5.4 Second Law of Thermodynamics Irreversible Flow 5

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1 5.4 Second Law of hemodynamics Ievesile Flow Second Law of hemodynamics Ievesile Flow he second law of themodynamics fomalizes the notion of loss. he second law of themodynamics affods us with a means to fomalize the equality ǔ ǔ q net 0 (5.90) fo steady, compessile, one-dimensional flow with fiction see Eq In this section we contue to develop the notion of loss of useful o availale enegy fo flow with fiction. Mimization of loss of availale enegy any flow situation is of ovious engeeg impotance Semi-fitesimal Contol Volume Statement of the Enegy Equation If we apply the one-dimensional, steady flow enegy equation, Eq. 5.70, to the contents of a contol volume that is fitesimally th as illustated Fig 5.8, the esult is m c dǔ d a p d av (5.9) g dzd dq net Fo all pue sustances cludg common engeeg wokg fluids, such as ai, wate, oil, and gasole, the followg elationship is valid see, fo example, Ref. 3. ds dǔ pd a (5.9) whee is the asolute tempeatue and s is the entopy pe unit mass. Comg Eqs. 5.9 and 5.9 we get m c ds pd a d ap d av o, dividg though y m and lettg dq net dq g dz d dq net net m, we ota d av g dz ds dq net (5.93) 5.4. Semi-fitesimal Contol Volume Statement of the Second Law of hemodynamics A geneal statement of the second law of themodynamics is D Dt sys dq net s dv a a sys (5.94) z Semi-fitesimal contol volume θ x g θ d Flow Figue 5.9 Semi-fitesimal contol volume. 5 his entie section may e omitted without loss of contuity the text mateial.

2 Chapte 5 Fite Contol Volume Analysis he second law of themodynamics volves entopy, heat tansfe, and tempeatue. o wods, the time ate of cease of the sum of the atio of net heat entopy of a system tansfe ate to system to asolute tempeatue fo each paticle of mass the system eceivg heat fom suoundgs he ight-hand side of Eq is identical fo the system and contol volume at the stant when system and contol volume ae cocident; thus, a adq net (5.95) a a dq net sys cv With the help of the Reynolds tanspot theoem Eq. 4.9 the system time deivative can e expessed fo the contents of the cocident contol volume that is fixed and nondefomg. Usg Eq. 4.9, we ota D s dv 0 s dv (5.96) Dt 0t sys sv nˆ da cs Fo a fixed, nondefomg contol volume, Eqs. 5.94, 5.95, and 5.96 come to give 0 0t s dv cs sv nˆ da a a dq net cv (5.97) At any stant fo steady flow 0 0t s dv 0 (5.98) If the flow consists of only one steam though the contol volume and if the popeties ae unifomly distiuted one-dimensional flow, Eqs and 5.98 lead to m dqnet s out s a (5.99) Fo the fitesimally th contol volume of Fig. 5.8, Eq yields m dqnet ds a (5.00) If all of the fluid the fitesimally th contol volume is consideed as eg at a unifom tempeatue,, then fom Eq we get he elationship etween entopy and heat tansfe ate depends on the pocess volved. ds dq net o ds dq net 0 (5.0) he equality is fo any evesile fictionless pocess; the equality is fo all ievesile fiction pocesses Comation of the Equations of the Fist and Second Laws of hemodynamics Comg Eqs and 5.0, we conclude that c d av g dz d 0 (5.0)

