Chapter 2: Energy and the 1 st Law of Thermodynamics. The Study of Energy in Closed Systems

Transcription

1 Chapter 2: Energy and the 1 st Law of Thermodynamics The Study of Energy in Closed Systems

2 Topics 2.1 Mechanical Concepts of Energy 2.2 Broadening Understanding of Work 2.3 Broadening Understanding of Energy 2.4 Energy Transfer by Heat 2.5 Energy Accounting: Energy Balance for Closed Systems 2.6 Energy Analysis of Cycles

3 Mechanical Concepts of Energy

4 Work and Kinetic Energy F = ma dv ds dv dv F = m = m = mv dt dt ds ds V V s 2 2 mvdv = F ds = Work s 1 1 s Where: V V 2 1 V ( ) 2 1 mvdv = mv m V V 2 = 2 V 1 2 s2 = s = s ( 2 2) 2 1 mv V Fds Work Work done ON the system increases its Kinetic Energy (KE) KE is extensive property 1

5 1 2 s2 = s = s ( 2 2) 2 1 mv V Fds Work 1 ( ) F = R mg Potential Energy R z ( 2 2) 2 1 s s s m V V = F ds = Rdz mgdz 1 2 s s s s ( 2 2) 2 1 ( 2 1) mv V + mg z z = Rdz KE + PE = Work done by Applied Force, R s s 2 1 z mg z 1 PE = PE PE = mg z z ( ) PE is Extensive Property

6 References for kinetic and potential energy. Properties because doesn t matter how attained the velocity or height

7 Energy: The Property Kinetic Energy KE = 1 2 m( V 2 2 V1 2 ) Potential Energy PE = m g( z 2 z1) Common Units J(N m) or kj ft lbf Btu

8 Examples What happens to KE if mass moved up from Z 1 to Z 2 for: R = mg R 1 2 ( 2 2) 2 1 s s s m V V = F ds = Rdz mgdz s s s s z 2 What happens if mass is at Z 2 with V 2 = 0 and then mass is released (R = mg -> 0) z mg z 1 Problem 2.13: Show that 2 bodies with different masses falling from same height hit ground with same velocity

9 Broadening Understanding of Work

10 Work Thermodynamic definition of Work: Work is done BY a system ON its surroundings if the sole effect on everything external to the system COULD HAVE BEEN the raising of a weight [may not be obvious] W s s 2 = F ids Work is form of energy transfer. Appears during a process. Systems DO NOT possess Work. Systems possess energy. Magnitude of Work depends on details of interactions between system and surroundings, i.e. How does force vary with displacement? Thus, work is NOT a property. 1 s

11 Types of Work Expansion/Compression Work (Moving Boundary Work) V p dv V 2 1 Elongation of a solid bar Stretching of a Liquid Film Rotating Shaft Electric Polarization/Magnetization Work is process (path) dependent, and is NOT a property of the system

12 Two Examples of Work W s 2 = F ids s 1 s System A: Torque and angular displacement. F & ds System B: Current and voltage driving force Less obvious that this is consistent with eq above Battery COULD drive motor that could lift weight instead of driving paddle wheel. Thus F & ds

13 Sign Convention and Notation W > 0: Work done BY the system W < 0: Work done ON the system Convention from Heat Engines- Convert heat to work Heat Work Sometimes this convention will be changed. Arrow used to show positive direction on system diagram Energy conservation equation must be consistent

14 Energy Transfer Across a Boundary Energy Transfer by Work F = Force S = Path over which force acts W S b = F a b S a ds Work = Area Under F-S diagram No Work since ds = 0

15 Integral is INEXACT because depends on path 2 W = δw W W Note δ notation for INEXACT differentials In contrast, integral of differential of a PROPERTY (such as Volume below) is EXACT. Does not depend on details of process between states (ie path) V 2 V 1 dv = V V 2 1

16 Power Many thermodynamic analyses concerned with time rate of energy transfer, i.e. Power W = FiV Force * Mass/time = Work/time= Power t t 2 2 W = Wdt = FiVdt 1 1 t t SI: J/s (Watt) kw UNITS [ English: ft lbf/s, BTU/h, hp

