Pressure drop for flow of air and grain mixtures in circular pipes of two diameters
by John J Cassidy
A THESIS Submitted to the Graduate Faculty i n partial fulfillment of the requirements for the degree
of Master of Science in Civil Engineering at Montana State College
Montana State University
© Copyright by John J Cassidy (1960)
Abstract:
The results of experimental measurements made on the pressure' drop incurred during the flow of
mixtures of air and various weight rates of wheat flow in circular pipes of two diameters are given in
this paper in both graphical and tabular form.
The pressure drop due to the air alone and wheat alone are separated and the calculated values found
for the pressure drop due to the solid particles alone is tabulated in graphical form for each of the
experimental 'measurements.
An equation which seems to account for the pressure drop due to the solid particles alone is derived by
dimensional analysis. Friction factors for use with this equation are calculated from the original
measurements and are given graphically for use in the derived equation.
The results of this research are compared with that of Mr. J. W. Crane who conducted a similar project.
The work of other researchers along the lines of this project has been reviewed and discussed.
The physical properties of the wheat used in this research are listed in an appendix.
PRESSURE DROP FOR FLOW
OF
AIR AND GRAIN MIXTURES
IN
CIRCULAR PIPES OF TWO DIAMETERS
by
JOHN J . CASS I DY
A THESIS
S u b m itte d t o th e G raduate F a c u lt y
in
p a r tia l
f u l f i l l m e n t o f t h e req u ire m e n ts
f o r th e degree o f
M aste r o f S cience in C i v i l E n g in e e rin g
at
Montana S t a t e C o ! Iege
Approved
Head', M a jo r Department
. I \ .v .
Bozeman, Montana
June I960
5
d > 13f
2
ACKNOWLEDGEMENTS
T h is t h e s i s was accom plished under t h e s u p e r v is io n o f D r. Eldon
R . Dodge, whose guidance and many h e l p f u l
acknowledged.
su g g e s tio n s are g r a t e f u l l y
Thanks are due t o M r s . G e t t i e Tschache and my w i f e ,
A l i c e , who rendered v a lu a b le a s s is ta n c e in t y p in g th e m a n u s c r ip t .
Thanks a re a ls o due t o th e Farmers Union G ra in Term inal A s s o c ia t io n
who fu r n is h e d th e wheat which was used d u r in g th e a c tu a l e x p e rim e n ts .
Funds f o r t h e e x p e rim e n ta l equipment were p ro vid e d by th e C i v i l
E n g in e e rin g Department o f Montana S t a t e C o lle g e .
141118
3
TABLE OF CONTENTS
-JUl InD
Page
VOOOOD
ACKNOWLEDGMENTS
LIST OF ILLUSTRATIONS
ABSTRACT
I NTRODUCTI ON
Purpose
I importance
P re v io u s Work
28
29
RATE OF FLOW OF AIR, CONTROL AND MEASUREMENT
O r ific e
Theory o f t h e o r i f i c e e q u a tio n
E ffe c ts o f c o m p r e s s ib ility
33
34
41
SYSTEM FOR MEASURING PRESSURE DROP ALONG THE TEST SECTION
P ressure ta p s and c o n t r o l panel
C o n tro l panel and manometer
C o n tro l o f r a t e o f f lo w o f g r a in
49
51
53
EXPERIMENTAL PROCEDURE
The g r a in used
D e s c r i p t i o n o f procedure used
I n i t i a l ru n n in g
03
DESIGN AND CONSTRUCTION OF EXPERIMENTAL APPARATUS
Components o f th e general system
The Feeder
P ipe T e s t S e c tio n s
t h e Blow er
'
58
60
61
EXPERIMENTAL RESULTS
C o l l e c t i o n o f data
Accuracy o f data recorded
C a lc u la tio n o f a i r v e lo c it y
A c c e le r a t io n losses
63
64
66
69
GENERAL THEORY
Measurement o f v e l o c i t y o f s o l i d p a r t i c l e s .
T heory developed '
Dimensional a n a ly s i s
A n a ly s is o f t h e r e s u l t s
71
72
76
87
CONCLUSI ONS
97
SUGGESTIONS FOR ADDITIONAL RESEARCH
99
4
Page
APPENDIX
Symbols and s u b s c r ip t s
A b b r e v ia t io n s used
P h y s ic a l p r o p e r t ie s o f
Summary data shee ts
LITERATURE CONSULTED
used
t h e wheat used
100
IOI
102
103
I 10
5
LIST OF ILLUSTRATIONS
Page
2Q
F ig .
I
Drawing o f General System
F ig .
2
Drawing o f Feeder
22
F ig .
3
Photograph Showing G ra in Hopper and Feeder
23
F ig .
4
Graph f o r A i r Leakage Through Feeder f o r 1.-1 / 2 Inch
T e s t P ipe
25
F ig .
5
Graph For A i r Leakage Through Feeder For 2 Inch T e s t Pipe
26
F ig .
6
Photograph Showing T ra n s p a re n t T e s t S e c tio n
3Q
Fig.-
7
Photograph Showing Equipment For C a l i b r a t i n g Hopper
31
F ig .
8
Photograph Showing th e Blower
35
F ig .
9
Drawing o f O r i f i c e
36
F ig .
IO
Photograph Showing th e Manometer and C o n tro l M a n if o ld
37
F ig .
II
Photograph Showing a P ressure Tap
50
F ig .
12
Drawing o f M a n if o ld Layout
52
F ig .
13
Photograph Showing t h e O r i f i c e in th e Bottom o f th e
G ra in Hopper
55
In s ta lla tio n
F ig .
14
Graph f o r G ra in D ischarge From Hopper
56
F ig .
15
Graph o f P ressure Drop Due t o A i r and G ra in in 1 -1 /2
Inch Pipe
73
Graph o f P re ssu re Drop Due t o A i r and G ra in in 2 Inch
Pipe
74
Graph o f F r i c t i o n F a c to rs f o r P ressure Drop Due t o A i r
Alone
78
Graph o f P re ssu re Drop Versus W eight Rate o f Flow o f
Wheat in 1 - 1 /2 Inch Pipe
84
Graph o f P ressure Drop Versus W eight Rate o f Flow o f
Wheat in 2 Inch P i pe
85
F ig .
F ig .
F ig .
F ig .
16
17
18
19
6
Page
F i g . 20
F i g . 21
F i g . 22
F i g . 23
F i g . 24
F i g . 25
-Graph o f F r i c t i o n F a c to rs f o r P ressure Drop Due To
Wheat Alone
90
Graph o f P re ssu re Drop Due t o Wheat AJone f o r 1 -1 /2
inch Pipe
92
Graph o f P re ssu re Drop Due t o Wheat Alone f o r 2 Inch
Pi pe
93
Graph o f P ressure Drop Due t o Wheat Alone From C ra n e 's
Resu I t s
94
L o g a r it h m ic P l o t o f P re ssu re Drop Due t o Wheat and
A i r f o r 1 - 1 /2 Inch P ipe
95
L o g a r it h m ic P l o t o f Pressure Drop Due t o Wheat and
A i r f o r 2 Inch P i pe
96
7
ABSTRACT
The r e s u l t s o f e x p e rim e n ta l measurements made on t h e p re s s u re drop
in c u r r e d d u r in g t h e f lo w o f m ix tu re s o f a i r and v a r io u s w e ig h t r a te s o f
wheat f lo w in c i r c u l a r p ip e s o f two d ia m e te rs a r e given in t h i s paper in
both g r a p h ic a l and t a b u l a r form .
The p re s s u re drop due t o t h e a i r a lo n e and wheat a lo n e a re separated
and t h e c a l c u l a t e d v a lu e s found f o r t h e p re s s u re drop due t o t h e s o l i d
p a r t i c l e s a lo n e i s t a b u l a t e d in g ra p h ic a l form f o r each o f t h e e x p e r t mental ‘measurements.
An e q u a tio n which seems t o account f o r th e p re s s u re drop due t o
t h e s o l i d p a r t i c l e s a lo n e i s d e r iv e d by dim ensional a n a l y s i s . F r i c t i o n
f a c t o r s f o r use w ith t h i s e q u a tio n a re c a l c u la t e d from t h e o r i g i n a l
measurements and are given g r a p h i c a l l y f o r use in th e d e r iv e d e q u a tio n .
The r e s u l t s o f t h i s research are compared w ith t h a t o f M r. J . W.
Crane who conducted a s i m i l a r p r o j e c t . The work o f o t h e r re s e a rc h e rs
along t h e li n e s o f t h i s p r o j e c t has been reviewed and d is c u s s e d .
The p h y s ic a l p r o p e r t ie s o f t h e wheat used in t h i s re search a re l i s t e d
in an a p p e n d ix .
8
INTRODUCTION
PURPOSE
Theory c o n c e rn in g t h e p re s s u re lo sses in c u r r e d in t h e pneumatic con­
v e y in g o f g r a in have been p r a c t i c a l Iy n o n - e x i s t e n t . Pneumatic g r a in con­
v e y in g i n s t a l l a t i o n s have been designed by e m p ir ic a l
form u la s o r by r u le s
o f thumb.
The purpose o f t h i s t h e s i s was t o s tu d y t h e p re ssu re drop
en­
co un tered in t h e f lo w o f a i r and wheat m ix tu re s and t o develop a th e o r y
which would account f o r th e s e p re s s u re lo s s e s .
The t h e o r y as developed
was s u b s t a n t ia t e d o v e r th e range o f e x p e rim e n ta l
I t was planned t h a t t h e i n i t i a l
r e s u lts .
c o n s t r u c t i o n o f t h i s equipment
would p r o v id e t h e b e g in n in g o f a c o n t in u in g gradu ate research program
a t Montana S t a t e C o lle g e in t h e f lo w o f a i r and s o l i d m ix t u r e s .
I MPQRTANCE
The use o f pneumatic methods in conve ying g r a in and o t h e r s o l i d s has
in crea sed r a p i d l y d u r i n g . t h e p a st se ve ra l
yearsJ
The ty p e s o f m a t e r ia ls
b e in g handled by pneumatic processes has a ls o in crea sed w i d e l y . These
m a t e r i a ls v a r y w id e ly in p a r t i c l e s i z e , p a r t i c l e shape, and d e n s it y .
T h is in c re a se d use o f pneumatic co n ve ying w i l l
r e q u i r e in cre a se d r e ­
search a c t i v i t y c o n c e rn in g t h e t h e o r y in v o lv e d in t h e p ro c e s s , i f pneu­
m a tic co n ve ying systems a re t o be designed w ith t h e c o n fid e n c e which a '
f i r m knowledge o f w e ll
I,
founded t h e o r y can g iv e t o a d e s ig n .
Rhodes, T . W ,, " A i r As A Conveying Medium For H a n d lin g M a t e r i a l s . "
i n d u s t r i a l and E n g in e e rin g C h e m is tr y , (O ctober 1953). Page I-19-1 20.
9
P re se n t emp'pri cal methods o f design r e q u i r e th e knowledge o f th e
c h a r a c t e r i s t i c s o f some p r e v io u s ly c o n s t r u c te d system.
s ig n in g a system which w i l l
convey a m a te r ia l
For someone de­
which has n o t been handled
p n e u m a tic a lly p r i o r t o t h a t tim e i t would-be d i f f i c u l t ,
i f n o t im p o s s ib le ,
f o r him t o design a system which would o p e ra te as designe d.
If,
however,
a t h e o r y c o u ld be e s t a b lis h e d which would e x p la in th e c h a r a c t e r i s t i c s o f
pneqm atic conve ying th ro u g h t h e use o f t h e v a r ia b le s concerned, then th e
d e s ig n e r would need o n l y to. determ ine th e n a tu re o f th e v a r ia b le s con­
cerned in h i s p a r t i c u l a r problem and a p p ly t h i s t h e o r y .
T h is would un­
d o u b te d ly save much money s p e n t on t h e design o f systems which may prove
t o be uneconomical a f t e r being p u t i n t o o p e r a t i o n .
PREVIOUS WORK
The v a s t m a j o r i t y o f t h e work which has been done in i n v e s t i g a t i n g '
th e p re s s u re drop due t o t h e pneumatic conveying o f s o l i d s has been o f
an e m p ir ic a l
s itu a tio n
n a tu re .
Many o f t h e i n v e s t i g a t o r s have had a p a r t i c u l a r
in mind d u r in g t h e i r i n v e s t i g a t i o n s and have come up w ith r e ­
s u l t s which f o r t h a t p a r t i c u l a r s i t u a t i o n have been s u f f i c i e n t f o r them
t o p e rfo rm th e r e q u ir e d d e s ig n .
The most comprehensive work o f t h i s n a tu r e was t h a t o f S e g le r
S e g le r perform ed la b o r a t o r y work o v e r a p e r io d o f se v e ra l
years d u rin g
which he c o l l e c t e d data on such g r a in s as o a t s , wheat, b a r le y , and f l a x . 2
2.
S e g le r , D r . - I n g . G . 1951 . "Pneum atic G ra in C o n v e y in g ." N a tio n a l
I n s t i t u t e o f A g r i c u l t u r a l E n g in e e r in g . Braunsweig, Germany.
I
IO
He measured p re s s u re drops in s t r a i g h t p ip e s , around bends, and d u rin g
p e r io d s o f p a r t i c l e a c c e l e r a t io n a f t e r t h e g r a in had been in tro d u c e d i n t o
t h e a i r stre a m .
S e g le r presented, t h e f o l l o w i n g e m p ir ic a l
hs -
fo rm u la .
L V2
In t h i s e q u a tio n , hs i s t h e p re s s u re drop due t o t h e sim ulta n e o u s flo w
o f a i r and g r a i n ,
(L )
is t h e p ip e l e n g t h , ( ? ) is t h e a i r v e l o c i t y , and
( 7 \ ) i s a c o n s t a n t depending upon th e t y p e o f g r a i n , t h e p ip e d ia m e te r,
and t h e w e ig h t r a t e o f f lo w o f t h e g r a i n .
He has l i s t e d v a lu e s o f (?) )
f o r many d i f f e r e n t g r a i n s , p ip e s iz e s , and a i r v e l o c i t i e s .
No th e o r y has
beet] p resen ted in s u p p o rt o f t h i s e q u a tio n .
S e g le r a ls o l i s t e d v a lu e s o f e q u iv a le n t p ip e le n g th s which can be
useci t o p r e d i c t p re s s u re drops around bends and p re s s u re drops d u rin g
a c c e l e r a t io n o f p a r t i c l e s a f t e r th e y have been in tr o d u c e d i n t o th e a i r
s trq a m .
He suggested t h a t th e p re s s u re drop be c a l c u la t e d f o r these
e q u iv a l e n t p ip e le n g th s by th e above e q u a t i o n .
S e g le r a ls o p resen ted a r a t h e r th o ro u g h d is c u s s io n o f t h e o p e r a tin g
c h a r a c t e r i s t i c s o f v a r io u s ty p e s o f f e e d e r s .
Al I o f S e g le r ’ s work has been e m p ir ic a l
in n a t u r e , and th e values
he has suggested f o r ( A ) a re t h e r e s u l t s o f many t e s t s on many pipe
diam e te rs u s in g v a r io u s g r a in s , v a r y in g a i r v e l o c i t i e s , and v a r y in g
w e ig h t r a t e s o f f lo w o f g r a i n .
I
I'I
Vogt and W hite^ have done c o n s id e r a b le la b o r a t o r y work on th e
p re s s u re lo s s due t o t h e conveyance o f s o l i d s in an a i r stre a m .
They have suggested t h e f o l l o w i n g e q u a tio n f o r th e d e te r m in a tio n o f
t h i s head lo s s .
wherje
K
i s th e r a t i o o f t h e head: lo s s due t o t h e flo w o f a i r and s o l i d s
t o th e head lo ss due t o t h e f I ow~ o f a i r a lo n e , and “ a" and " k " are
f u n c t io n s o f t h e dimension I ess g r o u p /
D p A -U
V alyes f o r Ca) and ( k ) have been l i s t e d g r a p h i c a l l y in t h e i r paper f o r
b o ty v e r t i c a l
and h o r iz o n t a l
f lo w .
(Dp) is t h e e f f e c t i v e dia m e te r o f t h e
!
p a r t i c l e s and has been d e fin e d as t h e d ia m e te r o f a sphere o f.e q u a l
s u r fa c e a re a , ( p s) i s t h e mass d e n s it y o f th e s o l i d m a t e r i a l ,
i s f h e mass d e n s it y o f t h e conveying f l u i d ,
p i p s , (g )
(D) i s th e d ia m e te r o f th e
i s t h e a c c e l e r a t io n due t o g r a v i t y ,
(R) i s t h e r a t i o o f th e
w e ig h t r a t e o f s o l i d flo w t o t h e w e ig h t r a t e o f f l u i d
t h e v i s c o s i t y o f th e .c o n v e y in g f l u i d .
( ^ f)
f lo w , and ( / X . ) i s
The d e te r m in a tio n o f (Dp) has p r e ­
sented c o n s id e r a b le d i f f i c u l t y , s in c e f o r i r r e g u l a r p a r t i c l e s
e x tre m e ly d i f f i c u l t t o measure s u r fa c e a re a .
it
is
The work o f Vogt and White
has been used q u i t e e x t e n s i v e ly in t h e Chemical E n g in e e rin g I n d u s t r y .
37
V o g t , ' E . G ., and W h ite , R. R-. " F r i c t i o n in th e Flow o f Suspensions'1
■ i n d u s t r i a l and E n g in e e rin g C h e m is tr y . Vol . 40, Page 17 3 ! - 1 738, 1948.
12
P in ku s^ was t h e f i r s t t o a tte m p t t o develop a t h e o r y c o n c e rn in g th e
p re s s u re drop due t o th e flo w o f a i r and s o l i d m ix t u r e s .
In h i s work he
atte m p te d t o use t h e Fanning E q u a tio n ^ t o account f o r t h e p re s s u re d ro p .
He s t u d ie d t h e p re s s u re drop due t o t h e f lo w o f a i r and sand m ix tu re s in
a 2 .00 inch s t e e l
p ip e .
H is o b s e r v a tio n s proved t h a t t h e Fanning E q ua tio n
Wa=? n o t v a l i d w i t h i n t h e range o f h i s e x p e rim e n t.
From h i s o b s e r v a tio n s ,
P inkus concluded t h a t t h e p re s s u re drop due o n ly t o t h e s o l i d s phase o f
t h e f lo w was l i n e a r l y p r o p o r t io n a l t o t h e v e l o c i t y o f t h e s o l i d p a r t i c l e s .
He a ls o concluded t h a t t h e r a t i o o f t h e v e l o c i t y o f th e s o l i d s t o th e
v e l o c i t y o f t h e co n ve ying f l u i d was a c o n s t a n t .
However, he was unable
t o measure th e v e l o c i t y o f t h e s o l i d p a r t i c l e s and th u s was unable t o
prove t h i s
l a s t c o n c lu s io n .
P in k u s ' re se a rch was done u s in g a s in g l e
h o r iz o n t a l
p ip e o f 2 inch d ia m e te r.
feqd sand i n t o t h e a i r stre a m .
w it h t h i s t y p e f e e d e r .
He used an auger ty p e fe e d e r "to
C o n s id e ra b le t r o u b l e was experienced
The fe e d e r was s u b j e c t t o p lu g g in g and b in d in g
W hile c a r r y i n g t h e s o l i d m a t e r i a l .
P inkus graphqd t h e head lo ss due t o t h e s o l i d p a r t i c l e s alo n e a g a in s t
a i r v e l o c i t y , o b t a i n in g a l i n e a r r e l a t i o n s h i p .
He was a b le t o s o lv e f o r
t h e p re s s u re drop due t o t h e s o l i d p a r t i c l e s alo n e by s u b t r a c t i n g th e
p re s s u re drop dtie t p t h e a i r a lo n e .
.
In d o in g t h i s , he assumed t h a t th e
■
p re s s u re drop due t o t h e f l u i t phase was u n a ffe c te d by t h e a d d i t i o n o f
)
t h e s o l i d s and th u s was.a f u n c t io n o f t h e r a t e o f f l u i d f Ibw a lo n e .
'
4.
5.
P in k u s , 0 . "P re s s u re Drops in Pneumatic Conveyance o f S o l i d s . "
Journal o f Appl ied M echanics. V o I . 19, Page 525-531, Dec. 1952.
F a n n in g , J . T . "A T r e a t i s e on H y d r a u lic and Water Su pply E n g in e e r­
i n g . " 4 th Ed. D. Van N o strand , New Y o rk .
1884.
12
In h i s o b s e r v a t io n s , P inkus found t h a t t h e p re ssu re drop due t o
t h e a c c e le r a t io n o f t h e s o l i d p a r t i c l e s a f t e r t h e i r i n t r o d u c t i o n t o th e
a i r stream o c c u rre d f o r o n ly a d is ta n c e o f a p p ro x im a te ly 8 inches down­
stream in t h e conve ying p ip e .
