Method and electronic device for determining a shape of fins of a finned tube for use in a heat exchanger, related computer program and manufacturing method

Abstract

The invention concerns a determination method for determining a shape of fins (15) of a finned tube, comprising: + for an initial shape of the fin and for each value of a set of predefined values of an external radius of the fin: - computing, for several values of an angular section of the fin: + a simulated heat flux from the first fluid (F1 ) to a group of fin(s) and a tube (14), via a thermal model coupled with a CFD model; + a heat efficiency depending on the heat flux and a dimensional variable of the group of fin(s) and the tube, - assessing the highest heat efficiency and the associated angular section value; + selecting the external radius value (Re_opt) corresponding to the maximal value among the highest heat efficiencies, and then the angular section value (0opt max) associated to said maximal heat efficiency, + calculating a final shape (Sfinai) of the fin by modifying the initial shape according to the selected external radius and angular section values.

Description

Method and electronic device for determining a shape of fins of a finned tube for use in a heat exchanger, related computer program and manufacturing methodFIELD OF THE INVENTIONThe present invention concerns a determination method for determining a shape of fins of a finned tube for use in a heat exchanger.The invention also relates to a computer program including software instructions which, when executed by a processor, implement such a determination method.The invention also concerns an electronic determination device for determining a shape of fins of a finned tube for use in a heat exchanger.The invention also relates to a manufacturing method for manufacturing a finned tube for use in a heat exchanger, a shape of each fin being determined using such a determination method.The invention concerns the field of heat exchangers and more particularly of heat exchangers with multiple rows of finned tubes, such as heat exchangers used in heat-recovery systems, for example for recovery of heat from gas turbines.BACKGROUND OF THE INVENTIONSuch heat-recovery systems recover the heat contained in the hot flue gases leaving gas turbines, furnaces or boilers. The heat is exchanged between a gas at near atmospheric pressure (flue gas side) and a working fluid to be heated or vaporized (liquid or steam water, supercritical CO2, synthetic oil, etc.). By nature, exchanges with an atmospheric gas lead to low heat exchange coefficients, very large exchange surfaces and ultimately very high volumes of equipment, and therefore important mass of equipment.This size and the weight is generally detrimental to the deployment of such heatrecovery systems, particularly in offshore oil and gas production. The heat recovery system can be separated into several components that each contribute to the total weight. Increasing the compactness of the whole is achieved by increasing the compactness of the individual components. Here the finned tubes is a critical component whose purpose is to retrieve a maximum amount of heat available in the hot fumes from the gas turbine and to transfer it to heat up a colder fluid. The finned tube is a tube surrounded by fins spaced at regular intervals. For tubes with a standard diameter (approximately 50 mm in this type of heat exchanger), the mass of the fins alone could count for as much or more than the mass of the bare tubes.Inefficient or non-optimized fins obviously contribute to the degradation of the heat flux density ratio per unit of equipment mass.The article “Shape optimization of air-cooled finned-tube heat exchangers” from H. Nemati and al, published in 2019, indicates that the use of annular fins in air-cooled heat exchangers is a well-known solution, commonly used in air-conditioning and heat-recovery systems, for enhancing the air-side heat transfer. Although associated with additional material and manufacturing costs, custom-designed finned-tube heat exchangers can be cost- effective.In this article, the shape of the annular fins in a multi-row air heat exchanger is optimized to enhance the performance without incurring a manufacturing cost penalty. The air-side heat transfer, pressure drop and entropy generation in a regular, four-row heat exchanger are predicted using a steady-state turbulent CFD model and validated against experimental data. The validated simulation tool is then used to perform model-based optimization of the fin shapes. This article then describes optimizing the shape of each fin row individually, resulting in a non-homogenous custom bundle of tubes.However, the shape of the fins may be further optimized.SUMMARY OF THE INVENTIONAn object of the invention is therefore to provide a method and an associated electronic device for determining a shape of fins of a finned tube, which allows a better efficiency by optimizing the heat flux density per mass unit of the fin, in particular by improving the ratio of the heat flux over the volume of the fin, as compared to an initial shape.For this purpose, the subject-matter of the invention is a determination method for determining a shape of fins of a finned tube for use in a heat exchanger, the finned tube including a tube and several fins and being configured to transfer heat between a first fluid flowing around the fins and a second fluid flowing within the tube, the tube extending in a longitudinal direction corresponding to the flowing direction of the second fluid, each fin being arranged parallel to a transverse plane perpendicular to the longitudinal direction, each fin having an external radius and an internal radius in the transverse plane, the internal radius of the fin being equal to an external radius of the tube in the transverse plane, the tube having an internal radius in the transverse plane, the first fluid flowing substantially perpendicular to the longitudinal direction, the determination method being implemented by an electronic determination device and comprising the following steps:+ for an initial shape of the fin and for each value of a set of predefined values of the external radius of the fin:- computing, for several predefined values of an angular section of the fin along the transverse plane:+ a simulated heat flux transferred from the first fluid to a group of fin(s) and the tube, via a thermal model of heat transfer in the group of fin(s) and the tube, coupled with a computational fluid-dynamic model of the first fluid flow external to the tube; and then+ a heat efficiency depending on the computed simulated heat flux and a dimensional variable of the group of fin(s) and the tube,- assessing, among the predefined values of angular section, the highest computed heat efficiency and the associated angular section value;+ selecting, among the set of predefined values of the external radius, the external radius value corresponding to the computed heat efficiency which is maximal among the assessed highest computed heat efficiencies, and then the angular section value associated to said computed maximal heat efficiency,+ calculating a final shape of