Probabilistic Study of a Refrigerator Steel Heat Loop Tube Joint

Transcription

1 Probabilistic Study of a Refrigerator Steel Heat Loop Tube Joint Kenneth J. Rasche, P.E. Whirlpool Corporation Abstract A probabilistic study was performed on a refrigerator heat loop joint to determine the relative significance of design changes and geometry process variation on the stress/strain at the tube joint. The heat loop is an extension of the condenser that is routed behind the front flange of a refrigerator cabinet to prevent condensate from forming on the flange. During assembly of the refrigerator, the heat loop is brazed to the condenser tube. Currently, the heat loop is copper for North American Products. Changing the material from copper to steel is potentially a large material cost savings. Unfortunately, brazing the two steel tubes together sometimes results in a crack in the heat loop tube due to Liquid Metal Embattlement (LME). In order to understand what is causing the cracks, the process variation had to be modeled using probabilistic methods. The probabilistic study indicated that both the design and the process variables are significant to the cracking of the joint. This understanding will aid the implementation of the design change while maximizing the material savings, which has a potential in excess of $800,000/year. Introduction A heat loop is a small diameter tube that is used to heat the front flange of a refrigerator cabinet. The heat is supplied by the warm refrigerant that has just exited the condenser. The heat loop replaced electric heaters in the late 1980's and early 1990's and significantly reduced energy consumption of the refrigerator. Heating the flange is necessary in order to prevent condensation on the flange in high humidity environments. Heat loops were first implemented with copper tubing. Replacing the copper tubing with steel tubing is an obvious cost reduction if quality and production rates can be maintained. Most of the challenges have been met with the exception of the braze joint of the condenser tube to the heat loop tube which is shown in Figure 1. Trial runs with steel heat loop tubes had a failure rate of 1-3%, which is too high considering the production rates. The failure (crack) was located at the braze joint and was due to Liquid Metal Embrittlement (LME).

2 Figure 1 - Braze Joint of the Condenser Tube to the Heat Loop Tube in a Product LME is a function of a contaminant, temperature and stress/strain. The brazing process obviously requires high temperature and solder (the contaminant). Testing did show that the amount of energy used to heat the joint and the location of the heat did effect the LME phenomenon; however, changing these variables would result in a slower process time. Stress/strain in the joint is not required for the brazing process. Thus the only contributor to LME that was evaluated for this study was the stress/strain at the joint. Figure 2 shows a joint before and after brazing (the tube shown did not crack). Figure 2 - Yielded (not cracked) Braze Joint of the Condenser Tube to the Heat Loop Tube in a Fixture

3 The stress/strain in the joint is due to the geometry of the parts and the process variation (if the tubes were perfectly formed and perfectly aligned, there would be no stress/strain at the joint). As the joint area is heated, the stress in the tube is relieved as the material strains (bends at the joint) due to its lowered yield stress at the elevated temperature. Initially, traditional static analyses were employed to evaluate design changes that were aimed at minimizing or eliminating the failures. Finite element models were built with the geometry varying from the print. This method quickly showed that there were many combinations of process variables that could interact to both increase and decrease strain in the joint after brazing. In order to address the problem without excess conservatism (increased process or material costs) probabilistic methods were needed to understand the important variables and important interactions. Additionally, the probabilistic methods allow consideration of the probability for a specific value of a variable Analysis The variation in the tube angles and tube locations made a non-probabilistic analysis ineffective. Structural models made to drawing dimensions would result in the strain in the brazed joint location to approach zero. Conversely, structural models made to worst case dimensions would result in very high strains without any estimate for the probability of that condition happening. For this reason, the variation of the significant variables and their impact on the stress/strain in the joint had to be understood and modeled. Finite Element Model In order to incorporate the probabilistic methods, the model(s) had to be created parametrically with APDL (Ansys Parametric Design Language). This was necessary since the probabilistic study had to regenerate the varying geometries hundreds of times. The condenser was modeled from where it is stiffened by its heat transfer wires to the end that gets brazed to the heat loop. Figure 3 shows the production condenser in the High Side subassembly. Figure 3 - Production Condenser High Side Subassembly

