A system consists of 2 kg of carbon dioxide gas initially at state 1, where p1 = 1 bar, T1 = 300 K. The system undergoes a power cycle consisting of the following processes: Proces... A system consists of 2 kg of carbon dioxide gas initially at state 1, where p1 = 1 bar, T1 = 300 K. The system undergoes a power cycle consisting of the following processes: Process 1-2: Constant volume to p2 = 6 bar. Process 2-3: Expansion with pv^1.4 = constant. Process 3-1: Constant-pressure compression. Assuming the ideal gas model and neglecting kinetic and potential energy effects, calculate thermal efficiency.

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Understand the Problem

The question asks to determine the thermal efficiency of a power cycle. The system consists of 2 kg of carbon dioxide, initially at a known state (pressure and temperature). It undergoes three processes: constant volume, expansion following a polytropic process, and constant pressure compression. The ideal gas model is assumed, and kinetic and potential energy effects are neglected. We need to combine the equations for each process with the ideal gas law to find the temperatures and pressures at each state, and then the heat transfer during each process. Finally, the thermal efficiency can be calculated from the net work and heat input.

Answer

$\eta = 48.14 \%$
Answer for screen readers

$\eta = 0.4814 = 48.14 %$

Steps to Solve

  1. Determine $T_2$ using the ideal gas law for the constant volume process 1-2

Since $V_1 = V_2$, we have: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

$T_2 = T_1 \frac{P_2}{P_1} = 300 \text{ K} \cdot \frac{6 \text{ bar}}{1 \text{ bar}} = 1800 \text{ K}$

  1. Determine $T_3$ and $V_3$ using the polytropic process 2-3 and the ideal gas law

For the polytropic process $P_2V_2^{1.4} = P_3V_3^{1.4}$ Since $P_3 = P_1$, $V_2 = V_1 = \frac{mRT_1}{P_1}$, we can find $V_3$: $V_3 = V_2 \left( \frac{P_2}{P_3} \right)^{1/1.4} = \frac{mRT_1}{P_1} \left( \frac{P_2}{P_1} \right)^{1/1.4}$

We can now use the ideal gas law to find T3: $T_3 = \frac{P_3V_3}{mR} = \frac{P_1}{mR} \frac{mRT_1}{P_1} \left( \frac{P_2}{P_1} \right)^{1/1.4} = T_1 \left( \frac{P_2}{P_1} \right)^{1/1.4}$ $T_3 = 300 \text{ K} \cdot (6)^{1/1.4} = 300 \text{ K} \cdot 3.684 = 1105.2 \text{ K}$

  1. Calculate the heat transfer for each process. Use $c_v = 0.657 \text{ kJ/kgK}$ and $c_p = 0.846 \text{ kJ/kgK}$

Process 1-2 (constant volume): $Q_{12} = mc_v(T_2 - T_1) = 2 \text{ kg} \cdot 0.657 \frac{\text{kJ}}{\text{kgK}} \cdot (1800 \text{ K} - 300 \text{ K}) = 1.314 \frac{\text{kJ}}{\text{K}} \cdot 1500 \text{ K} = 1971 \text{ kJ}$

Process 2-3 (polytropic): $Q_{23} = \frac{n(P_3V_3 - P_2V_2)}{1-n} = \frac{mR(T_3 - T_2)}{1-n} = \frac{2 \text{ kg} \cdot 0.1889 \frac{\text{kJ}}{\text{kgK}} \cdot (1105.2 \text{ K} - 1800 \text{ K})}{1-1.4} = \frac{0.3778 \frac{\text{kJ}}{\text{K}} \cdot (-694.8 \text{ K})}{-0.4} = \frac{-262.52}{-0.4} \text{ kJ} = 656.3 \text{ kJ}$

Process 3-1 (constant pressure): $Q_{31} = mc_p(T_1 - T_3) = 2 \text{ kg} \cdot 0.846 \frac{\text{kJ}}{\text{kgK}} \cdot (300 \text{ K} - 1105.2 \text{ K}) = 1.692 \frac{\text{kJ}}{\text{K}} \cdot (-805.2 \text{ K}) = -1362.4 \text{ kJ}$

  1. Calculate the net work $W_{net} = Q_{net} = Q_{12} + Q_{23} + Q_{31} = 1971 \text{ kJ} + 656.3 \text{ kJ} - 1362.4 \text{ kJ} = 1264.9 \text{ kJ}$

  2. Calculate the thermal efficiency $\eta = \frac{W_{net}}{Q_{in}} = \frac{W_{net}}{Q_{12} + Q_{23}} = \frac{1264.9 \text{ kJ}}{1971 \text{ kJ} + 656.3 \text{ kJ}} = \frac{1264.9 \text{ kJ}}{2627.3 \text{ kJ}} = 0.4814$

$\eta = 0.4814 = 48.14 %$

More Information

The thermal efficiency of the power cycle is approximately 48.14%. This value indicates the fraction of heat input that is converted into net work output during the cycle.

Tips

A common mistake is not using the correct specific heat values ($c_v$ and $c_p$) for the constant volume and constant pressure processes, respectively. Another mistake is using the wrong sign convention for heat transfer (heat added is positive, heat rejected is negative). For the polytropic process, students may use the wrong equation for heat transfer, or incorrectly calculate work.

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