Material Transport with Air Jet

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1 Material Transport with Air Jet Dr. István Patkó Bdapest Tech Doberdó út 6, H-1034 Bdapest, Hngary Abstract: In the field of indstry, there are only a very few examples of material transport with air jet, and one of these is the air jet loom. In this weaving technology, the weft (the transversal yarn of the fabric) is shot by air jet. This paper will set p the mathematical model of yarn end movement. For a special case, I will specify a soltion of the model. 1 Brief Description of Air Jet Looms In air-jet looms, the weft is introdced into the shed opening by air flow. The energy reslting from air pressre is converted into kinetic energy in the nozzle. The air leaving from the nozzle transfers its plse to stationary air and slows down. To this end, in order to achieve a larger rib wih, V. Svaty developed in 1947 a confser, which maintains air velocity in the shooting line. The confser drop wires are profiles narrowing in the direction of shoot, and they are of nearly circlar cross section open at the top. These drop wires are fitted one behind the other as densely as possible. Therefore, they prevent in the shooting line the dispersion of air jet generated by the nozzle. Fig. 1.1 shows the arrangement and design of the confser drop wires applied in machines of the P type, as well as the arrangement schematic of weft intake. The nozzle (1) is secred to the machine frame, and the confser drop wires () and the sction pipe (3) are fixed to the loose reed. The confser drop wires are profiles narrowing in the flow direction, and they have a conicity of 6. These profiles (section A) may be made of metal (Fig. 1.1 a) or plastic (Fig. 1.1 b). To be considered almost as a closed ring from the aspect of flow, a baffle plate of nearly circlar cross section is placed on top of the latter. 53

2 I. Patkó Material Transport with Air Jet In the top part in comparison with the metal confser they sbstantially redce the air otflow, and therefore the redction of air jet velocity will be smaller in the direction of shoot in the confser drop wires. The slay (1) is oscillated by a specific drive mechanism, to make sre that dring the shoot, the swinging motion of confser drop wires does not possibly inflence the conditions of flow. This is becase in case the displacement of the confser drop wires is large dring the shoot, the air flow conditions are nfavorable from the aspect of introdcing the weft into the shed, and hence the warps may reach into the inner space of the confser. Figre 1.1 Arrangement and design of the confser drop wires applied in a type P machine a/ metal confser and its fixing, b/ plastic confser In a type P machine, the nozzle is secred to the machine frame, while the confser drop wires swing with the rib (Fig. 1.). Dring the shoot, the nearly stationary position of the slay is ensred by an eccentric articlated movement. In machine P 165 mentioned as an example, the introdction of the weft is carried ot in 0.08 sec, for times in a second. 54

3 Figre 1. Movement of the confser drop wires By the application of the confser drop wires, a rib wih of b=165 cm was achieved. This is where the name of loom type P 165 comes from. The confser drop wires cover 75 to 85% of the rib wih. The design of the loom nozzle is shown in Fig. 1.3, indicating the velocity patterns of the air leaving the nozzle, in addition to the weft. Figre 1.3 Velocity distribtions evolving in the nozzle In front of the nozzle there is a yarn box, the fnction of which is to store one shoot of yarn. The box is designed in a way that the yarn can be removed from it almost withot resistance. One type of these boxes is the pnematic yarn box (Fig. 1.4). 55

4 I. Patkó Material Transport with Air Jet Figre 1.4 Pnematic yarn box The pnematic yarn box has a simple design, and its stores the yarn of specified length in a tbe with a slow airflow, in the form of a loop. For introdcing the weft, depending on the strctre of the yarn, compressed air of 1.5 to 3.0 bar pressre is reqired. Relationship between Yarn and Air The eqilibrim of forces imposed on the yarn placed into the airflow: d I = 1 df 1 + F s x= 0 (.1) where I: plse of the yarn F 1 : force reslting from the liqid friction of yarn and air F s : force reslting from the friction between yarn and a different solid body. Hereinafter the friction force F s will be disregarded, becase the yarn comes ot of the storing box almost withot friction. Inside the nozzle, the yarn proceeds in the yarn gide, and it is only exposed to the carrying air after leaving the nozzle. The paper [33] deals with nozzles, in which the yarn proceeds along the nozzle over a distance of approx. 100 to 150mm, and the yarn is already exposed to the air within the nozzle. This paper only deals with the relationship between the flow otside the nozzle and the yarn. Frthermore, sing the consideration as a point of 56

