In the industrial heat-exchangers, there are many cases of unsteady heat transfer 10 entry length with temperature distributions of pipe-walls along the direction of axis and pressure-drop from the inlet to the outlet, and in the LMFR atomic reactors, the heat is generated from the inside or the fluid. Therefore, in these cases, the research or unsteady neat transfer which generates heat from the inside of the fluid is necessary. In this paper, theoretical solutions of unsteady heat transfer in entry length of vertical double pipes are derived. In these solution, such cases of unsteady heat transfer in entry length of vertical double pipes are tried that heat generates from the inside of the fluid, free convection and laminar flow coexist at the same time, pressure drops from the inlet to the outlet, and temperature distributions of pipe-walls vary in the direction of pipe-axis. In the solutions, the finite Hankel transform, the finite Fourier transform, the Laplace transform and Bessel functions are used. But, there is no experiment on such a case. Theoretical solutions are reduced to the steady heat transfer and compared with experiment at results.