Question

Steam enters the high-pressure turbine of a steam power plant that operates on the ideal reheat Rankine cycle at 800 psia and \(900^{\circ} \mathrm{F}\) and leaves as saturated vapor. Steam is then reheated to \(800^{\circ} \mathrm{F}\) before it expands to a pressure of 1 psia. Heat is transferred to the steam in the boiler at a rate of \(6 \times 10^{4}\) Btu/s. Steam is cooled in the condenser by the cooling water from a nearby river, which enters the condenser at \(45^{\circ} \mathrm{F}\). Show the cycle on a \(T\) -s diagram with respect to saturation lines, and determine ( \(a\) ) the pressure at which reheating takes place, \((b)\) the net power output and thermal efficiency, and \((c)\) the minimum mass flow rate of the cooling water required.

Short answer

#tag_title# Question #tag_content# Calculate the pressure at which reheating takes place, the net power output, the thermal efficiency, and the minimum mass flow rate of cooling water for a steam power plant operating on the ideal reheat Rankine cycle, with given initial and final pressures, temperatures at different points, and a heat transfer rate of \(6 \times 10^{4}\) Btu/s in the boiler.

Step-by-step solution

Determining the properties of the steam at various states

For this step, let's define the states: 1. Inlet of high-pressure turbine 2. Outlet of high-pressure turbine (saturated vapor) 3. Inlet of low-pressure turbine (after reheating) 4. Outlet of low-pressure turbine (1 psia) We will use saturation tables and interpolation to find the enthalpy and entropy values at each point.

Calculate the turbine and pump work rates

For an ideal turbine and pump, isentropic processes take place. We need now to calculate the work rates of the high and low-pressure turbines and the pump. For the turbines, the work per unit mass is given by: $$W_{T} = h_{in} - h_{out}$$ For the pump, the work per unit mass is: $$W_{P} = v_{f} \cdot (P_{2} - P_{1})$$

Calculating the heat transfer rates

We will calculate heat transfer rates in the boiler and the reheat section which will be denoted by \(Q_{in1}\) and \(Q_{in2}\), respectively. For the boiler, we have: $$Q_{in1} = h_{1} - h_{4_{s}}$$ For the reheat section, we have: $$Q_{in2} = h_{3} - h_{2}$$

Determine the pressure at which reheating takes place

Now, we have the heat transfer rates and turbine work rates. We can calculate the pressure at which reheating takes place using the reheating temperature \(T_{3} = 800^{\circ} \mathrm{F}\), which will be the pressure at state 3. We can find the corresponding pressure using the steam tables.

Calculate the net power output

The net power output is given by the difference between the total rate of work output by the turbines and the work required to operate the pump. The net power output can be defined as: $$W_{net} = W_{T1} + W_{T2} - W_{P}$$

Calculate the thermal efficiency

The thermal efficiency of the cycle can be defined as the ratio of the net power output to the total heat input, which includes both boiler and reheat sections: $$\eta = \frac{W_{net}}{Q_{in1} + Q_{in2}}$$

Calculate the minimum mass flow rate of the cooling water

We know that heat is transferred to the steam in the boiler at a rate of \(6 \times 10^{4}\) Btu/s. To find the minimum mass flow rate of cooling water, we can apply a simple energy balance, considering the heat input to the cycle and the heat rejected in the condenser: $$Q_{in} - Q_{out} = m_{steam} \cdot W_{net}$$ where \(m_{steam}\) is the mass flow rate of steam, and \(Q_{out}\) is the heat rejected by the condenser. To find \(Q_{out}\), we need to determine the temperature and enthalpy change for the cooling water: $$Q_{out} = m_{water} \cdot c_{p} \cdot \Delta T$$ where \(c_{p}\) is the specific heat of water and \(\Delta T\) is the temperature difference for the cooling water. By substituting the known values, we can solve for the minimum mass flow rate of the cooling water.