ONE-DIMENSIONAL ANALYSIS OF STEADY-FLOW AIR-WATER
MIXTURES IN PIPES
by
MERSON BOOTH, LIEUTENANT, U. S. NAVY
B.S., U. S. NAVAL ACADEMY
(1946)
SUBMITTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE
DEGREE OF NAVAL ENGINEER
at the
MASSACHUSETTS IINSTITUTE OF TECHNOLOGY
May,
Signature
1953
of Author
Department of Naval Architecture
Certifiedby -
7
and Marine Engineering
May 25, 1953
-
_
oa e .SUpervisor
-- --Accepted by
Chairin,q
epartmental ommttee
oneuraamume Students
I
ONE-DIMENSIONAL
ANALYSIS
OF STEADY-FLOW AIR-WATER
MIXTURESIN PIPES
By
Merson Booth, Lieutenant, U. S. Navy
On
and MarineEngineering
Submitted to the Department of Naval Architecture
May 25, 1953, in partial
fulfillment
of the requirements
for the degreeof
Naval Engineer.
ABSTRACT
The object of this thesis is to investigate a number of simple flow
examples of air-water mixtures and to present the results in a simple quick
reference form which can be valuable for engineering use in design and underThe problems
standing the phenomena of flow of two-component mixtures.
analyzed are:
1.
The one-dimensional trajectory of droplets accelerated in a
gas stream
(YI;(·
2.
The variation of stream properties during the droplet acceleration process
3.
The variation of stream properties due to wall friction with
water present in the stream.
The procedure used was entirely analytical.
-a·
and plotted
tabulated,
problemwere calculated,
The results of the first
in dimensionless
form for
various water and air velocities, droplet diameters, distance moved down the
duct, and water and air properties.
The plots should be valuable for engi-
use in determining the one-dimensional trajectory of spherical partineering
cles in a gas stream. An iteration procedure using the plots can be used where
air velocity, droplet diameter, or stream properties vary during the acceleration.
The results of the second and third problem were calculated, tabulated,
and plotted in dimensionless form for various water to air velocity ratios,
water to air mass rates, air Ivach numbers, and stream property ratios. The
results should be valuable for engineering use in all regions except near
choking conditions in the stream. Near this condition the air velocity increases
too rapidly for the droplets to accelerate with the air. Since this was one
of the assumptions in the analysis the plots are not correct in this region.
The analysis of the third problem incorporates in it a pseudo-frictional
term accounting for the effect of momentum exchange of droplets with the duct
wall. The exact nature of this effect is not known, but some correlation with
experimental data is given.
Thesis Supervisor: A. H. Shapiro
Title: Professor of Mechanical Engineering
Cambridge, Massachusetts
May 25, 1953
Professor Earl B. Millard
Secretary of the Faculty
Massachusetts Institute of Technology
Cambridge, Massachusetts
Dear Sir:
In accordance with the requirements of the degreeof NavalEngineer,
I herewith submit a thesis entitled,
"One-Dimensional Analysis of Steady-
Flow Air-Water Mixtures in Pipes."
Respectfully,
Merson Booth
Lieutenant, U. S. Navy
II
I
TABLE OF CoNTENTS
I·
INTRODUCTION .
II.
DROPLET TRAJECTORY ANALYSIS .........
...
.....
........
.......-..
.....
......
Procedure .......
* .
.................
Results
...
.
.
.
.
.
.
.
..
.
III.
Oillee
.
DISCONTINUITY AALYSIS
.
*
.
...
0
.
.
.
.
..
.
..
.
.
.
.
.
.
.
2
e
e
e..
.
2
........
*
eeeoleeeee
2
........
e
..
......
4.....
Conclusion
..
1
.0.e..09
o
- eoeeee
4
eelcoo
9
cleo
..........
.....
Procedure .............
.
.
.-
9
4.....
0
m·e4
Results ...............
.
.
..
4 4 4 4 4 4 4
.
40 4
9
......
44444444
004®
.....
Conclusion ............
* e e
.....
l$
10
*
4.......
.....
IV.
4@4@44*4
MODIFIED FANNO LINE ANALYSIS ...
17
4.....
.4...
..
Procedure ...................
e.
.....
......
*
17
.....
.....
.........
Results ............
Conclusion
.........
*
.
4 4 4 4
...................
.
0
.
.
.
4··4·4O·
......
17
.
19
.....
3.......
V.
APPENDIX ........................
......
..............
*44·e4
Symbols .....................
.
Details - Droplet Trajectory
44
28
.
Analysis
........
Details - Discontinuity Analyrsis
...........
Details - Modified Fanno Line Analysis
.....
27
29
.
.
......
39
.
..
......
co.o..e.l
55
Modified Fanno Line - Low Velocity Analysis ..
61
Bibliography .................................
66
TABLE OF FIGURES
v
a(
Figure I
Droplet Trajectory,
Figure II
Droplet Trajectory,
Vw>Va
Figure III
Droplet Trajectory,
Cold .......
·
Figure IV
Droplet Trajectory,
Hot
Figure V
Discontinuity,
M2 Vs M1
Figure VI
Discontinuity,
Pol
Figure VII
Discontinuity,
402
Discontinuity,
e
.
ee.
..
..
*-
.eeeeeee
e..............
*.
0
*
.........
5
6
......................
7
8
.................................
a .9
.
0..
.......
1%.*.*.9**f.*.
* * . 4. . .. .
o*. w*9. . e e
12
VS ...................... e
13
Vs M
M
Pol
Figure VIII
*.
P1
14
2
Figure IX
Discontinuity,
T1
VS ......................
15
l
V2
Figure X
Discontinuity,
Vs
V1
Figure XI
Friction,
Figure XII
Friction,
L4
p
P
0
+ KJL
0
Friction,
Figure
XV
Friction,
pP
T
Friction,
Figure XVI
Friction,
Figure XVII
Friction,
T*
V
V*
F
F*
....
*..
vs M
..
20
*..............
eeoeeeeel 21
..
VS
M
vs
M ....................
0
Figure XIV
max
vs M oee
0
Figure XIII
16
Ml ..................
.....
22
23
VS M ......................
24
vs M
25
VS M
...................
·
...................
26
4
..
TABLE OF TABLES
Table I
Values of CD = f(R) for spherical particles ............... 32
Table II
Values of
Table III
nd .JrdR
RP
Values of
X
R
Z
vs
Paa
dV rw
.
V
Table V
Discontinuity,
V
Vw
a
=0
-=
-= .1, V,Va = 1
V
V
35
a
w'V
a
Discontinuity, all
33
-eeee..e
e...e
ooloooooo
-_Roooooeo
V
w
Table IV
*eeee
*
*
a
.e...A.o.eee*e
*.
.
.
42
e
eoeee
44
e*
a
V
Table VI
Discontinuity,
w
Table VII
Discontinuity,
vU
V
Discontinuity, tW =
a
e- ..........ee.....
48
V
V
W
Table IX
1 **..
= .3,
a*
Table VIII
46
= 1 ............................
.2, Va
a
1
49
.....
...
....................
50
a
v
Discontinuity, M 2
........
...............
52
a
Table X
Friction, Coefficients of Friction Term ................... 58
Table XI
· ....
Friction, Formulas in Integral Form ..................
Table XII
Friction,
59
V
= .1
-
........
62
.......
...............
a
U
Table XIII
w = .2
Friction,
...
.........
w
63
va
Table XIV
Friction, -
W
64
****'*****************************
= .3
a
w
Table XV
Friction,
w
a
=
.4*OOOO.O...*
OO
60
OO
OOO**SO
**O t
65
-1-
I.
INTRODUCTION
The object of this thesis is to investigate a number of simple flow
examples of air-water mixtures.
(1)
The problems to be analyzed are:
the one-dimensional trajectory of droplets
accelerated in a gas stream,
(2)
the variation of stream
properties during the
acceleration process, and
(3)
the variations of stream properties due to
wall friction with water present in the stream.
