NASA Contractor Report 3384 

NASA 

CR 

3384 

c. 1 

Zero-Gravity Aerosol Behavior 

Harry W. Edwards 

CONTRACT NASs-31673 

JANUARY 198 1 
NASA Contractor Report 3384 

Zero-Gravity Aerosol Behavior 

Harry W. Edwards 

Co Zoru do State Uvrrversity 

Fort Colliifs, Colorudo 

Prepared for 

Marshall Space Flight Center 

under Contract NASS-31673 

National Aeronautics 

and Space Administration 

Scientific and Technical 

information Branch 

TECH LIBRARY KAFB, NM 

IlnlllIIllIIIImlwIR

lJlJb22b5 

1981 

Table of Contents 

Page 

CoverPage .......................... i 

Abstract ......................... ..ii 

Table of Contents ..................... iii 

I. Introduction ...................... 

II. Objectives ....................... 

III. Theoretical Feasibility ................. 

IV. Technological Feasibility ................ 

V. Scientific Benefits ................... 

VI. Conclusions ....................... 

VII. References ....................... 

Appendix I by B. J. Benedict . . . . . . . . . . . . . . . . . 

iii 
1 

I. Introduction 

An orbiting laboratory provides the unusual scientific opportunity 

to carry out experiments in the absence of gravitational effects. Potential benefits in the field of aerosol science are the absence of sedimentation and convection. In an experiment carried out under terrestrial 

conditions, sedimentation is important for larger areosol particles. 

Preventing spurious convective currents is often troublesome in terrestrial aerosol experiments. In addition to these potential benefits, 

the absence of a gravitational field may provide theoretical simplification. The feasibility of an orbital aerosol experiment is therefore 

of scientific interest. This preliminary investigation was undertaken 

to examine the feasibility and scientific benefits for a zero-gravity 

aerosol behavior experiment in an orbiting laboratory. 

II. Objectives 

A. Examine the theoretical feasibility for a. zero-gravity 

aerosol behavior experiment. 

B. Examine the technological feasibility for a zero-gravity 

aerosol behavior experiment. 

C. Identify potential scientific benefits for the experiment. 

D. Present conclusions. 

III. Theoretical Feasibility 

In the evaluation of experiments concerned with the disappearance 

of particles from a confined aerosol, one must in general consider the 

simultaneous concentration changes due to coagulation, diffusion, and 

sedimentation. The mathematical complexity of this problem has frustrated 

attempts to devise totally satisfactory analytical models, even 

-- -;--- --. 
2 

for initially monodisperse particles in the absence of electrostatic 

and convective effects. The conduct of such experiments under the 

essentially zero-gravity conditions of an orbiting space laboratory 

would provide data in the absence of sedimentation, normally an important 

depletion mechanism for particles larger than a few tenths of one 

micrometer in radius. A key issue is whether the absence of sedimentation results in a significant reduction in the mathematical complexity 

of the problem. 

Considerable success has been achieved in describing the behavior 

of dilute aerosols in which effects due to coagulation are absent. For 

the.one-dimensional problem, the following partial differential equation 

giving the numerical parti,cle concentration 11 as a function of time t 

and vertical position z has been solved analytically by Davies (1) and - 

verified experimentally by Richardson and Wooding (2): 

an at = D$ - ve (11 

The aerosol considered by these investigators was monodisperse and 

confined between perfectly absorbing parallel surfaces of large horizontal extension such that effects due to the vertical walls were 

negligible. The particle diffusion coefficient D and settling velocity 

v may be evaluated from theoretical considerations given by Fuchs (3). 

A major problem associated with achieving a general solution which 

includes'coagulation is the evolution of the particle size distribution 

with time. Hidy and Brock (4) reviewed many investigations of coagulation 

and found that the only exact solution is due to Smoluchowski (5) in 

which the coagulation constant is independent of time. In the absence 

of particle depletion due to diffusion and sedimentation, the basic 
3 

equation of coagulation is 

dn dt= -Q-p (2) 

where K is 8rRD for spherical particles of radius R. Integration of - 

equation (2) gives 

1 1 

n - .ii- 3 Kt 1 or : = 1 + 

0 0 noKt (3) 

where no is the initial particle concentration at t = 0. Numerous . 

experimental studies reviewed.by Fuchs. (3) establish the linear relationship between n and.time although values for &tend to be somewhat 

larger than computed values. Hidy and Brock (4) suggested that the 

lack of quantitative agreement is attributable to electrical effects, 

spurious air currents, increasing polydispersion, and the need for a. 

slip correction. The effect of increasing polydispersion was investigated 

by Hidy (6) who numerically solved a set of simultaneous nonlinear 

differential equations for a discrete particle size distribution. The 

numerical results give values for n(t) which closely match those given 

by equation (3), although changes in the relative concentrations in 

individual particle size categories differ from those for constant 

particle collision parameter. Various authors have suggested that the 

increase in mean particle size to decrease Kis offset by the increasing 

polydispersion with its opposite effect. The linear relationship 

between n -1 and time is used in the present work to develop a macroscopic model for the combined effects of diffusion and coagulation. 

The equation for the general case which includes coagulation gives 

nky,zA by 
_ _.-- - _ - -.-.--A. - ---- 

where g, 2, and K are assumed independent of time and position. The - 

assumptions of perfectly absorbing walls and uniform initial particle 

concentration n o give n(x,y,z,t) = 0 at the walls and n(x,.y,z,O) = no in 

- 

the chamber. Equation (4) has been solved analytically by Wilhelm (7) 

for the mathematically similar problem of contained plasma particles 

undergoing simultaneous recombination, diffusion, and convection. 

Wilhelm's approach, which involves a transformation to a differential 

equation in which the nonlinear term becomes a small perturbation, was 

shown to apply to plasma particles confined by nonreflecting walls. 

Application to the coagulating aerosol undergoing simultaneous 

depletion by diffusive deposition was made by Benedict (8). The 

details of this treatment are given in Appendix I. 

The theoretical feasibility for a zero-gravity aerosol study 

was examined by carrying out simulated experiments with models developed 

by Benedict (8). The purpose of the computations was to determine 

whether the requirements for a zero-gravity aerosol study are, at 

least in principle, compatible with the time and space limitations 

for an orbital experiment. Input to the models consists of the 

particle properties and initial concentration, gas properties, and 

chamber geometry. Output consists of the particle concentration 

as a function of time and location in the chamber. Numerical 

results are given in Appendix I. 

Two conclusions arise from the theoretical feasibility study. 

Firstly, the time and physical space limitations for an orbital 

experiment are not prohibitive in terms of obtaining kinetic data. 

._. - .._. .-_ --T?~- --.- Y-- ---- - --T---- - . -.--- .- -.. - 

-_’ . 

: , _. ., 
II ----. 

5 

For example, experiments of a duration of l-2 hours in a cylindrical 

chamber (r = 25 cm, h = 50 cm) can produce meaningful data on the 

history of the aerosol confined in the absence of gravitational 

effects. Secondly, the model is internally consistent and produces 

physically reasonable results. 

While the solution procedure resulted in substantial simplification 

of the problem, the reduction in complexity is only partly attributable 

to the absence of the gravitational term. The nonlinear coagulation 

term necessitates a transformation of variables, whether or not the 

gravitational term is present. 

A final point concerns possible limitations of the models 

employed in the theoretical analysis. Both D and K have been used - - 

as ensemble parameters. The major limitation of this approach 

is that the models do not provide direct information on the evolution 

of the particle size distribution. It should also be emphasized that 

while equation (1) has been verified experimentally, equation (4) 

must still be regarded as a postulate. The experimental conditions 

under which it is permissable to treat both D and K as ensemble - - 

parameters in equation (4) are considered in Appendix I. 

IV. Technological Feasibility 

The technological feasibility was examined by addressing the 

following issues: 

A. Experiment Definition 

B. Requirements for Aerosol Generation 

C. Requirements for Measuring Particle Concentrations 

D. Requirements for Data Analysis 

The major findings are summarized in a previous report (9). Two 

areas of concern have been identified. The first is the current absence 
6 

of a single,. totally satisfactory experimental technique for 

determining aerosol particle concentrations over the particle size 

range of 10 -7 cm to 10 -3 cm. Either complementary experimental 

techniques would be required or, alternatively, the size range 

accessible must be narrowed to be compatible with a single experimental 

technique. In view of the potential problems associated with 

calibration of complementary techniques under orbital conditions, 

the latter alternative seems more practical. 

The second area of concern is that of determining the particle 

concentration to the accuracy required. For example, in the absence 

of electrostatic and convective effects, calculated and measured 

values of the coagulation constant K differ by perhaps 10%. - Clearly, 

the experimental technique selected must be capable of resolving 

differences in particle concentrations smaller than 10% in order 

to make meaningful comparisons between experimental and theoretical 

data. While considerable progress has been made in recent years 

in refining experimental methodologies in aerosol science, the required 

accuracy may not be available for the ranges of particle sizes and 

concentrations of interest. However, recent developments with 

electrostatic classifiers and light-scattering techniques are particularly 

encouraging. The required instrumental capabilities may not 

be far away, but the situation seems borderline at this time. 

V. Scientific Benefits 

Potential scientific benefits of a zero-gravity aerosol study 

include validation of theoretical models for aerosol kinetics and 

measurement of D and K in the absence of convective effects. However, - - 

because of existing gaps in both the theoretical and experimental 
7 

aspects, such an experiment would probably be premature at this time. 

In order for the potential benefits to be fully realized, more detailed 

theoretical models for the combined effects of coagulation and diffusion 

are needed. Moreover, additional refinements in certain experimental 

methodologies would be helpful to assure accurate measurements over 

the ranges of particle sizes and concentrations of interest. 

