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Response surface methodology (RSM) based on desirability function approach (DFA) is applied to obtain an optimal design of the impeller geometry for an automotive torque converter. The relative importance of six design parameters including impeller blade number, blade thickness, bias angle, scroll angle, inlet angle and exit angle is investigated using orthogonal design approach. The impeller inlet angle, exit angle and bias angle are found to exert the greatest influence on the overall performance of a torque converter, with two flow area factors being considered, namely 17% and 20%. Then, RSM together with central composite design (CCD) method is used to in-depth evaluate the interaction effect of the three key parameters on converter performance. The results demonstrate that impeller exit angle has the strongest impact on peak efficiency, with larger angles yielding the most favorable results. The stall torque ratio maximization is attainable with the increase of impeller bias angle and inlet angle together with smaller exit angle. In the end, an optimized design for the impeller geometry is obtained with stall torque ratio and peak efficiency increased by 1.62% and 1.1%, respectively. The new optimization method can be used as a reference for performance enhancement in the design process of impeller geometry for an automotive torque converter.

Torque converters are widely used in vehicle power transmission systems and transfer the torque and power from engine to transmission by the hydraulic operation of transmission oil. Despite of its ability to provide good acceleration performance and absorb excessive vibration, its mechanical efficiency is relatively low. Nowadays, a few new techniques are in use to overcome the low efficiency. The application of the locking clutch of the turbine to the impeller and the optimized design of shape of torus and blades to enhance the hydraulic performance are the two representative methods. The first technique is very effective at the wider range of vehicle speed, but not easy to obtain an optimized shape since a large number of design parameters are involved. Typically, several torque converters are designed and tested by computational fluid dynamics (CFD) and experiment until the target performance is met [

It is generally been accepted that while the accuracy of CFD analyses has not yet achieved a level that is equivalent to experimental techniques, its ability to correctly predict the direction of any changes is reliable [

The traditional one-factor-at-a-time approach has been widely used for evaluation or optimization of these parameters [

In this present paper, the main objective is to improve the overall performance of automotive torque converters by means of impeller geometry optimization using DOE and CCD methods. A new parametric geometric design method of impeller is proposed by using parametric equations and Creo software. A DOE array is used to reduce the number of analyses required at each flow area factor, with two area factors being investigated, namely 17% and 20%. The DOE post-processing analysis is also used to rank the relative importance of the geometric parameters for both area factors. Then, the CCD method is applied to in-depth evaluate the interaction effect of the three most important parameters and the optimization is done to have the best overall performance of a torque converter.

The impeller parametric design starts with the definition of the torus shape. Various design parameters including active diameter, aspect ratio and two arc radii are needed to determine the shell profile of an automotive torque converter torus. As the flow area in the circular path, in the proposed torque converter model, is assumed constant, only the design parameter area factor f a is used to determine the design path and core of torus. Consequently, by defining the impeller inlet radius R 1 and outlet radius R 2 , along with the shell and core, the torus profile of impeller can be obtained as shown in

Each impeller blade profile (

A x 2 + B y 2 + C x + D y + E x y = 0 , (1)

where A, B, C, D and E are the coefficients of the variables. It should be noted that the value of D is 1.0 and the parametric equation of the 2D design curve can be obtained as

A x 1 2 + B y 1 2 + C x 1 + y 1 + E x 1 y 1 = 0 , (2)

A x 2 2 + B y 2 2 + C x 2 + y 2 + E x 2 y 2 = 0 , (3)

tan ( α 2 − 90 ) + C = 0 , (4)

( E x 2 + 2 B y 2 + 1 ) tan ( α 1 − 90 ) + 2 A x 2 + E y 2 + C = 0 , (5)

where α 1 is the exit angle of impeller, and α 2 is the inlet angle of impeller. In the present study, four design parameters including exit angle α 1 , inlet angle α 2 , offset size d, and conic factor f c are provided to calculate the 2D design curve. The coordinates of P c and P 1 can be obtained as

y c = x c tan ( α 2 − 90 ) , (6)

y c = y 2 − ( x 2 − x c ) tan ( α 1 − 90 ) , (7)

x 1 = x 2 / 2 + ( x c − x 2 / 2 ) f c , (8)

y 1 = y 2 / 2 + ( y c − y 2 / 2 ) f c , (9)

where P c ( x c , y c ) is the intersection of the two tangent lines that across the curve’s starting and ending points, respectively (

