Non-isothermal capillary flow of plastics related to their thermal and rheological properties

Non-isothermal capillary flow of plastics

related to their

Non-isothermal capillary flow of plastics

related to their

thermal and rheological properties

PROEFSCHRIFT

ter verkrijging van de graad van doctor

in de technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de rector magnificus,

Prof. Dr. J.M. Dirken,

in het openbaar te verdedigen

ten overstaan van

het college van de dekanen

op donderdag 28 februari 1985 te 14.00 uur

door

Jozua Laven

scheikundig doctorandus

geboren te 's-Gravenhage

Dit proefschrift is goedgekeurd door de promotor

Prof. Dr. H. Janeschitz-Kriegl

Van velen in mijn omgeving, zowel bij de Technische Hogeschool te Delft als bij het Unilever Research Laboratorium te Vlaardingen, heb ik hulp ge­ had of steun ondervonden bij het tot stand komen van dit proefschrift. Hen allen zeg ik hartelijk dank.

Een paar zaken mogen echter niet. onvermeld blijven. De heren H.C. Nieuwpoort, R. Verhoeven en G. de Vos hebben een groot deel van de vaak moeilijke meetopstellingen gebouwd. Een aanzienlijk deel van de thermofy-sische gegevens dank ik aan nauwkeuring meten door de heer A. Suurland. De heer J . Verbeek dank i k , behalve voor constructie-adviezen, voor de buitengewone inzet waarmee hij en de heer H.G. Langer de vele tekeningen hebben vervaardigd. I r . J . de Graaf heeft, als afstudeerder, wezenlijk b i j ­ gedragen aan het totstandkomen van het calorimetrisch capillair. I n g . R. van Donselaar ben ik erkentelijk voor de veelvuldige instrumenteel-fysische ondersteuning.

Van de hulp die ik bij Unilever Research heb ondervonden bij het op papier krijgen van mijn resultaten dank ik in het bijzonder de heer D.M. Bancroft en D r . S. de Jong voor hun vele taalkundige adviezen. Niet onver­ meld mag verder blijven de redactionele hulp van de heer E . T . J . Eikema en de verzorging van het door de vele formules lastige typewerk door de dames M.L. Smiet-Kuiken en C. Verwol.

Met genoegen denk ik terug aan de samenwerking en nuttige discussies over dit proefschrift die ik gehad heb met o.a. Dr. I r . A . K . van A k e n , I r . D.W. de Bruijne, Dr. I r . J . van Dam, Drs. B. Koeman, Dr. I r . F.H. Gortemaker en Dr. I r . K. te Nijenhuis. De onvoorwaardelijke steun van de promoter bij het afronden van de promotie heb ik bijzonder gewaardeerd.

Tenslotte dank ik naast mijn ouders, die door de geboden opvoeding en scholing deze promotie mede mogelijk maakten, mijn vrouw. Josien, jij weet als geen ander hoe moeilijk het soms was. Zonder jouw opoffering zou nie­ mand nu dit boekje lezen.

C O N T E N T S

1 . I N T R O D U C T I O N 1

2 . THE H E A T C O N D U C T I V I T Y OF POLYMER MELTS 3

2.1 I n t r o d u c t i o n 3 2 . 2 S e l e c t i o n of e x p e r i m e n t a l method f o r heat c o n d u c ­ t i v i t y m e a s u r e m e n t s 3 2 . 3 S u r v e y of l i t e r a t u r e on heat c o n d u c t i v i t y in p o l y m e r s 6 2 . 4 T h e o r y of t h e q u a s i - s t e a d y - s t a t e h o t - w i r e method 8 2 . 5 L i m i t a t i o n s a n d c o r r e c t i o n s of t h e selected method 9 2 . 5 . 1 C o r r e c t i o n f o r f i n i t e t h i c k n e s s of w i r e 9 2 . 5 . 2 C o r r e c t i o n f o r t h e f i n i t e t h i c k n e s s of t h e s u r r o u n d i n g medium 10 2 . 5 . 3 C o r r e c t i o n f o r t h e f i n i t e l e n g t h of t h e w i r e 11 2 . 5 . 4 T h e o n s e t of n a t u r a l c o n v e c t i o n 11 2 . 5 . 5 T h e s i g n i f i c a n c e of t r a n s f e r of r a d i a n t heat 14 2 . 6 I n s t r u m e n t a l 21 2.7 E x p e r i m e n t a l 26 2 . 8 R e s u l t s a n d D i s c u s s i o n 28 2 . 9 R e f e r e n c e s 32 A 2 . 1 A p p e n d i x : T h e N u s s e l t n u m b e r of a c y l i n d e r i n t h e c e n t r e of a s l i t 36

3. THE T H E R M A L D I F F U S I V I T Y OF POLYMER MELTS 39

3 . 1 I n t r o d u c t i o n 39 3.2 L i t e r a t u r e s u r v e y 39 3.3 T h e o r y of t h e t e m p e r a t u r e o s c i l l a t i o n m e t h o d 42 3 . 4 I n s t r u m e n t a l 45 3.5 E x p e r i m e n t a l 47 3.6 R e s u l t s a n d d i s c u s s i o n 48 3.7 R e f e r e n c e s 52

4. THE D E N S I T Y A N D S P E C I F I C H E A T OF POLYMER M E L T S ; C O N S I S T E N C Y OF T H E MEASURED T H E R M O P H Y S I C A L D A T A 55 4 . 1 I n t r o d u c t i o n 55 4 . 2 S p e c i f i c heat 55 4 . 2 . 1 M e t h o d o l o g y 55 4 . 2 . 2 R e s u l t s and d i s c u s s i o n 56 4 . 3 D e n s i t y 58 4 . 3 . 1 M e t h o d o l o g y 58 4 . 3 . 2 R e s u l t s a n d d i s c u s s i o n 59 4 . 4 I n t e r r e l a t i o n of t h e i n v e s t i g a t e d t h e r m o p h y s i c a l p r o p e r t i e s 61 4 . 5 R e f e r e n c e s 62 5. N O N - I S O T H E R M A L C A P I L L A R Y FLOW: A T H E O R E T I C A L A P P R O A C H 63 5 . 1 I n t r o d u c t i o n 63 5.2 D i s c u s s i o n of r e l e v a n t r e s u l t s f r o m l i t e r a t u r e 64 5 . 2 . 1 Basic r e s u l t s 64 5 . 2 . 2 T h e o r e t i c a l r e s u l t s 67 5 . 2 . 3 N u m e r i c a l r e s u l t s 70 5.3 V e l o c i t y a n d t e m p e r a t u r e d i s t r i b u t i o n s f o r c a p i l l a r y f l o w of a l i q u i d w i t h a t e m p e r a t u r e and p r e s s u r e -i n d e p e n d e n t p o w e r - l a w v -i s c o s -i t y 72 5 . 4 V e l o c i t y a n d t e m p e r a t u r e d i s t r i b u t i o n s f o r c a p i l l a r y f l o w of a l i q u i d w i t h a t e m p e r a t u r e a n d p r e s s u r e -d e p e n -d e n t p o w e r - l a w v i s c o s i t y 79 5.5 R e s u l t s : c a l c u l a t i o n s of c h a r a c t e r i s t i c s of t h e d e ­ v e l o p e d model 83 5.6 D i s c u s s i o n of t h e n u m e r i c a l r e s u l t s of § 5 . 5 91 5 . 6 . 1 Range of v a l i d i t y of t h e model 94 5 . 6 . 2 T h e c a l c u l a t e d t e m p e r a t u r e p r o f i l e s 95 5 . 6 . 3 P r e s s u r e g r a d i e n t a n d r a d i a l heat f l u x t h r o u g h t h e wall 96 5 . 6 . 4 T h e i n f l u e n c e of t h e p a r a m e t e r s 3 a n d $ on t h e t h r o u g h p u t 97 5 . 6 . 5 T e s t i n g t h e model 99 5 . 7 R e f e r e n c e s 100

