At a steam power plant, steam engines work in pairs, the output of heat from one being the approximate heat input of the second. The operating temperatures of the first are 520°C and 320°C, and of the second 310°C and 170°C. If the heat of combustion of coal is 2.8x107 J/kg, at what rate must coal be burned if the plant is to put out 1494 MW of power? Assume the efficiency of the engines is 65% of the ideal (Carnot) efficiency.
Well, that's quite a steamy question you've got there! Let me calculate the coal burning rate for you.
First, let's calculate the total power output of the steam engines. We know that the plant is putting out 1494 MW of power, so the power output of each engine will be half of that, which is 747 MW.
Now, we need to take into account the efficiency of the engines, which is 65% of the ideal efficiency. The ideal efficiency is given by the Carnot efficiency formula, which is (1 - Tc/Th), where Tc is the cold temperature and Th is the hot temperature.
For the first pair of engines, the hot temperature (Th) is 520°C, and the cold temperature (Tc) is 320°C. So, the efficiency of the first pair would be (1 - 320/520).
For the second pair of engines, Th is 310°C, and Tc is 170°C. So, the efficiency of the second pair would be (1 - 170/310).
To find the total efficiency, we take the average of the efficiencies of both pairs of engines.
Now, let's calculate the coal burning rate. We can use the formula:
Power output = (Coal burning rate) x (Heat of combustion) x (Efficiency)
Plugging in the values we have:
747 MW = (Coal burning rate) x (2.8x10^7 J/kg) x (Average efficiency)
From here, we can solve for the coal burning rate:
(Coal burning rate) = (747 MW) / ((2.8x10^7 J/kg) x (Average efficiency))
So, once we calculate the average efficiency and do the math, we can find the rate at which the coal must be burned.
To find the rate at which coal must be burned, we need to use the given information about the operating temperatures and the efficiency of the steam engines.
Step 1: Calculate the ideal (Carnot) efficiency of the steam engines.
The Carnot efficiency, denoted by ηc, is given by the formula:
ηc = 1 - (T2 / T1), where T1 is the higher operating temperature and T2 is the lower operating temperature.
Given:
T1 (first engine) = 520°C = 520 + 273 = 793 K
T2 (first engine) = 320°C = 320 + 273 = 593 K
T1 (second engine) = 310°C = 310 + 273 = 583 K
T2 (second engine) = 170°C = 170 + 273 = 443 K
Applying the formula to each engine:
ηc (first engine) = 1 - (593 / 793)
ηc (second engine) = 1 - (443 / 583)
Step 2: Calculate the efficiency of the steam engines.
Given that the efficiency of the engines is 65% of the Carnot efficiency, we can calculate the actual efficiency (η) using the formula:
η = ηc * (actual efficiency percentage)
Given:
ηc (first engine) = 1 - (593 / 793)
ηc (second engine) = 1 - (443 / 583)
Actual efficiency percentage = 65%
Calculating the actual efficiency for each engine:
η (first engine) = ηc (first engine) * 0.65
η (second engine) = ηc (second engine) * 0.65
Step 3: Calculate the total efficiency of the steam power plant.
Since the engines work in pairs and the output of heat from one is the approximate heat input of the second, the combined efficiency (η_total) is calculated using the formula:
η_total = η1 * η2
Given:
η (first engine) = ηc (first engine) * 0.65
η (second engine) = ηc (second engine) * 0.65
Calculating the total efficiency of the plant:
η_total = η (first engine) * η (second engine)
Step 4: Calculate the rate at which coal must be burned.
The rate of burning coal, denoted by Q, is given by the formula:
Q = power output / (efficiency * heat of combustion)
Given:
Power output of the plant = 1494 MW = 1494 * 10^6 W
Heat of combustion of coal = 2.8 * 10^7 J/kg
η_total = η (first engine) * η (second engine)
Calculating the rate at which coal must be burned:
Q = Power output of the plant / (η_total * heat of combustion)
Note: Make sure to convert the power output to the same units as the heat of combustion (Joules).
Please substitute the given values and calculate the rate at which coal must be burned.
To solve this problem, we can use the first law of thermodynamics, which states that the net work output of a heat engine is equal to the difference between the heat input and the heat rejected. In this case, since the two steam engines work in pairs, we can analyze each one separately and then combine the results.
First, let's calculate the heat input and heat rejected in the first engine.
The heat input for the first engine can be calculated using the formula:
Q₁ = m₁ * Cp₁ * (T₁h - T₁c)
Where:
Q₁ = Heat input for the first engine
m₁ = Mass flow rate of steam in kg/s
Cp₁ = Specific heat capacity of steam in J/(kg·°C)
T₁h = High temperature in °C for the first engine
T₁c = Low temperature in °C for the first engine
Similarly, the heat rejected for the second engine can be calculated using the formula:
Q₂ = m₂ * Cp₂ * (T₂h - T₂c)
Where:
Q₂ = Heat input for the second engine
m₂ = Mass flow rate of steam in kg/s
Cp₂ = Specific heat capacity of steam in J/(kg·°C)
T₂h = High temperature in °C for the second engine
T₂c = Low temperature in °C for the second engine
Next, we can calculate the net work output for each engine:
W₁ = Q₁ * η₁
Where:
W₁ = Net work output for the first engine
η₁ = Efficiency of the first engine
W₂ = Q₂ * η₂
Where:
W₂ = Net work output for the second engine
η₂ = Efficiency of the second engine
Since the engines work in pairs, the net work output of the second engine is equal to the net work input of the first engine.
So, W₂ = W₁
Now, let's calculate the mass flow rate of steam (m₁ and m₂) using the given power output of the plant:
Power output = W₁ + W₂
1494 MW = W₁ + W₁
1494 MW = 2W₁
W₁ = 747 MW
Now, let's substitute the values of W₁, T₁h, T₁c, T₂h, T₂c, η₁, and η₂ to calculate the mass flow rates of steam (m₁ and m₂).
Finally, we will calculate the rate at which coal must be burned.
Mass flow rate of coal = (Power output) / (Heating value of coal)
Where,
Power output = 1494 MW
Heating value of coal = 2.8x10^7 J/kg
Substitute the given values to calculate the rate at which coal must be burned.