Engineering Thermodynamics. Chapter 4. The First Law of Thermodynamics

Transcription

1 Chater 4 The First Law of Thermodynamics It is the law that relates the arious forms of energies for system of different tyes. It is simly the exression of the conseration of energy rcile The first law of thermodynamics, also known as the conseration of energy rcile, roides a sound basis for studyg the relationshis among the arious forms of energy and energy teractions. Based on exerimental obserations, the first law of thermodynamics states that Energy can be neither created nor destroyed durg a rocess; it can only change forms. A major consequence of the first law is the existence and the defition of the roerty total energy E. Figure 4.1 Different forms of energies For the system shown aboe, the conseration of energy rcile or the first law of thermodynamics is exressed as Total Energy Total Energy The Change Total = Enterg the System Leag the System Energy of the System E E Esystem (4.1) This relation is often referred to as the energy balance and is alicable to any kd of system undergog any kd of rocess. Normally the stored energy, or total energy, of a system is exressed as the sum of three searate energies. The total energy of the system, Esystem, is gien as Comiled by Yidnekachew M. Page 1 of 1

2 E system Where: = Internal Energy+ Ketic Energy+ Potential Energy E system =U+KE+PE (4.) U is the sum of the energy contaed with the molecules of the system and is called the ternal energy. The ketic energy KE and the otential energy PE are gien by mv KE = PE mgz Or U m( u u ) 1 1 KE m ( V V1 ) PE mg( z z ) 1 Now the conseration of energy rcile, or the first law of thermodynamics for closed systems, is written as E E U+ KE+ PE (4.3) If the system does not moe with a elocity and has no change eleation, it is called a stationary system, and the conseration of energy equation reduces to E E U (4.4) The mechanisms of energy transfer at a system boundary are: Heat, Work, mass flow. Only heat and work energy transfers occur at the boundary of a closed (fixed mass) system. Oen systems or control olumes hae energy transfer across the control surfaces by mass flow as well as heat and work. Comiled by Yidnekachew M. Page of 1

3 4.1 Mechanisms of Energy Transfer, E and E Heat Transfer, (Q) Heat transfer to a system (heat ga) creases the energy of the molecules and thus the ternal energy of the system and heat transfer from a system (heat loss) decreases it sce the energy transferred as heat comes from the energy of the molecules of the system. Q is zero for adiabatic systems. Work Transfer, (W) Work transfer to a system (i.e., work done on a system) creases the energy of the system, and work transfer from a system (i.e., work done by the system) decreases it, sce the energy transferred as work comes from the energy contaed the system. Car enges and hydraulic, steam, or gas turbes roduce work while comressors, ums, and mixers consume work. Mass Flow, (m) When mass enters a system, the energy of the system creases because mass carries energy with it ( fact, mass is energy). Likewise, when some mass leaes the system, the energy contaed with the system decreases because the leag mass takes some energy with it. The energy balance can be written more exlicitly as E E ( Q Q ) ( W W ) ( E E ) E (4.5) mass, mass, System Figure 4. The energy content of a control olume can be changed by mass flow as well as heat and work teractions. E E Net energy transfer E kj System Change ternal, ketic, by heat, work and mass otential, etc..energies Comiled by Yidnekachew M. Page 3 of 1

4 Or on a rate form, as E E E kw Rate of net energy transfer by heat, work and mass System Rate change ternal, ketic, otential, etc..energies For constant rates, the total quantities durg the time teral Δt are related to the quantities er unit time as QQ t, W W t and E E t ( kj ) (4.6) The energy balance may be exressed on a er unit mass basis as e e esystem ( kj / kg ) (4.7) 4. The first law and a closed system For the closed system where the mass neer crosses the system boundary, then the energy balance is Closed system undergog a cycle Q-Q + W -W = E system (4.8) For a closed system undergog a cycle, the itial and fal states are identical, and thus Figure 4.3 For a cycle E = 0, thus Q = W. E E E system E 1 0 E 0 Comiled by Yidnekachew M. Page 4 of 1

