Aron-Wisnewsky Judith

Karine Clément

A place for vitamin supplementation and functional food in bariatric surgery ?

Keywords: bariatric surgery, vitamins, functional food, microbiota

 Purpose of review: This article summarizes the recent literature concerning vitamin deficiency and required supplementation post bariatric surgery (BS), with a focus on vitamin (and associated clinical adverse effect on bone loss) and on the potential implication of the gut microbiota.

 Recent findings: BS improves health outcomes in terms of weight loss and metabolic alterations yet with major inter-individual variability. Indeed, although BS is efficient in most patients, some display poor response (i.e. patients with the lowest weight loss at one year or weight regain afterwards, or patients who do not undergo metabolic disease remission). Furthermore, despite systematic vitamin supplementation post-BS, some patients develop vitamin deficiencies leading to poor clinical outcomes. Specifically, a major recently described adverse outcome is vitamin D deficiency associated in part with observed bone mass loss and fractures. Recent mechanistic studies have been performed to better understand the involved physiopathology. Furthermore, different intervention studies tested on top of BS (using vitamin supplementation, diet or nutrients acting as functional food) have evaluated whether they could improve these nutritional adverse outcomes and are summarized herein. Importantly, gut microbiota involved in food digestion and metabolization as well as vitamin synthesis, is largely perturbed during severe obesity and is partially restored post-BS, yet again with large inter-individual variability.

Whether differential modification of the gut microbiota could be associated with those vitamin

deficiencies is an open question.

 Summary: Future clinical research studies will need to evaluate whether add-on intervention to BS using vitamin, diet or specific food items could help prevent nutritional deficiencies and improve clinical response observed post-BS. More importantly, personalizing the add-on intervention after BS upon gut microbiota composition should be tested in predicted poor responders to BS as already performed during diet intervention to further improve metabolic health.

Introduction

Obesity progresses worldwide. Thus the number of bariatric surgeries (BS) reserved for the most severe cases (i.e. body mass index (BMI)≥40kg/m² or ≥35kg/m² associated with specific comorbidities [START_REF] Patel | Micronutrient Deficiencies After Bariatric Surgery: An Emphasis on Vitamins and Trace Minerals [Formula: see text[END_REF]) increases in parallel [START_REF] Angrisani | IFSO Worldwide Survey 2016: Primary, Endoluminal, and Revisional Procedures[END_REF]. Several BS procedures exist among which vertical sleeve gastrectomy (VSG) and Roux-en-Y gastric bypass (RYGB), the two most commonly performed [START_REF] Angrisani | Bariatric Surgery and Endoluminal Procedures: IFSO Worldwide Survey 2014[END_REF], followed by adjustable gastric banding (AGB) and biliopancreatic diversion (BPD), respectively. However, if BS decreases weight and improves metabolic diseases and global health [START_REF] Adams | Weight and Metabolic Outcomes 12 Years after Gastric Bypass[END_REF], it also leads to secondary effects among which frequent vitamin deficiencies [START_REF] Patel | Micronutrient Deficiencies After Bariatric Surgery: An Emphasis on Vitamins and Trace Minerals [Formula: see text[END_REF]. As reviewed in length in [START_REF] Doré | The importance of the gut microbiota after bariatric surgery[END_REF], the mechanisms involved in vitamin and mineral malabsorption include reduced gastric pouch and caloric intake and rearrangement of the digestive tract inducing pH modification (both in VSG and RYGB) and decreased nutrient absorption in RYGB. Modification of the gut microbiota could also be involved as discussed later in more details.

The American Society for Metabolic and Bariatric Surgery (ASMBS) has reviewed the current literature for vitamin deficiencies. They observed large interindividual variability among studies and patients [START_REF] Tabesh | Nutrition, Physical Activity, and Prescription of Supplements in Pre-and Post-bariatric Surgery Patients: a Practical Guideline[END_REF]. Monitoring vitamin deficiencies before and mostly post-BS as well as prescribing systematic vitamin and mineral supplementation is largely recommended post-BS.

The gut microbiota represents a recently described organ involved amongst others in digestion, intestine homeostasis, vitamin biosynthesis and metabolite production all of which playing important roles for the host health [START_REF] Rowland | Gut microbiota functions: metabolism of nutrients and other food components[END_REF][START_REF] Leblanc | Beneficial effects on host energy metabolism of short-chain fatty acids and vitamins produced by commensal and probiotic bacteria[END_REF]. Murine studies exploring gut microbiota composition as well as experiments using transfer of microbiota from mice or humans into germ-free mice have demonstrated that the gut microbiota is a novel factor involved in obesity pathophysiology as reviewed in details in [START_REF] Zmora | Transforming medicine with the microbiome[END_REF][START_REF] Clément | The gut microbiome, diet, and links to cardiometabolic and chronic disorders[END_REF][START_REF] Tilg | The intestinal microbiota fuelling metabolic inflammation[END_REF]. Human studies have reported that obesity is characterized by a major gut microbiota dysbiosis (i.e. large perturbation of the gut microbiota composition and functionality). In particular, obesity is characterized by decreased microbial gene richness and modified composition when compared to lean individuals [START_REF] Cotillard | Dietary intervention impact on gut microbial gene richness[END_REF][START_REF] Chatelier | Richness of human gut microbiome correlates with metabolic markers[END_REF]. This dysbiosis is exacerbated during severe obesity [START_REF] Prifti | Major microbiota dysbiosis in severe obesity: fate after bariatric surgery[END_REF]. Interestingly, BS has been shown to dramatically modify gut microbiota composition [START_REF] Doré | The importance of the gut microbiota after bariatric surgery[END_REF][START_REF] Debédat | Impact of bariatric surgery on type 2 diabetes: contribution of inflammation and gut microbiome?[END_REF][START_REF] Debédat | Gut Microbiota Dysbiosis in Human Obesity: Impact of Bariatric Surgery[END_REF] in general post-RYGB [START_REF] Prifti | Major microbiota dysbiosis in severe obesity: fate after bariatric surgery[END_REF][START_REF] Tremaroli | Roux-en-Y Gastric Bypass and Vertical Banded Gastroplasty Induce Long-Term Changes on the Human Gut Microbiome Contributing to Fat Mass Regulation[END_REF] but also in some cases post-VSG [START_REF] Liu | Gut microbiome and serum metabolome alterations in obesity and after weight-loss intervention[END_REF]. However, after RYGB the rescue of gut microbiota is incomplete [START_REF] Prifti | Major microbiota dysbiosis in severe obesity: fate after bariatric surgery[END_REF]19]. Importantly, some changes in gut microbiota composition are associated with improvement in clinical outcomes (such as improved corpulence and reduced inflammation). Yet, half of these associations are dependent on calorie and nutrient consumption [START_REF] Furet | Differential adaptation of human gut microbiota to bariatric surgery-induced weight loss: links with metabolic and low-grade inflammation markers[END_REF][START_REF] Kong | Gut microbiota after gastric bypass in human obesity: increased richness and associations of bacterial genera with adipose tissue genes[END_REF] suggesting the important role of food items in both the shaping of gut microbiota and also on the host's metabolic health.

