Fundamental Laws of Motion for Particles, Material Volumes, and Control Volumes

Transcription

1 Funamental Laws of Motion for Particles, Material Volumes, an Control Volumes Ain A. Sonin Department of Mechanical Engineering Massachusetts Institute of Technology Cambrige, MA 02139, USA August 2001 Ain A. Sonin Contents 1 Basic laws for material olumes 2 1 Material olumes an material particles 2 Laws for material particles 3 Mass conseration 3 Newton s law of (non-relatiistic) linear motion 3 Newton s law applie to angular momentum 4 First law of thermoynamics 4 Secon law of thermoynamics 5 Laws for finite material olumes 5 Mass conseration 5 Motion (linear momentum) 6 Motion (angular momentum) 6 First law of thermoynamics 7 Secon law of thermoynamics 8 2 The transformation to control olumes 8 The control olume 8 Rate of change oer a olume integral oer a control olume 9 Rate of change of a olume integral oer a material olume 10 Reynols material-olume to control-olume transformation 11 3 Basic laws for control olumes 13 Mass conseration 13 Linear momentum theorem 14 Angular momentum theorem 15 First law of thermoynamics 16 Secon law of thermoynamics 16 4 Proceure for control olume analysis 17

2 2 1 Basic Laws for Material Volumes Material olumes an material particles The behaior of material systems is controlle by uniersal physical laws. Perhaps the most ubiquitous of these are the law of mass conseration, the laws of motion publishe by Isaac Newton's in 1687, an the first an secon laws of thermoynamics, which were unerstoo before the nineteenth century ene. In this chapter we will reiew these four laws, starting with their most basic forms, an show how they can be expresse in forms that apply to control olumes. The control olume laws turn out to be ery useful in engineering analysis 1. The most funamental forms of these four laws are state in terms of a material olume. A material olume contains the same particles of matter at all times 2. A particular material olume may be efine by the close bouning surface that enelops its material particles at a certain time. Since eery point of a material olume s bouning surface moes (by efinition) with the local material elocity r (Fig. 1), the shape of the olume at all other times is etermine by the laws of ynamics. Fig. 1 A material olume moes with the material particles it encloses. 1 For a historical note on control olume analysis in engineering, see Chapter 4 of Walter G. Vincenti s What Engineers Know an How They know It, John Hopkins Uniersity Press, A material olume is the same as a close system in thermoynamics.

3 3 Laws for material particles The simplest forms of the four basic laws apply to an infinitesimal material particle that is so small that the elocity, ensity, thermoynamic temperature, an other intrinsic properties are essentially uniform within it. An obserer moing with a particle ( sitting on it, as it were) woul see its properties change with time only (Fig. 2). Fig. 2 Motion of a material particle between time t an time t+ t For a material particle with infinitesimal olume V(t), ensity t, an elocity, the four laws hae the following familiar forms: Mass conseration t ( V ) = 0 (1) This law asserts that the mass M = V of a material particle remains inariant. (The prefix inicates quantities that are of infinitesimal size, an the prefix refers to changes that occur in the inicate property in time t.) Newton s law of (non-relatiistic) linear motion ( V) t = F, or t ( V) = F (2)

4 4 Newton s law states that, relatie to an inertial reference frame 3, the prouct of a particle s mass an acceleration is at eery instant equal to the net force F (t) exerte on it by the rest of the unierse, or alternatiely, that the rate of change of a particle s momentum (a ector quantity) is equal at eery instant to the force applie to the particle by the rest of the unierse. (Actually the law states that the rate of change of momentum is proportional to the applie force, with the coefficient being uniersal, but in most systems of measurement the uniersal coefficient is set equal to unity, which etermines the units of force in terms of those of acceleration an time.) Newton s law applie to angular motion t ( r V) = r F (3) This law figures in rotary motion. The rate of change of a particle s angular momentum (the quantity in brackets on the left sie of (3), r (t) being the particle s position ector) is at eery instant equal to the net torque exerte on the particle by the rest of the unierse. This is not a new law, but one that follows from Eq. (2). Equation (3) is obtaine by taking the cross prouct of r (t) an Eq. (2), using Eq. (1), an noting that r t = = 0. Like the law it is erie from, Eq. (3) is ali only in inertial reference frames. Actually the law states that the rate of change of momentum is proportional to the applie force, with the coefficient being uniersal, but in most systems of measurement the uniersal coefficient is set equal to unity, which etermines the units of force in terms of those of acceleration an time. First law of thermoynamics ( e t V ) = W + Q (4) The increase of a material particle s total energy in a time interal t (e t is its total energy per unit mass, internal plus kinetic plus potential) is equal to the work W one in the interal t by forces exerte by the rest of the unierse on the material 3 An inertial reference frame is one in which the particle woul moe at a perceptibly constant elocity if all the forces acting on it were remoe.

