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Numerical Analysis of Fillet Shape and Molten Filler Flow during Brazing in theAlSi Alloy of Automotive Radiator
Hirokazu Tanaka+
Research & Development Division, UACJ Corporation, Nagoya 455-8670, Japan
In the brazing process of an automobile radiator, eutectic melting of the AlSi filler alloy of the clad sheets, fillet formation in the brazedjoints, and the flow of the molten filler metal on the solid core metal between joints occur continuously. With regard to designing heatexchangers, the optimum placement of the filler metal, considering the flow of the molten filler metal, is an important subject. In this study, a newnumerical analysis was applied to predict the shape of the fillets and the molten filler flow. The molten filler flow was calculated using thedifference method applied to a one-dimensional unsteady flow that is assumed to be a uniform flow of incompressible viscous fluid between twoparallel plates. Among some numerical results, when the hydraulic mean depth of the flow path was reduced 0.3 times, the calculation resultswere consistent with the actual values. It was necessary to narrow the flow path because it was calculated assuming that the flow of brazing wasbetween two parallel flat plates, whereas the actual flow path of the molten filler metal was not flat and a large flow resistance was produced.[doi:10.2320/matertrans.L-M2020874]
(Received September 1, 2020; Accepted November 19, 2020; Published February 19, 2021)
Keywords: automotive radiator, brazing, fillet, numerical analysis, clad sheet
1. Introduction
Aluminum heat exchangers for automotive applicationsare widely manufactured using a vacuum or controlledatmosphere brazing.1) The brazing sheets comprising an AlMn series core alloy cladded with an AlSi series filler alloyare used in tubes and fins components. In recent years, sincethe thickness of the brazing sheets has been decreasing withthe weight reduction of the heat exchangers,2,3) the quantityof filler metal supplied to the joint has been decreasing. Inaddition, even if a sufficient quantity of filler metal would beplaced near each joint prior to brazing, the individual size ofthe fillet at a joint would be insufficient because the moltenfiller metal does not stay in the intended joint but flows toother joints. Therefore, in the design and manufacture of heatexchangers, the optimum placement of the filler metal,considering the molten filler flow, is an important subject toform sufficient fillets in the brazed joints. However, the fillerflow during brazing between joints is not fully understood.To overcome this problem, it is effective to predict thequantity of the molten filler and the fillet shape at each jointduring brazing. Furthermore, the prediction of the filler flowis very useful to optimize the brazing conditions and reducebrazing defects.
Concerning the prediction of the fillet shape, there havebeen some numerical analyses based on the finite elementmethod, in which the summation of surface energy, potentialenergy, and interface energy of the fillet was minimized.46)
According to these results, the actual values and thecalculated values of the fillet shape were in good agreement.For the prediction of the molten filler flow, a study examinedthe phenomenon in which molten AA4343 filler flowedthrough the artificial micro groove formed on the AA3003substrate.7) The results showed that the actual flow distanceof the molten filler was smaller than that of the calculatedvalue based on the modified version of the capillary flowmodel of Washburn,8) advanced by Yost et al.9) In this study,
since the single groove model simplified the molten fillerflow, it was impossible to predict the complicated moltenfiller flow of the actual heat exchangers.
Computational fluid dynamics has been used and appliedin various technical fields, and there has been an example ofnumerical analysis of water flow in underground diversionchannels.10) The primary feature of this particular study isthe unsteady flow of water in the main tunnel and thesimultaneously changing water level in the vertical shaftsconnected to the main tunnel. The molten filler flow isconsidered to be similar to the flow in underground diversionchannels. This is because the molten filler flow on the surfaceof the tube material is equivalent to the water flow in themain tunnel, and the transformation of the fillet shape isequivalent to the changing water level in the vertical shaft.Although it is difficult to calculate the molten filler flow usingthe same analytical model of the underground diversionchannel, it is possible to apply the discretization method ofthis model to the molten filler flow model.
