Int. J. mech. 3ci. Pergamon Press. 1971. Vol. 13, pp. 803-812. Printed in Great Britain

DIESEL FUEL PROPERTIES FOR COMBUSTION CALCULATIONS

T. J. WILLIAMS University of Wales Institute of Science and Technology

(Received 16 Apr i l 1971, and in revised form 5 Ju ly 1971)

Summary--The composition of diesel fuel in relation to its thermodynamic properties is discussed. I t is indicated that there is less correlation between the fuel specification and the properties than in the case of petrols.

Maxwell's correlation methods and hydrocarbon mixture data are employed to deter- mine the properties of a light diesel off. Methods of approximation and reduction of this data are presented which give rapid methods of estimation of specific enthalpy and volume with allied properties without excessive computer storage requirements. The methods cover the whole range of stabes required, including the neighbourhood of the critical point.

A computer program for evaluation of the properties is described.

N O T A T I O N

P pressure V volume

R M universal gas constant T absolute temperature i ~ P V / R M T

T~ critical temperature Pc critical pressure Pa P/Pc C~ molar concentration of component i of mixture of all components except fuel CF molar concentration of fuel H molar specific enthalpy

Cp molar specific heat at constant pressure Cv molar specific heat at constant volume V I molar specific volume

I N T R O D U C T I O N

CURRENT research into the combus t ion processes in diesel engines requires a knowledge of the s ta te and proper t ies of the fuel droplets arising f rom the inject ion processes. These droplets no rma l ly occur ini t ial ly a t near-cr i t ical pressures and a t t e m p e r a t u r e s s om ewha t below the critical. E v a p o r a t i o n h is tory computa t ions carr ied out b y Wieber 1 indicate t h a t the droplets of ten reach s ta tes in the v ic in i ty of the critical point , as indeed would be expected. I n this region the specific volume, en t ha l py and allied proper t ies cannot be ob ta ined b y the simple procedures usual ly appl ied in regions well r e m o v e d f rom the critical point , i.e. the use of cons tant , p ressure- independent specific a n d l a t en t hea ts and cons tan t specific volumes. I t is thus necessary to ob ta in and store a more deta i led and comprehens ive descr ipt ion of the fuel proper t ies and to devise a means of comput ing values app rop r i a t e to pa r t i cu la r s tates .

803

804 T . J . WILLIAMS

This requirement leads to two difficulties which might be named "thermo- dynamic" and "descriptive". The thermodynamic difficulties arise from the fact tha t a mixture without a constant boiling point and hence saturation line is dealt with and concern is with pressure/temperature regions where the fuel cannot be treated as an ideal gas. Fig. ] illustrates the variation in the thermo-

o..

140

130

120

IiO

% "Non ideal" gas region

50

40

30

20 Ideal Critical pressure Critical pressure

gas region I NO vapour data in

,0~ this region ~"~-'~-------"t" /

200 400 600 800 1600

Temperature, K Critical

temperature

FIG. l . G a s - p h a s e v a r i a t i o n o v e r t h e r e l e v a n t p r e s s u r e - t e m p e r a t u r e f ield.

dynamic characteristics of the fuel over the field of pressures and temperatures encountered in diesel combustion problems. The descriptive difficulties arise because the usual specification characteristics, e.g. specific gravity, carbon: hydrogen ratio, 90 per cent volume temperature, cetane number, etc. cannot be correlated with the thermodynamic properties. This situation may be contrasted with tha t for the petrols and lighter distillation fractions where it is possible to assume tha t the fuel is a paraffin, calculate its average molecular weight from the carbon:hydrogen ratio and use the properties of the nearest pure paraffin (usually heptane or octane).