5 departre that the magnitde of friction force does not depend on the size of the srfaces in friction and that the weft intake is a qasi-stationary process, in addition to concentrating strictly on the relationship between the yarn and the air, the friction force F s is disregarded. Therefore, the model to be set p will not be comprehensive, bt it will be appropriate for defining the basic characteristics of the yarn/air relationship evolving along the axis of intake. Hence, the mechanical eqilibrim describing the relationship between the weft of the pnematic loom P165 and the weft intake flow is as follows: d I = 1 df 1 x= 0 The relationship between the yarn and the air is shown in Fig..1. (.) Figre.1 Relationship between yarn and air : yarn end velocity at the examined point v: air velocity at the examined point m: mass of yarn srronded by air jet ρ l : air density A: length of yarn protrding from the nozzle at the start of intake C: yarn resistance factor D f : characteristic yarn diameter ρ f : yarn density Using the symbols above, eqation (.) is: 57

6 I. Patkó Material Transport with Air Jet d where (m) = x df f x= 0 (.3) ρ1 Ff = CDf π( v ) x (.4) with differential rates: ρl dff = CDf π( v ) dx (.5) the yarn mass is: D π m = f ρf x 4 (.7) sbstitting (.7) and (.6) into (.): d D x f π x ρl ρ f = CDf ( v ) dx 4 π (.8) x = 0 by introdcing the following symbols: D f π ρl ρ f = Z and CDfπ = B 4 the new form of eqation (.8) is: d Z x ( x) = B ( v ) dx x = 0 after performing the specified derivation Z dx + x d x = B x = 0 ( v ) dx (.9) (.10) (.11) dx = The eqation describing the movement of yarn end is: 58

7 Z x d + x = B x = 0 ( v ) dx (.1) After making the eqation dimensionless by the highest flow rate (V) prevailing in the entrance cross section of the drop wires, and by the length of protrding yarn end (A), the dimensionless kinetic eqation of the yarn end is: V + x A d V = tv d A AB Z x A x = 0 A v V The dimensionless qantities featring in (.1) are: x = ; = ξ ; V A v = v V ; V tv = T ; A x d A With the new symbols, the dimensionless kinetic eqation is: ξ ( v ) AB = K Z (.1) d + ξ = K dξ (.13) ξ= 0 3 Soltion of the Dimensionless Eqation Describing the Yarn Movement ξ ( v ) d + ξ = K dξ (3.1) ξ= 0 On the right-hand side of the kinetic eqation, the integrated sqare of relative velocity is featred. De to sqaring, the sqare of relative velocity is always positive, which cold lead to calclation trobles. In the corse of a compter soltion, in order to avoid sign problems reslting from evental atomatisms, it is advisable to modify (3.1) as follows: d + ξ = K ξ ξ= 0 ( v ) ( v ) dξ (3.) 59

8 I. Patkó Material Transport with Air Jet The dimensionless flow rate (v) featring in the eqation (3.) is only sbject to the place, i.e.: v = f () ξ while the yarn end velocity is sbject to the time, too, i.e.: ( T ) = f ;ξ If it is assmed that the yarn behaves as a rigid body dring its movement, when calclating the right-hand side integral in (3.) f ( ξ ) i.e. the yarn end velocity is not sbject to the place, therefore = f ( T) in the corse of solving (3.), the initial condition is: = 0 ; ξ = 1 that is d = K ξ v ξ = 0 dξ (3.3) By nmeric integration from (3.), at time T = 0, at place ξ= 1 (at the moment of d starting the yarn), the initial acceleration can be calclated. After a time ΔT has passed dring which the rate of acceleration is assmed to be constant the position of the yarn end 1 d ξ = 1+ ΔT (3.4) and the velocity of the yarn end d = ΔT (3.5) can be calclated in a rogh correlation. 60

9 d And then, from (3.), the rate of ξ acceleration applying to place calclated by the following formla: d K = ξ ξ ξ= 0 ( v ) ( v ) dξ ξ can be (3.5) the new position of the yarn end is: 1 d ξ = ξelõzõ + ΔT + elõzõ ΔT (3.7) and here the velocity of the yarn end is: d = elõzõ + ΔT (3.8) where the previos index is a distingishing symbol of the place of calclation directly preceding the actal calclation and of the velocity applying there. In the corse of a nmeric soltion, the eqation mst be solved by the rate ΔT = constant, and the change in the velocity of the yarn end sbject to the place is to be determined: = f 1 ( ξ) Next, by redcing the vale ΔT, a new velocity distribtion can be defined: = f ( ξ) The soltion is to be repeated always with a smaller ΔT ntil the sqare sm of the deviations of the two soltions following each other will be smaller than a prespecified rate (H). The vale of constants reqired for the soltion is: on the basis of mine measring [ ]. K ρ 1 CDfπ AB ρ1 AC = = A = (3.9) D ρ f π f Df the rate of K is, if: ρ l = 1.kg/m3 ρ ρ f = 800kg/m3 4 61

