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A FORTRAN program for calculating the evaporation rates in diesel engine fuel sprays
Advances in Engineering Software 15 (1992) 67-71
Technical Note A FORTRAN program for calculating the evaporation rates in diesel engine fuel sprays D.A. Kouremenos, C.D. Rakopoulos* & E.A. Yfantis National Technical University of Athens, Mechanical Engineering Department, Thermal Engineering Section, 42 Patission Street, Athens 106 82, Greece
A FORTRAN program based on a theoretical model for predicting the evaporation process of liquid fu.el sprays in diesel engines is presented. The injected liquid fuel is assumed to break up into droplets in a high pressure, high temperature environment, just as encountered in a diesel engine cylinder. The complete 'history' of the droplets is described by the present program, following their motion as well as the heat absorption and evaporation mechanisms. The theoretical model is based on Newton's Second Law of Motion and the equations expressing the heat and mass balance between droplets and surrounding gas. Accurate correlations are taken into account for the prediction of Sauter Mean Diameter and the thermodynamic and transport properties of fuel. Key words: droplet evaporation, fuel spray, diesel engine.
NOTATION
Ap Pressure drop across the nozzle orifice (N/m 2) 2 Thermal conductivity (W/m/K) v Kinematic viscosity (mZ/s) p Density (kg/m 3) a Surface tension (N/m)
c Discharge coefficient (--) co Drag coefficient (--) cp Specific heat capacity under constant pressure (J/kg/K) do Injector nozzle orifice diameter (m) Dr Mass diffusivity (m2/s) H Fuel latent heat of vaporization (J/kg) rn Mass (kg) M Molecular weight (kg/kmole) N Engine rotational speed (RPM) p Pressure (N/m 2) q Heat transported to droplet per unit surface (W/ m 2) r Droplet radius (m) S M D Sauter Mean Diameter (m) t Time (s) T Absolute temperature (K) v Velocity (m/s) Y Mass fraction of vaporized fuel (--) Y~ Mass fraction of vaporized fuel at droplet surface
Subscripts a Air f Fuel 1 Liquid fuel (droplet) v Vapor inj Injection conditions
INTRODUCTION The evaporation of liquid fuel sprays is a very important process of diesel engines operation, since it relates strongly to the subsequent combustion process and so to its power and efficiency. The injected liquid fuel, after being atomized into small droplets near the nozzle exit to form a spray, must first evaporate before it can be mixed with the surrounding air and then burnt.~ Three phenomena determine the history of a liquid fuel droplet, with temperature close to the ambient, when
(--) *Corresponding author. Advances in Engineering Software 0965-9978/92]$05.00 © 1992 Elsevier Science Publishers Ltd.
67
68
D.A. Kouremenos, C.D. Rakopoulos, E.A. Yfantis
injected into an air environment having typical end-ofcompression phase engine conditions: 2 (a) Deceleration of the droplet due to aerodynamic drag; (b) Heat transfer to the droplet from the surrounding air; (c) Mass transfer of vaporized fuel away from the droplet surface. For the accurate determination of the fuel vaporization rate, within a diesel fuel spray, the simultaneous solution of the relevant conservation equations for the droplet and the surrounding air is needed. Models, which treat explicitly the two-phase structure of the diesel spray, describing the spray behaviour in terms of the conservation equations in differential form, for mass, momentum and energy, have appeared in the literature. E1 Wakil et al. 3 have considered the changes in droplet and gas temperatures and the fuel vapor concentration in accordance with heat absorption and evaporation processes varying in time. In the model proposed by Hiroyasu and co-workers, 4 the injected fuel spray is divided into several elements. These elements entrain air, vaporize, and then react after they ignite. During injection and combustion, these elements expand and entrain air. After ignition, the evaporation, mixing, and reaction processes may continue. Bracco 5 presented a model for full-cone sprays, where the break up and development regions are the most relevant to diesel engine applications. The works mentioned above constitute only a part of published works concerning the prediction of the droplet evaporation mechanism. The present work gives a fast and accurate F O R T R A N subprogram which can be used as a part of a general computer program modelling the entire thermodynamic cycle of a diesel engine. 6-8 It is developed for use even on a PC and is related to previously published works by the present research group. Kouremenos et al. 6-8 have considered the following two distinct periods in the fuel droplet lifetime: (a) Sensible heating period from the initial droplet temperature to the saturation temperature; (b) Evaporation period with constant droplet temperature equal to the saturation temperature.
fuel droplet, is given by the expression: II Vinj = C . /2Ap ~/ P~
Kuo ~2 recommends c = 0.39, a value based on comparisons made between numerical predictions and experimental data. A dimensionless analysis of the measurement results on the Sauter Mean Diameter (SMD) leads to the following equation concerning the various liquid fuel properties and injection conditionsg: = 0.38 /~einjn 0.25 WeinjO.a2(vl/Va)O.37(pl/Pa)
(SMD),
0.47d0
(2a) = 4.12 _e~.jR -°'2 We~.i°W(v,/v.)°S4(p~/p,)°"~d o
(SMD)2
(2b) where subscripts 1 and 2 denote 'complete' and 'incomplete' sprays respectively, while the Reynolds and Weber numbers are respectively Rein j
Vinj do , VI
=
Wein j
__
V~njdop 1 O"
The value of the initial droplet diameter is the maximum from the two values given by expressions (2a) and (2b). The instantaneous velocity is determined by the solution of the conservation of momentum equation: dv _
~z
2 CDt2paVr2el
ml dt
(3)
where Vre~is the relative velocity between the droplet and the surrounding gas. The drag coefficient is given by the following expression: 5 f - ~ e ( l + O.166 Re 2/3) if Re < 1000 CD
= I
0.424
if Re > 1000
(4)
where the Reynolds number Re
-
2rVrel
Va
The rate of change of the droplet radius is: 5 dr
-
In this work evaporation begins immediately after fuel injection, when the droplet appears in the combustion chamber, which obviously represents more realistically the actual phenomenon. Furthermore accurate correlations are taken into account for the prediction of the Sauter Mean Diameter 9 and the thermodynamic and transport properties of the fuel) °
(1)
-
~
dt
Pa DI Sh In (1 + B)
(5)
p~ 2r
where the Sherwood and Schmidt numbers are respectively 2 + 0.6 Rel/2Sc ~/3, Sc = v~/Df while B
=
(Y~- Y)/(1
=
M~pv M~p~ + M , ( p - P v )
-
Y~), Y
=
Pr~/P.
and MODEL
DEVELOPMENT
The injection velocity, which is the initial velocity of the
r~
To obtain the expression for the droplet surface mass
69
A F O R T R A N program for calculating the evaporation rates in diesel engine fuel sprays
fraction Ys, it has been assumed that the droplet temperature is uniform and that the partial pressure of fuel vapor at the droplet surface equals the equilibrium vapor pressure. The energy balance equation states: 5 dT~ mlCpl --~
dr 4nr~p~H d-~ = 4nr2q
15
._j
co10 ___ rm <~ rY ~.-L..aJ ._J a_ 5 0 rY ~
(6)
2 a ( T - T,) In (1 + B) 2r Nu B
and the Nusselt and Prandtl numbers are respectively Nu
= 2 + 0.6 Re ~/2 Pr ~/3, Pr
-
,'
v
where q -
OLD METHOD NEW METHOD
SPEED=1500 RPM II~J.ANOLE~34 J.ANGLE 54 CA B;DC B DC
VaPaCpa
\ \
,
J'a
Equation (6) is a statement of the fact that the heat transported to the droplet increases its temperature (heat up) and supplies heat for vaporization. The thermodynamic and transport properties of the surrounding gas are taken from Ref. 13 at a temperature value corresponding to (T + T~)/3. The thermodynamic and transport properties of the fuel are taken from Ref. 10 at a temperature value corresponding to T~. As in most diesel engine cycle simulations, N-Dodecane is assumed to represent here the typical diesel fuel.
