ixing and Distribution of liquids in High-Velocity Air Streams *
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JOHN P. LONGWELL AND MALCOLM A. WEISS Standard Oil Development Co., Box I2 I , linden, N. 1.
T
HE mixing and distribution of sprayed liquids in high-
velocity streams are of importance in several practical applications. The products of some petroleum cracking operations, for example, give high-velocity gas streams at high temperature. It is often desirable to quench these streams as rapidly as possible, and to t h a t end water injection is used. The rapidity of cooling obviously depends on how quickly the water distributes throughout the hot stream. ilnother application occurs in jet engines, particularly ramjets, The performance of any flame-stabilizing element in the engine depends on the concentration of fuel in air a t the element. Fuel is usually injected some distance upstream of the flame stabilizers and thus local concentration a t the stabilizers depends on how the fuel has been distributed throughout the stream between the injection and stabilizer stations. In both these cases, the ultimate mixing process (on a molecular scale) is accomplished by molecular diffusion. However, molecular diffusion is so slow as to be practical over only short distances; injection points cannot be spaced as closely as these distances require. When matclrial (liquid or gaseous) is sprayed into a gas stream, one must rely on the initial spreading and turbulent flow to provide the coarse mixing, distributing the material well enough so that molecular diffusion can rapidly complete the process. I n this work, a study was made of initial spreading and turbulent mixing of hydrocarbon fuels injected into air streams. The ultimate goals were t o predict the downstream distributions of various injector configurations for various operating conditions; and, the converse, to predict the injector configuration required for a given distribution.
Theoretical Mixing and Diffusion
4
Eddy Diffusion of Gases. Several factors must be .considered in describing the mixing process in a high-velocity air stream. Transport of material within the stream is a result of eddy diffusion. (Molecular diffusion is of the order of 1% or less of eddy diffusion and thus can be neglected.) Eddy diffusion occurs because, in turbulent flow, small volumes of gas have a continuous random motion, which is superimposed on the time average velocity of the stream and acts t o spread the diffusing material throughout the stream. This random motion is characterized by an average velocity (dZ2, the intensity of turbulence) and an average length (1, the scale of turbulence). The nature of turbulence and these characterizing factors are described by Taylor (7), Dr.yden ( 2 ) , and others. It is sufficient for the purpose here to note that, with this picture of turbulence, the eddy diffusion process results in a transport rate analogous to other diffusional processes-that is,
Equation 1 states simply t h a t the rate of transport of mass, w , in the x direction is proportional t o the concentration gradient of that material and t o the area, A , for diffusion. The proportionality constant, -E, known as the eddy diffusivity, is usually given by E = Id12and is thus proportional. to the product of
scale and intensity of turbulence. Unfortunately, specific knowledge of the turbulence t h a t exists in any given flow system is usually meager. Therefore, one must measure diffusivities directly rather than calculate them from the turbulence parameters. Eddy Diffusion of Drops. Although the definition of E above is accurate for transport of a gas ( d i o s e density is equal to or close to that of the turbulent stream), i t is not adequate for describing the transport of a drop. I n practice, liquid injection results in the formation of drops which may or may not evaporate. If the drops do not evaporate, they cannot be carried along with the random motion of the eddies as easily as a gas can. Drop inertia interferes. The diffusion rate, through E, may thus be reduced. A very rough illustration of this effect can be made by assuming that the velocity fluctuations in turbulent flow are sinusoidal and that Stokes' law applies to the drag on the drop. The equation of motion of the drop is then:
where Z is the drop displacement from its mean position and uo is the peak velocity of the turbulent air fluctuation relative t o its 3Tpdo time average velocity. Letting b = -, a solution to Equam
tion 2 is:
B y differentiating Equation 3 the maximum amplitude, Z,, of Z can be found. Dividing 2, by Zo,the maximum amplitude of the turbulent air fluctuation (where ZO = U O J W ) , gives:
(4) However, the frequency of motion of the drop is the same as that of the gas and therefore its velocity is reduced in proportion t o its amplitude. Because the eddy diffusion coefficient is proportional to the product of velocity and amplitude,
(5) where E/Eo is the ratio of diffusivity of the drop to t h a t of a gas which could follow the motion of the air stream exactly. To illustrate this effect, assume a typical drop (45 microns in diameter) of kerosene in a 300-feet-per-second air stream a t atmospheric pressure. The frequency, w , was estimated a t 300 radians per second for a 6-inch diameter duct with fully developed pipe turbulence. For these conditions, Equation 5 predicts E/Eo = 0.35. This reduction in diffusitivity for a liquid drop relative t o a vapor is of practical importance and is supported by experimental data cited in another section of this paper. Initial Spending. Any final distribution depends not only on the diffusion process within the stream b u t on the manner in which the liquid is initially injected into the stream. The distribution given by a high-pressure wide-angle spray nozzle will be different from t h a t given b y a simple low-velocity injection
667
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INDUSTRIAL AND ENGINEERING CHEMISTRY
tube. Assuming that the injection device does not greatly disturb the flow, the diffusion process and the eddy diffusivity remain the same, so that the differences in distribution must result from the initial conditions. It has been found that for many of the situations encountered in practice, much more of the mixing results from the initial spreading than from eddy transport. Initial spreading must be accounted for as a boundary condition in solutions of the differential equation of diffusion. POINT-SOURCE SOLUTIONS OF DIFFUSIOX EQCATION. Equation l, the basic equation for eddy diffusion, can be solved for sweral boundary conditions to give time average fluid concentrations a t any point. This information is useful for many purposes, but i t should be kept in mind that nothing is specified as to instantaneous variations of fluid concentration. Information about variations with time may be required for precisely evaluating the mixing for some combustion purposes. However, time average roncentrations have heen found useful in most applications.
