Theoretical Investigation of Hollow Fibres Structure: Relations between Fibre Inner Diameter, Wall Thickness, and Conventional Structural Indices

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1 ISSN MATERIALS SCIENCE (MEDŽIAGOTYRA). Vol. 9, No Theoretical Investigation o Hollow Fibres Structure: Relations between Fibre Inner Diameter, Wall Thickness, and Conventional Structural Indices D. Petrulis Department o Textile Technology, Kaunas University o Technology, Studentų 56, LT-3031 Kaunas, Lithuania Received 30 September 003; accepted 0 October 003 Some theoretical investigations o morphological structure o hollow ibres o the same linear density are developed in the current study. The structural indices o hollow ibres, namely ibre inner diameter and ibre wall thickness are systematically examined to show their relations with our conventional structural indices o ibres. Among these indices a hollow ibre outer diameter and a hollow ibre outer lateral area are used. Also such two indices as a ratio o hollow ibre outer lateral area to hollow ibre whole volume and a ratio o this area to hollow ibre mass are considered. In order to show the main structural peculiarities o the hollow ibres properly, the proposed relations cover a wide range o inner diameter and wall thickness up to the values or solid products. The comments on these relations or polyamide and polypropylene ibres with linear density o 3.9 dtex are presented. Keywords: hollow ibre, ibre inner diameter, ibre structure, ibre wall thickness. INTRODUCTION The innovations o ibres developed to meet the present day requirements are very signiicant. In recent years, considerable changes have taken place in the development o various morphological modiications o ibres. Among these textile products hollow ibres may be mentioned as a right example. Through the unique structural peculiarities the hollow ibres became recognised in rather dierent ields o textile use, also in products o special purposes. Hollow ibres enable us to reduce a heat transer, to clean various liquids and gases, to obtain higher moisture absorption, etc. Usually the hollow ibres are compared with solid ibres in some respects. On one side, it is worth to note that principles, which are applied or spinning o solid ibres, are also applied in manuacture o hollow ibres. On the other side, hollow ibres are manuactured by means o the spinnerets o special construction. These spinnerets have segmented oriice with open slits [1]. Such construction must be able to orm the hollow space within the ibre. Moreover, sometimes it is important to obtain an excellent uniorm geometrical size in the cross-section o the ibre. For this case the spinnerets o the tube-in-oriice type are used [1]. Thus, structure o the hollow ibres signiicantly diers rom a structure o solid ibres. For example, such speciic structural indices as ibre inner diameter and ibre wall thickness are used only in a case o hollow ibres. Moreover, an ater-treatment o the hollow ibres as well as o solid ibres is necessary in dierent orms such as drawing, heat treatment, dissolving, coating, etc [1 7]. According to [1], these processes o hollow ibre manuacturing are more extensive and more complicated than those o solid ibres. For instance, ater-treatment technologies are carried out sometimes in order to orm a porous structure in the wall o the hollow ibre [1, 7]. Coresponding author. Tel.: ; ax: address: Donatas.Petrulis@.ktu.lt (D. Petrulis) As a rule, the publications concerning hollow ibres include processing and device patents also the speciic technological and application aspects. Despite the importance o hollow ibres structural properties, the laws and relations between these parameters are scarcely described. Among the numerous reerences connected with hollow ibres only a small number o theoretical investigations could be ound. A noticeable theoretical work was done by Gupta [8]. The changes o various properties o hollow ibre and comparative analysis with cylindrical compact ibre o the same outer diameter were perormed. According to [8], the ratio between inner and outer diameters o hollow ibre signiicantly inluences the weight o this ibre. For instance, the eect o this ratio on mechanical rigidity is smaller as compared with the weight index. The authors o research project (see Web site regarding the integration o microsystems in hollow ibre reinorced composites presented a relation between capillarity and ibre geometry. According to this source, the capillarity is higher or hollow ibres with thinner wall. The other reerences are based on experimental investigations. The comments on ibre wall thickness are also presented in [1, 9 13]. Such index as ibre inner diameter was mentioned in reerences [1, 10 1, 1]. For instance, the polypropylene hollow ibres with an inner diameter o 00 µm are used in membrane or denitriication o contaminated drinking water [1]. The other undrawn samples o hollow ibres or the purpose o iltration have an inner diameter 165 µm approximately [1]. Ater spinning o the unoriented ibres a drawing process in several steps is necessary to obtain the desired microporous structure in the ibre wall [1]. Technology in [7] proposes polypropylene hollow ibres with inner diameter ranging rom 180 up to 50 µm. These products are used or the purpose o iltration. Diameter o the hollow core o ibres used in laserapparatus is 700 µm (see inormation in Web site The theoretical relations in 3