D E S C R I P T I O N OF T H E P R O G R A M - C A S E
STUDY-DISCUSSION The computer program is written in FORTRAN 77 language and its listing is given in Appendix A. The program requires as its input the engine rotational speed, the nozzle orifice diameter, the pressure drop across the nozzle orifice, the temperature, pressure and volume of the combustion chamber, as well as the initial droplet temperature, the air velocity and a factor which affects the determination of the time step for the computational procedure. After the estimation of the droplet initial characteristics, immediately after injection, the program predicts the complete 'history' of the fuel droplet as part of an iterative procedure. At the end of every time step, the droplet characteristics (radius, mass, temperature, velocity) are computed as well as the fraction of mass of the initial droplet which has been evaporated. A typical output listing is given in Appendix B. An experimental investigation has been conducted, at the authors' laboratory, on a single cylinder, naturally aspirated, four stroke, Ricardo E-6, indirect injection diesel engine fitted with a Comet MK.V swirl prechamber. 7 The engine bore is 76.2mm, the stroke 111.2 mm and the compression ratio 21.5. The pintle type injector nozzle is located in the swirl prechamber. The engine was operated at full load, a rotational speed of
I
0
I
I
10 20 5O 4O CRANK ANGLE FROM INJECTION
Fig. 1. Droplet radius variation with crank angle (from the moment of injection). 1500 RPM (revolutions per minute), and a static injection timing of 34 degrees CA BTDC (crank angle, before top dead centre). The program was applied considering the engine operating conditions mentioned above. In order to examine the differences between the two methods, i.e. the old one proposed in Ref. 7 and the new one presented here, both were applied for the same engine operating conditions. The comparison is effected by considering the predictions of the two procedures, concerning the variation of droplet radius and evaporated fuel mass with time. Figure 1 shows the variation with time (expressed in degrees CA) of the fuel droplet radius. The broken line
100,; v
/
_~ -75i,i D L~ rm ~,, 50 F-< rr 0 EL
OLD METHOD NEW METHOD
X 25 uJ
/
,,' LOAD= 1 O0 • SPEED=IS00 RPM ./ .... ./ INJ.ANGLE=34 CA BTDC
/
,;
I
140
I
I
I
160 180 200 220 CRANK ANGLE.(DEG)
I
240
Fig. 2. Fuel droplet mass evaporated (as a percentage) with crank angle.
70
D.A. Kouremenos, C.D. Rakopoulos, E.A. Yfantis
represents the result obtained with the old method, described in detail in Ref. 7, while the solid one represents the result obtained with the new one, as exposed in this work. Figure 2 shows the variation with time (expressed in degrees CA) of the evaporated fuel mass (expressed as a percentage of total). The old method results in an evaporation time lag at the beginning of the droplet 'history' inside the combustion chamber. This lag is caused by the lack of modelling of the evaporation at the very first period of the process. In the present method evaporation begins immediately after fuel injection, when the droplet appears in the combustion chamber. This more realistic representation of the actual phenomenon leads to a faster disappearance of the droplet, corresponding to about 3 degrees CA. This delay period could be proved very important when modelling, since it strongly influences the subsequent processes (mixing with air, ignition, reaction) and so the power and efficiency as well as the pollutants formation of the diesel engine.
REFERENCES 1. Heywood, J.B. Internal Combustion Engine Fundamentals, McGraw-Hill Book Co., New York, 1988. 2. Browne, K.R., Partridge, I.M. & Greeves, G. Fuel property effects on fuel/air mixing in an experimental diesel engine, SAE Paper 860223, 1986. 3. El Wakil, M.M., Myers, P.S. & Uyehara, O.A. Fuel vaporization and ignition lag diesel combustion, in Burning a Wide Range of Fuels in Diesel Engines, SAE Progress in Technology, 1967, 11, 30-44. 4. Hiroyasu, H., Kadota, T. & Arai, M. Development and use of a spray combustion modelling to predict diesel engine efficiency and pollutant emissions: Part 2. Computational procedure and parametric study, Bulletin of the JSME, 1983, 26(214), 576-583. 5. Bracco, F.V. Modelling of engine sprays, SAE Paper 850394, 1985. 6. Kouremenos, D.A., Rakopoulos, C.D. & Karvounis, E. Thermodynamic analysis of direct injection diesel engines by multi-zone modelling, ASME-WA Meeting, Boston MA., Dec. 13-18, AES, 1987, 3(3), 67-77. 7. Kouremenos, D.A., Rakopoulos, C.D. & Hountalas, D.T. Thermodynamic analysis of indirect injection diesel engines by two-zone modelling of combustion, Journal of Engineering for Gas Turbines and Power, Trans. ~1~ the ASME, 1990, 112(1), 138-149. 8. Kouremenos, D.A., Rakopoulos, C.D. & Hountalas, D.T. Computer simulation with experimental validation of the exhaust nitric oxide and soot emissions in divided chamber diesel engines, ASME-WA Meeting, San Francisco California, Dec. 10-15, AES, 1989, 10(1), 15-28. 9. Hiroyasu, H., Arai, M. & Tabata, M. Empirical equations for the Sauter Mean Diameter of a diesel spray, SAE Paper 890464, 1989. 10. Kouremenos, D.A., Rakopoulos, C.D. & Yfantis, E.A. A FORTRAN program for calculating the thermodynamic and transport properties of diesel fuel, Advances in Engineering Software, 1990, 12(4), 190-196. 11. Faeth, G.M. Evaporation and Combustion of Sprays, Progress in Energy and Combustion Science, 1983, 9, !-76.
12. Kuo, T.W. Evaluation of a phenomenological spray combustion model for two open-chamber diesel engines, SAE Paper 872057, 1987. 13. Pflaum, W. & Mollenhauer, K. Wi~rmeiibergang in der Verbrennungskraftmaschine, Springer-Verlag, Wien/New York, 1977.