The n e x t a tte m p t t o d e r iv e a t h e o r y t o e x p la in p re s s u re drop was
done in 1956 by G rane^.
Crane p a t t e r n e d h i s t h e o r y a f t e r t h a t o f P in k u s .
However, in p la c e o f t h e Fahning1E q u a tio n , Crane used t h e Darcy Weisbach e q u a tio n t o account f o r th e p re s s u re .d ro p due t o th e g r a in
c o l l i d i n g w ith t h e p ip e w a ll and w it h o t h e r g r a in p a r t i c l e s .
He a ls o
in c lu d e d a te rm t o account f o r th e p re s s u re drop due t o t h e in c re a s e in
e le v a t io n o f t h e s o l i d p a r t i c l e s
in an i n c l i n e d o r v e r t i c a l
p ip e .
The
e q u a tio n s which Crane developed are as f o l l o w s .
An e q u a tio n f o r p re s s u re drop:
Au - fs IkLGs
n h ~ 2 D3S Yh 1O
Gs L sin 9
j
r
Vs Kh zO
f a UVa
2 D j
(I)
An e q u a tio n f o r v e l o c i t y o f t h e s o l i d p a r t i c l e s :
Wl%
cap
__
.V 5 = ___ 2W p______ /
Ka -C A p Va .
IihC A pgS inQ , Vaz fs IfA C A p
q ! f a ) fs s / n &
ZW p
y
Z Wp
4 D d Wp
~
Tp J
ZD
And an e q u a tio n f o r t h e f r i c t i o n
fa c to r ( fs ) :
. _ D[_Th CApjsJb'‘ Vs)2' -ZgVpTp Sin©]
6.
V p T R ,# 6
Crane, J . W. " P r e d i c t i n g P ressure Drop in th e Pneumatic Conveying
o f G r a i n s . " A T h e s is . M ic h ig a n S t a t e U n i v e r s i t y , 1956.
/
!4
Where ( A h )
i s the- p re s s u re drop in f e e t o f w a te r, (Vs) is- th e
/
v e lo c it y o f the s o lid p a r t ic le s , (L )
•
.
i s t h e le n g th o f t h e p ip e , (Gs)
.
'■
■ ;-*
■is t h e w e ig h t r a t e o f flo w o f s o l i d s per square f o o t o f t h e c r o s s I
s e c t io n a l
area o f th e
v e y in g p ip e , ( ( f H2 O)
conve ying p ip e , (D) i s th e dia m e te r o f th e con­
is t h e s p e c i f i c w e ig h t o f w a te r , ( 0 ) i s th e
ancjle o f i n c l i n a t i o n o f t h e conveying p ip e , ( f ^ )
is t h e f r i c t i o n
fa c to r
in th e Darcy Weisbach
e q u a tio n , ( ^ ) i s t h e s p e c i f i c w e ig h t o f th e
co n ve ying f l u i d , '(V/\)
is t h e v e l o c i t y o f t h e conveying f l u i d ,
( "Zf p)
is
t h e a b s o lu te s p e c i f i c w e ig h t o f t h e p a r t i c l e s , (Wp) i s t h e w e ig h t per
p a r t i c l e o f t h e g r a i n , (C) i s th e drag c o e f f i c i e n t f o r a sphere w ith a
d ia m e te r equal t o th e s m a lle s t d ia m e te r o f a wheat p a r t i c l e ,
(Ap) is
t h e s m a lle s t p r o je c te d area o f a wheat p a r t i c l e , and (Vp) i s th e
a b s o lu te volume o f a wheat p a r t i c l e .
Crane measured t h e p re s s u re drop due t o th e conveying o f s o f t
w h ite s p r in g wheat in a 3 .5 8 inch d ia m e te r aluminum p ip e a t v a r io u s .
w e ig h t r a t e s o f s o l i d f lo w and a t v a r io u s p ip e i n c l i n a t i o n s .
B is r e ­
s u l t s seemed t o s u b s t a n t i a t e h i s t h e o r y , excep t a t a i r v e l o c i t i e s over
100 f p s .
In t h e d e r i v a t i o n o f th e s e e q u a tio n s . Crane assumed t h a t t h e drag
c o e f f i c i e n t s f o r wheat p a r t i c l e s
th o s e f o r sphe re s.
in an a i r stream were t h e same as
Crane computed Reynolds Numbers f o r t h e wheat
p a r t i c l e s by u s in g th e s m a lle s t o f th e m in o r diam e te rs o f t h e wheat
p a rtic le
f o r "D" in t h e e q u a tio n ,
/V# = ^ D ^
.
He th e n s e le c te d
15
drag c o e f f i c i e n t s
f o r spheres u s in g th e s e Reynolds Numbers.
appears a p p ro x im a te ly c o r r e c t i f a l l
w ith t h e i r
T h is
wheat p a r t i c l e s were o r ie n t e d
lo n g i t u d i n a l axes p a r a l l e l t o t h e f lo w .
However, t h i s
assumption seems d i f f i c u l t t o j u s t i f y s in c e in th e flo w o f t h e a i r
and s o l i d p a r t i c l e s , th e p a r t i c l e s a re c o n t i n u a l l y c o l l i d i n g w ith one
a n o th e r and w it h t h e p ip e w a l l .
In h i s t h e s i s . Crane has g iv e n v a lu e s o f h is c o e f f i c i e n t ( f s )
f o r v a r io u s a i r v e l o c i t i e s .
These v e l o c i t i e s are in a range o f
Reynolds Numbers from 80,000 t o 17 0 ,0 0 0 .
The c o e f f i c i e n t s a re f o r one
ty p § o f l i g h t s p r in g wheat o n l y .
Crane used a r e v o l v i n g bucket ty p e fe e d e r w ith c o n s id e r a b le success
H i s s o l i d p a r t i c l e s were in tro d u c e d i n t o th e a i r stream th ro u g h f ix e d
p la te o r if ic e s .
He made a tte m p ts t o measure th e v e l o c i t y o f t h e s o l i d
p a r t i c l e s by p h o to g ra p h ic means and by a tte m p ts t o measure t h e d e n s ity
o f d is p e rs e d p a r t i c l e s w i t h i n th e - p i pe.
in both a t te m p t s .
However, he was unsuccessful
In t h e photography a t t e m p t , he found t h a t in th e
h ig h speed photographs which he to o k o f t h e f lo w , t h e r e were so many
p a rtic le s
in t h e p i c t u r e t h a t i t was im p o s s ib le t o t r a c e a s i n g l e one
and th u s compute i t s v e l o c i t y .
H is a tte m p t t o measure th e d e n s it y o f t h e d is p e rs e d s o l i d s was. made
by i n s t a l l i n g two a u to m a tic gates which c lo s e d in s ta n ta n e o u s ly and .si mu I
ta n e o u s iy ,. th u s i s o l a t i n g a 5 f o o t le n g th o f p i p e . - He th e n removed t h i s
s e c t io n and weighed th e g r a in content,;
e rra tic .
H is r e s u l t s were extreme I y - f
Crane assumed t h a t h i s gates were n o t c l o s i n g s im u lta n e o u s ly ;
16
however, t h e v is u a l o b s e r v a tio n s made o f t h e s o l i d s f lo w , done in t h i s
t h e s i s , show t h a t f o r a i r v e l o c i t i e s
in t h e range o f th o s e re p o r te d by
C raqe, t h e p a r t i c l e s a re n o t d is p e rs e d u n i f o r m ly w i t h i n t h e p ip e .
H a r j u and M oIsta d T ' measured th e p re s s u re drop due t o v a r io u s
cheirjical
powders in v e r t i c a l c o n v e y in g .
However, th e y unknowingly, i n ­
clu d e d p re s s u re lo sses due t o t h e a c c e l e r a t io n o f th e conveyed p a r t i c l e s
as th e y l e f t a v e r t i c a l
bend in t h e i r a p p a ra tu s .
T h e ir measurements
were made in 1 / 4 . inch and 1/2 inch d ia m e te r g la s s tub e s which were 12
inches lo n g .
Belden and K a s s e l ^ p a d e exp e rim e n ta l measurements which c o r r e la t e d
;
w e ll
w ith th o s e o f H a riu and M o ista d f o r p a r t i c l e s up t o 0 .1 0 inches in
dia m e te r conveyed in l i n e s o f la rg e d ia m e te rs .
Here a ga in t h e i r approach
t o t h e problem' was e m p i r i c a l .
Many o t h e r e x p e rim e n te rs have a tta c k e d t h e problem o f e x p l a i n in g
p re s s u re lo sses due t o t h e flo w o f a i r and s o l i d m i x t u r e s .
a il
P ra c tic a lly
o f th e s e e x p e rim e n te rs have a tta c k e d t h e problem in an e m p ir ic a l
manner.
The work o f th e s e e x p e rim e n te rs i s much to o volum inous t o be
d e s c rib e d here; however, a r a t h e r th o ro u g h b i b li o g r a p h y has been in c lu d e d
a t t h e end o f t h i s t h e s i s .
7.
Hari u, 0 . H ., and X M c d s ta d T lT C . , "P re s s u re Drop
in T r a n s p o r t o f S o lid s by G ases."
I n d u s t r i a l and
Chemis t r y V o l ♦ 4 1 , Page I 148-1 I 60, 1949.
8 . Bel den, D. H ., and K a s s e l, L . S . , " P re s s u re Drops
C a r r y in g P a r t i c l e s o f Large D iam eters in V e r t ic a l
- i n d u s t r i a l and Chemical E n g in e e r in g . June 1949,
i n V e r t i c a l Tubes
E n g in e e rin g '
Encountered in
T ra n s fe r L in e s ."
1(7
To d a te , no t h e o r y has been evolved which w i l l e x p l a i n a l l o f th e
f a c t o r s in v o lv e d in pneumatic co n v e y in g ,
Rouse^ has a llu d e d t o t h i s
in h is t r e a tm e n t o f th e problems in v o lv e d
in t r a n s p o r t a t i o n o f d i s c r e t e m a t t e r .
The v a r ia b le s in v o lv e d in sediment
t r a n s p o r t a t i o n a re b a s i c a l l y t h e same as th o s e in th e problem o f pneu­
m a tic c o n v e y in g .
He says t h a t g e n e r a lly speaking the problem i s a
f u n c t io n o f th e f o l l o w i n g t y p e .
f ( l , V, P, P -, d , Cd,
Where .(L) i s a l i n e a r te rm t o d e f in e t h e boundary s c a le , (V) i s th e f l u i d
v e lo c ity ,
( /2 ) i s t h e mass d e n s it y o f t h e f l u i d ,
o f th e f l u i d ,
(d )
is
( /X ) is th e v is c o s ity "
a te rm t o account f o r p a r t i c l e s i z e ,
((T d )
is th e
s ta n d a rd d e v i a t io n about t h e p a r t i c l e s i z e , ( A T f ) i s t h e e f f e c t i v e
s p e c i f i c w e ig h t o f t h e d i s c r e t e m a t t e r , and (C) i s th e c o n c e n tr a t io n o f
t h e d i s c r e t e m a tt e r in t h e f l u i d .
Rouse w r i t e s ;
"W ith i t s e i g h t v a r i a b l e s , such a f u n c t io n
is o b v io u s ly
t o o c o m p lic a te d t o s tu d y as a whole, even though i t r e p r e s e n ts about as
s im p le an a spe ct o f such m otion as can be fo u n d .
The s o l u t i o n is
seldom a com plete one, however, and d e v i a t io n s are t o be expected as
t h e range o f t h e i n v e s t i g a t i o n
is exceeded.
It
is f o r t h i s reason t h a t
such o cc u rre n c e s as sedim ent t r a n s p o r t a t i o n along th e bed o f a channel
have been among t h e l a s t t o be b roug ht o u t o f th e realm o f pure e m p ir ic is m . "
gZ
Rouse, H u n te r. "Advanced Mechanics o f F l u i d s . "
and Sons, New Y o r k .
1959.
John WI l e y '
18
DESIGN AND CONSTRUCTION.OF EXPERIMENTAL APPARATUS
■COMPONENTS OF THE GENERAL SYSTEM
A lm o st any ty p e o f s o l i d m a te r ia l
which can be conveyed pneu­
m a t i c a l l y c o u ld have been used in t h e e x p e rim e n ta l re se a rch r e p o rte d
h er^.
Wheat was chosen p r i m a r i l y because many farm in g concerns have
been u s in g pneumatic conveyors t o handle wheat in t h i s w estern area
f o r some t im e .
I t was a ls o d e s ir a b le t o use wheat because o f i t s
r e l a t i v e c l e a n l in e s s d u r in g co n v e y in g .
The e xpe rim e nta l equipment
was s e t up in t h e C i v i l E n g in e e r in g L a b o r a to ry in Ryon L a b o r a to ry a t
Montana S t a t e C o lle g e .
L a b o r a to ry c la s s e s and o t h e r p r o j e c t s were
being conducted in t h e la b o r a t o r y s im u lta n e o u s ly w ith t h i s research
which made a m a te r ia l t h a t was r e l a t i v e l y c le a n d e s i r a b le .
The equipment w ith which th e e x p e rim e n ta l
was b u i l t d u r in g t h e summer o f 1959.
research was conducted
The b a s ic system c o n s is te d o f
t h e f o l l o w i n g components:
1. )
A c o n ic a l shaped hopper which d isc h a rg e d t h e g r a in i n t o
th e r o ta r y fe e d e r.
2. )
A r o t a r y - a i r - l o c k ty p e fe e d e r which re c e iv e d wheat from
t h e hopper mounted above and d isc h a rg e d i t
i n t o t h e con­
vey! hg p ip e .
3. )
Two com plete s e t s o f t e s t p ip e s , one w ith an in s id e d iam e te r
o f 2 .06 in c h e s , and t h e o t h e r w it h an in s id e d ia m e te r o f
I .63 in c h e s .
19
4 .)
A 3450 rpm, e l e c t r i c a l l y d r iv e n ,
im p e l le r ty p e b lo w e r, ra te d
a t 240 cfm o f a i r , w ith d is c h a rg e p re s s u re o f 2 7 .7 inches o f
w a te r.
The general
l o c a t i o n , arra n g e m e n t, and o v e r a l l
e x p e rim e n ta l system a re shown in F i g .
dim ensions o f th e
I oh Page 20.
Because t h e le n g th o f tim e r e q u ir e d t o make t h e necessary measure­
ments d u r in g an exp e rim e n ta l run was q u i t e lo n g , i t was decided t o
■
c o n s t r u c t t h e system so t h a t t h e wheat would c i r c u l a t e c o n t in u o u s ly ,
th e r e b y making t h e manual r e f i l l i n g o f t h e hopper unnecessary.
The
system was t h e r e f o r e designed so t h a t t h e wheat was blown th ro u g h th e
t e s t p ip e and then back i n t o t h e hopper f o r r e c i r c u l a t i o n .
THE FEEDER
A r o t a r y ty p e fe e d e r was used in t h e re se a rch r e p o r te d h e re , t h i s
t y p e was s e le c te d p r i m a r i l y because o f i t s s u cce ssfu l
use by Crane*0 .
The c h a r a c t e r i s t i c s o f t h i s ty p e o f fe e d e r have been d e s c rib e d in
d e ta il
by S e g le r * ^
a r e v o lv i n g d o o r,
The p r i n c i p a l o f t h e fe e d e r i s b a s i c a l l y t h a t o f
i t a llo w s th e s o l i d s t o e n t e r t h e fe e d e r from a low
p re s s u re area above t h e fe e d e r and then d is c h a rg e s them i n t o t h e high
p re s s u re area o f t h e co n ve ying p ip e , t h e o r e t i c a l l y w ith m inor losses
o f q ir.1
0
10.
Il.,
C rane, J . C. Op. C i t . P. 4 5 .
S e g le r , G .
Op. C it . P . 73.
Tfrs /
Sec -hion
M a n o rD e re r
,
A '- C
I---------- 1
_
C O rvrA O L
PA/VB L ^ l
I
ZO '-O "
First pressure. TapP ressure Taps..
r /o v s ^
/ o '- o *
LAYOUT
__
s '- o "
FIG. I
P LA N V IE W
OF E X P E R IM E N T A L A P P A R A TU S
■S C A L E D bJO N E
DRAWN BYi JJC
S - 4*60
/
21
A draw ing o f +he general design o f t h e fe e d e r i s shown in F i g . 2,
Pagie 22.
The assembled fe e d e r i s a ls o shown in F i g . 3 .
The body o f +he
fe e d e r was made from a 1 2 -in ch le n g th o f 10-inch dia m e te r sta n d a rd s te e l
p ip e .
The rem ainder o f t h e fe e d e r p a r t s a re d e s c rib e d in t h e drawing in
F ig . 2.
I t was decided t h a t t h e opening in th e top o f t h e fe e d e r thro ugh
which t h e hopper discharged, t h e g r a in i n t o t h e fe e d e r sho u ld be made
l a r g e r than t h e bottom o f t h e hopper.
Any a i r which leaked th ro u g h th e
fe e d e r would th e r e b y pass around t h e hopper and would n o t a f f e c t th e
w e ig h t r a t e o f d is c h a rg e o f t h e g r a in from t h e hopper as i t would i f
th is
leakage o f a i r escaped upward th ro u g h t h e h o le in t h e hopper thro ugh
which t h e g r a in d is c h a rg e d .
The fe e d e r was p r o p e lle d by a I /2 -h o rs e p o w e r, MO v o l t e l e c t r i c
m o to r.
The d r i v e c o n n e c tio n was made by u s in g ty p e A V - b e l+ s .
An
1 8 -in ch d ia m e te r V - b e l t p u l l e y was a tta c h e d t o th e fee d e r s h a f t .
The
motor drove t h e fe e d e r th ro u g h a r e d u c in g i d l e r s h a f t .
re ­
The t o t a l
d u c tio n in speed between t h e m otor and t h e fe e d e r was 19.3 t o
I , w ith
t h e m otor t u r n i n g a t 1750 rpm, w h ile t h e fe e d e r tu rn e d a t 9I rpm.
D u r in g . o p e r a t io n
i t was found t h a t t h e feeder leaked e x c e s s iv e ly
and si neb t h e q u a n t i t y o f a i r f lo w in g was measured ahead o f t h e fe e d e r,
t h i s was a s e r io u s problem .
With t h i s
leakage, t h e a i r v e l o c i t y in th e
p ip e c o u ld n o t be o b ta in e d w ith any a c c u ra c y .
The feeder was r e b u i l t ,
several t im e s , b u t t h e problem o f leakage was n o t overcome.
F in a lly i t
was decided t h a t t h e amount o f leakage th ro u g h th e fe e d e r would have t o
be measured.
a no.
Pulley
A
Lam m aied Ffr F n d Pfece, One
H otcl
A i F a c f F r d -----------------------R o ia iin g In n e r Body (woodl
L o c k k i n g A i F o tc h F re J
\
j o"Diatn. SiandarcJ
\
R u fto e r G a s k e is — 7
S le e l P ip e --------v
Leolher Gaskeis
T Pl
EXPLODED
E l- E V A T IO N
DETAILS
<£
S ection
SCALED NONE
DRAW N
ev:
OF
W C
FEEDER
5'4-6
/
i
F ig . 3.
G ra in Hopper in P lace and th e D r iv e System f o r th e Feeder.
24
The fe e d e r was then run under two c o n d i t i o n s .
F i r s t t h e to p o f th e
fee d e r was se aled so t h a t no le a k in g c o u ld ta k e p la c e .
Under t h i s
c o n d i t i o n , th e in ta k e q u a n t i t y o f a i r t o th e blower was measured, and
th e p re s s u re a t th e f i r s t p re s s u re ta p on th e t e s t p ip e was measured.
The lo c a t i o n o f t h i s p re s s u re ta p is shown in F i g .
fee d e r was then removed, and again the
I.
The seal on th e
in ta k e o f a i r t o t h e blower and
th e p re s s u re a t t h e f i r s t downstream p re s s u re ta p was re c o rd e d .
process was c o n tin u e d o v e r th e f u l l
T h is
range o f p o s s ib le a i r v e l o c i t i e s .
The in ta k e o f a i r w ith t h e feeder n o t le a k in g , and th e in ta k e o f a i r
w ith th e fe e d e r le a k in g , were then p l o t t e d on lo g a r it h m ic paper a g a in s t
t h e p re s s u re a t t h e f i r s t downstream p re s s u re t a p .
shown in F i g . 4 and F i g . 5 .
These graphs are
From th e se graph s, e q u a tio n s were d e riv e d
t o p r e d i c t t h e leakage o f a i r th ro u g h t h e fee d e r in term s o f th e
p re s s u re a t th e f i r s t p re s s u re t a p .
These e q u a tio n s were d e r iv e d as
f o l I ows* ^ .
