1. Description of Finite Heat Release Function

Size: px

Start display at page:

Download "1. Description of Finite Heat Release Function"

Imogene Porter
6 years ago
Views:

1 ME 40 Day 27 Desription of Finite Heat elease Funtion - SI Engines Differential Equations to Moel Cyle Software Implementation in EES Questions that an be answere. Desription of Finite Heat elease Funtion This is an attempt to moel more realistially the burn rates of fuel in the yliner uring the working yle. The key is to speify heat aition rate as a funtion of rank angle. Naturally, heat aition rate is zero for muh of the yle. We start with an S ure or Weibe funtion whih esribes the fration of fuel that has been burne. s Here is the math funtion whih esribes the burne fration. n () x s b exp a The shape is etermine by the parameters a an n. alues of a 5 an n 3 hae been foun to agree well with experiments. See pages 766 to 769 in the Text. Another ref: Internal Combustion Engines by Ferguson an Kirkpatrik, Wiley 200.

2 How it an be use. Q Q * x b Q Q an * ( x ) b s n This is a ery useful esription of how heat input rate aries with uring ombustion. Naturally, when the ombustion proess is not ourring, this eriatie is zero. 2. Differential Equations to Moel Cyle Start with the Ieal Gas Law mt an ifferentiate with respet to q. m T Next, write the first law of thermo for this lose system in rate form. δ Q δw U an make the following substitutions. δ W U m T

3 After some han waing, Q Next sole for the eriatie of pressure with respet to. Q Q Q p p γ γ Q Next, we nee the following expressions from kinematis. See our text, page 44. ( ) () () 2 2 sin os r 2 Now no longer stans for the gas onstant, it now stans for the ratio of onneting ro length to rank raius. is the learane olume an the olume of the yliner at. Sine ( ) r we an write this as

4 ( r ) 2 os 2 2 () sin () The rate at whih olume is swept out as a funtion of rank angle, we get by ifferentiating with respet to. () () () sin os sin Finally, we reall the efinition of work gies us W 3. Software Implementations We oneptualize this problem mathematially as 4 simultaneous ifferential equations in the inepenent ariable q an haing epenent ariables,, T an W. These equations must then be sole numerially using a time stepping sheme. The integral funtion in EES inokes the time stepping sheme whih is esribe in the software help. The EES file is inlue in these notes.

5 "Finite Heat elease Calulations" "eferene: Internal Combustion Engines by Colin Ferguson an Allan Kirkpatrik, 2n Eition. John Wiley 200. ages " "The heat release ours oer a finite time gien by a Weibe funtion. The heat release rate is alulate an use to write the first law of thermo in rate form. (ate of hange with respet to rank angle) The ieal gas law is use to relate pressure an temperature rates. The kinematis of the engine gie olumetri rates. The result is a set of ifferential equations whih may be integrate to esribe the yle." "Funtion to alulate burn fration s. rank angle. Crank angle aries from -pi at start of yle to 0 at TC to pi at the en of the yle. Exhaust proesses are not moele." funtion xb(theta) { theta_s is start of ombustion in raians, theta_ is the uration, a an n are the Weibe parameters.} $ommon theta_s, theta_,a,n if (theta < theta_s) then xb : 0 else if (theta > theta_s theta_) then xb : else xb : -exp(- a*((theta-theta_s)/theta_)^n) enif enif en "Funtion to alulate heat release rate" funtion Qtheta(theta) { theta_s is start of ombustion in raians, theta_ is the uration, a an n are the Weibe parameters. Q is the total heat release } $ommon theta_s, theta_,a,n,q

6 if (theta < theta_s) then Qtheta : 0 else if (theta > theta_stheta_) then Qtheta : 0 else Qtheta : n*a*q/theta_*(-xb(theta))*((thetatheta_s)/theta_)^(n-) enif enif en "Funtion to alulate olume rate wrt rank angle" funtion theta(theta) $ommon _, r_, theta : _/2 * sin(theta)*(os(theta)/sqrt(^2 - sin(theta)^2)) en "Funtion to alulate olume at a gien rank angle" funtion (theta) $ommon _, r_, : _/(r_-) _/2 *( - os(theta) - sqrt(^2- sin(theta)^2)) en "Combustion Data" theta_s -pi/9 theta_ 4*pi/9 a 5 n 3 Q.8 "Engine Data" bore 0. stroke 0.

7 _ 0.3 r_ 0 _ pi /4 * bore^2 * stroke L *L/stroke gamma.4 T_ 300 _ (theta_) m _*_ / T_ Mw 29 theta_ -pi theta_f pi (theta) "The eriaties of ressure an Temperature, Work an ol with respet to rank angle" theta -gamma*/(theta) * theta(theta) (gamma- )/(theta)*qtheta(theta) Ttheta /m * ( * theta(theta) (theta)*theta ) Wtheta * theta(theta) oltheta theta(theta) "We an now integrate the aboe ifferential equations." _ integral(theta,theta,theta_,theta_f) T T_ integral(ttheta,theta,theta_,theta_f) W integral(wtheta,theta,theta_,theta_f) ol _ integral(oltheta,theta,theta_,theta_f) $IntegralTable theta:0.,,t,w,ol "erformane Calulations" eta_f W / Q imep W / _

8 4. Questions that an be answere This moel is useful in the sense that it allows us to stuy the effets of spark timing on engine performane.

MODELS FOR VARIABLE RECRUITMENT (continued)

MODELS FOR VARIABLE RECRUITMENT (continued) ODL FOR VARIABL RCRUITNT (ontinue) The other moel ommonly use to relate reruitment strength with the size of the parental spawning population is a moel evelope by Beverton an Holt (957, etion 6), whih