particle dynamics in the mixing of a particle-laden stream
TRANSCRIPT
Brigham Young UniversityBYU ScholarsArchive
All Theses and Dissertations
1970-6
Particle Dynamics in the Mixing of a Particle-Laden Stream with a Secondary Stream in a DuctWilliam Doyle CranneyBrigham Young University - Provo
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BYU ScholarsArchive CitationCranney, William Doyle, "Particle Dynamics in the Mixing of a Particle-Laden Stream with a Secondary Stream in a Duct" (1970). AllTheses and Dissertations. 7115.https://scholarsarchive.byu.edu/etd/7115
PARTICLE DYNAMICS IN THE MIXING OF A PARTICLE-LADEN
2 2 (5. onc jYI/ j 1 £
STREAM WITH A SECONDARY STREAM IN A DUCT
J
A T hesis
P resen ted to the
D epartm ent of M echanical Engineering
Brigham Young U niversity
In P a rtia l Fu lfillm ent
of the R equirem ents fo r the D egree
M aste r of Science
by
W illiam Doyle C ranney
June 1970
This th e s is , by W illiam Doyle C ranney, is accepted in its p re sen t
form by the D epartm ent of M echanical Engineering of Brigham Young U niver
sity a s satisfy ing the th e s is req u irem en t fo r the degree of M aste r of Science
/ U yu J 2-Q; )47DDate
Typed by K atherine Shepherd
ACKNOWLEDGMENTS
H ie au thor would like to thank those who have contributed to the
w riting of th is th e sis in any m an n er. Special thanks go to D r. John M.
Sim onsen fo r h is tim e req u ired to be the chairm an of the com m ittee , and
to Dr. L. Douglas Smoot for his many helpful suggestions and fo r the su p
p o rt received while pursu ing the re se a rc h of the th e s is m a te r ia l .
G rateful apprecia tion is a lso ex p ressed to my fam ily fo r th e ir
patience and support during the p a s t y ear, fo r th e ir encouragem ent was
alw ays th e re when needed .
iii
TABLE OF CONTENTS
CHAPTER PAGE
I. INTRO D U CTIO N ............................................................................................... 1
II. PARTICLE DRAG............................................................................................... 3
III. THERMAL RESPONSE............................................................................. 14
IV. PARTICLE DIFFU SIO N .................................................................................... 21
V. SIMILARITY AND CONTINUUM ....................................................... 27
VI. PRESENT M O D E L S .......................................................................................... 33
VII. CONCLUSIONS AND RECOMMENDATIONS............................................. 35
LIST OF FIGURES
FIGURE PAGE
1. Model C onsidered in T h e s i s ..................................................................... 1
2 . D rag Coefficient v e rsu s Reynolds N u m b e r ......................................... 6
3 . S im ilarity between D rag Coefficient and N usselt N u m b e r ........... 19
4 . M ass Ratio v e rsu s Mean D istance between P a rtic le s . . . . . . . . 31
v
CHAPTER I
INTRODUCTION
The purpose of th is th e s is is to review the h is to ry and d iscu ss
the p re sen t work being done in p a r tic le -g a s flow, with application to a i r -
augm entation of a p a r tic le -c a rry in g , fu e l-r ich gas je t . A draw ing of the
m odel considered is shown in F igure 1, with the a re a of in te re s t lab eled .
The th e s is is intended as a thorough lite ra tu re review and an a ly s is .
F ig . 1. - -Model considered in th esis
The re su lts of th is work a re to be used as a sou rce fo r the beginning
1
2
of an en tire model developm ent study. While th e re a re m any th eo re tica l
tre a tm en ts of p a rtic le dynam ics, the m ain so u rces for th is th e sis is e x p e r
im ental w o rk . These so u rces a re used a s e ith e r supporting o r refu ting
given analyses o r tre a tm en ts p re sen te d .
The thesis is the result of work supported by the Naval Weapons
Center at China Lake, California, under Contract No. N 60530-68-C-0626
to Brigham Young University. The contract deals with, specifically, the
mixing and combusting region of an air-augmented, particle-carrying gas
stream .
In th is co n trac t, re s e a rc h e rs deal with various a sp ec ts , including:
gaseous je t m ixing, p a rtic le dynam ics (rep resen ted by th is th e s is ) , p a rtic le
and drop le t com bustion, and the e n tire p rob lem as a whole, that is , m ixing
and com bustion of a p a rtic le -lad en je t with a secondary s tream in an enclosed
d uc t.
R elated to the trea tm en t of p a rtic le dynam ics a re sev e ra l im portan t
a r e a s . T hese include p a rtic le d rag , heat t ra n s fe r analy sis and p red ic tion ,
diffusion of p a r tic le s into the secondary s tre a m , s im ila rity of the p a rtic le
velocity and concentration p ro file s and valid ity of the continuum flow a ssu m p
tion, and finally , the p re sen t m odels availab le , including th e ir assum ptions
and trea tm en ts of the p reced ing to p ic s . Each one of th ese topics w ill be
covered a s individual ch ap te rs with a final ch ap te r to conclude and su m m a r
ize .
CHAPTER II
PARTICLE DRAG
The study of p a rtic le d rag has long been pursued ; yet in the a re a s
of tu rbu lence, com bustion, and acce le ra tio n , little knowledge o r ag reem en t
has been re a liz e d . The analysis seem s to be an ex trem ely com plicated
one. C onsisten t and com plete physical m easu rem en ts lim it any d rag a n a l
y sis in an effo rt to com e up with a clean analysis.
In the li te ra tu re , the d rag fo rce upon a sphere is m ost often
d esc rib ed as a non-dim ensionalized "d rag coefficient, " Cd . It is usually
defined, and w ill be in th is th e s is , as:
Cd : D / I V2 A 2
w here C^ = D rag coeffic ien t.
D = D rag fo rce on sphere due to m otion with re sp e c t to su rro u n d ing fluid, not including buoyant, g rav ita tio n a l, e le c tro s ta tic , o r o ther non-velocity dependent fo rc e s .
= D ensity of the surrounding flu id .
V = V elocity of sphere re la tiv e to undisturbed free s tre a m .
9A = P rojected fron ta l a re a of sp h ere , IT r .
The d rag fo rces upon a sphere have been defined and sep ara ted from
3
4
other fo rce s , such a s g rav ity and buoyant fo rce s , by allow ing only those
fo rces which a re a re su lt of the re la tiv e fluid sp h ere m otion . In th is con
tex t, the fo rces acting upon the sphere at r e s t a re not included as d rag
fo rce . This d rag is then te rm ed by Olsen (1)* a s "dynam ic d ra g ."