This investigation is undertaken as part of the Aerothermopressor
Project at Massachusetts Institute of Technology.
The Aerothermopressor
is a device to increase the stagnation pressure of a gas stream by reducing
the stagnation temperature through evaporation of water injected into the
flow.
In the work on the Aerothermopressor, there has
been no quick refer-
ence guide to aid in design or in understanding the phenomena of flow of
air-water mixtures.
The purpose of this thesis is to analyze these separate
effects and to present the results in a simple quick reference form which
can be valuable for engineering use in problems concerning flow of airwater mixtures.
-2-
IIs
DROPLET TRAJECTORY
Procedure
The example studied is the injection of one droplet at a given initial
velocity into a moving gas stream.
The problem is to determine the subse-
quent droplet velocities as a function of distance moved downstream and the
properties of the gas and droplet.
The assumptions made are:
(1)
the gas velocity is constant,
(2)
the droplet is spherical and of constant
diameter during the acceleration, and
(3)
the table of drag coefficient vs. Reynolds
number as given in reference (1) and repeated
in Table I is correct.
The details of this analysis are given in the appendix.
It was necessary to
perform a graphical integration of a function of Reynolds number.
of one of these integrations is contained in reference (1).
gration was done using the trapozoidal rule.
The results
The results
The other inte-
of both these inte-
grations are given in Table II.
Results
Figures I and II are non-dimensional plots of droplet velocity vs. distance moved downstream as a function of gas velocity, droplet diameter, and
the properties of the gas and droplet.
Table III.
This same information is given in
Figure I is for droplet velocity less than gas velocity; Figure
II is for droplet velocity greater than gas velocity.
The plots in Figures I
and II are normalized to specific initial droplet velocities.
For droplet
-3-
velocity less than gas velocity, the initial droplet velocity is assumed zero.
For droplet velocity greater than gas velocity, the initial droplet velocity
is assumed twice the air velocity.
For an initial droplet velocity other
than the normalized conditions, it is necessary to subtract the distance
read at the initial condition from that of the final condition.
This is
shown in more detail in the appendix.
Figures I and II are used in the following manner:
From given initial
droplet velocity, gas velocity, droplet diameter, and gas and droplet properV
ties, calculate
-
. Enter the plot with these quantities to deter-
and
a
a
V
mine the initial point at the intersection of ordinate -W and the curve
corresponding to - ·
a
follow along the same
V
desired -V o
X Pa
a
d
W
Read on the absicca the initial value
Vd
a
.
Now
w
w
curve to the intersection of that curve and the
ra distance
from the initial
W
dX__ which gives the distance from the initial
ww
V
the value of
jr
w
point.
Read off the value of
point with the required VWor
Va
at that distance respectively.
a
Figures III and IV are plots of distances required to reach specific
V
values of
a
= .:.1 as a function of droplet
diameter, air Machnumber, and
given air properties assuming initial droplet velocity is zero.
Figure III
is for air stagnation temperature of 700°Fand stagnation pressure of 1 atmosphere.
Figure IV is for air stagnation temperature of 1500°R and stagna-
tion pressure of 1 atmosphere.
If the gas velocity, droplet diameter, or the stream and droplet properties are not constant, the solution can be approximated by assuming the variation to occur in a stepwise fashion and apply the plots successively over
each step.
This can be done in the case of evaporation in which a curve of
droplet diameter vs. droplet velocity or distance may be assumed, or in the
-4-
case of a converging or diverging section in which, because of known area
changes, a curve of gas velocity vs. distance can be assumed.
Conclusion
The results
manner as possible.
determining
obtained are given in dimensionless form in as simple a
The plots should be valuable for engineering use in
one-dimensional
trajectory
of spherical particles in a gas stream.
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DSCONTINUITY
III,
ANALYSIS
Procedure
The example studied is the flow of an air-water mixture past two sections
1 and 2 in the flow.
The problem is to determine the stream properties at
section 2 as a function of the water velocity at 2, the water rate, and the
stream properties at section 1.
The assumptions made are:
::
(1)
Constant area flow
(3)
AdIb~ahatiJunn
(3)
No evaporation of water droplets
(4)
No change in temperature of droplets
(5)
Perfect gas relation holds for air.
nhana
in
anation
Tmne
The details of this analysis are given in the appendix.
.rarq
The general method
of performing the analysis is given in reference (2).
Results
o
T
P
P
w
were calculated for V = 0, 1 for
To
0
a
various water rates and Mach numbers. The star condition is a normalized
Values of
,
,
, and
V
condition corresponding to
= 0 and M = 1 and-does not depend on the water
a
rate. These values are tabulated in Tables IV through VIII. The guiding prinTo
ciple in using the tables is that -T
is constant in going between tables.
0o
Figures V through X show the effect of droplet acceleration on stream
V
VbW
w
=
properties assuming that at section l, V = , and at section 2,
for
a
a
various initial Mach numbers and water rates. The numbers on each curve indi-
w
cate the water rate to air rate ratio, w
,
and the letter S on some branches
a
of the curves indicate that a normal shock in the flow is necessary to reach
these states.
-10-
Figures V through X are used in the following manner:
Knowing the
stream properties at section 1, the point of injection of the water, enter
the required plot with the
and
.
Read on the ordinate the value of the
a
ratio of the desired property at section 2 to that at section 1.
Conclusion
The plots given are easy to use and are correct within the assumptions.
However, near choking conditions the air velocity increases very rapidly.
It
is erroneous to assume that the droplets will be able to accelerate with the
stream in the absence of very large drag coefficients.
Therefore, near
choking conditions the plots will not give a correct answer to the variation
of stream properties due to droplet acceleration.
However, the value of M1
where choking of the flow at section 2 is indicated is approximately correct.
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IV.
MODIFIED FANNO LINE ANALYSIS
Procedure
mixture in a pipe with
The example studied is the flow of an air-water
wall friction and droplets striking the wall.
The problem is, knowing the
initial stream properties and friction factor, to determine the stream properties at any section downstream.
The assumptions made are:
(1)
Constant area flow
(2)
Adiabatic-no
(3)
No evaporation of water droplets
(4)
No change in temperature of droplets
(5)
Perfect gas relation holds for air
(6)
For droplet entrained in air stream; V
(7)
For droplet
(8)
In each differential section an amount of water
change in stagnation temperature
V
V
on the wall;
= 1
a
= 0
leaves
the stream and hits the wall, and an equal amount of
water leaves the wall and is picked up by the stream
(9) That this amount of water is proportional to the
length of section
The details of this analysis are given in the appendix.
The general method
of performing the analysis is given in reference (2).
Results
Values
of
+
o Values
~~~~K
* R
a
'
canoralculated
fondirvarious waterrates
P
T
V
F
P* ' T* ' V*
and F were
Mach numbers
rates
ater
and
and Mach numbers. The star
a normalized condition corresponding to choking of the flow.
onditlon is
These values,
to slide rule accuracy, are tabulated in Tables XII through XV and plotted in
Figures XI through XVII.
These figures show the effect of friction and
droplet momentum exchange with the walls on the stream properties.
The numbers
w
on each curve indicate the water rate to air rate ratio, W
The symbol K in the term
+ K
Lm
incorporates as a pseudo-fric-
tional term the effect of momentum exchange of droplets with the wall.
term is a function of the water rate to air rate ratio,
W
, the duct diameter,
and the droplet diameter and perhaps other stream properties.
of this function is not known.
This
The nature
However, using the data of reference (3)
for sand together with the low Mach number analysis described in the appendix, it appears that K
ww
.4
for 1" diameter pipe and sand diameter of
a
200 p.and 450 p.
It appears that the constant of proportionality is slightly
smaller for smaller sand diameters.
The effect of duct diameter is no known.