Figure I.
Geometry of Ring Source
A solution to Equation 1 can be obtained by assuming a duct large enough so that wall effects are negligible. Let an air stream flow through the duct with a velocity, u, which is constant a t all points, If we assume steady state, a constant, E, and negligible mixing in the mean flow direction, z,compared to mixing perpendicul:ir, r , to the flow direction, then,
Vol. 45, No. 3
by a material balance in cylindrical coordinates.
In Equatioii 6,
f is the fuel-air ratio, the local mass ratio of fuel to air. As the injected liquids in this work were always one of two hydrocarbon fuels, the terms “fuel” and “injection liquid” are used interchangeably in what follows. The termf differs slightly from a true fuel concentration. However, it is sufficiently accurate (over the range of interest) for this work and, because of its convenience, is used throughout this report. A particular integral of Equation 6 can be found using the following boundary condition, a material balance across :illy section :
The result is then -7,KZ
Equation 8 describes t,he fuel distribution downstream of a point source. R is the radial distance from the survey point ( a t which the local fuel-air ratio is desired) to the axis of the injection point, and z is the distance, measured along the injection point axis, from the injection point to the survey point. A point-source equation which differs slightly from Equation 8 has been proposed by other investigators-e.g., ( 9 ) . However, for the range of conditions of interest here, the variation betwepn these equations is usually of the order of a few tenths of 1%; although Equat,ion 8 is riot exact, its accuracy is more than adequate here and, in the opinion of the authors, i t is considerably simpler to use than the exact equation of ( 9 ) . SOXPOIKT SOURCE SOLCTIONS.I t was noted above that a point source is often a very poor approximation to an actual fuel injection device. However, any nonpoint source may ho pictured as consisting of an infinite number of point sources properly arranged. Suppose then t,hat a ring source exists, defining a ring source as a circular line source of radius Ro; liquid is injected uniformly
R/ R.
Figure 2.
Dimensionless Ring Ccurce Function, 4
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1953 0.4
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Figure 3.
I
I
+ R,2 - 2RiRo
COS
S = R8 [l
I
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I
I
I
I
Dimensionless Disk Source Function, $
@roundthe ring. This situation can be handled by first assuming t h a t N equal small arcs exist, making up a circle of radius Ro. If W, is the total fuel rate t o the circle, the rate to each prc is clearly W,Ap/2r, where A p is the angle subtended by each Frc. From Figure 1 it can be seen that the distance R varies @roundthe circle but can be given by
R2 = R?
I
669
+ (2)' - 2 cos @]
(9)
As N -+ m , each arc approaches a point source and the distribuiion is given by the limit of the following sum:
Specific application of this equation, and the point-source and disk-source equations, is discussed a t a later point. A procedure exactly analogous to deriving the ring-source equation from the point-source equation can be used t o obtain a disk-source equation from the ring-source equation. Assume a disk source to exist, defining a disk source as the entire area bounded by a circle of radius M ; liquid is injected uniformly over the disk area. Consider the area t o consist of N annular rings, each of average radius Ro. Sum up the contribution of each ring as N + m. This procedure gives the integral
Unfortunately, this integral probably cannot be evaluated in closed form with familiar-Le., tabulated-functions. However, Equation 14 can be written as This limit is the integral where $, a function of the dimensionless groups P (=uM2/2 E x ) and R / M , is plotted in Figure 3 from computer-calculated tables. Equation 15 then is a simple expression for the fuel distribution downstream of a disk source.
gYhose solution is, letting uR?/4Ex = K ,
Although the disk and ring sources are sufficient for this work, i t is evident that the above techniques are applicable t o sources of any plane geometry.