5 E E (4.9) Notg that a closed system does not ole any mass flow across its boundaries, the energy balance for a cycle can be exressed terms of heat and work teractions as W Q or Wnet, Qnet, (4.10) net, net, If the total energy is a combation of ternal energy, ketic energy and otential energy i.e E U KE PE (4.11) mv ( V1 ) Q1 W1 U U1 mg( Z Z1) (4.1) For negligible changes ketic and otential energy Internal energy and Enthaly Internal energy Q U U W (4.13) The ternal energy cludes some comlex forms of energy show u due to translation, rotation and ibration of molecules. It is designated by U and it is extensie roerty. Or er unit mass as, secific ternal energy,, is an tensie roerty of the system like P, V, and T but not measurable. The ternal energy exression can be used to determe the state of the substance if one additional roerty is known. If we take two hase as liquid and aor at a gien saturation ressure or temerature Or U U U (4.14) f g mu m f u f mgug (4.15) Fally u u xu (4.16) f fg Comiled by Yidnekachew M. Page 5 of 1

6 Enthaly It is another extensie roerty which has a unit of energy and it is denoted by H. The enthaly is a conenient groug of the ternal energy, ressure, and olume and is gien by H U PV (4.17) The enthaly er unit mass is,, which is said to be secific enthaly and h=u+p Consider a iston cylder assembly where we hae a contuous suly heat so that the boundary changes for the rocess is Q U U W (4.18) (In the aboe equation we are neglectg change ketic energy and otential energy) If we are assumg the rocess is at constant ressure W Pd P( V V ) 1 1 (4.19) W1 PV PV1 (4.0) Hence Q U U PV PV (4.1) Q U PV U PV (4.) If we take two hase at a gien saturation state Q1 H H1 (4.3) H H H (4.4) f fg hh xh f fg (4.5) 4.3 Secific Heat It is an tensie roerty of a substance that will enable us to comare the energy storage caability of arious substances. The unit is. Comiled by Yidnekachew M. Page 6 of 1

7 It defed as; the energy required to raise the temerature of a unit mass of a substance by one degree. In general, this energy deends on how the rocess is executed. (heat is ath deendent roerty) In thermodynamics, we are terested two kds of secific heats: secific heat at constant olume and secific heat at constant ressure. The secific heat at constant olume can be iewed as the energy required to raise the temerature of the unit mass of a substance by one degree as the olume is mataed constant. Here the boundary work is zero because the olume is constant From first law δq du (4.6) Per unit mass q du but q CdT CdT du (4.7) du C (4.8) dt (Change ternal energy with temerature at constant olume) The secific heat at constant ressure C can be iewed as the energy required to raise the temerature of the unit mass of a substance by one degree as the ressure is mataed constant. From first law δq du PdV du PV dh (4.9) Per unit mass q dh, but q CdT CdT dh (4.30) C dh dt (4.31) (Change enthaly with temerature at constant ressure) Secific heats are sometimes gien on a molar basis. They are then denoted by and and hae the unit kj/kmol C or kj/kmol K. 4.4 Internal Energy, Enthaly, and Secific Heats of Ideal Gases Comiled by Yidnekachew M. Page 7 of 1

8 We defed an ideal gas as a gas whose temerature, ressure, and secific olume are related by P RT (4.3) It has been demonstrated mathematically (the comg chaters) and exerimentally that for an ideal gas the ternal energy is a function of the temerature only. That is, U U( T) (4.33) Usg the defition of enthaly, we hae Combg the aboe to equation h u P but P RT h u RT (4.34) This shows that h h( T) From the secific heat relation du C ( T ) dt (4.35) u u1 CdT (4.36) Or takg aerage alue of secific heat for narrow temerature difference u u C ( T T) (4.37) 1 ae, 1 By the same argument, dh C T dt (4.38) And, h h1 CdT (4.39) h h1 C ( ) ae, T T1 (4.40) Hence, u C T (4.41) h C T (4.4) Relation between CP and CV for Ideal Gases Comiled by Yidnekachew M. Page 8 of 1