Figures XI through XVII are used in the following manner:
Knowing the
initial stream properties enter Figure XI with the initial Mach number, M1 ,
and W and read off the value [4f
the length of duct from
[f +
K
·
]
Lmx
[4f + K
Thenknowing
and
the initial section 1 to any other section 2, calculate
1
2.Then
calculate[4 +
Enter Figure XI with[4
D
Enter L Fgr and
Ii
the desired plot of properties with M,
and read
f + KJ L
off M2 .
- [4 Df+ K] h2
Then entering
and i , read off the value of
a
the normalized property at sections 1 and 2. From this form the ratio of the
property at 2 to that at 1.
M
=
2
Knowing the value of the property at 1, the vlue
at 2 can be calculated.
Figure XVII is -used to determine the Mach number after a normal shock
knowing the Mach number and
w
Just before
Vw
the shock and assuming V = i on
asides
the
ofthe
plot
shock
with
nter
initial
M and
both sides of the shock.
Enter the plot with initial M and
a
- to find initial
Wa
F
point.
Follow across horizontally with constant
until reaching the same
w
Read the value of M after the shock.
_w curve.
a
Conclusion
Prw4vr4And
m,-r4
,frma+4l-
n
ho -kA ndrAnl
r
yRf
+1hn nia.a--,-f,4 r
tional term, K, the plots given can be valuable for engineering use.
This
could well form the basis of a thesis to determine whether this simple model
of the
f
ow is correct and to correlate data in the field to two-component
flows.
In addition, very near choking conditions this model of the flow is not
correct since the air velocity increases very rapidly in a very short length.
It is erroneous to assume that the droplets can accelerate with the stream
in the absence of very large drag coefficients.
Therefore, near choking
conditions the plots will not give a correct answer to the variation of
stream properties due to friction.
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L. LL.
WX
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Cj
-27-
APPEMiDIX
SYMBOLS
AREA
A = CROSS 5EcrIONAL
C = 5 PEED OF SOUND
CD= DRAG COEFFICIENr
A COvs rANT PRE5SURE
Cp= SPECIFIC HEA
D HYDRAULIC DIAMETER OF DUCT
dur= DROPLET DIAMETER
FRICTION COEFFICIENrOF DUCT
f
F= IMPULSE FUNCTiON (PA*tfAV 2 )
h=
SPECIFIC
K:
RArio
LMAx=
ENTHALPY
PECIFIC HEAT5 (CP/Cv)
OF
MAXIMUM
OF DUCT FOR
LGTH
M
MACH NUMBER
m=
MAS5 OF PROFPLEr
CHOKING
FLOW
P= STATIC PRE-ssURE
rq Po= 15ENfrROPIC STAGNATION PRESSURE
NUMBER/
R = REYNOLDS
TE4vPEfATURE (ABSOLUTE)
T
To - ISENTROPIC
t = TIlE
SA GNATIOIV TEMPERATURE(ABSoLUTt)
V = VELOCITY
RATE OF FLOWVV
DISTANCE ALONG DLcr
X
P= MASS DENSITY
cUr=MASS
I) = KINEMATIC VISCOSITY
()*-= REFERS T A NORMALIZED
FEFER5 To AIR
( )o-
CHOKIlI6
CONDITIOON
)ur= REFER5 TO WA1ER
( )=
REFERS
r
= REFERS ro
ISENTROPIC
STo
STAGNIATO/V COND/TON
I
( ) = REFERS To SECTION Z
()'R-
REFERS
TO CONDIlr/ON RELATIVE
TO AIR
21
DROPLET TRAJECTORY- DEFAILS OF ANALYSIS
IIVOIVIO AL
DROPL'T5 /NJrCTED INTO MOV1Nd AIR
'=
Va
LE :
TREAM
Co/vSrANr
VR=
Vp-
,
Va > 1/UW)
(FOR
(roe
V 4 -V.,
Vw.V4
( 4-)
VW
dCL
FORCE EQUArIoN ON DOPLEr
m
_
dt
coD
= _ ,Zt, A
.gR.
z d
co R-1
d^
--
a
t
", a
u
t
? _a_ 4=
dR
6 0TD
m
lt
/4A
_
C Io sP.i
0
A
-
D
PM
T
a
1 2-
,prdar
4
Ra
3
dVR
dt
°o
oo
d R
c R,*
, d xR
= - --Pa.A-
z
d YR
(I)
COff"T
CONT
DOROPLETT ;TRAJ-C TOP Y)
TRAJEC TOP Y
DROPLET
d R
X
4
,,
P
d X6
dw-
4 ftP.
P.
-T~~
-
OF REAL
f'a
3
-
.
-
3
0
(2)
dC
CotI
D ISrANCE 5
-
= /a t, 2 NOTE:
X,
(3)
/IFVa.> VLur X /5 /
N r /AT'1
V5,-Dt REC r/'OV
IN EQIUArO/ (4) SI&N OFLAS
TERM /5
+ /F Va< V
-
F
V,. > Ver Kti)
Xi, 2
_t4
f,,.
- 4
0
aL
VLdJr
Juw
/ dR
/
co
i
_
re
JZ7_/Co
R
NORMALIZING EQUATION (4) so rTHAt /I/r/ALLY,
AT POINT I
V
= Va- i; s O THAT:
vu- oO Fol
VW
=ZV,
Va > V-
FOR V
< V
LET 7T/sNI6O1iALIZEC,Pos'or/ON BE PoINT 0, A NO
A IYY
OT
PO/NT DO WN rREAM BE PG/Mr X.
-
DROPLET TRAJECTOR
Y
&
Lar
90---
f
Xo x
-CON'T
d RI,-
~/o dLw'
, /dR
co IRJq
fur
2.
I
dur
J
. va-
xV_ J
Y.
FOR ANY
IN/JECrEO
NORMAL/ZED
.
Xx,
V,, OrER rHAN THF
COIVDiITION:
p,.
I
Ur
Xow
)(o, X
f~r
.
HAS 8Z
V"
Pof-rED
dw
P
1,
f
Au PLOt.
Atvo
J,
tEN
fl
jS
CALtCULATFED
(Xo,
OX
0
d(r
Ipa. )
(7-ABE
M-)
fU
(
(o
UR~F 7ANDT)
FROM ,T
Sl 7- 2000
0dR VAL UEJ
oF
ANOD R FRO .OI To /.
Table
Values of CD = f(R) for spherical particles
R
CD
0o2
240
120
0.3
O. S
49.5
0.1
80
CD R2
24.0
2.4
24.0
24. 0
4.8
7.2
24.8
12.4
0.7
1.0
36.5
26o5
26.5
2
14.4
10.4
28.S
57o6
31.
93.*7
34.5
173
265
3
5
7
6.9
10
20
30
50
70
100
25.6
37.o8
41.0
410
2.55
51.0
1.02 x 103
1.80
1.27
75.0
89.0
2.00
1.50
1.07
60.0
107
0.77
154
.65
.55
.50
.46
195
2000
3000
5000
.42
.40
7000
.390
.405
10000
2 x 104
.O385
.45
.47
.49
3
5
7
10
105
26.5
5.4
4.1
200
300
500
700
1000
2
R
275
3.75
6.23
10.7
3.08
5.85
x 104
13.75
24.5
350
460
46.0
840
1200
1920
lo68 x 106
3.60
9.60
2730
4050
9000
14,200
19.1
40.5
.180 x 109
.426
.50
.48
24,500
1. 23
35,000
48,000
4.8
.42
84, OO
16.8
2.45
x 109
I
I
I
Table
II
Values of X = f
(R),
Y = f
(R)
2x 105
1
=J2
o
dR
2
CR,,n
R
dR
Y =
0
R,
CDR
Y
X
2 x 105
17. 947
C
1
1.*09 x 10-5
16.311
7 x 104
2.04
15.570
5
3
3.18
5.82
9.42
x 10 - 5
14o876
13.764
12.856
11.066
1
21052
7 x 103
5
3.23 x 1
4.71
8.14
12.22
3
2
-4
10.146
9.259
7.905
6.894
5.2119
23°9
1
700
500
300
200
100
70
50
30
4.40
6.62
8.88
1. 809
2.214
1.
