Chapter 13. Gas Mixtures. Study Guide in PowerPoint. Thermodynamics: An Engineering Approach, 5th edition by Yunus A. Çengel and Michael A.

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1 Chapter 3 Gas Mxtures Study Gude n PowerPont to accopany Therodynacs: An Engneerng Approach, 5th edton by Yunus A. Çengel and Mchael A. Boles

2 The dscussons n ths chapter are restrcted to nonreactve deal-gas xtures. Those nterested n real-gas xtures are encouraged to study carefully the ateral presented n Chapter. Many therodynac applcatons nvolve xtures of deal gases. That s, each of the gases n the xture ndvdually behaves as an deal gas. In ths secton, we assue that the gases n the xture do not react wth one another to any sgnfcant degree. We restrct ourselves to a study of only deal-gas xtures. An deal gas s one n whch the equaton of state s gven by PV RT or PV NRT u Ar s an exaple of an deal gas xture and has the followng approxate coposton. Coponent % by Volue N 78.0 O 0.95 Argon 0.9 CO + trace eleents 0.03

3 Defntons Consder a contaner havng a volue V that s flled wth a xture of dfferent gases at a pressure P and a teperature T. A xture of two or ore gases of fxed checal coposton s called a nonreactng gas xture. Consder gases n a rgd contaner as shown here. The propertes of the xture ay be based on the ass of each coponent, called gravetrc analyss, or on the oles of each coponent, called olar analyss. gases T T P P V V The total ass of the xture and the total oles of xture N are defned as and N N 3

4 The coposton of a gas xture s descrbed by specfyng ether the ass fracton f or the ole fracton y of each coponent. Note that f and y N N f and y The ass and ole nuber for a gven coponent are related through the olar ass (or olecular weght). N M To fnd the average olar ass for the xture M, note NM N M Solvng for the average or apparent olar ass M N M M ym ( g / ol) N N 4

5 The apparent (or average) gas constant of a xture s expressed as Can you show that R s gven as R Ru ( J / g K) M R fr To change fro a ole fracton analyss to a ass fracton analyss, we can show that ym f ym To change fro a ass fracton analyss to a ole fracton analyss, we can show that f / M y f / M 5

6 Volue fracton (Aagat odel) Dvde the contaner nto subcontaners, such that each subcontaner has only one of the gases n the xture at the orgnal xture teperature and pressure. Aagat's law of addtve volues states that the volue of a gas xture s equal to the su of the volues each gas would occupy f t exsted alone at the xture teperature and pressure. Aagat's law: V V ( T, P ) The volue fracton of the vf of any coponent s and vf V ( T, P ) V vf 6

7 For an deal gas xture V NRT u and V P Tang the rato of these two equatons gves vf V N V N N RT P y u The volue fracton and the ole fracton of a coponent n an deal gas xture are the sae. Partal pressure (Dalton odel) The partal pressure of coponent s defned as the product of the ole fracton and the xture pressure accordng to Dalton s law. For the coponent P yp Dalton s law: P P ( T, V) 7

8 Now, consder placng each of the gases n a separate contaner havng the volue of the xture at the teperature of the xture. The pressure that results s called the coponent pressure, P '. Note that the rato of P ' to P s NRT u P ' and P V P ' V N P V N y N RT V u For deal-gas xtures, the partal pressure and the coponent pressure are the sae and are equal to the product of the ole fracton and the xture pressure. 8

9 Other propertes of deal-gas xtures The extensve propertes of a gas xture, n general, can be deterned by sung the contrbutons of each coponent of the xture. The evaluaton of ntensve propertes of a gas xture, however, nvolves averagng n ters of ass or ole fractons: and U U u Nu H H h Nh S S s Ns (J) (J) (J / K) u fu and u yu (J / g or J / ol) h fh and h yh (J / g or J / ol) s fs and s ys (J / g K or J / ol K) C fc and C yc C fc and C yc v, v, v, v, p, p, p, p, 9