This paper describes a unique numerical model of thefillet shape and the molten filler flow during brazing for anautomotive radiator with reference to the water flow inunderground diversion channels. Subsequently, the filletshape of the actual radiator was compared with the calculatedvalue. Based on the results, the growth of the fillet withincreasing temperature during brazing is discussed.
2. Experimental Procedure
2.1 Measurement of fillet shape of the actual radiatorThe fillet shape was measured using a radiator that was
brazed in an inert gas atmosphere while measuring thetemperature of the plate and the tube. The radiator consistsof a press-formed header plate, a welded oval tube, and acorrugated fin, as shown in Fig. 1. The header plate wasmade of a brazing sheet with a thickness of 1.2mm,composed of a 3000 series alloy core material cladded withAA4343 (Al7.5mass%Si) of 10% clad ratio. The tubewas also made of the brazing sheet above mentioned with a+Corresponding author, E-mail: [email protected]
Materials Transactions, Vol. 62, No. 4 (2021) pp. 498 to 504©2021 The Japan Institute of Light Metals
thickness of 0.23mm, and with AA4045 (Al10.0mass%Si)of 13% clad ratio. The fin was made of a 3000 series baresheet with a thickness of 0.05mm. The brazing conditionwas an oxygen concentration of 100 ppm or less and themaximum temperatures of 867K and 869K for the plate andthe tube in the center of the core, respectively.
The tube at the center of the core was cut out along withthe plate and fins, and the cross-section microstructure ofthe plate-tube joint and the fin-tube joint were observed at thecenter of the tube width. The curvature radius of the filletwas measured by image analysis of the cross-sectionmicrostructure. The curvature radii of the twenty fin-tubefillets located within 50mm from the plate-tube joint weremeasured continuously. In addition, the curvature radii of thefillet at the fin-tube every 25mm, located 50200mm fromthe plate-tube joint, were also measured. Figure 2(a) showsthe fillet at the plate-tube joint, and Fig. 2(b) and (c) show thefillets at the fin-tube joints, located at 15.5mm and 200.0mmfrom the plate-tube joint. The numbers in the figure showthe measured curvature radius of each fillet. The near and farsides of the fin-tube joint in Fig. 2(b) demonstrate thepositional relationship from the plate-tube joint.
2.2 Numerical model of the fillet shape and molten fillerflow
Numerical analysis of the fillet shape and the molten fillerflow was performed according to the flow chart in Fig. 3.These calculations were based on the following assumptions:a) The effect of gravity is negligible.b) The solidification shrinkage is negligible.c) The motion of the molten filler in the fillet is negligible.d) The effect of flux is negligible.
2.2.1 Materials, joint shape, and brazing conditionsThe specifications of the plate, tube, and fin materials are
the same as those of the actual radiator, as shown above. Theshapes of the plate-tube and fin-tube joints are traced fromthe cross-sections shown in Fig. 2 by image analysis. Withregard to the shapes of fin-tube joints, since the fin shapeswere different depend on each joint, these shapes wereadjusted to the one average shape. The brazing temperature atthe plate-tube joint TP and the fin-tube joint TF was obtainedfrom the actual measurement results shown in Fig. 4. Tosimplify the calculations, the time elapsed during the coolingphase has been removed from the brazing temperature cycle.2.2.2 Fillet shape
It is assumed that the fillets are formed by absorbing themolten filler through capillary action along the gap betweenthe two members forming the joint. Figure 5 shows theschematic of the fillet between part 1 and part 2. Theschematic of the fillet surface, defined by a circular arcwhose center is located at coordinates (X0,Y0), can bedescribed as
X ¼ R cos ª þX0 ð1ÞY ¼ R sin ª þ Y0 ð2Þ
where R represents the curvature radius of the membrane ofthe fillet. The coordinate of the point of intersection of part 1and the membrane is given as follows:
Xpart1 ¼ R cos ª1 þX0 ð3Þfpart1ðXpart1Þ ¼ R sin ª1 þ Y0 ð4Þ
The inclinations of the membrane at fXpart1; fpart1ðxÞg andfXpart2; fpart2ðxÞg are given as
PlateResin Tank
Side Plate
Tube
Fin
Fig. 1 Components of an automotive radiator.