4 The heat loop was modeled from its end where the braze is made to 50 mm into the foam of the cabinet. The foam supporting/constraining the heat loop was modeled with 10mm long beams using link8s. Both the heat loop and the condenser were modeled with beam 188s. Figure 4a, 4b and 4c show an ISO, rear and side view of the finite element model, respectively. Figure 4a - ISO View of Structural Model Figure 4b - Rear View of Structural Model

5 Figure 4c - Side View of Structural Model Loading Loading of the tube model was difficult. In the physical process of tubing up the product, the tubes are aligned and an expanded portion of the condenser tube overlaps the end of the heat loop stub. The tubes are then released and both tubes attempt to unload, but the overlap prevents this and stress develops at the joint location. The two tubes are then brazed together with the solder material sealing the gap between the inside diameter of the upset portion of the condenser tube and the outside diameter of the heat loop stub. Figure 2 shows a joint before and after brazing is performed in a test fixture. In order to model this physical loading, four displacement points are defined on the tube model. Points 1 and 2 are located on the heat loop at the joint location and the end of the heat loop stub, respectively. The location of point 1 relative to the end of the heat loop stub varies due to manufacturing variation. This dimension is a design variable and was labeled OLAP (overlap). Points 3 and 4 are on the condenser and are located at the condenser tube end and a distance equal to the overlap (OLAP) from the end of the condenser tube, respectively. Figure 4b shows the location of the four displacement points. Load Step One The model was built such that the tubes were aligned with each other and the global Y-axis if there was no discrepancy from print dimensions. The APDL input file adjusted the geometry as it was built according to the variation being evaluated and resulted in the tubes not being aligned with global Y-axis. The heat loop stub, however, always protruded from the deck at X and Z equal to zero in the global coordinate system (variation of the location of the heat loop stub protruding from the deck was modeled by moving the condenser tubing in the opposite directions). The horizontal coordinates (X and Z) for the four load points were obtained with APDL. The four points were then displaced such that they aligned with the global Y- axis and were therefore aligned with each other. All nodes in the model are set to 75 degrees F. Load Step Two The new vertical locations of points 1 and 3 are determined using APDL (original vertical coordinate + vertical displacement). Point 3 is then displaced vertically so that it is coincident with point 1. The

6 horizontal displacements are maintained and result in point 1 and 3 being coincident and point 2 and 4 being coincident. Load Step Three The reaction forces at the applied displacements (at points 1,2,3 and 4) are obtained with APDL. Points 1 and 3 are coupled in the horizontal degrees of freedom (X and Z). Points 2 and 4 are coupled in the 3 translational degrees of freedom. The applied displacements at the four points are deleted and the reaction forces are applied to the master nodes of the coupled sets. Load Step Four, Five, Six and Seven The applied reaction forces are reduced in each successive load step 4-6 until they are completely removed in load step 7. The bending, shear and axial stresses at the joint (point 1) are obtained with APDL. These components are then combined into a total stress at the joint. Load Step Eight This load step attempts to model the heat applied by the brazing operation. Elevated material properties were estimated in "Elevated Temperature Material Property Estimation". The maximum temperature in the braze area is estimated to be 1500F. Nodes within 4mm of the center of the braze location are set to 1500F. Nodes between 4mm and 20mm away from the center of the braze area are set to 1500F F depending on their distance from the center. Nodes farther than 20 mm from the center of the braze area are left at 75F (set in load step One). The material in the heated area yields due to the elevated temperature and the percent strain is obtained with APDL. Elevated Temperature Material Property Estimation Elevated temperature material properties were needed in order to evaluate the strain in the joint when the heat from the brazing process was applied. The temperature at the braze location was estimated to be 1500F. Properties at this temperature for commercial quality steel tubing were not available. Physical tests were performed on a fixture (figure 5a). Through trial and error experimentation, it was found that the test specimen would usually crack if an elastomer was used to load the joint in bending. The initial load of the elastomer was approximately 4N and the stiffness of the elastomer was N/mm.

