Manage Inverse Heat Conduction: Ill-Posed Problems - Chapter 8

280

CHAP.7

MULTIPLE HEAT FLUX ESTIMATION

form. Give the gain coef1icients Ij and 2j (forj= I, 2. 3) in terms of the influence functions cPj; wherej = I. 2 refers to the sensors a n d ; = I. 2 . . . . + r - I refers to time. 7.2. Write a computer program for the algorithms of Problem 7.1. 7.3.

a.

b.

7.4.

Using exact data from Table 1.1 at x/ and I with dimensionless time steps of 0.01, lind estimates of the heat fluxes at x/L=O and using the computer program for Problem 7.1. Repeat part (a) for the sensors at x/L=0.25 and 0.75. Also solve parts for ill + = I.

a.

Fo Problem 7.1 derive digital digital lilter algorithms to obtain estimates of q l ( M ) and (/2(M). Let the tilter coefticients be .Ij;. j= I. 2; ;=

b.

Obtain numeral values for the filter coefficients .Ij;. for the flat plate of Table 1.1. Use il =0.05 and sensors at x= and L.

CHAPTER

HEAT TRANSFER COEFFICIENT ESTIMATION

-1.0.1 .....

7.5.

Use the zeroth-order sequential regularization method with r= to derive algorith ms for estimating two heat flux histories. q l ( M ) and q2(M). Let (Xu be an input.

7.6.

Write a computer program for the algorithms of Problem 7.5.

7.7.

a.

b.

Using the exact data from Table 4.1 atx/L=Oand withill+=O.OI. find estimates of the heat fluxes at x/L=O and l;use the computer program from Problem 7.5. Repeat part (a) (a) for the sen sors at x/L=0.25 and 0.75.

8.1

INTRODUCTION

The estimation of the heat transfer coefficient, h, from transient temperature measurements has aspects of both the inverse heat conduction problem and parameter estimation.

An example of the treatment as an IHCP is that of a one-dimensional case with a known ambient temperature, Too(t), such as the transient determination of boiling heat transfer coefficients using an initially ho spherical copper solid suddenly immersed in water at its saturation temperature. From transient temperatures measured insid or at the surface of the copper body, the methods of the IHCP can be used to estimate the surface heat flux, qM, and the surface temperature, tOM; the definition of the heat transfer coefficient, h, can be used to obtain the estimate of given by qM TooM

-O.5(tOM

(8.1.1)

In this expression tOM is the estimated surface temperature at time qM, TooM, are usually most accurately evaluated at An example of a case that can be treated as parameter estimation problem is a flat plate over which a fluid is flowing at constant temperature, Too; see Figure 8.1. If the plate is suddenly heated by some electric heaters inside the plate, the plate temperature begins to rise and the heat transfer coefficient is strong function function of position from the leading edge ofthe plate. In some cases the hex), time variatio n is small and the basic form of is a function of x; that is, is known, such as and

h=pX-

(8.1.2) 281

CHAP. 8

282 Fluid

HEAT TRANSFER COEFFICIENT ESTIMATION

conditions. Section 8 2 covers som e sensitivity coefficient Section 8 3 discusses methods for lumped bodies an Section 8.4 briefly covers methods for bodies

Temperature sensor

- f 177 = ' e ~  77tE>drio+ /777 77777/ FIGURE 8.1

...

Electrically heated ftat plate.

The determination of

utilizing utilizi ng various time- and space-dependent measure measure ments of in the solid an Tao in the fluid is a parameter estimation problem Fo cases similar to the two just given, basic solution techniques are known. These examples illustrate the large diversity of problems associated with determini ng the heat transfer coefficient. coefficient. Fo this reason a discussion of various types of problems is given next. The heat t ransfer coefficient can be: 1.

A constant (indepen (independent dent of

3.

A function of only; that is, h(l) A function of only; that is, hex).

4.

A function of

and t).

t; that is, hex, t).