Plate
Tube
0.296mm
0.109mm 0.116mmFin
Tube
(a)
(b)
0.400mm
(c)0.070mm 0.073mm
Near Side Far Side
Fig. 2 Cross sections and curvature radii of fillets at joints. (a) Plate-tubejoint. Fin-tube joints located at (b) 15.5mm and (c) 200.0mm from theplate-tube joint.
YESNO
Start
Calculation of initial conditions at each joint1) Shape of the fillet
2) Pressure acting at fillet
time > brazing cycle
End
Out put: fillet shape at each joint
Numerical calculations of molten filler flow1) Dispersion of molten filler in each joint
2) Flow of molten filler between joints
Input: materials, shape at each joint, temperature at each joint,
surface tension, viscosity coefficient,and flow factor
tn+1=tn+⊿t
Calculation of conditions at each joint1) Shape of the fillet
2) Pressure acting at fillet
Fig. 3 Flow chart for numerical analysis of the fillet shape and the fillerflow.
Numerical Analysis of Fillet Shape and Molten Filler Flow during Brazing in the AlSi Alloy of Automotive Radiator 499
�1= tan ª1 ¼ dfpart1ðXpart1Þ=dxþ tan ªpart1 ð5Þ�1= tan ª2 ¼ dfpart2ðXpart2Þ=dx� tan ªpart2 ð6Þ
where fpart1(x) and fpart2(x) represent the shape functions ofpart 1 and part 2, respectively, and ªpart1 and ªpart2 denotethe contact angle between the membrane and each part. Thecross-sectional area of the fillet is calculated as follows:
S ¼ 1=2Xn
j¼1ðXj �Xjþ1ÞðYj þ Yjþ1Þ
������
� R2=2
Z ª2
ª1
cos2 ªdª ð7Þ
Through eq. (7), the cross-sectional area of the fillet iscalculated by subtracting the area within the membrane arcunder (X0,Y0), from the quadrilateral shaped by the fourpoints: (0, 0), ðXpart1; Ypart1Þ, ðX0; Y0Þ and ðXpart2; Ypart2Þ. Thevolume of the fillet is calculated as
V ¼ S � wjoint ð8Þwhere wjoint represents the width of the fillet. Five unknownquantities X0, Y0, ª1, ª2, and R of the equation of the circulararc can be obtained by solving the simultaneous eqs. (3)(7).An initial condition of the membrane of the filletis shown in Fig. 6. This shape was calculated under thecondition that the volume of the fillet has the minimumvalue. During brazing, the fillet shape was calculated usingthe fillet volume that constantly changes due to the inflow
and outflow of the molten filler to the joint. The numer-ical model of the molten filler flow is shown in the nextsection.2.2.3 Numerical calculation of molten filler flow
Figure 7 shows a schematic model of a tube in the radiatorwith a fin and plate, for numerical calculation. This is a half-tube model based on the longitudinal symmetry of a realradiator tube. There are one plate-tube joint and sixty-sevenfin-tube joints. As shown in Fig. 8, the region was divided sothat each element included only one joint. The element ati = 0 is the plate-tube joint and the elements at i = 1 to 67 arethe fin-tube joints. A continuity equation calculates the filletvolume at each element under the conditions that give theinflow and outflow of the molten filler at the joint. Thus, thecontinuity equation can be described as
dVi=dt ¼ Qmelti þQi �Qiþ1 ð9Þwhere Qmelti is the inflow rate of the molten filler from the
840
850
860
870
0 120 240 360 480 600 720
Tem
pera
ture
, T/ K
Time, t / s
TF
850K: Solidus of filler alloyｓ
TP
Fig. 4 Temperature-time curve during brazing, excluding the coolingphase.