9 Usg the defition of enthaly (h = u + P) and writg the differential of enthaly, the relationshi between the secific heats for ideal gases is h u RT (4.43) dh du RdT (4.44) Relacg dh by CdTand du bycdt we hae CdTCdT RdT (4.45) C C R (4.46) Or er mole (molar basis) C C R (4.47) Where is the uniersal gas constant =8.314 KJ/Kmol At this ot, we troduce another ideal-gas roerty called the secific heat ratio k, defed as K C (4.48) C C KC (4.49) Combg equation (4.46) and (4.49) KC C R (4.50) C R K 1 (4.51) and C C (4.5) K R C K R K 1 (4.53) Internal Energy, Enthaly, and Secific Heats of Solids and Liquids Comiled by Yidnekachew M. Page 9 of 1

10 A substance whose secific olume (or density) is constant is called an comressible substance. The secific olumes of solids and liquids essentially rema constant durg a rocess. Therefore, liquids and solids can be aroximated as comressible substances. It can be mathematically shown that the constant-olume and constant-ressure secific heats are identical for comressible substances The secific heat can be exressed as C C C (4.54) 4.5 The First Law and the Control Volume The conseration of mass and the conseration of energy rciles for oen systems or control olumes aly to systems hag mass crossg the system boundary or control surface. In addition to the heat transfer and work crossg the system boundaries, mass carries energy with it as it crosses the system boundaries. Thus, the mass and energy content of the oen system may change when mass enters or leaes the control olume. Figure 4.4 Tyical control olume or oen system Hence the conseration of mass rcile can be used to relate mass which enterg and leag a system. It can be exressed as The net mass transfer to or from a control olume durg a rocess (a time teral t) is equal to the net change (crease or decrease) the total mass with the control olume durg that rocess ( t). That is, Comiled by Yidnekachew M. Page 10 of 1

11 Total mass enterg Total mass leag Net change mass - = the CV durg Δt the CV durg Δt with the CV durg Δt It can also be exressed rate form as m m mcv ( kg ) (4.55) m m dm / dt ( kg / s ) (4.56) CV Where and are the total rates of mass flow to and of the control olume, and dmcv/dt is the time rate of change of mass with the control olume boundaries. Some time we also use olume flow rate which dicates the olume of the fluid flowg through a art er unit time and denoted by, m V (4.57) Thermodynamic rocesses olg control olumes can be considered two grous: steadyflow rocesses and unsteady-flow rocesses. Steady state rocess The flow through a control olume is at steady state if, the roerty of the substance at a gien osition with or at the boundaries of the control olume do not change with time. Durg a steady-flow rocess, the total amount of mass contaed with a control olume does not change with time (mcv= constant). Then the conseration of mass rcile requires that the total amount of mass enterg a control olume equal the total amount of mass leag it. Mass the control olume is constant dm CV dt m 0 (4.58) CV m m (4.59) m m (4.60) V V (4.61) V V A V A comressible assumtion V (4.6) (4.63) Comiled by Yidnekachew M. Page 11 of 1

12 Unsteady state rocess The roerties with the control olume change with time but rema uniform at any stant of time. The roerties at the flow areas do not change with time although the mass flow rates may change with time. Tyical examle:- fillg and emtg rocesses where most of the cases aerage alue of roerties must be used. For such cases dm c dt 0 (4.64) dt And for sgle streams, dm c mi e m (4.65) dm dt c mi me (4.66) 4.6 Flow Work and The Energy of a Flowg Fluid Unlike closed systems, control olumes ole mass flow across their boundaries, and some work is required to ush the mass to or of the control olume. This work is known as the flow work, or flow energy, and is necessary for matag a contuous flow through a control olume. Figure 4.5 Schematic for flow work Comiled by Yidnekachew M. Page 1 of 1

