CALCULATION OF PARTICLE TRAJECTORIES C. E. LAPPLE AND C. B. SHEPHERD'
_____
E. I. du Pont de Nemours & Company, Inc., Wilmington, Del.
exists an intermediate or transition region, in which the flow partakes of both forms to some extent. At Reynolds numbers over 200,000 the fluid surrounding the body breaks away from the surface of the body with a sudden drop in the drag coefficient. The region a t which this occurs, however, is important only in ballistics and in aircraft design. In chemical engineering unit operations, values of Reynolds numbers greater than 10,000 are rarely encountered. Where the particle diameter becomes comparable in magnitude to the mean free path of the fluid molecules, the drag coefficient will have values somewhat lower than those predicted by Stokes' law. Cunningham (6) developed a correction to Stokes' law for this case. This correction, however, is not important unless the particle diameter is less than 10-4 cm. for gases or cm. for liquids. A particle moving in a fluid with an initial velocity will be decelerated by the action of frictional forces and will be either accelerated or decelerated by the action of external forces such as gravity. The net force acting on a particle in a given direction will be the vectorial sum of the frictional force and the external forces. The frictional drag in each case will depend inherently on the instantaneous velocity of the par-
Equations are developed for calculating the paths taken by bodies or particles undergoing accelerated motion, taking into account the effect of fluid friction. In several cases curves have been prepared to simplify the computation and eliminate graphical integrations. The equations are summarized in Tables IV and V. Available experimental data on particle trajectories show excellent agreement with calculated values. Applications to spray cooling or crystallization, cyclone dust collection performance, classification, dust sampling, and ventilation are presented and discussed, together with illustrative numerical calculations.
I
N MANY unit operations involving particulate systems,
it often becomes desirable to determine the paths described by the component parts either for purposes of design or improved operation. Outstanding in this respect are classification, spray drying or cooling, centrifugal and gravity collection or separation, spray absorption, and smoke abatement. The fundamentals of particle motion are generally expressed in a drag coefficient (C) us. Reynolds number (Re) plot, first suggested by Rayleigh. Such curves have been determined fairly completely for spheres, disks, cylinders, and miscellaneous shapes by numerous experimenters (4, 11, 13, 1.4, 16, 19, ,% 21). Figure I , 1 presents average curves for various shaped particles. The values upon which the curve for spherical particles in Figure 1 is based are given in Table I. Over the range of Reynolds numbers generally encountered in unit operations, the flow around suspended bodies may be divided into three regions-streamline, intermediate, and turbulent. At low Reynolds numbers streamline or viscous flow exists around the particle. The drag coefficient in this region, except that for cylinders, may be approximated as C = k/Re
TABLEI. VALUES OF DRAQCOEFFICIENT C Re
10
(1-4
which is an alternate method of expressing the well-known law of Stokes. At high Reynolds numbers the flow around the particles is completely turbulent and the drag coefficient is approximately constant. For spherical particles over a Reynolds number range of 500 to 200,000, C has an average value of 0.44, which represents the basis and the limitations of what is often termed Newton's law of particle motion. Between the region of stramline and turbulent motion there
SPHERICAL
C
4.1
C Reg
CRe
41.0
410
840
0.47
1,200 1,920 2,730 4,050 9.000 14,200
1 . 8 8 X 10s 3.80 9.80 19.1 40.5 180 426
50,000 70,000 100,000 200,000 300,000 400,000 600,000 1,000,000
0.49 0.50 0.48 0.42 0.20 0.084 0.10 0.13
24,500 35,000 48,000 84,000 60,000 33,600 60,000 130,000
1 . 2 3 X 103 2.45 4.8 18.8 18.0 13.4 38.0 130
3,000,000
0.20
600,000
2,000 3,000 5,000 7,000 10,000 20,000 30,000
(1)
For spherical particles in streamline motion, C = 24/Re
FOR
PARTICLES'
0.42 0.40 0.385 0.390 0,405 0.45
1.8
x
10"
These are average vslues based on the combined data of Allen, Arnold, Bacon and Reid, Dorr and Roberts Flachsbart, Jacoba. Knodel Krey Liebster, Lunnon, McKeehan, Millik'sn, Schmidt, Schmiedel, Shahespesr: Silvey, Wieselsberger. For values of Re leas than 0.1: C 24/Re; C Re 24; C Re2 = 24 Re. a