* 5585
x 10 -
10
5*244
6o161
7.077
0
I
10.00
12.S0
~
12o495
x
lO. 038
7.038
-o 979
3.704
In
2.3505
14.38
20
5
4.4573
3.8080
2 9316
3.27 x 10 - 3
5.32242
3. 0242
x 1-2
%,'7
17070
10.966
7. 28
4.0134
0
-
-34-
Table II - Cont.
R
Rl.
~~x
¥
.7
.1387
28.655
.5
.3
20697
.1
15I7
.1727
.1900
.2185
*05
.2472
20.83
o01
.005
10-3
.3432
.4102
o2
.3142
0-a
I03- 5
k/
n
x 10 - 2
12.500
8.333
4. 66
10 - 4
4.166
20.33 X 1O
4 166
.5062
4.166
x 10-
. 6022
4o166
x 10-7
oo
A)t
U
Table III
Values of Z = f (r)
for various values o
Ri
Va
Ri= PaVadw
pa
Z
-
for
dw PwZ2
w
Ri
Vr
Va
.50
*.373
.35
.25
.15
2.210
4.360
.10
.05
.035
.025
.015
.010
Z2 for Vw >
9 *.115
140366
27.973
37.452
47.*921
67.916
87.002
Va.
Ri = 1000
= 2000
Z1
Va
<
> a
for
Z2
5.356
8.709
12.586
19 .678
26.477
42.196
52.499
63.624
84.419
103.990
Vr
zi
Va
.70
.50
*
Idv
.30
020
.10
.07
.05
.03
.02
.01
.1672
.3079
2.5990
4.3369
11o2688
15.6462
20.7174
30.5149
39.9373
60.1723
2. 1789
4.5507
8.6782
12.4654
203.8527
26. 2113
31.9361
42.5335
52.4396
73.2610
-36-
Table
Ri
III - Cont.
700
Ri = 500
Vr
Z2
Va
vr
Va
.714
.428
.286
143
.1889
I.*0921
2.4263
605026
.100
.0714
.0428
0286
.0143
.0100
9.5525
13.0041
19.7424
26.2656
40,3420
48.7970
1.9199
5,1596
.60
Z1
.3114
.40
1. 0431
8.0431
.20
14 2308
.14
18.1045
22.2111
.10
29 . 7492
.04
3.6531
5.7139
8.0857
12.7845
17.3749
36.7560
51.4190
60.0771
.06
.02
.014
.01
Ri = 300
27.3457
33.3559
39.3871
Z
72
2. 6479
4.9288
9 *.6503
12.5349
15.5617
21. 0603
26.1343
36.6916
42.9050
49.0841
R i = 200
a
Z2
Vr
Z1
Va
.667
0333
.233
.167
.100
0 667
.0333
.0233
.0167
.01
.
-
;, '-
1.2729
203446
3.6367
6. 2960
8.9535
14.8187
18. 3842
21.09733
28.2704
1.6784
4.9336
6.8 291
8.7762
12 .2353
15.3764
21.8281
25.5968
29.3339
35.7936
.50
.35
.25
.15
o10
005
0035
.025
.015
.01
.4106
.9878
2. 5220
3.9230
1.7400
3.3796
5.0706
8.8830
14.3432
11. 2261
16o8895
13.5942
17.7545
21.2558
5o3303
7 * 7697
9o443
19.4055
23.7285
27.3188
-37-
Tbjei$ - Cont.
Ri =
Z!
d~1
.1-7
05
.03
.0O
001
Ri = 70
00
.0826
.2950
.9148
1.6395
3.3990
4.5198
5.6668
7.7063
9.4347
120.7206
Z2
Vr
Va
.9060
.714
6. 7478
.143
.100
1.7738
3.1935
4.4017
8.0717
9.3667
11.5688
13.3863
16.7654
.428
.286
.0714
.0428
.0286
.0143
.01
Z1
.0504
.3643
.7990
1.9427
2.6968
3.4775
4.8817
6. 0782
8. 3627
9.6203
Z2
.7054
1.8191
2,7374
4.4677
5.4249
6.3536
7,9204
9. 2060
11.5870
12.8861
Ri = 30
Vr
Zi
Va
.6
.4
"10
i
.06
04
. O2
.014
.01
Z2
Vr
Z1
Z2
Va
.1100
.3514
1. 0845
1.5941
2.1307
3.1138
3.9557
5.5710
3.4688
7.3247
.9 098
1.6348
2.9545
3.6672
4,3517
5 4974
6.4284
8.1402
9.0687
9.9456
.667
*333
.*233
. 167
.100
.0667
.0333
.0233
.0167
.c01
.0481
. 3707
.6358
.9282
1.4835
1. 9708
2.9227
3.4553
3.9646
4. 7935
.5317
1.4408
1.9091
2,3494
3, 0674
3.6438
4.6922
5. 2554
5. 7858
6,6365
Table III - Cont.
1~ 0
Ri=2
R. - 10
Vr
Z1I
Z2
Va
.1173
.2602
.4304
.7747
1.0849
1.7032
.50
.35
25
.15
.10
.05
035
2. C0520
, 025
2.3881
2.9370
3. 3926
015
.01
.7038
Vr
Va
.7
1. 0499
1.3681
1.8753
2.2742
2.9890
3.3685
3.7256
4.2 964
4.7632
**'
.1
.07
.05
.03
Z1
1.4416
.01
1.8159
.0115
* 0789
286
.1585
143
.100
.0714
.3-436
0428
.2241
.4199
.7142
.9359
1. 3174
1.5148
1o6986
1.9895
2 2257
2 6112
2.6112
z2
V!r
Z1
Va
.428
.0286
.0143
1.2166
Z2
Ri =5
Va
. 714
.0204
.0688
.2002
.3331
.6181
.7849
.9476
.02
=7
Vr
Z1
.4562
.5668
.7518
.9077
1.1680
.1594
.3895
.6
.5581
.8396
.9 827
.14
1.1145
1.3214
1.4884
1.7598
.4
.2
.10
.06
.04
.02
.01
.0247
.0688
.1872
.2631
.3392
.4685
.5782
.7626
.9511
.1873
. 3205
.5353
.6419
.7391
.8902
1.0111
1. 2066
1.4007
DISCONTINUITY ANALYSIS
-
EFFECT OF DROPLEr ACC:ELERATI 2N ON AIR STREAM
ASSUMP/oNS:
I. ADIABATIC - NO CHAffM6c IN STAGNi4TION TEIVPERATURE
a. OVEVAPORATION OF WATER DROPL TS
3. NO CHAN6E /N
EMPER AT URE OF WArER DR6PLETS
6AS RELATION
4, PERFECT
/OLD5 FO
AR
CONTINUITY:
fa.
A
Va
=
pP2 M
I
FT-
RI
P,
M2
1;
Pa
MOMENTUM!
Va.
A
'1
A
CW-
P, + 1+
AQ
V.' fw V,= P+
ar.
BUT.:V
Va2.
L7'
Vz
A
fca
Va
2 = K PM 2'
FL
o
=P'Z *
VI;Al
km,2-
I
WI
.
-
I+(I +
WLru
vurzx m
-
ENER G Y:
Ato
7T Cp +
,'f'
BUT.'
+
L~rm'
(LY' .'
"
+Ld
h urol
WAa
la
WCLI
C
T C +
K R T1'
*
/AR
)
2 Cp
harol
Va.
(LT 4r
2
Kzl
21
INU/IT
Y ANALYSIS
DIS CON
- CON 'r
12
o T
Vu7)A12
(A,
I +1
~~~AzL
~~'*
VL
-~
A
-
zf
DEFINE:
up",
TO
=T
COMBINING:
. ._
. .t
Toe
km# 2
Vur
-
,
~~~~W..
,"'--I=+
)
uLre.