13 If the fluid ressure is P and the cross-sectional area of the fluid element is A, the force alied on the fluid element by the imagary iston is F PA (4.67) To ush the entire fluid element to the control olume, this force must act through a distance L. Thus, the work done ushg the fluid element across the boundary (i.e., the flow work) is Wflow FL PAL PV Per unit mass ( kj ) (4.68) wflow P (4.69) Hence, w P and w P (4.70) flow, i i flow, exit e e Rate exression of flow work W ( P ) m and W ( P ) m (4.71) flow, i i i flow, exit e e e The total flow work is, W m P P m (4.7) flow e e e i i i The total work of the system is W W flow W c (4.73) Deeloment of energy balance The general reresentation of the first law of thermodynamics Q1 W1 E E1 (4.74) The first law for oen system will also hae the same form, but W1 Wflow Wc (4.75) Comiled by Yidnekachew M. Page 13 of 1

14 The total energy of a simle comressible system consists of three arts: ternal, ketic, and otential energies E = Internal Energy+ Ketic Energy+ Potential Energy (4.76) Per unit mass E =U+KE+PE V euke eu gz (4.77) The fluid enterg or leag a control olume ossesses an additional form of energy, the flow energy P, as already discussed. Then the total energy of a flowg fluid on a unit-mass basis (denoted by ) becomes P e P ( u ke e) (4.78) But the combation P + u has been reiously defed as the enthaly h. So the relation the aboe equation reduces to V hke eh gz ( kj / kg ) (4.79) For let V i ei P i i ui gzi P i i (4.80) V i ei P i i hi gzi (4.81) For let V e ee P e e he gze (4.8) General equation E E de / dt system 0( steady) 0 (4.83) (4.84) Q W m Q W m V V Q W mh ( gz) Q W mh ( gz) (4.85) Comiled by Yidnekachew M. Page 14 of 1

15 In such cases, it is common ractice to assume heat to be transferred to the system (heat ut) at a rate of, and work roduced by the system (work ut) at a rate of, and then sole the roblem. The first-law or energy balance relation that case for a general steady-flow system becomes V V 1 QW mh h1 g( z z1) Diidg by gies the energy balance on a unit-mass basis as (4.86) V V1 qwh h1 g( z z1) (4.87) When the fluid exeriences negligible changes its ketic and otential energies (that is, = 0, e = 0), the energy balance equation is reduced further to qwh h1 (4.88) ke 4.7 Some Steady-Flow Engeerg Deices Nozzles and Diffusers Nozzles and diffusers are commonly utilized jet enges, rockets, sacecraft, and een garden hoses. A nozzle is a deice that creases the elocity of a fluid at the exense of ressure. A diffuser is a deice that creases the ressure of a fluid by slowg it down. That is, nozzles and diffusers erform oosite tasks. The cross-sectional area of a nozzle decreases the flow direction for subsonic flows and creases for suersonic flows. The reerse is true for diffusers. Figure 4.6 Schematic diagrams of Nozzles and diffusers Comiled by Yidnekachew M. Page 15 of 1

16 For flow through nozzles, the heat transfer, work, and otential energy are normally neglected, and nozzles hae one entrance and one exit. The conseration of energy becomes m m (4.89) m1 m m (4.90) E E (4.91) V i V e Qnet mihi gziw net mehe gze let exit (4.9) V i V e mihi mehe (4.93) V ( h h ) V (4.94) e i e i Turbes In steam, gas, or hydroelectric ower lants, the deice that dries the electric generator is the turbe. As the fluid asses through the turbe, work is done agast the blades, which are attached to the shaft. As a result, the shaft rotates, and the turbe roduces work. Figure 4.7 Schematic diagram of Turbes Comiled by Yidnekachew M. Page 16 of 1