Diesel fuels however consist of mixtures of paraffins, cyclo paraffins and aromatics in varying proportions as shown in Table 1 condensed from data give by Sacharen7

The cetane number is not a reliable guide to the composition because of the practice of using additives to natural or cracked fractions having a low cetane

Diesel fuel properties for combustion calculations 805

number. We are thus not able to choose a representative pure substance but shall adopt instead the technique initiated nearly 40 years ago by Watson and Nelson 3 and culminating in the comprehensive data book on the properties of petroleum fractions by Maxwell. 4 By these methods the thermodynamic data

T A B L E 1. C O ~ Y P O S I ~ O ~ OF QAS OIL FRACTIONS F R O ~ U . S . A . ]FIELDS

Range of paraiTms Range of cycle-paraffins Range of aromatics % % %

0-48 80-40 13-40

a r e c o r r e l a t e d a g a i n s t v o l u m e a v e r a g e , w e i g h t a v e r a g e , m o l a l a v e r a g e a n d

m e a n a v e r a g e bo i l i ng p o i n t ( o b t a i n e d f r o m t h e c r u d e oi l a s s a y cu rve ) a s wel l a s

specif ic g r a v i t y a n d " c h a r a c t e r i s a t i o n f a c t o r " . Th i s l a t t e r f a c t o r is de f i ne d as

(bo i l ing p o i n t o f fraction)½ x specif ic g r a v i t y . T a b l e 2 c o n d e n s e d f r o m N e l s o n ' s 5

d a t a shows t h e v a r i a t i o n o f t h i s f a c t o r for a r a n g e o f d iese l fuels o f m i d bo i l i ng p o i n t 560 K w i t h o u t a d d i t i v e s .

TABLE 2. VARIATION OF CHARACTERIZATION FACTOR

Characterization factor Cetane number Base

12.9-12.15 72-59 Paraffin 12 .1-11 .5 58-42 Intermediate 11.45-10.5 40-20 Naphthenic

E V A L U A T I O N O F T H E P R O P E R T I E S O F A L I G H T D I E S E L F U E L

The following da ta were taken as representative of a light diesel fuel: Specific gravi ty 0.84, i.e. 37 ° API , Volume averaged boiling point 598 K, Slope of assay curve 4.6°C/~/o, Carbon/hydrogen rat io 0.639.

Applicat ion of the empirical correlations of Maxwell to this da t a gives the following prel iminary information:

Molal average boiling point 521 K, Mean average boiling point 551 K, Weight average boiling point 600 K, Characterization factor 12.0, Molecular weight 222. Pseudo-critical temperature 700 K, Pseudo-critical pressure 16.9 bar,

The la t ter two characteristics are used for correlations where critical temperature and pressure would be used for pure substances. These pseudo values were found by K a y e to give bet ter correlations for mixtures, where the t rue critical conditions are obviously a function of the highest fraction of the mixture.

Wi th this prel iminary information we may now continue to use Maxwell 's da ta to determine specific volume and enthalpy.

S P E C I F I C V O L U M E

Available liquid volume data a t the critical pressure stops short of the critical tempera- ture but is here extrapolated linearly to the known specific volume a t the critical point . Specific volumes above and below the critical pressure are obtained by linear interpolat ion

55

806 T . J . ~'VILLIA,MS

for p ressure fo rm t h e cr i t ical p ressure curve . 8I~/8P is t a k e n as a l inear fmlc t ion of t em- p e r a t u r e h a v i n g one of two forms d e p e n d i n g on w h e t h e r t h e t e m p e r a t u r e is a b o v e or be low 500 K. A " s a t u r a t i o n l ine" is a d d e d in a s o m e w h a t a r b i t r a r y fash ion b y d r awing a s t r a i g h t l ine f rom t he l imi t ava i l ab le v o l u m e d a t a a t one b a r p ressure to t he cri t ical po in t . P o i n t s to t h e left of t h i s l ine will be well a b o v e t he boi l ing po in t of m o s t of t he m i x t u r e a n d i t seems reasonab le to a s sume t h a t u n d e r m o s t c i r cums tances t h e v a p o u r phase will ex is t here . Specific l iqu id vo lume is o b t a i n e d b y l inear i n t e r p o l a t i o n in a t ab l e of v o l u m e vs. t e m p e r a t u r e a t t he cr i t ical pressure . Th i s va lue is " c o r r e c t e d " tbr pressure b y m e a n s of a l inear f unc t i on of pressure . This f unc t i on ha s one of two slopes as a n a p p r o x i m a t i o n to a s m o o t h curve . T he d i s c o n t i n u i t y occurs a t 500 K. Tab le 3 con ta ins