the fin by modifying the initial shape according to the selected external radius value and to the selected angular section value.According to other advantageous aspects of the invention, the determination method comprises one or several of the following features, taken individually or according to any technically possible combination:- during computing the simulated heat flux, a velocity of the first fluid, a temperature of the first fluid and a pressure of the first fluid are predefined values;- during computing the simulated heat flux, a temperature on the inner radius of the tube is predefined;- during computing the simulated heat flux, a temperature on the inner radius of the tube is also computed via the thermal model coupled with the computational fluid-dynamic model;- during computing the simulated heat flux, the group of fin(s) includes only one fin, and the computational fluid-dynamic model is a Reynolds Averaged Navier-Stokes model;- during computing the simulated heat flux, the group of fin(s) includes several fins, and the computational fluid-dynamic model is a Large Eddy Simulations model;- during computing the simulated heat flux, the heat flux is supposed symmetric with respect to a median plane of the finned tube, the median plane being perpendicular to the transverse plane and dividing the angular section into two angular half-sections, and the simulated heat flux is computed for only one of the two angular half-sections;- the heat efficiency is a heat mass efficiency, and the dimensional variable is a mass of the group of fin(s) and the tube; the heat mass efficiency being preferably equal to a ratio of the computed simulated heat flux divided by the mass of the group of fin(s) and the tube;- the heat efficiency is a heat volume efficiency, and the dimensional variable is a volume of the group of fin(s) and the tube;the heat volume efficiency being preferably equal to a ratio of the computed simulated heat flux divided by the volume of the group of fin(s) and the tube;- the computational fluid-dynamic model is an unsteady-state model;- the initial shape is selected from the group consisting of: circular; elliptical; and serrated; and- along a longitudinal plane, perpendicular to the transverse plane and including the longitudinal direction, the fin has a longitudinal cross-section, and the longitudinal crosssection has a shape selected from the group consisting of: rectangular; triangular; and pointed with half-parabolic side edges.The subject-matter of the invention is also a computer program including software instructions which, when executed by a processor, implement a determination method as defined above.The subject-matter of the invention is also an electronic determination device for determining a shape of fins of a finned tube for use in a heat exchanger, the finned tube including a tube and several fins and being configured to transfer heat between a first fluid flowing around the fins and a second fluid flowing within the tube, the tube extending in a longitudinal direction corresponding to the flowing direction of the second fluid, each fin being arranged parallel to a transverse plane perpendicular to the longitudinal direction, each fin having an external radius and an internal radius in the transverse plane, the internal radius of the fin being equal to an external radius of the tube in the transverse plane, the tube having an internal radius in the transverse plane, the first fluid flowing substantially perpendicular to the longitudinal direction, the electronic determination device comprising:- a computing module configured, for an initial shape of the fin, for each value of a set of predefined values of the external radius of the fin, and for several predefined values of an angular section of the fin along the transverse plane, for computing:+ a simulated heat flux transferred from the first fluid to a group of fin(s) and the tube, via a thermal model of heat transfer in the group of fin(s) and the tube, coupled with a computational fluid-dynamic model of the first fluid flow external to the tube; and then+ a heat efficiency depending on the computed simulated heat flux and a dimensional variable of the group of fin(s) and the tube;- an assessing module configured, for each value of the set of predefined values of the external radius, for assessing, among the predefined values of angular section, the highest computed heat efficiency and the associated angular section value;- a selecting module configured for selecting, among the set of predefined values of the external radius, the external radius value corresponding to the computed heat efficiency whichis maximal among the assessed highest computed heat efficiencies, and then the angular section value associated to said computed heat efficiency,- a calculating module configured for calculating a final shape of the fin by modifying the initial shape according to the selected external radius value and to the selected angular section value.The subject-matter of the invention is also a manufacturing method for manufacturing a finned tube for use in a heat exchanger, the finned tube including several fins and being configured to transfer heat between a first fluid flowing around the fins and a second fluid flowing within the tube, the tube extending in a longitudinal direction corresponding to the flowing direction of the second fluid, each fin being arranged parallel to a transverse plane perpendicular to the longitudinal direction, each fin having an external radius and an internal radius in the transverse plane, the internal radius of the fin being equal to an external radius of the tube in the transverse plane, the tube having an internal radius in the transverse plane, the first fluid flowing substantially perpendicular to the longitudinal direction, wherein a shape of each fin is determined using a determination method as defined above.BRIEF DESCRIPTION OF THE DRAWINGSThe invention will be better understood upon reading of the following description, which is given solely by way of example and with reference to the appended drawings, wherein:- Figure 1 is a perspective partial view of a heat exchanger with several finned tubes, each finned tube including several fins and being configured to transfer heat between a first fluid flowing around the fins and a second fluid flowing within the tube, the tube extending in a longitudinal direction corresponding to the flowing direction of the second fluid, each fin being arranged parallel to a transverse plane perpendicular to the longitudinal direction, each fin having an external radius and an internal radius in the transverse plane, the internal radius of the fin being equal to an external radius of the tube in the transverse plane, the tube having an internal radius in the transverse plane, the first fluid flowing substantially perpendicular to the longitudinal direction;- Figure 2 shows different forms for an initial shape of the fin, namely circular fin (left) and serrated fin (right);- Figure 3 shows different forms for a longitudinal cross-section of the fin, along a longitudinal plane, perpendicular to the transverse plane and including the longitudinal direction;- Figure 