A 5 . 1 A p p e n d i x : C a l c u l a t i o n of t h e f i r s t - o r d e r c o r r e c t i o n s on p r e s s u r e p r o f i l e and t h r o u g h p u t 102 A 5 . 2 A p p e n d i x : C a l c u l a t i o n of t h e t e m p e r a t u r e and t h e r a d i a l heat f l u x t h r o u g h t h e w a l l , c o r r e c t e d a p p r o x ­ i m a t e l y t o f i r s t o r d e r 106 6. T H E C A L O R I M E T R I C C A P I L L A R Y : AN E X P E R I M E N T A L S T U D Y OF VISCOUS H E A T I N G EFFECTS 109 6 . 1 I n t r o d u c t i o n 109 6 . 2 T h e t e m p e r a t u r e d e p e n d e n c e of t h e v i s c o s i t y 109 6 . 2 . 1 T i m e - t e m p e r a t u r e e q u i v a l e n c e p r i n c i p l e 109 6 . 2 . 2 E x p e r i m e n t a l d e t e r m i n a t i o n of t h e t e m p e r a t u r e d e p e n d e n c e of t h e z e r o - s h e a r v i s c o s i t y 111 6 . 2 . 3 L i t e r a t u r e data on t h e t e m p e r a t u r e d e p e n ­ dence of t h e z e r o - s h e a r v i s c o s i t y 114 6 . 2 . 4 T h e t e m p e r a t u r e d e p e n d e n c e of t h e v i s c o s i t y at c o n s t a n t shear r a t e 116 6.3 T h e p r e s s u r e d e p e n d e n c e of t h e v i s c o s i t y 118 6 . 3 . 1 T h e i n f l u e n c e of p s e u d o p l a s t i c i t y 118 6 . 3 . 2 E x p e r i m e n t a l methods 118 6 . 3 . 3 T h e v a l u e of t h e p r e s s u r e d e p e n d e n c e of t h e v i s c o s i t y 121 6.4 Flow b e h a v i o u r of p o l y m e r melts u n d e r c o n s t a n t s h e a r r a t e c o n d i t i o n 122 6 . 4 . 1 T h e c a p i l l a r y r h e o m e t e r 122 6 . 4 . 2 Flow c u r v e s : r e s u l t s and d i s c u s s i o n 124 6 . 4 . 3 T h e o c c u r r e n c e of melt f r a c t u r e 130 6.5 T h e c a l o r i m e t r i c measurement of e n e r g y d i s s i p a t i o n in p o l y m e r melt f l o w t h r o u g h a c a p i l l a r y 132 6 . 5 . 1 E a r l i e r a t t e m p t s t o measure v i s c o u s h e a t i n g b y t h e c a l o r i m e t r i c p r i n c i p l e 132 6 . 5 . 2 T h e c a l o r i m e t r i c c a p i l l a r y 133 6 . 5 . 3 C a l i b r a t i o n and t e s t p r o c e d u r e 136 6 . 5 . 4 R e s u l t s and d i s c u s s i o n 140 6.6 R e f e r e n c e s 149 S u m m a r y 151 S a m e n v a t t i n g 154 L i s t of s y m b o l s 157 S u b j e c t i n d e x 153

1 . I N T R O D U C T I O N

The flow of a liquid through a pipe is a classical subject in the science of transport phenomena. This science deals with the transportation of mass and heat. The migration of both of these can proceed by convection as well as by diffusion ( o r , as it is often called in case of heat, by conduction). Additionally, heat can also be transported by radiation.

In many cases, observations in the area of transport phenomena can be understood with the aid of relatively simple models which, nevertheless, incorporate the essential physical mechanisms. The classical result for the flow of liquids is the one obtained by Hagen and Poiseuille around 1840 for liquid flow through a pipe, which relates the pressure drop to the t h r o u g h ­ p u t . Exactly one century ago Graetz published his famous study about heat transfer in pipe flow from wall to liquid and vice versa. This technologically extremely important process has since been the subject of numerous studies in which Graetz' solution was extended, in order to account also for viscous dissipation and for non-Newtonian flow. A historical survey is given in Chapter 5. Most of the studies were performed after 1950. An important motivation for these studies was the development of highly viscous, non-Newtonian, thermoplastic materials. Since computer facilities came available, these heat transfer studies were also extended to industrially important complex geometries like those of extruders and injection moulding equipment. In such systems accurate knowledge of local temperatures is particularly important if thermosetting materials or thermally unstable products must be processed. However, almost no studies are available in which theoretical predictions are directly related to experimental observations. This leads to the situation that people often ignore specific effects in their calculations because they are not really aware of the importance of these effects in practical situations ( e . g . the compression heating effect, see Chapter 5 ) . This study is intended to contribute to filling this gap.

That, intention puts several requirements on the present study. Three of them are mentioned here explicitly. First, the experiments should cover industrially relevant processing conditions such as the use of highly viscous, non-Newtonian liquids, flowing at high shear rates under high pressure conditions, while the temperature field is not f u l l y developed. As a model we chose the flow through a circular cylinder. Second, the theoretical model to be tested should be powerful enough to cope with these rather extreme experimental conditions. In principle, two theoretical approaches are possible once the mathematical equations covering all relevant physical aspects are formulated. These approaches are obviously the construction of an analytical solution and the numerical evaluation of the problem. We selected the analyt­ ical route because it often gives more insight in how the various physical mechanisms interact and contribute to the experimentally observed behaviour. Nevertheless, we fully admit that a numerical approach also has advantages. For instance, it allows for a rather easy incorporation of the pressure and temperature dependences of physical parameters. In principle, a more or less accurate numerical solution can always be obtained whereas an analytical solution may appear to be unattainable. Finally, our intention also requires accurate knowledge of thermophysical properties like heat conductivity and thermal d i f f u s i v i t y (for instance, an experienced temperature increase is approximately proportional to the inverse of the heat, c o n d u c t i v i t y ) . How­ ever, literature values for these quantities show large scatter.

Therefore, this book deals with the following subjects. In Chapter 2 the measurement of heat conductivity in polymer melts is discussed. Results of measurement of this quantity are given as carried out with equipment constructed in our laboratory. Attention is also paid to the increase of heat conductivity due to Boltzmann radiation. In Chapter 3 the thermal d i f f u s i v i t y is dealt w i t h . In Chapter 4 the consistency of our measurements of heat conductivity and thermal d i f f u s i v i t y is checked. Chapter 5 deals with the theoretical analysis of polymer melt flow through a capillary. In Chapter 6, after the rheological characterization of the liquids studied, an equipment by which the viscous heating effect can be quantified is described. Results obtained are discussed in relation to theoretical predictions.

2. THE HEAT CONDUCTIVITY OF POLYMER MELTS

2.1 Introduction

A quantitative analysis of the heat transfer in liquid flow as discussed in Chapters 5 and 6 requires data on the heat conductivity of the l i q u i d . In as much as data are available for molten polymers, there are rather large differences between data from different sources. Therefore, we present here a study of the heat conductivity of molten plastics. F i r s t l y , we consider the various possible methods together with a review of published investiga­ tions. Thereafter, we describe the quasi-steady-state hot-wire and discuss possible sources of error which occur in non-ideal conditions. Finally, we give a description of the experimental set-up together with the results of our measurements for polystyrene and for low- and high-density poly­ ethylene.

2.2 Selection of experimental method for heat conductivity measurements

The measurement of heat conductivity requires the determination of an energy f l u x , which is the power that flows per cross-sectional area through a medium under the influence of a temperature gradient. The main problem is in the measurement of the energy f l u x , not in that of the temperature gradient. The energy flux can, in most cases, only be determined indirectly via the calculation of the electrical power supply. One has to take consider­ able precautions to prevent the electrical heat from leaking away along other routes than through the medium under investigation. This d i f f i c u l t y determines, to a large extent, the usefulness of a given geometry ( e . g . bar, plate, cylinder, sphere) as a measurement set-up.

From this point of view a system of concentric spheres, in which the inner one is heated, is the best one. Practical problems in connection with alignment, filling of the shell-shaped slit (especially in the case of solid media), energy supply to the inner sphere, seem to form the reasons why this geometry is rarely used.

After t h i s , the cylindrical geometry has the fewest problems with heat-leakage compensation. As a special case of a cylindrical geometry the "hot­ wire" should be mentioned. If this wire is thin enough, axial heat-leakage is neglibible. For this technique a wire of high electric resistance is placed into the medium and heated by an imposed electric c u r r e n t . The temperature rise within the wire or at a distance r of the line source is then measured.

In the case of a flat-plate geometry much heat-leakage can be avoided by placing layers of the medium on both sides of the "hot" plate and en­ caging this sandwich between "cold" plates on either side. In this way, the double-platen method is realized. The heat-leakage from the rim of the "hot" plate can, in principle, be eliminated with a compensation heater. However the practical realization of this construction is d i f f i c u l t .