In this context, the field of functional food has emerged over the past ten years; functional foods are defined as biologically active ingredients considered functional since they are associated with physiological health benefits related to the maintenance or improvement of health (such as for example fibers, flavonoids or polyphenols). Amongst them, some specific food or ingredients can even be related to prevention or even treatment of several chronic diseases [START_REF] Crowe | Academy of Nutrition and Dietetics: Position of the academy of nutrition and dietetics: functional foods[END_REF]. Importantly, the digestion of some food items requires the gut microbiota to be metabolized into metabolites with beneficial effects for health. A most commonly cited example lays with fibers transformed into short chain fatty acids (SCFA) [START_REF] Leblanc | Beneficial effects on host energy metabolism of short-chain fatty acids and vitamins produced by commensal and probiotic bacteria[END_REF]) [START_REF] Hills | Gut Microbiome: Profound Implications for Diet and Disease[END_REF]. Thus, functional food seems partly related to the gut microbiota activity as discussed in [START_REF]You are what you eat: diet, health and the gut microbiota[END_REF]. This review will address the current knowledge on vitamin deficiencies and supplementation required post-BS and its most recent raised adverse outcomes in particular impacting the bone. We will discuss how and which diet items (if any) considered as functional food could help prevent or improve BS-induced nutritional complications or even further enhance beneficial clinical outcomes observed post-BS. A pending question is whether the modified gut microbiota composition could be involved in vitamin deficiencies observed post-BS. [START_REF] Parrott | American Society for Metabolic and Bariatric Surgery Integrated Health Nutritional Guidelines for the Surgical Weight Loss Patient 2016 Update: Micronutrients[END_REF].

Vitamin deficiency and need for supplementation following bariatric surgery:

Recent guidelines recommend [START_REF] Tabesh | Nutrition, Physical Activity, and Prescription of Supplements in Pre-and Post-bariatric Surgery Patients: a Practical Guideline[END_REF] systematic vitamin supplementation following BS in order to reach an adequate nutritional intake and to limit vitamin deficiencies [START_REF] Coupaye | Comparison of nutritional status during the first year after sleeve gastrectomy and Roux-en-Y gastric bypass[END_REF][START_REF] Verger | Micronutrient and Protein Deficiencies After Gastric Bypass and Sleeve Gastrectomy: a 1-year Follow-up[END_REF][START_REF] Verger | Nutritional and Protein Deficiencies in the Short Term following Both Gastric Bypass and Gastric Banding[END_REF] on top of the usual diet. In principle, such vitamin supplementation should prevent vitamin-deficiency associated complications which may impair overall health. Nevertheless, despite good compliance, some vitamin deficiencies and protein malnutrition can still be observed [START_REF] Coupaye | Comparison of nutritional status during the first year after sleeve gastrectomy and Roux-en-Y gastric bypass[END_REF][START_REF] Verger | Micronutrient and Protein Deficiencies After Gastric Bypass and Sleeve Gastrectomy: a 1-year Follow-up[END_REF][START_REF] Verger | Nutritional and Protein Deficiencies in the Short Term following Both Gastric Bypass and Gastric Banding[END_REF]. Recently, the ASMBS has reviewed the current knowledge concerning BS vitamin follow-up and supplementation after BS and proposed an updated version of these recommendations including 79 new proposals [START_REF] Parrott | American Society for Metabolic and Bariatric Surgery Integrated Health Nutritional Guidelines for the Surgical Weight Loss Patient 2016 Update: Micronutrients[END_REF]. However, the level of evidence upon which these recommendations have been proposed is in general intermediate or weak due to the lack of sufficiently powered studies or randomized control studies (RCT). Overall, ASMBS experts propose to systematically supplement patients with folate, iron, calcium, B vitamin (B1 Vitamin/thiamin and cobalamin/vitamin B12), lipid soluble vitamin (A, D, E, K), zinc and copper, whatever the surgery type. Noteworthy for most malabsorptive surgery models (most particularly BPD and to a lesser extend RYGB), an increase in the dosage of the systemic vitamin and mineral supplementation is recommended for calcium, vitamin A, K, iron, copper and zinc [START_REF] Parrott | American Society for Metabolic and Bariatric Surgery Integrated Health Nutritional Guidelines for the Surgical Weight Loss Patient 2016 Update: Micronutrients[END_REF], in order to meet the official daily requirement intakes thus taking into account BS induced malabsorption. New forms of multivitamins actually contain the adequate proportion of these micronutrients and vitamins needed post-BS and seem to be associated with less occurrence of vitamin deficiencies during follow-up [START_REF] Schijns | Do specialized bariatric multivitamins lower deficiencies after RYGB?[END_REF]. New means of delivery are also being tested (i.e. transdermal vitamin patch) but only one small powered study has evaluated its efficacy to prevent vitamin deficiencies [START_REF] Saurabh | Is Transdermal Multivitamin Patch Effective in Gastric Bypass Patients?[END_REF]. These new forms of multivitamins will need evaluation on larger sample size to examine both reduced vitamin deficiencies and reduction of related clinical events. Importantly, recent retrospective studies have evaluated the prevalence of nutritional deficiencies despite the use of oral systematic supplementation. Vitamin deficiencies are found after both RYGB and VSG [START_REF] Antoniewicz | Nutritional Deficiencies in Patients after Roux-en-Y Gastric Bypass and Sleeve Gastrectomy during 12-Month Follow-Up[END_REF] independently of the amount of weight-loss. Importantly, studies reveal that vitamin D deficiency is the most common one during the first year post-BS and increases thereafter in both VSG and RYGB, despite systematic supplementation [START_REF] Johnson | Analysis of vitamin levels and deficiencies in bariatric surgery patients: a single-institutional analysis[END_REF]. As a consequence, increased parathormone levels were observed post-RYGB [START_REF] Antoniewicz | Nutritional Deficiencies in Patients after Roux-en-Y Gastric Bypass and Sleeve Gastrectomy during 12-Month Follow-Up[END_REF], in line with previous data [START_REF] Patel | Micronutrient Deficiencies After Bariatric Surgery: An Emphasis on Vitamins and Trace Minerals [Formula: see text[END_REF]. Noteworthy, transdermal vitamin patch delivery performs even worse than oral pills to prevent vitamin D deficiency [START_REF] Saurabh | Is Transdermal Multivitamin Patch Effective in Gastric Bypass Patients?[END_REF]. In studies with longer follow-up (i.e. 5 [START_REF] Wei | High Incidence of Secondary Hyperparathyroidism in Bariatric Patients: Comparing Different Procedures[END_REF] to 10 years [START_REF] Blom-Høgestøl | Bone metabolism, bone mineral density and low-energy fractures 10 years after Roux-en-Y gastric bypass[END_REF]), the prevalence of vitamin D insufficiency or deficiency reaches up to 60% of patients [START_REF] Wei | High Incidence of Secondary Hyperparathyroidism in Bariatric Patients: Comparing Different Procedures[END_REF][START_REF] Blom-Høgestøl | Bone metabolism, bone mineral density and low-energy fractures 10 years after Roux-en-Y gastric bypass[END_REF]. More worryingly, this is present in a large proportion of patients having a bone mineral density (BMD) below the normal range, or even osteopenia and osteoporosis, thus increasing the risk of low-energy bone fractures [START_REF] Blom-Høgestøl | Bone metabolism, bone mineral density and low-energy fractures 10 years after Roux-en-Y gastric bypass[END_REF]. This adverse effect represents one of the recent concerns arising in the mid and long term post-BS.

