

nsfer, 1.

However, 2 will be considered justifiable

If <1 then we have an insulator instead of a heat fin

f

f

f

4.7 … Fin Efficiency and Effectiveness

Fin Effectiveness (contd...)

To increase f, the fin’s material should have higher thermal conductivity, k.

It seems to be counterintuitive that the lower convection coefficient, h, the higher f. But it is not because if h is very high, it is not necessary to enhance heat transfer by adding heat fins. Therefore, heat fins are more effective if h is low. Observation: If fins are to be used on surfaces separating gas and liquid. Fins are usually placed on the gas side. (Why?)

fC C

kP k PhA h A

, ,

, ,

( ) /( ) ( ) /

f f b t f t hf

C b b t h t f

q q T T R Rq hA T T T T R R

The fin efficiency is defined as the ratio of the energy transferred through a real fin to that transferred through an ideal fin. An ideal fin is thought to be one made of a perfect or infinite conductor material. A perfect conductor has an infinite thermal conductivity so that the entire fin is at the base material temperature.

Fin Efficiency

q

q

qq

h P k A m Lh P L

rea l

idea l

c L

L

tan h ( )

Tb

x

Real situation

x

Ideal situation

Tb

q

q

k Ah P

m LL

m Lm L

c L

L

tan h ( ) tanh ( )

Thermal Resistance Concept

T1

T

TbT2

L1 t

A=Ab+NAb,f

Rb= t/(kbA)

T1TTbT2

R1=L1/(k1A) )/(1, OtOt hAR

1 1

1 ,b t O

T T T TqR R R R

4.8 … Estimation of Error in Temperature Measurement in a Thermometer Well

A thermometer well is defined as a small tube welded radially into a pipeline through which a fluid whose temperature is to be measured is flowing.

Let,l= length of the well/tubed= internal diameter of well/tubeδ= thickness of well/tubetf= temperature of the fluid flowing through pipeto= temperature of the pipe=wall ta= ambient temperaturetl= temperature at bottom of well

When tf> ta , the heat flows from fluid to towards the tube wall along the well. The temperature at the bottom of well becomes colder than fluid flowing around, obviously the temperature shown by the thermometer will not be the temperature of the fluid. This error may be calculated by assuming the well to be a spine protruding from the wall of a pipe in which fluid is flowing.

Note: Assumption for simplicity, that there is no flow of heat from tip (i.e. insulated tip)

Temperature distribution at any distance x,

At x=l,

= Thermometric Error

Perimeter

Cross section Area

)cosh(1

)cosh()](cosh[

)cosh()](cosh[

mlmlllm

tttt

mlxlm

tttt

fo

fi

fo

fx

o

x

qq

kh

kAhPm

dd

AP

dAddP

cs

cs

cs

1

)2(

Thus, the temperature measured by the thermometer is not affected by the diameter of the well.

From previous equation, observed that in order to reduce the temperature measurement error, ml should be large necessitating the following: (i) Large value of h, (ii)Small value of k & (iii) Long and thin well

Definition:

An extended surface (also know as a combined conduction-convection

system or a fin) is a solid within which heat transfer by conduction is

assumed to be one dimensional, while heat is also transferred by convection

from the surface

Extended surfaces may exist in many situations but are commonly used as

fins to enhance heat transfer by increasing the surface area available for

convection.

Straight fin of uniform cross section

Straight fin of non uniform cross section

Annular fin Pin fin

They are particularly beneficial when heat transfer coefficient (h) is small as

for a gas and natural convection. Some typical fin configurations:

Equation for Extended Surfaces

x

T∞, h

Ac(x)dx

dx

Tb

cond,inq cond,outq

conv,outq

cond,in cond,out conv,outq q q

dAs(x)

dx

Ac(x)dAs(x)

cond,inq cond,outq

conv,outq

cond,in cond,out conv,outq q q

x

( ) ( )c cdT d dTkA x kA x dxdx dx dx

( ) ( )shdA x T x T

cond,inq ( )x cdTq kA xdx

cond,outq x dxq

conv,outq

T∞, h

T(x)

convdqEnergy Balance:

if k, A are all constants.

x

C

q q dq qdqdxdx hdA T T

kA d Tdx

dx hP T T dx

x dx conv xx

S

C

( )