Another aspect, which plays a role in the selection of a measuring method, is the shape stability of the medium. With a solid, the flat-plate geometry can be used as well as the bar geometry, in which the temperature gradient is along the bar axis. A concentric cylinder geometry is less suit­ able for solids because of the difference between thermal expansion coeffi­ cients of medium and measuring cell. This gives rise to heat contact prob­ lems. The influence of a gap developing between medium and cell wall can be minimized by the introduction of a liquid of comparable or higher heat conductivity (with plastics, silicone oil is often used). At v e r y low tempera­ t u r e s , at which these liquids also solidify, other solutions have to be found.

In those cases one may consider the use of helium gas which has a reason­ ably high thermal conductivity ( 1 ) . Nevertheless, also at low temperatures f l a t plates are preferred to concentric cylinders, provided the plates are flattened well: care has to be taken because plastic solids tend to warp u n ­ der large temperature changes ( 1 ) . Here, helium gas can be of use as well.

With liquids, contrary to solids, a concentric cylinder geometry is more suitable than the plate or bar geometry because the liquid has to be en­ closed completely. The walls needed in addition to the hot and cold walls, also conduct heat away from the sample. Especially with plastic melts and organic liquids, the wall conductance is higher than that, of the medium i t ­ self. With a suitable choice of the ratio of length to gap width the influence of additional walls can effectively be suppressed with concentric cylinder geometry.

Hence, using the various available geometries, several initial and boundary conditions are encountered. First of a l l , measurements can be performed in the steady-state, i.e. with temperature and temperature g r a ­ dient being constant with respect to time throughout the medium. With

quasi-steady-state meausurements, the temperature gradient is constant but the temperature itself rises continuously with time, either linearly (2) (flat plate) or logarithmically (3) (hot w i r e ) . A t h i r d possibility is given by the application of what has been called the regular thermal state, often applied by Russian investigators, and described, for example, by Kondratyev and Golubev ( 4 ) . In this method, the inner one of two concentric cylinders (both being initially at temperature T ) is heated up q u i c k l y . Afterwards, the stored heat flows to the outer cylinder, which is thermostated at T . During a part of this cooling process the temperature in each point of the medium obeys the regular-thermal-state condition:

1/CT-T ) • 3 ( T - T )/9t = constant.

Generally, a quasi-steady-state measurement (especially with the hot wire) requires less time than a steady-state measurement and thereby has less stringent requirements with respect to temperature control. However, steady-state measurements are more suitable in situations in which the specific heat of the medium changes, which is the case with phase t r a n s i ­ tions.

In a non-isothermal low-viscosity medium, heat transfer is not only effectuated by contacts between adjacent atoms ( " p u r e " or "molecular" condition) but also by natural convection and, with increasing temperatures, also by Boltzmann or black-body radiation. In such cases the hot-wire technique is recommended because

( i ) convection is avoided when measurements are taken rapidly after the start of this quasi-steady-state experiment, that is, before the convection is developed, and

( i i ) the influence of radiation is minimized when use is made of a v e r y t h i n wire, the temperature gradient near this wire being so large that the c o n t r i ­ bution of radiation from the wire surface is low due to its small surface area. The interplay of these two phenomena is discussed in more detail in Section 2.5.

We selected the hot-wire technique for our experimental programme be­ cause heat-losses at the wire ends and natural convection are practically absent whereas filling of the measuring equipment does not pose too large problems, notwithstanding the high viscosity of the (melted) samples. The transfer of radiant energy could not be suppressed because the heating wires had to have a minimum thickness for mechanical reasons in view of the highly viscous polymers. However, this is no real drawback, as is explained in Section 2 . 5 . 5 .

2.3 Survey of literature on heat conductivity in polymers

In the past, several investigations have been published concerning heat conduction in plastics, especially since the sixties. This literature will be reviewed here. Generally one has no possibility to judge whether the published data are reliable or not. Consequently, for the majority of p u b l i ­ cations mention is made only of the measuring method and the materials investigated. Exceptions are made only for a few cases. Discussions on reliability are concentrated in Section 2.8.

In 1958, Holzmüller and Münx used a steady-state double-platen appa­ ratus to measure the heat conduction coefficient A. of plastics such as PVC and PS between 25°C and 80°C ( 5 ) . In 1959, Cherkasova, applying the "regular thermal state", measured values of A for PS, LDPE, PU and paraffin wax, in a temperature range from 25°C to 90°C ( 6 ) . In 1960, Underwood and IVlcTaggart. used the quasi-steady-state hot-wire method to measure A. of LDPE and PS between 20°C and 200°C ( 7 ) . Their measuring technique was relatively primitive and resulted in a rather large scatter of the results (± 10%). In 1961, Kline used a steady-state concentric cylinder apparatus for Admeasurements on PS, LDPE, PTFE and on a few epoxy resins between 0°C and 100°C ( 8 ) . Shoulberg and Shetter determined A of PMMA between 20°C and 160°C with the aid of a double-platen apparatus ( 9 ) . They noticed large differences in published data on A of PMMA ( v a r y i n g as much as 150%, as based on the lowest value).

In Darmstadt (Federal Republic of Germany), in the sixties, Eiermann, Hellwege, Hennig, Knappe, and Lohe et al. performed a large number of apparently accurate A-measurements (on e . g . HOPE, LDPE, PC, PMMA, PS, PTFE, PP, PIB, PVC) at temperatures below and just above the softening temperature by using several versions of double-platen instruments (both steady-state and quasi-steady-state) (10-14), and above the softening temper­ ature (between 150°C and 240°C, at pressures between 20 and 300 bar) by using a steady-state concentric cylinder apparatus (15-17). These authors also developed theoretical heat conduction models in order to explain the influence of parameters such as c r y s t a l l i n i t y , strechting ratio, number and size of side chains, degree of polymerization, degree of cross linking and chain degradation (16-23). These models are based on the picture that heat flows from atom to atom and that the type of the bond between the atoms determines the thermal resistance between the atoms. The resistance is small if the spring constant related to such a bond is large. Consequently the resistance of a Van der Waals-bond is approximately ten times larger than

that of a bond along the main chain of the polymer molecule. A survey of the influence of these parameters in polymer systems has been given in a review by Knappe (24).

Bil and Avtokrativa (25) carried out measurements with a number of thermoplastics and thermohardeners using a quasi-steady-state platen method applying a linearly increasing temperature f i e l d , as described by Knappe ( 2 ) . With the aid of a steady-state double platen apparatus, in 1965, Hansen and Ho carried out a few measurements with PS and PE between 60°C and 160°C (26). A variant of this steady state technique has been developed in 1975 by Hands and Horsfall (33a) who determined A of HDPE between ambient and 250°C. Especially at the highest temperatures, the values given seem to be too low. The physical mechanisms involved and experimental techniques used in thermal conductivity and d i f f u s i v i t y of polymers and rubbers have been reviewed by Hands (33b). Greig and Sahota, in 1979, used a compara­ tive single platen method between - 50°C and + 90°C for the measurement of PE which had been stretched by extrusion (27). Furthermore, the steady--state concentric cylinder technique have been used in 1965 by Sheldon and Lane (28) (for several types of PE and plastisized PVC between 20°C and 100°C), by Fuller and Fricke (29) in 1971 (for PE, PS, nylon between 150°C and 230°C) and by Ramsey, Fricke and Caskey (30) in 1973 ( f o r well defined commercial PE samples, over the range 160°C-290°C). The results of the latter two investigations show much scatter.

More recently, Cocci and Picot (31) used a hot-wire around which the medium flows in axial direction in order to analyse the influence of molecular orientation on A of silicone o i l . As outlined before, with axial orientation one would expect A to decrease in radial direction; however, an increase was f o u n d . This may be related to so-called cluster rotation of liquid r e ­ ported by Mooney (32).

In 1977, Karl, Asmussen, Wolf and Ueberreiter measured k of PE in an indirect way, using thermal d i f f u s i v i t y a;d e n s i t y p and specific heat C of

the sample, at T = 178°C and at pressures between 320 bar and 1600 bar (34). In the rather unconventional method that they used, they suddenly compressed a polymer melt (already being at a high pressure) s l i g h t l y , resulting in a somewhat higher pressure. The temperature response in the sample, which was kept in a thermostatic vessel, was used to derive a value of a; from the maximum temperature rise after compression and from the measurement of p as a function of temperature at a fixed pressure they calculated C = - T • p"2 • O p / 9 T ) • ( d T / d P ) ,. , and thereby

P P duI3D. A = p • Cp • a.

