Determining entrance conditions from downstream measurements

INT. COMM. HEAT MASS TRANSFER Vol. 20, pp. 173-183, 1993 Printed in the USA

0735-1933/93 $6.00 + .00 Copyright°1993 Pergamon Press Ltd.

DETERMINING ENTRANCE CONDITIONS FROM DOWNSTREAM MEASUREMENTS

Rarnanuj a m Raghunath CIMAS/RSMAS, University of Miami, Miami, F L 33149

(Communicated by J.P. Hartnett and W.J. Minkowycz)

ABSTRACT The temperature profile in the entrance region of a thermally developing hydrodynamically developed laminar flow between parallel plates is derived f r o m temperature measurements downstream of the entrance using inverse analysis techniques similar to those used in inverse heat conduction studies. Solutions are obtained for various distributions of measurements. For all cases studied we are able to derive the entrance temperature. In some of the cases we attempted to determine the fiow rate also from temperature measurements. This was successful when sul~cient measurements were available. The sensitivity of the solution to the distribution and the quantity of measurements is also studied. As expected, we find that measurements closer to the entrance region are more important in determining the temperature profile at the entrance. W e also find that the location of measurements is more critical than the number of measurements.

INTRODUCTION Inverse techniques have been used in a variety of studies to determine unknown pararaeters or other quantities of interest from measurements of different quantities which are related to the unknowns of interest through a mathematical model. This is motivated by constraints in making measurements. Sometimes there is no access to the region where the measurements are needed and other measurements have to be used to infer the quantities that are of interest. Other times, it might be necessary not to interfere with the flow as it is very sensitive to minor fluctuations, as in the entrartce region. Also it is possible that no measurement techniques are available to measure the quantity of interest. In all these situations inverse techniques provide a good alternative. 0zi§ik and others [1,2,3] have used 173

174

R. Raghunath

Vol. 20, No. 2

inverse techniques in heat conduction to determine diffusivities, boundary conditions, etc. and in a recent article Huang and Ozi§ik [4] determine the unknown heat flux at the boundary from interior measurements of temperature in a thermally developing hydrodynamically developed laminar flow between parallel plates. Estimation of model parameters based on measurements is also referred to as "calibration of the model" by many modellers. The trial-and-error approach is the most straightforward and often used method of calibration. Various "intelligent guesses" are made for the value of the unknown parameter and that value of the parameter which when used in the model run provides the best fit of model simulation to the measurements is used subsequently as the best estimate for the value of the parameter. This approach works well if there is only one unknown parameter and if we have an a-priori estimate of the parameter. But as the number of unknowns increase this approach fails miserably. Also, there is no guarantee that we will ever estimate the unknown parameter accurately. A more sophisticated approach is similar to the "Newton-l~.aphson method" of obtaining the root of an equation. Again, we guess a value for the parameter and run the model. As in the trial-and-error technique, the measurements made and the corresponding model simulations would likely be different because the initial guess of the unknown parameter is not identical to its correct value. However, rather than choosing an arbitrary value for the next guess, information about the sensitivity of the measurement to the unknown parameter is determined from the model and the next guess is made in the steepest descent direction. This method repeated for several iterations ensures an optimal value for the parameter provided the measurements are sensitive to model parameter. If the measured quantities are not sensitive to the unknown parameter other measurements would be required to estimate the parameter. A further improvement is to use Conjugate Gradient Methods as in Huang and ()zi§ik [4]. Conjugate gradient methods use not only the gradient for that particular iteration but also the gradients for all the previous iterations to determine the next guess. In effect it uses curvature information in addition to gradient information. In this present study we use a limited memory quasi-Newton conjugate gradient method (which is a special case of conjugate gradient method) available in the NAG library [5] to obtain the temperature profile at the entrance of a thermally developing hydrodynamically developed laminar flow between parallel plates. The flow is steady and incompressible. Model genexated temperature measurements with no errors are available downstream of the entrance. The focus of this study is how close to the entrance these meaJurements need to be and how many are needed to obtain a satisfactory result. In some of the experiments we also estimate the mean velocity from temperature measurements. Prof. Ozi§ik has worked in the area of inverse problems for many years and has coauthored several articles. As a tribute to his seminal work in this area and as a courtesy to