5 5 olume s bounary (that is, not counting work one by olumetric boy forces), plus the heat Q ae to the particle at its bounary uring this interal. (Equation (4) is one part of the efinition of the quantity we call heat.) Secon law of thermoynamics ( s V) Q T (5) The increase of a particle s entropy (s represents the particle s entropy per unit mass) in a time t is greater than or equal to the heat ae to the particle at its bounary uring this interal iie by the absolute (thermoynamic) temperature, T. 4 The equality sign applies in the limit of a reersible process. Laws for finite material olumes From Eqs (1)-(5), which apply to an infinitesimal material particle, we can erie the laws for a finite material olume like the one sketche in Fig. 1. This is accomplishe by applying a particular law to each of the material particles that comprise the olume uner consieration, an summing. In the limit of a continuum, the sum can be iewe as an integral oer the olume of material properties which are expresse as fiels (that is, as functions of position r an time t ), consistent with the Eulerian way of escribing material flows. The result is the following set of rate equations 5 for a material olume s mass, momentum, energy, an entropy: Mass conseration t ( r,t)v = 0. (6) MV (t ) 4 Accoring to the Secon Law the temperature in Eq. (5) shoul be that of the reseroir from which the heat is supplie to the material particle. In this case the heat comes from the material that bouns the infinitesimal particle, where the temperature iffers infinitesimally from the particle s own aerage temperature T. 5 The usual term conseration equation is a bit of a misnomer, since mass is the only one of these quantities that is actually consere.

6 6 The mass containe in a material olume remains inariant. ( r,t) is the material s ensity fiel, V=xyz represents a olume element insie the material olume, an MV(t) uner the integral sign signifies integration oer the material olume at the instant t. Motion (linear momentum) t ( r,t) ( r,t)v = F MV (t). (7) MV (t ) This is Newton s law of motion: The rate of increase of a material olume's momentum, ealuate by integrating the local momentum per unit olume oer the material olume, is at eery instant equal to the ector sum F MV (t) of all the forces exerte on the material olume by the rest of the unierse. This force inclues boy forces acting on the material within the olume an surface forces acting at the bounary, but not the forces exerte between particles within the olume, which cancel each other out when the sum oer all the constituent parts is taken (action of one particle on another is exactly oppose by the reaction of the other on the first). It is unerstoo that Eq. (7) applies only in inertial (unaccelerating) reference frames uner non-relatiistic conitions. Motion (angular momentum) t MV (t ) r V = T MV (t) = r i F i (8) i This equation is obtaine by summing the angular momentum law for a material particle, Eq. (3), oer all the particles that comprise a finite material olume. The law states that the rate of increase of a material olume s angular momentum, expresse as the integral oer the olume of the angular momentum per unit olume, is equal to the ector sum T MV (t) of all torques exerte by the rest of the unierse on the material olume. This form of the law assumes that the torques exerte between two particles within the olume are equal an opposite, or zero, which is the case except in rare circumstances. Note again that Eq. (8) is not a new law, but a corollary of Newton s law of motion an subject to the same restrictions.