VI. Conclusions 

The theoretical feasibility for a zero-gravity aerosol study 

has been examined. The mathematical complexity of the problem is 

discouraging if one attempts to retain a detailed picture of the 

combined effects of coagulation and diffusion upon the evolving 

particle size distribution. However, an analytical solution is 

possible if one considers only the particle concentration n - 

and treats D and K as ensemble parameters. - - Experimental studies 

support use of K as an ensemble parameter in many cases. Treating - 

D as an ensemble parameter imposes some limitations, however. The - 

results of the macroscopic treatment show that an aerosol decay 

experiment is feasible in a compact chamber for a time duration of 

the order of hours. It is concluded that the limitations of physical 

space and time for an orbital experiment are not prohibitive in terms 

of conducting an aerosol experiment. 

Because of the present mathematical difficulties associated 

with treating the combined effects of coagulation and diffusion, 

reservations are expressed about the scientific urgency for a zerogravity aerosol study at this time. The experiment would also 

appear to stretch existing capabilities for characterization of 

aerosol particles. Nevertheless, the need for reliable and accurate 
8 

aerosol behavior data in the absence of convective effects is 

recognized. Periodic re-examination of the need for a zero-gravity 

aerosol study is therefore recommended. When such a study is 

planned, it will be important to include a complementary terrestrial 

investigation. The difficulties associated with carrying.out an 

experimental aerosol study under favorable terrestrial laboratory 

conditions can be formidable. The success of an orbital experiment 

will be highly dependent upon managing these difficulties and 

anticipating additional problems posed by the orbital situation. 

VII. References 

(1) C. N. Davies, Proc. Royal Sot. Lond., A200, 100 (1949). 

(2) J. F. Richardson and E. R. Wooding, Chem. Engr. Sci., 1, 51 (1957). 

(3) N. A. Fuchs, "The Mechanics of Aerosols," Pergamon, New York, 1964. 

(4) G. M. Hidy and J. R. Brock, "The Dynamics of Aerocolloidal 

Systems, " Pergamon, Oxford, 1970. 

(5) M. v. Smoluchowski, Z. Physik., 17, 557 (1916). - 

(6) G. Hidy, J. Colloid Sci., 20, 123 (1965). 

(7) H. E. Wilhelm, J. Chem. Phys., 53, 1677 (1970). 

(8) B. J. Benedict, M.S. Thesis, Colorado State University, 1977. 

(9) H. W. Edwards, Progress Report, NAS8-31673, May, 1977. 
9 

Appendix I 

THESIS 

THEORETICAL BEHAVIOR OF A CONFI'NED AEROSOL 

Submitted by 

Bruce John Benedict 

In partial fulfillment of the requirements 

for the Degree of Master of Science 

Colorado State University 

Fort Collins, Colorado 

Fall, 1977 
10 

COLORADO STATE UNIVERSITY 

Fall, 1977 - 

WE HEREBY RECOMMEND THAT THE THESIS PREPARED UNDER OUR SUPERVISION 

BY BRUCE JOHN BENEDICT ENTITLED THEORETICAL BEHAVIOR OF A 

CONFINED AEROSOL BE ACCEPTED AS FULFILLING IN PART REQUIREMENTS 

FOR THE DEGREE OF MASTER OF SCIENCE . 

Committee on Graduate Work 

Adviser 
11 

ABSTRACT OF THESIS 

'TEEORETICAL BEHAVIOR OF A CONFINED AEROSOL 

Coagulation, sedimentation and diffusive deposition are the primary 

removal mechanisms for an aerosol confined in a chamber on earth. The 

equation describing the depletion rate due to these mechanisms is a 

second order nonlinear partial.differential equation. For an aerosol 

in a zero-gravity environment t,he sedimentation term drops out, but 

this does not change the basic nature of the equation. An analytical 

solution to the resulting equation is presented and particle concentrations are computed as a function of time and location ina cylindrical chamber. 

The equation is also solved for an aerosol under the influence of 

a gravitational field. There are some difficulties with this solution 

G;:'re the removal mechanisms are operating at similar rates. These are 

overcome by modelling the decay process as if diffusion were not present. 

Results of the two models indicate that sedimentation is the most important of the removal mechanisms. Coagulation is next in importance and 

diffusion is negligible except within 1 centimeter of the chamber wall. 

Bruce John Benedict 

Mechanical Engineering Department 

Colorado State University 

Fort Collins, Colorado 

Fall, 1977 
12 

ACKNOWLEJIGENENTS 

The author-wishes to express his gratitude to his major professor, 

Dr. H. W. Edwards, for the guidance and concern he has shown throughout 

the theoretical developement and thesis preparation, also to Dr. C. E. 

Mitchell for assistance in the mathematical developement and finally to 

the other members of his committee, Dr. R. B. Kelman and G. R. Johnson 

for their suggestions during the preparation of this thesis. 

Acknowledgement is also given for financial support by the National 

Aeronautics and Space Administration (Marshall Space Blight Center) 

under contract No. NAS8-31673. 

-; -.- ----. : 

__ ~. - .I_-- ---. .-_. -- 

: .-; -.’ 

., 
13 

TABLE OF CONTENTS 

we 

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . 

NOMENCLATURE...................... 

Chapter 

I. 

II. 

III. 

IV. 

v. 

EQUATION .................... 

Introduction .................. 

Equation .................... 

Coefficients .................. 

Solution .................... 

ZERO-GW,WY SOLUTION .............. 

Equation .................... 

Solution .................... 

Inner-Outer Expansion ............. 

GRAVITY SOLUTION ................ 

Equation .................... 

Solution .................... 

Solution Near the ML1 ............. 

RESULTS AND DISCUSSION ............. 

Zero-Gravity .................. 

Gravity .................... 

CONCLUSIONS ................... 

BIBLIOGRAPHY...................... 

Appendix 

A. Program for Zero-Gravity Solution ........ 48 

B. Program for Gravity Solution .......... 55 

14 

15 

17 

17 

18 

18 

20 

23 

23 

23 

25 

28 

28 

28 

29 

31 

31 

41 

44 

47 

-‘-----7 

-__ _ - .- .- --ccI-7- __ .._- - 

, ,, ; . . ,’ . . 

; ,. :,; ” 7:.------T=, I,) ,.. ,:..’ ,’ ;.: 

--- 

._ .. .,- 
_ __ 

...   .

 . 

14 

LIST OF FIGURES 

Figure page 

1. 

2. 

3. 

4. 

5. 

6. 

7. 

8. 

9. 

Particle Settling Velocity . . . . . . . . . . . . . . . . 32 

Particle Depletion in the Absence of Gravity . . . . . 

Particle Depletion in the Absence of Gravity . . . . . 

Particle Depletion in the Absence of Gravity . . . . . 

Particle Depletion in the Absence of Gravity . . . . . 

Particle Concentration Near the Wall Due to Diffusion 

Alone . . . . . . . . . . . . . . . . . . . . . . 

Particle Concentration Near the Wall Due to Diffusion 

Alone . . . . . . . . . . . . . . . . . . . . . . . 

Particle Depletion in the Presence of Gravity and. 

Coagulation . . . . . . . . . . . . . . . . . . . 

Particle Depletion in the Presence of Gravity and 

Coagulation . . . . . . . . . . . . . . . . . . . 

. . 33 

. . 34 

. . 35 

. . 36 

. . 38 

. . 39 

. . 42 

43 

. . 

_ _ _. _ I _c .-..?- -- _ ._ . . 

- ,- - --_.- )-- 

, I 

-.- -- 
15 

NOMENCLATURE 

A 

Aij 

b 

B 

C 

'd 

f 

j 

JO 

J1 

k 

K 

L 

n 

n' 

n 

0 

Q 

r 

r' 

Constant in mobility equation 

Fourier coefficients for solution 

Constant in mobility equation 

Particle mobility 

Cunningham slip Correcti'on 

Particle drag coefficient 

Variable to which u is transformed 

Acceleration of gravity 

Height of chamber 

Index for eigenvalues associated with Jo 

Index for eigenvalues associated with sine 

Zero order Bessel function 

First order Bessel function 

Boltzmann's constant 

Particle coagulation coefficient 

Dimensionless chamber height 

Numerical concentration of particles 

Dimensionless concentration 

Initial concentration of particles 

Constant in mobility equation 

Radial coordinate 

Dimensionless radial coordinate 
16 

NOMENCLATURE (cant') 

R 

R* 

Re 

s (xl 

t 

tl 

U 

'r 

% 

Ut 

V 

V 

X’ 

z 

2’ 

B 

Y 

6 

e 

V 

u 

x i 

P 

pP 

lJ 

Particle radius 

Chamber radius 

Particle Reynolds number 

Unit step function 

Time 

Dimensionless time 

Variable to which n is transformed 

Function of r' only used in separation of variables 

Function of z' otily used in separation of variables 

Function of t' only used in separation of variables 

Terminal settling velocity of particle 

Particle velocity before applying Cunningham slip correction 

Distance from wall 

Vertical coordinate 

Dimensionless vertical coordinate 

Dimensionless coefficient 

Dimensionless coefficient 

Mean free path of air molecules 

Perturbation on diffusion 

First separation constant 

Second separation constant 

Eigenvalue associated with Jo 

Density of air 

Density of particle 

viscosity of air 
17 

CHAPTER I 

EQUATION 

Introduction 

The principal removal mechanisms for an aerosol confined in a 

chamber which is located in a gravitational field are sedimentation, 

coagulation, and diffusive deposition at the wall. In most situations 

of interest sedimentation is the most important. An experiment carried 

out in the zero-gravity environment of an orbiting space craft would 

provide a unique opportunity to study aerosols. By placing an aerosol 

in such an environment it is possible to obtain measurements of the 

coagulation coefficient and diffusion coefficient of the particles. 

