1.9.1 Graphical representation
The sim plest form of representing the droplet size distribution is by m eans of a histogram . This m ay be in the form of a drop size histogram , w ith each bar giving the number of drops in that size band, or a volum e based histogram , show ing the volume of drops in each size band. W hen the tw o are com pared, the latter will be skewed to the right d u e to the w eighting of larger drops. The values of the individual bars are often expressed as a fraction of the total num ber (or volum e) m easured, such that the total area under the graph equals 100%.
If the num ber of drops m easured is large, and the size bands are m ade sufficiently narrow , a frequency d istrib u tio n curve m ay be draw n. The cum ulative d istrib u tio n m ay then be calculated, this being the plot of the integral of the frequency curve. From this can be deduced the percentage of the total drops or volum e of spray below (or above) a given drop diam eter.
G raphical representation shows the distribution in good detail, as it w as m easured, b u t it is difficult to com pare one spray w ith another. Some form of num erical description, such as a
m athem atical function which describes the distribution, or som e m ean dro p let diam eter and w idth of distribution, w ould be desirable to com pare sprays.
1.9.2 Mathematical distribution functions
These are m athem atical expressions w hose p aram eters can be o btained from a lim ited n u m b er of dro p size m easurem ents. Lefebvre [44] suggested th a t th ey should have the follow ing features;
1. Provide a satisfactory fit to drop size data.
2. Allow extrapolation to drop sizes outside the range of m easured values.
3. Allow easy calculation of m ean and representative dro p diam eters, an d other param eters of interest.
4. Provide a means of consolidating large am ounts of data.
5. Furnish some insight into basic mechanisms involved in atom ization.
N um erous functions have been proposed for various spray applications, based on probability or em pirical considerations. The latter often contain one or m ore constants w hich are obtained by fitting the function to the data available.
The one w hich is used most commonly is the Rosin-Rammler distribution, p u t forw ard in the 1920's by tw o G erm an researchers to describe the d istrib u tio n of particle size after the crushing of coal. This has been successfully applied to fuel sprays and m ay be expressed as;
-W'
1 - Q = e (1.13)
w here Q is the fraction of the total volum e contained in droplets of diam eter less than d, and X and q are constants for a given spray. Exponent q provides a m easure of the spread of dro p sizes, w ith a higher q value indicating a m ore uniform spray. For a m onosize spray, q w ould be infinite, w hilst for m ost fuel sprays the value of q lies betw een 1.5 and 3.0.
A lthough the Rosin-Ramm ler d istribution assum es an infinite range of d ro p sizes, the expression has the virtue of simplicity. Lefebvre [44] pointed out th at it also perm its data to be extrapolated into the range of very fine drops, w here m easurem ents are m ost difficult and least accurate. A good fit to experimental data w ould be required before it could be used w ith this confidence, though.
For well fitted data, the plot of drop diam eter versus In(l-Q)"^ should give a straight line. The g radient of this line gives the value q. X is a representative d iam eter (often referred to as a characteristic diam eter), given by the value of d for w hich 1-Q=e'^. Solution of this equation yields the result that Q=0.632, that is, X is the dro p diam eter such that 63.2% of the total liquid volum e is in drops of smaller diam eter.
O th er d istrib u tio n s include the N ukiyam a-T anasaw a. N orm al, Log-norm al an d Log- probability. M ugele and Evans [57] carried o u t a review of several of these (including the Rosin-Rammler) and also proposed an upper-lim it equation based on their analyses. Elkotb
[20] attem pted to m atch diesel injection data to several distributions, including tw o further ones, the Chi-square and Tanasawa-Tesima.
Basically, none is better than the other, and the selection of an appropriate distribution is alm ost entirely dep en d an t on finding one w hich fits experim ental d ata for the particular system u n der observation. This view is shared w ith Lefebvre [44].
1.9.3 M ean droplet diam eter
There are various m ean d roplet diam eters th at can be selected to characterize the droplet size distribution. They can all be defined from the general m ean diam eter equation;
^ab -
lN,d,
b(1.14)
W here i denotes the size range considered. Ni is the num ber of drops in size range i, and di is the m iddle diam eter of size range i.
Two examples of m ean diam eters used are: d3o, volum e (or mass) m ean diam eter (diam eter of a drop having volum e equal to the average volum e per droplet of the entire spray), and d^o, length (or num erical, arithmetic or linear) m ean diam eter.