7 7 First law of thermoynamics t e t MV ( t ) V = Q MS (t) + W MS (t), (9) This law is obtaine by summing Eq. (4) oer all the particles that comprise the material olume an noting that the particle-to-particle heat transfer an work terms cancel for all particles insie the material olume when the sum is taken (what comes from one goes into the other). The law states that the rate of increase of a material olume's energy ( e t is the total energy per unit mass internal plus kinetic plus graitational) is equal to the sum of two source terms which represent interactions with the rest of the unierse at the olume s bounary. The first source term is the net heat flow rate into the material olume across its bouning surface Q MS (t) = q n A, (10) MS (t) where q = k T (11) is the conuctie heat flux ector at a point on the material olume s bounary, k isthe material s thermal conuctiity, T is its local thermoynamic temperature, n is the outwar-pointing unit ector at the bouning surface, an A is an elemental area on the bouning surface. The symbol MS(t) enotes integration oer the close bouning surface of the material olume at time t. The secon source term in (9) is the rate at which work is one by the rest of the unierse on the material olume at its bounary. This may be ealuate as W MS (t) = A (12) MS(t ) were is the ector stress exerte on the bounary by the rest of the unierse an is the material s local elocity at A. The quantity A is the force exerte by the rest of the unierse on the surface element A of the control olume.

8 8 Equation (9) thus has the form t MV (t ) e t V = q n A + MS(t ) A (13) MS(t ) Secon law of thermoynamics t q n A sv (14) T MV ( t ) MS (t) The rate of increase of a material olume s total entropy is greater than or equal to the sum of all the local heat inflows at the bounary when each contribution is iie by the local thermoynamic (absolute) temperature at the point on the material olume s surface where the transfer takes place. This law proies a bouning alue of the rate of entropy increase, but not the actual alue, an is less useful in ynamics than the other laws. It oes, howeer, hae some important uses in ynamics. One can for example iscar from the ynamically possible solutions (those that satisfy mass conseration an the equation of motion) those that are unrealizable because they iolate the Secon Law, an one can preict the entropy change in limiting cases of negligible issipation, where the equality sign applies. 2 The Transformation to Control Volumes The control olume Equations (6)-(8) an (13)-(14) state uniersal laws that apply to all material istributions. They are, howeer, in a form that makes them ill suite for applications. Each equation contains a term of the form in which a quantity t ( r,t)v (15) MV (t ) r,t that represents something per unit olume mass, momentum, energy, or entropy is first integrate oer a material olume an the result then ifferentiate with respect to time. When the material is flowing an eforming, the

9 9 olume s bounary moes with it an is not known as a function of time until the problem is sole. It seems, therefore, that one must know the solution before one can apply these laws to fin the solution. Clearly, we nee to fin a way of applying the basic laws to systems of our own choice, that is, to control olumes. A control olume is an arbitrarily efine olume with a close bouning surface (the control surface) that separates the unierse into two parts: the part containe within the control olume, an the rest of the unierse. The control surface is a mental construct, transparent to all material motion, an may be static in the chosen reference frame, or moing an expaning or contracting in any specifie manner. The analyst specifies the elocity c ( r,t) at all points of the control surface for all time. We shall show next how the uniersal laws for a material olume can be rewritten in terms of an arbitrarily efine control olume. This opens the way to the application of the integral laws in engineering analysis. Rate of change of a olume integral oer a control olume We begin by consiering a time eriatie like Eq. (15) for a control olume rather than a material olume. The time rate of change of the integral of some fiel quantity ( r,t) oer an arbitrarily efine control olume CV(t) is by efinition t CV( t ) V = lim t 0 ( r,t + t)v ( r,t)v CV ( t). (16) t CV ( t + t ) The first integral on the right han sie is ealuate at the aance time oer the aance olume, an the secon is ealuate at time t oer the olume at time t (Fig. 3). At any point r we can write for small t ( r,t + t) = ( r,t) + t t. (17) Inserting this into Eq. (15) we see immeiately that t V = CV( t ) CV (t ) ( r,t) V t + lim t 0 ( r,t)v ( r,t)v CV (t + t ) t CV ( t) (18)

