Nominal System Modeling

Fig. 4 lists our physics-based lumped-parameter state-space model of the WRS.

A Pump j installed between two points with pressures q_{i} and q_{j}, respectively, is modeled to boost the pressure at its input by its boost pressure p_{Pumpj}, i.e., the pressure difference between these two points with the pump in between is p_{i} + p_{Pumpj} - p_{j} . The boost pressure p_{Pumpj} is considered as an input to the system. The boost pressures for Pump 1, Pump 2, Pump 4, and RO Pump are denoted by p_{Pump1}, p_{Pump2}, p_{Pump4}, and p_{ROPump}, respectively.

Given two points in a hydraulic system, with pressures p_{i} and p_{j}, the volumetric flow rate of fluid between these two points is

where R_{ij} is the coefficient of flow for q_{ij}.

For a Tank i having input and output flow rates, q_{in} and q_{out}, pressure p_{Tanki} is

Osmosis is driven by the difference in solute concentrations across the membrane that allows the solvent molecules to pass, but rejects most solute molecules and ions. The general equation describing water transport in FO and RO is

- J
_{w}is the water flux (rate of flow of water per unit cross sectional area), - A is the water permeability constant of the membrane (i.e., the measure of the transport flux of material through the membrane per unit driving force per unit membrane thickness),
- σ is the reflection coefficient (i.e., measure of how much a membrane can “reflect” solute particles from passing through),
- Δπ is the osmotic pressure differential, and
- ΔP is the applied (hydraulic) pressure differential.

Osmotic pressure is the pressure that would prevent the transport of solvent across the membrane, when applied to the more concentrated solution. The hydraulic pressure is generated by pumps that are responsible for maintaining the needed pressure differential. Therefore, in the equation above, P≈0 for FO and P>Δπ for RO.

In addition to the hydraulic dynamics, we also model the reduction of solute molecules in the OA over time. The amount of NaCl in the OA is denoted by x_{NaCl}. Nominally, we assume that we start with 10 gL^{-1} of NaCl in the OA. During nominal operation of the WRS, some NaCl is lost through the membranes (we assume the rate of loss of salt to be -1.11x10^{-5}gL^{-1}s^{-1}). Now, the osmotic potential Δπ is directly proportional to the difference in concentration on the two sides of the semi-permeable membrane. Also, the goal of the controller is to maintain the flow through the FO membrane at approximately 155 Lh^{-1}. To maintain the osmotic pressure difference, and hence, the rate of flow of water through the FO membrane, the controller adds additional amounts of NaCl, represented by x_{NaCl} , to the OA. However, the total amount of NaCl in the OA cannot be more than 30 gL^{-1}, and hence the maximum value of x_{NaCl} can be 20 gL^{-1}. This x_{NaCl} also affects the flow of water through the RO membrane.

The inputs to the WRS nominal model shown in Fig. 4 include u_{Pump1}, u_{Pump2}, u_{Pump4}, and u_{ROPump} the input signal to switch the corresponding pump on or off. The input signals u_{FO} and u_{RO} basically indicate when the FO and RO modules are filled with feed and osmotic agent (we assume there is no water in any of the tanks or plumbings at the start of the simulation), and hence, FO and RO can begin, respectively.

The sensors of the model include q_{Pump2}, q_{Filt1}, q_{Filt2}, q_{Pump4}, q_{ROPump}, q_{Prod}, p_{WT}, p_{FT1}, p_{FT2}, p_{Prod}, p_{Filt1}, p_{Filt2}.

All flows are expressed in Lh^{-1} and all pressures are expressed in psi.