"July 1998."This report summarizes analyses of the effects of heat conduction in a copper block on the heat flux to a coolant flowing axially in the block. Heat is assumed to be added through one side of the block corresponding to conditions that may arise in fusion reactors or particle accelerator targets. It is found that three dimensional analysis of the heat transport will be required to accurately describe the heat flux at the wall of the coolant channel. The effects of axial and azimuthal heat conduction in the copper block depend on the block width to channel diameter ratio and the BIOT number of the channel

Lanza, Nina. Massachusetts Institute of Technology

Wang, Chunyun, 1968-

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MASSACHUSETTS INSTITUTEOF TECHNOLOGYMITNE-314GEOMETRICAL EFFECTS ON AXIAL &AZIMUTHALVARIATIONS OF HEAT FLUX TO COOLANT INASYMMETRICALLY HEATED CHANNELSChunyun WangEdited by Nina LanzaJuly 1998NUCLEAR ENGINEERINGMITNE-314GEOMETRICAL EFFECTS ON AXIAL &AZIMUTHALVARIATIONS OF HEAT FLUX TO COOLANT INASYMMETRICALLY HEATED CHANNELSChunyun WangEdited by Nina LanzaJuly 1998AbstractThis report summarizes analyses of the effects of heat conduction in a copper blockon the heat flux to a coolant flowing axially in the block. Heat is assumed to beadded through one side of the block corresponding to conditions that may arise infusion reactors or particle accelerator targets. It is found that three dimensionalanalysis of the heat transport will be required to accurately describe the heat flux atthe wall of the coolant channel. The effects of axial and azimuthal heat conductionin the copper block depend on the block width to channel diameter ratio and theBIOT number of the channel.Geometrical Effects on Axial & Azimuthal Variations of Heat Flux toCoolant in Asymmetrically Heated Channels1. PurposeThere are only few empirical relations of heat transfer in a high-velocity flow (i.e. v = 10-20 m/s) of sub-cooled water with extremely high wall heat flux (i.e. q" > 6 MW/m2). Inorder to study this relationship, an experimental apparatus was built to investigate heattransfer under these conditions, and an analysis of the geometrical effects on the coolantheat flux has been made in order to guide the experimental design.In the past, a copper block was uses for this experiment; however, this time anINCONEL 600 alloy tube was used as the test section. With the help of ADINA, thetemperature distributions for two and three-dimensions were calculated for both thecopper block and the INCONEL 600 tube at the given heat flux and heat.ADINA is a finite element software developed by ADINA R&D Inc. which is installedon Athena of MIT; it can be run at an SGI workstation. The ADINA system includes thepre-processor ADINA-IN, the structural analysis program ADINA, the heat transferprogram ADINA-T, the fluid flow program ADINA-F, and the post-processor ADINA-PLOT. The pre-processor ADINA-IN was used to define the geometry of the model, todiscover the material properties and loads, and to create meshes, elements and nodes.Using the data file created by ADINA-IN as the input file of ADINA-T, the out fileswere obtained.2. 2D and 3D ADINA models2.1 Models of the Copper BlockFig. 