ye n Y

Inverse heat transfer problem

Multi-layered tube dri w e w tio g e adv of a pr of the portan ounda els to solve the direct heat conduction problems. Instead, estimat(IHCP).

Analytical or numerical studies have been conducted to solve the inverse heat conduction problems. Analytic solutions were derived by using the integral or Laplace transform technique [1–5]. The analytic solutions are very efficient in the view of computation and are of fundamental importance for investigating basic properties, but are limited to simple geometries. Numerical information. In addition, if the physical system can be modeled, orithm is efficient es. rse heat c tion analysis. In previous studies, the system modeling wa ducted using the finite element method [19–22]. The element method expresses approximate functions from un variables and determines small element values using the weighted residual method. The finite element method is suitable for solving problems with complex boundaries, but has disadvantage that numerical cost is high because of large amount of computation.

In the meantime, the finite differential method expresses derivative terms using Taylor series and thus has advantage that numerical cost is relatively low. Therefore, in an effort to reduce the ⇑ Corresponding author. Tel.: +82 2 2220 0422; fax: +82 2 2220 2299.

E-mail address: ysjnuri@hanyang.ac.kr (S.-J. Yook).

International Journal of Heat and Mass Transfer 89 (2015) 1016–1023

Contents lists availab

H .eing unknown temperature and heat flux on one boundary of the system from known conditions on other boundaries of the system can be performed by solving the inverse heat conduction problem the Kalman filter and recursive least-squares alg for solving the problems with complex geometri

The system modeling is required for the invehttp://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.06.009 0017-9310/ 2015 Elsevier Ltd. All rights reserved.onducs confinite knownfied, then the temperature distribution in the tube wall can be obtained by solving the direct heat conduction problem (DHCP).

However, if the measurement of physical parameters such as temperature and heat flux on one of the boundaries of a system is not possible as in the case of propellant combustion in a tube, then the system consisting of the tube cannot be analyzed by utilizing modposed based on the concept of the Kalman filter technique and the least-squares estimation of recursive processing [15–18]. The

Kalman filter is a set of mathematical equations providing an efficient computational solution of the least-squares method. The

Kalman filter technique is simple and efficient, takes explicit measurement uncertainty incrementally, and can consider a priori1. Introduction

Heating of a tube can induce som performance. Especially, combustion tor for melt, crack, erosion, and wear and temperatures. Therefore, it is im distribution in the tube wall. If all berse effects on system opellant is a major factubes at high pressures t to know temperature ry conditions are specimethods including the sequential estimation technique were developed [6–9]. The conjugated gradient method was shown to be a straightforward and powerful iterative technique for solving linear and nonlinear inverse problems of parameter estimation [10–14]. Due to the necessity to estimate the history of unknown properties in real time in engineering applications, the recursive input estimation algorithm of digital estimation theory was pro-Thermal resistance network method

Kalman filterInverse heat transfer analysis of multi-la resistance network and Kalman filter

Jung-Hun Noh a, Won-Geun Kim a, Ki-Up Cha b, Se-Ji a School of Mechanical Engineering, Hanyang University, Seoul 133-791, South Korea b5-1, Agency for Defense Development, Daejeon 305-600, South Korea a r t i c l e i n f o

Article history:

Received 9 April 2015

Received in revised form 2 June 2015

Accepted 3 June 2015

Available online 18 June 2015

Keywords: a b s t r a c t

In this study, a hollow cylin resistance network method on the inner wall of the tub the recursive input estima

Assuming various operatin evaluated.

International Journal of journal homepage: wwwred tube using thermal ook a,⇑ cal tube with a coating layer on its inner wall was considered. The thermal as employed to solve the heat conduction in the tube. Unknown heat flux as estimated from measured temperature on the outer wall of the tube by n algorithm consisting of the Kalman filter and real-time least squares. conditions, the performance of the model developed in this study was 2015 Elsevier Ltd. All rights reserved. le at ScienceDirect eat and Mass Transfer l sevier .com/locate / i jhmt

M sensitivity matrix eat[M] conductance matrix n element number for steel layer

O order of error

P filter’s error covariance matrix

Pb error covariance matrix q heat flux, W/m2 q^ estimated input vector, W/m2

Q process noise covariance r radius, m

R measurement noise covariance s innovation covariance t time, sNomenclature

A surface area, m2

B sensitivity matrix [C] capacitance matrix

Cp specific heat, J/kg K

E total element number {F} thermal load vector h convectional heat transfer coefficient, W/m2 s

H measurement matrix

I identity matrix k thermal conductivity, W/m K

K Kalman gain

Kb steady-state correction gain m element number for chrome layer

J.-H. Noh et al. / International Journal of Hamount of computation, the thermal resistance network method, which is based on the energy balance for control volumes, was used in heat transfer analysis [23,24].

In this study, a numerical model was developed to predict unknown heat flux on the inner wall of a tube from known temperature on the outer wall of the tube. The thermal resistance network method was employed to solve heat conduction in the tube. Using the recursive input estimation algorithm consisting of the Kalman filter and the real-time least squares, the unknown heat flux was estimated. Then, the validity of the model was evaluated by assuming various operating conditions. 2. Model description 2.1. Multi-layered tube problem