Unsteady heat transfer process and the developed temperature field inside hot water storage tanks with a buoyant inflow jet can be described by a coupled first order partial and a first order ordinary differential equation system (PDE-ODE). Non-polynomial series solutions of this PDE-ODE system for an initial-boundary condition have been presented. The different solutions have been compared to the numerical solution of the PDE-ODE system and to the relevant measurement. The rate of the convergence of the analytic solutions is also investigated. Linear equation systems are constructed to calculate the constant coefficients of the series solution. Additionally the entire 3D flow and temperature field of the investigated thermo physical process have been calculated numerically and compared to the results of the analytic solution of the simplified model. Besides theoretical importance, the solutions can be benchmark solutions to develop the computational heat transfer.