It w ould seem that, for the sim ple case above, one could assum e that the d^ law w as valid from tim e zero w ith little error, thus providing a very sim ple relationship betw een droplet
size and tim e elapsed. The slope of the d^ line w ould be d ep en d en t on a com bination of the am bient tem perature and fuel properties. D eveloping this one step furtiier, if w e w ere to take a num ber of different initial droplet sizes, they w ould produce a fam ily of lines as show n in figure 2.3. If a sp ra y could be represented by such a set of discrete dro p let sizes, w ith a w eighting given to each size to indicate the proportion of the spray represented by that size, then a n evaporation history for the com plete spray could be calculated. It w as decided to take this concept as the basis of the required com puter program to evaporate the calculated d ro p -size d istrib u tio n s on a ro u tin e basis. The follow ing d escrib es the d e sig n a n d im plem entation of such a model.
2,4,1 Evaporation according to the d-squared law
The p ro g ram takes as its in p u t the b oundaries of n size classes into w hich the M alvern divides its spray m easurem ents, together w ith the fraction of the total spray w ithin each of these classes. The geom etric m ean of each size class is then calculated, such th at di,o is the geom etric m ean d iam eter of sizeband i a t tim e zero (to), w hen zero % of the spray has evaporated. Wi o is the fraction (by volum e) of drops in that sizeband at that time.
By definition, the total fraction at tim e zero.
(2.11) N ow a t tim e ty, w hen y% by volum e of the spray has evaporated according to the d^ law.
(2.12) w here di,y is the new geometric m ean diam eter of sizeband i , and K is the slope constant. Therefore, the fraction rem aining in each band at tim e ty is given by.
W | , y = W , _ „ X i,y
(2.13) The fraction of the total spray rem aining is tiien given by,
n W ,„ .,y = S W , _ ^
b u t w e also know that.
i = 1
=1 —---
100
(2.14)
(2.15)
Therefore w e can find the percentage of the spray that has evap o rated at any value of time. If plotted, the inform ation gained w ould provide a curve of percentage evaporated versus tim e, w ith the tim ebase solely d e p e n d en t on the g radient of d ^ /tim e line; ie, the value of K. The process described is show n in the first three stages of figure 2.4.
As alread y stated, K w ould be related to the fuel properties and the am bient conditions. W hilst one could pick a value for a particular case, results in term s of an arbitrary timescale could be m istaken for actual spray evaporation times. The preferred alternative is to convert the 'tim es' back to a form of equivalent diam eter, so that the results become independent of K.
2,4.2 E q u iv alen t diam eter, d ^ l y l
H aving obtained a relationship betw een percentage evaporated and tim e', it is possible to construct an equation sim ilar to (2.12), using the sam e value of K a n d a p air of equivalent diam eters. If w e take de,x a nd d^.y , equivalent diam eters w hich could represent the entire spray at tim es t^ and ty respectively, then.
2 2
d e, y — de, X ~ t y — t x) (2.16)
The fraction of the total spray rem aining at tim e ty is given in (2.14) above, and the fraction rem aining at t% can be found similarly. Thus, w e can also construct a n equation sim ilar to (2.13). This gives: ^ tot ,y W tot X X e,y V ^ e , x J T herefore, d . , y 1 / 3 w tot , X1 / 3
Com bining (2.16) and (2.18), gives:
1 - W t o t , y ' ' ' K ( t y - t x ) W 2/ 3 tot , X (2.17) (2.18) (2.19)
From (Z 15), l - W ^ ^ ^ y = — and s im ila rly , l " ^ t o t , x = ^ Therefore w e can form the equation:
/ K ( t y - t x ) V 1 - z w h ere. z =
.
2 / 3 1 - 100 (2.20) (2.21)E quation (2.20) gives us an equivalent diam eter, w hich d ep en d s up o n the tim e elapsed to evaporate a spray from %% to y%. The above procedure is show n in stages 4 and 5 of figure 2.4. Even though the equation contains the term K, the resulting diam eter is, in fact, independent of K as it will cancel out providing that the sam e value of K is used in the com putation of t*
and ty (ie in stages 2 and 4 of figure 2.4). Extending the notation used in (2.20), it can be stated th at de,x[y] is the equivalent diam eter of a m ono-size sp ray th a t w o u ld take the sam e X com p u ted tim e to evaporate from x% to y% as the real spray, assum ing a sim ilar constant d ^ /tim e gradient throughout.