Question

A simple ideal Rankine cycle which uses water as the working fluid operates its condenser at \(40^{\circ} \mathrm{C}\) and its boiler at \(300^{\circ} \mathrm{C}\). Calculate the work produced by the turbine, the heat supplied in the boiler, and the thermal efficiency of this cycle when the steam enters the turbine without any superheating.

Short answer

Based on a simple ideal Rankine cycle using water as the working fluid and given temperatures at the condenser and the boiler, the steps involved in finding the work produced by the turbine, the heat supplied in the boiler, and the thermal efficiency of the cycle include determining the state of water at each point in the cycle, calculating the enthalpy changes between the components, finding the work and heat interactions in the components, and finally, determining the thermal efficiency of the cycle using the net work output and heat supplied in the boiler.

Step-by-step solution

Determine the state of the water

To begin, we will determine the state of the water at each of the four main points in the cycle. We will use the given temperatures for the condenser and the boiler, and we will determine the pressures and specific enthalpies at these points by consulting the steam table. 1. State 1: At the entrance of the turbine (before expanding), the water is saturated steam at \(300^{\circ} \mathrm{C}\): Pressure, \(P_1 = P_{sat} (300^{\circ}\mathrm{C})\) Specific enthalpy of water, \(h_1 = h_{g}(300^{\circ}\mathrm{C})\), where \(h_{g}\) is the specific enthalpy of saturated vapor. 2. State 2: At the exit of the turbine (after expanding), the water is saturated at \(40^{\circ} \mathrm{C}\): Pressure, \(P_2 = P_{sat} (40^{\circ}\mathrm{C})\) Specific enthalpy of water, \(h_2 = h_{f} + x_2(h_{g} - h_{f})\), where \(h_{f}\) is the specific enthalpy of saturated liquid and \(x_2\) is the quality of the steam at state 2. To find \(x_2\), we will use isentropic expansion from state 1 to state 2, which means \(s_1 = s_2\) where \(s\) is the specific entropy. 3. State 3: At the entrance of the pump (before compressing), the water is a saturated liquid at \(40^{\circ} \mathrm{C}\): Pressure, \(P_3 = P_{sat} (40^{\circ}\mathrm{C})\) Specific enthalpy of water, \(h_3 = h_{f}(40^{\circ}\mathrm{C})\). 4. State 4: At the exit of the pump (after compressing), the water is a compressed liquid. We will assume that the pumping process is isentropic, which implies constant specific volume \(v\): Pressure, \(P_4 = P_{sat} (300^{\circ}\mathrm{C})\) Specific enthalpy of water, \(h_4 = h_3 + v_3 (P_4 - P_3)\). Now that we have defined the state of the water at each point in the cycle, we can proceed to calculate the enthalpy changes and determine the work and heat interactions.

Calculate enthalpy changes

To calculate the enthalpy changes within each component of the cycle, we will use the specific enthalpies defined above: 1. Turbine: \(\Delta h_{turbine} = h_1 - h_2\) 2. Condenser: \(\Delta h_{condenser} = h_2 - h_3\) 3. Pump: \(\Delta h_{pump} = h_4 - h_3\) 4. Boiler: \(\Delta h_{boiler} = h_1 - h_4\)

Find work and heat interactions

We can determine the work and heat interactions for each component of the cycle using their respective enthalpy changes: 1. Turbine: \(W_{turbine} = - \Delta h_{turbine}\) 2. Condenser: \(Q_{condenser} = \Delta h_{condenser}\) 3. Pump: \(W_{pump} = - \Delta h_{pump}\) 4. Boiler: \(Q_{boiler} = - \Delta h_{boiler}\) Neglecting the work input required by the pump, we can calculate the net work output of the cycle as: \(W_{net} = W_{turbine} + W_{pump}\)

Determine the thermal efficiency

Finally, we can determine the thermal efficiency of the cycle using the net work output and the heat supplied in the boiler: \(\eta = \frac{W_{net}}{Q_{boiler}}\) With these steps, we have calculated the work produced by the turbine, the heat supplied in the boiler, and the thermal efficiency of the simple ideal Rankine cycle when the steam enters the turbine without any superheating.

Key concepts

Understanding Saturated Steam Properties

Saturated steam is a key term in the context of the Rankine cycle, describing a condition where water and steam exist in equilibrium at a given pressure and temperature. In other words, any additional heat added to saturated steam without raising the pressure will convert some of the liquid water into more steam (a process known as vaporization), and any removal of heat will cause some steam to condense back into water.

For any given pressure, saturated steam has specific properties, including a certain temperature, specific enthalpy (both for the liquid part, known as 'h_f', and the vapor part, known as 'h_g'), and entropy. These properties are crucial for calculating the enthalpy changes in the various components of the Rankine cycle, such as the boiler and the turbine. When steam enters the turbine in the stated problem, it is at the saturated condition at the high temperature of the cycle, which simplifies the extraction of enthalpy values from the steam tables for further calculations.

The Process of Isentropic Expansion

Isentropic expansion is a hypothetical process in which fluid expands in such a manner that entropy remains constant. This concept is paramount for the operation of turbines within the Rankine cycle. During isentropic expansion in an ideal turbine, steam expands from a high pressure to a lower pressure, which results in the production of work.

In a practical sense, this process is an idealization as real turbines cannot achieve perfectly isentropic expansion due to friction and other irreversibilities. However, for calculations, we assume that the turbine is isentropic, enabling us to equate the entropy at the inlet (just before expansion) and the outlet (just after expansion) of the turbine. This simplifies the process of finding the end state of the steam after expansion by using properties of saturated steam at the starting and finishing pressures.

Specific Enthalpy in the Rankine Cycle

Specific enthalpy is a measure of the energy content per unit mass of a substance and is a pivotal parameter in thermodynamics and power cycle analysis. In the Rankine cycle, specific enthalpy helps in determining the energy exchanges that occur in various components, such as the amount of thermal energy added in the boiler, and the work extracted by the turbine.

In the given Rankine cycle problem, specific enthalpy values for steam and water at different states are obtained from steam tables associated with the respective temperatures and pressures. These values are crucial for the calculations of the work produced by the turbine and the heat added or removed in various cycle processes.

Calculating Thermal Efficiency of Power Cycles

Thermal efficiency is a dimensionless measure of the effectiveness of a power cycle in converting heat into work. It is defined as the ratio of net work output of the cycle to the heat input at the high temperature. The higher the thermal efficiency, the better the power plant's performance in terms of energy conversion.

In the Rankine cycle, the thermal efficiency calculation relies on the difference in specific enthalpy values between key points of the cycle. By analyzing the amount of heat energy supplied in the boiler and the net work done by the system, which includes the work produced by the turbine and the work consumed by the pump, one can evaluate the cycle's efficiency. This is especially important in designing and optimizing power plants to ensure that they provide the maximum output with the least possible fuel consumption.