The d rag fo rces a re com posed of two physically d ifferen t fo rce s ,
a p re s su re fo rce and a sh ea r fo rce . The p re s su re fo rces ac t perp en d icu la r
to the su rface and m ust be in teg ra ted around the su rface of the sphere to
get the to ta l p re s su re fo rc e . The sh ea r fo rce s , the re su lt of the v iscous
natu re of the accom panying fluid, ac t p a ra lle l to the su rfa c e . As in the case
of the pressure forces, the shear forces must be integrated to get the result
ant sh ea r fo rc e . The to ta l dynamic d rag is , then, the sum of the in teg ra ls
of v iscous sh ea r fo rce and p re s su re force .
In 1850, G. G. Stokes o rig inated a pure ly th eo re tica l solution of
the d rag coefficien t a s a function of Reynolds n um ber. He solved a much
sim plified N avier-S tokes equation which In tu rn ex p ressed that the d rag force
is com posed of on e-th ird p re s su re fo rce and tw o -th ird s sh ea r fo rc e . His
solution neglected the in e rtia l te rm s in the equation, which becom e im p o r t
ant a t h igher v e lo c itie s (2). Stokes* analysis re su lted in a d rag coefficient
equal to tw enty-four divided by the Reynolds num ber based on sphere d iam e te r,
C j = 24 / R ejy The generally accepted range of accuracy is fo r Reynolds
num bers le s s than one, R e^ < 1 . 0 . In addition, S tokes’ flow assu m es that
^N um bers in p a ren th eses re fe r to lis t of re fe re n c e s .
the flow is incom pressib le and la m in a r.
5
In p ra c tic e , the m ost often used and b est known d rag coefficient is
the steady sta te d rag , usually p resen ted a s a "standard d rag" curve (2).
It re p re se n ts the re su lts of experim en tal data co llected fo r many y e a r s .
Though its values w ere generated only fo r steady sta te conditions, i ts a p p li
cations a re found in many non-steady sta te conditions.
Many re su lts of the significant studies of sphere d rag coeffic ien ts,
including Stokes' and the steady s ta te d rag cu rv e , a re shown in F ig u re 2 .
As noted e a r l ie r , some of these studies show a wide varia tion in co effic ien ts .
One of the early experimental investigators was R, G. Lunnon (3).
His w ork involved the dropping of sp h eres of va rio u s density and d iam e te rs
down a mine shaft. The d isp lacem ents and co rrespond ing tim es w ere
reco rd ed and p resen ted in an a r tic le published in 1926 (3). Since then,
sev e ra l investiga to rs have analyzed h is data in an effo rt to p red ic t the m o
tion of p a rtic le s subjected to d ra g .
As technology w as p ro g ress in g , industry becam e in te re s ted in the
flow of so lid -liqu id m ix tu res in p ip e s . The m ain concern was that of p r e
d icting the in c reased p re s su re lo ss due to the p re sen ce of the solid p a r t ic le s .
A com plicated an aly sis was not d e s ired no r req u ired in o rd e r to com e up
with the d esired p red ic tions . Soo (4) g ives a b rie f d iscu ssio n of th is e ra
in gas -so lid flow .
With the advance of rocket technology and in c reased th ru s t, a g rea t
im petus w as given to the study of sphere d ra g . The advent of c e rta in m eta ls
Fig. 2
.--Drag coefficient versus R
eynolds number
RUDINGER
O'REYNOLDS NUM BER, Re
to solid rocket fuels produced a p a rtic le -la d en exhaust s tre a m . These
p a rtic le s lagged the changing velocity and tem p era tu re of the gas in the
acce le ra tin g and cooling nozzle flow . This then caused lo sse s in expected
th ru s t . Because of the p a rtic le lo s se s , nozzle sections had to be designed
in a m anner that would b est optim ize the g a s-so lid flow . A knowledge of
the d rag upon p a rtic le s by the gas becam e im portan t.
The next parag rap h s d iscu ss , in chronological o rd e r , the s ig n ifi
cant w ork in p a rtic le d rag since 1954. Most of the following w ill a lso be
p resen ted in F igure 2 .
D rag coeffic ien ts for burning kerosene drops w ere rep o rted in 1954
by Bolt and Wolf (5). In th e ir experim en t, how ever, the Reynolds num bers
w ere ve ry low, le ss than 1 .0 . T here is a significant am ount of s c a tte r
p re sen t in th e ir da ta , a s shown in F igure 2 . The experim en t, how ever,
showed a slight d ec rea se in d rag coefficient from steady state fo r burning
d rops as com pared to non-burning d ro p s .
Among the f ir s t in v estiga to rs to rea lize the significance of lo sse s
generated in rocket nozzles due to so lid -gas lag w ere G ilbert, D avis, and
Altm an (6). In th e ir work, they analyzed the flow p ro p e rtie s of lin ea rly
acc e le ra te d g a s -p a rtic le flow s. The flow was analyzed to determ ine the
propulsion lo sse s resu ltin g from the velocity lag between p a rtic le s and the
accom panying g a s . They assum ed that the p a r tic le lag, due to the a c c e le r
ating gas s tre am , followed accord ing to S tokes’ d rag law . T h e ir m odel was
a lso one-d im ensional and th e re w ere no in te rac tio n s between the w alls and
8
the flow. It was an e a r ly f i r s t step in rea liz in g the im portance of tw o-phase
flow lo sse s in n o zz le s .
Ingebo, in 1955, published experim en tal w ork which was concerned
with the d rag of a cloud of p a r tic le s acce le ra tin g in an a i r s tre am (7). In
h is w ork he used both solid and liquid sp h e re s . The acce le ra tio n s reached
by the p a rtic le s approached upw ards of 60,000 feet p e r second sq u a red .
The m ajo r conclusion of the experim ent w as that fo r a Reynolds num ber
betw een six and four hundred, and a sphere d iam e te r between 20 and 120
m ic ro n s , the em p irica l d rag coefficient is the following:
Cd = 27 / ReD 0 .8 4
His curve app ears with those of o ther im portan t ex p erim en te rs in F igure 2.
As stated p rev iously , the d rag of a sphere is due to two fo rce s , a
sh ea r fo rce and a p re s su re fo rc e . The sh ea r e x is ts due to the viscuous
natu re of the fluid with which the sphere is flow ing. In the case of the p r e s
su re d rag , the flow surrounding the sphere is unable to com pletely re c o v e r
the p re s su re on the back side of the sphere a s w as p re sen t on the front s id e .
This d ifference in p re s su re cau ses a ne t fo rce in the opposing d irec tio n .
A m ain fac to r in the degree of p re s su re reco v ery is the ab ility of the flow to
rem ain attached around the p eriphery of the sphere a t in c reasin g v e lo c itie s .
