Heat Engines, Thermal Efficiency, & Energy Flow Diagrams - Thermodynamics & Physics Problems
TLDRThe video script delves into the concept of heat engines, explaining their function as devices converting heat energy into mechanical work. It illustrates this through a series of problems, calculating the work performed, thermal efficiency, and power ratings of different engines. The script methodically breaks down how to use energy flow diagrams and equations to determine the amount of heat absorbed, work output, and heat discarded, ultimately providing a comprehensive understanding of heat engine dynamics and efficiencies.
Takeaways
- 🔥 A heat engine is a device that converts heat energy into mechanical energy.
- 📈 The efficiency of a heat engine is calculated by the formula: efficiency = (mechanical work / heat absorbed from the hot reservoir) × 100%.
- 🔄 The first example shows a heat engine absorbing 4500 J from a hot reservoir and expelling 2500 J to the environment, performing 2000 J of work.
- 🚀 The second example involves a steam engine performing 6000 J of work and discarding 14000 J to a cold reservoir, with an absorbed heat energy of 20000 J per cycle.
- 🔩 The power rating of an engine is calculated by dividing the work done by the time taken to complete a cycle.
- 🐎 The power rating can be expressed in watts or horsepower, with 1 horsepower equivalent to 746 watts.
- ✅ A jet engine with 18% thermal efficiency absorbs 30833.3 J from the hot reservoir per cycle and performs 1500 J of mechanical work.
- 💡 The mechanical work performed by the jet engine over 200 cycles is 300,000 J.
- 🚗 A gasoline engine burning 1.5 kg of fuel per hour, with each gram releasing 450,000 J, absorbs 112,500 J per cycle.
- 🔄 For the gasoline engine, 30,000 J of mechanical work per cycle implies that 82,500 J of heat energy is discarded per cycle.
- 🌡️ The power rating of the gasoline engine is 50 kW and 67 horsepower, with a thermal efficiency of 26.7%.
Q & A
What is a heat engine and how does it function?
-A heat engine is a device that converts heat energy into mechanical energy. It operates on the principle that heat naturally flows from a higher temperature to a lower temperature, and some of this energy can be harnessed to perform mechanical work.
How much work is performed by a heat engine that absorbs 4500 joules from a hot reservoir and discards 2500 joules into the environment?
-The work performed by the heat engine is the difference between the heat absorbed and the heat discarded. In this case, it's calculated as 4500 joules (absorbed) minus 2500 joules (discarded), which equals 2000 joules of work done.
What is the thermal efficiency of the heat engine described in the example, and how is it calculated?
-The thermal efficiency of the heat engine is the ratio of the mechanical work done to the heat energy absorbed from the hot reservoir, expressed as a percentage. It is calculated by dividing the mechanical work (2000 joules) by the heat absorbed (4500 joules) and multiplying by 100, resulting in an efficiency of 44.4%.
How can the power rating of an engine be determined?
-The power rating of an engine can be determined by dividing the work done by the time it takes to complete a cycle. Since power is work done per unit of time, knowing the work output and the cycle duration allows us to calculate the power in watts.
What is the relationship between watts and horsepower?
-One horsepower is equivalent to 746 watts. Therefore, to convert watts to horsepower, you divide the number of watts by 746.
How much heat energy is absorbed by the hot reservoir per cycle if a steam engine performs 6000 joules of work and discards 14000 joules into a cold reservoir?
-The heat energy absorbed by the hot reservoir per cycle (qh) can be found by adding the work done (w) to the heat discarded (qc). So, qh = w + |qc|, which equals 6000 joules (work) + 14000 joules (discarded), resulting in 20000 joules absorbed per cycle.
What is the thermal efficiency of a steam engine that performs 6000 joules of work and discards 14000 joules into a cold reservoir per cycle?
-The thermal efficiency is calculated by dividing the work output (6000 joules) by the heat absorbed from the hot reservoir (20000 joules) and then multiplying by 100 to get a percentage. This results in an efficiency of 30%.
How much mechanical work can a steam engine perform in one hour if it has a power rating of 24000 watts?
-Since power is the rate of doing work, and 1 watt equals 1 joule per second, a steam engine with a power rating of 24000 watts performs 24000 joules of work per second. To find the work done in one hour, multiply by the number of seconds in an hour (3600 seconds), resulting in 86,400,000 joules or 8.64 × 10^7 joules.
Given a jet engine with a thermal efficiency of 18%, how much heat energy is absorbed from the hot reservoir per cycle if it performs 1500 joules of mechanical work?
-Using the efficiency formula (efficiency = w/qh × 100%), we can rearrange to find qh (heat absorbed). With an efficiency of 18%, or 0.18, and w (work done) being 1500 joules, qh is calculated as 1500 / 0.18, which equals approximately 8333.3 joules.
If a gasoline engine burns 1.5 kilograms of fuel per hour and the fuel has a combustion energy of 450,000 joules per gram, how much thermal energy is absorbed per cycle if the engine goes through 100 cycles per minute?
-First, convert the fuel burned to joules: 1.5 kg/hour × 1000 g/kg × 450,000 J/g. Then, divide this energy by the number of cycles per minute (100) and by 60 (to convert minutes to seconds), resulting in 112,500 joules absorbed per cycle.