4 is a schematic view of an electronic determination device, according to the invention, for determining a shape of fins of the finned tube;- Figure 5 is an organigram of a determination method, according to the invention, for determining the shape of fins of the finned tube, the determination method being implemented by the electronic determination device of Figure 4;- Figure 6 shows respectively the external radius, a simulated angular heat flux, a simulated heat flux, and a heat efficiency, computed for a first value of a set of predefined values of the external radius of the fin;- Figures 7 and 8 are views similar to the one of Figure 6 for two others values of the set of predefined values of the external radius;- Figure 9 shows a curve representing the heat efficiencies computed for the successive values of the set of predefined values of the external radius; and- Figure 10 is a perspective view of a final shape of the fin calculated by modifying the initial shape of Figure 2, according to a selected external radius value and to a selected angular section value.In the following of the description, the expression "substantially equal to" defines a relationship of equality plus or minus 10%, preferably plus or minus 5%.In Figure 1 , a heat exchanger 10 comprises a plurality of finned tubes 12, substantially parallel to each other along a longitudinal direction X. In other words, the heat exchanger 10 comprises multiple rows of finned tubes 12, said rows extending along the longitudinal direction X.Each finned tube 12 includes a tube 14 and several fins 15 fixed perpendicularly to the tube 14. The fins 15 are for example welded to the tube 14. Each finned tube 12 is configured to transfer heat between a first fluid F1 flowing around the fins 15 and a second fluid flowing within the tube 14, as shown in Figure 2.The first fluid F1 flows substantially perpendicular to the longitudinal direction X. The first fluid F1 is typically a gas at near atmospheric pressure, such as a leaving a gas turbine, a furnace or a boiler.The second fluid F2 flows substantially along the longitudinal direction X. The second fluid F2 is typically a working fluid to be heated or vaporized, such as liquid or steam water, supercritical CO2, synthetic oil, etc.The tube 14 extends in the longitudinal direction X corresponding to the flowing direction of the second fluid F2. The tube 14 is typically made of metal, for example of steel, such as steel of P11 grade which usually contains carbon, chromium, manganese or molybdenum and which provide strength, resistance against wear, hardenability as well as corrosion resistance properties.The tube 14 has an internal radius Ro in the transverse plane T and an external radius corresponding to an internal radius R, of the fin 15 in the transverse plane T, as shown inFigure 2. For sake of simplicity, the external radius of the tube 14 is noted R,, likewise the internal radius R, of the fin 15.The external radius of the tube 14, is strictly greater than to the internal radius Ro of the tube 14. The external radius of the tube 14 is, for example, comprised between 20 mm and 28 mm, preferably substantially equal to 24 mm. The internal radius Ro of the tube 14 is, for example, comprised between 16 mm and 24 mm, preferably substantially equal to 20 mm.Each fin 15 is arranged parallel to a transverse plane T perpendicular to the longitudinal direction X. Each fin 15 is therefore substantially perpendicular to the tube 14 to which it is attached. Each fin 15 is typically made of metal, for example of steel, such as steel of high conductivity. The conductivity of the steel for the fin 15 is preferably greater than the conductivity of the steel for the tube 14.Each fin 15 has an external radius Re, visible in Figure 2, and the internal radius R, in the transverse plane T, the internal radius R, of the fin 15 being equal to the external radius of the tube 14 in the transverse plane T, as above mentioned.The external radius Reof the fin 15 is greater than or equal to the internal radius R, of the fin 15. Typical values of the external radius Reof the fin 15 will be defined hereinafter. The internal radius R, of the fin 15 is, for example, comprised between 20 mm and 28 mm, preferably substantially equal to 24 mm.Along a longitudinal plane L (shown in Figure 2), perpendicular to the transverse plane T and including the longitudinal direction X, each fin 15 has a longitudinal cross-section. The longitudinal cross-section is for example rectangular, corresponding to a first cross-section CS1 ; or triangular, corresponding to a second cross-section CS2; or else pointed with halfparabolic side edges, corresponding to a third cross-section CS3.In figure 3, the form of the fin 15 is shown from a lateral point of view for the first CS1 , second CS2 and third CS3 cross-sections.Each fin 15 has a width W along the longitudinal direction X. When the fin 15 has the first rectangular longitudinal cross-section CS1 , the width W is for example comprised between 1.0 mm and 1.5 mm, and preferably substantially equal to 1.3 mm. When the fin 15 has the second triangular longitudinal cross-section CS2 or the third pointed longitudinal cross-section CS3, the width W typically varies from a first width W1 at the internal radius R, of the fin 15 to a second width W2 external radius Reof the fin 15. The first width W 1 is for example comprised between 1.5 mm and 2.5 mm, and preferably substantially equal to 2 mm. The second width W2 is smaller than the first width W1 , and the second width W2 is for example comprised between 0.25 mm and 0.75 mm, and preferably substantially equal to 0.5 mm.The fins 15 of a respective finned tube 12 are advantageously evenly distributed along the longitudinal direction X. The distance E between two successive fins 15 along thislongitudinal direction X is, for example, comprised between 3 mm and 8 mm, preferably substantially equal to 5 mm.Initially, before determining an optimized shape according to the invention, each fin 15 has an initial shape in the transverse plane T. Said initial shape is typically chosen among a first shape S1 which is circular, a second shape S2 which is serrated and a third shape which is elliptical. The first shape S1 and second shape S2 are shown in Figure 2.In Figure 4, an electronic determination device 20 for determining a shape of fins 15 of the finned tube 12 comprises a computing module 30, an assessing module 32, a selecting module 34 and a calculating module 36.In the example of Figure 4, the electronic determination device 20 includes a processing unit 40 formed for example of a memory 42 and of a processor 44 coupled to the memory 42. In this example, the electronic determination device 20 also includes a display screen and input / output means, not shown, such as a keyboard and a mouse, each being connected to the processing unit 40.In the example of Figure 1 , the computing module 30, the assessing module 32, the selecting module 34 and the calculating module 36 are for example each realized, i.e. implemented, as a software executable by the processor 54. Accordingly, the memory 42 of the processing unit 40 is adapted to store a computing software, an