A Study on an Air-Jet Loom with Substreams Added

Part 1: Deriving the Equation of Motion for Weft

By Minoru Uno, Member, TMSJ

Kyoto Institute of Technology, Kyoto

Basedon the Journalof the TextileMachinerySocietyof Japan, Transactions
, Vol. 25, No. 3, T48-56 (1972)

Abstract

For scrutinizing the motion of weft gushed out and propelled by air stream, the physical
(especially aerodynamic) properties of air jet and weft yarns are examined. An equation of
motion for weft on an air jet loom is then derived, leading to the numerical solutions in
several cases. The followings are some of the results :
1) The axial velocity of air jet is approximated fairly well by the following formula excepting
the core region:
V=BV a/(x-A)
where A and B are constants, V U the initial jet velocity, x the axial distance from the
nozzle exit to any point.
2) The coefficient of friction of spun yarns in air stream is approximated by
C f=0.02+1/(V'+2)
where V' is the relative air velocity to the yarn in m/sec.
3) In case where an air jet loom utilizes a measuring drum for yarn reserving, the drag due
to yarn ballooning is as large as the air thrust. In order to minimize this harmful effect, a
suitable cover should be set around the drum.
4) As the time necessary for the yarn to travel for some given distance is rather long in the
region where the starting acceleration is not large, the initial jet velocity Vo should be
quite large, say over 200 m/sec.

KEY WORDS: LOOMS, AIR JET LOOMS, PICKING (WEAVING), AIR DRAG, YARN DIAMETER

1, Introduction
It has been thought difficult to widen the working width
of the air jet loom, in which the weft is gushed out by air
stream and carried through the shed without the aid of a
shuttle. It is because the air speed decreases quickly, even
if the compressed air is let out at high speed, as shown in
Fig. 1, in contrast with the gradual speed-up of the pro-
pelled weft and its overruning of the air speed, resulting in
the yarn buckling and entangling of itself and to warps.
However, if such an air stream is sustained as its speed is
a little bit higher than the yarn speed as shown in a dotted
line in Fig. 1, the weft will be kept under tension and run
straight through the wide shed. This paper treats of an
air jet loom which is aided with such substream added
along the shed. In Part 1, the equation of motion for weft
is derived after various factors affecting the weft motion are
Fig. 1 Relation between yarn velocity and air velocity
considered, and then applied to a loom having a measur-
ing-drum type reserving system.

Vol.18 No. 2 (1972) 37

 2. Equation of Motion for Weft 2.2 Assumption
2.1 Symbols The equation of motion for weft will F e derived on
the following assumptions:
1) Weft is propelled straight on x-axis with no crook.
2) Air thrust dFacting on small yarn element dx is given by

dF=Cf1/2po(V-v)2rd • dx (1)

when air velocity V is higher than yarn velocity v. If the

condition is adverse, dF is put equal to zero.
3) Air drag dD acting vertical to yarn axis is given by
Fig. 2 Symbols
dD=Cd1/2po ( relative air velocity)2 • d • dl •• (2)

4) Friction at Guides is governed by the factor e'°.