[}+(
Vur -
;
Va-
T
- M
t+K
(,+! -
M= I
a)M
vQ)M
,,'
]
11w-
ML
(K/)
T
I
KM
TO 7 COD/n7/oN
iOR MrALIZING ALl EUAT/ON
WHERE;
[_L
UrlL VZa M
-
'
2 (K+/)
-
i
_
M7
(+ arur
iiii
E+K
ISErttROPIC
UTr'
],2]
)
5 TA 6NAT/ION CONO(TIO#5:
I. AS 5M/G
&-I(
DROPLErTS
po = PZ
Po°
oo"---"
(l+
-7+
-
L[
DEC ELR A rE- /Is5,lTRoPil1AZZ
2 -1
-ir7 Ji M
-I,
•,
z~~~lurg,.
k (/* W-"r
lAro-
y,
VL
A·
)
M
1-1I Y-'
2
I
L
I
-CON T
LAN4ALS I
DISCONTINUITY
NOrl DECELERATED
Z. ASSUMAI6 DROPLErS ARE
_ p
M2
m
*/
K1
o f
2 /-
k
=~~P
vm..2 _ Kl X-
I'+k"
VALUES
TABLES
ur
Po
OF ro
oi )
£
T
VALUE5
OF /", "I2.
"Pol
A SS UMI/G
= 0AND
V.
Va
=
2;
I K-=.4.
JT2
pi
T I
(N F16URE5
V,
ivVI
THROUGH X
V/5 M, IN TALB 6g-
/15 TABULATr
M, FROl O ro 2;
TABULATED IN
p 100
PL OTTFO V3 M,
AND M
/RE
FOR M FROM O
W rTHlROUGH
I 2s is* j) a
ura.
_
.
=o, . I
AND
= /.
V
Va. a
.3,.4-) H=I +j
FOiR
-42-
Discontnity I Analysi s
Discontinuity Analysis
For all values of J
VW -
Wa
M
0
.10
020
.30
0
10
O
035
I
II
I
I
I
I
.40
.42
.44
.46
.48
.50
...
*
2I,
') 4
.56
o58
.60
°62
.64
LO
P
T0
go
P0
0
1.2679
.04678
.17355
.34686
.43894
.52903
.56376
°81892
.83982
°68
.70
.90850
.72
.74
.76
.8
1.9608
1.1140
1.1059
.85920
.87709
°89350
066
1 1566
1.1985
.69136
.71990
°77248
.79647
.0 2212
93442
°94546
. 95528
Gas,
K = 1.4
1o 0979
1. 09010
1.08255
1. 07525
1.06821
1.06146
1.05502
1. 04890
1.04310
1 .03764
I.03253
1.02776
1. 02337
7
P
1.-1779
1. 2346
.63007
.66139
.74695
Perfect
2.4000
2.3669
2.2727
2.1314
2.0487
1.2 591
1.1480
1.1394
1.1308
1.1224
.59748
0,
a
1 .9247
1.8882
1.8515
1.8147
1.7778
1.7410
1.7043
1.6678
1o6316
..A-
1. 2000
1. 1976
1.1905
1o1788
1i 1713
1.1628
1 *.1591
1.1553
1.1513
1.1471
1.1429
1.1384
1.1339
1.1292
1.1244
1.5957
1.5603
1.5253
1.4908
1.4569
1.1194
1.1144
1.1091
1.10383
1.4235
1.3907
I. 09290
I. 3585
1.3270
1.2961
1. 0984 2
1.08727
.L
CalE
1. 07573
1.06982
T
-43-
Tabe
IV - Cont.
PO
T0
P.
p*
T
1.2658
1.2362
1.2073
1.1791
1.1515
1.06383
1,05775
1.05160
1.1246
1.C3270
m*
P. *019
.96394
.97152
.86
.88
.98363
.98828
1.0124
1 00951
1.00698
o90
.99207
.99506
.99729
.99883
.99972
1.00485
1.00310
1.00174
1.00077
.84
.92
.94
096
*98
t1.00
1.05
1.10
101
4.
o5
0
.L o-1Q
1
1.0 1934
.82
.S0
25
1.30
lo40
97807
11,0000
.99838
.99392
o
-.8 21
1. 011569
1.07285
1. C4792
1 .00019
o
1.02364
1. 00000
1. 00121
1. C0486
1.o 00000
.01092
.94358
. 89086
.84166
.79576
1.02627
1. C1978
1.01324
1. 00664
1.00000
. 8320
.96618
.94899
.97872
1.01941
o96886
.95798
.93425
1.03032
1.04365
1.07765
1.1215
.75294
.9 14 29
.64102
.57831
089686
.86207
.82759
.52356
.47563
.43353
.39643
o36364
.79365
.76046
.72816
.69686
.66667
1.50
.9O928
1.60
1.70
1.80
1.90
.88419
.85970
1.1756
.8 3628
1.3159
1.4033
1.5031
00co
1 ,09842
1. 04537
1.03907
.81414
.79339
1. 2402
.71301
.9,3168
T
-44-
Discontinuity Analysis
vw
wa
Wa
M
To
To
= 1, Perfect
PO
90+
Po*
Gas,
K = 1.4
p
P
TX*
j
0
I
II
I
1]
~
t
~
.17186
1 2315
.33983
1.1918
.35
.42738
1 1667
2.0191
1.1895
1.1767
1.1685
.40
.42
o51 76
o54392
57493
.60467
.63305
1 * 1482
1. 1391
1.o255
1.1592
1. 8873
1.1552
1.1302
1. 8488
1. 1510
1 1127
1.7715
1.1466
1.1421
.65999
.68543
1.10413
1 .09584
1.08786
1 .7329
1.07994
1. 1226
.44
.46
.50
.52
.58
.60
.62
.64
.66
.68
.70
I
1.2000
1. 2583
.54
.56
~~~.76
2.400
2.3636
2.2607
.04666
.48
1
lo2679
.10
.20
.30
I
0
T
.74
.78
.70934
.73172
1.1215
2.1079
1i8101
1.6944
1.6562
1. 1974
1.1374
1 1326
1.1277
75 254
1.07233
1.6184
1.5810
.77183
.78960
1. 06504
1. 5440
1o1119
1.05830
1. 5076
1.1064
o 80590
.82075
.83421
°84634
.85717
.866 79
.87523
.88258
1.05154
1.04524
1.03937
1. C3399
1. 2872
1.02433
1. C2002
1 01586
1.4717
1 o.4364
1.4018
1.3678
1.1173
1.1008
1.0951
1. 0892
1.0832
1. 3346
1.0772
1.2020
1. C710
1. 2702
! 1239 1
1. 0647
1. 0583
-45-
baY-
pa
po*
.84
o88889
.89422
.89863
.88
. 90218
.90494
.90
.92
.90695
.90828
i .86
094
.96
,1
T
Pgo
M
.80
.82
on1.
.90896
.90O906
.98
.90862
1.00
090768
.05
.9 0344
1.01261
1.00946
1.00679
1.00467
P*
p*
1 .2087
T
1. 0519
1 1791
1.0454
1.0388
o1.00301
1. 09460
1.0253
1.00156
1.00058
1.0185
1.o00014
1.C06790
1 04191
1. C1663
1 o00045
.99204
°96813
1.000016
1 .1502
1.1220
.0321
1. C116
1. 0047
.9977
.9907
.89694
1.00119
1.00512
1.01148
.87914
1. 02078
1. 03245
.86861
.85741
1. 04708
1m.30
1.40
1.50
o .80988
1.10657
1.1601
.53751
.8385
.8027
1.60
.78636
102252
.48559
.7677
1
1.20
102
.88870
o83386
-.
1.70
o
1.80
1.90
cO
.7638C
.74250
.72259
.70410
1.0 6413
1. 3023
1.3921
1.4954
1.6133
094488
.88960
.83816
.79034
.74590
.70459
.66619
.59725
.44032
.40069
89
. 65,
.335 20
.9836
.9658
.9477
.9295
.9113
.8930
.8748
.7336
.7006
.6688
.6383
-46Table Vi
Discontinuity Analysis
VW
w
i-= !
