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Abstract
The sequential estimation of surface heat flux from discrete and noisy data of surface temperature is an ill-posed problem. From Duhamel's theorem and Fourier's law and using a stabilization technique based on the sequential application of the ordinary least squares (SOLS), we obtain a relatively simple but effective method. As the SOLS method uses a least squares fit over r-future and r-past temperatures, this method can be compared with the well-known function specification method (FSM). FSM is a more general method, but the numerical validations considered in this study reveal that the SOLS method gives better results. It is also shown that SOLS cannot be guided by the residual principle and consequently cannot be used in a practical scope. The ultimate goal of this study is to obtain a reliable estimation of heat flux history in an on-line industrial process where the tuneable parameter should be a time-variable r-value and it must be automatically updated from the experimental data. This requires that (i) the corresponding SOLS algorithm must be rewritten in recursive form, (ii) the classical definition of the residual principle is rewritten in recursive form, and (iii) the estimates are obtained from a hybrid procedure based on SOLS and FSM.
Nomenclature
| Ap | = | amplitude of the perturbation |
| Bi | = | Biot number |
| cp | = | specific heat |
| C | = | constant |
| D | = | estimate of bias |
| F | = | dimensionless factor, Eq. (32c) |
| F | = | row vector of dimensionless factors, Eq. (32a) (1 × m) |
| Fred | = | reduced F vector, Eq. (34) (1 × n) |
| FSM | = | function specification method |
| g | = | primitive function, Eq. (19) |
| h | = | heat transfer coefficient |
| k | = | thermal conductivity |
| L | = | thickness of the slab |
| M | = | total number of time step |
| N | = | number of time step included in the perturbation period |
| n | = | reduced number of components included in the heat flux past history |
| p | = | perturbation function |
| q | = | dimensionless heat flux |
| qn | = | nominal value of heat flux |
| q | = | heat flux vector |
| = | estimated vector of heat flux history (m × 1) | |
| = | reduced vector of heat flux history (n × 1) | |
| SOLS | = | sequential ordinary least squares method |
| T | = | dimensionless temperature |
| T0 | = | initial condition |
| TS | = | surface temperature |
| T∞ | = | fluid temperature |
| r | = | number of future (or past) time steps |
| t | = | dimensionless time |
| S | = | estimate of total error |
| u | = | response to a unit step change of surface temperature |
| ux | = | spatial derivative of function u |
| x | = | dimensionless coordinate |
| Y | = | measured temperature |
| = | time derivative of surface temperature measurement | |
| R | = | residual sum |
| Z | = | lower triangular matrix of Eq. (24) |
| Z | = | coefficient of the matrix Z |
Greek Symbols
| α | = | thermal diffusivity |
| βm | = | mth eigenvalue, Eq. (12) |
| Δt | = | time step size |
| δ | = | small tolerance |
| ΔΦ | = | response to a unit heat flux pulse |
| ϵ | = | random error |
| η | = | preassigned error |
| λ | = | dummy variable of integration |
| σ | = | standard deviation |
| σq | = | estimator of amplification of measurement errors |
| ρ | = | density |
| ρm | = | residual contribution at the current time step |
Subscripts
| m | = | at the current time step |
| past. | = | previous components |
Superscripts
| ′ | = | dimensional variable |
| ∧ | = | estimated |
| ∧* | = | estimated and based on reduced heat flux history |
| T | = | transpose |
Additional information
Notes on contributors
José M. Gutiérrez
José M. Gutiérrez is professor of applied physics at the University of Cádiz. He obtained a Ph.D. from the University of Cadiz in 1990. His work and publications have been developed in the following fields: effective thermal conductivity of fibrous composites, thermal simulation of Gleeble systems by finite element method, NDT by infrared thermography, and sequential algorithms for solving inverse heat conduction problems using SVD.
José M. Aguado Teixe
José M. Aguado Teixe is an energy manager and process optimization engineer at one of CEPSA's refineries in Spain. He has a B.S. and M.S. in physical sciences from the University of Granada and an M.S. in photovoltaic technology from the International University of Andalucía. He is currently working toward a Ph.D. in computational engineering at the University of Cádiz. His current research interests are related to the inverse heat conduction problem and combustion monitoring in process furnaces.
Juan A. Martín
Juan A. Martín received his Ph.D. degree in industrial engineering from the University of Cádiz in 2004. Since 1995, he has been with the Department of Electrical Engineering at the University of Cádiz, where he is currently a professor. He is interested in numerical methods applied to electrical and thermal simulations.
Paloma R. Cubillas
Paloma R. Cubillas is professor of thermodynamics and heat transfer in the Thermal Machines Department at the University of Cádiz. She received her Ph.D. in industrial engineering from the University of Cádiz in 2008, with a thesis titled “Building Thermal Simulation Model Centered on HVAC Coupling.” Her current research continues in the same field of knowledge, developing new building and systems simulation models. She worked at BRE (Building Research Establishment) in Watford, UK, within the scheme of an exchange agreement.