Control Volume Derivations for Thermodynamics

Transcription

1 Control olume Derivations for Thermoynamics J. M. Powers University of Notre Dame AME 327 Fall 2003 This ocument will give a summary of the necessary mathematical operations necessary to cast the conservation of mass an energy principles in a traitional control volume formulation. The analysis presente has been amalgamate from a variety of sources. Most irectly, it is an specialization of my course notes for AME Basic mathematical founations are covere well by Kaplan. 2 A very etaile a reaable escription, which has a stronger emphasis on flui mechanics, is given in the unergrauate text of Whitaker. 3 A very rigorous treatment of the evelopment of all equations presente here is inclue in the grauate text of Aris. 4 Popular mechanical engineering unergrauate fluis texts have closely relate expositions. 5 6 However, espite their etail, these texts have some minor flaws! The treatment given in the AME 327 text by Sonntag, Borgnakke, an an Wylen 7 SBW) is not as etaile. This ocument will use a notation generally consistent with SBW an show in etail how to arrive at its results. 1 Relevant mathematics We will use several theorems which are evelope in vector calculus. Here we give short motivations an presentations. The reaer shoul consult a stanar mathematics text for etaile erivations. 1.1 Funamental theorem of calculus The funamental theorem of calculus is as follows x=b ) x=b ψ φx) x = x = ψb) ψa). 1) x=a x=a x It effectively says that to fin the integral of a function φx), which is the area uner the ψ curve, it suffices to fin a function ψ, whose erivative is φ, i.e. = φx), evaluate ψ at x each enpoint, an take the ifference to fin the area uner the curve. 1 J. M. Powers, 2003, Lecture Notes on Intermeiate Flui Mechanics, University of Notre Dame, http : // powers/ame.538/notes.pf. 2 W. Kaplan, 2003, Avance Calculus, Fifth Eition, Aison-Wesley, New York. 3 S. Whitaker, 1992, Introuction to Flui Mechanics, Krieger, Malabar, Floria. 4 R. Aris, 1962, ectors, Tensors, an the Basic Equations of Flui Mechanics, Dover, New York. 5 F. M. White, 2002, Flui Mechanics, Fifth Eition, McGraw-Hill, New York. 6 R. W. Fox, A. T. McDonal, an P. J. Pritchar, 2003, Introuction to Flui Mechanics, Sixth Eition, John Wiley, New York. 7 R. E. Sonntag, C. Borgnakke, an G. J. an Wylen, 2003, Funamentals of Thermoynamics, Sixth Eition, John Wiley, New York. 1

2 1.2 Gauss s theorem Gauss s 8 theorem is the analog of the funamental theorem of calculus extene to volume integrals. Let us efine the following quantities: t time, x spatial coorinates, a t) arbitrary moving an eforming volume, A a t) bouning surface of the arbitrary moving volume, n outer unit normal to moving surface, φx, t) arbitrary scalar function of x an t Gauss s theorem is as follows: φ = φn A 2) at) A at) The surface integral is analogous to evaluating the function at the en points in the funamental theorem of calculus. Note if we take φ to be the scalar of unity whose erivative must be zero), Gauss s theorem reuces to 1) = 1)n A, 3) at) A at) 0 = 1)n A, 4) A at) n A = 0. 5) A at) That is the unit normal to the surface integrate over the surface, cancels to zero when the entire surface is inclue. 1.3 Divergence theorem The extension of Gauss s theorem 2) to vector functions is the ivergence theorem an is as follows: α = α n A. 6) at) Here α is an arbitrary vector function. We will use the ivergence theorem 6) extensively. It allows us to convert sometimes ifficult volume integrals into easier interprete surface integrals. It is often useful to use this theorem as a means of toggling back an forth from one form to another. 8 Carl Frierich Gauss, , Brunswick-born German mathematician, consiere the founer of moern mathematics. Worke in astronomy, physics, crystallography, optics, biostatistics, an mechanics. Stuie an taught at Göttingen. 2 A at)