In Fig. 2, Exit 1 of Nozzles 3 is set at the origin (x = 0),
5) Centrifugal force of yarn around the instantaneous
and air is let out in the positive direction of x-axis. Weft 4, center of motion is neglected.
wound-out from Measuring Drum 7, runs through Guides
Then the equation of motion for weft is:
6, 5, and enters the slender Guide Pipe 2 in Nozzle 3, being
then gushed out with air stream. -~
d (mv)
= ,~dF-S (3)
Symbols are summaried as follows :
where S is the total of drag forces hindering the weft
a : Distance between Guide 5 and Guide 6
motion, such as air drag, balloon tension, guide friction
b : Normal distance between Guide 6 and the plane
and so forth.
from which the yarn is wound out
Cd : Air drag coefficient of weft 2.3 Mass of Weft in Motion
C f : Air frictional drag coefficient of weft In Fig. 2, it is assumed that yarn is initially pulled back
D : Nozzle inner diameter and reserved between Guides 5 and 6 by the amount of 14.
d : Aerodynamic diameter of weft Thus, l5 has nothing to do with the weft motion before 14is
I : Distance between Nozzle Exit and Weft-end reduced to zero. So, there are two cases to be considered.
10 : Core length of jet stream (i) Before 14 = 0, namely, if 1 < 11 +- 14 - a, then,
11 : Yarn length initially protruded from Nozzle Exit m__ (l /112/2 213 l4) .,....(4)
12 : Distance between Nozzle Exit and Guide Pipe Exit
13 : Yarn length lying between Guide Pipe Exit and here, 11, /2, /3 are on x-axis, but l4 is not. But its mass
Guide 5 exerts influence upon the weft motion. In this case the yarn
14 Yarn length initially reserved between Guides 5 part which runs at the same speed of weft, v, is included in
and 6 m directly. Only one half of the mass of yarn element is
15 : Yarn length lying between Guide 6 and the point considered in m, if the element moves at 1/2 v. The latter
on the measuring drum from which the yarn is belongs to the lower part 5' 6' in Fig. 3 where yarn is bent
wound out at right angles at points 5' and 6'. Thus, between Guides 5
m : Mass of Weft in motion and 6, m is given by
: Coefficient of friction between Guides and Weft
m(5,6)=(14+l1-1)/2 ••(5)
N : Rpm of the loom
R : Radius of Measuring Drum Eq. (4) has been derived from this point of view. The
p : Linear density of Weft velocity condition and the yarn configulation above dis-
po : Density of air cussed will be applied to the air drag calculation, too.
01 : Angle of contact of yarn on Guide 5
(ii) After l4 = 0, namely, if h +- 14 - a < 1, then
02 : Angle of contact of yarn on Guide 6
V : Air velocity at x m=p(t+12+13+a+15) (6)
Vo : Air velocity at Nozzle Exit
V1 : Air velocity in the direction of x axis, given by Here, as will be indicated afterwards,
substream
15=R+b ......(7)
V2 : Air velocity for reserving the yarn between Guides
5 and 6 2.4 Air Velocity Distribution
V : Yarn velocity The velocity distribution of air jet has been studied con-
W : Working width of the loom siderably fully. For instance, the velocity V on x-axis is

38 Journal of The Textile Machinery Society of Japan

 Fig. 3 Configuration of yarn in loop
LiJYYLaaVV as Vaaa aav a.u.v va.. -, v...

Fig. 5 Velocity distribution of free air jet

given by Schlichting with laminar three-dimensional jet
as~2'

V _ 83rr Qx
' ...... (g)

where p = Coefficientof kinematic viscosity of the fluid

K - (Momentum of the fluid)/(Densityof the fluid).
With turbulent three-dimensionaljet, Reichard gave~3'

v=_3-K_(9)
8rr eox
where eo=0.0161y~K

These show that V is proportional to (1 /x). However, in the

usual jet, there are the potential core 1, which has the
uniform air velocity, and the mixing zone 2, as shown in
Fig. 4. So, after shifting the origin by length lo, which is
roughly equal to (3- . 5)D, V is assumed here to be

v = xBV
-A0 (10) Fig. 6 Measuring tube

The air velocity is put equal to Vi after V decreases and

meets Vi. So, hereafter V is put as

V =- 3-13- -Vo ......(11)

x+ 1
Duxbury gave such a distribution as~4~

V - Voe3a (12)

There is large discrepancy between eqs. (11) and (12),

partly because Duxbury derived the distribution to put the
observed air thrust equal to the calculated thrust.
3.2 Air Frictional Drag Coefficient, C f
3. Determination of Constants Air frictional drag was observed with Acryl 52/2 yarn,
vertically suspended in the tube shown in Fig. 6. C f ob-
3.1 Constants in Air Velocity Distribution, A, B
tained within the range of V < 20 m/sec is
Fig. 5 shows the air velocity distributions on the axis of a
cylindrical nozzle having the inner diameter of 8 mm. For Cf=-V
+2 +0.02 ......(13)
both initial conditions of Vo = 160 and 130 m/sec, curves
are quite well fitted with It is shown in Fig. 7, together with other results by
A = -1/3, B = 13/3. Selwood~57, Anderson~61, both of whom observed with

Vol.18 No. 2 (1972) 39

a

Fig. 9 Assumption of balloon shape

Fig. 7 Frictional drag coefficient of yarns in air stream
where w is the angular velocity of yarn when it is unwound
from the drum. If the drum turns thrice for one pick,

w _!L-10 R
v_ (16)

4. Solution of the Equation of Motion

4.1 Formulation
Sin Eq. (3) is, when l4 >_ 0,

Si=e~0I[ Cd -p0(V2+
2 )2d•dx+

Fig. 8 Aerodynamic diameter of yarns (Mack)