The a c tu a l
s lo p e o f th e two curve s shown in F i g . 4 i s 0.650 as
measured on t h e g raph .
The v a lu e o f Q ( fe e d e r n o t le a k in g ) when t h e p re s s u re a t th e f i r s t
p re s s u re ta p is equal t o one inch o f w a te r , i s 0 .3 0 9 c f s .
v a lu e f o r Q ( fe e d e r le a k in g )
The s i m i l a r
is 0 .3 3 5 c f s .
L e t th e p re s s u re a t ta p #1 in inches o f H2 O = P , t h e a i r in ta k e in 1
2
12.
S o k o ln i k o f f , J . S . , and E . S . "H ig h e r Mathematics f o r Engineers
and P h y s i c i s t s . " M c G r a w - H ill. 1941.
2.S
DRAWN
Q V iJJC
—f-r!
7,0 ^
FlG
4-5 rg
VOLUME
'■:.r i‘ : r
RATE Q ft F L O W Of AIR
VERSUS PRESSURE A T THE F I R S T .
PRESSURE
T A P FOR
I i INCH PIPE I
I :
j
OJ / /
Or 4
OS <x€>
; i
26
27
e f s w ith t h e fe e d e r n o t le a k in g = Q-j-, and t h e a i r in ta k e in c f s w ith
• t h e feeder le a k in g = Q[_.
The l o g a r it h m ic e q u a tio n s f o r th e s e two curves are
log Qf = 0.650 log P + 0.309
lo g Q|_ = 0 .6 5 0 log P + 0 .3 5 5
Changing th e s e e q u a t io n s - t o r e c t a n g u la r form , we have
Qt = 0.309 p0 *650
(2 )
Ql = 0 .3 3 5 P0 *650
(3 )
| f t h e v a lu e o f (P) i s t h e same in both e q u a tio n s , th e n t h e
d i f f e r e n c e in Q 's f o r t h i s case w i l l
be t h e amount o f leakage through
t h e fe e d e r in c f s .
Q leakage = Ql - Q+ = (0.3 35 - 0;3 0 9 ) PV *D3U
Q leakage ( c f s ) = 0 . 0 2 6 'P0 -650
(4 )
A s i m i l a r e q u a tio n was d e r iv e d f o r t h e 2 .0 6 inch p ip e .
shown in F i g . 5 .
T h is e q u a tio n i s
The p re s s u re a t ta p #1 was measured in a l l
experim ental
r u n s , and t h e volume r a t e o f flo w o f a i r in t h e t e s t p ip e was computed
by c o r r e c t i n g t h e a i r in t a k e t o t h e b lo w e r, u s in g th e s e e q u a tio n s .
I f :the f lo w o f t h e a i r
le a k in g th ro u g h t h e fe e d e r had been o f a
t u r b u l e n t n a t u r e , t h e exponent; o f P shou ld have been a p p ro x im a te ly 0 . 5 .
I f t h e f lo w had been la m in a r in n a tu r e , t h e exponent o f P shou ld have
been I .00..
T h i s fe e d e r had c le a ra n c e s between t h e r e v o lv i n g bucket wheel
and t h e fe e d e r body which were q u i t e s m a l l .
From t h i s s t a n d p o in t ,
appeared t h a t t h e exponent o f 0.650 f o r P was re a s o n a b le .
it
28
PIPE TEST SECTIONS
1\ ‘ •
.
S tandard t h i n - w a lI e l e c t r i c a l c o n d u it was used f o r both s iz e s o f con­
ve y in g p ip e used in t h i s p r o j e c t .
Several
d i f f e r e n t ty p e s o f p ip e were,
c o n s id e r e d , i n c lu d in g aluminum t u b in g and sta n d a rd s te e l
p ip e .
T h in -
w a lI e l e c t r i c a l c o n d u it was chosen because o f economy and ease o f
h a n d lin g .
The p ip e came in IO - f o o t s e c t io n s .
These s e c t io n s were
j o i n e d , u s in g sta n d a rd w a t e r - t i g h t nEMT" c o u p lin g s .
P a rtic u la r
a t t e n t i o n was used in s e l e c t i n g . t h e c o u p lin g s which jo in e d th e two le n g th s
o f p ip e which formed t h e t e s t s e c t i o n , t o in s u re t h a t a smooth j o i n t was
formed.
Bends in t h e p ip e were accom plished u s in g c o m m e rc ia lly a v a i l a b l e
90 degree bends.
These bends had a r a d iu s o f c u r v a t u r e o f 10 in ches,
and were o f c o n s t a n t s e c t io n t h r o u g h o u t.
The cbnve ying tu b e was jo in e d t o t h e 2 - 1 / 2 inch s ta n d a rd s te e l p ip e
which was w e ld e d .to th e bottom o f th e fe e d e r by u s in g a s ta n d a rd "EMT"
w a t e r - t i g h t c o n n e c te r and b e l l
re duce r c o u p lin g s .
The maximum o p e r a tin g
p r e s s u re in s id e t h e p ip e was a p p ro x im a te ly 20 inches o f w a t e r .
The e n t i r e pi p in g .s y s te m f u n c tio n e d s a t i s f a c t o r i l y , and was
a b s o lu t e l y a i r t i g h t o v e r t h e range o f p re s s u re s t o which i t was exposed.
. The t e s t s e c t io n o f t h e p ip e was made up o f two 1 0 -f o o t s e c tio n s o f
th e c o n d u i t .
The lo c a t i o n o f t h i s t e s t s e c t io n is shown in F i g .
I.
S p e c ia l c a re was e x e r c is e d t o make Sure t h a t th e t e s t s e c t io n was p e r­
f e c t l y h o r iz o n t a l and in good a lig n m e n t.
A t t h e end o f t h e t e s t s e c t io n .
29
a 4 - f o o t long s e c t io n o f t r a n s p a r e n t a c r y l i c t u b in g was i n s t a l l e d in
o r d e r t h a t t h e f lo w c o u ld be observed d u r in g t h e e x p e r im e n t a t io n .
The
in s id e d ia m e te r o f t h i s t u b in g matched t h e in s id e d ia m e te r o f t h e p ip e
b eing t e s t e d , so t h a t no change in th e f lo w p a t t e r n o c c u rre d as th e
g r a in passed from t h e t e s t s e c t io n i n t o t h e t r a n s p a r e n t s e c t i o n . T h is
o b s e r v a tio n s e c t io n proved t o be q u i t e v a lu a b le d u rin g t h e a c tu a l ex­
p e r im e n t a t io n .
Two s iz e s o f t e s t p ip e were used in t h e p r o j e c t .
These s iz e s were
1 - 1 /2 inch and 2 - in c h ."EMTm. c o n d u it which had measured in s id e diam eters
o f 1.63 inches and 2 .06 in c h e s , r e s p e c t i v e l y .
■; P o r t io n s o f th e conveying p ip e a re v i s i b l e in F ig s . 6 and 7.
P ressure ta p s were i n s t a l l e d a t i n t e r v a l s along, th e t e s t s e c t io n in
order th a t d if f e r e n t ia l
measured.
p re s s u re s a lo n g t h e t e s t s e c t io n c o u ld be
These p re s s u re ta p s are d e s c rib e d on Page
49 .
THE:BLOWER
To p r o v id e th e necessary a i r s u p p ly f o r t h e e x p e r im e n ta tio n , an
e l e c t r i c a l l y d r iv e n blow er was i n s t a l l e d
in t h e system .
The blower was
a c e n t r i f u g a l ty p e d r iv e n by a d i r e c t coup led tw o - h o r s e p o w e r ,.220 v o l t ,
A.C. m o to r .
The m otor ran a t 3450 rpm.
graph on Page
The blower i s shown in th e pho to­
35.
T h is blow e r was loaned t o th e C i v i l E n g in e e rin g Department by th e
Chemical E n g in e e r in g D epartm ent.
No s p e c i f i c data was a v a i l a b l e on th e
c h a r a c t e r i s t i c s o f t h i s b lo w e r; however, i t s maximum d e l i v e r y w h ile being
F ig . 6.
Downstream end o f th e t e s t p ip e showing th e t r a n s p a r e n t
s e c t io n o f a c r y l i c t u b in g used f o r o b s e rv in g th e f lo w .
F ig . 7.
Upstream end o f th e t e s t s e c t io n o f th e conveyi nr p ip e .
The drum was s e t in p o s i t i o n as i t was used t o c a l i b r a t e
th e w e ig h t r a t e o f flo w o f g r a in thro ugh th e h o p p e r.
32
used was NO cfm a t a p re s s u re o f 20 inches o f w a te r .
I t was I earned
e a r l y in t h e e x p e rim e n t t h a t t h e blow e r d id n o t have adequate c a p a c it y t o
p r o v id e t h e d e s ire d a i r v e l o c i t y range; however, a blow er w ith th e
d e s ire d c h a r a c t e r i s t i c s was n o t a v a i l a b l e f o r use.
In o r d e r t o measure t h e a i r in ta k e t o t h e system, a c i r c u l a r sharp
edged o r i f i c e was i n s t a l l e d on t h e i n l e t s id e o f th e b lo w e r.
C o n sid e r­
a b le amount o f t h r o t t l i n g was-- n o tic e d when a 2 .0 inch o r i f i c e was i n ­
s t a l l e d a t a d is ta n c e o f 8 , 0 inches from t h e eye o f th e blow er im p e l le r .
I t was decided t h a t t h i s t h r o t t l i n g was due t o t h e s h o r t d is ta n c e between
t h e o r i f i c e and t h e i m p e l l e r .
In a d i s ta n c e . t h i s s h o r t , t h e j e t o f a i r
would n o t have t im e t o expand and f i l l
'
5
t h e eye o f th e i m p e l l e r .
An
e x te n s io n t o t h e blow er i n l e t was b u i l t , and th e o r i f i c e r e i n s t a l l e d
a t a d is ta n c e o f .24 inches' from t h e eye o f t h e i m p e l l e r .
e f f e c t was observed due t o t h i s
No t h r o t t l i n g
in s ta lla tio n .
The o u t l e t end o f t h e blower was equipped w ith a 2-1 / 2 inch diam eter
th re a d e d f la n g e .
A 2 - in c h brass body g a te v a lv e was i n s t a l l e d in t h i s
o u t l e t l i n e between t h e blow e r and fe e d e r in o r d e r t h a t M f h e y r a t a . of
a i r c o u Id be c o n t r o I l e d i
33
RATE OF FLOW OF AIR, CONTROL AND MEASUREMENT
ORIFiCE
A sharp-edged c i r c u l a r o r i f i c e was i n s t a l l e d a t th e i n l e t t o th e
blow er in o r d e r t h a t t h e r a t e o f flo w o f a i r c o u ld be measured'.
in s ta lla tio n
has been mentioned b r i e f l y
i n s t a l l a t i o n on Page 29 o f t h i s p a p e r.
i
T h is
i n ' t h e d e s c r i p t i o n o f t h e blower
The o r i f i c e
i n s t a l l a t i o n can be
seen in F i g . 8 .
A I .65 in ch d ia m e te r o r j f i c e p l a t e was i n s t a l l e d
s h e e t metal p ip e qn t h e i n l e t s id e o f t h e b lo w e r.
was 0 .1 0 inches t h i c k and was sh arp-edg ed.
in t h e 6 .0 inch
T h is o r i f i c e p l a t e
A s e c t io n o f t h e 6 .0 inch
d ia m e te r s h e e t metal p ip e 1.0 f e e t long was i n s t a l l e d in f r o n t o f th e
o r i f i c e t o e l i m i n a t e any p o s s ib le r e - e n t r a n t e f f e c t s 13.
The o r i f i c e was i n s t a l l e d between tw o 3 /4 inch t h i c k plywood f la n g e s .
A s e c t io n o f 1/4 inch O.D. copper t u b in g was i n s t a l l e d
shown in F i g . 9 .
in each fla n g e as
These tu b e s were used as f la n g e ta p s in o r d e r t o
measure t h e p re s s u re d i f f e r e n t i a l
a cross t h e o r i f i c e p l a t e .
The t u b in g
was s o ld e re d t o th e shee t metal p ip e in o r d e r t o p r o v id e a t i g h t
c o n n e c tio n .
The ends o f t h e t u b in g led t o t h e c e n t r a l c o n t r o l
m a n ifo ld shown in F i g .
10.
A 60 inch U -tu b e manometer f i l l e d
panel
w ith o i l
o f s p e c i f i c g r a v i t y 0.824 was used t o measure th e p re s s u re d i f f e r e n t i a l
a cro s s t h e o r i f i c e p l a t e .
TT.
K in g , W is l e r , and Woodburn.
Sons, New Y o rk . P . 145.
"H ydraul i c s . "
1948, John W ile y and
34
THEORY OF THE ORIFICE EQUATION
■.....
X
An e q u a tio n was developed t o determ ine th e r a t e o f f lo w o f a i r - i n ­
t o t h e b lo w e r.
T h is e q u a tio n depends upon t h e p re s s u re d i f f e r e n t i a l
a cross t h e o r i f i c e p l a t e .
The o r i f i c e t h e o r y i s as f o l l o w s * ^ .
In F i g . 9 on Page 36, P o i n t I i s ta ke n as some p o i n t f a r enough
up stream from th e o r i f i c e p l a t e t h a t t h e stream li n e s o f t h e flo w a re
p a r a lle l.
P o in t 2 i s a p o i n t in th e vena c o n t r a c t s downstream from th e
o r i f i c e pi a te *
W r i t i n g th e c o n t i n u i t y e q u a tio n between P o in t s I and 2 , we have:
Qj = X A M
~
$2
A 2. Va
(5 )
Assuming t h a t
X, = Y z
, we can w r i t e th e f o l l o w i n g equ a tio n
f o r t h e v e l o c i t y o f t h e a i r a t P o in t I .
^
A
v* =
n
( 6)
W r i t i n g t h e B e r n o u l l i e q u a tio n 3 between I and 2, and n e g le c t in g
head loss,, we have
^
+I
+ 2|
=
Is
^
4
<- z *
Assuming P o in t s I and 2 are a t t h e same e l e v a t i o n , and s u b s t i t u t i n g
t h e v a lu e o f Vj
14.
15'.
from (6 ) above, th e f o l l o w i n g i s o b ta in e d : 1
4
I b id . - P. 123.
I b i d . P. 94.
F ig . 8.
In ta k e t o th e blower showing th e wooden
fla n g e s used f o r mounting th e I .65 inch
o rific e .
36
* 4 / / F /a n c fe s
M/tth
B o lte d
To o n 't ic e
"K 2 U B o lts
B lo w e r I n l e t
O n 't I ce
M a n o m e te r
Copper Tubinp
Plate
3
/ 6 G a S h e e l M e ta /
p /p e —
_________
P
F
H
I ":v
l o w
P
P la n g e s
y
r-9 "
g
c f
fr
I
/'- o "
SE C TIO N
S C A L E D NONE
FIG. 9
D E T A I L OF
ORi ri CE
I NSTAL LATION
DRAWN
BV:
S - 4 - 60
,Q1
%
'o
F ig .
10.
At L e ft;
The manometer used in measuring
p re ssu re d i f f e r e n t i a l along th e t e s t s e c t io n .
At R ig h t :
The c o n t r o l m a n ifo ld c o n n e c tin g t o
th e p ressure t a p s .
38
4
W
-
which cqn be r e w r i t t e n as
(7 )
T h is i s th e e q u a tio n f o r t h e t h e o r e t i c a l
con t r a c t a :
v e l o c i t y in t h e vena
However, due t o energy losses., V2 ( a c t u a l ) w i l l
than V2 ( t h e o r e t i c a l ) :
be less
To g e t Vg ( a c t u a l ) , we must m u l t i p l y t h e above
e q u a tio n by a c o e f f i c i e n t (Cv ) c a l l e d t h e v e l o c i t y c o e f f i c i e n t .
^ JL — 5 ^ =
L e ttin g
H, E q u a tio n 6 i s r e w r i t t e n t o form
29 H
V1 = C v
-(A
z/ a
)
(8 )
In o r d e r t o determ ine Q, t h e volume r a t e . o f f lo w , we must m u l t i p l y
E q u a tio n (8 ) by t h e c r o s s - s e c t i o n a l area o f t h e vena c o n t r a c t s , s in c e
Vg i s now t h e a c tu a l v e l o c i t y in t h e vena c o n t r a c t a .
area o f t h e vena c o n t r a c t a w i l l
The c r o s s - s e c t ionaI
be le s s than t h a t o f t h e o r i f i c e .
The
area o f t h e vena c o n t r a c t a i s d e fin e d as (Cc/\) where A „ i s t h e area o f
the o r i f i c e .
E q u a tio n ( I ) now becomes:
Q
=
CcCvK
where A0 i s t h e c r o s s - s e c t i o n a l
z
area o f t h e o r i f i c e o p e n in g .
r e w r i t t e n as
(9 )
T h is can be
39
L e t t i ng t h e q u a n t i t y
-------^ - C, t h e d is c h a rg e c o e f f i c i e n t ,
—
C c K /4 j*
we have
O =
C /W z s H
cio)
where t h e u n i t s o f H a r e f o o t pounds per pound o f a i r , o r s im p ly f e e t
o f a ir.
S in c e in t h i s r e s e a r c h , "H" was measured in inches o f w a fe r,
i t was d e s i r a b le t o m o d ify E q u a tio n ( IO ) in o r d e r t h a t Q c o u ld be
computed in c f s w h ile w o rkin g w ith "H" in inches o f w a t e r .
In o rd e r
t o accom plish t h i s , we w r i t e t h e f o l l o w i n g e q u a t i o n .
H (in c h e s o f w a te r) = H ( f e e t o f a i r X 12 X Y a
6 2.4
where 6 2;4 i s t h e s p e c i f i c w e ig h t o f w a te r .
But
(II)
a, th e s p e c ific
w e ig h t o f a i r , v a r i e s w it h p re s s u re and te m p e ra tu re a c c o rd in g t o th e
gas la w *6 .
In E q u a tio n ( 1 2 ) , "P " t h e p re s s u re i s
in pounds per squa re f o o t
a b s o lu t e , T i s t h e te m p e ra tu re in degrees F a h re n h e it a b s o lu t e , and R
( t h e gas c o n s t a n t ) = 5 3 .3 f o r a i r .
S o lv in g E q u a tio n ( I l )
f o r H ( f e e t o f a i r ) and s u b s t i t u t i n g f o r
y a a c c o rd in g t o E q u a tio n ( 1 2 ) , we h a v e '- ''
H ( f e e t o f a i r ) = 276.8 TH (in c h e s o f w a te r)
P ■
S u b s titu tin g t h is
16.
(13)
i n t o E q uation ( 1 0 ) , we o b ta in t h e f o l l o w i n g .1
6
Venriard, J . K. "E le m e n ta ry F l u i d M e c h a n ic s ."
and Sons, New Y o rk , P . 6.
1958.
John Wi le y
40
e q u a tio n f o r t h e volume r a t e o f flo w o f a i r
t e m p e ra tu r e .
in term s o f p re s s u re and
_________
Q -
.
CA;
(14)
A therm om eter and a barom eter were s t a t i o n e d near t h e I n l e t t o
t h e blow e r so t h e a i r te m p e ra tu re and p re s s u re c o u ld be recorded d u rin g
t h e e x p e rim e n t.
The barom eter recorded in inches o f m e rc u ry .
Re­
w r i t i n g E q u a tio n (1 4 ) so t h a t P' can be- in s e r t e d in t h e e q u a tio n in
inches o f m ercu ry, we have
<P
_____ ______ _____ _
CA0
“
Zg(\Z)(2.7$.8)
TH
6 Z . 4 0 3 '6 )
p'
where 6 2.4 = s p e c i f i c w e ig h t o f w a te r, and 13.6
= s p e c ific g ra v ity o f
m e rc u ry .
L e t t i n g g equal 3 2 .2 f p s 2 , t h i s e q u a tio n reduces t o
p
= is . a s C A c- J r S '
(15)
The c r o s s - s e c t i o n a l area o f th e o r i f i c e used was 0 . 0 i4 8 square f e e t .
The c o e f f i c i e n t o f d is c h a rg e is n o t o n ly dependent upon t h e geo­
m e t r ic p r o p e r t ie s o f t h e o r i f i c e , but e x p e rim e n te rs have shown t h a t i t
i s a ls o dependent on Reynolds Number!7
Reynolds Number f o r th e o r i f i c e
was computed from th e e q u a tio n
where Q i s t h e volume r a t e o f f lo w , D i s t h e d iam e te r o f t h e o r i f i c e , and7
1
17.
I b i d . . P. 302.