A P P E N D I X A: P R O G R A M LISTING PROGRAM TESTING O P E N ( 3, F I L E - 'EV. n A T ', S T A T U S = 'O L D ' ) OPEN (4,FILE='EV.RES' ,STATUS='NEW' ] R E A D (3,*) R P M , X R E A D [3,*) D P E E S , D I N J R E A D [3,*) T E M P , P R E S , V O L R E A D 13,*) T D R O P R E A D (3,*) A I R V E L T E M P C - T D P ~ O P 273. 15 C A L L L H V A P (T E M P C , HV ] C A L L V A P R E S (T E M P O , P V ) C A L L D E N S L I Q (T E M P C , P R E S , D E N S L ) CALL SURTEN (TEMPC, SURT ) C A L L DVI SC ( T E M P C , P R E S , VI S C L } C A L L C P L I Q (T E M P C , C P L ) T E M P O = (T E M P + 2. * T D R O P ) / 3. - 2 7 3 . 1 5 C A L L P R O P A I R ( T E M P O , C O N D A , VI BCA, C P A ) RGAS-8314 ./28.96 DENSA=PRES/RGAS/TEMP PC=3.14159 O ............................................... C . . . . C a l c u l a t i o n of the droplet initial characteristics . . . . . . . . . . . C ............................................... C A L L I N J E C T ( D P R E S , D I N J, S U E T , D E N S A , D E N S L , VI SCA, +VISCL, UINJ, SMD) GASMASS =DENSA*VOL VAPMASS-0.0 VDROP=UINJ RDROP=SMD/2. D R O M 0 = 4 . * P C * R D E O P * * 3. * D E N S L / 3 . R E L V E L - A B S (A I R V E L - V D R O P ) I=l WRITE (4,14) 14 FORMAT ********************************************************** WRITE (4,16) 16 F O R M A T ( IX, ' D R O P L E T I N I T I A L C H A R A C T E R I S T I C S '] WRITE (4,14) W R I T E (4,3) R D R O P * I . E 6 3 FORMAT (iX,'Radius [ m]=',F6.2) W R I T E (4,4) D R O M O * I 0 0 O . 4 FORMAT (iX,'Mass [gr]=',ES.4) W R I T E (4,5) T D R O P - 2 7 3 . 1 5 5 FORMAT (iX,'Temperature [C]=',F6.2) W R I T E (4,6) V D R O P 6 FORMAT IiX,'Velocity [m/s]=',F6.2) C ............................................. C Calculation of the droplet evaporation history . . . . . . . . C .......................................... C A L L E V A P O R (R P M , R D R O P , R D R O P N S , V D R O P , V D R O P N S , D R O M 0 , D R O M N S , X, I, + D M A S S, T D R OP, T D R O P NS, G A S M A S S, V A P M A S S, T E M P , P R E S , VOL, A I R V E L , R E L V E L ) WRITE (4,14) WRITE (4,17) 17 FORMAT (IX,'DROPLET CHARACTERISTICS AT THE END OF THE TIME STEP'] WRITE (4,14) W R I T E (4,3) R D R O P N S * I . E 6 WRIT~ ~4,4 DR~3~NS*] 00O.
i[4
WEIT~ ~, V D R O F N $ WR:TE 4,14) P E R C E N T 1 0 0 * V A F M A S $ . DRd'M0 IF{ZERCENT.GT.99: G O T O 124 WE]TE :4,~ [ER:ENT,I F O R M A T ~ I X , ~ F E R C E N T A G E OF E V A P O R A T E D WRITE (4,14 I F ( R D R O P N S .LE.< . G O T O 124 RDROP=RDROPNS VDROP-VDROFNS T[:RC ? - T D R O P N S R E L V E L ;AZS ~V [ , ~ 0 ~ - A I R V E L ) CLOSE 3) CLOSE 4, 3T,3[ E}.S
MASS
',F6.2,'%
STEP
',12~
BLOC/ DATA COMMON/CRIT/RCR,TCR,VCR,ZCR,TB
COMMON/BBB1/BOO,BOI,BO2,RO3,BIO,BII,BI2,BI3,B20,B21 COMMON/BBB2/B22,B23,B30,R31,B32,B33,B40,R41,B42,B43 C(~MMON/CONS/A,B,C,D,AA,BB,CC,DD D A T A PCR,TCR,VCR,ZCR,TB/264.0,1184.9,0.0669,0.237,881.O/ DATA R00,B01,B02,B03,BI0,RII,BI2,BI3,B20,B21/ +i.b368, 1.g695,2.4638,-I.5841,-0.04615,0.21874,-0.36461, ÷0.25136,2.1138R-3, 8.0028E-3/ D A T A B22,B23,B~O,B31,B32,B33,B40,B41,B42,B43/ *11.8763E , I I . 3 8 0 5 E 3,-0.7845E-5,-8.2328E-5,14.805SE 5, , 9 . 5 6 7 2 E 5, 0 . 6 9 2 3 E 6 , 5 . 2 6 0 4 E - 6 , - 8 . 6 8 9 5 E 6 , 2 . 1 8 1 2 E 6/ DATA A,S,C,D,AA,BR,CC,DD/0.84167,-I.4704,1.67165,-0.59198, + 0.00382C, 0.000747,0.041126, 0.01395/ E~ D
~ U B R U U T ~ N E E V A P O R (RRM, R D R O P , R D R O P N S , V D R O P , V D R O P N S , D R O M 0 , D R O M N S , X, • i, D M A S S , T D R OP, T D R O P N S, G A S M A S S , V A P M A S S, T E M P , P R E S , V O L , AI R V E L , •R E L V E L ) ~ O M M O N ,OR I T / P C R , TCR, V C R , ZCR, TB PC-5.14159 DI2-I.E-6 ~{(3A~:83 I 4 . / 9 8 . 9 6 bEN:ZA=PRES/RGAS/TEMP M W A - 28 .96 MWF-170.3~ DT:I. (X*RPM~
A FORTRAN program for calculating the evaporation rates in diesel engine fuel sprays C ~
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C Cai~ula~ion o~ the t h e r m o d y n a m i c a n d t r a n s p o r t p r o p e r t i e s . . . . . . . C .... of the l i q u i d fuel a n d t h e s u r r o u n d i n g a i r . . . . . . . . . . . . . . . C. . . . . . . . . . . . . . . . . . . . TEMPC:TDROP 273.15 C A L L L H V A P ~ T E M P C , HV~ CALL VARRES~TEMPC,f~V) CALL DEN~L~,2(TEMPC,PRRg,DENSL) CALL CPLIQ(TEMPC,CPL~ TEMPC=(TEMP+2.*TDROP)/3.-273.15 CALL PROPAIR(TEMPC,CONDA,VISCA,CPA) C ............................................................... C ..... C a l c u l a t i o n of t h e d i m e n s i o n l e s s numbers .................... C ...................................................... REYN=2.*DENSA*RDROP*RELVEL/VISCA SC=VISCA/(DENSA*DI2) PE=VISCA*CPA/CONDA REYNI2=REYN**[I./2.) REYN23=REYN**(2./3.) SC13=SC**(I./3.) PRI3=PR**(I./3.) C ....................................................... C .... Calculation of t h e m a s s f r a c t i o n of v a p o u r i z e d f u e l . . . . . . . . . . . . . . C .................................................. Y=VAPMASS/GASMASS IF(TDROP.GE.600.) G O T O 20 G O T O 30 20 PV=PRES 30 YSUR=MWF/IMWF+MWA*[(PRES/PV]-I.)} IF(I.-YSUR.LT.I.E-6) G O T O 50 C ........................................................... C ..... C a l c u l a t i o n of t h e S h e r w o o d a n d N u s s e l t n u m b e r s . . . . . . . . . . . . . . . . . C .................................................................. BY=(YSUR-Y)/II. YSUR) BYB=ALOG(I.+BY)/BY SHER=(2.+0.6*REYNI2*SCI3]*BYB
C
C
500
600 C C
SUBROUTINE VAPRES(TEMPC,PV) COMMON/CRIT/PCR,TCR,VCR,ZCR,TB TT=TEMPC*9./5.*32. TT=TT+459.7 TT=TT/TCR PRLN0=5.92714-6.09648/TT-I.28862*ALOG{TT]+0.169347*TT**6. PRLNI=I5.2518-15.6875/TT-13.4721*ALOG(TT)+0.43577*TT**6. PRLN=PRLN0+0.5622*PRLNI PVR=EXP(PRLN) PV=PVR*PCR PV=PV*6894.7591 RETURN END C C SUBROUTINE
COMMON/BBB2/B22,B23,B30,B31,B32,B33,B40,B41,B42,B43 TT=TEMPC*9./5.*32. TT~TT+459.7 TT=TT/TCR PP=PRES/6894.7591 PP=PP/PCR DENSI=0.7526*999.024 A02=B00+BI0*PP+B20*PP**2.+B30*PP**3.+B40*PP**4. AI2=B01+BII*PP+B21*PP**2.+B31*PP**3.+B41*PP**4. A22=B02+BI2*PP+B22*PP**2.+B32*PP**3.+B42*PP**4. A32=B03+BI3*PP÷B23*PP**2.+B33*PP**3.+B43*PP**4. CC2=A02+AI2*TT+A22*TT**2.+A32*TT**3. CCI=I.I1342 DENSL=DENSI*CC2/CCI BETURN END
RDEOPNS=AAR**(I./2.) G O T O 60 50 AAR=O.0 RDROPNS=0.0 G O T O 80 C ..................................................... C ..... Calculation of the n e w d r o p l e t t e m p e r a t u r e . . . . . . . . . . . . . . . . . . . . . C ......................................................... 60 AAS=RDROP+RDROPNS
AAT=AAg*CPL/6. AAV=CONDA*NUSS*DT/[AAS*DENSL) TDROPNS=(HV*(RDROPNS-RDROP)+TEMP*AAV+TDROP*AAT)/(AAT+AAV) ....................................................
