Tolerance Stack Analysis in Francis Turbine Design

Transcription

1 ITB J. En. Sci. Vol. 42, No. 1, 2010, Tolerance Stack Analysis in Francis Turbine Desin Indra Djodikusumo, Koko Suherman & Paskalis Bowo A. Oken Faculty of Mechanical Enineerin and Aerosace, Institute of Technoloy Bandun Abstract. The tolerance stackin roblem arises in the context of assemblies from interchaneable arts because of the inability to roduce or to join arts exactly accordin to nominal dimensions. Either the relevant art s dimension varies around some nominal values from art to art or the act of assembly that leads to variation. For examle, as runner of Francis turbine is joined with turbine shaft via mechanical lock, there is not only variation in the diameter of runner and the concentricity between the runner hole and turbine shaft, but also the variation in concentricity between the outer arts of runner to runner hole. Thus, there is the ossibility that the assembly of such interactin arts won t function or won t come toether as lanned. Research in this area has been conducted and 2 mini hydro Francis turbines (800 kw and 910 kw) have been desined and manufactured for San Sarino and Sawi Dao 2 in Central Sulawesi. Exeriences in analyzin the tolerance stacks have been documented. In this aer it will be demonstrated how the requirements of assemblin erformance are derived to be the desined tolerances of each interactin comonent, such a way that the assemblin would be functionin and come toether as lanned. Keywords: assemblin requirements; eometric dimensionin and tolerancin; tolerance stacks; variation of feature eometry; worst case method. 1 Introduction The utilization of renewable enery as an alternative to fosil enery has been romoted all around the world, includin Indonesia. Since then, the demand of mini hydro turbines in Indonesia increases very fast. Around 40 mini hydro turbines are required in Indonesia each year until 2020 [1]. The ower for the required turbines is around 500 kw until 2 MW. Most of them are Francis tye and some others are from other tyes such as Kalan for low head otential sites and Pelton for hih head otential sites. This is a very ood oortunity for Indonesia to develo their own mini hydro ower lant usin their caacity, not only in the enineerin, rocurement and construction (EPC), but also in the suly of the required equiments such as turbines, enerators, control anels, transformers and others. PT. Ganesha Reverse Enineerin and Toolmakin (GREAT) is one of the tenant in Industry and Business Incubator at the Institute of Technoloy Bandun, that desin mini hydro turbines, esecially Francis, Kalan and Pelton Turbines. With some artners in the manufacturin industry, Received December 10 th, 2009, Revised March 17 th, 2010, Acceted for ublication March 20 th, 2010.

2 74 Indra Djodikusumo, et al. mostly small and medium size comanies, PT. GREAT starts manufacturin mini hydro turbines. Eventhouh it is not easy to enetrate the market, some comanies such as the National Electricity Provider (PT. PLN) starts orderin mini hydro turbines from PT. GREAT. It is the objectives of PT. GREAT to roduce more mini hydro turbines that fulfill some criterias as follows: Good erformance Good relliability Cometitive delivery time Cometitive rice Ease of oeration Ease of maintenance Good after sale service by rovidin its comonents usin art number, so the concet of interchaneability should be imlemented in the desin and manufacturin Many efforts have been conducted by PT. GREAT durin the develoment of mini hydro turbines, and one of them is in the field of Geometric Dimensionin and Tolerancin (GDT). In order to achieve ood efficiency, the losses esecially due to water leakae should be avoided. Water should flow throuh runner blades, not throuh the as between runner and its covers and also between uide vanes and its covers. As the result, a very small as between runner and covers and between uide vanes and covers are required [2-6]. This small as require small tolerances for the eometric of its related turbine comonent features. Moreover, the objective to achieve the interchaneability of its sare arts usin art number by after sales service also require the GDT theory. The GDT method that has been utilized for analyzin the tolerance stacks will be demonstrated and some of its results will be shown in this aer. 2 Stacked Tolerance Analysis Methodoloy Stacked tolerance analysis is the rocess of breakin down comonents in assembly in order to take known tolerances each comonent and analyzin the combination of these tolerances at an assembly level. This analysis is done only at critical features in assembly. The first ste in the rocess is to identify the requirements for the system or we can say to identify features that have bi contribution for fit and function of the roduct. These features are said critical for assembly.