This study provides a model with which such an experiment can be designed. A model which describes the depletion of an aerosol stored in a 

gravitational field is also presented. 

For the purposes of this modelling effort the following assumptions 

were made: 1. The particles are unit density spheres. 2. The system 

is monodisperse. 3. Initially, the particles are uniformly distributed 

within the chamber. 4. The coagulation coefficient is constant with 

respect to time. The particle formed when two particles stick together 

should have a coagulation coefficient which is larger than that of a 

single particle because the combined particle is larger. However, Fuchs 

[l] states that this effect is almost balanced by the decreased diffusivity of the larger particle. 5. All collisions, particle-particle 
18 

and particle-wall, have a sticking coefficient of unity. Fuchs [l] 

indicates that as long as the only driving forces in the system are 

thermal, the sticking coefficient is nearly one. 

Equation 

Richardson and Wooding 123 present an equation describing the 

depletion of a monodisperse aerosol which is confined in a chamber in a 

gravitational field. 

an an at= DV2n .- vz - ~2 

where n is the numerical concentration of particles, t is time, D is the 

particle diffusion coefficient, v is the terminal settling velocity of 

the particle, z is the vertical coordinate of the chamber, and K is the 

particle coagulation coefficient. The boundary conditions are implicit 

in the assumptions. Because the sticking coefficient for particles 

colliding with walls is unity, a particle which strikes a wall sticks to 

it and is thus removed from the 

n(r,z,t) = 0 at the walls. The 

uniform throughout the chamber, 

n 

0’ 

bulk of the chamber. Thus we have 

initial concentration was assumed to be 

thus the initial condition is n(r,z,O) = 

Coefficients 

Strauss [3] gives a procedure for calculating the terminal settling 

velocity for aerosol particles in terms of the particle Reynolds number, 

Rey and the particle drag coefficient, Cd. The procedure is to first 

calculate the product, CdRi, using properties of the particles and the 

fluid in which they are dispersed. 
19 

CdRE = 

32~(~ - P)R2g 

3112 

(2) 

where p is the fluid density, p P is the particle density, g is the 

acceleration of gravity, R is the particle radius and u is the viscosity 

of air. Davies [43 gives a series of empirical formulas for the Reynolds 

number in terms of the product CdRz. Once the Reynolds number is determined the product can be used to find a value for the drag coefficient. 

These values are then used in Stokes law to find a velocity, V, 

16Rpg(p - P> 

V= (3) 

3CdR,p 

The terminal settling velocity is determined from V by applying the 

Cunningham slip correction, 

C = 1 + $1.257 + 0.400exp(-l.lO$) 

where C is the Cunningham slip correction, 6 is the mean free path of 

the fluid molecules. Now it is possible to determine a value for the 

terminal settling velocity, v. 

v = cv 

The diffusion coefficient is given in Fuchs [l] 

D = kTB 

(5) 

(6) 
20 

where k is Boltzmann's constant, T is absolute temperature, and B is the 

particle mobility also given by Fuchs IlJ. 

B= 

l+~+Q$xp(+ 

67TRl.l (7) 

The constants A, Q, and b are given in Millikan [5] for oil drops in air 

at 23Q and 1 atmosphere. They are A = 0,864, Q = 0.290, and b = 1.25. 

The coagulation coefficient is in Fuchs [lj as follows: 

K= ~ITRD 03) 

Solution 

The nondimensional variables, rt = r[R*, z' = z/R*, n' = n/n 

0 

, and 

t.' = tD/R*2 are introduced. R* is the radius of the chamber. 

Substituting these into equation (1) results in 

ant an1 

at!’ 

V2n’ - fi- 7- ad yn 

r2 

where 

R*v B-, R*2KIl Y= Do 

(9) 

The boundary conditions become n'(r',z',t') = 0 at the walls and 

the initial condition is n'(r',z',O) = 1. 

The nonlinear coagulation term in eqZlation (9) presents the 

greatest difficulty in obtaining a solution thus it is desirable to 

remove it using a transformation. This is done by using the solution of 
-- 

2 1. 

the Smoluchowski coagulation equation as was done by Wilhelm [6]. The 

transformation used was 

n'(r',z',t') = 

( ',z',t') 

1+ yt:u(r',z',t') 

This results in the equation 

au 

atl =V2u-g+E 

where 

2yt' au 

E = 1.+ yt’u ((artI + (a=1 i&-)2) 

(11) 

(12) 

(13) 

The boundary conditions remain the same for the transformed variable, 

u, as for the variable, n. That is u(r',z',t') = 0 at the walls and 

u(r',z',O) = 1. 

The times of 1 and 2 hours, which are used in the calculations, 

are very short as compared to the characteristic time for diffusion. 

This is born out by the values of t', which range from 7.0 X 10 -8 for 

large particles to 1.7 X low3 for small particles. The values of y are 

relatively large, ranging from 15 for small particles to 1.6 X lo6 for 

large particles. However, the values are such that for a given particle 

the product yt' is of order 1 or smaller. It is further anticipated 

that except near the walls the spatial derivatives will be small. Thus 

the value for E will be small. 

Equation (12) can now be solved by a successive approximation 

technique. To do this E is set equal to 0 and equation (12) is solved. 
22 

This is the zeroth approximation. Using the solution for the zeroth 

approximation a value can be calculated for E. This value is then put 

back into equation (12) as a constant and the equatian is solved, 

resulting in the first approximation. This process is repeated using 

the value of E calculated from the previous approximation until the 

desired accuracy is obtained. 
23 

CHAPTER II 

ZERO-GRAVITY SOLUTION 

Equation 

The simplest case for obtaining a solution is the zero-gravity 

situation. This is because with zero-g,ravity the terminal settling 

velocity of the particle is zero, thus 6 = 0 and equation (12) reduces 

to 

au 

at( = v2u - E 

Setting E = 0 for the zeroth approximation results in the diffusion 

equation which for a cylindrical chamber is 

au i au a2u a2u 1= p- ar.l at -+m+ azf’ (15) 

Solution 

The solution for equation (15) can be found using the technique of 

Separation of Variables as given by Wylie [7J. For this it is assumed 

that the solution is the product of three functions, a function of r’ 

only, a function of z' only, and a function of t' only. Thus 
24 

u(r',z',t') = Ur(r')Uz(z')Ut(t') 

Substituting this into equation (15) gives 

aut aur a2u a2u 

+ p 

r Z 

att +arl'+azi'=v -- 

ut 'r '1: U 

Z 

(16) 

(17) 

One side of this equation is a function of t' only, while the other is 

a function of r' and z'. The only way this can be true is if both sides 

are equal to a constant, V. Using this fact it is possible to solve 

for Ut. That solution is 

Ut = exp(vt') 08) 

This equation implies that v < 0 because the solution for u(r',z',t') 

cannot increase exponentially with time. Now it is possible to take 

the part of equation (17) which is a function of r' and z' and rearrange 

it so that there is an equation which is a function of r' equal to a 

function of z' which again must be equal to a constant. 

au a2u a2u 

1 ar: + y+ -v,= c Q+ '0 

P-----i-- r 'r uz 

Taking the z' equation it is possible to solve for U . 

Z 

(19) 

U = 

Z 

sin(Gz') (20) 
25 

From the boundary conditions on the ends of the chamber, o can be determined. 

jrR* u= ( 2 

H ) 

where H is the height of the chamber. 

The solution of the remaining equation is Ur. 

'r = Jo(hir') 

(21) 

(22) 

where the Xi are the zeros of Jo(Xi) = 0. Combining all of these solution results in the complete solution results in 

u(r',z',t') = Y ? A J (X r')sin(yA')exp(-(Xi f (y)2)t1) i=l j=l ij 0 i 

(23) 

Using the initial condition, A.. can be found. J-J 

A 

/'jLrl.To(hir')sin(~')dz'dr' 

= O" ij I~I~r'J~(Air')sin2(~')dz'dr' 

= 4(1 - (-l>j) 

jnXiJ1 (Ai) 

where L = H/R*. 

Inner-Outer -Expansion 

Results of the solution given by equation 

is almost a constant for most of the chamber, 

(23) are that u(r',z',t') 

indicating that coagulation is much more important as a removal mechanism for the bulk of the 

aerosol than is diffusive deposition. The fact that the higher order 
_---..-.. ---.-.-. -- 

26 

approximations are very difficult to solve indicates that the solution 

might be found using an inner-outer expansion technique. The innerouter expansion technique is used on problems which have several 

regions in which the processes occurring are of different nature. The 

classical example of such a problem is a boundary layer problem such 

aa illustrated here. In this technique a solution is found describing 

the dominant process in each region. The boundary conditions for the 

solution in one region are then modified by using the boundary conditions which were shown to exist in the adjacent region. These modifications are repeated until the boundary conditions for adjacent regions 

match each other. 

For the outer expansion, diffusion can be ignored in equation (9). 

This leads to 

(25) 

which is the well known equation for coagulation developed by Smoluchowski and presented in Fuchs [l]. The solution for this equation is 

n'(t') = 

1 

1+yt' (26) 

The inner expansion begins with the assumption that near the wall 

the only removal mechanism of any significance is diffusive deposition 

on the wall. It is further assumed that the particles are diffusing 

only along the x coordinate. These assumptions seem reasonable due to 

the extremely large gradient which exists at the walls of the chamber. 

