Methods, apparatus, systems and articles of manufacture to monitor health of a closed loop system in a turbine engine are disclosed. An example apparatus includes non-linear solver circuitry to: iteratively determine a volume of a gas in a system of a turbine engine based on a pressure measurement, a temperature measurement and a constant that corresponds to the gas, the non-linear solver circuitry to iteratively determine the volume of the gas using a thermodynamic state model; and determine a mass of the gas based on the volume of the gas and a volume of the system; optimization circuitry to iteratively determine a mass loss of the gas based on the mass of the gas using a system of equations; and component control circuitry to adjust use of the system of the gas based on the mass loss of the gas.