Experimental Procedures
lo is a modified Bessel function of the first kind, zero order. T h e subscript of RI has been dropped for convenience betw'een Equations 11 and 12; R is now defined as the radial distance from the axis of any injection source to the survey point. Equation 12 may be simplified to
where 6, a function of only the dimensionless groups K and @/Ro,is plotted in Figure 2. A simple expression, Equation 13, is thus available for the fuel distribution downstream of a ring pource.
Mixture Sampling Methods. The performance of a system is influenced not only by distribution, the local mass ratio of injected fluid to air, but by the condition of the fluid-i.e., liquid or gaseous. Two different experimental techniques have thus been used in this work, one for measuring the distribution of liquid fuels (as droplets) and one for measuring fuels which are chiefly gaseous at the measuring station. The former technique, of course, corresponds to injection of a nonvolatile liquid and the latter to injection of a very volatile liquid. The distribution for any intermediate (partially vaporizing) liquid will be somewhere between the volatile and nonvolatile extremes; the intermediate distribution can thus be a t least bracketed b y the two
INDUSTRIAL AND ENGINEERING CHEMISTRY
610
techniques used here. I n addition, the nonvolatile measuring technique, responsive only to liquid droplets, offers a method for determining the degree of vaporization of fuels of intermediate volatility. This is done simply by comparing the mass of collected fuel with the mass of injected fuel. Both liquid and gaseous sampling methods involve removal of stream samples through total-head probes arranged in a rake. I n both cases a careful sampling technique was required to obtain truly representative samples of the stream. Liquid “Spillover” Technique. The collection of liquid droplets a t high velocity results from the fact t h a t the momentum of the droplets causes some of them to strike an obstruction in the stream instead of being deflected around i t with the gas stream lines. If the obstruction is a hollow probe, the droplets can be withdrawn and measured. The percentage of drops flowing through the upstream projected area of the probe that enter the probe is the collection efficiency. As shown by Langmuir and Blodgett (Q), this fraction increases as droplet size increases, droplet velocity increases, and probe size deceases. For example, the collection efficiency of a probe 3/16 inch in diameter is probably greater than 9Syo for all droplets larger than 10 microns and velocities greater than 100 feet per second.
-
quantity of air was permitted t o flow through the sample tubes, the probe did not pick up sufficient liquid; this may be the result of a liquid meniscus forming and blocking the tube. The gas flow was accomplished by bleeding air out of the collection bottle through a small length of hypodermic tubing (see Figure 4). A tube of size (the order of 0.04 inch in inside diameter) sufficient to bleed 1 or 2% of the intercepted air flow was satisfactory. Too much bleed gas carried away fuel in a fine mist unless claborate precautions were taken. Extensive tcsts with this sampling method gave results believed accurate within 5%
Table
1.
Properties of Hydrocarbon so. 1
Specific gravity (60’ F./60’ F.) Composition Paraffins, liq. vol. 70 Iiauhthenes Aromatics AST31 distillation Initial boiling point, F. liq. vol. % over a t , F.
l4
0
4 . f DETAIL
OF SAMPLING T I P
n
Solvent Naphtha 0.707
Diesel Fuel 0.840 2 71 27
66 33
0 186 149 156 160
228 262
346 370 412 442 465 485 496 510 524 539 556 583 612 630
HYPODERM I C TUB1 NG
/
dlf
BLEED
RUBBER STOPPER
LIQUID COLLECTING BOTTLE
Figure 4.
