Hydrodynamically and thermally fully developed, steady, incompressible laminar flow with constant physical properties in eccentric Curved annular square duct was investigated numerically. Inner and outer walls were assumed to tic isothermal, but at different temperatures. For the Cartesian coordinate system, the continuity , momentum and energy equations included the curvature ratio were discretized by using control volume finite difference method and the dependent variables in the governing equations were solved by ADI method use; the TDMA. The Stone's method was employed to solve the pressure-correction equation instead of ADI method. The upwind scheme and the central difference scheme were employed to represent the convection and diffusion terms, respectively. Solutions were obtained for air (Pr=0.7). Secondary flow streamlines, velocity and temperature fields, velocity profiles, the function coefficients and average and local Nusselt numbers were presented depending on Dean number and annulus dimension ratio (a/b). It was observed that secondary flow resulting from centrifugal force highly affects the velocity and temperature fields. It has seen that curvature, annulus dimension ratio and core position affect heat transfer and function factor. With the increasing annulus dimension ratio, it has been shown that the convective heat transfer is remarkably enhanced at both the inner and the outer walls.