2 . 4 T h e o r y of t h e q u a s i - s t e a d y - s t a t e h o t - w i r e method

T h e q u a s i - s t e a d y - s t a t e h o t - w i r e t e c h n i q u e i n v o l v e s t h e measurement of t h e t e m p e r a t u r e i n o r near a line s o u r c e of h e a t , e m b e d d e d i n t h e medium t o be i n v e s t i g a t e d . T h i s t e m p e r a t u r e is m o n i t o r e d f r o m t h e moment t h a t t h e line s o u r c e is h e a t e d e l e c t r i c a l l y at a c o n s t a n t r a t e . In p r i n c i p l e t h e l i n e a r line s o u r c e is c o n s i d e r e d to be i n f i n i t e l y t h i n and long a n d embedded in an i n f i n i t e m e d i u m . As a f i r s t s t e p i n t h e c o n s t r u c t i o n of t h e mathematical s o l u t i o n f o r t h e t e m p e r a t u r e f i e l d a r o u n d a c o n t i n u o u s s o u r c e of h e a t , t h e s o l u t i o n f o r an i n s t a n e o u s line s o u r c e is p r e s e n t e d . S u p p o s e t h i s line s o u r c e , a c t i n g o n l y at t = t , is p o s i t i o n e d at ( x , y , z ) , w i t h - w <z< » . A c c o r d i n g l y , t h e t e m p e r a ­ t u r e T i n t h e s u r r o u n d i n g medium is d e s c r i b e d b y t h e d i f f e r e n t i a l e q u a t i o n 1 3T 92T 32T _ . _ = + [ 2 . 4 - 1 ] a 3 t 9x 3y w i t h t h e i n i t i a l c o n d i t i o n T=0 at t = tQ a n d ( X - XQ)2 + ( y - yo)2 > £ w i t h e0+ 0 , [ 2 . 4 - 2 a ] a n d w i t h t h e b o u n d a r y c o n d i t i o n f o r t h e " s t r e n g t h " of t h e s o u r c e +00 +00 f f T d x d y = at m [ 2 . 4 - 2 b ] 1 ' pC ° - 0 0 - 0 0 P w h e r e Q is t h e heat g e n e r a t e d i n t h e w i r e p e r l e n g t h . E q . 2 . 4 - 1 can be s o l v e d v i a F o u r i e r t r a n s f o r m a t i o n on x a n d y . T h e n , a f t e r s u b s t i t u t i n g 2 2 2 r - x +y a n d t a k i n g x =y =0, t h e classical s o l u t i o n is o b t a i n e d f o r t h e t e m p e r a t u r e T a t time t and d i s t a n c e r f r o m t h e i n s t a n t a n e o u s l i n e s o u r c e a c t i n g at t = t w i t h s t r e n g t h Q / ( p - C ) : u p 2 Q / T ( r , t - t . ) = e 4 a<t _ t 0 ) . [ 2 . 4 - 3 ] ° 4 n \ ( t - tQ) T h e t e m p e r a t u r e f i e l d a r o u n d a c o n t i n u o u s line s o u r c e of heat w i t h s t r e n g t h q / ( p C ) , at time t a f t e r t h e s t a r t of h e a t i n g , f o l l o w s f r o m s u b s t i -t u -t i o n of Q b y ( q - d -t ) in E q . 2 . 4 - 3 and i n -t e g r a -t i o n o v e r -time -t b e -t w e e n 0 a n d t . I n t h i s w a y one o b t a i n s :

T = - — E i f - — ) , [2.4-4] 4nA \ 4at /

where the exponential integral is defined as:

Ei(-x) - ƒ

u

For small values of x , the exponential integral E i ( - x ) can be written 2 3

as E i ( - x ) = -y + l n ( x ) - x + \x + 0 ( x ) where Y = 0,5772 represents the 2 Euler number. In this way for large values of the Fourier number Fo = a t / r Eq. 2.4-4 becomes:

q ( /4at\ r2 >

T ( r ) = ] - ï + i n ( -7) + + . . . [2.4-5]

4nA ( \ r' 4at

By plotting T ( r ) against log(t), the slope gives 2,3 q/(4rtA.), provided Fo > 25. Whereas Underwood and McTaggart measured the temperature at a small distance from the line source in the polymer with the aid of a thermo­ couple ( 7 ) , in the present investigation use is made of a platinum wire for the " h o t - w i r e " . This wire could be used both as electrical heating source and as thermometer. In the past this method was applied to low viscosity liquids, for instance by Horrocks and McLaughlin (35).

2.5 Limitations and corrections of the selected method

2.5.1 Correction for f i n i t e thickness of wire

The result of Eq. 2.4-5 has been derived for a line source, i.e. for an infinitely thin w i r e , whereas, in practice the wire used has a finite thickness. Carslaw & Jaeger (36, 35) give an expression for the temperature

2 of a cylindrical source with radius R for 4at/R » 1:

q

4n\

where w = 2p C / ( p , C ,) is a thermophysical material constant, with sub-m sub-m w w

scripts m and w denoting the medium and the wire. Eq. 2.5-1 shows that the temperature of a cylindrical source with a finite diameter deviates from that of an ideal line source only shortly after the start of the experiment.

The influence of u> on the temperature of the wire (35) is shown in Fig. 2 . 5 - 1 . In our experiments such effect is negligible.

2.5.2 Correction for the finite thickness of the surrounding medium

The finite radial extension of the medium around the wire causes the temperature field to deviate from Eq. 2.4-5 at long times. For a medium enclosed in a concentric cylinder with radius b, Fischer has derived an equation for the temperature T ( r ) in the medium (37), which, in view of the previous Section, is only valid for not too small values of time:

T ( r ) =

4rtA

2

at.(x

/br

[ 2 . 5 - 2 ]

in which x are the roots of the zeroth order Bessel function J0( x ) , and

N (x ) is the corresponding modified Bessel function of the zeroth order. This equation is only valid if the required number of elements in the series development is limited such that the largest eigenvalue needed is x « b / R .

2

For values of at/b <0,12 the difference in temperature between the results of Eq. 2.4-5 and 2.5-2 is less than 0,01% but for at/b2=0,25 the difference

becomes already 0,4% (35). Such deviation in temperature for a heated wire in a bounded medium is illustrated in Fig. 2 . 5 - 1 .

t

—

Fig. 2.5-1 Reduced temperature versus reduced time in a transient hot-wire experiment, in which a wire of radius R is embedded in a medium bounded by an iso­ thermal, concentric cylinder surface of radius b. As parameters a series of values of the thermophysical material constant m and of the thickness ratio b/R are selected (after Horrocks and McLaughlin (35)).

2.5.3 Correction f o r the f i n i t e length of the wire

Horrocks and McLaughlin have derived formulas for a temperature cor­ rection due to the finite length of the line source (35). They discriminate between the abrupt discontinuity in the heating and the heat leakage over the metal wire. On the basis of their theorectical result and our e x p e r i ­ mental observations (Section 2.6) it can be concluded t h a t , with our equip­ ment, no correction for f i n i t e wire length is needed. Therefore we shall not discuss this problem in more detail.

2.5.4 The onset of natural convection

It is stated sometimes that natural convection around thin heating wires may lead to a considerable increase in heat t r a n s f e r , even with highly v i s ­ cous polymer melts. This problem will be considered here in three different ways. F i r s t l y , it is discussed on the basis of straightforward information, based on correlation functions for natural heat transfer as found in the l i t e r a t u r e . This will be shown to be irrelevant for our particular geometry, and thus no conclusion can be drawn. Thereafter, two alternative approaches (one from literature) will be presented, both indicating that natural convection is negligible in our equipment.

( i ) Method based on published correlation functions for natural heat t r a n s ­ f e r . From measurements of the heat transfer between horizontal concentric cylinders with diameters D. and D (with D >D.) an apparent heat con­ d u c t i v i t y A can be found which is larger than the real conductivity A because, apart from conduction, natural convection and radiation (not taken into account here) can also take place. The ratio £ = A /A > 1 in a

steady-a

state situation can formally be written (38, 39) as

I = f ( G r , P r , D0/ D . ) [2.5-3]

3 ? where Gr is the Grashoff number Gr = D.g e ( T . - T )/v and Pr is the Prandtl number Pr = \)/a = C r\/\. Here, g is the intensity of the g r a v i ­ tational f i e l d , e is the cubic thermal expansion coefficient of the medium, v and rj are the kinematic and dynamic viscosities and T. and T are the temperatures of the inner and outer cylinders. Neglecting the inertial forces with respect to the viscous forces, Eq. 2.5-3 can be simplified to ( 3 9 ) :

i = f ( R a , D0/ D . ) , [ 2 . 5 - 4 ]

w h e r e Ra = G r - P r is t h e R a y l e i g h n u m b e r .