Vol. 20, No. 2 ENTRANCE CONDITIONS FROM DOWNSTREAM MEASUREMENTS 175

the readers the terminology and notation used is similar to his. Differences in approach are made clear as they occur. DIRECT PROBLEM The governing equation for the hydrodynamically developed, thermally developing laminar flow of a constant property fluid flowing between two parallel plates is: ka2T(=,y) , , ,~ O T ( x , y ) OY2 - uLy)pu, p ~

in

0 < y < L,

(la)

0< x < b

T = T¢

at

y= 0

(lb)

T=T h

at

y= L

(lc)

x=0

(lg)

T=ToCy)

.t

where, Tc and T h are known and To(y) is the temperature profile at the entrance (to be determined from the measurements). The fully developed velocity u(y) is given by: Y uCy) = 6 urn L(1 - ~)

C2)

where, u m is the mean velocity (assumed to be known in some cases and to be determined from the temperature measurements in certain other cases). Fig. 1 describes the notation.

/

/

/ Tt~,y)

T~ t FIG. 1 Geometry and coordinates.

Finite difference methods are used to solve the above equations numerically. An explicit upwind differencing in the x-direction and central differencing in the y-direction is used and

176

R. Raghunath

Vol. 20, No. 2

the discretized equation for obtaining the temperature at a location (i,j) is: k

Ax

1.

~.

T(i,j)=T(i-l,j)+pOp~y2u~.)[T(~-l,j+l)-2T(i,j)+T(i-l,j-1)]

(3)

Since the boundary temperatures are known the above equation is sufficient to determine the temperature profile downstream of the entrance region if the temperature profile at the entrance

(To(y))and the mean velocity (urn) are known. Our objective here, however, is to

determine the temperature profile at the entrance and the mean velocity. This is achieved by solving the inverse problem which is also referred to as the adjoint problem. INVERSE PROBLEM We define a cost function (J) as a measure of the misfit between the model simulation and the corresponding measurements:

J:Z

1 wtCm) (T(ra) - Y(m)) 2

(4)

rn=l,M where, the summation is carried over all the M available measurements. In general, the measurements need not be located at the grid points of the model. But here, for simplicity, we will assume that the measurements are located at the grid points. The vector wt(m) contains the inverse of the variance of each measurement and is a measure of the coincidence in each measurement. It is essentially the diagonal of the inverse of the covariaace matrix if the measurements are uncorrelated. If the measurements are correlated, however, the inverse of the covariance matrix has non-zero off-diagonal terms also and equation (4) becomes more messy; but the method remains the same. The optimum value for the unknown parameters is obtained by minimizing the cost-function J. There are several minimization techniques available in mathematical libraries such as NAG [5] and IMSL [6]. There are routines available that adjust the unknown parameters by making repeated evaluations of the cost-function and determining the sensitivity of the cost-function to the unknown parameters by finite-difference methods. Where possible it is preferable and economical to supply gradient information explicitly. In this study we supply the gradient information to the minimization routine. To obtain the gradient of the costfunction nut to the unknown entrance temperature it is possible to repeatedly substitute for W(m), the upstream temperatures using equation (3). The process is cumbersome and error-prone and Lagrange Multipliers are often used to aid in the book-keeping. Details of this technique may be found in several text books [7,8]. Thus, in the inverse problem knowing the temperature field corresponding to certain estimates of our unknown parameters, we obtain the model-data misfit (and hence the costfunction) and use this to obtain the gradient of the cost-function un-t the unknown parameters. Huang and 6zi§ik [4] refer to this as the sensitivity problem. In this study subroutine

