上海交通大学：《热力学 Thermodynamics（I）》课程教学资源（课件讲义）Lecture 15_Polytropic process

上游充通大￥ SHANGHAI JIAO TONG UNIVERSITY Engineering Thermodynamics I Lectures 15 Spring,3/25/2019 Prof.,Dr.Yonghua HUANG 强 月是A http://cc.sjtu.edu.cn/G2S/site/thermo.html 1日G

Engineering Thermodynamics I Lectures 15 Spring, 3/25/2019 Prof., Dr. Yonghua HUANG http://cc.sjtu.edu.cn/G2S/site/thermo.html

Polytropic Process Defining a polytropic process of a closed system: 400 300 燃烧结束 Logog'气，p=consn log p 200 点火 Any gas/liquid 100 =1.30 60 膨胀 State1→State2: 40 压缩 1=1.29 30 20 排气 p网p,”or 大气 10 8 index 6 56810 203040V constant.-o∞~+o∞ In(p2/p) n= Exp.observation log V In(v/v2) 上游充通大 March 25,2019 2 SHANGHAI JIAO TONG UNIVERSITY

March 25, 2019 2 Polytropic Process Defining a polytropic process of a closed system: n pV constant  constant: -∞ ~ + ∞ State 1State 2: 1 1 2 2 n n p V p V  or 2 1 1 2 n p V p V        log p Exp. observation log V     2 1 1 2 ln / ln / p p n v v  Any gas/liquid log p=-nlogV+c index

Polytropic process Ideal gas Any gas/liquid P2 P V 1- (Ideal gas EoS) i.e. pyPVz T Ideal gas n-1 only or P2 上游充通大学 March 25,2019 3 SHANGHAI JIAO TONG UNIVERSITY

March 25, 2019 3 Polytropic process + Ideal gas 1 1 2 2 1 2 pV p V T T  (Ideal gas EoS) 2 2 2 1 1 1 T p V T pV  1 2 2 1 1 n n T p T p         2 1 1 2 n p V p V        or Ideal gas only 1 1 1 2 1 1 1 2 1 2 2 n n n n n V V V V V V V V              Any gas/liquid 1 2 1 1 2 n T V T V         i.e

Four particular cases n=0 D=const. (isobaric process;constant pressure process) n=1 pv =const. (isothermal process;constant temperature process) pV"=p2'2” n=K DyA =const. (isentropic process;reversible adiabatic process) For ideal gas:adiabatic index K =k(specific heat ratio) n=士o∞ V=const. (isometric/isochoric process;constant volume process) 上游充通大学 March 25,2019 4 SHANGHAI JIAO TONG UNIVERSITY

March 25, 2019 4 Four particular cases 1 1 2 2 n n p V p V  n p   0 const. n pv   1 const. n pv const.     n v   const. (isobaric process; constant pressure process) (isometric/isochoric process; constant volume process) (isentropic process; reversible adiabatic process) (isothermal process; constant temperature process) For ideal gas: adiabatic index = k (specific heat ratio) 

Work of a polytropic process w- Vn (constant）Vg-"-(constant0）V-n (n≠1) 1-n (p2V)vI-"-(piV)VI-" (Any fluids) 1-n mR(T2-T) (ideal gas,.n≠1) 1-n or w= R(T-T) for unit mass,ideal gas 1-n isothermal [v:dv W constant v= (n=1) (Any fluids) (ideal gas,n 1) or w-RTn业 mRT In V for unit mass,ideal gas y 上降文通大学 March 25,2019 5 SHANGHAI JIAO TONG UNIVERSITY

March 25, 2019 5 Work of a polytropic process isothermal (Any fluids) (Any fluids)   2 1 1 R T T w n    or for unit mass, ideal gas for unit mass, ideal gas 2 1 RT ln v w v  or