Fuels
... ... ... 178 ... ... iifi
DIRECTION OF GAS FLOW
P
Vol. 45, No. 3
Probe Tube and Liquid Collection Bottle
The individual probes are total-head tubes grouped in a bank and extending forward of a streamlined probe holder. The probe tips should extend forward into a region in which the flow lines are not disturbed by the holder. I n the work here, banks of from 9 to 15 tubes were used, mounted on radii increments t h a t gave approximately equal increments of radius squared. Figure 4 is a diagram of a single tube and its collection bottle. Figure 5 illustrates a 15-tube bank, with collection bottles attached, mounted in a spool section of 10-inch pipe. The probe tubes form gradual 90” bends within the probe holder and emerge through the bottom wall of the spool section for bottle connection. The probe tubes were of various diameters; steel tubing 0.10 inch in inside diameter, or smaller hypodermic tubing, was satisfactory. I n operation, droplets enter the tube tip (ground to form a sharp conical lip) and flow by gravity down through the tubes into the collection bottles. Total collection is determined by weighing the bottles before and after a timed run. From the total fuel in each bottle, a n average fuel-air ratio at the probe tip is calculated (as the mass of air in the upstream projected area of the probe is known). A fuel-air ratio curve across the entire duct can then be plotted, and on integrating, the total fuel passing the probe station calculated. The material balance check compares this fuel with the fuel actually injected. Material balances of this kind were made on cold Diesel fuel injection a t an air temperature of 200 O F. Under these conditions vaporization of the fuel was negligible (see Table I for physical properties of this fuel). Experience showed t h a t unless a small
Vapor Technique. The second method used in this work for measuring fuel concentrat,ion is applicable to completely vaporized fuels, or t o partially vaporized fuels which can be completelv vaporized before a concentration measurement is made (in a thermal conductivity meter). The technique thus differs, in the first instance, in prohibiting spillover a t the probe tip of either air or fuel. The total stream in the projected probe area must be collected. This requirement occurs because in most systems, using practical fuels and operating conditions, there is usually some small fraction of fuel in droplet form. Even with volatile fuels, thcse residual drops result from the highest-boiling components of the fuel and/or very large initial (on injection) drops. When the velocity of air through the probe is small, some of the air in the probe path necessarily spills over and around the tip. However, the residue of liquid drops enters the probe (because of drop momentum) and, when evaporated into the collected air, gives erroneously rich readings on the conductivity meter. Therefore, probe spillover must be kept a t a minimum unless it is positive t h a t the fuel to be sampled is completely evaporated. Spillover can be minimized by withdrawing sample through the probe at a velocity as close t o the main stream velocity as is practicable. This behavior is illustrated by the following data, in which material balances were calculated as described in the preceding section: Sample Velocity i n Probe T i p , Feet/Sec.
hIaterial Balance,
yo of Injected Fuel
50 225 110 100 100 160 Operating conditions Fuel. No. 1 solvent naphtha Fuel injection rate. 0.102 pound per second Air rate. 8 27 pounds per second Air temp. 300’ F. Air velocity. 270 feet, per second Pressure. Atmospheric
These data were taken in 10-inch pipe with the probe located 35 inches downstream of a central point source of injection of cold fuel. The effect of sample velocity (measured by a wet-test meter) on the material balance is clear; 100% is reached when
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INDUSTRIAL AND ENGINEERING CHEMISTRY
the ratio of sample t o stream velocity is about 0.5. However, a ratio of 1 is usually desirable. Velocity through the probe was controlled by varying the vacuum to which the sample was withdrawn. A proper material balance can also be obtained by assuring evaporation, although this is not always practicable. I n tests under conditions otherwise similar to those above, naphtha was injected at 0.20 pound per second into a n air stream and samples were withdrawn through the probe at 45 feet per second. The results were: Air Temperature, F. 300 400 500
Iraterial B a h n r e , oi Injecrcd Fual
5;
190
125 100
i
Again, the effect of complete vaporization (by raising air temperature) on the material balance is clear. Once the sample is withdrawn through the probe a t the proper velocity, i t must be led t o the thermal conductivity meter. Tubing may be of any convenient material with an inside diamto 3 / 1 6 inch. Precautions must be taken t o eter of about assure t h a t no fuel condenses in the lines, and t h a t residual liquid drops are evaporated and thoroughly mixed with the sample stream. Liquid drops in the meter cell will cause erratic readings. It is advised t h a t periodic calibration checks of the meter be made with the specific fuel under test; in addition, humidity must be corrected for. Meter readings can be converted directly in terms of fuel-air ratio. The mechanics of the probe design are the same for the vapor technique as for liquid spillover sampling. Throughout this work i t was found convenient to build the probes in multiple form as fixed-position rakes. However, if designed according to the same principles, a single traversing probe tube may be more convenient in some applications. Equipment and Test Procedure. The experimental work described here was performed at the Esso Laboratories in straight sections of 6- and 10-inch pipe. A compressor and furnace are available for supplying up to 11 pounds of air per second at pressures from atmospheric t o 100 pounds per square inch gage and temperatures of 150' t o 500' F. I n a smaller facility, equipped with a steam ejector, subatmospheric pressures may be obtained with mass flows up t o about 1 pound of air per second ( a t '/s atmosphere).