assessing software, a selecting software and a calculating software. The processor 44 of the processing unit 40 is then configured to execute the computing software, the assessing software, the selecting software and the calculating software.As a variant not shown, the computing module 30, the assessing module 32, the selecting module 34 and the calculating module 36 are each in the form of a programmable logic component, such as a Field Programmable Gate Array or FPGA, or in the form of a dedicated integrated circuit, such as an Application Specific integrated Circuit or ASIC.When the electronic determination device 20 is in the form of one or more software programs, i.e. in the form of a computer program, it is also capable of being recorded on a computer-readable medium, not shown. The computer-readable medium is, for example, a medium capable of storing electronic instructions and being coupled to a bus of a computer system. For example, the readable medium is an optical disk, a magneto-optical disk, a ROM memory, a RAM memory, any type of non-volatile memory (for example EPROM, EEPROM, FLASH, NVRAM), a magnetic card or an optical card. A computer program with software instructions is then stored on the readable medium.The computing module 30 is configured, for an initial shape S it of the fin, for each value of a set of predefined values of the external radius Reof the fin, and for several predefined values of an angular section 0 of the fin 15 along the transverse plane T, for computing a simulated heat flux transferred from the first fluid F1 to a group of fin(s) 15 and the tube 14,via a thermal model of heat transfer in the group of fin(s) 15 and the tube 14, coupled with a computational fluid-dynamic model of the first fluid F1 flow external to the tube 14.The thermal model is adapted to model the thermal transfer in the solid part of the finned tube 12, namely in the solid part of the group of fin(s) 15 and the tube 14. The thermal model is known per se and is for example provided in any one of the following tools:- the tool Ansys Fluent provided by the company Ansys, Ansys Fluent is a fluid simulation software with physics modeling capabilities;- the tool COMSOL Mutliphysics ® provided by the company COMSOL. COMSOL Multiphysics® is a simulation software bringing fully coupled multiphysics and single-physics modeling capabilities; and- the tool AVTP provided by CERFACS (Centre Europeen de Recherche et de Formation Avancee en Calcul Scientifique). AVTP is a heat transfer solver for solids.The computational fluid-dynamic model, called hereinafter CFD model, is adapted to model the flow of the first fluid F1 which is external to the tube 14. The CFD model is known per se and is for example provided in any one of the following tools:- the tool Ansys Fluent provided by the company Ansys;- the tool Ansys CFX provided by the company Ansys. Ansys CFX is a CFD software for turbomachinery applications;- the tool YALES2 provided by CNRS (Centre National de la Recherche Scientifique). Yales2 software package, developed by the Coria laboratory with academic and industrial partners, allows to simulate complex fluid dynamics phenomena such as unsteady turbulent or two-phase flows;- the tool Open Foam which is Open FOAM is an open source CFD software developed primarily by OpenCFD Ltd;- the tool Simcenter STAR-CCM+ provided by the company Siemens. Simcenter STAR- CCM+ is a multiphysics computational fluid dynamics (CFD) simulation software; and- the tool AVBP provided by CERFACS. AVBP is a simulation software aimed at solving the compressible Navier-Stokes equations for the fluid flow.According to a first example of computation, the group of fin(s) 15 includes only one fin, and the CFD model is a Reynolds Averaged Navier-Stokes model, also called RANS model. The RANS model is advantageously a steady-state model.According to a second example of computation, the group of fin(s) 15 includes several fins 15, and the CFD model is a Large Eddy Simulations model, also called LES model. The LES model is advantageously an unsteady-state model. According to this second example, the group of fin(s) 15 typically includes at least 4 fins. According to this second example, periodic boundary condition or symmetry boundary conditions are advantageously defined on lateral faces of a computation domain.For each example of computation, the computation domain for modeling is typically in the form of a rectangular parallelepiped extending along the direction of flow of the first fluid F1 , with inlet and outlet faces substantially perpendicular to said flow direction, the inlet face being located upstream of the group of fin(s) 15 with respect to the flow orientation of the first fluid F1 , and the outlet face being downstream of the group of fin(s) 15. The lateral faces of the computation domain are substantially parallel to the transverse plane T and arranged on either side of the fin group. Horizontal faces of the computation domain are substantially parallel to the longitudinal plane L and arranged above and below the group of fin(s) 15, when the computation is made for the global angular section 0. The skilled person will note that when the computation is carried out for an angular half-section 0, that will be described in more detail hereinafter, then the horizontal faces of the computation domain are located above and in the middle of the group of fin(s) 15. The horizontal face in the middle of the group of fin(s) 15 then corresponds to the median plane MP, shown in Figure 10.The coupling of the thermal model with the CFD model is also known per se and is for example done by any one of the following coupling tools:- the tool CWIPI (Coupling With Interpolation Parallel Interface) provided by ONERA. CWIPI is a coupling library adapted for coupling the thermal model with the CFD model; and- the tool OpenPALM provided by CERFACS and ONERA.The skilled person will observe the coupling tool is necessary only if two different tools are used for the CFD model and thermal model. No coupling tool is needed if the thermal and CFD models are solved within the same tool (as with the tool Ansys Fluent for example).During computing the simulated heat flux <t>, a velocity U1 of the first fluid F1 , a temperature T1 of the first fluid F1 and a pressure P1 of the first fluid F1 are advantageously predefined values.During computing the simulated heat flux <t>, a temperature on the inner radius Ro of the tube 14 is for example predefined.Alternatively, the computing module 30 is further configured for computing also a temperature on the inner radius Ro of the tube 14 via the thermal model coupled with the CFD model.The computing module 30 is advantageously configured for firstly computing a fin angular flux cp corresponding to an angular intensity of the flux, namely a heat flux received by an angular portion with a predefined angle at a given angle value a. The fin angular flux cp is typically expressed in Watt per degree, also denoted W / deg or W / °. The value of the predefined angle is for example comprised between 0.5° and 2°, and preferably substantially equal to 1 °.The fin angular flux cp verifies for example the following equation:

[0001] where cp represents the fin angular flux, a represents the given angle value where the angular portion is position with respect to the global angular section 0,Rerepresents the external radius of the fin 15, i represents the internal radius of the fin 15,<p represents the heat flux received by the angular portion, the heat flux being obtained via the thermal model coupled with the CFD model.The computing module 30 is advantageously configured for secondly computing the simulated heat flux corresponding to the integrated angular flux intensity as a function of the angular section 0. The simulated heat fluxis typically expressed in Watt, also denoted W.The simulated heat flux verifies for example the following equation:[2]where represents the simulated heat flux,0 represents the angular section, cp represents the fin angular flux, such as the one according to equation [1], a represents the angle value where the angular portion (for which the fin angular flux was computed) is position with respect to the global angular section 0.According to an optional addition, during computing the simulated heat flux <t>, the heat flux is supposed symmetric with respect to a median plane MP of the finned tube 12, shown in Figure 10, the median plane MP being perpendicular to the transverse plane T and dividing the angular section 0 into two angular half-sections, and the simulated heat flux is computed for only one angular half-section, denoted 0, of the two angular half-sections.According to this optional addition, the simulated heat fluxverifies for example the following equation:[3]where represents the simulated heat flux,0 represents the angular half-section, cp represents the fin angular flux, such as the one according to equation [1], a represents the angle value where the angular portion (for which the fin angular flux was computed) is position with respect to the angular half-section 0.The skilled person will therefore understand that the simulated heat fluxrepresents the flux received by the fin 15 from the reference zero angle corresponding to the upstream up to an azimuthal angle corresponding the angular half-section 0, the azimuthal angle being also equal to 0. The difference between this azimuthal angle and the zero angle indeed corresponds to the angular half-section 0.The computing module 30 is then configured, for the initial shape Sinit of the fin 15, for each value of the set of predefined values of the external radius Reof the fin 15, and for several predefined values of the angular section 0 of the fin 15 along the transverse plane T, for computing a heat efficiency q depending on the computed simulated heat fluxand on a dimensional variable of the group of fin(s) 15 and the tube 14.The skilled person will observe that when the group of fin(s) 15 includes several fins 15, the computed simulated heat fluxused for computing the heat efficiency q is a flux for a single fin 15. Accordingly, the computed simulated heat fluxis typically averaged, to be reduced to a unitary flow for a single fin, and the heat efficiency q then depends on this averaged flow, i.e. on the unitary flow for a single fin.The heat efficiency q is for example a heat mass efficiency q i, and the dimensional variable is a mass of the group of fin(s) 15 and the tube 14. The heat mass efficiency qw is typically expressed in kiloWatt per kilogram, also denoted kW / kg.According to this example, the heat mass efficiency qw is preferably equal to a ratio of the computed simulated heat flux divided by the mass of the group of fin(s) 15 and the tube 14.According to this example, the heat mass efficiency qw typically verifies the following equation:[4]where qw represents the heat mass efficiency,0 represents the angular section, represents the simulated heat flux,Ptube, resp. pnn, represents the density of the tube 14, resp. of the fin 15,Rerepresents the external radius of the fin 15,Ri represents the internal radius of the fin 15,Ro represents the internal radius of the tube 14,E represents the distance between two successive fins 15 along the longitudinal direction X,W represents the width of the fin 15 along the longitudinal direction X.According to the optional addition with computation for the angular half-section 0, the heat mass efficiency QM typically verifies the following equation:[5]where 0 represents the angular half-section, instead of the angular section 0, the other parameters being unchanged with respect to the previous equation [4], Alternatively, the heat efficiency q is a heat volume efficiency qv, and the dimensional variable is a volume of the group of fin(s) 15 and the tube 14. The heat volume efficiency qv is typically expressed in MegaWatt per m3, also denoted MW / m3.According to this alternative, the heat volume efficiency qvis preferably equal to a ratio of the computed simulated heat fluxdivided by the volume of the group of fin(s) 15 and the tube 14.According to this alternative, the heat volume efficiency qv typically verifies the following equation:[6]where qv represents the heat volume efficiency,0 represents the angular section, represents the simulated heat flux, such as the one according to equation [2],Rerepresents the external radius of the fin 15, i represents the internal radius of the fin 15,Ro represents the internal radius of the tube 14,E represents the distance between two successive fins 15 along the longitudinal direction X,W represents the width of the fin 15 along the longitudinal direction X.According to the optional addition with computation for the angular half-section 0, the heat volume efficiency qv typically verifies the following equation:[7]where 0 represents the angular half-section, instead of the angular section 0, represents the simulated heat flux, such as the one according to equation [3], the other parameters being unchanged with respect to the previous equation [6].The assessing module 32 is configured, for each value of the set of predefined values of the external radius Re, for assessing, among the predefined values of angular section 0, the highest computed heat efficiency r|opt and the associated angular section value 0opt.The assessing module 32 is typically configured, for each predefined value of the external radius Re, for comparing all the heat efficiency values computed for all the predefined values of angular section 0 and for keeping the greatest value among the compared ones. The highest computed heat efficiency r|opt is therefore this greatest value, and is also called optimal heat efficiency r|opt-According to the optional addition with computation for the angular half-section 9, the assessing module 32 is configured, for each value of the set of predefined values of the external radius Re, for assessing, among the predefined values of angular half-section 9, the highest computed heat efficiency n(9opt) and the associated angular half-section value 9opt.The selecting module 34 is configured for selecting, among the set of predefined values of the external radius Re, the external radius value Re-opt corresponding to the computed heat efficiency which is maximal among the assessed highest computed heat efficiencies r|opt, said maximal computed heat efficiency being denoted r|max. The selecting module 34 is then configured for selecting the angular section value associated to the maximal computed heat efficiency r|max, said angular section value corresponding to the maximal computed heat efficiency r|max being denoted 0opt max.The selecting module 34 is typically configured for comparing all the optimal heat efficiencies r|opt computed for all the predefined values of the external radius Reand for keeping the greatest value among the compared ones.According to the optional addition with computation for the angular half-section 9, the selecting module 34 is configured for selecting, among the set of predefined values of the external radius Re, the external radius value Re-opt corresponding to the computed heat efficiency nmax(9opt) which is maximal among the assessed highest computed heat efficiencies n(9opt), and then the angular section value 9opt max associated to said computed heat efficiency r|max(9oPt). The selected external radius value is also called optimal external radius value Re-opt.The calculating module 36 is configured for calculating a final shape Sfinai of the fin 15 by modifying the initial shape Sinit according to the selected external radius value Re-opt and to the selected angular section value 0opt max.According to the optional addition with computation for the angular half-section 9, the calculating module 36 is configured for calculating the final shape Sfinai of the fin 15 by modifying the initial shape Sinit according to the selected external radius value Re-opt and to the selected angular half-section value 9opt max, and by proceeding symmetrically with respect to the median plane MP.The operation of the electronic determination device 20 according to the invention will now be explained in view of Figure 5 representing a flowchart of a determination method according to the invention for determining a shape of fins 15 of a finned tube 12 for use in a heat exchanger 10, the determination method being implemented by the electronic determination device 20.Initially, the determination method comprises a computation phase 100 carried out for each value of the set of predefined values of the external radius Reof the fin 15. The computation phase 100 includes a computing step 110 and then an assessing step 120, these two steps 1 10 and 120 being carried out