M.S.thesis Zhong_Inverse Algorithm for Determination of Heat Flux

INVERSE ALGORITHM FOR DETERMINATION OF HEAT FLUX

A Thesis Presented to The Faculty of the Fritz J. and Dolores H. Russ College of Engineering and Technology Ohio University

In Partial Fulfillment of the Requirements for the Degree Master of Science

BY Rong Zhong June, 2000

OHlO UblIVERSITY LIBRARY

Acknowledgement

I wish to express my deep gratitude and sincere appreciation to my advisor, Dr.

Khairul Alam, Moss Professor, for his valuable guidance, encouragement and support throughout this research. I would like to thank Professor Hajrudin Pasic for his support and assistance in

this research and help with numerical methods. I wish to thank Dr. Lany Snyder, Professor, Mathematics Department, for his help and suggestions. My parents, brother and sister have also been a source of encouragement to me throughout the time I have studied at Ohio University. Their encouragement has kept me going during the times I felt overwhelmed. Finally I would like to thank June and Warren Crockett, Lori and Mark Tyler, Grace and Eddie Liu and all my h e n d s for their encouragement and patience during this time.

Table of Contents

Chapter 1 Introduction ......................................................................................................1 1.1

The Inverse Problem ..............................................................................................1

1.2

Objective of Current Research .............................................................................. 3

1.3

Thesis Overview .................................................................................................... 3

Chapter 2 Background and Relevance .............................................................................. 5 2.1

Introduction ...........................................................................................................5

2.2

Literature Review ..................................................................................................6

2.3

Polynomial Solution .............................................................................................. 9

2.4

Theory .................................................................................................................. 10

2.5

Solution for Polynomial q(t) .............................................................................. 12

2.6

The Spline Interpolation Model ........................................................................... 20

Chapter 3 Mathematical Model ...................................................................................... 21 3.1

Introduction ......................................................................................................... 21

3.2

Definition of the Problem .................................................................................... 21

3.3

Analytical Solution for the Direct Problem ......................................................... 23

3.4

Validation of Analytical Solution ........................................................................27

3.4.1

Checking the Governing Equation ................................................................ 27

3.4.2

Checking for the Boundary Conditions......................................................... 29

3.4.3

Checking for the Initial Condition ................................................................ 30

3.5

Comparison with Alternate Solutions.................................................................. 31

3.5.1

Validation for q(t) = constant ...................................................................... 31

3.5.2

Validation for q(t) = f (t) ............................................................................. 32

3.6

Inverse Solution ................................................................................................... 32

3.6.1

Cubic Spline Interpolation ............................................................................ 36

3.6.2

Minimization Algorithm (IMSL Software) .................................................. 36

3.6.3

Program Flowchart ........................................................................................ 40

Chapter 4 Experimental Results...................................................................................... 42 4.1

Quenching ............................................................................................................42

4.1.1

Introduction ...................................................................................................42

4.1.2

Cooling Curve ............................................................................................... 43

4.2

Experimental Studies ...........................................................................................44

4.2.1

Experimental Apparatus ................................................................................ 44

4.2.2

Experimental Procedures...............................................................................47

4.3

ExperimentalResults ........................................................................................... 48

Chapter 5 Results ............................................................................................................ 51 5.1

Introduction ......................................................................................................... 51

5.2

Results from Polynomial Solution....................................................................... 51

5.3

IHCP Algorithm with Cubic Spline..................................................................... 57

5.4

Heat Transfer Coefficient .................................................................................... 77

5.5

Effect of a Shortened Vapor Blanket Stage ......................................................... 78

Chapter 6 Discussion and Conclusions...........................................................................82

6.1

Conclusion ........................................................................................................... 82

6.2

Future Work .........................................................................................................84

Reference .......................................................................................................................... 85 Appendix ...........................................................................................................................

89

List of Tables

Table 3.1 Dimensionless Temperature Values, T' (x' ,t' ) , for Various Dimensionless Time and Distances for a Plate Heated at x = L and Insulated at x = 0 .................................................. 33 Table 3.2 Dimensionless Temperature Values, T' (x' ,t ' ) , for Various Dimensionless Time and Distances for a Plate Heated at x = L and Insulated at x = 0 .................................................. 34 Table 3.3 Comparison of Analytical and Finite Difference Solutions............................. 35

List of Figures

Figure 2.1 One-dimensional plate with boundary conditions ..........................................11 Figure 3.1 Flowchart for IHCP ........................................................................................ 41 Figure 4.1 Quench probe .................................................................................................. 45 Figure 4.2 Picture of the quenching system showing the quench probe suspended over a quench tank. .........................................................................................

47

Figure 4.3 Cooling curve for 304 stainless steel for 10-degree quench angle test .......... 49 Figure 4.4 Cooling curve for 304 stainless steel for 60-degree quench angle test .......... 50 Figure 5.1 Comparison of experimental results with analytical profile obtained by using a 6thdegree polynomial approximation of the heat flux function (quench angle = 10"). ...................................................................................... 52 Figure 5.2 Heat flux polynomial obtained by least-square matching of the analytical and experimental temperature profiles (quench angle = 10"). ....................... 53 Figure 5.3 Comparison of experimental results with analytical profile obtained by using a 6thdegree polynomial approximation of the heat flux function (quench angle = 60 " ). .......................................... .. .. .. . . .. .. .. . .. ......... . 54 Figure 5.4 Heat flux polynomial obtained by least-square matching of the analytical and experimental temperature profiles (quench angle = 60" )........................ 55 Figure 5.5 Comparison of analytical and experimental temperature profiles using a 8-interval cubic spline approximation for the heat flux function

(quench angle = 10" , 2 time intervals in vapor blanket stage). ...................... 58 Figure 5.6 Heat flux profile obtained by using a 8-interval cubic spline (quench angle = 10" , 2 time intervals in vapor blanket stage). ...................... 59 Figure 5.7 Heat transfer coefficient profile obtained by using a 8-interval cubic spline (quench angle = 10" , 2 time intervals in vapor blanket stage). ...................... 60 Figure 5.8 Comparison of analytical and experimental temperature profiles using a 15-interval cubic spline approximation for the heat flux function (quench angle = 10" , 3 time intervals in vapor blanket stage). ......................61 Figure 5.9 Heat flux profile obtained by using a 15-interval cubic spline (quench angle = 10" , 3 time intervals in vapor blanket stage). ......................62 Figure 5.10 Heat transfer coefficient profile obtained by using a 15-interval cubic spline (quench angle = l o 0 , 3 time intervals in vapor blanket stage). .......... 63 Figure 5.1 1 Comparison of analytical and experimental temperature profiles using a 16-interval cubic spline approximation for the heat flux function (quench angle = 10" , 4 time intervals in vapor blanket stage). ..................... 64 Figure 5.12 Heat flux profile obtained by using a 16-interval cubic spline (quench angle = 10" , 4 time intervals in vapor blanket stage). .....................65 Figure 5.13 Heat transfer coefficient profile obtained by using a 16-interval cubic spline (quench angle = 10" , 4 time intervals in vapor blanket stage). ..........66 Figure 5.14 Comparison of analytical and experimental temperature profiles using a 9-interval cubic spline approximation for the heat flux function