If these clinical observations are frequently observed at mid-term post-RYGB [START_REF] Zhang | A meta-analysis of the effects of bariatric surgery on fracture risk[END_REF][START_REF] Yu | Fracture Risk After Bariatric Surgery: Roux-en-Y Gastric Bypass Versus Adjustable Gastric Banding[END_REF], the precise mechanisms underlying this BMD loss and deleterious changes in bone microarchitecture [START_REF] Schafer | Effects of Gastric Bypass Surgery on Bone Mass and Microarchitecture Occur Early and Particularly Impact Postmenopausal Women[END_REF] are not yet completely understood and probably go beyond vitamin deficiency. A recent mouse study demonstrated that VSG per se may induce both loss of trabecular and cortical bone mass and bone weakness, independently of weight-loss, diet, sex or gut hormones [START_REF] Li | G-CSF partially mediates effects of sleeve gastrectomy on the bone marrow niche[END_REF]. Furthermore, VSG may induce a reduction of bone formation and mineralization as well as a myeloid cell expansion and an increase of circulating G-CSF both in mice and humans post-VSG. The increase of G-CSF per se recapitulated at least partially the BMD decrease in mice. Indeed, other mechanisms independent of G-CSF are also involved in BMD loss post-VSG since G-CSF KO mice undergoing VSG were not completely protected against bone loss [START_REF] Li | G-CSF partially mediates effects of sleeve gastrectomy on the bone marrow niche[END_REF]. Importantly, the effect of vitamin D deficiency on top of these myeloid modifications were not evaluated in this study and further explorations are needed. A number of other factors are involved in this BMD loss as discussed in detail in [START_REF]Trim the gut, lose the weight -and the bone[END_REF]. Among them, the gut microbiota might also be involved since germ-free mice models display increased bone mass due to reduction in bone resorption [START_REF] Ohlsson | Effects of the gut microbiota on bone mass[END_REF]. Another indirect proof of the link between gut microbiota and PTH-vitamin D status lays in a recent retrospective study which reports a positive correlation between increased PTH levels (secondary to decreased calcium absorption) and loose stools post-BS [START_REF] Schijns | Loose and frequent stools and PTH levels are positively correlated post-gastric bypass surgery due to less efficient intestinal calcium absorption[END_REF]. Loose stools have been associated with a specific dysbiotic microbiota composition and reduced microbial gene richness with contribution of a proinflammatory context [START_REF] Vandeputte | Stool consistency is strongly associated with gut microbiota richness and composition, enterotypes and bacterial growth rates[END_REF]. One could thus hypothesize that gut microbiota changes might be involved in increased bone resorption function in some individuals post-BS.

Another concern following BS is the deficiency in vitamin B1 thus leading to Gayet-Wernicke encephalopathy, which can occur specifically in patients vomiting due to surgical complications post-BS [START_REF] Oudman | Preventing Wernicke Encephalopathy After Bariatric Surgery[END_REF]. Interestingly, if Gayet-Wernicke encephalopathy was initially described post-RYGB, a number of cases were also more recently reported post-VSG [START_REF] Oudman | Preventing Wernicke Encephalopathy After Bariatric Surgery[END_REF][START_REF] Chang | Wernicke encephalopathy concurrent with polyradiculoneuropathy in a young man after bariatric surgery: A case report[END_REF][START_REF] Kröll | Wernicke Encephalopathy: a Future Problem Even After Sleeve Gastrectomy? A Systematic Literature Review[END_REF] which is the major BS worldwide [START_REF] Angrisani | Bariatric Surgery and Endoluminal Procedures: IFSO Worldwide Survey 2014[END_REF].

Importantly, it was recently demonstrated that some gut bacteria were able to produce and to provide some vitamins (in particular thiamin) to the host [START_REF] Sannino | The Drosophila melanogaster Gut Microbiota Provisions Thiamine to Its Host[END_REF]. One could thus hypothesize that some individuals are more prone to thiamin deficiency according to their gut microbiota profile and to differential post-BS changes across individuals. This research area has not yet been properly evaluated. Other deficiencies from the B vitamin group may also occur in particular for B12 after the first year [START_REF] Antoniewicz | Nutritional Deficiencies in Patients after Roux-en-Y Gastric Bypass and Sleeve Gastrectomy during 12-Month Follow-Up[END_REF] which may also involve the gut microbiota [START_REF] Degnan | Vitamin B12 as a modulator of gut microbial ecology[END_REF].

Functional changes in gut microbiota after bariatric surgery: links with vitamins and nutrients.

Literature has now repeatedly reported that gut microbiota composition is modified post-BS, whatever the surgery type, as reviewed in [START_REF] Doré | The importance of the gut microbiota after bariatric surgery[END_REF][START_REF] Debédat | Impact of bariatric surgery on type 2 diabetes: contribution of inflammation and gut microbiome?[END_REF][START_REF] Debédat | Gut Microbiota Dysbiosis in Human Obesity: Impact of Bariatric Surgery[END_REF]. More importantly than microbial composition per se, metagenomic studies post-SG have also observed switches in functional capacities of gut microbiota as compared to the ones present in the severely obese state. More importantly, the observed functional capacities of the gut microbiota 3 months post-SG were similar to those characterizing lean individuals [START_REF] Liu | Gut microbiome and serum metabolome alterations in obesity and after weight-loss intervention[END_REF] suggesting that BS could at least partially restore gut microbiota dysbiosis. As such, 3 months post-VSG, a number of enzymatic functions are modified upon BS-induced weight loss, including vitamin pathways (i.e. a decrease in thiamin metabolism and folate biosynthesis and an increase in riboflavin (B2) metabolism) but also impaired nutrient fermentation (decrease in carbohydrate metabolism) and amino acids biosynthesis and degradation [START_REF] Liu | Gut microbiome and serum metabolome alterations in obesity and after weight-loss intervention[END_REF]. Longer-term studies have observed a shift of intestinal fermentation towards reduced production of SCFAs and increased amino acid fermentation post-BS due to reduced carbohydrates intake and increased protein fermentation [START_REF] Tremaroli | Roux-en-Y Gastric Bypass and Vertical Banded Gastroplasty Induce Long-Term Changes on the Human Gut Microbiome Contributing to Fat Mass Regulation[END_REF]. These changes seem to occur independently of diet modifications. These switches in functional gut microbiota capacities could in part be secondary to the universally demonstrated decrease in calorie intake post-BS especially with a massive reduction in fiber intake [START_REF] Verger | Micronutrient and Protein Deficiencies After Gastric Bypass and Sleeve Gastrectomy: a 1-year Follow-up[END_REF][START_REF] Verger | Nutritional and Protein Deficiencies in the Short Term following Both Gastric Bypass and Gastric Banding[END_REF]. Nonetheless, the microbiota may adapt to compensate for these nutritional changes. For example, the phosphotransferase system increases post-BS and represents a transport system for sugars and their derivatives, uniquely present on bacteria, that enable the microbes to assimilate sugars despite the reduced calorie intake observed post-BS [START_REF] Tremaroli | Roux-en-Y Gastric Bypass and Vertical Banded Gastroplasty Induce Long-Term Changes on the Human Gut Microbiome Contributing to Fat Mass Regulation[END_REF][START_REF] Palleja | Roux-en-Y gastric bypass surgery of morbidly obese patients induces swift and persistent changes of the individual gut microbiota[END_REF]. Likewise, bacteria display increased transport system for different vitamin and minerals including (vitamin B1, vitamin B12, manganese, iron, and zinc) probably to increase their availability post-BS since they are also essential for microbes functioning [START_REF] Palleja | Roux-en-Y gastric bypass surgery of morbidly obese patients induces swift and persistent changes of the individual gut microbiota[END_REF]. Overall, on top of digestive tract modifications, BS induces modification of the gut microbiota functional capacities which per se could play a role in the modifications of nutrient intake and vitamin status observed post-BS.

