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ABSTRACT
Slab surface radiative emissivities severely affect the radiative heat transfer in a reheating furnace, as well as the slabs’ coupled conduction, convection, and radiation. Accurate evaluation of these parameters is of significance to ensure the high accuracy of the mathematical model for a reheating furnace, which is beneficial to the energy saving. However, it is difficult to directly and accurately measure these parameters. In this article, slab surface radiative emissivities in a reheating furnace are estimated by solving a nonlinear inverse problem, which is an inverse coupled conduction, convection, and radiation problem. An efficient and accurate gradient method, i.e., Levenberg–Marquardt algorithm, is applied to obtain the solution of the inverse problem. First, a finite difference method and the complex-variable-differentiation method are used for sensitivity analysis, and the inversion accuracy coupled with the efficiency is demonstrated. Then, effects of initial guesses, measurement errors, and measurement locations on estimated slab surface radiative emissivities are investigated in detail. Finally, conclusions are drawn based on the results and analysis.
Nomenclature
| A | = | area, m2 |
| c | = | mass specific heat, J/ (kg K) |
| d | = | half depth of a slab, m |
| Eb | = | blackbody emissive power, W/m2 |
| F | = | objective function |
| f | = | nonlinear function |
| G | = | gas |
| = | total radiative exchange area of G to S, m2 | |
| H | = | imaginary part |
| h | = | convective heat coefficient, W/(m2 K) |
| = | total radiative exchange area of G to W, m2 | |
| J | = | sensitivity coefficients matrix |
| i, j | = | no. of the inverted parameter |
| K | = | total number of inverted parameters |
| k | = | kth inverted parameter |
| M | = | total number of measured temperatures |
| n | = | power |
| P | = | iteration number |
| S | = | slab |
| = | total radiative exchange area of S to G, m2 | |
| = | total radiative exchange area of S to W, m2 | |
| T | = | temperature, K |
| t | = | temperature,°C |
| W | = | wall |
| = | total radiative exchange area of W to G, m2 | |
| = | total radiative exchange area of W to S, m2 | |
| x | = | vector of inverted parameters |
| Greek symbols | = | |
| α | = | absorptivity |
| δ | = | updated vector of inverted parameters |
| ε | = | emissivity |
| ζ | = | measurement error |
| η | = | random number |
| μ | = | the damping factor |
| ξ | = | tolerance |
| ρ | = | density, kg/ m3 |
| φ | = | view factor |
| Subscripts | = | |
| a | = | air |
| b | = | bottom |
| c | = | convection |
| com | = | combustion |
| exact | = | exact |
| g | = | medium/gas |
| r | = | radiation |
| s | = | surface |
| store | = | stored |
| u | = | upper |
| w | = | wall |
| Superscripts | = | |
| * | = | measured |
Nomenclature
| A | = | area, m2 |
| c | = | mass specific heat, J/ (kg K) |
| d | = | half depth of a slab, m |
| Eb | = | blackbody emissive power, W/m2 |
| F | = | objective function |
| f | = | nonlinear function |
| G | = | gas |
| = | total radiative exchange area of G to S, m2 | |
| H | = | imaginary part |
| h | = | convective heat coefficient, W/(m2 K) |
| = | total radiative exchange area of G to W, m2 | |
| J | = | sensitivity coefficients matrix |
| i, j | = | no. of the inverted parameter |
| K | = | total number of inverted parameters |
| k | = | kth inverted parameter |
| M | = | total number of measured temperatures |
| n | = | power |
| P | = | iteration number |
| S | = | slab |
| = | total radiative exchange area of S to G, m2 | |
| = | total radiative exchange area of S to W, m2 | |
| T | = | temperature, K |
| t | = | temperature,°C |
| W | = | wall |
| = | total radiative exchange area of W to G, m2 | |
| = | total radiative exchange area of W to S, m2 | |
| x | = | vector of inverted parameters |
| Greek symbols | = | |
| α | = | absorptivity |
| δ | = | updated vector of inverted parameters |
| ε | = | emissivity |
| ζ | = | measurement error |
| η | = | random number |
| μ | = | the damping factor |
| ξ | = | tolerance |
| ρ | = | density, kg/ m3 |
| φ | = | view factor |
| Subscripts | = | |
| a | = | air |
| b | = | bottom |
| c | = | convection |
| com | = | combustion |
| exact | = | exact |
| g | = | medium/gas |
| r | = | radiation |
| s | = | surface |
| store | = | stored |
| u | = | upper |
| w | = | wall |
| Superscripts | = | |
| * | = | measured |