Temperature and density relaxation close to the liquid-gas critical point:: An analytical solution for cylindrical cells -: art. no. 041201

We present a study of the temperature and density equilibration near the liquid-gas critical point of a composite system consisting of a thin circular disk of near-critical fluid surrounded by a copper wall. This system is a simplified model for a proposed space experiment cell that would have 60 thin fluid layers separated by perforated copper plates to aid in equilibration. Upper and lower relaxation time limits that are based on radial and transverse diffusion through the fluid thickness are shown to be too significantly different for a reasonable estimate of the time required for the space experiment. We therefore have developed the first rigorous analytical solution of the piston effect in two dimensions for a cylindrically symmetric three-dimensional cell, including the finite conductivity of the copper wall. This solution covers the entire time evolution of the system after a boundary temperature step, from the early piston effect through the final diffusive equilibration. The calculation uses a quasistatic approximation for the copper and a Laplace-transform solution to the piston effect equation in the fluid. Laplace inversion is performed numerically. The results not only show that the equilibration is divided into three temporal regimes but also give an estimate of the amplitudes of the remaining temperature and density inhomogeneity in each regime. These results yield characteristic length scales for each of the regimes that are used to estimate the expected relaxation times in the one- and two-phase regions near the critical point.