Vol. 20, No. 2 ENTRANCE CONDITIONS FROM DOWNSTREAM MEASUREMENTS 177

E 0 4 D G F in N A G library [5] is used for minimization. It is a limited-memory quasi-Newton conjugate gradient method. RESULTS We illustrate our solution using a coarse grid with 11 grid points in the stream-wise direction and 11 grid points between the plates. The distance between the plates is of nondimensional length 1.0 and the grid length Ay is thus 0.1. To satisfy the stability criteria we choose Az to be 1./300. With only 11 grid points in the stream-wise direction we are only 1./30. non-dimension~l lengths from the entrance and thus well within the thermally developing r e , o n . Rather than ~ n g

the thermal diffusivity a to be 1.0 as we have, we can

assume that a = 1./300. and then the eleventh grid point is ten non-dimensional lengths from the entrance. Our conclusions will not be altered by using a fine grid with many more grid points but the the computational needs would increase substantially. Solutions are obtained for four different cases. Case-A: Measurements of the temperature profile at only one stream-wise location are available and the temperature profile at the entrance is to be determined from these measurements, But the location of this measurement is moved from the second grid point to the eleventh gad point in the stream-wise direction. The same first guess of the entrance temperature profile is used in all cases; a linear variation from one boundary temperature to the other (the fully developed profile). In Fig. 2 we show the results for Case-A. All the cases converged to a solution not far from the exact solution. But, when the measurements were closer to the entrance section (2a-e) the recovery of the temperature profile at the entrance was much better than when it is further away (2f-j). We do not use the number of iterations required for convergence as a criteria in this study. A systematic study of the number of iterations and CPU time required for different minimization techniques will be presented in a future article. Case-B:

Measurements of the temperature profile are available at more than one stremm-wise location and again the temperature profile at the entrance is to be determined from these measurements. The recovery obtained using the same first guess as in Case-A is shown in Fig. 3. In Fig. 3a we use 10 temperature profiles, corresponding to measurements of the temperature profileat stream-wise locations I=2 to I=11 to recover the temperature profile at the entrance (I=I). In Fig. 3c we use 9 temperature profiles (I=3-11), in Fig. 3d we use 8 temperature profiles(I=4-11) and so on. In Fig. 2a on the other hand only the temperature profile at I=2 is used as measurements. By comparing Figs. 3a and 2a, it is apparent that

178

R. Raghunath

~

I

I

I

C'4

I

I J

-..

.,.:

~

,.£

.~

I

I

"o

Vol. 20, No. 2

6~I-i

r,--~

~

~_.~,

oO o

~

.,..:

~0

1" ~3

, ~

, ~

, ,~

oO o

"~ co

~.

j

~q

~£

~

~

q0

,,g

~.o

~g

oO o

o

I

...:

~q

~

.~

~

oo o

I

,(

,,(

I

j .z

6

o~._. c"4

~.

o

o

~ ' ' ' t X

~ o

o

~q

,..£

.~

~

oO o

,~.~ o---

~Oo~ o~

.~

~ ~.~.-.

oO o

Vol. 20, No. 2 E N T R A N C E CONDITIONS FROM D O W N S T R E A M M E A S U R E M E N T S 179

'.

i) f'O

00

'

'

~.

.~.

.

'

~

."~.

b,c'-

oO

,.~

~

'

'

'

~

~

~

t "~ oO ) • 1"4_ 1.4 .e--~

ID

m

IP W

. ~

.~os&-

I

I

I

I

~';

~,~"

¢p

I-a

ii ~

,

P'~

~D el')

c~

o

x

x

.~.~

,.~

oq

~

~:

~

oO

,.~

~0

~

~.

~.

o

oC:J o

,,[

x -...~-. ~

@~.~

".2, ~ '~ "~ ,-q

,~ ~ .~ ~-'~" I ,.~

~ ~

~ ~-I-..-~ .~. ~

I ,.~

-I qO

~.