Wa
Perfect
Gas,
K
1.4
a
Po
M
L
T
T*
0
.10
.2679
2.4000
1.1637
2.2489
2.0848
1.9904
.2574
1 0 285
1. 18695
035
.04654
°17020
.33301
.41630
.40
.49538
1.1402
1.1308
.20
* i
.30
.42
.52517
.55370
.46
.48
.5 8087
44
*50
54
.56
58
e 'JQ
60
.62
.64
66
.68
070
.74
076
.78
2 .3603
1 .8916
*
1 .8513
.60660
.11245
1.10357
1.8110
1.7706
1.7303
.63083
A.09486
1.6901
lo1216
.65353
.67468
.69429
1, 07839
.71237
1.06o307
1.5334
.72894
.74405
.75773
.77004
.78104
. 05592
1°04919
. 04278
1.4955
1.4583
o79078
°79933
°80676
.81312
.81848
.08649
* 07056
1.03691
1o6503
1.6109
1.5719
1.
*4217
I- 2000
1 .1971
1 .1886
1o1746
1.1657
1.1556
1.1513
1.1467
101420
1. 1371
1.1321
1.1269
1.1215
1.1160
1.1104
1o1046
1 0986
1.0926
1. 03133
1.3858
1.3507
1. 0864
1. C2624
1.3164
100737
11 2828
1. 0672
1 606
1 .02155
1.C1731
!362/
1L.0
1. C1031
1. 2500
1. 2180
1.1869
1,0801
1.0539
1.C0471
i
I1,e, VI - Cont.
)
p0
I'0
N11
P 0*
o*
e0
-)
1. 0744
°82648
P
pe
1.15655
1.1270
T
1 . 0402
1.0333
.84
.86
.88
*. 82923
03 1 2
.83255
1.00503
1..00311
L.00
jOl 64
1.00066
.90
.oo322
1.00012
1. C01660
.83330
1. 00005
. 9974
.9900
OS2
0
r)
d
.83124
1 . 09819
1.0262
1.07022
10o 4303
1o0191
1o0047
1.0119
.83284
1. 0046
.33189
1. 00182
.99094
.96601
.94181
°83050
1.00264
.91832
o9752
loCO
L. o
.82869
1. 0443
1.05
i.10
.8 2265
.81482
°80567
.79556
1. 04801
.89552
.84146
.79135
.74493
.70192
.9489
.9 299
1.15
I.01096
1.02041
1.03276
.78478
1.06619
.66207
.8727
.77359
1. 08732
062513
.8 537
1.1386
1.50
.75067
°72793
1
.70601
1 .2787
°68528
1 .3689
* 4
.94
.96
1o20
.1.SO
1.40
60
1.70
1.80
1 090
2.00
6691
.64797
.63143
1. 2022
I.4737
1
940
1. 7313
.55908
050209
.45276
.40989
.37249
. 33971
.31075
09827
o9677
.9109
.8918
.8161
°7792
.7433
.7085
.6751
6429
.6122
-4.8-
T
tabl
VIa
Discontinuity Analysis
ww
wa
Wa
M
0
O 0
010
.20
.30
V
w = i,
-
3.o
Po*
0
1o2 679
. C4642
.16856
.32642
°40567
.40
.47982
.44
.46
.48
.5 0743
.53370
.55853
.58187
.50
.60368
052
o62394
.54
.56
Q
0
o35
.42
Perfect Gas,
K = 1.4
a
Va
.64266
.65985
1,2566
1.2255
1.1813
I
p*
2°4000
2.3571
2.2 371
2. C0622
1. 1570
2.9625
1.1325
1.1227
1.8587
1.8167
1o1133
1.1041
1. 7747
1 7327
1. 09502
1.6909
1.08630
1.07779
1. 6495
1. 06976
1. 6084
105679
.67554
1. 05466
1.5279
1.4886
.68977
1. 04767
1, 04119
o64
.71404
1.4500
1.4121
1o03511
1.3750
.68
.73314
.70
.74091
.74758
75321
.60
.62
.66
o72
.74
076
78
.70258
.72421
.75788
.76166
1. C6202
1.02950
1.02440
1.01972
1.01550
1.01184
1.00864
1,00600
1.3387
1o3032
1. 2686
1o 2349
1o2020
1.1700
1.1389
I*
1o2000
1. 1969
1.1876
1.1726
1.1630
1.1521
1.1474
1.1425
1.1374
1.1322
1.1268
1.1212
1.1154
1.1095
1. 1035
1.0973
1.0910
1.0845
1.0779
1. C712
1. 0644
1. 0575
1 0504
1 o0433
1.0361
-49-
Table VII - Cont
M
'0
.86
.76460
.76677
.76822
.76902
.88
o 76922
os80
.82
.84
o
Po
PO
P 0*
T
P*
7*
1.00375
1,10865
1,00205
1. 0091
1. C7925
1,o 05070
1000018
1.00010
.90
.76888
1 o 00035
094
,76803
.76673
1.00255
.9
p
100 0116
1 02299
..
.99610
1.0288
1,0C214
1. 0140
1.0065
.9989
.99610
.94472
.92019
.89642
.9912
.9836
.9758
.9680
.9602
o76502
.0.C0435
.76293
1.00673
.87339
.76053
.75326
.74460
1o00955
1.01890
1.03136
.9524
1.15
1.20
.73495
1. C4701
.85106
°79826
.74948
.70444
.72462
1.06582
o 66284
.8731
1.25
.71386
.70288
1.08781
1.1130
.8533
. 337
1.50
.68086
.65942
.62439
.58884
.5 2549
1.2475
.47105
1.60
063906
1.3361
104400
.7205
1.80
.60239
.42409
. 8340
1.5*606
1.90
.58617
.5713
1. 6990
1 8568
.28986
.6190
.5882
* - zO
o98A1.00
1.05
.10
1 J
1.30
1.40
1o70
2.0co
.62002
1.1734
.34799
.31703
.9327
.9128
. 8930
.7949
.7571
.6852
.6513
-50I
Table VIII
Discontinuity Analysis
Wa
M
VW
.4,
Va
Perfect Gas,
= 1,
0
0
.10
.20
.30
.35
0
.04630
.16744
o32002
P
Po
Po*
To
1. 2679
1 * 2557
10 243
K
P*
2.4000
2.3539
2.2 288
o9547
1.1759
1.1505
40
.40
.46502
1 .1 252
1.
.44
.46,
.46
.48
.49063
.51482
.53753
1.1153
1.10562
.55871
1.08711
1,7834
1.7398
1.6964
1.6534
.55783
1o07831
" ds
.4'2
.42
.50
.521
* wZ
.54
.56
.58
.60
.62
.64
.66
.68
.70
.72
.74
.76
078
.59644
.61301
.62807
.64168
1, 09624
1. 06985
1.06182
2, 0401
1.9353
8270
196107
1.5686
105272
1.4
T
T*
1i.2000
1.1966
1.1867
1.1705
1o1602
1.1485
1.1435
1.1383
1.1329
1.1273
1.1215
1.1155
1.1094
1. 05427
1. 04704
1.4864
1.4464
*65388
.66473
.67430
.68265
.68984
1 . 04035
1. 03418
1.0C2842
1.0901
1.0C1848
1.4071
1.3688
1.3313
1.2947
1.2590
.69596
1
1 .2242
1.0552
o70107
1.01064
1.1904
1.1576
1.1257
1. C479
.70524
1. 02320
01431
1. 00751
.7 0853
1
°71102
1. C0289
ff00494
1.0C9466
1.1031
1.0967
1,C834
1. 0765
1.0695
1.0624
1. 0405
1.0329
I. 0253
-51-
Table VIII - Conto
M
To
T
.71277
.80
o
8
.