( ) ,2

2 0

When k = constant,

( ) ( ) ( ) 0c sd dTkA x dx hdA x T x Tdx dx

( ) ( ) 0sc

dAd dTkA x h T x Tdx dx dx

( ) ( ) 0sc

dAd dT hA x T x Tdx dx k dx

Fins of Uniform Cross-Sectional Area

Ac(x) = constant,

and dAs = Pdx

P: fin perimeter

x

Ac

Tb

L

dAsdx

,T h

( ) ( ) 0sc

dAd dT hA x T x Tdx dx k dx

2

2 ( ) 0c

d T hP T x Tdx kA

P

boundary conditionsat x = 0:

excess temperature : q(x) = T(x) - T∞

2

2 ( ) 0c

d T hP T x Tdx kA

22

2 0d mdxq q

where 2

c

hPmkA

dxx

Tb, T h

(0) bT T (0) (0)T Tq bT T bq

L

T(x)

1 2( ) mx mxx C e C eq 1 2sinh( ) cosh( )D mx D mx

Boundary ConditionsSeveral boundary conditions are typically employed:• At the fin base– Specified temperature boundary condition, expressed

as: q(0)= qb= Tb-T∞

• At the fin tip1. Specified temperature2. Infinitely Long Fin3. Adiabatic tip4. Convection (and

combined convection).

Adapted from Heat and Mass Transfer – A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.

Temperature distribution for fins of different configurations

Case

Tip Condition Temp. Distribution Fin heat transfer

A Convection heat transfer: hq(L)=-k(dq/dx)x=L

mLmkhmL

xLmmkhxLm

sinh)(cosh

)(sinh)()(cosh

MmLmk

hmL

mLmkhmL

sinh)(cosh

cosh)(sinh

B Adiabatic (dq/dx)x=L=0 mL

xLmcosh

)(cosh mLM tanh

C Infinitely long fin q(L)=0

mxe M

bCbb

C

hPkAMTT

kAhPmTT

qqq

q

,)0(

, 2

Fin Effectiveness Adapted from Heat and Mass Transfer – A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.

• The performance of the fins is judged on the basis of the enhancement in heat transfer relative to the no-fin case.

• The performance of fins is expressed in terms of the fin effectiveness fin defined as

fin finfin

no fin b b

Q QQ hA T T

Heat transfer rate

from the surface of area Ab

Heat transfer rate from the fin of base

area Ab

Fin Effectiveness (contd...)

, ,

, ,

The effectiveness of a fin can also be characterized as( ) /

( ) ( ) /

It is a ratio of the thermal resistance due to convection to the thermal resistance of a fin

f f b t f t hf

C b b t h t f

q q T T R Rq hA T T T T R R

. In order to enhance heat transfer,the fin's resistance should be lower than that of the resistancedue only to convection.

Fin Design

Total heat loss: qf=Mtanh(mL) for an adiabatic fin, or qf=Mtanh(mLC) if there is convective heat transfer at the tip

C C

C,

,

hPwhere = , and M= hPkA hPkA ( )

Use the thermal resistance concept:( )hPkA tanh( )( )

where is the thermal resistance of the fin.

For a fin with an adiabatic tip, the fin

b bc

bf b

t f

t f

m T TkA

T Tq mL T TR

R

q

,C

resistance can be expressed as( ) 1

hPkA [tanh( )]b

t ff

T TRq mL

Tb

T

Fin Efficiency

Define Fin efficiency:

where represents an idealized situation such that the fin is made upof material with infinite thermal conductivity. Therefore, the fin shouldbe at the same temperature as the temperature of the base.

f

qq

q

q hA T T

f

f b

max

max

max ( )

T(x)<Tb for heat transferto take place

Total fin heat transfer qf

Real situation Ideal situation

For infinite kT(x)=Tb, the heat transferis maximum

Ideal heat transfer qmax

Tb x x

Fin Efficiency (contd…)

Fin Efficiency (cont.)