successively for each value of the set of predefined values of the external radius Re.During the computing step 1 10, the electronic determination device 20 computes, via its computing module 30 and for several predefined values of the angular section 0 of the fin 15 the simulated heat fluxtransferred from the first fluid F1 to the group of fin(s) 15 and the tube 14, via the thermal model of heat transfer in the group of fin(s) 15 and the tube 14, coupled with the CFD model of the first fluid F1 flow external to the tube 14; and then the heat efficiency q depending on the computed simulated heat flux and the dimensional variable of the group of fin(s) 15 and the tube 14.The heat efficiency q is for example the heat mass efficiency QM, and the heat mass efficiency qMis for example computed according to equation [4], or according to equation [5] in case of the optional addition with computation for the angular half-section 0.Alternatively, the heat efficiency q is the heat volume efficiency qv, and the heat volume efficiency qv is for example computed according to equation [6], or according to equation [7] in case of the optional addition with computation for the angular half-section 0.Figures 6, 7 and 8 show respectively the external radius Re, the simulated angular heat flux cp, the simulated heat flux , and the heat efficiency q, computed for three different values of the set of predefined values of the external radius Reof the fin 15.The example of Figures 6, 7 and 8 corresponds to the simulation results obtained with the velocity U1 of the first fluid F1 substantially equal to 15 m.s-1, the temperature T1 of the first fluid F1 substantially equal to 773 Kelvins (K) and the pressure P1 of the first fluid F1 substantially equal to 101325 Pascal (Pa). In this example of Figures 6, 7 and 8, a temperature To of the tube 14 on the internal radius Ro is fixed substantially equal to 423 K. In this example of Figures 6, 7 and 8, the tube 14 and the group of fin(s) 15 is made of steel with a thermal conductivity A substantially equal to 35 W / (m.K).In this example of Figures 6, 7 and 8, the set of predefined values of the external radius Reincludes six successive values, namely 24 mm, 28 mm, 32 mm, 36 mm, 40 mm and 44 mm. Figure 6 corresponds to the simulation results obtained with the minimal value of this set of predefined values, namely the external radius Reequal to 24 mm. Figure 7 corresponds tothe simulation results obtained with the optimal external radius value Re-opt for this set of predefined values, namely the external radius Reequal to 40 mm, as it will be further explained hereinafter. Figure 8 corresponds to the simulation results obtained with the maximal value of this set of predefined values, namely the external radius Reequal to 44 mm.In Figures 6, 7 and 8, the external radius Recorresponds to first curves 200, 300, 400; the simulated angular heat flux cp corresponds to second curves 210, 310, 410; the simulated heat flux corresponds to third curves 220, 320, 420; and the heat efficiency q corresponds to fourth curves 230, 330, 430. These first, second, third and fourth curves 200 to 420 are curves which are function of the azimuthal angle 0 varying from the 0° angle corresponding to the upstream of the first fluid F1 flow to the 180° angle corresponding to the downstream of the first fluid F1 flow.In this example of Figures 6, 7 and 8, the heat efficiency q is the heat volume efficiency qv, typically corresponding to equation [7],During the assessing step 120, the electronic determination device 20 assesses, via its assessing module 32 and among the predefined values of angular section 0, the highest computed heat efficiency qoptand then the associated angular section value 0opt.According to the optional addition with computation for the angular half-section 0, the assessing module 32 assesses, among the predefined values of angular half-section 0, the highest computed heat efficiency q(0opt) and the associated angular half-section value 0opt.Accordingly, in the example of Figures 6, 7 and 8, the assessing module 32 assesses the highest computed heat volume efficiency qv(0oPt) and the associated angular half-section value 0opt. In Figure 6, the highest heat volume efficiency qv(0oPt) is equal to 6.58 MW / m3and the optimal angular half-section value 0opt for this external radius Reequal to 24 mm is 180°, i.e. the full fin. In Figure 7, the highest heat volume efficiency qv(0oPt) is equal to 15.80 MW / m3and the optimal angular half-section value 0opt for this external radius Reequal to 40 mm is 110°. In Figure 8, the highest heat volume efficiency qv(0oPt) is equal to 15.12 MW / m3and the optimal angular half-section value 0opt for this external radius Reequal to 44 mm is 112°.For this example, the simulation results according to the invention are summarized in below table 1 .[Table 1]In comparison, the simulation results according to the state-of-the-art are summarized in below table 2.[Table 2]In a next step 130, the electronic determination device 20 selects, via its selecting module 34 and among the set of predefined values of the external radius Re, the optimal external radius value Re-opt corresponding to the computed heat efficiency which is maximal among the highest computed heat efficiencies qopt assessed during the assessing step 120, said maximal heat efficiency being denoted r|max.In the selecting step 130, the selecting module 34 then selects the angular section value associated to the maximal computed heat efficiency r|max, said angular section value corresponding to the maximal computed heat efficiency r|max being denoted 0opt max.According to the optional addition with computation for the angular half-section 0, the selecting module 34 selects, among the set of predefined values of the external radius Re, the optimal external radius value Re-opt corresponding to the computed heat efficiency r|max(0opt) which is maximal among the highest computed heat efficiencies q(0opt) assessed during the assessing step 12O.Then, the selecting module 34 selects the angular section value 0opt max associated to said computed heat efficiency q max (0OPt).In Figure 9 related to the example of Figures 6-8, a fifth curve 500 corresponds to the computed heat volume efficiency qv with the angular half-section 0 equal to 180°, namely for the full fin, i.e. for the initial shape Smit of the fin 15, and this for each external radius Revalue of the set of predefined external radius values. In other words, the fifth curve 500 shows the computed heat volume efficiency qv for the state-of-the-art shape of the fin 15.In Figure 9, a sixth curve 510 corresponds to the computed heat volume efficiency qv with each optimal angular half-section 0opt. In other words, the sixth curve 510 shows the computed heat volume efficiency qv for the optimal shape of the fin 15 according to the invention.The skilled person will understand that the fifth curve 500 directly derives from the last column of Table 2, while the sixth curve 510 derives from the last column of Table 1 .The skilled person will observe that the maximal computed heat efficiency nmax(0oPt) is substantially to 15.80 MW / m3for the external radius Reequal to 40 mm. The optimal externalradius value Re-opt is therefore equal to 40 mm. Turning back to Figure 7 or Table 1 , the angular section value 0opt max associated to said computed heat efficiency r|max(0oPt) is equal to 110°. Accordingly, the optimal angular half-section 0optfor the optimal external radius value Re-opt is equal to 110°.In a last step 140, the electronic determination device 20 calculates, via its calculating module 36 the final shape Sfinai of the fin 15 by modifying the initial shape Sinit according to the selected external radius value Re-opt and to the selected angular section value 0opt_max.In the calculating step 140, according to the optional addition with computation for the angular half-section 0, the calculating module 36 calculates the final shape Sfinai of the fin 15 by modifying the initial shape Sinit according to the selected external radius value Re-opt and to the selected angular half-section value 0opt_max, and by proceeding symmetrically with respect to the median plane MP.The selected angular section value 0opt max corresponding to an upstream part of the fin 15 is kept, while the downstream part of the fin 15 outside the selected angular section value 0opt max is removed. In other words, the external contour of the fin 15, i.e. the final shape Sfinai of the fin 15, is obtained by keeping only the zones with high flux intensity, i.e. with high heat efficiency, and by removing the zones with low flux intensity, i.e. with low heat efficiency.The final shape Sfinai of the fin 15 calculated in the example of Figures 6-8 is shown in Figure 10, where the upstream and the 0° angle are on the left of Figure 10, while the downstream and the 180° angle are on the right.The skilled person can easily observe the gain in efficiency achieved with the invention, by comparing curve 500 according to the state of the art with curve 510 according to the invention in Figure 9, or by comparing the last column of Table 2 according to the state of the art with the last column of Table 1 according to the invention.Thus, the electronic determination device 20 and the determination according to the invention allow a better efficiency by optimizing the heat flux density per mass unit of the fin 15, in particular by improving the ratio of the heat flux over the volume of the fin 15, as compared to the initial shape Sinit.

