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Lecture 11: Thermodynamic cycles – The Otto and Diesel cycles Air standard cycles • there are several standard cycles, using air as the working fluid, that are used to model thermodynamic cycles. • assumptions common to the analysis of air standard cycles: 1. 2. 3. 4. air is the working fluid combustion is represented by heat transfer to air air goes through a closed cycle air is returned to its original state by heat transfer from the air. Ideal Otto cycle • the Otto cycle is the ideal cycle for a gasoline fuelled, spark ignited, internal combustion engine. The vast majority of passenger vehicles on Australian roads have engines of this kind, in so-called ‘four stroke’ configuration. • the basic operation of a four stroke engine: • the Otto cycle in p − v and T − s diagrams: • processes: 5→1 1→2 2→3 3→4 4→5 1→5 inlet isentropic, adiabatic compression constant volume heating isentropic, adiabatic expansion constant volume cooling exhaust • in order to calculate the thermal efficiency, we must note that this air standard cycle considers a fixed mass of fluid. We therefore use the non-flow energy equation (NFEE), q − w = ∆u = cv ∆T • it follows that ηth = = = Wout Qin Q 23 −Q 41 Q 23 mcv [(T 3 −T 2 ) − (T 4 −T 1 )] mcv (T 3 −T 2 ) T 4 −T 1 T 3 −T 2 = 1− = 1− T 1 (T 4 / T 1 −1) T 2 ( T 3 / T 2 −1) • but, for an adiabatic, ideal gas p 2 ⎛ V 1 ⎞ pV =⎜ = const. ⎟ , p1 ⎝ V 2 ⎠ T γ • thus T 2 ⎛V 1 ⎞ =⎜ ⎟ T 1 ⎝V 2 ⎠ γ −1 = T3 T4 ⎛ V V ⎞ = rv(γ −1) ⎜ rv = 1 = 4 ⎟ V2 V3⎠ ⎝ • where rv is termed the ‘compression ratio’. This gives ηth = 1 − T1 1 = 1 − (γ −1) T2 rv T4 T3 = and hence T1 T 2 • so ηth is a function of rv and γ only, 90 Ideal cycle efficiency % 80 70 60 50 40 30 20 10 0 0 5 10 15 20 25 30 35 40 Volumetric compression ratio r v Ideal Diesel cycle • Diesel engines do not have spark plugs. Instead, the fuel is injected into the gas during the latter stages of compression, and undergoes spontaneous ignition. This is because the gas is hot enough to initiate combustion of the fuel. • the ideal Diesel cycle is similar to the Otto cycle, but with constant pressure heat addition (combustion). • note the compression ratio r V = • thermal efficiency: η TH = V V1 and the cut-off ratio rc = 3 V2 V2 W OUT Q 23 −Q 41 (NFEE) = Q IN Q 23 • for the constant pressure heat addition (control volume boundary is moving): Q23 = mCP (T3 − T2 ) • note that this looks like the SFEE mistakenly applied to this non-flow problem. It is not! It is a particular statement of the NFEE applied to a constant pressure expansion (or compression) of a fixed mass. The work is non-zero. See the appendix to these notes for further explanation. • for the constant volume heat extraction (control volume boundary is stationary): Q41 = mCV (T4 − T1 ) • thus: η TH = CP (T 3 −T 2 ) − CV (T 4 −T 1 ) CP (T 3 −T 2 ) CV (T 4 −T 1 ) CP (T 3 −T 2 ) = 1− = 1− = 1− 1 (T 4 −T 1 ) C ∵γ = P γ (T 3 −T 2 ) CV 1 T1 γ T2 ⎡ T 4 / T 1 −1 ⎤ ⎢ T / T −1 ⎥ ⎣ 3 2 ⎦ • for adiabatic processes with an ideal gas pν pvγ = const , = const. T • thus T T2 T = rvγ −1 , 4 = r cγ , 3 = r c T1 T1 T2 • and it follows that η TH = 1 − 1 rν γ −1 ⎡ r c γ −1 ⎤ ⎢ ⎥ ⎣ γ (r c −1) ⎦ • note: 1. the expression for the diesel cycle efficiency is similar to the Otto cycle, but with added term in square brackets. 2. Thus, for a given r V , η DIESEL < η OTTO . However, in reality r v DIESEL > r v OTTO since diesel combustion usually has lower flame temperatures. This leads to η DIESEL > η OTTO . 3. expansion ratio (V 4/V 3 ) < compression ratio (V 1/V 2 ) Appendix - The NFEE during constant pressure expansion or compression • using only dq − dw = du, NFEE (1st law) dw = pdv, h = u + pv, definition of enthalpy • it follows that dh = du + pdv + vdp • thus dq = du + dw = du + pdv • = dh − pdv − vdp + pdv = dh − vdp. so, for a constant pressure process with constant specific heats dq = dh = c p dT . note that whilst this last equation looks like the steady flow energy equation for a zero work device, it is not. It is a particular statement of the non-flow energy equation applied to a constant pressure expansion (or compression) of a fixed mass. The work is non-zero. •

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posted: | 7/19/2009 |

language: | English |

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