7 Figure 5a - Yielded (not cracked) Braze Joint of the Condenser Tube to the Heat Loop Tube in a Fixture Figure 5b shows the finite element model for this evaluation. The model was loaded by first stretching the elastomer (stiffness equal to N/mm) in order to give the desired load (4 N). The model was then loaded by increasing the temperature of nodes in the braze area as described in "Load Step Eight". The material properties at the elevated temperature were then modified in an attempt to duplicate the deformation seen in the physical tests. Figure 5b - Finite Element Model of Brazing Fixture

8 Unfortunately, the model was not able to perfectly duplicate the physical test. In the physical tests, the tubes deflected to approximately 25 degrees out of alignment. This resulted in a strain value of approximately 9%. With the model, the elevated temperature material properties could only be softened to the point where the tubes deflected to approximately 16 degrees (5.9% strain). This probably could have been overcome with a more refined model, but these results were judged to be sufficient for comparison studies. The elevated material properties determined from this exercise are: Yield stress: Tangent E: 6 Mpa 500 Mpa Failure Criteria Initially, the failure criteria was going to be the stress at the joint prior to heat application. However, after some testing performed on various physical fixtures it was determined that the strain in the joint after brazing was not a function of only stress. The application of the load in the product that causes the stress is also a factor in the strain in the joint after brazing. For example, the condenser tube bent from side to side may result in the same value for the stress in the joint as the condenser tube bent from front to back. However, the condenser tube's stiffness will be different when bent in the two different directions because of geometry differences. This difference in stiffness will result in the load "dropping off" at a different rate as the heated material yields, and will therefore result in a different strain value after brazing. For this reason, the percent strain in the joint AFTER the heat is applied was chosen as the failure criteria. In order to evaluate the strain in the joint after the heat is applied, material properties for the tubes at the elevated temperature had to be estimated. Estimation of the elevated material properties is described in "Elevated Temperature Material Property Estimation". The estimated values for the elevated temperature material properties were used consistently throughout this probabilistic study. Correlation of Probabilistic Model to Trial Run Results Before the finite element models were employed, several trial runs were performed on the assembly lines. The most recent trial run used over 400 products and had a failure rate of approximately 3%. The production condenser and the production variation were modeled to simulate the trail run. These conditions were evaluated with a probabilistic analysis. The strain distribution from this evaluation was used to calculate a 3% population failure strain (3% of the products would have a strain value higher than this number). The 3% population failure strain was calculated by using equation 1. Eq. 1: ε 3 % ε avg. + (1.88) ε st. dev. = [1] The 3% population failure strain was calculated to be 2.97%. This value is used as the failure strain. Probabilistic Model The forming and assembly processes of both the heat loop and the condenser were evaluated in order to find sources of variation. The sources of variation in the geometry of the parts (which drive the stress/strain in the heat loop at the joint) are too numerous to list, so only the sources judged to be significant will be addressed. Twenty-one sources of variation are included in the probabilistic model. Figure 3 is a picture of the "high side" subassembly with the production condenser. This image shows the locations of the various bends in the production condenser that are most significant to the variation in condenser stub location and orientation. Figure 6 is an image of the 2599 condenser and it shows the location of the significant condenser bends. The production condenser and the 2599 condenser are the same except that the production condenser is missing "bend 2".

9 Figure 6 - Condenser 2599 and Location of Bend 2 Sources of Variation This section describes each source of variation that was considered to be significant to the resulting stress/strain in the brazed joint. All of the variables are assumed to have Gaussian distributions except where noted. The variable name used in the parametric model is listed first, followed by the full name and then followed by a brief description STLEN - Heat Loop Stub Length - The length of the heat loop stub that protrudes from the deck. During the forming process of the heat loop, any length variation ends up at the end that protrudes from the deck. During the cabinet foaming process, the foam flow and expansion can move the heat loop and affect the length of the stub. The distribution for this variable was measured on physical products. STANGF - Heat Loop Stub Angle (front to back) - The angle the heat loop stub is off of vertical in the direction from front to back of the product. During the foaming process, the foam flow and expansion can impact this variable. The distribution for this variable was measured on physical products. This variable is shown on figure 4c. STANGS - Heat Loop Stub Angle (side to side) - The angle the heat loop stub is off of vertical in the direction from side to side of the product. During the foaming process, the foam flow and expansion can impact this variable. The distribution for this variable was measured on physical products. This variable is shown on figure 4b. OLAP - Heat Loop Stub overlap - The length of heat loop stub that is "overlapped" by the condenser tube upset at the joint. The two tube ends are "tubed up" just prior to being brazed. The stub fits into the "upset" portion of the condenser tube. The length of the overlap can vary from a maximum of being bottomed out in the upset to some value greater than 0. This variable uses the truncated Gaussian distribution. The distribution for this variable was estimated after inspection of physical parts.