If the flow is able to attach to the r e a r su rface , the d rag is d ra s tic a lly
reduced . On any aerodynam ic body the d e s ired response is that of the flow
rem ain ing attached to the r e a r su rface of the object, which red u ces the d rag
to a m inim um . A erodynam ic bodies a re tapered in o rd e r to reduce th is
9
se p a ra tio n .
As in the c ase of the tran s itio n from a lam in a r to a tu rbu len t bound
a ry lay er on a fla t p la te , the sam e event occurs on the su rface of a sp h e re .
The effect of a tu rbu len t boundary la y e r on a sphere a t a Reynolds num ber of
approx im ately 10^ is to cause the flow around the sphere to rem ain attached
fu r th e r around the p e rip h e ry . This allow s a g re a te r p re s su re reco v ery ,
which reduces the d rag fo rc e . This is the explanation fo r the sudden drop
in d rag coefficient v e rsu s Reynolds num ber fo r steady sta te d rag at a value
of Reynolds num ber approxim ately equal to 10^, a s shown in F igure 2.
Roughness on the su rface of a sphere can cause th is sam e t r a n s i
tion to occur, but e a r l ie r than o th erw ise . F o r the p ro p e r range of Reynolds
n u m b ers , roughness can have the effect of reducing the d rag on a sp h e re .
The p rim e exam ple is the golf b a ll. The dim ples cause the boundary lay er
to becom e tu rbu len t e a r l ie r than would a sm ooth sphere , and thus, fo r the
p a r tic u la r range of Reynolds num bers encountered by the ball, the d rag is
reduced , whereby the ball goes much fa r th e r in fligh t.
In addition to surface roughness, the action of the free s tre am tu rb u
lence in troduces turbulence into the boundary la y e r , the sam e effect as s u r
face rou g h n ess. F ree s tream turbulence cau ses the boundary lay er on the
sphere to becom e tu rbu len t e a r l ie r than n o rm a l. Realizing th is , Torobin
and Gauvin (8) attem pted to find possib le c o rre la tio n between free s tream
turbulence in ten s itie s and p a rtic le d ra g . In th e ir a r t ic le , published in 1960,
10
they sta te that in te re s t in th is a re a of p a rtic le d rag w as due to a m ajo r d iff i
culty encountered in d rag ex p e rim en ts . Investigato rs could not reproduce
p revious re su lts with any significant degree of a ccu racy . In m ost of these
c a se s , the effect of free s tream turbulence had been neg lected .
As might be suspected , the re su lt of th e ir work with turbulence
showed that " in c reasin g in ten sitie s cause a sy stem atic re g re ss io n of the
tran s itio n region of the d rag coefficien t curve tow ards low er Reynolds num
b e rs , together with a m oderate in c rease of the d rag coeffic ien ts fo r both
the su b critica l and su p e rc ritic a l Reynolds num bers" (8). F igure 2 shows the
range of th is transition region described by Torobin and Gauvin,
A study of the d isp lacem ent and sh a tte rin g of p a r tic le s w as w ritten
by Ranin, e t a l . (9) in 1960. In th e ir experim en t, a shock tube w as used to
de term ine the p a rtic le b reak -u p c h a ra c te r is t ic s and position data . The
re su lts showed that the d rag coefficient w as reasonab ly c lose to that of steady
s ta te fo r sm all p a r tic le s , but p a r tic le s with la rg e r d iam e te rs had a s ig n ifi
cant in c rease in d rag . It was shown p ic to ria lly tha t these la rg e r p a r tic le s
w ere deform ing into d is c s . This deform ation began to occur a t sphere d ia m
e te r s g re a te r than one hundred m ic ro n s . The re su ltin g d rag coeffic ien ts fo r
the la rg e r p a r tic le s w ere shown to approach the coefficien ts fo r that of d iscs
of the sam e frontal a re a . T his fact w as noted in the a r t ic le , but is o v e r
looked by many who quote h is data a s a com parison fo r d rag c o e ffic ien ts .
The effect of com bustion and acce le ra tio n of the d rag coefficient
was dealt with by Crow e, N icholls, and M orrison in 1963 (10). T h e ir
11
experim ent consisted of burning p a rtic le s (gunpowder) and non-burning
p a rtic le s flowing in a shock tu b e . F o r th e ir experim en t they s ta te that the
ra te of burning did not produce any significant change in d rag coeffic ien ts
of the burning p a r t ic le s . The sam e was tru e of the effect of acce le ra tio n
upon d rag , w here little change was no ted . At the conclusion of th e ir a r tic le ,
they com pare th e ir data and re su lts with o ther prom inent investig a to rs in
the field of d rag co effic ien ts . The w ork of Ingebo and of Torobin and Gau-
vin ag reed reasonab ly with th e ir conclusive position , while Rabin, et a l . ,
was sta ted to have been in some e r r o r . The au tho rs fa il to rea lized , a s has
been pointed out p rev iously , that much of Rabin's data is for deformed
sp h eres which have a la rg e r d rag coefficient than the s tandard d rag .
Rudinger (11), in 1963, rep o rted some additional experim en ts with
clouds of p a rtic le s in a shock tu b e . His data and co rrespond ing an aly sis
give a resu ltin g d rag coefficient of:
Cd = 6000 Rep)’ 1 *7
A plot of th is equation ap p ears with that of o th e rs in F igure 2.
A very recen t a r t ic le , w ritten in 1968 by Selburg and N icholls (12),
d iscu sses experim en tal w ork with burning and in e r t sp h e re s . A shock tube
was used to determ ine the d rag coefficien ts of the p a r tic le s under o b se rv a
tion . In th e ir experim ent, the d rag coefficien ts calcu la ted w ere slightly
h igher than the standard d rag cu rv e , e sp ec ia lly fo r the case of the burning
sp h e re s . In analyzing the reaso n s fo r the d ifference in d rag over the
12
correspond ing steady state curve fo r smooth sp h eres , the au th o rs concluded
by photographic exam ination that the p a r tic le s have rough su rfa c e s . On th is
point an assum ption was made that quite possib ly the su rfaces on p a rtic le s
in rocket m otors have a roughness, a lso causing an in c re a se in expected
d rag of the p a r t ic le s . This would be advantageous, because the p a r tic le s
would then not lag the acce le ra tin g gas s tream to the sam e degree a s would
smooth sp h e re s .
D uring the sea rch of the li te ra tu re for significant a r t ic le s on p a r
tic le d rag , two survey a r t ic le s w ere found to be very u se fu l. Hoglund (13)
w rote of recen t advances in g a s-p a rtic le nozzle flow . His a r tic le was pub
lished in 1962 and co v ers the period until approxim ately that tim e . The
a r tic le is quite generally re fe ren ced by o ther m ore recen t a r t ic le s on the
su b jec t. In 1965, S. L . Soo (4) w ro te about the h is to ry and app lications of
so lid -gas flow. In th is publication, the a r t ic le s re fe ren ced and d iscu ssed
by Soo a re "intended to be illu s tra tiv e ra th e r than com prehensive" (4).
