We develop a coupled thermomechanical model, that includes mould taper, for the formation of air gaps in the vertical continuous casting of round billets. The system is very sensitive to the small width of the air gap. Mould tapers are used to mitigate the contraction of the solidified shell during cooling. We apply numerical and perturbation methods to show that a small mould taper significantly reduces the insulating effect of the air gap. The analysis is presented in a more transparent and less computationally expensive way than earlier, fully numerical models. We also consider a theoretical ideal taper, which eliminates the air gap altogether. The air gap is found to be quite robust; increasing the size of the taper does not constitute an equal reduction in the air gap size. Sample computations are carried out using parameters for the continuous casting of a pure metal (copper), although the framework can easily be extended to the continuous casting of alloys. (C) 2015 Elsevier Inc. All rights reserved.