71363
p
P
Po*
P*
T*
1.06458
1.0176
1 C.03542
1 o0099
.00143
1. 0047
.64
.71427
1. C0002
l.00714
.86
.71414
1. 00019
.97975
.9941
.88
. 71349
I. 00084
.95320
, 9862
.71237
1.00208
.92750
.9782
.70891
.70665
1. 00616
1.C00900
.85521
., 39
.98
o70408
1.01238
. 83264
.9457
1. 00
.70124
1. 01633
.81081
.9375
1. 05
.69317
.68402
.67415
. 6382
1.02851
1.04415
1.0o6314
1.08560
.75928
.71183
.66813
.62788
.9169
. 963
8757
.8552
1 .30
.65325
.64262
.62161
1.50
.6014.7
.59077
55653
.*
49570
.44362
. 348
.8146
1o40
i.1115
1. 1410
1.2108
1 95
1. 60
1.70
1.80
. 58258
0.56507
.54899
1.3970
. 5154
1 6526
.39883
.36012
.3 2651
.6990
.6633
. 6292
1. 8
.29707
.5968
1 9898
.2 7149
o 660
.
n
020
.701
. 2 .71082i. 00385 .90261
.9539
.87852
.94
. 96
1.10
1.15
1. 20
.25
1.90
2. 00
.53428
.52087
°
07748
.7362
A-
-
Table
x
Discontinuity Analysis
Values of MI
2 = f
,
M_
%!N
Perfect Gas, K = 1.4
for
Ww
=
Wa
.1
.3
.4
o201
0252
*203
.304
308
127
0
o 312
.31
.32
.315
.325
.319
331
.324
.332
*342
-34
.336
.346
.356
o353
.348
.360
.372
.367
.376
.387
. 393
.398
.384
.397
.408
.410
.422
.423
.437
o436
.452
.422
.433
.444
.454
.467
.433
.449
.464
.467
.436
.520
542
.478
.498
.489
,500
0512
0 ' 1 25
.526
.562
.586
.513
. 41
,.
.35
.36
.37
.38
o39
.40
.41
.42
.43
.44
.45
.46
.47
.48
.49
.50
.378
0388
.399
.411
.525
2
254
. 364
.45
.472
. 4 , ,"
. 5j 7
. 204
o
c57
.336
.478
.493
.~00
.205
*260
o 317
.328
.342
.354
0367
.380
.407
.421
.484
e.502
.612
. 645
.603
*
"1
0"
08
.. ;
*. e
. !:' 6.O
.687
Table IX
M 2 for
Ml1
.537
.51
-
Cnt,
ww =
Wa
.2
.3
4
573
.626
.745
845
o 516
.52
.549
.589
.651
.53
.563
.607
.683
°54
.575
.625
.718
.55
.588
.644
.770
.601
.665
.557
.877
.56
.57
.614
.688
.58
.629
.716
.59
°
642
.60
.657
.61
.673
.744
.782
.847
.913
.614
.62
.688
.63
.707
.64
.725
.65
743
.66
.67
.765
.789
.68
.69
.70
.827
.855
.925
.702
.953
- 54-
Table
M2
IX - Cont.
w =
for
.1
Wa
Shock
Shockless
1.50
1.52
.953
* 870
1.050
1.*56
.817
1.54
.953
1.085
*842
1.58
1.60
1.123
*793
.782
1.158
1.62
1.64
1.66
1.68
* 769
1* 200
*756
.734
1.227
1.250
1.270
1.70
.723
1 294
1.72
1.74
1.76
1.78
1*80
*716
.707
1.310
.700
1.349
1 * 369
1.389
1.177
*745
1.330
.692
.685
M2 for
Ww =
Wa
.4
.1
Shock
Shockless
1.815
1.82
1.84
1.86
1.88
1.90
1.92
1.94
1.96
1.98
2.00
Shock
Shockless
.913
.663
.656
.649
1.444
1.482
.779
.761
.913
.948
1.010
1.045
1.077
1.106
*.643
1.500
1.520
1.537
1.553
1.571
.748
.736
.725
.715
.706
1.131
1.153
1.175
1.193
1.210
.677
.669
.638
.633
.628
.624
1.406
1.426
1.463
.877
.825
.800
FANNO L INE ANALY51
MODIFIED
CONSrANT-AREA.
FRICrIONAL, ADIABATIC,
:
AsSuMErI/oN
CH4N6E IN 5A6NATION
I. ADIABArIlC- MVO
2.NO fEVAPORATIOV OF
T,PERATRE
WATER DROPLETS
3.N CA/6t IN TtMPERATURE OF WATER DROPf5
4, PERFECt GAS RELATION HOLD5 FOR AIR
=I
. FOR DROPLEr E-NRAINED /N AIR STREAM;
L
N THE WALL;
6. FOR DROPLer
7. IN EACH D/FFfRERNrIA L ECTI/ON,LEN6r dx AA
dwu ) LEAVE5 TE STEReAM
AMOUNT OF WAre,
ANC HIrJ
WATeA, )
LIP
ThE
dU'&,
Y rE
IN ADDITION AN AMOUNT OF
.EAVE5 THE WA£L AND 15 P/CK5D
WALZ.
STREAtlM.
CONIvIlrTy:
pV= coNsrANr
v
f
EQUAT/ON OF SrATe:
P= fRT
P_f
D£F//r/ol1(
OF
.m__
7
:
dY
_
(j)
MODIFIED FANNO LINE ANAL Y5
-
'
CON
-
ENER6 Y:
wo(ha * d h)
{(W'
tTuf
2U
(VXz
uroo W
_ (dr -dwA)(L~
.+(ura-
DIFFREIAL TERM
DROPPIV6 2 0
OROIE/
REARRAN61
NOTINGTAT:
NG6
dha
- CPadr
£2.
z
i2
=
+
AND
)
KR
)
zcf
R
ZCp
._L
I
d >- d ,
Ith
i--Ij(
L(1it CZ )M
IN
rSTtA
) V STATE
IS
AT ANY
-
1+
9-1
z
(I
f
2
/iV)M
WALLS,
IS CONS AIYr w /H
WITH WATrFR ON TE
SECTION
AND THE FILM
NEG L I61BLE.
a0o4
(4)
_
TlE FILM T41C4NES5
ASSUME
rIM
u0
a~~
0TTr 't'
ur )
z 42.
dAlM
d
0
VELOCITY
MODIFIED
FANNO LINE ANALYS)5-
mOMENrUM:
FomLNrTM
FLUX
FORC ES
PA
CON'T
_. (
Ir - -
[PdP)A
I
$
+du2)
(Wlk
(V+dV)
_F
r.,rDdx
d,
(
-A dP - rw TrD dx
P
4r
d2(o)
urw) dV+dw
dx = (I-
C.)
D
IP
DEFIYE :
=
fTu
-
dV ;
dP *(I+ S, )
ASSUME
=~-k, x
Iz
Urx
ura.
REs ULT5 OF 5lMILrAFVEOU5
0) HRoilH )
zP
m2
SPECIFIC CASE
TO THE
PROPORrTIONAL
V
PMZ
2
-5
THAT IN AY
O's
"dli
dV
±tIJ4f dx
WATE&R PIC/AED UP BY THE SrREA1
IS
ar,-dr,)d
V
A MOUNT'
FRO/yl THE
-r
arc
0
(5)
OF
WALLS
LENGTH.
- Mdx
SOLUTION
A RE PR E5ENTED
Of FQUATION
IN rALE
I
MODIF/ED FANNO
L IUNE A NA l YSI5 -
TABLE
Co rrFIClV
IENr
l
CON'r
I-
I-* K] dX
OF FRICT/Of TERIfJ ,
JA<
Mz
-
Y+i (I e @),z 7
KMZ
',r+ m - -
/r7z
1- (+
i-C
,WarM
-)
VZ~
I~~~
1! .r-2
I
,
OF
IN rTE6RA rIojv
_H91(-
L
P
,
)\
- 1,4M;
+ ,)M
4QUATIOLII
1
d--Z-
-
r1--l
; M=MI
WHEN:
XL=
[C/fotnIla %) fcoinoN7
MAA
2
Ar
rM
-
L MA A
ddl
I
Aid
IA1
RS ULT-5 OF IlrE6RAT rON
ARE
GIVEN IN
TA BLE
AND
ITE6FAAL
RELArTIONS
557
-
LINE ANALYSIS - CON'T
MODIFIED FANNO
TABLE ICT
FORMULAS IN INTEGRAL
f + La-
14lo
t
FORMI
/*(
-
-
T'*
2
-
-
a
AI]
K .-
-L
f
VY
Lt~L~if~
'a
2 g+'
-~
~~K
I
ASSUMING
I
f
Al .