(quench angle = 60" , 2 time intervals in vapor blanket stage) ...................... 68 Figure 5.15 Heat flux profile obtained by using a 9-interval cubic spline (quench angle = 60" , 2 time intervals in vapor blanket stage)...................... 69 Figure 5.16 Heat transfer coefficient profile obtained by using a 9-interval cubic spline (quench angle = 60°, 2 time intervals in vapor blanket stage). .......... 70 Figure 5.17 Comparison of analytical and experimental temperature profiles using a 13-interval cubic spline approximation for the heat flux function (quench angle = 60" , 3 time intervals in vapor blanket stage) ...................... 71 Figure 5.18 Heat flux profile obtained by using a 13-interval cubic spline (quench angle = 60" , 3 time intervals in vapor blanket stage). ..................... 72 Figure 5.19 Heat transfer coefficient profile obtained by using a 13-interval cubic spline (quench angle = 60" , 3 time intervals in vapor blanket stage). .......... 7 3 Figure 5.20 Comparison of analytical and experimental temperature profiles using a 14-interval cubic spline approximation for the heat flux function (quench angle = 60" , 4 time intervals in vapor blanket stage)...................... 74 Figure 5.21 Heat flux profile obtained by using a 14-interval cubic spline (quench angle = 60" , 4 time intervals in vapor blanket stage)...................... 75 Figure 5.22 Heat transfer coefficient profile obtained by using a 14-interval cubic spline (quench angle = 60°, 4 time intervals in vapor blanket stage). .......... 76 Figure 5.23 Comparison of analytical and experimental results for 10-degree quench angle test by cubic spline solution ................................................................. 79

Figure 5.24 Heat flux curve for 10-degree quench angle test by cubic spline solution.... 80 Figure 5.25 Heat transfer coefficient curve for 10-degree quench angle test by cubic spline solution ........................................ ... ... . ...... ..... .. .. ............... ......... 8 1

Chapter 1 Introduction 1.I The Inverse Problem Most heat conduction problems are concerned with the determination of temperature distribution inside the solid body when certain initial and boundary conditions are given, such as temperature or heat flux, which are known as a function of time. These problems belong to the class of "direct problems." A direct problem has a unique solution, because the solution involves a direct integration of differential equations with known initial and boundary conditions. In practice, however, the surface heat flux cannot be determined experimentally; but the temperature in the solid can be monitored by sensors. In theory, a surface heat flux history can be calculated from a set of temperature values, and this is called the "inverse heat conduction problem (IHCP)." However, because of the errors that inevitably exist in the experimental data, the IHCP often has non-unique solutions, leading to instabilities and convergence problems. The inverse heat conduction problem is particularly difficult because it is extremely sensitive to measurement errors (Beck, et al., 1985). Inverse heat conduction problems arise because measurements can only be made in easily accessible locations, or perhaps a desired variable can only be measured indirectly. For example, we may want to estimate the temperature history on the inside of a pressure vessel, but it is difficult to measure the temperature on the inside surface.

Therefore we use temperature profiles on the outside surface to estimate the temperature and heat flux inside the vessel. In high-speed flight, such as spacecraft re-entry, shock wave interaction with the spacecraft can produce enormous heat fluxes, which can damage aerospace vehicles.

To study these fluxes, the phenomenon is studied in

hypersonic wind tunnels. However, we cannot measure the flux in these situations. Therefore, the temperature is measured and computations are done to recover the heat flux. We know that effective heat treatment and thermal processing of metals and alloys is an essential component in the production of dependable components. More exacting metallurgical specifications call for greater precision in heat treatment and thermal processes. This has resulted in the need for accurate prediction and simulation of these processes through experimentation and computer aided design. A critical case in the simulation of quenching process is the determination of the surface heat transfer coefficient, which is required as an input parameter. The accuracy of the simulation and prediction is dependent on the surface heat transfer values. Quenching experiments are often carried out to determine the surface heat fluxes from experimental measurements of the temperature history. Therefore, this is a typical IHCP, and the solution algorithms require special consideration. The goal of IHCP algorithms for quenching is to determine the surface heat flux history from a set of measured temperature histories inside a heat-conducting body. This is obviously an estimation of the true values, since errors that are always present affect the accuracy of the heat flux calculation.

1.2 Objective of Current Research The objective of this research is to estimate the surface heat flux of a onedimensional plate whose temperature history is obtained by a quenching experiment. In this study we first develop an analytical solution to the direct problem, which consists of determining the temperature in a one-dimensional plate for a given time-dependent heat flux. The direct solution is determined by an approach based on the separation of variables. To solve the inverse problem, we minimize the difference between the experimental values and the analytical temperature calculated by assuming a surface heat flux. The heat flux input at the surface is assumed to be a function q(t) whose values are supposed to be known at a set of times t,, t, , . . . , tn. In our case, q(t) is approximated as a cubic spline.

A least square method is used to fit the analytical data to the

experimental solution and then evaluate the value of the heat flux q(t) , at the times t,, t,, ... , tn. The heat flux history is the cubic spline fitted through these q(t) values. Experimental data were obtained from experiments conducted at Ohio University by Zajc (1998).

1.3 Thesis Overview Chapter 1 introduces the inverse heat conduction problem associated with quenching. Chapter 2 gives the background of the IHCP problem.

In chapter 3 a

mathematical description of the IHCP is displayed, the direct solution and inverse solution of the problem are also shown. Chapter 4 explains the experimental setup,

procedure, and the experimental results obtained at Ohio University. Chapter 5 discusses the results of application of the IHCP algorithm. Chapter 6 presents a discussion of results, the conclusions and suggestions for future work. The above is a brief summary of the direct and inverse problems and the importance of the inverse heat conduction problem. The next chapter provides a review of the literature related to the IHCP problems.

Chapter 2 Background and Relevance 2.1 Introduction The "direct problem", in which the governing differential equation with known initial and boundary conditions is solved by integration, generally has unique solutions. In the "inverse problem", the surface flux history is to be calculated from a set of temperature values, and this should be possible in an ideal experiment. But the inverse problem is extremely sensitive to any error in the input data. Because of the errors that invariably exist in the experimental data, the IHCP often has non-unique solutions, leading to instabilities and convergence problems. Therefore, the inverse heat conduction problem is much more difficult to solve, both analytically and numerically, than the direct problem. There are two main reasons for the additional difficulty in solving the inverse problem. In the direct problem, the high-frequency components of the applied heat flux are damped as the heat flow diffuses through the solid medium. In the inverse problem, the opposite occurs. The highfrequency components or noise in the measurements will be amplified in the projection to the surface, and the resulting surface condition estimations can be easily overwhelmed by the noise in the interior measurement. The inverse problem is also made difficult by the factor that the physics of heat conduction introduces a natural lag between the applied heat flux and the temperature

response away from the flux. Thus, a step change in the surface heat flux will not be fully felt in the interior until a finite amount of time has passed. There is a choice between relatively difficult measurements or a difficult analytical problem. An accurate and tractable inverse problem solution would thus minimize both disadvantages simultaneously.

2.2 Literature Review There have been many different approaches to the inverse heat conduction problem. One of the earliest studies concerned with the calculation of heat transfer rates during quenching of bodies of simple finite shapes was published by Stolz (1960). Analytical solution techniques for IHCP were proposed by Burggraf (1964), Imber and Khan (1972), Langford (1967), and Kover'yanov (1967). As we will discuss below, such techniques have limited use for realistic problems, but they can give considerable insight into the IHCP. Tikhonov (1977) introduced a regularization method to reduce those sensitivities of ill-posed problems to measurement errors. Stolz (1960) developed the analysis specifically for spheres and other simple shapes. The system he chose was treated as a linear problem, permitting use of the superposition principle. The essence of his method is the numerical inversion of a suitable direct problem: given a surface heat flux versus time, find an interior temperature versus time. He solved the problem in a sequential manner that did not change the basic physical treatment of the problem. He did not consider the lag and damping of the measurements, which result in the problem becoming ill-posed.