Thus the equation describing the depletion of the aerosol near the wall 
27 

is 

i3nr a2n1 

F’ ax,’ 

where x is the distance from the wall. Carslaw and Jaeger [8] give a 

solution to this equation. 

n'(x',t') = erf(*) 

This solution technique is presented as a possible alternative to 

the successive approximation technique presented earlier. However, 

the inner-outer expansion technique is a somewhat less desirable approach. For the inner-outer expansion technique it is necessary to 

solve two problems while for the successive approximation technique 

only one problem need be solved. Further, matching the two solutions 

repeatedly is likely to be a problem whose difficulty is comparable to 

that of solving for the higher order approximations in the successive 

approximation technique. 
28 

CHAPTER III 

GRAVITY SOLUTION 

Equation 

A somewhat mOre difficult case is the solution of equation (12) in 

the presence of a gravitational field. A transformation is made which 

removes the sedimentation term. This transformation is due to Wilhelm 

[6] and is 

u(r',z',t') = f(r',z',t')exp(F' - p') 

This transformation yields 

af 

atl = V2f - E: exp(- $z' + $') 

(29) 

(30) 

The boundary conditions then become f(r',z',t') = 0 at the.wall and 

f(r',z',O) = exp(- $z'). 

Solution 

Equation (30) is solved by separation of variables in the same manner as was equation (15), resulting in 

f(r',z',t') = i=l j=l ij 0 i ?!! ?, A J (X r')sin(F')exp(-(Xi + (pr)2) 
29 

where 

Aij = 

I:!ir ‘exp (- @zl)Jo(Xir')sin(~')dz'dr' 2 

I~~~r'J~(Xir')sin2(~')dz'dr' 

4j7T(l - (-l)jexp(- 9)) 

= 

AiJl(Xi) (q + j2n2) 

(32) 

Solution Near the Wall --- 

There are several areas in which this solution has difficulties. 

An examination of equation (29) will show where these difficulties ocValues for B range from 3.58 X 10 -3 

cur. for small particles to 

1.11 x 1o12 for large particles. Thus for certain particle sizes, the 

transformation given by equation (29) becomes so large that the results 

are meaningless. The solution does work for particles smaller than 

0.04 micrometers where sedimentation is much less important that diffusion, and for particles larger than 0.7 micrometers where diffusion 

is much less important than sedimentation and for long times when sedimentation has essentially depleted the chamber. 

Results from the zero-gravity solution indicate that the effects of 

diffusive deposition are limited to a thin boundary layer. Thus diffusive deposition has a negligible effect on the concentration of the 

bulk of the chamber and if seems reasonable to expect that the problems 

of the above solution can be overcome by ignoring diffusion altogether. 

This leads to the equation 

an’ an1 

at” -BP-F 12 (33) 
30 

now applying the transformation in equation (11) the result is 

au au 

atl’ -?g (34) 

The solution to this is in terms of a unit step function, s(x). 

The unit step function is defined as follows 

0 xc0 

s(x) = { 

1 x>o 

So the solution to equation (34) is 

u(r',z',t') = s($ - z' - Bt') (35) 

This solution can be combined with the inner expansion solution of 

equation (28) to give the solution for the entire chamber. 
31 

CHAPTER IV 

RESULTS AND DISCUSSION 

Zero-Gravity 

Figure 1 gives the results of the settling velocity calculations 

for unit density spheres in air at 23C and 1 atmosphere. 

Figures 2 - 5 give the results of equation (23) which describes the 

depletion of a confined aerosol which is acted on only by coagulation 

and diffusive deposition. The first two figures give the concentration 

as a function of particle size for a location near the center of the 

chamber and times of 1 and 2 hours. These figures do not show any effects of diffusive deposition on the aerosol concentration and are 

therefore identical to a curve for coagulation alone. Figures 4 and 5 

give similar information but for a location near the wall of the chamber. The dotted lines on the curves for the smaller particles show what 

the concentration would be if only coagulation were acting on the aerosol at that point. Thus it can be seen that diffusive deposition is 

not very effective as a removal mechanism for the bulk of the chamber. 

Near the wall, however, the situation is reversed. There is a steep 

gradient in concentration which drives diffusive deposition. Thus for 

a thin boundary layer the dominant removal mechanism is diffusive deposition. The steep gradient near the wall will reduce equation (14) 

to a one dimensional equation except at the corners. The solution to 
32 

10. 

1. 

10 .l 

1O-2 

Settling 

Velocity, 

cmlsec. 

1o-3 

1o-4 

1o-5 

0.01 0.1 1.0 10. 

Particle Radius, micrometers 

Figure 1. PARTICLE SETTLING VELOCITY. 
0.8 

0.7 

0.6 

0.5 

n - 

n 

O 0.4 

0.3 

0.2 

0.1 

0.0 

crf 

Unit density spl-eres in still air at 23C and 1 atm. 

Cylindrical chamber: radius = 25cm. height = 50cm. 

Location: R = 5cm. z = 25cm. t=lhr. 

I 1 I I I IIII I I I I Illll I I I I I lII( 

0.01 0.1 1.0 10. Particle Radius., Micrometer 

Figure 2. PARTICLE DEPLETION IN THE ABSENCE OF GRAVITY: IMPORTANCE OFXOAGULATION. 
1.0 

0.8 

0.7 

0.6 

0.5 

n - 

n 

O 0.4 

0.3 

0.2 

0.1 

0.0 

CI r\ w 3 

n = 105/cm3 

0 

. 

Unit density spheres in still air at 23C and 1 atm. 

Cylindrical chamber: radius = 25cm. height = 50cm. 

Location: R = 5cm. z = 25cm. t = 2hr. 

I I I IIIlll I I I I I II11 I I I I IlId 

0.01 0.1 1.0 10. 

Particle Radius, Micrometer 

Figure 3. PARTICLE DEPLETION IN THE ABSENCE OF GRAVITY: IMPORTANCE OF COAGULATION. 
- 

1.0 

0.9 

0.8 

0.7 

0.6 

0.5 

n - 

n 

O 0.4, 

0.3 

0.2 

0.1 

0.0 

. fi h - A ” ” ” v a 

n 

0 

= 104jcm3 

h 

n f? ” 0 ” 

n = 105/cm3 

0 

Unit density spheres in still air at 23C and 1 atm. 

Cylindrical chamber: radius = 25cm. height = 50cm. 

Location: R = 22.5cm. 2 = 25cm. t = lhr. 

I I I Illjlf 1 I I I Illll I I I I Illl~ 

- _- 0.01 0.1 1.0 10. 

Particle Radius, Micrometer 

Figure 4. PARTICLE DEPLETION IN THE ABSENCE OF GRAVITY: IEIPORTANCE OF COAGULATION. ' 
0.9 

0.8 

0.6 

0.5 

n - 

n 

O 0.4 

0.3 

0.2 

0.1 

0.0 

Unit density spheres in still air at 23C and 1 atm. 

Cylindrical chamber: radius = 25cm. height = 50cm. 

Location: R = 22.5cm. z = 25cm. t = 2hr. 

0.1 1.0 

Particle Radius, Micrometer 

Figure 5. PARTICLE DEPLETION IN THE ABSENCE OF GRAVITY: IMPORTANCE OF COAGULATION. 
37 

the one dimensional diffusion equation (28). Figures 6 and 7 are concentration profiles for the region near the wall of the chamber and 

show the extent of the action of diffusive deposition for times of 1 

and 2 hours respectively. Thus for the zeroth approximation, equation 

(15) gives the concentration for the entire chamber. 

The zeroth approximation is sufficiently accurate for all but the 

most demanding applications. The higher order solutions are very difficult to obtain and contain only a small correction to the solution 

given by equation (23). Table I gives the values of the terms of equation (14) for various locations within the chamber and times of 1 and 2 

hours. The values for locations near the center of the chamber are 

large relative to the other terms in equation (14) but are quite small. 

The size of E for these locations is a further indication that there 

are no gradients and thus diffusive deposition is not important. The 

values are so large because of truncation error in evaluating equation 

(23) and not any fluctuations in aerosol concentration at those points. 

If the truncation error were not present it is expected that the value 

of E would be zero. Substituting E = 0 into equation (14) for the first 

approximation would yield exactly the same result as the zeroth approximation, Thus for those situations where E = 0 the zeroth approxiamtion 

is exact. 

The inner-outer expansion solution embodied in equations (25) 

through (28) give further evidence that the zeroth-approximation is accurate because this solution technique gives the same results as those 

of equation (15). That one dimensional diffusion is the dominant removal mechanism near the wall is also shown by the fact that the solution to equation (27) predicted the difference between the curves for 
n 

n 0 

0.8 

0.6 

0.5 

0.4 

0.3 

0.2 

0.1 

0.0 

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0..40 0.45 0.50 

Distance from wall, centimeters 

Figure 6. PARTICLE CONCENTRATION NEllR WALL DUE TO DIFFUSION ALONE. 

Unit density shperes in still air at 23C and 1 atm. 

Cylindrical chamber: radius = 25cm. height = 50cm. 

Location: 2 = 25cm. t = lhr. 
1.0 

0.9 

c 

0.8 

0.7 

0.6 

n 0.5 

- 

n 

O 0.4 

0.3 / 

O.lum 

0.2 

Oil 

I, ,, J. _ i 

R= 

yy y-yy 

O.Olym 

-- 

0.00 0.05 0.10 G.ij 0.20 0.25 0.3c1 0.35 0.40 0.45 0.50 

Distance from wall, centimeters 

Figure 7. PARTICLE CONCENTRATION NEAR WALL DUE TO DIFFUSION ALONE. 

Unit density shperes in still air at 23C and 1 atm. 

