1. AME 436 Energy and Propulsion Lecture 7 Unsteady-flow (reciprocating) engines 2: Using P-V and T-s diagrams

2. 2 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Outline  Air cycles  What are they?  Why use P-V and T-s diagrams?  Using P-V and T-s diagrams for air cycles  Seeing heat, work and KE  Constant P and V processes  Inferring efficiencies  Compression & expansion component efficiencies  Correspondence between processes on P-V and T-s  Hints & tricks

3. 3 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Air-cycles - what are they?  In this course we will work primarily with “air cycles” in which the working fluid is treated as just air (or some other ideal gas)  In air cycles, changes in gas properties (C P, M i, , etc.) due to changes in composition, temperature, etc. are neglected; this greatly simplifies the analysis and leads to simple analytic expressions for efficiency, power, etc.  Later we’ll examine “fuel-air cycles” (using GASEQ) where the “real” gases are considered and the properties change as composition, temperature, etc. change (but still can’t account for slow burn, heat loss, etc. since it’s still a thermodynamic analysis that tells us nothing about reaction rates, burning velocities, etc.)  In addition to the analytical results, P-V and T-s diagrams will be used extensively to provide a visual representation of cycles

4. 4 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Why use P-V diagrams?  Pressure vs. time and cylinder volume vs. time are easily measured in reciprocating-piston engines  ∫ PdV = work  ∫ PdV over whole cycle = net work transfer + net change in KE + net change in PE = net heat transfer  Heat addition is usually modeled as constant pressure or constant volume, so show as straight lines on P-V diagram  Note that V on P-V diagram is cylinder volume (m 3 ) which is NOT a property of the gas, whereas specific volume (v) (m 3 /kg) IS a property of the gas (we need to use V since cycle work = ∫ PdV; can’t use v since mass changes during intake/blowdown/exhaust)

5. 5 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Why use T-s diagrams?  Idealized compression & expansion processes are constant S since dS ≥  Q/T; for adiabatic process  Q = 0, for reversible = (not >) sign applies, thus dS = 0 (note that dS = 0 still allows for any amount of work transfer to occur in or out of the system, which is what compression & expansion processes are for)  For reversible process, ∫TdS = Q, thus area under T-s curves show amount of heat transferred  ∫ TdS over whole cycle = net heat transfer = net work transfer + net change in KE + net change in PE  T-s diagrams show the consequences of non-ideal compression or expansion (dS > 0)  For ideal gases,  T ~ heat xfer or work xfer or  KE (see next 2 slides)  Efficiency can be determined by breaking any cycle into Carnot- cycle “strips,” each strip (i) having  th,i = 1 - T L,i /T H,i

6. 6 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams T-s & P-v for control mass: work, heat & KE  For an ideal gas with constant C P & C v :  h = C p  T,  u = C v  T and the 1st Law says, for a control mass with  PE = 0 (in internal combustion engines we can almost always neglect PE) dE = d(U + KE) = d[m(u + KE)] =  q -  w (note the use of “KE” rather than “u 2 /2” to avoid confusion between u {internal energy} and u {velocity}) q = Q/m (heat transfer per unit mass) w = W/m (work transfer per unit mass)  If no work transfer (dw = 0) or KE change (dKE = 0), du = C v dT = dq  q 1  2 = C v (T 2 - T 1 )  If no heat transfer (dq = 0) or  KE, du = C v dT = dw  w 1  2 = -C v (T 2 - T 1 )  If no work or heat transfer   KE = KE 2 – KE 1 = -C v (T 2 - T 1 )  For a control mass containing an ideal gas with constant C v,  T ~ heat transfer - work transfer -  KE  Note that the 2nd law was not invoked, thus the above statements are true for any process, reversible or irreversible

7. 7 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams T-s & P-v for control volume: work, heat & KE  1st Law says, for a control volume, steady flow, with  PE = 0  For an ideal gas with constant C P, dh = C P dT  h 2 - h 1 = C P (T 2 - T 1 )  If 2 = outlet, 1 = inlet, and noting  If no work transfer (dw = 0) or KE change (dKE = 0), du = C v dT = dq  q 1  2 = C P (T 2 - T 1 )  If no heat transfer (dq = 0) or  KE, du = C v dT = dw  w 1  2 = -C P (T 2 - T 1 )  If no work or heat transfer   KE = KE 2 – KE 1 = -C P (T 2 - T 1 )  For a control volume containing an ideal gas with constant C P,  T ~ heat transfer - work transfer -  KE (Same statement as control mass with C P replacing C v )  Again true for any process, reversible or irreversible

