An electric generating station is designed to have an electric output power of 1.40 MW using a turbine with two-thirds the efficiency of a Carnot engine. The exhaust energy is transferred by heat into a cooling tower at 110°C. (a) Find the rate at which the station exhausts energy by heat as a function of the fuel combustion temperature Th. (b) If the firebox is modified to run hotter by using more advanced combustion technology, how does the amount of energy exhaust change? (c) Find the exhaust power for Th = 800°C. (d) Find the value of Th for which the exhaust power would be only half as large as in part (c). (e) Find the value of Th for which the exhaust power would be one-fourth as large as in part (c).
(a)
The rate at which the station exhaust energy by heat as a function of the fuel combustion temperature
Answer to Problem 15P
The rate at which the station exhaust energy by heat as a function of the fuel combustion temperature
Explanation of Solution
The rate of work output of the engine is
Write the formula to calculate the carnot efficiency of the engine.
Here,
The actual efficiency of the engine is equal to two-thirds of the efficiency of the carnot engine.
Here,
Substitute
Write the formula to calculate the rate of heat input to the engine.
Here,
Write the formula to calculate the rate at which the station exhaust energy by heat as a function of the fuel combustion temperature
Here,
Substitute
Substitute
Conclusion:
Substitute
Thus, the rate at which the station exhaust energy by heat as a function of the fuel combustion temperature
(b)
The effect on the amount of the energy if the firebox is modified to run hotter by using more advanced combustion technology.
Answer to Problem 15P
The amount of the energy exhaust decreases as the fire box temperature increases.
Explanation of Solution
If the firebox is modified to run hotter by using more advanced combustion technology, the exhaust power increases by a factor of
Conclusion:
The amount of the energy exhaust decreases as the fire box temperature increases.
(c)
The exhaust power for
Answer to Problem 15P
The exhaust power for
Explanation of Solution
The rate of work output of the engine is
From equation (IV), write the formula to calculate the exhaust power for
Conclusion:
Substitute
Thus, the exhaust power for
(d)
The value of
Answer to Problem 15P
The value of
Explanation of Solution
The rate of work output of the engine is
Write the expression for the exhaust power whuch would be only half as large as in part (c).
Here,
Substitute
Thus, the exhaust power whuch would be only half as large as in part (c) is
From equation (IV), Write the formula to calculate the value of
Conclusion:
Substitute
Thus, the value of
(e)
The value of
Answer to Problem 15P
No temperature value will provide an exhaust power of one-fourth of the value in part (c).
Explanation of Solution
The rate of work output of the engine is
Write the expression for the exhaust power whuch would be one-fourth as large as in part (c).
Here,
Substitute
Thus, the exhaust power whuch would be one-fourth as large as in part (c) is
Conclusion:
From equation (IV), Write the formula to calculate the value of
Thus, the given exhaust power is lesser than the minimum possible exhaust power
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