41
i s th e k in e m a tic v i s c o s i t y o f t h e f l u i d .
o r i f i c e ranged from 30,000 t o 7 0 ,0 0 0 .
numbers, w ith t h e o r i f i c e
Reynolds numbers f o r th e
For t h i s range o f Reynolds
i n s t a l l e d as shown w ith fla n g e t a p s , B u cking­
ham'® suggests a d is c h a rg e c o e f f i c i e n t "C" from 0.6063 t o 0 .6 0 2 2 .
was decided t o use an average v a lu e o f 0.6 0 4 2 in E q u a tio n ( 1 5 ) .
s titu tin g
f o r "C1' and f o r t h e c r o s s - s e c t i o n a l
It
Sub­
area in E q u a tio n ( 1 5 ) ,
we o b t a i n t h e f o l l o w i n g e q u a tio n .
( 16)
In E q u a tio n ( 1 6 ) , Q i s in c f s o f a i r , T i s t h e a i r te m p e ra tu re in
degrees F a h re n h e it a b s o lu t e , and P 1 i s t h e a i r p re ssu re in inches o f
m e rcu ry.
I f t h e e f f e c t s o f c o m p r e s s i b i l i t y a re c o n s id e re d n e g l i g i b l e ,
then E q u a tio n (1 6 ) can be used t o compute t h e volume r a t e o f flo w
th ro u g h t h e in t a k e o r i f i c e .
EFFECTS OF COMPRESSIBILITY
T hroughout th e p re c e d in g d e r i v a t i o n , th e f lo w o f a i r was con­
s id e re d t o be a p p l i c a b l e t o t h e same laws which a p p ly t o a non-compressib le f lu i d .
S in ce a i r i s c o m p re s s ib le , t h e e r r o r due t o t h i s assumption
was. a n a ly z e d .
I t i s c o n v e n ie n t t o c o n s id e r t h e f lo w process th ro u g h t h e o r i f i c e
as i-s e n tro p i c .
In a c t u a l i t y t h e r e i s some energy lo ss th ro u g h th e
o r i f i c e , and due t o t h i s
1.8-.
lo s s , t h e r e w i l l
be some heat exchange.
How-*
Buckingham, E,. " H i s t o r y o f O r i f i c e M ete rs and th e C a l i b r a t i o n ,
C o n s t r u c t i o n , and O p e ra tio n o f O r i f i c e s f o r M e t e r i n g . " American
S o c ie t y o f Mechanical E n g in e e r s . New Y o rk . 1935. P . 123.
i
42
e v e r , assuming f h a t t h e f l u i d
in g development, and w i l l
i s I d e a l> w i l l
ease th e work in t h e f o l l o w ­
n o t g iv e a s e r io u s e r r o r in c a l c u l a t i o n , s in c e
t h e f lo w process th ro u g h t h e o r i f i c e o cc u rs r a p i d l y and t h e r e i s
little
t im e f o r h e a t exchange.
In F i g . 9 ,
l e t P o in t I be a p o i n t in t h e o r i f i c e
in ta k e p ip e f a r
enough upstream from t h e o r i f i c e t h a t t h e stream li n e s o f t h e flo w are
p a r a lle l.
L e t P o in t 2 be a p o i n t in t h e vena c o n t r a c t a .
W r i t i n g t h e c o n t i n u i t y e q u a tio n between I and 2, we have
G = A| Vj (f I = AgVg^f 2
The general
p o in t s
form o f th e B e r n o u ll i e q u a tio n can be w r i t t e n between
I and 2 as
I9
4-
vr
4
a
%
f
z,
29
20
(17)
In t h e e q u a tio n s used here> G i s t h e w e ig h t r a t e o f f l o w , Aj i s t h e
c ro s s -s e c tio n a l
o r ific e ,
area o f t h e p ip e , Ao i s t h e c r o s s - s e c t i o n a l area o f th e
i s t h e s p e c i f i c w e ig h t o f t h e f l u i d , k is t h e a d i a b a t i c co­
e f f i c i e n t f o r t h e f l u i d , V i s th e average v e l o c i t y , P i s t h e p re s s u re in
t h e f lo w , T " i s t h e u n i t s h e a rin g s t r e s s , L i s t h e le n g th o f p ip e , and R
i s t h e h y d r a u lic r a d iu s o f t h e p ip e .
S in ce i t has been a ssum ed .that t h e f l u i d
fric tio n
19.
lo ss between I and 2 .
is id e a l, th e re w ill
be no
T h e r e f o r e t h e l a s t term, i s dropped from 1
9
Vennard, J . K . / Op. G i^ . P'. 71'. -
43
t h e B e r n o u ll i
e q u a tio n above, s in c e t h i s te rm accounts f o r energy loss
due t o f r i c t i o n .
Remembering t h a t P o in t s I and 2 a re a t equal e l e v a t i o n s . Equation
(17) i s r e w r i t t e n as
(18)
c o n s ta n t = C '.
For an I s e n t r o p i c process
S o lv in g t h i s
fo r
, the
f o l l o w i n g i s o b t a in e d .
S u b s titu tin g t h is
fo r
0 in E q u a tio n . ( 1 8 ) , i n t e g r a t i n g and r e a r ra n g ­
in g t e r m s , t h e f o l l o w i n g e q u a tio n can be w r i t t e n .
VT
- i2gf .
h'
'
(' t""X
1-
( * ) v J
(19)
When t h e c o n t i n u i t y e q u a tio n i s s o lv e d f o r V| and t h i s value sub­
s titu te d
i n t o E q u a tio n ( 1 9 ) , t h e f o l l o w i n g e x p re s s io n i s o b ta in e d
2 ^
o -7
. o v r
2?
But
X *-
/ o \ K -J
( 20 )
&
and
^
S u b s t i t u t i n g th e above va lu e s f o r Z z
and
and s o l v in g Equation
(2 0 ) f o r V2» E q u a tio n (2 1 ) i s o b t a in e d .
■
M
m
F
V
h
B
W
W
L
\4
(21)
44
S u b s t i t u t i n g t h i s v a lu e o f V'2 i n t o t h e e q u a tio n CG =
2 % 2^ > th e f o l I ow-
in g is o b f a i n e d .
( 22 )
, E q uation (2 2 ) can be r e w r i t t e n as
L e t t i ng
6
=
(23)
T h is e q u a tio n g iv e s t h e t h e o r e t i c a l
v a lu e o f G .
As was s t a te d e a r l i e r in
t h i s s e c t i o n , t h e f l u i d d e a lt w ith i s n o t i d e a l , and some f r i c t i o n
does e x i s t across t h e o r i f i c e .
In o r d e r t o account f o r t h i s
lo ss
fric tio n
lo s s , we must m u l t i p l y E q u a tio n (23) by a v e l o c i t y c o e f f i c i e n t Cv t o
reduce th e t h e o r e t i c a l
j e t v e l o c i t y t o t h e a c tu a l
je t v e lo c ity .
P o in t
2 has been taken as a p o i n t in th e vena e o n tr a c t a which has a c r o s s s e c t io n a l area which i s
le s s than t h a t o f t h e o r i f i c e .
In o r d e r t o
c o r r e c t E q u a tio n (23) t o a l lo w f o r t h e decrease in area o f t h e j e t a t
t h e vena e o n t r a c t a , we must s u b s t i t u t e CcAo ^o r ^2 ‘ n E q u a tio n ( 2 3 ) ,
where A0 .is t h e area o f t h e o r i f i c e , and Cc i s th e c o n t r a c t i o n co­
e ffic ie n t.
The area o f t h e vena e o n tr a c t a th e n is d e fin e d as CcA0 -
T h is i s i d e n t i c a l t o t h e o r y used in in c o m p re s s ib le f lo w .
'ib le
For compress-
f lo w , however, t h e v a lu e o f Cc does n o t depend on t h e r a t i o Ag/Aj
a lo n e , b u t depends a ls o upon t h e r a t i o Pg/Pj » Cc in c re a s e s as th e r a t i o
p 2/P |
dee re a se s.M u fttip ty in g E q u a tion (2 3 ) by CvCc and s u b s t i t u t i n g A0 f o r
A2 , we have
45
Ce Cv A o
2 ^ 5
"
l / K' M
y , ' L
■ ''''
^
<24)
T h is e q u a tio n i s much t o o complex f o r u 'c p h v e n ie n t,.d s # < rn r'm e t^ r-",.
in g .
soI I
I t has been expressed in a much s im p le r form by Daugherty and Sngeras
!
G
-
_______
^
(25)
where ( C = CvCc ) and has t h e same v a lu e as t h e d is c h a rg e c o e f f i c i e n t (C)
f o r an o r i f i c e w i t h In c o m p re s s ib le f l u i d
f lo w i n g .
The f a c t o r "Yn is an
expansion f a c t o r which accounts f o r t h e e f f e c t s o f c o m p r e s s i b i l i t y .
If
E q ya tIo n (2 5 ) i s equated t o E q u a tio n 24 and t h i s , e q u a l i t y i s so lve d f o r
” YH, t h e f o l l o w i n g r e s u l t i s found
v„A
y_1/
m
i m
d
m
b - m i
,
I U
7 W
i b - m
F
(26)
m
For a n o z z le o r a v e n t u r i tu b e where Cc = I , i t can be seen t h a t Y
can be computed from t h e above e q u a tio n ) however, where t h e c o e f f i c i e n t
Cc i s changing w ith t h e r a t i o E g /P ], th e n e xpe rim e nta l methods must be
used t o f i n d t h e v a lu e s o f " Y " .
e m p ir ic a l
Buckingham2
21 suggests t h e f o ll o w i n g
0
e q u a tio n f o r " Y " .
R -fc
y _ I _ [0.41 + 0 .3 5 ( § ) 4] [ Tepr1]
(27)
20.
21.
D a u g h e rty, R . L . , and Vngerso1.1 , A.C. " F l u i d Mechanics With
E n g in e e rin g A p p l i c a t i o n s . "
1954. McGraw-Hi I I « New Y o r k . P. 150.
Buckingham,. E . Op. G i t . P . 28.
•
46
where D0 I s t h e d ia m e te r o f th e o r i f i c e .
Buckingham s t a t e s t h a t t h i s e q u a tio n i s v a l i d f o r f la n g e ta p s o r
vena c o n t r a c t a t a p s .
E q u a tio n s (2 5 ) and (2 7 ) can now be used t o compute t h e r a t e o f flo w
o f a i r th ro u g h t h e o r i f i c e .
T h is r e s u l t can then be compared w ith th e
r a t e o f f lo w o f a i r computed by E q u a tio n (1 6 ) which n e g le c ts t h e e f f e c t s
o f c o m p r e s s ib ility .
To make t h i s com parison, th e maximum r a t e o f flo w
was chosen, s in c e th e e f f e c t s o f c o m p r e s s i b i l i t y shou ld b e -a maximum a t
th is ra te .
T h e re fo r e t h i s comparison w i l l
I n d i c a t e r t h e maximum..error
in v o lv e d in com puting t h e r a t e o f f lo w o f a i r w h ile n e g le c t in g th e
e ffe c ts o f c o m p r e s s ib ility .
In t h i s
i n s t a l l a t i o n , th e o r i f i c e diam eter
was 1.65 in c h e s , and t h e p ip e dia m e te r was 6 .0 in c h e s .
For t h e maximum r a t e o f f lo w , t h e f o l l o w i n g values were re co rd e d .
Tem perature = 75° F (The te m p e ra tu re i s assumed t o be
th e same a t P o in ts I and 2 ) .
Pj
= 25.62 inches o f m e rcu ry.
Pj - P2
= 10.00 inches o f w a te r.
For a i r t h e a d i a b a t i c c o e f f i c i e n t K = I .4 0 , and th e gas c o n s t a n t R = 5 3 . 3 .
From t h e above f i g u r e s , t h e f o l l o w i n g com p u ta tio n s can be made.
/4 c =
J2& =
Di
R -P a
IO
2 .0 /4 6
L if =
6 .0 0
= 0 .0 Z 8 7
0
. 2 73"
Pl
(2 S l4 z )(l2 ,£ )(U -4 )
= 0' 063£ /hs* Pef
r,
47
V=
!“
[? ’4 I f
. 0 .9 9 IS S
These r e s u l t s can now be s u b s t i t u t e d i n t o E q uation (2 5 ) t o c a l c u l a t e
th e r a t e o f f lo w o f a i r .
The d is c h a rg e c o e f f i c i e n t used w i l l
be taken as
used in t h e development o f E q uation (1 6 ) on Page 41 .
t h e same as t h a t
i
6 - 0.6042 fa 9 0 /S 5 ~ )(0.0/48)
0 . CO8 8 7 - / 2 - / 4
-
p o t CZ A )
7
UJ2Z-O
0 - 4;
C-L74)V
0 ./3 0
/ / s . p e r r Z V n iZ ^ e
When t h e above va lu e s are s u b s t i t u t e d i n t o E q u a tio n ( 1 6 ) , t h e volume
r a t e o f f lo w i s o b ta in e d f o r t h e case when c o m p r e s s i b i l i t y e f f e c t s are
n e g le c te d .
From E q u a tio n (16,)
_ __
2. O S ' c f s
0 =
I f t h i s v a lu e o f Q i s m u l t i p l i e d by. ^
, . a w e ig h t r a t e o f flo w is
fo u n d , and t h i s v a lu e can be compared t o t h a t o b ta in e d by in c lu d in g
e ffe c ts o f c o m p r e s s ib ility .
G = 2.05 ( . 0 6 3 5 ) =
0.1305 l b s . per m in.
The e r r o r in G in c u r r e d by n e g le c t in g th e e f f e c t s o f c o m p r e s s i b i l i t y can
th e n be expressed as
;
E r r o r = F®♦ 1305
L
-OJ 500 lx 100
J
0.1300
In t h e 2.06 inch p ip e , t h i s
v e l o c i t y o f 0 .3 4 f p s .
wouId be 0.55 f p s .
For t h e
—
0*385%
i s e q u iv a l e n t t o an e r r o r in a i r
1.63 inch p ip e , th e e r r o r in v e l o c i t y
48
S in ce t h e e r r o r in th e r a t e o f f lo w o f a i r due t o n e g le c t in g
c o m p r e s s i b i l i t y e f f e c t s was s m a ll. E q u a tio n (1 6 ) was used t o compute
t h e volume r a t e o f a i r flo w th ro u g h o u t t h i s p r o j e c t .
49
SYSTEM FOR MEASURING PRESSURE DROP ALONG THE TEST SECTION
PRESSURE TAPS AND CONTROL PANEL
Two com plete s e ts o f t e s t p ip e s were u se d .
p ip e s are given on Page
29 -
P re ssu re ta p s were i n s t a l l e d
along th e p ip e .
The dimensions o f the se
in t h e t e s t p ip e a t 3 - f o o t i n t e r v a l s
Seven ta p s in a l l
were i n s t a l l e d
in th e t e s t s e c t i o n .
ThQ f i r s t ta p was lo c a te d 2 f e e t from th e end o f t h e 90° bend in theconve ying p ip e which marked th e b e g in n in g o f th e t e s t s e c t i o n .
photograph o f one o f th e s e p re s s u re ta p s can be seen in F i g .
A d e ta il
II.
The p re s s u re ta p was made by f i r s t d r i l l i n g a I /6 4 inch diam eter
h o le th ro u g h t h e p ip e w a l l .
S p ecia l c a re was taken t o in s u r e t h a t the se
h o le s were normal t o th e p ip e w a l l .
in ch brass f l a r e ty p e f i t t i n g
h o le .
S p ecia l
A f t e r t h e h o le was d r i l l e d , a 3/16
was welded t o t h e p ip e o v e r t h e d r i l l e d
ca re was used in removing the; b u r r around t h e d r i l l e d h o le
on th e in s id e o f th e p ip e so t h a t none o f t h e v e l o c i t y head would be
c o n v e rte d t o n e g a tiv e pressure- head as t h e a i r flo w passed o v e r th e
b u rr.
To accom plish t h i s , a long dowel
wrapped w it h f i n e emery c l o t h
end.
I i-JZz inches in dia m e te r was
f o r a d is ta n c e o f about one f o o t from one
The dowel was th e n in s e r t e d i n t o th e t e s t p ip e , and th e b u rr a t
each p re s s u re ta p was ground smooth.
A f t e r one o p e r a tio n w it h th e dowel
and emery c l o t h , t h e ta p s were r e d r i l l e d , u s in g th e same s iz e d r i l l
b it.
Fol lo w ing t h i s , t h e ta p s were ground smooth oi"Ke<#or%^&tng,.t h e d'o.we,t-an-di.
emery c l o t h .
Next 3 /1 6 inch O.D. copper t u b in g was i n s t a l l e d from t h e
F ig .
II.
Pressure ta p on th e t e s t p ip e .
51
p re s s u re ta p t o t h e c o n t r o l
p a n e l.
Only one p re s s u re ta p was i n s t a l l e d a t each 3 - f o o t i n t e r v a l .
Care,
was ta k e n t o see t h a t each ta p was i n s t a l l e d e x a c t ly in t h e to p c e n t e r o f
t h e p ip e .
O the r i n v e s t i g a t o r s 22 have used m u l t i p l e ta p s a t each s e c t io n
w i t h th e s e ta p s spaced a t equal
i n t e r v a l s around th e p ip e d ia m e te r. How­
e v e r , th e ta p s used here seemed t o f u n c t io n q u i t e w e l l , and v e ry l i t t l e
f l u c t u a t i o n was n o t ic e d on t h e manometer when p re s s u re re a d in g s were'
being t a k e n .
The d i f f e r e n t i a l
p re ssu re s b e tw e e n 'th e ta p s w e r e ' s a t i s ­
f a c t o r i l y u n ifo rm ,when ste a d y s t a t e c o n d it io n s o f flo w e x i s t e d .
CONTROL PANEL AND MANOMETER
A c o n tro l
panel was s e t up so t h a t t h e d i f f e r e n t i a l
p re s s u re between
p re s s u re ta p s c o u ld be measured w i t h o u t d is c o n n e c tin g t h e manometer from
each p a i r o f c o n s e c u tiv e t a p s .
a tta c h e d i s v i s i b l e
panel
in F i g .
10.
T h is c o n t r o l
panel w ith t h e manometer
The la y o u t o f t h e p ip in g f o r t h e c o n tro l,
i s shown in th e drawing on" Page 52.
An i n c l i n e d tu b e t y p e manometer was used t o measure t h e d i f f e r e n t i a l
pressures.
0 .8 2 4 .
The manometer used a red o i l
w ith a s p e c i f i c g r a v i t y o f
The manometer s c a le read d i r e c t l y in inches o f w a te r , and was
graduated t o read a maximum o f 4 inches o f w a ter by g r a d u a tio n s o f 0 .0 2
in c h e s .
However, i t was found t h a t manometer d e f l e c t i o n s o v e r 3 .0 InchSs
o f w a ter were t o o u n s ta b le t o a f f o r d a c c u r a te r e a d in g s .
A t t h e high
w e ig h t r a t e s o f g r a in f lo w more than 3 .0 inches o f d e f l e c t i o n was r e q u ire d
22.
Crane, J . W. Op. C i t . p T l s I
~~
’ ""
~
Orifice
vVVe:
AU Va/ves
B ra s s
T y zP f s.
VALVE
PRESSURE
DRAWAZ
FIG. IZ
M ANIFO LD
FOR
MEASUREMENTS
S C A LE ~
none:
e v : J.J.C.
5 - 4 - eo
t o read t h e d i f f e r e n t i a l
p re s s u re between t a p #1 and ta p § 1 .
To remedy
t h i s , t h e two v a lv e s marked A and B in t h e drawing on Page 52 were i n ­
s t a l l e d so t h a t t h e d i f f e r e n t i a l
p re s s u re between ta p #4 and ta p #5,
# 6 , and #7 c o u ld be measured a f t e r t h e d i f f e r e n t i a l
ta p #1 and ta p # 2 , # 3 , and #4- had been measured.
p re ssu re s between
A fte r t h is c o rre c tio n ,
t h e manometer was capa ble o f measuring any r e q u ir e d d i f f e r e n t i a l
p re ssu re
w it h n e g l i g i b l e f l u c t u a t i o n .
The two f la n g e ty p e p re s s u re ta p s on e i t h e r s id e o f t h e in ta k e
o r ific e ,
m a n if o l d .
p r e v io u s ly d e s c r ib e d , were a ls o connected t o t h e c o n t r o l panel
A v e rtic a l
U -tube manometer f i l l e d
w ith red o i l
g r a v i t y 0.824 was used t o measure th e d i f f e r e n t i a l
o r if ic e p la te .
F lu c tu a tio n
o f s p e c ific
p re s s u re a cross th e
in t h i s manometer d u rin g measurement was
n e g l i g i b l e , and i t was p o s s ib le t o e s tim a te t h e manometer re a d in g t o th e
n e a re s t 0 .0 2 inches o f o i l .
CONTROL OF RATE QF FLOW OF GRAIN
The e x p e rim e n ta l
p ressure' drops were measured a t v a r y in g a i r
v e l o c i t i e s w it h a c o n s ta n t w e ig h t r a t e o f g r a in f lo w .