This com pletes the l i te ra tu re review section of p a r tic le d ra g . The
rem a in d er of th is chap ter w ill be devoted to the recom m endation of a d rag
coefficien t su itab le to a ir-au g m en ted com bustion and m ixing.
Though the problem of a suitable d rag coefficien t has been an old
one, it s ti l l rem ains a form idable one. The decision to accep t any d rag
coefficient is dependent upon the range of Reynolds num bers encoun tered .
In addition, item s such a s the turbu lence and burning c h a ra c te r is t ic s of the
gas and p a rtic le s play a significant ro le in de term in ing the b est co effic ien t.
13
The f i r s t im portan t m a tte r is then to find the range of Reynolds
n u m b ers . In the scien tific l i te ra tu re , th e re ex ist sev e ra l one-d im ensional
m odels of g a s -p a rtic le flow sy s te m s . These m ay be used to p red ic t the v e l
ocity lug and co rrespond ing Reynolds num ber between the p a rtic le and the
gas s tre a m . One com puter p rog ram w as used which involved an an aly sis
of rocket nozzle perfo rm an ces with solid p a r tic le s p re sen t in the exhaust
(14). T his p ro g ram shows that fo r a m axim um lag between the p a rtic le s
and the gas, the re la tiv e Reynolds num ber is about equal to tw en ty -four.
This is then a rough num ber to w ork with in evaluating a suitable d rag co effi
c ien t.
T rad itionally , in the field of rocke t propulsion , the d rag coefficient
used fo r liquid d rop le ts is that of Ingebo (7). The steady state d rag curve
and the d rag form ula of Ingebo follow a lm o st exactly a t v e ry low Reynolds
num bers, which is the range in our p rob lem accord ing to the p rev ious d is
cu ssio n .
R ese a rch e rs in the field com pare and d iscu ss v a ria tio n s between
th e ir own individual re su lts and the steady s ta te d rag cu rv e . An inheren t
conclusion is that the steady s ta te d rag curve is s till considered a s the valid
d rag coeffic ien t. Because of the c lo sen ess of the steady s ta te curve to that
of Ingebo, and the need fo r a form ula which can be calcu la ted , the choice of
d rag form ula fo r our m odel is that of Ingebo, = 27 R e ^ ”®*®^. A close
exam ination of F igure 2 w ill b e a r out th is conclusion, noting the range of
Reynolds num bers up to approxim ately tw enty-five.
CHAPTER III
THERMAL RESPONSE
In the analy sis of the a ir-au g m en ted m odel, tem p e ra tu re response
of the p a r tic le s is im portan t. In o rd e r for the p a r tic le s to burn they m ust
f i r s t heat up, then ignite and continue to b u rn . The ignition and com bustion
p ro c e sse s a re not covered in th is th e s is ; only a considera tion of the heating
period w ill be analyzed .
At the beginning of the section being analyzed, the fu e l-r ic h , g a s -
p a r tic le s tream s ta r ts to mix with the ox id izer (the secondary s tre a m ) . As
the fu e l-r ich gas m ixes with the ox id izer it com busts, ra is in g the te m p e r
a tu re of the gas m ix tu re . Because of the p a r tic le m ass and a sso c ia ted heat
capacity , they w ill not change tem p era tu re a s rap id ly as the surrounding
gas and, th e re fo re , exhibit a tem p e ra tu re d ifference, o r, a s com m only
re fe r re d to, a th e rm a l lag . This is analogous to the velocity lag caused
by the d rag , p reviously d iscussed in th is th e s is . D rag involved a tra n s fe r
of m om entum due to a velocity d ifference while the heat tra n s fe r involves
a t ra n s fe r of energy due to a th e rm a l d iffe ren ce . It w ill be shown la te r how
the s im ila rity between the two co m p ares .
S im ila r to the non-dim ensionalized d rag coeffic ien t, C^, the heat
14
15
tra n s fe r to a p a rtic le is usually c h a rac te riz ed by a d im ension less num ber
called the N usselt num ber, Nu . F o r the p a r tic u la r c a se of the heat t r a n s
fe r to a sph ere , the N usselt num ber is defined as the following:
Nu = hc D / kf
w here hc
D
kf
q
The convective heat tra n s fe r coeffic ien t, q / A (Tg - T^)
The d iam e ter of the s p h e re .
The th e rm al conductivity of the surrounding flu id .
The ra te of heat t ra n s fe r , BTU / H r.
A ; The surface area of the sphere,
T s = The su rface tem p e ra tu re of the sp h ere ,
Tf * The tem p era tu re of the surrounding flu id .
Because many d ifferen t convective heat t ra n s fe r coeffic ien ts ex is t,
it is noted that th is coefficient is based upon the d ifference in tem p era tu re
between the instantaneous tem p e ra tu re of the sphere , taken a s a lumped
th e rm a l capacitance, and the bulk tem p e ra tu re of the surrounding flu id .
The heat t ra n s fe r w ill vary as the tem p e ra tu re of the sphere v a r ie s .
If the fluid is considered as infinite and a t re s t , an analy sis s im ila r
to the d rag calcu lation fo r Stokes flow gives a value fo r the N usse lt num ber
equal to 2 .0 (15). The N usselt num ber equal to 2 .0 is som etim es re fe rre d
to as N usselt num ber based upon Stokes flow . Just as fo r the d rag co effi
c ien t in Stokes flow, the heat t ra n s fe r coefficien t is a ccu ra te only fo r v e ry
low Reynolds n u m b e rs .
16
One of the ea r ly c o rre la tio n s of heat tra n s fe r data w as m ade by
F ro ess lin g (16) and w as la te r m odified by McAdams (17) to give an e m p ir
ica l relationship between the Nusselt number, Reynolds number, and
Prandtl num ber fo r lam in ar flow . The c o rre la tio n s w ere m ade so that fo r
v e ry low Reynolds num bers the N usselt num ber approached 2 .0 , ju s t a s
theory p red ic ted . The following re p re se n ts the co rre la tio n of M cAdams:
0 .5 0 .33Nu = 2 .0 + 0 .6 Re Pr
where; Re u The Reynolds number based upon sphere diameter.
Pr - The P randtl n um ber.
Hsu and Sage (18), in 1957, and la te r Sato and Sage (19), in 1958,
ran experim en ts to de term ine heat t ra n s fe r to s ilv e r sp h eres behind a
g rid and a lso in the wake of a free je t . They stated tha t turbulence in tensity
has a m arked effect upon the ra te of heat tra n s fe r a t Reynolds num bers
g re a te r than about four thousand. The re su lts in th e ir a r t ic le s showed as
m uch a s a tw o-fold in c rease fo r turbulence in ten s itie s of 15 p e r c en t.
