Entropy balance Outline Closed systems Open systems Reversible steady flow wor Minimizing compressor wor Isentropic efficiencies Examples Entropy balance Sin Sout + Sgen = Ssys Entropy balance Entropy can be changed by Heat transfer between system and surroundings (here modeled as isothermal) Mass transfer (only for open systems (CVs)) Irersibilities (for all actual systems) ds dt CV Qɺ ɺ ɺ = + misi mese + Sgen T ɺ
Entropy balance special forms dscv Qɺ = + mɺ isi mɺ ese + Sɺ gen dt T Qɺ + mɺ isi mɺ ese + Sɺ gen = 0 steady T Qɺ + m ɺ ( s i s e) + S ɺ gen = 0 SISO T closed system Q = = + s s s Sgen T Quasi (QE) process Recall moving boundary wor formula w = Pdv Valid for closed system undergoing quasi compression or expansion A quasi process involves small departures from All states through which quasi process passes can be considered states Initial state Near Near Near Final state QE process is ersible Reversible process must proceed through a series of states Otherwise there would a tendency for system to change spontaneously which is irersible Hence, a quasiequilbrium process is an internally ersible process
Wor for internally ersible process Consider st law in differential form on a per unit mass bass flowing through a CV for an internally ersible process δ q δ w = dh + de + dpe δ q = Tds δ q = dh vdp Tds = dh vdp δ w = vdp + de + dpe w = vdp e pe Special cases For incompressible fluid, we have: w = v( P P ) e pe For devices with no wor interaction: v( P P ) + e + pe = 0 Above is called Bernoulli s equation (fluid mechanics) When one can neglect inetic and potential energy changes we have: w vdp = Implications w What does this equation tell us? The larger the specific volume, the larger the ersible wor produced or consumed by steady flow devices Conclusion is valid for actual steady flow devices Hence, we desire specific volume to be as small as possible during compression processes and as large as possible during expansion processes This tell us to compress (pump) liquids and expand gases (minimum wor input and maximum wor output) It also tells us to cool during compression and heat during expansion = vdp 3
What is the difference between these two formulas? w = Pdv w One is for closed system undergoing QE (int..) boundary wor The other is for open system, steady flow, internally ersible process Don t confuse the two! Stop! Warning! = vdp Proof that steady flow devices deliver most wor and consume least wor when process is ersible δ q δ w = dh + de + dpe act act δ q δ w = dh + de + dpe δ q δ w = δ q δ w act act δ w δ w = δ q δ q δ q act act = Tds δ w δ wact δ qact = ds 0 T T δ w δ w w w act act Minimizing compressor wor w, in = vdp How? Approach ersible process by minimizing irersibilities due to friction, turbulence, non- QE compression Keep specific volume of gas as small as possible during compression For gases, volume is proportional to temperature So, eep temperature as low as possible e.g. cool as you compress 4
Wor comparison Isentropic compression ( ) / R( T T ) RT P wcomp, in = vdp = = Pv = const P Polytropic compression w ( n ) / n nr( T T ) nrt P comp, in = vdp = = n Pv = const n n P Isothermal compression P wcomp, in = vdp = RT ln Pv= const P On P-v diagram area to left of curve is ersible steady flow wor Adiabatic compression requires maximum wor Isothermal compression requires minimum wor Motivates use of cooling water jacets around compressor casing to approach isothermal Wor comparison Multistage compression with intercooling Maintaining near isothermal conditions using cooling jacets is not always practically possible Gas is compressed in stages and cooled between each stage as it passes through a heat exchanger called an intercooler Ideally intercooling taes place at constant pressure and gas is cooled to initial temperature upon entry to intercooler 5
P-v and T-s diagrams for two-stage steady flow compression process What is best intercooler pressure? To minimize compression wor during two-stage compression, the pressure ratio across each stage of the compressor should be the same e.g. wor across each stage is identical w = w + w comp, in comp comp I, in II, in minimize ( n ) / n ( n ) / n x x nrt P nrt P = + n P n P P P P = ( PP ) or = x / x P Px Turbine Compressor Pump Isentropic efficiencies Actual turbine wor wa h h ηt = = Isentropic turbine wor w h h s a s Isentropic compressor wor ws h h ηc = = Actual compressor wor w h h a s a Isentropic pump wor ws v( P P ) ηp = = Actual pump wor w h h a a 6
Illustration Turbine Compressor Isentropic efficiency of nozzles η = N Actual KE at nozzle exit Isentropic KE at nozzle exit V h h = V h h a a s s Note exit pressure is same for both actual and isentropic processes But exit state is different To be done in class Examples 7
Summary Ideal compressor or turbine operation under ersible conditions To minimize actual compression worinput eep specific volume small; cooling for gases Ideally isothermal; practically multistage comrpession with intercooling and equal pressure ratios To maximize actual expansion wor output eep specific volume large; heating for gases; multistage expansion with reheating 8