In e s t i m a t i n g | f o r o u r h o r i z o n t a l - h o t - w i r e a p p a r a t u s t h e p r o b l e m a r i s e s t h a t t h e e x i s t i n g c o r r e l a t i o n s f o r % (see e . g . K u e h n a n d G o l d s t e i n , who g i v e a c o r r e l a t i o n in t h e f o r m | = f ( R a , P r , D / D . ) , i n w h i c h Pr has o n l y a s l i g h t i n f l u e n c e ( 4 0 ) ) almost always have been measured w i t h gases

o ( P r ~ 1 ) . T h e h i g h level o f t h e P r a n d t l n u m b e r f o r p o l y m e r s (~ 10 ) , h o w e v e r , makes t h e a p p l i c a t i o n of t h e s e c o r r e l a t i o n s q u e s t i o n a b l e . A n o t h e r o b j e c t i o n is t h a t t h e c o v e r e d r a n g e of Ra n u m b e r s ( f r o m 10 t o 10 ) is n o t w i d e e n o u g h ( 4 0 ) . In t h e p r e s e n t i n v e s t i g a t i o n Ra n u m b e r s d o w n t o 10 a r e of i n t e r e s t . T h e c o n d i t i o n s chosen b y K r a u s s o l t a r e more a p p r o p r i a t e t o t h e p o l y m e r melt s i t u a t i o n because he i n v e s t i g a t e d u p t o Pr = 3000 ( 3 9 ) . H o w e v e r , he o n l y i n v e s t i g a t e d f o r D / D . < 3 , w h e r e a s a v a l u e of 200 is r e p r e s e n t a t i v e f o r o u r e q u i p m e n t . I f one i g n o r e s t h e above m e n t i o n e d o b j e c t i o n s t h e way t o a c o n c l u s i o n is as f o l l o w s . A c c o r d i n g t o K u e h n a n d G o l d s t e i n t h e c o n v e c t i v e heat t r a n s f e r f o r D / D . = 200 is o n l y s l i g h t l y d i f f e r e n t f r o m t h a t f o r D / D . = » ( 4 0 ) . For t h i s s i t u a t i o n , h o w e v e r , more s u i t a b l e i n f o r m a t i o n is a v a i l a b l e . A r a j s and M c L a u g h l i n h a v e g i v e n a s u r v e y of all p u b l i s h e d r e s u l t s on n a t u r a l - c o n v e c t i o n heat t r a n s f e r f r o m f r e e h o r i z o n t a l w i r e s ( 4 1 ) . On t h e basis o f t h e i r s t u d y we have used t h e c o r r e l a t i o n p r o p o s e d b y M o r g a n ( 4 2 ) .

-15 -3 - 1 - 4 I n o u r e x p e r i m e n t s Gr ~ 10 ( b e c a u s e E ~ 10 K ; D. = 10 m ; +8 - 7 AT ~ 10 K ) a n d Pr ~ 10 , l e a d i n g to Ra ~ 10 . A c c o r d i n g t o M o r g a n , N u ( R a = 1 0 "7) = 0 , 2 5 w h e r e Nu = h D j A = q / { n \ ( T j - T ) } . T h i s c o n v e c t i o n a l N u s s e l t n u m b e r s h o u l d be compared w i t h t h e one c a l ­ c u l a t e d f o r p u r e , s t e a d y - s t a t e c o n d u c t i o n . A s an a p p r o x i m a t i o n one may use t h e e x p r e s s i o n f o r a c o n c e n t r i c s l i t ( 4 3 ) : Nu . = 2 / l n ( D / D - ) . H o w e v e r , i t is of c o u r s e b e t t e r t o c o n s i d e r t h e g e o m e t r y a c t u a l l y used i n t h e p r e s e n t i n v e s t i g a t i o n , w h e r e a h o r i z o n t a l w i r e is placed in t h e m i d d l e of a v e r t i c a l s l i t w i t h w i d t h L, f o r w h i c h we have c a l c u l a t e d (see A p p e n d i x A 2 . 1 ) : N uc o n d = 2/ £| n(L/D) + 0 , 2 4 7 } . S e t t i n g D / D . and D / L e q u a l t o 200 g i v e s t h e v a l u e s 0,377 a n d 0,364 f o r Nu ., w h i c h are o n l y s l i g h t l y h i g h e r t h a n t h e a b o v e m e n t i o n e d v a l u e of Nu f o r n a t u r a l c o n v e c t i o n a r o u n d a h o r i -c o n v 1 / 1 5 z o n t a l w i r e . From t h e r e l a t i o n s h i p N u . . . = ( N u ,+ Nu ) , as g i v e n r t o t a l cond c o n v 3 b y K u e h n a n d G o l d s t e i n ( 4 0 ) , i t f o l l o w s t h a t 4 = N I K . , / N u . ~ 1 , w h i c h w o u l d i m p l y t h a t n a t u r a l c o n v e c t i o n is n e g l i g i b l e . H o w e v e r , we t h i n k t h a t i n v i e w of all t h e a s s u m p t i o n s and a p p r o x i m a t i o n s made, no r e l i a b l e c o n c l u s i o n

( i i ) Method of Horrocks and McLaughlin. A better method for estimating the influence of natural convection is given by Horrocks and McLaughlin (35). These authors start from the conclusion of Kraussold (39) that stationary free convection in horizontal cylindrical slits is only significant if Grf i-Pr > 1000, where G rg is defined as Grg = 63ge (Tj-T0)/v> with

6 = (D - D . ) / 2 . They calculate how far in a certain time interval the heat emerging from the line source of their hot-wire experiment has penetrated into the surrounding medium, say over a distance r*. Assuming a quasi-steady-state temperature field within a cylinder with radius r*, they cal­ culate at what time after the start of the line source with strength q/pC

3 ? P the value of Gr*-Pr becomes 1000, where Gr* = 6*-e-g ( T - - T )/v with

2 o I o

6* = 2 ( r * - R . ) and R. = D./2. If at/R. » 1 , the c r i t e r i o n , as developed by

I I I o i J, 2

!-these authors, can be written as x - l n ( x ) = R* with x = (4at) 2/ ( Y R;) 2 and

•3 3 t

R* = 2*10 nA.r|a/(gqR-£p ). This c r i t e r i o n , when applied to our situation,

i rn j

-would allow a measurement time of ~ 10 s. As the time of measurement in the present investigation is less than 200 s, one may conclude t h a t , despite some questionable approximations, natural convection is negligible.

( i i i ) Alternative method. A more elegant method of answering the problem of whether natural convection occurs or not is the following. In a time interval t after the start of the hot-wire experiment (for not too small a time (Sec­ tion 2.5.1) and not too long a time (Section 2 . 5 . 2 ) ) an amount of heat q * t - L flows out of the wire with length L. As this heat will diffuse radially over a certain distance corresponding to an original volume V of the s u r ­ rounding medium, this volume V will have expanded to V +AV at time t . At time t=0 the mass of V was pV . Because the mean temperature rise in V „

0 0 O

at time t is qtL/(V pC ), the volume expansion is AV = e q t L / ( p C ). This represents a loss of mass s q t L / C in the volume V . From t h i s , the d r i v i n g force for natural convection as given by Archimedes' law can never be more than geqtL/C , irrespective of whether one considers V or a part of it having a local temperature which is higher that the mean one in V .

The volume V (or an a r b i t r a r y part of i t ) tends to rise according to Archimedes. This movement is opposed more or less by two effects, f i r s t l y , by the flow resistance around the wire, which for the moment is assumed to be fixed and, secondly, by the shear stress between V and the wall of the

t According to these formulas, the times after which natural convection occurs appears to be ~ 1/3 of the times measured by these authors, and not, as they state, comparable to these times.

measuring cell. If one supposes the second effect to be negligible, then the whole upward force is exerted upon the wire. The problem is then exactly equal to that of a horizontal cylinder which, due to g r a v i t y , moves vertical­ ly through a l i q u i d . White has calculated the velocity of a horizontal cylinder, moving vertically in the middle of a vertical slit ( 4 4 ) , and ex-pressed this in a f r i c t i o n coefficient 4 which is defined in F = L-A-pv /2 where F is the force responsible for the movement and A = 2RL is the projected area of the cylinder with radius R. He found that 4 = f ( R e , L / R , b / R ) , where 2b is the slit, width and Re = 2vRp/n is the Reynolds number. For small values of Re his result is

C. + C9 12,6 R R/L

4 = —- - with C1 = and C„ = 40 - (1 + ). ' Re ' log(0,85- b/R) d L b/R - 1

According to Jones and Knudsen this result is valid for L/R > 32 and b/R > 9,17 (45). They checked it for 0,01 < Re < 2. Applying White's result to the present problem, one obtains

2 geqt

c

n(c

c

) '

-3 -1 In the experiments described later in this Section e ~ 10 K , q ~ 1 W/m, C ~ 2-103 J / ( k g - K ) , n ~ 104 Pa-s, b/R = 2-102 and

3 P

L/R ~ 2*10 . This implies t h a t , during an experiment of 200 s, the wire will not move more than 8 um. Taking into account t h a t , in r e a l i t y , the wire is free to move over such a distance together with the surrounding r u b b e r - l i k e liquid because it is not s t i f f l y f i x e d , we must conclude t h a t , also according to this reasoning, natural convection is completely negligible.

