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A theoretical study of the heat and momentum transfer resulting from a flow of power plant condenser effluent discharged vertically to shallow, quiescent coastal receiving water is presented. The complete partial differential equations governing steady, incompressible, turbulent flow driven by both initial momentum and buoyancy are solved using finite-difference techniques to obtain temperature and velocity distributions in the near field of the thermal discharge. The method of steady-flow vorticity transport was deemed the most attractive approach for this numerical study. A partial differential equation for buoyancy transport was used as a direct couple to the vorticity transport equation, and related the effluent temperature and salinity to buoyancy through an equation of state for sea water. Three-dimensional formulations along with two-dimensional translent methods were investigated at the outset of this research. However, in view of excessive computation requirements, two-dimensional steady flow techniques were found to be satisfactory and computationally more attractive to meet objectives of this study. Turbulent quantities were treated through the use of Reynolds stresses with further simplification utilizing the concept of eddy diffusivities computed by Prandtl's mixing length theory. A Richardson number correlation was used to account for the effects of density gradients on the computed diffusivities. Results were obtained for over 100 cases, 66 of which are reported, using the computer program presented in this manuscript. These results ranged from cases of pure buoyancy to pure momentum and for receiving water depths from 1 to 80 discharge diameters deep. Various computed gross aspects of the flow were compared to published data and found to be in excellent agreement. Data for shallow water plumes and the ensuing lateral spread are not readily available; however, one computed surface temperature distribution was compared to proprietary data and found also to be in excellent agreement. It is concluded that the numerical techniques presented in this study comprise an accurate and practical method for thermal analysis of the type of discharge cited. Although Prandtl's theory was used in this study with good success, it was found that modeling eddy transport coefficients is an area of considerable weakness and research is needed for general numerical fluid dynamic applications.