Inverse solutions of the diffusion equation used for the analysis of photothermal erosion (abstract)
Authors: Gibkes J. , Fotsing J. L. N. , Dietzel D. , Pelzl J. , Bein B. K. 
Published: January 2003

Absorption of intense pulses of electromagnetic radiation can increase the temperature at the surface and inside heat shield plates until erosive processes appear comprising phase changes and material removal. During heating up the initially very low heat losses at the front and rear surface of the heat shield due to radiation and conduction/convection increase continuously. When the surface temperature required for phase transition and thermal erosion is reached, the surface temperature increase can temporarily slow down, due to the surface heat sinks related to the phase transitions and thermal erosion. In the present work, the inverse solutions of the surface heating problem have been derived, by which the net surface heat flux Fs(t) of arbitrary time dependence can be calculated from the measured time-dependent surface temperature evolution. The time evolution of the surface temperatures at the front Ts(t) and rear surface Tr(t) have been measured by means of an IR camera. The power balance at the front surface, which governs the heating process, is given by Finc(t)−Fref(t)=Frad(t,Ts)+Fcon(t,Ts)+Fs(t,Ts)+Fsink/source(t,Ts). Here Finc(t) is the power of the incident electromagnetic radiation and Fref(t) its reflected part. Frad(t,Ts) and Fcon(t,Ts) are the heat losses due to radiation and conduction/convection in the surrounding atmosphere, Fs(t,Ts) is the resulting net surface heat flux contributing to heating up of the plate, and Fsink/source(t,Ts) are the heat sinks/sources due to exo- or endothermic processes. The terms Frad(t,Ts) and Fcon(t,Ts) are functions of the measured surface temperature Ts(t) and thus are well known. If we assume that the incident and reflected radiation remain unchanged during the transition from heating up to erosion, the difference Finc(t)−Fref(t) can be determined during the first phase, in the absence of any heat sinks/sources related to thermal erosion: Finc(t)−Fref(t)=Frad(t,Ts)+Fcon(t,Ts)+Fs(t,Ts). For the quantitative interpretation of the power balance in the form (1) or (2), a solution of the heat diffusion equation of the plate with respect to the surface heat flux Fs(t)=Fz(z→0,t) is required, with the measured surface temperature Ts(t) as the boundary condition. In principle, such solutions exist, e.g., for the semi-infinite solid even in analytical form:1 Fs(t)=0.5λρc/π{∫τ=0tdτ[Ts(t)−Ts(τ)]/(t−τ)3/2+2[Ts(t)−Ts(t=0)]/t1/2}, with the disadvantage, however, that such solutions can become unstable due to the limited time resolution and fluctuations of the temperature measurement. To overcome these difficulties, expansion solutions have been derived, which are both numerically stable and appropriate to resolve small peaks of power deposition. Solutions for different types of boundary conditions are presented: (1) Front and rear surface temperature prescribed by time-dependent measurements; (2) measured front surface temperature and shifting rear surface temperature with negligible heat losses at the rear; and (3) measured front surface temperature and shifting rear surface temperature combined with conduction/convection and radiation heat losses at the rear.