3 Tolerance Stack Analysis in Fancis Turbine Desin 75 This stacked tolerance analysis will be divided into dimensional tolerance analysis and eometrical tolerance analysis. 2.1 Dimensional Tolerance Analysis Accordin to Paul Drake [7], the rocess of analysis for tolerance stacks could follow the followin traditional aroach (Fiure 1). Each activity will be resented into more detail in the followin sub chater. Establish the Performance Requirements Draw a Loo Diaram Convert All Dimensions to Mean Dimension with an Equal Bilateral Tolerance Calculate the Mean Value for the Performance Requirement Determine the Method of Analysis Calculate the Variation for the Performance Requirement Fiure 1 Tolerance Stack Analysis Process [7] Establish the Performance Requirements First, identify all the requirements for assembly system that will lead to the success of roduct s erformance or ease of assembly. Then, flow down these requirements to each comonent. Finally, convert all erformance requirements into a requirements for assembly Drawin of a Loo Diaram The loo diaram is a rahical reresentation of each analysis. Each requirement requires a searate loo diaram. There are two tyes of loo diaram, vertical and horizontal. Laws for drawin loo diaram are: For horizontal dimension loos, start at the surface on the left of the a. Follow a comlete dimension loo, to the surface on the riht. For vertical dimension loos, start at the surface on the bottom of the a. Follow a comlete dimension loo, to the surface on the to.

4 76 Indra Djodikusumo, et al. Usin vectors, create a closed loo diaram from the startin surface to the endin surface. Do not include as when selectin the ath for the dimension loo. Each vector in the loo diaram reresents a dimension. Use an arrow to show the direction of each vector in the dimension loo. Identify each vector as ositive for dimension followed from left to riht or from bottom to to, and as neative for dimension followed from riht to left or from to to bottom. Assin a variable name to each dimension in the loo. Record sensitivities for each dimension. The manitude of the sensitivity is the value that the a chanes, when the dimension chanes 1 unit. For examle, if the a chanes 1 mm when the dimension chanes 1 mm, then the manitude of sensitivity is 1 (1 mm/1 mm). On the other hand, if the a chanes 0.5 mm when the dimension chanes 1 mm, then the manitude of sensitivity is 0.5 (0.5 mm/1 mm). Usually, the manitude of sensitivity is 0.5 for comonents involvin diameter. Determine whether each dimension is fixed or variable. A fix dimension is one in which we have no control, such as a vendor art dimension. A variable dimension is one that we can chane to influence the outcome of the tolerance stack, such as custom made comonents (made by order) Convertin All Dimensions to Mean Dimension with an Equal Bilateral Tolerance Next, all tolerances in loo diaram should be chane into equal bilateral tolerance, where uer and lower tolerance are equal. As a rule, desiner should use equal bilateral tolerances, excet if usin this equal bilateral tolerances may force manufacturin to use nonstandard tools. This chane is based on fact that manufacturin rocess are normally distributed, where manufactured roduct s dimension will vary around its mean nominal dimensions. If the desiner uses uniteral tolerances for the roducts, then most of roducts will be rejected. Stes for convertin to an equal bilateral tolerance are: Calculate the uer and lower limit dimension. Subtract the lower limit from the uer limit to et the total tolerance band, and then divide the tolerance band by two to et an equal bilateral tolerance. Add the equal bilateral tolerance to the lower limit to et the mean dimension. Alternately, subtract the equal bilateral tolerance from the uer limit.