C - - C a l c u l a t i o n of t h e d r o p l e t e v a p o r a t i o n r a t e . . . . . . . . . . . . . . . . . C .......................................................... 80 RDROP3=RDROP**3. RDROPNS3=RDROPNS**3. DROM=4.*PC*RDROP3*OENSL/3. DMASS=4.*PC*[RDBOP3-RDROPNS3}*DENSL/3. GASMASS=GASMASS÷DMASS VAPMASS=VAPMASS+DMASS C ............................................................ C Calculation of t h e n e w d r o p l e t v e l o c i t y . . . . . . . . . . . . . . . . . . . . . . . . . . c .............................................. CO=24.*(I.+REYN23/6.]/REYN IF(REYN.GE.I000.1GOTO 90 G O T O 10O 90 CD=0.424 100 AAW=DENSA/DENSL VDROPNS=VDROP-0.375*CD*AAW*RELVEL*RELVEL*DT/RDROPNS C .................................................................. C ..... Calculation of t h e n e w d r o p l e t m a s s a n d r a d i u s . . . . . . . . . . . . . . . . . . . C ................................................................... DEOMNS=DROM-DMASS TEMPC=TDNOPNS-272.15 CALL DENSLIQ(TEMPC,PRES,DENSL) R~ROPNS3=3.*DROMNS/14.*PC*DENSL) RDROPNS=RDROPNS3**(I./3.) RETURN END C
150 200
C C SUBROUTINE
SUBTEN(TEMPC,SURT)
COMMON/CRIT/PCR,TCR,VCR,ZCR,TB TT=TEMPC*9./5.+32. TT=TT+459.7 TR=TT/TCE SURT=(I.-TR}**I.232 SURT=0.6737*SURT/12.74 RETURN END C C SUBROUTINE DVISC(TEMPC,PRES,VISCL) TT=9.*TEMPC/5.+32. PP=PRES/6894.7591 DVSI=3.21248-3.81521E-2*TT+2.40018E-4*TT**2. DV02=-~.3371?E-7*TT**3.+I.4875E-9*TT**4. DV03:-I.05978E-12*TT**5. DV0=DV01+pv02+DV03 DVOE=0.0239+0.01638*DV0**0.278 ALOGV=PP*DVOE/1000. DVRED=I0.**ALOGV VISCL=DVRED*DV0/1000. RETURN END C
C
300 400
DENSLIQ(TEMPC,PRES,DENSL)
COMMON/CRIT/PCR,TCR,VCN,ZCR,TB COMMON/BBBI/BOO,BOI,BO2,BO3,BIO,BII,BI2,BI3,B20,B21
NUSS=(2.+0.6*REYNI2*PRIS)*BYB AAR=RDROP**2.-DT*DENSA*DI2*SHER*BY/DENSL IFIAAR.LE.0.0} GO TO 5~
C
SUBROUTINE LHVAP{TEMPC,HV) TEMPF=9.*TEMPC/5.+32. TR=(TEMPF+459.7)/II84.9 IF(TR.GE.0.4) G O T O 500 TRA=(725.2-TEMPF)/303.9 TRAD=TRA**S.38 HV=366095,*TRAD GOTO B00 POLYI=666.511-7457.69*TR+35956.7*TR**2. POLY2=-95009.2*TR**3.+I48446.*TR**4. POLY3=-I37210.*TR**5.+69506.4*TR**6.-14897.7*TR**7. HVBED=POLYI+POLY2+POLY3 HV=32113.6*EVRED RETURN END
C
SUBROUTINE INJECT(DPRES,DINJ,SURT,DENSA,DENSL,VISCA, +VISCL,UINJ,SMD) UINJ=0.29*(2.*DPRES/DENSL)**0.5 RE=UINJ*DENSL*DINJ/~ISCL WE=UINJ*UINJ*DINJ*DENSL/SURT VISRED=(DENSA*VISCL)/(DENSL*VISCA) DENRED=DENSL/DENSA REI=RE**0.12 REC=RE**0.25 WEI=I./WE**0.75 WEC=I./WE**S.32 VISREDI=VISRED**0.54 VISREDC=~ISRED**0.37 DENREDI=DENRED**0.18 PENREDC=I./DENRED**0.47 SMDI=4.12*REI*WEI*VISREDI*DENREDI*DINJ SMDC=0.38*REC*WEC*VISREDC*DENREDC*DINJ IF(SMDI.GE.SMDC) G O T O 300 SMD=SMDC S O T S 400 SMD=SMDI RETURN END
SUBROUTINE PROPAIR(TEMPC,CONDA,VISCA,CPA) TK=TEMPC+273.15 CONDA=3.17*I.E-4*TK**0.772 VISCA=0.612*I.E-6*TK**0.609 IF(TK.GE.700.) G O T O 150 CPA=573.*TK**0.097 SOTS 300 CPA=392.*TK**0.155 RETURN END
SUBROUTINE
CPLIQ(TEMPC,CPL)
COMMON/CRIT/PCR,TCR,VCR,ZCR,TB COMMON/CONS/A,B,C,D,AA,BB,CC,DD TT=TEMPC*9./5.+32. TT-TT+459.7 TT=TT/TB CPI-A+B*TT+C*TT**2.+D*TT**3. CP2=AA+BB*TT+CC*TT**2.+DD*TT**3. CPL=CPI+12.*CP2 CPL=(5.*I054.35071*CPL/(9.*0.453592371 RETURN END
APPENDIX B: PROGRAM OUTPUT DROPLET INITIAL CHARACTERISTICS *************************************************** Radius i~a]= 13.94 Mass [grl=.8142E-08 Temperature [CI= 7 6 . 8 5 Velocity la/s]= 92.11 DROPLET
CHARACTERISTICS
AT THE
END
OF
THE TIME
STEP
Radius [~m]= 14.35 Mass igrl-.8141E-0S Temperature [C]-172.54 Velocity ~m/sl= 73.98 PERCENTAGE
OF EVAPORATED
MASS
=
0.02%
-
STEP=
1
71
Technical Note A FORTRAN program for calculating the evaporation rates in diesel engine fuel sprays D.A. Kouremenos, C.D. Rakopoulos* & E.A. Yfantis National Technical University of Athens, Mechanical Engineering Department, Thermal Engineering Section, 42 Patission Street, Athens 106 82, Greece
A FORTRAN program based on a theoretical model for predicting the evaporation process of liquid fu.el sprays in diesel engines is presented. The injected liquid fuel is assumed to break up into droplets in a high pressure, high temperature environment, just as encountered in a diesel engine cylinder. The complete 'history' of the droplets is described by the present program, following their motion as well as the heat absorption and evaporation mechanisms. The theoretical model is based on Newton's Second Law of Motion and the equations expressing the heat and mass balance between droplets and surrounding gas. Accurate correlations are taken into account for the prediction of Sauter Mean Diameter and the thermodynamic and transport properties of fuel. Key words: droplet evaporation, fuel spray, diesel engine.