8. 8 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams T-s & P-v diagrams: work, heat & KE 1 2 3 4 q 2  3 = C v (T 3 -T 2 ) (const. V) q 2  3 = C P (T 3 -T 2 ) (const. P) w 1  2 + (KE 2 – KE 1 ) = -C v (T 2 -T 1 ) (control mass) w 1  2 + (KE 2 – KE 1 ) = -C P (T 2 -T 1 ) (control vol.) w 3  4 + (KE 4 – KE 3 ) = -C v (T 4 -T 3 ) (control mass) w 3  4 + (KE 4 – KE 3 ) = -C P (T 4 -T 3 ) (control vol.) Case shown: cons. vol. heat in, r c = r e = 3,  = 1.4,  T comb = fQ R /C v = 628K, P 1 = 0.5 atm

9. 9 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams T-s & P-v diagrams: work, heat, KE & PE  How do I know that work shown on the P-v (via ∫PdV) diagram is the same as that shown (via C v  T) on the T-s diagram? As an example, for isentropic compression

10. 10 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams T-s & P-v diagrams: work, heat, KE & PE  Going back to the 1st Law again  Around a closed path, since E = U + KE + PE; since U is a property of the system,, thus around a closed path, i.e. a complete thermodynamic cycle (neglecting PE again) But wait - does this mean that the thermal efficiency No, the definition of thermal efficiency is  For a reversible process,  Q = TdS, thus  q = Tds and  Thus, for a reversible process, the area inside a cycle on a T-s diagram is equal to (net work transfer + net KE) and the net heat transfer

11. 11 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams T-s & P-v diagrams: work, heat & KE Net heat transfer = net work +  KE Heat transfer in  Animation: using T-s diagram to determine heat & work Heat transfer out

12. 12 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Constant P and V curves  Recall for ideal gas with constant specific heats (1st lecture)  If P = constant, ln(P 2 /P 1 ) = 0  T 2 = T 1 exp[(S 2 -S 1 )/C P ]  If V = constant, ln(V 2 /V 1 ) = 0  T 2 = T 1 exp[(S 2 -S 1 )/C v ]   constant P or V curves are growing exponentials on a T-s diagram  Since constant P or V curves are exponentials, as s increases, the  T between two constant-P or constant-V curves increases; as shown later, this ensures that compression work is less than expansion work for ideal Otto or Brayton cycles  Since C P = C v + R, C P > C v or 1/C P C v or 1/C P < 1/C v, constant v curves rise faster than constant P curves on a T-s diagram  Constant P or constant V lines cannot cross (unless they correspond to cycles with different C P or C V )

13. 13 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Constant P and V curves 1 2 3 4 T(s) = T 1 exp[(s-s 1 )/C v ] (const vol.) T(s) = T 1 exp[(s-s 1 )/C P ] (const press.) T(s) = T 2 exp[(s-s 2 )/C v ] (const vol.) T(s) = T 2 exp[(s-s 2 )/C P ] (const press.) Constant v or P curves spread out as s increases  T 3 - T 4 > T 2 - T 1

14. 14 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Constant P and V curves 1 2 3 4 T 2 - T 1 ~ Work input + KE decrease during compression “Payback” for compression work and KE decrease (T 3 - T 4 ) - (T 2 - T 1 ) ~ net work + KE increase Net work + KE decrease ~ (T 3 - T 4 ) - (T 2 - T 1 ) = (T 2 - T 1 )[exp(  s/C v ) - 1] > 0 or (T 2 - T 1 )[exp(  s/C P ) - 1] > 0

15. 15 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Constant P and V curves Constant-v curves are steeper than constant-P curves on the T-s Both cases: T 2 /T 1 = 1.552,  = 1.4, fQ R = 4.5 x 10 5 J/kg, P 1 = 0.5 atm The two cycles shown also have the same thermal efficiency (  th ) Double-click chart to open Excel spreadsheet

16. 16 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Inferring efficiencies Carnot cycles appear as rectangles on the T-s diagram; any cycle can be broken into a large number of tall skinny Carnot cycle “strips,” each strip (i) having  th,i = 1 - T L,i /T H,i Double-click chart to open Excel spreadsheet Carnot cycle “strip”  th,i = 1 - T L,i /T H,i T H,i T L,i

17. 17 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Compression & expansion efficiency  If irreversible compression or expansion, dS >  Q/T; if still adiabatic (  Q = 0) then dS > 0  Causes more work input (more  T) during compression, less work output (less  T) during expansion  Define compression efficiency  comp & expansion efficiency  exp

18. 18 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Compression & expansion efficiency  Control volume: replace C v with C P, but it cancels out so definitions are same  These relations give us a means to quantify the efficiency of an engine component (e.g. compressor, turbine, …) or process (compression, expansion) as opposed to the whole cycle

19. 19 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Compression & expansion efficiency  Animation: comparison of ideal Otto cycle with non-ideal compression & expansion  Same parameters as before but with  comp =  exp =0.9 2 charts on top of each other; double-click each to open Excel spreadsheets

20. 20 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Correspondence between P-V & T-s Pressure increasing Volume increasing

21. 21 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Correspondence between P-V & T-s Temperature increasing Entropy increasing