5 Tolerance Stack Analysis in Fancis Turbine Desin Calculate the Mean Value for the Performance Requirement The mean value of the requirements (a) is calculated by [7-9]: d n a d (1) i i i Determine the Method of Analysis There are 3 tyes of tolerancin models to analyze the variation at the a, which are worst case (WC) model, root sum of the square (RSS) statistical model and the combination of WC and RSS that is modified root sum of the square (MRSS) statistical model. WC model verifies all comonents will erform their intended function 100% of the time. This is a conservative aroach and used for retail roduction or roduction by order. RSS model assume that most of the manufactured arts all centered on the mean dimension. This is used for mass roduction. MRSS model is created to bride WC model which is too tiht with RSS model which is too loose. The comarison of these 3 models can be seen at Table 1. Table 1 Comarison of Analysis Models [7]. Consideration WC Model RSS Model MRSS Model Risk of defect Lowest Hihest Middle Cost Hihest Lowest Middle Assumtions about comonent rocesses Assumtions about drawin tolerances Assumtion about exected assembly variation None Dimensions outside the tolerance rane are screened out. 100% of the arts are within the maximum and minimum erformance rane. The rocess follows a normal distribution. The mean of the rocess is equal to the nominal dimension. Processes are indeendent. The tolerance is related to manufacturin rocess caability. Usually the tolerance rane is assume to be the +/- 3 limit of the rocess. Assembly distribution is normal % of the assemblies will be between the minimum and maximum a. The rocess follows a normal distribution. The mean of the rocess is not necessarily equal to the nominal dimension. The tolerance is related to manufacturin rocess caability. Usually the tolerance rane is assume to be the +/- 3 limit of the rocess % of the assemblies will be between the minimum and maximum a. The correction factor (C f ) is a safety factor.

6 78 Indra Djodikusumo, et al Calculate the Variation for the Performance Requirement Worst Case Model The followin equation calculates the exected variation at the a [7-9]: t wc n a t (2) i1 i i The minimum and maximum as are equal to: Minimum Ga = d - t wc Maximum Ga d t wc If either the value of minimum a or maximum a doesn t conform to the requirement, then the tolerance value of each comonent in the assembly needs to be by usin: F wc d - - a t m j jf j1 q k 1 at k kv The resize factor is only alied to variable comonent by usin: t F t (5) kv, wc, wc. kv Then exected variation after resizin is: t a t a t wc, j jf k kv, wc, j1 k1 q (6) (3) (4) So the new minimum and maximum as are: Minimum Ga d - t Maximum Ga d t wc, wc, (7) RSS Model The exected variation is calculated by [7-9]: t a t a t a t a t (8) rss n n The minimum and maximum as are equal to:

7 Tolerance Stack Analysis in Fancis Turbine Desin 79 Minimum Ga = d - t rss Maximum Ga d t rss (9) If either the value of minimum a or maximum a doesn t conform to the requirement, then the tolerance value of each comonent in the assembly needs to be by usin: F rss 2 2 m j jf j1 q 2 ( at k kv ) k 1 ( d - ) ( a t ) The resize factor is only alied to variable comonent by usin: kv, rss, rss. kv (10) t F t (11) Then exected variation after resizin is: t a t a t rss, j jf k kv, rss, j1 k1 q (12) So the new minimum and maximum as are: Minimum Ga d - t Maximum Ga d t rss, rss, (13) MRSS Model The exected variation is calculated by [7-9]: Where: t C a t a t a t a t (14) mrss f n n C f 0.5 twc - trss 1 (15) t n - 1 rss The minimum and maximum as are equal to: Minimum Ga = d - t mrss Maximum Ga d t mrss (16) If either the value of minimum a or maximum a doesn t conform to the requirement, then the tolerance value of each comonent in the assembly needs to be by usin:

8 80 Indra Djodikusumo, et al. Where: F mrss 2 - b - b - 4ac (17) 2a q 2 q q q a 0.25 a t a t 3 n a t - n a t k1 k1 k1 k k kv k kv k kv k kv (18) b 0.5 a t a t a t d - - n a t d - k1 j1 k1 k1 q q q k kv j jf k kv m k kv m (19) 2 c 0.25a t d n d - n d - j1 a jt jf d - m - n a jt jf d - m a jt jf j jf j jf j jf m m m j1 j1 j n a t - n a t j1 j1 2 (20) The resize factor is only alied to variable comonent by usin: t F t (21) kv, mrss, mrss. kv Then exected variation after resizin is: t a t a t mrss, j jf k kv, mrss, j1 k1 q (22) So the new minimum and maximum as are: Minimum Ga d - t Maximum Ga d t mrss, mrss, (23) 2.2 Geometrical Tolerance Analysis Beside dimensional tolerance, eometrical tolerance in a comonent is also need to be analyzed in stacked tolerance. Geometric tolerance will control form, orientation and location of the feature. GDT controls are enerally used only in worst case analysis. Since WC model assumes 100% insection, so GDT control will influence the a variation. In a statistical analysis, GDT doesn t influence to a variation because manufacturin rocesses themselves are sources for variation.

9 Tolerance Stack Analysis in Fancis Turbine Desin 81 Rules in eometrical tolerance analysis [7,10,11]: Location control on a feature in the loo diaram is usually included in the analysis. Orientation control on a feature in the loo diaram is included in the analysis as lon as the location of the feature is not a contributor to the requirement. Form control on a feature in the loo diaram is included in the analysis as lon as the location, orientation or size of the feature is not a contributor to the requirement. Geometric form and orientation controls on datum features are usually not included in the loo diaram since datum is startin oint for measurement and considered as TGC. If form or orientation control is used in the loo diaram, then it is modeled with a nominal dimension equals to zero and an equal bilateral tolerance equals to form or orientation tolerance. For location control at RFS, feature s size and location are treated indeendently. Meanwhile for MMC or LMC condition, the size and location dimension can t be treated indeendently. For MMC or LMC condition, first it is necessary to calculate the larest outer boundary and smallest inner boundary allowed by the dimensions and tolerances. Formulas to calculate these boundaries can be seen at Table 2. Table 2 Formulas to calculate outer and inner boundary for location control at MMC or LMC condition [7]. Feature Condition Inner Boundary Outer Boundary External MMC LMC Tolerance at LMC MMC + Geometric Tolerance at MMC LMC LMC Geometric Tolerance at LMC MMC + Tolerance at MMC Internal MMC MMC Geometric Tolerance at MMC LMC + Tolerance at LMC LMC MMC Tolerance at MMC LMC + Geometric Tolerance at LMC Next, convert the inner and outer boundary into a nominal diameter with an equal bilateral tolerance by usin: Nominal Diameter = (outer boundary + inner boundary) / 2 (24) Equal Bilateral Tolerance = (outer boundary - inner boundary) / 2 (25)

10 82 Indra Djodikusumo, et al. Beside those three controls above, there is run-out control. Analyzin run-out control in tolerance stacks is similar to analyzin location control at RFS, where size and run-out tolerance are treated indeendently. Run-out tolerance can be modeled with a nominal dimension equals to zero and an equal bilateral equals to run-out tolerance/2. And also there is concentricity control which is treated similar to location and run-out control. Concentricity tolerance can be modeled with a nominal dimension equals to zero and an equal bilateral equals to concentricity tolerance/2. 3 Case Study Fiure 2 shows a cross section of a Francis turbine assembly made by PT. GREAT. From this examle, it will be demonstrated the stacked tolerance analysis. On this examle, there are several erformance requirements, which are: Requirement 1. The a between runner s cone and enerator side cover must always be reater than zero to ensure that the runner can rotate freely but it shouldn t be too lare to revent leakae so the erformance is still okay. Requirement 2. The a between runner s rin and intermediate rin must always be reater than zero to ensure that the runner can rotate freely but it shouldn t be too lare to revent leakae so the erformance is still okay. Requirement 3. The a between uide vanes and enerator side cover must always be reater than zero to ensure that the uide vanes can rotate freely but it shouldn t be too lare to revent leakae so the erformance is still okay. Requirement 4. The a between uide vanes and draft tube side cover must always be reater than zero to ensure that the uide vanes can rotate freely but it shouldn t be too lare to revent leakae so the erformance is still okay. Next, convert each requirement into an assembly a requirement as follow: Requirement 1: 0 < a Requirement 2: 0 < a Requirement 3: 0 < a Requirement 4: 0 < a For this occasion, author only resents requirement 1 to reresent requirement for radial rotor direction. Stes for analysis are described in followin sub chater.

