EXPERIMENTAL ESTIMATION OF A HEAT FLUX IMPOSED BY …...dependent transient heat flux, imposed on the top surface of a thin plate by a laser diode. Infrared thermography measurements

3rd Thermal and Fluids Engineering Conference (TFEC) March 4–7, 2018

Fort Lauderdale, FL, USA TFEC-2018-20899

*Corresponding Author: [email protected]

1

EXPERIMENTAL ESTIMATION OF A HEAT FLUX IMPOSED BY A

LASER DIODE WITH THE STEADY STATE KALMAN FILTER

Henrique M. da Fonseca1*, Cesar C. Pacheco1, Helcio R.B. Orlande1, Olivier Fudym2

, George S. Dulikravich3

1 PEM/COPPE/UFRJ – Department of Mechanical Engineering, Cid. Universitaria, Cx. Postal: 68503, Rio

de Janeiro, RJ, 21941-972, Brazil 2 CNRS Office for Brazil and the South Cone, Avenida Presidente Antônio Carlos, 58 / Sala 416, Rio de

Janeiro 20020-010, Brazil 3MAIDROC Laboratory, Department of Mechanical and Materials Engineering, Florida International

University, Miami, FL, USA

ABSTRACT

In this paper, we deal with the solution of an inverse problem involving the real-time identification of a position-dependent transient heat flux, imposed on the top surface of a thin plate by a laser diode. Infrared thermography measurements of the temperature of the plate, at the same surface where the heat flux is imposed, are used in the inverse analysis. The physical problem involves two-dimensional transient heat conduction in a plate with constant thermophysical properties, initially at a uniform temperature. A partial lumping across the plate's thickness is used for the mathematical formulation of the physical problem. We use the Steady State Kalman Filter (SSKF) for the solution of the inverse problem in real-time. The Steady State Kalman Filter results are compared to those obtained with the Classical Kalman Filter (CKF), in experiments where a laser diode at the 830 nm wavelength is used to heat the samples. KEY WORDS: Steady State Kalman Filter, Infrared Thermography, Inverse Problems

1. INTRODUCTION The estimation of parameters or functions appearing in the formulation of heat transfer problems can be very challenging when measurement techniques like infrared cameras are used, because they can provide experimental data with high frequency and large spatial resolution. Different techniques have been developed/applied by the authors for dealing with the huge amount of data available in such cases. In reference [1], the nodal strategy was used in conjunction with a Markov Chain Monte Carlo (MCMC) method for the estimation of the spatially distributed thermal diffusivity, in a one-dimensional heat conduction problem. This approach was then applied for the simultaneous estimation of the spatially distributed thermal diffusivity and heat flux in a two-dimensional heat conduction problem [2]. In reference [3], an experimental heat flux was estimated, validating the theoretical development of our previous work [2]. In reference [4], the inverse problem of estimating a transient heat flux that also varies spatially, in a two-dimensional linear heat conduction problem, was solved. More recently, we presented the estimation of the transient heat flux in real-time, by using an asymptotic version of the Kalman filter with simulated transient temperature measurements [5]. In this work, experimental temperature data collected with an infrared camera is used for the real-time estimation of a transient spatially varying heat flux [1-5]. The main objective of this paper is to validate the Steady State Kalman Filter with actual temperature measurements, for the estimation of the surface heat flux

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in controlled experiments. The results obtained with the Steady State Kalman Filter are compared to those obtained with the Classical Kalman Filter.

2. PHYSICAL PROBLEM AND MATHEMATICAL FORMULATION The physical problem examined in this paper involves two-dimensional transient heat conduction in a plate with constant thermophysical properties, initially at a uniform temperature T0. The lateral and bottom boundaries of the plate are considered insulated. The top boundary is subjected to a transient spatially varying heat flux and to convective heat losses, as presented by Fig. 1. The mathematical formulation for this problem is given in terms of the mean temperature across plate, obtained by partial lumping [6] in the z direction as [1-5]:

( ) ( , , )T T T h q x y tC k k T Tt x x y y e e∞

∂ ∂ ∂ ∂ ∂= + − − + ∂ ∂ ∂ ∂ ∂

in 0 < x < a, 0 < y < b, for t > 0 (1)

0Tx

∂=

∂ at x = 0 and x = a, for t > 0 (2)

0Ty

∂=

∂ at y = 0 and y = b, for t > 0 (3)

0T T= for t = 0, in 0 < x < a and 0 < y < b (4) Therefore, the mathematical model presented by Equations (1)-(4) is a simplification of a more complex 3-D non-linear formulation with temperature dependent thermophysical properties. The average temperature in the z direction is defined as:

0

1( , , ) ( , , , )e

z

T x y t T x y z t dze =

= ∫ (5)

In order to write the linearized lumped formulation above, the thermophysical properties were supposed available at a reference temperature [5]. In the direct problem associated with the mathematical formulation given by Equations (1)-(4), the thickness of the medium, thermophysical properties, initial temperature, heat transfer coefficient and the spatial and time variation of the wall heat flux are known. The objective of the direct problem is then to determine the transient temperature variation within the medium.

