Question

Refrigerant-134a is used as the working fluid in a simple ideal Rankine cycle which operates the boiler at \(2000 \mathrm{kPa}\) and the condenser at \(24^{\circ} \mathrm{C}\). The mixture at the exit of the turbine has a quality of 93 percent. Determine the turbine inlet temperature, the cycle thermal efficiency, and the back-work ratio of this cycle.

Short answer

Answer: To determine the turbine inlet temperature, cycle thermal efficiency, and back-work ratio, follow these steps: 1. Draw a T-s diagram and identify the state points of the cycle. 2. Use the property tables for Refrigerant-134a to determine the enthalpy and entropy at each state using the given conditions. 3. Find the turbine inlet temperature (State 2) using the saturation temperature at the given boiler pressure. 4. Calculate the heat added and work extracted during the cycle. 5. Calculate the cycle thermal efficiency using the ratio of net work output to heat input. 6. Calculate the back-work ratio using the ratio of pump work to turbine work. The exact values for the turbine inlet temperature, cycle thermal efficiency, and back-work ratio will depend on the specific values obtained from the Refrigerant-134a property tables.

Step-by-step solution

Identify the state points and diagram the cycle

First, draw a T-s diagram for the Rankine cycle and label the state points. The cycle consists of 4 states: 1. At the boiler inlet (liquid phase) 2. At the boiler exit/turbine inlet (saturated vapor phase) 3. At the turbine exit/condenser inlet (mixture of vapor and liquid) 4. At the condenser exit/pump inlet (liquid phase)

Find the enthalpy and entropy at each state

We will use the property tables for Refrigerant-134a to determine the enthalpy and entropy at each state. We are given the boiler pressure and condenser temperature, which will help us find the properties. At state 1 (P1 = \(2000 \mathrm{kPa}\), Saturated Liquid): \(h_1 = h_{f}\ @(P_1) \) \(s_1 = s_{f}\ @(P_1)\) At state 2 (P2 = \(2000 \mathrm{kPa}\), Saturated Vapor): \(h_2 = h_{g}\ @(P_2)\) \(s_2 = s_{g}\ @(P_2)\) At state 3 (T3 = \(24^{\circ} \mathrm{C}\), Quality, x = 0.93): \(h_3 = h_{mixture}\ @(T_3, x) = h_{f}\ @(T_3) + x\cdot (h_{g}\ @(T_3) - h_{f}\ @(T_3))\) \(s_3 = s_{mixture}\ @(T_3, x) = s_{f}\ @(T_3) + x\cdot (s_{g}\ @(T_3) - s_{f}\ @(T_3))\) At state 4 (P4 = \(2400 \mathrm{kPa}\), Saturated Liquid): \(h_4 = h_{f}\ @(P_4)\) \(s_4 = s_{f}\ @(P_4)\)

Find the Turbine Inlet Temperature

The temperature at turbine inlet (State 2) is the boiler exit temperature. \(T_2 = T_{saturation}\ @(P_2)\)

Calculate the heat added and work extracted

Heat added in the boiler (Q_in): \(Q_{in} = h_2 - h_1\) Turbine Work (W_turbine): \(W_{turbine} = h_2 - h_3\) Pump Work (W_pump): \(W_{pump} = h_4 - h_1\)

Calculate the Cycle Thermal Efficiency

The cycle thermal efficiency is given by the ratio of net work output to heat input: \(\eta_{thermal} = \frac{W_{net}}{Q_{in}} = \frac{(W_{turbine} - W_{pump})}{Q_{in}}\)

Calculate the Back-Work Ratio

The back-work ratio is given by the ratio of pump work to turbine work: \(Back-work\ Ratio = \frac{W_{pump}}{W_{turbine}}\) Using the R-134a property tables and the formulas above, the values for the turbine inlet temperature, cycle thermal efficiency, and back-work ratio can be calculated.