1 shows the schematic of the copper block: D denotes the channel diameter, Hdenotes height, W denotes width, and L denotes length. Because the copper block issymmetrical, only half the block was taken as the calculation model. The 2D ADINAmodel of the copper block is shown in Fig.2; looking at this figure, it is possible to seethat the meshes close to the channel are denser than those in other places. An incidentheat flux q;, was imposed at to the left side of the block, and heat convection was allowedbetween the top surface of the block and the air, as well as between the channel surfaceand coolant. In order to keep the node distribution agreement in each element, 9 nodeswere created for each conduction element and 3 nodes were created for each convectionelement. Loads and boundary conditions are shown in Table 1.The thermal conductivity of copper varies with temperature, and the valuesused in this experiment are shown in Table 2.1Table 1 Loads and Boundary Conditions in the 2D ADINA Model of the Copper BlockHeat flux q, 6.5 MW/m2Heat transfer coefficient at 50,000 W/(m2 *C)channel surfaceHeat transfer coefficient at top 5 W/(m2 oC)surfaceBulk coolant temperature 64 *CCoolant pressure 3 MPaAir temperature 20 *CTable 2 Values of Copper's Thermal Conductivity, as Used in the ADINA ModelTemperature (*C) k (W/m *C) Temperature (*C) k (W/m *C)0 401 527 37127 398 627 36477 394 727 357127 392 827 350227 388 927 342327 383 1027 334427 377 1085 329.36The 3D ADINA model of the copper block is shown in Fig.3. In this model, the heattransfer coefficient of the channel surface and the bulk coolant temperature are variedaxially. The other loads and boundary conditions are the same as those of the 2D model(see Table 1). The block was divided into 4 azimuthal sections of equal length; the heattransfer coefficient of the channel surface applying to each section, along the entire lengthfrom entrance to exit, are 100,000 W/(m2 *C), 83,000 W/(m2 *C), 67,000 W/(m2 *C) and50,000 W/(m2 *C) respectively. The bulk coolant temperatures for each section are 23 *C,36 *C, 50*C and 64 *C respectively. Table 3 shows the loads and boundary conditions.2.2 INCONEL 600 Tube ModelsFigures 4 and 5 show the 2D ADINA models of asymmetrical and symmetricalINCONEL tubes. As Fig.4 shows, Ri is the channel radius, Rol is the outside surfaceradius, and Ro2 is the outermost surface radius. Unlike the copper block, there is noincident heat flux in the INCONEL models, but an internal heat generation is applied. Theheat transfer coefficient at the channel surface as well as the bulk coolant temperature in2the channel are given as boundary conditions. The thermal conductivity of INCONEL isalso varied with the temperature as shown in Table 4.Table 3 Loads and Boundary Conditions in 3D ADINA Model of Copper BlockSection I Section II Section III Section IV(Entrance) (Exit)Heat flux q; (MW/m 2) 6.5 6.5 6.5 6.5Heat transfer coefficient at 100,000 83,000 67,000 50,000channel surface (W/m2 *C)Heat transfer coefficient at 5 5 5 5top surface (W/m 2 *C)Bulk coolant temperature 23 36 50 64(*C)Coolant pressure (MPa) 3.0 3.0 3.0 3.0Air temperature (*C) 20 20 20 20Table 4 INCONEL Thermal Conductivity Used in the ADINA ModelTemperature ( *C) k (W/m *C) Temperature (*C) k (W/m *C)-150 12.5 400 20.5-100 13.1 500 22.1-50 13.6 600 23.920 14.9 700 25.7100 15.9 800 27.5200 17.3 900 29.3300 19.0 1000 31.13. Results3.1 Flux Peaking Factor of the 2D Copper BlockFor the copper block, the peaking factor is defined as PF = qL, the ratio of heat flux atqinthe channel surface as qh , and the incident heat flux as q. . The azimuthal angle 0 isshown in Fig. 2. Fig. 6 depicts the dependence of the peaking factor on the azimuthalangle 0 for three cases in which the channel radii are varied; both the height and width ofthe copper block are two times the diameter of its channel. If the peaking factor is definedas PF2= h r-, then q,,, is the average heat flux at the channel surface.qch,av3Fig. 7 shows the PF2 versus the azimuthal angle 0 for three cases with different channelradii. In Figures 6 and 7, one should note that the azimuthal variation increases as theradius of the channel increases; because the geometrical properties, loads, and boundaryconditions are proportional in the three cases, the average heat flux at the