John Paull

Rudolf Koberwitz

George Steiner

Adams Kaufmann

Lili Kolisko

Marna Pease

The Pioneers of Biodynamics in Great Britain: From Anthroposophic Farming to Organic Agriculture (1924-1940)

Keywords: ), England, Scotland, Wales

Organic agriculture is the direct descendent of biodynamic agriculture; and biodynamic agriculture is the child of Dr Rudolf Steiner's Agriculture Course presented at Koberwitz (now Kobierzyce, Poland) in 1924. Rudolf Steiner founded the Experimental Circle of Anthroposophic Farmers and Gardeners towards the end of that course. The task of the Experimental Circle was to test Steiner's 'hints' for a new and sustainable agriculture, to find out what works, to publish the results, and to tell the world. Ehrenfried Pfeiffer published his book Bio-Dynamic Farming and Gardening in 1938, thereby fulfilling Steiner's directive. Two years later, from Steiner's characterisation of 'the farm as an organism', the British biodynamic farmer Lord Northbourne coined the term 'organic farming' and published his manifesto of organic agriculture, Look to the Land (1940). In the gestational period of biodynamics, 1924-1938, 43 individual Britons joined the Experimental Circle. Each received a copy of the Agriculture Course. Copies were numbered individually and inscribed with the name of the recipient. These 43 members were the pioneers of biodynamics and organics in Britain, and finally their names and locations are revealed. Of the 43 individuals, 11 received their Agricultural Course in German, 27 in English, and five received copies in both German and English; one couple shared a single copy. Of the 50 Agriculture Course copies supplied to Experimental Circle members in Britain, 17 copies were in German, while 33 were in English. The membership of the Experimental Circle comprised both men (n=21) and women (n=22). Members were domiciled in England (n=39), Scotland (n=3) and Wales (n=1) (Dr Lili Kolisko received her Agriculture Course in Stuttgart, Germany, migrated to England in 1936, and is tallied here as 'England'). The revelation of the earliest pioneers of biodynamics, and thus organics, in Britain provides 43 starting points for further research.

Introduction

In June 1924 the Austrian New Age philosopher, Dr Rudolf Steiner (1961Steiner ( -1925)), called for the development of an agriculture differentiated from chemical agriculture, a natural agriculture not dependent on synthetic inputs. At the behest of Count Carl von Keyserlingk (1869-1928) (Figure 1), Rudolf Steiner travelled to the obscure village of Koberwitz near Breslau, Germany (now Kobierzyce, near Wrocław, Poland), to deliver eight lectures, the Agriculture Course (7-16 June) [START_REF] Steiner | Agriculture Course. "Printed for private circulation only[END_REF][START_REF] Paull | Attending the first organic agriculture course: Rudolf Steiner's Agriculture Course at Koberwitz[END_REF]. Agriculture Course at his agricultural estate at Koberwitz [START_REF] Mier | Man and the Earth: A study of the timing and significance of Rudolf Steiner's lectures at Koberwitz, Whitsuntide[END_REF].

On the journey back to his home in Switzerland, Rudolf Steiner declared: "Now we have accomplished also this important work" [in 4, p. 552]. "Seldom have I seen him so joyfully moved after the completion of a task as in this moment after the agricultural conference" (p. 552).

Rudolf Steiner's health was fading by this time. He pushed himself hard, with a punishing schedule of lectures, travel, and writing. On this trip, he lectured on agriculture at Koberwitz during the days, and he lectured on Anthroposophy at Breslau each night. Three months later, his strength gave out. He withdrew from public life on 28 September 1924, never to return, and he passed away on 30 March 1925 [START_REF] Collison | Rudolf Steiner. X a.m. 30th[END_REF]. Despite his intentions otherwise, the Agriculture Course was never repeated or elaborated. What was said at Koberwitz was to be Rudolf Steiner's last word on the subject.

At the Koberwitz course, there were 111 attendees, including estate managers, farmers, gardeners, and a variety of other professions [START_REF] Paull | Attending the first organic agriculture course: Rudolf Steiner's Agriculture Course at Koberwitz[END_REF]. Several attendees took notes in shorthand.

Steiner's drawings to illustrate the course were on blackcard and were preserved [START_REF] Steiner | Wandtafelzeichnuchen zum Vortragswerk[END_REF]. Towards the end of the Agriculture Course, Rudolf Steiner founded a group, the Experimental Circle of Anthroposophic Farmers and Gardeners. Steiner's vision was of an environmentally sound agriculture. His Agriculture Course was an offering of "hints" as to how that might be achieved. To that end the Experimental Circle was charged with putting the "hints" to the test, finding out what works, and getting the results into a publishable form that could be broadcast to the world. Steiner later wrote that: "the lectures should be considered first of all as hints, which for the present should not be spoken of outside this circle, but looked Northbourne took the core concept of Rudolf Steiner's outlook, that the farm is an organism and, from that characterisation, he coined the term 'organic farming' [START_REF] Mirbt | The Farm as a Living Organism. Huby Experiments. Anthropsophical Methods on Animals and Produce[END_REF][START_REF] Paull | The farm as organism: the foundational idea of organic agriculture[END_REF].

Northbourne was an excellent wordsmith, in contrast to Ehrenfried Pfeiffer who was a plodding writer (and was perhaps also not well served in translation). Northbourne published a compelling manifesto of organic agriculture, Look to the Land [START_REF] Northbourne | Look to the Land[END_REF]. He pitted 'organic farming versus chemical farming'. He foresaw that such a contest might be waged "for many decades, perhaps for centuries" [13, p. 115

Methods

Each [START_REF] Pfeiffer | Bio-Dynamic Farming and Gardening: Soil Fertility Renewal and Preservation[END_REF].

Two separate sets of documents provided the data for the present paper: (a) The Agreements, the originals of the confidentiality agreements that were signed and submitted by each incoming member of the Experimental Circle, and (b)

The Directory, the ongoing tally of issued numbered Agriculture Courses, a loose-leaf compilation, 'Verzeichnis der Besitzer des landwirtschaftlichen Kursus der von Dornach ausgegeben wurde' (Directory of the owners of the agricultural course issued by Dornach). These documents are held in the Archives of the Goetheanum.

The Directory is a progressive register, compiled by many hands over multiple decades, with some entries typewritten and some handwritten. It begins on 28 May 1926, with a record of the destinations of the first 88 copies of the Agriculture Course, beginning with the record that Dr Steiner was allocated copies Nos. 1-3. Neither of the document sets (the Agreements & the Directory) is wholly complete. Some of the names identified in the present paper appear in both the Agreements and in the Directory, others in one or the other.

The Directory appears to be missing the destinations of the German language books, No. 89 -No. 149 entirely (perhaps a single missing page). Since some of the English editions were issued from England and some from Dornach, both the Agreements and the Directory may be more prone to omissions of British data than German data. The details of the British biodynamics pioneers were transcribed, and reconciled where possible between the Agreements and the Directory, and are now revealed (Table 1).

In general, titles declared by members (e.g. Mr, Mrs, Miss) in their Aggrement forms have been retained in Table 1 (most did not offer a title). In the period under examination, all copies of the Agriculture Course were numbered. Copies were issued only in German or English. The German copies are printed books and they bear the number and the name of the member. The English copies are typed (not printed) and bear the number accompanied by an 'E' as a prefix or a suffix, and the name of the member. For the purposes of Table The Agriculture Course was issued in German before the English edition, so members who were familiar with German had the opportunity of being an early adopter and opting for the German edition with the option of an English language edition later if required. Members are listed in bold in Table 1 where they make a re-appearance in the list. Note that there are 51 entries in Lili Kolisko (DE17 in Marna Pease made it her life's work to promote biodynamics. She was the secretary and the driving force of the Anthroposophical Agriculture Foundation (AAF) (Figure 3). The typed English issues of the Agriculture Course distributed around the world in the gestational period of biodynamics were typed up by Marna Pease [START_REF] Mier | Marna Pease[END_REF]. The whereabouts of the records of the AAF, and if they have survived, are unknown to the present author.

Results and Discussion

It is not known if the British Experimental Circle members were diligent in, firstly, experimenting to test Rudolf Steiner's 'hints', and, secondly, in reporting the results back to the Goetheanum. The cache of records that might have existed have not been located. In the event that the members were doubly diligent in this manner, then the results would, at some time, have passed through to Ehrenfried Pfeiffer. It is unresolved by the present author if such a cache of surmised documents has survived. There are no records surviving in the Natural Sciences Section of the Goetheanum. Such records were not located in the Archives of the Goetheanum. The records appear not to be in the USA. where some of Pfeiffer's papers have survived. There was a certain amount of bad blood between Pfeiffer and the Natural Sciences Section and there may have been no effort to protect his legacy and records. In any event, when Pfeiffer visited Dornach after the war, with the view to relocating, he was not made welcome, and he returned to the USA disappointed [START_REF] Selawry | Ehrenfried Pfeiffer: Pioneer of Spiritual Research and Practice[END_REF]. Pfeiffer spent the rest of his life in the USA as an advocate for Anthroposophy and biodynamics (BD). He advised Jerome Rodale (1898-1971), the US organics pioneering advocate and publisher, and was a mentor for BD farmers and he was an informant for Rachel Carson (1907[START_REF] Carson | Silent Spring, Readers Union 1964 members[END_REF] when she was writing Silent Spring [START_REF] Carson | Silent Spring, Readers Union 1964 members[END_REF][START_REF] Paull | The Rachel Carson letters and the making of Silent Spring[END_REF].

Conclusions

The revelation of the earliest pioneers of biodynamics and organics in Britain provides 43 starting points for further research. The numbers, names, dates, and places are keys to unlock the stories of the early history of organics in Britain, and the world, that has been heretofore largely overlooked.

Figure 1 .