Cylindrical chamber: radius = 25cm. height = 50cm 

Location: z = 25cm. t = 2hr. . 
40 

Table I -- 

VALUES OF TERMS IN EQUATION.(14) 

au t(hr) w V2U E 

lE-6 5 1 0. 3.05E-6 3.05E-6 

lE-6 22.5 1 18.96 18.97 5.803-3 

13-6 5 2 0. 4.18E-16 4.183-16 

lE-6 22.5 2 87.69 88.03 0.340 

lE-5 5 1 0. 4.09E-9 4.09E-9 

1E-5 22.5 1 0. 4.09E-9 4.093-g 

lE-5 5 2 0. 2.26E-12 2.263-12 

lE-5 22.5 2 0. 2.26E-12 2.26E-12 

1E-4 5 1 0. 5.Jl8E-6 5.183-6 

1E-4 22.5 1 0. 5.40E-6 5.403-6 

lE-4 5 2 0. 6.47~-6 6.473-6 

lE-4 22.5 2 0. 6.48E-6 6.483-6 

lE-3 ,5 1 0. 7.46~-6 7.46E-6 

lE-3 22.5 1 0. 1.35E-5 1.35E-5 

lE-3 5 2 0. 1.41E-5 1.41E-5 

lE-3 22.5 2 0. 2.24E-5 2.243-5 
I -- 

41 

coagulation alone, depicted by the dotted lines, and the curves for 

both coagulation and diffusive deposition in figures 4 and 5 to within 

1%. 

Table I shows that the values for E for locations near the wall are 

quite large relative to those far from the wall. Comparison with the 

values of the other terms in equation (14) shows that they are negligible. Thus the approximation calculated by ignoring E for those regions was good. 

Gravity 

Equation (33) is the model of a confined aerosol which is acted on 

by sedimentation as well as coagulation and diffusive deposition. Figures 8 and 9 give the results of this model. Figures 8 and 9 are 

graphs of the concentration as a function of particle size for a location near the bottom of the chamber. An inportant result of this model 

is the sharp gradient which appears in the vertical extent of the chamber as a result of the action of gravity. This gradient moves downward 

with a velocity equal to the terminal settling velocity of the particles. 

Above the line of the gradient there are no particles. Below the line 

the concentration changes as if the particles were being acted on by 

coagulation and diffusive deposition. 
0.8 

0.6 

0.5 n - 

n 0 

0.4 

0.3 

0.2 

0.1 

0.0 

0.1 

Particle Radius, micrometers 

Unit density spheres in still 

air at 23C and 1 atm. 

Cylindrical chamber: radius 

= 25cm. height = 50cm. 

Location: R = 5cm. 2 = 5cm. 

t = lhr. 

Figure 8. PARTICLE DEPLETION IN THE PRESENCE OF GRAVITY AND COAGULATION. 
0.8 

0.6 

0.5 

n - 

n 

0 

0.4 

0.3 

0.2 

0.1 

0.0 

Unit density spheres in still 

air at 23C and 1 atm. 

Cylindrical chamber: radius = 

25cm. height = 50cm. 

Location: R = 5cm. z = 5cm. 

t = 2hr. 

P 

w 

Ij_i 1.0 10. 

Particle Radius, micrometers 

Figure 9. PARTICLE DEPLETION IN THE PRESENCE OF GRAVITY AND COAGULATION. 
44 

CHAPTER V 

CONCLUSIONS 

The aerosol depletion model given by equation (14) represents the 

action of a confined aerosol both in the presence and absence of gravity 

for the entire chamber. It was found that in the presence of a gravitational field, sedimentation was the dominant mechanism, even for small 

particles. When there is no gravitational field, coagulation is the 

most significant mechanism for removing particles from the bulk of the 

chamber. 

An advantage of the zero-gravity model is that it embodies the solution for the entire chamber. The more conventional technique of inner-outer expansions requires that two different problems be solved and 

the solutions then matched. The matching problem is likely to present 

difficulties comparable to those of solving for the higher order approximations. 

The results presented are those of the zeroth approximation. 

However, the higher order approximations would only yield a small correction to the zeroth approximation and thus may not be worthwhile in 

most cases. 

The situation in the presence of gravity yields an important result in the sharp boundary which moves downward through the chamber. 

This result is not unreasonable and is substantiated by the results of 

the one dimensional diffusion work. This work indicated that aerosol 
45 

particles do not diffuse very far from the location of the gradient 

which drives the diffusion. Nonetheless, diffusion is present and the 

boundary would be more or less blurred by it. 

The model is not complete in that it does not directly deal with 

the shifting size distribution of particles caused by coagulation. 

Fuchs [l] indicates that there has been no significant time dependence 

in the coagulation coefficient detected in the terrestrial experiments 

done to date. This is because the expected increase in the coagulation 

coefficient due to the increased particle size is almost exactly offset 

by the decrease in diffusivity of the larger particle. The diffusion 

coefficient for the new particle would be less, but this would have the 

effect of decreasing the effect of diffusive deposition on the particle 

concentration. Thus this effect should not require a major correction. 

The terminal settling velocity of the new particles would be greater 

and this could cause a problem for the gravity model. It is expected 

that for the lower initial concentrations and short times that the results of the gravity model would be valid because there would only be a 

few of these larger particles formed. An attempt to deal with this 

problem would follow the lines of research presented by Tolfo [9]. This 

would involve writing a model such as this for each size class and solving them simultaneously. Some further work that would be instructive 

would be to find the higher order approximations, although it is expected that they will not provide much additional information. 

There is much experimental work that needs to be done. Data are 

needed to validate this model. These data should take the form of 

aerosol concentrations for various times and locations within the chamber, both in the presence and absence of gravity. The values of the 
46 

coagulation coefficient and diffusion coefficient used in this model 

were theoretical. A zero-gravity experiment would be instrumental in 

obtaining measurements of these quantities. These measurements are 

severely hampered in terrestrial experiments by the action of gravity. 

Zero-gravity experiments could be designed which would provide direct 

measurements of these quantities, thus validating that portion of this 

mode& and advancing our knowledge of aerosol science. 
47 

BIBLIOGRAPHY 

[II 

[21 

[31 

141 

[51 

161 

173 

181 

r91 

Fuchs, N. A., The Mechanics of Aerosols, Permagon Press, New York, - 

1964. 

Richardson, J. F., Wooding, E. R., Chem. Eng. Sci., I, 51-59, 1957. 

Strauss, W., Industrial Gas Cleaning, Permagon Press, Oxford, 1966. 

Davies, C. N., Proc. Phys. SOC., 57, 4, 259-270, 1945. 

Millikan, R. A., Phys. Rev., 22, 1, l-23, 1923. 

Wilhelm, H. E., 3. Chem. Phys., 53, 5, 1677-1682, 1970. 

Wylie, C. R., Advanced Engineering Mathematics, McGraw-Hill, New 

York, 1975. 

Carslaw, H. S., Jaeger, J. C., Conduction of Heat in Solids, --- Clarendon Press, Oxford, 1959. 

Tolfo, Flavio, J. Aerosol Sci., S, 9-19, 1977. 
48 

AJ?J?ENDLXA 
49 

PWOORAM SOLUTN ~IN~~UT~O~JTPUT~IAPE~=INPUT~TAPE~=OU~PUT~ 

C 

~....,****..*******. l **..**.******.* 

C PROGRAM CALCULATES THE ZERDTH APPROXIMATION TO THE SOLUTION 

C OF THE AEROSOL DECAY EQUATION FOR A LEN0 GRAVITY ENVIRONMENT 

~.***..*.*~**.*~******.*~~*~~***~~** 

C 

C 

~-~~~:R~:~:,VARIABLES ARE INITIALIZE0 AND PHOGRAH OPERATION 

C 

c l l 

C 

10 

20 

30 

40 

COHMO3I/RH/TICE 

l l l * INITIALIZE 

TIHPRH=3600. 

CttHMAD=25, 

Ct’t’wT=SO. 

COhCl=l.E4 

IF=11 

IZ=ll 

RINC=.l 

ZItdC=.2 

PI=4.0ATAN(l.) 

PIO2=PI/2. 

PI02SG=P102*PI02 

ITEWuL=350 

ITERMR=500 

CONTINUE 

IPASS= 1 

HEAD (5,201 PIVICOAGID 

FORpAT l4FlO.O) 

IF 1EOFtS)l 270,30,270 

CONT IkUE 

WRITE (6,401 R,V~DtCOAG*TIHPWMrCONCI 

SOL 

SOL 

l SOL 

SOL 

SOL 

l SOL 

SOL 

SOL 

l SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

FORMAT ~llil~//~lOX~ 1UHlHIS OUTPUT IS FDR*//rl5X* 13HPARTICLE SIZESOL 

1*15X*1PE@.1*/r15X* 17HSETTLING vE~OCITY~~IX~I~E~~.~~/~~~X~ ZlHDIFFSOL 

ZUSION COEFFICIENT~7X~IPElI.4~/~15X~ 23HCOAGULATlON COEFFICIENJ*~XISOL 

31PEll.4*/*15X* 4HTIHE124X~OPFB.l~/~15X, 2lHINITIAL CONCENTRATIONISOL 

47X*IPCB.l.//t SOL 

C 

c l l 

C 

50 

60 

C 

c l l 

C 

70 

C 

c l l 

RETA=CHHHAD“V/D 

FAt~pA=C~HRAD*CHHRAD*COAG’CONCI/D 

TI~C=TIHPR~*D/ICHHRAD~CHMRAD) 

tlT4SO=RET4’Rt~TA/4. 

RTAOV2=RETA/Z. 