Burggraf (1964)

presented an approximate solution for unsteady conduction with unknown surface boundary conditions. He approached the problem by assuming that both the temperature T ( t ) and heat flux q(t) were known functions of time at a single sensor location inside

the medium. The temperature field was represented in terms of an infinite series of both T ( t ) and q(t) and their derivatives and the solution was found for some very simple

geometry, such as a circular cylinder or a sphere. Kover'yanov (1967) developed results for the hollow cylinders and spheres. Imber and Khan (1972) obtained an exact solution for the temperature field using Laplace transforms when the temperature was known at two distinct interior points.

This method allows for the replacement of the input

thermocouple data by a temporal power series and a second series of error functions weighted by powers of time. The resultant expression for the prediction temperature is in the form of a summation of the repeated integrals of the error function. This method can be used to determine boundary conditions at either face of a finite slab or hollow sphere. Tikhonov regularization method and iterative regularization method are usually presented as whole domain methods in which all the heat flux components are simultaneously estimated for all times. Two advantages of these methods are that they have had rigorous mathematical investigation and can be applied very generally. Numerical methods have been the focus of recent studies in IHCP. Tervola (1989) developed a numerical method to determine thermal conductivity from measured

temperature profiles. He defined the problem as an optimization problem where the heat equation appears as a constraint. This optimization problem is solved with the DavidonFletcher-Powell method, and in each iteration the heat equation is solved by finite

element techniques with the predictor-corrector method. The boundary element method (BEM) was used by Lesnic et al. (1998) to determine the boundary conditions in a transient conduction problem where energies are specified in two areas of a onedimensional slab. Tseng et al. (1995), Hunag et al. (1995), and Keanini (1998) describe applications of IHCP to manufacturing process.

One-dimensional finite element

procedure was studied by Tseng et al. (1995) to predict the circumferential heat flux during water-cooled hot rolling. Keanini (1998) developed a finite element solution of a time-dependent, axially varying surface heat flux distribution during rolling. Numerical solutions have an advantage over analytical solutions in that thermal property variation is accounted for in the solution process. However, inverse calculations based on numerical direct solutions tend to be computationally intensive and often have convergence problems.

The typical inverse algorithm uses an error minimization

approach, which is sensitive to the initial guess for the heat flux. The attraction of an exact analytical solution is that it can provide a fast, approximate solution that can be used as an intermediate step (or an initial guess) in a detailed and more accurate inverse calculation. This would improve the efficiency of iterative numerical solutions, which are highly sensitive to the initial guess. The instability problem associated with IHCP has been discussed in detail by Beck (1985). The instability arises from the fact that arbitrarily small differences in the input data can produce arbitrarily large differences in the output values. A classic problem in ICHP is the determination of surface heat flux in a one-dimensional slab. For

the direct problem with a known constant heat flux, the temperature solution is known. If the heat flux varies with time, Duhamel's theorem can be used to find a solution.

2.3 Polynomial Solution Alam et al. (1999) describes an analytical solution to the direct problem, which consists of finding the temperature in a one-dimensional plate for a given heat flux which is assumed to be a polynomial function of time. The solution is based on separation of variables, and is described in detail by Kumar (1998). To solve the inverse problem, a least square method is used to minimize the difference between the analytical prediction and experimental temperature profiles.

The analytical approach is tested by using

experimental data that is obtained by quenching a special probe, which produces a good approximation to the boundary conditions in the analytical solution. A least square method is then used to produce best fit to the experimental data and determine the polynomial coefficients of heat flux function. The main idea is to assign heat flux as a k'th degree polynomial in time t , q ( t ) = a,

+ a,t + a2t2+ + aktk, where a,, a,

, + . a ,

a, are unknown parameters. With this

information the heat transfer equation can be solved to yield a curve of temperature versus time. By taking the experimental temperature at n times, t, ,t , ,.

a ,

t, , we obtain

n equations for the determination of the k + 1 unknowns.

Quench probes have been used to collect temperature data in controlled quenching experiments; the data is then used to deduce the heat transfer coefficients in the quenching medium. The process of determination of the heat transfer coefficient at

the surface is the inverse heat conduction problem, which is extremely sensitive to measurement errors. This thesis reports on an experimental and theoretical study of quenching that is carried out to determine the surface heat flux history during an experimental quenching process by an inverse algorithm based on an analytical solution. The algorithm is applied to experimental data from a quenching experiment carried out at Ohio University.

The surface heat flux is then calculated, and the theoretical curve

obtained from the analytical solution is compared with experimental results. The inverse calculation appears to produce fast, but approximate results. These results can be used as the initial guess to improve the efficiency of iterative numerical solutions, which are sensitive to the initial guess.

2.4 Theory Consider a one-dimensional heat conduction problem through a uniform plate, as shown in Figure 2.1. At one end, x = 0 , the surface is insulated, and at x = L it is subjected to heat flux q(L,t) , which is an unknown function of time. The governing heat conduction equation describing the temperature distribution in the plate is

with the initial condition: T(x,O) = To(x), Although the initial temperature is assumed to be uniform, the solution method described below'will apply even when the initial temperature of the plate is a function of

x . The boundary conditions correspond to the insulated left-end of the plate and the heat flux input at the other end:

where:

a - Thermal difhsivity k - Thermal conductivity q - Heat flux

T - Temperature

x - Space coordinate L - Thickness of plate

Figure 2.1 One-dimensional Plate with Boundary Conditions

The IHCP problem to be solved may be stated as follows. Given a certain, measured temperature T(0,t) profile as a function of time at x = 0 , find the heat flux q(L,t) at x = L which produces the experimental temperature history. The solution will also determine the temperature distribution as a function of space and time, i.e. T(x,t) over the entire plate at any instant. The analytical solution to the above problem for the special case of q = constant is known to be the following (Beck et al., 1985): 2

9

[[

cosnn: 1--

;))I

(2.5)

2.5 Solution for Polynomial q(t) It is now necessary to determine the analytical solution for the case of timevarying q(t). This is done by making the boundary conditions homogeneous by introducing the following transformation:

Without loss of generality, it is assumed that the heat flux q(L, t) can be expressed as a polynomial function of time:

In the above equation am (m = 0,1,2, ... ,j, ... , p )

are p + 1 undetermined

coefficients, which will be evaluated by minimizing the error between the experimental and analytical solutions. Differentiating equation (2.6) with respect to x, we get the following:

while differentiation of equation (2.6) with respect to t produces

Written in terms of y~ ,the governing equation (2.1) now becomes nonhomogeneous

The boundary conditions of this equation are now homogeneous: at x = O ; dT(x, t) = 0 , therefore, dx atx=L;k

aw (x, t) = o ax

W x , t) = q(L, t, a m) ,therefore, dx

aw (x, t) = o ax

The initial condition at t = 0 changes to

Now one needs to solve equation (2.10) subjected to the boundary conditions in equations (2.1 1) and (2.12), and the initial condition in equation (2.13). This was done

nn by using the separation of variables method and using eigenvalues h = - and L

(nr)

eigenfunctions cos

-

of the homogeneous problem. We assume that the solution to

the nonhomogenous equation (2.10) is given by:

where Cn are coefficients that are to be determined. It should be noted that all these coefficients ( C , ) are functions of time.