10 10 where the integrals on the right are ealuate base on the alues of / t an at time t. In the limit t 0, the ifference between the two olume integrals in the secon term can be ealuate (see Fig. 3) by means of an integral oer the material surface at time t: ( r, t)v ( r,t)v = ( r,t) CV ( t + t ) CV (t) CS( t ) c t n A. (19) Here c ( r,t) is the elocity of the control surface element A, n is the outwarlyirecte unit normal ector associate with A, an c n ta is the control olume size increase in time t ue to the fact that the surface element A has moe in that time interal. The integral on the right sie is taken oer the entire (close) bouning surface CS(t) of the control olume. Fig. 3 Motion of a control olume between t an t+ t for small t. olume CV(t), Substituting Eq. (19) into Eq. (18), we obtain for an arbitrarily chosen control t CV( t ) V = ( r,t ) V + ( r,t) c n A t (20) CV (t) CS( t ) Rate of change of a olume integral oer a material olume The corresponing equation for a material olume MV(t) can be obtaine simply by noting that a material olume is a control olume eery point of which moes with the material elocity. Equation (20) thus applies to a material olume if we set the control

11 11 olume elocity equal to the material elocity, r c = r, an ientify the limits of integration with the material olume. This yiels for a material olume t MV (t ) V = ( r,t) V + ( r,t) n A t (21) MV ( t ) MS ( t ) Reynols' material-olume to control-olume transformation theorem Reynols transformation theorem proies a recipe for transforming the funamental laws in Eqs. (6)-(8) an (13)-(14) to control olumes. The transformation theorem is obtaine by consiering a control olume at time t an the material olume which coincies with it at that instant. The control olume CV(t) is chosen arbitrarily by efining its close bouning surface CS(t). The material olume is comprise of all the matter insie the control olume at time t (Fig. 4). The two olumes will of course ierge with time since the material olume wafts off with the particles to which it is "attache" an the control olume moes accoring to our specification. This is of no consequence since we are consiering only a frozen instant when the two olumes coincie. Fig. 4 The control an material olumes in the transformation theorem We apply Eq. (20) to our CV an Eq. (21) to the MV that coincies with it at time t, an note that because the olumes coincie, the integrals on the right-han sie of Eq. (21) may be ealuate oer either the CV instea of the MV. This yiels two alternatie equations for the time eriatie of an integral oer a material olume, expresse in terms of a CV that coincies with the material at the time inole:

12 12 Form A Form B t t V + ( r MV ( t ) V = t CV (t ) V = V + t MV( t ) CV ( t) r c ) n r A (22) CS (t) r n r A (23) Equation (22) is obtaine by subtracting Eq. (20) from Eq. (21). Equation (23) is Eq. (21) with the integrals referre to the CV instea of the MV, the two being coincient. Recall that r is the local material elocity, r c is the local control surface elocity at the surface element A, an n r is the outwar-pointing unit normal ector associate with A.. Both forms A an B are ali for arbitrarily moing an eforming control olumes (i.e. control olumes that may be expaning, translating, accelerating, or whateer), an for unsteay as well as steay flows. The two forms express exactly the same thing, but o the bookkeeping in ifferent ways. Remember that represents something per unit olume. Both forms express the material-olume time eriatie on the left as a sum of two terms that refer to the control olume that coincies with the material olume at the instant t. In form A, the first term on the right is the rate of change of the amount of insie the control olume at time t (the olume integral is ealuate first, then the time eriatie), an the secon term is the net rate of outflow of across the control olume's bounary. In form B, the first term on the right is the olume integral of the partial time eriatie of oer the control olume at time t (the CS is hel fixe at its position at time t while the integration is performe). The secon term accounts for the fact that the material olume s bounary (on the left) oes not in fact maintain the shape it has at time t, but enelops more olume (an more of the quantity ) when it expans, eery point moing with the local material elocity. The control surface elocity oes not enter at all in form B. We shall see that Form A is usually more conenient in unsteay applications than Form B. This is particularly true in cases where t is singular at some surface insie the control olume (as it is at a moing flame front insie a soli-propellant rocket, for example, if is the material ensity istribution in the rocket), in which case it is ifficult to ealuate the olume integral in Form B. The olume integral in From A, on the other han, can be calculate straightforwarly an then ifferentiate with respect to time.