Since the gut microbiota capacity to metabolize and ferment food is modified post-BS, preferring food items capable of being utilized by the post-BS microbiota might enhance clinical benefits on top of those observed post-BS. Although appealing, the field of precision nutrition in the bariatric surgery context remains scarce and still hypothetic [START_REF] Nicoletti | Bariatric Surgery and Precision Nutrition[END_REF]. Noteworthy, tools have been developed in clinical research during fiber-enriched diet intervention programs to help prescribe the adequate diet regimen for a specific individual, based on its metagenomic profile. For example, the casino tool box is a computational platform able to model the metabolomic effect of a diet regimen in a specific gut microbiota composition context (i.e. it predicts the level and type of circulating metabolites obtained after the ingestion of a known diet based upon the known composition of the gut microbiota). More importantly, its use could, in principle, help prescribe a personalized diet regimen to an individual in order to reach a healthier metabolic state and healthier gut microbiota based upon his specific baseline microbiota composition [START_REF] Shoaie | Quantifying Diet-Induced Metabolic Changes of the Human Gut Microbiome[END_REF]. More recently, Korem at al observed large inter-individual variability in glycemic response upon bread consumption. Specifically, they showed that baseline metagenomic profile (i.e. microbiome based classifiers) enables to predict the glycemic response of an individual after the ingestion of a specific bread type [START_REF] Korem | Bread Affects Clinical Parameters and Induces Gut Microbiome-Associated Personal Glycemic Responses[END_REF][START_REF] Zeevi | Personalized Nutrition by Prediction of Glycemic Responses[END_REF]. These tools should be tested in the BS field to evaluate whether prescribing functional foods may improve clinical outcomes and possibly reduce secondary effects such as vitamin deficiencies. Research in this field is urgently needed.

Effects of supplementation on vitamin status and clinical outcomes

Different types of interventions have been tested to evaluate their efficacy in reducing adverse bone effects post-BS. Increasing the supplementation of vitamin D by adding 50,000 IU cholecalciferol monthly, enabled a decrease in the prevalence of patients presenting vitamin D deficiency, yet some were still observed [START_REF] Smelt | The Influence of Different Cholecalciferol Supplementation Regimes on 25(OH) Cholecalciferol, Calcium and Parathyroid Hormone after Bariatric Surgery[END_REF]. However, this study did not assess whether this increased supplementation had any effect on BMD or the number of fractures. Another 12 weeks duration RCT comparing increased dosage of calcium supplementation (i.e. 1600mg daily) composed of citrate or carbonate forms, did not show any significant difference in PTH and vitamin D levels. Noteworthy, the citrate form was associated with reduced bone turnover markers suggesting less bone resorption [START_REF] Ring Madsen | Effect of calcium citrate vs calcium carbonate on elevated parathyroid hormone after Roux-en-Y gastric bypass. A double-blinded, randomized trial[END_REF], yet BMD was not assessed.

Interestingly, more complex interventions seem to prevent BMD loss. Indeed, a recent RCT trial comparing usual follow-up (without any supplementation) to the effects of 1000mg daily citrate calcium, vitamin D and protein supplementation (adjusted to BMI and body resting energy expenditure) associated with an aerobic exercise program, demonstrated subnormal systemic vitamin levels, reduced BMD loss and bone turnover in the intervention group. More importantly, although participants were of pre-menopausal age, atraumatic fractures only occurred in the non-intervention group [55]. Finally, a recent RCT comparing the addition of exercise training to usual follow-up post-RYGB, showed a reduced loss, yet no improvement, of BMD in the exercise group. However, fractures were not evaluated in this cohort composed mostly of pre-menopausal women [START_REF] Murai | Exercise mitigates bone loss in women with severe obesity after Roux-en-Y gastric bypass: a randomized controlled trial[END_REF].

Importantly, the authors point at a very low compliance of vitamin and mineral supplementation during post-BS follow-up. Indeed, 16% of patients drop out of the RCT [55] and only one third of the patients reported regular calcium and vitamin D intake in retrospective studies [START_REF] Blom-Høgestøl | Bone metabolism, bone mineral density and low-energy fractures 10 years after Roux-en-Y gastric bypass[END_REF]. Furthermore, the prevalence of patients undergoing systematic vitamin dosage dropped dramatically at one year post-BS which confirms previous findings of a decreased compliance to follow-up after the first years [START_REF] Thereaux | Long-term follow-up after bariatric surgery in a national cohort[END_REF].

Overall, adherence to vitamin intake and compliance to follow-up are major issues post-BS.