 1 Figure 1. Count Carl von Keyserlingk (1869-1928) hosted Rudolf Steiner's Agriculture Course at his agricultural estate at Koberwitz [3].

  With the exception of Dr LiliKolisko (1889Kolisko ( -1976)), the biodynamic experiments of the members of the British contingent of the Experimental Circle are generally or wholly unknown. No documentation regarding their experiments and results have survived in the Natural Science Section at the Goetheanum, and none were located in the Archives of the Goetheanum.

Figure 2 .

 2 Figure 2. Annual number of members in Britain joining the Experimental Circle during the gestation period of biodynamics (1924-1938) and organic agriculture (1924-1940) (Each member is accounted for once only, on the year of first joining) (∑=43).

Figure 3 .

 3 Figure 3. Marna Pease (1867-1947) was the secretary of the Anthroposophical Agriculture Foundation (AAF) and she made it her life's work to promote biodynamics.Ehrenfried Pfeiffer migrated to the USA about the time of the publication of his Bio-Dynamic Farming and Gardening.There was a certain amount of bad blood between Pfeiffer and the Natural Sciences Section and there may have been no effort to protect his legacy and records. In any event, when Pfeiffer visited Dornach after the war, with the view to relocating, he was not made welcome, and he returned to the USA disappointed[START_REF] Selawry | Ehrenfried Pfeiffer: Pioneer of Spiritual Research and Practice[END_REF]. Pfeiffer spent the rest of his life in the USA as an advocate for Anthroposophy and biodynamics (BD). He advised Jerome Rodale (1898-1971), the US organics pioneering advocate and publisher, and was a mentor for BD farmers and he was an informant for RachelCarson (1907[START_REF] Carson | Silent Spring, Readers Union 1964 members[END_REF] when she was writing Silent Spring[START_REF] Carson | Silent Spring, Readers Union 1964 members[END_REF][START_REF] Paull | The Rachel Carson letters and the making of Silent Spring[END_REF].

  The shorthand notes were used as the basis of the publication of the Agriculture Course, firstly in German (as a published book), and, secondly, translated into English by George Kaufmann and issued in original typescriptThe publication of the Agriculture Course enabled those who had not attended at Koberwitz to join the Experimental Circle. The translation into English enabled an Anglo-membership for the Experimental Circle. Members of the Experimental Circle were enjoined to keep the contents of the Agriculture Course and their experiments confidential and confined to the Experimental Circle.

upon as the foundation for experiments and thus gradually brought into a form suitable for publication"

[7, p. 10]

. Sixty attendees joined the Experimental Circle as the foundational research group

[START_REF] Paull | Attending the first organic agriculture course: Rudolf Steiner's Agriculture Course at Koberwitz[END_REF]

.

At Dornach, Switzerland, Ehrenfried Pfeiffer (1899-1961) coordinated the work of the Experimental Circle, at the Natural Science Section of the Goetheanum. In 1938, he published his book Bio-Dynamic Farming and Gardening. Pfeiffer wrote in German, it was his first language. His book was translated and appeared in English, Dutch, French, and Italian, as well as the original German

[START_REF] Paull | Biodynamic Agriculture: The journey from Koberwitz to the World, 1924-1938[END_REF]

. Ehrenfried Pfeiffer oversaw the evolution of Steiner's 'Anthroposophic farming' through to 'biodynamic farming'. An English confederate took the next step. Lord Northbourne (1896-1982), a biodynamic farmer from Kent, an agriculture graduate of Oxford University, and the Provost of Wye Agricultural College, travelled to Switzerland to meet Ehrenfried Peiffer. Northbourne prevailed on Pfeiffer to present a Biodynamics Summer School at his estate in Kent, Betteshanger. The Bettteshanger Conference was presented in 1939 (1-9 July) by Ehrenfried Pfeiffer and several other biodynamics advocates from the Continent [9, 10].

Shortly after the close of the Betteshanger Conference, Europe was again at war. The opportunity for proselytising in Britain for Germanic ideas was swept away, as a new maelstrom of death and catastrophe engulfed Europe and the world.

Table 1 .

 1 Experimental Circle beyond the group who attended his course at Koberwitz, and then beyond Europe. Those attending at the Koberwitz course (n=111) were from Germany (n=61), Poland (n=30), Austria (n=9), British members who joined the Experimental Circle of Anthroposophic Farmers and Gardeners, in chronological order of joining(1926)(1927)(1928)(1929)(1930)(1931)(1932)(1933)(1934)(1935)(1936)(1937) (EN=English language 'Agricultural Course', DE= German language; A=Agreement, D=Directory; names in bold are repeats).

	listing to

British members of Rudolf Steiner's Experimental Circle of Anthroposophic Farmers and Gardeners appear in

Table

1

. British-domiciled members joined from 1927. Dr Lili Kolisko (1889-1976) appears in Table 1 as the first entry because, although she received her Agriculture Course in Stuttgart, Germany, she subsequently migrated to England. Lili Kolisko received one of the earliest printed versions (#17) of the Agriculture Course. She is the only person in the The Agriculture Course was originally delivered by Rudolf Steiner as a series of eight lectures with some question and answer sessions, at Koberwitz, near Breslau. The Course was transcribed in shorthand by several course attendees, including by Dr Lili Kolisko

[START_REF] Steiner | The Agricultural Course[END_REF]

. These transcripts enabled the subsequent spread of the In total, 43 British members joined the Experimental Circle during the gestation period of biodynamics

(1924)(1925)(1926)(1927)(1928)(1929)(1930)(1931)(1932)(1933)(1934)(1935)(1936)(1937)(1938)

. Between them, they received 50 copies of the Agriculture Course. Members received copies in German (n=17) or English (n=33). Eleven members received a copy in German only, and 27 members received a copy in English only (

Table 1 )

 1 . Five members received copies in both German and English (Margaret Cross, Edward Maurice Wood, Mr H J Heywood-Smith, William Wilkinson, & Harry Collison). Of these, Margaret Cross received two German editions (DE171 & DE315) and two English editions (EN64 & EN65), and Harry Collison received two German (DE228 & DE630) and one English (EN60). One couple, Mr & Mrs Heywood-Smith, shared a single copy of the German edition (DE327) (Table1).

The membership of the Experimental Circle comprised both men (n=21) and women (n=22). Members were domiciled in England (n=39), Scotland (n=3) and Wales (n=1) (Dr Lili Kolisko received her Agriculture Course in Stuttgart, Germany, migrated to England in 1936, and is tallied here as 'England'). There were no members from Ireland, neither Northern Ireland nor the Republic.

Table 1 ,

 1 one book was shared between two members, so there are 50 Agriculture Course books to account for. Note that of the 51 entries, eight names are repeated, so there are 43 individuals to account for. Enlistment of new Experimental Circle members peaked in 1931 (Figure2).No further British members joined the Experimental Circle in 1938, 1939 or 1940 (Figure2). This result contrasts with USA, Australia and New Zealand, each of which attracted new Experimental Circle members during those years. There were ongoing power struggles at Anthroposophy headquarters in Dornach, Switzerland in the 1930s. In April 1935 the Anthroposophy Society of Great Britain was expelled from the Swiss-based General Anthroposophical Society. Many prominent Anthroposophists were targets of the purge, not only British Anthroposophists. The Anthropops in USA, Australia, and New Zealand appear to have entirely escaped the wrath of the Dornach purgers.

Table 1

 1 GeorgeKaufmann (1894Kaufmann ( -1963) (DE354 in Table1) was responsible for the translation of the Agriculture Course into English. He was not at Koberwitz, but he was very experienced at extemporaneously rendering Rudolf Steiner's lectures into English. For example Kaufmann rendered into English, Steiner's lectures in Britain at Oxford in 1922[START_REF] Paull | Rudolf Steiner and the Oxford Conference: The birth of Waldorf education in Britain[END_REF] and at Torquay in 1924[START_REF] Paull | Torquay: In the Footsteps of Rudolf Steiner[END_REF]. Later, George Kaufmann changed his name to George Adams, adopting his mother's maiden name during World War 2, to avoid the impediment of bearing a Germanic name in Britain at war (though in reality, his father was Australian and his mother was English). There is an extensive file of Kaufmann's mathematical papers at the Archive of the Mathematics and Astronomy Section at the Goetheanum, but the whereabouts of Kaufmann's other papers, or if they have survived, is unknown to the present author. Rudolf Steiner's final trip to Britain was to the International Summer School of the Anthroposophical Society held at Torquay over twelve days[START_REF] Mirbt | The Farm as a Living Organism. Huby Experiments. Anthropsophical Methods on Animals and Produce[END_REF][START_REF] Paull | The farm as organism: the foundational idea of organic agriculture[END_REF]. Rudolf Steiner's health was faltering, but he had been keen to reach an Anglo audience and Britain was his gateway to fulfil that. From Koberwitz in June, Steiner had retreated directly to Dornach. After Koberwitz, Torquay was the final and only venture beyond Dornach that Rudolf Steiner undertook. Although there were no lectures on agriculture at Torquay, there was a side excursion to Tintagel[START_REF] Paull | Tintagel: In the footsteps of Rudolf Steiner[END_REF].

	Included in that excursion
	were Dr Guenther Wachsmuth (1893-1963) and Dr Elizabeth
	Vreede (1879-1943), from the Goetheanum, who had both
	attended the Koberwitz Agriculture Course, and the English
	sisters, Eleanor Merry (1873-1956) and Marna Pease
	(1867-1947). Whatever was discussed that day -besides the
	Arthurian legends of Tintagel -both Eleanor Merry (DE232 in
	Table 1) and Marna Pease (EN11) joined the Experimental
	Circle and went on to play prominent roles in the development
	of biodynamics in Britain.

