You have been requested to design a Stirling heat engine that uses sunlight to heat 2.7 kg of helium gas (molar mass-4.0026 g) initially at 6.06 MPa, to 102°C and a cold mountain stream to cool the gas back down to 2°C. The working fluid undergoes a Stirling cycle that begins with ((sochorie solar thermal heating of the fluid followed by sothermal expansion of the fluid to a pressure of 4.65 MPa. Next, the fluid is restored to its original temperature by anusochoric process. Finally, its original volume and pressure are restored by undergoing isothermal compression.
a. By defining a heat engine, distinguish internal and external combustion engines.
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b. Modeling the working fluid as an ideal gas, determine the values for the state
variables (P. V and T) at the end of each thermodynamic process in the cycle. The
ideal gas constant, R, is 8.31 J.mol' K
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c. Assuming the engine is reversible and 4184 J of heat is absorbed from the high temperature reservoir in one cycle, how much work is done per cycle?
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d. You intend to use the energy produced by the solar engine to work a refrigerator
that has a coefficient of performance of 2.75. What mass of water, originally at 0°C, can be converted into ice at 0°Cper cycle? Specific latent heat of fusion of water is 334 kJ.kg¹.
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e. By explaining the processes involved, distinguish the Stirling engine described above from Diesel and Otto Engines.
a. A heat engine is a device that converts thermal energy into mechanical work. An internal combustion engine is a type of heat engine where the combustion of fuel takes place internally, within the engine itself. The fuel is burnt in a combustion chamber and the resulting hot gases expand and push a piston, generating mechanical work. An external combustion engine, on the other hand, is a type of heat engine where the combustion of fuel takes place outside the engine. In this case, the heat from the combustion is transferred to a working fluid, which expands and performs mechanical work in the engine.
b. To determine the values for the state variables at the end of each thermodynamic process in the cycle, we'll use the ideal gas law:
PV = nRT
where P is the pressure, V is the volume, n is the number of moles of gas, R is the ideal gas constant, and T is the temperature.
1. Solar thermal heating (sochorie process): The gas is heated from an initial temperature of 6.06 MPa to the final temperature of 102°C. The pressure remains constant, so the process is isobaric. Using the ideal gas law, we can determine the volume:
V1 = (nRT1) / P1
2. Isothermal expansion process: The gas expands isothermally from the pressure of 6.06 MPa to 4.65 MPa. Since the process is isothermal, the temperature remains constant. Using the ideal gas law, we can determine the volume:
V2 = (nRT1) / P2
3. Anusochoric process: The temperature remains constant, but the volume changes. Since the process is not isobaric, we can't use the ideal gas law directly. Instead, we need additional information or assumptions to determine the change in volume.
4. Isothermal compression process: The gas is compressed isothermally from the pressure of 4.65 MPa to the initial pressure of 6.06 MPa. Since the process is isothermal, the temperature remains constant. Using the ideal gas law, we can determine the volume:
V4 = (nRT1) / P4
c. To determine the work done per cycle, we can use the equation:
W = Qh - Qc
where Qh is the heat absorbed from the high temperature reservoir and Qc is the heat rejected to the low temperature reservoir. Since the engine is reversible, the heat transferred in each process is proportional to the corresponding temperature change:
Qh = nCv(T3 - T2)
Qc = nCv(T4 - T1)
where Cv is the molar specific heat at constant volume.
d. To determine the mass of water that can be converted into ice per cycle, we need to calculate the heat removed from the water. The heat removed is given by:
Qc = mcΔHf
where mc is the mass of water and ΔHf is the specific latent heat of fusion of water. We can then equate this to the heat rejected by the engine:
Qc = nCv(T4 - T1)
Solving for mc:
mc = (nCv(T4 - T1)) / ΔHf
e. The Stirling engine described above is different from Diesel and Otto engines in several ways:
1. Working principle: The Stirling engine operates on the Stirling cycle, which involves cyclic compression and expansion of a fixed amount of gas. The expansion and compression are driven by changes in temperature. In contrast, Diesel and Otto engines operate on the principles of internal combustion, where fuel is combusted inside the engine to create high-pressure gases that drive the piston.
2. Fuel type: The Stirling engine can operate on any external heat source, including solar energy, biomass, or fossil fuels. Diesel engines operate on diesel fuel, while Otto engines operate on gasoline.
3. Efficiency: Stirling engines tend to have lower thermal efficiency compared to Diesel and Otto engines, primarily due to the temperature differences involved in the Stirling cycle. Diesel and Otto engines can achieve higher thermal efficiency through direct combustion of fuel.