Functional food, health outcomes and bariatric surgery: perspectives

The European commission defines functional food as "a food that beneficially affects one or more target functions in the body, beyond adequate nutritional effects, in a way that is relevant to either an improved state of health and well-being and/or reduction of risk of disease. It is part of a normal food pattern" [START_REF] Crowe | Academy of Nutrition and Dietetics: Position of the academy of nutrition and dietetics: functional foods[END_REF]. Functional food can include dietary intake compounds or physiologically active nutrients (i.e. antioxidant vitamins, fibers for example) or non-nutrients (i.e. flavonoids for examples) with proven health benefits. Some pieces of evidence show that intervention programs using functional food are beneficial to improve T2D control [START_REF] Zmora | Transforming medicine with the microbiome[END_REF][START_REF] Alkhatib | Functional Foods and Lifestyle Approaches for Diabetes Prevention and Management[END_REF] (i.e. Mediterranean diet component or polyphenols), or some obesity related traits [START_REF] Song | The modulatory effect and the mechanism of flavonoids on obesity[END_REF][START_REF] Thaiss | Persistent microbiome alterations modulate the rate of post-dieting weight regain[END_REF] (using flavonoid). Nevertheless further larger studies using precision nutrition should be undertaken before recommending a wide use in routine care [START_REF] Zmora | Transforming medicine with the microbiome[END_REF]. However, the use of these specific functional food (for example flavonoid, polyphenols or Mediterranean diet) after bariatric surgery has not yet been evaluated. Importantly, defining the targeted patients who will mostly benefit from such add-on therapies is also mandatory (i.e. whether in all patients undergoing BS or in selected patients predicted to have a poor clinical response post-BS as defined earlier). Even though BS is effective in inducing weight loss, results display, in fact, major interindividual variability both in the first year [START_REF] Bel Lassen | The FAT score, a Fibrosis score of Adipose Tissue: predicting weight loss outcome after gastric bypass[END_REF] as well as in the longer-term [START_REF] Courcoulas | Seven-Year Weight Trajectories and Health Outcomes in the Longitudinal Assessment of Bariatric Surgery (LABS) Study[END_REF], with some patients even regaining weight. The same is observed for metabolic improvement. Indeed, while some patients undergo diabetes remission post-BS (i.e. normalization of glycemic parameters in the absence of glucose lowering drugs) [START_REF] Sokolovska | The advanced-DiaRem score improves prediction of diabetes remission 1 year post-Roux-en-Y gastric bypass[END_REF], others remain diabetic and still need treatments despite major improvement in Hba1c target. Finally, some patients even relapse their diabetes in the mid-term [START_REF] Debédat | Long-term Relapse of Type 2 Diabetes After Roux-en-Y Gastric Bypass: Prediction and Clinical Relevance[END_REF]. Whether interventions proposed on top of BS could enhance its clinical benefice is an open question, specifically using functional food.

Nevertheless, studies evaluating whether functional food could be added to bariatric surgery in order to further improve its outcomes have not yet been performed to our knowledge. Retrospective studies have observed that patients maintaining their weight post-BS compared to those regaining weight after 18 months were those consuming more fibers and having a global healthier behavioral lifestyle [START_REF] Masood | Dietary and Lifestyle Factors Serve as Predictors of Successful Weight Loss Maintenance Postbariatric Surgery[END_REF]. Some studies have further tested whether dietary interventions associated to BS would decrease weight regain in unselected patients undergoing BS. A recent RCT [START_REF] Hanvold | Does Lifestyle Intervention After Gastric Bypass Surgery Prevent Weight Regain? A Randomized Clinical Trial[END_REF] where patients were randomized post-BS to usual follow-up or to a lifestyle intervention program (i.e. increasing healthy food such as fruits and vegetables and improving food choices and quantities) did not evidence any difference in weight regain between groups. However, at the end of the study period, both groups displayed similar food pattern. Likewise, another recent RCT comparing protein supplementation with or without resistance training to usual follow-up in an unselected bariatric cohort did not evidence any difference in weight or body composition outcomes [START_REF] Oppert | Resistance Training and Protein Supplementation Increase Strength After Bariatric Surgery: A Randomized Controlled Trial[END_REF]. Overall, these data could suggest that synergic interventions post-BS should be proposed in selected individuals predicted to display a poor response (either poor weight-loss [START_REF] Bel Lassen | The FAT score, a Fibrosis score of Adipose Tissue: predicting weight loss outcome after gastric bypass[END_REF] or non-diabetes remission or diabetes relapse [START_REF] Debédat | Long-term Relapse of Type 2 Diabetes After Roux-en-Y Gastric Bypass: Prediction and Clinical Relevance[END_REF]) or at risk to display adverse surgery outcomes (bone fractures) to evaluate whether it improves further health outcomes.

Conclusion:

Bariatric surgery represents an effective therapeutic tool to treat obesity and its related diseases, yet with major inter-individual variability in clinical outcomes and secondary effects in particular due to vitamin deficiencies. The mechanisms involved in differential improvement between individuals remain incompletely understood and involve many factors. Among them, the gut microbiota might play a role. Current knowledge is growing regarding the implication of gut microbiota in health outcomes. Its role in vitamin biosynthesis and degradation has been shown but further studies are needed to evaluate whether microbiota changes post-BS could be linked to vitamin deficiencies and adverse medical outcomes. Indeed, despite systematically prescribed vitamin and mineral supplementation post-BS, nutritional deficiencies still occur. This underlines the need of close biological monitoring, systematic vitamin supplementation (which needs to be increased in more malabsorptive procedures). Some interventions using vitamin or nutrients have been performed to evaluate their benefice in risk reduction. Nevertheless, to date, precision medicine proposing specific vitamin supplement to patients predicted at risk of vitamin deficiencies has not been evaluated. With increasing knowledge on gut microbiota and its interaction with the host, this field probably represents the future research perspective. Likewise, if functional food has demonstrated its benefice in chronic metabolic diseases, its use post-BS has not been tested to improve specific health outcomes and thus warrants clinical studies before its prescription in routine care.

 Key points:

o Bariatric surgery is commonly associated with vitamin deficiencies despite systematic vitamin and mineral supplementation.

o

  Bariatric surgery decreases bone mass density which can be attenuated by sufficient calcium and vitamin D supplementation associated with exercise training. o Gut microbiota is largely disturbed in severe obesity and only partially recovers after bariatric surgery. o The modified gut microbiota post-BS is involved in vitamin pathways and nutrient metabolism switch and could thus be involved in vitamin deficiencies post-BS. o Functional food (such as flavonoid, polyphenols or Mediterranean diet component) improves certain health outcomes in metabolic diseases, yet their use has not been tested after BS to further improve clinical outcomes in poor responders.  Acknowledgements, Financial Support and Sponsorship, and Conflicts of Interest: Financial Support: none, we do not have any Conflicts of Interest.


M Belghazi

A Bontemps
email: bontempsan@chartreuse.cea.fr

C Marvillet

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An experimental study has been carried out to investigate the characteristics of condensation in a bundle of horizontal tubes. These tubes have different types of external surfaces: smooth surface (1D), low trapezoidal fins with several fin pitches (2D) and specific fins (3D, C+ tube). The used fluids are either pure refrigerant (HFC 134a) or binary mixture of refrigerants (HFC 134a/HFC23).

 19, 26, 32 and 40 fins/inch) and a comparison has been made with the Gewa C+ tube. The results were analysed with the Beatty and Katz theory and compared to a specific model, taking into account both gravity and surface tension effects, developed for the Gewa C+ tube.

For the bundle and for a pure fluid, the inundation of the lowest tubes has a strong effect on the Gewa C+ tubes performances contrary to the finned tubes. For the mixture the heat transfer coefficient decreases dramatically especially for Gewa C+ tube.

