There has been increasing demand for the development of lumped-parameter models that can be used for real-time control design and optimization for laser-based additive manufacturing (AM) processes. Our prior work developed a physics-based multivariable model of melt-pool geometry and temperature dynamics for a single-bead deposition in a directed energy deposition process and validated the model using experimental data on deposition of single-bead Ti-6AL-4V (or Inconel®718) tracks on an Optomec® laser engineering net shaping (LENS) system. In this paper, we extend such model for melt-pool geometry on a single-bead single-layer deposition to a multi-bead multi-layer deposition and use the developed model on melt-pool height dynamics to predict part height of three-dimensional builds. Specifically, the extended model incorporates temperature history during the built process, which is computed using temperature field generated from super-positioning of Rosenthal's solution of point heat sources, with one heat source corresponding to one bead built before. The proposed model for part height prediction is then validated using a single-bead thin wall structure built with Ti-6AL-4V using an Optomec® LENS MR-7 system. The model prediction shows good agreement with measurement of part height with less than 10% error rate.