Neutral Instability and Optimum Convective Mode in a Fluid Layer with PCM Particles

Linear stability analysis is performed to determine the critical Rayleigh number for the onset of convection in a fluid layer with phase-change-material particles. Sine and Gaussian functions are used for describing the large variation of apparent specific heat in a narrow phase changing temperature range. The critical conditions are numerically obtained using the fourth order Runge-Kutta-Gill finite difference method with Newton-Raphson iteration. The critical eigenfunctions of temperature and velocity perturbations are obtained. The results show that the critical Rayleigh number decreases monotonically with the amplitude of Sine or Gaussian function. There is a minimum critical Rayleigh number while the phase angle is between $π∕2$ and $π$⁠, which corresponds to the optimum experimental convective mode.