Known An ideal reheat cycle operates with steam as the working fluid. Operating pressures and temperatures are specified, and the net power output is given.
Find Determine the thermal efficiency, the mass flow rate of the steam, in kg/h, and the heat transfer rate from the condensing steam as it passes through the condenser, in MW. Discuss.
Schematic and Given Data:
Engineering Model
1. Each component in the cycle is analyzed as a control volume at steady state. The control volumes are shown on the accompanying sketch by dashed lines.
2. All processes of the working fluid are internally reversible.
3. The turbine and pump operate adiabatically.
4. Condensate exits the condenser as saturated liquid.
5. Kinetic and potential energy effects are negligible.
Analysis To begin, we fix each of the principal states. Starting at the inlet to the first turbine stage, the pressure is 8.0 MPa and the temperature is 480°C, so the steam is a superheated vapor. From Table A-4, h_{1}=3348.4 kJ / kg \text { and } s_{1}=6.6586 kJ / kg \cdot K.
State 2 is fixed by p_{2}=0.7 MPa \text { and } s_{2}=s_{1} for the isentropic expansion through the first-stage turbine. Using saturated liquid and saturated vapor data from Table A-3, the quality at state 2 is
x_{2}=\frac{s_{2}-s_{ f }}{s_{ g }-s_{ f }}=\frac{6.6586-1.9922}{6.708-1.9922}=0.9895
The specific enthalpy is then
\begin{aligned}h_{2} &=h_{ f }+x_{2} h_{ fg } \\&=697.22+(0.9895) 2066.3=2741.8 kJ / kg\end{aligned}
State 3 is superheated vapor with p_{3}=0.7 MPa and T_{3}=440^{\circ} C, so from Table A-4, h_{3}=3353.3 kJ/kg and s_{3}=7.7571 kJ/kg ⋅ K.
To fix state 4, use p_{4}=0.008 MPa \text { and } s_{4}=s_{3} for the isentropic expansion through the second-stage turbine. With data from Table A-3, the quality at state 4 is
x_{4}=\frac{s_{4}-s_{ f }}{s_{ g }-s_{ f }}=\frac{7.7571-0.5926}{8.2287-0.5926}=0.9382
The specific enthalpy is
h_{4}=173.88+(0.9382) 2403.1=2428.5 kJ / kg
State 5 is saturated liquid at 0.008 MPa, so h_{5}=173.88 kJ/kg. Finally, the state at the pump exit is the same as in Example 8.1, so h_{6}=181.94 kJ/kg.
a. The net power developed by the cycle is
\dot{W}_{\text {cycle }}=\dot{W}_{ t 1}+\dot{W}_{ t 2}-\dot{W}_{ p }
Mass and energy rate balances for the two turbine stages and the pump reduce to give, respectively,
\begin{aligned}&\text { Turbine 1: } \quad \dot{W}_{ t 1} / \dot{m}=h_{1}-h_{2}\\&\text { Turbine 2: } \quad \dot{W}_{ t 2} / \dot{m}=h_{3}-h_{4}\\&\text { Pump: } \quad \dot{W}_{ p } / \dot{m}=h_{6}-h_{5}\end{aligned}
where \dot{m} is the mass flow rate of the steam.
The total rate of heat transfer to the working fluid as it passes through the boiler–superheater and reheater is
\frac{\dot{Q}_{\text {in }}}{\dot{m}}=\left(h_{1}-h_{6}\right)+\left(h_{3}-h_{2}\right)
Using these expressions, the thermal efficiency is
\eta=\frac{\left(h_{1}-h_{2}\right)+\left(h_{3}-h_{4}\right)-\left(h_{6}-h_{5}\right)}{\left(h_{1}-h_{6}\right)+\left(h_{3}-h_{2}\right)}
=\frac{(3348.4-2741.8)+(3353.3-2428.5)-(181.94-173.88)}{(3348.4-181.94)+(3353.3-2741.8)}
=\frac{606.6+924.8-8.06}{3166.5+611.5}=\frac{1523.3 kJ / kg }{3778 kJ / kg }=0.403(40.3 \%)
b. The mass flow rate of the steam can be obtained with the expression for net power given in part (a).
\dot{m}=\frac{\dot{W}_{\text {cycle }}}{\left(h_{1}-h_{2}\right)+\left(h_{3}-h_{4}\right)-\left(h_{6}-h_{5}\right)}
=\frac{(100 MW )\left|3600 s / h \| 10^{3} kW / MW \right|}{(606.6+924.8-8.06) kJ / kg }=2.363 \times 10^{5} kg / h
c. The rate of heat transfer from the condensing steam to the cooling water is
\dot{Q}_{\text {out }}=\dot{m}\left(h_{4}-h_{5}\right)
=\frac{2.363 \times 10^{5} kg / h (2428.5-173.88) kJ / kg }{\left|3600 s / h \| 10^{3} kW / MW \right|}=148 MW
To see the effects of reheat, we compare the present values with their counterparts in Example 8.1. With superheat and reheat, the thermal efficiency is increased over that of the cycle of Example 8.1. For a specified net power output (100 MW), a larger thermal efficiency means that a smaller mass flow rate of steam is required. Moreover, with a greater thermal efficiency the rate of heat transfer to the cooling water is also less, resulting in a reduced demand for cooling water. With reheating, the steam quality at the turbine exhaust is substantially increased over the value for the cycle of Example 8.1.
Skills Developed
Ability to…
• sketch the T–s diagram of the ideal Rankine cycle with reheat.
• fix each of the principal states and retrieve necessary property data.
• apply mass and energy balances.
• calculate performance parameters for the cycle.
Quick Quiz
What is the rate of heat addition for the reheat process, in MW, and what percent is that value of the total heat addition to the cycle? Ans. 40.1 MW, 16.2%.
