One Dimensional Non-Homogeneous Conduction Equation

P M V Subbarao

Associate Professor

Mechanical Engineering Department

IIT Delhi

Another simple Mathematical modification…..But finds innumerable number of Applications….

Further Mathematical Analysis : Homogeneous ODE

• How to obtain a non-homogeneous ODE for one dimensional Steady State Heat Conduction problems?

• Blending of Convection or radiation effects into Conduction model.

• Generation of Thermal Energy in a solid body.• GARDNER-MURRAY Ideas.

0

drdrdT

Ad

Blending of Convection or Radiation in Conduction Equation

Body to gain or loose heat

Extended surface

Continuous Convection or Radiation heat transfer to/from fin surface

Conduction heat transfer to /from body.

Conduction through the fin is strengthened or weakened by continuous convection or radiation from/to fin surface.

Mathematical Ideas are More NaturalAn optimum body size is essential for the ability to regulate body temperature by blood-borne heat exchange. For animals in air, this optimum size is a little over 5 kg. For animals living in water, the optimum size is much larger, on the order of 100 kg or so.

This may explain why large reptiles today are largely aquatic and terrestrial reptiles are smaller.

Mathematical Ideas are More Natural

• Reptiles like high steady body temperatures just as mammals and birds.

• They have sophisticated ways to manage flows of heat between their bodies and the environment.

• One common way they do this is to use blood flow within the body to facilitate heat uptake and retard heat loss.

• Blood flow is not effective as a medium of heat transfer everywhere in the body.

• Body shape also enters into the equation. • It also helps expalin the odd appendages like crests and

sails that decorated extinct reptiles like Stegosaurus or mammal-like reptiles like Dimetrodon.

• Theoretical Biologists did Calculations to show these structures could act as very effective heat exchange fins.

• These fins are allowing animals with crests to heat their bodies up to high temperatures much faster than animals without them.

Amalgamation of Conduction and Convection/Radiation

Heat Conducitonin

Heat Conducitonout

Heat ConvectionIn/out

profile

PROFILE AREA

cross-section

CROSS-SECTION AREA

Basic Geometric Features of Fins

Innovative Fin Designs

Single Fins :Shapes

Longitudinal or strip

Radial Pins

Anatomy of A STRIP FIN

thickness

x

x

Flow

Dire

ctio

n

GARDNER-MURRAY ANALYSIS : ASSUMPTIONS

Steady state one dimensional conduction Model. No Heat sources or sinks within the fin . Thermal conductivity constant and uniform in all directions. Heat transfer coefficient constant and uniform over fin faces. Surrounding temperature constant and uniform. Base temperature constant and uniform over fin base. Fin width much smaller than fin height. No bond resistance between fin base and prime surface. Heat flow off fin proportional to temperature excess.

Slender Fins

thickness

x

x

1 b

orL

orL

D

Steady One-dimensional Conduction through Fins

qx qx+dx

qconv or qradiation

Conservation of Energy:

radiationxxcondxcond qqq ,,

convectionxxcondxcond qqq ,,

OR

)()( TTxPhTTAhq radradradiation

222 TTTTh guessguessrad

Where

)()( TTxPhTThAqconvection

OR

Substituting and dividing by x:

0)(,,

TThP

x

qq xcondxxcond

Taking limit x tends to zero and using the definition of derivative:

0)( TThPdx

dq

Substitute Fourier’s Law of Conduction:

0)(

TThPdx

dxdT

kAd c

0)(

TThPdx

dxdT

kAd c

Fins with Cartesian Geometry Heat Transfer

Straight fin of triangular profile rectangular C.S.

b

xLxA )(

Straight fin of parabolic profile rectangular C.S.

b

L

x=0b

x=b

bx

qb

L

b

qb

b

x=b x=0

xb

2

)(

b

xLxA

0)(

TThPdx

dxdT

kAd c

Fins with Cartesian Geometry Heat Transfer

Straight fin of triangular profile Circular C.S.