2.5.5 The significance of transfer of radiant heat

Heat conductivity is defined as the heat flux under the influence of a gradient in temperature. The temperature is related to the average kinetic energy of the molecules. Temperature differences vanish by kinetic energy transfer between the molecules. According to the kinetic theory of heat conduction this transfer proceeds by particles which transfer an amount of energy (heat) over a given distance (46) (the free path length of such particles). Two types of particles are important in polymer melt systems: molecules and photons (emitted by the molecules). The wave length

depen-dent energy density of photons in the bulk of a medium is directly given by the Stefan-Boltzmann description of black-body radiation. In liquids, the mean free path length of the molecules (A ) is of atomic size, say

-10 -9 _ 10 -10 m, while that of the photons (A ) is much larger (Saito and Venant report, for hexane (presumably at room temperature), a value of ~ 0,5 mm ( 4 7 ) ) .

In the remainder of this section, f i r s t l y data for A will be p r o v i d e d , both for polyethylene and for polystyrene. Thereafter, we shall discuss under which conditions a n d , if so, in what way radiant heat transfer can be treated as part of the total thermal conductivity of a bulk material. Problems arising at boundaries are also discussed. Finally, we shall discuss these effects in relation to the hot-wire cell and make a conclusion about the importance of radiant heat transfer in our measurements.

^ v / * * >

1500 1000 WAVENUMBER/lcm-'l

Fig. 2.5-2 Infrared transmission spectra of Manolène 6050 (thickness 0,28 mm) and Hostyren N4000 (thickness 0,16 mm), at 298 K.

Free path length for IR radiation. For an estimate of A , data on the i n f r a ­ red transmission were needed. These data were kindly offered to us by Mr. F.A. ten Haaft (Koninklijke Shell Laboratorium, Amsterdam) and by Mr. M.A. van Schaick (Unilever Research Laboratorium, Vlaardingen). In Fig. 2.5-2, IR-spectra, taken with a Perkin-Elmer-580 IR-spectrometer, are shown for a high density polyethylene (HOPE) sample (Manolène 6050) and a polystyrene (PS) sample (Hostyren N4000). These are t r u e transmission spectra, i.e. reflection at the sample surface is suppressed* by placing the polymer sample between KBr windows while using paraffin oil as a contact liquid. Additonally, the sprectra were corrected for the absorption in the window material. In separate experiments, without KBr windows, we investi­ gated the influence of temperature on the absorption. It appeared that the absorption changed only a few percent when HOPE and PS were heated from 25°C to 120°C and 105°C respectively. For our purposes we thus assumed that the IR-spectra are independent of temperature.

The mean path length A in a non-scattering medium can be calculated with a method described by Rosseland (49, 50):

1 3E

ƒ

ƒ

where the frequency dependent absorption coefficient, a ■ is defined as a . = - log(transmission)/(sample thickness), where a is the mean absorption coefficient and E, is Planck's result for the emissive energy flux of a black body at frequency \), namely:

2hn2 \l3

E

^ c2 ( e x p ( h v / k T ) -1)

The calculated values of A for HDPE and PS at several temperatures are given in Table 2 . 5 - 1 . In accordance with its higher transmittance, the

* The reflected part of a light beam at r i g h t angles to an interface is given 2 2

by ( n - - n2) / ( n . + n?) where n.. and n? are the refractive indices at both

sides of the interface. In our experiments, n(PE) =1,51, N(PS) = 1,59, n ( K B r ) = 1,56, n ( p a r a f f i n ) = 1,43 (48).

— -4

Table 2.5-1 Values of A /(10 m) at several temperatures as calculated with Eq. 2 . 5 - 1 .

material 298 K 400 K 500 K

polystyrene 2,2 3,8 4,5 polyethylene 16 11 8,3

figures of A f o r PE are much larger than for PS. The dependence of A on temperature is due to the shape of the weighting function E , as plotted against \). The function E ■, which approaches zero at very high and very low frequencies, has a maximum value at \/ = 2,8215-kT/h thus shifting from \) = 2,35 • 101 3 s "1 (or UJ = 784 cm"1) at 400 K to \f = 2,94-101 3 s~1 (or

_ i

ui = 979 cm ) at 500 K, u> being the corresponding wavenumber. The almost transparent character of PE at low frequencies has more influence on A at 400 K than at 500 K, thus resulting in higher A" at lower temperatures. With PS the dependence of A on temperature, however, is d i f f e r e n t .

Radiant heat transfer in the bulk as intrinsic part of the thermal conduc­ t i v i t y . Whether, for the bulk of a medium, the radiation may be incorporated in the thermal c o n d u c t i v i t y , depends on the magnitude of A relative to the size of the system under consideration.

( i ) If A is relatively small, the so-called "optically-thick" approximation is allowed, provided ( d T / d x ) is not too large. The radiant and molecular thermal conductivities A and A are indiscernable and both contribute to

r m

the total conductivity: A. = A + A . This the case in many practical

situa-' t r m situa-' r

tions and also in most devices for the measurement of thermal c o n d u c t i v i t y . Under such circumstances, A is given by (46, 51)

Ar - (16/3) n2 Ap o, T3 [ 2 . 5 . 2 ]

-8 -2 -4

where a = 5,67 • 10 W-m -K is the Stefan-Boltzmann constant. Values for A of polystyrene and polyethylene, using Table 2 . 5 - 1 , are given in Table 2 . 5 - 2 . At high temperatures, especially, A may become a substantial part of A.. The validity of Eq. 2.5-2 is limited to low values of the tempera­ ture gradient. At too high levels of this g r a d i e n t , i.e. if the condition

Table 2.5-2 Estimates of the contributions \ /(W-m~ • « " ) by radiation,to the total heat conductivity of polystyrene and polyethylene at several temperatures.

material 298 K 400 K 500 K polystyrene 0,004 0,019 0,043 polyethylene 0,029 0,049 0,072 9T A dA r m 3x A dT m « 1

_[2.5.3a]

is not f u l f i l l e d , the thermal conductivity cannot be considered constant over a distance A as is presupposed in the derivation of Eq. 2.5-2. In that derivation it is also assumed t h a t , over a length A in the medium, the temperature difference AT = T . . - T - is small enough to j u s t i f y the s u b s t i t u ­ tion of O j C T4- ^ ) by 4ajT3-AT where f = ( T]+ T2) / 2 . This assumption implies

a second condition for the validity of Eq. 2.5-2:

A_ 3T

ax

« 1 [ 2 . 5 , 3 b ]

(ii) If A is relatively large the optically-thin approximation is allowed, so the radiation cannot, be incorporated in a total heat conductivity as was done in the case of the optically-thick situation. In this case the stationary radiant heat transfer between e . g . opaque parallel boundaries at a mutual distance Ax with temperatures T1 and T? and emissivity 0 i e S 1 can be

approximated with (52)

j>r = 0l£ . ( T1- T 2 ) ( 1 - A x / Ar)

[2.5-4]

In this steady-state case, the importance of radiation is equivalent to an aeT.Ax.(1-Ax/A ). This expression increase of the thermal conductivity by

clearly shows that the radiantnheat transfer cannot be incorporated in the

total conductivity because A would be dependent on the distance between the plates. Such dependence on Ax has previously been demonstrated by

experiments of Poltz and Jugel (46).

( i i i ) In the case that the size of the medium and Ar are comparable, the

situation is more complex. Several attempts have been made to analyse the interaction of radiation and conduction in this case, both in planar ( e . g . Refs. 5 1 , 52) and in cylindrical geometry ( e . g . Refs. 47, 53). Of these, Refs. 51 and 53 also consider the fact that the absorbtion coefficient (the inverse of the free path length of photons) is a function of the wavelength. One of the reasons for this careful isolation of Am from kt as presented in

the literature is the verification of theories on the kinetics of molecular motion (see for instance Ref. 54).