10 SYIELD - Yield strength of the Heat Loop tube - Yield strength of the commercial quality steel tubing from which the heat loop is made. The distribution for this variable was measured. STUBLOCZ - Location of Stub Protruding from the deck (front to back) - Front to back location of where the stub protrudes from the deck of the product relative to the mounting plate mounting holes. The location is driven by the hole in the deck/back panel. The location of the hole can vary and the location of the deck/back panel can vary relative to the mounting plate mounting holes. The distribution for this variable was estimated. STUBLOCX - Location of Stub Protruding from the deck (side to side) - Side to side location of where the stub protrudes from the deck of the product relative to the mounting plate mounting holes. The location is driven by the hole in the deck/back panel. The location of the hole can vary and the location of the deck/back panel can vary relative to the mounting plate mounting holes. The distribution for this variable was estimated. STUBLOCY - Location of Stub Protruding from the deck (up and down) - Up and down location of where the stub protrudes from the deck of the product relative to the mounting plate mounting holes. The up and down location is driven by foam growth/bulging. The deck bulge at the stub location will vary from foam fixture to foam fixture. The distribution for this variable was estimated. CDLEN - Condenser Tube Length - The length of the condenser tube after bend 3. This is the vertical portion that gets brazed to the heat loop. During the forming process of the condenser, any length variation ends up at this end of the condenser. The standard deviation for this variable was assumed to be the same as STLEN. CNANGF Condenser tube angle (front to back) - The angle the condenser tube is off of vertical in the direction from front to back of the product. This angle is sometimes manipulated by the assembly operators as they perform various process functions near this tube. The distribution for this variable was measured on physical products. CNANGS - Condenser tube angle (side to side) - The angle the condenser tube is off of vertical in the direction from side to side of the product. This angle is sometimes manipulated by the assembly operators as they perform various process functions near this tube. The distribution for this variable was measured on physical products. CONDLOCZ - Location of the condenser (front to back) - Front to back location of where the condenser is mounted relative to the mounting plate mounting holes. The condenser is snapped into grommets and clips. The clips are formed in the mounting plate and their front to back location will vary. The grommet dimensions may vary and drive variation in the condenser location. The distribution for this variable was estimated after inspection of physical parts. CONDLOCX - Location of the condenser (side to side) - Side to side location of where the condenser is mounted relative to the mounting plate mounting holes. The condenser is snapped into grommets and clips. The condenser and the clips are designed such that the grommets can slide side to side on the tube and the grommet can slide side to side in the clip. The distribution for this variable was estimated after inspection of physical parts. CONDLOCY - Location of the condenser (up and down) - Up and down location of where the condenser is mounted relative to the mounting plate mounting holes. The condenser is snapped into grommets and clips. The height of the clips will vary and the grommet thickness will vary to drive this variation. The distribution for this variable was estimated. MPLOCZ - Location of the mounting plate (front to back) - Front to back location of how the mounting plate is attached to the product. The mounting plate holes are on folded and toggle locked tabs of the mounting plate. The front to back location of the tabs will vary. The distribution for this variable was estimated. MPLOCX - Location of the mounting plate (side to side) - Side to Side location of how the mounting plate is attached to the product. The mounting plate holes are on folded and toggle locked tabs of the mounting plate. The side to side location of the holes in the tabs will vary.. The distribution for this variable was estimated.