In this list, is a coordinate parallel to the heated surface; it can also be general ized to two surface coordinates such as and z are hex, z) where both parallel to the heated surface. In these problems the ambient temperature, and Tao is assumed to be known but it can also be a function of A related problem is the determination of the ambient temperature, Tao, with known The same four categories of (1) Too = C (2) Tao Too(t), (3) Too Tao(x), and (4) Tao Too(x, t) can be listed. These four estimation problems are linear if the heat conduction equation and boundary conditions are linear. An example of a problem wherein the ambient fluid temperature is needed is in the extrusion of plastics The temperature of the molten plastic is critical

with interior temperat ure gradients.

8.2

SENSITIV ITY COEFFICIENTS

In this section t he sensitivity coefficients coefficients for the h eat tr ansfer coefficient, coefficient, h, are

investigated for two cases The first case is for a lumped body-that is, one in which the temperature is a function of time only. Sensitivity coefficients are given for constant over the complete time domain and for finite time intervals. Also the sensitivity coefficient of the ambient temperature, Tao, is found, both for Too constant with time an for Too approximated by a number of constant segments. The second case for a semi-infinite body that is suddenly exposed to a fluid Th heat transfer coefficient is constant with time.

8.2.1

Lumped Body Case

The differential equation for a lumped body which is suddenly exposed to

fluid at a temperature

the solid mold, not in the flowing plastics. A more complicated problem is to simultane simultaneously ously estimate the heat transfer coefficient and the ambient temperature There are many concepts and techniques of the IHCP and parameter estima tion that can be utilized in the solution of these problems. On concept is that the sensitivity coefficients can be employed to gain much insight into the estimation problems. Another concept is that the use of a sequential procedure can be advantageous in terms of computation speed an for insight. These concepts are briefly explored in this chapter. Due to the large variety of cases in connection with the determination of the heat transfer coefficient, only a few can be treated. The case of h(l) is the main one covered. The basic concepts

Tao

is dT

(8.2.1)

pcV Tt=hA(Too-T)

where is the volume and is the heated surface area of the lumped body. convenience ience in notation, the ratio of divided by is denoted L, For conven (8.2 2)

L= initiall temperat ure of To and with both For an initia

an

Tao

independent of time,

the solution of Eq. (8.2.1) is

and

\:Y,

283

SENSITIVITY SENSITIV ITY COEFFI CIENTS

and examples serve to illustrate procedures which can be modified for different

at T.

an

SEC 8.2

-T T+=-=l-exp ( - h t )

pcL

Too- To

(8

3)

Note that T+ can be plotted as a function of the single dimensionless time +,

where

ht pcL Fo a constant h, the

(8.2.4)

step sensitivity coefficient, coefficient, Zh, is given by Zh

iJT iJh

=(Too-To)-exp(-t peL

(8.2.5)

284

CHAP. 8

A dimensionless

HEAT TRANSFER COEFFICIENT ESTIMATION

step sensitivity coefficient is Z.

aT

+

Too-To ah

exp(-t

(8.2.6)

which is also a function of t+. The distinction between the pulse and step sensi tivity coefficients is discussed in Section 6.2. The Tao step sensitivity coefficien is aT

ZTao=aT = l - e x p ( - t

(8.2.7)

00

which is dimensionless and thus rearrangement of terms is not needed. Notice that T+ =Z

oo

(8.2.8)

and see the upper curve in Figure 8.2 As the temperature rise becomes large for increasing t+, the Tao sensitivity coefficient, ZT oo increases in exactly the same manner. As a parameter's sensitivity coefficient becomes large, more information can be gained about the parameter. Consequentl Tao is relatively easy to measure for a constant Tao over the complete time range. Since the Tao sensitivity coefficient becomes largest as 1 - + 0 0 , the most information regarding Too is found by using measured temperatures at large times; that is, t+ 3. Also shown in Figure 8.2 is the dimensionless heat transfer sensitivity co efficient, increases and is nearly equal Fo the small values of t+ " pe

j=

(8 3.9b)

..

As a consequence Eq. (3 2.29) gives AI

tAl+i-d411= .. ·=o=To+q,1

qj.

j=1.2

..