Differentiating the above equation and

combining with equation (2.10) the following equation is obtained:

- - q ( L y t y a m((jx )

- 2 ~+-)1

kL2

a

( x 3 - L X ~ ) aq(L,t, a,)

k~'

at

>

(2.15)

where C,( t ) is the time derivative of C , ( t ) . The coefficient C , ( t )can be determined by multiplying eqution (2.15) by the eigenfunctions and integrating over the domain. For n = 0 this produces the following equation: L dq(L,t , a,) c,( t ) = - q ( L ykLt ay , )a +-12k dt Integrating equation (2.16) with respect to time we get:

a a,tm+' L +q(L, t, a,) + const. co (t) = kL C-,=, rn + l 12k -

From the initial condition it is seen that:

Lao Co(0) =-+To 12k

For n greater than zero, we get the following set of equations:

12q(L,t,am' k~n'n

(cos(nn ) - 1)- 2L aq(L' "am ) ((n 'n h4n at

' - 6)cos(nn ) + 6)

Integrating the above equation with respect to time, one obtains:

kLn 2n ' 2L aq(Ly at

(cos(nn ) - 1))

((n 'n

' - 6)cos(nn ) + 6)

; : :( ;ldt]

- a

where El, is a constant of integration. P

The heat flux is assumed to have the form q(L,t,a,)

=za,tm m=O

equation (2.20) becomes

. Therefore,

where

+-2L -- ((n 2 n 2 - 6) cos(nn ) + 6 k?1471:

The two constants can be combined into one by taking En = E,, Therefore, equation (2.14) can be written as:

+ E2, .

+'

n=l

I

dq(L,t, a,

l 2 a (cos(nn - 1)) q(L, t, a,) kLn2n2

-

-~ d2q(L,t, a ,

)

dt2 3

cos(y)

Applying initial condition on equation (2.22) to determine En leads to the following result for the analytical temperature profile:

" -

2L

--n=l

kn 4~

((n 'n

- 6) cos(nn )

+6

Having determined the temperature profile in terms of the heat flux, the final step is the determination of the heat flux by using experimental data. This is the inverse problem, and one now must find q(t), the polynomial hnction. This is now essentially the problem of finding the unknown coefficients , a l ,a

m in the function

This is accomplished by applying the least-squares method to the temperature profile from experiments, i.e.,

where T is the analytically determined temperature profile given by equation (2.23), while Te is the experimentally determined temperature. If 's' values of temperature data are obtained from experiments, equation (2.24) requires that 2% (T - T,(ti))dr i=o a'm

=0

for m = 0 , 1 , 2,...,j ,...,p

Using the analytical solution for T(t) , it is possible to find the derivatives with respect to the coefficients a o , a l,..., a j ,...,a,. Let the derivatives be designated as rj(t) ,

'T(0, t) = rj (t) .j

This produces the following set of ' p ' equations from equation (2.26), with

Solving the above linear algebraic system in the coefficients by the Gauss elimination method (or any other matrix inversion algorithm), one can find the polynomial coefficients a,, a, ,...,a ,...,a, that are necessary for describing the heat flux as a function of time. This is the heat flux sought that produces the given experimental temperature profile at x = 0 .

2.6 The Spline Interpolation Model To develop the analytical solution for the heat flux by the inverse algorithm outlined above, the heat flux was assumed by Kumar (1998) to be a sixth-degree polynomial fbnction of time, i.e.:

In the present study, we assign the heat flux to be a cubic spline function in time t . The cubic spline is fitted to the experimental data such that the least square error

between the cubic spline and the experimental data is minimized. After the heat flux is calculated in the cubic spline form, it is possible to compare the analytical value of the temperature with the experimental data.

Chapter 3 Mathematical Model

3.1 Introduction In chapter 2 we described an analytical solution based on a sixth-degree polynomial. It will be shown later that a single polynomial cannot adequately match the heat flux in typical quenching processes. We will now develop the inverse solution on the basis of a cubic spline match to the experimental heat flux. We will first define the mathematical problem, and then derive the analytical solution for the direct problem with the boundary heat flux expressed as a cubic spline. The inverse problem consists of calculating the analytical temperature history in the quenched part by using a cubic spline form of the boundary heat flux and then comparing the analytical and the experimental temperature profiles. The heat flux values at the node points of cubic spline are obtained by minimizing the difference between the analytical and experimental temperature profiles.

3.2 Definition of the Problem The inverse problem is to estimate the surface heat flux function utilizing measured interior temperature history. Our approach is to first develop a mathematical

model and the solution of the direct problem. The direct problem was defined in chapter 2. We briefly summarize the direct problem below Consider a one-dimensional heat conduction problem through a plate as shown in Figure 2.1. It is assumed that the plate consists of a single material, homogeneous and isotropic. At one end x = 0 , the bar is insulated, and at x

=L

it is subjected to a heat

flux q(t), which is an unknown function of time. The initial temperature of the plate T(x,O) is assumed to be a known function of x . For constant thermal properties, the governing heat conduction equation describing the temperature distribution in the bar is:

subject to the initial condition, i.e.

The boundary conditions are given by the insulated left-end of the plate and the heat flux input at the other end (Figure 2.1).

3.3 Analytical Solution for the Direct Problem Kumar (1998) used a sixth-degree polynomial for q ( t ) to solve the above problem. In this study we will first assume a general form for q ( t ) , and finally use a cubic spline to express q ( t ) . As was done by Kumar (1998), the boundary condition is made homogeneous by

taking a transformation as following:

Now the governing equation (3.1) becomes nonhomogeneous:

while the boundary conditions become homogeneous: dT(x t) x = 0 ; ----1--= 0 , therefore, h y ( x 7 t ) = 0 dx ax x=L; k

dx

t , = q ( t ) , therefore,

QJ

( x , t ) =O dx

The initial condition at t = 0 changes to:

w (x,o) = T(x70)- q(0) (x3 - L x 2 ) = f ( x ) The homogeneous equation can be solved subject to the boundary conditions in equations (3.7) and (3.8), and the initial condition in equation (3.9) by using the separation of variables method. The homogeneous solution is given as:

nn where hn = - are eigenvalues and cosh,x are eigenfunctions of the homogeneous

L

problem. Now we assume that the solution to the nonhomogenous equation (3.6) is given

y (x,t ) =

C en(t)e"';'

cos hnx

Differentiating the above equation and combining with equation (3.6) the following equation is obtained: 1 "

--

a

.=o

4(t) 1 (x3 - L X ~dq(t) ) C, (t)e-"':' cos hnx= - --(6x - 2 ~+ ) k~~ a k ~ ~ dt

We multiply both sides cos h,x and integrate fiom 0 to L :

There are two different solutions depending on whether the integer n is zero or non-zero. Case 1: n = 0

a L ddt) co( t ) = kL q(t) + -12k dt -

Integrating equation (3.14)with respect to time we get:

Case 2: n > 0

where A, =

12a (cos nn kLn2n

-1)

Again integrating equation (3.16)with respect to time we get:

Finally, the general solution of equation (3.6) is:

( x ,t ) = Co( t )+

C, (t)e"r' cos h,x

Therefore the solution of the governing heat conduction equation subject to the boundary condition is :

m

(0 ( x 3 C C , (t)e"^:' cos hnx+ 4kL2 -

- LX')

,=I

where C , ( t ) is given by equation (3.17). Now we apply initial condition to equation

(3.9)to determine Eo and E n .

z m

Since T(x,O) = f (x) =

d o ) (x3 - Lx2) Cn(0) cos hnx+ n=O k~~

Therefore, m

z c n ( 0 ) c o s h n x= f (x) - m ( x 3 - Lx2) n=O kL2 We multiply both sides by cos hmx and integrate from 0 to L to find Eo and En :

After carrying out the integration and simplifying, the following expressions are obtained for Eo and En :

E

=

I

l L

[ f (x) - ( x 3 - Lx2) ni

0

Case 2: n = 1,2,... :

Based on the above equations, the analytical temperature solution is given by:

+ [ n=l

1

12a(cosnn -1) ' 14(7 )eagTddr e"gt cos hnx kLn2n2 0

+

m

n=l

[-

2L [(n' n - 6)cos nn

kn4n -

+ 61 01 dz

nn where An = - , n

L

=1,2,..

