Numerical method for hyperbolic inverse heat conduction problems

Int. Comm. Hat

Pergamon

Mass Transfec Vol. 28, No. 6. pp. 847-856, 2001 Copyright 0 2001 Elsevier Sctence Ltd Printed m the USA All rights reserved 0735.1933/01/$-see front matter

PII: SO7351933(01)00288-3

NUMERICAL METHOD FOR HYPERBOLIC INVERSE HEAT CONDUCTION

PROBLEMS

H.T.Chen, S.Y. Peng,P.C.Yang National Cheng Kung University Department of Mechanical Engineering Tainan, Taiwan 701, R.O.C. L.C. Fang Chinese Military Academy Fengshan, Kaohsiung Taiwan 830. R.O.C.

by K. Suzuki and S. Nishio)

(Communicated ABSTRACT

The Laplace transform technrque and control volume method III conjunction \vlth the hyperbolic shape function and least-squares scheme are applied to estimate the unknown surface condltlons of one-dImensIonal hyperbolic Inverse heat conduction problems In the present study. the expressIon of the unknown surface condltlons IS not given a prIorI To obtam the more accurate estimates. the whole time domam IS dlvlded mto several anal!.sls sub-time Intervals .4fterward. the unknown surface condltlons In each analysis Interval are estimated To evidence the accuracy of the present method. a comparison betn-ten the present estlmatlons and exact results IS made Results show that good estlmatlons on the unknown surface condltlons can be obtained from the transient temperature recordings onI> at one selected locatlon even for the cases with measurement errors 0 2001 Elsevier Science Ltd Introduction The Inverse lnltlal

condltlons

InsIde

the material

measurements surface known

problems

lnvolvmg

the estlmatlon

as well as thermal

propertles

of a body

are

condltlons

have

various

fxst

fltted

appllcatlons and

then

may be estunated

from these

short the

period

of time

circumstances.

thermal

wa\c

methods

ha\ c been developed

will

become

the

or for ver!

curve-fltted

with

the

dominant

To

date.

finite

condltlons

These temperature

various

and

measurements

near

or

temperatures

quantltles

unkno\vn It IS ~211

heat flow absolute

propagation

of the parabolic 847

fields

for the transient

10~ temperatures

theory

for the analysis

physlcal

down

of surface

from measured

In engmeerlng

unknown

that the classIcal Fourier’s model can break

extremcl! Under

thermal

analytlcal znverse

In an

zero

[I ]

velocity

of

the

and

numerlcal

heat conductlon

848

H.T. Chen et al.

problem

(IHCP)

temperatures

mvolvtng

mslde

the

csttmation

the material knowledge,

there

IHCP

non-Fourter

heat

observed

on the

from Refs. [6. 71 that there

and tnteractlon problems

feature

surface

condtttons Fourrer’s

IS no report

of such

flux

tn the

model

can exist

tn the numertcal

Thus the hyperboltc

of

based on the classical

best of the authors’ based

Vol. 28, No. 6

the sharp

solutton

IHCP IS more

from

model

a study open

of the

[2-j]

To the

for the hyperbolic

literatures

drsconttnutty.

difficult

measured

It can

reflection

hvperboltc

heat

to be solved

be

feature

conductton

than the paraboltc

IHCP It can be found

Ref

by the frntte-difference an optimum

approsrmatton

combtnatlon

numerical

solutton

transform

techntque

of the ttme

one-dimenstonal

m

errors

concept

dtmenstonal

IHCP

Chen

the Laplace

transform function

present

study

applies

concept

to estimate

present

to

using

the

size

applied

the

same

the unknown

or uncertainty

the hyperbolic technique surface

In conJunction

on the estimation

of the unknown

surface

condtttons

of the

surface wrthout

scheme

the hybrid

wtth

appltcatton

problems

with

a

III two-

in conJunctron

of

wtth a Thus.

the

a sequential-In-time

of the present

measurements,

control

temperature

heat conductron

condittons

of the temperature

stmtlar

method

exists

of the Laplace

the unknown

surface

volume

there

temperatures

the

and Ltn [6. 71 have ever developed

to analyze

method

to estimate

unknown

and the control

for error

measured

[j]

the ttme dertvattve

In general,

the hybrtd

method

et al.

estimate

technique

shape

errors

Chen

data

errors

and the mesh [4] applied

IHCP

Recently.