J0Cf-2-po(V2+v)2d
•dx
monofilament-yarns. Therefore, for spun yarn, eq. (13) The first term in the right hand side of the above formula

will soundly be applied. is the air drag on the head of the loop pulled back, the
second being the drag on the leg of the loop. While inte-
3.3 Cd, Air Drag Coefficient, and d
grating the formula, yarn velocity v can be put in
From Mack's experiment~77, Cd can be put equal to 1.5,
constant. So,
as the maximum value. He also gave the aerodynamic
diameter of yarn as shown in Fig. 8, where the solid line is Si=ePel[1
4-CdPOd'a(V2+
y 1 ~
2 )2 + 4 Cfporrd
drawn by the author who will use it to get the yarn diameter
for various yarn count. The dotted line is given by (-l+ll+l4-a)(V2+v)2J ......(17)
d= 25
1 ...(14) Thus, the equation of motion for weft when 14 >_ 0 is
.8/Yarn Count
which is obtained when yarn is wound tightly on a rod,
and would not be used here. dt (mv)--'2CfZ po(V-v)2rcd•dx-Si
=fi ,.....(181
3.4 Frictional Coefficient between Guide and Yarn, p
p is put equal to 0.27 after refering to Harris Table~81. where the integration should be performed while V < v •
3.5 Balloon Tension C' f is the frictional coefficient of yarn in the back region
In Fig. 9, yarn is assumed to start from the surface of the of the nozzle.
measuring drum 7 and reach Guide 6 in a crooked state. After 14 = 0,
To consider the most disadvantageous condition, yarn is
thought parallel and then vertical to the axis of the measur- S2 - pR
2 (b+R) (n 1 $ -- R
v )2efL2+
ing drum. So, the yarn tension at the entrance of Guide 6 is
f6c'-i pov2~rd
•dx
of 2
To- pR w2(b R) ,.....(15) 2
where 8i is put equal to 0. If the air thrust in Guide Tube is

40 Journal of The Textile Machinery Society of Japan

 Fig. 11 Weft motion when Va =100 m/sec, V1=50 m/sec.
(Dotted lines are in case of Drum Cover added)

Fig. 12 Thrust and drag on yarn

Fig. 10 Flow chart for calculation

canceled with the air drag of yarn between Guide 5 and the 4.2 Example of Calculation
inlet of Guide Tube, S is integrated as before:
An example of calculation is shown in Fig. 11 on the
condition shown in Table 1. In this case, as the air velocity
S2=-s--(b+R)
(1k-__)2eP°2
~v+
10 R
1-C'porrd.av2
2 V is always higher than the yarn velocity, there is no
....fig.). concern as to the buckling of the weft pulled by air. If no
Thus, the equation of motion for weft after l4 = 0, and
substream is added, the air velocity decreases as shown in a
when there is a balloon, is:
dotted line and meets the yarn velocity at x = 20 cm.
Thereafter, yarn would have to run in the wind relatively
slower than the yarn.
= J1
dt (mv) ' 12 po(V-v)2nd.dx-S2=f2
_12 ;20) There is a sharp change in yarn velocity, which cannot
reach 27 m/sec. Time necessary for the yarn to travel
These formulas are transformed, after putting 2.25 cm (= Working width 2 m + Pulled back length
l4-a = 0.25 m) is 0.108 sec. If one pick should be done
4(mv)=m--+
ddt vdm
dt d=v{l d
m dv+ dm
t} within 1/3 cycle of loom revolution, it corresponds to only
185 rpm, too slow to increase the weaving efficiency.
for eq. (19), (21) dv = f' dl-v dm ......(21) The slow yarn velocity is partly due to the large drag by
my m
balloon as shown in Fig. 12, which cancels the air thrust.
for eq. (20), (22)dv =-f ~ dl-v dm (22) Therefore, in the next section, the remedy to improve this
my m will be considered.
These are calculated by the flow chart shown in Fig. 10, in
which x = 0.04 shows the core length of the jet from a
nozzle having the inner diameter of 8 mm. A is the position 5. Remedy to improve the Weaving Efficiency
where V equals Vi, B being the position of weft end when 5.1 Decreasing the Drag by Balloon
l4 = 0. Large balloon tension can be partly decreased by a

Vol.18 No. 2 (1972) 41

 Table 1 Condition for Calculation

smooth drum cover 8 shown in Fig. 13. In this case, the

yarn tension just before Guide 6 is

T,=2 p(RH1
e)2w2
1-bp(R+e)w2v23)
where e = Distance between Cover and the surface of
Drum
v = Frictional coefficient between yarn and Cover.
Comparing with eq. (15), it gives
T (R ; e) (R--e 2bv)
To - R(R)

Fig. 14 Effect of v on T1/To ratio

Fig. 14 shows this ratio when R = 0.106 cm and e = 5 mm,

indicating no useful effect when such a cover is used as
having so high frictional coefficient as 0.27 seen in Table 1.
However, small v can decrease the drag by balloon con-
siderably. The dotted line in Fig. 12 is obtained with v =
0.10 and b = 5 cm, showing desirable effect. But even in
Fig. 15 Weft motion when the initial air velocity is
this case, the yarn velocity and the total time show little
enhanced (V0 =200 m/sec. V1 = 60 m/sec,
improvement as shown in Fig. 11. This is due to the slow with drum cover added)
initial air velocity, resulting in slow starting of yarn.

