Question

A heat engine operates in a Carnot cycle between 84.0∘C and 345∘C. It absorbs 20,000 J of energy per cycle from the hot reservoir. The duration of each cycle is 5.00 s. (a) What is the mechanical power output of this engine? kW (b) How much energy does it expel in each tycle by heat? k)

Answer 1

The mechanical power output of the Carnot engine is 1.688 kW, and it expels 11,560 J of energy in each cycle by heat. The efficiency of the engine is used to determine the work done per cycle, and from there, the power and expelled energy are calculated.

Step-by-step explanation:

The Carnot engine operates between two heat reservoirs at different temperatures. The efficiency (e) of a Carnot engine is given by:

e = 1 - (Tc/Th)

where Tc is the temperature of the cold reservoir (in kelvins) and Th is the temperature of the hot reservoir (in kelvins).

We have:

Cold reservoir temperature (Tc) = 345 + 273 = 618 K

Hot reservoir temperature (Th) = 84 + 273 = 357 K

Heat absorbed from the hot reservoir (Qh) = 20,000 J per cycle

Duration of each cycle = 5.00 s

Using the formula for efficiency:

e = 1 - (Tc/Th) = 1 - (357/618) = 0.422

(a) To find the mechanical power output, we need to calculate the work done per cycle (W) using the efficiency:

W = e × Qh = 0.422 × 20,000 J = 8,440 J per cycle

The power (P) is given by:

P = W/time = 8,440 J/5.00 s = 1,688 W or 1.688 kW

(b) The amount of energy expelled in each cycle by heat (Qc) is the difference between the heat absorbed and the work done:

Qc = Qh - W = 20,000 J - 8,440 J = 11,560 J

In conclusion, (a) the mechanical power output of this Carnot engine is 1.688 kW and (b) it expels 11,560 J of energy in each cycle by heat.

Answer 2

The mechanical power output of the Carnot engine is 4 kW. It expels 8,000 J of energy in each cycle by heat.

Step-by-step explanation:

To find the mechanical power output of the Carnot engine, we can use the formula:

Power = Energy absorbed per cycle / Duration of each cycle

Power = 20,000 J / 5.00 s = 4,000 W = 4 kW

To determine the amount of energy expelled in each cycle by heat, we can use the first law of thermodynamics:

Energy expelled per cycle = Energy absorbed per cycle - Mechanical work done per cycle

The mechanical work done per cycle is given by the equation:

Mechanical work = Efficiency * Energy absorbed per cycle

Since the Carnot engine operates at an efficiency of 0.60, we have:

Mechanical work = 0.60 * 20,000 J = 12,000 J

Therefore, the energy expelled by heat in each cycle is:

Energy expelled per cycle = 20,000 J - 12,000 J = 8,000 J