..~

~

~

,,l-

o

,<

~

oO o

,{

o~

o 2t

"~.~ ~ ~

o X

x

180

R, Raghunath

Vol. 20, No. 2

increasing the number of measurements does little to improve the fit although the convergence maybe faster. Figs. 2j and 3j are identical as they use the same measurements. By providing measurements at I=11 in addition to measurements at I=10 we do not improve the fit (compare Figs. 2i and 3i). In both Fig. 2i and Fig. 3i the fit is not exact and there are some differences from the correct solution but providing additional measurements did not improve the fit in Fig. 3i. Case-C: In Case-C, as in Case-A measurements of the temperature profile are available at only one stream-wise location but now we try to determine the mean velocity um in addition to the temperature profile at the entrance. Thus, we have 10 unknowns and only nine measurements, an under-determined problem. The correct value for Urn is 1.0 and the number at the bottom right comer of each figure shows the value of Urn at convergence. The initial guess for Urn was 2.0 and it is clear that the mean velocity field has not been adjusted by the temperature measurements. Instead the temperature profile at the entrance has been adjusted with the measurements and the converged To is surprisingly close to the exact solution, although closer examination would reveal that the converged solution is not as close to the exact solution as in Case-A. Because the problem is under-determined there exists a null space and which parameters are in the null space may be affected by preconditioning the gradients. In this case, we do not precondition the gradient and urn is in the null space. Case-D: Here as in Case-B, measurements are available at more than one location but as in Case-C we determine the temperature profile at the entrance and also the the mean velocity U~rr,. The same initial guess is used as in Case-C. The converged values of um are in the bottom right comer if Figs. 5a-Sj. We find that both the temperature profile at the entrance and the mean velocity um are recovered well. Only one temperature profile (1=11) is available for Fig. 5j and as in Fig. 4j the mean velocity u m could not be recovered also because the problem is under-determined CONCLUSIONS Using simple examples we show that the temperature profile at the entrance of

a

thermally developing hydrodynamically developed laminar flow between parallel plates can be obtained from temperature measurements downstream of the entrance. As anticipated we found that measurements closer to the entrance are more useful than measurements farther away. Also, as anticipated we found that measurements too close to each other do not provide additional information (other than reducing the error estimate of the fit, which we did not consider here). We also found that when suf~cient temperature measurements are available we can recover the mean velocity and the temperature profile at the entrance

,

/

/

,

i

,

//

,

2-0

4O

,

i

/

~

4f

,

Z.O

,

>,

/

I

/ ,

/

/

I

ii

,

2.0

4b I

,

I

,

II

2.0

49

.7 .8 .9 1. temp

,

I

I

.7 .8 .9 temp

,

I

/ I

0.5.6

.2

.4

.6

0.0

,

,

0.5.6

.8

I,

0.0

/

.2 - /

.4

.6

.8

1.

>~

>,

/

/

/

/ 2-0

0.0

/ i

!

/

I

I

i i

~.0

4h

/ I

FIG. 4

temp

0.5 . 6 . 7 . 8 . 9 1 .

.4

.6

I

I I I I 0.0 0.5 . 6 . 7 . 8 . 9 1 . temp

.4

.6

4c i i i i

/ ,

/

~

0.5.6

.2

0.0

f

i

!

/

i

,

2.0

4d

,

i

,

i

,

,

.7 .8 .9 I. temp

~

2.0

4i

0.5 .6 .7 .8 . 9 1 . temp

.6

0.0

.2

.4

I

I

4e

I I

,

2.0

i

,

i

"/ ,

/

I

4j

/ -1-"

I I

,

2.0

I

,

I

0.5 .6 .7 .8 . 9 1 . temp

.2

.4

.6

0.0

,

I

i

0.5 .6 .7 .8 . 9 1 . temp

.8

I.

0.0

i

/

.2 - /

.4

.6

.8

I.