NOTATION
Ap Pressure drop across the nozzle orifice (N/m 2) 2 Thermal conductivity (W/m/K) v Kinematic viscosity (mZ/s) p Density (kg/m 3) a Surface tension (N/m)
c Discharge coefficient (--) co Drag coefficient (--) cp Specific heat capacity under constant pressure (J/kg/K) do Injector nozzle orifice diameter (m) Dr Mass diffusivity (m2/s) H Fuel latent heat of vaporization (J/kg) rn Mass (kg) M Molecular weight (kg/kmole) N Engine rotational speed (RPM) p Pressure (N/m 2) q Heat transported to droplet per unit surface (W/ m 2) r Droplet radius (m) S M D Sauter Mean Diameter (m) t Time (s) T Absolute temperature (K) v Velocity (m/s) Y Mass fraction of vaporized fuel (--) Y~ Mass fraction of vaporized fuel at droplet surface
Subscripts a Air f Fuel 1 Liquid fuel (droplet) v Vapor inj Injection conditions
INTRODUCTION The evaporation of liquid fuel sprays is a very important process of diesel engines operation, since it relates strongly to the subsequent combustion process and so to its power and efficiency. The injected liquid fuel, after being atomized into small droplets near the nozzle exit to form a spray, must first evaporate before it can be mixed with the surrounding air and then burnt.~ Three phenomena determine the history of a liquid fuel droplet, with temperature close to the ambient, when
(--) *Corresponding author. Advances in Engineering Software 0965-9978/92]$05.00 © 1992 Elsevier Science Publishers Ltd.
67
68
D.A. Kouremenos, C.D. Rakopoulos, E.A. Yfantis
injected into an air environment having typical end-ofcompression phase engine conditions: 2 (a) Deceleration of the droplet due to aerodynamic drag; (b) Heat transfer to the droplet from the surrounding air; (c) Mass transfer of vaporized fuel away from the droplet surface. For the accurate determination of the fuel vaporization rate, within a diesel fuel spray, the simultaneous solution of the relevant conservation equations for the droplet and the surrounding air is needed. Models, which treat explicitly the two-phase structure of the diesel spray, describing the spray behaviour in terms of the conservation equations in differential form, for mass, momentum and energy, have appeared in the literature. E1 Wakil et al. 3 have considered the changes in droplet and gas temperatures and the fuel vapor concentration in accordance with heat absorption and evaporation processes varying in time. In the model proposed by Hiroyasu and co-workers, 4 the injected fuel spray is divided into several elements. These elements entrain air, vaporize, and then react after they ignite. During injection and combustion, these elements expand and entrain air. After ignition, the evaporation, mixing, and reaction processes may continue. Bracco 5 presented a model for full-cone sprays, where the break up and development regions are the most relevant to diesel engine applications. The works mentioned above constitute only a part of published works concerning the prediction of the droplet evaporation mechanism. The present work gives a fast and accurate F O R T R A N subprogram which can be used as a part of a general computer program modelling the entire thermodynamic cycle of a diesel engine. 6-8 It is developed for use even on a PC and is related to previously published works by the present research group. Kouremenos et al. 6-8 have considered the following two distinct periods in the fuel droplet lifetime: (a) Sensible heating period from the initial droplet temperature to the saturation temperature; (b) Evaporation period with constant droplet temperature equal to the saturation temperature.
fuel droplet, is given by the expression: II Vinj = C . /2Ap ~/ P~
Kuo ~2 recommends c = 0.39, a value based on comparisons made between numerical predictions and experimental data. A dimensionless analysis of the measurement results on the Sauter Mean Diameter (SMD) leads to the following equation concerning the various liquid fuel properties and injection conditionsg: = 0.38 /~einjn 0.25 WeinjO.a2(vl/Va)O.37(pl/Pa)
(SMD),
0.47d0
(2a) = 4.12 _e~.jR -°'2 We~.i°W(v,/v.)°S4(p~/p,)°"~d o
(SMD)2
(2b) where subscripts 1 and 2 denote 'complete' and 'incomplete' sprays respectively, while the Reynolds and Weber numbers are respectively Rein j
Vinj do , VI
=
Wein j
__
V~njdop 1 O"
The value of the initial droplet diameter is the maximum from the two values given by expressions (2a) and (2b). The instantaneous velocity is determined by the solution of the conservation of momentum equation: dv _
~z
2 CDt2paVr2el
ml dt
(3)
where Vre~is the relative velocity between the droplet and the surrounding gas. The drag coefficient is given by the following expression: 5 f - ~ e ( l + O.166 Re 2/3) if Re < 1000 CD
= I
0.424
if Re > 1000
(4)
where the Reynolds number Re
-
2rVrel
Va
The rate of change of the droplet radius is: 5 dr
-
In this work evaporation begins immediately after fuel injection, when the droplet appears in the combustion chamber, which obviously represents more realistically the actual phenomenon. Furthermore accurate correlations are taken into account for the prediction of the Sauter Mean Diameter 9 and the thermodynamic and transport properties of the fuel) °
(1)
-
~
dt
Pa DI Sh In (1 + B)
(5)
p~ 2r
where the Sherwood and Schmidt numbers are respectively 2 + 0.6 Rel/2Sc ~/3, Sc = v~/Df while B
=
(Y~- Y)/(1
=
M~pv M~p~ + M , ( p - P v )
-
Y~), Y
=
Pr~/P.
and MODEL
DEVELOPMENT
The injection velocity, which is the initial velocity of the
r~
To obtain the expression for the droplet surface mass
69
A F O R T R A N program for calculating the evaporation rates in diesel engine fuel sprays
fraction Ys, it has been assumed that the droplet temperature is uniform and that the partial pressure of fuel vapor at the droplet surface equals the equilibrium vapor pressure. The energy balance equation states: 5 dT~ mlCpl --~
dr 4nr~p~H d-~ = 4nr2q
15
._j
co10 ___ rm <~ rY ~.-L..aJ ._J a_ 5 0 rY ~
(6)
2 a ( T - T,) In (1 + B) 2r Nu B
and the Nusselt and Prandtl numbers are respectively Nu
= 2 + 0.6 Re ~/2 Pr ~/3, Pr
-
,'
v
where q -
OLD METHOD NEW METHOD
SPEED=1500 RPM II~J.ANOLE~34 J.ANGLE 54 CA B;DC B DC
VaPaCpa
\ \
,
J'a
Equation (6) is a statement of the fact that the heat transported to the droplet increases its temperature (heat up) and supplies heat for vaporization. The thermodynamic and transport properties of the surrounding gas are taken from Ref. 13 at a temperature value corresponding to (T + T~)/3. The thermodynamic and transport properties of the fuel are taken from Ref. 10 at a temperature value corresponding to T~. As in most diesel engine cycle simulations, N-Dodecane is assumed to represent here the typical diesel fuel.
