Extended thermodynamics derivation of energy dissipation in unsteady pipe flow

Extended irreversible thermodynamics (EIT) provides a framework for derivin g 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 pip e 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 proportion al to the local (i.e., temporal) acceleration; and (3) a term that is propo rtional to the product of the velocity and the convective (i.e., spatial) a cceleration. The form of this FIT-based wall shear stress formula shows tha t EIT provides a physical basis for instantaneous acceleration based unstea dy friction formulas. It also illustrates the limitations and underlying as sumptions of these models. For example, instantaneous acceleration based un steady friction formulas are limited to fast transients (i.e., transients i n which the water hammer timescale is significantly smaller than the diffus ion timescale). A characteristics solution for unsteady pipe flow is propos ed in which the phenomenological equation is used to model energy dissipati on. Comparison of numerical test results with measured data from upstream a nd downstream valve closure laboratory experiments shows excellent agreemen t.