sequenttal-tn-ttme

the error

step

and the fintte-difference

temperature

measurement

can lead to large

Thus Chen and Chang

measurement

suitable

[3] that for “notsy”

problem

the effect 1s also

Owmg

to

of measurement

tnvesttgated

in the

analysis Mathematical Formulation The hyperboltc

be written

heat

equation

wtth

constant

thermal

properttes

can

as [I, 61:

8T’ ~-+-=~(g2 where

conductton

x’ and t’ respectively

ttme assumed

to be constant.

i?T’

d’T*

at*

ax**

denote

the spatial

a is the thermal

(1) coordinate diffusivtty.

and time Formally,

r ts a relaxatton the sttuation

for r

Vol. 28, No. 6

-

0 leads

HYPERBOLIC INVERSE HEAT CONDUCTION

to tnstantaneous

dtffusron

with the classical

thermal

For convcntcnce

of numertcal

defined

as

where

c = (dry”

dlmenslonless

at infinite

dtffuston

1s the

the followtng

propagation

speed

form ofEq. (I) can be written

with the drmenstonless

boundary T= F,(t)

d’T T-Y

condttlons or

of

Q, = --

discrete

temperature

thermocouple = I.

measurements

temperature

the

htstory

practtcal

osctllations Tr

owing

obtained

the dtrect

applications,

number

of the

small

random

can be expressed

w represents

0 05 tn the present

the averaged study

condttlons

the addltronal

ttme

history t,

discrete

informatron taken

IS denoted

of

from the

by T,“‘“. m

measurement

times

The

as

at x=x,

Thus,

errors

(7)

profiles

often

the simulated

to the exact

exhibit

data

value

from

m = 1.

,M

random

of measurements the solutton

of

as Tr=

where

(3)

(6)

can be expressed

measured

errors

t,

(3

The temperature

at x =x,

actual

to measurement

by adding

problem

the

the

= 0

T= T,(t) In

Thus.

at s=l

dtmenstonless

T,(t)

are

at x=0

= F,(t) f% IX=”

F,(t) or F,(t).

specific

wave

mtttal

IS required

A4 denotes

cornctdes

parameters

thermal

and dlmenstonless

g(O,x)

functtons

at the successtve

. Ad.where

dtmenstonless

the unknown

the

I. o
aT 0

T(O.x)=O.

dtmenstonless

,n 05x<

-= 3X

To estimate

whtch

as

+,r3T -x=

dt’

velocity

849

theory

analysts,

d’-r

propagation

PROBLEMS

random

The polynomtal

T:=(I error

+a),

and IS assumed

function

IS often

to be within

used to approximate

(8) -0.05

to

these

850

H.T. Chen et al.

measured

temperatures

standard respect

del,ratrons

usmg

the

least-squares

of the measured

to the exact

Vol. 28, No. 6

scheme

temperatures

[S]

cr,,,r and

and calculated

CT,~ are

temperatures

transform

of Eqs

(3)-(j)

A4

0, = i!:(s)

dT -_=O dx .Y IS the Laplace

1’ =s’+

transform

T(x,s)

2s and

=

It IS easy to obtarn

cT(x,

at x = 0

(11)

at x=1

parameter.

The

(1-a

parameter

1’ and

T are defined as

t)e-“dt

the drscretrzed

T!_,- 2cosh(he)?,

form of Eqs

(lo)-(

+?*,, = 0 i

T, = F,(s)

or

12) as

i=l.2,

(13)

.n

;I -T

-AiL=F2(s)

(14)

21

T”_,= T”+I where

n denotes

the number

The rearrangement

where

drmensronless Gaussran

nodal

elrmlnatron

are applred

to invert

temperatures algorrthm

divided

Into some

the followrng

( 7 ; IS an n xl

matrix,

In the

and the numerical

analysrs

functron ranges.

and inversron

equation

representmg (f}

IS an II xl of Laplace

the

unknown

matrix.