11 Tolerance Stack Analysis in Fancis Turbine Desin 83 Requirement 3 Requirement 4 Requirement 2 Requirement Notes: 1. Shaft 5. Guide vane 2. Runner 6. Draft tube side cover 3. Generator side cover 7. Intermediate rin 4. Stay rin 8. Draft tube Fiure 2 Cross Section of a Francis Turbine Assembly made by PT. GREAT. 3.1 Establish the Performance Requirements Refers to erformance requirement, a 1 which is a between runner s cone and enerator side cover should be between 0 mm and 0.6 mm. 3.2 Drawin of a Loo Diaram It is necessary to determine base (or stoin oint) so that the loo doesn t have to involve all comonents. For this analysis, assume turbine s stay rin (comonent number 4 in Fiure 2) as the base. In analysis, base comonent is always considered ideal. So, in real, this stay rin must be manufactured and

12 84 Indra Djodikusumo, et al. assembled roerly to resemble ideal condition. Detail A on Fiure 3 shows the loo diaram for a 1. DETAIL A DETAIL A Ga 1 A B C D Fiure 3 Turbine Assembly Detail A.

13 Tolerance Stack Analysis in Fancis Turbine Desin 85 The exlanations for each vector are as follow: A is a vector that reresents concentricity tolerance of runner s cone. Its nominal value is 0 and its equal bilateral tolerance is half from concentricity tolerance. A = 0 ± mm. Its sensitivity factor is 1. A is a variable comonent because its value can be adjusted by order. B is a vector that reresents outside diameter of runner s cone. Its value is 556 ± 0.05 mm. Its sensitivity factor is B is a variable comonent because its value can be adjusted by order. C is a vector that reresents outside diameter of enerator side cover. Its value is 9457 ( 945 ) mm. Its sensitivity factor is 0.5. C is a fixed comonent because it fits with another comonent so that it can t be chaned to kee the erformance. D is a vector that reresents thickness of enerator side cover until the a. Its value is 9457 mm (557 ± 0.05) mm. Its sensitivity factor is D is a variable comonent because its value can be adjusted by order. 3.3 Convertin All Dimensions to Mean Dimension with an Equal Bilateral Tolerance Table 3 shows the summary of vectors for a 1. Descrition Concentricity tolerance of runner s cone Outside diameter of runner s cone Outside diameter of enerator side cover Thickness of enerator side cover Table 3 Summary of Vectors for Ga 1. Name Mean Dimension Sensitivity Fixed/ Variable +/- Equal Bilateral Tolerance A 0 mm 1 Variable mm B 556 mm -0.5 Variable 0.05 mm C mm 0.5 Fixed mm D mm -0.5 Variable mm 3.4 Calculate the Mean Value for the Performance Requirement The mean value of the a is:

14 86 Indra Djodikusumo, et al. n d a d i i i1 d (1) A (-0.5) B 0.5 C (-0.5) D d d (1)(0) (-0.5)(556) (0.5)(944.93) (-0.5)(387.93) 0.5 mm 3.5 Determine the Method of Analysis For this case, worst case analysis model is chosen because PT. GREAT manufactures 1 turbine only and it is necessary to assure the correctness of each dimension 100%. 3.6 Calculate the Variation for the Performance Requirement The a variation is: n t a t wc i i i1 t t wc wc (1)(0.025) (-0.5)(0.05) (0.5)(0.045) (-0.5)(0.095) 0.12 mm So the maximum and minimum as are: Maximum a = d + t wc = = 0.62 mm Minimum a = d t wc = = 0.38 mm As exlained before in erformance requirement, maximum a must not exceed 0.6 mm. So it is necessary to do an adjustment. To reach 0.6 mm maximum a, the value of a variation should be 0.1 mm. So, the value of m in Equation 4 must be 0.4 mm. Then the resize factor is: F F F wc wc wc d - - a t m j jf j1 q k at k kv

