A novel method for estimating the distribution of convective heat flux in ducts: Gaussian Filtered Singular Value Decomposition

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1 A novel method for estimating the distribution of convective heat flux in ducts: Gaussian Filtered Singular Value Decomposition F Bozzoli 1,2, L Cattani 1,2, A Mocerino 1, S Rainieri 1,2, F S V Bazán 3 1 Department of Industrial Engineering, University of Parma, Parma, Italy 2 SIEIA.PARMA Interdepartmental Centre, University of Parma, Parma, Italy 3 Department of Mathematics, Federal University of Santa Catarina, Florianópolis/SC, Brazil Corresponding author: fabio.bozzoli@unipr.it Abstract. A new regularization approach for the unconstrained linear least square problem is proposed and assessed. his approach improves the classical methods based on the singular value decomposition approach by employing the Gaussian filter, a kind of filter which has been proved to be effective in several application related to noise suppression. In particular, as benchmark, it is hereby considered the estimation of the heat flux at the internal wall surface in a forced convection problem in ducts, by solving the inverse heat conduction problem in the solid wall only using temperature distribution available at the exterior boundary. Results on both synthetic and experimental data are reported with the aim of discussing the effectiveness of the proposed method in comparison to other similar solution approaches. 1. Introduction Singular value decomposition (SVD), in the truncated formulation [1], is considered one of the most popular method for the regularization of ill-posed linear least square problems. Although this technique is very effective, some Authors have proposed improvements in order to overcome some of its acknowledged limitations. In particular, in the classical approach, the truncation of the SVD expansion is not very conservative when the signal to be estimated is supposed to be a wide-spectrum one. For this reason, it has been proposed a modified approach called damped SVD which damps the small singular values instead of neglecting them, [2]. It means that, while the truncated SVD filters the singular value series by a sharp ideal high-pass filter, the damped SVD smooths the singular value series by a first-order filter. his work proposes and assesses a new regularization approach, that refines the damped SVD method by employing the Gaussian filter, a kind of filter which has been proved to be effective in several application related to noise suppression with regards to inverse problems [3]. In particular, as benchmark, it is hereby considered the estimation of the heat flux at the internal wall surface in a forced convection problem in ducts, by solving the inverse heat conduction problem in the solid wall using only temperature distribution available at the exterior boundary [4]. Results developed on both synthetic and experimental data are reported to discuss the effectiveness of the proposed method in comparison to other similar solution approaches.

2 2. Regularization methods his paper deals with methods for solving the unconstrained linear least square problem: min x b Ax 2, A R m n, m n (1) It is well known that finding the minimizer is equivalent to solving the square n n system: A Ax = A b (2) Furthermore, when the columns of A are linearly independent, it turns out that A A is invertible, and so x is unique and given by x = (A A) 1 b (3) In the majority of the practical cases, A A is not invertible and the solution can be found by the Singular Value Decomposition approach (SVD): the matrix A (m n) can be decomposed it into three matrices U, Σ and V: A = UΣV = n i=1 u i σ i v i (4) where U=(u 1,u 2,,u n). and V=(v 1,v 2,,v n) are matrices with orthogonal columns, and where the matrix Σ has diagonal form: Σ = diag(σ 1, σ 2,, σ n ) (5) he matrix Σ has non-negative diagonal elements, called the singular values of A, appearing in the following non-increasing order: σ 1 σ 2 σ n (6) Based on SVD, the solution to the least-squares problem (1) is: x V 1 U b u b n i vi i1 i (7) It can be observed from Eq.(7) that those terms with small singular value σ i become large. Since the noise level in b is generally greater than the smallest non-zero singular values, it turns out that the problem ill-posed in the sense that a small perturbation of b leads to a large perturbation of the solution. An interesting and important aspect of discrete ill-posed problems is that ill-conditioning of the problem doesn t mean that a meaningful approximated solution cannot be computed. Rather, the ill-conditioning implies that standard methods in numerical linear algebra are unsuccessful, more sophisticated methods must be applied. he regularization method proposed in the present paper, named Gaussian Filtered Singular Value Decomposition (), is based on SVD and on the idea that a filter factor f i should be added in Eq.(7) in order to filter out the noise corresponding to the small singular values. he solution is then expressed by: x n i1 f i, ui b v i i (8) In particular, a Gaussian function is adopted. Gaussian filter is widely adopted in the enhancing of image quality within graphics software and its property to smooth noisy data has been experimented [3,5].