) wrote: "In 1924 Rudolf Steiner, well known as scientist and founder of the Anthroposophical Movement, entrusted me personally with the task of making all the necessary scientific investigations in connection with his Agricultural Course. Since 1924 I have studied all his suggestions for regenerating Agriculture… the innumerable personal suggestions I received from Rudolf Steiner… Rudolf Steiner meant his suggestions for the whole world, not for a small group of privileged farmers. Only a real understanding of the wonderful organism 'Agriculture' will make it possible to grow healthy food" [1939 in 17, p.x]. Lili Kolisko carried out her biodynamics research at Stuttgart, then at Bray (near Maidenhead) from 1936, then at Edge (near Stroud) from 1940. She published extensively in both German and English [e.g. 17, 19, 20].

  What copies of the Agriculture Course of these pioneers have survived, if any? Were their copies of the Agriculture Course marked-up or annotated? The Agreement that they signed stated, of the numbered copy of Steiner's Agriculture Course that each member received: "I accept it on loan for my personal use in carrying out the experiments undertaken by <me> within the Agricultural Experimental Circle of the General Anthroposophical Society". Within the agreement, there was no explicit commitment to remit the results of experiments back to the Goetheanum (or elsewhere).Whether the British Experimental Circle members recorded their BD experiments and sent the results to Dornach is unknown, but is perhaps discoverable. By publishing his Bio-Dynamic Farming and Gardening in 1938, Ehrenfried Pfeiffer arguably met Steiner's injunction that the 'hints', having been practically tested, be published and shared with the world -and arguably extinguished Steiner's injunction to secrecy thenceforth. Pfeiffer migrated to USA about this time. A collection of Pfeiffer's papers does not appear to have survived in the archives of the Natural Science Section nor of the Goetheanum, and it appears that Pfeiffer's papers surviving in the USA do not include reports from Experimental Circle members.The agreement of Experimental Circle members was that their numbered copy of the Agriculture Course was to be returned to the Goetheanum should the recipient leave the Experimental Circle or the General Anthroposophical Society, or on the death of the recipient. In this way, some returned copies were reissued, and later, copies returned to the Goetheanum were destroyed. So, it is unclear how many, if any, of the original 50 numbered copies issued to British members may have survived.The 43 British members of Rudolf Steiner's Experimental Circle of Anthroposophic Farmers and Gardeners identified in the present paper hold secrets of the earliest history of the biodynamics and organics movement in Britain and the world. The details revealed invite further research to examine the individual and collective early initiatives in the domain of biodynamics and organics which have long been forgotten. The seminal role of the Experimental Circle members in advocating for biodynamics and organics deserves crediting in the history of biodynamics and organics.

	the globe. The novelty of a globally distributed citizens'
	science project in the 1920s and 1930s warrants further
	scholarship.
	There are many unanswered questions. Were the British
	members of the Experimental Circle, farmers or gardeners?
	What were their experiences of biodynamics? Did they fulfil
	their commitments to test and experiment? Were results
	reported to the Goetheanum? If reported, were they reported
	individually or collectively? Did the members continue with
	biodynamics, the Experimental Circle, and/or Anthroposophy?
	Did these pioneers leave written accounts of their
	biodynamics experiments and experiences? Such questions
	are opportunities for further investigation.
	The Experimental Circle was the world's first research
	organisation for organic agriculture. Its membership spanned

Acknowledgements

The kind assistance of Dr Uwe Werner, previous director, Peter Braithwaite, previous archivist, and Dr Johannes Nilo, director, of the Archive of the Goetheanum (Dokumentation am Goetheanum Bibliothek Kunstsammlung Archiv), Dornach, Switzerland, is acknowledged for their skill and helpfulness in locating the records from which the current account is drawn. A related paper, The Pioneers of Biodynamics in USA: The Early Milestones of Organic Agriculture in the United States appeared in the American Journal of Environment and Sustainable Development [28]. Previous papers identified biodynamics pioneers in Australia [29] and New Zealand [30].


R Himo

C Habchi

On the Validity of the Field Synergy Principle in Elliptic Flows

Keywords: Convective heat transfer, advection transport, diffusion transport, backward facing step, elliptic flows

Local analysis and visualization of convective heat transfer is fundamental for many engineering applications. In fact, the correlation between the flow structure and the transport of heat by advection and diffusion is crucial for designing and optimizing devices undergoing heat transfer, such as heat exchangers. The field synergy principle has been used for more than two decades in order to analyze and enhance the convective heat transfer in different applications. In this paper, the validity of the field synergy principle in elliptic flows is tested for both laminar and turbulent flow regimes. For this end, the flow past a backward facing step is considered for a wide range of Reynolds numbers based on the step height ranging from 36 to 36,000. It is shown that the synergy angle and synergy number both fail to represent the local physics of heat transfer process near the step. This is caused by the assumption of negligible molecular and turbulent thermal diffusivities in this principle. In this paper, it is shown that these diffusivities play a crucial role in heat transfer in elliptic flows and cannot be ignored. Moreover, a global analysis shows that the field synergy number could be used to predict the Nusselt number variation if the elliptic region in the flow is small however, the synergy angle fails here also to characterize the variation in Nusselt number.

Introduction

Convective heat transfer is found in natural phenomena or in man-made devices and it is thus essential to different fields in physics and engineering. The domain of applications spans over a wide range from simple domestic equipment, industrial devices to weather prediction [START_REF] Theodore | Heat Transfer Applications for the Practicing Engineer[END_REF][START_REF] Bejan | Convection Heat Transfer[END_REF].

Convective heat transfer is the resultant of advective and diffusive transport in a fluid flow. Therefore, it is crucial to understand and characterize locally the effect of the flow structure on the heat transfer phenomena occurring in a given application. This could be done for instance by using the famous "Heatlines" introduced by Kimura and Bejan [START_REF] Kimura | The "Heatline" visualization of convective heat transfer[END_REF] and based on superposing the heat conduction lines and enthalpy flow lines for a given fluid flow [START_REF] Morega | Heatline visualization of forced convection laminar boundary layers[END_REF]. Other visualization techniques exist in the open literature aiming to analyze the local flow and heat transfer physics such as the energy streamlines and energy flux vectors [START_REF] Mahmud | Visualizing energy flows through energy streamlines and pathlines[END_REF], field synergy principle [START_REF] Guo | A novel concept for convective heat transfer enhancement[END_REF][START_REF] Li | Explicit analytical solutions for fluid flow and temperature eld synergy in cylindrical coordinates[END_REF][START_REF] Ma | Physical quantity synergy analysis and efficiency evaluation criterion of heat transfer enhancement[END_REF] and the Lagrangian description [START_REF] Speetjens | A generalised lagrangian formalism for thermal analysis of laminar convective heat transfer[END_REF]. The readers can refer to Mahapatra et al. [START_REF] Mahapatra | Heatlines and other visualization techniques for confined heat transfer systems[END_REF] for a review on these different methods.

In this paper, we focus on the validity Field Synergy Principle introduced by Guo et al. [START_REF] Guo | A novel concept for convective heat transfer enhancement[END_REF] which is based on modifying the intersection angle between velocity vector and temperature gradient and its application to heat transfer enhancement. This principle has been widely used in the last two decades to study the heat transfer enhancement in parabolic and elliptic flows [START_REF] Zuo | Numerical investigation of helically coiled tube from the viewpoint of field synergy principle[END_REF][START_REF] Li | Numerical study and experimental verification of transverse direction type serrated fins and field synergy principle analysis[END_REF][START_REF] Chang | Numerical investigation into thermal mixing efficiency in y-shaped channel using lattice boltzmann method and field synergy principle[END_REF][START_REF] Cheng | Numerical simulation of conjugate heat transfer in electronic cooling and analysis based on field synergy principle[END_REF][START_REF] Guo | Numerical investigation of helically coiled tube from the viewpoint of field synergy principle[END_REF][START_REF] Lu | Numerical simulation on performances of plane and curved winglet pair vortex generators in a rectangular channel and field synergy analysis[END_REF][START_REF] Zhu | Improvement in field synergy principle : More rigorous application, better results[END_REF][START_REF] Mehra | Local field synergy analysis of conjugate heat transfer for different plane fin configurations[END_REF], in which most engineering applications involve elliptic flows. However, the use and validity of this principle has been the issue of much debate, especially when applied to elliptic flows [START_REF] Habchi | Entropy production and field synergy principle in turbulent vortical flows[END_REF]20,[START_REF] Bejan | Heatlines (1983) versus synergy[END_REF][START_REF] Awad | Field synergy number versus Stanton number[END_REF][START_REF] Bejan | Comment on Study on the consistency between fi eld synergy principle and entransy dissipation extremum principle[END_REF]. Hereafter, a brief introduction to the Field Synergy Principle is given.

In 1998, Guo et al. [START_REF] Guo | A novel concept for convective heat transfer enhancement[END_REF] published an article based on the analogy between conduction and convection heat transfer in the presence of a heat source. After, integrating the energy equation for a parabolic boundary layer flow, they obtained the following relation between the field synergy number F c = N u x / (Re x P r), introduced later by Guo et al. [START_REF] Guo | The field synergy (coordination) principle and its applications in enhancing single phase convective heat transfer[END_REF], and the dot product of the velocity vector and temperature gradient u • ∇T :

F c = δt 0 u • ∇T dy U ∞ (T ∞ -T w ) (1)
Or in other terms :

F c = δt 0 |u| |∇T | cos (β) dy U ∞ (T ∞ -T w ) (2)
It is worthy to note that the field synergy number F c is similar to the Stanton number widely used to quantify heat transfer problems [START_REF] Awad | Field synergy number versus Stanton number[END_REF][START_REF] Bejan | Comment on Study on the consistency between fi eld synergy principle and entransy dissipation extremum principle[END_REF].

According to this relation, Guo et al. [START_REF] Guo | A novel concept for convective heat transfer enhancement[END_REF] suggested three ways to enhance the heat transfer process : by increasing the Reynolds and/or Prandlt numbers, increasing the magnitude of the dot product and by increasing the synergy angle β between the velocity vector and the temperature gradient given in Eq. 2.