2 [9] or the control o hollow ibre geometry show technological possibility to adjust the ibre wall thickness independently o the ibre outer diameter. Thereore it is possible to manuacture the samples o the hollow ibres with rather dierent wall thickness and ibre inner diameter. For instance, two dierent ranges o inner diameter and wall thickness o hollow ibres produced on the basis o cellulose solution are indicated in [10]. One range or this diameter is µm; the wall thickness o these ibres is between 3 and 0 µm in [10]. The other range o inner diameter varies between 150 and 00 µm [10]. The wall thickness or these samples is 5 to 8 µm [10]. Similar data o inner diameter (00 µm) and wall thickness (5 µm) are presented in [1]. Thick polyethylene hollow ibre with inner diameter greater than 000 µm is proposed in [11]. Thickness o a porous wall o this ibre is more than 100 µm [11]. The above-presented survey shows a rather wide variety o data or speciic structural indices o hollow ibres. A question about relations between these indices and the conventional structural indices isn t suggested in a theoretical way as yet. The current study is ocused on the tendencies between some speciic structural indices o hollow ibres and their conventional structural indices. The latter indices are suitable or a wide variety o ibre structures. Here such products as the hollow ibres o dierent inner diameter and wall thickness, even the solid ibres are kept in mind. OBJECT OF THE INVESTIGATION AND ASSUMPTIONS The current investigation is made or polyamide () and polypropylene () ibres. The ollowing structural indices o these ibres used here are: T h - hollow ibre linear density; d o hollow ibre outer diameter; S h hollow ibre outer lateral area; s vh ratio o hollow ibre outer lateral area to hollow ibre whole volume; s mh ratio o hollow ibre outer lateral area to hollow ibre mass. Usually solid ibres may be described by means o analogous indices T, d, S, s v, and s m. Thereore the previous indices T h, d o, S h, s vh, and s mh because o their universality are named as conventional structural indices. In the current paper we investigate the relations between these indices and two earlier mentioned speciic indices, i.e., hollow ibre inner diameter d i and hollow ibre wall thickness b. The methodology o the research is based on such assumptions: the indices d i and b are variables; the indices T h and T remain at ixed level independently o variation o the indices d i and b, i.e., T h = T = const; the indices T h and T have equal values or and ibres. Index d i varied in such order: d i = 0, d i = d, d i = d and d i = 3d, where d solid ibre diameter. One series o computations is made or polyamide () hollow ibre with a linear density o 39 dtex. Kanebo (Japan) company (see Web site produces similar hollow ibre. Other series is presented or polypropylene () hollow ibre with the same linear density. The values o ibre density used in this research are the ollowing: ρ = 110 kg/m 3 or ibres, ρ = 910 kg/m 3 or ibres. Fibre length is equal to 1 m or all samples in this study. It is worth to point out that all the relations examined in this paper are suitable or round hollow ibres with a round concentric hollow space inside cross-section. RESULTS AND DISCUSSIONS Main equations Solid ibre linear density T and hollow ibre linear density T h are deined as ρ πd T =, (1) π ( d o d i ) ρ T h =. () Because o the second assumption, we have: πd ρ π ( d o d i ) ρ = (3) or o 1 ( d d i ) d = +. () From (), it is easy to express the ollowing relation between hollow ibre outer lateral area S h and inner diameter d i : S h i 1 = π ( d + d ) l, (5) where l is the ibre length. In a similar manner the equations or the indices s vh and s mh we re also proposed: s s vh mh =, (6) ( d + d 1 i ) 1 ( d + d i ) =. (7) d ρ The values o wall thickness b varied in accordance with the () modiied into the ollowing orm: b d =. (8) 1 [( d + d ) + d ] i i d Here or d i = 0 we have: b =. (9) In a case o d i = d : b d =. 1 ( + 1) (10)