The

transform

[9]

domarn

F,(t) or F,(t), where

vector-matrix

matrix

s domain

T to that rn the physrcal

To fit the unknown

(13

of nodes

of Eqs. (Id)-( 16) gives

IS an nxn

[k]

(9)

IS

T = F,(s)or

where

~rth

data [8]

m = 1, The Laplace

the

the whole

t, IS the rnrtial

time

domarn

measurement

t, It St, is time

In other

Vol. 28, No. 6

words,

HYPERBOLIC INVERSE HEAT CONDUCTION PROBLEMS

a sequential-in-time

conditions

Owing

procedure

IS introduced

to the application

IS not alwa! s the initial

time

of the Laplace

Assume

that time

2.

interval

F,(t) or F:(t) on each

approximated

analysis

by a polynomial

and can be e\rpressed

series

transform

the dimensionless

At, IS At, = Ct; - t,, )/M The discrete .M

to estimate

coordinate

t,

t, It

of time before

the unknown

851

surface

in the present

stud!-.

measurement

time

t,, step

IS t,,, = t,, + m At,. m = I.

< t,_,_,,

I = 0. N-2. 2(N-2).

performing

the inverse

can be

calculation

as N

F,(t)

xC,t'-'

=

F,(t) = .$)z$-’

or

where

C, IS the unknown

N denotes

the degree

squares

of

the

temperature E(C!,C,,

coefficient

and is estimated

of the polynomial

The least-squares

technrque

between

measurements

taken

using

the least-squares

method

series

mmimization

deviations

117)

t-1

,=I

the

from

IS applied

calculated

the

to minimize

temperatures

thermocouple

the sum of the and

The error

curve-fitted

m the estimates

,C,)

(18) o-1

is to be

minimized

measurements. value

The estimated

of E( C,,C,,

procedures

T,‘:, IS obtamcd

,C,)

for estimating

from

values

the

of C,,

IS minimum

J

=

curve-fitted 1, 2,

The detailed

C, can refer

to Ref

profile

of

, N, are determined

description

of the

temperature until

the

computational

[7]

Results and Discussion The present

numerical

results

are obtained

N = 8, n = I I and B = 0 1 The initial Example First.

I:

Dtrichlet

the Dirichlet

boundary boundary

guess

by using

of {C,,C,,.

t, = 1, t, = 8. At, = 1, M = I, ,C,)

is { I, I,

1)

condttron

condition

IS considered

0(0, t) = sin(t)

as:

(I’))

852

H.T. Chen et al.

Table

1 shows

the estrmatton

at x, = 0 5 and obtained

from

1 0 wrth

of Table

I

non-Fourrer

heat

model

for0

the paruboltc

respect

to u,,

flux

It can be seen

the estimated

1 t 1

of the surface temperature T(0. t) usrng = 0 and

tr

results

This rmpltes

IHCP

slrghtly for

the present from

that the hyperbolic

Thus discrepancy

rn the solutron

1 also shows

T-method

and IS 0 0072 for the Qr -method

of the hyperbolrc

0 008 ts 0 0219 for the T-method

and IS 0.021

the present

behavior

cstrmates

exhrbtt

stable

Comparison

1(0.6527)1

I(0 6499)l

value

for the

for varrous

estrmates

obtained

In the

from

the

for short

trmes

wrth the exact

results

dtfficult

to solve

than

of the Jump drscontrnurtv. heat conductton

problem

[6,

= 0 0075 IS 0 014 for the

at x, = 1 w& respect to b,r Qr-method

This

cr,r values

TABLE I of T(0, t) for Example

1(0.6538)1

present

results

IS more

o,r at x,= 0 5 wnh respect to cr,r The u,,

The

estimates

agreement

IHCP

T- and Qr-methods

~,,,r = 0 are Itsted

can be to the exrstence

71 Table

I

f 0

the exact

T = 0 are rn good

and rnteractron

that

that

devrate

reflectton

I

o,,

the classrcal Fourrer heat flux model ( s = 0) for

parentheses

However.

Vol. 28, No. 6

1

\(0.6505)1

rmplres

=

that

Example

853

HYPERBOLIC INVERSE HEAT CONDUCTION PROBLEMS

Vol. 28, No. 6

Neumann

2:

The unknown

boundary

boundary

condition

condition

of the second aT

--

example

IS assumed

as

= sm( t)

(20)

ax <=” It can be seen loom Table 2 that the present exact

for I i

solution

surface

Ho\\ever.