3.4 Validation of Analytical Solution 3.4.1. Checking the Governing Equation The above analytical solution to determine temperature profile can be checked by back substitution into equation (3.12). Since

Therefore:

Substitution into the equation (3.12)produces the following equation:

Equation (3.24) is valid if the following two equations are satisfied:

a -

kL

+

"

A, cos h,x

a

= -(6x

.=I

kL2

-21)

Equation (3.25) is satisfied if the left hand side is the Fourier cosine series

a

expansion of -(6x kL2

-2L)

with the first term

a -

kL

as the zeroth ( n

=0

) term of the

series. This is seen to be the case from the following orthogonality conditions: For n = O weget: For n = 1,2,

=-

kL

, we get:

The above value of A, is the same as in equation (3.16); therefor equation (3.25) is a valid solution.

In a similar manner, equation (3.26) is satisfied if the left hand side is the Fourier cosine series expansion of

-

x3 - Lx2 L with the first term - as the zeroth ( n = 0 ) term k~~ 12k

of the series. This is seen to be the case from the following orthogonality conditions: For n = 0 we get: For n

= 1,2,

, we get:

The above value of B, is the same as in equation (3.16);therefor equation (3.26) is valid. 3.4.2 Checking for the Boundary Conditions

From the analytical solution in equation (3.23),

2.=, C, ( t )h, e-"':' sin h,x + 4(t) (3x2 kL2 -

- 2Lx)

nrr Since h, = -, therefore, sin h,x = 0 at x

= 0, x = L

=

ax

L

at x = O ; dT(x,t) = 0 ax

. Consequently:

3.4.3 Checking for the Initial Condition Substituting t = 0 in equation (3.23) we get:

This can be written as:

From the above equation, we can see the first term on the right hand side is the Fourier cosine series expansion of f (x). The term within the second parenthesis is the 4 (0) (x3 - Lx2),therefore it cancels the last term of the right Fourier cosine expansion of kL2

side. Consequently, at t = 0 ; T(x,O) = f (x) . The above procedure proves that the analytical solution found is correct.

3.5 Comparison with Alternate Solutions 3.5.1 Validation for q(t) = constant

A solution of the above quenching problem for the case of q ( t ) = constant was evaluated by Beck et al. (1985). For the coordinates of current problem this solution can be written in non-dimensional form as follows:

T X Where T + = ---- , t + = -a t x + =L- . q- = - 4. L2 ' 9,L 1k 4,

In order to compare our analytical solution with the above solution we first nondimensionalize equation (3.23) to obtain the following:

-f

( ~ )+- q ( ~ ) ( ~- + X + '2 )

12(cosnn

+i[ n=l

n2n2

- 1) " nf

1e-nzn2t

q ( t) e n z z d t

f (x' ) - i j ( ~ ) ( x-+x~* ~ )

cos n m +

e-n2n2t+ cos n m +

To compare equations (3.27) and (3.28), the above was evaluated with q(t) = constant = q, at x'

= 1,

andx'

=0

as the insulated surface. The temperature

values as a function of time are shown in Table 3.1. The solution calculated from equation (3.27) as reported by Beck et al. (1985) is shown in Table 3.2 in the coordinates of the current problem. Comparison of Table 3.1 and Table 3.2 shows that the two solutions are in excellent agreement.

3.5.2 Validation for q(t) = f (t) For the case of a variable heat flux given by q(t) = 10000t2 at x = L , and T(x,O) = 400, we can obtain the temperature profile at x = 0 by using a finite difference method. A finite difference program was written to compare numerical and analytical solutions. Comparing the analytical profile with the results from finite difference method (Table 3.3), it can be seen that the analytical solution (3.28) matches the finite difference solution quite well.

3.6 Inverse Solution The IHCP problem to be solved may be stated as follows.

Given initial

temperature profile T(x,O) , and the measured temperature T(0, t) history as a function of time at the position x = 0 , find the heat flux q(t) at x = L which produces the experimental temperature history. The solution will also determine the temperature distribution as a function of space and time, i.e., T(x,t) over the entire plate at any instant.

Table 3.1: Dimensionless Temperature Values, T' (x' ,t' ) , for Various Dimensionless Time and Distances for a Plate Heated at x = L and Insulated at x = 0 . (Source: Beck et al., 1985)

Table 3.2: Dimensionless Temperature Values, T' (x' ,t' ) , for Various Dimensionless Time and Distances for a Plate Heated at x = L and Insulated at x = 0 .

Table 3.3 Comparison of Analytical and Finite Difference Solutions t 0.0 0.5 1.o

2.0 2.5 3.0 3.5 4.0 4.5 5'0

Finite Difference Method Analytical Solution Finite Difference Method Analytical Solution FiniteDifferenceMethod Analytical Solution Finite Difference Method Analytical Solution Finite Difference Method Analytical Solution Finite Difference Method Analytical Solution Finite Difference Method Analytical Solution Finite Difference Method Analytical Solution Finite Difference Method Analytical Solution Finite Difference Method Analytical Solution Finite Difference Method Analytical Solution

x =0

x = 0.25

x = 0.5L

x

400.0000 400.0000 400.0000 400.0109 400.0000 400.1246 400.3765 400.5320 401.9003 401.4413 403.4241 403.0617 406.4086 405.6022 410.2314 409.2720 415.3688 414.2795 422.0577 420.8332 430.4652 429.1411

400.0000 400.0000 400.0000 400.0121 400.0000 400.1446 400.4627 400.5927 401.9865 401.5632 403.6609 403.2637 406.6676 405.9020 410.6609 409.686 415.9697 414.8240 422.8306 421.5243 431.4326 429.9954

400.0000 400.0000 400.0000 400.0208 400.0027 400.2188 400.7213 400.8022 402.2650 401.9795 404.4127 403.9588 407.6538 406.9485 412.0363 41 1.1569 417.8438 416.7921 425.2805 424.0625 434.5639 433.1763

400.0000 400.0000 400.0000 400.0498 400.1196 400.3803 401.1524 401.2126 402.9855 402.7569 405.6795 405.2221 409.4989 408.8172 416.6008 413.7509 421.2423 420.23 18 429.6306 428.4681 439.9739 438.6680

= 0.75

x =L 400.0000 400.0000 400.0253 400.1 173 400.4086 400.6732 401.7902 401.8936 404.2209 403.9875 407.5948 407.1621 412.2865 411.6249 41 8.4412 417.5838 426.3033 425.2470 436.0784 434.8229 447.9499 446.5198

In order to develop the analytical solution for the heat flux by the inverse algorithm, the heat flux profile as a function of time was assumed to be a cubic spline. In the last few sections we developed the analytical temperature solution as a function of the heat flux. The next step is the determination of the heat flux by using experimental data. This is the inverse problem, and we must find theq(t), which is a piecewise cubic spline function. To find the heat flux, the difference between analytical temperature solution and the experimental temperature is to be minimized.

