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Dimensionless Length Dimensionless Length (case I) m LNCLN < m RCP (case II) m LNCLN > m RCP FIGURE 2. Schematic temperature distribution of liquid nitrogen (LN) and refrigerant (REF) in HX2, depending on the relative magnitude of capacity rate of two streams. depending upon the relative magnitude of capacity rate of two streams, as demonstrated in FIGURE 2. In case I, the LN capacity is smaller, thus the temperature difference has its minimum at cold end (e and 5). In case II, on the contrary, the refrigerant capacity is smaller, thus the temperature difference has its minimum at the warm end (i and 6). If the value of effectiveness is given for two heat exchangers, we have 7 unknowns and 6 equations in an ideal cycle, which means that only one variable is independent. We take the independent variable as the pressure ratio, rP, and then the system of equations can be solved for a complete cycle. FIGURE 3 is the plot of FOM as a function of rP for various values of ε = εHX1 = εHX2. The graph is divided into two regions (case I and case II) by a dashed curve. The dots indicating the maximum FOM are located in case II region. A few points should be noted in FIGURE 3. First, the selection of refrigerant from helium and neon does not affect the ideal Brayton cycle, since they have the same k value. There exists a unique optimum for rP to maximize the FOM for any given ε, because a smaller value of rP results in more entropy generation due to large ΔT at warm end of HX2 (case I), a larger value of rP also results in more entropy generation due to large ΔT at cold end of HX2 (case II) and after-cooler. As ε increases, the corresponding optimum of rP decreases, and gets close to the balanced case (dashed curve). The maximum FOM of ideal (internally reversible) Brayton cycle is 93.6% for this application. FIGURE 3. FOM of ideal Brayton cycle as a function of pressure ratio for various values of heat exchanger effectiveness. 1667PDF Image | Thermodynamic design of 10 kW Brayton cryocooler for HTS cable
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