than those

using

estimates

t S 7 even though

the estimated the T-method

results Results

Thus

method

approslmatlon

ImplIes

the sensor usmg

The

Mixed

mixed

applied

the accuracy

to estimate

the

to be more

accurate

ocT. at x5 = 0 5 with respect to g’mT The Gus value

at

that

can

the

present

hybrid

temperatures

unknown

provide

with the measurement

the

good

error

2

condition

condition

of the

method

TABLE 2 of T(0, t) forExample

in this

example

-- ZT = Isx x:0 To improve

seem

and IS 0 09 I for the Q, -

boundary

boundary

with the

= 0 0054 IS 0 0 17 for the T-method

even for the measured

3:

that

agree

1s far from the estimated

and IS 0 0086 for the T-method

Comparison

Example

locatlon

the Q,--method

also show

= 0 0055 IS 0 0139 for the Q,-method x5 = I \h~rhrespect to 6,,

usmg T- and Q, -methods

present surface

IS assumed

as

T(0, t)

estimates, temperature

(21) N = 3, N = 4 and T(0,

t) using

N = 5 are

the T-method,

854

H.T. Chen et al.

as shown m Table 3 Table

3 shows

that

increasing

of N

However,

In the

expected

beyond

observed

that

result

IN = 8 A suitable the

estimates

I shows

FIN

present

value

estimates

show

that

the

Other

different

such as (C,,C,,. agree

estimated

results

the estimated the

value

values

present

hybrid

method

sets

of the mltlal

well

with

are not

of the lnltlal

has

guesses

good

the estimates

using

shown

manuscript

m this

guesses

.,08}

{C,,C,,...,C,)={

on the estimates

The

T(0.

I. 1.

foregoing

IS not slgnlflcant

be

exact

and

the

I and FIN

good

Results

the

result

Table

to predict

, 0 5) and {O 8. 0.8,

It can

with

and

is not

4-6

exact

o’mr = 0

accuracy

are applied

range

the

with

estimates

agreement

t) between

for N = 5, x, = 0 5, At, = 0 5 and

of

In the

In good

of T(0,

can be Improved

improvement

of N seems

N = 5 are

comparison

.,C,} = (0 5, 0 5.

estimates

that the effect

the

for

Vol. 28, No. 6

1

rellabllrty t) for N = 8,

show

, I}

that their Thus.

statement

their shows

for the present

method.

Comparison

TABLE 3 of T(0, t) for Example

and various

N values

using

3, crmr=O, x, = 0 5

the T-method

Conclusions The accurate estimates for the Inverse hyperbolic heat conduction problems are obtained by ustng a hybnd application of the Laplace transform technique and the control volume method m conjunction with a hyperbolic shape function

The main difficulty in solvmg the Lnverse hyperbohc heat conduction

problems 1s that there emst the phenomena of jump discontinuity,

reflection and interaction. It is seen

from three different examples that the stab&y and accuracy of the es&mated results for various boundary

HYPERBOLIC INVERSE HEAT CONDUCTION PROBLEMS

Vol. 28, No. 6

conditions are good to compare with the exact result. Hence, the proposed numerical method IS efficient and applicable for solvmg the inverse hyperbolic heat conduction problems

FIG. I Comparison of T(0, t) Between the Present Estimates and Exact Results for Example 3,N=5. x, = 0 5 and b,,,r = 0

Nomenclature

e

distance

Qi

temperature

T

dimensionless

temperature

t!

dimensionless

final time

dimensionless

measurement

4

between

two neighboring

nodes

gradient in the transform

domain

time step

I

measurement

X,

location References

1

W W Yuan and S C Lee. J Heat Transfer

2

M N

3

J Taler.

4

H T Chen and S.M

Gzisik.

Heat Conduction.

Heat and Mass Transfer Chang,

Int

Chap

I1 1, 178 (1989)

14 Wylie,

New York (1993)

3 I, 105 (1996) J Heat Mass

Transfer

33, 621 (1990)

855

856

H.T. Chen et al.

5

H T Chen.

6

H.T Chen and J Y Lm, Int

7

H.T Chen and J Y Lm, Int. J Heat Mass Transfer

8

JP Holman and WI GaJda. Jr McGraw-Hill, New York (19X9)

9.

G. HomgandU.

S Y Lln and L.C

H1rdes.J

J

Fang,

Vol. 28, No. 6

Int. J. Heat Mass Transfer

Heat Mass Transfer

Comput

36.2891 (1993). 37, 153 (1994).

ExperImental

Appl.

Math

44, 1455 (2001)

Methods

for

Engineers.

p

37

IO. 113 (1984)

Received June I, 2001