17 Example Power Problem

18 Modeling Expansion or Compression Work δ W = F idx ON ON F ON = PA Expansion: δ W = F dx < 0 ON δ WON = PAdx= PdV Negative sign required since P & dv positive for expansion and δw ON < 0 Forces balanced in Quasi-equilibrium process, so: Check Compression Compression: δ W = F dx > 0 ON ON δwby = δwon = PdV > 0 ON δw = δw = PdV < 0 BY ON

19 Quasi-equilibrium Expansion or Compression Processes Note single-values of properties during process. Thus a single P value at each V Area under curve is Work Area (magnitude) same for compression (-) and expansion (+) But sign is different

20 Non-Quasi-equilibrium Expansion or Compression Processes P low P high Note pressure gradient- (property) non-uniformity Higher than average pressure during compression at piston- WORK IN P high Lower than average pressure during Expansion at piston- WORK OUT What Pressure to use? Is Net Work done during cycle Zero? Relaxation considerations? P low

21 Example Problem: 3 Process Cycle in Piston-Cylinder

22 Further Examples of Work Elongation of a solid bar F FON = σ A= A A σ is normal stress BY ON ON x x x 2 2 W = W = F dx= σadx x 1 1 Negative sign: Because work done ON system when dx is positive

23 Example Wire Extension Problem

24 Approximate Young's Moduli of Various Solids Material Young's modulus (E) in GPa Rubber (small strain) Low density polyethylene 0.2 Polypropylene Polyethylene terephthalate Polystyrene Nylon 2-4 Oak wood (along grain) 11 High-strength concrete (under compression) 30 Magnesium metal 45 Aluminium alloy 69 Glass (all types) 72 Brass and bronze Titanium (Ti) Carbon fiber reinforced plastic (unidirectional, along grain) 150 Wrought iron and steel Tungsten (W) Silicon carbide (SiC) 450 Tungsten carbide (WC) Single Carbon nanotube [1] approx. 1,000 Diamond 1,050-1,200 Young's modulus (E) in lb/in² (psi) 1,500-15,000 30, , , , , , , , ,000 1,600,000 4,350,000 6,500,000 10,000,000 10,400,000 17,000,000 15,000,000-17,500,000 21,800,000 30,000,000 58,000,000-59,500,000 65,000,000 65,000,000-94,000,000 approx. 145,000, ,000, ,000,000

25 Work W > 0 : Work done by the system W < 0 : Work done on the system Time rate of work is Power: W Heat Transfer Q > 0 : Heat transfer into the system Work & Heat Transfers Q < 0 : Heat transfer out of the system Rate of heat transfer: Q

26 Further Examples of Work Stretching of a Liquid Film Surface tension = τ Force/length = F/l Factor of 2 since two film surfaces (front/back) F = 2lτ δw = δw = F dx= 2lτ dx BY ON ON ( ) da = 2ldx A W BY A A 2 = τ da 1 Negative sign: Because work done ON system when dx is positive

27 Further Examples of Work V Electric Power: W BY = εii Watt = Volt-Ampere δw BY ε = idz + ε i - Charge carriers (current) moving from (-) to (+) will have work done ON the charge, dz [Halliday & Resnick, p.792-3]

28 M&S convention for current INTO + terminal is opposite to H&R convention of current OUT of + terminal Thus, opposite signs for Work done BY or ON system + ε i -

29 Further Examples of Work W Electric Power: ( ) BY = V i δ W BY Watt = Volt-Ampere = V dz ( ) V V is voltage difference/potential [Volt] dz is electrical charge flow into system i is the current (dz/dt) [Ampere] δw BY is the incremental work BY system Negative sign: work done ON system- for current IN at the positive terminal

30 Further Examples of Work Rotating Shaft Torque: τ = F R t Tangential force, F t Shaft radius, R Velocity V = Rω Power: ( )( ) W = FV τ t = R R ω = τω Positive sign: Because work done BY system