22. 22   Why do internal combustion engines compress before burning? Is it possible to produce work or thrust without compression? To generate positive area (thus net work) on a T-s diagram (left), the pre-heat- addition process must be more nearly vertical than the heat addition process, otherwise there is no area thus no work (middle). The best way to do this is with isentropic compression followed by heat addition (left) - but it’s not required. You could have (for example) constant-V heat addition followed by isentropic expansion back to ambient P (right) and work would be generated without a “real” compressor (although with lower efficiency.) AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Example

23. 23   Why is it necessary to add heat to generate work or thrust? Without heat transfer, dq = 0 and thus (for reversible cycles) TdS = 0, thus = = Net work = 0 (left figure). If the process is irreversible, TdS < 0, thus < 0 and thus = = Net work < 0 (right figure). AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Example

24. 24 Consider the “Lenoir cycle” engine shown on the P-V diagram a) a)Sketch the T-s diagram corresponding to the P-V diagram b) b)Sketch modified P-V and T-s diagrams if the initial temperature is increased by 10% (same P 1, V 1 and P 2 ) P P V 3 1 2 s = const T s 1 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Example 1  2 const. V, increasing P  heat addition, s increases 2  3 const. s, decreasing P  expansion, T decreases 3  1 const. P, decreasing v  heat rejection, s decreases P V 3 1 2 s = const T s 3 1 2 3’3’3’3’ 2’2’2’2’ 1’1’1’1’ P - V plot unchanged (but higher T means less mass processed) Higher T, same P  s 1, s 2 increases Still need 3  1 const. P P V P V P 2 3

25. 25 c) Sketch modified P-V and T-s diagrams if the “compression ratio” V 3 /V 1 is increased by 20% (same P 1, V 1 ) d) Sketch modified P-V and T-s diagrams if a gas with higher  is used (same P 1, V 1, P 2, R) AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Example 1  2 const. V, P 2 larger than base cycle On P-V, curves of const. T slope less steeply than const. s, thus T 3 ’ > T 3 Still need 3  1 const. P P V 3 1, 1’ 2 s = const T s 2’2’2’2’ 3’ 1,1’ 2 P V 3 2’ 3’ PV  curves steeper if  increases C V = R/(  -1); if R = const. and  increases, C V decreases, thus const.-v curves on T – s are steeper T 2 doesn’t change since V 1 = V 2, P 1, P 2 and R don’t change P V 3 1, 1’ 2, 2’ s = const T s 3’ 1,1’ 2 P V 3 2’ 3’

26. 26 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Using T-s and P-v diagrams - summary  Thermodynamic cycles as they occur in IC engines are often approximated as a series of processes occurring in an ideal gas  T-s and P-v diagrams are very useful for inferring how changes in a cycle affect efficiency, power, peak P & T, etc.  The  T (on T-s diagrams) and areas (both T-s & P-v) are very useful for inferring heat & work transfers  Each process (curve or straight line) on T-s or P-v diagram has of 3 parts  An initial state  A process (const. P, v, T, s, as shown in the previous slides), constant area (Rayleigh, Fanno or shock flow, discussed in propulsion section), etc.  A final state, which is usually »For compression and expansion processes in reciprocating piston engines, a specified volume relative to the initial state (i.e. a particular compression or expansion ratio) »For compressors in propulsion cycles, a specified pressure ratio »For turbines in propulsion cycles, a specified temperature that makes the work output from the turbine equal the work required to drive compressor and/or fan »For diffusers in propulsion cycles, a specified Mach number (usually zero) »For nozzles in propulsion cycles, the pressure after expansion (usually the ambient pressure) »For heat addition processes, either a specified heat input = ∫ Tds (i.e. a mixture having a specified FAR and Q R ) thus a given area on the T-s diagram, or a specified temperature (i.e. for temperature limited turbines in propulsion cycles) »The constant P and constant v exponential curves on the T-s diagram are very useful for determining end states

27. 27 AME 436 - Spring 2013 - Lecture 7 - Using P-V & T-s diagrams Using T-s and P-V diagrams - summary  Three or more processes combine to make a complete cycle  When drawing P-V or T-s diagrams, ask yourself  What is the P, V, T and s of the initial state? Is it different from the baseline case?  For each subsequent process »What is the process? Is it the same as the baseline cycle, or does it change from (for example) reversible to irreversible compression or expansion? Does it change from (for example) constant pressure heat addition to heat addition with pressure losses? »When the process is over? Is the target a specified pressure, volume, temperature, heat input, work output, etc.? »Is a new process (afterburner, extra turbine work for fan, etc.) being added or is existing one being removed? »In gas turbine cycles, be sure to make work output of turbines = work input to compressors and fans in gas  Be sure to close the cycle by having (for reciprocating piston cycles) the final volume = initial volume or (for propulsion cycles) (usually) the final pressure = ambient pressure