A study is made of the inverse free convection problem (IFCP) by adjoint equations and conjugate gradient (CG), to determine an unknown space and time dependent boundary heat flux on the side of an enclosure, from temperature measurements by sensors within the flow. The direct, sensitivity and adjoint set of equations for a Boussinesq fluid are solved by control volumes. Solutions are presented for different types of boundary conditions and a wide range of Rayleigh numbers, for a square enclosure. It is found that by placing sensors close enough to the active boundary, solutions may be achieved for Rayleigh numbers higher than in previous studies. Noisy data solutions are regularized by stopping the iterations according to the discrepancy principle of Alifanov, before the high frequency components of the random noises are reproduced. The accuracy of the solutions is shown to depend strongly on the Rayleigh number, the sensor's position and the type of boundary conditions imposed.