TABLE 3. VOLITME DATA

T e m p e r a t u r e Specific v o l u m e a t 16"9 b a r T (K) I,} (m~/kg mole)

250 0.2551 300 0-2656 35O O.2788 400 0"2945 450 0.3051 500 0.3314 550 0.3524 600 0.3866 650 0.4445 700 0.8442

Liquid: Specific g r a v i t y t a k e n as 0.84 a t 60°F 16.9 bar .

Specific v o l u m e a t p ressure P = VF + 0.0165 ( 1 6 . 9 - P ) * Z where Z --- 0 . 0 0 0 1 6 ( T - 2 5 0 ) , T < 500; Z = 0 . 0 0 0 4 ( T - 4 0 0 ) , T > 500.

d a t a for t h e cr i t ical p ressure curves a n d pressure cor rec t ion fac tors for l iquid. The a b o v e m e t h o d s are cons idered a~lequate for c o m b u s t i o n ca lcu la t ions where t h e a m o u n t of l iquid v o l u m e is n o t v e r y i m p o r t a n t . O t h e r types of ca lcu la t ions m a y w a r r a n t a more de ta i led speci f ica t ion of l iqu id vo lume.

D e v i a t i o n of t h e specific v o l u m e of t he v a p o u r f rom t h a t of a n ideal gas is cha rac t e r i zed b y / ~ def ined b y [ P V / R M T] , where P is pressure , V vo lume , T t e m p e r a t u r e a n d R M the u n i v e r s a l gas c o n s t a n t . Maxwel l ' s d a t a for /~ ha s b e e n r e p l o t t e d aga in s t t e m p e r a t u r e for a series of va lues of PR where PR is t h e r a t io of ac tua l p ressure to cr i t ical pressure . Below t h e cr i t ical p ressure i t was f o u n d possible to ut i l ize a single cu rve o f /~ b y p l o t t i n g i t aga in s t t h e p a r a m e t e r T-700 (PR) -°'25 (see Fig. 2). A b o v e t h e cr i t ical p ressure i t h a s b e e n necessa ry to use a n u m b e r of s t r a i g h t lines, t he pos i t ion a n d slope of wh ich are d e p e n d e n t on t h e pressure , a n d a s i n g l e / z - T curve as s h o w n in Fig. 3. The d o t t e d l ines a re a n enve lope s u r r o u n d i n g cu rves p l o t t e d f rom Maxwel l ' s da t a . The e r ror is l a rger t h a n t h a t of t h e sub-cr i t i ca l case b u t n o w h e r e g r ea t e r t h a n 5 pe r cent .

T h e m a n n e r in w h i c h t h e g r a p h s a n d express ions for v o l u m e cover t he pressure t e m p e r a t u r e field is s h o w n in Fig. 4. Maxwel l r e c o m m e n d s t h a t in m i x t u r e s of fuels a n d ideal gases t h e p ressure to be used in ca lcu la t ion of /~ is P ~]C, where C is m o l a r concen- t r a t i o n a n d P t h e t o t a l pressure . T he specific v o l u m e of t h e m i x t u r e is g iven b y

[RM T(( 1 -- ~Cf) - Z , C , ) /P ] ,

where C t is fuel concen t r a t i on . I n ca lcu la t ions i nvo lv ing a h o m o g e n e o u s m i x t u r e of fuel v a p o u r a n d a i r t h e m o l a r c o n c e n t r a t i o n would be a b o u t 0.01. A t a t o t a l p ressure of a b o u t 160 b a r a n d t e m p e r a t u r e of 1000 K / ~ would be in t he reg ion of 0.9; t h u s in v iew

Diesel fuel properties for combustion calculations 807

> 1 . - o_1c= Ill

1.0

O.E

0.6

0 4

0.2

FIG. 5.

