Not so black. Black Hole Thermodynamics. Nobel Prize 2017: Rainer Weiss, Kip Thorne, and Barry Barish.
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1 Not so black Nobel Prize 2017: Rainer Weiss, Kip Thorne, and Barry Barish. Black Hole Thermodynamics Zeroth Law of thermodynamics requires that black holes are round *. First law of thermodynamics requires that black holes have energy. Angular momentum, electrostatic charge, surface tension * strictly, The horizon has constant surface gravity + centrifugal force Second law of thermodynamics requires black holes have entropy. proportional to the surface area Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
2 ...Thermodynamics Engines convert (some) Heat to Work Second Law of thermodynamics CLAUSIUS: Heat can t flow from colder to hotter KELVIN/PLANCK: Cannot convert ALL heat to work CARNOT: Best possible heat engine between T 1 and T 2 is... a) Reversible b) Has all heat transfer at exactly T 1 and T 2. i.e. Reversible isothermal/adiabatic Carnot cycle Carnot cycle is not generally useful: isothermal = slow, so very low power Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
3 Engines (Finn: 4.7, 4.8, 4.9, 4.10) Carnot Efficiency Thermodynamic temperature scale. Carnot devices. A (nearly) real engine. Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
4 Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
5 Carnot Engine efficiency Is it consistent with ideal gas scale?... consider ideal gas as a working substance. Carnot efficiency is independent of working substance. Heat flows (Q 1 and Q 2 ) are determined solely by the reservoir temperatures (T 1 and T 2 ). Can define a temperature scale based on thermodynamic principles. Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
6 Heat flow and work done in a Ideal gas Carnot cycle Isothermal expansion ab : Ideal gas U = U(T ); First Law gives du = 0 = d q PdV =, q ab = Vb Va PdV = nrt 1 Vb Va dv V Similarly, isothermal compression cd : q cd = Vd Vc PdV = nrt 2 Vd Vc dv V = nrt 1 ln( V b V a ), which is positive. = nrt 2 ln( V d V c ), which is negative. Heat flows using engine sign convention. Q 1 = q ab ; Q 2 = q cd. Q 2 = nrt 2 ln(v c /V d ) Q 1 nrt 1 ln(v b /V a ) = T 2 ln(v c /V d ) T 1 ln(v b /V a ) Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
7 Q 2 = nrt 2 ln(v c /V d ) Q 1 nrt 1 ln(v b /V a ) = T 2 ln(v c /V d ) T 1 ln(v b /V a ) Adiabats T 1 V γ 1 b = T 2 Vc γ 1 and T 1 Va γ 1 = T 2 V γ 1 d So V b /V a = V c /V d = ln(v b /V a ) = ln(v c /V d ). Substitution of this result in the formula above gives Q 2 Q 1 = T 2 T 1 Conclusion Defining thermodynamic temperature T th via T th2 T th1 = Q 2 Q 1 makes the thermodynamic temperature scale the same as the ideal gas temperature scale. Henceforth we denote temperatures on both scales by T, measured in kelvin. Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
8 Efficiency for engines, refrigerators and heat pumps Can define ideal efficiency via of the temperatures of the reservoirs. Efficiency is always defined by (what you want out)/(what you put in). So For an Engine you put heat in and get work out. For a Refrigerator, put work in, take heat out (from the cold region). For a Heat Pump put work in, get heat out (into the warm region). There s no d in: refri d gerator Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
9 Heat Engine hot reservoir T 1 Q 1 W C Q 2 T 2 cold reservoir Refrigerator hot reservoir T 1 Q 1 W C Q 2 T 2 cold reservoir Engine, efficiency η = 1 Q 2 Q 1 Refrigerator, coefficient of performance η R = Q 2 W = Q 2 Q 1 Q 2 Heat pump efficiency: η HP = Q 1 W = Q 1 Q 1 Q 2 Carnot only: η C = 1 T 2 T 1 ; η HP C = T 1 T 1 T 2 η R C = T 2 T 1 T 2 Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
10 World s biggest Fridge Magnet? CERN uses about 1/3rd as much energy as Geneva. Of which.. LHC cryogenics 27.5 MW LHC experiments 22 MW Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
11 infinitely efficient? Heat Pump and Refrigerator: same device, different purpose. Refrigerator, coefficient of performance η R : η R = Q 2 W = Q 2 Q 1 Q 2 η R C = T 2 T 1 T 2 Heat pump, heat pump efficiency η HP : η HP = Q 1 W = Q 1 Q 1 Q 2 η HP C = T 1 T 1 T 2 QUESTION : Explain (using the engineering definition of the efficiency of a heat pump) why heat pumps are best used to produce domestic background heating. Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
12 Steam Engine Steam engine works with a liquid+vapour mixture, which combines big volume expansion (steam) and easy to pump/heat (water) Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
13 The Otto cycle: a nearly real engine Simplified two-stroke petrol engine. Assume single working substance with external heating Two adiabats and two isochores. Heat exchange takes place in the isochoric processes (i.e. not isothermal) Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
14 1 a b: reversible adiabatic compression (Piston moves in) T a V γ 1 1 = T b V γ b c: heat added (actually, combustion) at constant volume. Q 1 = C V (T c T b ) 3 c d: reversible adiabatic expansion (the power stroke : piston moves out) T d V γ 1 1 = T c V γ d a: heat rejected (actually exhaust) at constant volume. Q 2 = C V (T d T a ) Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
15 Otto efficiency The efficiency η for the engine has to be specified in terms of Q 1 and Q 2. From (4) and (2): η = 1 Q 2 Q 1 = 1 T d T a T c T b To get more insight into the factors controlling efficiency use (1) and (3) to give: ( ) γ 1 V2 η = 1 = 1 1 V 1 rc γ 1 where r c = V 1 /V 2 is called the compression ratio. If r c is 5, η 50%. Other considerations mean real engines are well below this. Four-stroke engines have exhaust and intake stages between da and ab. Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
16 Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
17 Fridge without work Consider three heat baths, Hot, Ambient, Cold Run An Engine between Hot and Ambient Use the work to run a Fridge between Cold and Ambient Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18
18 Graeme Ackland Lecture 6: EFFICIENCY OF ENGINES October 5, / 18 Patent for the patent clerk 1930 Einstein and Szilard patent a fridge. No moving parts. No work input. Butane as working fluid. Ammonia/water mixture pump. Energy supplied as heat to Ammonia/water
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