X
Y
0
S
θpart2
Part 1
(X0, Y0)
θ1
fpart1(x)
R
Part 2
θ2
fpart2(x)
θpart1
{Xpart2, fpart2(Xpart2)}
{Xpart1, fpart1(Xpart1)}
wjoint
V
Fig. 5 Schematic of fillet used for calculations.
0.000
0.500
1.000
1.500
-1.000 -0.500 0.000 0.500 1.000
Hig
ht fr
om c
ente
r of j
oint
/ m
m
Length from center of joint / mm
Tube surface
Fin
surfa
ce
0.000
0.050
0.100
-0.100 -0.050 0.000 0.050 0.100
Fillet membrane
Fig. 6 Initial condition of the membrane of fillet between fin-tube joints.
Plate
Tube
Fin
Front side
Rear side
Fig. 7 Symmetric model of the tube with fin and plate.
H. Tanaka500
tube surface on both sides of joint to the joint, Qi is the inflowrate of the molten filler from (i ¹ 1)th joint to ith joint, andQi+1 is the outflow rate of the molten filler from ith joint to(i + 1)th joint. The path of the molten filler flow of Qmelti, Qi
and Qi+1 is the residual filler on the tube surface, as shown inFig. 8. The residual filler is the molten metal remaining onthe surface after the filler metal has been absorbed intothe joints and its thickness is indicated di. The continuityequation of the flow rate of the molten filler and the motionequation can be described as
@Q=@x ¼ @ðuAÞ=@x ¼ 0 ð10Þµð@u=@tþ u@u=@xÞ ¼ �@P=@xþ ®@2u=@x2 ð11Þ
where A is the cross-sectional area of the path of the moltenfiller flow, u is the mean flow velocity, P is the pressure, µis the density, and ® is the viscosity coefficient. The moltenfiller flow is caused by capillary action between the moltenmetal and solid metal at the joints. P in eq. (11) indicatesthe pressure difference between the inside and outside of themolten metal membrane as a driving force of capillary actionand is calculated by Laplace’s equation,
P ¼ £=R ð12Þwhere £ represents the surface tension of molten aluminum.Note that P is a negative value because the center of thecurvature of the membrane is outside the molten metal. Thismeans that the pressure inside the molten metal at the jointis lower than that outside the membrane. It is assumed thatthe density and the surface tension of molten filler havevalue of 2700 kg/m3 and 0.914N/m,11) respectively, and areconstant with respect to the brazing temperature.
Assuming that the flow of the molten filler is a steady flowof incompressible viscous fluid between two parallel flatplates and the motion of the molten filler in the fillet isneglected, the average flow rate Qave obtained from eqs. (10)and (11) is described as
Qave ¼ wð¡dÞ3�P=12®L ð13Þwhere w and d are the width and depth of the flow path,respectively, and L is the distance between joints. ¦P isdifferent from P in eq. (11) and indicates the pressuregradient within and between elements for numerical analysis.The molten filler flow was calculated as the flow betweentwo parallel flat plates because it is assumed that the moltenfiller flows between the oxide film and the core substrate. A
correction coefficient ¡ is used for the channel depth d sincethe flow path is not stable, owing to the destruction of theoxide film, and the interface between the molten filler andthe core substrate is not flat. Qmelti and Qi obtained fromeq. (13) are as follows:
Qmelti ¼ �wð¡diÞ3Pi=12®Li � wð¡diþ1Þ3Pi=12®Liþ1 ð14ÞQi ¼ wð¡diÞ3ðPi�1 � PiÞ=12®Li ð15Þ
Equation (14) represents the inflow rate of the generatedmolten filler on the tube surface between the joints. Here, thefirst term on the right side represents the inflow between the(i ¹ 1)th and ith joints to the ith joint, and the second termon the right side represents the inflow between the ith and(i + 1)th to the ith joint. Equation (9) is rearranged based onthe time step tnþ1 ¼ tn þ�t:
Vnþ1i ¼ Vn
i þ ðQmeltni þQni �Qn
iþ1Þ�t ð16ÞSubstituting eqs. (14) and (15) into eq. (16), the following isobtained:
Vnþ1i ¼ Vn
i þ ½f�wð¡dni Þ3Pni =12®
ni Li
� wð¡dniþ1Þ3Pni =12®
ni Liþ1g
þ wð¡dni Þ3ðPni�1 � Pn
i Þ=12®ni Li
� wð¡dniþ1Þ3ðPni � Pn
iþ1Þ=12®ni Liþ1��t ð17Þ
The thickness of the flow path in eq. (17) is given as follows:
dnþ1i ¼ dni � f�wð¡dni Þ3Pn
i�1=12®ni Li
� wð¡dni Þ3Pni =12®
ni Lig�t=wLi ð18Þ
Since it is assumed that the molten filler flow path is theresidual molten filler on the tube surface, the thickness atthe next time step is obtained from the difference betweenthe thickness at the current time step and the thicknesscorresponding to the amount of molten filler absorbed by thejoints. The thickness of the flow path on the plate surfaceat (i = 0)th is also obtained from the same equation of thetube surface. The viscosity coefficients of AA4343 at i = 0and AA4045 at i ² 1 are given as follows:12)
®ni¼0 ¼ �0:0060Tn
P þ 3:6795 T � 850K ð19Þ®ni�1 ¼ �0:0175Tn
F þ 10:3104 T � 850K ð20ÞThese values are given at more than 850K that is the solidusof the filler alloy of the AA4343 and AA4045.
The quantity of molten filler at every temperature duringthe brazing process was calculated by multiplying the flow
P0
L0
Q1
L1 L2 L3
P1
Plate
Fin
V0 V１
i=1 i=2i=0 i=3
L4
P2V2 P3
V3
FinMolten filler
Core
Tube
Q2Q0 d1 d2 Q3d3
Qmelt1 Qmelt1
Fig. 8 Mesh division of joints for calculation.
Numerical Analysis of Fillet Shape and Molten Filler Flow during Brazing in the AlSi Alloy of Automotive Radiator 501
factor Kni by the filler volume prior to brazing. The flow
factor is calculated according to the Si concentration of thebrazing material using the Lagrange complement polynomial,based on the actual data at various Si concentrations shown inFig. 9.13) In the AA4343 plate filler case, the value of thedotted line is entered into Kn
i¼0. In the case of the AA4045tube filler, the corrected value of the dashed line is enteredinto Kn
i�1. The actual data were measured when the cladthickness of filler was 0.12mm, although the clad thicknessof the tube filler was 0.03mm, so the flow factorof AA4045 was reduced from the dotted line of AA4045 tothe dashed line of the corrected AA4045. This is because themolten filler amount decreases with the decrease of the fillerthickness owing to the diffusion of Si elements in the filleralloy to the core alloy during brazing. The total absorbedmolten filler does not exceed the quantity of the molten fillerobtained from the flow factor because the molten fillergenerated between the (i ¹ 1)th and ith joints is absorbed intoboth the (i ¹ 1)th and ith joints. This relation is described as
Kni wtclad;iLi �
Xnf�wð¡dni Þ3Pn
i�1=12®ni Li
� wð¡dni Þ3Pni =12®
ni Lig�t ð21Þ
where tclad,i is the thickness of the filler metal prior to brazing.The volume of the fillet at each element is calculated by
the one-dimensional forward difference method by combin-ing eqs. (17)(21) at every 0.02 sec time step, and then thefillet shapes are determined using the volumes of the fillets,as described in section 2.2.2. V 0
i and P0i as the initial
conditions are assigned the minimum value obtained from thecalculated minimum fillet shape of each joint, as shown inFig. 6. d0
i is the clad thickness of the plate and tube prior tobrazing. The boundary conditions are Qn
0 ¼ 0 and Qn67 ¼ 0,
indicating a zero flow rate at both ends of the tube. Thenumerical analysis was executed using a self-build code onVisual Basic for Applications software.
