![Sample our Engineering & Technology journals, sign in here to start your access, latest two full volumes FREE to you for 14 days]({"type":"image" , "src" : "/sda/5933/TOC-Subject-Sample-2021-Engineering-Tech.jpg"})
ABSTRACT
This paper utilizes concepts extracted from phase-plane and cross-correlation analyses for estimating the optimal regularization parameter required by a nonlinear space marching inverse heat conduction method. The regularization parameter for this approach is based on the cut-off frequency used in a low-pass Gauss digital filter for preprocessing the in-depth temperature data. The simulation uses two in-depth thermocouples which are representative of a physical experiment. Numerical results indicate a high level of robustness as the best prediction is extracted from the prediction space over the spectrum of cut-off frequencies. The feasibility and effectiveness of phase-plane and cross-correlation analyses are discussed.
Nomenclature
| b1, b2 | = | in-depth thermocouple probe positions, m |
| C | = | specific heat capacity, J/(kg · K) |
| Cn | = | power spectral density |
| f, fc | = | cut-off frequency, Hz |
| fc,DFT | = | cut-off frequency determined by DFT, Hz |
| fsampling | = | sampling frequency, Hz |
| fn | = | nth chosen cut-off frequency, Hz |
| k | = | thermal conductivity, W/(m · K) |
| L | = | thickness of the one-dimensional geometry, m |
| M | = | total number of spatial nodes in the space marching region x ∈ [0, b1] minus 1 |
| N | = | total number of temporal nodes past the initial condition |
| N* | = | number of data points cut in predicted surface heat flux and heat flux rate for phase-plane and the cross-correlation investigation |
| P | = | total number of chosen cut-off frequencies |
| q″ | = | heat flux, W/m2 |
| = | predicted surface heat flux under specific cut-off frequency f, W/m2 | |
| = | predicted surface heat flux under nth chosen cut-off frequency fn, W/m2 | |
| = | predicted surface heat flux rate under nth chosen cut-off frequency fn, W/(m2 s) | |
| = | discrete heat flux at xi = iΔx, tj = jΔt in the space marching region x ∈ [0, b1], W/m2 | |
| = |
| |
| = | maximum value of the double Gaussian heat flux defined in Eq. (6), W/m2 | |
| = | surface heat flux, W/m2 | |
| = | heat flux obtained by solving direct problem in x ∈ [b1, b2] using “thermocouple” data, W/m2 | |
| = | heat flux obtained by solving direct problem in x ∈ [b1, b2] using filtered “thermocouple” data by cut-off frequency fc, DFT determined by DFT, W/m2 | |
| rj | = | jth random number in the interval [−1,1] |
| Rq | = | heat flux cross-correlation coefficient defined in Eq. (5a) |
| = | heat flux rate cross-correlation coefficient defined in Eq. (5b) | |
| t | = | time, s |
| tk | = | kth temporal node, s |
| tmax | = | maximum data collection time, s |
| tj | = | jth temporal node, s |
| T | = | temperature, K |
| Tf | = | predicted surface temperature under chosen cut-off frequency f, W/m2 |
| T0 | = | initial temperature, K |
| = | Ttc(b1, tj), which is the “thermocouple” temperature at x = b1, K | |
| = | discrete temperature at xi = iΔx, tj = jΔt in the space marching region x ∈ [0, b1], K | |
| Ts | = | surface temperature, K |
| Ttc | = | “thermocouple” temperature, K |
| = | filtered “thermocouple” temperature by cut-off frequency denoted by fc, K | |
| = | filtered “thermocouple” temperature by cut-off frequency fc,DFT determined by DFT, K | |
| = | filtered “thermocouple” temperature by nth chosen cut-off frequency fn determined by DFT, K | |
| x | = | spatial coordinate, m |
| xi | = | xth spatial node, m |
| x0 | = | first spatial nodes at x = b1 in the space marching region x ∈ [0, b1] |
| xM | = | last spatial nodes at x = 0 in the space marching region x ∈ [0, b1] |
| Δf | = | the stepping frequency for forming a finite set of predictions, Hz |
| Δt | = | time step, s |
| Δx | = | space step, m |