Radiant heat transfer at boundaries. Near the boundaries of a medium, the net radiant energy transfer is usually lower than that in the b u l k , this r e ­ duction being maximal in the immediate vicinity of a wall, provided that it is not transmitting for l i g h t . In case of such an opaque wall, the photons have only one half-space at their disposal. From the kinetic theory of d i f f u s i o n , one can derive t h a t , if the opaque wall is black, the radiant heat transfer is locally reduced by a factor of two. If that boundary is grey with emissivity e < 1 , the reduction is more pronounced, the factor being 2/e (46) ( f o r platinum, from which our hot-wires were made, values of e between 0,05 and 0,2 have been reported ( 4 7 ) ) . This reduction occurs near the boundaries in a layer of thickness ~ A . The consequence of this is t h a t , in practice, the "overall" heat conductivity goes down with decreasing thickness of the slab over which the thermal conductivity is measured (46, 51, 55). This effect is negligible, however, if the slab thickness is much larger than A .

Significance of radiation in the hot-wire cell. Because of the advantages previously mentioned, we selected the hot-wire method f o r our experiments. This technique can be v e r y useful in minimizing the relative importance of the radiation, by using a wire of only a few micrometer thickness: in the quasi-steady-state situation and at a given temperature of the wire the radiant heat flux at the wire surface is independent of the wire radius, whereas the temperature gradient at the surface, and thus the molecular-conductivity f l u x , increases with decreasing wire thickness. Hence, this method is well suited for measuring A .

a m

For reasons of robustness (see Section 2.7) we could not use wires thinner than 10 m. Consequently, the radiant energy flux cannot be neglected a p r i o r i . Moreover, due to the cylindrical geometry of the e x p e r i

-mental set-up, the information given before for the one-dimensional, linear heat flow cannot simply be applied to our case. A thorough numerical analysis of radiation in a hot-wire experiment, using a finite differences method, has been developed by Saito and Venart, assuming a frequency independent absorption coefficient (47). However, their analysis is r e ­ stricted to R/A < 0,01 where R is the wire radius. In our experiments this ratio is 0,12 for PS and 0,05 for PE. Therefore, their analysis is not appli­ cable to our particular geometry.

The way to treat the radiation in our experiments can be deduced on more practical grounds. From the discussion given before on A , and also from numerical results of Saito and Venart (47), it can be deduced t h a t , in the course of a hot-wire t e s t , the apparent thermal conductivity (as derived from the slope of the curved line in a temperature versus log (time) plot. according to Eq. 2.4-5) increases in a complicated way from a value close to k at small times to the value k + k at very large times (provided the

m m r ' a ^

boundary of the measuring cell is far enough away). In other words, at times at which the line is c u r v e d , the apparent conductivity has a value be­ tween k and k + k , while at large times, if the line is s t r a i g h t , the slope corresponds to A. + k . This behaviour parallels the results on k by Poltz and Jugel who measured k in a parallel plate geometry under steady-state conditions, as a function of the distance between the plates (46).

One would expect the graph of temperature versus log(time) to become straight only when the temperature field has been extended radially over at least a distance A ( o r , perhaps better, some miltiple of A ). Taking

2

a t / r = 0,5 as a reasonable criterion for the radial extension r of the tem­ perature field as a function of time, this leads for r = A to a minimum time of 1,0 s for PS and 3,7 s for PE ( f o r the thermal d i f f u s i v i t y we used data provided in Chapter 3 ) . In practice, we always found that the lines were curved at small times but became straight at longer times, the transition from one region to the other being at times of 2-3 s in the case of poly­ styrene and of 6-12 s in the case of polyethylene. Because ( i ) we always used the slope of the straight part of the line for the determination of A (the temperature usually was monitored until about 100 s after the start of a test) and ( i i ) it can be shown that the requirements formulated in Eqs. 2.5-3a and 2.5-3b are satisfied in our experiment, we feel confident in claiming that the data to be presented in Section 2.8 are "engineering" values (k.), i.e. they contain both k and k , where the non-negligible contribution k is defined in Eq. 2.5-2. This is fortunate because in Chapter 3 (which describes the way in which thermal d i f f u s i v i t y is

measured) and in Chapter 6 (where the values of A are used in calculations of the temperature field in capillary flow) we are also interested in "engineering" values of A, namely A. = A + A .

Because of the frequency dependence of a . some precaution is required with respect to the formulation of conditions for the validity of the "optical­ l y - t h i c k " approximation in terms of A . However, no theoretical solution would appear to be available which clarifies this point. In view of the fact that curvature in the line of temperature versus log(time) was absent at the large times at which we determined the slope of the line, we assume that the longer free path lengths for the more transparent regions of the

IR spectrum do not invalidate our "optically-thick" approximation.

2.6 Instrumental

The measuring cell for the determination of the heat conductivity by the hot wire technique, as shown in Fig. 2 . 6 - 1 , was built to the specifica­ tions of the present author in the laboratory workshop of the Chemistry Department of Delft University of Technology. It consists of a brass cylinder A of which the ends are closed by hardened glas windows B ( S i l i d u r , thickness 8 mm). Viton seals, o-rings ( C ) , are used between window and cylinder. To keep the windows in position, screws D press the aluminium rings F against the windows. To avoid excessive stresses (which could arise, for example, from thermal expansion), the aluminium rings F are also fitted with Viton o-rings ( G ) . In the wall of the brass cylinder A, gas-tight passages for the electrical wire ( H - , H j were fitted at four positions. They worked satisfactory although they did not have a long lifetime. Such a passage consists of an aluminium tube containing the electrical lead-in J fused in a glass bead. Between the passages H - , a platinum wire L was placed as the heating wire with a thickness 10 m. This was then soldered to the lead-in using special solder (melting point 350°C). This wire (manufactured by pulling a thicker wire through p i n -holes) has a very uniform thickness, which is essential for realizing a uniformly distributed energy dissipation over its whole length (~ 10 m). Platinum wires (M) of equal thickness were fixed at the passages H~. The wires M were electrically spot-welded to wire L, using copper electrodes of electrolytical p u r i t y (when copper electrodes of technical quality were used, often the platinum sticked to the electrodes d u r i n g welding. Thereafter, the wire L was pulled between the lead-ins H1 until it was s t r a i g h t . Thereafter

(c):

immKD

0

L

®

(a):

voltage measurement

voltage measurement

©

© ®

®^nEr*iL'

Fig. 2.6-1 The cell for measuring the thermal conductivity with the hot-wire tech­ nique. ( a ) : vertical cross-section, ( b ) : horizontal cross-section, ( c ) : passage for electrical wire (symbols: see t e x t ) .

it was heated to red-hot for a short period in order to eliminate mechanical-stress-induced microstructural orientations which influence the temperature dependence of its electrical resistance. Furthermore, the measuring cell was equipped with a de-aeration capillary P and a handle (not drawn) by which the cell could be placed in an oil-filled thermostatic bath.

The electrical scheme used is shown in Fig. 2.6-2. The measuring cell A was placed in a Wheatstone b r i d g e , together with two precision resistors B. and B„ of 5000 fi, one precision resistor C (R = 1 Q; I = 0,7 A,

1 2 i f m a x

tolerance 10 ppm) and a slide potentiometer D (0-8 0; I = 1 6 A) made of manganin resistance wire (which ensures absence of temperature influence on its resistance at room temperature).

The electrical arrangement described was chosen for the following reason. When an electrical current flows through the measuring cell the platinum wire temperature increases as does its resistance. With a constant voltage over the cell the electrical dissipation would decrease. To avoid this we choose the present Wheatstone bridge design, keeping the voltage over the bridge constant. Resistor D is adjusted so that the sum of the resist­ ances C and D equals that of A. When the resistance of A then changes s l i g h t l y , say by a relative increase e, the dissipated power in A is lowered

o

only by e / 4 . To realize a constant voltage over the b r i d g e , a DC supply VOLTAGE SENSOR

rat

_m

w

té

^ S v i j ^

- /

i

Gp was used which had been equipped with external voltage sensors which were connected to points I , and I „ , where the c u r r e n t is supplied to the b r i d g e . With these precautions the power dissipation, which was at a level of ~ 1 W/m, never changed more than 0,1% during an experiment.

