# Performance of a Closed Cycle Power Takeoff for a Shore-Based Wave Energy Device
## 2 Materials and Methods  
### System Modelling  
The CCPTO is idealised as a closed pneumatic system, as shown in Figure 2. This is a similar approach to that previously adopted [15]. However, unlike the previous formulation in which the pressure in each chamber was the primary variable, in this formulation, the instantaneous mass in each chamber is determined directly from the sum of the mass flux through the valves and turbine. The mass flow rates are obtained from the volumetric flow rates, which depend on the instantaneous pressures in the system. In this way, mass is conserved, yielding a more robust model. This 1D model is crude as it excludes the full complexity of the internal flow and it is based on several assumptions:  
* isentropic compression/expansion is assumed in all chambers;
* the sea surface acts as a rigid piston, so no sloshing is considered;
* air density at valves and turbines is assumed to be equal to the upstream value;  
Figure 1: Closed cycle power takeoff (CCPTO) for a shore-based installation. Pneumatic path including non-return valves and turbine through the sea; high-pressure chambers and low-pressure chambers indicated. a rudimentary turbine model is adopted. In effect, the turbine is represented as an orifice, with an area \(A_{T}\) and a turbine coefficient, \(C_{T}\). This is justified by the detailed performance simulations of the turbine shown below. The mass in each chamber can be decomposed into a mean and fluctuating component:  
\[M_{i}=m_{i0}+m_{i} \tag{1}\]  
where \(i=S,H,L\) for sea, high-pressure chambers and low-pressure chambers, respectively. Assuming the system starts at atmospheric conditions, the initial mass in each chamber is given by:  
\[m_{i0}=\frac{V_{i0}}{\rho_{0}} \tag{2}\]  
The principal solution variables are given by the first-order ordinary differential equations, which simply enforce mass conservation (Equations (3)-(5)). The mass flow rates through each component in Figure 2 are denoted as \(q_{i}\), where \(i=(T,L,H)\) for flow through the turbine, low-pressure valve and high-pressure valve, respectively. Sea Chamber mass flux:  
\[\frac{dm_{S}}{dt}=q_{L}-q_{H} \tag{3}\]  
High-Pressure Chamber mass flux:  
\[\frac{dm_{H}}{dt}=q_{H}-q_{T} \tag{4}\]  
Low-Pressure Chamber mass flux:  
\[\frac{dm_{L}}{dt}=q_{T}-q_{L} \tag{5}\]  
The density in each chamber is simply the mass divided by the volume:  
\[\rho_{i}=\frac{M_{i}}{V_{i}} \tag{6}\]  
The volumes, \(V_{i}\), of the high- and low-pressure chambers are fixed and the volume of the Sea Chamber is determined by the instantaneous water level:  
\[V_{S}=V_{S0}+V_{S}(t) \tag{7}\]  
and:  
\[V_{H,L}=V_{H0,L0} \tag{8}\]  
The entire system is driven by the perturbation from the equilibrium of the sea volume in the sea chamber caused by the incoming waves. Figure 1:0 Schematic of lumped parameter model of CCPTO. The flow rates in the system are obtained by treating both valves and the turbine as a simple flow restriction. It is given by:  
Figure 1:1  
where \(C_{i}\) is a discharge coefficient and \(A_{i}\) is proportional to the open area of the component. Note that, if the pressure difference is negative (i.e., \(P_{i+1}>P_{i}\)), the flow rate is 0. For the valves, the value of \(C_{i}\) is 0.6, with a cross-sectional area \(A_{i}\) of 1.5 m\({}^{2}\). The corresponding values for the turbine are \(C_{T}=0.74\) and \(A_{T}=0.017\) m\({}^{2}\). The density, \(\rho\), is assumed to be that of the upstream chamber, but this could be relaxed to be either an average of up- and downstream values or simply set to the reference value. Note that no account is taken of the shape of the ducting close to the valves or turbine. The effect of turbulence, flow separation and irrecoverable pressure drop are embedded in the assumed discharge and turbine coefficients. The instantaneous pressure is required in each chamber to calculate the flow rates. This is calculated based on an isentropic process.