3. Results and Discussions
3.1 Fillet shape of the actual radiatorFigure 10 shows the curvature radii of the fillet at the
plate-tube and fin-tube joints. In Fig. 10, there are two dotsin the plate-tube joint on the front and rear side where themeasurement was made. In the fin-tube joints, the values
were also measured at both the front and rear side, and thesevalues are in average the curvature radii of the near and thefar side as shown in Fig. 2(b). The maximum value of thecurvature radius among all joints was 0.322mm measured atthe plate-tube joint. The curvature radii at the fin-tube jointswithin 75mm from the plate-tube joint decreased non-linearly from 0.180mm to 0.070mm with the distance fromthe plate-tube joint. The curvature radii at the fin-tube jointslocated at more than 75mm from the plate-tube joint becamesteady value around 0.070mm. According to these results, itis considered that the molten filler of the plate flowed throughthe tube surface within 75mm from the plate-tube joint, andthe fillets of fin-tube joints were formed from the filler ofboth plate and tube. On the other hand, for the fin-tube jointslocated at more than 75mm from the plate-tube joint, it isconsidered that the fillets were formed from only the filler ofthe nearest tube surface.
3.2 Comparison of actual value and numerical model offillet shape
The actual value and numerical model of the fillet shapeof the plate-tube and fin-tube joints are shown in Fig. 11(a)and (b), respectively. The actual values were measured fromFig. 2(a) and (c). The numerical models were determined byadjusting the fillet areas, considering the actual value; thecontact angles were 0°, 20°, and 40°, respectively. In theplate-tube joint, the fillet shape of each contact anglecorresponded to the actual value. The fillet shape correspond-ing to the contact angle of 0° was the best match among allthe contact angles for the actual value. The result for thefin-tube joint was the same as that of the plate-tube joint.According to these results, it is appropriate to represent themolten filler surface through a simple circular arc. In the caseof a curvature radius of less than 0.300mm, the bond numberBo, indicating the ratio of gravity to surface tension, is lessthan 0.01, and the effect of gravity becomes extremely small.The reason for the best match of the measurement value at acontact angle of 0° with the numerical model was attributedto the surface tension of solid aluminum being sufficientlylarger than that of liquid aluminum under good brazingconditions.
0.0
0.2
0.4
0.6
0.8
1.0
840 850 860 870 880 890 900 910
Flow
fact
or, K
Temperature, T / K
ActualCalculation
4.6Si
6.8Si
9.2Si
11.8Si
AA4045
AA4343
corrected AA4045
Fig. 9 Experimental and calculated flow factor of AlSi series filler alloy.
0.000
0.100
0.200
0.300
0.400
0 50 100 150 200
Cur
vatu
re ra
dius
of f
illet
, R /
mm
Distance from plate, L / mm
Plate-Tube joint
1st to 12th Fin-Tube joint
25mm Intervals Fin-Tube joint
Fig. 10 Distribution of curvature radius of the actual fillet at the plate-tubeand fin-tube joints.
H. Tanaka502
3.3 Comparison of actual values and the values obtainedfrom the numerical model of molten filler flow
The actual value and numerical model of the curvatureradii at each joint are shown in Fig. 12. The actual value is thesame as those shown in Fig. 10. The numerical model valueswere determined using the fillet volume Vi corresponding tothe molten filler flow for the following values of correctioncoefficient ¡: 1.0, 0.5, 0.4, 0.3, and 0.2. When ¡ was 1.0, andthe distance from the plate was 030mm, the curvature radiiof the numerical model were smaller than those of the actualvalue. On the contrary, when the distance from the plate wasmore than 30mm, the curvature radii of the numerical modelwere larger than those of the actual value. These resultsindicated that the molten filler flow was smaller than that ofthe numerical model. Because the rate of the molten fillerflow in the numerical model decreased with decreasing ¡, thecurvature radii of the numerical model in the case of ¡ = 0.3were in good agreement with the actual values in the samecondition. Furthermore, when ¡ was 0.2, and the distancefrom the plate was 0, the curvature radii of the numericalmodel were larger than those of the actual value. On thecontrary, when the distance from the plate was 1030mm, thecurvature radii of the numerical model were smaller than
those of the actual value. Therefore, ¡ = 0.2 was too small acorrection coefficient to be used in this model for obtainingaccurate results.