(8.3.9c)

294

C CHA HAP.8 P.8

HEAT TRANSFER TRANSFER COEFFICIENT ESTIMAT ION

With Eqs. (8.3.9a, b, c) used in Eq. (4.4.24), the equation for qM for the lumped bodies is

LUMPED BODY ANALYSES

SEC.8.3

for t> tM-I. The sensitivity coefficient, coefficient,

.II-I

(YM+)-I-4>1 ... qM

)=

.I

ZM+I-I

l- To)}

(8310) ..

.=

The equation for

is given by Eq. (8.3.2) with 0.5(1'.11_1 0.5(1'.11-1+1'.11)=4>1

1'.11)

.11+.-1-

given by

.II-I

I ~ I  ql+0.5qM +To

(8.3.11)

The algorithm for each time step is Eq. (8.3.10) followed by Eq. (8.3.2) which employs Eq. (8.3.11). Fo example 8.3, calculations for r=2 are displayed as circles in Figure 8.8. The results are less sensitive to small irregularities in the measurements than the exact matching procedure and yet follow the exact matching case when the latter moves regularly, as near the time step index of 20 in Figure 8.8. Even less sensitivity to random temperature errors can be achieved using r= 3 and 4. Some numerical values of qM, ~ M '  and 1'.11 are given in Table 8.1. In order to permit verification by the reader, values are given for M to 10.

8.3.4

Function Specifica tion Procedure

with

h=Constant

To estimate th e heat trans fer coefficient coefficient using the function specification method, a more direct procedure is to estimate without the intermediate calculations calculations for q(t). Unfortunately the nonlinearit of he problem enters into this approach. In this section a direct sequential estimation procedure for is investigated for the temporary assumption of constant The sum of squares, S,

s=

1=1

(8.3.12)

is minimized with respect to hM where is the number offuture times over which hM is temporarily held constant. Taking the partial derivative of S with respect to hM' replacing hM by ~ M '  and setting the equation equal to zero gives .L..

.=1

(Y.

.11+1-1-

t.M + I - d 01'.11+1-1 oh

(8.3.13)

.II

Expressions for the calculated temperatures 1'.11+1-1 can be found analogous to Eq. (8.2.9). To simplify the presentation, however, the ambient temperature, T",(t), is approximated by T""M for the r future times; for this case, the temper ature is given by

t..11+1

"".II exp

T"'M+( t..II-I

T"'M) exp

hM(tM+I_I - t M - d ]

pcL

(8.3.14)

(8.3.15)

ohM

[_hM(tM+I-I-tM-t>]tM+I-I-tM-I pc

(8316)

pc

which is valid for t> -I The 1'.11 -I value is the converged temperature for the solution for I. Because this coefficient is a function of hM' the estimation problem is nonlinear. Due to this nonlinearity, an iterative solution of Eq. (8.3.13) for hM is needed. One such procedure is the Gauss method. In the G auss linearization method, it is first first assumed that an estimate of hM st iteration and then an improved value hi;' is sought. is known for the (v The sensitivity coefficient given explicitly in Eq. (8.3.13) is evaluated with hM equal to h ~ - I )  and is denoted Z ~ ~ l ~  I. Moreover, the calculated temperature in Eq. (8.3.13) is approximated by the two-term Taylor series, 'Y) .11+1-1

t'Y-1) (h'Y)-h'Y-I) (h'Y)-h 'Y-I) .11+1-1 +Z,Y-I) .II .II .11+1-1

and Z is given by Eq. (8.3.16) with hM replaced by tions in Eq. (8.3.13) and solving for hi;' gives '"

L..

h ~ - I ) . 

(Y.

hi;' = h ~ - I ) = h ~1=- I ) 

.11+1-1 .11+1-1- t'Y-I) L..