D E S C R I P T I O N OF T H E P R O G R A M - C A S E
STUDY-DISCUSSION The computer program is written in FORTRAN 77 language and its listing is given in Appendix A. The program requires as its input the engine rotational speed, the nozzle orifice diameter, the pressure drop across the nozzle orifice, the temperature, pressure and volume of the combustion chamber, as well as the initial droplet temperature, the air velocity and a factor which affects the determination of the time step for the computational procedure. After the estimation of the droplet initial characteristics, immediately after injection, the program predicts the complete 'history' of the fuel droplet as part of an iterative procedure. At the end of every time step, the droplet characteristics (radius, mass, temperature, velocity) are computed as well as the fraction of mass of the initial droplet which has been evaporated. A typical output listing is given in Appendix B. An experimental investigation has been conducted, at the authors' laboratory, on a single cylinder, naturally aspirated, four stroke, Ricardo E-6, indirect injection diesel engine fitted with a Comet MK.V swirl prechamber. 7 The engine bore is 76.2mm, the stroke 111.2 mm and the compression ratio 21.5. The pintle type injector nozzle is located in the swirl prechamber. The engine was operated at full load, a rotational speed of
I
0
I
I
10 20 5O 4O CRANK ANGLE FROM INJECTION
Fig. 1. Droplet radius variation with crank angle (from the moment of injection). 1500 RPM (revolutions per minute), and a static injection timing of 34 degrees CA BTDC (crank angle, before top dead centre). The program was applied considering the engine operating conditions mentioned above. In order to examine the differences between the two methods, i.e. the old one proposed in Ref. 7 and the new one presented here, both were applied for the same engine operating conditions. The comparison is effected by considering the predictions of the two procedures, concerning the variation of droplet radius and evaporated fuel mass with time. Figure 1 shows the variation with time (expressed in degrees CA) of the fuel droplet radius. The broken line
100,; v
/
_~ -75i,i D L~ rm ~,, 50 F-< rr 0 EL
OLD METHOD NEW METHOD
X 25 uJ
/
,,' LOAD= 1 O0 • SPEED=IS00 RPM ./ .... ./ INJ.ANGLE=34 CA BTDC
/
,;
I
140
I
I
I
160 180 200 220 CRANK ANGLE.(DEG)
I
240
Fig. 2. Fuel droplet mass evaporated (as a percentage) with crank angle.
70
D.A. Kouremenos, C.D. Rakopoulos, E.A. Yfantis
represents the result obtained with the old method, described in detail in Ref. 7, while the solid one represents the result obtained with the new one, as exposed in this work. Figure 2 shows the variation with time (expressed in degrees CA) of the evaporated fuel mass (expressed as a percentage of total). The old method results in an evaporation time lag at the beginning of the droplet 'history' inside the combustion chamber. This lag is caused by the lack of modelling of the evaporation at the very first period of the process. In the present method evaporation begins immediately after fuel injection, when the droplet appears in the combustion chamber. This more realistic representation of the actual phenomenon leads to a faster disappearance of the droplet, corresponding to about 3 degrees CA. This delay period could be proved very important when modelling, since it strongly influences the subsequent processes (mixing with air, ignition, reaction) and so the power and efficiency as well as the pollutants formation of the diesel engine.
REFERENCES 1. Heywood, J.B. Internal Combustion Engine Fundamentals, McGraw-Hill Book Co., New York, 1988. 2. Browne, K.R., Partridge, I.M. & Greeves, G. Fuel property effects on fuel/air mixing in an experimental diesel engine, SAE Paper 860223, 1986. 3. El Wakil, M.M., Myers, P.S. & Uyehara, O.A. Fuel vaporization and ignition lag diesel combustion, in Burning a Wide Range of Fuels in Diesel Engines, SAE Progress in Technology, 1967, 11, 30-44. 4. Hiroyasu, H., Kadota, T. & Arai, M. Development and use of a spray combustion modelling to predict diesel engine efficiency and pollutant emissions: Part 2. Computational procedure and parametric study, Bulletin of the JSME, 1983, 26(214), 576-583. 5. Bracco, F.V. Modelling of engine sprays, SAE Paper 850394, 1985. 6. Kouremenos, D.A., Rakopoulos, C.D. & Karvounis, E. Thermodynamic analysis of direct injection diesel engines by multi-zone modelling, ASME-WA Meeting, Boston MA., Dec. 13-18, AES, 1987, 3(3), 67-77. 7. Kouremenos, D.A., Rakopoulos, C.D. & Hountalas, D.T. Thermodynamic analysis of indirect injection diesel engines by two-zone modelling of combustion, Journal of Engineering for Gas Turbines and Power, Trans. ~1~ the ASME, 1990, 112(1), 138-149. 8. Kouremenos, D.A., Rakopoulos, C.D. & Hountalas, D.T. Computer simulation with experimental validation of the exhaust nitric oxide and soot emissions in divided chamber diesel engines, ASME-WA Meeting, San Francisco California, Dec. 10-15, AES, 1989, 10(1), 15-28. 9. Hiroyasu, H., Arai, M. & Tabata, M. Empirical equations for the Sauter Mean Diameter of a diesel spray, SAE Paper 890464, 1989. 10. Kouremenos, D.A., Rakopoulos, C.D. & Yfantis, E.A. A FORTRAN program for calculating the thermodynamic and transport properties of diesel fuel, Advances in Engineering Software, 1990, 12(4), 190-196. 11. Faeth, G.M. Evaporation and Combustion of Sprays, Progress in Energy and Combustion Science, 1983, 9, !-76.
12. Kuo, T.W. Evaluation of a phenomenological spray combustion model for two open-chamber diesel engines, SAE Paper 872057, 1987. 13. Pflaum, W. & Mollenhauer, K. Wi~rmeiibergang in der Verbrennungskraftmaschine, Springer-Verlag, Wien/New York, 1977.