VI
[Tt
S (lt r)
mr:
l
/tl (/tc)
--
DROPLET5
.6.
P [ I+
"-
-
MJt
(
V2 (K+l)[,
D ECELER A rE
15ENTROPICAL
Y:.
I (I,
* r)M]
Pc
I
rO
L
_
1
--
_
DO NOT DEC EL IERATfE:
A SS UMING DROPL ET
-R-1
~~
1~33~~~~
m
id-
.
,p
Po.-
__
.=
-S~
vi
ItJ (i
.
.
a4m1AI
r,
)y
a
I
60
-
MODIFIED FANIVOLINE ANALYSIS VALUES
OF E4+LMAA
; po
UUrr=.i.
;.
AND
j.
P OTTED
/=/.
P
.1
. V/F.
' To, *-P , we*,
OF AiFRM 0 r 2 a
10F
4
ARE 61 VEN FOR V4 UES
.r
CON'T
4
IN TABLES Z
THROUGH
IN F/6URES xir rt#RdOU6
UIV.
MODIFIED FANNO LINE -LOW
Mt <<1 . Pi.
ASSUME:
-_ I
2
D,
P,
0
KM 2 7
L1*
K
.+±+
if t
400
LI
K J
VELOCI1Y ANALYS1
I
+(a(t kwL N ..
,A
-
~=/VZ
fin
ur f
*
iL
F
- KM,
lz
E4--
/~L , 2
-
_P6
*oa,-P
f
K]Lt,2
D
pi
Poz
PALO
M K4
eKHJL,,2
i9J I
M,K4
2 ml /8
0
z M72f
-
-
~'~/\
PCf
-
*tK
·
.
]
I
-62-
Table XII
Frictional, Adiabatic, Constant Area Flow
Perfect Gas, K = 1.4,
M
+
Lmax
Po
p
0
co
0
.!
.2
.3
Po
p ,*
66.8
14.24
5.15
5*54
2.91
1.958
1 *°58
2.82
1.940
2.186
.978
1.298
1.511
1.271
1 . '1"7
.6
.425
.1651
1.149
1.127
1.0C67
1. 040
1.001
.983
.955
1.001
.946
1.01 1.049
1.097
1.152
1.229
.951
.0472
o 00480
1.0
.00288
1.1
.0231
1. 2
00565
1.4
1 .5
1.6
1.7
1.8
1.9
2.0
Pp
.0948
.1370
.1784
.2201
.260
.297
.333
.366
10531
1.020
1.315
1.420
1.538
1.670
1,834
vL = p
V*
00
0* a
.5
1.3
l
p
m
co
.4
.7
.8
.9
=_
2 563
2. 030
1,200
1.197
1.190
1.177
1.159
1.137
1,675
1.419
1.0C83
10.40
5o19
3.45
1.221
070
.946
L.
1.112
1.052
l.C19
0
.1147
.2286
0340
.450
.558
.663
.764
.860
.951
.984
1.041
* 844
.758
.948
1.122
.687
.624
.570
.875
.839
.803
.522
.481
O768
.444
.410
.701
1.448
1.572
o381
3970
1.003
1.049
1.103
1.171
1. 253
1.345
.911
.734
.669
.638
1.201
1.276
1.o348
1.411
1.470
1.530
1.581
1.630
1.676
-63-
Table XIII
Frictional,
Adiabatic, Constant-Area Flow
Perfect Gas, K
A [
0
.1
o3
.4
+D Lma
x
o
66.2
602
14.10
4.95
2.062
1.4,
p0
P0*
0
P0*
WW =
Wa
p
P
~0
5.29
2.70
1.871
5.28
1 ,471
1.442
2.69
1.350
9.92
4.97
1.105
1.0C74
.691
.367
.1276
.0282
.000323
1.123
1.076
1, 47
o.989
1.010
1.001
1o008
.936
1.600
1.350
1.164
.907
1.017
1.029
10071
1.124
1.199
1.287
.01108
1.1
. 0421
.0842
1.2
1.3
14
.1304
.1781
1.5
.2252
1.6
1.7
1.8
.271
1.9
2.0
.314
.355
.395
.431
1 .392
1.502
1.649
1.810
1.989
0
.1201
1.132
2.450
.6
1 *(D
1o197
P*
P
1.941
3029
1.215
1.O
1.200
=
.2388
1.252
.9
V*
1.189
1.175
1.156
.883
.83
V
T*
co
.-
.7
T
1.040
1.005
.356
.470
.583
.794
*.893
.898
.968
.989
. 078
.902
.800
.930
1.162
.919
.949
.718
.892
1,243
*.649
.988
.588
854
.816
.779
1,318
1.386
1.451
743
1.513
.709
.675
. 643
1.621
1.672
.898
1.040
.537
1.105
1.178
.493
.452
1.263
1.362
1.479
.417
0386
.358
.612
1.569
1.719
-64-
Table XIV
Frictional, Adiabatic, Constant-Area Flow
Perfect Gas, K = 1.4,
M[r+K]max
Po
0
0
.!1
.2
.3
.4
.
r,J
65.8
13.84
4.80
1.943
.810
.7
.313
. C964
.8
.9
.01430
.001172
Po
Po*
5.12
2.61
1,807
1.430
1.228
5.10
2.59
9.58
4.78
3.17
2.352
1.861
1.C29
.858
.874
Of
1.782
1.384
1.165
1.0
.0236
1.1
1.2
1 3
.0647
1.043
.1152
.1682
.222
.274
1. 97
1.163
10248
1.346
.900
.940
.986
.325
.372
1.458
1.603
1.043
.416
.457
.495
1. 765
I
1.4
1.5
1.6
1.7
1.8
1.9
2.0
P
P
1.107
1. 034
1.006
1.001
1.016
.6
= = 3
wa
1.938
2.153
.947
.895
.866
1.530
1.292
1.112
.970
1.200
1.197
1.188
1.173
1.152
10127
1.097
. 064
1.029
.991
.952
.856
1.112
1.191
1,281
1.388
T
T*
.913
.873
- P_
v__ =px
V*
p
0
.1246
.2486
o371
.489
,605
o716
.823
.925
1.022
1.112
.559
o795
.509
.757
1.199
1.279
1.355
1.422
1o489
.466
.427
*721
1o551
.685
1.608
1.658
1,704
1.750
.761
.683
.616
.393
.363
o 336
*.834
.651
.619
.588
-65-
Table XV
Frictional, Adiabatic, Constant-Area Flow
Perfect Gas, K = 14,
=
W
4
Wa
M
0
.1
P_
65,5
13.58
Po
P 0*
0*
c0
QO
4.95
2.53
4.92
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BIBLITGRAPh{Y
(1)
Lapple,
. E.
INTSTRIAL
(2)
and. Shepard, C.
A
., "Calculation of Particle Trajectories,"
ENGI.NEERING CHEMISTPY, ivay194C0, p
60-617.
Sh.piro, A. H. and Hawthorne, W. R., "The Mechanics and Thermodynamics
of Steady, One-Dimensional Gas Flow," JOURNAL OF APPLIED 'MECHANICS,
Pecember 1947, pp. A-317 to A-336.
3) Pinkus, 0., "Pressure Drops in the Pneumatic Conveyance of
JOURNAL OF APPLIED MECHAINICS,December 1952, pp.
25-31.
olidst