To m eter t h e w e ig h t r a t e o f flo w o f g r a i n ,
v a r i a b l e area o r i f i c e .
i t was decided t o use a
T h is o r i f i c e was c o n s tr u c te d in t h e form o f a
s l i d i n g aluminum gate in t h e bottom o f t h e g r a in hopper.
A r e c ta n g u la r
sharp-edged h o le 1,0 inches wide and 3 . 0 inches long was machined in
t h e c e n t e r o f t h i s g a te .
T h is gate i s shown in F i g .
13,
As t h e gate
was pushed f u r t h e r i n t o th e s l o t . u n d e r t h e hopper, t h e h o le opened
lo n g e r , a l lo w in g more wheat t o flo w th ro u g h th e o r i f i c e .
54
A system o f h o le s d r i I led th ro u g h t h e p l a t e was devised so t h a t
t h e gate c o u ld be locked in 6 d i f f e r e n t p o s i t i o n s .
The hopper was then
c a l i b r a t e d in a s t a t i c p o s i t i o n o f f t h e exp e rim e n ta l equ ipm ent.
The
w e ig h t r a t e o f f lo w o f g r a in th ro u g h each p r e s e t o r i f i c e opening was
c a l i b r a t e d by t a k i n g t h e average o f 20 tim e d r a t e s o f f lo w f o r each
p o s itio n .
The r e s u l t s o f t h i s c a l i b r a t i o n are shown in Fig.. 14.
As
can be seen, th e w e ig h t r a t e o f f lo w when p l o t t e d a g a in s t t h e measured
gate opening produces a s t r a i g h t l i n e .
mined by t h e methods used on Page
The e q u a tio n o f t h i s
lin e d e te r­
27 is
Gs =64.37 X 1 '95
(28)
where Gs i s th e w e ig h t r a t e o f flo w o f g r a i n in pounds per m inute and X
i s th e g ate opening in in c h e s .
T h is e q u a tio n i s v a l i d o n ly f o r a gate
opening 1.0 in ches w id e , f o r wheat w ith t h e p r o p e r t ie s g ive n in th e
a p p e n d ix, and f o r a "head" o f a t le a s t 6 inches o f wheat above th e
o r ific e .
A f t e r t h e hopper was p laced back in t h e e xpe rim e nta l
c a l i b r a t e d aga in w ith t h e system r u n n in g .
system, i t was
A d i f f e r e n c e o f a p p ro x im a te ly
0.1 I pounds per m in u te g r e a t e r f lo w p e r o r i f i c e opening was observed in
th is te s t.
By v a r y in g t h e volume r a t e o f f lo w o f a i r ,
i t was found t h a t
t h e a i r le a k in g from t h e fe e d e r d id n o t a f f e c t t h e w e ig h t r a t e o f flo w o f
g r a in from t h e hopper.
The in c re a s e in w e ig h t r a t e o f f lo w was a t t r i b u t e d
t o t h e v i b r a t i o n o f t h e equipment w h ile r u n n in g .
I t was noted d u rin g th e
c a l i b r a t i o n t h a t when th e g r a in was k e p t a t a depth g r e a t e r than 6 inches
F ig .
13. Bottom view o f v a r i a b l e o r i f i c e
in bottom o f th e g r a in hopper.
56
•riiri::
tL :: :
F IG .
14
: T ': ^
- I : ''
W E I G H T R A T E OF FLOW OF L I G H T SPRI NG
W H E A T THRU
l-»R. W / D E R E C T A N G U L A R
ORIFICE
IN B O T T O M OF G R A I N H OP PE R
/
a3
O R IF IC E
asr CU
ap
OPEN/IN'G
/.J
/5"
(in ches)
2 ,c
3
57
above t h e o r i f i c e p l a t e , t h e w e ig h t r a t e o f flo w o f g r a in th ro u g h th e
o r i f i c e was independent o f t h e depth o f g r a in above th e o r i f i c e opening
The g r a in hopper i t s e l f was c o n ic a l
in shape and was b u i l t from
'0 .1 2 5 inch t h i c k g a lv a n iz e d sheet s t e e l .
The 11 inch high metal rim
v is ib le
in F i g . 3 was added a f t e r e x p e rim e n ta tio n began t o p re v e n t th e
escape o f wheat g r a in s from t h e hopper.
The hopper and o r i f i c e
f u n c tio n e d w e ll
d u rin g th e e x p e r im e n ta tio n .
P e r i o d i c checks were run on t h e hopper c a l i b r a t i o n t o make sure t h a t
t h e w e ig h t r a t e o f flo w o f g r a in had n o t d e v ia te d from t h e c a l i b r a t e d
ra te .
No d e v ia t io n s g r e a t e r than 0 .1 0 pounds per m inute were n o te d .
In t h e 1 - 1 /2 inch t e s t p ip e , 5 d i f f e r e n t w e ig h t r a t e s o f g r a in
' f lo w were used, and d u r in g t h e t e s t o f t h e 2 inch p ip e , 6 d i f f e r e n t
r a t e s were used.
The w e ig h t r a te s o f g r a in f lo w t h a t were used were
3 .3 6 pounds per m in u te , 6 .67 pounds per m in u te , 10.35 pounds per m inute
15 .98 pounds per m in u te , and 17.1.0 pounds per m in u te .
The f i r s t f i v e
r a t e s were th o s e used w h ile t e s t i n g t h e 1 - 1 /2 inch p i p e , w h i l e a l l
were used in t e s t i n g t h e 2 inch p ip e ;
s ix
58
EXPERIMENTAL PROCEDURE
THE GRAIN USED
Four hundred pounds o f l i g h t s p r in g wheat were o b ta in e d f o r use in
t h e e x p e rim e n ta l work in t h i s p r o j e c t .
lo c a l
g r a in e l e v a t o r . . A l l
from t h e same b i n .
wheat fan m i l l
The wheat was o b ta in e d from a
f o u r hundred pounds were take n a t one tim e
The wheat was cle aned in a commercial ty p e seed
a t t h e g r a in e l e v a t o r .
To remove th e f o r e ig n m a t t e r , th e
wheat was cle aned a g a in , t h i s tim e u s in g a small k i c k e r - t y p e c le a n e r ,
which i s n o r m a lly used t o determ ine t h e p e r c e n t 11dockage”
in wheat'. T h is
o p e r a tio n cle aned t h e wheat s a t i s f a c t o r i l y .
The s p e c i f i c g r a v i t y o f t h e wheat was determined by measuring th e
volume o f a known w e ig h t o f t h e wheat.
A small
beaker was f i r s t f i l l e d
number o f p a r t i c l e s
The f o l l o w i n g proced ure was used.
w ith wheat and weighed.
in t h e beaker were c o u n te d .
Then th e
A f t e r c o u n t in g , th e
Wheat p a r t i c l e s were poured i n t o a g ra d u a te c o n t a in in g a known volume o f
w a te r .
The c o n te n ts o f t h e grad u a te were s t i r r e d v ig o r o u s ly w h ile th e
wheat p a r t i c l e s were being in tr o d u c e d , in o r d e r t o d is lo d g e t h e a i r
bubbles which seemed t o c l i n g t o t h e m in u te f i b e r s around th e wheat,
p a rtic le .
The in c re a s e in volume due t o t h e a d d it io n o f t h e wheat
p a r t i c l e s was n o te d .
From t h e above in f o r m a t i o n ,
i t was p o s s ib le t o
compute t h e w e ig h t per p a r t i c l e , t h e volume per p a r t i c l e , and th e a b s o lu te
s p e c i f i c w e ig h t o f t h e p a r t i c l e s in pounds per c u b ic f o o t .
f o r t h e wheat a re g iv e n in t h e a p p e n d ix .
These values
The average v a lu e o b ta in e d f o r
59
th e a b s o lu te s p e c i f i c ' g r a v i t y o f th e wheat was compared w ith an average
f i g u r e from t h e G ra in L a b o r a to ry a t Montana S t a t e C o lle g e .
t a i n e d in t h i s paper was 8 7 .8 pounds p e r c u b ic f o o t .
The va lu e ob­
The range o f values
f o r t h i s ty p e o f wheat grown in t h e G a l l a t i n V a lle y as g ive n by th e
G ra in L a b o r a to r y were from 82.5 t o 8 8 .0 pounds per c u b ic fo ot..
87.8
pounds per c u b ic f o o t seemed rea so n a b le inf t h i s com parison.
No data Was a v a i l a b l e f o r comparison "o f th e w e ig h t per p a r t i c l e and.
t h e v q Iume per p a r t i c l e .
Ten p a r t i c l e s were measured, u s in g a micrometer-.
These p a r t i c l e s
were measured on t h e i r m ajor a x is and on both m inor a x e s .
1
A shape
f a c t o r o f C/-{~ab-} as suggested by Corey2 3 , was determined f o r each
p a r tic le .
I n . t h i s e x p r e s s io n , "C" i s t h e le n g th o f t h e m ajor a x i s ,
and " a ” and " b " a r e " t h e le n g th s o f th e m inor axes.
t h i s shape f a c t o r was 0 .6 0 0 .
The average va lu e o f
The d e v i a t io n o f th e in d i v i d u a l p a r t i c l e s
from 0 .6 0 0 was cjui'te s m a ll, even though some o f the: p a r t i c l e s v a r ie d
c o n s id e r a b ly in s i z e .
S p e c ia l c a re was used in exam ining t h e wheat d u r in g th e a ctu a l t e s t
runs..
When t h e wheat began t o show s ig n s o f damage, i t was removed and
an unused charge o f wheat was placed in t h e hopper.
pounds o f wheat were used in each c h a rg e .
A p p ro x im a te ly f o r t y
The average t im e o f use f o r
each charge o f wheat, b e fo re n o t ic e a b le damage t o th e p a r t i c l e s began
23.
C o re y, A. T . "Drag C o e f f i c i e n t s o f N o n-u niform P a r t i c l e s . "
T h e s is , C o lorado S t a t e U n i v e r s i t y , 1954. P. 20.
A
60
t o o c c u r , was a p p ro x im a te ly te n h o u rs .
T h is would have re p re s e n te d an
average o f 142 com plete c i r c u l a t i o n s th ro u g h t h e system, assuming th e
average w e ig h t r a t e o f f lo w was t h e average o f th e 6 r a t e s used.
DESCRIPTION OF PROCEPORE USED
The s te p by step proced ure fo llo w e d d u r in g each e x p e rim e n ta l run
was t h e same f o r each p ip e s i z e .
For t h e I .63 inch p ip e , o n ly f i v e
w e ig h t r a t e s o f f lo w o f wheat were used, w h ile in th e 2.06 inch p ip e
6 w e ig h t r a t e s o f f lo w o f wheat were u se d .
These r a te s o f flo w have been
I i s t e d on Rage 5?.
The e x a c t procedure f o llo w e d d u r in g t h e runs was as f o l l o w s .
1. )
The blower was t u rn e d on, and t h e 2 - in c h a i r c o n t r o l v a lv e
was opened c o m p le te ly .
2. )
The fe e d e r was s e t i n t o o p e r a t io n , . .
3. )
The v a r i a b l e o r i f i c e
in t h e bottom o f th e g r a in hopper was
opened t o t h e d e s ire d w e ig h t r a t e o f flo w and locked in t o
p o s itio n .
4. )
The re a d in g o f t h e v e r t i c a l
U -tube manometer connected across
t h e in ta k e o r i f i c e was re c o rd e d .
5.
)
6. )
The a i r te m p e ra tu re and th e b a r o m e tric p re ssu re were re co rd e d .
The p re s s u re in s id e t h e t e s t p ip e a t t h e f i r s t p re s s u re tap
was re c o rd e d .
7. )
The d i f f e r e n t i a l
pressu re s between p ressure ta p s alo n g th e
t e s t s e c t io n were recorded
61
8. )
The v e r t i c a l U-tuti'e was a g a in connected t o t h e in t a k e o r i f i c e
and t h e manometer re a d in g a cross t h e o r i f i c e was checked.
If
t h i s re a d in g and t h a t ta k e n in ste p (4 ) d i f f e r e d , t h e run was
r e p e a te d .
I f t h e re a d in g checked t h a t in step ( 4 ) , step (9)
was i n i t i a t e d .
9. )
The opening o f t h e tw o - in c h a i r c o n t r o l v a lv e was reducedu n til
a r e d u c t io n in th e in t a k e o r i f i c e manometer was n o te d .
F o llo w in g s te p ( 9 ) , t h e proced ure from ste p (4 ) th ro u g h step ( 9 )i was
repea ted th ro u g h o u t t h e f u l l
range o f f l o w .
The lower l i m i t o f flo w
o c c u rre d when t h e conve ying p ip e began t o b lo cks
B e fo re b lo c k in g
o c c u r r e d , t h e f lo w became i r r e g u l a r , and t h e s e n s i t i v e manometer used
t o measure d i f f e r e n t i a l
p re s s u re s along t h e t e s t pipe began t o f l u c t u a t e .
A n o t ic e a b l e f l u c t u a t i o n o f t h e manometer used t o measure d i f f e r e n t i a l
p re s s u re a cross t h e in ta k e o r i f i c e was a ls o e v id e n t when a i r v e l o c i t i e s
approached t h e v e l o c i t y a t which b lo c k in g occurred'.
A f t e r t h e c o m p le tio n o f th e e x p e rim e n ta l
run f o r each s o l i d r a t e o f
f lo w , t h e leakage from t h e fee d e r was checked as d e scrib e d on Page 22.
The procedure d e s c rib e d above was repea ted f o r each w e ig h t r a te
o f s o l i d f lo w f o r both p ip e s iz e s .
A p p ro x im a te ly two and o n e - h a lf hours
o f c o n tin u o u s t e s t i n g were, necessary t o com plete th e e n t i r e run f o r each
w e ig h t r a t e o f s o l i d f lo w .
IN ITIA L RUNNING
B e fo re t h e e x p e rim e n ta l
data c o u ld be re c o rd e d , i t was found t h a t t h e
-
.
62
system had t o be o p e ra te d f o r a p p ro x im a te ly s i x hours b e fo r e th e
p re s s u re drop a long t h e t e s t p ip e began t o be a c o n s ta n t between
t h e p re s s u re t a p s .
T h is change in p re s s u re drop was due t o t h e
c o n s ta n t p o l i s h i n g o f t h e p ip e i n t e r i o r by th e wheat p a r t i c l e s .
No data was recorded f o r t h e p re s s u re drop due t o a i r a lo n e b e fo re
t h e p ip e became p o lis h e d ; however, i t can be re a so n a b ly assumed
t h a t t h e p ip e Was tra n s fo rm e d frorn.-one-which behaved as a rough
p ip e t o a p ip e which was h y d r a u l i c a l l y smooth w i t h i n t h e range o f
R e yn o ld ’ s numbers r e p o r te d in t h i s r e s e a r c h .
63
EXPERIMENTAL RESULTS
COLLECTION OF DATA
P re ssu re drop alo n g t h e t e s t s e c t io n was measured t h ro u g h o u t th e
fu ll
v e l o c i t y range o f t h e exp e rim e n ta l syste m .
by two s e p a ra te c o n d i t i o n s .
T h is range was l i m i t e d
The maximum a i r v e l o c i t y a t t a i n a b l e was
l i m i t e d by t h e c a p a c it y o f t h e b lo w e r.
The maximum a i r v e l o c i t y
a t t a i n e d was 84 fps in t h e I .63 inch p ip e , and 80 fp s in t h e 2.06 inch
p ip e .
Both v a lu e s were a t t a i n e d w i t h a w e ig h t r a t e o f f lo w o f wheat o f
3 .3 6 pounds p e r m tn u te .
The- lo w e r l i m i t i n g v e l o c i t y o f t h e e xpe rim e nta l
range was governed by t h e b lo c k in g te n d e n c ie s o f th e t e s t p i p e .
An ex­
p e rim e n ta l run was always begun u s in g t h e maximum a i r v e l o c i t y a t t a i n I
a b l e . - The f lo w o f a i r was then t h r o t t l e d down in small increm ents u n t i l
i t reached t h e p o i n t a t which th e system began t o b l o c k .
A f t e r ' some ex­
p e r ie n c e , i t was found t h a t b lo c k in g was impending when t h e s e n s i t i v e
manometer used t o measure p re s s u re d i f f e r e n t i a l
ta p s began t o f l u c t u a t e .
t h a t even a v e ry small
between th e pressure
Once t h i s c o n d i t i o n was reached, i t was found
decrease in a i r f lo w would cause b l o c k i n g .
The
r a t e a t which t h e b lo c k in g became com plete was dependent upon how f a r
below t h e b lo c k in g l i m i t t h e r a t e o f a i r f lo w was c u t .
When b lo c k in g
was t a k i n g p la c e , th e system behaved in t h e f o llo w in g 'm a n n e r .
F i r s t t h e d i s t r i b u t i o n o f t h e s o l i d s in th e flo w ceased t o be u n ifo r m .
Large s lu g s o f g r a in c o u ld be seen passing t h e . t r a n s p a r e n t s e c t io n .
e x t e n t o f th e b lo c k in g c o n tin u e d , th e g r a in began t o flo w in surges.
As t h e
Flow
64
o f g r a in would s to p t e m p o r a r i l y , c a u s in g a b u ild u p o f p re s s u re w i.thin th e
t e s t p ip e .
When th e p re s s u re became la r g e enough, a surge o f g r a in was
blown v i o l e n t l y th ro u g h t h e system, r e s u l t i n g in a decrease in p r e s s u r e .
T h is process re p e a te d i t s e l f u n t i l com plete blockage o c c u r r e d .
p lu g t h e t e s t p ip e , t h e g r a in hopper o r i f i c e was c lo s e d .
To un­
The c y c le o f
b lo c k in g j u s t d e s c rib e d would then re p e a t i t s e l f in re v e rs e process
u n til
a ll
t h e g r a in in t h e system had been d is c h a rg e d .
The maximum g r a in c a p a c it y o f t h e app a ra tu s was 16.05 pounds per
m in u te w it h an a i r v e l o c i t y o f 5 8 .0 fps in t h e I .63 inch p ip e , and 18.03
pounds per m in u te w ith an a i r v e l o c i t y o f 5 9 .9 fps in th e 2.06 inch p ip e .
D u rin g each e x p e rim e n ta l
t h e f o l l o w i n g p re s s u re t a p s :
6, and 4 and 7 .
r u n , t h e p re s s u re drop was measured between
I and 2, I and 3 ,
I and 4 , 4 and 5 , 4 and
The in crem en t o f p ip e le n g th between s u c c e s s iv e p ressure
ta p s was 3 .0 0 f e e t .
E xp erim e ntal
ment used.
data was read as a c c u r a t e l y as p o s s ib le w it h th e e q u ip ­
In some measurements ta k e n d u r in g th e ru n s , where t h e 15.98
and 17.10 pounds per m inute s o l i d s r a t e s were used, t h e manometer was
f l u c t u a t i n g as much as 0 .0 4 inches o f w a te r .
The s o l i d f lo w d u rin g th e s e
measurements was n o t u n ifo r m , and t h e a i r v e l o c i t y was app ro achin g th e
minimum necessary t o s u s t a in f l o w .
can be seen in F i g .
These measurements were e r r a t i c , as
16 .
ACCURACY OF DATA RECORDED
The accuracy o f t h e data taken d u r in g th e s e measurements was i n -
v
65
f Iuenced by se v e ra l
fa c to rs .
Al I b u t one o f th e se f a c t o r s in flu e n c e d
t h e accuracy o f t h e com putation o f t h e v e l o c i t y o f a i r in t h e t e s t p ip e .
The one f a c t o r which .was n o t in v o lv e d in t h e c a l c u l a t i o n o f t h e a i r
v e l o c i t y was t h e measurement o f d i f f e r e n t i a l
s e c tio n .
pressures alo n g t h e t e s t
T h is measurement was dependent upon two t h i n g s , th e c h a r a c t e r ­
i s t i c s o f t h e p re s s u re t a p s , and t h e accuracy o f t h e manometer.
The
manometer re a d in g s Were e s tim a te d a c c u r a t e l y t o t h e c lo s e s t 0.01
inch
o f w a te r .
The c h a r a c t e r i s t i c s o f th e p re s s u re ta p s were unknown; how­
e v e r, the d i f f e r e n t ia i
p re s s u re s between them and n o t t h e p re ssu re s a t
t h e ta p s were o f prim e im p o rta n ce , so s p e c ia l ca re was ta ke n t o make th e
p re s s u re ta p s as n e a r ly u n ifo rm in c o n s t r u c t i o n as p o s s i b l e , so t h a t
d iffe re n tia l
p re s s u re s would be measured a c c u r a t e l y .
An e r r o r in a i r v e l o c i t y o c c u r r e d , due t o t h e measurement o f th e
r a t e o f f lo w o f a i r , a t t h e in ta k e o r i f i c e .