13 13 3 Basic laws for control olumes The basic physical laws expresse by Eqs (6)-(8) an (13)-(14) in materialolume terms are transferre to a control olume as follows. We transform the left sies by setting equal to either,, r, e t, or s, in form A or form B of Reynols transformation theorem [ Eqs. (22) an (23)]. The right han sies are transforme by noting that since the MS an CS coincie at the instant being consiere (see Fig. 4), the force, torque, an heat flow terms on the right han sie of Eqs (7)-(8) an (13)-(14) are the same for the CV as for the MV. Note, howeer, that the rate at which work is being one on the CS is not equal to the rate at which work is being one on the MS because these surfaces moe at ifferent elocities. Two alternatie forms are obtaine for each equation, epening on whether Form A [Eq. (22)] or form B [Eq. (23)] of the transformation theorem is use. The alternatie forms are expressions of the same physical law, state in somewhat ifferent terms. Both apply to any control olume at eery instant in time no matter how the control surface is moing an eforming, proie the reference frame is one where the basic equations apply. We remin the reaer (see Fig. 3) that in what follows, n = r r n = cos (24) is the outwar normal component of the material s absolute elocity at the control surface, being the angle between r an the outwar-pointing normal unit ector n r, an rn = ( r r c ) r n = n cn (25) is the outwar normal component of the material's elocity relatie to the control surface, cn being the outwar normal component of the control surface's elocity. Mass conseration Setting = in Eqs. (22) an (23), we transform Eq. (6) into two alternatie forms for a CV:

14 14 Form A t V + rn A = 0 CV( t ) CS(t ) (26A) Form B V + n A = 0 t CV ( t ) CS ( t ) (26B) Equation (26A) states the mass conseration principle as follows: The rate of increase of the mass containe in the CV, plus the net mass flow rate out through the (generally moing) CS, equals zero at eery instant. Equation (26B) states the same principle in ifferent but equally correct terms: The rate of increase of the mass containe in the fixe olume efine by the control surface at time t, plus the net mass outflow rate through the fixe bouning surface of that olume, equals zero at all times. Linear momentum Putting = in Eqs. (22) an (23) an substituting into (7), we obtain the following alternatie forms for the equation of motion applie to a CV: Form A t V + CV (t ) CS (t) rn A = F CV (t) (27A) Form B ( ) V + t CV ( t ) n A = F (t) (27B) CV Here, F CV (t) is the ector sum of all the forces exerte at time t by the rest of the unierse on the control olume, incluing olumetric forces an stresses exerte on the control olume s bounaries. For a continuous istribution of surface an boy forces, F CV (t) = A + G V (28) CS CV Equation (27A) states that the rate at which the linear momentum containe in the CV increases with time, plus the net flow rate of linear momentum out through the control surface, is equal at eery instant to the force exerte by the rest of the unierse on the material within the control surface.

15 15 Equation (27B) states it in ifferent terms: The rate of increase of the momentum containe in a fixe olume ientical with the control surface at time t, plus the net mass outflow rate through the fixe bouning surface of that olume, is equal at all times to the force exerte by the rest of the unierse on the material in the control olume. Angular momentum Setting = r in either (22) or (23) an substituting into (8) yiels the angular momentum theorem for a CV in two alternatie forms: Form A t ( r ) V + ( CV(t ) CS( t ) r ) rn A = T CV (t) (29A) Form B ( r ) V + ( t CV (t ) r ) n A = T CV (t) (29B) Here r is the position ector from an arbitrary origin, T CV (t) is the sum of all the torques (relatie to the chosen origin) that the rest of the unierse exerts on the control olume, incluing those resulting from both surface forces (pressure an shear) an olumetric boy forces (e.g. graity). An inertial reference frame is presume. For a continuous istribution of surface an boy forces, T CV (t) = r A + r G V. (30) CS CV where is the ector stress exerte on the bounary element A by the rest of the unierse, an G is the boy force exerte by the rest of the unierse on unit mass of material within the olume. Equation (29A) states the following: The rate at which the angular momentum insie the control olume increases with time, plus the net rate at which angular momentum flows out of the control surface, is equal to the net torque exerte by the rest of the unierse on the matter in the control olume (on the bounary as well as on the mass within). The reaer will be able to interpret (29B) base on the comments been mae aboe with reference to (26B) an (27B).