While the first two statements are evident, the third one lacks some accuracy. In fact, and according to Eq. 1 and Eq. 2, the intersection angle between the velocity vector and the temperature gradient β should be far from nπ/2, with n an odd number, and closer to (n -1) π/2 in order to enhance the heat transfer process.

A subsequent study by Tao et al. [START_REF] Tao | Field synergy principle for enhancing convective heat transfer its extension and numerical verification[END_REF] suggests the extrapolation of this principle to elliptic flows to cater for most heat transfer problems in engineering. In this study, Tao et al. [START_REF] Tao | Field synergy principle for enhancing convective heat transfer its extension and numerical verification[END_REF] proposed integrating the energy equation in the longitudinal and transverse directions while neglecting the streamwise diffusive term in the fluid k∂ 2 T /∂x 2 . Following this method, only global parameters are obtained without gaining insight into the local physics of the heat transfer process. Moreover, we show in the present paper that the effect of streamwise diffusion in the fluid on the heat transfer process is significant for local analysis and thus it cannot be neglected.

Wu and Tao et al. [START_REF] Wu | Numerical study on laminar convection heat transfer in a rectangular channel with longitudinal vortex generator. Part a : Verification of field synergy principle[END_REF] later performed a similar study on the flow and heat transfer downstream threedimensional longitudinal vortex generators where no further improvement is provided on the local variation of the advective transport neither on the local variation of the synergy angle.

As a summary, the field synergy angle, which is the essence of the field synergy principle, can never be used alone to describe the heat transfer process. In this paper we present a detailed and substantial discussion of the validity of the field synergy principle by taking into consideration the effect of the fluid diffusion term in the energy equation, which has been neglected in previous works. Moreover, we perform a local analysis of the different physical parameters participating in the heat transfer phenomena to show that the field synergy principle fails to describe the local physics of convective heat transfer. For this end, numerical simulations are performed for a flow past a backward facing step. The results are first validated against experimental data then the analysis is performed for a wide range of Reynolds numbers from laminar to turbulent flow regimes. The eddy diffusion is also taken into consideration for the turbulent flow and we show that its effect is significant on the field synergy principle.

The paper is organized as follows, in Section 2 we define the computational domain and numerical method adopted, the local analysis of the field synergy principle is given in Setion 3.1 for laminar flow regime and in Section 3.2 for turbulent flow. The global analysis of the Nusselt number and field synergy parameters are given in Section 3.3. Section 4 is devoted for the concluding remarks.

2 Problem Description

Computational domain and boundary conditions

The computational domain is based on a benchmark provided by the National Aeronautics and Space Administration (NASA) [START_REF] Rumsey | 2D backward facing step[END_REF] as shown in Fig. 1 for a flow past a backward facing step. The numerical results will be validated against NASA's numerical results and experimental data obtained by Driver and Seegmiller [START_REF] Driver | Features of reattaching turbulent shear layer in divergent channel flow[END_REF] in a subsequent section 2.3.

As stated in the introduction (Section 1), the axial diffusive term in the fluid region ∂T 2 /∂ 2 x was neglected in previous studies [START_REF] Guo | A novel concept for convective heat transfer enhancement[END_REF][START_REF] Tao | Field synergy principle for enhancing convective heat transfer its extension and numerical verification[END_REF]. To consistently test the validity of the field synergy principle and the assumption that the diffusive transport in the flow direction is insignificant in elliptic flows, the wall heat flux must only emanate vertically in y direction (keeping in mind that heat flux is normal to the boundary). Consequently, all horizontal walls are set to a constant temperature T w = 350 K whereas the only vertical wall, namely the backward facing step, is thermally insulated.

The working fluid is air entering with a uniform velocity profile at constant temperature T in = 300 K.

Governing equations and numerical procedure

The laminar flow field is governed by the two-dimensional (2D) Navier-Stokes equations. The continuity and momentum equations for an incompressible Newtonian fluid are as follows :

∇ • u = 0 (3) u • ∇u = - 1 ρ ∇p + 1 ρ ∇ • S (4)
In the linear momentum equation S is the viscous stress tensor :

S = µ ∇u + ∇u T (5)
The energy equation is given by :

ρc p ∇ • (uT ) = k f ∇ 2 T (6)
where c p is the specific heat and k f is the molecular thermal conductivity.

The turbulent flow is modeled by the two-dimensional (2D) Reynolds Averaged Navier-Stokes (RANS) equations. While the continuity equation remains unchanged, the momentum equation for turbulent incompressible Newtonian fluid flows becomes as follows :

u • ∇u = - 1 ρ ∇p + 1 ρ ∇ • S -u ∇u (7)
The term u ∇u is the Reynolds stress tensor resulting from the averaging procedure on the nonlinear convective terms in the momentum equations. This term is computed by the Prandtl closure hypothesis with the k -ω SST (Shear Stress Transport) turbulence model [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF].

The energy equation is also computed in the fluid domain and is given by :

ρc p ∇ • (uT ) = ∇ • (k ef f ∇T ) (8)
where k ef f = k f + k t to account for the effect of turbulent fluctuations as modeled by the turbulent Prandtl analogy embedded in k -ω SST model.

The solver used for the flow computation is the CFD code ANSYS Fluent 18.0 [30], which is based on cell-centered finite volume discretization method. The governing equations are solved sequentially with double precision and a second-order upwind scheme [START_REF] Warming | Upwind second-order difference schemes and applications in aerodynamic flows[END_REF] for spatial discretization of the convective terms. The diffusion terms are central differenced and second-order accurate. Pressure-velocity coupling is achieved by the SIMPLE algorithm [30].The residual value 10 -6 is set as the convergence criterion for the solutions of the flow and energy equations. Beyond this value no significant changes were observed in the velocity and temperature fields.

Meshing and experimental validation

A non-uniform structured two-dimensional mesh is generated while paying special attention to the nearwall refinement at all solid boundaries and near the backward facing step, so as to take into account the high velocity and temperature gradients in these regions.

The different mesh densities and their main features are given in Table 1. To determine the appropriate mesh density for solution grid independence, the solver is run with increasing mesh densities until no significant effect on the results is detected. The mesh validity verification is performed by using the method proposed by Celik [START_REF] Celik | Procedure for estimation and reporting of uncertainty due to discretization in CFD applications[END_REF] where the grid convergence index (GCI) and the apparent order of convergence p c can be obtained. The mesh validity verification is applied to the global Nusselt number N u (Eq.( 9)) representing the ratio of convective to conductive heat transfer :

N u = hH k f ( 9
)
where h is the overall convective heat transfer coefficient (W/m 2 K), H(m) the step height and k f (W/mK) is the thermal conductivity of the working fluid.

In Eq.( 10), h is calculated from the logarithmic mean temperature difference :

h = q A∆T lm ( 10
)
where q is the overall rate of heat transfer (W) defined in Eq.( 11), A (m 2 ) is the heat transfer area and ∆T lm is the logarithmic mean temperature difference defined in Eq. [START_REF] Li | Numerical study and experimental verification of transverse direction type serrated fins and field synergy principle analysis[END_REF].

q = ṁc p ∆T ( 11
)
where ṁ is the mass flow rate (kg/s) and ∆T = T out -T in is the bulk temperature difference between the inlet and outlet.

∆T lm = (T w -T in ) -(T w -T out ) ln( Tw-T in Tw-Tout ) (12)
As shown in Table 1, the uncertainty in the fine-grid solution is found to be about 0.02% and the order of convergence is 7.6. Therefore, the last mesh is considered for all simulations. For more details about the calculation of the parameter GCI and p c , the readers can refer to Celik [START_REF] Celik | Procedure for estimation and reporting of uncertainty due to discretization in CFD applications[END_REF]. It should be noted that the mesh study presented here is performed for the highest Reynolds number i.e. Re H = 36, 000 representing the critical case due to highest fluctuations and gradients. The fine mesh is then used for the lower Reynolds numbers insuring good accuracy of the numerical results.

The numerical simulations are performed on eight parallel Intel® Core™ i7-7700 2.80 GHz processors.

Each run takes around one day to converge. The numerical method is validated with experimental data obtained by Driver and Seegmiller [START_REF] Driver | Features of reattaching turbulent shear layer in divergent channel flow[END_REF] using Laser Dopler Velocimetry and with numerical data reported by NASA [START_REF] Rumsey | 2D backward facing step[END_REF] using SST turbulence model [START_REF] Menter | Two-equation eddy-viscosity turbulence models for engineering applications[END_REF] for a Reynolds number of 36,000. The axial velocity profiles at several x/H positions from the actual study are compared to those obtained from experimental and numerical simulations as shown in Fig. 2. From this figure, it could be concluded that a fair agreement is obtained between actual CFD results and those obtained in the literature. The maximum relative error did not exceed 4% between the present study and NASA's numerical results. In this figure, U ref is the reference velocity at the centerchannel near x/H = -4 used to normalize the velocity.

Results and Discussion

In this section, the Nusselt numbers are calculated using three different methods. To avoid redundancy and confusion, we denote :

• N u q the Nusselt number calculated based on the wall heat flux obtained directly from the CFD results on the solid boundaries.

• N u s the Nusselt number calculated based on the Field Synergy principle from Eq.( 13).

• N u c the Nusselt number calculated based on the corrected formulae from Eq.( 16) for laminar flows and Eq.( 22) for turbulent flows.