3 I d i = d then d b =. (11) 1 (5 + ) Using d i = 3d, the wall thickness becomes d b =. (1) 1 (10 + 3) One additional equation, which was widely used in this research, is 00 o = b d i. (13) d + For instance, application o (13) and (8) enables us to compute the values o the hollow ibre outer diameter d o : d 1 i ) d o = + d ( d + d + d i i. (1) Similarly, the indices S h, s vh, and s mh or hollow ibres, wall thickness b o which luctuates at the inspected range, have been computed. hollow ibre mass. This index increases with the increase o S h (µm 10 6 ) d i (µm) Fig.. Relations between index d i and index S h. polyamide ibre, polypropylene ibre 65 Graphical representation o relations A graphical view o the relations between hollow ibre inner diameter d i and hollow ibre outer diameter d o is shown in Fig. 1. The index d o increases as the value o index d i grows. Moreover, the values o the index d o are higher or ibre compared with ibre because o dierence o the density. s vh (µm ) d i (µm) 0 d o (µm) Fig. 3. Relations between index d i and index s vh. polyamide ibre, polypropylene ibre d i (µm) Fig. 1. Relations between index d i and index d o. polyamide ibre, polypropylene ibre The tendencies or relations between hollow ibre inner diameter d i and hollow ibre outer lateral area S h presented in Fig. are similar to the situation described earlier or the index d o. A linear relation between the index d o and the index S h determines the similarity between these graphs. Dierent types o relations are obtained or the index s vh (Fig. 3) and index s mh (Fig. ). In the case o the increase o the d i, the ratio o hollow ibre outer lateral area to hollow ibre whole volume (index s vh ) has tendency to decrease. Another situation is determined or the index s mh, i.e., or the ratio o hollow ibre outer lateral area to s mh (µm g ) d i (µm) Fig.. Relations between index d i and index s mh. polyamide ibre, polypropylene ibre d i. These contrary tendencies may be explained in such a way. Firstly, the outer lateral area o hollow ibre increases slowly compared with the increase o its whole volume including volume o hollow space. Thereore, the highest index s vh (see Fig. 3) was computed or a solid ibre. Secondly, according to the assumption about ixed index 5

4 d o (µm) b (µm) Fig. 5. Relations between index b and index d o. polyamide ibre, polypropylene ibre Meanwhile Fig. 3 shows reverse tendency on this point: the index s vh or ibre is above this index or ibre. Another series o relations is computed or the variable o wall thickness b. These relations are shown or the indices d o, S h, s vh, and s mh in Fig. 5 8, respectively. s mh (µm g ) b (µm) S h (µm 10 6 ) b (µm) Fig. 6. Relations between index b and index S h. polyamide ibre, polypropylene ibre s vh (µm ) b (µm) Fig. 7. Relations between index b and index s vh. polyamide ibre, polypropylene ibre T h, we also have a ixed mass o hollow ibre. Thus, index s mh has the same tendency o increase (see Fig. ) as previously described indices d o and S h. Also, the index s mh is higher or ibres compared with ibres (Fig. ) as in the graphs or the indices d o and S h (Fig. 1 and ). Fig. 8. Relations between index b and index s mh. polyamide ibre, polypropylene ibre Contrary to previous tendencies shown or the index d i in Fig. 1,, and, an increase o the index b causes a decrease o the indices d o, S h, and s mh (Fig. 5, 6, and 8). The relations or the index s vh (see Fig. 7) also have a reverse character compared with the previous relations shown in Fig. 3. It is also necessary to point out that dierences between the structural indices or and ibres in Fig. 5 8 are identical to previously noted or these ibres in Fig. 1, respectively. For instance, since o dierent ibre density, the values o the d o, S h, and s mh (Fig. 5, 6, and 8) or ibres are higher the same indices or ibres in any case o ibre wall thickness. Due to the same reason, the index s vh (see Fig. 7) is higher or samples compared with ibres. The results o the current paper are applicable or predicting various properties o the hollow ibres and other materials produced on a basis o these ibres. CONCLUSIONS On the basis o made assumptions and proposed relations such generalisations or the eect o hollow ibre indices (inner diameter and wall thickness) up on conventional structural indices d o, S h, s vh, and s mh are done: the hollow ibres dier essentially rom the solid ibres with respect to the indices d o, S h, s vh, and s mh ; the hollow ibres with greater inner diameter exhibited greater indices d o, S h, and s mh ; the greater hollow ibre wall thickness, the less are the indices d o, S h, and s mh ; the highest values o index s vh are characteristic to hollow ibres with minimal inner diameter, also to products with maximal wall thickness; the indices d o, S h, and s mh or hollow ibres are higher the same indices or ibres, while an opposite tendency is valid or index s vh. 6

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