1 Present address, E. I. du Pont de Nemours & Company, Inc., Niagara Falls. N . Y .
605
-
-
INDUSTRIAL AND ENGINEERING CHEMISTRY
606
VOL. 32, NO. 5
But by definition,
and
F C
A (4) f(Re) = f ( D u p / p ) (4-4)
= (1/2) p u2 C =
Also from the relation between vectors, = uh/u sin a = U,/U
(5-4)
u2 = u;
(5B)
COS ff
REYNOLDS N U M B E R ,
FIGURE 1. DRAGCOEFFICIENT .4s
A
Re
+ uf
(5)
Hence Equations 3 and 3A become:
=";e
FOKCTION OF REYNOLDS NUMBER(TABLE I)
ticle as defined through the medium of the Reynolds number in Figure 1. In the subsequent paragraphs the equations for motion under various conditions will be developed. The general procedure involves the determination of a velocitytime relation for the particle motion in component directions. Once this has been obtained, the distance traveled in a component direction may be obtained as a function of the time by a simple graphical integration under a curve in which the respective velocity component is plotted against the time, according to the relation, s = !"dl
where s and u are vectors in the same direction. Unfortunately, the solution of the general equations for two-dimensional motion to obtain a velocity-time relation involves some method of incremental approximation. For the general cases of one-dimensional motion where tedious graphical integrations are implied, design curves have been prepared to simplify the calculations. For specific cases, houever, the equations may be integrated directly t o derive a velocity-time relation. It should also be pointed out that for a given material, particle size, and fluid condition, the velocity is related to the Reynolds number by a constant, and a Reynolds number-time relation is then the equivalent of a velocity-time relation. In the development of the theoretical equations, the corresponding Reynolds number is introduced for the velocity in order to present the manner in which the C vs. Re curve enters into the general solution. While the methods of derivation are presented in the following paragraphs, the final formulas are summarized in Tables IV and V and a knowledge of the derivation is not essential for the complete understanding and interpretation of the subsequent paragraphs on experimental verification and application.
Two-Dimensional Motion i n a Gravitational Field For a particle moving in the presence of a gravitational field with an initial velocity uo,it follows from a balance of forces that
In interpreting these equations, the direction of the horizontal component of uo is taken as positive for horizontal vectors, and the downward direction is positive for vertical vectors; the net velocity, u, is a scalar quantity and is always regarded as positive. From the definition of directional signs, it also follows that for practical application Uh will always be positive. These general equations are too complex to solve in general form and as such can be solved only by a method of incremental approximation. For special cases, however, the equations can be integrated directly, as will be shown in subsequent sections. Both the horizontal motion and the vertical motion are dependent on the resultant particle velocity and thus on the velocity at right angles to each, unless C is inversely proportional to u. This is usually the case for streamline motion. Since for streamline flow, C may usually be expressed as (7)
and by definition in the streamline flow region
and Equations 6 and 6A become
Thus for streamline flow the motion in a given direction is shown t o be independent of the motion in a direction perpendicular to the first, and the net trajectory will be that of one superimposed on the other. Equations 8 and 8A lend themselves to direct integration, as will be shown in subsequent paragraphs (Equations 13 and 19).
SEPARATION OPERATIONS
M A I , 1940
607
One-Dimensional Motion in Absence of Gravitational Field Although the case of motion in the absence of a gravitational field is not encountered practically, it is approached when u, is very large compared to u, a t all times under consideration and becomes important in evaluating two-dimensional flow for streamline motion. For this case u = uh and Equation 6 simplifies to dt
2M REYNOLDS NUMBER,
By substituting the identity FIGURE 2.
VALUES OF
IRe 2
x
105 dRe C
~ AS A 2
Re
=*
P
FUNCTION O F RDYNOLDS Y U M B E R (TABLE 11)
(10)
u = Rep/Dp
which for spherical particles (Stokes’ l w ) , becomes
in Equation 9 and rearranging,
t =
which can be integrated for the limits t Re = Re, (u = uo),Re = Re (u= u), &*t
=
=
0. t
=
t , and
SRjie0g2
For turbulent motion, C is constant and Equation 12 can also be integrated directly to give
which for spherical particles (Newton’s law: C comes t=---“ 3 0 3 p D 1
This can be written
(u -
P
2
TABLE 11. VALUESOF
is plotted against Re for spherical particles (values for Figure 2 are given in Table 11). Here Re, was arbitrarily taken as 2 X 105 and the value of the integral was calruiated on the basis of the C vs. Re values for spherical particles given in Table I. In employing Figure 2, a value of u is chosen and the corresponding value of Re calculated (Re, is determined by uo). The right-hand side of Equation 12A is determined by subtracting the respective integrals as determined from Figure 2. The value of t corresponding to the assumed u can then be calculated directly. By repetition of this procedure a time-velocity relation can be determined. From this relation a distancetime relation is obtainable by a simple graphical integration implied by f u d l
Jo
=
t)
0.44), be-
(14.4)
For motion in the intermediate range between streamline and turbulent flow, it is necessary in the general case to resort to a graphical integration of Equation 12, but for spherioal particles Equation 12A may be employed in conjunction with Figure 2.
