A scale analysis of the transient discharge of a fully mixed thermal storage vessel with an immersed single-tube heat exchanger is extended to provide a generalized expression for the transient natural convection Nusselt number for heat exchangers comprising many tubes. The transient Nusselt number is expressed in terms of the Rayleigh number at the initiation of the discharge (or charge) process and easily measured geometric parameters. Nusselt numbers measured for a 240-tube heat exchanger immersed in a fully mixed 126 L storage vessel are well correlated in the proposed form. The applicability of the approach to thermally stratified storage fluids is evaluated for both a single-tube and the 240-tube bundle. For heat exchangers of practical size for solar systems, for example the 240-tube bundle, buoyancy driven flow within the storage is sufficient to mix an initially stratified fluid. In this case, Nusselt numbers during the discharge process are predicted accurately by the proposed transient formulation. However if the storage fluid remains stratified during discharge, as is the case for an initially stratified vessel with a single-tube heat exchanger, the transient formulation is not recommended.