The 2D design curve can be easily constructed by having the torus and the four design parameters defined. After the definition of 3D curves of shell and core, another design parameter bias angle β is used to obtain the blade profile. Given the impeller blades are constant-thickness stamped sheet metal, the blade thickness t is defined. Once the torus shape and five-blade design parameters including exit angle α 1 , inlet angle α 2 , offset size d, conic factor f c , bias angle β and blade thickness t are determined, the basic blade geometry of impeller is generated by means of parametric equations and Creo software. The schematic representation of the construction of impeller blade is shown in

After the definition of the torus and a blade, the construction of the whole impeller geometry is easy and is based on the rotation of the generic blade around the axis; for doing so the number of blades z should be provided by the user. The completed impeller parametric geometry is shown in

No. | Description | Parameter |
---|---|---|

1 | Exit angle of impeller at shell (˚) | α s 1 |

2 | Inlet angle of impeller at shell (˚) | α s2 |

3 | Conic factor of impeller blade at shell | f cs |

4 | Offset size of impeller blade at shell (mm) | d s |

5 | Exit angle of impeller at core (˚) | α c 1 |

6 | Inlet angle of impeller at core (˚) | α c 2 |

7 | Conic factor of impeller blade at core | f cc |

8 | Offset size of impeller blade at core (mm) | d c |

9 | Bias angle of impeller (˚) | β |

10 | Blade number of impeller | z |

11 | Blade thickness of impeller (mm) | t |

An automotive torque converter is selected as a reference to generate the parametric model. The reference torque converter has an active diameter of 250 mm and the number of blades in the impeller, turbine, and stator are 31, 29, and 21, respectively. STAR-CCM + software is used to generate the computational mesh and perform the internal flow calculations in an appropriate way. The computational mesh is given in

When the circuit size is determined with satisfying the performance requirements, the flow cross-section can be determined with a rule of thumb that the flow area is uniform throughout the blades. In the present study, three toque converter parametric models with different flow areas are designed and simulated to investigate the effects of flow area on their overall performance including efficiency and impeller torque factor. For comparison, the flow area is altered to 17%, 20%, and 23% of area of a circle represented by the converter diameter (f_{a} = 17%, 20%, and 23%). The design parameters are unchanged as much as possible except the area factors and simulation models with varying area factors are shown in

In

DOE is a collection of mathematical and statistical techniques to reduce the number of experiments in order to find the effect of parameters affecting a response in a process, thereby aiming for a reduction in both costs and time [

The scroll angle can be determined by four design parameters including exit angle α 1 , inlet angle α 2 , conic factor f c , and offset size d, provided that the torus is defined. Matlab software is used to develop a simple-to-use GUI to calculate scroll angle as shown in

The blade scroll angles at shell and core can be represented by the blade scroll angle on the design path. Similarly, the inlet angles and exit angles at shell and core can be represented by the inlet angle and exit angle on the design path. Finally, the 11 design parameters of impeller are translated into 6 parameters including exit angle α 1 , inlet angle α 2 , scroll angle γ , bias angle β , blade thickness t, and blade number z. The selected factors and levels in DOE are listed in

For this paper, six main geometrical parameters mentioned above are selected as design variables (factors) and five different values (levels) are assigned for each design parameter. So 25 (L_{25}[5^{6}]) configurations with different combinations are generated for DOE. Stall torque ratio T r 0 and peak efficiency η ∗ are used as the dynamic characteristic and economic characteristic, respectively, to evaluate the performance of torque converters. Two area factors including 17% and 20% are considered. The final design matrix in DOE is presented in

One aspect of the DOE method is utilized response averages, calculated for each response and area factor in relation to a specific geometry parameter variable, to provide detail relating to the influence of the geometric factors on the performance of a torque converter. For example, to calculate the response averages relating to the 27 blade number, the average of the stall torque ratio and peak efficiency would be calculated from case number 1, 2, 3, 4 and 5. The calculated response average values for each level and for all parameters are shown for each area factor in

