Department of Mechanical Engineering ME 322 Mechanical Engineering Thermodynamics. Ideal Gas Mixtures II. Lecture 32

Transcription

1 Departent of Mechanical Engineering ME 322 Mechanical Engineering Therodnaics Ideal Gas Mixtures II Lecture 32

2 The Gibbs Phase Rule The nuber of independent, intensive properties required to fix the state of a therodnaic substance is... f C P 2 Nuber of coponents in the ixture Nuber of phases present Single phase pure fluid: f 2 2 Single phase binar ixture of A + B: f T, P, Wh is B not included here??? A 2

3 Ideal Gas Mixture Properties We have previousl seen that, b w b or b b Consider the internal energ and enthalp of an ideal gas ixture. The coponents of the ixture exist at the sae teperature as the ixture. Therefore, according to the expressions above, or u T w u T u u T T T or h T w h T h h T T T 3

4 Ideal Gas Mixture Properties What about the entrop of an ideal gas? s w s or s s We now that the entrop of an ideal gas is a function of teperature and pressure.,,? s T P w s T Which pressure is used here? To answer this question, we need to go bac to the ideal gas equation of state. 4

5 Ideal Gas Mixture Properties Consider a ixture of ideal gases inside of a fixed volue container. T, P, V na,, n,, nn n n PV RT PV RT The ixture ust obe, P V nrt The ixture coponents ust obe, P V n RT Question: Can the coponents of the ixture exist at the sae teperature, pressure, and volue of the ixture siultaneousl? Answer: NO! If this were true, the nuber of oles of each coponent would have to be the sae as the nuber of oles of ixture. Therefore, n n i i Which we now is incorrect! 5

6 Dalton s Law of Partial Pressures Dalton s Law of Partial Pressures states that the ixture coponents exist at the ixture teperature and occup the sae total volue as the ixture. Then, for the ixture and a coponent in the ixture, For the following ratio... P n RT / V and P n RT / V P P n RT / n RT / V V n n P is called the partial pressure of the coponent in the ixture 6

7 What is a Partial Pressure? The partial pressure of a coponent in an ideal gas ixture is the pressure the coponent would attain if it was at the sae teperature and occupied the sae volue as the ixture. n A T, V T, V, n,, n, B nn n n i i P n RT n RT P V V 7

8 Dalton s Law of Partial Pressures The partial pressures of the coponents in the ixture ust su up to the total pressure of the ixture, P P P P P P P P P P Now, we have the basis for evaluating the ideal gas coponent entrop,,, or,, s T P w s T P s T P s T P partial pressure of the coponent 8

9 Ideal Gas Mixture Properties Suar... Using Dalton s Law... or u T w u T u u T or h T w h T h h T or c T w c T c c T p p p p or c T w c T c c T v v v v,, or,, s T P w s T P s T P s T P P P 9

10 Exaple Given: A copressor is being used to ove a binar ixture of ethane and ethane (50/50 b oles). The ixture enters the copressor at 20 psia, 70 F and leaves the copressor at 80 psia. The copressor is being odeled as isentropic. P T 20 psia 70F P2 80 psia CH CH Find: (a) The wor required (Btu/lb) (b) The wor required as a function of ixture coposition w c 0

11 Exaple P2 80 psia w c P 20 psia T 70F CH4 CH The First Law applied to the copressor gives, w c h h 2 M Btu/lbol lb/lbol Btu lb

12 Exaple P2 80 psia The olecular ass of the ixture is, M M P 20 psia T 70F CH4 CH w c The olar enthalp of the ixture at the inlet to the copressor can be deterined using ideal gas ixing, h h At the copressor exhaust, the entrop of the ixture is the sae as at the inlet. To calculate entrop values, partial pressures are needed at each state. P / P 2

13 Exaple P2 80 psia The olar entrop at the inlet to the copressor is,,, s T P s T P P 20 psia T 70F CH4 CH w c Since the copressor is isentropic,,,, s T P s 2 T2 P2 s2 T2 P2 The onl unnown in this set of equations is the copressor discharge teperature, T 2 3

14 Exaple P2 80 psia w c Now that the copressor exit teperature has been found, the olar enthalp can be found, h h 2 2 P 20 psia T 70F CH4 CH Solution (All variables) 4

15 w c [Btu/lb] Exaple Paraetric Stud [] (Methane) 5

16 Exaple Paraetric Stud Does this trend ae sense? For an ideal gas undergoing an isentropic copression, Pv constant The wor done during this process is, w P v Pv 2 2 R T T 2 R T T M 2 As the olecular ass of the substance increases, the aount of wor required to copress fro P,T to P 2,T 2 decreases! Therefore, the trend seen on the previous slide is correct! 6