According to these results, it is considered that the flowresistance of the actual flow was larger than that calculatedfrom the numerical model. In the numerical model, the moltenfiller flow was calculated as the flow between two parallel flatplates because it is assumed that the molten filler flowsbetween the oxide film and the core substrate. In the case ofthe actual molten filler flow, the path of the molten filler metalwas not flat because of the partially destroyed oxide film orthe unevenness of the substrate owing to erosion of the corematerial, thus, a large flow resistance was produced. There-fore, it is considered that the actual flow could be simulatedby increasing the flow resistance and using a correctioncoefficient ¡ for the channel depth d in the numerical model.In addition, the motion of the molten filler in the fillet andthe flux among the neglected factors that were shown in thesections 2.2, and the erosion might affect to ¡. However, theeffect of gravity and the solidification shrinkage might notaffect to ¡, because the effect of gravity was sufficiently smallfrom the Bo and the effect of the solidification shrinkagewas less than that of the flow resistance.
The curvature radius variations, calculated by thenumerical model with increasing temperature, is shown inFig. 13. These were calculated under the condition of ¡ =0.3. When the plate and tube temperatures were elevatedto 850K and 851K, respectively, just above the solidustemperature of the filler alloys AA4343 and AA4045 in 225 s(n = 11250), as shown in Fig. 4, the curvature radii of thefillet at each joint were approximately 0.050mm each. Thesefillets were formed by only the tube filler since K11250
i¼0 wasapproximately zero for the plate at 850K and K11250
i�1 was 0.32for the tube at 851K. When the plate and tube temperatureswere elevated to 853K and 852K in 285 s (n = 14250), andthe flow factors were K14250
i¼0 ¼ 0:18 and K14250i�1 ¼ 0:35, the
curvature radius of the fillet at the plate-tube joint increasedrapidly. Further, when the plate and tube temperatures wereelevated to 859K in 430 s, and the flow factors wereK21500
i¼0 ¼ 0:25 and K21500i�1 ¼ 0:46, the curvature radius of the
fillet at the plate-tube joint reached the maximum value. The
0.000
0.500
1.000
1.500
0.000 0.500 1.000 1.500
Hig
ht fr
om c
ente
r of j
oint
/ m
m
Length from center of joint / mm
Calclated by θpart1=θpart2=0°Calclated by θpart1=θpart2=20°Calclated by θpart1=θpart2=40°Actual
Plate surface
Tube surface
(a)
0.000
0.200
0.400
0.000 0.200 0.400
Hig
ht fr
om c
ente
r of j
oint
/ m
m
Length from center of joint / mm
Calclated by θpart1=θpart2=0°Calclated by θpart1=θpart2=20°Calclated by θpart1=θpart2=40°Actual
Fin surface
Tube surface
(b)
Fig. 11 Comparison of actual and calculated data for the fillet membrane at(a) plate-tube and (b) fin-tube joints.
0.000
0.100
0.200
0.300
0.400
0.500
0 50 100 150 200
Cur
vatu
re ra
dius
of f
illet
, R/ m
m
Distance from plate, L / mm
Calculated by α=1.0Calculated by α=0.5Calculated by α=0.4Calculated by α=0.3Calculated by α=0.2Actual
i=0： Plate-Tube joint
i=1 to 67： Fin-Tube joint
Fig. 12 Comparison of actual and numerical model value for thedistribution of the curvature radius of the fillet membrane at each joint.