3 A II A I w II w I I a II n I n II I t) a t+ t) a II Figure 1: Sketch of the motion of an arbitrary volume a t). The bounaries of a t) move with velocity w. The outer normal to a t) is A a t). Here we focus on just two regions: I, where the volume is leaving material behin, an II, where the volume is sweeping up new material. 1.4 Leibniz s theorem Leibniz s 9 theorem relates time erivatives of integral quantities to a form which istinguishes changes which are happening within the bounaries to changes ue to fluxes through bounaries. Let us consier the scenario sketche in Figure 1. Say we have some value of interest, Φ, which results from an integration of a kernel function φ over a t), for instance Φ = φ. 7) at) We are often intereste in the time erivative of Φ, the calculation of which is complicate by the fact that the limits of integration are time-epenent. From the efinition of the erivative, we fin that Now we have Φ t = φ = lim t at) t 0 φt + t) at+ t) at) φt). 8) t a t + t) = a t) + II t) I t). 9) 9 Gottfrie Wilhelm von Leibniz, , Leipzig-born German philosopher an mathematician. Invente calculus inepenent of Newton an employe a superior notation to that of Newton. 3

4 Here II t) is the amount of new volume swept up in time increment t, an I t) is the amount of volume abanone in time increment t. So we can break up the first integral into at+ t) φt+ t) = which gives us then t lim t 0 at) φ = at) φt+ t) + φt+ t) φt+ t), 10) II t) I t) φt + t) + at) II t) φt + t) I t) φt + t) at) t Rearranging 11) by combining terms with common limits of integration, we get t at) φ = lim t 0 + lim t 0 Let us now further efine at) φt + t) φt)) t II t) φt + t) I t) t w velocity vector of points on the moving surface a t), φt).11) φt + t). 12) Now the volume swept up by the moving volume in a given time increment t is an the volume abanone is II = w }{{ n} t A II = w II t A II, 13) positive istance I = w n negative t A I = w I t A I. 14) istance Substituting into our efinition of the erivative, we get t + lim t 0 Now we note that at) φt + t) φt)) φ = lim t 0 at) t A II t) φt + t)w II t A II + A I t) φt + t)w I t A I t 15) We can use the efinition of the partial erivative to simplify the first term on the right sie of 15), The time increment t cancels in the area integrals of 15), an A a t) = A I + A II, 4

5 so that φ = t at) total time rate of change φ t at) intrinsic change within volume + φw n A. 16) A at) net flux into volume This is the three-imensional scalar version of Leibniz s theorem. We can also apply the ivergence theorem 6) to Leibniz s theorem 16) to convert the area integral into a volume integral to get ) φ φ = t at) at) t + φw). 17) Say we have the very special case in which φ = 1; then Leibniz s theorem 16) reuces to t at) = t at) = at) A at) t A 1) + 1)w n A, 18) at) w n A. 19) This simply says the total volume of the region, which we call a t), changes in response to net motion of the bouning surface. Leibniz s theorem 16) reuces to a more familiar result in the one-imensional limit. We can then say x=bt) x=bt) φ φx, t) x = t x=at) x=at) t b a x + φbt), t) φat), t). 20) t t As in the funamental theorem of calculus 1), for the one-imensional case, we o not have to evaluate a surface integral; instea, we simply must consier the function at its enpoints. Here b a an are the velocities of the bouning surface an are equivalent to w. The terms t t φbt), t) an φat), t) are equivalent to evaluating φ on A a t). 1.5 General Transport Theorem Let B be an arbitrary extensive thermoynamic property, an β be the corresponing intensive thermoynamic property so that B = βm. 21) The prouct of a ifferential amount of mass m with the intensive property β give a ifferential amount of the extensive property. Since m =, 22) where is the mass ensity an is a ifferential amount of volume, we have B = β. 23) 5

6 If we take the arbitrary φ = β, the Leibniz theorem becomes our general transport theorem: β = t at) at) t A β) + β w n) A. 24) at) Applying the ivergence theorem to the general transport theorem, we fin ) β = β) + βw) t at) at) t 1.6 Reynols transport theorem. 25) We get the well known Reynols 10 transport theorem if we force the arbitrary velocity of the moving volume to take on the velocity of a flui particle, i.e. take w = v 26) In this case, our arbitrary volume is no longer arbitrary. Instea, it always contains the same flui particles. We call this volume a material volume, m t). The proper way to generalize laws of nature which were evelope for point masses is to consier collections of fixe point masses, which will always resie within a material volume. That sai, it is simple to specialize the general transport theorem to obtain the Reynols transport theorem. Here we give two versions, the first using area integrals, an the secon using volume integrals only: t t β = β = 1.7 Fixe control) volumes t A β) + β v n) A, 27) mt) ) β) + βv). 28) t If we take our arbitrary volume to be fixe in space, it is most often known as a control volume. For such volumes w = 0. 29) Thus the arbitrary volume loses its time epenency, so that an the general transport theorem reuces to a t) =, A a t) = A, 30) β = t β). 31) t 10 Osborne Reynols, , Belfast-born British engineer an physicist, eucate in mathematics at Cambrige, first professor of engineering at Owens College, Manchester, i funamental experimental work in flui mechanics an heat transfer. 6

