Chapter 4. Thermodynamic Variables and Relations

Transcription

1 hapter hermodynamic ariables and elations ssoc. rof. Dr. Mazlan bdul Wahid Faculty of Mechanical ngineering Universiti eknologi Malaysia UNIII KNOLOGI MLYI - dvanced hermodynamics - Mazlan 03 hermal xpansion oefficient olumetric hermal xpansion oefficient (,, X... ) f Linear hermal xpansion oefficient L L 3 L L Isotropic Material (I Units K - ) (I Units K - ) L L L3 UNIII KNOLOGI MLYI - dvanced hermodynamics - Mazlan 03

2 oefficient of ompressibility olumetric oefficient of ompressibility β β f (,, X...) Note negative sign in definition. pproximately: β 3 / β j / β i lastic Modulus / i (I Units atm - ) j UNIII KNOLOGI MLYI -3 dvanced hermodynamics - Mazlan 03 eat apacity t onstant ressure: δq rev, d (I Units J/mole-K) f(,,x, ) mpirical Fit: () a + b + c/ t onstant olume: δq rev, d (I Units J/mole-K) f(,,x, ) In General: > & /β UNIII KNOLOGI MLYI - dvanced hermodynamics - Mazlan 03

3 UNIII KNOLOGI MLYI Internal nergy du δq + δw +δw / δq rev d δw rev d st & nd Laws: du d - d +δw / oefficient relations: U Maxwell relation: U s -5 dvanced hermodynamics - Mazlan 03 nthalpy Defining an energy & state function: U + Differentiating: d du + d + d ubstituting for du: d d-d+dw / +d+d st & nd Laws: d d + d +δw / Good for isobaric processes: d 0 oefficient relations: UNIII KNOLOGI MLYI Maxwell relation: -6 dvanced hermodynamics - Mazlan 03 3

4 elmholtz Free nergy F (Deoff) or (rbeiten) Defining an energy & state function: F U Differentiating: df du d d ubstituting for du: df d-d+dw / -d-d st & nd Laws: df - d d +δw / Good for isothermal processes: d 0 oefficient relations: F F Maxwell relation: UNIII KNOLOGI MLYI -7 dvanced hermodynamics - Mazlan 03 Gibbs Free nergy G (Deoff) or F (Others) Defining an energy & state function: G - Differentiating: dg du+d+dd-d ubstituting for du: dg d-d+dw / +d+d-d-d st & nd Laws: dg d + d +δw / Good for isothermal/isobaric processes: d0, d0 oefficient relations: G G Maxwell relation: UNIII KNOLOGI MLYI -8 dvanced hermodynamics - Mazlan 03

5 tate Functions (able.) tate ariables emperature ressure olume nergy Functions Internal nergy U nthalpy elmholtz Free nergy F Gibbs Free nergy G ntropy UNIII KNOLOGI MLYI -9 dvanced hermodynamics - Mazlan 03 Internal nergy nthalpy nergy Functions elmholtz Free nergy Gibbs Free nergy U U + F U G UNIII KNOLOGI MLYI -0 dvanced hermodynamics - Mazlan 03 5

6 6 dvanced hermodynamics dvanced hermodynamics - Mazlan 03 Mazlan 03 UNIII KNOLOGI MLYI UNIII KNOLOGI MLYI - ombined st & nd Laws du d d + δw / d d + d + δw / df d d + δw / dg d + d + δw / dvanced hermodynamics dvanced hermodynamics - Mazlan 03 Mazlan 03 UNIII KNOLOGI MLYI UNIII KNOLOGI MLYI - oefficient elations G G F F U U

7 7 dvanced hermodynamics dvanced hermodynamics - Mazlan 03 Mazlan 03 UNIII KNOLOGI MLYI UNIII KNOLOGI MLYI -3 Maxwell elations s dvanced hermodynamics dvanced hermodynamics - Mazlan 03 Mazlan 03 UNIII KNOLOGI MLYI UNIII KNOLOGI MLYI - tate Functionsf(,) (able.5) d d dg +

8 olume elations to emperature & ressure d d ( ), d + d d βd UNIII KNOLOGI MLYI -5 dvanced hermodynamics - Mazlan 03 tate Functionsf(,) (able.5) d d βd UNIII KNOLOGI MLYI dg d + d -6 dvanced hermodynamics - Mazlan 03 8