A P P E N D I X A: P R O G R A M LISTING PROGRAM TESTING O P E N ( 3, F I L E - 'EV. n A T ', S T A T U S = 'O L D ' ) OPEN (4,FILE='EV.RES' ,STATUS='NEW' ] R E A D (3,*) R P M , X R E A D [3,*) D P E E S , D I N J R E A D [3,*) T E M P , P R E S , V O L R E A D 13,*) T D R O P R E A D (3,*) A I R V E L T E M P C - T D P ~ O P 273. 15 C A L L L H V A P (T E M P C , HV ] C A L L V A P R E S (T E M P O , P V ) C A L L D E N S L I Q (T E M P C , P R E S , D E N S L ) CALL SURTEN (TEMPC, SURT ) C A L L DVI SC ( T E M P C , P R E S , VI S C L } C A L L C P L I Q (T E M P C , C P L ) T E M P O = (T E M P + 2. * T D R O P ) / 3. - 2 7 3 . 1 5 C A L L P R O P A I R ( T E M P O , C O N D A , VI BCA, C P A ) RGAS-8314 ./28.96 DENSA=PRES/RGAS/TEMP PC=3.14159 O ............................................... C . . . . C a l c u l a t i o n of the droplet initial characteristics . . . . . . . . . . . C ............................................... C A L L I N J E C T ( D P R E S , D I N J, S U E T , D E N S A , D E N S L , VI SCA, +VISCL, UINJ, SMD) GASMASS =DENSA*VOL VAPMASS-0.0 VDROP=UINJ RDROP=SMD/2. D R O M 0 = 4 . * P C * R D E O P * * 3. * D E N S L / 3 . R E L V E L - A B S (A I R V E L - V D R O P ) I=l WRITE (4,14) 14 FORMAT ********************************************************** WRITE (4,16) 16 F O R M A T ( IX, ' D R O P L E T I N I T I A L C H A R A C T E R I S T I C S '] WRITE (4,14) W R I T E (4,3) R D R O P * I . E 6 3 FORMAT (iX,'Radius [ m]=',F6.2) W R I T E (4,4) D R O M O * I 0 0 O . 4 FORMAT (iX,'Mass [gr]=',ES.4) W R I T E (4,5) T D R O P - 2 7 3 . 1 5 5 FORMAT (iX,'Temperature [C]=',F6.2) W R I T E (4,6) V D R O P 6 FORMAT IiX,'Velocity [m/s]=',F6.2) C ............................................. C Calculation of the droplet evaporation history . . . . . . . . C .......................................... C A L L E V A P O R (R P M , R D R O P , R D R O P N S , V D R O P , V D R O P N S , D R O M 0 , D R O M N S , X, I, + D M A S S, T D R OP, T D R O P NS, G A S M A S S, V A P M A S S, T E M P , P R E S , VOL, A I R V E L , R E L V E L ) WRITE (4,14) WRITE (4,17) 17 FORMAT (IX,'DROPLET CHARACTERISTICS AT THE END OF THE TIME STEP'] WRITE (4,14) W R I T E (4,3) R D R O P N S * I . E 6 WRIT~ ~4,4 DR~3~NS*] 00O.
i[4
WEIT~ ~, V D R O F N $ WR:TE 4,14) P E R C E N T 1 0 0 * V A F M A S $ . DRd'M0 IF{ZERCENT.GT.99: G O T O 124 WE]TE :4,~ [ER:ENT,I F O R M A T ~ I X , ~ F E R C E N T A G E OF E V A P O R A T E D WRITE (4,14 I F ( R D R O P N S .LE.< . G O T O 124 RDROP=RDROPNS VDROP-VDROFNS T[:RC ? - T D R O P N S R E L V E L ;AZS ~V [ , ~ 0 ~ - A I R V E L ) CLOSE 3) CLOSE 4, 3T,3[ E}.S
MASS
',F6.2,'%
STEP
',12~
BLOC/ DATA COMMON/CRIT/RCR,TCR,VCR,ZCR,TB
COMMON/BBB1/BOO,BOI,BO2,RO3,BIO,BII,BI2,BI3,B20,B21 COMMON/BBB2/B22,B23,B30,R31,B32,B33,B40,R41,B42,B43 C(~MMON/CONS/A,B,C,D,AA,BB,CC,DD D A T A PCR,TCR,VCR,ZCR,TB/264.0,1184.9,0.0669,0.237,881.O/ DATA R00,B01,B02,B03,BI0,RII,BI2,BI3,B20,B21/ +i.b368, 1.g695,2.4638,-I.5841,-0.04615,0.21874,-0.36461, ÷0.25136,2.1138R-3, 8.0028E-3/ D A T A B22,B23,B~O,B31,B32,B33,B40,B41,B42,B43/ *11.8763E , I I . 3 8 0 5 E 3,-0.7845E-5,-8.2328E-5,14.805SE 5, , 9 . 5 6 7 2 E 5, 0 . 6 9 2 3 E 6 , 5 . 2 6 0 4 E - 6 , - 8 . 6 8 9 5 E 6 , 2 . 1 8 1 2 E 6/ DATA A,S,C,D,AA,BR,CC,DD/0.84167,-I.4704,1.67165,-0.59198, + 0.00382C, 0.000747,0.041126, 0.01395/ E~ D
~ U B R U U T ~ N E E V A P O R (RRM, R D R O P , R D R O P N S , V D R O P , V D R O P N S , D R O M 0 , D R O M N S , X, • i, D M A S S , T D R OP, T D R O P N S, G A S M A S S , V A P M A S S, T E M P , P R E S , V O L , AI R V E L , •R E L V E L ) ~ O M M O N ,OR I T / P C R , TCR, V C R , ZCR, TB PC-5.14159 DI2-I.E-6 ~{(3A~:83 I 4 . / 9 8 . 9 6 bEN:ZA=PRES/RGAS/TEMP M W A - 28 .96 MWF-170.3~ DT:I. (X*RPM~
A FORTRAN program for calculating the evaporation rates in diesel engine fuel sprays C ~
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C Cai~ula~ion o~ the t h e r m o d y n a m i c a n d t r a n s p o r t p r o p e r t i e s . . . . . . . C .... of the l i q u i d fuel a n d t h e s u r r o u n d i n g a i r . . . . . . . . . . . . . . . C. . . . . . . . . . . . . . . . . . . . TEMPC:TDROP 273.15 C A L L L H V A P ~ T E M P C , HV~ CALL VARRES~TEMPC,f~V) CALL DEN~L~,2(TEMPC,PRRg,DENSL) CALL CPLIQ(TEMPC,CPL~ TEMPC=(TEMP+2.*TDROP)/3.-273.15 CALL PROPAIR(TEMPC,CONDA,VISCA,CPA) C ............................................................... C ..... C a l c u l a t i o n of t h e d i m e n s i o n l e s s numbers .................... C ...................................................... REYN=2.*DENSA*RDROP*RELVEL/VISCA SC=VISCA/(DENSA*DI2) PE=VISCA*CPA/CONDA REYNI2=REYN**[I./2.) REYN23=REYN**(2./3.) SC13=SC**(I./3.) PRI3=PR**(I./3.) C ....................................................... C .... Calculation of t h e m a s s f r a c t i o n of v a p o u r i z e d f u e l . . . . . . . . . . . . . . C .................................................. Y=VAPMASS/GASMASS IF(TDROP.GE.600.) G O T O 20 G O T O 30 20 PV=PRES 30 YSUR=MWF/IMWF+MWA*[(PRES/PV]-I.)} IF(I.-YSUR.LT.I.E-6) G O T O 50 C ........................................................... C ..... C a l c u l a t i o n of t h e S h e r w o o d a n d N u s s e l t n u m b e r s . . . . . . . . . . . . . . . . . C .................................................................. BY=(YSUR-Y)/II. YSUR) BYB=ALOG(I.+BY)/BY SHER=(2.+0.6*REYNI2*SCI3]*BYB
C
C
500
600 C C
SUBROUTINE VAPRES(TEMPC,PV) COMMON/CRIT/PCR,TCR,VCR,ZCR,TB TT=TEMPC*9./5.*32. TT=TT+459.7 TT=TT/TCR PRLN0=5.92714-6.09648/TT-I.28862*ALOG{TT]+0.169347*TT**6. PRLNI=I5.2518-15.6875/TT-13.4721*ALOG(TT)+0.43577*TT**6. PRLN=PRLN0+0.5622*PRLNI PVR=EXP(PRLN) PV=PVR*PCR PV=PV*6894.7591 RETURN END C C SUBROUTINE
COMMON/BBB2/B22,B23,B30,B31,B32,B33,B40,B41,B42,B43 TT=TEMPC*9./5.*32. TT~TT+459.7 TT=TT/TCR PP=PRES/6894.7591 PP=PP/PCR DENSI=0.7526*999.024 A02=B00+BI0*PP+B20*PP**2.+B30*PP**3.+B40*PP**4. AI2=B01+BII*PP+B21*PP**2.+B31*PP**3.+B41*PP**4. A22=B02+BI2*PP+B22*PP**2.+B32*PP**3.+B42*PP**4. A32=B03+BI3*PP÷B23*PP**2.+B33*PP**3.+B43*PP**4. CC2=A02+AI2*TT+A22*TT**2.+A32*TT**3. CCI=I.I1342 DENSL=DENSI*CC2/CCI BETURN END
RDEOPNS=AAR**(I./2.) G O T O 60 50 AAR=O.0 RDROPNS=0.0 G O T O 80 C ..................................................... C ..... Calculation of the n e w d r o p l e t t e m p e r a t u r e . . . . . . . . . . . . . . . . . . . . . C ......................................................... 60 AAS=RDROP+RDROPNS
AAT=AAg*CPL/6. AAV=CONDA*NUSS*DT/[AAS*DENSL) TDROPNS=(HV*(RDROPNS-RDROP)+TEMP*AAV+TDROP*AAT)/(AAT+AAV) ....................................................