Laminar flow

Starting from the following Eq.( 13) which is the y integration of the energy equation for a laminar parabolic flow :

ρc p δ th 0 u ∂T ∂x + v ∂T ∂y dy = -k f ∂T ∂y w (13)
Eq.13 could also be written as follow :

ρc p δ th 0 (|u| • |∇T | • cos (β)) dy = q w (14)
Guo's "Novel concept for convective heat transfer" suggested three possible ways to increase the heat transfer and they are :

• Increasing the Reynolds and Prandlt numbers • Increasing the fullness of dimensionless velocity and temperature profiles which he defined as U = U U∞ and ∇T = ∇T (T∞-Tw)/δ th • Increasing the angle between the velocity and temperature gradient vectors. (Later published by Tao [START_REF] Tao | Field synergy principle for enhancing convective heat transfer its extension and numerical verification[END_REF] as the decrease in the angle which will cause the increase in cos(β))

While the first two suggestions are self evident, the last one presents an obvious overlooking of the fact that the simplified version of the energy equation becomes obsolete when the synergy angle is changed, since the diffusive transport in the axial direction can no longer be neglected due to the fact that the whole flow structure has changed locally.

A more correct representation of the heat transfer in an elliptic flow should be as follows. Starting from the complete energy equation in a two dimensional flow :

ρc p u ∂T ∂x + v ∂T ∂y = ∂ ∂x k f ∂T ∂x + ∂ ∂y k f ∂T ∂y (15)
Isolating and integrating in the y direction :

q w = k f ∂T ∂y w = ρc p δ th 0 u ∂T ∂x + v ∂T ∂y dy - δ th 0 ∂ ∂x k f ∂T ∂x dy (16)
or,

q w = k f ∂T ∂y w = ρc p δ th 0 (|u| • |∇T | • cos (β)) dy - δ th 0 ∂ ∂x k f ∂T ∂x dy (17)
Eq. 17 shows that the heat flux, i.e. the Nusselt number, depends on both : advection and diffusion. The advection term was described by Tao et al. [START_REF] Tao | Field synergy principle for enhancing convective heat transfer its extension and numerical verification[END_REF] as the Synergy, and the diffusion is totally neglected. Now we will show how that the field synergy principle and the synergy angle β are not representative of the local heat transfer in elliptic flows.

Calculating the local Nusselt number based on the step height H from q w according to Eq. ( 18), the validity of the field synergy principle can be put to the test.

N u x = q w H k f (T w -T b ) (18)
where T b is the fluid bulk temperature at given x/H.

The results of the local Nusselt are shown in Fig. 3 from the wall heat flux obtained from CFD results (as N u q x in red), from synergy in Eq.13 (as N u s x in black) and from the present study's modified equation in Eq.16 (as N u c

x in blue). From this figure, it could be observed that the Field Synergy principle's assumption is valid throughout the entrance and the developed region of the flow, except near the proximity of the step. That's where the synergy angle is varied greatly, hence the claim that the convective heat transfer can be improved by increasing the synergy angle becomes unreliable since the equation defining this principle is not fully representative of the heat transfer. A close-up on the variation of the Nusselt numbers near the step is shown also in Fig. 3 to the right side, where it is clear that the behavior of the transport described in Eq.13 (using the Field Synergy principle) is completely different than the actual phenomenon. However, the present study's claim of the importance of the diffusive transport in the streamwise direction upon the change in the flow structure is clearly representing the transfer of the thermal energy.

In addition, from Fig. 3, one can evaluate the claim that the increase in the synergy angle is a way to increase the heat transfer is misleading, once the Nusselt number is juxtaposed with the synergy angle.

In fact, an opposite behavior is witnessed after x = 0 where the increased in β is mirrored by a decrease in N u q x and vice versa. The reason behind this discrepancy can be simply explained by referring to the energy equation (Eq.6) where the dot product of the advective term can be rewritten as the multiplication of both magnitudes of u and ∇T with the cosine of the synergy angle. Consequently, it is more fair to analyze the heat transfer vis-à-vis the variation in cos(β) as shown in Fig. 4. Obviously, cos(β) in the advective term is not solely responsible for the heat transfer since it did not reflect the increase in the local Nusselt number at x < 0, due to the presence of the prominent diffusive term especially when the synergy angle is changing, which further supports the claim in this manuscript. This issue will be further elaborated by showing the contribution of each of the advective and diffusive terms given by the following equations respectively and shown in Fig. 5.

ψ = ρc p δ th 0 u ∂T ∂x + v ∂T ∂y dy (19)
ξ = δ th 0 ∂ ∂x k f ∂T ∂x dy (20)
The results in Fig. 5 further support the claim that diffusion cannot be neglected in elliptic laminar flows especially near the step where the flow will be modified. Hence, the effect of the diffusion term is responsible for the increase in the local Nusselt number near the step as shown in Fig. 4.

F 5 -Variation of ψ and ξ terms with X

Turbulent flow

In this section we will consider turbulent flow regime. The energy equation computed by k -ω SST including the turbulent prandtl analogy is as follows :

ρc p u ∂T ∂x + v ∂T ∂y = ∂ ∂x k f ∂T ∂x + ∂ ∂y k f ∂T ∂y + ∂ ∂x k t ∂T ∂x + ∂ ∂y k t ∂T ∂y (21)
where k t is the turbulent eddy conductivity.

Isolating and integrating in the y direction :

q w = k f ∂T ∂y = ρc p δ th 0 u ∂T ∂x + v ∂T ∂y dy- δ th 0 ∂ ∂x k f ∂T ∂x dy- δ th 0 ∂ ∂x k t ∂T ∂x + ∂ ∂y k t ∂T ∂y dy (22)
When the local Nusselt number based on the step height H is calculated from q w , the validity of the Field Synergy principle for elliptic turbulent flows can be put to the test. The results of the local Nusselt are shown in Fig. 6 from the wall heat flux, from Eq.13 and from the present study's claim in Eq.22.

F 6 -Local Nusselt number for the different methods and the synergy angle for Re = 36, 000

Similar to the laminar case, the claim of the Field Synergy principle is compromised near the step, where the synergy angle is varied greatly as shown in Fig. 6. On the other hand, when the full energy equation is taken into account such as in Eq.22 the heat transfer is described with good accuracy. This means that the diffusive and turbulent transports of the thermal energy are contributing to the heat transfer and could not be neglected as assumed by Tao et al. [START_REF] Tao | Field synergy principle for enhancing convective heat transfer its extension and numerical verification[END_REF] in elliptic flows.

The contribution of each of the advective ψ, diffusive ξ and turbulent transport ξ t , given in Eq. ( 23), are shown in Fig. 7. This figure highlights the importance of diffusive and turbulent transport terms relative to the advective term. Thus neglecting these terms induces variance between the computed Nusselt number and that obtained using the field synergy principle. The ratio of the Nusselt number derived from the synergy (N u s x ) over the actual span-averaged local Nusselt number obtained from CFD or full energy equation (N u q

ξ t = δ th 0 ∂ ∂x k t ∂T ∂x + ∂ ∂y k t ∂T ∂y dy (23)
x ) is shown in Fig. 8. It could be observed that near the step, where the flow is classified as elliptic flow, the field synergy principle induces a relative error of about 32%, even in a highly turbulent case with a high Péclet number, in opposition to the claim of Tao et al. [START_REF] Tao | Field synergy principle for enhancing convective heat transfer its extension and numerical verification[END_REF] who claimed that the field synergy principle works well for high Péclet numbers.

Global analysis

In this section of this manuscript, the global Nusslet number and synergy angle variation with Reynolds number is analyzed as shown in Fig. 9. As shown in this figure, the Nusselt number increases with increasing Reynolds number while cos(β) varies slightly between 0.048 and 0.056 and shows a maximum at a Reynolds of 3,600. Beyond this Reynolds number cos(β) decreases opposing to the variation in Nusselt number. Thus it could be concluded that cos(β) cannot be used alone to represent the global heat transfer as for the local analysis.

In addition, the Stanton number, sometimes referred to as the Field synergy number, shows a decreasing behavior with increasing Reynolds number, due to the increase of Re at a higher rate than the increase of the global Nusselt number. The concluding remarks are summarized as follows :

• The assumption that the molecular thermal diffusivity in the streamwise direction is negligible proves to be invalid for elliptic flows. Therefore, the field synergy number and synergy angle cannot be used to describe the local heat transfer process.

• In turbulent flow regimes, neglecting the turbulent thermal diffusivity led to misleading results in the field synergy parameters and thus this principle fails to describe accurately the physics of heat transfer near the step.

• If the elliptic region in the flow is small relative to the overall computational domain, the field synergy number can be used to predict the variation in the overall Nusslet number. However, special care should be taken into consideration when dealing with complex geometries in which elliptic regions are dominant.

• The field synergy angle cannot be used alone to describe local and global variation in the Nusselt number or convective heat transfer coefficient.

F 2 -

 2 Axial velocity profiles at x = 1 (a), x = 4(b) and x = 10 (c)

F 3 -

 3 Local Nusselt number from the different methods along with local the synergy angle

F 4 -

 4 A close up on the Local Nusselt number for the different methods and cos(θ)

F 7 -

 7 Variation of ψ, ξ and ξ t terms with x

F 8 -

 8 Variation of N u s x

F 9 - 4 Conclusion

 94 Global N u and cosine of the average Synergy angle vs. Re H Field synergy principle was extensively used to study parabolic and elliptic convective heat transfer. In this paper we study the validity of this principle by considering the flow past a backward facing step for both laminar and turbulent flow regimes.


Rawad Himo
email: rawad.himo@univ-nantes.fr

Charbel Habchi

Heat Transfer Enhancement by Electromagnetic Forcing in Duct Flow

Electromagnetic forcing could be used to produce streamwise vortices in order to enhance the heat and mass transfer processes in ducts and pipes. In our current study we take this novel method to a more practical use, to suggest a Magneto hydrodynamic heat exchanger in which several transverse Lorentz forces are arranged along the streamwise direction to produce secondary flow structures and enhance mixing of hot fluid particles near the wall with cold fluid particles near the center. The Reynolds number based on the mean flow velocity and hydraulic diameter is taken 1000. The flow structures, the local heat transfer and the thermal performances are quantitatively analyzed to show the effectiveness of this concept. Increases by more than 100% in the global Nusselt number and the thermal performance were detected. Several configurations of forcings were tested, where it was found that the increase in local and global Nusselt is more efficient with subsequent forces of the same direction, rather than changing directions.