Numerical Analysis of Fillet Shape and Molten Filler Flow during Brazing in the AlSi Alloy of Automotive Radiator 503
growth of the fillet of the plate-tube joint affects the increasein Kn
i¼0 in the plate with increasing temperature. At thistemperature, the curvature radii of the fillets at the fin-tubejoint within 75mm from the plate-tube joint are graduallyreduced with the distance from the plate-tube joint. It can beseen that the filler of the plate was flowing in the longitudinaldirection of the tube. When the temperature was furtherelevated, the curvature radius of the fillet at the plate-tubejoint started to decrease. On the contrary, the curvature radiiof the fillet at the fin-tube joint gradually increased forevery joint. In particular, the curvature radii of the fillet atfin-tube located within 75mm from the plate-tube joint wereincreased significantly with temperature.
Since a lot of information can be obtained from thenumerical calculations, it is useful to understand the behaviorof the molten filler flow during brazing. Furthermore, in thisnumerical model, it is possible to input various joint shapes,brazing heat cycles, and the contact angle considering theactual brazing atmosphere and flux condition. Theseapproaches might contribute to the optimum placement ofthe filler metal and the prediction of brazing defects, andeventually to clarify brazing phenomenon in more detail.
However, the present research could not accurately predictthe actual molten filler flow because this numerical modelsimplifies the flow as a uniform flow of incompressibleviscous fluid between two parallel plates and the calculationsunder the assumption stated in section 2.2. Therefore, one ofthe subjects to improve this numerical model is to reflectupon these neglected factors. To predict the fillet shape andfiller flow more accurately, it is necessary to improve thenumerical model by experimentally understanding andsimulating the flow path of the molten filler, and bymeasuring the change in the fillet shape with increasingtemperature. Furthermore, it is necessary to measure thesurface tension, viscosity coefficient, and flow factor moreaccurately, as these factors change with the brazing cycle,atmosphere, and fluxing. On the other hand, since the filletshape was determined as a three-dimensional shape and theflow of the molten filler metal was considered as a one-dimensional unsteady flow in this numerical model, its
application on more complex shape heat exchangers is quitedifficult. Thus, to expand the application of this numericalmodel, a three-dimensional numerical model of the moltenfiller flow is necessary for the future.
4. Conclusion
In this study, a unique numerical analysis method wasapplied to an automotive aluminum radiator to predict thefillet shape and molten filler flow in the brazing joint and theactual fillet shape of the brazed radiator was compared withthe numerical analysis results. Although it was necessary tonarrow the flow path in this calculation, the results wereconsistent with the actual values. This prediction of the filletshape is very useful in determining the optimum placementof the filler metal, the adequate material choice in the designof the heat exchangers, and the optimization of brazingconditions for the formation of sufficient fillets in the brazedjoints.
In order to further increase the accuracy in predicting afillet shape, it is effective to reflect measurements of theactual flow path of the molten filler, surface tension, viscositycoefficient, and flow factor to the numerical model iseffective. On the other hand, given that the fillet shape wasdetermined as a three-dimensional shape and the molten fillerflow was ascertained as a one-dimensional unsteady flow inthis numerical model, its application on more complex shapeheat exchangers is quite difficult. Hence, it will be necessaryto enhance the numerical model of the molten filler flow to athree-dimensional model taking into account the neglectedfactors such as gravity.
Acknowledgments
The assistance of Mr. Itoh and his colleagues at the Ikedaplant of DENSO Corporation in providing aluminumradiators and advice on brazing technology is gratefullyacknowledged.
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0.000
0.100
0.200
0.300
0.400
0.500
0 50 100 150 200
Cur
vatu
re ra
dius
of f
illet
, R /
mm
Distance from plate, L / mm
Plate: 850K, Tube: 851KPlate: 853K, Tube: 852KPlate: 859K, Tube: 859KPlate: 862K, Tube: 864KPlate: 866K, Tube: 869K
i=0： Plate-Tube Joint
i=1 to 67： Fin-Tube Joint
Fig. 13 Change in the curvature radius distribution of the fillet membraneat each joint during brazing; values were obtained from the numericalmodel, calculated using ¡ = 0.3.
H. Tanaka504