where Re, is any arbitrary base. In Figure 2 the value of the integral
s =
(134
In (uo/u)
(2)
For streamline flow, Equation 12 can be integrated directly for limits t = 0, t = t , and u = u., u = u, to give a final time-velocity relation,
x
10’ dRe FOR SPHERICAI.
700 500 300 200 100 70 50 30 20 10
7 5 3
2 1 0.7 0.5
0.3 0.2 0.1 0.05 0.01 0,005 0.001
0.0001 0.00001
0.1000 0.1250 0.1387 0.1517 0.1727 0,1900
0.2185 0.2472 0.3142 0.3432 0.4102 0,5062 0.6022 ~~
One-Dimensional Motion in the Presence of a Gravitational Field The case of a particle moving in a vertical direction is frequently encountered-e. g., in drop shotting, certain forms of spray cooling or drying, and in spray towers. For this case u = u,, and Equation 6A simplifies to
VOL. 32. NO. 5
INDUSTRIAL AND ENGINEERING CHEMISTRY
608
By substituting the relation of Equation 10 for u and rearranging, dRe
PA
by means of Equation 21, letting Re, = 0. Equations 21 and 21A may also be expressed in terms of velocity: for downward motion (positive values of Re and Re,) :
mdt = b - CReRe where
An analysis of the function 4 reveals that it is the maximum value of C Rea which a particle of size D can attain when falling freely in a gravitational field of magnitude g. For spherical particles, 4 =
4g
3 P2
D3
P)
P (ps
(16H)
for upward motion (negative values of Re and Re.):
where
urn is
obtainable by:
By integrating Equation 16 between the limits t = 0, t = t , and Re = Re,, Re = Re,
Leo Re
2
G
t =
6
dRe
- CReRe
If the particle starts from rest, Re, Since for streamline motion, CRe = k
=
0. (7)
Equation 17 can be integrated directly for this case to yield
By eliminating Re and 4 by means of Equations 10 and 16A, Equation 18 may be expressed directly in terms of velocity
In Equations 18 to 22 directions are automatically accounted for by using the correct vectorial signs for the various velocities and Reynolds numbers indicated as vectors. Where no vectors are indicated, however, the absolute values should be used regardless of the sign of the corresponding vector. By direct graphical integration of Equation 17, charts could be drawn up which would permit a direct numerical solution. The required range of coordinates would, however, be very wide. By the use of a simple expedient involving the specific solution for streamline motion, it is possible to simplify the general solution of Equation 17 and obtain a factor, K , which shows only a small variation in magnitude over the entire flow region. If in Equation 18, C Re is substituted for its corresponding value, k , the following expression results : ~P A L ...- 1
where u, is the free-falling velocity in a gravitational field. For the case of streamline motion, u, is defined by
2
D M
CReln
4
[
- C6 Re, Re,
.$
- C Re Re
1
which, strictly speaking, applies only for the case of streamline motion. Now let K be defined by the expression: dRe - C Re Re
For spherical particles in streamline motion, Equation 19 reduces to
The general relation (Equation 17) may then be written in terms of K :
[44 --C,C Re,Re, ReRe 1
Since, for the turbulent region, C is constant, Equation 17 can be integrated directly for motion in that region to Sive
P A t =- K __ 2 D ;If C Re In
for Re and Re, 2 0 (i, e., positive):
where Z
for Re and Re,, 5 0 (i. e., negative):
(C R e 2 ) / ( C , Re;,), which means that the quantity 22 gives the ratio of C Re2 a t any selected velocity to the value of C Re2 a t the terminal settling velocity of the particle. Since C Re2 increases with Re a t all values of Re, Xz may be inter-
=
Re
m+
From the above definition of Z it is apparent that
These equations indicate that, where a particle has an initial downward velocity, Equation 21 may be used directly. Where a particle has an initial upward velocity, however, Equation 21A should be applied to calculate the time-velocity relation during the ascension period. Thereafter the timevelocity relation during the descension period is computed Since the preparation of this manuscript, L. M. K. Boelter has pub:ished an article covering this case for liquid droplets [ H e a l i n g , P ~ p i n gA i r Conditioning, 11, 639-47 (1939)l.