The voltage control device w i l l , before a test is started, be completely out. of its control range due to the fact that at the points I1 and I? no cor­

rect voltage is measured. With the power supply G~ used (Solartron, type A5-1412), it took 5 s after the start of a measurement for the power supply to become completely stable. To reduce this complication, the voltage sensor leads were connected in such a way, via the magnetic switch P, that before the start of a test, the voltage was sensored before the mercury switch K. We used all three parallel connections of switch P to minimize possible contact resistances in the sensor c i r c u i t . With these precautions the stabili­ zation time of the c i r c u i t r y is about one second.

The measurement of the platinum wire resistance d u r i n g a test involves the measurement of both the c u r r e n t and the voltage over the measuring section of the heating wire. These two quantities were measured in two separate tests. The measurement mode ( c u r r e n t or voltage) is selected by switch H-. The voltage was then determined via the c u r r e n t - f r e e platinum wires M ( F i g . 2 . 6 - 1 ) , the current being measured by a voltage measurement. over resistor C ( F i g . 2 . 6 - 2 ) . An experiment was started by pressing switch Hp, which closed ( i ) the magnetic mercury switch K, so allowing the c u r r e n t to flow through the bridge and which started ( i i ) the integrator L to inte­ grate a constant voltage signal of DC-supply IVL. This integrated signal is amplified logarithmically in N and f e d , together with either the voltage or c u r r e n t signal of the cell, into an X-Y recorder giving a plot of the voltage or the c u r r e n t against the logarithm of the time.

With this measuring device the thermal conductivity was determined ac­ cording to

2,303 / I dR\ / 1 dR \1

K = q. \ . , [2.6-1]

4rt \R d T / \R d l o g t '

where R is the resistance of the wire over the length L. The measurement of R and of d R / d T was carried out in the following way together with a series of tests. After rewiring the cell and f i l l i n g it with polymer, tests were carried out at a series of set temperatures. Before each test the resistance at the set temperature was measured using a 0,1 V DC calibration source G-. The low voltage of G.., ensures negligible heating of the wire

d u r i n g calibration. The temperature of the thermostatic bath was measured with a calibrated Pt-thermometer in the oil bath. It appeared that R/T is extremely constant over the temperature range investigated (~ 380-500 K ) , which implies t h a t , with high accuracy, (1/R ) ( d R / d T ) = ( 1 / T ) .

In order to be sure of proper functioning of the equipment the f o l ­ lowing points were checked, using water-free glycerol as a test l i q u i d : ( i ) There is no influence of the length of the test section of the heating wire (3 cm v s . 6 cm) on the value obtained for \.

( i i ) The value of A. as obtained according to Eq. 2.6-1 is not influenced by the voltage sensing wires M ( F i g . 2.6-1) which, in principle, may disturb the cylindrical temperature field around the heating wire. Even welding 5 extra platinum wires (each with length 2 cm) on the test section of the heating wire (~ 6 cm length) did not influence the results.

( i i i ) The absolute level of the obtained values for A. has been checked by comparing our results in Table 2.6-1 with results of a comparative study of 24 different investigations of \ j iv c e r oi / as carried out by Touloukian, Liley

and Saxena (55). They calculated a correlation formula on the basis of the eight most reliable sets of data. The mean deviation in their correlation is 1,7% and the maximum 4,7%. In view of the agreement of our results with their findings we conclude that the absolute level of our A measurements is very satisfactory.

( i v ) In Table 2.6-1 is indicated the time at which a measured plot of voltage or current versus log(time) starts to show curvature due to the onset of convection. Also given is the prediction of this time according to the method of Horrocks and McLaughlin (35) as described in Section 2 . 5 . 4 . Experimental times are a factor 2-4 shorter than the predicted ones, this factor being

Table 2.6-1 Values of the thermal conductivity \ of glycerol (measured and from liter­ ature) and of the time t after which convection occurs in a hot-wire

conv

experiment (both experimental and predicted according to the method of Horrocks and McLaughlin).

_ ,„ exp. .. , . . lit. ,. , , s exp. , pred.,

T /

°

*

|yc/<

/

/

> ^glyc.^

) W

W

140 - 0,303 4,0 9 100 0,303 0,298 8 28

the highest at the highest viscosity (n , , = 10 Pa.s at 60°C). At the

M M _a 'glycerol

lowest viscosity (n , , = 10 Pa.s at 140°C) the factor is about 2. Our ' 'glycerol

results suggest that the dependence of the induction time on the viscosity is somewhat different from what they suppose. This makes the extrapolation of their method to the very high viscosities encountered with polymer melts doubtful. Besides, in view of our findings it is not clear why Horrocks and McLaughlin, as mentioned in Section 2 . 5 . 4 , found the experimental values for the time of the onset of convection ( e . g . 14 s with toluene) to be much larger than the ones predicted with their method ( e . g . 2,0 s with toluene; the value 7,0 s as given by them is apparently i n c o r r e c t ) .

2.7 Experimental

The plastics investigated were available in granular form. In order to avoid inclusions of a i r , the samples were pressed into plates before putting them into the test cell. For this purpose a moulding plate was used with a thickness (1 cm) half that of the inner distance between the glass windows of the test cell. In the moulding plate a cylindrical hole had been machined with a radius equal to that of the test cell. This plate was placed in a thermostatic platen-press and the hole was filled with the granular sample. After heating for 30 min at 190°C (polystyrene) or at 150°C (polyethylene) the sample was pressed under a load of ~ 2.10 N and thereafter cooled under load to ambient temperature in about 4 h. The disk-shaped sample was sawn into two equal half circles which fitted on both sides of the platinum wire in the test cell.

Although after production the polymer plates did not contain visible gas bubbles, bubbles appeared after remelting. In order to avoid the forma­ tion of such bubbles in the sample while in the cell, we extracted the gas (probably included air or low-molecular residues in the polymer) by f i r s t storing the plates for at least two days just, below the weakening temperature

o

(~ 90°C) at a low pressure (< 10 Pa). At room temperature, the two half disks were placed in the lower part of the measuring cell, at both sides of the platinum wire. Thereafter the cell was fixed in the thermostatic bath and heated up slowly while keeping the pressure low (< 10 Pa) until the two half circles fused. Then the low pressure was replaced by an atmos­ pheric nitrogen pressure, to ensure that no oxidation could take place. Otherwise bubbles would appear, as was typical for one of the PS-batches. In some cases the sample still contained a few air inclusions. In such a case, the temperature was set at 200°C for a while in order to reduce the

p o l y m e r v i s c o s i t y . T h i s h e a t i n g was c o n t i n u e d u n t i l t h e i n c l u s i o n s had r i s e n to t h e s u r f a c e . With t h e PPMA samples w h i c h w e r e t o be i n v e s t i g a t e d , we s t o p p e d o u r t r i a l s because we c o u l d n o t f i n d ways of a v o i d i n g e x c e s s i v e b o i l i n g in t h e p o l y m e r w h i l e i t was b e i n g r e m e l t e d . T h i s f o a m i n g led t o b r e a k a g e of t h e p l a t i n u m w i r e . T h e p o l y m e r s i n v e s t i g a t e d w e r e i n d u s t r i a l g r a d e : a HDPE (Manolène 6050, a p r o d u c t i d e n t i c a l t o M a r l e x 6 0 5 0 ) , a LDPE ( S t a m y l a n , s u p p l i e d b y D u t c h S t a t e M i n e s ) and t w o PS b a t c h e s h a v i n g t h e same e x t r u s i o n c h a r a c ­ t e r i s t i c s , b u t b e i n g p r o d u c e d i n d i f f e r e n t ways ( b o t h H o s t y r e n N4000, b y H o e c h s t ; a s u s p e n s i o n p o l y m e r i s a t e p r o d u c e d i n 1966 a n d a mass p o l y m e r i s a t e

Table 2.7-1 Characteristics of the polymers investigated.

Polymer Mn/10'J W £. Meltindex ^ ' d e c a l i n / ( m3/ k g ) Manolène 6050 (HDPE, identical to Marlex 6050) 100 5,0 0,153 DSM-polymer (LDPE) 22* 240* 630 2,0 0,133 0,112 (TCB) Manolène 6001 (HDPE) 0,1 Hostyren N4000V glasklar-01 (PS) (suspension poly­ merisation product) 87 240 495 Hostyren N4000-1 (PS) (mass polymeri­ sation product) 107 316 717 Diakon LG 156 (PMMA) 80 16,0 (10 kg)

* Figures obtained by gel permeation chromatography. Calibration made with HDPE; M is considerably smaller than the real molecular mass M^.