Consider a diesel-fired steam power plant that produces 400 MW of electric power. The turbine inlet conditions of 5 MPa and 500°C and a condenser pressure of 25 kPa. The state of the water at the inlet of the pump is saturated liquid. The steam has quality of (x = 0.88) at the inlet of the condenser. Water from a nearby river is used to cool the condenser. To prevent thermal pollution, the cooling water is not allowed to experience more than 8°C as it flows through the condenser. The diesel has a heating value (energy released when the fuel is burned) of 40,100 kJ/kg. Assuming that 80 percent of this energy is transferred to the steam in the boiler and that the electric generator has an efficiency of 95 percent, determine (a) the overall plant efficiency (the ratio of net electric power output to the energy input as fuel), (b) the required rate of diesel supply and (c) the minimum mass flow rate of the cooling water from the river.
Given: W ̇=400MW, Q=40100 kJ/kg, η_generator=95%=0.95,η_Boiler=80%=0.80
Turbine inlet condition
P=5MPa,T_3=500℃
Using Properties table for steam
T_3>T_Saturation (=264℃), Hence state is superheated
h_3g=3433.8kJ/kg,s_3g=6.976kJ/(kg-K)
Turbine outlet or inlet to condenser condition
P=25 kPa=0.25 MPa,x=0.88
h_4f=271.9 kJ/kg, h_4fg=2618.2 kJ/kg
h_4=h_4f+xh_4fg
h_4=271.9+0.88(2618.2)
h_4=2575.92 kJ/kg
Condensor outlet or inlet to pump condition
P=25 kPa,Staturation state (Given)
〖h_f=h〗_1=271.9 kJ/kg, v_f=0.001020 m^3/kg
Work done by pump,W_p=-∫▒〖v_f dP〗
W_p=0.001020 (5000-25)=5 kJ/kg
Pump outlet or inlet to boiler condition
h_2=h_1+W_p
h_2=271.9 +5
h_2=276.9
Heat supplied to the boiler
〖Q=(h〗_3g-h_2)=3433.8-276.9=3156.9
Work done by turbine
W_T=h_3g-h_4=3433.8-2575.92=857.88
Cycle efficiency
η_cycle=(W_T-W_P)/Q
η_cycle=(857.88-5)/(3156.9)
η_cycle=27%
Overall efficiency,(η)= η_cycle×η_generator× η_Boiler
η=0.27×0.95×0.80
η=20.52%
Diesel supply ((m_f ) ̇)
(m_f ) ̇=(power generated(W ̇))/ηQ
(m_f ) ̇=(400×1000)/((0.2052)(40100))
(m_f ) ̇=48.61 kg/s
Mass flow rate of cooling water ((m_w ) ̇)
Heat rejected by gases = heat taken by water
Temperature at state 4, T_4=65℃ (using steam table)
(m_f ) ̇(h_4-h_1)= (m_w ) ̇c_P (T_4-T_w)
(m_w ) ̇=(48.61(2575.92-271.9) )/(4.18×(65-8))
(m_w ) ̇=470 kg/s
To solve this problem, we need to consider several steps. Let's go through them one by one:
Step 1: Determine the overall energy input to the power plant
The overall energy input will be the energy transferred to the steam in the boiler, which is 80% of the heating value of the diesel fuel. The heating value is given as 40,100 kJ/kg. Therefore, the overall energy input per unit mass of diesel fuel can be calculated as:
Energy input = 80% * 40,100 kJ/kg = 32,080 kJ/kg
Step 2: Determine the mass flow rate of the diesel fuel
To calculate the required rate of diesel supply, we need to know the net electric power output of the plant. It is given as 400 MW. The net electric power output can be calculated by dividing the energy output by the efficiency of the electric generator:
Energy output = Net electric power output / Electric generator efficiency
Energy output = 400 MW / 95% = 421.05 MW
Now, we can determine the mass flow rate of the diesel fuel using the formula:
Mass flow rate of diesel fuel = Energy output / Energy input
Mass flow rate of diesel fuel = 421.05 MW / 32,080 kJ/kg
Step 3: Determine the energy rejected by the condenser
To calculate the minimum mass flow rate of cooling water from the river, we need to determine the energy rejected by the condenser. First, we need to calculate the specific enthalpy at the inlet and outlet of the condenser.
Specific enthalpy at the inlet of the condenser:
From the given steam quality (x = 0.88) at the inlet of the condenser, we can use the steam tables to find the specific enthalpy at that condition.
Specific enthalpy at the outlet of the condenser:
The condenser pressure is given as 25 kPa, which is below the saturation pressure corresponding to the inlet temperature of 500°C. Therefore, the water leaving the condenser will be in the subcooled liquid state. We can find the specific enthalpy of subcooled water from the steam tables at the condenser pressure.
Step 4: Determine the minimum mass flow rate of the cooling water
To calculate the minimum mass flow rate of the cooling water, we need to calculate the energy rejected by the condenser and divide it by the temperature rise allowed for the cooling water.
Energy rejected by the condenser = Mass flow rate of cooling water * (Specific enthalpy at the inlet of the condenser - Specific enthalpy at the outlet of the condenser)
Now we can rearrange the equation to solve for the mass flow rate of the cooling water:
Mass flow rate of cooling water = Energy rejected by the condenser / (Temperature rise allowed for cooling water * Specific heat capacity of water)
Note: The specific heat capacity of water is approximately 4.18 kJ/kg°C.
Step 5: Calculate the overall plant efficiency
The overall plant efficiency is the ratio of net electric power output to the energy input as fuel:
Overall plant efficiency = Net electric power output / (Mass flow rate of diesel fuel * Energy input)