In our study, we use an

optimization routine from International Mathematics and Statistics Library (IMSL) to find a best fitting cubic spline curve to match the experimental profile. The results will

be an approximation to the heat flux profile in the form of a cubic spline. This is described in the following sections. 3.6.1

Cubic Spline Interpolation We sometimes know the value of a function f (x) at a set of points x, ,x, ,

a ,

x, ,

but we don't have an analytical expression for f (x) that lets us calculate its value at an arbitrary point. We want to estimate f (x) for arbitrary x by drawing a "smooth curve" through data points xi. Mathematically, it is possible to construct cubic functions S, (x) on each interval [x,, x,,,] so that the resulting piecewise curve y = S(x) and its first and second derivatives are all continuous on the large interval [x,, x,]. In this study we use a natural cubic spline for which the second derivative of the interpolating curve vanishes at the first and the last data points of the original set. In the computer program, which is included in the appendix, there are two subroutines (SPLINE, SPLINT), which allow the user to interpolate a value given a set of data points yi = f (xi). These subroutines require the value yi at the point xi. During the minimization process, the value of y, are obtained so that the cubic spline based on yi = f ( x i ) represents the heat flux which produces minimum error between the analytical and experimental temperatures. 3.6.2

Minimization Algorithm (IMSL Software) The IMSL software that was used in this calculation is a library of C language

subroutines useful in scientific programming. Each fbnction is designed and documented to be used in general programming tasks.

In this study a number of routines from this library were used. One is called "bounded -least-squares", which can solve a nonlinear least-squares problem subject to simple bounds on the variables using a modified Levenberg-Marquardt algorithm. The following synopsis provides directions for using this routine: Synopsis: #include float *imsl~f~bounded~least~squares(void fcn(), int m, int n, int ibtype, float xlb[], float xub[], ... , 0 ) Required Arguments: void fcn (int m, int n, float x[], float

fl]) (InputIOutput)

This is a user-supplied hnction to evaluate the function that defines the leastsquares problem where x is a vector of length n at which point the fhnction is evaluated, and f is a vector of length m containing the function values at point x. int m (Input) Number of functions (number of experimental data points). int n (Input) Number of variables (number of cubic spline points), where n I m. int ibtype (Input) Scalar indicating the type of bounds on the variables. float xlb[] (Input, Output, or Input/Output) Array with n components containing the lower bounds on the variables.

If there is no lower bound on a variable, then the corresponding xlb value should be set to -lo6. float xub[] (Input, Output, or InputIOutput) Array with n components containing the upper bounds on the variables. If there is no lower bound on a variable, then the corresponding xub value should be set to lo6. Return Value: A pointer to the solution x of the nonlinear least-squares problem. To release this space, use free. If no solution can be computed, then NULL is returned. Synopsis with Optional Arguments: #include float *imsl~f~bounded~least~squares (void fcn(), int m yint n, int ibtype,float xlb[], float xub [I, IMSL-XGUESS,float xguess[], IMSL-JACOBIAN, void jacobiano, IMSL-SCALE,float xscale[],

. . ., 0) x[] in this function represents the heat flux values at n data points, fi] is the difference between the experimental temperature and the analytical temperature at every iteration. The total number of experimental data points used in this calculation is 120 (m=120). The number of cubic spline intervals are 9, 15, 16 and 23 (n=10, 16, 17,24).

The analytical solution requires integration of certain functions. This is done by using a number of routines. A typical subroutine is the "int-fcn".

This routine integrates

a function by using a globally adaptive scheme based on Gauss-Kronrod rules.

It

subdivides the interval [a, b] and uses a (2k + l)-points in each subinterval. The following is the synopsis of this routine as given in IMSL manual: Synopsis:

float imsl-f-int-fcn

(float fcn(),float a,float b, ... , 0)

Required Arguments: float fcn (float x) (Input) User-supplied function to be integrated. float a (Input) Lower limit of integration. float b (Input) Upper limit of integration. Return Value: b

The value of fcn(x)dx is returned. If no value can be computed, then NaN is returned. a

Synopsis with optional Arguments:

float imsl-f-int-fcn

(float fcn(float x),float a,float b,

IMSL-RULE, int rule,

IMSL-ERR-ABS,float err-abs, IMSL-ERR-REL, float err-rel,

. . ., 0) 3.6.3

Program Flowchart The flowchart for IHCP solution is given in Figure 3.1. The primary inputs are

experimental temperature data points, the cubic spline intervals, and material properties.

The upper bound was made approximately 0, and no limit was set for the lower bound.

Start

1

Input Experimental time Experimental temperature

Initialize 1.Parameters of one-dimensional problem 2.Heat flux cubic spline function 3 .The calculation tolerance 4.Special parameters of routines 5 .The upper and lower bounds for solution

Calculate The analytical temperature profile

Minimize The difference between the experimental Data and the analytical solution

No

Output 1.The analytical temperature 2.Heat flux solution 3 .Heat transfer coefficient

Stop

Figure 3.1 Flowchart for IHCP

Chapter 4 Experimental Results

In this chapter the quenching process will be briefly described, along with the experimental setup which is used to determine the temperature history during quenching. These experiments were carried out in research studies (Zajc, 1998; Kumar, 1998) at the Department of Mechanical Engineering at Ohio University.

4.1 Quenching 4.1.1 Introduction Many metal alloys that are currently being manufactured are subjected to heat treatment before being placed in service. They are heat treated in order to improve certain properties, such as hardness, strength, toughness, ductility and corrosion resistance, as well as to increase uniformity of the properties. Quenching is a particular type of thermal treatment process that involves rapid cooling of metal alloys for the purpose of hardening. Alloys of iron are quenched to prevent the austenite from decomposing under equilibrium conditions thereby producing microstructures that alter the mechanical properties of the materials. The word quenching, when applied to the heat treatment of steel, covers the process of cooling the steel from the austenitizing temperature at a rate

such that decomposition of the austenite will occur at sub-critical temperatures. The quenching medium can be a gas, liquid, or solid. The quenching problem is one of heat transfer. The quenching medium extracts heat from the surface of the steel which results in steep temperature gradients being set up in the body of the steel. 4.1.2 Cooling Curve

Among the various experimental results that have been used to predict the ability of quenching process to get the desirable properties, the cooling curve is the most popular. Cooling curves are obtained experimentally using an apparatus that primarily consists of an instrumented quench probe to monitor the interior temperature response and a data acquisition system to collect and display the sampled data. If a heat-treated steel part is continuously cooled from a solution treating temperature in a quenching medium that rapidly extracts heat, four stages of heat transfer will take place. These stages characterize the four different cooling mechanisms that occur during quenching. The four cooling stages are: (i) the initial liquid contact or wetting stage, (ii) the vapor blanket or film boiling stage, (iii) the nucleate boiling stage, and, (iv) the convective heat transfer stage. Except the first stage, other three stages are recognizable on cooling curve obtained by experimental procedures. The first stage lasts for an extremely short time, and is generally not detected in the experimental cooling curve.

