Optimal Engine Design Using Nonlinear Programming and the ...

More documents

Recommendations

Info

In the monotonicity table the columns are the design variables and the rows are theobjective and constraint functions, the entries in the table being the monotonicities of each functionwith respect to each variable. Positive (negative) sign indicates increasing (decreasing) function, Uindicates undetermined or unknown monotonicity. An empty entry indicates that the function doesnot depend on the respective variable, so the table acts also as an incidence table.The Monotonicity Principles can be more readily applied using the monotonicity table.These principles state the following (Papalambros and Wilde 1988):First Monotonicity Principle (MP1): In a well-constrained objective function every increasing(decreasing) variable is bounded below (above) by at least one active constraint.Second Monotonicity Principle(MP2): Every monotonic variable not occurring in a wellconstrainedobjective function is either irrelevant and can be deleted from the problem together withall constraints in which it occurs, or is relevant and bounded by two active constraints, one fromabove and one from below.Based on these principles and Table 5 the following conclusions are reached. By MP1 wrtdI at least one of g3 or g4 must be active. We examine these two cases. (i) If g3 is active, then byMP2 wrt dE g4 must be active as well. (ii) If g4 is active, then by MP2 wrt dE g3 and/or g5 mustbe active. But g5 and g4 cannot be simultaneously active because they represent upper and lowerbounds on the same quantity. Therefore in case (ii) g4 must be active also. We see that in eithercase, both g3 and g4 must be active at the optimum. This is a necessary optimality condition thatmust be satisfied by any subsequent numerical results. Solving for dE and dI in terms of theoptimal value b * (asterisk denotes optimal value) we getdE * = b * K3K4/(1+K4)dI * = b * K3/(1+K4) (44)a solution that is acceptable provided K4 ≤ K5. Note that g5 is then inactive. Using the second ofEquation (44) to eliminate dI from the objective and remaining constraints Model A1 is nowreduced toMODEL A2 (45)min f(cr, w, b) = K0w [FMEP(cr, w + , b - ) - P 0 η t (cr, b - ) η v (w, b + )]subject tog1(b + ) = b - P1 ≤ 0g2(b - ) = P2 - b ≤ 0g6(w + , b - ) = P 3 w - [K3/(1+K4)] 2 b 2 ≤ 0g7(cr + , b + ) = cr - 13.2 + 0.045 b ≤ 0min bore wall thicknessmax engine heightmax port/chamber Mach Indexknock-limited compression ratio12
g8(w + ) = w - K7 ≤ 0max torque converter rpmg9(cr, b + ) = P 4 - 0.8595(1- c r- 0.33 ) + Sv (cr, b + ) ≤ 0min fuel economy at part loadwhere asterisks are dropped for convenience. There are only three design variables left and at mostthree constraints from the six in Model A2 can be active.We proceed with monotonicity analysis for Model B. We now haveFMEP = FMEP(cr , Vp + (w + , b - , v + )) = FMEP(cr, w + , b - , v + )η t = η t (cr + , S - v (c r, b + , v - )) = η t (cr, b - , v + ) (46)η v = η v (w, d + I , v - )v d = v d (b, d - I , dE - , r)The algebraic expression for v d in Equation (37) is not easy to use for proving the monotonicitieswrt d I and d E indicated above, but the geometry in Figure 2 shows that this is evidently the case.Eliminating variable h and constraint h2 we getMODEL B1 (47)min f(cr, w, b, d - I , v) = K 0wv [FMEP(cr, w + , b - , v + )- P 0 η t (cr, b - , v + ) η v (w, d + I , v - )]subject toh1(cr + , v - , vc - ) = - 1 + cr - (v /Ncvc) = 0 head geometrywhere vc = πK11b 3 /4 + vd(b, d - I , dE - , r)g1(b + ) = b - P1 ≤ 0min bore wall thicknessg2(b - , v + ) = P2C v 1/2 - b ≤ 0max engine heightg3(b, dE + , d + I , r) ≤ 0valve geometry and structureg4(dE - , d + I ) = K4 dI - dE ≤ 0min valve diameter ratiog5(dE + , d - I ) = dE - K5 dI ≤ 0max valve diameter ratiog6(w + , d - I , v + ) = P3C wv - d I2 ≤ 0max port/chamber Mach Indexg7(cr + , b + ) = cr - 13.2 + 0.045 b ≤ 0knock-limited compression ratiog8(w + ) = w - K7 ≤ 0max torque converter rpmg9(cr,b + ,v + )=P 4 - 0.8595(1- c r- 0.33 )+Sv (cr,b + ,v + ) ≤0min fuel economy at part load13
Page 1 and 2: This is an unofficial copy of a tec
Page 3 and 4: 1. IntroductionEngine modeling acti
Page 5 and 6: sufficiently determines whether the
Page 7 and 8: 2.2.2 The Karush-Kuhn-Tucker (KKT)
Page 10 and 11: inducted into the combustion chambe
Page 12 and 13: design model stoichiometry will be
Page 14 and 15: parameter implies Cs is independent
Page 16 and 17: A typical design specification on t
Page 18 and 19: 3.2 Monotonicity AnalysisIn many de
Page 22 and 23: g10(v + ) ≤ 0g11(v - ) ≤ 0where
Page 24 and 25: file to run ESA with the appropriat
Page 26 and 27: Iteration HistoryPredicted Brake Po
Page 28 and 29: specified explicitly for the engine
Page 30 and 31: Optimal Peak Power(kW/l)68666462608
Page 32 and 33: 70Optimal Peak Power(kW/l)68666462f
Page 34 and 35: 45Peak Power(kW/l)444342f*Optimal41
Page 36 and 37: 48Optimal Peak Power(kW/l)464442403
Page 38 and 39: Campbell, A.S., 1979, Thermodynamic
Page 40: Taylor, C.F., 1985, The Internal Co

Optimal Engine Design Using Nonlinear Programming and the ...

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?