Question

One of the most efficient engines built so far has the following characteristics: combustion chamber temperature = 1900°C, exhaust temperature = 430°C, 7.0 × 109 cal of fuel produces 1.4 × 1010 J of work in one hour.(a) What is the actual efficiency of this engine?(b) What is the Carnot efficiency of the engine?(c) What is the power output, in hp, of this engine?

Answer 1

The actual efficiency of the engine is 47.86%, the Carnot efficiency is 77.37%, and the power output of the engine is 5207 hp.

Step-by-step explanation:

(a) To calculate the actual efficiency of the engine, we first need to determine the heat input and the work output of the engine. In one hour, the engine produces 1.4 × 10¹⁰ J of work, which is equivalent to 1.4 × 10¹⁰/4.184 = 3.35 × 10⁹ cal. The heat input is given as 7.0 × 10⁹ cal. Therefore, the actual efficiency can be calculated as:

Actual Efficiency = (Work Output / Heat Input) x 100% = (3.35 × 10⁹ cal / 7.0 × 10⁹ cal) x 100% = 47.86%

(b) The Carnot efficiency can be calculated using the temperatures of the hot and cold reservoirs. The hot reservoir temperature is given as 1900°C and the cold reservoir temperature is given as 430°C. The Carnot efficiency can be calculated using the formula:

Carnot Efficiency = 100% - (Cold Reservoir Temperature / Hot Reservoir Temperature) x 100% = 100% - (430°C / 1900°C) x 100% = 77.37%

(c) The power output of the engine can be calculated using the formula:

Power Output = Work Output / Time = 1.4 × 10¹⁰J / 3600s = 3.89 × 10⁶ W

Converting the power output to horsepower:

Power Output (in hp) = Power Output (in W) / 746 = 3.89 × 10⁶ W / 746 = 5207 hp

Answer 2

actual efficiency is 47.78 %

Carnot efficiency is 67.65 %

power output is 5.20 × 10^3 hp

Step-by-step explanation:

given data

temperature = 1900°C = 1900+ 273 K = 2173 K

exhaust temperature = 430°C = 430 + 273 K = 703 K

fuel = 7.0 × 10^9 cal

work = 1.4 × 10^10 J

to find out

actual efficiency and Carnot efficiency and power output of engine

solution

first we find actual efficiency that is = work / heat input

put the value and

input energy = 7.0 × 10^9 cal (4.184 J/1 cal) = 29.29 × 10^9 J

actual efficiency = 1.4 × 10^10 / ( 29.29 × 10^9 )

actual efficiency = 0.4778

actual efficiency is 47.78 %

and

Carnot efficiency is = 1 - ( 703 / 2173 )

so Carnot efficiency is = 0.67648

Carnot efficiency is 67.65 %

and

power output = work / time

power output = 1.4 × 10^10 / 3600 sec

power output = 3.88 × 10^6 W

power output = 3.88 × 10^6 W / 746 hp

so power output is 5.20 × 10^3 hp