3 5.4 Second Law of hemodynamics Ievesile Flow 3 he equality is fo any steady, evesile fictionless flow, an impotant example eg flow fo which the Benoulli equation Eq. 3.7) is applicale. he equality is fo all steady, ievesile fiction flows. he actual amount of the equality has physical significance. It epesents the extent of loss of useful o availale enegy which occus ecause of ievesile flow phenomena cludg viscous effects. hus, Eq. 5.0 can e expessed as c d av g dz d dloss ds dq net (5.03) he ievesile flow loss is zeo fo a fictionless flow and geate than zeo fo a flow with fictional effects. Note that when the flow is fictionless, Eq multiplied y density,, is identical to Eq hus, fo steady fictionless flow, Newton s second law of motion see Section 3. and the fist and second laws of themodynamics lead to the same diffeential equation, d av g dz 0 (5.04) If some shaft wok is volved, then the flow must e at least locally unsteady a cyclical way and the appopiate fom of the enegy equation fo the contents of an fitesimally th contol volume can e developed statg with Eq he esultg equation is c d av g dz d dloss dw shaft net (5.05) Equations 5.03 and 5.05 ae valid fo compessile and compessile flows. If we come Eqs. 5.9 and 5.03, we ota dǔ pd a dq net dloss Fo compessile flow, d 0 and, thus, fom Eq. 5.06, dǔ dq net dloss Applyg Eq to a fite contol volume, we ota ǔ out ǔ q net loss (5.06) (5.07) which is the same conclusion we eached ealie see Eq fo compessile flows. Fo compessile flow, d 0, and thus when we apply Eq to a fite contol volume we ota out ǔ out ǔ pd a (5.08) q net loss dicatg that u out u q net is not equal to loss. Zeo loss is associated with the Benoulli equation Application of the Loss Fom of the Enegy Equation Steady flow along a pathle an compessile and fictionless flow field povides a simple application of the loss fom of the enegy equation Eq We stat with Eq and tegate it tem y tem fom one location on the pathle, section, to anothe one downsteam, section. Note that ecause the flow is fictionless, loss 0. Also, ecause the flow is steady thoughout, w shaft net 0. Sce the flow is compessile, the density is constant. he contol volume this case is an fitesimally small diamete steamtue Fig he esultant equation is p V gz p V gz which is identical to the Benoulli equation Eq. 3.7 aleady discussed Chapte 3. (5.09)

4 4 Chapte 5 Fite Contol Volume Analysis If the fictionless and steady pathle flow of the fluid paticle consideed aove was compessile, application of Eq would yield (5.0) V gz V gz o cay out the tegation equied, a elationship etween fluid density,, and pessue, p, must e known. If the fictionless compessile flow we ae consideg is adiaatic and -, volves the flow of an ideal gas, it is shown Section. that p constant (5.) k whee k c p c v is the atio of gas specific heats, c p and c v, which ae popeties of the fluid. Usg Eq. 5. we get hus, Eqs. 5.0 and 5. lead to k k ap p (5.) k p V k gz k p V k gz (5.3) Note that this equation is identical to Eq An example application of Eqs and 5.3 follows. E XAMPLE 5.9 Enegy Compaison of Compessile and Incompessile Flow GIVEN Ai steadily expands adiaatically and without fiction fom stagnation conditions of 00 psia and 50 R to 4.7 psia. FIND Deteme the velocity of the expanded ai assumg (a) compessile flow, () compessile flow. SOLUION (a) If the flow is consideed compessile, the Benoulli equation, Eq. 5.09, can e applied to flow though an fitesimal coss-sectional steamtue, like the one Fig. 5.7, fom the stagnation state () to the expanded state (). Fom Eq we get o 0 ( is the stagnation state) p V gz p V gz 0 (changes gz ae negligile fo ai flow) V a p p B We can calculate the density at state () y assumg that ai ehaves like an ideal gas, p 00 psia44. /ft R 76 ft l/slug R50 R 0.06 slug/ft 3 () () hus, 00 psia 4.7 psia44. ft V B 0.06 slug ft 3 3 l s slug ft4 40 ft s (Ans) he assumption of compessile flow is not valid this case sce fo ai a change fom 00 psia to 4.7 psia would undoutedly esult a significant density change. () If the flow is consideed compessile, Eq. 5.3 can e applied to the flow though an fitesimal coss-sectional contol volume, like the one Fig. 5.7, fom the stagnation state () to the expanded state (). We ota 0 ( is the stagnation state) k p V (3) k gz k p V k gz 0 (changes gz ae negligile fo ai flow)

5 5.4 Second Law of hemodynamics Ievesile Flow 5 o k V (4) B k ap p Given the polem statement ae values of p and p. A value of was calculated ealie (Eq. ). o deteme we need to make use of a popety elationship fo evesile (fictionless) and adiaatic flow of an ideal gas that is deived Chapte ; namely, p constant (5) k whee k.4 fo ai. Solvg Eq. 5 fo we get o a p k p 4.7 psia slug ft 3 c 00 psia d slug ft 3 hen, fom Eq. 4, with p 00 l. 44. ft 4,400 l ft and p 4.7 l. 44. ft 7 l ft, o V B.4.4 a 4,400 l ft 0.06 slug ft 3 7 l ft slug ft 3 60 l ft slug 3 slug ft s l4 V 60 ft s (Ans) COMMEN A consideale diffeence exists etween the ai velocities calculated assumg compessile and compessile flow. In Section 3.8., a discussion of when a fluid flow may e appopiately consideed compessile is povided. Basically, when flow speed is less than a thid of the speed of sound the fluid volved, compessile flow may e assumed with only a small eo.

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