Fig. 1 Physical problem.

a x

z

y

e

b

Medium with constant propertiesC and k h

h

h

h

h

Ambient temperature Tinf

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3. INVERSE PROBLEM

The objective of this work is to estimate the heat flux q(x,y,t) at z = e. Temperature measurements taken at the same surface where the heat flux is imposed are used for the solution of the inverse problem. This nonstationary inverse problem is solved within the Bayesian framework [7,8] with a filtering approach. For the solution of inverse problems related to the partial lumped system given by Equations (1)-(4), the Classical Kalman Filter (CKF) has been applied together with the Nodal Strategy in references [1-4,9]. The main idea of this strategy is to rewrite the forward model in a non-conservative form and solve the resulting system of equations using a predictive error model, instead of an output error model. In the predictive model, the experimental measurements and their related uncertainties are used for the calculation of the sensitivity coefficients and the inverse problem then becomes linear [7,8]. Hence, the solution of the inverse problem of estimation of several model parameters could be performed in small computational times, even with the huge amount of data generated by infrared cameras [10]. In the Bayesian Filter approach, the estimation of the nonstationary function is performed with the definition of evolution and observation models. Here, these models are assumed as linear, with additive and Gaussian uncertainties, respectively in the forms [11-15]: 1 1k k k k+ += +x F x v (6) 1 1 1 1k k k k+ + + += +z H x n (7) where the subscripts denote time instants, while F and H are known matrices for the linear evolution of the state vector x and of the observations z, respectively. By assuming that the noises v and n have zero means and covariance matrices Q and R, respectively, the Kalman filter gives the optimal solution of this state estimation problem. In the Kalman filter, the prediction and update steps are given by [8,11,13,16]: Prediction: 1 1ˆk k k

−+ +=x F x (8)

T1 1 1 1k k k k k

−+ + + += +E F E F Q (9)

Update:

( ) 1T T1 1 1 1 1 1 1k k k k k k k

−− −+ + + + + + += +K E H H E H R (10)

1 1 1 1 1 1ˆ ( )k k k k k k− −

+ + + + + += + −x x K z H x (11)

( )1 1 1 1k k k k−

+ + + +=E I - K H E (12) The matrix K is called Kalman’s gain matrix, which is a weighting factor for the difference between the measurements 1k+z and the measurement model 1 1k k

−+ +H x (see Equations 7 and 11). For the present inverse

problem, the vector of state variables x is composed by the temperatures and heat fluxes at each point of the discretization mesh, given by the vectors T and P, respectively, that is:

kk

k

=

Tx

P (13)

with covariance matrix Qk given by [4]

kT

k kP

=

Q 0Q

0 Q (14)

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where in Eq (14) 0 is a matrix of zeros. The matrices kTQ and k

PQ can be found in [4,5]. The vector n is Gaussian and with a constant covariance matrix R, which results from the calibration of the infrared camera used to obtain the measurements. The matrix F results from the finite volume discretization of equations (1) – (4), which constitutes the evolution model for temperature, and by a random walk Gaussian evolution model for the heat flux. This random walk model is given by [4,5]: 1

1k

k k P+

+ +P = P v (15) where ~ ( , )k k

P PNv 0 Q . The measured temperatures are assumed available at the discretization nodes that coincide with the infrared camera pixels, so that the matrix H is written as: [ ]k =H I 0 (16) The measurement uncertainty n is Gaussian and with a constant covariance matrix R, which results from the calibration of the infrared camera used to obtain the measurements. 3.1 Steady State Kalman Filter The application of the Kalman Filter as the optimal solution for state estimation problems relies on strong hypotheses, like the linearity of the evolution and observation models, and additive and Gaussian uncertainties. The evolution and observation matrices, as well as the noise covariance matrices, can be functions of time. However, for time-invariant systems, like the present one, these matrices that define the evolution and observation models are constant, that is,