42 Journal of The Textile Machinery Society of Japan

5.2 Increasing the Initial Air Velocity 6.2 Effect of Initial Yarn Velocity
Fig. 15 shows the case when the initial air velocity is To start the numerical calculation of eqs. (21) and (22),
enhanced from 50 m/sec to 200 m/sec, with a drum cover it is necessary to put the initial yarn velocity vo equal to
added and on the following condition : some value. Fig. 17 shows this effect when h = 2 cm and
Vi = 60 m/sec, h = 0.02 m, l2 = 0.04 m l2 = 4 cm on the condition shown in Table 1. If vo is high-
Yarn velocity goes up to 37 m/sec already before the er than 1 m/sec, each solution roughly coincides with each
balloon begins, giving the total time of only 0.063 sec for other and shows little discrepancy. But if it is too small,
the yarn to travel 2.25 m, corresponding to 320 rpm, the increment of the next step is so large that, in an extra-
which is high enough to be desired. ordinary case, it may exceeds the air velocity. So, care
should be taken in choosing the initial yarn velocity.
6. Discussion
6.1 Magnitude of Cf 7. Conclusion
The author already reported the air thrust on static yarns 1) The condition for calculating the weft motion is de-
as shown in Fig. 16.X97 To compare with this, the air thrust termined as follows :
on cotton 20's is calculated according to C f obtained from i) The air velocity on the axis of the nozzle is, excepting
Fig. 7, when Vo = 300 and l1 = l2 = 0. The agreement is the core region,
quite well, considering that the air velocity in the former
experiment was ambiguous, and that Cf shown in this V = 300
13Vo
x±1
report is obtained below 100 m/sec.
where Vo is the initial air velocity in m/sec, x being the
distance from nozzle exit in m, when the core length is
0.04 m.
ii) Air frictional coefficient for spun yarn is

C1-O.02-I V 1
L
where V' is the relative air velocity to yarn in m/sec.
iii) Air drag coefficient Cd = 1.5
iv) Aerodynamic diameter of yarn is given in Fig. 8.
v) Frictional coefficient between yarn and guides p _
0.27.
2) Availing of this condition, the equation of motion for
weft is derived on the assumption that yarn would fly
straight along the nozzle axis with no crook when adequate
substream is added. As this equation contains every factor
affecting the weft motion, it can be used to compare various
Fig. 16 Comparls ~,n of observed and calculated air thrust
weaving conditions.
3) The equation of motion thus obtained is applied to an
air jet loom which has a measuring drum to reserve the
weft. Some calculations show that
i) The drag by balloon tension is considerably large as
compared with the air thrust.
ii) A smooth cover set around the measuring drum can
decrease this drag.
iii) However, if the initial air velocity is below 100
m/sec, the accelerating thrust is so small that it takes
much time for the yarn to run through the shed..
iv) On the contrary, high air velocity at start can avoid
this trouble easily. For instance, the initial air velocity
of 200 m/sec can give 300 RPM in case of 20's cotton
yarn and the working width of 2 m.
The author wishe3 to express his thankfulness to Mr.
Fig. 17 Effect of initial yarn velocity v0 on weft motion Zenpei Tachibana and Yosuke Ichino for their prelimi-
nary experiment and calculation.

Vol. 18 No. 2 (1972) 43

 Literature Cited [6] S. L. Anderson, R. Stubbe; J. Text. Inst., 49, T. 53 (1958)
[1] Uno, et al; J. Text. Mach. Soc. Japan, 14, 141 (1961) [7] C. Mack, J. L. Smart; J. Text. Inst., 45, T. 349, (1954)
[2] H. Schlichting; ZAMM,13, No. 4, 260 (1933) [8] M. Harris; Handbook of Textile Fibers, p.183, Harris
Res. Lab., Washington, (1954)
[3] H. Reichardt; VDI Forschungsheft, 414 (1942)
[4] V. Duxbury; J. Text. Inst., 50, T. 558 (1959) [9] Uno, et al; J. Text. Mach. Soc. Japan, 13, 872 (1960)
[5] A. Selwood; J. Text. Inst., 53, T. 576 (1962)

44 Journal of The Textile Machinery Society of Japan