31 Generalized Forces & Displacements ( ) ( ) i ( ) ( )... δw = PdV σd Ax τ da V dz E d VP µ Hd VM + ( ) ( ) i ( ) ( )... δw = PdV σd Ax τ da V dz E d VP µ Hd VM + Align dipoles in elec or mag field o o Generalized force Generalized displacement Force-displacement relationship needed to integrate Work is NOT a property Usually one characteristic/domain dominates (eg PdV) Simple compressible substance (PdV work mode)

32 Broadening Understanding of Energy

33 Types of Energy so far: KE, PE, Work (various forms) Note that work can be done on 3 systems below & energy stored However, KE, PE not increased (How do we know?) New energy form is INTERNAL ENERGY, U (extensive property) Thermal Electrical Elastic

34 Energy Changes Total Energy: An extensive property of a system Kinetic Energy (Mechanical) Potential Energy (Mechanical) Internal Energy: U or u Represents all other forms of energy Includes all microscopic forms of energy E = KE + PE + U

35 Change in Total Energy of a System E = KE + PE + U For example, change in total energy from state 1 to state 2 E E = KE KE + PE PE + U U ( ) ( ) ( ) Note possibilities for energy conversion and conservation Identification of Internal Energy important energy form distinct from mechanics forms of KE and PE Ch 3: will learn how to evaluate changes in U for gases, liquids and solids using empirical data

36 Internal Energy Translational Rotational Vibrational Chemical, Nuclear & KE and PE: Macroscopic Properties Electrical Internal Energy (U): Microscopic Property SI Units: Joules(J) Like kinetic and potential energies USCS Units: British Thermal Unit (Btu) Different than kinetic and potential energies (ft lbf)

37 To Further Understand Internal Energy Microscopic Interpretation of Internal Energy in a Gas Temperature Related to Internal Energy in a Gas

38 Energy Transfer by Heat

39 Energy Transfer as Heat Energy transfer to gas that can t be characterized as work Occurs as a result of temperature differences Joule research, early 1800s Heat Transfer Q > 0 : Heat transfer into the system Q < 0 : Heat transfer out of the system Rate of heat transfer: Q Will sometimes change this convention in this text/course Most importantly, remain self-consistent with such conventions and statements of energy conservation (1 st Law)

40 Heat is NOT a system Property 2 Q= δq Q Q Limits mean from State 1 to State 2. Does NOT mean system has Q 2 at state 2 and has Q 1 at state 1 Integral is INEXACT because depends on path Heat energy appears during transfer process, not at equilibrium Systems DO NOT possess HEAT (Internal Energy) Heat is NOT a system Property

41 Heat Transfer: Rates & Amounts Rate of heat transfer denoted as: (energy/time, e.g. J/s, W) Q Amount of energy transferred by heat from t 1 to t 2 denoted as: (energy, e.g. J) Q = t t 1 2 Qdt Heat flux: q Energy/area/time Rate of heat transfer over all area, A, denoted as: Q (energy/time, e.g. J/s) = A qda

42 Heat Transfer Modes (AME432) Conduction Radiation Convection dt x = κa dx 4 e = εσ ATb Q Q Q = ha( T T ) c b f Photo courtesy of Mike Benson

43 Thermodynamics & Heat Transfer IN Q STORED W OUT Q = (E 2 E 1 ) + W de Q W dt = Quasi-equilibrium: No Gradients in Properties No relationship between energy flows/fluxes and gradients in properties dt x = κa dx Q Non-equilibrium: Gradients in Properties of interest Direct relationship between energy flows/fluxes and gradients in properties

44 Example Problem: Insulation Thickness (Prob 2.53)

45 [Text version] Conservation of Energy: The 1 st Law of Thermodynamics for Closed Systems Change in amount of energy STORED within the system during some time interval = Net amount of energy transferred IN across the system boundary by heat transfer during the time interval - Net amount of energy transferred OUT across the system boundary by work during the time interval E 2 E 1 = Q W KE + PE + U = Q - W IN Q STORED E W OUT

