Heat Machines (Chapters 18.6, 19)

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1 eat Machines (hapters 8.6, 9) eat machines eat engines eat pumps The Second Law of thermodynamics Entropy Ideal heat engines arnot cycle Other cycles: Brayton, Otto, Diesel

eat Machines Description The principles of thermodynamics are directly relevant when applied to the conversion of energy in heat machines: heat engines convert partially an input of heat into work,

2 eat Machines Description The principles of thermodynamics are directly relevant when applied to the conversion of energy in heat machines: heat engines convert partially an input of heat into work, while heat pumps use work to transfer heat A heat engine circulates a working substance through a cyclic process in which the process originates and ends in the same state, such that E Q W W 0 in out Any such cycle must include:. A process in which energy is transferred from a hot reservoir at a high temperature (Q ) 2. A process in which energy is partially converted in work done by the engine (W out ) 3. A process in which energy is expelled to a cold reservoir at a lower temperature (Q ) Therefore, the work by the machine is given by Wout Qnet Q Q Ex: Energy transfer diagram of a heat engine

eat Machines Thermal efficiency of heat engines The quantity rating the thermal efficiency of a heat engine is defined as the ratio of the work done by the engine to the energy absorbed at the higher

3 eat Machines Thermal efficiency of heat engines The quantity rating the thermal efficiency of a heat engine is defined as the ratio of the work done by the engine to the energy absorbed at the higher temperature: W Q Q Q out Q Q Q Notice that η = (00% efficiency) only if Q = 0: no energy flow to cold reservoir Exercise: A heat engine circulates 2 mols of ideal gas around the cycle represented on the figure: state is at temperature T = 600 K and states 2,3 are on the isotherm T 2 = 2400 K a) In which process is the heat received? alculate it. b) In which process is the heat expelled? alculate it. (0 5 a) W out >0 for Q 3 (m 3 ) T 2 cycle c) What is efficiency of the cycle? W out Q T

eat Machines eat pumps eat pumps such as refrigerators and air conditioners are built by reversing the cycle of a heat engine: work is done on the gas and the cycle extracts energy from the cold

4 eat Machines eat pumps eat pumps such as refrigerators and air conditioners are built by reversing the cycle of a heat engine: work is done on the gas and the cycle extracts energy from the cold reservoir and transfers it to the hot reservoir The performance of the heat pump is given by the coefficient of performance (K) which in general is larger than. Ex: Energy transfer diagram of a heat pump When running in cooling mode: K cooling Q W in eat pump When running in heating mode (heat is transferred inside from the colder outside): K heating Q W in

5 Quiz:. Two engines work based on the shown cycles. Which engine has a larger efficiency? 2 Engine A Engine B 2 3 a) Engine A b) Engine B c) They have the same efficiency 4 2 2

The 2 nd Law of Thermodynamics Statements The second law of thermodynamics describe the limits of converting energy into useful forms, and has several formulations which can be shown to be analog 2

6 The 2 nd Law of Thermodynamics Statements The second law of thermodynamics describe the limits of converting energy into useful forms, and has several formulations which can be shown to be analog 2 nd Law, Kelvin-lank: No cyclic heat engine can absorb energy from a reservoir and use it entirely for the performance of an equal amount of work Therefore, the work W out can never be equal to Q : that is, always Q 0 and the efficiency of a thermal engine can never be 00% 2 nd Law, lausius #: It is not possible to transfer spontaneously the heat from a body of lower temperature to a body of higher temperature Ex: The two forms can be shown to be equivalent: imagine an heat engine supplying the work input to a refrigerator across the same reservoirs. If Q were entirely converted to work by the engine, this would result into an unassisted transfer of heat from cold to hot ideal heat engine + refrigerator = heat flow from cold to hot In summary, while the First Law states that we cannot get a greater amount of energy out of a cyclic process than we put in, without setting limits on how much of the input can be converted into useful work, the Second Law sets the limits by stating that we can t break even some of the input will always be lost

7 The 2 nd Law of Thermodynamics Reversible and irreversible processes The 2 nd Law works because the transfer of heat from hot to cold is an example of an irreversible process, a process that can happen only in one direction Most natural processes are irreversible since the energy is dissipated, or degraded, in forms which cannot be completely recovered spontaneously A reversible process is one in which every state along some path is an equilibrium state: in this cases, the system can be returned to its initial state along the same path Reversible process are an idealization, but some real processes can be approximated as being reversible is the system responds faster than the applied change Ex : Reversible processes: The isobaric, isochoric and isothermal processes are considered reversible since each state along the path is considered as having a well defined set of thermodynamic parameters: in order to emulate such a reversible process it must be very slow like the isothermal quasistatic frictionless compression of a gas. Ex 2: Irreversible processes: a) If the process is fast, like the adiabatic expansion, it is irreversible since the intermediary states do not have sets of parameters uniquely characterizing the gas. b) Any process including quasistatic ones are irreversible if friction is involved doing nonconservative work So, the irreversibility of natural processes is related to the limits set by the 2 nd Law in building a perfectly efficient heat engine

