**Objective**

State an n-dimensional PDE, with appropriate BCs, describing an engineering analysis problem of your interest. Briefly describe features of its solution prompting your interest, and anything you know about available solution processes (for the simplified or real problem).

**Problem Statement**

My research interest involves the study of liquid entrainment in the boiler of a liquid potassium Rankine cycle in a micro-gravity environment. This cycle could serve as a power conversion system for a nuclear reactor capable of producing levels of electricity far beyond those seen in current space applications.

In a basic sense, the goal of this boiler design is to elongate the annular flow regime while reducing the drop flow regime. By doing this, the walls of the boiler are wetted for as long as possible to ensure good heat transfer to the remaining liquid potassium. This allows for high-quality flow at the boiler exit. Thus liquid entrainment, which “steals” liquid from the annular film and embeds it as droplets in the gas core flow, should be avoided.

The phenomenon of entrainment deals with the dynamics between the annular liquid film and the continuous vapor core. The studies of Ishii and Grolmes have found that the dominant mode of entrainment for low viscosity flows is the shearing-off of roll wave crests by a turbulent gas flow.

To make the problem of modeling the interaction between more tractable, we will assume only 1-D flow. For a single-component fluid, the integro-differential equation of motion is

For a two-phase annular flow, we must introduce the volume fraction. With this defined, the differential equation becomes

and under steady-state conditions

The appropriate boundary conditions for this type of PDE would include Dirichlet or no-slip conditions along the wall of the pipe, a Neumann or no-acceleration condition at the pipe exit, and a specified velocity distribution at the pipe inlet.

The conservation of mass principle supplies some aspect of closure to the equation of motion. From the integro-differential equation for mass conservation of a single-phase liquid

the differential equation can be derived for a two-phase annular mixture after again invoking the concept of the void fraction

Finally, with a steady-state assumption, the mass equation reduces to

Challenges present to the solution of this problem include the fact that the phases are changing due to latent heating effects. Thus there is a momentum credit to the vapor phase while the liquid phase is debited. Also, this equation fails to include the friction forces due to shear between the liquid and vapor interface which is of extreme importance to the entrainment process. Finally, the first step to a numerical solution of this equation should involve changing to a cylindrical coordinate system.

**References**

Ishii, M., and M.A. Grolmes. “Inception Criteria for Droplet Entrainment in Two-Phase Concurrent Film Flow,” __AIChE Journal__ **21** (1975): 308-318.

Wallis, Graham B, __One-Dimensional Two-Phase Flow__. New York: McGraw-Hill, 1969. pp. 61-66.

Todreas, Neil E., and Mujid S. Kazimi, __Nuclear Systems I: Thermal Hydraulic Fundamentals__. New York: Hemisphere Publishing Corporation, 1990. pp. 144-146.