11 MPLOCY - Location of the mounting plate (up and down) - Up and down location of how the mounting plate is attached to the product. The mounting plate holes are on folded and toggle locked tabs of the mounting plate. The up and down location of the holes in the tabs will vary. The distribution for this variable was estimated. DLEN2 - Location of Bend 2 - This is the location of bend 2 relative to the front of the condenser. This bend is put into the condenser second to last and its location will vary. The distribution for this variable was estimated. This variable is only applicable to the 2599 condenser. ANG1A - Bend 1 Angle - This variation is included in the analysis, but the variation is tied to the yield strength of the condenser tubing (angle is "open" if yield strength is high in the distribution, and the angle is "closed" if the yield strength is low in the distribution). The distribution for this variable was estimated. ANG2A - Bend 2 Angle - This variation is included in the analysis, but the variation is tied to the yield strength of the condenser tubing (angle is "open" if yield strength is high in the distribution, and the angle is "closed" if the yield strength is low in the distribution). The distribution for this variable was estimated. This variable is only applicable to the 2599 condenser. CYIELD - Yield strength of the Condenser tube - Yield strength of the commercial quality steel tubing from which the heat loop is made. The distribution for this variable was measured. All of above twenty-one variables were used in each of the probabilistic evaluations. The production distributions for these variables are shown in Table 1.

12 Table 1: Sources of Variation and Their Production Distributions Units Average Production Variables Standard Dev. 1 STLEN - Heat Loop Stub Length mm STANGF - Heat Loop Stub Angle (F-B) Deg STANGS - Heat Loop Stub Angle (S-S) Deg OLAP - Heat Loop Stub overlap [1] mm SYIELD - Yield strength, heat loop tube MPa STUBLOCZ - Stub Location (F-B) mm STUBLOCX - Stub Location (S-S) mm STUBLOCY - Stub Location (U-D) mm CNANGF Condenser Tube Angle (F-B) Deg CNANGS Condenser Tube Angle (S-S) Deg CONDLOCZ - Condenser Location (F-B) mm CONDLOCX - Condenser Location (S-S) mm CONDLOCY - Condenser Location (U-D) mm MPLOCZ - Mounting Plate Location (F-B) mm MPLOCX - Mounting Plate Location (S-S) mm MPLOCY - Mounting Plate Location (U-D) mm DLEN2 Location of Bend 2 mm ANG1A - Bend Angle 1 [2] Deg ANG2A - Bend Angle 2 [2] Deg CYIELD - Yield strength, condenser tube MPa Notes: [1] This variable uses the truncated Gaussian distribution. The maximum value is the average plus 2 mm. The minimum value is 5 mm. [2] These variables are tied to the variable CYIELD (yield strength of condenser tube). A high value for the CYIELD variable will result in an open value for the bend angle variables (because of more spring back). A low value for the CYIELD variable will result in a closed value for the bend angle variables (because of less spring back). Description of Design and Process Modification Sets Two basic condenser models were used for this evaluation. The Production Condenser is a model of the condenser geometry as used in production at the time of this evaluation. The 2599 Condenser is a model of condenser geometry as depicted on the production drawing for the condenser. The only difference between the two is that the production condenser does not have bend 2 (see figures 3 and 6). This results in the condenser tube being significantly offset from alignment with the heat loop stub (even without much variation).

13 Several design and process modification sets were tried on the two condenser models. For all of the design and process modification sets, the majority of the production variables were unchanged from the values shown in table 1. Table 2 shows the production variables that were changed for each of the design and process modification sets evaluated on one or both condenser models. Values that were changed from the production variables in table 1 are highlighted in blue. Table 2: Design and Process Modification Sets Design and Process Modification Sets Variable Production DC2 DC3 D6P5 D7P5 D8P5 STLEN (avg.) STLEN (st.dev.) CDLEN (avg.) CDLEN (st.dev) CNANGS (avg.) CNANGS (st.dev.) CNANGF (avg.) CNANGF (st.dev.) A description of each set of design and process modification sets are listed in the following subsections. DC2 This modification was based on "Engineering Judgement". The stub length (STLEN) average was increased by 30mm and the condenser tube length (CDLEN) was decreased by a corresponding 30mm. The assumption was that the longer stub length would allow the stub to yield where it protrudes from the deck because of the longer moment arm. This would allow the stub to better align with the condenser tube and result in a lower stress at the joint prior to heating and a lower percent strain after heating. Table 2 highlights the changes relative to the "production variables". DC3 This modification was based on disappointing results from DC2. The stub length (STLEN) average was decreased by 20mm and the condenser tube length (CDLEN) was increased by a corresponding 20mm relative to the production variables. This change is in the opposite directions of DC2 because DC2 resulted in worse results (higher 3sigma strain after heat application). Table 2 highlights the changes relative to the "production variables". D6P5 This modification was intended to reduce the response variable for the 2599 condenser. Relative to the production variables, the condenser tube angle front to back (CNANGF) average was decreased to -6.0 degrees. The CNANGF and CNANGS standard deviations were reduced to 1.0 degrees. These changes were based on the results of the evaluation of the 2599 condenser with the production variables. Table 2 highlights the changes relative to the "production variables".

