Detailed Derivation of the Scalar Explicit Expressions Governing the Electric Field, Current Density, and Volumetric Power Density in the Four Types of Linear Divergent MHD Channels Under a Unidirectional Applied Magnetic Field

Abstract

The current study belongs to the field of applied mathematics in plasma physics and electric power, where mathematical analysis of the algebraic equations governing the electric field vector, and the electric-current density field vector within a Magnetohydrodynamic (MHD) linear two-dimensional divergent supersonic channel is utilized to derive analytical expressions for these important fields, as well as closed-form equations for the volumetric power density (output electric power per unit volume of the plasma channel). The expressions presented here describe analytically the operation of the MHD channel as an electric power source within an Open-Cycle Magnetohydrodynamic (OCMHD) generator. The four common types of the MHD linear channels are covered here: namely, (1) continuous-electrode Faraday channel, (2) linear Hall channel, (3) segmented-electrode Faraday channel, and (4) diagonal-electrode channel. The mathematical results, their detailed derivation, and the companion graphical illustrations aid in making a proper decision regarding which channel type is the most suitable for a given application.Under typical operational conditions of 5 S/m plasma electric conductivity, 5 T magnetic field, and 2,000 m/s plasma speed, as well as an optimized load factor of 0.5, we estimate the following numerical values (unsigned magnitudes) for the continuous-electrode Faraday channel (with a Hall parameter of 1): useful electric field (across the external electric load): 5 kV/m, useful electric current-density (between the terminal electrodes within the channel): 12.5 kA/m2 , volumetric power density (dissipated by the load per unit volume of plasma): 62.5 MW/m3 , and electric efficiency (for the electric field or voltage): 50%. For the Hall linear channel (with a Hall parameter of 5), these quantitative performance values become 25 kV/m, 4.808 kA/m2 , 120.19 MW/m3 , and 46.30%. For either an infinitely segmented-electrode Faraday channel or an on-design diagonal-electrode linear channel (and irrespective of the Hall parameter), these quantitative performance values become 5 kV/m, 25 kA/m2, 125 MW/m3, and 50%.