T- O0 t 100 200 3'0° 4'00 Critical Reduced temperature, K temp

Ideal gas deviation for sub-critical pressures.

0 9

0,7

0 5

0.3

i " " - - - Approximation

r r ' ~ L i m i t s of actual data

Y I I I I I I 700 SO0 900 I000 I I 00 1200 1400

Temperature, K

FIG, 3. Ideal gas deviation for super-critical pressures.

of the small concentration of fuel the error in taking the mixture as an ideal gas would be small and becomes smaller at higher temperatures. In this case, the use of the volume data presented here would not be warranted. I n droplet combustion, on the other hand, calculated values of/z can be as low as 0.25.

In Fig. I the "saturation" line is tabulated at intervals of 50 K whilst the compressed liquid and superheated vapour regions are described by a small number of straight lines. The temperature-enthalpy diagram is divided into a number of regions in each of which the enth~lpy is obtained by linear interpolation for temperature and pressure. The values of OH/OT and ~H/~P required for interpolation in regions RI , R4 and R5 are quoted in Table 5 as S1-$6. The data have been converted from a datum of liquid fuel at 60°F to elements at 0 K by taking the heat of combustion of liquid fuel at 60°F as 10, 151, 873 kJ/kg mole and using enthalpy data for the elements of the J.A.N.A.F3 tables. All relevant enthalpy values are given in Tables 4 and 5.

808 T. ,1. WILLIAMS

5~

o

E #-

725

700 (critical)

650

600

o) "--.. -- .~ n C r e ~ .

/ ~ - ' ~ - ~ --

# / '- # / / Z ~ ' Region R5 7 ~ / / / ~

T,650, P~ 50/ . .~'//~T>725,

" / ¢°//T P~,

. . . . . . ----~ --21c--- ---/--/---- 650<T t / W / D , o > . ,

Region R2 / / ~ PNI

-all p, T.< 6 5"-'-0 - - / 7r'rRegio~-" n RWgo t % '-''z-~/''z-'r ~ liquid if P>I and -£<600

Liquid / Vapour / ~ o o / Values tabulated

~ at 50K ° intervals

--Constant press, lines

Enthalpy

FIe. 4. Diagrammatic representation of approximate temperature-enthalpy diagram.

TABLE 4. "SATuI~ATION ~ LINES

Temperature K

t t s

Liquid (kJ/kg mole)

H g

Vapour (kJ/kg mole)

250 300 350 400 450 500 550 600 650 700

337,469 358,010 381,246 404,483 431,335 461,284 492,783 528,413 567,140 608,450

415,637 430,302 450,957 469,030 492,267 518,602 544,420 572,304 600,705 608,450

Diesel fuel properties for combustion calculations 809

T A B L E 5. ~ ~SUPERHEAT" DATA

S1 797.4 kJ/kg mole K $2 738.0 kJ/kg mole K $3 796.5 kJ/kg mole K $4 635.7 kJ/kg mole K $5 509.6 kJ/kg mole bar $6 102.5 kJ/kg mole bar

Enthalpy at 16.9 bar, 725 K 643,167.3 kJ/kg mole

Molecular weight 222

Once specific volume and enthalpy have been obtained it is a simple matter to obtain internal energy as H - P V and specific heat as enthalpy or internal energy difference over an interval of 1 K.

Since the mathematical methods employed are of an elementary nature the pro- gramming of the arithmetic operations is a simple matter. Construction of the logic for choice of region requires some care, however, and appropriate logic diagrams for the liquid and vapour phase are given in Figs. 5 and 6 respectively. I t may be noted that these diagrams overlap in some cases and a choice of phase is made irrespective of that initially specified. Fig. 7 shows a logic diagram for computation of specific volume.

True

Folse

FIG. 5.