Factors | Levels | ||||
---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | |

z | 27 | 29 | 31 | 33 | 35 |

t/(mm) | 0.9 | 1.0 | 1.1 | 1.2 | 1.3 |

β /(˚) | −1.5 | 0 | 1.5 | 3 | 4.5 |

γ /(˚) | −0.28 | 0.72 | 1.72 | 2.72 | 3.72 |

α 1 /(˚) | 103 | 108 | 113 | 118 | 123 |

α 2 /(˚) | 45 | 50 | 55 | 60 | 65 |

Case number | Factors | Responses | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

z | t/(mm) | β /(˚) | γ /(˚) | α 1 /(˚) | α 2 /(˚) | f a = 17 % | f a = 20 % | |||

T r 0 | η ∗ /(%) | T r 0 | η ∗ /(%) | |||||||

1 | 27 | 0.9 | −1.5 | −0.28 | 103 | 45 | 1.715 232 | 79.865 50 | 1.809 260 | 78.503 70 |

2 | 27 | 1.0 | 0 | 0.72 | 108 | 50 | 1.702 694 | 80.233 73 | 1.810 344 | 80.110 81 |

3 | 27 | 1.1 | 1.5 | 1.72 | 113 | 55 | 1.748 850 | 81.292 62 | 1.866 738 | 82.279 18 |

4 | 27 | 1.2 | 3 | 2.72 | 118 | 60 | 1.769 172 | 80.974 52 | 1.880 240 | 81.483 16 |

5 | 27 | 1.3 | 4.5 | 3.72 | 103 | 65 | 1.798 510 | 79.840 45 | 1.918 926 | 79.157 27 |

6 | 29 | 0.9 | 0 | 1.72 | 118 | 65 | 1.744 180 | 81.482 67 | 1.831 391 | 81.165 12 |

7 | 29 | 1.0 | 1.5 | 2.72 | 123 | 45 | 1.713 947 | 79.865 50 | 1.817 788 | 84.493 67 |

8 | 29 | 1.1 | 3 | 3.72 | 103 | 50 | 1.795 316 | 80.233 73 | 1.865 114 | 79.053 13 |

9 | 29 | 1.2 | 4.5 | −0.28 | 108 | 55 | 1.777 395 | 81.292 62 | 1.897 089 | 80.721 67 |

10 | 29 | 1.3 | −1.5 | 0.72 | 113 | 60 | 1.693 166 | 80.974 52 | 1.772 419 | 80.656 17 |

11 | 31 | 0.9 | 1.5 | 3.72 | 108 | 60 | 1.768 039 | 79.840 45 | 1.866 011 | 80.304 73 |

12 | 31 | 1.0 | 3 | −0.28 | 113 | 65 | 1.750 658 | 81.482 67 | 1.844 729 | 81.120 89 |

13 | 31 | 1.1 | 4.5 | 0.72 | 118 | 45 | 1.733 369 | 83.626 48 | 1.842 111 | 83.468 03 |

14 | 31 | 1.2 | −1.5 | 1.72 | 123 | 50 | 1.694 157 | 79.178 20 | 1.805 450 | 84.588 20 |

15 | 31 | 1.3 | 0 | 2.72 | 103 | 55 | 1.755 297 | 79.719 79 | 1.843 188 | 80.388 25 |

16 | 33 | 0.9 | 3 | 0.72 | 123 | 55 | 1.714 483 | 80.939 80 | 1.826 645 | 83.849 77 |

17 | 33 | 1.0 | 4.5 | 1.72 | 103 | 60 | 1.806 527 | 80.095 42 | 1.924 480 | 79.972 57 |

18 | 33 | 1.1 | −1.5 | 2.72 | 108 | 65 | 1.709 343 | 80.472 92 | 1.775 109 | 78.728 82 |

19 | 33 | 1.2 | 0 | 3.72 | 113 | 45 | 1.681 214 | 83.000 76 | 1.762 818 | 82.019 11 |

20 | 33 | 1.3 | 1.5 | −0.28 | 118 | 50 | 1.709 789 | 83.692 89 | 1.787 212 | 82.959 86 |

21 | 35 | 0.9 | 4.5 | 2.72 | 113 | 50 | 1.754 020 | 81.025 56 | 1.871 894 | 83.023 52 |