67 1
Lfmited cell length made it impossible t o have a horizontal pipe run long enough t o assure normal pipe turbulence. I n order t o damp out the effects of an elbow immediately upstream of the pipe run, a perforated plate was installed in the line. A few velocity traverses across the pipe a t a typical sampling station showed the velocity distribution to be very close t o t h a t of Nikuradse (6) for equilibrium turbulence. The fuel injection station was located about 15 pipe diameters downstream of the perforated plate for both sizes of pipe used. The fuel injector was mounted in a ring which clamped between two pipe flanges at the injection station, A typical injector is illustrated in Figure 6. The collars on the cross tube prevent injected fuel from running along the tube. Obviously such an injector can be used for costream or contrastream injection. Downstream of the injector a probe rake of the type described above was mounted, By inserting pipe sections between the injector and probe stations, t8hediffusion distance could be varied from 3 inches t o 6 feet.
Figure 6.
Typical Fuel Injector Mounted in Pipe Section
When nonvolatile fuels were sampled, all samples were collected simultaneously in the rake tube bottles. The procedure during running was as follows: With the bottles removed, fuel and air rates were set and allowed t o come to equilibrium; the fuel was then shut off and the tubes were allowed t o drain for 1 minute before the bottles were attached. After attachment, the fuel was turned on for a measured time interval. At the end of the interval the fuel was turned off again and the tubes were allowed t o drain into the bottles for 1 minute before being removed in the same order as used for attachment. This procedure was found to give consistent and reproducible results. When vapors were sampled, the individual rake tubes were connected to the thermal conductivity meter one a t a time. ( I n some runs, several meters were used and thus several tubes sampled simultaneously.) The procedure here was t o check the meter zero with normal air flow but no fuel injection. With fuel injection, the air-fuel mixture was admitted t o the cell, allowing sample flow for about 3 to 5 seconds. A reading was taken after galvanometer stabilization. On completion of the run, the zero was rechecked with no fuel injection and corrections for a n y zero drift were noted. Experimental Results
Figure 5.
Fifteen-Tube Probe Bank with Collection Bottles Attached
Single Injector Point Source. A fuel injection source consisting of a small tube through which fuel is ejected at low velocity can be logically considered t o act as a point source. B y inspection of Equation 8, one finds t h a t a point source gives a fuel distribution which will plot on semilog paper as a straight line; this plot requires f as the ordinate (logarithmic scale) and the square of the diffusion radius, R2, as the abscissa. The slope of such a straight line equals, from Equation 8, u/4 Ex and
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
672
the intercept equals Wf/4rWdEr. Thus, by measuring thc'distribution downstream of an injection source one can determine if the source behaves as a point. If i t does, then the eddy diffusivity can also be directly determined from the slope or intercept or both. The results of a typical run with contrastream low-velocity injection of Diesel oil through a small tube are illustrated in Figure 7 .
,004
,001
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1
h
Figure 7.
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\I 12
- INCHES'
Point Source injection
of Diesel Fuel
For this run, in a 10-inch pipe, air velocity was 290 feet per second a t 180" F. and atmospheric pressure. The distribution was measured 34 inches downstream of the source, a tube 7/82 inch in inside diameter through which fuel was injected a t 7.5 feet per second. The data are very well represented by a straight line. In this run a8 in others cited here, the data are corrected (usually less than 10%) for material balance and for spray asymmetry (usually less than 0.10 inch); the distribution for one half the duct is superimposed on the other half, so t h a t a single average curve can be drawn. From the slope in Figure 7 , E calculates to 0.46 square foot per second, and from the intercept, 0.47 square foot per second. Thus it is shon-n t h a t this injector behaves as a point source and that the diffusivities measured from slope and intercept are consistent. By means of tests of the type above, the effect on E of several variables was investigated briefly. I n some cases, the sources deviated slightly from true point behavior. For these runs, appropriate disk-source corrections were made as described below. The influence of duct size on E was determined by comparing the distributions obtained in 6- and 10-inch pipes under otherwise identical conditions. Solvent naphtha was injected, a t three rates, through a tube "16 inch in outside diameter by ? / 3 2 inch in inside diameter, 1 inch long. Injection was contrastream t o an air velocity of 286 feet per second a t atmospheric pressure and 300" F. At a diffusion distance of 35 inches, the following results were obtained: Fuel Injection Rate, Lb./Sec.