A v e ry recen t a r t ic le , w ritten in 1967 by L avender and Pei (20),
adds significantly to the field of heat tra n s fe r to a sphere in a tu rbu len t
flow fie ld . In th e ir w ork an experim en ta l apparatus w as built to c o rre la te
and p red ic t heat t ra n s fe r ra te s from sp h eres in tu rbu len t flow f ie ld s .
In an e ffo rt to include the effects of a ll possib le v a r ia b le s , a g re a t
17
num ber of physical m easu rem en ts w ere m ade in th e ir ex p erim en ts . The
turbulence in tensity was found to be a dom ineering fac to r in considering
the ra te of heat tra n s fe r from sp h e re s . They analyzed the data from which
developed an equation which p red ic ted the heat tra n s fe r fo r th e ir own e x p e r
im ents .
In form ing the form ula fo r c o rre la tio n , the au thors s ta r te d with
the basic equation for lam in ar flow of McAdams (17):
Nu = 2 .0 + 0 .6 Rg0 ,5 Pr 0 ,33
To the Reynolds number the authors added a turbulence factor, the turbu
lence intensity, to get an equation in the following form:
Nu = 2 .0 + A Re0 *5 Pr 0 ,33 ReTB
w here: A and B a re constan ts to be fitted to the d a ta .
Rg,p = The tu rbu len t Reynolds num ber, Re , x Lp.
Iy = The turbulence in tensity , / Vi
V| = The instan taneous velocity of the s tre a m .
V = The average of the instan taneous v e lo c ity .
Just as Torobin and Gauvin (8) found that the tran s itio n from lam in a r
to tu rbu len t flow affected the d rag quite d ra s tic a lly , Lavender and Pei found
that the heat tra n s fe r was a lso affected by th is tra n s itio n . Two se ts of
values fo r A and B had to be used, one fo r the heat tra n s fe r before tran s itio n
18
and the o ther se t a f te r tra n s itio n . A plot of the re su ltin g equations and the
data, in th e ir a r t ic le , showed that the p red ic tion of heat t ra n s fe r had a
range of e r r o r of ^ 10 p e r c en t.
D uring these te s ts made by Lavender and Pei, the P randtl num ber
w as v e ry c lose to constant in v a lu e . T h e re fo re , the au thors included th is
value into the re su ltin g coeffic ien ts, A and B. If the m issing Prand tl num
b e r w ere to be re in s ta ted into the equation, the resu ltin g values fo r A and
B w ere:
A = 0.717 B - 0 .035 Rprp < 1000
A = 0 .1 6 6 B = 0 .225 ReT > 1000
An im portan t com parison between the d rag coeffic ien t and the N us-
se lt num ber can be made a t low Reynolds n u m b ers . If, in each c ase , the
actual coeffic ien ts can be divided by the Stokes coeffic ien ts, the re su lt
would give a com parison of the two m odes of t ra n s fe r , energy and m om entum .
The re su lts of ju s t such a com parison appear in F igure 3 .
In observ ing the plot of these two ra tio s , a definite s im ila rity is
seen to e x is t . This s im ila rity does not continue a t h igher values of R ey
nolds num ber, how ever, because the m echanics of the two m odes of t r a n s
fe r change ap p rec iab ly .
In a d iscussion of the heat t ra n s fe r to a sp h ere , there ex ists not
only the convective heat t ra n s fe r to co n sid er, but a lso the in te rn a l te m p e ra
tu re d is tribu tion with its a sso c ia ted heat t ra n s fe r an a ly s is .
In m ost a r t ic le s review ed, the su rface and c en te r te m p e ra tu re s
Cd
stea
dy
stat
e
Cd
stok
es
19
F ig . 3 . - -S im ila rity between d rag coefficien t and N usselt num ber
RE
YN
OLD
S
NU
MB
ER
20
w ere usually tre a ted as being equal. This tre a tm en t w as usually re fe rre d
to as a lumped capacitance m odel, w hereas the en tire m ass of the p a rtic le
was considered to be at one tem p era tu re only, a t any in stan t in tim e .
The following m athem atica l an a ly sis w as m ade to d isco v e r the
valid ity of the assum ption of a lumped th e rm a l cap ac itan ce . F o r alum inum
2p a rtic le s with a d iam e te r equal to 10 jx and a th e rm a l diffusivity of 1 .6 cm /
sec , the tim e req u ired fo r the cen te r of the p a rtic le to reach 99 p e r cent of
the su rface tem p era tu re with a step in c rease in tem p era tu re a t the su rface
-8w as 8 .15 x 10 seconds.
C hannapragada, e t a l . (21), rep o rted that in th e ir th eo re tica l model
-4of an a ir-au g m en ted rocket m oto r, the mean residence tim e w as l x 10
sec o n d s . In considering the sh o rtn ess of tim e req u ired fo r the p a r tic le s to
assu m e a c en te r tem p era tu re the sam e as tha t tem p era tu re a t the su rface ,
the assum ption of a lumped th e rm al capacitance is considered a valid one.
CHAPTER IV
PARTICLE DIFFUSION
P artic le diffusion in a tu rbu len t gas s tre a m is one of the im portan t
top ics re la ted to p a rtic le dynam ics. C onsidering the system of two coaxial
je t s tre a m s , p a r tic le s which em erge from the c en te r o r p rim ary s tream
w ill sp read o r diffuse into the ou ter o r secondary s tream as they p ro g re s s
down the duct length . The am ount of sp read w ill de term ine v ario u s concen
tra tio n s a s a function of positio n . It w ill a lso determ ine the am ount of oxy
gen availab le to the burn ing p a r t ic le s . This an a ly s is , then, w ill affect the
n e ce ssa ry duct length req u ired fo r the com bustion of the p a r t ic le s .
The diffusion of p a r tic le s is accom plished through tu rbu len t m ix
ing . In re g a rd to the phenomena of tu rbu lence, little is known of the causes
w hile m uch is being learned about the e f fe c ts . T urbulence has usually , in
tim es p a s t, been tre a ted as a c h a ra c te r is tic random ness asso c ia ted with
the fluid velocity and d ire c tio n . It g re a tly enhances any m ixing p ro c e ss ,
such a s the one being co n sid ered .
In the g a s -p a rtic le li te ra tu re the diffusion of p a r tic le s is usually
tre a ted a s a function of the diffusion of the accom panying g a s . The m ost
common p o rtray a l of p a rtic le d iffusivity is in the form of a ra tio of the
21
22
diffusivity of the p a r tic le s to that of the g a s . If the p a r tic le s follow the gas
exactly in its diffusion, then the ra tio of each diffusivity is one. If, how
e v e r , the p a rtic le s exhibit a c h a ra c te r is tic lag , due to the d rag and in e rtia
of the p a rtic le m a ss , then the ra tio m ay u ltim ate ly approach z e ro . A m a th e
m a tica l ex p ression of the p reced ing is as follows:
0 < £ p / £ g < 1
w here: Gp = the eddy diffusivity of the p a r tic le s , and
£ g = the eddy diffusivity of the g a s .