Claims

CLAIMS1 . A determination method for determining a shape of fins (15) of a finned tube (12) for use in a heat exchanger (10), the finned tube (12) including a tube (14) and several fins (15) and being configured to transfer heat between a first fluid (F1) flowing around the fins (15) and a second fluid (F2) flowing within the tube (14), the tube (14) extending in a longitudinal direction (X) corresponding to the flowing direction of the second fluid (F2), each fin (15) being arranged parallel to a transverse plane (T) perpendicular to the longitudinal direction (X), each fin (15) having an external radius (Re) and an internal radius (R,) in the transverse plane (T), the internal radius (R,) of the fin (15) being equal to an external radius of the tube (14) in the transverse plane (T), the tube (14) having an internal radius (Ro) in the transverse plane (T), the first fluid (F1 ) flowing substantially perpendicular to the longitudinal direction (X), the determination method being implemented by an electronic determination device (20) and comprising the following steps:+ for an initial shape (Sinit) of the fin (15) and for each value of a set of predefined values of the external radius (Re) of the fin (15):- computing (110), for several predefined values of an angular section (0) of the fin (15) along the transverse plane (T):+ a simulated heat flux ( ) transferred from the first fluid (F1 ) to a group of fin(s) (15) and the tube (14), via a thermal model of heat transfer in the group of fin(s) (15) and the tube (14), coupled with a computational fluid-dynamic model of the first fluid (F1 ) flow external to the tube (14); and then+ a heat efficiency (q) depending on the computed simulated heat flux ( ) and a dimensional variable of the group of fin(s) (15) and the tube (14),- assessing (120), among the predefined values of angular section (0), the highest computed heat efficiency (qopt) and the associated angular section value (0opt);+ selecting (130), among the set of predefined values of the external radius (Re), the external radius value (Re-opt) corresponding to the computed heat efficiency (r|max) which is maximal among the assessed highest computed heat efficiencies (r|opt), and then the angular section value (0opt max) associated to said computed maximal heat efficiency (r|max),+ calculating (140) a final shape (Sfinai) of the fin (15) by modifying the initial shape (Sinit) according to the selected external radius value (Re-opt) and to the selected angular section value (0opt_ max)-2. Method according to claim 1 , wherein during computing (1 10) the simulated heat flux (<t>), a velocity of the first fluid (F1 ), a temperature of the first fluid (F1 ) and a pressure of the first fluid (F1 ) are predefined values.