2 2
=
preted as representing a measure of the fractional approach to the terminal velocity. Therefore Z determines what proportion of the curve of C us. Re (from Re = Re, t o Re = Re,) is used in performing the integration of Equations 17 and 24. A comparison of Eqbations 17, 23, and 24 will show that K , thus defined, gives the fraction by which a calculated time of fall based on Stokes' law must be multiplied to include the effect of deviations from Stokes' law during the time of fall. It is thus apparent that K = 1 for motion in the streamline region. By reference to Equations 21 and 21A, it can be shown that for motion in the turbulent region,
MAY, 1940
SEPARATION OPERATIONS
for positive values of Z and Z,:
609
given downward velocity (which in turn determines values Re, C Re, and Re in any flow region without resorting to graphical integration. Although these curves were calculated for spherical particles, they are probably sufficiently accurate for most purposes when other shaped particles are considered, provided the appropriate C 2‘“2 us. Re curve, corresponding to the particle shape FIGURE 3. VALUESOF K FOR SPHERICAL PARTICLES FOR 0 5 2 6 1 ONLY considered, is used (TABLE 111) in calculating values of C Re and C ReZ. Where the particle has an initial downward velocity less than u, (i. e., 0 5 2, < l),a method of integration by difference (similar to that employed in obtaining Equation 12A) yields the following equation:
l/q
for negative values of Z and Z,:
I n order to understand the physical significance of the above equations, thepctual motion for each case should be considered. W re a particle has an initial upward velocity, the par e is gradually decelerated until i t comes to rest. The motion a t the instant before coming to rest will in any case be in the streamline region. Where the motion is partly in the turbulent region, this period is described analytically by Equation 218. Equation 19 gives the equivalent expression when the motion is entirely in the streamline region; for motion in the intermediate flow region, graphical integration of Equation 17 must be resorted to. After coming to rest, the particle will be accelerated in a downward direction as described by Equations 19 and 21 for streamline and turbulent motion, respectively. The particle will fall a t increasing speed and will gradually approach a constant speed, u,. A particle projected downward with an initial speed less than u, will accelerate until the speed u, is approached. On the other hand, a particle projected downward with an initial speed greater than u, will be decelerated, approaching constancy at u,. The practical utility of K lies in the fact that, where the motion is apt to be in the intermediate region or partially in the streamline or turbulent regions, the time-velocity relation may be computed directly without recourse to graphical integration of Equation 17, provided K is known. For the simple case where the particle starts from rest, it can be shown from Equation 26 that, regardless of the region of motion, K has a maximum range of 1 to 0.5. Table 111 and Figure 3 give values of K computed over the entire flow region. These values were calculated by actual graphical integration of Equation 24 for the case of a spherical particle starting from rest. By means of Figure 3 and Equation 25 it is thus possible to calculate the time corresponding to a
,xp”
TABLE111. VALUESOF K Z=
10
50
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.85 0.90 0.95 0.98 0.99 1.00
1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00
1,000 1.000 0.999 0.995 0.991 0.987 0,982 0.976 0.968 0.964 0.960 0.953 0.948 0.946 0.943
FOR
SPHERICAL PARTICLES (FOR
0 5 Z 4 1 ONLY)= -Value of K a t Following Value of 0: 100
1000
10’
lo6
lo6
1.000 0.998 0.995 0.990 0.984 0.976 0.967 0.955 0.941 0.934 0.924 0.913 0.905 0.902 0.900
1.000 0.995 0.990 0.982 0.973 0.961 0.947 0.928 0.905 0.890 0.876 0.856 0.841 0.836 0.830
1.000 0.990 0.983 0.975 0.965 0.950 0.933 0.909 0.877 0.858 0.835 0.807 0.785 0.775 0.760
1.000 0.987 0.976 0.963 0.947 0.930 0.907 0.880 0.845 0.826 0.800 0.768 0.743 0.730 0.700
1.000 0.985 0.970 0,952 0.933 0.912 0.886 0.857 0.821 0.799 0.770 0.734 0.700 0.680 0.600