How much heat energy is discarded per cycle by the gasoline engine described in the example?
-Using the formula w = qh - qc, and knowing that the mechanical work (w) is 30,000 joules and the heat absorbed (qh) is 112,500 joules per cycle, we can solve for qc. Thus, qc = qh - w, which equals 112,500 - 30,000, resulting in 82,500 joules discarded per cycle.
What is the power rating of the gasoline engine in kilowatts and horsepower, given the information in the example?
-The mechanical work done per cycle is 30,000 joules, and with 100 cycles per minute, the power rating in watts is 30,000 joules/cycle × 100 cycles/minute = 3,000,000 joules/minute. Converting this to kilowatts (1 kW = 1000 W) gives us 3,000,000 J/minute ÷ 60 seconds/minute = 50,000 W, or 50 kW. To convert to horsepower, 50,000 W ÷ 746 W/hp ≈ 67 hp.
What is the thermal efficiency of the gasoline engine described in the example?
-The thermal efficiency can be calculated using the formula efficiency = w / qh × 100%. With w being 30,000 joules and qh being 112,500 joules per cycle, the efficiency is (30,000 / 112,500) × 100%, which equals approximately 26.7%.
Outlines
🔥 Heat Engine Fundamentals and Energy Conversion
This paragraph introduces the concept of heat engines, explaining their function as devices that convert heat energy into mechanical energy. It describes the natural flow of heat from high to low temperature and how some of this energy can be harnessed to perform mechanical work. A specific problem is discussed, where a heat engine absorbs 4,500 joules from a hot reservoir and discards 2,500 joules into the environment, with the work performed by the engine calculated to be 2,000 joules. An energy flow diagram is used to illustrate the process. The paragraph also covers the calculation of thermal efficiency, which in this example is found to be 44.4%.
🌬️ Steam Engine Energy Flow and Efficiency
The second paragraph focuses on a steam engine, detailing its energy flow and efficiency. It describes how the steam engine performs 6,000 joules of work and discards 14,000 joules into a cold reservoir per cycle. The heat energy absorbed by the hot reservoir is calculated to be 20,000 joules per cycle. The thermal efficiency of the steam engine is then calculated to be 30%. The paragraph further explores the power rating of the engine in watts and horsepower, with the engine's power output calculated as 24,000 watts or 32.2 horsepower. Lastly, the mechanical work performed by the engine in one hour is calculated to be 86 million joules.
🚀 Jet Engine Efficiency and Work Calculation
This paragraph delves into the workings of a jet engine, given its thermal efficiency of 18% and the mechanical work of 1,500 joules per cycle. The heat energy absorbed from the hot reservoir is calculated to be 38,333.3 joules. The heat energy discarded to the environment per cycle is found to be 6,833 joules. The paragraph then calculates the total mechanical work performed over 200 cycles, which amounts to 300,000 joules.
🚗 Gasoline Engine Combustion and Power Output
The fourth paragraph discusses a gasoline engine, detailing its fuel consumption and power output. The engine burns 1.5 kilograms of fuel per hour, with the combustion energy of the fuel being 450,000 joules per gram. The thermal energy absorbed per cycle is calculated after converting the fuel mass to grams and considering the number of cycles per minute. The energy per cycle is found to be 112,500 joules. Given that 30,000 joules of mechanical work is performed per cycle, the heat energy discarded per cycle is calculated to be 82,500 joules. The power rating of the engine is then determined to be 50 kilowatts or 67 horsepower, with the thermal efficiency of the engine calculated to be 26.7%.
Mindmap
Keywords
💡Heat Engine
💡Mechanical Energy
💡Thermal Efficiency
💡Energy Flow Diagram
💡Hot Reservoir
💡Cold Reservoir
💡Work
💡Power Rating
💡Joule
💡Steam Engine
💡Jet Engine
Highlights
Heat engines convert heat energy into mechanical energy.
Heat flows naturally from high to low temperature, and some of this energy can be used to perform mechanical work.
A heat engine absorbs 4500 joules from a hot reservoir and discards 2500 joules into the environment, performing 2000 joules of work.
The energy absorbed by the engine must equal the total energy released to maintain its temperature.
Thermal efficiency is calculated as the mechanical work divided by the heat energy absorbed from the hot reservoir, multiplied by 100.
The example engine has a thermal efficiency of 44.4%.
A steam engine performs 6000 joules of work and discards 14000 joules into a cold reservoir per cycle, absorbing 20000 joules from the hot reservoir.
The thermal efficiency of the steam engine is 30%.
The power rating of the steam engine is 24000 watts or 32.2 horsepower, with a cycle time of 0.25 seconds.
The steam engine performs 8.64 x 10^7 joules of mechanical work in one hour.
A jet engine with 18% thermal efficiency absorbs 30833.3 joules from the hot reservoir per cycle.
The jet engine discards 6833 joules of heat energy to the environment per cycle.
Over 200 cycles, the jet engine performs 300000 joules of mechanical work.
A gasoline engine burning 1.5 kg of fuel per hour absorbs 112500 joules per cycle.
If the gasoline engine performs 30000 joules of mechanical work per cycle, it discards 82500 joules per cycle.
The power rating of the gasoline engine is 50000 watts or 67 horsepower.
The thermal efficiency of the gasoline engine is 26.7%.
Transcripts
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