Phase change materials (PCMs) are widely used in thermal energy storage systems, as they can absorb and release a large amount of heat during the phase change process. Numerical simulations can be used for parametric studies and analysis of the thermal performance of the PCM heat exchanger (HX) to produce an optimal design. Among various numerical methods, the lattice Boltzmann method (LBM), a mesoscopic approach that considers the molecular interactions at relatively low computation costs, offers certain key advantages in simulating the phase change process compared with the conventional Navier-Stokes-based (NS-based) methods. Moreover, LBM is ideal for parallel computing, by which numerical analysis can be efficiently performed. Therefore, a comprehensive solid-liquid phase change model is developed based on LBM which is capable of accurately and efficiently simulating the process of convective PCM phase change with and without porous media in both Cartesian and axisymmetric domains. Double distribution functions (DDF) coupled with a multi-relaxation-time (MRT) scheme are utilized in the LBM formulation for the simulation of the fluid flow and the temperature field. A differential scanning calorimetry (DSC) correlated equation is applied in LBM to model enthalpy, by which the solid-liquid interface can be automatically tracked. The source term in the MRT scheme is modified to eliminate numerical errors at high Rayleigh numbers. Moreover, the conjugate thermal model is adopted for the consideration of heat transfer fluid (HTF) flow and conducting fins. The new model is verified and validated by various case studies. The results indicate that the new model can successfully predict the process of PCM phase change with errors confined to less than 10%. Parametric studies are then performed using the validated model to quantitatively evaluate the effect of convection on PCM melting, from which the acceleration rates ((a_c)) of PCM melting and the threshold Rayleigh numbers ((Ra_{dc})) at various aspect ratios are defined and quantified. Furthermore, PCM melting in porous cylindrical HX is also investigated. The results indicate that the acceleration of melting could reach 95% compared to that in pure PCM at 60% energy storage. Moreover, the negative effect of uneven temperature distributions on thermal performance of the HX caused by convection is quantified and analyzed. A modified cylindrical HX that offsets this negative effect by varying the geometry is also evaluated. The results indicate that the modified geometry can successfully enhance heat transfer and balance the uneven temperature distributions.