It is proposed to use a standard gasoline engine with a compression ratio of 6:1 for an industrial operation. A test is conducted to determine the efficiency of the engine. Gasoline with a density of 0.70 an a heating value of 44.432 MJ/kg is used for the test. A torque device measures 200 N at a distance of one meter at an engine speed of 1500 r/min. The engine uses 3-3/4 liters of gasoline in 15 min. while developing that torque. Which of the following most nearly equals the thermal efficiency of the engine?
To determine the thermal efficiency of the engine, we can use the formula:
Thermal Efficiency = Work output / Heat input
Let's calculate each of these values step by step:
1. Work Output:
The work output of the engine can be calculated using the torque and engine speed. The formula for work is:
Work = Torque x Distance x Revolutions
In this case, the torque is 200 N, the distance is 1 meter, and the engine speed is 1500 r/min. So, the work output is:
Work = 200 N x 1 m x 1500 r/min = 300,000 Nm/min
2. Heat Input:
The heat input is equal to the energy content of the gasoline consumed by the engine during the test. To calculate the heat input, we need to determine the energy content of the gasoline used.
The volume of gasoline used is 3-3/4 liters, which is equal to 3.75 liters.
The density of the gasoline is 0.70, so the mass of the gasoline used is:
Mass = Volume x Density = 3.75 liters x 0.70 kg/liter = 2.625 kg
The heating value of the gasoline is 44.432 MJ/kg, so the heat input is:
Heat Input = Mass x Heating Value = 2.625 kg x 44.432 MJ/kg = 116.519 MJ
Converting the heat input from MJ to J:
Heat Input = 116.519 MJ x 10^6 J/MJ = 116519000 J
Now, we can calculate the thermal efficiency using the formula:
Thermal Efficiency = Work output / Heat input = 300,000 Nm/min / 116519000 J
Simplifying the units:
1 Nm/min = 1 J/s
1 J/min = 1 J/s
So, Thermal Efficiency = 300,000 J/min / 116519000 J = 0.00257417
Rounding the value to four decimal places, the thermal efficiency of the engine is approximately 0.0026.
To determine the thermal efficiency of the engine, we need to calculate the amount of energy input and the amount of useful work output.
Step 1: Calculate the energy input.
The energy input can be calculated using the amount of gasoline consumed and its heating value:
Energy input = Amount of gasoline consumed * Heating value of gasoline
Since the engine uses 3-3/4 liters of gasoline in 15 minutes, we need to convert it to kg and seconds:
Amount of gasoline consumed = 3.75 liters = 3.75 kg (since the density of gasoline is 0.70 kg/l)
Time = 15 minutes = 900 seconds
Energy input = 3.75 kg * 44.432 MJ/kg = 166.44 MJ
Step 2: Calculate the useful work output.
The useful work output can be calculated using the torque and engine speed:
Useful work output = Torque * Distance * Engine speed
Torque = 200 N
Distance = 1 meter
Engine speed = 1500 r/min = 1500 rev/min * (2π rad/rev) * (1 min/60 s) = 157.08 rad/s
Useful work output = 200 N * 1 m * 157.08 rad/s = 31416 J
Step 3: Calculate the thermal efficiency.
The thermal efficiency is given by the ratio of useful work output to energy input:
Thermal efficiency = (Useful work output / Energy input) * 100%
Thermal efficiency = (31416 J / (166.44 MJ * 10^6 J/MJ)) * 100%
Thermal efficiency = 0.0188%
Therefore, the most approximate value of the thermal efficiency of the engine is 0.0188%.
To find the thermal efficiency of the engine, we need to calculate the useful work output and the energy input.
First, let's calculate the useful work output:
The torque device measures 200 N at a distance of one meter, so the work done is given by:
Work = Force x Distance = 200 N x 1 m = 200 J
Next, let's calculate the energy input:
The engine uses 3-3/4 liters of gasoline in 15 minutes. To calculate the energy input, we need to convert the volume of gasoline used to its mass.
1 liter of gasoline weighs approximately 0.70 kg (given the density of gasoline is 0.70).
Mass of gasoline used = (3-3/4 liters) * (0.70 kg/liter) = 2.75 kg
The energy input is given by the heating value of the gasoline, which is 44.432 MJ/kg. Converting the mass to joules:
Energy input = (44.432 MJ/kg) * (2.75 kg) * (10^6 J/1 MJ) = 121.96 MJ
Now we can calculate the thermal efficiency:
Thermal efficiency = Useful work output / Energy input
Thermal efficiency = 200 J / 121.96 MJ = 200 J / (121.96 * 10^6 J) ≈ 1.64 x 10^(-6)
Therefore, the thermal efficiency of the engine is approximately 1.64 x 10^(-6) (or 0.00000164).