3. Method according to claim 1 or 2, wherein during computing (110) the simulated heat flux (<t>), a temperature on the inner radius (Ro) of the tube (14) is predefined.

4. Method according to claim 1 or 2, wherein during computing (110) the simulated heat flux (<t>), a temperature on the inner radius (Ro) of the tube (14) is also computed via the thermal model coupled with the computational fluid-dynamic (CFD) model.

5. Method according to any one of claims 1 to 4, wherein during computing (110) the simulated heat flux (<t>), the group of fin(s) (15) includes only one fin, and the computational fluid-dynamic (CFD) model is a Reynolds Averaged Navier-Stokes (RANS) model.

6. Method according to any one of claims 1 to 4, wherein during computing (110) the simulated heat flux (<t>), the group of fin(s) (15) includes several fins, and the computational fluid-dynamic (CFD) model is a Large Eddy Simulations (LES) model.

7. Method according to any one of the preceding claims, wherein during computing (110) the simulated heat flux (<t>), the heat flux (<t>) is supposed symmetric with respect to a median plane (MP) of the finned tube (12), the median plane (MP) being perpendicular to the transverse plane (T) and dividing the angular section (0) into two angular half-sections, and the simulated heat flux (<t>) is computed for only one (0) of the two angular half-sections.

8. Method according to any one of claims 1 to 7, wherein the heat efficiency (q) is a heat mass efficiency (qM), and the dimensional variable is a mass of the group of fin(s) (15) and the tube (14); the heat mass efficiency (HM) being preferably equal to a ratio of the computed simulated heat flux ( ) divided by the mass of the group of fin(s) (15) and the tube (14).

9. Method according to any one of claims 1 to 7, wherein the heat efficiency (q) is a heat volume efficiency (qv), and the dimensional variable is a volume of the group of fin(s) (15) and the tube (14); the heat volume efficiency (qv) being preferably equal to a ratio of the computed simulated heat flux ( ) divided by the volume of the group of fin(s) (15) and the tube (14).

10. Method according to any one of the preceding claims, wherein the computational fluid-dynamic model is an unsteady-state model.

11. Method according to any one of the preceding claims, wherein the initial shape (Sinit) is selected from the group consisting of: circular; elliptical; and serrated.

12. Method according to any one of the preceding claims, wherein along a longitudinal plane (L), perpendicular to the transverse plane (T) and including the longitudinal direction (X), the fin (15) has a longitudinal cross-section, and the longitudinal cross-section has a shape selected from the group consisting of: rectangular (CS1 ); triangular (CS2); and pointed with half-parabolic side edges (CS3).

13. A computer program including software instructions which, when executed by a processor, implement a method according to any one of the preceding claims.

14. An electronic determination device (20) for determining a shape of fins (15) of a finned tube (12) for use in a heat exchanger (10), the finned tube (12) including a tube (14) and several fins (15) and being configured to transfer heat between a first fluid (F1 ) flowing around the fins (15) and a second fluid (F2) flowing within the tube (14), the tube (14) extending in a longitudinal direction (X) corresponding to the flowing direction of the second fluid (F2), each fin (15) being arranged parallel to a transverse plane (T) perpendicular to the longitudinal direction (X), each fin (15) having an external radius (Re) and an internal radius (R,) in the transverse plane (T), the internal radius (R,) of the fin (15) being equal to an external radius of the tube (14) in the transverse plane (T), the tube (14) having an internal radius (Ro) in the transverse plane (T), the first fluid (F1 ) flowing substantially perpendicular to the longitudinal direction (X), the electronic determination device (20) comprising:- a computing module (30) configured, for an initial shape (Sinit) of the fin, for each value of a set of predefined values of the external radius (Re) of the fin, and for several predefined values of an angular section (0) of the fin (15) along the transverse plane (T), for computing:+ a simulated heat flux ( ) transferred from the first fluid (F1 ) to a group of fin(s) (15) and the tube (14), via a thermal model of heat transfer in the group of fin(s) (15) and the tube (14), coupled with a computational fluid-dynamic model of the first fluid (F1 ) flow external to the tube (14); and then+ a heat efficiency (q) depending on the computed simulated heat flux ( ) and a dimensional variable of the group of fin(s) (15) and the tube (14);- an assessing module (32) configured, for each value of the set of predefined values of the external radius (Re), for assessing, among the predefined values of angular section (0), the highest computed heat efficiency (qopt) and the associated angular section value (0opt);- a selecting module (34) configured for selecting, among the set of predefined values of the external radius (Re), the external radius value (Re_oPt) corresponding to the computed heat efficiency (r|max) which is maximal among the assessed highest computed heat efficiencies (r|opt), and then the angular section value (0opt max) associated to said computed heat efficiency (r|max),- a calculating module (36) configured for calculating a final shape (Sfinai) of the fin (15) by modifying the initial shape (Sinit) according to the selected external radius value (Re-opt) and to the selected angular section value (0opt max).

15. A manufacturing method for manufacturing a finned tube (12) for use in a heat exchanger (10), the finned tube (12) including several fins (15) and being configured to transfer heat between a first fluid (F1 ) flowing around the fins (15) and a second fluid (F2) flowing within the tube (14), the tube (14) extending in a longitudinal direction (X) corresponding to the flowing direction of the second fluid (F2), each fin (15) being arranged parallel to a transverse plane (T) perpendicular to the longitudinal direction (X), each fin (15) having an external radius (Re) and an internal radius (Rj) in the transverse plane (T), the internal radius (Ri) of the fin (15) being equal to an external radius of the tube (14) in the transverse plane (T), the tube (14) having an internal radius (Ro) in the transverse plane (T), the first fluid (F1 ) flowing substantially perpendicular to the longitudinal direction (X), wherein a shape of each fin (15) is determined using a determination method according to any one of the claims 1 to 12.

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