Modeling transient fully-developed natural convection in vertical ducts – Method of eigenfunction superposition

Abstract

• Explicit solutions are found for transient natural convection in ducts. • Eigenfunction superposition method is well suited for fully-developed flow. • Rise time to steady state depends on the first eigenvalue of the Helmholtz equation . This paper models the transient natural convection problem for the hydrodynamical and thermal fully-developed open vertical ducts. The powerful method of eigenfunction superposition yields analytic solutions for general duct cross sections. It is found that the induced flow rate approaches a constant in the steady state, but the transient depends on the Prandtl number (Pr). The rise time to reach 95% of steady state is 3 / ( Pr λ 1 ) if Pr < 1, and is 3/λ 1 if Pr > 1, where λ 1 is the lowest eigenvalue of the Helmholtz equation. The theory is then applied to the circular, semi-circular, rectangular and equilateral triangular ducts. For low Pr at small times, the transient induced velocity is larger in the velocity boundary layer near the walls but the maximum velocity is near the corners of the duct.