Diffusivity, Sq.Foot/Bec. 6-inch pipe 10-inch pipe
0.102
0.55
0.160
0.55
0,202
0.55
turbulence in the center of a pipe is, according t o Taylor ( 8 ) , approximately proportional to pipe diameter. Recalling that the diffusivity should be proportional to the scale of turbulence, one tyould expect the diffusivity t o vary directly with pipe diameter. However, the scale of turbulence is very sensitive t o duct length, for short ducts, and to the nature of the flo-w preceding the section of interest. Therefore, one cannot automatically assume a direct change of diffusivity with duct diameter. This is borne out by the data presented here, in which only a small increase was observed. Some tests under similar conditions in a duct approximately 2 feet in diameter resulted in diffusivities no larger than those observed in the 10-inch pipe. There are t v o factors responsible for the small effect of duct diameter on diffusivity. (1) The scale of turbulence in these tests probably depended primarily on the upstream perforated plate and not on duct size (as there was insufficient length for normal pipe turbulence to develop). (2) At these high velocities and short diffusion distances, there is enough time for only about one "cycle" of the turbulent air fluctuation. In some cases, time was insufficient for even one cycle. Thus, one would expect scale to be less important than turbulent intensity in determining eddy diffusivity.
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SURVEY RADIUS'
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0.58 0 58
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Diffusivity in the 10-inch pipe is only slightly greater than diffusivity in the 6-inch pipe. Data of Towle and Sherwood (9) on gaseous injection in long 6- and 12-inch ducts showed diffusivity to be proportional to duct size, as contrasted to the results here. Confirming their data is the fact that the scale of normal
AIR VELOCITY
Figure 8.
- q - FTJSEC.
Diffusion Parameter, Elu, for Two Liquids
Data of Little and Wilbur ( 5 ) show that the intensity of turbulence (in normal pipe flow) is approximately a constant fraction of the mean flow velocity, u. Because the diffusivity, E, is proportional to the turbulent intensity, one would expect 13 to vary directly with the flow velocity-Le., E / u should be a constant. The ratio, E / u , the "diffusion parameter," was determined a t several air velocities for point injection of both naphtha and Diesel oil. In these tests, a t atmospheric pressure, distributions were measured in a 6-inch pipe 17 inchcs downstream of the injector. The results, in Figure 8, show that the diffusion parameters approach each other a t low velocities but that the parameter for Diesel oil becomes much lower a t high velocities. Recalling that the Diesel oil does not evaporate a t all, the data show the decrease in diffusivitg of liquid drops relative to a vapor (most of the naphtha was evaporated under the test conditions) as ve1ocit.y increases. This is in accordance with Equation 5 , which predicts such a falling off as the frequency of turbulent fluctuation (proportional to average velocity) increases. Some falling off of the naphtha parameter is also evident. This results from the fact that the napht,ha, is only chiefly, not completely, evaporated. Thus the diffusion parameter is not constant with velocity because of the presence of some liquid drops. The data of Toivle and Sherwood (9) on gaseous injection (of carbon dioxide in air) show some decrease of the diffusion parameter with increasing velocity. In the 6-inch duct, E / ! / dropped from about 0.0011 to 0.00056 foot when vdocity \ m s
INDUSTRIAL AND ENGINEERING CHEMISTRY
March 1953 0.2
0.1
.06 .04
Ep4
.02 DISK SOURCE
P
?
d
a
.01 .006
.004
.002
,001
.oom
2
4
6
-
RADIUS~
Figure 9.
8
10
INCHES~
Contrastream Tube Injection of Diesel Fuel
increased from about 4 to 39 feet per second. In the 12-inch duct, E / u decreased from 0.0018 to 0.0013 foot when velocity was raised from 7 t o 30 feet per second. These values of the diffusion parameter agree fairly well with those for naphtha of Figure 8 despite an order of magnitude difference in air velocity. T h e influence of static pressure on the diffusivity parameter of naphtha was measured in 6-inch ducts a t three air pressures. The results were as follows: Air Velocity, Feet/Sec. 310 286 285
*
e
Sampling Distance, Inches 22 35.5 22
0.102 0.160 0.202
due t o fuel rate. However, as shown below, fuel rate does affect the initial spreading and must be accounted for as a boundary condition in the diffusion equations. All of the diffusivities cited above should be considered relative values only. Eddy diffusivity is a very sensitive function of the nature of the air flow. Ideal normal pipe turbulence is rarely attained in practice and therefore the diffusivity applicable to any actual system may depart widely from the values given above for apparently similar conditions. I n particular, the presence of objects in the stream which disturb flow may have a marked effect. Diffusivities have been observed in disturbedflow systems which are of the order of ten times those cited in the preceding data. For example, the frames on which injectors are mounted may themselves introduce considerable disturbance. Usually, one has no specific knowledge about these disturbances and the only resort is to measure diffusivities experimentally as above, by the distribution downstream of a point source In general, diffusivity will also vary throughout any given cross section, but an average value can be used with confidence in most applications. Fortunately, the diffusivity need not be known with high accuracy in many practical applications. In such cases, most of the mixing results from initial spreading due t o the injector. T h e distribution is then relatively insensitive t o the value of E. Single Injector Nonpoint Source. The diffusivities measured above were obtained from the distributions downstream of what were effectively point sources (in most cases). However, many actual liquid injectors are not effectively points. Even the simple tube injector used in the experiments of the preceding section does not behave as a point if the liquid is injected through
Pressure, Lb./Sq.