EXPRTA=ErP(-PETA’CHMHl/(2.*CHMRAD)l 

HTATIH=RTASO~TIHE 

XPTIt’E=EXP(RTATIH) 

GMATIM=GAtW4’TIHE 

TOGMAT=E..GMATIW 

l l l * SUM SERIES FOR 2 

z(lb=‘o. 

DO SO I=2312 

ZIIl=Z(I-l)*LINC 

CONTINUE 

00 60 I=l.IZ 

CALL SINES ~Z~l~rSERSIN~I~rIfEAHZ~ 

CONTINUE 

l l l l SUM SERIES FOR R 

RAD-0. 

DO 10 I=l*IH 

CALL RESSEL IRAOoSEHJO~I)rITERHH) 

RAO=RAD*HINc 

CONTINUE 

l l l l CALCULATE TOTAL SOLUTION 

SO; 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SDL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

c I-., -)VL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

10 

20 

30 

40 

50 

h0 

70 

80 

90 

100 

110 

120 

130 

140 

150 

160 

170 

180 

190 

200 

210 

220 

230 

240 

250 

260 

270 

280 

290 

300 

310 

326 

330 

340 

350 

360 

370 

380 

390 

400 

410 

420 

430 

440 

4so 

460 

470 

480 

490 

500 

510 

520 

530 

540 

55u 

560 

5 7 0 

580 

590 

600 

610 

620 

f-70 1-1 

640 

650 

660 

67U 

680 

b90 

700 

7-l u 

720 
50 

C <OL 

DO 110 I=lrIR SOL 

00 100 J=l*Iz SOL 

F~I*J~=SEH.J0~I~*SEPSINO SOL 

XEXP=BTAOV2*Z(J1-BTATIH SOL 

IF (XEXP.GT.600.1 GO 10 60 SOL 

tJlI~J)=F(I*Jl.EXP(XEYP) SOL 

CO~ICN~I~J~=U~I,J~/~~.*GMA~IM*U~I~J~~ SOL 

GO JO 100 SOL 

80 CON1 INUE SOL 

YHITF (6990) I,JIFIIIJ)~XEXP SOL 

90 FORMAT (15x* 4HI = */3r3Xe 4HJ = ,13*3X, YHFII*Jl = rlPE11.4,JXSoL 

I* 7btXEXP = tlPE11.4) SOL 

IPA.SS=-1 SOL 

100 CONTINUE SOL 

110 CONTINUE SOL 

IF fIRASS.LT.0) GO TO 200 SOL 

C SOL 

C l l l l l l CALCULATE DERIVATIVES FOR EPSILON SOL 

C SOL 

DO 120 I=l,IR SOL 

0U0R(l~I)=(U(191l-U(ZII))/(-RINC) SOL 

120 CON1 INUE SOL 

DO t40 1=2,IH SOL 

730 

740 

750 

760 

770 

7tlo 

790 

800 

810 

820 

630 

840 

850 

860 

870 

El80 

89U 

YOO 

YAO 

Y20 

Y30 

Y40 

9so 

Y60 

DO 130 J=l*IZ sol 970 

DIIDRII~J)=(U(I-~.J)-U(IIJ))~(-RI~C) SOL 980 

130 CONTINUE SOL Y90 

140 CONTINUE SOL 1000 

DO 150 I=l,IZ SOL 1010 

DUOz~I,l~=~U~I~l~-U(IIZ))/(-ZINC) SOL 1020 

150 CON1 INlIE SOL 1030 

DO 170 I=l,IR SOL 1040 

00 160 J=ZvIZ SOL lOS0 

DUDZtI~JJ=~U~I~J-l1-U~IIJ))I(-ZINCl SOL 1060 

160 CONTINUE SOL 1070 

170 CORTINUE SOL 1080 

C SOL 1090 

C l l l l l l CALCULATE EPSILON SOL 1100 

C SOL 1110 

DO 190 I=IeIR SOL 1120 

00 lb0 J=lrIZ SOL 1130 

EPSLON(I~JI=~TOGMAT/~I .*GMATIH*U~I~J~~~*~DUDR~I~J~*DUOR~~~J~S~L 1140 

1 l DIIDL(I*J)YDUDL(I,J)) SOL 1150 

180 CONTINUE SOL 1160 

190 CONTINUE SOL 1170 

C SOL 1180 

C l . l l l l OUTPUT VARIABLES SOL 1190 

C SOL 1200 

200 CONTINUE SOL 1210 

*F’ITE t6*21Ol SOL 1220 

210 FORMAT (15x1 SHCONCN 1 SOL 1230 

CALL TERMS (IR,IzvCONCNl 

WRITE (he2201 

220 FORMAT 115x1 7HEPSILONt 

CALL TERMS (IHIIZIEPSLON~ 

hRIiF: l6*230) 

230 FORMAT (15x9 1liU) 

CALL TERMS (IR,IZIUI 

YRITE (61240) 

240 FORMAT (15x1 4HDUDR 1 

CALL TERMS (IR,IZ~DUDR) 

VRITE (6,Z’SOl 

250 FORMAT (15x9 4HDUDZ 1 

CALL TERMS (IWeIZtDUOZl 

CRITE (612601 

260 FORMAT (15x1 IHF) 

CALL TFRMS (IR,IZtF) 

GO TO 10 

270 CONTIUUE 

STOP 

C 

EN0 

SOL 

SOL 

SOL 

SOL 

SOi 

SOL 

SOL 

SOL 

SOL 

SOL 

1240 

1250 

1260 

1270 

1280 

lZY0 

1300 

1310 

1320 

1330 

sol 1340 

SOL 1350 

SOL 1360 

SOL 1370 

SOL 13AO 

SOL 1390 

SOL 1400 

SOL 1410 

SOL 1420 

SOL 1430 

‘SOL 1440 
51 

C 

c l 

C 

c l 

C 

C 

c l 

C 

C 

c l 

C 

C 

c l 

C 

C 

c l 

C 

C 

SUBROUTINE BESSL (GNUwXeBESSELT BES IO 

l3L.s 20 

.a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..“ES 3u 

THZS SR CALCULATLS THE HESSEL FUNCTIONI J(X)* OF ORDER GNU HES 4u 

. . . . . . . . . . ..Y...Y ..(I . . . . . . . . . . . . ..“~S 5u 

LIES 6U 

DATA AA21/.25/mAA41/-. 375/,AA42/.kSb2S/rAA61/1.875/~ BE5 7d 

1 AA62/-1.15625/rAA63/.1171~‘75/,AAU1/-19.b~75/~AA~~/14.2265625/~ EES 00 

2 AAB3/-2.3H671~75/rAAd4/.U9~214~437/*AAlUl/3S4.3?~/~ BES 9u 

3 AA102/-2T7.87~/rAAIU3/5~.224bOY~H/~AAlO4~-4.~UU~MS93U~* HES 100 

4 ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ HES 110 

X ;~~EI’;.75/dM2/ -.35/r~963/.0125/rHHh~~-S.6~~/*~~~2/3.b026f~857l/*HES 120 

-.4241071429/rHR84/~.~60357k4~/~H~101/78.7~~~~~102/-~~./~ HE5 130 

7 60109/10.28645H33/rR0104/-. 34722222222L/~l3~1Ut~/3.03tTl94444E-3/ RES 140 

HES ISi, 

l l l l l INITIALIZE HES 16U 

HES 170 

PI=4.*ATAN(l.T HES 180 

PIOZ=PI/2. BES 190 

ALPRA’CNU*GNU-.25 RES 200 

T-l ./X HE5 210 

lSD~T*T RES 220 

RES 230 

l l l * l CALCULATE COEFFICIENTS BES 240 

AES 250 

P2=ALPHA@A421 #ES 260 

A4=tAA42”bLPHAeAA41) *ALPHA HE-3 270 

Ah=t~AAh3~ALPHA+AAb2)~ALPtiA*AA6l~*ALPHA HES 280 

A~=((IAAR4OAL~‘HA’AA~3)aALPHA~AA~2)*ALPHA~AA~~)oAL~HA HE5 2YO 

A10=~I((AA1050ALPHA*AAAO4~*ALP~A~AAlO3~~ALPHA*AAlU2~*ALP~A*AAlOll*BES 300 

1 ~l.Pti~ HES 310 

H2=HRZl’ALPHA RES 320 

R4=~RRL2”AL~HA*H841)*AL~HA HES 330 

R6=fIWHb3~PLPHA~BR62)“ALP~A*BH61~*ALPHA RES 340 

AB=f((HR~4*PLPHA+HBB3)~ALPHA*~0~2)’PLPHA*BR~l~~AL~~A RES 350 

B10=~~~~RR1050ALPHA*BRlO4~UALPHAIHB103)’ALPHA~~~lU2~~ALPHA*~BlOl~o~~S 360 

~ALPHA BES 370 

RES 380 

l l l l l CALCULATE INTERMEDIATE FUNCTIONS EES 390 

BE5 400 

8=~1~~AlO*TSfl~A~~~TS~*Ab~~TSD*A4~*TSD*A2~*TSD+l. PES 410 

PRTP~l=~~~~R10’TS0~8~~~TS~*~6~~TS~~B4~*lS~*~2l~lS~*I. BES 420 

PHI=PRTPHI/T-(GNU*.S)OPIOZ BES 430 

BES 440 

. l . l l CALCULATE BESSEL FUNCTION HES 450 

HES 460 

EIESSEL=R*SW7T~T/PIO2)QCOSo BE5 410 

RETURN RES 480 

HES 49u 

END HE5 500 
52 

SUORDUTINE HLAHDA (XLAHOAtITEWM) 

C 

C................*............. 

