# Thermodynamic Analysis of Partitioned Combined Cycle using Simple Gases
## 3 Model Description
### Heat Transfer Rates in Combustion Chamber
A moderate factor of (1 + 2 S) in flame length reduction was thus used [62; 63], where 'S' is the swirl number. Once the flame length is calculated then boundaries for the heat balance equations can be defined, as it is assumed that all the heat released by chemical reaction/combustion is in the flame and the rest of the chamber length only acts like a simple heat exchanger. The heat dissipation within the flame length can be estimated by an exponential decay function, as provided in Equation (5), where 'q(L)' is the heat released at any flame length and 'Q\({}_{\rm o}\)' is the total heat released. Partitioned fluid temperature thus measured updated the solution and the flow rates adjustment algorithm was repeated before Equation (4) was solved with new values. Tolerance limits of 0.1 \({}^{\circ}\)C and 10 kW were chosen for partitioned and main fluids flow adjustment. \[q(L)=\ \frac{4.6\ Q_{o}}{F_{L}}exp\left[\frac{-4.6L}{F_{L}}\right] \tag{5}\]  
\[Nu=\ 0.023\ Re^{0.8}\times Pr^{0.4} \tag{6}\]  
The Dittus-Boelter equation (Equation (6)) was used to estimate local heat transfer coefficients in each differential length. The Prandtl number, Reynold number, and L/D ratio were within the specified range of 0.6 to 16, \(>\)10,000, and \(>\)10, respectively. The log mean temperature difference (LMTD) was taken equal to adiabatic temperature and partitioned fluid inlet temperature difference. This introduced an error in the model and was dealt with by decreasing the step size, causing a maximum temperature error of 0.04 \({}^{\circ}\)C. Combined with the heat capacity balance equation, Equation (4) was thus solved. The exchanger total length was fixed to 7 m with 100 tubes. The pressure drop estimation details can be found in the relevant text [64]. Pressure drop can have a drastic effect in some fluids, as discussed earlier. As the output drops because of the pressure drop, if the change in the gas turbine output is more than 10 kW the whole algorithm restarts by adjusting the main fluid (air + fuel) mass flow rates. As the Rankine cycle was chosen as a bottoming cycle in CCGT, the partitioned cycle Rankine cycle conditions must be defined before presenting any comparison. The pressure of the Rankine cycle was adjusted (according to the temperature) as described in previous section. However, the optimum temperature was required. To completely harness the advantage of cycle partitioning, HRSG must be designed to keep the main fluid outlet temperature as low as possible. The HRSG area was kept same as that for the main cycle. However, the flow rates were lesser, if not halved, compared to the original cycle. In order to keep the overall heat transfer coefficient in the same range, the velocities (Reynolds number) of the fluids were adjusted. Thus, after fixing 'U' and 'A', sensitivity analysis was conducted on temperature with respect to overall efficiency. Figure 5: Combustor chamber design and HRSG convergence algorithm. Discretized length = 0.001 m. \(\mathrm{T_{p,i}}\), \(\mathrm{T_{p,o}}\), \(\mathrm{m_{p}}\) = inlet temperature, outlet temperature, and mass flow rate of partitioned fluid, respectively. \(\mathrm{T_{m,i}}\), \(\mathrm{T_{m,o}}\), \(\mathrm{m_{m}}\) = inlet temperature, outlet temperature, and mass flow rate of main fluid, respectively. \(\mathrm{Q_{m}}\) and \(\mathrm{Q_{p}}\) are the heat absorbed or liberated by the main and partitioned fluid, respectively. \(\mathrm{F_{L}}\) = flame length. \(\mathrm{d_{j}}\) = jet injection diameter. dL = step length. Seed values for the algorithm were obtained by equating the mass flow rates in heat capacity equation using 1435 \({}^{\circ}\)C as the partitioned fluid outlet temperature.