The v a ria b le s affecting the value of the ra tio should include the
sam e v a riab les which affect the d rag coefficient of the p a r t ic le s . Some of
these include: the size of the p a r tic le s , the density of both the gas and the
p a r tic le s , the v isco sity of the gas s tre a m , and the size and frequency of
the fluctuations in the turbulent gas s tre a m .
One of the e a r l ie r tre a tm en ts of the diffusion of p a r tic le s in a
tu rbu len t gas s tre am was that of Longwell and W eis (22). T h e ir a r t ic le ,
published in 1953, m akes an approxim ation to the velocity fluctuations by
assum ing a sinusoidal fluctuation of gas v e lo c ity . The d rag of the p a rtic le
is a ssum ed to be given by Stokes d rag la w . Without going into the theory
of the approxim ation h e re , the re su lt of Longwell and Weis shows that the
ra tio of the eddy diffusivity of the d rop , £ , to that of the gas, £ , is equalJr o
to:
23
fcp / £ g = b2 / (w2 + b2)
w here: b = 3 x /a x dia . of p a rtic le / m ass of p a r tic le .
w - Tlie frequency of osc illa tions of the p a r tic le s and the g a s .
p ; The v isco sity of the g a s .
The a r tic le il lu s tra te s the use of th is equation by assum ing a k e ro
sene drop d iam e te r of 45 m icrons flowing in a fully developed a ir s tream
with a velocity of 300 feet p e r second. The frequency is ca lcu la ted , fo r a
s ix -inch d iam e te r duct, as equal to approxim ately 300 rad ians p e r second .
With a ll of the above p a ra m e te rs ca lcu la ted , the ra tio of d iffusiv ities tu rn s
out to be equal to 0 .3 5 . L a te r , c lacu lations using th is technique w ill be
shown which cover the range of p ra c tic a l im portance to the p a r tic u la r a i r -
augm ented model under s tu d y .
R egarding the diffusivity of the p a r tic le s , Longwell and Weis d is
covered that th e re is not any "su b stan tia l change of diffusivity with a fo u rteen
fold change in absolute p re s su re " and that the "Reynolds num ber is not a
p ro p e r c o rre la tin g function fo r eddy diffusivity" (22).
Soo, Ih rig , and Kouh (23) w ro te , in 1960, about tu rbu len t two-
phase m otion . They experim ented with the diffusion of solid p a r tic le s in
a duct of th re e -sq u a re -in c h c ro s s sec tio n . In th e ir w ork, the ve lo c ities
ranged from twenty to one hundred feet p e r second . F o r p a r tic le s of the
size of 100 and 200 m icrons in d iam e te r , and a t the la rg e r v e lo c ities
24
encountered , the ra tio of p a r tic le eddy diffusivity to gas eddy diffusivity
w as found to be approxim ately equal to 0 .0 2 . The re la tiv e ly low value was
the re su lt of la rge in e rtia s of the p a r tic le s which tend to make the p a rtic le s
follow the bulk flow of the s tream ra th e r than the tu rbu len t fluctuations
p re s e n t. One of the re su lts of the te s t was noted in that fo r p a rtic le loadings
of up to 0 .06 pounds of p a r tic le s p e r pound of gas the p a rtic le s have little o r
no effect upon the flow of the gas s tre a m . The gas then behaves a s though
the p a rtic le s w ere not p re sen t at a l l .
One of the m ore recen t a r t ic le s was w ritten in 1966 by Goldschm idt
and Eskinazi (24). T h e ir a r tic le dealt with the diffusion of solid p a r tic le s
in a plane j e t . At f i r s t it seem ed to be a significant contribution to the
p rob lem , but proved of little value because of the range of v a riab les in the
te s t . The p a rtic le s used had an average d iam e te r of th re e m icrons and
the velocity of the gas a t the g re a te s t value w as approxim ately 35 feet p e r
second . Because of the re la tiv e ly low velocity and sm all p a rtic le d ia m e te rs ,
the p a r tic le s tended to follow the s tre am exactly during the diffusion p r o
c e s s . T his gave a value fo r the ra tio of eddy diffusivity of the p a r tic le s to
that of the gas equal to o n e .
G enerally , in the a re a of p a rtic le diffusiv ity , ve ry little experim en tal
data have been found to c h a ra c te r iz e p a rtic le eddy diffusion coeffic ien t.
As sta ted p rev iously , Longwell and W eiss theorized that the ra tio
of the eddy diffusivity of the p a r tic le s to that of the gas could be es tim ated ,
using th e ir assum ptions lis ted (22). U sing th e ir re su ltin g equation given
25
prev iously , a gen era l deg ree of ab ility to p red ic t th is ra tio can be a tta ined
fo r an a ir-augm en ted ro ck e t.
If a frequency of 1000 rad ians p e r second w ere to be im p ressed
upon both a one - and ten -m icro n d iam e te r p a r tic le , the ra tio of d iffusiv i-
t ie s would be as follows:
Size- P — g-
0.735
10 ^ 0 .004
The ratio, 6p/€g> is approximately inversely proportional to the
fourth power of the d iam e te r, by the analy sis of Longwell and W eiss . The
trend only can be pred ic ted and proven technique p resen tly ex is ts fo r p r e
dicting the p a rtic le eddy diffusivity with random turbulence . No e x p e r
im ental data a re availab le which encom pass the range of Reynolds n u m
bers in th is th e s i s .
F o r confined je t so lutions, w here is a req u ired p a ra m e te r , it
is recom m ended that the ra tio of €.p/ fcg be v a ried between ze ro and unity
until optimum agreem ent with m easu red system p a ra m e te rs is obtained.
If th is p roves to be too la rge a ta sk , the m ethod of Longwell and W eiss can
be used to estim ate £ p /£ g . In using th is m ethod, the frequency could be
found on the assum ption that the la rg e s t eddy of concern w ill be equal in
s ize to the p rim ary flow channel d ia m e te r . Com bining with the c o r r e s
ponding velocity of the je t w ill then produce a frequency req u ired in the
calculation of d iffusiv ity .
26
In conclusion, w ork is now only being undertaken (25) and m ay at
som e la te r date contribute to the p red ic tion of the eddy diffusivity of p a r
tic le s in the spread ing of p a r tic le s en tering from one duct and diffusing into
ano ther confined duct.