22 | 35 | 1.0 | −1.5 | 3.72 | 118 | 55 | 1.696 652 | 82.713 15 | 1.747 687 | 82.212 35 |

23 | 35 | 1.1 | 0 | −0.28 | 123 | 60 | 1.707 872 | 79.185 77 | 1.805 092 | 83.913 13 |

24 | 35 | 1.2 | 1.5 | 0.72 | 103 | 65 | 1.734 484 | 80.007 38 | 1.851 991 | 79.621 78 |

25 | 35 | 1.3 | 3 | 1.72 | 108 | 45 | 1.746 645 | 82.465 14 | 1.808 241 | 80.599 8 |

Responses | Influence level | Factors | |||||
---|---|---|---|---|---|---|---|

z | t/(mm) | β /(˚) | γ /(˚) | α 1 /(˚) | α 2 /(˚) | ||

Stall torque ratio T r 0 | K 1 | 1.746 891 | 1.739 190 | 1.701 710 | 1.732 189 | 1.761 371 | 1.718 081 |

K 2 | 1.744 801 | 1.734 095 | 1.718 251 | 1.715 639 | 1.740 823 | 1.731 195 | |

K 3 | 1.740 304 | 1.738 950 | 1.735 022 | 1.748 072 | 1.725 581 | 1.738 535 | |

K 4 | 1.724 271 | 1.731 284 | 1.755 255 | 1.740 356 | 1.730 632 | 1.748 955 | |

K 5 | 1.727 934 | 1.740 681 | 1.773 964 | 1.747 946 | 1.725 794 | 1.747 435 | |

Peak efficiency η ∗ /% | k 1 | 80.441 37 | 80.630 80 | 80.640 86 | 81.103 89 | 79.984 37 | 81.764 68 |

k 2 | 80.769 81 | 80.878 10 | 80.724 54 | 81.156 38 | 80.860 97 | 80.872 82 | |

k 3 | 80.769 52 | 80.962 30 | 80.939 77 | 80.902 81 | 81.555 23 | 81.191 59 | |

k 4 | 81.640 36 | 80.890 70 | 81.219 17 | 80.411 66 | 82.497 94 | 80.214 14 | |

k 5 | 81.079 40 | 81.338 56 | 81.176 11 | 81.125 71 | 79.801 94 | 80.657 22 |

Responses | Influence level | Factors | |||||
---|---|---|---|---|---|---|---|

z | t/(mm) | β /(˚) | γ /(˚) | α 1 /(˚) | α 2 /(˚) | ||

Stall torque ratio T r 0 | K 1 | 1.857 101 | 1.841 040 | 1.781 985 | 1.828 676 | 1.858 806 | 1.808 043 |

K 2 | 1.836 760 | 1.829 005 | 1.810 566 | 1.820 702 | 1.831 359 | 1.828 002 | |

K 3 | 1.840 298 | 1.830 833 | 1.837 948 | 1.847 260 | 1.823 719 | 1.836 269 | |

K 4 | 1.815 253 | 1.839 517 | 1.844 994 | 1.837 644 | 1.817 728 | 1.849 648 | |

K 5 | 1.816 981 | 1.825 997 | 1.890 900 | 1.832 111 | 1.834 780 | 1.844 429 | |

Peak efficiency η ∗ /% | k 1 | 80.306 82 | 81.369 37 | 80.937 85 | 81.443 85 | 79.507 89 | 81.816 86 |

k 2 | 81.217 95 | 81.582 06 | 81.519 28 | 81.541 31 | 80.093 17 | 81.947 10 | |

k 3 | 81.974 02 | 81.488 46 | 81.931 84 | 81.720 97 | 81.819 77 | 81.890 25 | |

k 4 | 81.506 03 | 81.686 79 | 81.623 48 | 81.623 48 | 82.257 71 | 81.265 95 | |

k 5 | 81.874 12 | 80.752 27 | 81.268 61 | 80.549 32 | 83.200 41 | 79.958 78 |

One other important aspect of the DOE method is the ability to calculate the percentage contribution of a geometric parameter to a specified performance characteristic. The percentage contributions are calculated from an analysis of variance that effectively measures how far the performance characteristic values for a specific geometry variable vary from the mean. The amount that the high and low parameter levels vary from the mean provides a measure of that parameter’s influence on a particular performance characteristic. This is converted into a percentage value to provide a measure of the contribution relative to the other parameters. Again percentage contributions have been calculated for each area factor. A summary of the percentage contributions for stall torque ratio T r 0 and peak efficiency η ∗ at area factors 17% and 20% is shown in

