A heat engine operates in a Carnot cycle between 78.0°C and 340°C. It absorbs 22,000 J of energy per cycle from the hot reservoir. The duration of each cycle is 3.00 s.
(a) What is the mechanical power output of this engine? (answer in kW)
(b) How much energy does it expel in each cycle by heat? (answer in kJ)
To solve this problem, we can use the Carnot cycle efficiency formula and the given information.
(a) To find the mechanical power output, we need to calculate the work done by the engine per cycle.
The Carnot cycle efficiency formula is:
Efficiency = (1 - Tc/Th) * 100%
where Tc is the temperature of the cold reservoir and Th is the temperature of the hot reservoir.
Given: Tc = 78.0°C = 78.0 + 273.15 = 351.15 K
Th = 340°C = 340 + 273.15 = 613.15 K
Using the efficiency formula, we can find the efficiency of the Carnot cycle:
Efficiency = (1 - Tc/Th) * 100%
= (1 - 351.15/613.15) * 100%
≈ 42.7%
The efficiency of the Carnot cycle gives the fraction of energy absorbed from the hot reservoir that is converted into useful work.
The energy absorbed per cycle is given as 22,000 J.
So, the work done per cycle (W) can be calculated as:
W = Efficiency * Energy absorbed
= 0.427 * 22,000 J
≈ 9,394 J
The duration of each cycle is given as 3.00 s.
Therefore, the mechanical power output (P) can be calculated as:
P = W / Time
= 9,394 J / 3 s
≈ 3,131 W = 3.131 kW
Therefore, the mechanical power output of this engine is approximately 3.131 kW.
(b) To find the energy expelled by heat per cycle, we can subtract the work done per cycle (W) from the energy absorbed per cycle.
The energy expelled per cycle (Q) can be calculated as:
Q = Energy absorbed - Work done
= 22,000 J - 9,394 J
≈ 12,606 J = 12.606 kJ
Therefore, the engine expels approximately 12.606 kJ of energy in each cycle by heat.
To solve this problem, we can use the formulas for the efficiency and the work done by the engine in a Carnot cycle.
The efficiency of a Carnot cycle is given by the formula:
η = 1 - (Tc/Th)
where η is the efficiency, Tc is the temperature of the cold reservoir, and Th is the temperature of the hot reservoir.
(a) To find the mechanical power output of the engine, we need to calculate the work done per cycle. The work done per cycle is given by the formula:
W = Qh - Qc
where W is the work done, Qh is the heat absorbed from the hot reservoir, and Qc is the heat expelled to the cold reservoir.
We know that the engine absorbs 22,000 J of energy per cycle from the hot reservoir (Qh = 22,000 J), and we need to find Qc.
Using the formula for efficiency, we can rearrange it to solve for Qc:
Qc = (1 - η) * Qh
We are given the temperature of the hot reservoir as 340°C, which we need to convert to Kelvin by adding 273.15:
Th = 340°C + 273.15 = 613.15 K
We are also given the temperature of the cold reservoir as 78.0°C, which we again need to convert to Kelvin:
Tc = 78.0°C + 273.15 = 351.15 K
Now we can calculate the efficiency of the engine:
η = 1 - (351.15 K / 613.15 K)
Next, we can calculate Qc:
Qc = (1 - η) * 22,000 J
Finally, we can calculate the work done per cycle:
W = Qh - Qc
(b) Given that the duration of each cycle is 3.00 seconds, we can find the mechanical power output of the engine:
Mechanical power output = W / (duration of each cycle)
Let's calculate the answers step by step:
(a) First, let's calculate the efficiency of the Carnot engine:
η = 1 - (351.15 K / 613.15 K)
η ≈ 0.428 (rounded to 3 decimal places)
Now, let's calculate Qc:
Qc = (1 - η) * 22,000 J
Qc ≈ 0.572 * 22,000 J
Qc ≈ 12,584 J (rounded to the nearest whole number)
Finally, let's calculate the work done per cycle:
W = Qh - Qc
W = 22,000 J - 12,584 J
W ≈ 9,416 J (rounded to the nearest whole number)
Now, let's calculate the mechanical power output:
Mechanical power output = W / (duration of each cycle)
Mechanical power output = 9,416 J / 3 s
Mechanical power output = 3,138.67 W (rounded to 2 decimal places)
Mechanical power output = 3.14 kW (rounded to 2 decimal places)
(b) To find the energy expelled by heat in each cycle, we can simply convert the value of Qc from joules to kilojoules:
Energy expelled by heat = Qc / 1000
Energy expelled by heat = 12,584 J / 1000
Energy expelled by heat = 12.58 kJ (rounded to 2 decimal places)
So, the answers are:
(a) The mechanical power output of the engine is 3.14 kW.
(b) The engine expels 12.58 kJ of energy by heat in each cycle.