C v & Thermodynamics Relationships
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1 Mathematical heorems hermodnamics Relations Dr. M. Zahurul Haq rofessor Department of Mechanical Engineering Bangladesh Uniersit of Engineering & echnolog BUE Dhaka-1000, Bangladesh ME 6101: Classical hermodnamics c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 If there eists a relation among, & ; f,, 0 1 f,, 0,,,,,, d 1 d + d Md + Nd For continuous functions, 2 2 M N c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 Mawell Relationships Internal Energ, du q + w ds d for re. process Enthalp, h u + dh du + d + d ds + d Helmholt Energ, f u s df d sd Gibbs Energ, g h s dg dh ds sd d sd For continuous functions, 2 2 M 1 du +ds d s 2 dh +ds + d 3 df sd d 4 dg sd + d s + + N A B C D C & hermodnamics Relationships du ds d C C [ ] 2 s using Mawell s relation: + [ ] 2 2 If -- data or mathematical relationship is aailable, it is possible to ealuate 2 / 2, and then C /. Ideal gas: R R & 2 0 C 2 f an der Wall s gas: R b a 2 R b & 2 0 C 2 f c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22
2 C & hermodnamics Relationships dh ds + d C C [ ] [ ] 2 using Mawell s relation: 2 Ideal gas: R & 2 0 C 2 f an der Wall s gas: 2 2 R2 2a 3 6ab 4 R a 2 1 2b a 2 1 2b 3 C f d ds Relationships s s, ds d + d C & +, Mawell s Relation C ds C d + d 1 st ds s s, ds d + d C &, Mawell s Relation D ds C d d 2 nd ds s s, ds d + d C & C ds C d + C d 3 rd ds c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 Internal Energ, u u u, du d + d C d + d du ds d C d + d d 1st ds Eq. Ideal gas: [ R ] u f : an der Waals gas: du C d f 0 0 u f R b a 2 du C d + a 2 d f, Enthalp h h h, dh d + d C pd + d dh ds + d C d d + d 2nd ds Eqn. Ideal gas: R 0 h f h f : 0 h f dh C d f an der Waals gas: h f, c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22
3 C C 011 for isobar: dh C d h 2 h 1 h 2 h2 + h 2 h 1 + h 1 h 1 2 [ 0 d + ] C o d 1 [ 0 ] d 1 for isotherm: dh [ ] d s s, ds d + d + [ C C V f,, C C C, C 2 1 For liquids & solids, 0 C C C 2 2 is +e & is -e for all known substances, C C 3 as 0,C p C, at 0,C C ] c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 Isothermal compressibilit, k 1 Volume epansiit, β 1 C C 2 β2 k Ideal gas: R k 1, β 1 C p C R for liquid water at 1 bar: Eample: Liquid Water 1 atm & 20 o C C C β2 k K K bar kg m 2 C C J kg.k, C 4188 J kg.k C C, d d N m 2 1bar d β k d 1 k d d β k d 1 k d 012 β 0.04 o C for water. If liquid water temperature is raised form 19.5 to 20.5 o C at constant olume: d β k d β k K bar bar 10 2 ka 1K 450 ka c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22
4 Applications of ds Relationships 2 nd ds : ds C d d Reersible isothermal change in pressure: ds d q d βd β w d d 2 1 k d k for a liquid & solid,, β & k are insensitie to change in Reersible adiabatic change in pressure: d C d β C d β C Eperiments show that C hardl changes for a solid & liquid een for an increase of 10,000 bar. Eample: 15 cm 3 o C & 1 bar 1000 bar at 0 o C V m 3 C 28.6J/K β K 1 k bar 1 isothermal compression: 1 1 bar, bar. Q mq mβ V β 78.2 J W mw V k J U Q + W J 78.2 J heat is liberated but onl 2.91 J work is done. he etra amount of heat comes from the store of the internal energ. For a substance with a negatie epansiit, heat is absorbed and the internal energ is increased. isentropic compression: 1 1 bar, bar. β C 2.55 K c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 Clausius-Claperon Equation for hase Change s 2 s : phase change sat const. d d, for miture of 2 phases, f d d s 2 s h 2 h d d Claperon Eqn. sat h Eample: using onl -- data, estimate h fg of R-134a at 20 o C fg g f sat,20 o C m 3 kg d d sat,20 o C sat,20 o C sat,24 o C sat,16 o C 24 0 C 16 o C ka/k d h fg fg d kj/kg sat,20 o C abulated alue of h o C is kj/kg. 2 >> 1 2 R d ln d1/ sat h fg R Clausius-Claperon Eq. ln 2 h 1 1 sat R ln sat A + B + C ln + D, widel used apour-pressure Eq. c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 Low emperature Refrigeration c & c of Common Substances e789 If the temperature and pressure of a gas can be brought into the region between the saturated liquid and saturated apour lines then the gas will become wet and this wetness will condense giing a liquid. c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22
5 Liquefaction b Cooling Liquefaction b Cooling Liquefaction b Cooling Liquefaction b Epansion his method is satisfactor if the liquefaction process does not require er low temperatures. Eample butane, propane, Eamples of these are the hdrocarbons butane and propane, which can both eist as liquids at room temperature if the are contained at eleated pressures. Mitures of hdrocarbons can also be obtained as liquids and these include liquefied petroleum gas LG and liquefied natural gas LNG. e792 e790 c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 1 Compress isentropicall to 2, where 2 > c 2 As 2 > a, cool it to a using ambient sources, and further cool to 3 using aailable cold sources. 3 Epand isentropicall form 3 to 4 liquid formation. c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 Liquefaction b Cooling Gas Epansion & Joule-homson coefficient Liquefaction b Cooling he temperature behaiour of a fluid during a throttling process is described b Joule-homson coefficient, µ J. e772 µ J h e : temperature increase 0 : temperature same +e : temperature drop c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 e773 A cooling effect cannot be achieed b throttling unless the fluid is below its maimum inersion temperature. For hdrogen its alue is -68 o C and hdrogen must be cooled below this temperature if further cooling is to be achieed. e769 c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22
6 Liquefaction b Cooling Liquefaction b Cooling Simplified Linde Liquefaction lant e793 R b a 2 µ J 1 C [ R + a 2 b 2a Maimum inersion temperature 6.75 c 3 Minimum inersion temperature 0.75 c If air is compressed to a pressure of 200 bar and a temperature of 52 o C, after the throttling to 1 bar it will be cooled to 23 o C. In case of helium, throttling from the came condition will result in 64 o C. c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22 ] e796 e797 wo performance parameters: Yield, : mass of liquid produced per unit mass of gas compressed. Sp. work required, w : work per unit mas of liquid produced. m h 7 h 2 h 7 h 5 w W in c Dr. M. Zahurul Haq BUE hermodnamics Relations ME / 22
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