The axially-symmetric solutions to the Navier-Stokes equations coupled with the heat conduction are considered. in a bounded cylinder Ω ⊂ R 3 . We assume that v r , v φ , ω φ vanish on the lateral part S 1 of the boundary ∂Ω and v z , ω φ , ∂ z v φ vanish on the top and bottom of the cylinder, where we used standard cylindrical coordinates and ω = rot v is the vorticity of the fluid. Moreover, vanishing of the heat flux through the boundary is imposed. Assuming existence of a sufficiently regular solution we derive a global a priori estimate in terms of data. The estimate is such that a global regular solutions can be proved. We prove the estimate because some reduction of nonlinearity are found. Moreover, we need that f ( p ) ≡ ∥ v φ ∥ L t ∞ L x p / ∥ v φ ∥ L t ∞ L x ∞ is bounded from below by a positive constant. The quantity f( p) is close to 1 for large p because f(∞)=1. Moreover, deriving the global estimate for a local solution implies a possibility of its extension in time as long as the estimate holds.