3 he filter factor assumes the following expression: 1 ( ) f i = e σ i 2 2 λ 2 (9) where λ is the regularization parameter that should be chosen adequately. he application of a Gaussian factor has the effect of reducing large 1/σ i values in order to overcome the ill-posedness of the problem. he proposed approach is compared to other three approaches that share the idea of smoothing the matrix Σ obtained by SVD: runcated Singular Value Decomposition (SVD) method, Damped Singular Value Decomposition (DSVD) method and ikhonov Regularization (R) method. In particular SVD can be reduced to Eq.(8) with a step function as filter factor: f i = {, σ i > λ; 1, σ i λ. (1) where λ is the regularization parameter. Analogously, the Damped Singular Value Decomposition (DSVD) method use the filter factors defined as follows: f i = σ i σ i +λ ikhonov Regularization (R) method is usually formulated in a different way but, as shown in [6,7], when its zero derivative formulation is considered, it could be easily expressed by Eq.(8) with filter factors defined as follows: f i = σ i 2 σ i 2 +λ 2 (12) he trend of the four considered filter functions are reported in Fig. 1., where the cut-off value σ c is introduced, according to the classical definition; σ c is the value at which f is equal to his figure highlights that the four filter functions defined in Eqs.(9-12) have a similar behavior but totally different shapes: SVD is the steepest one while DSVD is the most gradual. (11) 1 DSVD R SVD f / c Figure 1. Filter functions for the four considered approaches. It is well known that the effectiveness of all regularization approaches strongly depends on the choice of a proper value of the regularization parameter [8,9]. In the present analysis, to make the comparison

between the considered regularization techniques more straightforward, the criterion provided by the discrepancy principle, originally formulated by Morozov [1], was adopted for all the techniques.

4 between the considered regularization techniques more straightforward, the criterion provided by the discrepancy principle, originally formulated by Morozov [1], was adopted for all the techniques. his principle, for the four approaches considered, suggests computing λ in such a way to satisfy the non linear equation: b A n f ui b v i, i i1 i 2 b N (13) where N is the size of the vector b and η b is the standard deviation of the measurement error in b. 3. est case he comparison of the four above described techniques (i.e.,, SVD, DSVD, R) was performed considering a well-known inverse heat conduction problem: the estimation of the local convective heat flux density inside a duct, heated by Joule effect, from the temperature distribution acquired on the external wall surface of the pipe. Figure 2. Geometrical domain with coordinate system. In the 2-D solid domain (sketched in Fig. 2), the steady state energy balance equation is expressed in the form: k 2 + g = (14) where g is the heat generated per unit volume and k is the tube wall thermal conductivity. he following two boundary conditions completed the energy balance equation: k r = ( env) R env (15) that is applied on surface S 1 and where R env is the overall heat transfer resistance between the tube wall and the surrounding environment with temperature env ; k = q (16) r

5 that is applied on surface S 2 and where q is the local convective heat flux at the fluid-internal wall interface, assumed to be varying with the angular coordinate and taken positive when it is released by the wall into the fluid. In the discrete domain, the direct problem, which is linear with respect to the heat flux q, can be described as follows: q Xq (17) where is the vector of the discrete temperature data at the external pipe surface, q is the heat flux vector at the fluid-internal wall interface, q= is a constant term and X is the sensitivity matrix. he inverse problem becomes: min q Y q= Xq 2 (18) where Y is the measured temperature distribution on the external tube s wall. his equation is equal to the problem expressed by Eq.(1). 4. Numerical comparison he comparison of the proposed techniques with the others described previously was first performed, within the Matlab environment, by adopting synthetic data. he q= and X terms of Eq. (17) were calculated by the finite element method implemented in Comsol Multiphysics. By imposing a known distribution of q, a synthetic temperature distribution on the external wall surface was then obtained. he physical and geometrical parameters used in this work correspond to a stainless steel tube with an internal radius of 7 mm and an external radius of 8 mm. For the internal convective heat flux, two meaningful distributions, suggested by the data of Bozzoli et al. [4], were adopted: 2 q a ( / 1) 2a ( / 1) a / 2 (19.a) b, α π/3 q (19.b) b/2, α π/3 In particular a=36 W/m 2 and b= W/m 2 were taken in order to have two distributions, compared in Fig. 3, with the same average value but totally different profiles: the first one is a typical parabolic function while the second is a step function. It is well known that effectiveness of inverse problem solution methods depends on the kind of function that has to be estimated, for this reason testing the proposed approach on different distributions is fundamental for performing a robust comparison.

6 Eq.(19.b) Eq.(19.a) (rad) Figure 3. est convective heat flux distributions. hen the synthetic temperature distribution Y on the external wall surface, deliberately spoiled by random noise, was used as the input data of the inverse problem. In particular, a white noise characterized by a standard deviation μ Y ranging from.1 K - 5 K was considered. In order to quantify the effectiveness of the two approaches at different signal to noise level an error analysis could be performed by plotting the global relative estimation error, defined as follows: E = (q) restored (q) exact 2 q exact 2 (2) versus the standard deviation of the measurement error. Since the added noise depends intrinsically on the random sequence generated by Matlab, the estimation procedure was repeated for 1 different random sequences of noise added to the simulated measurements and an averaged value E avg was calculated for each noise level. Eq.(19.a) - parabolic distribution Eq.(19.b) - step distribution E avg (%) DSVD SVD R E avg (%) DSVD SVD R Y Y Figure 4. Average estimation error of the four consider techniques at different noise level, for the two considered numerical test case.