7 2 Conservation axioms A funamental goal of mechanics is to take the verbal notions which emboy the basic axioms into usable mathematical expressions. First, we must list those axioms. The axioms themselves are simply principles which have been observe to have wie valiity as long as length scales are sufficiently large to contain many molecules. Many of these axioms can be applie to molecules as well. The axioms cannot be proven. They are simply statements which have been useful in escribing the universe. A summary of the axioms in wors is as follows Mass conservation principle: The time rate of change of mass of a material region is zero. Linear momenta principle: The time rate of change of the linear momenta of a material region is equal to the sum of forces acting on the region. This is Euler s generalization of Newton s secon law of motion. Angular momenta principle: The time rate of change of the angular momenta of a material region is equal to the sum of the torques acting on the region. This was first formulate by Euler. Energy conservation principle: The time rate of change of energy within a material region is equal to the rate that energy is receive by heat an work interactions. This is the first law of thermoynamics. Entropy inequality: The time rate of change of entropy within a material region is greater than or equal to the ratio of the rate of heat transferre to the region an the absolute temperature of the region. This is the secon law of thermoynamics. Here we shall systematically convert two of these axioms, the mass conservation principle an the energy conservation principle, into mathematical form. 2.1 Mass Mass is an extensive property for which we have B = m, β = 1. 32) The mass conservation axiom is simple to state mathematically. It is m = 0. 33) t A relevant material volume is sketche in Figure 2. We can efine the mass enclose within a material volume base upon the local value of ensity: m =. 34) 7

8 A w = v n Figure 2: Sketch of finite material region m t), infinitesimal mass element, an infinitesimal surface element A with unit normal n, an general velocity w equal to flui velocity v. So the mass conservation axiom is Invoking the Reynols transport theorem 27), t we get = 0. 35) t [ ] = mt) [ ] t + A mt) v n[ ]A, = t t A + v n A = 0. 36) mt) Now we invoke the ivergence theorem, Eq. 6) t) [ ] = At) n [ ]A, to convert a surface integral to a volume integral to get the mass conservation axiom to rea as t + v) = 0, 37) mt) ) t + v) = 0. 38) Now, in an important step, we realize that the only way for this integral, which has absolutely arbitrary limits of integration, to always be zero, is for the integran itself to always be zero. Hence, we have + v) = 0. 39) t This is the very important ifferential form of the mass conservation principle. We can get a very useful control volume formulation by integrating the mass conservation principle 39) over a fixe volume : ) t + v) = 0. 40) 8

9 Now the integral of 0 over a fixe omain must be zero. This is equivalent to saying b a 0x = 0, where the area uner the curve of 0 has to be zero. So we have ) t + v) Next apply the ivergence theorem 6) to 41) to get = 0. 41) t A + v n A = 0. 42) Applying now the result from 31) to 42), we see for the fixe volume that + v n A = 0. 43) t A We note now that at an inlet to a control volume that v points in an opposite irection to n, so we have v n < 0, at inlets. 44) At exits to a control volume v an n point in the same irection so that v n > 0, at exits. 45) If now, we take the simplifying assumption that an v have no spatial variation across inlets an exits, we get for a control volume with one inlet an one exit that + e v e A e i v i A i = 0. 46) t Here the subcript i enotes inlet, an the subscript e enotes exit. Rearranging 46), we fin = i v i A i e v e A e. 47) t We now efine the mass in the control volume m cv as m cv =. 48) Here 48) is equivalent to the equation on the top of p. 164 of SBW. If we make the further simplifying assumption that oes not vary within, we fin that m cv t rate of change of mass = i v i A i mass rate in e v e A e. 49) mass rate out Here m cv is the mass enclose in the control volume. If there is no net rate of change of mass the control volume is in steay state, an we can say that the mass flow in must equal the mass flow out: i v i A i = e v e A e. 50) 9