(27) where 2 is the time required for the particle to accelerate from velocity u. to velocity u, and K and K , are factors obtained from Figure 3 for values of Z and Z,, respectively. Interpreted physically, Equation 27 represents the difference in the time required by a particle starting from rest to reach a velocity of u and u,, respectively. Charts similar to Figure 3 can be drawn for cases where 2 is negative or greater than unity. I n general, for values of 4 well in the turbulent region (4 > l O 7 ) , the function K will be 0.5 for 2 = 2, and 2. # 0; for negative values of 2,K will decrease to zero as 2 approaches zero, gradually a t first and then more rapidly as 2 comes very close to zero; for values of 2 greater than unity, K will decrease as 2 decreases, gradually a t first and then more rapidly, but as 2 approaches unity, K will VALUES increase very rapidly to a value of 0.5 a t 2 = 1. The curves of K for values of 4 less than lo7 will lie between those described above for 4 > lo7 10’ and the line R = 1. 1.000 0.982 0.964 0.945 0.924 0.900 0.872 0.840 0.801 0.778 0.748 0.708 0.668 0.845 0.500
0 The values of,K were calculated for a spherical body starting from rest by graphical integration as indicated by the following expression:
Two-Dimensional Motion i n a Centrifugal Field A knowledge of particle motion in a centrifugal field is important for a critical evaluation of the performance of centrifugal separators and classifiers which are involved in many industrial processes. The fundamental equations for such motion will be briefly summarized here. By vector analysis it can be shown that the general equations for the motion of a particle in a fluid stream, undergoing simple rotational motion, are
INDUSTRIAL AND ENGINEERING CHEMISTRY
VOL. 32, KO. 5
0
I1
t
-
rn
5-2 I
+ I I
*
t
E c
a 0
I1
-
u
7 C
N,
I
II 1
I
.-
C
bo
e s
5:
u
c
E
e
n
c Y
5
-
j E
.c?
a
w
I
+
O
N
E
:I; N L
J
t
,
!
MAY, 1940
SEPARATION OPERATIONS
61 1
I u; = uf
c
=
+ ( V , - ut)*
f(D
(28B)
P/P)
For spherical particles moving according to Stokes' law, these equations reduce to
n
n
SI
a-
2
+/P $1
It is possible for some purposes to neglect the tangential and radial acceleration, especially in cases where ut is close to V,and u,is comparatively small. For such a case Equation 28 would reduce to
.=.-
For spherical particles in streamline motion, Equation 30 becomes
which is a form of Stokes' law for constant settling rate in a centrifugal field.
3 5.
E
z
Equations of Motion
n 01
h
I
+-
$1 U J
-
E
=I
21s
The equations developed in the previous paragraphs are summarized in Tables IV and V. In certain cases they have been written in somewhat different form for simplicity. All equations, except those for motion in a centrifugal field, are developed for motion 'of particles in a fluid a t rest. If the fluid medium is in motion, these equations will still apply, provided all particle-velocity terms are redefined as relative to the fluid. In that case all distances computed on the basis of these velocities will also be relative to the fluid. Simple vectorial addition of the corresponding distances moved by the fluid in the same time will then give the absolute trajectory of the particle. The equations for motion in a centrifugal field are developed for a fluid undergoing simple rotational motion with no radial velocity component, and particle velocities are absolute unless specifically defined otherwise. It should also be pointed out that in all integrated equations the distance is taken as zero a t zero time whereas the velocity a t zero time is taken as u,,which may or may not be zero. The equations do not make allowance for agglomeration or "particle clouds"; the former result from combinations of small particles to form, in effect, larger particles, and the latter result from relatively close spacing of particles due to high particle concentration. Agglomeration is primarily a function of the physical and chemical characteristics of the particles and is usually important only in very small particle sizes. Particle clouds introduce complications only when very high dispersoid concentrations are involved (6, 17). Both of these phenomena may result in higher settling velocities and smaller decelerations than are computed for completely dispersed particles in dilute concentrations. "Hindered settling" (6, 17) is another complicating phenomenon associated with high particle concentration. Equations 6 and 6A show that the motion of a particle in any direction will, in general, be dependent not only on the velocity component in that direction but also on the resultant particle velocity. Thus a particle falling a t its constant terminal velocity in still air will fall more slowly if a horizontal air blast is applied to the particle. The particle will not attain its initial falling velocity again until it has attained
612
INDUSTRIAL AND ENGINEERING CHEMISTRY
u
c,
8
c
n
3
91;
v
u
L"
t u)
E
VOL. 32, NO. 5
MAY, 1940
S EPARATION OPERATIONS
TABLE VI. EXPERIMENTAL TESTD.4TA
613
ON PARTICLES COLLECTED IN
BEAKERS (FIGURE 6)
Predicted w t . of Theoretioal Material 7 Analysis of Material Collected in Beaker, % on Screen Av. Particle Beaker Collected, 14" 20 28 35 4s 65 100