Inch Abs. 4 15 55
0.00204 0.00213 0.00200
Although these data are of a low order of accuracy, they are sufficient t o indicate no substantial change of diffusivity with a fourteenfold change in absolute pressure. The data above have an important corollary. It is clear that the negligible influence of air density-i.e., air static pressure-means that Reynolds number is not a proper correlating function for eddy diffusivity. Of the four variables involved in u, p, and p - t h e first three have been the Reynolds number--2m, examined here. A fourteenfold change in p and a 67% change in ro had negligible effect on E, although E varies about linearly with u. Thus, the grouping of these variables in the Reynolds number will not correctly describe their effect on diffusivity. Distributions from a naphtha injector were measured a t two different locations (17.5 and 35.5 inches downstream of the injector) with the following results: Fuel Injection Rate, Lb./Sec.
673
Diffusivity, Sq. Foot/Sec. At 35.5 inches 0.55 0.55 0.55
At 17.5inches 0.46 0.46 0.46
These data were taken in a 6-inch pipe a t atmospheric pressure and a n air velocity of 286 feet per second. There is about R 20'% increase in diffusivity on doubling the diffusion distance. This may be the result of build-up of normal pipe turbulence and/or evaporation of the residue of liquid drops in the longer distance. From the data above and the data previously cited on the effect of duct size, it is apparent that there is no effect on diffusivity
\
-1
a2
.01 ,008
I
EQUATION, M.3.6"
\
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II
,006 POINT SOURCE EQUATION
,004
.0020 1
\ 5
10 SURVEY RADIUS'-
Figure 10.
15
20
5
INCHES'
Contrastream Spra Nozzle Injection Diesel Fuey
of
the tube with any appreciable velocity. When liquid flows through the tube (contrastream) at low velocity it merely folds back over the end of the tube, a very small circle, thus behaving as a point source. However, when liquid is injected contrastream a t high velocity, the liquid jet penetrates the air stream. Before the liquid velocity can be stopped and reversed, the jet has spread out radially, forming a cone (or frustum) whose apex is a t the tube end and whose base is a circle which is now the effective source. A similar result is obtained with pressure atomizing nozzles; here the behavior is more obvious. I n either case the source is no longer a point but can now be approximated by a disk. Thus the disk source equation (Equation 15) is applicable. I n applying this equation, the diffusivity to be used is t h a t diffusivity measured from a point-source distribution (as above) for otherwise identical conditions.
INDUSTRIAL AND ENGINEERING CHEMISTRY
674
Distribution data from two nonpoint sources are illustrated in Figures 9 and 10. I n Figure 9, the measured points are given for contrastream tube injection of Diesel oil. Conditions for this run were identical t o those of Figure 7, except t h a t the fuel injection velocity vias high a t approximately 12’feet per second. Two curves are presented. One, corresponding t o the point source equation using an E of 0.35 square foot per second (found to apply in point-source injection under similar conditions), does not fit the data. The second curve, corresponding t o a disk source equation with the same diffusivity, is successful. The disk radius, M , chosen in this case was 1 inch (as given by Equation 16, below). I n Figure 10, the distribution data 35 inches downstream of a Diesel oil spray nozzle are illustrated. Conditions were the same as for Figure 9 except for the source. This source was a typical hollow-cone spray nozzle operated at a pressure drop of approximately 100 pounds per square inch gage. Disk-source and point-source equations are again shown. The disk radius in this case was 3.5 inches. Again it can be seen t h a t the disk-source equation successfully represents the data when a predicted eddy diffusivity is used.
I00
50
20
10
.
5 TO 5 8 0 FT/SEC
e ‘01
Equation 16 is a reliable correlation for contrastream tubes. Another common type of injector, the spray nozzle, is more complex. I n addition to fuel and air velocities and densities, and orifice radius, there are the variables of spray pattern, pressure drop, cone angle, and nozzle body size. All these factors can be expected to affect the correlating disk radius. The data on nozzles are too scant to permit generalization. A first approximation for M is the empirical factor 1.55 times the maximum observed spray radius. However, a few available data show t h a t this first approximation is often a poor one and that the proper 31 is often somenhat larger than the 31 so calculated.