C FINDS THE FIRST ITERM ROOTS DF THE ZERO OHDER HESSEL 

~.......*.......U.............. 

C 

‘DIMENSION XLAMDA~ITERM) 

C 

C l **ooo XNITlALILE 

C 

PI=Q.‘ATAN(l.) 

A=2.4 

ERROR=1 .F-6. 

DO 60 I=lrITEWH 

R=P*l. 

X=B 

IF (A.LE.10.) CALL HESJ (A*O,XJOFAtl.E-6*IEH) 

IF (H.LE.lO.1 CALL HESJ (R~O~XJDFHI~.E-~~IEH~ 

IF (A.GT.10.) CALL HESSL (O..A+X.JDFA) 

IF tR.GT.IO.1 CALL BESSL (U.rR4XJOFH) 

10 CONT INUF 

IF tX.GT.10.) CALL HESSL (b.tX+XJOFXT 

IF (X.LE.10.) CALL RESJ (X~OBXJOFXI~.E-~,IERJ 

IF (XJOFA*XJOFX) 20150~30 

20 CONTINUE 

e-x 

XJOFR=XJOFX 

GO TO 40 

30 CO~JT INUE 

A=X 

XJOFA-XJOFX 

40 CONTINUE 

IF (ARS(XJOFX).LE.ERROH) GO TO 50 

X=(A”XJOFH-H*XJOFA)/(XJOFB-XJOFtT-XJOFAb 

GO TO 10 

50 CONTINUE 

A=Xej. 

XLAMDA(I)=X 

60 CONTINUE 

RETURN 

L 

EkD 

HLA 10 

HLA 20 

l l l l l HLA 30 

FTN RLA 40 

l l l l l HLA 50 

HLA 60 

HLA 70 

RLA 80 

HLA 90 

HLA 100 

RLA 110 

HLA 120 

HLA 130 

RLA 140 

RLA 150 

RLA 160 

BLA 170 

QLA 180 

HLA 190 

ALA 200 

HLA 210 

RLA 220 

HLA 230 

HLA 240 

HLA 250 

HLA 260 

HLA 270 

HLA 280 

HLA 290 

HLA 300 

HLA 310 

RLA 320 

HLA 330’ 

HLA 340 

HLA 350 

HLA 360 

RLA 370 

HLA 380 

HLA 390 

HLA 400 

BLA 410 

HLA 420 

SUBROUTINE SINES (Z,F,ITERM) SIN 10 

c c SIN 20 

C...................................~~t, 30 

C 11-11s SR CALCULATES 1tiE SIN SERIES, Fe FOR POSITION 2 SIN 4U 

C...........................-........~~~ 50 

C 

DIMENSION A(350 )rSINEt350 )vTERM(350 1 

COMMON/RR/TIME 

COMHON/ZZ/HETA~PItPIO2~PIO2SQ~HTASQ~EXPHTA 

F=O: 

XHNI=l. 

DO 10 ITRH=lrITEHH 

XMNl=XMNl*(-1.) 

XTRH=FLOAT (ITRH) 

~XPT~~~=fXP~-XTRM*XTHM*P102S~*1~M~1 

AWG=XTRY*l’IO2~2 

AIITRH~zXTHH*PIO2*~I.- XMN1*EXPHTA)/(HTASCl*XTRM”XTRMoPI02SO~ 

SINE~ITWM~=SIN~AHGl 

T~RM~ll~~Ml=AIITHM~*SINE~~THH)‘EXPTIH 

F=F*TERM(ITRM) 

10 CONTINUE 

RETURN 

C 

END 

SIN 60 

SIN 70 

SIN 80 

SIN 90 

SIN 100 

SIN 110 

SIN 120 

SIN 130 

SIN 14u 

SIN 150 

SIN 160 

SIN 170 

SIN 180 

SIN 1YO 

SIN 200 

SIN 21u 

SIN 22U 

SIN 230 

SlN 240 
53 

SUBROUTINE BESSEL (R~FIITERM~ nss 

C tss 

c**...**..*.**.**.*. . . l * l Q . l . l . . l l l l fjss 

C THUS SW CALCULAIES IHE JU SERIESI Fe FOR POSillON I? ass 

c l l l ..* l l l l 0 l l ..*.* l *...* **.*.*.4*.*~ss 

C HSS 

-OIH~STON.~XLWDA~~OO 1 *HE5SL1~500 ) *HESSLO ISOU I rTERMt500 19 RSS 

1 A1500 ) HSS 

COMHDN/RR/TIHE HSS 

F=O; 05s 

ERQDR=l.E-6 HSS 

CALL BLANDA (XLHDA~ITERH) RSS 

DO 80 ITRH=l~ITEHM HSS 

IF lXLHDA(IlRH).GT.lO.~ GO TO 20 HSS 

CALL RESJ lXLMDA~ITRH~*l*HESSLYorEHHOHIIEM~ RSS 

IF fIEW.NE.01 WRITE IbtlOT IEP~XL~~A~ITHH~~HESSLl~ITRMl 055 

10 FORMAT (10X, SHIER= 913~ YM LAHDA q rlPEll.4r bti Jl = rlPEll.41HSS 

GO TO 30 RSS 

20 CON1 IIIUE HSS 

CALL MESSL ~1.~XLHDA~ITHH~rBESSLl~ITRM~l tjss 

30 CONT INUF. HSS 

ARO=R’XLHOA I ITRH) HSS 

IF fPRG.ED.0.) GO TO 60 HSS 

IF (ARG.GT.lO.) GO TO 50 PSS 

CALL RESJ ~PRG~O,HESSLO~ITHM~~ERHOR,IEW~ BSS 

;F (1ER.NE.01 kHITE (6,4(J) IEHsAHG~BtSSLO(ITHH1 ASS 

40 FORMAT (10X* 6HIEH = 9139 TH AA0 = rlPE11.4~ 6ti JO = rlPE11.4) RSS 

GO TO 70 HSS 

50 CONTINUE HSS 

CALL PESSL (O.rARGsHESSLO(ITRH)T RSS 

GO TO 70 RSS 

60 CONTINUE 05s 

BESSl.O(ITRH)=l. BSS 

70 CONTINUE RSS 

HO CONTINUE RSS 

00 90 I=lrlTEti~ P s s 

A(1)=2./(HESSLl~I)*XLMDA~I11 PSS 

90 CON1 INUF HSS 

DO 100 ITRp=l~ITERH HSS 

~xRT~H=~xP(-XLMOA~ITRM~*XLMDA~ITHH)V~IHE~ RSS 

TFRH(~TH~)=A~ITHM)*HESSLOO”EXPTIW HSS 

F=F*TFRMtITHMT RSS 

100 CONTINUE RSS 

RETURN. RSS 

C RSS 

10 

20 

30 

40 

50 

60 

70 

80 

90 

1ou 

110 

120 

130 

14u 

IS0 

160 

170 

180 

190 

200 

210 

220 

230 

240 

250 

260 

270 

2HU 

290 

300 

310 

320 

330 

34d 

350 

360 

370 

3HO 

390 

400 

410 

420 

430 

440 

450 

HSS 460 ._ 
54 

SUBROUTINE TERMS (ITERHvJTERHeA) 

DIMENSION AfITERHwJTERH) 

hRITE (6.10) 

10 FORMAT 11) 

DO 30 K=lrJTERH 

rRIT@ .(6r20) fA(J*K)rJ=l~ll) 

20 FORMAT (3X.llllPE9.2~3Xl) 

30 CONTINUE 

IF !IlERt’.LE.ll) GO TO 70 

hHITE (6tlO) 

DO 40 K=~*JTERH 

kRfTE (6120) (A(JeK) eJ=l2*22) 

40 CONTINUF 

IF tITERH.LE.22) GO TO 70 

kAITE l6.10) 

DO 50 K=lrJTEHH 

kRITF (6920) tA(JsK) rJ=23,331 

50 CONTINLJF 

IF tITERH.LE.33) GO TO 70 

WRITE (6910) 

DO 60 K=l*JTERM 

WRITE (6120) (A(JvK) sJ=34r44) 

60 CONTINUE 

70 CONTINUE 

WHITE (6~80) 

no FORMAT i//b 

RETURN 

C 

EhD 

TER 10 

TER 20 

TEH 30 

TER 40 

TER 50 

TER 60 

TER 70 

TER 80 

TEH YO 

TEH 100 

TER 110 

TEH 120 

TEH 130 

TER 140 

TER 150 

TEH 160 

TER 170 

TEH 180 

TER 190 

TER 200 

TEH 210 

TEW 220 

TER 230 

TER 240 

TER 250 

TER 260 

TER 270 

TER 280 

TER 290 
r55 

APPENDIX B 
56 

PRDQRAM SOLUTN ~INPUT~OUTPUT,TAPE5=XNPUl~TAPEbmOUTPUTPUT~ SOL 

C SOL 

C.o***...‘* I) l ..***.******* l o.** o*o*.so~ 

: 

PROGRAM CALCULAlES THE ZEWOTH API’ROXIHATION 70 THE SOLUTION SOL 

OF THE AEROSOL DECAY EQUATION FOR A GRAVITY ENVIRONHfNT’ SOL 

cooo+ooooo’*o*.*****.*oboo+ooooooooooso~ 

C IN THIS PART VARIABLES ARE INIT.IALIZED AND PROORAH OPERATION SOL 

C tS CONTROLLED SOL 

c l o*.*.o*o*ooooo*o***oooo l oooooo*oo*soL 

C 

DIHCNSION FlIl~ll~~U~1l*Il~~EPSLON(llrll)r~(ll~rCONCNlll~ll~~ 

1 SEUSl~~ll~*DUOL~11*1l~*~UD~~ll,ll)rSERJO~ll~ 

COHMGN/ZZ/RETA,PI~PIO2~PIOi?SOtEXPHTA 

COMPON/RR/TIPE 

C 

C l l l l l * INITIALIZE 

C 

71HPR~=7200. 