7 his comparison, reported in Fig. 4, highlights that, SVD and R show similar performance while DSVD is unable to estimate the unknown heat flux distribution, also at low noise levels. Moreover, when a parabolic heat flux distribution is considered, and R worked slightly better than SVD. In order to improve the analysis on these three approaches, in Fig. 5 the local heat flux distributions restored by, SVD and R are compared with the exact one for two representative noise levels. SVD, for the parabolic distribution, overestimated the maximum value when the noise level is high. For the step distribution, SVD strongly suffers of ringing artifacts at low noise level and of excessive smoothing at high noise level. Although M and show a similar behavior in terms of estimation error, if step distribution with high noise level is considered, GSFD is more effective in locating the plateau while R catches better the background. Y =.5 K exact SVD R Y =.5 K exact SVD R (rad) (rad) Y = 1 K exact SVD R Y = 1 K exact SVD R (rad) (rad) Figure 5. Exact and reconstructed heat flux distribution for the two test cases with different noise levels.

8 4. Experimental comparison he proposed estimation technique was compared to the others more promising ones (i.e., SVD, R) through their application to a set of experimental data obtained in [4]: it was considered the forced convection problem within a stainless steel coiled tube under the prescribed condition of uniform heating generated by Joule effect in the wall. Ethylene Glycol was used as working fluid and laminar flow regime in coiled tubes was investigated. More details on the experimental facilities and procedures can be found in [4]. In order to apply the four estimation procedures, the noise level in the acquired data has to be estimated; this task was performed by measuring the surface temperature distribution while maintaining the coil wall under isothermal conditions. For a representative Reynolds number value, the distributions of the convective heat transfer coefficient restored by R, SVD and are compared in Fig. 6. SVD R (rad) Figure 6. Exact and reconstructed heat flux distribution for the experimental case. he comparison between the three restored distributions of the convective heat flux underlines that, for the case here investigated, and R give equivalent results while SVD suffers of ringing artifacts. 5. Conclusions A new regularization approach for the unconstrained linear least square problem was proposed and assessed. In particular, this approach, named Gaussian Filtered Singular Value Decomposition (), is based on the singular value decomposition approach smoothed by Gaussian filter. was compared to other three approaches that share the idea of inverting SVD: runcated Singular Value Decomposition (SVD) method, Damped Singular Value Decomposition (DSVD) method and ikhonov Regularization (R) method. As benchmark, it was considered the estimation of the heat flux at the internal wall surface in a forced convection problem in ducts. Results, on both synthetic and experimental data, highlighted the goodness of in solving this kind of inverse problem and the limits of SVD that, although it is probably the most widely used approach, could suffer of ringing artifacts.

9 6. References [1] Hanson, R. J. (1971). A numerical method for solving Fredholm integral equations of the first kind using singular values. SIAM Journal on Numerical Analysis, 8(3), [2] Ekstrom, M. P., & Rhoads, R. L. (1974). On the application of eigenvector expansions to numerical deconvolution. Journal of Computational Physics, 14(4), [3] Bozzoli, F., Pagliarini, G., & Rainieri, S. (213). Experimental validation of the filtering technique approach applied to the restoration of the heat source field. Experimental hermal and Fluid Science, 44, [4] Bozzoli, F., Cattani, L., Rainieri, S., Bazán, F. S. V., & Borges, L. S. (214). Estimation of the local heat-transfer coefficient in the laminar flow regime in coiled tubes by the ikhonov regularisation method. International Journal of Heat and Mass ransfer, 72, [5] Murio, D.A. (1993). he Mollification Method and the Numerical Solution of Ill-Posed Problems, John Wiley and Sons, New York [6] Hansen, P. C. (1987). he truncatedsvd as a method for regularization. BI Numerical Mathematics, 27(4), [7] Wei,., Hon, Y. C., & Ling, L. (27). Method of fundamental solutions with regularization techniques for Cauchy problems of elliptic operators. Engineering Analysis with Boundary Elements, 31(4), [8] Bazan, F.S.V., & Borges, L.S. (21). GKB-FP: an algorithm for large-scale discrete ill-posed problems. Bit Numerical Mathematics, 5(3), [9] Beck, J.V., Balckwell, B., Clair, C.R.St. (1985). Inverse Heat Conduction Ill-posed Problems, Wiley-Interscience, New York, USA. [1] ikhonov, A.N., Arsenin V.Y. (1977). Solution of Ill-Posed Problems, Winston & Sons, Washington DC, USA.

Passive techniques for the enhancement of convective heat transfer in single phase duct flow

Journal of Physics: Conference Series OPEN ACCESS Passive techniques for the enhancement of convective heat transfer in single phase duct flow To cite this article: S Rainieri et al 2014 J. Phys.: Conf.