Some Useful Formulae

Transcription

1 ME - hrmodynamcs I Som Usful Formula Control Mass Contnuty Equaton m constant Frst Law Comprsson-xpanson wor U U m V V mg Z Z Q W For polytropc procs, PV n c, Scond Law W W PdV P V P V n n P V ln V V n For sothrmal procss For rvrsbl procss For adabatc procss hrfor, for a rvrsbl adabatc procss S S δq δq Q S gn 0 Q 0 S gn S S 0 s s 0 hrfor, a rvrsbl adabatc procss s an sntropc procss.

2 Control Volum. Stady Stat Stady Flow (SSSF) Contnuty ṁ ṁ 0 Frst Law ṁ h V gz ṁ h V gz Q cv Ẇ cv 0 Scond Law ṁ s ṁ s j Q j j Ṡ gn 0 Rvrsbl procss Ṡ gn 0 Adabatc procss Q 0 For on nlt-on outlt dvc, a rvrsbl adabatc procss s thrfor an sntropc procss, wth s s. Unform Stat Unform Flow (USUF) Contnuty m m m m Frst Law m u V gz m u V gz m h V gz m h V gz Q cv W cv Scond Law m s m s m s m s 0 t Q cv dt S gn Gbbs Equaton ds du Pdv dh vdp hs quaton holds tru for all smpl comprssbl substancs.

3 Proprts of Pur Substancs. Vapor-Lqud Phas Equlbrum For a spcfc proprty φ (such as h,u,v,s tc) undr th dom φ φ f xφ f g. Idal Gas Idal Gas Equatons of Stat Pv R Spcfc Hats and Idal Gas Constants du C v d u u C v d C v f C v s constant dh C p d h h C p d C p f C p s constant Entropy Rlatonshps R C p C v R C p C v R M d ds C p R dp P s s d C p Rln P P C p ln Rln P P s 0 s 0 Rln P P f constant C p othrws d ds C v R dv v s s d C v C v ln Rln v v Rln v v f constant C v

4 Isntropc Procss for Idal Gas For varabl spcfc hats P P v v P r P r v r v r For constant spcfc hats Pv constant P v P v h abov four rlatonshps also hold for a rvrsbl polytropc procss wth th polytropc xponnt n rplacng.. Incomprssbl Substanc Equaton of Stat P P v v Entropy Rlatonshps du dh Cd u u h h Cd du ds s s C d C f C s constant Cln C=constant can usually b assumd for ncomprssbl substancs. f C s constant

5 5 Hat Engns, Hat Pumps and Rfrgrators hrmal ffcncy of hat ngn η th W nt Q L Coffcnt of Prformanc (C.O.P) of Hat Pump β W nt Q L Coffcnt of Prformanc (C.O.P) of Rfrgrator β Q L W nt Q L Q L Carnot Cycl Q L H L η th L H β H H L β L H L 6 Isntropc Effcncy of Engnrng Dvcs η turbn η comprssor w w s w s w η nozzl V V s 5

6 7 Irrvrsblty and Avalablty Avalablty: Ψ h 0 s h r f 0 s r f Irrvrsblty: İ 0 Ṡ gn Wor n an SSSF procss wth hat gan from H and hat loss Q 0 to 0 : Ẇ nt 0 H ṁ Ψ Ψ İ Scond Law Effcncy (Effctvnss) urbn: η Ẇ nt ṁ Ψ Ψ Comprssor: 8 Powr and Rfrgraton Cycls 8. Idal Rann Cycl η ṁ Ψ Ψ Ẇ nt s -:Isntropc comprsson n pump -: Constant prssur hat addton n bolr -: Isntropc xpanson n turbn -: Constant prssur hat rjcton n condnsr Worng flud s usually watr or othr two-phas substanc 6

7 Wor and hat transfr formula: w p h h v P P q b h h w h h q c h h Also s s s s 8. Idal Ar Standard Brayton Cycl s -:Isntropc comprsson n comprssor -: Constant prssur hat addton n hat xchangr -: Isntropc xpanson n turbn -: Constant prssur hat rjcton n hat xchangr Worng flud dalzd to b ar Wor and hat transfr formula: w C h h q h h h w h h q c h h Also s s s s 7

8 For constant C p w C C p q h C p w C p q c C p Also P P P P For varabl C p hrmal ffcncy of Brayton cycl (For constant C p ) P r P P r P P r P P r P η th P P 8. Idal Ar Standard Otto Cycl v =c v = c s Otto cycl s a pston-cylndr cycl (control mass) Modls spar gnton (SI) ngns -:Isntropc comprsson 8

9 -: Constant volum hat addton -: Isntropc xpanson -: Constant volum hat rjcton Worng flud dalzd to b ar Procss -: Procss -: s s u u P P V V w C v u u q C v Procss -: Procss -: s s u u P P V V w C v hrmal ffcncy of Otto cycl (For constant C p ) u u q C v whr r v s th comprsson rato. Man Effctv Prssur: η th V V r v mp w nt v v 9

10 8. Idal Ar Standard Dsl Cycl v = c s Dsl cycl s a pston-cylndr cycl (control mass) Modls comprsson gnton (CI) ngns -:Isntropc comprsson -: Constant prssur hat addton -: Isntropc xpanson -: Constant volum hat rjcton Worng flud dalzd to b ar Procss -: Procss -: Procss -: s s u u P P V V w C v h h q C p s s u u P P V V w C v 0

11 Procss -: hrmal ffcncy of Otto cycl (For constant C p ) 8.5 Idal Vapor Rfrgraton Cycl u u q C v η th C v C p s Smply Rann cycl run bacwards Not how ponts,,, hav bn shftd -:Isntropc comprsson n comprssor -: Constant prssur hat loss at hgh-tmpratur sourc -: Isntropc xpanson n xpanson valv -: Constant prssur hat gan n vaporator Worng flud s usually rfrgrant or othr two-phas substanc