Fk = F , Hk

= H , Qk = Q and Rk = R (17)

Important quantities for the implementation of the Kalman Filter, like the Kalman gain matrix, Kk, as well the prior and posterior covariance matrices, 1k

−+E and 1k+E , respectively, are functions of Fk, Hk, Qk and Rk and of

the initial posterior covariance error matrix, 0kE . Therefore, Kk, 1k

−+E and 1k+E evolve in time, such as the state

variables, but might become time-invariant after a finite number of steps depending on the evolution and observation models. Thus, one can calculate these matrices at the steady state, aiming to use them as an approximation during the whole recursive estimation, in order to reduce the associated computational time. This technique is known as the Steady-State Kalman Filter (SSKF) [16], where steady-state refers to the time-invariant matrices Kk, 1k

−+E and 1k+E . Although this filter is no longer optimal [12], it is still very attractive

because of the clear reduction of the number of operations in comparison with the Classical Kalman Filter. In the SSKF method, we thus approximate [16]:

Kk ≈ K∞ and 1 1k k−+ + ∞= ≈E E E (18)

The equations for the SSKF are obtained by substituting (17) and (18) into the equations of the Classical Kalman Filter, thus resulting in:

( ) 1T T T T−

∞ ∞ ∞ ∞ ∞= − + +P FP F FP H HP H R HP F Q (19)

( ) 1T T −

∞ ∞ ∞= +K P H HP H R (20)

1 1ˆ ˆ( )k k k+ ∞ ∞ += − +x I K H Fx K z (21) The major advantage of the Steady-State Kalman Filter is that Equations (19) and (20) can be calculated offline, that is, before any estimation takes place, while only Equation (21) needs to be calculated recursively. This

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represents a large reduction of computational times in comparison with the original Kalman Filter. Equation (19) is a Discrete Algebraic Riccati Equation (DARE) [16], which can be readily solved by using several available packages [15]. We note that the nodal strategy described above [1-4,9] was applied only for the Classical Kalman Filter.

4. EXPERIMENT In this work, the temperature measurements were obtained with a 560M Titanium infrared camera, manufactured by Cedip Infrared Systems. The electrical signal given by the camera is a function of the photons received by the sensor of the camera. This electrical signal is converted to a Digital Level (DL). The transfer functions for photons/volts and volts/DL are linear increasing. The camera’s detector has 640 x 512 pixels and is sensitive between 1.5 µm and 5.0 µm. A sample made of epoxy, heated by a laser diode, was used in this work. The resulting temperature rise, due to the laser flux imposed on the top surface of the sample, was recorded with the infrared camera. The laser diode is composed by a single emitter semiconductor laser from Spectra-Physics, model SFB 060, connected to a fiber optic cable. A NewPort F-H10 collimator was connected to the other end of the fiber optic, resulting in a Gaussian beam with 6 mm diameter. The laser diode was controlled by a 525B NewPort controller. The laser emits at the wavelength of 830 nm, with a maximum power of 0.7 W that gives a heat flux of approximately 25 kW/m2. The absorption of this energy depends on the radiative properties of the sample at the laser´s wavelength. The sample has known thermophysical properties, obtained experimentally by the Flash method, with the Netzsch LFA 447/1 Nanoflash apparatus. For the epoxi used in this work, three tests were made at 20oC. The mean diffusivity retrieved with the LFA 447/1 was 0.24 mm2/s, with a standard deviation of 0.001mm2/s. The transient heat flux is obtained by connecting the laser driver to a function generator. The current of the laser is controlled by the function generator and the output power is a function of the applied current. An Agilent 33220a function generator is used to obtain the desired shape of the waves used for the laser, as illustrated by Fig. 2.

Fig. 2 Experiment.

5. RESULTS AND DISCUSSIONS

In order to challenge the two approaches used for the estimation of the boundary heat flux, either with the Classical Kalman Filter or with the Steady State Kalman Filter, two time variations of the heat flux imposed by the laser diode were used, in the form of a square pulse and a sinusoidal wave, both with frequencies of 1 Hz. For the square pulse, its duration was chosen as 500 ms and the current in the laser diode controller was set to 900 mA. For the sinusoidal wave, a current of 1050 mA was used. In both cases, the amplitude of the signal from the function generator was in the 0 to 5 V range.