The accuracy o f t h e measure­
ment here was d i r e c t l y dependent upon t h e accuracy o f t h e manometer
r e a d in g .
o il
The manometer used here was read t o t h e n e a re s t 0 .0 2 inches o f
o f s p e c i f i c g r a v i t y 0 .8 2 4 .
In th e o r i f i c e e q u a tio n ( 1 6 ) , which was
used t o compute t h e volume r a t e o f flo w o f a i r , t h i s amounts t o an
accuracy o f p lu s o r minus 0.005 c f s a t a manometer re a d in g o f te n inches
o f w a te r , which was a p p ro x im a te ly t h e maximum re a d in g used.
The maximum e r r o r in th e volume r a t e o f flo w o f a i r due t o n e g le c t ­
ing t h e e f f e c t s o f c o m p r e s s i b i l i t y was computed as 0.0079 c f s on Page 4 7 The method used in d e te rm in in g t h e leakage th ro u g h t h e fe e d e r was
66
a ls o a p o t e n t i a l source o f e r r o r . ■The o n l y p o s s ib le e s tim a te o f t h e
a ccuracy o f t h i s method i s t h a t which was in h e r e n t in t h e manometer read­
in g s .
The manometer was read t o t h e n e a re s t 0.05 inches o f o i l o f s p e c i f i c
g r a v i t y 0 .8 2 4 .
Using t h e maximum p re s s u re measurement, which was 15 inches
o f w a te r, t h i s accuracy o f measurement would have g ive n an e s tim a te d e r r o r ,
c a l c u la t e d by E q u a tio n ( 3 ) , o f 0 .0 0 2 c f s .
Assuming t h a t a i l
o f th e s e e r r o r s o c c u r in t h e same d i r e c t i o n g iv e s
an e s tim a te o f t h e maximum e r r o r in c u r r e d in computing th e r a t e o f flo w
o f a ir.
T h is maximum e s tim a te d e r r o r i s t h e sum o f th o s e l i s t e d above.
Q (maximum e r r o r ) = 0.008 +• 0.005 + .002 = 0.015 c f s
T h is e r r o r in Q was e q u iv a l e n t t o an e s tim a te d maximum e r r o r in a i r
v e l o c i t y in t h e t e s t s e c t io n o f I .04 fps in th e I .63 inch p ip e , and .0 .64
fp s in t h e 2.06 inch p ip e .
The in f l u e n c e o f c o m p r e s s i b i l i t y on t h e f lo w has a lr e a d y been d e t e r ­
mined on Page 4 7 .
A lth o u g h t h i s e f f e c t was n e g l i g i b l e , t h e v e l o c i t y o f
t h e f lo w in th e t e s t s e c t io n was computed from t h e w e ig h t r a t e o f flo w
o f a i r u s in g t h e c o n t i n u i t y e q u a tio n .
The p re s s u re in th e t e s t s e c tio n
was measured d u r in g e v e ry r u n , and from t h i s v a lu e , t h e s p e c i f i c w e ig h t
o f the a i r
in t h e t e s t s e c t io n was computed.
CALCULATION OF A l R -VELOCITY
in c a l c u l a t i n g t h e v e l o c i t y o f t h e a i r in th e t e s t s e c t i o n , th e
presence o f t h e wheat was n e g le c te d .
i t s e f f e c t was n e g l i g i b l e .
T h is was an a p p r o x im a t io n . ' However,
In o r d e r t o c a l c u l a t e th e e x a c t a i r v e l o c i t y .
67
i t would have been necessary t o know t h e v e l o c i t y o f th e s o l i d p a r t i c l e s .
S in c e i t was n o t f e a s i b l e t o measure th e v e l o c i t y o f t h e s o l i d p a r t ic le s , i t was decided t o n e g le c t t h e i r e f f e c t ,
The e f f e c t o f t h i s assumption can be seen in th e f o l l o w i n g sample
c o m p u ta tio n s ;
The r a t e o f a i r f lo w in a 2.06 inch d ia m e te r p ip e has been
determ ined t o be I .95 c f s .
T h is corresponds t o an a i r v e l o c i t y in t h e
t e s t p ip e o f 8 4 .0 f p s .
Assume t h a t t h e v e l o c i t y o f t h e s o l i d p a r t i c l e s
is 40 f p s .
T h is
assum ption seems rea so n a b le a f t e r o b s e rv in g the flo w o f t h e s o l i d p a r t i c l e s
in t h i s expe rim e nt.-
A ls o assume t h a t t h e w e ig h t r a t e o f flo w o f s o l id s
was 17.10 pound's per m in u te , o r 0.285 pounds per second.
t h e cro s s s e c t io n a l area o f a p a r t i c l e
w e ig h t o f a p a r t i c l e
i s 7 .3 0 x IO"
5
Assume a ls o t h a t
i s 0.000255 square f e e t > and t h e
pounds.
These were p r o p e r t ie s o f th e
average wheat p a r t i c l e used.
The number' o f p a r t i c l e s f lo w in g per second would be
0.285
3900 p a r t i c l e s per second.
7 .3 0 x IO" 5
The number o f p a r t i c l e s per f o o t o f p ip e would be
3900 p a r t i c l e s per second
40 f e e t p e r second
The volume o f th e p a r t i c l e s
= 97.5 p a r t i c l e s / f t .
in one f o o t o f p ip e can be found by m u l t i ­
p l y i n g t h e number o f p a r t i c l e s in one f o o t o f p ip e by th e volume o f .one
p a r t i c l e , which can be assumed as 8.79 x 10“ ^ c u b ic f e e t .
Volume o f p a r t i c l e s per f o o t o f p ip e = 97.5 (8.7 9x10
c u b ic f e e t .
= 8.57x10
5
68
The volume o f a i r in dfie f o o t o f p ip e i s t h e volume o f "the;, p ip e minus
th e volume o f t h e p a r t i c l e s .
Volume o f a i r per f o o t o f p ip e =
2 .06
12
TT
4
= .0231453 -
-
.000857
.0000857
= .0230596 c u b ic f e e t
The average a i r v e l o c i t y
is t h e v e l o c i t y o f t h i s volume o f a i r
necessary t o produce th e given volume r a t e o f flo w o f a i r .
s e c t io n a l
The n e t c r o s s -
area o f th e p ip e a v a i l a b l e f o r t h e flo w o f a i r i s t h e volume o f
a i r per c u b ic f o o t o f p ip e d iv id e d by t h e le n g th o f p ip e which we have
co n s id e re d t o be one f o o t .
The average a i r v e l o c i t y can be computed by d i v i d i n g th e volume r a t e
o f f lo w o f a i r by th e a v a i l a b l e c r o s s - s e c t i o n a l
Average A i r V e l o c i t y =
,
I .95
.0230596
a rea.
= 8 4 .7 0 fps
The c r o s s - s e c t i o n a l area o f t h e p ip e a v a i l a b l e f o r a i r f lo w ,
wheat volume i s n e g le c te d , i s 0.0231453 square f e e t .
i f th e
I f th e a i r v e l o c i t y
i s computed from t h i s c r o s s - s e c t i o n a l a re a u s in g th e same volume r a t e o f
a i r f lo w as b e f o r e , we o b t a in
Average a i r v e l o c i t y =
I .95 = 8 4 .2 0 fps
.0231453
The e r r o r due t o t h i s a p p ro x im a tio n can be expressed p e rce n ta g e -w ise
as
8 4 .7 0 - 84.20 x 100 = 0.590#
84.70
69
T h is e r r o r i s on Iy "'-slight I y more th a n t h e p o t e n t i a l e r r o r in th e
c a l c u l a t i o n o f t h e vo I umeS'rate o f . flo w o f :a i r .
The p o t e n t i a l
e r r o r due t o
in s tru m e n ta l accuracy was c a l c u la t e d p r e v io u s ly on Page 65.
S in ce t h e e r r o r due t o t h e above assumption was so s m a ll , average
a i r v e l o c i t i e s were c a l c u la t e d n e g le c t in g t h e g r a in volume th ro u g h o u t
t h i s p ap er.
ACCELERATION LOSSES .
The two e x p e rim e n ta l measurements Which were o f prim e im portance were
t h e a i r v e l o c i t y in t h e p ip e and t h e p re s s u re drop per u n i t le n g th o f p ip e .
in a n a ly z in g t h e r e s u l t s ,
i t was found t h a t a lo ss due t o a c c e l e r a t io n
o f t h e a i r and t h e s o l i d p a r t i c l e s was o c c u r r in g a f t e r t h e f lo w had
n e g o tia t e d th e 90 degree bend which e n te re d t h e t e s t s e c t i o n .
I t was ob­
served from t h e e x p e rim e n ta l measurements t h a t t h i s e f f e c t d ie d o u t some­
where between t h e second and t h i r d p re s s u re t a p s .
T h is was a d is ta n c e o f
a p p ro x im a te ly e i g h t f e e t from t h e end o f t h e bend a t t h e b e g in n in g o f th e
t e s t s e c tio n .
No a tte m p t was made t o measure these lo s s e s .
To p re v e n t t h e in c l u s i o n o f th e s e a c c e l e r a t io n
losses in th e data
r e p o r t e d , p a r t i c u l a r c a re was given t o a n a ly z in g th e measured pressure
drops between p re s s u re t a p s .
The p re s s u re drops between ta p s became
constan t-d o w n stre a m from t h e #3 p re s s u re t a p ,
In o b t a i n in g t h e pressure
drop per u n i t le n g th o f p ip e , o n ly th e s e ste a d y s t a t e p re s s u re drops were
averaged i n t o t h e f i n a l
fig u re s .
The average v a lu e f o r t h e p ressure drop
which was used in t h i s a n a ly s i s was o b ta in e d by d i v i d i n g t h e average o f th e
70
p re s s u re drops between p re s s u re ta p s by t h e le n g th o f th e p ip e between t h e
ta p s .
Thus t h e r e s u l t s o f t h i s re search p e r t a i n o n ly t o s te a d y s t a t e flo w o f
a i r and wheat m ix t u r e s .
The problems o f a c c e le r a te d f lo w o f a i r and s o l i d
m ix tu r e s i s a n o th e r complex problem w ith which t h i s paper is not concerned.
71
. GENERAL THEORY
MEASUREMENT OF VELOCITY OF SOL I D: RARTI CLES
.
' "
04
"
The th e o r y o f Crane
has been m entioned p r e v io u s ly in t h i s paper.
When t h i s p r o j e c t was f i r s t begun, i t . was t h e i n t e n t i o n o f t h e w r i t e r t o
work along t h e same l i n e s as Crane in d e v e lo p in g a th e o r y t o cove r th e head
lo s s .
In t h i s
l i g h t , c o n s id e r a b le th o u g h t and p la n n in g was done toward th e
problem o f measuring s o l i d v e l o c i t i e s in th e f l o w .
any f u r t h e r t h e o r y u s in g C ra n e 's o r i g i n a l
In o r d e r t o develop
id e a s , i t would have been alm ost
a n e c e s s it y t o measure t h e v e l o c i t i e s o f this s o l i d p a r t i c l e s , s in c e th e
drag f o r c e on t h e wheat p a r t i c l e
i s r e l a t e d t o t h e r e l a t i v e v e l o c i t y o f th e
a i r and t h e g r a in o r (Va-V s ) , where t h e s u b s c r i p t , a, i n d ic a t e s a i r , and
the s u b s c rip t, s,
i n d ic a t e s s o l i d s .
The p o s s i b i l i t y o f c o a t in g some o f th e wheat p a r t i c l e s w ith a r a d i o ­
a c t i v e m a te r ia l
was c o n s id e r e d .
I f t h i s had been done, i t would have been
t h e o r e t i c a l l y p o s s ib le t o i n s t a l l t h e necessary G e ig e r tu b e s and o th e r
e l e c t r o n i c equipment t o have tim ed t h e p a r t i c l e in i t s
p o in ts .
However, upon f u r t h e r c o n s i d e r a t io n ,
f l i g h t between two
i t was decided t h a t th e convey­
in g p ip e would und o u b te d ly become co n ta m in a te d from t h e r a d i o a c t i v e .m a te ria l
which rubbed o f f t h e wheat p a r t i c l e s .
any p a r t i c u l a r p a r t i c l e
im p o s s ib le .
T h i s would have made t h e t im in g o f
;
Some c o n s id e r a t io n was given t o t h e grow ing o f r a d i o a c t i v e wheat
24.
Crane, J . W.
Op. C i t . P. 40.-'
72.
p a r tic le s .
However, i t seeded t h a t t h i s process would o n l y lengthen th e
tim e r e q u ir e d f o r t h e system t o become c o n ta m in a te d .
The p o s s i b i l i t y o f ph o to g ra p h in g t h e p a r t i c l e s
speed m otion p i c t u r e camera was a ls o c o n s id e r e d .
in m otion w ith a high
However, such equipment
was found t o be v e ry expe nsive and was n o t a v a i l a b l e l o c a l l y .
THEORY DEVELOPED
A f t e r t h e e x p e rim e n ta l measurements had been made, graphs o f th e
p re s s u re drop p e r f o o t o f p ip e were p l o t t e d a g a in s t th e computed a i r
v e lo c ity .
The graphs f o r both p ip e s iz e s a re shown in F i g s . 15 and 16.
These appeared t o be a f a m i l y o f c u rve s which c l o s e l y p a r a l l e l e d t h e c u rve
f o r t h e p re s s u re drop due t o a i r a lo n e .
The e q u a tio n o f t h e curve f o r th e
p re s s u re drop due t o t h e a i r alo n e i s t h e f a m i l i a r ' Darcy. Weisbach e q u a t i o n ^ .
From a s tu d y o f F i g .
15 and F i g .
C29.)
16, i t was decided t h a t t h e f a m i ly o f
c u rve s c o u ld be re p re s e n te d by an e q u a tio n .
(30)
where K i s a p r o p o r t i o n a l i t y f a c t o r which i s dependent upon t h e w e ig h t r a t e
o f f lo w o f s o l i d s .
However, i t appeared e q u a lly p o s s ib le t h a t th e f a m i l y o f curves in p i g .
15 and F i g .
25.
16 c o u ld be re p re se n te d by an e q u a tio n o f t h e form ,
Vennard, J , K. Op. p i t . P.. 192.
'
'
73
%
JTffiBt
M E A a iife E P
PRESSU
V E R S US A I R V PLOC
i 4 t e N E : f fe e
PIPE
74
M E A all RED
AIR ALO NE
F?RE
AN> FOR
A lF
A tiIl
:
V A i i o D i L .RATjEJS .01
S D L t D B r L d w j : ;■:::
• :[•;•:
• I: j'|
IN
I: fBilhil:
z . 0 6 IMCTr C'T7 .T9 Vr
75
r ^
TT
2J
^
(3 ,)
In E q u a tio n ( 3 1 ) , t h e f i r s t term on t h e r i g h t was- assumed t o account f o r
t h e p re s s u re drop due to. t h e a i r a lo n e , and t h e second te rm was assumed t o
account f o r t h e p re s s u re drop due t o t h e s o l i d m a te ria l a lo n e ,
in o r d e r t o
account f o r t h e p re s s u re drop due t o t h e s o l i d m a t e r i a l s , t h e second term
on t h e r i g h t o f E q u a tio n (31) must be a f u n c t i o n o f a t I e a s t th o s e v a r i ­
a b le s shown which a re :
V = v e l o c i t y o f th e conveying f l u i d ;
d e n s it y o f t h e co n ve ying f l u i d ;
= mass
= mass d e n s it y o f t h e s o l i d s ; ' ^
=
v i s c o s i t y o f t h e f l u i d ; D = d ia m e te r o f t h e conveying p ip e ; Gs = w e ig h t
r a t e o f f lo w o f t h e s o l i d m a t e r i a l ; and S
= a f a c t o r t o account f o r th e
shape and s iz e o f t h e s o l i d p a r t i c l e s .
I f E q u a tio n (31) proved v a l i d , then t h e f o ll o w i n g c o n c lu s io n c o u ld be
made.
The p re s s u re drop due t o th e f lo w o f a i r and s o l i d m ix tu r e s can be
d iv id e d i n t o t h e f o l l o w i n g components.
1 .)
The p re s s u re .drop due t o th e a i r a lo n e .
2 .)
The p re s s u re drop due t o t h e i n t e r a c t i o n o f t h e a i r and th e
s o l i d p a r t i c l e s , th e i n t e r a c t i o n o f th e s o l i d p a r t i c l e s , and
t h e i r a c t i o n upon th e p ip e w a l l s .
Number 2 above i s a d e f i n i t i o n o f t h e p re ssu re drop due t o th e s o l i d
m a te r ia l .
In regard t o th e c o n c lu s io n above. E q u a tio n (31) im p lie d t h a t th e
76
p re s s u re d r o p ^ to t h e a i r alone was u n a ff e c t e d by t h e a d d i t i o n o f th e s o l i d
m a te r ia l .
In o r d e r t o t e s t t h e v a l i d i t y o f E q u a tio n ( 3 1 ) ,
i t was necessary t o
determ ine t h e v a lu e o f t h e p re s s u re drop due t o - t h e s o l i d s a lo n e .
I t was
assumed t h a t t h i s c o u ld be accom plished by s u b t r a c t i n g v a lu e s of: p ressure
drop due t o a i r a lo n e from measured v a lu e s o f p re ssu re drop due t o a i r and
s o lid s .
The m agnitude o f t h e p re s s u re drop due t o a i r a lo n e was computed
from E q u a tio n ( 2 9 ) .
Here i t was assumed t h a t th e f r i c t i o n
fa c to r
was
u n a ffe c te d by t h e a d d i t i o n o f th e s o l i d s t o t h e f lo w .
DIMENSIONAL ANALYSIS
With t h e above t h e o r y in m ind, i t was decided t o p e rfo rm a dim ensional
a n a ly s i s o f t h e v a r ia b le s in v o lv e d in t h i s p a r t i c u l a r r e s e a r c h .
I t has
a lr e a d y been p o in te d o u t by Rouse, as d e s c rib e d on Page 17, t h a t a l l o f
t h e p o s s ib le v a r ia b le s in v o lv e d in t r a n s p o r t a t i o n : o f d i s c r e t e m a tte r are
to o numerous t o a f f o r d e x a m in a tio n .
The v a r i a b l e s which are concerned in t h i s research are th e f o l l o w i n g .
1. ) ^
—
D e n s ity o f th e f l u i d
2. ) A
—
D e n s ity o f t h e s o l i d s in s lu g s per c u b ic f o o t .
3 . ) yU
— V is c o s ity o f t h e . f l u i d
in s lu g s per c u b ic f o o t .
in pound seconds p e r square f o o t .
4 . ) D — Diam eter o f t h e p ip e in f e e t .
5 . ) Vf
6
. )
R1
—
V e l o c i t y o f th e f l u i d
in f p s .
— R a tio o f th e w e ig h t r a t e o f flo w o f s o l i d m a te r ia l t o
t h e w e ig h t r a t e o f f lo w o f conveying f l u i d .
77
I,)
h|_^— P re ssu re drop due t o s o l i d s in pounds per square f o o t
per f o o t o f p ip e .
The r a t i o " l f ' above was in c lu d e d a f t e r several a tte m p ts had a lre a d y
been made a t dim ensional a n a ly s i s in v o l v i n g o t h e r v a r i a b l e s t o account f o r
t h e amount o f s o l i d m a te r ia l which was p r e s e n t in t h e f l o w .
F ric tio n
f a c t o r s f o r t h e Darcy Weisbach e q u a tio n were computed f o r
t h e p re s s u re drop due t o a i r alo n e in both p ip e s .
These f r i c t i o n
have been p l o t t e d a g a in s t Reynolds number in th e graph in F i g .
e xa m in a tio n o f t h e g raph , i t was found t h a t th e f r i c t i o n
B i a s i u s 1 e q u a tio n 2 6 f o r a h y d r a u l i c a l l y smooth p ip e .
w r itte n
17.
fa c to rs
From
fa c to rs s a tis fie d
B l a s i u s ' e q u a tio n i s
in t h e f o l l o w i n g form .
0. 3/6
Tf
The l i n e shown in F i g .
'
Npa2ir
(32)
17 i s th e c u rv e re p re se n te d by E q uation (32)
T h is r e s u l t meant t h a t t h e p re s s u re drop due t o th e a i r a lo n e conveyed in
t h e p ip e was independent o f th e p ip e roughness.
T h e r e f o r e i t was assumed
t h a t s in c e p ip e roughness had not a f f e c t e d t h e p ressure drop due t o a i r
a lo n e , i t would n o t a f f e c t t h e p re s s u re drop due t o th e s o l i d p a r t i c l e s
in
t h e a i r strea m , and a c c o r d in g ly no param eter f o r p ip e roughness was i n ­
clu d e d in t h e dim ensional a n a l y s i s .
No v a r i a b l e was p u t in t h i s a n a ly s is t o account for. shape o f th e
p a r tic le s .