Use an adiabatic rectangular fin as an example:

max

f

max

,,

( ) tanhtanh( ) ( )

tanh tanh (see Table 3.5 for of common fins)

The fin heat transfer: ( )

, where 1/( )

f c bf

f b b

c

f f f f b

b bf t f

f f t f

q hPkA T T mLM mLq hA T T hPL T T

mL mLmLhP L

kAq q hA T T

T T T Tq RhA R

,

, , b

1

Thermal resistance for a single fin.1As compared to convective heat transfer:

In order to have a lower resistance as that is required to enhance heat transfer: or A

f f

t bb

t b t f f f

hA

RhA

R R A

Overall Fin Efficiency

Overall fin efficiency for an array of fins:

Define terms: Ab: base area exposed to coolant

Af: surface area of a single fin

At: total area including base area and total

finned surface, At=Ab+NAf

N: total number of fins

qb

qf

Overall Fin Efficiency (contd…)

( ) ( )

[( ) ]( ) [ (1 )]( )

[1 (1 )]( ) ( )

Define overall fin efficiency: 1 (1 )

t b f b b f f b

t f f f b t f f b

ft f b O t b

t

fO f

t

q q Nq hA T T N hA T T

h A NA N A T T h A NA T T

NAhA T T hA T T

ANAA

Heat Transfer from a Fin Array

,,

,

,

1( ) where

Compare to heat transfer without fins1( ) ( )( )

where is the base area (unexposed) for the fin

To enhance heat transfer Th

bt t O b t O

t O t O

b b b f b

b f

t O

T Tq hA T T RR hA

q hA T T h A NA T ThA

A

A A

Oat is, to increase the effective area .tA

Heat Transfer through Extended Surface or Fins

Prepared by: Nimesh Gajjar

Bare surface Finned surface

Definition:

An extended surface (also know as a combined conduction-convection

system or a fin) is a solid within which heat transfer by conduction is

assumed to be one dimensional, while heat is also transferred by convection

from the surface

Extended surfaces may exist in many situations but are commonly used as

fins to enhance heat transfer by increasing the surface area available for

convection.

Straight fin of uniform cross section

Straight fin of non uniform cross section

Annular fin Pin fin

They are particularly beneficial when heat transfer coefficient (h) is small as

for a gas and natural convection. Some typical fin configurations:

Equation for Extended Surfaces

x

T∞, h

Ac(x)dx

dx

Tb

cond,inq cond,outq

conv,outq

cond,in cond,out conv,outq q q

dAs(x)

dx

Ac(x)dAs(x)

cond,inq cond,outq

conv,outq

cond,in cond,out conv,outq q q

x

( ) ( )c cdT d dTkA x kA x dxdx dx dx

( ) ( )shdA x T x T

cond,inq ( )x cdTq kA xdx

cond,outq x dxq

conv,outq

T∞, h

T(x)

convdqEnergy Balance:

if k, A are all constants.

x

C

q q dq qdqdxdx hdA T T

kA d Tdx

dx hP T T dx

x dx conv xx

S

C

( )

( ) ,2

2 0

When k = constant,

( ) ( ) ( ) 0c sd dTkA x dx hdA x T x Tdx dx

( ) ( ) 0sc

dAd dTkA x h T x Tdx dx dx

( ) ( ) 0sc

dAd dT hA x T x Tdx dx k dx

Fins of Uniform Cross-Sectional Area

Ac(x) = constant,

and dAs = Pdx

P: fin perimeter

x

Ac

Tb

L

dAsdx

,T h

( ) ( ) 0sc

dAd dT hA x T x Tdx dx k dx

2

2 ( ) 0c

d T hP T x Tdx kA

P

boundary conditionsat x = 0:

excess temperature : q(x) = T(x) - T∞

2

2 ( ) 0c

d T hP T x Tdx kA

22

2 0d mdxq q

where 2

c

hPmkA

dxx

Tb, T h

(0) bT T (0) (0)T Tq bT T bq

L

T(x)

1 2( ) mx mxx C e C eq 1 2sinh( ) cosh( )D mx D mx

Boundary ConditionsSeveral boundary conditions are typically employed:• At the fin base– Specified temperature boundary condition, expressed

as: q(0)= qb= Tb-T∞

• At the fin tip1. Specified temperature2. Infinitely Long Fin3. Adiabatic tip4. Convection (and

combined convection).

Adapted from Heat and Mass Transfer – A Practical Approach, Y.A. Cengel, Third Edition, McGraw Hill 2007.