14 D7P5 This modification was intended to reduce the response variable for the production condenser. The condenser tube angle side to side (CNANGS) was reduced to -4.0 degrees and the standard deviation was reduced to 1.0 degrees. The condenser tube angle front to back (CNANGF) average was decreased to -2.0 degrees and the standard deviation was reduced to 1.0 degrees. These changes were based on the results of the evaluation of the production condenser with the production variables. Table 2 highlights the changes relative to the production variables. D8P5 This modification was intended to reduce the response variable for the 2599 condenser. The condenser tube angle side to side (CNANGS) average was increased to 1.5 degrees. The CNANGF average and the CNANGF and CNANGS standard deviations were kept the same as D6P5 (-6, 1.0 and 1.0 respectively). These changes were based on the results of the evaluation of the 2599 condenser with the D6P5 variables. Table 2 highlights the changes relative to the production variables. Discussion The "production variables were used for the first evaluations of the heat loop brazed joint. Both the production condenser and the 2599 condenser were evaluated with the production variables. The results of the production condenser with production variation evaluation were used to help determine failure criteria (see section Failure Criteria ). Initially, a couple of simple design changes were attempted on both condenser designs. "Engineering Judgement" indicated that the heat loop stub length may have an effect on the response variable in the joint. The design and process modification set DC2 (see "Description of Design and Process Modification Sets" for a description of DC2) was evaluated on both condenser designs. The strain distributions for each condenser design got worse than the production evaluation (see Table 3 and Table 4). Since lengthening the heat loop stub (DC2) made matters worse, it was hoped that shortening it (DC3) would improve the strain distributions. This change did improve the strain distributions for the 2599 condenser, but the improvement was not as significant as hoped (see Table 4). This change had a negligible effect on the production condenser (set Table 3). Because the simple design changes were not working, it was decided to evaluate each condenser separately. Also, because both condensers had been evaluated with production variables, the results of those evaluations would be used to develop the next design and process modification set for evaluation. Production Condenser A Rank-Order Correlation Sensitivity plot was used to determine what variables were significant to the response variable (percent strain at the joint after heat applied). The rank-order correlation sensitivity plot for the production variables applied to the production condenser are shown in figure 7a. This plot indicates that the significant variables are CNANGS, STANGS, CNANGF and STANGF. Scatter plots were made to show the relationship between these input variables and the response variable. These scatter plots are shown in figures 7b - 7e.

15 Figure 7a - Correlation Sensitivities for Production Condenser with Production Variation Figure 7b - Scatter Plot of Percent Strain vs. CNANGS for Production Condenser with Production Variation

16 Figure 7c - Scatter Plot of Percent Strain vs. STANGS for Production Condenser with Production Variation Figure 7d - Scatter Plot of Percent Strain vs. CNANGF for Production Condenser with Production Variation