26.
The same wheat was used th ro u g h o u t th e e x p e rim e n ta tio n so t h e r e
Vennard, J .■K. Op. C i t . P. 194.
78
nSBACH EQUATION
I T T t r r COMPUTED FRCM THE M EASJR R D P R$ SSVKE DROPS
■I
1
DU E T C FLO W OF . M R ALO M E IN :T h E T jE S T P lE E S
*
;
-V lt
I
4 i' T “I l t T i ' f I'Tl [Til
—t- f-4
— I ^
iI
* 4- »
-J--4 - 1 | - r
• • *
T M : 'T l
t— •
-]— I
• -
4- 4 -
- «-
TL1.. .REVNOLDfS N U M 6$.
z
!;;t
-
:j
1
•; 4-
Ir
j—
•- • <j f-4~rf•
hi ‘ ’ jj i f i
r
TlTT
I ■* Tf
l±*j tt*’ I n i t j f 1IitI41
# I 4HiP4Jst I
T 4I i T #
S
ft
It4-I
79
was no v a r i a t i o n
in m a te r ia l t e s t e d .
U ndoubtedly th e shape and s iz e o f th e
s o l i d p a r t i c l e s play's some r o l e in th e r e l a t i o n s which account f o r th e
p re s s u re drop due t o pneumatic conveying o f s o l i d s ; however, t h i s f a c t o r
was n o t i n v e s t i g a t e d in t h i s p ap er.
The above v a r i a b l e s were combined t o form t h e f o l l o w i n g f u n c t i o n .
Hu5=
R >, R
(33)
The v a r i a b l e , h[_, was moved t o the,, o p p o s it e s id e o f t h e e q u a tio n in
t h e f o l l o w i n g manner.
The above e q u a tio n was w r i t t e n
in t h e f o ll o w i n g form
(34)
which rendered both s id e s o f t h e e q u a tio n d im e n s io n le s s .
In t h e dimensional':':
a n a l y s i s , th e fo o t-p o u h d -se co n d system was u s e d .
Under t h i s system , th e u n i t s o f Mass CM), Length CL), and Time (T) are as
f o 1 1 ows:
M = s lu g s
L = fe e t
T = seconds
Using t h e above u n i t s , th e v a r ia b le s were expressed in dimensional
e q u a l i t i e s as f o ll o w s :
80
Rf = Di mens ion I ess v a r i a b l e
Sr
D=L
Vf= L /T
■ hLr _M
T2L 2
S u b s t i t u t i n g t h e v a r ia b le s as expressed above i n t o E q u a tio n (34) r e s u lt e d
in t h e f o l l o w i n g e x p r e s s io n .
(35)
S in ce both s id e s o f - E q u a t io n (3 5 ) a re d im e n s io n le s s , t h e sum o f th e
exponents o f l i k e q u a n t i t i e s must be equal t o zero.
E q u a tin g t h e sum o f
t h e exponents o f t h e mass ( M ), Length ( L ) , and Time (T) t o ze ro gave th e
f o l l o w i n g t h r e e e q u a t io n s .
a + b + d + g=
0
~ 3a ~ 3b - d + e + f ~ 2g = 0
- a - f -
2
g
(36)
= 0
The above t h r e e e q u a tio n s t o g e t h e r are a system o f t h r e e e q u a tio n s in
s i x unknowns.
The f i r s t o f th e t h r e e e q u a tio n s can be s o lv e d f o r "a " t o
g iv e
a = * - b - d ' i- g
(37)
The t h i r d e q u a tio n can be so lve d f o r f t o g iv e th e f o l l o w i n g :
f = - d - 2g
Wf ■
\
(38)
81
The second o f th e t h r e e e q u a tio n s can be s o lv e d f o r " e " and th e values o f
" d " and " f "
in E q u a tio n s (37) and (38) s u b s t i t u t e d i n t o t h i s e x p re s s io n t o
o b t a i n th e f o l l o w i n g e q u a tio n .
e =
d + g
(39)
The va lu e s o f " a " , " f " , and " e " i n E q u a tio n s ( 3 7 ) , ( 3 8 ) / and (39)
can th e n be s u b s t i t u t e d i n t o E q u a tio n ( 3 4 ) .
T h is e lim in a t e s t h r e e o f t h e
s i x unknown exponents and g iv e s th e f o l l o w i n g .
(Vf ) - " '
ChJ9] 3.
(40)
C o l l e c t i n g q u a n t i t i e s w it h
l i k e exponents g iv e s r i s e t o t h e f o ll o w i n g
e q u a tio n .
T h is e q u a tio n may be r e w r i t t e n t o g iv e t h e f o l l o w i n g e q u a tio n where "K"
is a p r o p o r t io n a lity c o n s ta n t.
(42)
E q u a tio n (42) can then be solved f o r "h[_" t o g iv e t h e f o l l o w i n g .
lo L =
K,
$
W
(
V
L e t t i ng £ = x , £ = Y, g, = £ ,
b
c
d
'
in g e q u a tio n .
)
W
K f = ^ f , and R = Gs /G f produces th e f o l low-?
g
^
82
(44)
S ince in +he b e g in n in g o f t h e a n a l y s i s , p re ssu re drop was expressed
as pounds per square f o o t per f o o t o f p i p e . E q uation (44) may be m u l t i ­
p l i e d by L t o g iv e t o t a l
h .,
K1
i
r
The q u a n t i t y
p re s s u re drop in pounds per square f o o t .
( * ) y (
ft DVf
/W.
W
(45)
i s re cogn ized, as. Reynolds number.
Using t h a t
i d e n t i t y . E q u a tio n (45) may be r e w r i t t e n t o g iv e th e f o llo w ! n g j.
(46)
E q u a tio n (3 7 ) d iv id e d by T
gi ves t h e p re s s u re drop in f e e t o f w a te r.
Sn o r d e r t o e v a lu a te t h e exponent " y " , a graph o f th e p re s s u re drop due t o
a i r p lu s s o l i d s versu s t h e w e ig h t r a t e o f f lo w o f s o l i d s was drawn f o r
each p ip e s i z e .
These graphs a re shown in F i g .
o f th e s e graphs shows t h a t p re ssu re drop i s
18 and F i g .
19.
A study
I i n e a r ly p r o p o r t io n a l
t o th e
w e ig h t r a t e o f f lo w o f s o l i d s .
From t h i s e v id e n c e , i t can be assumed t h a t t h e exponent " y "
E q u a tio n (47) must be equal t o
I.
in
Using t h i s assum ption, ahd l e t t i n g
G f = ( GTfVAg,), E q u a tio n (47) can be r e w r i t t e n as
83
(48)
C a n c e llin g o u t t h e "V" in t h e den o m in a to r, s u b s t i t u t i n g TtD 2 f o r th e
4
area o f t h e p ip e , and com bining c o n s t a n t s , t h i s equation' i s again r e ­
w r i t t e n as
| t was then assumed t h a t E q ua tio n (4 9 ) was t h e second te rm on th e
r i g h t in E q u a tio n ( 3 1 ) .
S u b s t i t u t i n g E q u a tio n (49) f o r t h i s te rm in
E q u a tio n (31) produces th e f o ll o w i n g e q u a tio n .
z
l l L5=
^
^
r J
2O ( i f 1*
L Vf (js
d
35
(50)
E q u a tio n (49) can be w r i t t e n as
h Ls-
fs
Lm Vf D6*g
?
K
( 51)
where
fs
s
^3
f" § )
(52)
and
K3 a c o n s ta n t
Y g = s p e c i f i c w e ig h t o f t h e s o l i d p a r t i c l e s
= s p e c i f i c w e ig h t o f t h e co n ve yin g f l u i d
)
-FO-R
V A R IO U S
• FO R .
__ - T u T A L
1.6 3
A IR V E l ( 0 C l t n E 5 . ..
INCH
P R ESSU R E
P ipe
DROP
____ .
I
i
L
tv.-.- :-T-
P £ R „ E £ .o i:_ O F .P.iPE (if). W; 0 )
.
Co
86
Nr = ReynoIds number
The f a c t o r " f s " can be c a l l e d a f r i c t i o n
Examining E q u a tio n ( 5 1 ) , i t
fa c to r fo r the s o lid m a te ria l.
i s e v id e n t t h a t i f th e e q u a tio n were v a lid ?
then t h e p re s s u re drop due t o t h e sol i d s 'a l o n e i s a l i n e a r f u n c t io n o f
t h e v e l o c i t y o f t h e conve ying f l u i d .
I t was expected then t h a t a graph
o f t h e p re s s u re drop due t o t h e s o l i d s a lo n e versus th e average v e l o c i t y
o f - t h e conve ying f l u i d would produce a f a m i ly o f s t r a i g h t l i n e s .
E q u a tio n (5 0 ) can be r e w r i t t e n u s in g E q u a tio n (51) t o produce
(53)
where h|_ i s t h e p re s s u re drop in f e e t o f w a t e r .
T h is e q u a tio n i s s i m i l a r t o t h a t o f Crane given on Page 15 o f t h i s
p a p e r.
For a h o r iz o n t a l
i s E q u a tio n ( I ) ,
p ip e . C ra n e 's e q u a tio n f o r p re s s u re d ro p , which
reduces t o
f-S VsL ($5 _
Z -D 3S
VHtP
CD
where t h e m id d le te rm became z e r o , s in c e f o r a h o r iz o n t a l
p ip e s in
6
=
0
.
In h i s e q u a tio n , Crane used t h e s o l i d s v e l o c i t y - i n t h e second term
o f h i s e q u a tio n , w h ile in E q u a tio n ( 4 4 ) , t h e v e l o c i t y o f t h e conveying
flu id
i s used.
C ra n e 's e q u a tio n a ls o has a f a c t o r o f 2 in th e denominator
T h is would i n d i c a t e t h a t t h e f r i c t i o n
be t w ic e th o s e f o r E q u a tio n ( I )
both e q u a tio n s are v a l i d .
fa c to rs ( fs )
in E q u a tio n (44) should
f o r t h e same flo w c o n d i t i o n s , p rovid ed
87
E q u a tio n (52) shows t h e v a r ia b le s upon which th e f r i c t i o n
th e s o lid s v a r ie s .
A p lo t o f the f r i c t i o n
fa c to r fo r
f a c t o r a g a in s t t h e Reynolds
number s h o u ld show a d e f i n i t e r e l a t i o n s h i p ,
i f E q uation (5 2 ) i s v a l i d .
W ith t h e above e q u a tio n s developed, expe rim e n ta l r e s u l t s were used
t o determ ine t h e i r v a l i d i t y .
ANALYSIS OF THE RESULTS
A f t e r t h e e x p e rim e n ta l
r e s u l t s had been o b ta in e d , t h e graphs shown
in F i g . 15 and F i g i 16 were drawn.
L in e s o f b e s t f i t were c o n s tr u c te d
th ro u g h t h e e x p e rim e n ta l p o i n t s p l o t t e d f o r t h e v a r io u s f lo w s .
The cu rves appeared t o be a p p ro x im a te ly p a r a l l e l , and upon a n a ly s is
as d e s c rib e d on Page
QZ,
were found t o be spaced a t i n t e r v a l s which were
l i n e a r l y p r o p o r t io n a l t o t h e w e ig h t r a t e s o f s o l i d f lo w .
In o r d e r t o determ ine t h e p re s s u re drop due t o th e s o l i d s a lo n e , th e
o r d i n a t e o f t h e c u rv e f o r t h e p re s s u re drop due t o t h e a i r a lo n e was sub­
t r a c t e d from t h e o r d i n a t e o f th e curve f o r t h e p re ssu re drop due t o th e
a i r and wheat combined.
. The c u rv e s f o r t h e p re s s u re drop due t o th e g r a in a lo n e are shown in
F i g . 21 and F i g . 22.
A fric tio n
I
f a c t o r r,f s " f o r th e e q u a tio n
r
L \ 4 &S _
IlLf fs D*g KHlO
(51)
was determ ined f o r each e xpe rim e nta l p o i n t .
An. a tte m p t was made t o p l o t
th e s e c o e f f i c i e n t s a g a in s t Reynolds number.
However, i t was found t h a t
■4
88
t h i s p l o t formed a s e p a ra te graph f o r each p ip e d ia m e te r.
T h is r e s u l t
seemed t o prove t h a t t h e e q u a tio n
(52)
was n o t v a l i d , a t I e a s t , w i t h i n t h e range o f Reynolds numbers encountered
in t h i s r e s e a rc h .
An a tte m p t was then made t o p l o t t h e f r i c t i o n
a g a in s t t h e a i r v e l o c i t i e s
t h e graph shown in F i g . 20.
in th e co n ve ying p ip e .
fa c to rs fo r the s o lid s
T h is a tte m p t produced,
T h is graph showed t h a t t h e f r i c t i o n
c a l c u l a t e d f o r t h e two p ip e d iam e te rs p l o t t e d as one l i n e .
fa c to rs
There was con­
s id e r a b le spread among t h e p l o t t e d p o i n t s ; however, i t was a pp arent from
t h i s graph t h a t w i t h i n t h e range o f th e s e r e s u l t s , t h e f r i c t i o n
fa c to r
" f s " was n o t dependent upon Reynolds number, b u t was dependent upon a i r
v e l o c i t y r e g a r d le s s o f p i p e d ia m e te r.
The f r i c t i o n
f a c t o r s were c a l c u la t e d so t h a t u sin g Gs in pounds per
second,- and .th e rem ainder o f t h e term s in foot-p o u n d -se co n d u n i t s , h|_ in
E q u a tio n (4 2 ) i s c a l c u la t e d in f e e t o f w a te r per f o o t o f p i p e .
' The graph o f t h e f r i c t i o n
tre n d .
The f r i c t i o n
f a c t o r " f s " versus v e l o c i t y shows a d e f i n i t e
f a c t o r in cre a se s w i t h decrea sing v e l o c i t i e s i
The spread o f t h e e x p e rim e n ta l p o i n t s in F i g . 20 i s l a r g e , w ith th e
g r e a t e s t spread o c c u r r in g a t t h e lower v e l o c i t i e s .
As has been p r e v io u s ly
s t a t e d , t h e f lo w in t h e t e s t s e c t io n ceased t o be u n ifo rm a t th e s e lower
v e l o c i t i e s , and as a r e s u l t , t h e measured p re s s u re drop was s u b je c t t o
89
c o n s id e r a b le f l u c t u a t i o n .
The s c a t t e r o f th e s e p o in t s c o u ld be decreased by u s in g a lo n g e r t e s t
s e c t io n w ith ' g r e a t e r spacing between p re s s u re t a p s .
With t h e graph f o r " f s " , i t was p o s s ib le t o o b t a in p r e d ic te d va lu e s
o f t h e p re s s u re drop due t o t h e s o l i d p a r t i c l e s f o r any s iz e p i p e ; Values
f o r t h e p re s s u re drop due. t o t h e s o l i d p a r t i c l e s were computed f o r each
p ip e d ia m e te r and then p l o t t e d on F i g . 21 and F i g . 22.
T h is p l o t shows t h a t t h e p re s s u re drop as c a l c u la t e d u s in g Equation
(51) and v a lu e s o f " f s " from F i g . 20, agrees q u i t e w e ll w i t h t h e measured
p re s s u re drops a t a l l
t h e s o l i d f lo w r a t e s .
The p o o re s t agreement occu rs
in t h e 2.06 inch p ip e a t th e 17.10 l b . / m i n u t e s o l i d f lo w r a t e .
T h is la ck
o f agreement can be e x p la in e d by t h e o c c u rre n c e o f non-uni form flo w and
"s lu g g in g * ' c o n d it io n s a t t h i s s o l i d flo w r a t e .
The c u rv e f o r " f s " and E q u a tio n (51.) were then used t o compare t h e
r e s u l t s o f Crane ^ 7 w ith th o s e o f t h i s r e s e a r c h .
The r e s u l t o f t h i s com p a ri­
son i s shown in F i g . 2 3 . . The r e s u l t s do n o t a gree .
However, C ra n e 's
r e s u l t s were o b ta in e d w it h s m a ll e r , le s s dense wheat p a r t i c l e s , a h ig h e r
a tm o sp h e ric p r e s s u r e , and f lo w c o n d it io n s a t Reynolds numbers beyond t h e
range o f th o s e r e p o r te d h e r e .
I f E q u a tio n (52) i s v a l i d a t a l l
fo r " f s " ,
th e n t h e d i f f e r e n c e in t h e v a r ia b le s in c lu d e d in E q u a tio n (4 3 ) between
C ra n e 's r e s u l t s and th o s e re p o r te d here c o u ld account f o r th e la ck o f agree­
ment shown in F i g . 23.
27.
Crane, J . W. Op. C i t . P. 100.
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The p r e s s u re drops due t o t h e a i r and s o l i d s t o g e t h e r were a ls o
p l o t t e d a g a in s t a i r v e l o c i t y on lo g a r it h m ic graphs f o r both p ip e diam e te rs
These graphs a re shown in F i g . 24 and Fig.. 25.
in te r e s tin g p ic tu r e .
These graphs p re s e n t an
As can be seen on t h e g ra p h , th e cu rv e s a re a p p r o x i­
m a te ly s t r a i g h t l i n e s c o n v e rg in g in t h e d i r e c t i o n o f in c r e a s in g v e l o c i t i e s
S in ce i t seems u n l i k e l y t h a t th e p re s s u re drop due t o a i r p lu s s o l i d s w i l l
e v e r become le s s than t h e p re s s u re drop f o r a i r a lo n e , t h e curve s in F ig s .
24 and 25 would l o g i c a l l y be a s y m p to tic t o t h e p re ssu re drop cu rv e f o r
a i r a lo n e .
92
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CONCLUSIGNS
The f o l l o w i n g c o n c lu s io n s can be reached as a r e s u l t o f th is , r e ­
search .
I.)
v a ilin g
W it h in t h e l i m i t s o f a i r v e l o c i t i e s and Reynolds numbers p re ­
in t h i s re s e a rc h t h e p re s s u re drop due t o wheat p a r t i c l e s ( w it h
t h e p r o p e r t ie s o f th o s e used here) being conveyed by an a i r stream in a
h y d r a u l i c a l l y smooth p ip e can be computed by t h e e q u a tio n .
(51 )
where h|_ i s t h e p re s s u re drop in f e e t o f w a ter per f o o t o f p ip e .
2.
)
The p re s s u re drop due t o t h e wheat p a r t i c l e s a lo n e is alm ost
independent o f t h e a i r v e l o c i t y .
T h is c o n c lu s io n is sup p o rte d by th e
graphs o f p re s s u re drop due t o s o l i d s in F i g . 21 and F i g . 22, which
show th e p re s s u re drop cu rv e s t o be a lm o s t h o r i z o n t a l .
3.
)
The f r i c t i o n
f a c t o r " f s " in t h e above e q u a tio n i s n o t expressed
by t h e e q u a tio n ,
(52)
w i t h i n t h e range o f t h i s research s in c e i t has been shown t h a t " f s "
dependent n o t on R e y n o ld 's number, b u t on t h e a i r v e l o c i t y .
s u f f i c i e n t data was n o t a v a i l a b l e t o dete rm in e i f " f s "
y s .
is
However,
i s dependent, on
Comparison o f th e data o b ta in e d here w ith t h a t o b ta in e d by
J . W. Crane i n d ic a t e s t h a t " fs". is dependent on some v a r i a b l e o t h e r than
98
+he a i r v e l o c i t y .
4.
)
The f r i c t i o n
c r e a s in g v e l o c i t i e s .
fa c to r " fs "
in E q u a tio n (5.1) in c re a s e d w ith de­
T h is f a c t can be e x p la in e d in th e f o l l o w i n g manner.
As t h e v e l o c i t y o f th e conve ying f l u i d
p a r t i c l e s e x p e rie n c e le ss l i f t .
becomes s m a lle r , t h e in d iv id u a l
With t h i s
decrease in l i f t ,
th e
p a r t i c l e s c o n c e n tr a te a t t h e i n v e r t o f t h e conveying p ip e , and th e
spacing o f t h e p a r t i c l e s
in th e flo w becomes s m a ll e r .
In c r e a s in g amounts
o f energy are then d i s s ip a t e d in overcom ing th e f r i c t i o n o f t h e p a r t i c l e s
a g a in s t t h e p ip e w a l l , and in c r e a s in g amounts o f energy are d is s ip a te d
due t o t h e in cre a se d freq uency o f p a r t i c l e c o n t a c t .
5.
)
The q u a n t i t i e s
in v o lv e d in d e te rm in in g th e p re s s u re drop due t o
t h e wheat p a r t i c l e s a lo n e are q u i t e s m a ll, and in o r d e r t o o b t a in b e t t e r
v a lu e s f o r " f s " ,
improved te c h n iq u e s and equipment are needed.
