DYNAMIC OBSERVERS

BASED ON GREEN’S

FUNCTION APPLIED

TO 3D INVERSE

THERMAL MODELS

Federal University of Uberlândia

School of Mechanical Engineering

INTRODUCTION

Based on Blum & Marquardt (1997) work focused on the one-dimensional linear case

Here it is extended to solve an inverse

multidimensional heat conduction problem.

That means: to apply the technique to solve 2D

and 3D thermal problems.

The Thermal Model: 1D

q(t) = ?

L

Isolated surface

x

0t,Lx0x

1

t 2

2

0t?)t(qx

k0x

0t0x Lx

Lx00)0,x(

Heat difusion equation:

Boundary condition:

Initial condition:

Fundamentals

Taking the Laplace transform of the spatially

discretized system :

The heat transfer function defined by an input/output

system obtains:

equationdDiscretizesQ

sa

sb

sxTnh

i

i

i

mh

i

i

i

j

)(),(

0

0

functiontransferheatsQ

sT

sa

sb

Gnh

i

i

i

mh

i

i

i

H

)(

)(

0

0

Fundamentals

GH TM q N

Gc

ĜH

T*M

+

+ +

TM ^

q ̂

q ^

NqGNTT HM

*

M

*

1ˆ

M

Hc

c TGG

Gq

The thermal model can be represented by

a dynamic system given by a block diagram

It can be observed from the block diagram that:

i) the unknown heat flux is applied to the conductor (reference model), GH,

ii) and results in a measurement signal T*M corrupted by noise N

q̂

Fundamentals

NQ GG

Hc

c

Hc

Hc NGG1

Gq

GG1

GGq̂

the estimate value is computed from the input data . Thus, the estimator can be represented in a closed-loop transfer function of the feedback loop as

H

Q

NHQNG

GGorGGG 1It can be observed that:

01 NQ GandG

It can be observed that if the algorithm estimates the heat flux correctly, GQ is equal to unity or GQ = 1

Fundamentals

The heat transfer function is known (GH) :

Frequência (rad/s)

Fas

e (

°)M

agni

tud

e (

dB

)

-600

-400

-200

0

200

10-8

10-6

10-4

10-2

100

102

104

-1080

-900

-720

-540

-360

-180

0

Phase

Modulu

s

Frequency, hz

Fundamentals

1- If , the noise can be amplified and this

depends of the choice of GQ:

10-8

10-6

10-4

10-2

100

102

104

-400

-300

-200

-100

0

100

200

300

400

Frequencia (rad/s)

Mag

nitu

de (

dB)

GH

GQ

GN

1

HNQ jGjG,1G

1-

01 NQ GandG

Modulu

s

Frequency, hz

Fundamentals

.0GQ

.000 HQN GthanfasterGG

2- To determine the C (cut frequency)

10-8

10-6

10-4

10-2

100

102

104

-1200

-1000

-800

-600

-400

-200

0

200

400

Mag

nitu

de (

dB)

Frequencia (rad/s)

GH

GQ

GN

C

Modulu

s

Frequency, hz

Fundamentals

)II()ik(q̂a)ik(q̂b)k(q̂nn

1ii

nm

0ii

In this case, any on-line estimator involves a phase

shift or lag. To remove this lag Blum and Marquardt

proposes a filtering procedure that can be resumed

in the use of two discrete-time difference

equations:

)I()ik(q̂a)ik(Tb)k(q̂nn

1ii

nm

0i

*Mi

The technique proposes to obtain GH by using

Green’s Function

The solution of a 3D-transient heat conduction

equation can be given in terms of Green’s function

as follows

)(),(,x qtxGtT h

The equation above, in the Laplace domain, can be written as the convolution product

d)(q)/t,z,y,x(Gt,z,y,Tt

H

0

x

Obtaining of Gh by using Green’s function

)s(q)s,x(Gs,T 1hx

This dynamic system can be represented as shown in Fig. Convolution equation can also be evaluated in the Laplace domain as a single product

q(s)

T(xi,s)

)s,x(Gh

The heat transfer function can, then, be obtained through the auxiliary problem which is a homogenous version of the original problem for the same region with a zero initial temperature and unit impulsive source located at the region of the original heating

Obtaining of Gh by using Green’s Function

X(t) = 1

Yi(t) = T+ (x,t)

),( txGh 1)t,x(Gt,T h x

s,Ts)s,x(Gh x

If the dynamic system is linear and physically invariable the response function

is the same, independently of the pairs input/output and can be obtained by

s

1)s,x(Gs,T h x

Results: GH comparison – 1D case

Frequencia(rad/sec)

Fas

e (

°)M

agni

tud

e (

dB

)

-600

-400

-200

0

200

Método Clássico

Este trabalho

10-8

10-6

10-4

10-2

100

102

104

-1080

-900

-720

-540

-360

-180

0

Phase

Modulu

s

Frequency, hz

Blum & Marquardt

this work

RESULTS AND DISCUSSION Heat flux estimation using temperature data without noise error.

a) sinusoidal;

0 20 40 60

0

2000

4000

6000Heat flux imposed

Ref. [8]

This work

hea

t flu

x [

W/m

2]

time [s]

RESULTS AND DISCUSSION Heat flux estimation using temperature data without noise error.

b) triangular;

0 20 40 60

0

2000

4000

6000Heat flux imposed

Ref. [8]

This work

hea

t flu

x [

W/m

2]

time [s]

Results: 3D simulated case

K=401W/mK = 117 10-06m2/s

Heat flux estimation using simulated temperatures with

i = 1.50C: a) flux sinusoidal (test 3D)

Results: 3D simulated case

0 10 20 30 40 50

0

20000

40000

60000

Heat flux imposed

Sensor 1

Sensor 2

Sensor 3

Sensor 4

hea

t flu

x [W

/m2]

time [s]

Heat flux estimation using simulated temperatures with

i = 1.50C: b) flux triangular (test 3D)

Results: 3D simulated case

0 20 40

0

20000

40000

60000

80000Heat flux imposed

Sensor 1

Sensor 2

Sensor 3

Sensor 4

hea

t flu

x [

W/m

2]

time [s]

Experimental Results: 3D

z

x

y

12.7 mm

12.7 mm

4.7 mm

10mm

10mm

T2(3.5,8.9,4.7)

T1(4.3,3.5,4.7)

q(t)=?

Two thermocouples were brazed on the bottom surface of the sample

opposite to the surface that receives the heat, at the points T1(t) and T2(t)

Experimental Results: 3D Comparison between the measured and

estimated heat flux using Y1(t)

0 10 20 30

0

10000

20000

30000

Observer

Beck

Heat flux imposed

time [s]

Hea

t fl

ux

[W

/m2]

Experimental Results: 3D

0 10 20 30

0

2000

4000

6000

8000

Observer error

Beck error

time [s]

Hea

t fl

ux r

esid

ual

s [W

/m2]

Residuals of heat flux

estimated and measured

Conclusions

The observers method was shown efficient for the

inverse heat conduction problem.

The proposal of obtaining numerically the heat

function transfer based on Green´s function gives

a great flexibility to the technique allowing to deal

with 3D transient problems both simulated and

experimental.

Acknowledgements

UFU