(Comments to Wolfgang Dreybrodt remark "On feasibility of condensation processes in caves", Speleogenesis and Evolution of Karst Aquifers, 1 (2), 2003, www.speleogenesis.info)
Audra, Bigot and Mocochain (2003) proposed an explanation for the development of a hydrothermal cave in Provence (France), referring to the Szunyogh model (1989). Dreybrodt (2003) then shows by calculations that this model is unlikely. We will discuss Dreybrodt’s answer here. Our conclusions will emphasise that Dreybrodt’s hypothesis (transient conduction in a semi-infinite solid) is not the only possibility. When other conditions are considered (steady-state conduction with constant temperature at a finite distance), this cupola-development model can be valid.
W. Dreybrodt calculation The corrosion mechanism by degassing of thermal water and condensation on the ceiling supposes, as Dreybrodt mentioned, a wall-cooling process (condensation <=> Watt’s cold wall). In order to test the model proposed by Audra et al. (2003), Dreybrodt stated the implicit hypothesis that cooling proceeds in an infinite limestone mass around the ceiling cupola. This allows him to use a classical analytical solution (transient conduction in a cylindrical infinite mass). The flux at the wall diminishes approximately according to the square root of time and this flux rapidly becomes too weak to produce any noticeable condensation. Thus, the process gradually stops and the amount of limestone corrosion is too small to validate the development of cupolas.
Choice of boundary conditions. This calculation is perfectly correct. However, it is based on the hypothesis that the limestone host rock is, at the start of the process, in a steady isothermal state, meaning that the heat sink is located in an infinite distance from the cupola. In real karst this never occurs. If the ceiling cupola is located at a moderate distance below the ground surface and in a karst massif, it is normal for seepage from rainfall to occur rather close to the hydrothermal flow. This seepage water provides a source of cooling at a moderate distance from the cupola. The difference ∆T between hydrothermal and meteoric seepage temperatures, at any distance ∆x between both flows, generates a temperature gradient ∆T/∆x, which can be chosen at about 1 °C/m.
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