4.2 Experimental Studies 4.2.1 Experimental Apparatus In the quenching experiment the following equipment was used: quench probe, quenching probe support, quench tank, data acquisition hardware, data acquisition software and computer. This is briefly described below: Quench Probe: Experiments were conducted with a stainless steel probe in the shape of a rectangular box, which is shown in the Figure 4.1. This probe is a rectangular box approximately 15cm-by-10 cm-by-2.5 cm. Each wall of this probe box is 6 mm thick (Zajc, 1998). The cover of the probe box was screwed on with a gasket to make it watertight and the inside surface of the largest side was instrumented with type K thermocouples. Since the box is empty inside, the inside surfaces can be assumed to be insulated (convection heat transfer to the air from the walls is negligible for the short duration of the quench test). This probe is assembled with the thermocouples that pass through the handle of the probe, and then the probe box is sealed with the cover. The probe was heated to a specified steady-state temperature (typically 400°C), and then quenched in water in a quenching tank. Quenching Probe Support: The quench probe is supported by a metal fixture that can lower the probe into the water at different angles. The angle that the probe is quenched is determined by the seven sets of mounting holes in the adjustment plate that are spaced in 10 degree

(b) Figure 4.1 Quench Probe: (a) Schematic of the box shaped probe showing the walls and the screw holes for attaching the top, (b) Picture of the probe used in the experiments -the cover has been removed to show the thermocouples connected to the inside of the probe wall

increments, therefore, the probe could be potentially quenched at any 10 degree interval between 0 and 60 degrees. Quenchant and Quench Tank: The quenchant used during the experiments was water. The water was held in a 100 gallon quenching tank shown in Figure 4.2 that is made of a high strength plastic. A plastic tank is used to ensure that a reaction does not occur between the quenchant and the tank that could potentially change the composition of the quenching medium. Data Acquisition Hardware: The data acquisition equipment consists of the following: (i) D.A.S. 1701 board made by Keithley Metrabyte. This board has 16 singleended or 8 differential channels with 12-bit resolution, 166.67 x 1o3 samples/second maximum throughput, and gains of 1, 5,50 and 250. (ii) The software package used to collect the data for analysis during the experiments is called TestPoint. Each of the thermocouples was sampled every 0.01 second for approximately 1 to 2 minutes. TestPoint has several capabilities which include: controlling external measurement devices, creating user interface item, displaying and analyzing data, creating files, and dynamically exchanging data with other software applications. Furthermore, TsetPoint applications have the ability to support sequential execution, repeating loops, and conditional statement. (iii) Computer: P5-200 Gateway 200 MHz Pentium processor, 32MB SDRAM, 3.0 GB hard drive.

Figure 4.2 Picture of the quenching system showing the quench probe suspended over a quench tank.

4.2.2 Experimental Procedures

The experimental procedures consisted of three steps: Step 1: Heat the probe up to the specified steady-state temperature. Step 2: Initialize the data acquisition system and remove the probe fi-om the furnace. Step 3: Lower the probe into the tank to quench the probe and collect the data for approximately 2 minutes.

4.3 Experimental Results The experimental results of temperature versus time are given in Figure 4.3 and Figure 4.4 for quenching experiments carried out at 10 degree and 60 degree angles. In each figure the two curves correspond to the data from two thermocouples which are attached to the inside surface (at the center and at the comer) of the 15cm-by-l0cm wall of the probe. Since the wall thickness (6mm) is much smaller than the wall dimensions, the one-dimensional solution is valid for this wall of the probe. This is verified by the close matching of the two temperature profiles. It can be observed that the thermocouple data indicate a rapid cooling period of about 10 seconds after approximately 2 seconds of quenching. This is characteristic of quenching curves (Totten et al., 1994). The first part of the curves are nearly horizontal; this period corresponds to the air cooling of the probe during transport from the oven to the quenching system, and the vapor blanket stage at the beginning of quenching. The rapid cooling part of the curves represents the nucleate boiling stage.

The final part of the cooling curves is the

convective cooling stage. The goal of the inverse heat transfer algorithm is to estimate the heat flux from the probe on the basis of this cooling curve. This is done in the following chapter.

Chapter 5 Results

5.1 Introduction In this chapter we present the results from the application of the IHCP algorithm with the cubic spline approximation of the heat flux history for the quenching process. The heat flux curves, heat transfer coefficient plots and the comparison of the analytical and experimental temperature curves will be presented.

5.2 Results from Polynomial Solution Figures 5.1 to 5.4 show the results obtained by using an IHCP code from a previous study (Alam et al., 1999). In this method the temperature solution is obtained by using a 6thdegree polynomial to estimate the heat flux during quenching of the probe. The analytical temperature history is obtained by carrying out a least-square error minimization of the experimental and analytical temperature history obtained from the 6th degree heat flux polynomial. The experimental data consists of 12,000 temperature data points over 120 seconds of quenching time. The average of every 100 points is computed in order to get a smooth curve containing 120 points, with adjacent points separated by 1

second.

It is seen that the temperature profiles compare reasonably well for the 10-degree and 60-degree quenching angles, but the analytical temperature profiles have a 'wavy' shape which is not representative of the continuous cooling of the probe.

The

temperature profile shows an increasing trend at several points in the analytical curve, for example after 40 seconds of cooling. This could be due to the fact that a single 6thdegree polynomial is being used. This produces the heat flux curves as shown in Figures 5.2 and 5.4. The heat flux curves demonstrate another problem in that the heat flux curves are very unrealistic after the first 60 seconds. In one case the heat flux shows a very high negative value, and in the other we see a very high positive value (the positive heat flux is obviously impossible for this process). This error is due to the fact that there is a significant diffusion time lag between the quenching surface at x = L , and the thermocouple position at x = 0 . This time lag has a characteristic time of L~/ a , and is of the order of 10 seconds. Consequently, the analytical temperature prediction during the last 10 to 30 seconds is quite insensitive to the heat flux prediction. The polynomial that results from this error minimization algorithm is not influenced by the heat flux curve for the final part of the quenching. These results in the mismatch observed in the heat flux curves. Therefore, to match the data set for 120 seconds with a polynomial form of the heat flux curve, additional data points should be included beyond the 120 seconds (either from experiment, or by simple extension of the experimental temperature history).

It can also be seen from the figure that a continuous polynomial profile for heat flux may not be very accurate when the temperature slope has a sudden change, such as near the end of the vapor blanket stage.