CONC1~1.F5 

CHHR4D=ZS. 

ClJ’HHT=-50. 

H=CHWT/CHHRAD 

IP=lJ 

IZ=ll 

PINC=.l 

ZINC=.2 

pI=S.*ATAN(l,) 

p102=pr/2. 

PrO2SO=PI02*PI02 

ITEQH2=350 

IlERHp=500 

Z(l)=O. 

DO 10 I=Z,IZ 

Z(I)=Z(I-ll*ZINC 

10 CONTINUE 

20 CONTINUE 

YEAD (Sr30) R*V*COAGeD 

30 FORMAT (4FlO.O) 

IF TEOF(5) 1 21004Ov210 

40 CONTINUE 

WRITE (6e50) R~V~D~COAG~TIHpHH~CONCI 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SO FORMAT (l~lr//vlOX~ 1HHTHIS OUTPUT IS FOHe//,I5X, 13HpARTICLE 51 ZESOL 

lrlSXe1PFR.lr/rlSXt 17HSETTLING VELOCITY~llX*lPElI.4l/r15X* EIHOIFFSOL 

2USlOfJ CDEFFICIENT~7X~lPEl1.4~~~T5X~ 23HCOAGULATION COEFFICIENTI~XISOL 

31PE~1,4*/*15X* ~HTIHE,Z~XI’JPFB.~~/.~~X~ ZIHINITIAL CONCENTRATIONISOL 

47XtlPER.l*//) SOL 

RETA=QWRAD’V/D SOL 

GAPCA=CHHHAD*CHHHAD’COAG*CONCI/D SOL 

TIHC=T~~PHH~D/~CHHRAD*CH~RAD) SOL 

RTPSQ~HETA’HETA/Q. SOL 

tiTAOV?=RfTA/i’. SOL 

EXPGTA=fXPl-HETA’CHHHT/(2.*CHHRAD)) SOL 

HlATlM=RTASP’T1HE SOL 

HTTIt’F=I~FTA’TIHE SOL 

XPlI~E=fXplGTATIY) SOL 

GPATIM=GAWM~~TIHE SOL 

TOGHAT=2.*Gt4ATIH SOL 

C SOL 

c 0 . 0 l l l CALCULATE TOTAL SOLUTION SOL 

C SOL 

no 79 r=l.T~q SOL. 

00 60 J-l.12 SOL 

STPFNl=ZlJ)-I~TTIHE SOL 

STPFNZ=STPFNI-H SOL 

If (STPFNl.LI .O.l SPFNl=O. SQL 

IF (STPFNI.GE.0.) SpFNl=l. SOL 

IF (STPFN2.LT.0.) SpFN2’0. SOL 

IF (STPFN2.tE.U.) SpFNZfl. SOL 

U(T ,J)=SPFNl-SpfN2 SOL 

CDNCN(IIJ)=U(I~J)/(~.*GMATIM*U(I*J)) SOL 

60 CON1 INUE SOL 

r,., 

70 CONTINUE 3”L 

10 

20 

30 

40 

so 

60 

70 

BU 

90 

100 

110 

120 

130 

140 

150 

160 

17u 

180 

190 

200 

210 

220 

230 

24U 

250 

260 

270 

280 

290 

300 

310 

320 

330 

340 

350 

366 

3?U 

300 

390 

400 

410 

420 

430 

440 

45u 

460 

470 

480 

490 

500 

510 

520 

530 

540 

550 

560 

570 

580 

590 

600 

610 

620 

630 

640 

650 

660 

670 

666 

696 

100 

710 

720 
57 

DO 80 I=l*ll 

U~ltll~=O. 

Utllrl)=O. 

CONCN(Iell)=O. 

COKN(llrIl=O. 

GO COR:.INUE 

C 

C l l l l l l CALCULATE DERIVATIVES FOR EPSILON 

1 L 

DO 90 l=l,IR 

DUOR~1.l~=~U~l,I~-U~2~I~~/~-RINC~ 

90 CONTINUE 

DO 110 1=2*IR 

00 100 J=leIZ 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

DUDR~I*J~“~U~l-l~J~-U~l~J~~~~-RINC~ soi 

100 CONTINUE SOL 

110 CONTINUE SOL 

DO 120 I=l.lZ SOL 

DUDZlI,l~=~U~I~1)-U~IIZ))/(-ZINC) SOL 

120 CONTINUE SOL 

DO 140 I=lsIR SOL 

DO 130 J=2rIZ SOL 

DUDZll~J~~~U~ItJ-l~-U~I1J))((-ZINC~ SOL 

130 GOIJT INUE SOL 

140 CONTINUE SOL 

C SOL 

C l l w l l ’ CALCULATE EPSILON SOL 

C SOL 

00 140 J=lrIR SOL 

DO 150 J=lrIZ SOL 

IF ~ABS~DUDW~I*J~).GE.l.E27O.DR.A~S~DUD~~I~J~~.GE.l.E270~ GOSOL 

1 TO 150 SOL 

EPSLON~I*Jl=ITOGHAT/~l.*GMATIM*U~l*J~~~’~DUDR~l~J~*DUDR~I*JlSOL 

1 l OUnZ(I*J)*DUDL(I*J)) SOL 

150 CON1 I NUE SOL 

160 CONTINUE SOL 

C SOL 

C l l l l l + OUTPUT VARIABLES SOL 

C SOL 

CALL TERMS (IRIIZICONCN) SOL 

kR17E (6e 170) SOL 

170 FORMAT 115x1 ‘IHEPSILON) SOL 

CALL TERMS IIRrIZeEPSLON) SOL 

WRITE (6.180) SOL 

190 FORMAT (15X. 1HU) SOL 

CALL TERMS (IA,lZ,Ul 

WRITE (he1901 

190 FOHCAT 115X* 4HDUDH 1 

CALL TERMS (IW~IZ~OUDR) 

*RITE lhr200) 

200 FORMAT (15Xv CHOUDZ J 

GILL 1ERHS ~IH*lZ,DUoL~ 

GO TO 20 

210 CONTINUE 

STOP 

c 

FNII 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

SOL 

zm 

730 

740 

750 

760 

770 

780 

790 

800 

810 

820 

030 

840 

850 

860 

tl70 

8t3U 

090 

900 

910 

920 

930 

940 

Y5U 

960 

970 

YBU 

990 

1000 

1010 

1020 

1030 

1040 

1050 

1060 

1070 

1080 

1090 

11’00 

1110 

1120 

1130 

1140 

1150 

1160 

1170 

1180 

1190 

1200 

1210 

1220 

1230 

1240 

1250 

1260 

1470 

12tlo 

1 ?on 
1. REPORT NO. 2. GOVERNMENT ACCESSION NO. 

NASA CR-3384 

4. TITLE AND SUBTITLE 

Zero-Gravity Aerosol Behavior 

3. RECIPIENT’S CATALOG NO. 

5. REPORT DATE 

January 1981 

6. PERFORMING ORGANIZATION CODE 

7. AUTHOR(S) 

Harry W. Edwards 

3. PERFORMING ORGANIZATION NAME AND ADDRESS 

Department of Mechanical Engineering 

Colorado State University 

Fort Collins, Colorado 80523 

2. SPONSORING AGENCY NAME AND ADDRESS 

8, PERFORMING ORGANIZATION REPOR r i 

(0. WORK UNIT NO. 

M-332 

11. CONTRACT OR GRANT NO. 

NAS8-31673 

13, T’YPE OF REPOR’; & PERIOD COVEREI 

Contractor Report 

National Aeronautics and Space Administration 

Washington, D.C. 20546 1.1. SPONSORING AGENCY CODE 

IS. SUPPLEMENTARY NOTES 

Marshall Technical Monitor: George Fichtl 

Final Report 

6. ABSTRACT 

The feasibility and scientific benefits of a zero-gravity aerosol study in an 

orbiting laboratory have been examined. A macroscopic model was devised to deal 

with the simultaneous effects of diffusion and coagulation of particles in the confined aerosol. The particle depletion rate is given by a second-order, nonlinear 

partial differential equation. An analytical solution was found by treating the 

particle coagulation and diffusion constants as ensemble parameters and employing 

a transformation of variables. The solution was used to carry out simulated zerogravity aerosol decay experiments in a compact cylindrical chamber. The results 

demonstrate that the limitations of physical space and time imposed by the orbital 

situation are not prohibitive in terms of observing the history of an aerosol confined 

under zero-gravity conditions. 

While the absence of convective effects would be a definite benefit for the 

experiment, the mathematical complexity of the problem is not greatly reduced when 

the gravitational term drops out of the equation. Since the present model does not 

deal directly with the evolution of the particle size distribution, it may be desirable to develop more detailed models before undertaking an orbital experiment. 

Moreover, it was found that the' accuracy of measurement required to validate various 

kinetic models may not be presently available for the full range of aerosol particle 

sizes and concentrations of interest. 

7. KEY WORDS 

Convection 

Gravitation 

Particle size 

Aerosol decay 

S. SECURITY CLASSIF. (of thh report) 

Unclassified 

18. DISTRIBUTION STATEMENT 

Unclassified - Unlimited 

Subject Category 47 

20. SECURITY CLASSIF. (of thh pew) 21. NO. OF PAGES 22. PRICE 

Unclassified 61 A04 

For sale by National Technical Information Service. Swingfield. Virginia 2 * 16 1 

NASA-Langley. 1981