8 The 2 nd Law of Thermodynamics Entropy. Macroscopic concept Entropy S is a physical quantity describing the thermodynamic state of a system: at a macroscopic scale and constant temperature, entropy can be defined using lausius s formulation: Let Q r be the energy absorbed or expelled during a reversible, isothermal process between two equilibrium states. Then the change in entropy during any isothermal process connecting the two equilibrium states can be defined by Q r S S SI T In general, if the process is actually irreversible, calculating the change of entropy requires finding a reversible path between the initial and final states: When energy is absorbed, Q > 0 and entropy increases When energy is expelled, Q < 0 and entropy decreases Entropy allows a mathematical formulation of the 2 nd Law of thermodynamics: 2 nd Law, lausius #2: The entropy of the universe increases in natural processes This doesn t mean that entropy cannot decrease: it means that, if entropy decreases in a system it must be accompanied by a larger increase of entropy in another system J K

9 The 2 nd Law of Thermodynamics Entropy. Microscopic concept Entropy can also be described microscopically within statistical mechanics, in terms of disorder: a disorderly arrangement is much more probable than an orderly one if the laws of nature are allowed to act without interference Entropy is a measure of disorder, as defined by Boltzmann s formulation If the probability of a system to be in a certain macroscopic configuration is given by a number W the number of microstates corresponding to the same macrostate the entropy of the respective macrostate is S k lnw B don t confuse with work Ex: If a molecule occupies a volume m in a container of volume i, the number of microstates is i / m. For N molecules, it is W = ( i / m ) N. The tendency of nature to move toward a state of disorder (more microstates for the same macrostate means a larger probability) affects a system s ability to do work: various forms of high grade energy can be converted into internal energy, but the reverse transformation is never complete (called degradation of energy) So, since in any process involving internal energy the energy degrades, there is less and less energy for doing work In order to keep doing work, the cycle of an engine must be periodically supplied with energy from outside

10 Quiz: 2. What happens with the entropy in a reversible adiabatic process? a) increases b) decreases c) stays constant 3. System A transfers heat spontaneously to system B. What happens with the entropy of system A? ow about the entropy of system B? ow about the entropy of the whole system A+B? roblem:. Macroscopic and microscopic approach to entropy: A quantity of n = mol of gas expands isothermally at temperature T = 300 K to 4 times its initial volume. a) What is the change in internal energy? What is the work and heat exchanged in this process? b) alculate the change in entropy using a macroscopic argument. c) onfirm the change in entropy by using a microscopic argument.

based on a cycle called arnot cycle an ideal, reversible, frictionless cycle between two reservoirs It is the most

nd Law, arnot: No real engine operating between two energy reservoirs can be more efficient than a arnot engine

with a movable frictionless piston The arnot engine cycles through two adiabatic processes between the hot and cold

11 Ideal eat Engines arnot engine. Operation The efficiency of real engines can be understood based on an ideal engine called arnot engine functioning based on a cycle called arnot cycle an ideal, reversible, frictionless cycle between two reservoirs It is the most efficient engine possible, since real engines operate via irreversible cycles affected by dissipative loses of energy 2 nd Law, arnot: No real engine operating between two energy reservoirs can be more efficient than a arnot engine operating between the same two reservoirs onsider an ideal gas as working substance in a thermally insulated cylinder with a movable frictionless piston The arnot engine cycles through two adiabatic processes between the hot and cold reservoir temperatures T and T, intercalated by two isothermal processes at these temperatures. Isothermal expansion Q>0, W<0 4. Adiabatic compression Q=0, W>0 T T 2. Adiabatic expansion Q = 0, W<0 3. Isothermal compression Q<0, W>0

12 Ideal eat Engines arnot engine. ycle and efficiency During the arnot cycle, heat is absorbed during the isothermal expansion from a hot reservoir at temperature T and released during the isothermal compression to a cold reservoir at temperature T Therefore, the arnot efficiency is given by Q Q Q nrt ln 2 Q nrt ln 4 3 T T T T ln ln T T 3 4 Q Q T T T T where the temperatures must be expressed in Kelvins. Notice that η = 00% only if T = 0 This equation can be applied to any cycle operating between two temperatures, and in order not to violate the 2 nd Law the arnot efficiency is the maximum achievable by a heat engine operating between the respective temperatures

13 Ideal eat Engine Other cycles max Q 2 3 Q 2 3 Q 2 4 min Brayton cycle Used by many ideal-gas heat engines, such as jet engines in aircraft The efficiency is given by 4 max min Q 3 T T min Otto cycle max Q Used by spark-ignition engines, such as gasoline engines in cars The efficiency is given by max min T T Diesel cycle 4 Q Used by compressionignition engines in diesel cars The efficiency is given by T T

14 Quiz: 4. The efficiency of a arnot heat engine working as indicated by the adjacent diagram is a) η < ½ b) η = ½ c) > η > ½ 5. The efficiency of a heat engine working based on the adjacent cycle is a) η < ½ b) η = ½ c) > η > ½ 6. The heat engine working as indicated by the adjacent diagram is K 50 K a) A reversible arnot engine b) An irreversible engine c) An impossible engine roblem: 2. Otto cycle: a) Derive explicitly the thermal efficiency of an ideal engine working based on the Otto cycle between volumes and r, where r is called compression ratio. b) What would be the coefficient of performance of a refrigerator based on this Otto cycle reversed?

Chapter 12. The Laws of Thermodynamics

Chapter 12. The Laws of Thermodynamics Chapter 12 The Laws of Thermodynamics First Law of Thermodynamics The First Law of Thermodynamics tells us that the internal energy of a system can be increased by Adding energy to the system Doing work