5.3 IHCP Algorithm with Cubic Spline In order that the errors caused by the use of a single polynomial can be avoided; we have expressed heat flux by a cubic spline, which is made up of a set of cubic polynomials defined over a set of time intervals. Therefore, our results depend on the number of intervals, and the width of the intervals used in the error minimization algorithm. In the calculations carried out to determine the heat flux, the number of intervals for the different stages of cooling was taken as a variable parameter. It was observed that the oscillations tend to occur near the end of the vapor blanket stage. This is due to the fact that there is a sharp drop in the temperature right after the vapor blanker stage. In the results described below, the total number of spline intervals were varied from about 8 to 16; with 2 to 4 intervals in the vapor blanket stage. The results of the comparison of the experimental data with analytical solution based on different time intervals for the cubic spline are shown in the following pages. Figures 5.5 to 5.13 show the comparison of analytical and experimental temperature profiles for 10 degree quench angle, the heat flux curve predicted by the IHCP algorithm, and the heat transfer coefficient obtained by the analysis. It can be seen that the vapor blanket stage lasts for about 15 to 20 seconds, and is followed by the nucleate boiling

b 0

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stage with high heat flux for about 10 seconds. The convective heat transfer stage sets in after approximately 30 seconds. In these figures, it can be observed that the analytical and experimental temperature profiles match reasonably well when sufficient number of intervals are used. The differences due to the number of intervals show up in the heat flux curves (and therefore in the heat transfer coefficient). As will be shown later, the results are very similar for 60 degree quench angle. The heat flux in the nucleate boiling stage is seen to have a maximum value of about 70 kw/m2 for 10 degree quench angle. The heat flux is not very accurate when only 8 intervals are used (Figure 5.5 -5.7); the temperature profile tends to be wavy and the maximum heat flux is 58 kw/m2. This is not surprising, since the inverse problem is very sensitive to input parameters. A similar variation is seen for the 60 degree quench angle (Figures 5.14 to 5.22). However, the 60 degree quench angle test appears to have a higher maximum flux (approximately 110 kw/m2). Therefore, it is quite critical that the number of time intervals be varied in order that the heat flux can be calculated with a good degree of confidence. Even though the optimization was carried out with the constraint of heat flux constrained to be negative in the IMSL code, it is seen that the code results contain positive heat fluxes which are typically 20 kw/m2 (about 20% of the maximum value). This error in the IMSL solution could not be resolved. In all curves, the final stages of the cooling show the expected decay towards zero heat flux. This is in sharp contrast to the single polynomial case discussed earlier.

approximation for the heat flux function (quench angle = 60°, 2 time intervals in vapor blanket stage).

Figure 5.14 Comparison of analytical and experimental temperature profiles using a 9-interval cubic spline

Time (second)

approximation for the heat flux function (quench angle = 60°, 4 time intervals in vapor blanket stage).

Figure 5.20 Comparison of analytical and experimental temperature profiles using a 14-interval cubic spline

(quench angle = 60°, 4 time intervals in vapor blanket stage).

Figure 5.22 Heat transfer coefficient profile obtained by using a 14-interval cubic spline

Time (second)

Therefore, the piecewise continuous cubic spline is not affected strongly by the diffusion time lag as was seen with the single polynomial. Since the vapor blanket stage ends in a sharp drop into the nucleate boiling stage, the (Figures 5.5, 5.14), the temperature in vapor blanket stage is not matched very well. The cubic spline tends to produce fluctuations in and near the vapor blanket stage. This is strongly influenced by the number of intervals used in the vapor blanket stage. When the number of intervals in the vapor blanket stage is increased (Figures 5.8 to 5.13, and 5.17 to 5.22), the temperature profiles show an improvement in the match with the experimental curve. However, the temperature profile in vapor blanket stage now has a sinusoidal variation, and that produces a sinusoidal variation in the vapor blanket stage of the heat flux curves.

5.4 Heat Transfer Coefficient According to the convection heat transfer equation, the heat transfer coefficient is given by:

From the analytical solution of the IHCP algorithm, we can get the heat transfer coefficient curves for each of the cases by using the heat flux curve ( q ) for the numerator, and the temperature solution ( T ) values in the denominator. The heat flux values will obviously reflect the fluctuations observed in the heat flux curves, and this can be observed in the heat transfer coefficient curves shown in

Figures.5.7, 5.10, 5.13, 5.16, 5.19, and 5.22. The heat transfer coefficients show large fluctuations when the heat flux curve oscillates. As the number of time intervals is increased, the heat transfer coefficient curves attain more stable values.

Therefore,

increasing the number of time intervals reduces the fluctuations in the heat flux and heat transfer coefficients. It appears that the problems in the IHCP algorithm originate from the sharp changes in the temperature and heat flux profiles. In order to examine this aspect, the following simulation was carried out.

5.5 Effect of a Shortened Vapor Blanket Stage To examine the effect of the vapor blanket stage curve, simulations were carried out after reducing the experimental data set so that only 3 seconds of the vapor blanket stage remained. The results are shown in Figures 5.23 to 5.25. The temperature profiles match very well. The heat transfer coefficient profile is not very smooth (23 intervals were used), but the values are reasonable. There is an improvement in the heat flux curve in that the heat flux history shows a consistent decrease after the first 3 seconds, and fluctuations are minimal.

Chapter 6 Discussion and Conclusions

6.1 Conclusion An inverse heat transfer algorithm was developed to find the surface heat flux in a quenching process. The algorithm uses an analytical solution to calculate the surface heat flux as a cubic spline containing a variable number of intervals, each of which is spanned by a cubic polynomial. The algorithm was then tested with experimental data. The experimental data consisted of 12,000 data points for 120 seconds of quenching. This data was used to determine the surface heat flux history. It was shown that a single polynomial provides reasonably accurate results for simple quenching histories, but is not able to match temperature jumps very accurately.

A piecewise continuous polynomial, such as a cubic spline, can represent any arbitrary heat flux quite accurately. Therefore a cubic spline was selected for the IHCP algorithm. It was observed that the cubic spline produces better match to the temperature history, and more accurate heat fluxes over a wider time span. However, even the cubic spline produces temperature and heat flux fluctuations that are not physically realistic. Increasing the number of time intervals can reduce these fluctuations. However, the basic problem is the acute sensitivity of the heat flux to the input data. The IHCP

problem tends to be highly unstable; consequently more stable methods have to be developed to reduce the fluctuations in the inverse solution. It may be possible that using piecewise continuous straight lines instead of cubic polynomials can improve the stability of the solution. When the heat flux curve is assumed to be a cubic spline, the curve is forced to be smooth because of the requirement that the first derivative is continuous at the interface between the intervals.

This

requirement tends to produce variations in the heat flux curve that are similar to sinusoidal curves. A curve made of piecewise continuous straight lines may produce a more stable, although not a highly accurate solution. It is also quite likely that the cubic polynomial estimation for the heat flux leads to a large number of degrees of freedom in the solution, because of which the solution curve may not converge to the shape that is physically more plausible. To improve the solution, more constraints may be needed. A constraint that could be applied is that the heat flux curve must have a negative or zero slopes at all times. This constraint was not applied in the IMSL code because the results from the code did not satisfy the first constraint that the heat flux be negative. The IMSL code did not produce satisfactory results; but this is not surprising. It should be noted that constrained optimization is a difficult problem, and global minimum is often not achieved. The primary advantage of this algorithm is that it appears to produce very fast results as compared to other solution schemes. This algorithm, if improved further, can be used as the initial guess to the numerical solution methods used in multi-dimensional solutions. It should be noted that multi-dimensional inverse problems sometimes do not

converge, or require hours of computational time because the solution steps are very sensitive to the input data, which includes the initial guess. A good initial guess in a complex multi-dimensional problem can reduce the solution time considerably.

6.2 Future Work Since this method is applicable to many functional representations of heat flux, future work should focus on using the inverse algorithm with a piecewise continuous polynomial of different degrees, including the straight line representation. Future work could also be extended to two- and three-dimensional analytical models and their experimental verifications. These models can then be integrated with finite element and finite difference inverse codes. The selection of a proper minimization/optimization code should also be investigated.

The IMSL optimization routine used for this study did not perform

satisfactorily. A more flexible routine that will allow more constraints may perform better for the IHCP problem.

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Appendix

....................................................................... / * This program is used to solve the inverse heat conduction / * problem. / * Written by Rong Zhong .......................................................................

*/ */ */

#include cstdio.h> #include cmath.h> #include