22

4)( x

b

DxA

Straight fin of parabolic profile rectangular C.S.

44

4)( x

b

DxA

0)(

TThPdr

drdT

kAd c

Fins with Cylindrical Geometry Heat Transfer

Circumferential fin of rectangular profile

rrA 2)(

Straight fin of triangular profile

22)( rR

rA

For a constant cross section area:

0)(2

2

TThPdx

TdkAc

0)(2

2

TTkA

hP

dx

Td

kA

hPm 2

Fin factor for pin Fin:4

& 2d

AπdP

kd

h

A

4

k

hP=m

Fin factor for strip Fin: LALP & 2

k

h

kL

Lh

A

22

k

hP=m

0)(22

2

TTmdx

Td

Define: TT

022

2

m

dx

d

At the base of the fin:

TTbasebase

Tip of A Fin

Linear Second order ODE with Constant Coefficients

• This equation has two linearly independent solutions.• The general solution is the linear combination of those two

independent solutions.• Each solution functionx and its second derivative must be

constant multiple of each other.• Therefore, the general solution function of the differential equation

above is:

0mxd

d 22

2

-mxmx eCeCxθ 21

At the base of the fin:

TTbasebase

Infinitely long fin:

LTT tipfintipfin as 0

x C e C e1mx

2-mx

Logic from Mathematics shows that C1 = 0 !

-mx2eCx

At the base of the fin:

02

mbasebase eCTT

mxbase eTTTxT

)(

mx

base

eTT

TxT

)(

For a strip fin:

x

k

h

base

eTT

TxT 2

)(

Rate of Heat Transfer in an Infinitely Long Strip Fin

TTmkAdx

dTkAQ basecs

xcs

0

TTkA

hPkA

dx

dTkAQ base

cscs

xcs

0

surfacefinA

surbasecsx

cs dAxhTThPkAdx

dTkAQ

0

Most Practicable Boundary Condition

Corrected adiabatic tip:

2

bb adicorr

thickness

x

x

bb

The boundary condition are:0=0)=q(x

Using these gives: (x = b) = C e C eb 1mb

2-mb

and q(x = 0) = k Ld

dxk Lm C e C e

q(x = 0) k Lm(C Cx=0

1mx

2-mx

x=0

1 2

) 0

The foregoing shows that: -mbmb

b eeC

CCC

=

21

b=b)=(x

Longitudinal Fin : Adiabatic Tip

With the general solution for the temperature excess

-mxmx eCeCxθ 21

And from the previous slide

-mbmbb-mbmb

b ee

θCee=Cθ

CCC

21

mb

mxθee

ee

θb

-mxmx-mbmb

b

cosh

coshx

The heat flow through the fin at any location x is:

mb

mxLmkxq

mb

mx

xLk

xLkxq

cosh

sinh

cosh

cosh

d

d

d

d

b

b

And at x=b (heat entering fin base): mbLmkq bb tanh

mbLhkq bb tanh2 21 For a strip fin:

The fin efficiency, , is defined as the ratio of the actual heat dissipation to the ideal heat dissipation if the entire fin were to operate at the base temperature excess

idealq

qactual

Efficiency of Strip Fin

bb

Lb

hLbhQ 2SURFACE2

ideal

For infinitely long strip fin:

TThPkAQ basecsact inf

For Adiabatic strip fin:

mbTThPkAQ basecsadiact tanh

Strip Fin: Infinitely Long

0

q

Q

inf

id

inf-actinf

bsurface

basecs

hA

TThPkA

Strip Fin: Adiabatic tip

mb

mbtanh

bm

mbtanhm

2hLb

mbtanhhPkA

q

Q

2

b

bcs

id

adi-act

adi

adi

adi

SUMMARYLongitudinal Fin of Rectangular Profile: adiabatic

tip

Temperature Excess Profile

xmx

mbbcosh

cosh Heat Dissipated = Heat Entering Base

mbLmθkq bb tanh

Fin Efficiency tanh mb

mb