A lo nger
t e s t s e c t io n w ith g r e a t e r d is ta n c e s between p re ssu re ta p s would tend t o
smooth o u t the. s c a t t e r o f t h e e x p e rim e n ta l p o in t s in t h e c u rv e f o r " f s "
versus a i r v e l o c i t y in F i g . 20.
H ig h e r a i r v e l o c i t i e s would a ls o g iv e more
r e g u l a r r e s u l t s , s in c e t h e combined f lo w o f t h e a i r and s o l i d s
form a t h ig h e r v e l o c i t i e s .
is more u n i ­
99
SUGGESTIONS FOR ADDITIONAL RESEARCH
The research and r e s u l t s shown here have in d ic a te d t h a t f u r t h e r r e ­
search in t h e t o p i c o f p re s s u re drop in t h e f lo w o f a i r and s o l i d m ix tu re s
i s needed.
The f o l l o w i n g t o p i c s were among th o s e in which i n t e r e s t was
aroused d u r in g t h e process o f t h i s p r o j e c t .
1. )
A s tu d y o f t h e f lo w o f a i r and s o l i d m ix tu re s s h o u ld be
c a r r i e d o u t o v e r a much w id e r range o f v e l o c i t i e s and
Reynolds numbers than th o s e which were p o s s ib le in t h i s
p r o je c t i
2. )
Study o f th e flo w o f a i r and s o l i d m ix tu re s u sin g several
ty p e s o f s o l i d s in v o l v i n g v a r i a t i o n
in p a r t i c l e s i z e , shape
and d e n s it y i s needed t o determ ine t h e e f f e c t o f t h e p h y s ic a l
c h a r a c t e r i s t i c o f t h e s o l i d m a t e r i a ls upon th e p re s s u re drop'.
3. )
Study o f t h e flo w o f a i r and s o l i d m ix tu re s in se ve ra l d i a ­
m eters o f p ip e s o t h e r than th o s e used here shou ld be made
t o determ ine b e t t e r t h e r e l a t i o n s h i p o f p ip e s iz e to, th e
fric tio n
4. )
lo s s .
The s tu d y o f th e above items s h o u ld be c a r r i e d o u t , using
more than one conveying f l u i d .
T h is would i n d i c a t e th e
r e l a t i o n s h i p o f th e f l u i d p r o p e r t ie s in th e e q u a tio n f o r
t h e p re s s u re drop due t o t h e s o l i d phase o f th e conveyed
m a te r i a I ■a l o n e .
IOO
APPENDIX
SYMBOLS AND SUBSCRIPTS USED
The. f o l l o w i n g symbols have been used th ro u g h o u t t h i s paper.
Symbol
Meaning
hL
P ressure drop
L
Length o f conve ying p ip e .
V
V e lo c ity .
y
S p e c if I c Weig h t
D
Diam eter o f th e co n ve ying p ip e .
P
Mass D e n s ity
g
A c c e le r a t io n due t o g r a v i t y .
Dp
E f f e c t i v e d ia m e te r o f a p a r t i c l e .
f
R a tio o f th e p re s s u re drop due t o f l u i d plus
s o l i d s , t o t h e p re s s u re drop due t o a i r a lo n e .
R'
R a tio o f th e w e ig h t r a t e o f flo w o f s o l i d s t o
t h e w e ig h t r a t e o f f lo w o f f l u i d .
R
Gas c o n s t a n t .
Np
R eynolds Number.
V is c o s lty o f f l u i d .
f
F r ic tio n
G
W eight r a t e o f f l o w .
6
A
WP
fa c to r.
. A ngle o f i n c l i n a t i o n o f th e conve ying d u c t.
G r o s s - s e c tio n a l a r e a .
W eight per p a r t i c l e .
IOI
Symbol
. M eaninq
C
D isch a rg e c o e f f i c i e n t .
P
P ressure.
Q
Volume r a t e o f f lo w .
2
E le v a tio n .
H
D iffe re n tia l
Ce
C o n t r a c t io n c o e f f i c i e n t .
0V
V e lo c ity c o e f f ic ie n t .
T
Tem perature.
X
Opening o f v a r i a b l e o r i f i c e
K
P ro p o rtio n a lity c o n s ta n t.
k
A d ia b a t i c c o e f f i c i e n t s
M
Mass
p re s s u re across th e o r I f i c e p l a t e .
in g r a in hopper.
The f o l l o w i n g s u b s c r ip t s were used in co n n e c tio n w it h t h e symbols
above „
s
S o lid s .
Qf
A ir.
D
Conveying d u c t .
f
F lu id .
P
P a rtic le
.
ABBREVIATIONS USED
fps
Feet per second'
cfm
Cubic f e e t per m in u te -
c fs
C ubic f e e t per second.
102
SUMMARY .DATA SHEET
PHYSI OAL PROPERTIES OF THE WHEAT■USED
S p e c i f i c g r a v i t y o f t h e wheat
I .375
S p e c i f i c w e ig h t o f t h e wheat p a r t i c l e s
8 7 .8 I b s Z f t 3
Weight per p a r t i c l e o f t h e wheat
7 5 .3 x IO"6' lb s .
Average volume per p a r t i c l e
8 .7 9 x IO"
Average m ajor a x i s le n g th ( c )
.0.231
Average m inor a x is le n g th Ca)
0.099 inches
Average m inor a x i s l e n g t h . (b )
0.118 inches
Average shape f a c t o r
0.600
B u lk w e ig h t per bushel o f wheat
56.1 pounds
Type o f'W h e a t
L i g h t s p r in g wheat
7
ft3
inches
103
SUMMARY DATA SHEET
P ip e D ia m e te r
T em perature
I .63 inches
75° F .
B a ro m e tric P ressure
25.62 inches o f m e rc u ry .
Average S p e c i f i c Weight o f a i r
Nb.
Ave.Affr ^
V e lo c ity .
ft/s e c „
A ve .P ressu re
Drop
Inches HoO/Ft.
Iigii
Wheat = .
l0 .3 0 # /m in .
Run
I
7 2 .2
3
4
5
68.3
64.9
62.2
59.6
6
56.4
7
54.8
52.7
2
8
9.
10
II
12
13
14
15
16
17
18
19
.
in t e s t s e c t io n
J
' 0.142
■"
0.141' '
0.134
49.8
4 7 .0
4 4 .6
41 . 8
'
Wheat =
6.67#/m ln.
2
75.8 '
73.5
72.0
8
9
. 0.117
10
0 .1 1 2
II
0 .1 2 0
12
:
0.201
..
13
14
15
16
17
18
0.209
0.194
0.185
0 .1 8 2
0 .1 7 3
79.5
3
4
5
6
0.178
.
IlgM
0.127
"
.
Ave. P ressure
Drop.
Inches H ? 0 /F t.
7
0.101
49.5
67.3
72.5
No.
Ave. A i r •
V e lo c it y
ft/s e c .
Run
I
■0.195
0.185
0.169
0.160
0.151
0 . 0 6 5 Q # / ft ^
69.1
.
' '
6 6 .8
6 4 .3
6 0 .2
5 6 .3
53.7
51 . 0
47.5
4 4 .8
42.9
6 2.3
4 9 .2
7 7.0
65.0
:
--
0.166
0.160
0.139
0.130
0 .1 2 0
0.113
0 .1 0 2
0.094
0.083
0.154
0.1 I I
0 .2 0 3
0.155
- :
-
104
SUylPlARY DATA-SHEET
P ipe Diam eter
Tem perature
I .63 inches
77° F
B a ro m e tric P re ssu re
25.68 inches o f .mercury
Average s p e c i f i c w e ig h t o f a i r in t h e t e s t s e c t io n
A ve . A i r
V e lo c ity
No. • F t /S e c . ■
A ve .P re ssu re
. Drop.
Inches H ? 0 /F t .
Run
"G"
I
2
3
4
5
6
7
8
9
IO
II
12
13
14
15
16
17
18
19
Wheat =
8 7 .2
8 4 .0
8 2 .2
76.7
7 3 .2
7 0 .0
■6 6 . 2
64.2
62.3
60.2
5 8 .3
5 5 .8
5 3.7
4 9.7
4 5 .2
'
3 .3 6 # /M in.
0.226
0,213
0.206
0.187
0.171
0.159
"
'
Run
No.
Ave. Ai r
V e lo c ity
F t./S e c .
Ai r
I
2
3
4
5
'
6
0.147
7
0.142
0.136
0.128
0.123
0.114
0.108
0.097
0.084
'
8
9
10
II
12
13
14
15
16
17
18
19
.0653 # / f t ^
Ave. P ressure
Drop
Inches H?0/FT.
Al one
4 5 .0
47.5
5 0 .0
5 5 .0
5 8.0
61 .5
6 3.3
0.066
0 .0 7 2
0.078
0.094
0,104
0 .1 1 2
0.119
.6 '
69.5
7 1 .3
75.0
77.5
• -80.0
0 .1 3 2
6 6
82.0
8 5 .0
8 7 .3
90.0
92.3
0.142.
0 .(5 0
0 .1 6 2
0.172
’
0.183
0 .1 9 2
0.202
G .2 li 0.223
0.230
•
105
SUMMARY DATA SHEET
P ip e D iam ete r
X
Tem perature
I .63 inches
75° F
B a ro m e tric P re ssu re
25.62 inches o f mercury
Average s p e c i f i c w e ig h t o f a i r
Run
No.
Ave. A ir .
V e lo c ity
F t./S e c .
Wheat =
5 8 .2
.2
5 7 .0
3
5 5 .2
4
5 6 .2
5
53.7
6'
5 2 .0
7 ' ■ 5 0 .4
"G"
I
8
48.6
9
5 9 .7
in t e s t s e c t io n
A ve .P re ssu re
Drop
Inches H g G /F t.
15.98 #/M in.
0.178
0.171
0.166
0.166
0 .1 6 2
0.158
0 .1 5 2 r
0.148
. 0.177
Run
No.
"G"
I
2
3
4
5
6
7
8
9
10
10
II
11
12
12
13
13
14
15
16
17
18
19
14
15
16
17
18
19
0.0652 # / f t " 5
Ave. A i r
V e lo c ity
F t./S e c .
Wheat =
6 3 .2 '
61 .5
59.4
60.5
5 8 .2
5 6.3
53.7
5 5.0
52.0
51 .4
4 9 .7
48.2
45.5
50.9
54.8
57.7
Ave. Pressure
Drop
Inches H2 0 / F t .
I3 .3 5 # /M in .
0 .1 8 2
0.173.
■ 0.167
0.171
■ 0.167.
0 .1 6 2
0 .1 5 2
0.156
0.147
0.147
0 .1 4 2
0.136
0.123
0.141
0.153
0 .1 6 2
106
SUMMARY DATA SHEET
P ip e D iam ete r
T em perature
2.0 6 inches
73° F
B a ro m e tric P re s s u re
25.60 inches mercury
Average s p e c i f i c w e ig h t o f a i r In t e s t s e c t io n
Run
No.
Ave. A ir
V e lo c ity
F t./S e c .
ii@it
Wheat
I
70.0
2
to
II
6 8 .3
66.5
6 7 .2 '
65.4
62.5
6 3.6
6 0.3
58.5
57.1
55.1
12
52.8
13
5 1.9-'
14
15
16
17
49.8
3
4
5
6
7
8
9
18
19
4 7 .3
48.3
=
A v e .P ressure
Drop
inches H z O /F t;
Run
No.
A ve . A i r
V e lo c ity
F t./S e c .
13.35 # / M i n .
HQ.I
Wheat
=
I
7 3 .3
71 . 8
7 0.8
.
0 .1 4 2
0 .I3 6
0.134'
0.136
0.129
2
3
4
5
0 .1 2 2
6
68.3
7
8
0 .1 1 0
9
6 7.3
6 5.3
6 5 .2
6 3.0
61 . 8
59.8
5 8 .2
0.089
0.087
0.088
10
II
12
13
14
15
16
17
18
I9
Ave. Pressure
Drop
Inches H zO /F t.
10.30 #/M in
69.8
69.4
0.127
0.116
0.106
• 0.104
0.095
0.095 '
■
.0653 # / f t ^
55.7
53.8
50.2
'
0.145 '
0.140
0.138
0.133
0.131
0.128
0.127
0.121
01120
II
0.113
0.103
0.098
0.1
0.095
0.088
0.079
.
107
SUMMARY DATA SHEET
P ip e D iam ete r
78° F
25.70 inches o f mercury
Averajge s p e c i f i c w e ig h t o f a i r in t e s t s e c t io n
Run
No.
"G"
I
2
3
4
5
6
7
8
A ve. A i r
V e lo c ity
' F t./S e c .
Wheat
7 8 .3
76.7
75.7
74.7
74.2
73.3
71 .3
6 9 .2
9
66.8
10
64.0
II
59.8
12
56.7
52.9
4 9 .3
13
14
15
16
17
18
A v e . P ressure
Drop
I nches'
.
=
6.67 # / M i n .
0.152
0.1 4 !
0.141 ■
0.139
0.135
Run
No.
"G"
I
2
0.0651 # / f t ^
A ve . A i r
V e lo c it y
F t,/S e c .
. Wheat
80.0
76.0
A v e . P ressure
Drop
i nches H2 Q / F t =
3 .3 6 # /M :n .
0.146
0.135
0.131
3 .
4 •
.
5
6 7
0.122
8
0.119
0.108
10
0.098
0.087
II
5 0.3
0.067
12
44.2
13
14
15
16
17
18
5 0 .0
0.054
0.067
0.070
0.076
0.134
44.2
0.078
0.071
0.064
45.1
0.063
9
.75 .5
73.5
71 .3
6 8 .4
65.5
62.3
58.5
53,8
52.3
5 4.8
64.8
0.133
0.127
0.121
0.113
0 .(0 5
0.095
0.088
0.075
1
B a ro m e tric P ressure
0
T em perature
2 .0 6 inches
.
108
SUMMARY DATA SHEET
P ip e D iam ete r
T em perature
2.0 6 inches
75° F
B a ro m e tric P re ssu re - 25.62 inches o f mercury
S p e c i f i c w e ig h t o f a i r in t e s t s e c t io n
Run
No.
A ve. A i r
V e lo c ity
F t./S e e .
3
4
5
Wheat
4 6 .8
64.9
61 .5
57.5
65.1
6
63.0
"G"
I
2
61 .3
7
' 6 0 .3
8
9
57.9
5 6 .3
10
II
53.5
12 '
51 . 0
5 0 .0
13'
' 5 2 .0
J4
15
55 -.5
16
58.5
63.7
17
=
A v e . P ressure
Drop
Inches HgO/Ft.
Run
17.10 # / M in .
"G"
I
0 .1 0 2
-
0.135
0.130
0.118
0 . 137
0.127
0.128
A ve . A i r
V e lo c i t y
F t./S e c .
No.
Wheat
64.3
63.2
2
3
4
5
6
61 . 0
5 9.7
5 7 .0
5 6 .2
7
54.8
0.121
8
0.1 17
9
52.9
51 . 2
49.1
4 6 .8
61 . 8
60.6
0.1
10
0.109
OiIOI
' 0.105
0.108 ■
0 . 1 16
0 .1 2 2
■
0.0652 0 / f t ^
0.136
10
11
12
13
14
15
16
17
55.3
51.2
53.9
■
A ve. Pressure
Drop
I nches Pl2 0 / F t .
=
15.98 # / M ln .
0.138
0.129
0.123
0.121
0.1 14
0.1 I I
Oi 109
0.106
0.107
0 .1 0 0
0.099
0.130
0.126
0.1 11
0.105
0.1
10
109
SUMMARY DATA SHEET
P ip e D iam ete r
Tem perature
2.06 inches
80° F
B a ro m e tric p re s s u re 25.59 inches mercury
Average s p e c i f i c w e ig h t o f a i r in t e s t s e c t io n
Ave. A ir
V e lo c ity
F t./S e c .
Run
No.
A ve . Pressure
Drop
Inches H g O /F t.
0.0648 # / f t ^
A ve. A i r
V e lo c ity
F t./S e c .
Run
No.
A ve . P ressure
Drop
Inches H2 0 / F t .
A i r Alone
I
4 0 .0
0.041
I
2
44.5
4 8 .3
51 .5
5 3 .3
55.0
0.047
0.056
0.063
3
4
3
4
5
6
■7 8
9
57.5
5 9 .0
61 .7
0.065
0.071
2
5
6
0.076
7
0.082
8
0.087
0.095
10
0.105
0.109
12
9
II
64.8
68.5
12
72.1
13
14
76.7
0.117
0.127
13
14
80.3
83.2
0.138
0.147
8 6 .0
0.155
15
16
17
18
19
10
15
16
17
18
19
73.0
■
II
,
|
LITERATURE CONSULTED
A l b r i g h t , C. W. and o t h e r s . PRESSURE DROP.I N FLOW OF DENSE COAL AIR
MIXTURES. I n d . and Eng. Chem.' 43» P. 1837-1840. (August 1951).
A l b r i g h t , C. W. and o t h e r s . PNEUMATIC FEEDER FOR FINELY DIVIDED SOL IDS.
Chem, Eng. 5 6 . P. 108-11 I . (June 1949).
Bel den, D . H. and K e s s e l, L . S . PRESSURE DROPS.
P. I 174-1178. (June 1949).
In d . and Eng. Chem. 41.
Buckingham, E . HISTORY OF ORIFICE METERS AND THE CALIBRATION, CON­
STRUCTION AND OPERATION OF ORIFICES FOR METERING. American
S o c ie t y o f Mechanical E n g in e e rs . New Y o rk .
(1 9 3 5 ).
Chal le y , H. THE PUMPING OF GRANULAR.SOLIDS IN FLUID SUSPENSION.
in g . 149. P. 230-231.
(March 19 4 0 ).
E n g in e e r­
C l a r k , R. H .j C h a r le s . D. E . , R ich a rd so n , J . F . , and Newit t , D. M.
PNEUMATIC CONVEYING,.PRESSURE DROP DURING HORIZONTAL CONVEYANCE.
I n s t n . Chem. E n q r s . T r a n s . V. 30 n . 4 » P . 209-233. (1 9 5 2 ),
V. 31 n . I . P. 94-95 (1953) . '
C o rey, A. T . DRAG COEFFICIENTS OF NON-UNIFORM. PARTICLES.
C o lorado S t a t e U n i v e r s i t y .
( 1 9 5 4 ).
Cramps, W.
PNEUMATIC TRANSPORT PLANTS.
P. 207-213.
( 1 9 3 5 ).
A T h e s is .
C h em istry and I n d u s t r y 44.
C rane, J . W. PREDICTING PRESSURE DROP IN THE PNEUMATIC CONVEYING OF GRAINS.
A T h e s is . M ich ig a n S t a t e U n i v e r s i t y .
(1 9 5 6 ).
D a v is , R. F . THE CONVEYANCE OF SOLID PARTICLES BY FLUID SUSPENSION.
E n g in e e rin g 140. P, I 24-125.
(A ugust 2, 1935).
Fanning, J . T . A TREATISE ON HYDRAULIC AND WATER SUPPLY ENGINEERING.
4 t h Ed. D. Van N o s tra n d , New Y o rk . ■ ( 1 8 8 4 ).
F is c h e r , J . PRACTICAL PNEUMATIC CONVEYOR DESIGN.
1958).
Chem. E n g.
(June 2,
H a r i u, 0 . H. and M o ls ta d , M. C. PRESSURE DROP IN VERTICAL TUBES IN
TRANSPORT OF SOLIDS BY GASES.
in d . and Eng. Chem. 4 1 .
P. I 148-1160. (1949).
J e n n in g s, M. PNEUMATIC CONVEYING IN THEORY AND PRACTICE.
150. P. 361 -3 63. (1940),
E n g in e e r in g .
K in g , W i s l e r , and Woodburn.
P. 145. ( 1 9 4 8 ).
HYDRAULICS.
John W ile y and Sons. New Y o rk .
K l e i s , R. W. OPERATING CHARACTERISTICS OF PNEUMATIC GRAIN CONVEYORS„ I I I
Ag. Exp. S t a . B u l l . 5 9 4 . P. 12.
(O c to b e r, 1955).
K o rn, A . H. HOW SOLIDS FLOW IN PNEUMATIC HANDLING SYSTEMS.
P. 108-111.. (March 1950).
Lapp Ie , C. E . and Shepherd, C. B .
I n d . and Eno^ Chem. '32,.
Che. Eng. 57
CALCULATION OF PARTICLE TRAJECTORIES.
P. 605 -6 16. (May, 1940).
Longhouse, A . D . and o t h e r s . APPLICATION OF FLUIDIZATION TO CONVEYING
GRAIN. Ag. Eng. 31 . 349.
( J u l y 1 9 5 0 )'.'
Longhouse, A. D . THE FLUIDIZED GRAIN CONVEYOR. W. Va. Ag. Exp. S t a .
B u l I e t i n 564. P . 19.
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MONTANA STATE UNIVERSITY LIBRARIES
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