15 Tolerance Stack Analysis in Fancis Turbine Desin 87 This resize factor is multilied to each variable s tolerance. So the new values are: A = 0 ± 0.02 mm B = 556 ± 0.04 mm C = ± mm D = ± mm The a variation after adjustment is: t a t wc, i i i1 t wc, t wc, n (1)(0.02) (-0.5)(0.04) (0.5)(0.045) (-0.5)(0.075) 0.1 mm Then the maximum and minimum as are: Maximum a = d + t wc resixed = = 0.6 mm Minimum a = d t wc, 4 Result = = 0.4 mm Accordin to this analysis, desin from PT. GREAT can t meet the erformance requirement. Desin from PT. GREAT creates a 1 that varies between mm. This fault can cause a decrease of turbine s efficiency and also difficulty in assembly rocess. To fulfill the erformance requirement 1 which demands 0 < a mm, comonents that build the assembly must have dimensions and tolerances as shown in Fiure 4. From this analysis, it can be concluded that as in assembly can be controlled since desin rocess. So desiners must consider this a analysis to assure roduct s erformance and assemblability.

16 Ø 9457 Ø 557 ± 0.03 Ø 556 ± Indra Djodikusumo, et al A A Fiure 4 Dimensions and Tolerances of Radial Comonents (Ga 1) After Adjustment. Nomenclature a i = sensitivity factor that defines direction and manitude for the i th dimension a j = sensitivity factor for the j th, fixed comonent in the stack u a k = sensitivity factor for the k th, variable comonent in the stack u C f = correction factor used in MRSS equation d i = the mean value of the i th dimension in the loo diaram d = the mean value at the a (ositive means clearance and neative means interference) F mrss = resize factor for MRSS model F rss = resize factor for RSS model F wc = resize factor for WC model

17 Tolerance Stack Analysis in Fancis Turbine Desin 89 m = minimum value at the a = 0 if no interference or clearance is allowed n = the number of indeendent variables (dimensions) in the stack u q t i t jf t kv = number of indeendent, fixed dimension in the stack u = number of indeendent, variable dimension in the stack u = equal bilateral tolerance of the i th comonent in the stack u = equal bilateral tolerance of the j th, fixed comonent in the stack u = equal bilateral tolerance of the k th, variable comonent in the stack u t mrss = maximum exected variation (equal bilateral) usin MRSS model t rss = maximum exected variation (equal bilateral) usin RSS model t wc = maximum exected variation (equal bilateral) usin WC model References [1] Investor Daily Friday, JAKARTA, 23 February [2] Bernhard Pelikan (President ESHA), Euroean Small Hydro Power Association, Guide on How to Develo a Small Hydroower Plant - Part 1, ESHA Publishin, [3] Bernhard Pelikan (President ESHA), Euroean Small Hydro Power Association, Guide on How to Develo a Small Hydroower Plant - Part 2, ESHA Publishin, [4] Hartwi Petermann, Stroemunsmaschinen, Sriner Verla, Berlin - Heidelber - New York, [5] Bernhard Pelikan, The President of Euroean Small Hydroower Association, Guide on How to Develo a Small Hydroower Plant, ESHA, [6] Arne Kjølle, The Develoment of Hydroower in Norway, Trondheim, [7] Paul Drake, JR., Dimensionin and Tolerancin Handbook, McGraw- Hill, ISBN , [8] Geor Henzold, Geometrical Dimensionin and Tolerancin for Desin, Manufacturin and Insection, Elsevier Publishin, ISBN-13: , [9] Hon-Chao Zhan, Advanced Tolerancin Techniques, Willey Interscience, ISBN , 1997.

18 90 Indra Djodikusumo, et al. [10] Gene R. Coorno, Geometric Dimensionin and Tolerancin for Mechanical Desin, McGraw-Hill, [11] Bryan R. Fischer, Mechanical Tolerance Stacku and Analysis, Marcel Dekker Inc.