CHAPTER V
SIMILARITY AND CONTINUUM
S im ilarity in p a rtic le dynam ics involves the ac t of non-dim ension -
a liz ing im portan t physical v a riab les so that item s such a s velocity v e rsu s
position o r concentration v e rsu s position can be plotted on one g ra p h . The
plot would co n sis t of only one line o r cu rve , rep re sen tin g the en tire range
of in te re s t , such a s position . This has been shown to be possib le a lready
in the gas phase fiow, w here velocity and position appear a s one p lo t. It
is d e s ired to find out, then, if the behavior of the p a r tic le s allow s them to
be non-dim ensionalized and shown as one common g raph .
The continuum assum ption has been accepted by many of the a r t i
c les review ed in th is th e s is . The p ro p e rty of continuum is tha t the p a rtic le s
act as though they w ere one body, evenly d isp ersed and con tinuous. The
p rev ious d iscussion on p a rtic le d rag and p a rtic le heat tra n s fe r is dependent
upon the continuum assum ption being tru e .
With th is b rie f introduction, s im ila rity and continuum w ill be
tre a ted in the pages following. F ir s t , s im ila rity w ill be d iscu ssed .
In 1966, Goldschm idt and E skinazi (24) ran te s ts with the object of
p red ic ting the sp read of tw o-phase flow in turbulent j e t s . In th e ir te s ts the
27
28
liquid phase w as com posed of Safflower oil because of its low ra te of evap
oration under the p a rtic u la r conditions of the te s t . Of concern to the p re sen t
an aly sis w as the fact that they w ere able to develop a s im ila rity p ro file fo r
the velocity of the p a r tic le s as well as fo r the g a s . S everal plots a re m ade
of velocity , position , and concen tration p ro f ile s . As stated e a r l ie r in the
section on diffusion, the velocity of the gas s tre am and the co rrespond ing
sm all s ize of the p a rtic le s caused the p a rtic le s to follow the gas s tream
exactly , during every tu rbu len t fluctuation . Since th e re a lread y ex ist p lo ts
of gas phase, the p lots of the p a rtic le velocity and concen tration p ro file s
com e a s no g rea t su rp rise .
The only conclusion possib le , concerning the a r tic le by Goldschm idt
and Eskinazi (24), is that although s im ila rity p ro files a re valuable in a i r -
augm ented com bustion, th e ir a r tic le is of little value because of the fact
that the p ro files of the p a r tic le s a re exactly the sam e as that of the g a s .
The low velocity and sm all size of the p a r tic le s involved a re not com parable
to a ir-a u g m e n ta tio n .
In search ing the experim en tal l i te ra tu re , v e ry little was found on
the subject dealing with p a r tic le s im ila r ity . A th eo re tic a l m odel, how ever,
w as found which d iscu ssed the very topic in question . Channapragada, et a l .
(26), w ro te in 1967 about a ir-au g m en ted rocket m o to rs . In th e ir th eo re tica l
tre a tm en t of p a rtic le s im ila rity they stated:
It is observed from F ig u res 7 -9 that the gas velocity and te m p e ra tu re d istribu tion p ro file s a re d is s im ila r from the n e a r to the fa r f ie ld . It is in te re s tin g to note that the p a rtic le ve locity lag and loading
29
d istribu tion vary appreciably a t d ifferen t ax ial locations, ind icating that the d is s im ila r na tu re of the flow field is v e ry im portan t on p a r tic le p ro p e rtie s but not on the gas p ro p e rtie s (26).
In the absence of any o ther so u rces , nam ely those involving e x p e r
im ental da ta, the conclusion reached is sim ple , and follows that of C hanna-
p ragada, et a l . (26). . Due to the uneven p a rtic le loading and the velocity lag
v aria tion a c ro ss the section of the d iam e te r of the duct, th e re ex ist no
s im ila rity p ro files of velocity , te m p e ra tu re , or concen tration d istribu tion ,
un less the p a rtic le s a re so sm all o r ve locities low enough to p e rm it the p a r
tic le s to flow exactly as the g a s .
Continuum is assum ed by many au th o ritie s a t p re sen t because one
of the following two conditions may ex ist: (a) the flow, in actuality , may be
a continuum , o r (b) fo r the case of sim plification , the flow is c lose to a
continuum and is th e re fo re assum ed as being so .
C arlson and Hoglund (27) w rote a th eo re tica l trea tm en t of the p a r
tic le heat t ra n s fe r and d rag in rocket nozzles in 1964. In th e ir a r t ic le , a
com puter p rog ram w as used to evaluate the continuum flow assum ption .
The conditions p re sen t w ere typical of a rocket no zzle . The nozzle ev a lu
ated w as d esc rib ed as one having "condensed-phase p a rtic le s (ranging from
1 to lO^k) in the nozzle of a sm all (1000 lb th ru s t) rocket m otor opera ting
a t 400 psia"(27). They showed four reg ions of flow p re sa it in th e ir an a ly sis :
f i r s t , th e re ex is ts the continuum flow; second, the slip flow regim e; th ird ,
the tran s itio n flow; and fourth , the free m olecule flow . In th e ir analy sis
the model never reach es the free m olecule re g im e . F o r th e ir given
30
d iam e te r p a r tic le s , the m odel reached the slip flow reg im e a t the th roa t
and moved into the tran s itio n reg im e in the d iverging section of the nozzle .
C onsidering the conditions fo r an a ir-augm en ted rocke t leads to the conclu
sion that our assum ing that continuum does indeed ex ist is t r u e .
In o rd e r fo r the p re sen t d iscussion to be m ore exact, an analy sis
w as m ade to find out in p re c ise te rm s the physical p a ra m e te rs constitu ting
the continuum flow reg im e . The following is a desc rip tio n of the an a ly s is .
The basic req u irem en t fo r a continuum is that the m inim um length
of in te re s t , 1, m ust be g re a te r than the average d istance between p a r tic le s ,
b . A ssum ing that the p a r tic le s a re evenly spaced sp h eres , the d is tance b
is given by the following equation:
. 3b = 4 * p
2 3 m r e g
w here: m r = M ass frac tion of p a rtic le to g a s .
f p = Density of p a rtic le m a te r ia l ,
e g z D ensity of gas .
dp z D iam eter of p a r tic le .
An exam ple of the p reced ing equation and its usefu lness is the follow
ing. F o r a gas at 3000°R and 50 psia containing alum inum p a r tic le s , the
plot of m ass ra tio v e rsu s m ean d istance between p a rtic le s is p lotted in F ig
u re 4. The mean d istance between p a rtic le s is the quantity re fe r re d to as b,
and should be much le ss than the m inim um length of in te re s t , 1, in o rd e r for
mea
n d
ista
nce
b
etw
een
p
arti
cles
,
cm.