channel surfacehappens at the same azimuthal angle. Fig. 8 shows the azimuthal temperature distributionat the channel surface; note that the highest temperature (204.3 *C) at the channel surfaceis much lower than the saturation temperature (233.84 *C) in these three cases.The Peaking Factor PF and Peaking Factor PF2 versus the azimuthal angle 0 are shown inFigure 9 and 10; these are for the copper block with a height of 19mm, a width of18.6mm, and a channel with a radius of 1.5 mm. The temperature at the channel surface isshown in Fig. 11; note that the azimuthal variation is within - 9%. When the block has aheight and width of 6mm each and a radius of 1.5mm, the azimuthal variation is a factor oftwo for both the peak and the minimum heat flux, and the temperature of any point onthe channel surface is at least 73 *C higher than the saturation temperature.3.2 Effect of the Biot Number on the 2D Copp2er BlockFigures 12 and 13 show the Peaking Factor (PF2) and the temperature distribution versusazimuthal angle for the following three cases: (1) Bi=150/k, Ri=4.5mm, W=18mm,H=18mm; (2) Bi=150/k, Ri=3mm, W=12mm, H=12mm; (3) Bi=225/k, Ri=4.5mm,H=18mm, W=18mm. Because the temperature on the channel surface did not vary much(less than 64.1 *C), the copper's thermal conductivity can be considered to be the same asat the channel surface in these three cases. Another consideration is that the Biot numberis same in the first case as in the second case; from Fig. 12, one can see that the PeakingFactor (PF2) curves coincide with the same Biot numbers. With the higher Bi number(250/k), the azimuthal variation is larger. From Fig. 13, one can see that the temperature atthe channel surface increases with an increasing channel radius, while at the same timekeeping the same Bi number.3.3 Effect of Three-Dimensional Heat Conduction of the Copper BlockBecause the heat transfer coefficient at the channel surface and the bulk coolanttemperature at the channel are varied axially, heat conduction exists axially in the copperblock. In the 3D model, the heat transfer coefficient is 100,000 W/m2 *C and the bulktemperature is 23 *C in the entrance section. They change from these values to 50,000W/m2 C, and 64 *C respectively in the exit section.Figures 14, 16, and 18 compare the 2D and 3D copper blocks' Peaking Factor (PF) forthree cases: with channel radii of 1.5mm, 3 mm, and 4.5mm respectively. Figures 15, 17,and 19 are the temperature distribution azimuthally. In the three cases, the curves for the3D block show the heat transfer situation in the exit section; one can see that the heat4transfer shape in the 3D exit section is same as that of the 2D model. However, the 3Dmodel's Peaking Factor is less than the 2D's and the temperature is also less than the 2Dmodel's at the same azimuthal angle; this indicates that a part of heat is transferred axiallyupstream through the copper block.Figures 20 and 21 compare the different lengths of the 3D copper blocks. One can seethat the longer copper block, the less heat will be transferred axially from the exit sectionto the entrance section.3.4 The 2D INCONEL Tube3.4.1 The 2D Symmetrical INCONEL TubeThree cases were calculated for the fully symmetrical INCONEL tube. Table 5geometric properties, loads, and boundary conditions. Table 6 shows theresults: the temperature at the channel surface is much higher than thetemperature in all three cases.shows thecalculationsaturationTable 5 Properties of the 2D Symmetrical INCONEL ModelsCase I Case II Case IIIInside surface radius Ri (mm) 1.5 3.0 4.75Outside surface radius Ro 2.0 3.5 5.13(mm)Internal heat generation q' 2.914 x 1010 3.138x10 1 0  4.289x10' 0(W/m 3 )Heat transfer coefficient 50,000 50,000 50,000at channel surface(W/(m2 *C))Bulk coolant