COMPUTATION OF THE BRAYTON CYCLE
The following subsections consider a Brayton cycle with a constant specific heat working fluid and an OCR defined by \(r_p\). The cycle includes \(N_t\) and \(N_c\) turbine and compression stages with isentropic efficiencies \(\eta_t\) and \(\eta_c\) (The associated \(N-1\) intercooling and reheating stages are implied). The cycle also includes a regenerator with effectiveness \(\epsilon\). Since \(r_p\) is known and \(P_{H} = r_pP_{L}\), we need only specify either \(P_{H}\) or \(P_{L}\). Finally, the max and min fluid temperatures are defined as \(T_{H}\) and \(T_{L}\). The individual compression ratios for compressors (\(r_{pc}\)) and turbines (\(r_{pt}\)) in a cycle can be found by:
- \(r_{pc} = r_p^{(\frac{1}{N_c})}\)
- \(r_{pt} = r_p^{(\frac{1}{N_t})}\)
Previously, we showed that for both compressors and turbines:
\[ \exists a_1 \in \mathbb{R} : T_2 = a_1T_1 \] \[ \exists a_2 \in \mathbb{R} : P_2 = a_2P_2 \]Where subscripts 1 and 2 correspond to their designation in section [braytonSection]. Similarly, in the case of multistaged cycles, we can say that:
\[ \begin{array}{l} \exists b_1 \in \mathbb{R} : b_1 = \frac{1}{a_1} \to T_3 = b_1T_2 \iff b_1a_1T_1 \\ \exists b_2 \in \mathbb{R} : b_2 = 1 \to P_3 = b_2P_2 \iff b_2a_2P_1 \end{array} \]For reasons shown in the following paragraphs, we will convert these vectors as diagonal matrices of the form:
\[ \begin{align*} \mathbf{A} &\equiv \text{diag}([a_1, a_2]) \\ \mathbf{B} &\equiv \text{diag}([b_1, b_2]) \end{align*} \]Next, let us define a general state vector for cycle index i:
\[ \theta_i \in \mathbb{R}^2 : \theta_i = \left[ {\begin{array}{c} T_{i} \\ P_i \end{array} } \right] \]We can then write the high and low temperature and pressure points in vector form, corresponding too:
\[ \begin{array}{ll} \theta_L = \left[ {\begin{array}{c} T_{L} \\ P_{L} \end{array} } \right] & \theta_H = \left[ {\begin{array}{c} T_{H} \\ P_{H} \end{array} } \right] \end{array} \]Then we can say that there is a linear transfer matrix relating the outlet state of an \(N\) stage compressor or turbine circuit to the inlet state vector:
\[ \theta_{\text{out}} = (\mathbf{AB})^{N-1}\mathbf{A}\theta_{\text{in}} \]For the compressor stages, \(\theta_{\text{in}} = \theta_L\) and for turbine stages \(\theta_{\text{in}} = \theta_H\). Furthermore, the outlet state vectors from the compressor and turbine side correspond to the inlet conditions \(T_{ci}\) and \(T_{hi}\) in section [regen]. Alternatively, we can phrase the equation in index form. For an \(N\) stage cycle, there are a total of \(2N-1\) individual states after the initial state 1. Therefore, for state \(i\), where \(i \geq 2\):
\[ \theta_{i} = \begin{cases} (\mathbf{AB})^{\frac{i}{2}-1}\mathbf{A}\theta_{1} & \text{i is even, after turbine or compressor} \\ (\mathbf{BA})^{\frac{i-1}{2}}\theta_{1} & \text{i is odd, after reheat or intercool} \end{cases} \]Equations [aInit] - [transform] can be used to find the hot and cold inlets \(T_{ci}, T_{ti}\) of the regenerator. Here, we elect to use subscripts c and t for compressor and turbine side. Then, from equation [simpEff] and some simple algebra, we can say:
\[ \begin{align} \left[ {\begin{array}{c} T_{co} \\ T_{to} \end{array} } \right] = \left[ {\begin{array}{cc} (1-\epsilon) & \epsilon \\ \epsilon & (1-\epsilon) \end{array} } \right] \left[ {\begin{array}{cc} T_{ci}\\ T_{ti} \end{array} } \right] \end{align} \]In a quasi-ideal Brayton cycle with steady flow processes, we can compute the regular and specific energy transfer rate with:
\[ \dot{E} = \dot{m}c_p(T_{f} - T_i) \] \[ \dot{e} = c_p(T_{f} - T_i) \]Where subscripts \(i\) and \(f\) correspond to initial and final. \(\dot{E}\) and \(\dot{e}\) are in units W and \(kJ/kg\) respectively. In a power cycle, we only need to know the outlet temperature of a single compression and turbine stage to calculate all of the work of the system. Remembering that for a single stage we can say \(T_f = a_1T_i\), we can re-write equation [edot] as:
\[ \begin{aligned} \dot{e} &= c_p(a_1T_i - T_i) \\ &= c_p(a_1 - 1)(T_i) \end{aligned} \]For the turbine and compressor (T\&C) circuits, let \(\mathbf{A_t}\) and \(\mathbf{A_c}\) correspond to their respective \(\mathbf{A}\) matrix from the previous section. Additionally, we identify \(a_{1t} \in \mathbf{A_t}\) and \(a_{1c} \in \mathbf{A_c}\) as the (1,1) elements in either matrix. We can then form a set of equations:
\[ \begin{aligned} \dot{e_c} &= c_p(a_{1c} - 1)(T_{L}) \\ \dot{e_t} &= c_p(a_{1t} - 1)(T_{H}) \end{aligned} \]We can then calculate the total specific work and heat loads for any compressor and turbine circuit with an arbitrary number of stages:
\[ \begin{aligned} \dot{w_c} &= N_c\dot{e_c} \\ \dot{q_c} &= -(N_c-1)\dot{e_c} \\ \dot{w_t} &= N_t\dot{e_t} \\ \dot{q_t} &= -(N_t-1)\dot{e_t} \end{aligned} \]To simplify the above statements, we write:
\[ \begin{aligned} \phi &\equiv \left[ {\begin{array}{c} T_{L} \\ T_{H} \end{array} } \right] \\ \mathbf{C} &\equiv \left[ {\begin{array}{cc} c_p(a_{1c}-1) & 0 \\ 0 & c_p(a_{1t}-1) \end{array} } \right] \\ \mathbf{N} &\equiv \left[ {\begin{array}{cc} N_c & 0 \\ (1-N_c) & 0 \\ 0 & N_t \\ 0 & (1-N_t) \end{array} } \right] \end{aligned} \]Now, it can easily be shown that the specific net work and heat transfer are calculated by:
\[ \left[ {\begin{array}{c} \dot{w_c} \\ \dot{q_c} \\ \dot{w_t} \\ \dot{q_t} \end{array}} \right] = \mathbf{N}\mathbf{C}\phi \]Finally, we calculate the thermal heat accepted and rejected from the cycle. Using variables defined in [regenEq]:
\[ \left[ {\begin{array}{c} \dot{q}_{primary} \\ \dot{q}_{rejected} \end{array}} \right] = \left[ {\begin{array}{c} T_{H} - T_{co} \\ T_{L} - T_{to} \end{array}} \right]c_p \]