17 Figure 7e - Scatter Plot of Percent Strain vs. STANGF for Production Condenser with Production Variation The process Engineers have the least control over the stub angle and its variation. For this reason, only the condenser angle variables were modified for further evaluation. The scatter plot for CNANGS (figure 7b) indicated that the response variable could be minimized if this variable was set to -4.0 degrees and the variation tightened. The best estimate on how tight the angle could be controlled was +/- 3 degrees, so the standard deviation was set to 1.0 degrees. The scatter plot for the CNANGF (figure 7d) variable indicated that the response variable could be minimized if this variable was set to -2.0 degrees and the variation tightened. The standard deviation was also set to 1.0 degrees for this variable. These changes resulted in the design and process modification set D7P5. See the section "Description of Design and Process Modifications" for a table of modification sets. The production condenser was then evaluated with the modification set D7P5. This modification set reduced the response variable significantly (see "Results"). A rank order correlation sensitivity plot (with the significance level at the default 2.5%) showed that there were no significant variables Condenser A Rank-Order Correlation Sensitivity plot was used to determine what variables were significant to the response variable (percent strain at the joint after heat applied). The rank-order correlation sensitivity plot for the production variables applied to the 2599 condenser are shown in figure 8a. This plot indicates that the significant variables are CNANGF and STANGF. Scatter plots were made to show the relationship between these input variables and the response variable. These scatter plots are shown in figures 8b - 8c. As mentioned in the previous section, the process Engineers have the least control over the stub angle and its variation. For this reason, the condenser angle variable was modified for further evaluation. The scatter plot for CNANGF (figure 8b) indicated that the response variable could be minimized if this variable was set to -6.0 degrees and the variation tightened. The best estimate on how tight the angle could be controlled was +/- 3 degrees, so the standard deviation was set to 1.0 degrees. The standard deviation for CNANGS was also tightened to 1.0 degrees. These changes resulted in the design and process modification set D6P5. See the section "Description of Design and Process Modifications" for a table of modification sets.

18 Figure 8a - Correlation Sensitivities for 2599 Condenser with Production Variation Figure 8b - Scatter Plot of Percent Strain vs. CNANGF for 2599 Condenser with Production Variation

19 Figure 8c - Scatter Plot of Percent Strain vs. STANGF for 2599 Condenser with Production Variation The 2599 condenser was then evaluated with the modification set D6P5. This modification set reduced the response variable significantly (see "Results"). The rank-order correlation sensitivity plot for the D6P5 variables applied to the 2599 condenser are shown in figure 9a. This plot indicates that the significant variables are now CNANGS, STANGS and STUBLOCZ. Scatter plots were made to show the relationship between the significant input variables and the response variable. These scatter plots are shown in figures 9b - 9d. Figure 9a - Correlation Sensitivities for 2599 Condenser with D6P5 Variation

20 Figure 9b - Scatter Plot of Percent Strain vs. CNANGS for 2599 Condenser with D6P5 Variation Figure 9c - Scatter Plot of Percent Strain vs. STANGS for 2599 Condenser with D6P5 Variation

21 Figure 9d: Scatter Plot of Percent Strain vs. STUBLOCZ for 2599 Condenser with D6P5 Variation By looking at the scatter plots for CNANGS, it was decided that the next evaluation should use and average CNANGS value of 1.5 degrees while maintaining all the other variables the same as in D6P5. The STUBLOCZ variable was not modified because tooling costs would be significant. STANGS was not modified for reasons stated in an earlier section. These changes resulted in the design and process modification set D8P5. See the section "Description of Design and Process Modifications" for a table of modification sets. The 2599 condenser was then evaluated with the modification set D8P5. This modification set reduced the response variable significantly (see "Results"). The rank-order correlation sensitivity plot for the D8P5 variables applied to the 2599 condenser are shown in figure 10a. This plot indicates that the significant variables are now STANGF, STUBLOCZ and CONDLOCY. Scatter plots were made to show the relationship between these input variables and the response variable. These scatter plots are shown in figures 10b - 10d. Modifications to the significant variables were not desirable for capital and material cost reasons. Also, the response variable and the 3-sigma strain value were already greatly reduced. For these reasons, additional modification sets were not pursued.