10 We efine the mass flow rate ṁ as ṁ = v A. 51) For steay flows with a single entrance an exit, we have ṁ = constant. 52) For unsteay flows with a single entrance an exit, we can rewrite 49) as m cv t = ṁ i ṁ e. 53) For unsteay flow with many entrances an exits, we can generalize 49) as m cv = i v i A i e v e A e, 54) } t {{} rate of change of mass mass rate in mass rate out m cv = ṁ i ṁ e } t {{} rate of change of mass mass rate in mass rate out 55) Note that 55) is fully equivalent to SBW s Eq. 6.1), but that it actually takes a goo eal of effort to get to this point with rigor! For steay state conitions with many entrances an exits we can say i v i A i = e v e A e, 56) ṁi = ṁ e. 57) Here 56) is the same as SBW s 6.9). 2.2 Energy For energy, we must consier the total energy which inclues internal, kinetic, an potential. Our extensive property B is thus B = E = U + 1 mv v + mgz. 58) 2 Here we have assume the flui resies in a gravitational potential fiel in which the gravitational potential energy varies linearly with height z. The corresponing intensive property β is β = e = u + 1 v v + gz. 59) 2 We recall the first law of thermoynamics, which states the the change of a material volume s total energy is equal to the heat transferre to the material volume less the work one by the material volume. Mathematically, this is state as E = δq δw 60) 10

11 We recall the total erivative is use for E, since energy is a property an has an exact ifferential, while both heat transfer an work are not properties an o not have exact ifferentials. It is more convenient to express the first law as a rate equation, which we get by iviing 60) by t to get E t = δq t δw t. 61) Recall that the upper case letters enote extensive thermoynamic properties. For example, E is total energy, inclusive of internal an kinetic an potential 11, with SI units of J. Let us consier each term in the first law of thermoynamics in etail an then write the equation in final form Total energy term For a flui particle, the ifferential amount of total energy is E = u + 1 ) 2 v v + gz, 62) = u + 1 ) 2 v v + gz. 63) mass internal +kinetic +potential Work term Let us partition the work into work W p one by a pressure force F p an work one by other sources, which we shall call W mv, where the subscript mv inicates material volume. Taking a time erivative, we get W = W p + W mv. 64) δw t = δw p t + Ẇmv. 65) The work one by other sources is often calle shaft work an represents inputs of such evices as compressors, pumps, an turbines. Its moeling is often not rigorous. Recall that work is one when a force acts through a istance, an a work rate arises when a force acts through a istance at a particular rate in time hence, a velocity is involve). Recall also that it is the ot prouct of the force vector with the position or velocity that gives the true work or work rate. In shorthan, we can say that the ifferential work one by the pressure force F p is δw p = F p x, 66) δw p t = F p x t = F p v. 67) 11 Strictly speaking our erivation will only be vali for potentials which are time-inepenent. This is the case for orinary gravitational potentials. The moifications for time-epenent potentials are straightforwar, but require a more nuance interpretation than space permits here. 11

12 δw t p = F v > 0 for expansion p. v n v v F = pan p Figure 3: Sketch of flui element oing work. Here W has the SI units of J, an F p has the SI units of N. Now let us consier the work one by the pressure force. In a piston-cyliner arrangement in which a flui exists with pressure p within the cyliner an the piston is rising with velocity v, the work rate one by the flui is positive. We can think of the local stress vector in the flui as pointing in the same irection as the flui is moving at the piston surface, so that the ot prouct is positive. Now we can express the pressure force in terms of the pressure by Substituting 67) into 68), we get F p = pan. 68) δw p t = pan v. 69) It is note that we have been a little loose istinguishing local areas from global areas. Better state, we shoul say for a material volume that δw p t = A mt) pn v A. 70) This form allows for p an v to vary with location. This is summarize in the sketch of Figure 3. 12

13 2.2.3 Heat transfer term If we were consiering temperature fiels with spatial epenency, we woul efine a heat flux vector. This approach is absolutely necessary to escribe many real-worl evices, an is the focus of a stanar unergrauate course in heat transfer. Here we will take a very simplifie assumption that the only heat fluxes are easily specifie an are all absorbe into a lumpe scalar term we will call Q mv. This term has units of J/s = W in SI. So we have then δq t = Q mv. 71) The first law of thermoynamics Putting the wors of the first law into equation form, we get u + 1 ) t 2 v v + gz = δq t δw t. 72) We next introuce our simplification of heat transfer 71) an partition of work 65) along with 70) into 72) to get u + 1 ) t 2 v v + gz = Q ) mv Ẇ mv + pn v A. 73) A mt) Now we bring the pressure work integral to the right sie of 73) to get u + 1 ) t 2 v v + gz + pn v A = Q mv Ẇmv. 74) A mt) We next invoke the Reynols transport theorem 27) into 74) to expan the erivative of the first integral so as to obtain u + 1 )) t 2 v v + gz + u + 1 )) A mt) 2 v v + gz v n A + pn v A = Q mv Ẇmv. 75) We next note that the two area integrals have the same limits an can be combine to form u + 1 )) t 2 v v + gz + u + p + 12 )) v v + gz v n A A mt) We recall now the efinition of enthalpy h, A mt) = Q mv Ẇmv. 76) h = u + p 77) Invoking 77) into 76), we get u + 1 )) t 2 v v + gz + A mt) 13 h + 1 )) 2 v v + gz v n A = Q mv Ẇmv. 78)