| β1 | = | parameter in the double Gauss function defined in Eq. (6), s |
| β2 | = | parameter in the double Gauss function defined in Eq. (6), s |
| εT | = | temperature noise factor |
| ρ | = | density, kg/m3 |
| σ1 | = | parameter in the double Gauss function defined in Eq. (6), s |
| σ2 | = | parameter in the double Gauss function defined in Eq. (6), s |
| ωc | = | circular cut-off frequency, Hz |
| Subscripts | = | |
| s | = | surface |
| tc | = | thermocouple |
Nomenclature
| b1, b2 | = | in-depth thermocouple probe positions, m |
| C | = | specific heat capacity, J/(kg · K) |
| Cn | = | power spectral density |
| f, fc | = | cut-off frequency, Hz |
| fc,DFT | = | cut-off frequency determined by DFT, Hz |
| fsampling | = | sampling frequency, Hz |
| fn | = | nth chosen cut-off frequency, Hz |
| k | = | thermal conductivity, W/(m · K) |
| L | = | thickness of the one-dimensional geometry, m |
| M | = | total number of spatial nodes in the space marching region x ∈ [0, b1] minus 1 |
| N | = | total number of temporal nodes past the initial condition |
| N* | = | number of data points cut in predicted surface heat flux and heat flux rate for phase-plane and the cross-correlation investigation |
| P | = | total number of chosen cut-off frequencies |
| q″ | = | heat flux, W/m2 |
| = | predicted surface heat flux under specific cut-off frequency f, W/m2 | |
| = | predicted surface heat flux under nth chosen cut-off frequency fn, W/m2 | |
| = | predicted surface heat flux rate under nth chosen cut-off frequency fn, W/(m2 s) | |
| = | discrete heat flux at xi = iΔx, tj = jΔt in the space marching region x ∈ [0, b1], W/m2 | |
| = |
| |
| = | maximum value of the double Gaussian heat flux defined in Eq. (6), W/m2 | |
| = | surface heat flux, W/m2 | |
| = | heat flux obtained by solving direct problem in x ∈ [b1, b2] using “thermocouple” data, W/m2 | |
| = | heat flux obtained by solving direct problem in x ∈ [b1, b2] using filtered “thermocouple” data by cut-off frequency fc, DFT determined by DFT, W/m2 | |
| rj | = | jth random number in the interval [−1,1] |
| Rq | = | heat flux cross-correlation coefficient defined in Eq. (5a) |
| = | heat flux rate cross-correlation coefficient defined in Eq. (5b) | |
| t | = | time, s |
| tk | = | kth temporal node, s |
| tmax | = | maximum data collection time, s |
| tj | = | jth temporal node, s |
| T | = | temperature, K |
| Tf | = | predicted surface temperature under chosen cut-off frequency f, W/m2 |
| T0 | = | initial temperature, K |
| = | Ttc(b1, tj), which is the “thermocouple” temperature at x = b1, K | |
| = | discrete temperature at xi = iΔx, tj = jΔt in the space marching region x ∈ [0, b1], K | |
| Ts | = | surface temperature, K |
| Ttc | = | “thermocouple” temperature, K |
| = | filtered “thermocouple” temperature by cut-off frequency denoted by fc, K | |
| = | filtered “thermocouple” temperature by cut-off frequency fc,DFT determined by DFT, K | |
| = | filtered “thermocouple” temperature by nth chosen cut-off frequency fn determined by DFT, K | |
| x | = | spatial coordinate, m |
| xi | = | xth spatial node, m |
| x0 | = | first spatial nodes at x = b1 in the space marching region x ∈ [0, b1] |
| xM | = | last spatial nodes at x = 0 in the space marching region x ∈ [0, b1] |
| Δf | = | the stepping frequency for forming a finite set of predictions, Hz |
| Δt | = | time step, s |
| Δx | = | space step, m |
| β1 | = | parameter in the double Gauss function defined in Eq. (6), s |
| β2 | = | parameter in the double Gauss function defined in Eq. (6), s |
| εT | = | temperature noise factor |
| ρ | = | density, kg/m3 |
| σ1 | = | parameter in the double Gauss function defined in Eq. (6), s |
| σ2 | = | parameter in the double Gauss function defined in Eq. (6), s |
| ωc | = | circular cut-off frequency, Hz |
| Subscripts | = | |
| s | = | surface |
| tc | = | thermocouple |