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6.1 Sinusoidal Heat Flux For the sinusoidal heat flux, the discretization of the plate was made with 32 nodes, where the values of the local parameters qi,j were estimated. The pixel in the image was a square of side 78.1 µm and the region observed in the sample was a square of side 0.00249 m. The experimental duration was 3.02s, and the images were acquired with a time step Δt = 0.005s, which satisfied the explicit finite difference stability criterion. Figure 3 shows the temperature field at time step 30 (which represents the physical time of 0.15s) and the location of some particular pixels chosen for plotting the results, at positions (i = 7, j = 19), (i = 17, j = 27), (i = 9, j = 28) and (i = 22, j = 6), which correspond to (x = 0.00054 m, y = 0.00150 m), (x = 0.00130 m, y = 0.00210 m), (x = 0.00070 m, y = 0.00220 m) and (x = 0.00170 m, y = 0.00047 m), respectively.

Fig. 3 Temperature distribution at time step 30 and location of selected pixels. Figure 4 presents the comparison between the measured and estimated temperatures at the points shown by Fig. 3. Estimated temperatures were obtained with the Classical and with the Steady State Kalman filters. The 99% confidence intervals are also shown in this figure, where CI SSKF stands for confidence interval for the SSKF estimation and CI CKF stands for confidence interval for the CKF estimation. Since the standard deviations of the experimental errors were of the order of 0.006 oC, the agreement between estimated temperatures and experimental data is excellent.

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Fig. 4 Sinusoidal heat flux: Measured and estimated temperatures at selected pixels. The temperature residuals, that is the difference between estimated and measured temperatures, are presented by figure 5 for the results obtained with both CKF and SSKF. Fig. 5 confirms the excellent agreement between the measured and estimated temperatures, although the residuals are correlated. Such correlation can be attributed to the filtering approach, where the instantaneous heat flux affects temperatures at later times, that is, the system response is lagged with respect to the system excitation. On the other hand, for the case of the sinusoidal heat flux, the residuals obtained with CKF and SSKF are quite similar. Therefore, the use of the asymptotic Kalman gain matrix does not affect the accuracy of the solution of the state estimation problem.

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Fig. 5 Residuals at selected pixels.

Figures 6 a-b present the spatial distribution of the estimated heat flux term at time step 30. It can be seen from these figures that the heat fluxes estimated with the Steady State Kalman Filter (Fig. 6-a) presented a spatial behavior closer to the experimental temperature distribution (Fig. 3) than the Classical Kalman Filter (Fig. 6-b). This is probably due to the fact that in the nodal strategy used for application of the Classical Kalman Filter the estimations are made pointwise for each pixel, independently of the estimations performed for the surrounding pixels.

(a)

(b)

Fig. 6 Estimated heat flux distribution at t = 0.15s: (a) Steady State Kalman Filter and (b) Classic Kalman Filter.

Figure 7 shows the time evolution of the estimated heat fluxes obtained with both the Classical and the Steady State Kalman filters. Figure 7 shows that the heat fluxes estimated with both methods are in excellent agreement.

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Also, the magnitude of the heat flux from the laser output (25 kW/m2) was closely retrieved. Negative heat fluxes observed between the heating periods can be attributed to additional heat losses not accounted for in the mathematical model.

Fig. 7 Estimated flux term for selected pixels.

6.1 Square pulse For the experiments with the square laser pulses, the region was also discretized with 32 internal nodes. The parameters qi,j were estimated for each of these nodes and time steps. Each pixel represents a square with side of 78.1 µm, and hence the dimensions of the cropped image of the plate was a square of side 0.00249 m. The final time was 4.4s and the chosen time step was 0.005 s. Figure 8 presents the temperature distribution at time step 30 (t = 0.15s) and selected points for the analysis of the results, that is, (i = 7, j = 19), (i = 17, j = 27), (i = 9, j = 28) and (i = 22, j = 6), which correspond to (x = 0.00054 m, y = 0.00150 m), (x = 0.00130 m, y = 0.00210 m), (x = 0.00070 m, y = 0.00220 m) and (x = 0.00170 m, y = 0.00047 m), respectively.

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Fig. 8 Temperature distribution at time step 30 and location of selected pixels

Figure 9 presents the comparison between the measured and estimated temperatures. Estimated temperatures were obtained with the Classical and the Steady State Kalman filters. The 99% confidence intervals are also shown in this figure. Such as for the sinusoidal heat flux, the agreement between estimated and measured temperatures is excellent; the residuals are very small but correlated (see Figure 10), due to the lag in the solution of the state estimation problem that is inherent to the Kalman filter.