9 ntropy elations to emperature & ressure d ( ), δqrev d d ( ) d d d d + d UNIII KNOLOGI MLYI -7 dvanced hermodynamics - Mazlan 03 hermodynamics tate Functions - able.5 d d βd d d d UNIII KNOLOGI MLYI ( ) d + ( β )d du d d + ( ) d df ( + ) d βd dg d + d -8 dvanced hermodynamics - Mazlan 03 9

10 elations Between tate ariables Identify the variables. ZZ(X,Y) Write the differential form. dzmdx+ndy onvert dx & dy in terms of d & d. dzm[x d+x d]+n[y d+y d] where dxx d+x d; dyy d+y d ollect terms. dz[mx +NY ]d+[mx +NY ]d Obtain: ZZ(,) & dzz d+z d et: MX +NY Z MX +NY Z olve for M & N, integrate between end points. UNIII KNOLOGI MLYI dzmdx+ndy -9 dvanced hermodynamics - Mazlan 03 xample: Find (,) Identify the variables. (,) earrange. (,) Write the differential form. d Md+Nd onvert d & d in terms of d & d by substituting d d & d d-βd d Md+N[d-βd] ollect terms. d [M+N]d-Nβd UNIII KNOLOGI MLYI -0 dvanced hermodynamics - Mazlan 03 0

11 xample: Find (,) d [M+N]d-Nβd Obtain: (,) & d( /)d-d et: [M+N]( /) & -Nβ - olve for M & N: M /( - /β) & N /β Insert M & N in differential form: dmd+nd d /( - /β)d+/βd Note the relation: /β d ( /)d+(/β)d UNIII KNOLOGI MLYI - dvanced hermodynamics - Mazlan 03 xample: Find (,) d ( /)d+(/β)d olve for d: d (/ )d-(/β )d For isentropic process: d (/β )d Integrating: ln β ( ) UNIII KNOLOGI MLYI exp β ( ) - dvanced hermodynamics - Mazlan 03

12 Ideal Gas n / β / Monatomic: 5/ 3/ Diatomic: 7/ 5/ U & depend only on temperature: U d d UNIII KNOLOGI MLYI -3 dvanced hermodynamics - Mazlan 03 β UNIII KNOLOGI MLYI Ideal Gas n n n n ( n ) - n n β ( n ) - dvanced hermodynamics - Mazlan 03

13 XML ompare the for the following processes. (a.) One gr-at of Ni is heated at atm from 300 K to 300K. Need (,) evaluated for constant. For constant, d0. ( ) d d d d ( )d For this state function, integrate between limits. ( ) d UNIII KNOLOGI MLYI -5 dvanced hermodynamics - Mazlan 03 ompare the for the following processes. (a.) One gr-at of Ni is heated at atm from 300 K to 300K. ( ) From the ppendix. Ni a b J + mol K Where a 7.0 & b a + b d [ aln + b] d UNIII KNOLOGI MLYI -6 dvanced hermodynamics - Mazlan 03 3

14 ompare the for the following processes. (a.) One gr-at of Ni is heated at atm from 300 K to 300K. [ aln b] + ubstituting values for Ni and limits ln ( ) 300 Where a 7.0 & b J 5. gr at UNIII KNOLOGI MLYI K -7 dvanced hermodynamics - Mazlan 03 ompare the for the following processes. (b.) One gr-at of Ni is heated at 300 K is isothermally compressed from atm to 00 kbars. Need (,) evaluated for constant. UNIII KNOLOGI MLYI ( ) d d d For constant, d0. d d ssume and are independent of pressure. d ( ) -8 dvanced hermodynamics - Mazlan 03

15 ompare the for the following processes. (b.) One gr-at of Ni at 300 K is isothermally compressed from atm to 00 kbars. From ppendices: Ni Ni 6 O 6.60 cc / mole 0 x 0 K ubstituting numerical values for and. 6 5 ccatm ( 0 ) 6. mol K UNIII KNOLOGI MLYI ( ) J mole K -9 dvanced hermodynamics - Mazlan 03 ompare the for the following processes. (c.) One mole of ZrO from 300 K to 300 K at atm. Need (,) evaluated for constant. UNIII 09/9/00 KNOLOGI MLYI ( ) d d d For constant, d0. d ( )d For this state function, integrate between limits. ( ) d -30 dvanced hermodynamics - Mazlan 03 5

16 ompare the for the following processes. (c.) One mole of ZrO from 300 K to 300 K at atm. From the ppendix. ZrO ( ) a+ b+ c J mol K Where a 69.6, b , c -.x0 5. a c + b+ 3 d c d aln + b UNIII 09/9/00 KNOLOGI MLYI -3 dvanced hermodynamics - Mazlan 03 ompare the for the following processes. (c.) One mole of ZrO from 300 K to 300 K at atm. [ aln + b+ ( c ) ] ubstituting values for ZrO and limits ln ( ) Where a 69.6, b , & c -.x0 5. UNIII 09/9/00 KNOLOGI MLYI 77 J gr at K -3 dvanced hermodynamics - Mazlan 03 6