Introduction

The ozone depletion and the earth global warming problems force the refrigeration industry to reconsider the nature of fluids used in the refrigerating units. New environment friendly refrigerants such as HydroFluoroCarbons are more and more used either as pure fluids or as mixtures. For example, R134a (pure fluid) is regarded as suitable substitute for R12, whilst R407C (zeotropic mixture) is an alternative to R22. This transition brought us to carry out experiments to determine the Heat Transfer Coefficient (HTC) during condensation of R134a and a zeotropic mixture of R23 and R134a, on horizontal tubes with enhanced surfaces. Two types of surfaces have been studied, integral-fin tubes with trapezoidal fins and tubes with notched fins.

Theoretical models to predict the heat transfer coefficient for single low-finned tubes have been well developed since the 1940s, in particular with the pioneering work of Beatty and Katz [START_REF] Beatty | Condensation of vapors on outside of finned tubes[END_REF]. Their model assumed that the condensate is drained only via gravity. They neglected surface tension forces, which have an important role at the fin tip, since they are responsible for the draining of condensate from the tip to the fin flanks, and because the major part of condensation occurs at the fin tip. The surface tension forces also have a non-beneficial effect because of condensate retention at the underside of the tube. Theoretical models combining both gravity and surface tension forces, are used to predict the HTC of a single horizontal tube with sufficient accuracy. Rose [START_REF] Rose | An approximation equation for the vapour-side heattransfer coefficient for condensation on low-finned tubes[END_REF] proposed a semi-empirical model for a horizontal tube having trapezoidal fins. He defined an enhancement ratio ε T , as the ratio of the heat transfer coefficient for a finned tube to that for a plain tube, based on plain tube area at the fin root diameter for the same T ( T = T sat -T w ). To compare his model to experimental results, Rose used various finned tubes with different pitches, heights and diameters as well as various fluids (water, ethylene glycol, methanol, R113, R11, R12, . . . ).

The foregoing models for a single tube are not directly applicable to tube banks, since the heat transfer in the lower rows is affected by condensate inundation and the HTC is lower for these tubes. In the literature there are There are few experiments on condensation of HFC134a on low finned tubes. In the work of Blanc et al. [START_REF] Blanc | Condensation heat transfer of HCFC22 and HFC134a outside a bundle of horizontal low finned tubes[END_REF] the HTC on trapezoidal fin tubes (K26) is compared with another type of finned tube together with current theories. Honda et al. [START_REF] Honda | Condensation of HFC 134a and HFC 123 in a staggered bundle of horizontal finned tubes[END_REF] measured the row-by-row heat transfer coefficients of HFC134a condensing on a bundle of tubes having 26 fins/inch and a diameter at the fin root of 15.8 mm. Their results are slightly lower than those of Blanc et al. Cheng and Wang [START_REF] Cheng | Condensation of R-134a on enhanced tubes[END_REF] and Agrawal et al. [START_REF] Agrawal | Enhancement of heat transfer rates during condensation of refrigerants over horizontal finned tubes[END_REF] conducted experiments on condensation of HFC134a using several types of low finned tubes. The former found that for a vertical column of three tubes with trapezoidal fins no significant inundation effect is observed. The latter have studied the variation of the HTC in function of T for a single tube.

Studies covering the condensation of zeotropic mixtures are essentially confined to flat plates and smooth tubes. Hijikata and Himemo [START_REF] Hijikata | Condensation of azeotropic and nonazeotropic binary vapor mixtures[END_REF] conducted experiments using horizontal finned tubes during condensation of the binary mixture (90% R113 + 10% R114), and they found that the tube with high fins (3 mm) is better than the one with small fins (0.8 mm). Honda et al. [START_REF] Honda | Condensation of downwardflowing zeotropic mixture HFC-123/HFC-134a on a staggered bundle of horizontal low-finned tubes[END_REF] conducted experiments during condensation of a downward-flowing zeotropic mixture HFC123/HFC134a (about 9% HFC134a at the test section inlet), on a 13 × 15 (columns × rows) staggered bundle of horizontal low finned tubes. Their experimental data show that both the heat and the mass transfer coefficients increased with the row number up to the third (or the second) row, then decreased monotonically with increasing row number, finally to increase at the last row.

Experimental apparatus and reduction of data

The experimental apparatus consists of a thermosiphon refrigerant loop and a forced circulation coolant (water) loop (Fig. 1). The test rig used in this investigation is basically the same as that used in a previous study [START_REF] Belghazi | Condensation heat transfer of a pure fluid and binary mixture outside a bundle of smooth horizontal tubes, Comparison of experimental results and a classical model[END_REF].

The fluids are either a pure fluid (HFC 134a) or a mixture (HFC 23/HFC 134a). The mixture concentrations indicated in the figures refer to the composition of the refrigerant charged into the boiler.

In the refrigerant loop the vapour is generated in a boiler heated with hot water which is itself heated with an electric heater. The vapour flows toward the test section, passes vertically downwards and condenses outside the water cooled tubes. The test section (Fig. 2) is a stainless steel duct and contains a staggered copper tube bank consisting of 13 rows, each of 2 (even rows) or 3 tubes (odd rows). The cross-hatched tubes are dummies (no heat exchange), while the others are active. Half tubes are attached to vertical walls of the test section in order to eliminate vapour by-pass. A metallic rod with a diameter of 11 mm was inserted in each active tube in order to increase the water velocity. In this way the heat transfer is enhanced on the coolant side. The horizontal tube pitch is 24 mm, whereas the vertical pitch is 20 mm. The length of the tubes is 300 mm. The characteristics of the tested tubes are given in Table 1.

The vapour temperature inlet was maintained at 40 • C and the vapour velocity is less than 2 m•s -1 . The vapour temperature is measured by five thermocouples (T06, T07, T08, T09 and T10) in the test section. These thermocouples indicate the same temperatures (T sat ) during the condensation of HFC134a, in contrast to the condensation of HFC23/HFC134a, where the vapour temperature decreases from the inlet to the outlet of the test section. The temperatures indicated by the five thermocouples are interpolated in order to obtain the vapour temperature in the neighbourhood of each row.

In this study, commercially tubes (Wieland-Werke AG) are used, having a wall thickness of about 1 mm. An indirect Wilson plot method which measures the vapour-side heat transfer coefficient without measuring the wall temperature is employed.

