Formulation of the Slip Loss in a Two-Phase Liquid-Metal Magnetohydrodynamic Generator

Abstract

Nicklin's derivation of the slip loss in a vertical two-phase flow has been extended to a two-phase LMMHD generator flow, where that loss can be very important. The slip loss was derived by considering energy exchange between liquid and gas. The energy exchange approach indicates somewhat larger slip losses. Comparison of two expressions shows that the energy exchange approach yields a more universal formulation of the slip loss. Loss of LMMHD generator efficiency is discussed in more detail. I . Introduction Although there are many experimental and theoretical studies of two-phase flows, only a few of them treat energy loss resulting from slip. This is probably because the presence of several interrelated loss mechanisms in two-phase flow makes it difficult to isolate and formulate the loss of energy caused by slip. Often researchers are primarily interested in heat transfer, in which case slip loss is not significant. However, because of the existence of pressure drops two to three orders of magnitude higher in a liquidmetal MHD (LMMHD) generator than in ordinary two-phase flows, the loss of energy (or generator efficiency) that rePaper presented at Third Beer-Sheva International Seminar on Magnetohydrodynamic Flows and Turbulence, Ben-Gurion University of the Negev, Beer-Sheva, Israel, March 23-27, 1981. This paper is declared a work of the U.S. Government and therefore is in the public domain. *Senior Scientist. (Present address: Combustion Dynamics and Propulsion Technology Division, Science Applications, Inc., Canoga Park, Cal i f . ) 216 SLIP LOSS IN A LIQUID-METAL MHD GENERATOR 217 suits from slip could be much more important. In addition, the slip itself could be greater because of the much higher pressure gradient [Fabris and Hantman (1976)] , although the most recent data show very small slips [Fabris et al . (1979)]. Thus, it is necessary to analyze and define clearly the slip loss mechanism in two-phase LMMHD flow. The only existing expression for the LMMHD slip loss is given in Cutting and Amend (1974) based on the derivation of Petrick et al „ (1970). Among the few papers which treat the energy balance in non-MHD vertical two-phase flow in some detail, two can be singled out. The first [Richardson and Higson (1962) ] presents extensive experimental data on eff iciencies and losses in an air-lift pump. However, it simply states that the slip loss is proportional to the additional airflow rate required by the existence of a slip velocity; no other physical reasoning about the slip loss is given. The second paper, by Nicklin (1962), presents an enlightening discussion on the decomposition of the pressure drop in a vertical twophase flow and derives an expression for the pressure drop resulting from slip. In this paper, Nicklin 's derivation will be used as a starting point and extended to MHD flow and the results will be compared with another derivation based on direct mechanical energy exchange between the two phases. II. Slip Loss in Vertical Two-Phase Flow after Nicklin Nicklin (1962) divides pressure drop in a vertical twophase flow into reversible and irreversible parts,