Nomenclature

A overall area of heat transfer, m 2 B magnetic field, T c heat capacity of the fluid, J/kgK F Lorentz force, N/m 3 GCI Grid convergence index h convective heat transfer coefficient, W/m 2 K J current density, A/m 3 k bulk fluid thermal conductivity, W/mK N u global Nusselt number N u z local Nusselt number p c apparent order of converge q overall rate of heat transfer, W S exchange area, m 2 T temperature, K ∆T lm log mean temperature difference, K u velocity vector field, m/s Greek symbols ρ density, kg/m 3 µ dynamic viscosity, Pa.s φ electric field scalar potential, V σ electric conductivity of the fluid, S

Introduction

Vortex Generators have been extensively studied and used in multifunctional heat exchangers/reactors to enhance heat and mass transfer. While, two main types of vortices can be distinguished, namely: longitudinal and transverse. Previous studies [1] proved that longitudinal VGs, having a principal axis in the direction of the flow, are more efficient since they induce swirling motions, not otherwise witnessed in the quasi-2D flows generated by transverse VGs.

Different shapes and angles of attack for the inserts have been addressed in the literature to generate longitudinal as well as transverse vortices [START_REF] Buren | Interaction between a vortex generator and a synthetic jet in a crossflow[END_REF][START_REF] Oneissi | Novel design of delta winglet pair vortex generator for heat transfer enhancement[END_REF][START_REF] Hamed | Vortical structures in the near wake of tabs with various geometries[END_REF]; hence, the two aforementioned control parameters (shape and angle of attack) affect the flow structure, in addition to the evident importance of the main flow's Reynolds number. Extensive attempts of studying the shapes and orientations of vortex generators remain mostly stochastic design problems that are relatively time consuming, experimentally and numerically. In the present work, we intend to generate vortices in the flows by using electromagnetic forcing to mimic vortex generators, this type of flows is also known as magneto-hydrodynamics (MHD). Consequently, we transform our control parameters from the geometric features of the vortex generators to the magnitude of the electrical potential and magnetic field; which in fact are more easily controlled by simply adjusting the intensities of the two entities [START_REF] Himo | Coherent flow structures and heat transfer in a duct with electromagnetic forcing[END_REF].

Many studies in the open literature were performed to analyze the convective heat transfer enhancement in MHD flows [START_REF] Huang | Heat transfer enhancement of free surface MHD-flow by a protrusion wall[END_REF][START_REF] Melnikov | Experimental investigation of MHD heat transfer in a vertical round tube affected by transverse magnetic field[END_REF][START_REF] Cassells | Heat transfer enhancement using rectangular vortex promoters in confined quasi-two dimensional magnetohydrodynamic flow[END_REF][START_REF] Alam | A comprehensive review on single phase heat transfer enhancement techniques in heat exchanger applications[END_REF]. Motozawa et al. [START_REF] Motozawa | Effect of magnetic field on heat transfer in rectangular duct flow of a magnetic fluid[END_REF] studied experimentally the convective heat transfer in rectangular smooth duct flow of a magnetic fluid at a Reynolds number of 780. As expected, the local Nusselt increased in the proximity of the external magnetic field. A maximum recorded increase of the Nusselt number was 20%. Considering the difficulty of visualizing the flow, the study did not include an analysis of the flow structure vis-à-vis the heat transfer enhancement. More specifically, an experimental study on MHD flows using a potassium hydroxide solution (KOH) revealed that for low Reynolds numbers the MHD effects are more prominent due to weaker production of turbulence [START_REF] Junichi | Experimental investigation of magnetohydrodynamic turbulent pipe flow of aqueous electrolyte solution[END_REF]. It is also important to mention that numerical studies have shown that the pressure drop for electrically insulated ducts is lower than electrically conducting walls due to eddy currents generated near the electrically conducting walls [START_REF] Bhuyan | Effect of magnetic field on MHD pressure drop inside a rectangular conducting duct[END_REF].

Our previous work showed that by using a transverse Lorentz force in a duct flow where the working fluid is an aqueous solution of 25% wt. KOH; heat transfer is enhanced locally and globally due to the coherent structures that were generated and were correlated to the increase in heat transfer [START_REF] Himo | Coherent flow structures and heat transfer in a duct with electromagnetic forcing[END_REF]. In this current study, we upgrade this novel method of heat transfer to cater for industrial applications, whereby we use a higher Reynolds number and different configurations of subsequent Lorentz forces to further increase the overall heat transfer.

The paper is organized as follows: the computational domain and numerical procedure are presented in section 2. , then the results are discussed in section 3. . Finally, section 4. is devoted for the concluding remarks.

Problem description 2.1. Governing equations

The flow field is governed by the three-dimensional (3D) steady Navier Stokes equations. The continuity and momentum equations for an incompressible Newtonian fluid in presence of electromagnetic forcing are as follows:

∇ • u = 0 (1) u • ∇u = - 1 ρ ∇p + 1 ρ ∇ • µ ∇u + ∇u T + F ρ (2)
where F = J×B is the Lorentz force, with J the current density and B the external magnetic field.

In the present study, the boundary conditions are for fixed electric potential. Thus Ohm's law Congrès Français de Thermique SFT 2019, Nantes, 3 -6 juin 2019 is used here to determine the current density as follows:

J = σ [-∇φ + (u × B)] (3)
where φ is the electric field scalar potential and σ the electric conductivity of the fluid.

The conservation of electric charge enforces:

∇ • J = 0 (4)
Therefore Eq. ( 3) reduces to

∇ 2 φ = ∇ • (u × B) (5)
The energy equation for forced convection is also computed in the fluid domain and is given by:

ρc p ∇ • (uT ) = k∇ 2 T + 1 σ J.J (6)
where c p is the specific heat and k is the bulk molecular thermal conductivity.

In the present study, the buoyancy effects are negligible compared to the inertial and viscous ones. Consequently, the convection remains forced as per the flow's Reynolds and Rayleigh numbers [START_REF] Holman | Heat Transfer[END_REF].

Numerical procedure

The solver used for the flow computation is the CFD code ANSYS Fluent 18.0 [START_REF][END_REF], which is based on cell-centered finite volume discretization method. The governing equations are solved sequentially with double precision and a second-order upwind scheme [START_REF] Warming | Upwind second-order difference schemes and applications in aerodynamic flows[END_REF] for spatial discretization of the convective terms. The diffusion terms are central differenced and secondorder accurate. Pressure-velocity coupling is achieved by the Coupled algorithm [START_REF][END_REF].

The walls are electrically insulated except for the conducting electrodes. The working fluid is an aqueous solution of 25% potassium hydroxide KOH, with dynamic viscosity varying with the fluid temperature [START_REF] Oxychem | Caustic Potash Handbook[END_REF] according to a polynomial function given in Eq. ( 7):

µ = 4.917 × 10 -7 T 2 -3.348 × 10 -4 T + 5.793 × 10 -2 (7)
Flow and heat-transfer simulations are carried out for a Reynolds number of 1000, based on the mean flow velocity and the hydraulic diameter, and for uniform wall temperature T w = 350 K. For Reynolds numbers above 1000, higher electrical potential is required to affect, making it less practical for industrial applications. The temperature of the inlet of the computational domain is uniform T in = 300 K. Atmospheric pressure is prescribed at the domain outlet. Noslip boundary conditions are prescribed on all solid surfaces. The residual value 10 -6 is set as the convergence criterion for the solutions of the flow and energy equations. Beyond this value no significant changes were observed in the velocity and temperature fields. It is equally important to note that this numerical method has been validated with DNS results in our previous study [START_REF] Himo | Coherent flow structures and heat transfer in a duct with electromagnetic forcing[END_REF].

Computational domain and meshing

The computational domain consists of a square duct of hydraulic diameter D h = 10mm, with four pairs of electrodes starting from the position z = 2.5 × D h and spaced by a distance of 5 × D h . The electrode width (in the flow direction) is equal to D h /2. The electrodes are embedded inside the duct material, therefore the computational domain has a constant cross sectional area. A uniform magnetic field |B 0,y | = 50 mT is applied in the streamwise direction starting at location where the electrodes are turned on. Therefore an electromagnetic force will be generated at such locations.

The arrangement and polarity of electrodes is chosen in order to obtain forces in the transverse direction to generate secondary flows and mimic the common flow up induced by the pressure gradients across vortex generators which are used for heat and mass transfer enhancement [START_REF] Habchi | Turbulence behavior of artificially generated vorticity[END_REF][START_REF] Hamed | Vortical structures in the near wake of tabs with various geometries[END_REF]. Five different configurations of electrodes are considered in this study and are all compared with the empty duct flow. The different configurations are summarized in Table 1. In this table configuration (6) corresponds to the empty duct flow without forcing. Configuration (1) is shown in Figure 2 (a Non-uniform structured three-dimensional meshes were generated while paying special attention to the near-wall refinement at all solid boundaries (duct wall and electrodes) so as to take into account the high velocity and temperature gradients in these regions. The different mesh densities and their main features are given in Table 2.

To determine the appropriate mesh density for solution grid independence, the solver is run with increasing mesh densities until no significant effect on the results is detected. The mesh validity verification is performed by using the method proposed by Celik [START_REF] Celik | Procedure for estimation and reporting of uncertainty due to discretization in CFD applications[END_REF] where the grid convergence index (GCI) and the apparent order of convergence p c can be obtained. The mesh validity verification is applied to the global Nusselt number N u obtained from Eq.8, representing the ratio of convective to conductive transfer:

N u = hD h k ( 8
)
where h is the overall convective heat transfer coefficient (W/m 2 K), D h the hydraulic diameter, and k is the bulk thermal conductivity of the working fluid.