17 If we neglect the changes ketic and otential energies as fluid flows through an adiabatic turbe hag one entrance and one exit, the conseration of mass and the steady-state, steadyflow first law becomes m m m1 m m E E V i V e Qnet mihi gziw net mehe gze let exit mh i i mh e e W (4.95) W m( hi he) (4.96) Comressors Comressors, as well as fans, are deices used to crease the ressure of a fluid. Work is sulied to these deices from an external source through a rotatg shaft. Therefore, comressors ole work uts. Een though these three deices function similarly, they do differ the tasks they erform. A fan creases the ressure of a gas slightly and is maly used to mobilize a gas. Figure 4.8 Schematic diagram of Comressors Comiled by Yidnekachew M. Page 17 of 1

18 If we neglect the changes ketic and otential energies as fluid flows through an adiabatic comressor hag one entrance and one exit, the steady-state, steady-flow first law or the conseration of energy equation becomes V i V e Qnet mihi gziw net mehe gze let exit W net m( he hi ) W net m( hi he) (4.97) Pums The work required when umg an comressible liquid an adiabatic steady-state, steadyflow rocess is gien by V V 1 QW mh h1 g( z z1) The enthaly difference can be written as h h1 u u1 P P 1 (4.98) For comressible liquids we assume that the density and secific olume are constant. The umg rocess for an comressible liquid is essentially isothermal, and the ternal energy change is aroximately zero (we will see this more clearly after troducg the second law). Thus, the enthaly difference reduces to the difference the ressure secific olume roducts. Sce = 1 = the work ut to the um becomes V V 1 W mp P1 g( z z1) (4.99) is the net work done by the control olume, and it is noted that work is ut to the um; so, =, um If we neglect the changes ketic and otential energies, the um work becomes W m P P 1 (4.100) W, um m P P 1 (4.101) We use this result to calculate the work sulied to boiler feed water ums steam ower lants. Throttlg Vales Comiled by Yidnekachew M. Page 18 of 1

19 Throttlg ales are any kd of flow-restrictg deices that cause a significant ressure dro the fluid. Some familiar examles are ordary adjustable ales, caillary tubes, and orous lugs. Unlike turbes, they roduce a ressure dro with olg any work. The ressure dro the fluid is often accomanied by a large dro temerature, and for that reason throttlg deices are commonly used refrigeration and air-conditiong alications. Figure 4.9 Schematic diagrams of Throttlg ales m m V i V e Qnet mihi gziw net mehe gze let exit mh mh (4.10) i i e e h i he (4.103) Mixg Chambers The mixg of two fluids occurs frequently engeerg alications. The section where the mixg rocess takes lace is called a mixg chamber. The ordary shower is an examle of a mixg chamber. Comiled by Yidnekachew M. Page 19 of 1

20 Figure 4.10 Schematic diagram of Mixg Chamber m m (4.104) m1m m3 (4.105) m m3 m1 (4.106) E E V i V e Qnet mihi gziw net mehe gze let exit Accordg to the sketched control olume, mass crosses the control surface. Neglectg ketic and otential energies and notg the rocess is adiabatic with no work, we hae for two entrances and one exit mh 1 1mh mh 3 3 (4.107) mh mmh mh (4.108) m ( h h ) m ( h h ) (4.109) m m ( h h ) ( h3 h) (4.110) Heat Exchangers Comiled by Yidnekachew M. Page 0 of 1

21 Heat exchangers are normally well-sulated deices that allow energy exchange between hot and cold fluids with mixg the fluids. The ums, fans, and blowers causg the fluids to flow across the control surface are normally located side the control surface. Figure 4.11 A heat exchanger can be as simle as two concentric ies. m m (4.111) For each fluid stream sce there is no mixg. m1 m mw (4.11) m3 m4 mr (4.113) E E V i V e Qnet mihi gziw net mehe gze let exit m1h1m3h3 mh m4h4 (4.114) mw( h h ) m ( h h ) (4.115) 1 R 4 3 Comiled by Yidnekachew M. Page 1 of 1