17 ompare the for the following processes. (d.) One mole of ZrO at 300 K is isothermally compressed from atm to 00 kbars. Need (,) evaluated for constant. UNIII 09/9/00 KNOLOGI MLYI ( ) d d d For constant, d0. dd ssume and are independent of pressure. d ( ) -33 dvanced hermodynamics - Mazlan 03 ompare the for the following processes. (d.) One mole of ZrO at 300 K is isothermally compressed from atm to 00 kbars. From ppendices: ZrO O 7.0 cc / mole ubstituting numerical values for and. UNIII 09/9/00 KNOLOGI MLYI ( ) ZrO 5 ( 0 ) x 0 K J mole K cc atm mol K -3 dvanced hermodynamics - Mazlan 03 7

18 ompare the for the following processes. (e.) One mole of O from 300 K to 300 K at atm. Need (,) evaluated for constant. UNIII 09/9/00 KNOLOGI MLYI ( ) d d d For constant, d0. d ( )d For this state function, integrate between limits. ( ) d -35 dvanced hermodynamics - Mazlan 03 ompare the for the following processes. (e.) One mole of O from 300 K to 300 K at atm. ( ) From the ppendix. O a b c J + + mol K Where a 30.0, b 0.00, c -.7x0 5. d a + b + c 3 d a ln + b c UNIII 09/9/00 KNOLOGI MLYI -36 dvanced hermodynamics - Mazlan 03 8

19 ompare the for the following processes. (e.) One mole of O from 300 K to 300 K at atm. [ ln ( c a + b+ ) ] ubstituting values for ZrO and limits ln ( ) Where a 30.0, b 0.00, & c -.7x0 5. UNIII 09/9/00 KNOLOGI MLYI J gr at K -37 dvanced hermodynamics - Mazlan 03 ompare the for the following processes. (f.) One mole of O at 300 K is isothermally compressed from atm to 00 kbars. Need (,) evaluated for constant. For constant, d0. ( ) d d d d d For mole of an ideal gas. UNIII 09/9/00 KNOLOGI MLYI d p d -38 dvanced hermodynamics - Mazlan 03 9

20 ompare the for the following processes. (f.) One mole of O at 300 K is isothermally compressed from atm to 00 kbars. d ln ubstituting numerical values. d 8.3 ln J mole K UNIII 09/9/00 KNOLOGI MLYI -39 dvanced hermodynamics - Mazlan 03 ompute the U when liters of r at 73 K and atm are compressed to 6 liters with final pressure 0 atm. (a.) Find UU(,) & integrate. U U (, ) du Md + Nd du Md + N ( d βd ) du N d + ( M N β )d ompare to. du d + β ( ) ( )d M N β ( β ) N β M UNIII 09/9/00 KNOLOGI MLYI XML N -0 dvanced hermodynamics - Mazlan 03 0

21 ompute the U when liters of r at 73 K and atm are compressed to 6 liters with final pressure 0 atm. (a.) Find UU(,) & integrate. β M β M ( ) Using: Using: + du UNIII 09/9/00 KNOLOGI MLYI d + N ( ) N ( ) d - dvanced hermodynamics - Mazlan 03 ompute the U when liters of r at 73 K and atm are compressed to 6 liters with final pressure 0 atm. (a.) Find UU(,) & integrate. U U ( ) U UNIII 09/9/00 KNOLOGI MLYI d + d + ( ) d + d d - dvanced hermodynamics - Mazlan 03 d imulate step process, constant pressure + constant volume.

22 ompute the U when liters of r at 73 K and atm are compressed to 6 liters with final pressure 0 atm. (b.) Use the temperature change. UNIII 09/9/00 KNOLOGI MLYI n n , 365 U 3 U 638 K ( ), -3 dvanced hermodynamics - Mazlan 03 For one mole of nitrogen gas compute and plot the surfaces that represent the variation with pressure and volume over the range ( atm,. l) to (0 atm, 8. l)of (a) the internal energy. Use the result from.6b: du d + UNIII 09/9/00 KNOLOGI MLYI d U(, ) U(, ) U(, ) U(, ) By analogy XML (, ) [ ( ) + ( )] [ ] [ ] - dvanced hermodynamics - Mazlan 03