In Eq.8 h is calculated from the logarithmic mean temperature difference:

h = q A∆T lm (9)
Cell number n 638, 880 1, 103, 937 1, 827, 011 Mesh size l = (V /n) where q is the overall rate of heat transfer (W) defined in Eq.10, A(m 2 ) is the heat transfer area and ∆T lm is the logarithmic mean temperature difference defined in Eq.11.

q = ṁc p ∆T ( 10
)
where ṁ is the mass flow rate (kg/s) and ∆T = T out -T in is the bulk temperature difference between the inlet and outlet:

∆T lm = (T w -T in ) -(T w -T out ) ln( T w -T in T w -T out ) (11)
As shown in Table 2, the uncertainty in the fine-grid solution is found to be about 0.17% and the order of convergence is 1.77. For more details about the calculation of the parameter GCI and p c , the readers can refer to Celik [START_REF] Celik | Procedure for estimation and reporting of uncertainty due to discretization in CFD applications[END_REF]. It should be noted that the mesh study presented here is performed for configuration (1) which induces high gradients considering the changing direction of the Lorentz forces.

The numerical simulations are performed on eight parallel Intel® Core™ i7-7700 2.80 GHz processors. Each run takes between 12 and 36 hours.

Results and discussion

Temperature distribution

The temperature distribution of the six tested configurations is shown in Figure 1 at four different cross section. It can be clear from configurations (1) and [START_REF] Oneissi | Novel design of delta winglet pair vortex generator for heat transfer enhancement[END_REF], where the forcing changes directions, that the flow structure and temperature distributions change with the changing direction of the imposed Lorentz forces. On the other hand, for configurations (2) and ( 4) the direction doesn't change. As a result, the flow structure and temperature distributions, highlighted by the primary vortex counter-rotating pair, is further energized after each force instead of being destroyed, leading to more growing secondary vortices near the corners and side walls. Consequently, the secondary flows enhance the advective heat transfer from the hot walls to the cold fluid.

Local and global results

The local Nusselt numbers of each configuration is shown in Figure 2 (a) where the effect of the forces on the heat transfer manifest clearly with local increases that begin at the location of the applied force and carry on along the streamwise direction. It is obtained from the spanaveraged heat flux at every streamwise position. From Figure 2 (a) the two configurations ( 1) and ( 2) can be compared, in addition to also [START_REF] Oneissi | Novel design of delta winglet pair vortex generator for heat transfer enhancement[END_REF] and (4). It is obvious that configuration (2) provide better heat exchange than (1), as is the case for configuration (4) compared to (3); despite having the same input electromagnetic force, where only the direction is varied.

One can therefore summarize those results in a plot showing the global Nusselt number vs the number of forces for both same and different directions of the forces. The global Nusselt is calculated from the log mean temperature method explained in section 2.3. . As vorticity is induced more heat is exchanged from to hot walls to the cold fluid, this will in turn raise the average temperature at the end of the duct. This plot in Figure 3 (a) confirms our conclusion from the local results, that having consecutive forces of the same direction is more efficient than changing directions. The thermal performance enhancement introduced in Equation 12based on the same pumping power [START_REF] Himo | Coherent flow structures and heat transfer in a duct with electromagnetic forcing[END_REF] is shown in Figure 3 (b) where it can be deduced that despite the increase in friction, thermal performance is improved up to 102% for the same direction forcings and up to 89% for the changing direction. Lastly, it is worthy to note that the difference between the changing and non-changing configurations increases as the number of forces increase and therefore this study leads to the recommendation of using non-changing transverse Lorentz forces for higher efficiency in industrial applications.

η = N u N u 0 • f f 0 -1 3 ( 12
)
Congrès Français de Thermique SFT 2019, Nantes, 3 -6 juin 2019

Conclusion

The effect of electromagnetic forces in conducting fluid duct flow on the heat transfer has been studied and the results can be summarized below:

• Transverse Lorentz forces in conducting fluid flows improves convective heat transfer by inducing vorticity.

• The global Nusselt number can be increased by more than 100% when using Lorentz forces as vortex generators • When using consecutive forces, having the same direction is more efficient than changing directions of the forces.

Figure 1 :

 1 Figure 1 : Temperature distribution for the different configurations at different streamwise locations z D = z/D h

Figure 2 :Figure 3 :

 23 Figure 2 : Setup for Configuration 1, with positive (in red) and negative (in blue) electrodes (a), and local Nusselt number for the different configurations (b)

Table 1 :

 1 ). Flow configurations

	Electrode Row	1 st	2 nd	3 rd	4 th
	Streamwise Location Configuration (1)	5D h 2 Flow up Flow left Flow down Flow right 16D h 27D h 38D h 2 2 2
	Configuration (2)	Flow up Flow up	Flow up	Flow up
	Configuration (3)	Flow up	off	Flow left	off
	Configuration (4)	Flow up	off	Flow up	off
	Configuration (5)	Flow up	off	off	off
	Configuration (6)	off	off	off	off

Table 2 :

 2 Mesh sensitivity analysis

Congrès Français de Thermique SFT 2019, Nantes, 3 -6 juin 2019


Rawad Himo
email: rawad.himo@univ-nantes.fr

Gérard-Philippe Z Éhil

Chady Ghnatios

2D modeling of the thermal conductivity of XLPE-modified concrete

Cross-linked polyethylene (XLPE) is a thermoset polymeric material extensively used in the industry, including in firefighting systems, hot water and sanitary piping, and for the insulation of electric cables. The cross-linking of polyethylene increases its performance while reducing its recyclability. A recent approach considers processing XLPE waste materials in the form of small flakes to partially replace natural aggregates in concrete mixtures. Ongoing experimental and numerical studies addressing the homogenized mechanical properties of such composite mixtures are showing their feasibility. Despite a decreased stiffness and strength due to the presence of XLPE inclusions (which are typically weaker than traditional aggregates) the modified concrete mixtures can still maintain working structural properties. In this vein, it is also interesting to characterize the role of XLPE inclusions in improving the thermal insulation of XLPE-modified concretes. A virtual concrete wall is thus modeled numerically, in 2D at the mesoscale, to study the statistical effect of varying the volume fractions of XLPE inclusions, and aggregates, on the homogenized thermal conductivity of the modified concrete. The equivalent thermal conductivity of the wall k eq typically drops as more aggregates are replaced by XLPE inclusions, thus achieving better thermal insulation. The dependence of k eq on the area fraction of XLPE is roughly linear, while the statistical variability is very low.

Introduction

Recycling has recently increased considerably as environmental awareness rises; yet, this remains insufficient since only 34.5% of municipal waste was properly recycled in 2012, as per the facts and figures of the US Environmental Protection Agency (EPA) [START_REF]Municipal Solid Waste Generation, Recycling, and Disposal in the United States: Facts and Figures for 2012[END_REF]. An actively researched method of recycling plastics consists of using these as replacement for natural aggregates in concrete mixtures. One of the most widely discussed plastic aggregate-replacement An interesting advantage of this plastic-waste recycling approach is the added benefit of improved thermal insulation, as discussed by Marzouk et al. [START_REF] Marzouk | Valorization of post-consumer waste plastic in cementitious concrete composites[END_REF] who studied PET aggregates in concrete mixtures. The study furthermore suggests that durability factors increase with the addition of PET, since the degradation of plastics is slow. Subsequent studies confirmed this hypothesis [e.g. [START_REF] Poonyakan | Potential use of plastic wastes for low thermal conductivity concrete[END_REF][START_REF] Yesilata | Thermal insulation enhancement in concretes by adding waste PET and rubber pieces[END_REF] by showing a reduced overall thermal conductivity. It is however important to note that the drop in thermal conductivity observed at a given replacement ratio depends in fact on the shapes of the inclusions [START_REF] Yesilata | Thermal insulation enhancement in concretes by adding waste PET and rubber pieces[END_REF]. Besides, despite addressing various geometries, previous works represent little data and therefore fail to establish a clear trend in the thermal effect and behavior of plastic aggregates in concrete mixtures. A statistical approach is thus retained in the current study: 30 different two-dimensional finite element simulations are run, at the mesoscale, for each volume fraction of inclusions considered, to build a representative set of data, from which clear conclusions can be drawn.

Cross-linked polyethylene (XLPE) is produced by forming molecular cross-links in the micro-structure of polyethylene to improve its physical and mechanical properties, including its stiffness and strength. XLPE is extensively used in the industry for its superior performance, including in firefighting systems, hot water and sanitary piping, and for the insulation of electric cables [e.g. [START_REF] Madani | Preparation of granular crosslinkable medium density polyethylene[END_REF][START_REF] Meola | Experimental Evaluation of Properties of Cross-Linked Polyethylene[END_REF]. The main drawback of XLPE corresponds to the increased difficulty to recycle thermosets. Replacing natural aggregates in concrete mixtures with XLPE waste shreds can thus be considered as a suitable method to reuse this type of plastic waste. This work complements our studies estimating the effect of XLPE shreds on the mechanical properties of XLPE-modified mixtures [START_REF] Zéhil | Determination of the elastic properties of cross-linked polyethylene mixed concrete using soft computing techniques (under review)[END_REF]. It models the effect of replacing aggregates by chopped XLPE flakes on the thermal insulation of the resulting concrete mixture.

Modeling

Computational domain and governing equation

In this work, we simulate the conductive heat transfer in a two-dimensional cross-section of a 5 cm thick and 10 cm high concrete wall. The wall's dimensions are retained following a set of preliminary iterations conducted to insure that the variability on the equivalent properties determined from the simulations would be suitably low. A 2D approach is adopted on the grounds that:

• Large depths (in the third dimension) are usually employed in walls while inhomogeneities in the meso-structure are distributed evenly, which justifies the assumption that the heat fluxes along the third dimension even out. • Given the wall dimensions, particle size-distributions and volume fractions involved in this work, it is assumed that artificially constraining the flow of heat to the plane of the model would not make the preferential paths of heat conduction significantly longer than these could be in three dimensions.

