Verified Experiment: Intelligent Fuzzy Weighted Input Estimation Method to Inverse Heat Conduction Problem

In this paper, the innovative intelligent fuzzy weighted input estimation method (FWIEM) can be applied to the inverse heat transfer conduction problem (IHCP) to estimate the unknown time-varying heat flux efficiently as presented. The feasibility of this method can be verified by adopting the temperature measurement experiment. We would like to focus attention on the heat flux estimation to three kinds of samples (Copper, Iron and Steel/AISI 304) with the same 3mm thickness. The temperature measurements are then regarded as the inputs into the FWIEM to estimate the heat flux. The experiment results show that the proposed algorithm can estimate the unknown time-varying heat flux on-line. Keywords—Fuzzy Weighted Input Estimation Method, IHCP and Heat Flux. NOMENCLATURE ( ) B k Sensitivity matrix ŝ q Estimated heat flux B ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ Gradient matrix R Measurement noise covariance p C Sample specific heat ( / J Kg K ⋅ ) s Innovation covariance H Measurement matrix t Time I Identity matrix f t Sampling time k Time (discretized) T Temperature k Sample thermal conductivity ( / ( ) J m s K ⋅ ⋅ ) 0 T Initial temperature K Kalman gain ( ) v t Measurement noise vector b K Correction gain ( ) X k State vector ( ) M k Sensitivity matrix ( ) Z k Observation vector Chen-Yu Wang is with the School of Defense Science, Chung Cheng Institute of Technology, National Defense University, Ta-His, Tao-Yuan, Taiwan, R.O.C. (e-mail: [email protected]). Tsung-Chien Chen is with the Department of Power Vehicle and Systems Engineering, Chung Cheng Institute of Technology, National Defense University, Ta-His, Tao-Yuan, Taiwan, R.O.C. (e-mail: [email protected]). Ming-Hui Lee is with the Department of Civil Engineering, Republic of China Military Academy, Ta-His, Tao-Yuan, Taiwan, R.O.C. (e-mail: [email protected]). Jen-Feng Huang is with the School of Power Vehicle and Systems Engineering Chung Cheng Institute of Technology, National Defense University, Ta-His, Tao-Yuan, Taiwan, R.O.C (e-mail: [email protected]). M ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ Global conductance matrix α Thermal diffusivity N Total number of nodes γ Weighting factor n Total number of time steps Γ Input matrix P Filter’s error covariance matrix δ Kronecker delta function b P Error covariance matrix ρ Density of the sample ( 3 / Kg m ) Q Process noise covariance p c ρ Heat-melting coefficient of the sample ( ) q k Heat flux Ω Coefficient matrix (̂ ) q k The unknown input estimated heat flux ω Process noise vector σ Standard deviation t Δ Sampling time interval Φ State transition matrix x Δ Discrete space interval Ψ Coefficient matrix