Growth of a new phase is experienced in several chemical engineering processes/operations such as polymer foaming, oil/gas transportation, bubbly columns, distillation towers, and petroleum recovery. Thus, it seems vital to understand the dynamics of transport phenomena and new phase (e.g., bubbles) evolution over the corresponding processes to efficiently design and operate the plant equipment. The present study introduces new analytical and approximate solutions for the growth of a new phase in a medium of finite extent. The governing conservation equations are solved by an effective mathematical approach through combination of variables and enhanced homotopy perturbation method (EHPM) where it is assumed that the mass transfer controls the rate of growth through both convection and diffusion mechanisms. The modeling outcome confirms the importance of the convection mass transfer in growth of the new phase in finite extent. The results show that the radius of the new phase is strongly dependent on the diffusivity, temperature, and initial void fraction. It is also found that the bubble evolution follows the power-law model. To examine the practical effectiveness of the developed mathematical models, the growth of water bubbles, hydrate particles (as a new phase), and bubbles (or gas) growth in oil reservoirs are studied. The models' results are compared with some experimental data, exhibiting a very good agreement (e.g., error percentage of <5%). This research work offers systematic strategies for investigation of bubble expansion/shrinkage phenomena where the bubble growth dynamics are observed in various processes/systems.