By Kyung Sub Kim and Steven C. Chapra. Published in the Journal of Hydraulic Engineering, January 1997. 

Abstract: The development and application of a one-dimensional heat transport model in highly transient streams governed by unsteady flow is described here. The resultant framework consists of two modules called the hydrodynamic and heat transport modules. In the hydrodynamic module, the hydraulic variables such as flow depth and velocity are simulated. Based on this information, the heat transport module is executed to calculate temperatures. A new approach—coupling heat transport in the surface water and diffusion in the sediment zone—is developed and applied for this module. In this approach, an interaction term accounts for the flux of heat energy between the water and sediments, which affects the distribution of water temperature in clear and shallow streams. Implicit finite difference methods called the Preissmann four-point and the Crank-Nicolson schemes are used to solve each module. Application of this framework demonstrates that the model effectively simulates the hydraulic variables and temperatures.