1=1

(8317)

Using these approxima

)Z,Y-I) .11+1-1

(8.3.18)

.11+.-1 )2

( z ' Y - ~ ) 

This equation is used in an iterative manner until the changes in hi;' are less than some small amount, such as h'Y)

.II

(YM+I-I-TM+I_d2

oT

is found from Eq. (8.3.14) to be

- - ( t o.II-I

)=

296

h'Y-I)1 .II hi;'

10-

(8.3.19)

After a converged value of hi;' is found, M is increased by one, 1'.11 is calculated, and the procedure is repeated for the new ~ M .  The same copper billet problem previously considered in this section is also investigated using this method. Table 8.2 gives some details of the calculation for the first two time steps for the case of hM C and r 2. The sensitivity co efficients are given as functions of M and the iteration index, v. The column labeled Numerator is the numerator of the fraction in Eq. (8.3.18) and Denomi nator is the denominator of Eq. (8.3.18). Notice that the numerator for fixed M rapidly decreases in magnitude with v, whereas whereas the denominator (sum of squares of Z's) approaches a constant. The initial guess for hi was 2.7 Btu/hr-ftl-F and the procedure rapidly converged to 2.3964 Btu/hr-ft -F. This converged value few values of of is used as the initial value for are plotted in Figure 8.8 as squares. The first few values are significantly different from the C,

296

ji

CHAP. 8

HEAT TRANSFER COEFFICIENT ESTIMATION

TABLE 8.2 Details of C Copper opper Billet Calculation fo S Specification pecification Method w i t h h",=C an ,=

Z"+I

II

Ii:

II'

1 2 2 2 2

:111

II:

II:, ;II'

-5.806

-10.652

-5.863

-10.862

0.3926

-5.862

-10.860

-1.8E-4

-5.862

-10.860

-1.4E-5

-5.430

-10.060

-3.99

-5.436

-10.080

-5.436

-10.080

3.05E-3 -1.56E-4

147.167 152.360 152.315 152.315

-45.05

171.832 133.452 131.153

Ii",

'Ii

BODIES WITH INTERNAL TEMPERATURE

Btu/hr-ft -F 2.7 2.3939 2.3964 2.3964 2.3964 2.3964 2.3659 2.3659 2.3659

In Section 8.3 it was found that the function specification method for qM = C gave nearly the same results as for the temporary assumption of hM C. One case of a lumped body does not prove that the temporary assumptions of constant and are equally valid for other cases. More research is needed to determine the relative merits. merits. Until more is known, however, the method is recommended because it is computationally simpler and does not involve iteration Furthermor e, existing existing programs for the IHCP can be employed. For these reasons, only the function specification method with the constant heat flux flux temporary assump tion is used in this section.

8.4.1 Analysis fo Future Temperatures Using q=C Function Specification Method one-dimensional heat conduction model is (8.4.1)

~ ~ ( r " a T ) = ~  aT

r" ar

ar

(X

at

where n=O is for a rectangular coordinate, n== is for a radial cylindrical coordinate, and = 2 is for a radial spherical coordinate. The boundary conditions are (8.4.2)

aT r ~ r o  =h(t)[TCX)(t)=h(t)[TCX)(t)- T(ro, t)] ar

(8.4.3)

aT

ar r ~ r l -

TABLE 8.3 Results for the Copper Billet Example Using th Function

and the initial temperature is T(r, t)

Specification Method w i t h h= an ,=

t,,(s) 1

..........

V)

-....J

7 9 10

96 192 288 384 480 576 672 768 864 960

Ii", Btu/hr-ft 2.39643 2.36590 2.32878 2.35714 2.32776 2.25087 2.18316 2.14357 2.13633 2.12164

-F

297

BODIES WITH INTERNAL TEMPERATURE GRADIENTS

GRADIENTS

results which are shown as circles. This can be verified by comparing the estimated hM values for M = to lOin Table 8.3 with those for C, = 2 1; for given in Table 8.1. The maximum difference is 3%, which is for =4, it is 0.3% and for M = 10 it is abou t 0.05% For this example, the hM = C and qM = C results are then very close; in general the results are closer than the variability due to small fluctuations in the measured temperature. The hM = C calculation is more expensive because iteration is needed. needed. The num ber of iterations for this case is two or three which is not large bu t even so there is more computation than without the iterations. In addition, the computer program for the hM analysis is more complicated than that for the = C analysis. Fo these reasons and at least for this billet example, the = C sequential analysis is preferred preferred over the = C method.

