Extended Thermodynamics Derivation of Energy Dissipation in Unsteady Pipe Flow

Extended irreversible thermodynamics (EIT) provides a framework for deriving extensions to phenomenological equations (e.g., Newton's law of viscosity, Fick's law of mass transport, and Darcy's law for porous media flow) for problems involving high frequencies (i.e., rapid transients). In this paper, a phenomenological equation is derived for energy loss in 1D unsteady pipe flow using an EIT formalism. The resulting wall shear stress is equal to the sum of (1) the steady-state shear stress; (2) a term that is proportional to the local (i.e., temporal) acceleration; and (3) a term that is proportional to the product of the velocity and the convective (i.e., spatial) acceleration. The form of this EIT-based wall shear stress formula shows that EIT provides a physical basis for instantaneous acceleration based unsteady friction formulas. It also illustrates the limitations and underlying assumptions of these models. For example, instantaneous acceleration based unsteady friction formulas are limited to fast transients (i.e., transients in which the water hammer timescale is significantly smaller than the diffusion timescale). A characteristics solution for unsteady pipe flow is proposed in which the phenomenological equation is used to model energy dissipation. Comparison of numerical test results with measured data from upstream and downstream valve closure laboratory experiments shows excellent agreement.