14 Next use the ivergence theorem 6) to rewrite 78) as u + 1 )) t 2 v v + gz + v h + 1 )) 2 v v + gz = Q mv Ẇmv. 79) Now, for convenience, let us efine the specific control volume heat transfer an work q mv an w mv, each with SI units J/kg such that Q mv = Ẇ mv = so that by substituting 80) an 81) into 79), we get u + 1 )) t 2 v v + gz + = t q mv), 80) t w mv), 81) t q mv) v h + 1 )) 2 v v + gz t w mv). 82) Now all terms in 82) have the same limits of integration, so they can be groupe to form u + 1 )) t 2 v v + gz + v h + 1 )) 2 v v + gz ) t q mv) + t w mv) = 0. 83) As with the mass equation, since the integral is zero, in general we must expect the integran to be zero, giving us u + 1 )) t 2 v v + gz + v h + 1 )) 2 v v + gz t q mv) + t w mv) = 0. 84) To get the stanar control volume form of the equation, we then integrate 84) over a fixe control volume to get u + 1 )) t 2 v v + gz + v h + 1 )) 2 v v + gz ) t q mv) + t w mv) = 0. 85) Now efining the control volume heat transfer rate an work rate, Q cv an Ẇcv, Q cv = Ẇ cv = t q mv), 86) t w mv), 87) 14

15 we employ 86) an 87) in 85) to get u + 1 )) t 2 v v + gz + v h + 1 )) ) 2 v v + gz = Q cv Ẇcv. 88) Applying the ivergence theorem 6) to 88) to convert a portion of the volume integral into an area integral, an 31) to bring the time erivative outsie the integral for the fixe volume, we get u + 1 ) t 2 v v + gz + A v n h + 1 ) 2 v v + gz A = Q cv Ẇcv. 89) We now efine the total energy in the control volume as E cv = u + 1 ) 2 v v + gz. 90) Next assume that all properties across entrances an exits are uniform so that the area integral in 88) reuces to v n h + 1 ) A 2 v v + gz A = ṁe h e + 1 ) 2 v e v e + gz e ṁ i h i + 1 ) 2 v i v i + gz i. 91) Substituting 90) an 91) into 89), we get E cv t + ṁ e h e v e v e + gz e ) ṁ i h i v i v i + gz i ) = Q cv Ẇcv. 92) Rearranging 92), we get E cv t rate of C energy change = Q cv C heat transfer rate Ẇ cv C shaft work rate + ṁ i h i + 1 ) 2 v i v i + gz i ṁ e h e + 1 ) 2 v e v e + gz e. 93) total enthalpy rate in total enthalpy rate out Here 93) is equivalent to SBW s Eq. 6.7). Note that the so-calle total enthalpy is often efine as h tot = h + 1 v v + gz. 94) 2 Employing 94) in 93), we fin E cv t = Q cv Ẇcv + ṁ i h tot,i ṁ e h tot,e. 95) Here 95) is equivalent to SBW s 6.8). 15

16 If there is a single entrance an exit, we lose the summation, so that 93) becomes E cv t = Q cv Ẇcv + ṁ i h i v i v i + gz i ) ṁ e h e v e v e + gz e ). 96) If the flow is steay, we have Ecv = 0 an ṁ t i = ṁ e = ṁ, so the first law with a single entrance an exit becomes 0 = Q cv Ẇcv + ṁ h i h e + 1 ) 2 v i v i v e v e ) + gz i z e ). 97) Defining the specific control volume heat transfer an work as an substituting 98) into 97), we get q = Q cv ṁ, w = Ẇcv ṁ, 98) 0 = q w + h i h e v i v i v e v e ) + gz i z e ), 99) Now 99) can be rearrange to form SBW s 6.14): q + h i v i v i + gz i = w + h e v e v e + gz e. 100) If the flow is aiabatic, steay, has one entrance an one exit, an there is no shaft work, we fin that the total enthalpy must remain constant: h i v i v i + gz i = h e v e v e + gz e 101) 16