16 16 First law of thermoynamics (energy equation) Setting = e t in Eqs. (22) an (23) an substituting into (13), we obtain two forms of the first law for a CV: Form A t e t V + e t rn A = q n A + A (31A) CV( t ) CS( t ) Form B CV ( t ) ( e t ) t V + CS ( t) e t n A = q n A + A (31B) Equation (31A) states that the rate at which the total energy containe in the CV increases with time, plus the net rate at which total energy flows out of the CS, is equal to the sum of two terms on the right. The first term is the rate at which heat is conucte into the CV ia the control surface. The secon is the rate at which the rest of the unierse oes work on the material olume whose bouning surface coincies with the CS at the instant in question. The work one at the control surface, W CS (t) = r r c A, (32) CS(t ) epens on the control surface elocity istribution, which is chosen at will by the analyst an obiously has no place in a uniersal law. Secon law of thermoynamics Form A t q n sv + s rn A T A (33A) CV( t ) Form B CV ( t ) ( s) t V + q n s n A T A (33B) Equation (33A) states that the rate of increase of the entropy containe in the CV (s is the entropy per unit mass), plus the net rate of entropy conection out of the control surface, neer excees the integral oer the control surface of the normal heat influx iie by the local absolute temperature.

17 17 4 Proceure for Control Volume Analysis The application of any one of the integral laws inoles consieration of the following nine steps: Step 1 Choose the reference frame in which the problem is iewe an elocity an other properties are measure. If Newton s law is inole in the problem, the reference frame must be an inertial (non-accelerating) frame. Step 2 Choose your control olume by specifying its (close bounin surface at some instant (e.g. t=0) an at all times thereafter. The control surface must be close. It may be multiply connecte. It may moe in the chosen reference frame an expan an istort as it oes so. All this is your choice. If the CS runs parallel to a flui-soli interface, take care to specify whether your control surface is just on the flui sie, or just on the soli sie. It must be on one sie or the other, so that quantities like, r, e t, etc. hae well efine alues. Step 3 Write own the integral law that you wish to apply. Step 4 Ientify the alues of the properties (,, c,e t,, q, an s, or whicheer of them figure in your problem) at eery element A of the control surface an calculate the surface integrals that appear in your integral equations. Select the control olume so that the bouning surface passes as much as possible through regions where you know the properties, or can easily euce them. Whereer you on t know some quantities, introuce them as unknowns, expecting to etermine them as you procee. Step 5 Ientify the alues of, r,,e t,s an G at eery olume element V insie the control olume, an ealuate the olume integrals in your integral equations. Step 6

18 18 Calculate the time eriatie of the olume integral that appears on left-han sie of your integral equation. Step 7 From steps 4, 5 an 6, substitute into your integral equations. Step 8 If you wish to sole a practical problem using the control olume theorems, you must write own enough equations to ensure that their number equals the number of unknowns in the equations. The four integral laws that we hae escribe are totally general an rigorous, but these laws themseles will not proie enough equations to sole for the unknowns. You will nee to raw on other physical laws (e.g. graitational theory to characterize the external boy force fiel) an constitutie equations (e.g. the thermoynamic equations of state). Aboe all you will nee to make simplifying approximations whereer they are appropriate. Uniform flow conitions oer any gien cross-sections ( quasi-one-imensional flow ) is a typical engineering approximation, for integral relations by themseles proie no information about elocity istributions. If you you hae reason to beliee that the flow may be approximate as inisci, you inoke Bernoulli s equation. If base on the equation of state you think ensity aries little, you write =constant. (Note: Bernoulli s equation is erie from Newton s equation of motion, just like the linear momentum theorem. By inoking Bernoulli, are we not simply writing own the same equation twice? We are not. The linear momentum equation applies generally. Bernoulli s equation introuces the aitional information that the flow is inisci.) Step 9 Sole for the unknowns. Step 10 Check, by suitable orer-of-magnitue estimates, that your solution is consistent with any approximations that you mae.