Fig. 9 Square wave: Measured and estimated temperatures for selected pixels.

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Fig. 10 Residuals for selected pixels. Figures 11 a-b present the spatial distribution of the heat fluxes estimated at the 30th time step, which corresponds to the physical time of 0.15s. It can be seen in this figure that the estimated heat flux obtained with the SSKF (Figure 11a) is smoother and in better agreement with the temperature distribution shown by Fig. 8, than the heat flux estimated with the CKF (Fig. 11b), such as for the sinusoidal heat flux.

(a)

(b)

Fig. 11 Estimated heat flux distribution at t = 0.15s: (a) Steady State Kalman Filter and (b) Classic Kalman Filter..

Figure 12 shows the time evolutions of the heat fluxes estimated with both the Classical and the Steady State Kalman filters, at the selected positions presented by Fig. 8. Estimations obtained with both techniques are in excellent agreement, while the estimated magnitudes during the application of the heat flux are quite close to the laser output (25 kW/m2). Figure 12 also shows that there is a spatial variation of the applied heat flux, which is in

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accordance with the observed temperatures (see also 8 and Fig. 11). This is due to the position of the laser with respect to the heated plate.

Fig. 12 Square wave: Estimated heat flux term for the selected pixels.

The standard deviation used in the state evolution model, Eq. (16), for the heat flux estimation with the CKF was 60% of the heat flux estimated at the previous iteration, that is 1

,0.6k kq i jqσ + = and for the temperature evolution

model the standard deviation was obtained by the uncertainty propagation due to the uncertainties in the Jacobian matrix, the temperature at the previous time step and the vector of parameters Pk. In the case of the SSKF, both the temperature and heat flux evolution models were simulated with uncorrelated noise with constant standard deviations, given by 0.01 ºC and 100 W/m², respectively.

7. CONCLUSIONS The present work aimed at the experimental estimation of a transient heat flux imposed on the surface of a plate. The heat flux also varied spatially. The heat flux was estimated through the solution of a state estimation problem, by using two implementations of the Kalman filter, namely: (i) the Classical Kalman Filter together with a nodal strategy to speed up the estimation procedure, and (ii) the Steady State Kalman Filter. These techniques were compared for two periodic heat fluxes, with sinusoidal and step variations. The results obtained with both strategies for the solution of the state estimation problems exhibit quite small temperature residuals, while the transient variations of the estimated heat fluxes are in accordance with the sinusoidal and step functions imposed by the laser diode. The amplitude of the retrieved heat fluxes were also in excellent agreement with that from the output of the laser diode. Therefore, both implementations of the Kalman filter are appropriate and robust for the estimation of fast variations of the imposed heat flux, even for cases with spatial distributions over pixels of small size. On the other hand, the Steady State Kalman Filter presented a spatial behavior closer to the experimental temperature distribution than the Classical Kalman Filter. This is probably due to the fact that in the nodal strategy used for application of the Classical Kalman Filter the estimations are made pointwise for each pixel, independently of the estimations performed for the surrounding pixels. Moreover, the Steady State Kalman filter can be orders of magnitude faster than the Classical Kalman filter, since its major computational

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burden can be performed offline, thus reducing the number of operations from the order of N3 to the order of N2 , where N is the number of state variables.

ACKNOWLEDGMENTS The support provided by CNRS (France), CNPq (Brazil), CAPES (Brazil) and FAPERJ (Brazil – State of Rio de Janeiro) is greatly appreciated.

NOMENCLATURE C Volumetric heat capacity (J/m3K ) T Temperature (oC) t Time (s) x Space coordinate (m) y Space coordinate (m) z Space coordinate (m) k Thermal conductivity (W/(mK)) h Heat transfer coefficient (W/(m2K) e Thickness of the plate (m) q Heat flux (W/m2) a Dimension in x direction (m) b Dimension in y direction (m)

T0 Initial temperature (oC) Tinf Room temperature (oC) x State vector (-) F Evolution Matrix (-) z Observation vector (oC) H Observation matrix (-) n Measurement error (oC) E State prediction covariance matrix (-) Q State evolution covariance matrix (-) K Kalman gain matrix (-) R Measurement covariance matrix (oC2)

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EXPERIMENTAL ESTIMATION OF A HEAT FLUX IMPOSED BY …...dependent transient heat flux, imposed on the top surface of a thin plate by a laser diode. Infrared thermography measurements