Additional Calculation Techniques
R~ULTISOURCE INECTION. It is rare in practical applications that a single injector is sufficient t o give desired distributions or injection mass flow. The problem then arises of taking account of these multisource injectors by means of the equations given above. Regardless of the number or nature of the sources, the differential equation for diffusion, Equation 6, is applicable. The number and nature of the sources forin only a boundary condition to a solution of this equation. Equation 6, as a linear homogeneous differential equation, has as a solution any linear combination of other solutions. Therefore, the distribution resulting from multisource injection can be expressed by summation of the equations for each individual source. I n physical terms, this nieans that the fuel in any volume of air is a summation of the fuel contributed by each injection source acting as though it were the only souice present. For each source, the disk or point-source equation is used as appropriate. I
z
=
Vol. 45, No. 3
02
05
I
2
5
1
10
p*qp*4 Figure 11.
Correlation of Disk Radii for Contrastream Tubes
Although the disk-source equation is useful in describing certain nonpoint injection sources, its utility depends on whether or not an appropriate disk radius is known. The proper diffusivity may often be estimated from point-source data, thus leaving M as the only unknown variable. A correlation is available for contrastream tubes giving M as a function of known variables. As noted above, a contrastream tube fed a t high velocity gives rise t o a disk source because of penetration and mushrooming of the fuel jet. One can observe this behavior b y watching the jet through a glass section while in operation. By visual observation, or from photographs, the actual size of themushroomed jet can be determined. It has been found t h a t the actual size of the jet so observed correlates well with the proper disk radius. The disk radius, M , was found t o be 1.55 times the maximum radius of the spray within 2 inches downstream of the injector tip. Many such observations were made under various operating conditions and are illustrated in Figure 11. Additional points on the curve were obtained by matching, by trial, disk-source equations t o observed distribution data. The ratio of disk radius, fV, to actual tube inner radius, Ri, depends on the ratio of air momentum t o fuel momentum. The fuel velocity is calculated by assuming the fuel t o fill the tube completely. The straight line in Figure 11 is given b y the equation
(-)uf &2
Ri
= 11.2
Pf
-113
Suppose then t h a t the fuel distribution is required in any particular survey plane located downstream from a bank of injectors arranged, for simplicity, in a single plane. For each poiiit taken in the survey plane, R must be computed for each source axis. Equation 8 or 15 is applied and the values of 4 so computed from each source are summed up to give the total J” a t the survey point. This is a very tedious procedure when more than a fen sources are present. The graphical method below is suggested for such a computation. The assumptions are that all the sources are identical, coplanar, and fed with equal amounts of fuel. If one or more of these assumptions are not valid, the method can be modified accordingly. The first step is preparation of a scale drawing of the cross section a t the injector station locating each of the injector points. On this same drawing, all the survey points are similarly plotted to scale. Choosing Equation 8 or 15 as appropriate to the injection devices used, a table is calculated o f f as a function of R. All other variables in the equations are fixed by the geometry and the operating conditions. A transparent (plastic) rule is then made up, to the same scale as t h a t used for the injector geometry, in which the divisions denote increments in R. However, for each R division the scale is marked with the corresponding f from the computed table. A pin hole is then located a t R = 0. Such a rule is illustrated in Figure 12. ,040“ PLASTIC STRIP
F U E L I A I R RATIO
Figure 12.
1
Scale for Graphical Determination of Liquid Distributions
A in is placed, through the scale hole, a t the survey point at whicg the fuel-air ratio is to be measured. The scale is then rotated about the pin and the reading for each injection point noted. These readings are tabulated and summed t o give the total fuel-air ratio. The process is repeated for as many other survey points as may be desired.
Ring-Source Injection. A common configuration for location of fuel injectors is equidistant spacing around a large circle. This arrangement can be handled adequately by the graphical method above, b u t the labor involved can be greatly
March 1953
INDUSTRIAL AND ENGINEERING CHEMISTRY
decreased if a ring-source equation is applicable. T h e ring source, as defined earlier, is approached if there is a very large number of injection points closely spaced on a circle. For fewer points, the ring-source equation obviously gives progressively greater errors. Figure 13 can assist in estimating the distribution error in using this equation. For a given K and R/Ro, the parametric curves represent a given number of points on the circle required for an error not exceeding 10%. The area below and t o the left of each curve is a region of < l o % deviation. For example, for a K of 5 and a n R/Ro oY 1.0, the injection ring must have a t least eight equally spaced points t o give
