## Background

This works offers a simplified embodiment of the Stationary liquid mass balance method for the continuous estimate of the OUR estimation by means of the use of inexpensive proportional valves and the monitoring of their control signals is introduced and compared with the Global mass balance and Dynamic methods.

As far said electrovalves can be considered to be linear, it can be demonstrated through the Mean value theorem for integrals that the mass balance equation for the liquid phase can be expressed as a first order differential equation:

Where:

C_{L} [mol/l]: DO concentration in the liquid phase

C*_{L} [mol/l]: DO concentration in equilibrium with the gas phase

k_{L}·a [1/h]: Mass transfer coefficient

a [ ]: PWM control signal or duty cycle

Now, considering the DO concentration control loop error signal for a given constant set point. The previous equation (i) can be rewritten as a function of the control loop parameters and a new expression of the OUR proportional to the duty cycle and the mass transfer coefficient can be proposed:

Where:

e[mol/l]: DO control loop error signal.

C^{sp}_{L}[mol/l]: DO set point