The differential equation governing the transfer of heat in steady-state is given by:

∂ ∂x k ∂T ∂x + ∂ ∂y k ∂T ∂y = 0, ( 1
)
where T is the temperature field and k the local thermal conductivity.

Congrès Français de Thermique SFT 2019, Nantes, 3 -6 juin 2019

Essential boundary conditions are prescribed on the right edge of the wall, i.e. T Right = 10 • C at x = 5 cm, whereas natural conditions are applied to the remaining boundaries: a uniform heat flux q Lef t = 500 W/m 2 is applied across the left edge of the wall, while the upper and lower boundaries are thermally insulated. The thermal conductivities of the individual material components retained in this work are: k mortar = 1.73, k aggregates = 2.70 and k XLP E = 0.45 W/(m. • C) for the mortar, the aggregates and the XLPE inclusions, respectively [START_REF] Khan | Factors affecting the thermal properties of concrete and applicability of its prediction models[END_REF][START_REF] Marie | Thermal conductivity of hybrid recycled aggregate -Rubberized concrete[END_REF][START_REF] Morgan | The thermal conductivity of crosslinked polyethylene insulation in aerial bundled cables[END_REF][START_REF] Md | Thermal conductivity of silane cross-linked polyethylene composites[END_REF][START_REF] Siwińska | Thermal conductivity coefficient of cement-based mortars as air relative humidity function[END_REF][START_REF] Thomas | Thermal conductivity of carbonate rocks[END_REF][START_REF] Zhang | Mesoscale model for thermal conductivity of concrete[END_REF].

This boundary value problem is solved numerically by means of a finite element scheme implemented on the commercial software Matlab. To this aim, the domain is discretized using typical four-node (linear) quadrilateral finite elements of size 0.25 mm. The mesh was retained following a convergence analysis considering the size of the particles in the system. The aggregates are added selectively inside a homogeneous mortar, one after another, followed by the inclusions. The algorithms for generating the natural aggregates and the XLPE flakes are similar. Briefly, these operate as follows:

Meso-structure generation

1. The total size-range [S min , S max ] of aggregates and inclusions is divided into sub-ranges [S k , S k+1 ] of aggregate diameters (S = D) and inclusion lengths (S = L). Each subrange is characterized by a corresponding target area fraction āf (k) determined from a given cumulative size-distribution, as shown in Figure 1, for the aggregates. Sub-ranges are filled in order of decreasing particle size. While filling the sub-range [S k , S k+1 ], particles of size S ∈ [S k , S k+1 ] are randomly generated and placed, one after the other, and the area fraction a f (k) is continuously updated.

2. Following the random selecton of a particle's size, the coordinates of its centroid x and y, and a factor α ∈ [0, 1] are also picked randomly, so as the particle is inside the domain.

The factor α concerns aggregates only: the size of an aggregate's trace in the plane of the wall's cross-section is given by d = αD.

3.

A test is performed to reject any new particle that overlaps with the ones already in place. This is practically achieved by comparing the number of mortar nodes inside the particle's projected location (A), to the total number of nodes inside the same (B). If A and B are equal, the projected location contains no previous particles and it can thus be considered. Otherwise, step (2) is repeated until the condition is satisfied.

4. The area fraction a f if updated when the new particle is retained/accepted.

5.

Only particles achieving a f (k) ≤ 1.01ā f (k) are accepted. This forces the random sizedistribution to fit within 1% of the prescribed distribution mentioned in step (1). The particle is otherwise rejected and the algorithm goes back to step (2).

6. When a f (k) ≥ āf (k) the algorithm starts filling the next sub-range of smaller size particles.

Component proportions and simulation plan

The control concrete mixture retained contains 1375 kg/m 3 of mortar (including cement, sand and fine aggregates) and 985 kg/m 3 of coarse aggregates, characterized by a density of 2600 kg/m 3 . The control mix is modified by replacing an increasing proportion of the coarse aggregates by an equivalent volume of XLPE inclusions whose density is 920 kg/m 3 . The mass of coarse aggregates in a modified mixture is thus given by:

m aggregates = 985 - 2600 920 m XLP E . (2)
In this work, the total mass of XLPE inclusions is varied from 0 to 230 kg/m 3 , by increments of 10 kg/m 3 , which corresponds to a total of 24 cases of XLPE content. In each case, 30 different random aggregate and inclusion structures are generated and a similar number of finite element simulations are run to build a statistical database of equivalent homogenized thermal properties. This corresponds to a total of 720 simulations.

Numerical results

Temperature distributions

The random distribution of aggregates and XLPE inclusions inside the mortar phase, as generated by the algorithm presented in section 2.2. , is illustrated for two selected examples, in Figure 2. The mortar appears in dark blue, the aggregates in yellow, while XLPE inclusions are shown in aqua blue. It can be noted that 190 kg/m 3 of XLPE corresponds to an area fraction of a f = 0.198.

The temperature fields corresponding to the random structures represented in Figure 2 are also shown in Figure 3. It is clear, in both cases, that the local temperatures are affected by the presence of particles of different thermal properties than those of the mortar phase. A closer observation of Figure 3 reveals that the presence of XLPE flakes creates temperature fluctuations with somewhat sharper contour lines. This can be attributed to the sharp geometry of the XLPE flakes combined with their relatively small thermal conductivity. The latter modify the path of the thermal bridges in the mortar phase to more irregular shapes, hindering the heat diffusion process.

Figure 4 shows quiver plots of the heat flux vector field defined by q = -k ∂T ∂x i + ∂T ∂y j , where i and j correspond to the unit vectors in directions x and y, respectively. It is interesting

Equivalent Thermal Conductivity

We designate by l x and l y the thickness and the height of the concrete wall, respectively. The temperature can be averaged over the left boundary of the wall, i.e.

T Lef t = 1 l y y T (0, y) dy. (3)
The equivalent thermal conductivity of the composite can be determined macroscopically as: The equivalent thermal conductivities k eq calculated according to equation (4) for the 720 simulations performed in this work are presented in Figure 5. For each area fraction of XLPE content, a f (XLPE), the average value of k eq is plotted, with error bars showing the standard deviation, over 30 simulations. The thermal conductivity of the wall drops as the proportion of XLPE inclusions is increased in the mixture. Replacing traditional aggregates by an equivalent volume of a less conductive waste material improves the thermal insulation of the modified concrete. The dependency of k eq on a f (XLPE) is clearly linear as it is well fitted by a straight line corresponding to the rule of mixtures.

k eq = q Lef t • l x T Lef t -T Right . (4
It is also interesting to note that the very small standard deviations obtained confirm that the size of the domain retained for the simulations is appropriate, i.e. of the order of magnitude of the representative elementary volume.

Conclusion

This study addressed the statistical influence of partially replacing typical aggregates in concrete mixtures by recycled XLPE flakes, on the thermal insulation of the modified concrete. This was achieved by means of a large number of 2D finite element simulations. The thermal properties and the shapes of the inclusions affected the diffusion of heat and led to local and global changes in the temperature field inside of an XLPE-modified concrete wall. As a result, the equivalent thermal conductivity of the wall dropped linearly as the area fraction of XLPE content was increased. The low variability between simulations confirmed that the size of the model is statistically sound. The results of this work must be confirmed by 3D simulations taking the third spatial dimension into consideration.

Nomenclature a f

  area fraction of inclusions d size of aggregate's trace in the domain, m D aggregate diameter, m k thermal conductivity, W/ (m. • C) L length of inclusions, m l x wall/domain thickness (x-direction), m l y Wall/domain height (y-direction), m m mass, kg q heat flux, W/m 2 T temperature, • C x position through wall/domain thickness, m y position through wall/domain height, m Greek symbols α cross-section/trace factor of aggregates Ω 2D wall domain Subscripts -(.) aggregate traditional aggregates -(.) Lef t left wall/domain boundary -(.) mortar mortar matrix -(.) Right right wall/domain boundary -(.) XLP E XLPE inclusions/flakes

Figure 1 :

 1 Figure 1 : Target and simulated cumulative size-distribution of aggregates.

  (a) m XLP E = 0 kg/m 3 (b) m XLP E = 190 kg/m 3

Figure 2 :

 2 Figure 2 : Random aggregate and inclusion structures for 0 and 190 kg/m 3 of XLPE.

  (a) m XLP E = 0 kg/m 3 (b) m XLP E = 190 kg/m 3

Figure 3 :

 3 Figure 3 : Temperature fields for 0 and 190 kg/m 3 of XLPE (cases of Fig. 2).

Figure 4 :

 4 Figure 4 : Heat flux vector for 0 and 190 kg/m 3 of XLPE (cases of Fig. 2).

Figure 5 :

 5 Figure 5 : Equivalent thermal conductivity versus area fraction of XLPE inclusions.

  Congrès Français de Thermique SFT 2019, Nantes, 3 -6 juin 2019 is polyethylene terephthalate (PET) waste. Several previous works have shown that plasticmodified mixtures have lower overall stiffness, compressive strength and density, considering that traditional aggregates are characterized by typically larger such properties [e.g. 2, 3].

  )

	2.05
					Numerical Results
	2				Fit: k eq	= (-1.77 af flake	)+ 2.03
	1.95
	1.9
	1.85
	1.8
	1.75
	1.7
	1.65
	1.6	0	0.05	0.1	0.15	0.2	0.25

Congrès Français de Thermique SFT 2019, Nantes,3 -6 2019

Congrès Français de Thermique SFT 2019, Nantes, 3 -6 juin 2019

Acknowledgements

Part of this work was jointly funded with the support of the National Council for Scientific Research in Lebanon CNRS-L and Notre Dame University -Louaize.