Spacecraft utilizing cryogenic propellants need accurate microgravity liquid mass gauging capability. In this paper we describe the spatial regularization method for providing estimations of liquid mass fraction for any liquid configuration in a propellant tank using capacitance measurements among electrodes surrounding the tank volume. The capacitance measurements, which interrogate the entire fluid volume, are weighted and summed to spatially regularize the capacitance sensitivity in the tank volume and to minimize the influence of the fluid configuration on the liquid mass fraction measurement. We begin by developing the mathematical foundation for the spatial regularization method for arbitrary tank geometries with continuous electrode distributions, including the theoretical limit on liquid mass fraction accuracy achievable with the method. We then discretize the mathematics, and we provide the solution for a cylindrical volume with 16 discrete electrodes that depends on self-capacitances. Finally, we demonstrate how to apply the discrete solution in a manner that uses only mutual capacitances. For both the continuous and discrete cases, accuracy predictions are made using finite element modeling and a variety of fluid configurations. Our analyses indicate that the spatial regularization method yields liquid mass fractions with mean accuracy greater than 98% for a cylindrical volume using 16 electrodes.
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