II

8.4

8.4

Function

Denominator

Numerator

SEC.

(8.4.4)

= constant

To

simplicity, the thermal conductivity and the density-speci density-specific fic heat prod uct pc are assumed to be independent of temperature so that nonlinearity does not enter the problem via the thermal properties. The heat transfer coefficient is assumed to vary only with time. time. Other "inactive" boundary cond itions at can be as readily treated as the insulation condition indicated in Eq. (8.4.3). Much more general conditions can be treated but they unduly complicate the

Fo

Iterations

presentation. For a single interior sensor, a function specification algorithm for the temporary assumption is Eq. (4.4.24), repeated here (YM+j-l-tM+j-d4 q

M

=

j=

!

.

-

.

.

.

:

~

-

-

-

-

cPJ

r

-

.. · ~ o ) c P j  -

-

-

-

-

-

=C

(8.4.5)

298

CHAP.8

HEAT TRANSFER COEFFICIENT ESTIMATION

where r future temperatures are used. This equation can be used to obtain a single qM after which hM is estimated. Also all of the qM components can be estimated before any of the hM values is estimated. A heat transfer coefficient is found from qM

t

nM

(8.4.6)

o,M-I)

TooM -O.5('t

The zero subscript is used to denote the surface at which the fluid is located. Unless the sensor is located at the heated surface, the calculated temperature in Eq. (8.4.5) is not the same as that with zero subscripts in Eq. (8.4 6) This is major difference between this case and that of a lumped body. Equations (8.4.5) and (8.4.6) are valid for both the Duhamel integral solut on and a difference method solution (e.g., FD, FE, FeV) of Eqs (8.4.1)-(8.4.3). In both cases tPj in Eq. (8.4.5) represents the temperature rise at time tj at the location of the sensor for a unit step increase in the surface heat flux [r=ro, for Eq. (8.4 2)]. accurate solutions of the heat conduction problem are ob tained in both approaches, the results for hM are nearly identical. For simplicity of presentation. presentation. Duhamel's theorem approach is selected. The calculated temperatures in Eq. (8.4.5) are given by Eq. (3.2.29) and can be written as

tMI,.,=o=qIAtPM-1 +q2 tPM-2+ .. +qM-1AtPl M-I qjAtPM-j+ To ;=-1

tM+d,.,=,., •• -o=qIAtPM+ .. +QM

AtP2+ o=

M-l i=1

SEC.8.4

BODIES WITH INTERNAL TEMPERATURE GRADIENTS

8.4.2

Examples

The case of a semi-infinite body suddenly exposed to a fluid at Too is considered. The exact solution for the case of constant is given by Eq. (8.2.14) and is plotted in Figure 8.6. Though is actually constant, it is estimated as though it is function o f time. Some results are displayed in Figures 8 9, 8.10, and 8.11 (Reference 5).

1.21h,.,LOr

-+/:V-.-

0-

0.6

CJ

05

oB.

FIGURE 8.9 Semi-infinite body example with cxaCI matching of temperature measure ments (41; =0 3)

5

° 0 ~ ~ - - ~ 2 - - ~ 3 ~ ~ 4 

To

(8.4 7a)

31

l:l.

/\ l\

qiAtPM i+,+T

qjAtPO.M-j+ To

YM-fMI,.,=o tPl

No comparable simplifications occur in Eqs. (8.4.6) and (8.4.8).

I/

•

hM

l o

(8.4.8)

\ \

~

'1 /.