10 These relatons are applcable to both deal- and real-gas xtures. The propertes or property changes of ndvdual coponents can be deterned by usng deal-gas or real-gas relatons developed n earler chapters. Rato of specfc heats s gven as Cp, C v, The entropy of a xture of deal gases s equal to the su of the entropes of the coponent gases as they exst n the xture. We eploy the Gbbs-Dalton law that says each gas behaves as f t alone occupes the volue of the syste at the xture teperature. That s, the pressure of each coponent s the partal pressure. For constant specfc heats, the entropy change of any coponent s C C p, v, 0

11 The entropy change of the xture per ass of xture s The entropy change of the xture per ole of xture s

12 In these last two equatons, recall that Exaple 3- P y P,,, P y P,,, An deal-gas xture has the followng voluetrc analyss Coponent % by Volue N 60 CO 40 (a)fnd the analyss on a ass bass. For deal-gas xtures, the percent by volue s the volue fracton. Recall y vf

13 Cop. y M y M f y M /M g/ol g/ol g/g N CO M Σy M 34.4 (b) What s the ass of 3 of ths gas when P.5 MPa and T 30 o C? R Ru ( J / g K) M J ol K J 0. 4 g g K ol PV R T 5. MPa( ) ( 04. J / ( g K))( ) K 045. g 0 J 3 MPa 3 3 3

14 (c) Fnd the specfc heats at 300 K. Usng Table A-, C p N.039 J/g K and C p CO J/g K C, fc p, ( )( 039. ) + ( 05. )( ) p J g K J Cv, Cp, R ( ) g K J g K 4

15 (d) Ths gas s heated n a steady-flow process such that the teperature s ncreased by 0 o C. Fnd the requred heat transfer. The conservaton of ass and energy for steady-flow are The heat transfer per unt ass flow s & & & h & & + Qn h & Q& h & ( h ) q n n C & ( T T ) p, Q& n C T T p, ( ) & J ( 0K) g K J. 8 g 5

16 (e) Ths xture undergoes an sentropc process fro 0. MPa, 30 o C, to 0. MPa. Fnd T. The rato of specfc heats for the xture s Cp, C Assung constant propertes for the sentropc process v, (f) Fnd S per g of xture when the xture s copressed sotherally fro 0. MPa to 0. MPa. 6

17 But, the copresson process s sotheral, T T. The partal pressures are gven by The entropy change becoes P yp For ths proble the coponents are already xed before the copresson process. So, Then, y y,, 7

18 s f s g J g N J CO ( )( ) + ( 05. )( 03. ) g g K g g K J 067. g K N Why s s negatve for ths proble? Fnd the entropy change usng the average specfc heats of the xture. Is your result the sae as that above? Should t be? (g) Both the N and CO are suppled n separate lnes at 0. MPa and 300 K to a xng chaber and are xed adabatcally. The resultng xture has the coposton as gven n part (a). Deterne the entropy change due to the xng process per unt ass of xture. CO 8

19 Tae the te to apply the steady-flow conservaton of energy and ass to show that the teperature of the xture at state 3 s 300 K. But the xng process s sotheral, T 3 T T. The partal pressures are gven by The entropy change becoes P yp 9

20 But here the coponents are not xed ntally. So, and n the xture state 3, Then, y y y y N N CO CO,, 3,,

21 Then, s f s gn J g J CO ( )( 05. ) + ( 05. )( 073. ) g g K g g K J 063. g K If the process s adabatc, why dd the entropy ncrease? Extra Assgnent N Ntrogen and carbon doxde are to be xed and allowed to flow through a convergent nozzle. The ext velocty to the nozzle s to be the speed of sound for the xture and have a value of 500 /s when the nozzle ext teperature of the xture s 500 C. Deterne the requred ole fractons of the ntrogen and carbon doxde to produce ths xture. Fro Chapter 7, the speed of sound s gven by CO C RT Mxture N and CO NOZZLE C 500 /s T 500 o C Answer: y N 0.589, y CO 0.4