10 I. Patkó Material Transport with Air Jet D f = 0.00m A= 0.0m C= 0.3 V= 7m/s K = T = 0 H = The soltion is shown in diagram 3.1. The soltion reslted in the rates ΔT = 6.5 and H= We have made the soltion of (3.) only p to the place (ξ) as long as the following sitation prevailed v It is shown by diagram 7.1 that the yarn end velocity in the first qarter of the trajectory reaches its nearly constant vale, the rate with which it covers most of the trajectory. Frthermore, in the case of the vale ξ k = ξ = 55, the flow rate of the yarn end and of the transport air will be eqal, that is: v = In the range 1 < ξ = ξ k, the velocity of the yarn end is lower than that of the air, and therefore the trajectory of the yarn is definite and straight. We have solved eqation (3.) for this range only, that is: ξ ξ k In diagram 3.1 in range ξ > ξ k, the rate has been shown for gidance only. In this range the yarn does not behave as a rigid body, and therefore the starting eqation (.) is not sitable for describing the motion. By means of the model set p, the yarn end velocity can be calclated p to the rate of ξ = ξ k. In the interval 1 < ξ ξ k = 55, the air velocity is higher than the yarn end velocity. Therefore, in this section the trajectory of yarn end is straight. In the range ξ > ξ k = 0.5 the yarn end velocity is higher than that of the transport air. Therefore, in this section of the intake, the trajectory of yarn end cannot be defined as it only depends on spontaneos circmstances. It freqently happens in this range that the yarn end comes back i.e. it is looped. If the sction apertre at the end of the rib is nable to straighten this loop, a weaving defect occrs. If the weaving wih is shorter than the critical distance (ξ k ) no weaving defect is generated de to flow technology reasons. 6

11 Diagram 3.1 The weaving wih of the machine P165 is 165 cm, which corresponds to ξ= 8.5 in or dimensionless system. Conseqently, on looms P 165, with technological data associated with K = and with the measred by me - fnction v = f ( ξ ), the weaving defect reslting from the weft intake may not be (flly) avoided. The only way to exclde a weaving defect is raising the critical distance to above the weaving wih (fabric wih). There are two possibilities to do so: - changing the rate of parameter K, - f ( ξ ) v = changing the fnction relationship. 4 Impact of K on Weft Movement According to (3.3), the material and flow characteristics of parts contribting to the movement featre in the dimensionless K, like the density of air, the density, diameter and air resistance factor of the yarn, as well as the length (A) of the yarn protrding from the nozzle in a stationary position ( v = 0 ). If A = 0 then K = 0, i.e. the yarn may not be started. The relationship between the initial acceleration valid at the moment of starting the yarn and K according to (3.3) is the following: d = KQ (4.1) 63

12 I. Patkó Material Transport with Air Jet Where the reslt of integration specified by Q is constant. By increasing the K rate, the initial acceleration of the yarn end can be increased, bt at the same time the risk that the yarn end will break off increases. In the calclation shown in diagram 3.1, at ξ= 1, the rate of initial acceleration is, which represents with the assmed vales the initial acceleration d m = 566 s of the yarn end. 5 Impact of Air Velocity (V) on Weft Movement According to diagram 3.1, the critical distance (ξ k ) increases, if the air velocity (v) rises along the shooting line. Increasing the air velocity may be provided by: - increasing the spply pressre - by designing an appropriate nozzle shape. Increasing the spply pressre may only be imagined within a short interval, and the extent of increase heavily depends on the material characteristics of the weft. This is becase in case the spply pressre is higher than permissible, the yarn will be torn off the nozzle. According to the description, increasing the flow rate is only possible with an appropriate nozzle shape (Laval-tbe). Conclsions I investigated the critical distance (ξ k ) of weaving wih. I speified the impact of K wich was fonded by me and the dimensionless air velocity (v) on weft movement. References [1] Patkó I.: Lamellák közötti áramlás tlajdonságainak meghatározása, Kandidártsi disszertáció, Bdapest, 1994 [] KMF Gépészeti Tanszék: A P165 tipsú szövőgép vetülékbeviteli folyamatának fejlesztése, Ktatási jelentés, 198 [3] Szabó R.: Szövőgépek, Műszaki Könyvkiadó,

13 [4] Alther R.: Atomatische Optimierng des Schsseintrages beim, Lftdüsenweben, Kandidátsi disszertáció, ETH, Zürich, 1993 [5] Lünenschloss J., Wahhod F. J.: Das Eintragsverhalten verschidener Filamentgarme beim, Idstriellen Lftweben, Textil Praxis Int,

5. The Bernoulli Equation

5. The Bernoulli Equation 5. The Bernolli Eqation [This material relates predominantly to modles ELP034, ELP035] 5. Work and Energy 5. Bernolli s Eqation 5.3 An example of the se of Bernolli s eqation 5.4 Pressre head, velocity