AI

0,

. ~ V ~ ' ~ - - Q 

-H 10

B.

where the subscript 0 for AtPO.M denotes the heated surface For the case of one future temperature, r= 1, the interior calculated temper ature is made to match the measured temperature. Fo this case, Eq. (8.4.5) simplifies to qM

-0

0.2

tP} are for the location of the sensor. All the tPj values (recall AtP tP The surface temperature denoted, is found in a similar manner to the calculation of an internal temperature; the equation is

;=

-0

+B, 0.1 oB.=

0.4

(8.4.7b)

to,M=

299

\

: \

=2

-2

-3

(8.4.9) FIGURE 8.10 (41; =0.5)

2

4

6

8

10

Semi-infinite body example with exact malching of temperature measurements

300

CHAP. 8

HEAT TRANSFER COEFFICIENT ESTIMAT ION

PROBLEMS

301

time period. The needed sensitivity coefficients are denoted and are for individual hAl components. (The Z coefficients coefficients are always as large or larger than the coefficients, however.) In order to take smaller time steps, future temperatures are needed in the function specification method. Even so, for large values of B", such as B" 10, the sensitivity coefficients are very small (see Figure 8.7) and thus accurate estimation is still difficult. This is verified by the results shown in Figure 8.11 which are for ~ t ;  =0.25, a value for which the Stolz method is unstable. Future time steps, such as r or 3, remove the instability but inaccurate results are is (Figure 8.7). A conclusion is that the sensor should be located as near the surface as possible, at least so that hxjk 1, where is the distance from the heated surfac surface. e.

To 37

l1

h.]

8.5

ESTIMATION OF CONTACT CONDUCTANCE

The estimation of the contact conductance from transient measurements is quite similar to the method for determining the heat transfer coefficient. Sensi tivity coefficients and optimal considerations are given in References 3 and 4.

Ii Ii

l>.B.

10. ,=

AB.=

10. ,20. ,= 20.

OB. -B

)! :1

REFERENCES 1.

10 FIGURE

8.11

(I1t: =0.25).

Semi-infinite body example using function specification algorithm with q=

Figures 8.9 and 8.10 are for exact matching of the simulated data. Figure 8 is for dimensionless time steps of ~ t :  =0. 3 which are relatively small (i.e., near the stability limit) for exact matching, which is equivalent to the Stolz method For Bx hxjk equal to 0.05 0.05 and 0.1, the estimated values of hAl are very good for all times, whereas for B" 1 there are some oscillations oscillations on the order of 20%. Evidently larger B" values cause increased difficulty for a fixed time step with exact data and exact matching of measurements. More results for exact match ing are shown in Figure 8.10 with the larger time step of ~ t :  =0.5. The value of B" 1 in this figure is more accurate than that shown in Figure 8.9. Conse quently increasing the time step can aid in reducing oscillations, but informat ion regarding changes in can be lost as becomes larger. Figure 8.10 also reaffirms the finding from Figure 8.9 that increasing B" makes the estimation process more difficult; difficult; this is consistent with the sensitivit coefficient coeff icientss becoming smaller as B" is increased, as shown in Figure 8.7. (This figure is for the sensitivity coefficient Z which is for constant over the entire

S.

Beck, J and Arnold, K J., Parameter Estimation in Engineering and Science, Wiley, New York,1977. Heat Transfer, 4th cd McGraw-Hili New York 1976 Holman, Beck, J. , Transient Sensitivity Coefficien Coefficients ts for the Thermal Cont act Conductance, Int J. Heat Mass Transfer 10,1615 1616 (1967). Beck, J ., Determination of Optimum, Transient Experiments for Thermal Contact Con ductance, Int J. Heat Mass Transfer 12, 621 633 (1969) Osman, A. Personal communication, Aug Aug., ., 1983

PROBLEMS

8.1.

Starting with Eq. (8.2.14), derive Eq. (8.2.16). Derive also an expression in terms of t; and B" for the maximum sensitivity coefficient, Z: for small values of B".

82

A thick concrete wall initially at 30 is suddenly exposed to a fluid at 80 K. Calculate and plot the four temperature histories at 1.2 cm from the heated surface if the heat transfer coefficient has the values of 9, 10, 90, and 100 Wjm -K. Relate the differences between the plots to the js. sensitivity coefficients. Use k= 1.2 Wjm-K and a=7.5 x 10-