The percentage contributions provide a great deal of information regarding the importance of the various geometric factors of impeller. As shown in

The DOE array provides useful information in the form of the percentage contributions, but not provide information on interaction effects between the geometric parameters. Later, a CCD technique will be used to gauge the interactive effect among these three dominant factors.

The CCD consists of a two-level full or fractional factorial design (corner points), an additional design (star points) and at least one point at the center of the design space (center points) (^{m})^{1}^{/4}, where m is the number of factors [

According to CCD, 15 torque converter cases should be modeled and their properties are presented in

In CCD, a polynomial model with quadratic order is applied to responses (stall torque ratio and peak efficiency). P-value Probs are estimated to be 0.0024 and less than 0.0001 while R-squared values are 0.8002 and 0.9909 for T r 0 and η ∗ , respectively, for 20% area factor. For 17% area factor, P-values are also estimated to be less than 0.0001 while R-squared values are 0.9596 and 0.9957 for T r 0 and η ∗ , respectively, which are meaningful. By using the quadratic equation for the surfaces, the stall torque ratio and peak efficiency can be estimated by the following equations for different area factors.

Factors | Levels | ||||
---|---|---|---|---|---|

−1.682 | −1 | 0 | 1 | 1.682 | |

α 1 /(˚) | 96.182 | 103 | 113 | 123 | 129.818 |

α 2 /(˚) | 33.182 | 40 | 50 | 60 | 66.818 |

β /(˚) | −3.363 6 | −2 | 0 | 2 | 3.363 6 |

Case number | Factors | ||
---|---|---|---|

α 1 /(˚) | α 2 /(˚) | β /(˚) | |

1 | 103 | 40 | −2 |

2 | 103 | 40 | 2 |

3 | 103 | 60 | −2 |

4 | 103 | 60 | 2 |

5 | 123 | 40 | −2 |

6 | 123 | 40 | 2 |

7 | 123 | 60 | −2 |

8 | 123 | 60 | 2 |

9 | 96.18 | 50 | 0 |

10 | 129.82 | 50 | 0 |

11 | 113 | 33.18 | 0 |

12 | 113 | 66.82 | 0 |

13 | 113 | 50 | −3.36 |

14 | 113 | 50 | 3.36 |

15 | 113 | 50 | 0 |

1) 17% area factor:

T r 0 = 3.142854 − 0.020606 × α 1 − 0.008621 × α 2 + 0.069109 × β + 0.000033 × α 1 × α 2 − 0.000614 × α 1 × β + 0.000228 × α 2 × β + 0.000077634 × α 1 2 + 0.000057431 × α 2 2 + 0.000088982 × β 2 (10)

η ∗ = 70.576115 + 0.063839 × α 1 − 0.052123 × α 2 + 0.459299 × β + 0.001304 × α 1 × α 2 − 0.003970 × α 1 × β + 0.000718 × α 2 × β + 0.000224 × α 1 2 − 0.001356 × α 2 2 − 0.053936 × β 2 (11)

2) 20% area factor:

T r 0 = 2.133640 − 0.003426 × α 1 − 0.001724 × α 2 + 0.003084 × β + 0.000040423 × α 1 × α 2 − 0.000061050 × α 1 × β + 0.000203 × α 2 × β − 0.000002332 × α 1 2 − 0.000018698 × α 2 2 + 0.001840 × β 2 (12)

η ∗ = 6.418795 + 0.844738 × α 1 + 0.751594 × α 2 + 0.373003 × β − 0.002840 × α 1 × α 2 − 0.004281 × α 1 × β + 0.001974 × α 2 × β − 0.002217 × α 1 2 − 0.004945 × α 2 2 − 0.017527 × β 2 (13)

By applying the above formula one can approximate the stall torque ratio and peak efficiency for all non-simulated cases.

