We describe an analytic approach to designing axially water-cooled Bitter-type electromagnets with an emphasis on heat dissipation considerations. The design method here described aims to enhance the efficiency of the design process by minimizing the role of finite element analysis (FEA) software. A purely analytic design optimization scheme is prescribed for establishing the cooling hole placement. FEA software is only used to check the accuracy of analytic predictions. The analytic method derived in this paper predicts the required heat dissipation rate by approximating the volumetric joule heating profile with a smooth, continuous profile. Equations for turbulent convective heat transfer in circular ducts are generalized to model the cooling capacity of elongated cooling passages. This method is currently in use at the University of Maryland Baltimore County Dusty Plasma Laboratory to design a Bitter magnet capable of generating fields of 10 T.