31
mg
F ig . 4 . - -M a s s r a t io v e r s u s m ean d is ta n c e b e tw e e n p a r t i c l e s
the continuum assum ption to hold tru e .
32
The m inim um length of in te re s t is then graphically rep resen ted
in F igure 4. This length is the P randtl m ixing length given by Schlicting
(2), The mixing length of Prandtl is a parameter used to describe the diffu
sion p ro cess of e ith e r a gas o r a system of p a r tic le s in a tu rbu len t s tre a m .
Mixing lengths m ay be m odeled in the continuing developm ent of a
m athem atical m odel. T his an a ly sis , then, m akes it possib le to calcu la te
the continuum c r i te r ia in a so lid -gas m ix tu re .
CHAPTER VI
PRESENT MODELS
T his ch ap te r deals with the p re sen t m odels availab le , including
th e ir im portan t sim plifying assu m p tio n s . The only a re a s d iscu ssed w ill
be those concerned in the an aly sis of the p a rtic le flo w . Because of the co m
plications involved in m odeling an a ir-au g m en ted com busting je t, a ll of the
a r t ic le s rep o rted used sim plifying assu m p tio n s .
Em m ons (28) in 1965 w rote of a th eo re tic a l analy sis of the mixing
of a i r with a reac tin g g a s-p a rtic le je t . The effect of the p a r tic le s , how ever,
was not co n sid e red . Emmons assum ed that the p a r tic le s w ere in both th e r
m al and dynamic equ ilibrium with the gas s tre a m . The p ap er does not p r e
d ict the diffusion of p a rtic lu e s , but a ssu m es that they diffuse a t the sam e
ra te a s the g a s . The a r tic le , though not d isc lo sing any experim en tal w ork
dealing with the g a s -p a rtic le in te rac tion , does s ta te that th is inform ation
ex is ts in the c lass ified l i te ra tu re .
A pap er d esc rib in g a ir-au g m en ted solid p ropellan t m ixing was
w ritten in 1967 by Channpragada, e t a l . (26). In th e ir an a ly s is , the p a r
tic le s w ere assum ed to have no velocity o r th e rm a l lags a t the s ta r t of m ix
ing . A ctually, the en tran ce section should probably have the la rg e s t lags
33
34
of any location . This assum ption , how ever, g rea tly sim plifies the m odel.
The tem p era tu re response of the p a r tic le s is calcu la ted using a gen era l
N usselt n um ber. T heo re tica lly , accord ing to the an a ly sis , the tem p e ra tu re
response should follow an exponential c u rv e . The an aly sis a ssu m es that
the N usselt num ber, density , and th e rm a l conductivity rem ain constant
over the range of conditions p re s e n t. The p a rtic le cloud is tre a ted as a
continuum , which ag rees with many a r tic le s read and our own p rev ious co n
clusion .
A recen t a r t ic le , w ritten in 1967 by Midgal and Agosta (29), does
an excellent job of developing equations for the model, The article states
concern ing the re la tio n s between the gas and p a rtic le s (d rag , N usselt n u m
b e r , e t c .) that, "It is not the intent h e re to recom m end the b est form of
th ese p a rtic le constitu tive eq u a tio n s ." This a r t ic le , then, p rov ides no ad d i
tional inform ation fo r th is th e s is .
CHAPTER VII
CONCLUSIONS AND RECOMMENDATIONS
The p reced ing ch ap te rs have d iscu ssed , in d e ta il, va rio u s im p o r t
ant a sp ec ts of p a rtic le d y n am ics . These w ere specifica lly re la ted to a i r -
augm entation of a p a r tic le -c a rry in g gas s tre a m .
It has been shown that, considering the range of re la tiv e Reynolds
num bers between the gas and the p a r tic le s , the c lo se s t equation following
. , . () §4the s tandard d rag curve was that of Ingebci, C j = 27 / Rg . F o r any
re la tio n req u irin g the fo rce between the p a r tic le s and the g as , it is re c o m
m ended that Ingebo's d rag form ula be u sed .
It is recom m ended that the heat tra n s fe r to a sphere in a turbulent
gas s tream use the re la tio n s of Lavender and Pei (20). This heat t ra n s fe r
re la tio n , in te rm s of the N usselt num ber, is
Nu = 2 .0 + A p 0 .5 p 0 .33 R B Be r r
w here the definition and explanation of te rm s appear in C hapter III, T herm al
R esponse .
P red icting the p a rtic le diffusivity is a form idable p ro b lem . The
b est e s tim a te , b a rr in g experim en tal da ta , is one using Longwell and W eiss '
35
36
(22) form ula which app ears in C hapter IV, P a rtic le D iffusion. T h is te c h
nique fo r determ in ing the diffusion of the p a r tic le s should be used only if
the diffusion proves to be v e ry significant and a ll o ther e ffo rts fa il, fo r it
w ill only give a range of probable diffusion r a te s .
The p re sen t m odels, which cover a ir-augm en ta tion of a p a r t ic le -
c a rry in g gas s tre a m , fa il to t re a t in any significant d e ta il any of the asp ec ts
of p a rtic le dynam ics . This was pointed out in C hapter VI, P resen t M odels.
In conclusion, th e re is sufficient inform ation on tra n sp o rt p ro p e r
tie s to recom m end p re lim in a ry values fo r com putation . However, fu tu re
w ork is d esirab le to expand the knowledge a v a ila b le ,
PARTICLE DYNAMICS IN THE MIXING OF A PARTICLE-LADEN
STREAM WITH A SECONDARY STREAM IN A DUCT
W illiam Doyle C ranney
D epartm ent of M echanical Engineering
M. S. D egree, June 1970
ABSTRACT
The purpose of th is th e s is w as to d iscu ss a ll phases of p a r tic le behav ior involved in the m ixing of the two s tre am s in an a ir-augm en ted solid propellan t ro ck e t.
Specific recom m endations fo r p a rtic le d rag coeffic ien t, N usselt num ber, eddy diffusivity coefficien t, along with the reaso n s for the a u th o r 's cho ices, w ere g iven . Also d iscussed w ere the s im ila rity p ro file s of velocity and tem p era tu re d is tr ib u tio n s . The assum ption of a p a r tic le -g a s continuum w as analyzed fo r app licab ility to the a ir-au g m en ted com bustion p ro b le m .
The th e sis rep re sen ted an in -dep th li te ra tu re sea rc h and analysis of any experim en tal w ork which would help p red ic t the m ixing p r o c e s s .
COMMITTEE APPROVAL:
LIST OF REFERENCES
AND
BIBLIOGRAPHY
LIST OF REFERENCES
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Solid Rockets with Boron P a r t ic le s ," AIAA Paper 67-481, 3 rd P ro pulsion Joing S pecialist C o n ference . W ashington, D . C ., July 17- 21, 1967.
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