:otse

True

Logic for liquid-phase enthalpy.

810 T.d. WILLIAMS

True

25~ True

False

I 0 ~ True m_

False

---olse

True /

:olse

True

~ F o l s e P,

True

L False

True m-

False

FIG. 6. Logic for vapour-phase enthalpy.

Diesel fuel properties for combustion calculations

False

True ~ False _ r

811

0 0 ~ True :

False

:alse

:olse

True ~

True ~

FIG. 7. Logic for specific volume computation.

C O M P U T A T I O N A L M E T H O D S

The methods described in the previous pages have been embodied in an Algol pro- cedure. In systems where volume is an independent variable it may be necessary to employ the procedure iteratively {which would not be the case for an ideal gas); it is thus important that computation should be rapid and it is for this reason that linear interpolation has been extensively employed. The procedure is a compromise between speed and storage requirements and could have been made faster by storing more /~ - P - T data and replacing the expression [700 ((PR) -° '2s- 1)] by linear interpolation. However, in its present form, since the amount of data is small and nearly all stored externally to the procedure, modification to suit fuels of different characteristics should be a fairly simple matter. I t should be noted, however, that for fuels of different characteristics, e.g. residual otis, it will be necessary to generate new correlations by the methods outlined here. I t will also be necessary to insert new numerical values for the Pc, To and inter- polation limits.

A P P L I C A T I O N

Substitution of numerical values for the formal parameters: phase, pressure, tempera- turc and concentration sets the variable FUEL PROP to the numerical value of the desired property. This property is specified by the formal parameter PROP which takes

812 T.J . WILLIAMS

the value of 1-6 to indicate internal energy, enthalpy, specific heat at constant pressure, specific heat at constant vohtme, specific heat ratio CP/CV and specific volume respec- tively. Thus fi)r the following conditions: vapour phase, 100 K, 100 kN/m s and 0.5 molar concentration; the specific volume would be obtained by writing FUEL PROP (2, 1000, 100, 0.05, 6). The formal parameters may be treated as variables if it is necessary t~, determine their values by iterative application of tile procedure.

A listing of the program can be obtained by application to the author.

C O N C L U S I O N S

The report has demons t ra ted a simple method of describing the thermo- dynamic properties of a hydro-carbon mixture fuel for the whole of the pressure- t empera tu re -phase range likely to be encountered during diesel combust ion calculations.

The accuracy of representat ion of the da ta is perhaps somewhat less t han tha t justified by the correlation techniques for a part icular, correctly specified, fuel, The technique can, however, be made more sophist icated by increasing the stored da ta wi thout great ly increasing the computa t iona l time. In most practical cases where the fuel specification is known bu t vaguely, the repre- sentat ion m a y be considered too detailed and some increase in da ta storage intervals is justified. Care mus t be t aken in this case, to ensure tha t the "coarsening" of the da ta does no t obscure the qual i ta t ive aspects of the fuel properties, par t icular ly in the region of the critical point.

Acknowledgements--This work was executed whilst the writer was seconded to the Depart- ment of Mechanical Engineering at U.M.I.S.T., Manchester. The author is grateful to Professor W. Johnson for the facilities made available there and to Professor R. S. Benson for help and guidance in preparation of the manuscript.

R E F E R E N C E S

1. P. R. WIEBER, z t I A A Jour~zal l, 2764 (1963). 2. A. N. SACHAREN, Chemistry of Petroleum Hydrocarbons. Reinhold, New York (1954). 3. K. M. WATSON and E. F. NET,SO~ ~, Ind. Chem. Eng. 25, 880 (1933). 4. J. B. MAXWELL, Data Book on Hydrocarbons. Van Nostrand, New York (1950). 5. W. L. N~SLSON, Petroleum Refinery Engineering. McGraw-Hill, New York (1958). 6. W. B. KAY, Ind. Eng. Chem. 28, 1014 (1936). 7. J.A.N.A.F. Thermochemical Data Tables. Dew Chemical Corporation (1960).