temperature (*C) 59 59 59Coolant pressure (MPa) 3.0 3.0 3.0Table 6 Results of the 2D Symmetrical INCONEL Models Calculations_____________________Case I Case II Case IIITemperature at inner surface 398.967 398.950 398.929(*C)Temperature difference between 183.087 187.992 147.693inner and outside surface AT(*C)Heat flux at the inner surface 1.7x10 7  1.7x10 7  1.7x10 7(W/m 2 )53.4.2 The 2D Asymmetrical INCONEL TubeThree cases were calculated for the asymmetrical INCONEL tube. Table 7 shows theirgeometric properties and loads. For INCONEL tube, the Peaking Factor is defined as PF2= h , which is the same as the PF2 for copper block. Fig. 22 depicts the azimuthalqch,avdistribution of the Peaking Factor (PF2); one can see that as the inner radius increases, theazimuthal variation increases for the peak heat flux, and the curve becomes steeper at theangles of 300 60*. This shows that the heat transferring from the thick cylindrical part tothe thin cylindrical part will decrease as the inner surface's radius increases. Fig. 23 showsthe temperature distribution azimuthally around the channel surface; note that thetemperature of the channel surface in the thick cylindrical part of the INCONEL tube ishigher than the saturation temperature by about 100 *C.Table 7 Properties of the 2D Asymmetrical INCONEL ModelsCase I Case II Case IIIInner surface radius Ri 1.5 3.0 4.75(mm)Outside surface radius 2.0 3.5 5.13Rol (mm)Most outside surface 2.74 4.33 6.13radius Ro2 (mm)Internal heat generation 2.914 x 1010 3.138x10 10  4.289x104'q.' (W/m3)Coolant temperature (*C) 74 59 59Coolant pressure (Mpa) 3.0 3.0 3.03.5 Effect of Three-Dimensional Heat Conduction in the INCONEL TubeFor a 3D asymmetrical INCONEL tube with the inner surface radius of 1.5mm, anoutside surface radius of 2.0mm, an outermost surface radius of 2.74mm, and a length of50mm, the heat transfer coefficient at the channel surface and bulk coolant61~~Table 8 Loads and Boundary Conditions in 3D ADINA Model of INCONEL TubeSection I Section II Section III Section IV(Entrance) (Exit)Internal heat generation q"' 9.714x10 9  9.714x10 9  9.714x10 9  9.714x10 9(MW/m3)Heat transfer coefficient at 100,000 83,000 67,000 50,000channel surface (W/m2 * C)Bulk coolant 23 40 57 74Temperature (*C)Coolant pressure (MPa) 3.0 3.0 3.0 3.0temperature vary axially. The heat transfer coefficient is decreasing from the entrancesection to the exit section: 100,000 W/(m2 *C), 83,000 W/(m2 *C), 67,000 W/(m2 *C) and50,000 W/(m2 *C) respectively. The bulk coolant temperature is increasing axially: 23 *C,40 *C, 57 *C and 74 *C respectively. Loads and boundary conditions in the 3Dasymmetrical INCONEL tube are shown in Table 8; this is the ADINA model.Figures 24 and 25 show the comparison of the 2D and 3D models' Peaking Factors (PF2),as well as the azimuthal temperature distribution of the asymmetrical INCONEL tube.The curves for the 3D model are the values in the exit section. The shape of the curves forthe 3D block nearly coincide with that of the 2D model; in fact, the temperaturedifference at any point between the 2D and 3D models on the channel surface with thesame azimuthal angle is less than 2 *C. The reason for these results can be explained bythe fact that the conducting ability of the INCONEL tube is much lower than that of thecopper blocks.7Lbiw~'1Fig.1 Schematic of the Copper BlockA D I N A - T TIME 1.000Fig. 2 2D ADINA Model for Copper Block81HIVIzW-ADINA-T TIME 1.000 zxFig. 3 3D ADINA Model for Copper BlockA D I N A - T TIME 1.000Fig. 4 20 ADINA Model for Asymmetrical INCONEL Tube9A D I N A - T TIME 1.000Fig. 5 2D ADINA Model for Symmetricel INCONEL Tube10zI- Y250200150100Temperautre Azimuthal Distributionat 2D Copper Block--- Ri=1.5mm,H=6.Omm W=6.Omm-- Ri=3.Omm,H=12mmW=l2mmRi=4.5mm,H=18mrpW=-8mm-saturaion.1.50 -0Azimuthal Angle50 100II150 200Fig. 7 Peaking Factor (PF2) Versus Azimuthal Angleat 2D Copper Block1.8___ -e- Ri=1.5mm,H=6.Omm,W=6.Omm1.6 -=- Ri=3.Omm,H=12mm,1.4 W=12mm1.2 - --- Ri=4.5mm,H=18mm,1__W=18mm- 0.80.6 --0.40.20 50 100 150 200Azimuthal AngleFig. 