As mentioned above, the interaction of each parameter with other parameters should be considered at the same time on stall torque ratio and peak efficiency. Therefore, the response surfaces are presented in

With CCD, optimization is based on a parameter called “desirability”. Desirability is an objective function ranging from 0.0 outside of the limits to 1.0 at the goal. The numerical optimization finds a point that maximizes the desirability function. The characteristics of the goal may be altered by adjusting the weight or importance. For several responses and factors, all goals get combined into one desirability function [

D ′ = ( d ′ 1 × d ′ 2 × ⋯ × d ′ n ) 1 n = ( ∏ i = 1 n d ′ i ) 1 n (14)

where n is the number of responses in the measure (in this case, n = 2). If any of the responses or factors fall outside their desirability range, the overall function becomes zero. For simultaneous optimization, each response must have a low and high value assigned to each goal. In this study, the goal parameter used is “maximum” (for both stall torque ratio and peak efficiency) as follows:

d ′ i = 0 Y i ≤ L o w i d ′ i = [ ( Y i − L o w i ) / ( H i g h i − L o w i ) ] w t i , L o w i < Y i < H i g h i d ′ i = 1 Y i ≥ H i g h i (15)

where Y_{i} is the ith response value and wt is the weight of that response. Weight adds emphasis to the goal. A weight greater than 1 (maximum weight is 10), emphasizes the goal and less than 1 (minimum weight is 0.1), deemphasizes the goal. In this paper, two responses are defined, the weights will be determined according to the designer’s performance demand.

After optimization analysis by above surface formula, CCD proposes three optimized cases for each area factor (

In the present study, optimized case 6 is selected as the final optimized case which is designed and calculated.

A new parametric design method of the impeller for an automotive torque converter has been used to conduct a parametric study covering 11 geometric parameters. The 11 geometric parameters can be represented by 6 design parameters including the impeller blade number z, blade thickness t, bias angle β , scroll angle γ , exit angle α 1 and inlet angle α 2 .

No. | f a | α 1 /(˚) | α 2 /(˚) | β /(˚) | Predicted T r 0 | Predicted η ∗ /% |
---|---|---|---|---|---|---|

1 | 17% | 96.180 | 33.180 | 3.36 | 1.823 2 | 79.458 8 |

2 | 17% | 129.82 | 43.142 | −0.233 | 1.698 5 | 85.172 5 |

3 | 17% | 97.771 | 33.180 | 3.36 | 1.812 9 | 79.677 1 |

4 | 20% | 96.180 | 66.820 | 3.36 | 1.900 6 | 77.162 8 |

5 | 20% | 129.82 | 38.104 | −3.069 | 1.765 1 | 86.294 5 |

6 | 20% | 122.57 | 54.098 | 3.36 | 1.841 6 | 83.659 3 |

Responses | Calculation data | Predicted by CCD |
---|---|---|

T r 0 | 1.841 955 | 1.841 60 |

η ∗ /% | 83.768 95 | 83.659 3 |

An L_{25}[5^{6}] DOE array has been successfully constructed for two responses including stall torque ratio and peak efficiency. The analysis of the array identified the dominant geometrical influences on the performance of the torque converter. In general, the impeller exit angle α 1 , inlet angle α 2 and bias angle β are the three strongest influences on the overall performance, with two area factors (17% and 20%) considered.

CCD method has been employed to investigate the interactive effect among the impeller inlet angle, exit angle and bias angle. Predictive equations are presented that can identify expected performance at specific arrangements different from those analyzed. Results show that larger impeller bias angle and inlet angle together with smaller exit angle bring about an increase of the stall torque ratio while a larger exit angle is favorable to increase the peak efficiency. A final optimized design of impeller is obtained based on desirability function approach together with the predictive equations. Compared with the original model, the stall torque ratio and peak efficiency of the optimized model are increased by 1.62% and 1.1%, respectively.

Performance prediction models with more prediction accuracy for a torque converter can be used as a further research direction.

The authors declare no conflicts of interest regarding the publication of this paper.

Chen, X. and Chen, J. (2020) Optimization of the Impeller Geometry for an Automotive Torque Converter Using Response Surface Methodology and Desirability Function. Open Journal of Applied Sciences, 10, 455-475. https://doi.org/10.4236/ojapps.2020.107032