From the measured water temperatures at the inlet and the outlet of each tube and from the measured vapour temperature in the neighbourhood of each row, the overall HTC is calculated as follows:

U = ṁc C p,c πD r ln T v,j -T c,j,in T v,j -T c,j,out (1)
where j refers to the row number. The vapour-side HTC α e is calculated by the following relation:

1 α e = 1 U - 1 α i D r D i - D r 2λ w ln D r D i ( 2
)
α i is the inner heat transfer coefficient determined with the Gnielinski correlation [START_REF] Gnielinski | New equation for heat and mass transfer in turbulent pipe and channel flow[END_REF]:

Nu G = (f/2)(Re c -1000)Pr c 1 + 12.7(f/2) 0.5 (Pr 2/3 c -1) ( 3
)
where f is the friction factor determined from the following expression:

f = 1.58 ln(Re c ) -3.28 -2 (4)
The Gnielinski correlation has been chosen because it has a wide range of applicability, 2300 < Re < 5 × 10 6 and 0.5 < Pr < 2000, thus, it covers both the transition and the turbulent flow regimes. The inner HTC α i is expressed as:

α i = B • Nu G • λ c D h (5)
where the coefficient B is determined by the Wilson plot procedure [START_REF] Wilson | A basis for rational design of heat transfer apparatus[END_REF]. Table 2 gives the B values for all tubes tested. The overall HTC U has a relative uncertainty of 15%, and the calibration of the inner HTC of K11, K19, K26, K32, K40 and Gewa C+ tubes leads respectively to relative uncertainties of 9.8%, 8.3%, 4.2%, 13% and 3%, 5%, determined using Moffat's method of sequential disturbances. The relative uncertainty of the vapour-side HTC is the quadratic mean of the inner and the overall HTCs, and is equal to 17.9%, 17.1%, 15.6%, 19.8%, 15.2%, 15.8% for the K11, K19, K26, K32, K40 and Gewa C+ tubes respectively.

Experimental results

Pure fluid (HFC 134a)

Single tube

Fig. 3 shows the evolution of the HTC of the first row with T , during condensation of pure HFC134a on tubes tested in this investigation and on smooth tube tested in a previous work [START_REF] Belghazi | Condensation heat transfer of a pure fluid and binary mixture outside a bundle of smooth horizontal tubes, Comparison of experimental results and a classical model[END_REF]. It can be seen that K32 have the best performances compared to the other trapezoidal fin tubes A fin spacing of about 0.6 mm is then an optimum to have the best heat transfer coefficient during condensation of HFC134a. Gewa C+, a notched fin tube which have almost the same fin pitch than K40, presents a HTC better than the K32 tube, because notches located at the middle of the fin height, enhance surface tension effects at the fin tip. Then, the film condensate is thinner and heat transfer performances are better.

To predict the Gewa C+ HTC a theoretical model has been developed. During condensation of pure fluids. It is based on the Nusselt approach but taking the surface tension into account.

Tube bundle

Fig. 4 shows the evolution of the heat transfer coefficient along the bundle. It can be seen that the inundation effect (impinging of deep rows by condensate flowing from upper rows) is important for Gewa C+ tubes in contrast to trapezoidal fin tubes [START_REF] Signe | Condensation de mélanges non azéotropes de fluides frigorigénes à l'extérieur d'un faisceau de tubes horizontaux[END_REF] where the deterioration of the HTC is not important. It is noticed also that the more important the Gewa C+ HTC deterioration, the more important T .

Mixture

Single tube

Fig. 5 shows the evolution of the HTC during condensation of two compositions (3% and 6% HFC23) of the binary mixture HFC23/HFC134a on K19 and Gewa C+ tubes. We note a deterioration of the mixture HTC because of the mass transfer induced by the more volatile component (HFC23), which accumulates in the liquid-vapour interface and constitutes a diffusion layer which acts as a thermal resistance.

Contrary to the case of the pure fluid it can be seen from Fig. 5 that the Gewa C+ has the same HTC as the one of the K19 tube. Indeed, as Gewa C+ tube has low fins compared to the other fin tubes, its fins are flooded by the diffusion layer (Fig. 6), and its performances are deteriorated. For the composition (6% HFC 23) and for low T 's, where the diffusion layer controls the heat transfer, the K19 and Gewa C+ HTC values tend to the smooth tube HTC values, because the diffusion layer is very thick and screens the fins, then the finned tube is seen as a smooth tube.

Tube bundle

During condensation of mixture outside Gewa C+ bundle the HTC may increase or decrease from the top to the bottom of the tube bank (Fig. 7), depending on the heat flux. Such effect has already been observed by Signe [START_REF] Signe | Condensation de mélanges non azéotropes de fluides frigorigénes à l'extérieur d'un faisceau de tubes horizontaux[END_REF] who noted that the HTC on a bundle of smooth tubes increases throughout the bundle. This effect was explained by the fact that the condensate formed on upper rows disturb the diffusion layer, which gives an amelioration of the heat transfer.

Theoretical model

As shown in Fig. 8, the Beatty and Katz model underestimates the HTC of the Gewa C+ tube, because it neglects surface tension effects, which are enhanced by notches located at the fin flanks (Fig. 10). To develop a model predicting the HTC on the Gewa C+ tube, the tube circumference have been divided into a flooded part and an unflooded part.

The flooded part located at the bottom of the tube is referred as an area submerged completely by the condensate because of the capillary retention. We assume that there is no heat transfer in this part of the tube. The retention angle Φ (Fig. 9) is calculated by the Rudy and Webb equation [START_REF] Rudy | An analytical model to predict condensate retention on horizontal integral-fin tubes[END_REF]:

Φ = cos -1 4σ cos θ ρgbD e -1 (6)
the unflooded region has been divided into four regions:

Region I: referred as the upper part of the fin, located above notches; Region II: fin area beneath notches; Region III: fin area located between notches; Region IV: the interfin channel.

It was assumed that the condensate is drained by surface tension in regions I and III, and by gravity in regions II and IV.

The model is based on the Nusselt expressions for condensation on a vertical plate (Eq. ( 7)), and on horizontal smooth tube (Eq. ( 8))

α plate = 0.943 λ 3 ρ (ρ -ρ v )g h v µ (T sat -T w )L 1/4 ( 7
)
α tube = 0.728

λ 3 ρ (ρ -ρ v )g h v µ (T sat -T w )D 1/4 ( 8
)
where L and D are plate length and tube diameter, respectively. Indeed, for regions I and III where only surface tension forces drain condensate, we modify Nusselt's expression (Eq. ( 7)) by replacing ρ g which is a gravity volume force, by an equivalent in terms of surface tension. Generally the condensation of a fluid having a liquidvapour interface radius, varying along the wall, induces a pressure variation along the liquid-vapour interface following the equation [START_REF] Gregorig | Film condensation on finely rippled surfaces with consideration of surface tension[END_REF]:

dP ds = d(σ/r) ds (9)
Eq. ( 9) has been expressed in term of a volume force and gives:

dF σ = σ d(1/r) ds dV ( 10
)
where dV is a condensate volume control.