22 Figure 10a - Correlation Sensitivities for 2599 Condenser with D8P5 Variation Figure 10b - Scatter Plot of Percent Strain vs. CONDLOCZ for 2599 Condenser with D8P5 Variation

23 Figure 10c - Scatter Plot of Percent Strain vs. STANGF for 2599 Condenser with D8P5 Variation Figure 10d - Scatter Plot of Percent Strain vs. STUBLOCZ for 2599 Condenser with D8P5 Variation Non-Converged Iterations Each evaluation attempted to perform 100 probabilistic iterations. However, every evaluation ended up with some iterations not converged (non-converged iterations were between 1 and 25). A comparison of variable values for converged and non-converged iterations was made, but no pattern was found. It was noticed that the largest strain distributions tended to have the larger number of non-converged iterations. It

24 is assumed that the non-converged iterations could have been addressed with better model refinement, but this was not attempted. Results The 3 sigma strain value (average + 3*sigma) was unexpectedly similar for both the production condenser and the 2599 condenser given production variation (see tables 3 and 4). The 2599 condenser was expected to have a lower 3 sigma strain value because it had better alignment of the tubes per the drawings. However, the probabilistic studies indicate that the alignment of the tubes is much more significant than the offset between the two tubes when the production condenser is used. Simple design changes (DC2 and DC3, lengthening and shortening the tubes) did not make the 3 sigma strain values better (see tables 3 and 4). The probabilistic results indicate that the alignment of the tubes is much more significant than the length of the tubes. The 3 sigma strain values were significantly improved when specific design and process modification sets were developed for each condenser design separately. The production condenser 3-sigma strain value was reduced from 3.63% to 1.37% when design and process modification set D7P5 was applied. The condenser 3-sigma strain value was reduced from 3.56 % to 1.58% when design and process modification set D8P5 was applied. The strain results for the production condenser and the 2599 condenser are shown in tables 3 and 4 respectively. Table 3: Production Condenser Strain Results with Various Design and Process Modifications Applied Design and Process Variation Modification Sets [1] Production DC2 DC3 D7P5 Number of Iterations [2] Avg. Strain at Joint (%) St. Dev. of Strain (%) Failure Strain (%) [3] sigma Strain (%) [4] CpK - Num. Of st.dev. between 3sigma and failure strain [5] Notes: [1] - See section "Description of Design and Process Modifications" for an explanation of how each set differs from the production variation set. [2] - Number of successful probabilistic iterations performed. Each set attempted 100 iterations, however some of the iterations did not converge. See section "Discussion" for more information. [3] - Failure Strain calculated in section "Failure Criteria". [4] - 3sigma strain calculated by adding three times the standard deviation to the average strain. [5] - CpK, number of standard deviations between 3sigma strain and failure strain. The larger the number the better.

25 Table 4: 2599 Condenser Strain Results with Various Design and Process Modifications Applied Design and Process Variation Modification Sets [1] Production DC2 DC3 D6P5 D8P5 Number of Iterations [2] Avg. Strain at Joint (%) St. Dev. of Strain (%) Failure Strain (%) [3] sigma Strain (%) [4] CpK - Num. Of st.dev. between 3sigma and failure strains [5] Notes: [1] - See section "Description of Design and Process Modifications" for an explanation of how each set differs from the production variation set. [2] - Number of successful probabilistic iterations performed. Each set attempted 100 iterations, however some of the iterations did not converge. See section "Discussion" for more information. [3] - Failure Strain calculated in section "Failure Criteria". [4] - 3sigma strain calculated by adding three times the standard deviation to the average strain. [5] - CpK, number of standard deviations between 3sigma strain and failure strain. The larger the number the better. Conclusions The Probabilistic results from this study indicate that the steel heat loop joint can be made successfully if the right design and process modifications are employed. These changes will be challenging because the company will have to exert greater control over the process variation. This is always difficult in a production process that must operate at high speed. The Probabilistic design module proved to be invaluable in evaluating design and process changes to help the steel heat loop cost reduction project. Standard single pass analyses or optimizations did not provide much insight in determining what the important variables were. Also, extensive physical trial runs and "DOEs" would not have been cost effective because the failure rate was so low (3%). The Probabilistic design module allowed the engineer to determine the significant variables and to develop modification sets to address them. References 1 R. E. Walpole and R. H. Myers, "Probability and Statistics for Engineers and Scientists", third edition, pg 574.