4. Emissions: Stirling engines have low emissions since the combustion of fuel occurs outside the engine, resulting in cleaner exhaust gases. In contrast, Diesel and Otto engines produce emissions as a result of the combustion process.
Overall, each type of engine has its advantages and disadvantages, and their suitability depends on the specific application and requirements.
a. A heat engine is a device that converts heat energy into mechanical work. Internal combustion engines and external combustion engines are two types of heat engines. The main difference between them is the location of the combustion process.
In an internal combustion engine, the combustion of fuel occurs internally within the engine cylinders. The fuel is ignited directly inside the engine, and the resulting high-pressure gases push the piston, generating mechanical work. Examples of internal combustion engines include gasoline engines and diesel engines.
In contrast, an external combustion engine, like the Stirling engine mentioned in the question, has the combustion process occurring externally to the engine. The working fluid is heated externally, and the heat is transferred to the engine to produce work. This allows for more control over the combustion process and enables the use of various heat sources, such as solar energy or fuel combustion outside the engine.
b. To determine the values for the state variables (P, V, and T) at the end of each process in the Stirling cycle, we need to analyze the cycle step by step.
1. Sosoric solar thermal heating: The gas is heated isochorically (constant volume) by solar energy until it reaches a pressure of 4.65 MPa. The volume remains constant, so V remains unchanged. The final pressure (P1) is 4.65 MPa. We can assume the temperature (T1) is the same as the initial temperature since it is not provided in the question.
2. Sothermal expansion: The gas expands isothermally (constant temperature) from a pressure of 4.65 MPa to the final pressure of 6.06 MPa. Since it is an isothermal process, T remains constant. We can use the ideal gas law equation to find the final volume (V2) at 6.06 MPa. The equation is PV = nRT, where n represents the number of moles of gas, R is the ideal gas constant, and T is the temperature in Kelvin.
3. Anusochoric process: The gas is restored to its original temperature (T1) by an anusochoric (non-constant volume) process. The volume (V3) changes, but the temperature remains constant.
4. Isothermal compression: The gas is compressed isothermally from the final volume (V3) to the initial volume. Using the ideal gas law, we can find the final pressure (P4) at the original volume (V).
c. Assuming the engine is reversible and 4184 J of heat is absorbed from the high temperature reservoir in one cycle, we can calculate the work done per cycle. In a heat engine, the work done is given by the difference between the heat absorbed and the heat rejected. Since the engine is reversible, all the heat absorbed is converted into work, so the work done per cycle is 4184 J.
d. To determine the mass of water that can be converted into ice per cycle, we need to consider the energy produced by the solar engine and the coefficient of performance of the refrigerator.
The energy produced by the solar engine is the work done per cycle, which we determined to be 4184 J.
The amount of heat extracted from the water to convert it into ice is given by the equation Q = m * L, where Q is the heat absorbed (equal to the energy produced by the solar engine), m is the mass of water, and L is the specific latent heat of fusion of water.
Setting up the equation, we have 4184 J = m * 334 kJ/kg. Converting the specific latent heat to joules, we have 4184 J = m * 334,000 J/kg.
Rearranging the equation to solve for m, we get m = 4184 J / 334,000 J/kg.
Substituting the values, we can calculate the mass of water that can be converted into ice per cycle.
e. The Stirling engine, Diesel engine, and Otto engine are all types of internal combustion engines, but they operate based on different cycles and have different processes involved.
The Stirling engine: As described in the question, the Stirling engine uses an external heat source, usually solar energy in this case. It operates on a Stirling cycle, which includes processes like sochoric solar thermal heating, sothermal expansion, anusochoric restoration, and isothermal compression. The working fluid is typically a gas, such as helium, which undergoes cyclic compression and expansion to convert heat energy into mechanical work.
Diesel engine: A diesel engine is an internal combustion engine that operates on the diesel cycle. In this cycle, air is compressed at a high ratio, resulting in high temperatures. Fuel is then injected into the combustion chamber, where it self-ignites due to the high temperature, leading to the expansion of gases that push the piston. Diesel engines are known for their efficiency and are commonly used in heavy vehicles and machinery.
Otto engine: An Otto engine is another type of internal combustion engine, commonly known as a gasoline engine. It operates on the Otto cycle, which consists of processes like intake, compression, combustion, and exhaust. Fuel-air mixtures are ignited by spark plugs, leading to rapid combustion and expansion of gases that push the piston. Otto engines are widely used in cars and light vehicles.
In summary, the Stirling engine differs from Diesel and Otto engines in terms of the heat source, the thermodynamic cycle it operates on, and the processes involved in converting heat energy into mechanical work.