There are at least five others, b u t the one m ost com m only used in the IC engine field is the Sauter M ean Diameter dg] (or Smd). This represents the droplet diam eter of a notional spray of uniform size, w ith the same overall volum e as the actual spray, and also the sam e overall surface area. In m athematical term s this is;
(1.15)
This has been selected in the past because the evaporation of sprays is of prim e interest, and the rate of evaporation is a function of surface area. Indeed, Smd is directly related to the Specific Surface Area (SSA) of the spray, such that;
SSA = ^
^32 (1.16)
1.9.4 Problem s w ith Sauter m ean diam eter
G reat care m ust be taken w hen using Smd, as serious problem s can arise w hen com paring values of Smd obtained by different m ethods, or w hen very sm all d ro p let diam eters are considered.
D ue to its definition, Smd is very sensitive to small d ro p let diam eters, as these have the largest surface areas. W hilst being an inherent feature of this m ean diam eter, this is also the root of the problem, which can lead to very m isleading results. An exam ple of tiie sensitivity of Smd to small droplets was given in the paper by M iller and N ightingale [56]. They found an investigation necessary d ue to the indication (by a laser diffraction instrum ent) of a small percentage of droplets in the sm allest b and size, w hen observing sprays from m ultipoint injectors. Here, they took a typical dro p let distribution from a m ultipoint fuel injector, and a d d e d 1% by volum e to each size band in turn, in order to dem onstrate the effect on Smd. W hilst for size bands above about 10 pm , the v alue of Sm d w as fairly insensitive to this change, for bands below 10 pm the calculated Smd fell dram atically.
Let us consider two different m easuring systems, one w ith range dow n to (say) 3 pm and the o th er w ith range d o w n to (say) 5 pm . Both system s are used to m easure the sam e (hypothetical) spray, in which 0.5% by volum e falls below 3 pm and 0.5% falls betw een 3 and 5 pm . The first system m ay successfully show this result, w hilst the second m ay sim ply see 1% by volum e below 5 pm. This m arginal difference will significantly affect the calculation of Smd, giving a distinctly low er value from the instrum ent w ith the 3 pm lower limit. Even if the percentage below 3 and 5 pm respectively is ignored in the Smd calculation, the 0.5% betw een these sizes m ay still cause a significant discrepancy. Thus, it is very im portant w hen specifying Smd that the range of the m easuring system is clearly stated. Single very large droplets m ay also cause a problem , so the u p per lim it of the m easurem ents is im portant as w e ll.
A nother point to consider is the very existence of the droplets in the sm allest size bands. Such low percentages, of drop sizes which are only usually detectable using some laser based optical m easuring system, cannot be verified by any other m eans, and could be contested as being spurious (due to background scattered light, etc). O ne m ight then consider a low cut-off droplet diam eter, below w hich the data is rejected if it is in sufficiently small am ounts.
Indeed, w hen further thought is p u t to the m atter, one m ight ask the question: Even if these sm all d ro p le ts do exist, w ith such sm all percentages, are they in fact of a n y practical significance to the m ixture by the time it enters the cylinder?
In an evaporating spray it seems reasonable to assum e th at sm all droplets will evaporate m ore rapidly. Therefore, the value of Smd m ay rise fairly quickly as these sm all droplets disap p ear, unless the distribution initially contains m arginally larger d roplets w hich m ay begin to evaporate, and take their place (not common in the case of injection systems at least). Thus, after some time has passed, the rem aining distribution m ay be dependent m ainly on the larger dro p lets in the original distribution, w hich will p ersist for longer (although they them selves will be evaporating, albeit m ore slowly).
It seems from the above that one m ight question the use of Smd as it is often used, that is, quoted as a single value to totally describe the spray, w ith no further qualification.
1.9.5 Representative diam eters and span
O ther diam eters sometimes quoted to describe the d ro p size distribution, are the so-called representative diam eters. The m ost com m on ones are d(v,0.1)/ d(V,0.5) and d(v,0.9)- These represen t the d ro p diam eter such th at 10%, 50% o r 90% respectively, of the total liquid volum e is in drops smaller them this value. d(V, 0.632) also used, tiiis being X in the Rosin- Ram mler distribution (63.2% of liquid volum e as droplets below this diam eter). This is also know n as the Mass M edian Diameter (MMD), n ot to be confused w ith m ass m ean diam eter dgo (N ote - If Rosin-Ramm ler distribution is used, all these diam eters are uniquely related to each other).
Some description of the dispersion of the dro p s can be very useful, and w hilst there are several w ays of doing this, probably the easiest is to use the Span or Relative Span Factor. This is found from the representative diam eters above, and is defined as;
Span.
‘^(v.o.^)-Vo.i)
^(V, 0.5) ( 1. 17)