Comparison of the exact solution (solid-line), the initial guess (dashed line) and the converged solution from the inverse problem (-t-) for Case-C. Mean velocity u m is also a parameter to be recovered from the measurements in this case. The converged value of um for each case is printed in each figure. The initial guess for um was 2.0. The same measurements as in Case-A were used here.

0.5 .6 .7 .8 .9 1. temp

.2

O.C

/

i

0.5 .6 .7 .8 .9 temp

.6

0.0

.2

.4

.6

.8

I.

<

Z

C

>

>

m

Z

o

©

r~

Z

0 2:

z¢b m

m

to

O

Z

bD

o

182

R. Raghunath

~" ...

~' r,.~

,

~

0"

V01.20,

"~"

'

,

,

i~

~q

~

~

06 o

~'

F

No. 2

C>

,

,

~

,t

, .t ~ 8 .-.e,,.~

.~

~0

,{

~

o<5 c5

.{

I:l ^'F--, e

~ ~,.=~ ~

70 u3

.I

I

I

I

t

"~

"-r'-~ "~

I,.~

,

~q

,

~,,~- . 1 ~~

~£

O

) up

,

,

oO o

7..,=1{ ,'~ :~

'~.1~

3.=

I~ e

"1:1

o

,{

,(

'-'"El

.a=~ •

e

i::l

~

Lr)

n

.._:

i

~q

L

L -"----f.~" .~ ~

~

J ,n oO o

,~

~q

~

,(

~

~

oO o

,.q

,(

.£'-.- =,.9 Lr}

--..

c3,

r.--~

r-.~

im

l/3

u ~ ~

o

~.~

.,..=me

o

o e ~

. ~ ~0 ~.

I

I

i

I

o

..~

~

~

.~. /(

~

oO o

I

~q

~

~

~

.~ ,{

+~'--~----~,

~

u

"J

oO c:;

~.

o

~

Vol. 20, No. 2 ENTRANCE CONDITIONS FROM DOWNSTREAM MEASUREMENTS 183

from the temperature measurements. The degradation of the fit as the measurements are further away from the entrance section may also be due to the numerical scheme used. Using implicit schemes such as CrankNicholson we have begun to study the sensitivityof such inverse problems to numerics.

NOMENCLATURE Cp J

heat capacity Cost Function

L Distancebetween the plates M Total number of measurements T(z, !/)Model Estimated Temperature Tc Temperature at lower wall Th Temperature at upper wall To(~t) Temperature at the Entrance Y( z, !t) measured temperature

b k ~n u(7/) wt(rn) z !/ p

Entrance Length Thermal Conductivity mean velocity fully developed velocity profde weight given to measurement m stream-wise direction transverse direction density

REFERENCES

.

Mikhailov, M. D. and M. N. Ozi~ik, Unified Analysis and Solutions of Heat and Mass Diffusion, John Wiley & Sons, NY, 1984.

.

A1-Najem, N. M. and M. N. Ozi~ik, A Direct Analytic Approada for Solving Linear Inverse Heat Conduction Problems, J. Heat Trans., 107, 700-703, 1985.

.

AI-Najem, N. M. and M. N. Ozi~ik, Estimating Unknown Surface Condition in Composite Media, Int. Comm. Heat and Mass Trans., 19, 69--77, 1992.

.

Huang, C. H. and M. N. Ozi§ik, Inverse Problem of Determining Unknown Wall Heat Flux in Laminar Flow through a Parallel Plate Duct, Num. Heat 7~ns., 21, 55--70, 1992.

.

NAG Fortran Library Concise Reference, Mark 15, Numerical Algorithrns Group Inc., Downers Grove, IL, 1991.

.

.

.

International Mathematical and Statistical Library Edition 10.0, Users' Manual, Math~Library Version 1.0, IMSL, Houston, TX, 1987. Gill, P.E., W. Murray and M. H. Wright, Practical Optimization, Academic Press, San Diego, CA, 1981. Kaplan, W., Adraneed Calcnlns, Addison-Wesley Publishing Company, Reading, MA, 1952.

Received October 28, 1992