For a condensate film surface having two different radius r t and r b as shown in Fig. 11, the curvature derivative can be written as:

d(1/r) ds ≈ 1 h 1 r b - 1 r t ( 11
)
where h is the fin height. Combining Eqs. ( 10) and ( 11) we have:

dF σ dV ≈ σ h 1 r b - 1 r t ( 12
)
The Gewa C+ HTC, calculated with reference to the surface of tube having D r as a diameter, is a function of the HTCs of regions I, II, III and IV and is given by:

α p A r = η(α I A 1 + α II A II + α III A III ) + α IV A IV ( 13
)
where η is the fin efficiency,

A I = 2 Φ π Kwe 1 /p ( 14
)
A II = 2 Φ π Kwe 2 /p (15) A III = 2 Φ π Kzh/p = 2 Φ π
Kz(e 1 + e 2 + e 3 )/p (16)

A IV = Φ π KπD r b/p (17) A r = πD r ,
where K is the notch number on one side of a fin flank and p is the fin pitch. Assuming that (Fig. 10):

r t = t/2, r b1 = b/4, r b2 = b/2
The acting force in the region I is given by:

dF σ dV Region I = σ e 1 1 r t + 1 r b1 = σ e 1 2 t + 4 b (18)
substituting ρg in Eq. ( 7) by Eq. ( 18) leads to:

α I = 0.943 λ 3 (ρ -ρ v ) h v µ (T sat -T w )e 1 1/4 σ e 1 2 t + 4 b 1/4 (19)
In the same manner in region III:

dF σ dV Region III = σ h 1 r t + 1 r b2 = 2σ h 1 t + 1 b (20) α III = 0.943 λ 3 (ρ -ρ v ) h v µ (T sat -T w )h 1/4 2σ h 1 t + 1 b 1/4 (21)
In region II, the condensate is drained by gravity and the HTC is given by:

α II = 0.943 λ 3 ρ (ρ -ρ v )g h v µ (T sat -T w )e 2 1/4 (22)
The HTC in the interfin channel is given using Eq. ( 8) as follows: Fig. 8 shows that the present model predicts very accurately the Gewa C+ HTC, without fitting any parameter. The deviation from experimental results is less than 10%.

α channel = 0.728 λ 3 ρ (ρ -ρ v )g h v µ (T sat -T w )D r 1/4 (23)

Conclusion

The present investigation brings us to draw the following concluding remarks:

Experimental data for condensation of HFC134a on five commercially available, copper, integral fin tubes, show an optimum fin spacing of about 0.6 mm. The Gewa C+ tube (notched tube) has been tested and gives an enhancement of about 30% of the best integral fin tubes (K32). Notches drawn on fin flanks enhance the surface tension effect. The inundation effect during condensation of HFC134a is important for high T .

For the mixtures, experimental results show that, due to the zeotropic character of the mixture tested (HFC23/ HFC134a), the HTC is deteriorated dramatically. Contrary to the pure fluid case, the Gewa C+ tube gives no enhancement compared to the K19 tube.

The model developed for a single Gewa C+ tube, takes into account surface tension effects, and predicts accurately the present experimental data.

Work is currently in progress to develop a model, based on the condensation curve method, predicting heat transfer during condensation of binary mixtures outside a bundle of Gewa C+.

Fig. 1 .

 1 Fig. 1. Test rig.

Fig. 2 .

 2 Fig. 2. Test section.

Fig. 3 .

 3 Fig. 3. HTC of the first row during condensation of HFC134a.

Fig. 4 .

 4 Fig. 4. Evolution of the Gewa C+ HTC along the bundle during condensation of HFC 134a.

Fig. 5 .

 5 Fig. 5. Evolution of mixture's HTC on Gewa C+ and K19 tubes.

Fig. 6 .

 6 Fig. 6. Flooding of fins by the diffusion layer.

Fig. 7 .

 7 Fig. 7. Evolution of the ratio α j /α 1 in function of the row number.

Fig. 8 .

 8 Fig. 8. Experimental and predicted HTC during condensation of HFC 134a on Gewa C+ and K32.

Fig. 9 .

 9 Fig. 9. Retention angle.

Fig. 10 .-where

 10 Fig. 10. The shape of Gewa C+ tube.

Fig. 11 .

 11 Fig. 11. Variation of the vapour interface radius.

  Nomenclature A heat exchange surface area . . . . . . . . . . . . . . . m 2 b fin spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m C p heat capacity . . . . . . . . . . . . . . . . . . . . J•kg -1 •K -1 D e diameter at the fin tip . . . . . . . . . . . . . . . . . . . . . m D h hydraulic diameter . . . . . . . . . . . . . . . . . . . . . . . . m D i inner diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . m D r diameter at the fin root . . . . . . . . . . . . . . . . . . . . m f friction factor F force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N g gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m•s -2 h fin height . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m K notches number on each fin flanks L plate length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m m mass flow rate . . . . . . . . . . . . . . . . . . . . . . . kg•s -1

			α p	predicted HTC . . . . . . . . . . . . . . . . . W•m -2 •K -1
			T h v	(T sat -T w ) or (T v,j -T w ) . . . . . . . . . . . . . . . . . K latent heat . . . . . . . . . . . . . . . . . . . . . . . . . . . J•kg -1
			Φ	retention angle . . . . . . . . . . . . . . . . . . . . . . . . . . rad
			Γ	condensate mass flow rate . . . . . . . W•m -1 •K -1
			µ	dynamic viscosity . . . . . . . . . . . . . . . . . . . . . . Pa•s
			ν	kinematic viscosity . . . . . . . . . . . . . . . . . . m 2 •s -1
			ρ	density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kg•m -3
			σ	surface tension . . . . . . . . . . . . . . . . . . . . . . N•m -1
			Indices
			I, II, III, IV region's index
			c	coolant
			e	external
		. . m	f	film condensate
	P	pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pa	in	water inlet
	r	curvature radius . . . . . . . . . . . . . . . . . . . . . . . . . . m	j	row index
	s	curvilinear coordinate . . . . . . . . . . . . . . . . . . . . . m		liquid
	t	fin thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m	out	water outlet
	U	overall HTC. . . . . . . . . . . . . . . . . . . . W•m -2 •K -1	v	vapour
	w	notch width . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . m	w	w all
	z	space between notches . . . . . . . . . . . . . . . . . . . . m	Dimensionless numbers
	Greek symbols	Nu	Nusselt number
	α e α i	vapour side HTC . . . . . . . . . . . . . . . W•m -2 •K -1 inner HTC . . . . . . . . . . . . . . . . . . . . . W•m -2 •K -1	Pr Re	Prandtl number Reynolds number
	several approaches to this problem. The simplest consists
	of multiplication of the heat transfer for a single tube by a
	factor less than unity, taking into account the row position in
	the bundle.

p fin pitch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 1

 1 Tube dimensions (mm)

	Tube	D r	D e	D i	p	h	t
	Smooth	16.8	16.8	14.2	-	-	-
	K11	16	18.9	14	2.31	1.45	0.38
	K19	16	18.9	14	1.34	1.45	0.33
	K26	15.8	18.8	14	0.97	1.5	0 .25
	K32	16.2	18.8	14	0.82	1.3	0 .2
	K40	16.3	18.9	14.4	0 .635	1.3	0 .16
	Gewa C+	17	18.9	14	∼0.7	0 .96	∼0.12

Table 2

 2

	B values
	Tube	Gewa C+	K40	K32	K26	K19	K11
	B	1.07	1.19	0.99	0.98	1.08	0.92