C - - C a l c u l a t i o n of t h e d r o p l e t e v a p o r a t i o n r a t e . . . . . . . . . . . . . . . . . C .......................................................... 80 RDROP3=RDROP**3. RDROPNS3=RDROPNS**3. DROM=4.*PC*RDROP3*OENSL/3. DMASS=4.*PC*[RDBOP3-RDROPNS3}*DENSL/3. GASMASS=GASMASS÷DMASS VAPMASS=VAPMASS+DMASS C ............................................................ C Calculation of t h e n e w d r o p l e t v e l o c i t y . . . . . . . . . . . . . . . . . . . . . . . . . . c .............................................. CO=24.*(I.+REYN23/6.]/REYN IF(REYN.GE.I000.1GOTO 90 G O T O 10O 90 CD=0.424 100 AAW=DENSA/DENSL VDROPNS=VDROP-0.375*CD*AAW*RELVEL*RELVEL*DT/RDROPNS C .................................................................. C ..... Calculation of t h e n e w d r o p l e t m a s s a n d r a d i u s . . . . . . . . . . . . . . . . . . . C ................................................................... DEOMNS=DROM-DMASS TEMPC=TDNOPNS-272.15 CALL DENSLIQ(TEMPC,PRES,DENSL) R~ROPNS3=3.*DROMNS/14.*PC*DENSL) RDROPNS=RDROPNS3**(I./3.) RETURN END C
150 200
C C SUBROUTINE
SUBTEN(TEMPC,SURT)
COMMON/CRIT/PCR,TCR,VCR,ZCR,TB TT=TEMPC*9./5.+32. TT=TT+459.7 TR=TT/TCE SURT=(I.-TR}**I.232 SURT=0.6737*SURT/12.74 RETURN END C C SUBROUTINE DVISC(TEMPC,PRES,VISCL) TT=9.*TEMPC/5.+32. PP=PRES/6894.7591 DVSI=3.21248-3.81521E-2*TT+2.40018E-4*TT**2. DV02=-~.3371?E-7*TT**3.+I.4875E-9*TT**4. DV03:-I.05978E-12*TT**5. DV0=DV01+pv02+DV03 DVOE=0.0239+0.01638*DV0**0.278 ALOGV=PP*DVOE/1000. DVRED=I0.**ALOGV VISCL=DVRED*DV0/1000. RETURN END C
C
300 400
DENSLIQ(TEMPC,PRES,DENSL)
COMMON/CRIT/PCR,TCR,VCN,ZCR,TB COMMON/BBBI/BOO,BOI,BO2,BO3,BIO,BII,BI2,BI3,B20,B21
NUSS=(2.+0.6*REYNI2*PRIS)*BYB AAR=RDROP**2.-DT*DENSA*DI2*SHER*BY/DENSL IFIAAR.LE.0.0} GO TO 5~
C
SUBROUTINE LHVAP{TEMPC,HV) TEMPF=9.*TEMPC/5.+32. TR=(TEMPF+459.7)/II84.9 IF(TR.GE.0.4) G O T O 500 TRA=(725.2-TEMPF)/303.9 TRAD=TRA**S.38 HV=366095,*TRAD GOTO B00 POLYI=666.511-7457.69*TR+35956.7*TR**2. POLY2=-95009.2*TR**3.+I48446.*TR**4. POLY3=-I37210.*TR**5.+69506.4*TR**6.-14897.7*TR**7. HVBED=POLYI+POLY2+POLY3 HV=32113.6*EVRED RETURN END
C
SUBROUTINE INJECT(DPRES,DINJ,SURT,DENSA,DENSL,VISCA, +VISCL,UINJ,SMD) UINJ=0.29*(2.*DPRES/DENSL)**0.5 RE=UINJ*DENSL*DINJ/~ISCL WE=UINJ*UINJ*DINJ*DENSL/SURT VISRED=(DENSA*VISCL)/(DENSL*VISCA) DENRED=DENSL/DENSA REI=RE**0.12 REC=RE**0.25 WEI=I./WE**0.75 WEC=I./WE**S.32 VISREDI=VISRED**0.54 VISREDC=~ISRED**0.37 DENREDI=DENRED**0.18 PENREDC=I./DENRED**0.47 SMDI=4.12*REI*WEI*VISREDI*DENREDI*DINJ SMDC=0.38*REC*WEC*VISREDC*DENREDC*DINJ IF(SMDI.GE.SMDC) G O T O 300 SMD=SMDC S O T S 400 SMD=SMDI RETURN END
SUBROUTINE PROPAIR(TEMPC,CONDA,VISCA,CPA) TK=TEMPC+273.15 CONDA=3.17*I.E-4*TK**0.772 VISCA=0.612*I.E-6*TK**0.609 IF(TK.GE.700.) G O T O 150 CPA=573.*TK**0.097 SOTS 300 CPA=392.*TK**0.155 RETURN END
SUBROUTINE
CPLIQ(TEMPC,CPL)
COMMON/CRIT/PCR,TCR,VCR,ZCR,TB COMMON/CONS/A,B,C,D,AA,BB,CC,DD TT=TEMPC*9./5.+32. TT-TT+459.7 TT=TT/TB CPI-A+B*TT+C*TT**2.+D*TT**3. CP2=AA+BB*TT+CC*TT**2.+DD*TT**3. CPL=CPI+12.*CP2 CPL=(5.*I054.35071*CPL/(9.*0.453592371 RETURN END
APPENDIX B: PROGRAM OUTPUT DROPLET INITIAL CHARACTERISTICS *************************************************** Radius i~a]= 13.94 Mass [grl=.8142E-08 Temperature [CI= 7 6 . 8 5 Velocity la/s]= 92.11 DROPLET
CHARACTERISTICS
AT THE
END
OF
THE TIME
STEP
Radius [~m]= 14.35 Mass igrl-.8141E-0S Temperature [C]-172.54 Velocity ~m/sl= 73.98 PERCENTAGE
OF EVAPORATED
MASS
=
0.02%
-
STEP=
1
71