46 [My version] Conservation of Energy: The 1 st Law of Thermodynamics for Closed Systems Net amount of energy transferred IN across the system boundary by heat transfer during the time interval = Change in amount of energy STORED within the system during some time interval + Net amount of energy transferred OUT across the system boundary by work during the time interval Q = (E 2 E 1 ) + W Q = ( KE + PE + U) + W IN Q STORED E W OUT

47 Other forms of Energy Balance Differential form: de = δq δw Rate form: Time rate of change in amount of energy STORED within the system at time t = Net rate at which energy is being transferred IN by heat transfer at time t de Q W dt = - Net rate at which energy is being transferred OUT by work at time t de dke dpe du = + + = Q W dt dt dt dt Convenience defines which from used Care with Signs and Units, Rates and Amounts

48 Example Problems: Equivalence of alternate sign conventions and self-consistent energy conservation statements Alternate choices for system boundaries (Fig 2.14) Importance of boundary at location to identify/quantify energy transfer of interest Transient (unsteady) problem

49 Text example probs 2.2 Cooling of gas in a piston-cylinder 2.3 Considering alternate systems 2.4 Gearbox at steady state 2.5 Silicon chip at steady state 2.6 Transient operation of a motor

50 Energy Analysis of Cycles

51 System executes cycle when system undergoes sequence of processes from initial state and returns to initial state Cycles important in development of thermodynamics as subject Cycles important for power generation and refrigeration Cycles are considered here relative to energy conservation Cycles can be studied in more detail when 1 st and 2 nd laws are combined

52 Cycle Energy Balance Qcycle = Ecycle + Wcycle [ In = Stored + Out ] OR Ecycle = Qcycle Wcycle For a Cycle, system which returns to initial state: No NET change in E Why? Because E is a Property Q cycle = W cycle Note: this must be satisfied for ANY cycle Thus, for ANY sequence of processes, ANY substance in system

53 Power Cycles T H Examples of Cycles 0 In = Stored + Out Refrigeration & Heat Pump Cycles T H Q in = W cycle + Q out T L Q in + W cycle = Q out T L Note: directions for work and heat transfer are chosen for convenience Directions do not always comply with previous convention Must write energy conservation equations to match convention chosen

54 Power Examples of Cycles Refrigeration Heat Pump Note Objective of each device in terms of energy conversions (Arrows)

55 Efficiencies of Cycles Power Desired result Cost to get desired result η = W cycle Q in

56 Efficiencies of Cycles Refrigeration Desired result Cost to get desired result β = Q W in cycle

57 Efficiencies of Cycles Heat Pump Desired result Cost to get desired result γ = Q W out cycle

58 Examples of Cycles: Power Cycles Power Cycles Note Q exchange occurs with hot & cold body 1) W cycle = Q in Q out T H 2) Thermal efficiency: Sub 1 into 2: η = W cycle Q in η = (Q in Q out )/Q in = 1 Q out /Q in Q in > Q out η larger as W cycle -> Q in ; Q out -> 0 T L Energy conservation says η < = 1 Experience says η is always less than 1 2 nd law: places limits on η [T H & T L ]

59 Examples of Cycles: Refrigeration Refrigeration objective: Q in per W cycle Wcycle = Qout Qin Refrigeration & Heat Pump Cycles Q Qin W Q Q in β = = cycle out in Since work done ON system: Q out Coefficients of performance defined as ratios of desired heat transfer compared to cost of work to get it. Thus, coefficient should be as large as possible > Q in Area cooled is in blue circle. Eg: Refrigerator Electrical work. Q out goes to room

60 Examples of Cycles: Heat Pump Heat Pump Objective: Q out per W cycle Wcycle = Qout Qin Heat Pump Cycles Since work done ON system: Q out > Q in Coefficient of Performance: Q Qout W Q Q out γ = = cycle out in Coefficients of performance defined as ratios of desired heat transfer compared to cost of work to get it Thus, coefficient should be as large as possible Area heated is in red circle. Electrical work. Q in comes from surroundings (e.g ground)

61 END