Numerically Stabilizing Ill-Posed Moving Surface Problems Through Heat-Rate Sensors

Nomenclature B2 M = deterministic bias, Eq. (10c) b = constant in Gaussian function, s, Eq. (7) cp = heat capacity, kJ/(kg◦C) I j (z) = j th modified Bessel function K = convolution kernel, Eq. (1c) k = thermal conductivity, W/(m◦C) M = number of data points Ma = convolution kernel, Eq. (5a) Mb = convolution kernel, Eq. (5b) N = convolution kernel, Eq. (6b) q ′′ = dimensional heat flux, W/m2 q ′′ s = surface heat flux, W/m 2 q ′′ 0 = maximum Gaussian heat-flux value, W/m 2 T = temperature, ◦C Ti = discrete temperature, ◦C Ts = surface temperature, ◦C T0 = initial temperature, ◦C t = time, s tmax = maximum time, s t0 = dummy variable, s u = velocity of moving surface, m/s v = dummy variable, Eq. (3) x = spatial variable, m z = dummy argument α = thermal diffusivity, m2/s β = constant, Eq. (1c) i = noise factor, Eqs. (8a) and (8b) θ = temperature difference, Ts − T0, ◦C λ = constant, Eq. (1c) ρ = density, kg/m3 σ = constant, Eq. (7) σ 2 M = variance, Eq. (10d)

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