8CLERi=1.5mm,Width=1 8.6mm,Height=19mmFig. 9 Peaking Factor (PF) Versus AzimuthalAngal at 2D Copper Block2.252.22.152.1L 2.0521.951.91.851.850 100Azimuthal Angal12150Fig. 10 Peaking Factor(PF2) VersusAzimuthal Angle at 2D Copper Block1.1 -- Ri=1.5mm,W=18.6mm,H=19mm04 1.05 _______________LI. 10.950.90 50 100 150 200Azimuthal Angle0 20013Fig. 11 Temperature Azimuthal Distribution at2D Copper Block360 W=18.6mm,H340 =19mm320-- Saturation320300280E 26002402202000 50 100 150 200Azimuthal AngleFig.12 Effect on Peaking Factor(PF2) byBiot Number at 2D Copper Blck-- Bi=150/k,Ri=4.5mm,W=18mm1.8,H=18mm1.8 -e- Bi=150/k,Ri=31.6 mm,W=12mm,1.4 H=12mm1.2 -- *-Bi=225/k,Ri=4.N 1 5mm,W=18mmLL _ _ _(L 0.8 H=18mm0.60.40.200 50 100 150 200Azimuthal Angle~*--Bi=150/k,Ri=4.5mm,W=8mm H=38mm-- Bi=15OIk,Ri=3mmr,W=12mm H=12mm--- Bi=221k1Ri--4.5mnm,-Saturation%-0IVTE50 100 150 200300250200150 -100 -50 - -0Azimuthal AngleFig. 14 Comparison 2D and 3D's Peaking Factor(PF) at Copper Block with Channel Radius 1.5mm0.92D,Ri=1.5mm.W=6mm,H-. =6mm3D,Ri=1.5mm,W=6mm0.7 ,H=6mm,L=50mmCL0.60.50.40 50 100 150 200Azimathul Angle14Fig. 13 Effect on Temperature by BiNumber at 2D Copper Block- Fig. 15 Comparion 2D and 3D's TemperatureDistribution at Copper Block with Radius 1.5mm260240220200180160140120100100Azimuthal Angle50_,_ 3D,Ri=1.5mm,W=6mmH=6mm, L=50mm2D,Ri=1.5mm,W=6mm,H=6mm-Saturation150'a15$D:3.6-0caEa)I--0Fig. 16 Comparion 2D and 3D's Peaking Factor(PF) at Copper Block with Channel Radius 3mmA2D,Ri=3mm,W=l2mm,H=12mm0.8 3D,Ri=3mm,W=12mmH=12mm,L=0.6 50mm0.40.200 50 100 150 200Azimuthal Angel."EA200250200150100Fig. 17 Comparion 2D and 3D's TemperatureDistribution at Copper Block with Radius 3mm3D,Ri=3mm,W=12mm,H=12mm,-- D mm,W=12mm,H=12mm-Saturation50200100500Azimuthal Angle16150E(DFig. 18 Compraion 2D and 3D's Peaking Factor(PF) at Copper Block with Channel Radius1.2 4.5mm i 2D,Ri=4.5mm,W=18mm, H=18mm0.8 3D, Ri=4.5mm,W=18mm, H=1a . 8mm,L=50mm0.4-0.200 50 100 150 200Azimuthal Angel17Fig. 20 Comparion Peaking Factor (PF) at3D Copper Block with Different Length1.21 -- 3D,Ri4.5mm,H=18mm,W=18mm,0.8 L=50mm3D,Ri=4.5mm, H=- 0.6 .18mn,W=18mm,L=75mm0.40.20-0 50 100 150 200Azimuthal AngleFig. 22 Peaking Factor(PF2) VersusAzimuthal Angal at 2D AsymmetricalINCONEL Tube~I1_Ri=1.5mmRi=3.5mmRi=4.75mm100Azimuthal Angle18Fig. 21 Comparion Temperature Distribution at3D Copper Block with Different Length250-- 3D,Ri=4.5mm,W=200 18mm,H=18mm,L=50mm--- 3DRi=4.5mmW=150_ 18mm,H=1 8mm,L0--aturation100-0 50 100 150 200Azimuthal Angle21.81.61.404u-1.210.80.60.40 50 150 200.r~-Fig. 23 Temperature Azimuthal Distribution at2D Asymmetrical INCONEL Tube450400. Ri=1.5mm,Tb=74 CS350. -i- Ri=3mm,Tb=59 C+-- R=4.5mm,Tb=59 C300- -- ------ Saturation0- 2502001500 50 100 150 200Azimuthal AngleFig. 24 Comparion 2D and 3D INCONEL Tube PeakingFactor (PF2) at Asymmetrical Cases withRi=1.5mm,Rol =2.Omm,Ro2=2.74mm,Tb=74 C2.11.91.71.5 A3D, L=50mmU.. 1.31.10.90.70.50 50 100 150 200Azimuthal Angle1920. Fig. 25 Comparion 2D and 3D INCONEL TubeTemperature at Asymmetrical Cases withRi=1.5mm,Rol=2.Omm,Ro2=2.74mm,Tb=74 C400--350 ~..2D) 3D,L=50mm300 .Saturation0250-E2001500 50 100 150 200Azimuthal Angie

Geometrical effects on axial & azimuthal variations of heat flux to coolant in asymmetrically heated channels

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Geometrical effects on axial & azimuthal variations of heat flux to coolant in asymmetrically heated channels

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