Heat engines and the second law of thermodynamics Thermodynamic cycles A thermodynamic cycle is a series of processes which change the volume temperature and pressure of a gas but which at the end return to the same conditions as at the start Thermodynamic cycles are important because if we can find one that does something useful for us it can be repeated indefinitely Understanding of thermodynamic cycles was extremely important to the industrial revolution and they remain key to most large scale manufacturing processes to most engines refrigerators and air conditioners Heat engines and refrigerators The basic principles which enable heat engines and refrigerators to work are as follows i heat engine If you allow a gas to expand by moving a piston then the gas does work on the piston In this way the kinetic energy in the gas is converted into useful work moving the piston When the gas expands it cools down so at the end of the expansion we have to heat the gas up again This is achieved by burning some kind of fuel e g gasoline coal nuclear fuel hydrogen etc ii refrigerator or heat pump If you want to take heat from a house and pump it outside you have to take heat from a colder region and move it to a hotter region This is achieved by using evaporation of a coolant liquid Evaporation of the coolant requires energy latent heat of vaporization and this energy is taken from the inside of the object or rooms which we want to cool Once the coolant has evaporated it is pumped to the compressor which liquifies the coolant again to complete the cycle In a refrigerator the compressor is inside the house while for an air conditioner the compressor is outside the house In designing a heat engine or heat pump many decisions have to be made e g what is the fuel what is the working gas what is the nature of the thermodynamic cycle Heat engines such as electrical generators and gasoline engines use an expanding gas to drive a piston or turbine In a gasoline engine the expanding gas is a mixture of the products of burning air and gasoline In a turbine steam is usually used In a refrigerator or air conditioner evaporation of a coolant gas is used e g Freon Ammonia FR 12 etc Heat engines work on a cycle where an incoming gas at low temperature Tc is heated to a higher temperature Th causing expansion of the gas which in turn does the mechanical work on the piston or turbine At the end of 1 the expansion the working gas still has some heat which is typically lost as waste heat We define the heat which is added to the gas to cause expansion to be Qh while the heat lost is Qc The efficiency of the engine is defined to be W e 1 Qh Since the working gas at the end of the cycle is at the same temperature at which it started the internal energy is the same at the beginning and at the end The work done by the engine is then Weng Qh Qc so the efficiency of the heat engine is given by e Qh Qc Qc W 1 Qh Qh Qh 2 The efficiency is in the range 0 e 1 E g for gasoline engines typical efficiencies are of order 0 25 or 25 percent A refrigerator is a heat engine in reverse so that work is done to extract heat from a cold reservoir This is achieved by evaporating a gas which extracts heat from its surroundings For a refrigerator or air conditioner the coefficient of performance is COP Qc W 3 The COP can be greater than one e g 3 is a typical number for a house air conditioner The Carnot cycle produces the most efficient heat engine possible The four steps in a Carnot cycle are Isothermal expansion Adiabatic expansion Isothermal compression Adiabatic compression For the Carnot cycle the efficiency is given by eC 1 Qc Tc 1 Qh Th 4 It is quite hard to prove that the Carnot cycle is the most efficient cycle possible By reversing the cycle the Carnot cycle can act as a refrigerator or air conditioner and in that case the Carnot coefficient of performance is given by Tc Qc 5 COPC Qh Qc Th T c 2 Human metabolism The average metabolic rate for people in the age range 20 30 is 40kcal m 2 hr hr for males and 37kcal m2 hr for females Note that multiplying this by 24 hours and an average surface area 1 9m2 for adult men and 1 6m2 for adult women gives the target consumption of an adult The metabolic rate of a person can be measured quite easily by measuring the amount of oxygen which is consumed per unit time The relation is VOxygen U 4 8 t t 6 where the result is in kcal s and the oxygen volume if is liters s Many performance enhancing drugs aim at increasing the oxygen transport rate through the blood system in order to increase metabolic rate Typical metabolic rates for some tasks are sleeping 80W walking 450W extreme sport 1600W A person s fitness can be measured though his her ability to use oxygen with a couch potato having a peak oxygen consumption rate of 28mL minkg and Lance Armstrong having a peak consumption rate of 70mL minkg Since by the equation above this translates directly into the energy available to the human body a high oxygen consumption rate is essential to high activity The metabolic rate is the amount of power available to the human body The efficiency of our metabolism is the rate at which mechanical work is done divided by the metabolic rate for the task ie e W U t t 7 The efficiency of humans at some tasks are approximately shovelling 0 03 bike riding 0 19 This means that the 1600W available for an extreme sportsperson translates into roughly 0 19 1600W 304W of mechanical work for a cyclist Entropy Entropy is a very important concept in thermodynamics and is also important in philosophical and religious discussions about whether life could possibly have originated spontaneously on our planet 3 Entropy is a measure of the amount of disorder in a system If a system has a lot of different possible configurations it has high entropy For example if 100 particles of an ideal gas move randomly in a volume V then the gas has entropy S If the same 100 particles are given the same kinetic energy and are placed in volume 2V then when the gas in the larger volume has higher entropy because it has a much larger number of possible configurations Boltzmann introduced a quantity which is the total number of states which are available to the gas It is evident that if the kinetic energy of a gas is fixed then V T 2V T In fact Boltzmann was able to prove a general relation that the entropy is given by S kB ln 8 where S is the …
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