MAXWELL–WAGNER OBSERVABILITY ANALYSIS IN TWO-FREQUENCY CAPACITIVE MEASUREMENT OF WATER CONTENT AND DENSITY IN HYDROCARBON EMULSIONS

Capacitive transducers are widely used for in-line control of fuels and other hydrocarbon media, but a single-frequency reading of effective permittivity cannot separate the water content of an emulsion from the density of the hydrocarbon phase because both quantities shift the same scalar output. This study formulates the separation task as a parameter-observability problem and shows that Maxwell–Wagner interfacial polarization provides the physical mechanism required to resolve it. A Maxwell–Garnett mixture model with complex phase permittivities was combined with the Clausius–Mossotti relation linking hydrocarbon density to permittivity, and the Jacobian of the two-frequency measurement map was analysed over a five-decade band. The determinant of the Jacobian vanishes identically when the two probing frequencies coincide, demonstrating that a second frequency is a necessary condition for observability rather than merely an accuracy refinement. Numerical analysis places the practical optimum at a low frequency below 0.05 of the relaxation frequency and a high frequency above ten times it; beyond this separation, the gain saturates within 1%. With a channel resolution of 1·10⁻⁴ in relative permittivity, the propagated standard uncertainties are 0.033% for water volume fraction and 1.07 kg/m³ for density; Monte Carlo inversion reproduced these values and revealed negligible bias. The scheme was tolerant to salinity: a 50% error in the assumed water conductivity displaced the density estimate by only 0.11 kg/m³.

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