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 Figure 1.

Each impeller blade profile (Figure 3(c)) is formed by a three dimensional (3D) curve of the shell and a 3D curve of the core. The 3D curve can be calculated from the torus and a two dimensional (2D) design curve using conformal transformation principle. Figure 2 shows the schematic representation of a 2D design curve. With the origin at the starting point of the curve, the coordinate of P 0 is (0, 0). The curve expression can be written with the following equation:

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 (Figure 2). According to the definition of 2D design curve, x 2 value and y 2 value equal to the length of torus l and offset size d, 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 Figure 3.

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 Figure 4. The design parameters considered for the parametric study are shown in Table 1, provided that the torus of impeller is determined.

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 Figure 5 where only one blade passage is modeled for each element to illustrate the mesh distribution in the computational field when approximately 111,000 grid cells in total are used. The polyhedral cells with five prism layers are used on the model. The leakage between the elements and also between an element and the core flow are disregarded. A cyclic boundary condition is imposed on both peripheral boundaries outside a blade passage. A no-slip wall boundary condition is also imposed on all the walls bounding the domain, with a spin applied as necessary. The interfaces between elements have been handled by using the mixing plane method. A second-order upwind differencing scheme is utilized and the SST k-w model is also used for the turbulence. Steady state simulations are performed for a range of speed ratios from 0.0 to 0.9 while maintaining an impeller speed of 2000 rpm.

Figure 6 compares the measured and calculated overall performance including efficiency η , torque ratio T r , and impeller torque factor λ   I for the parametric model of the torque converter. As indicated here, the tendencies of the experimental data correlated relatively well with the calculated results, confirming that the parametric model and computational method are valid in general. It is of note that the CFD underestimates the impeller torque factor λ   I at low speed ratios. The discrepancy between measurements and calculations are reasonable because of the overestimated incidence losses at low speed ratios.

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 Figure 7.

In Figure 8, the performance analysis shows the significant increases of impeller torque factor with the flow area increases, whereas efficiency calculation does not show significant increase as shown in Figure 9. It can be concluded that the impeller torque factor is more sensitive than efficiency to the torque converter’s flow area.

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 [27] [28] [29] . A DOE method sets out configurations (or arrangements) to be conducted using an appropriate orthogonal array; the terminology used in these arrays includes “factors”—an item that is to be varied during the simulations, “level”—the number of times a factor is to be varied during the simulations and “configuration number”—the number of simulations that are required to be run to complete the analysis [30] . The selection of an appropriate DOE array is dependent on the number of factors and the levels of the factors to be analyzed. As mentioned above, 11 design parameters are used to obtain an impeller parametric model. Blade scroll angle γ is introduced as another design parameter to investigate the effect of the impeller geometry on the performance of a torque converter. The scroll angle is defined as the angle between the two planes containing the intersection of the design path and the entering and leaving edges of the blade when that blade does not lie in one axial plane. Impeller blades with varying scroll angles are shown in Figure 10.

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 Figure 11.

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 Table 2. The underline sign in the table indicates the base value of impeller design parameters.

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 (L₂₅[5⁶]) 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 Table 3 along with the responses predicted by CFD.

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 Table 4, Table 5.

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 Figure 12.

The percentage contributions provide a great deal of information regarding the importance of the various geometric factors of impeller. As shown in Figure 12, exit angle α 1 has the strongest effect on peak efficiency at both 17% and 20% area factors although this effect reduces for the stall torque ratio. It should be noted that while the peak efficiency contributions for the bias angle β are relatively low, the contributions to the stall torque ratio at both area factors are surprisingly large. The inlet angle α 2 and blade number z show a lesser contribution while the blade thickness t generally has a significantly lesser effect. It can be concluded that the inlet angle α 2 , bias angle β , and exit angle α 1 exert the greatest influence on the overall performance of a torque converter.

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) (Figure 13). If the distance from the center of the design space to a factorial point is ±1 unit for each factor, the distance from the center of the design space to a star point is ± α with | α | > 1 . The α value equals to (2^m)^1/4, where m is the number of factors [31] [32] . In the present study, m equals to 3 so that the α value is 1.682. Therefore, each factor is studied in five different levels (−1.682, −1, 0, +1, +1.682), which is listed in Table 6.

According to CCD, 15 torque converter cases should be modeled and their properties are presented in Table 7. It should be noted that nine points are considered for central points and two responses are considered for the optimization, stall torque ratio and peak efficiency. The corresponding calculation results for two area factors are presented in Figure 14 and Figure 15. Figure 14 reveals that case 4 has the maximum stall torque ratio, but it has a low peak efficiency and case 10 has the maximum peak efficiency. As seen in Figure 15, case 10 also has the best peak efficiency, but it has the minimum stall torque ratio and case 9 has the maximum stall torque ratio. So, an optimization study is needed which is performed using CCD technique in the present study.

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.

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 Figure 16 and Figure 17 for each area factor, respectively, to show this interaction on T r 0 and η ∗ . These figures show the interaction between the three parameters for the optimized design. An increase in bias angle β and inlet angle α 2 together with an decrease in exit angle α 1 yield higher stall torque ratio amounts although the effect of bias angle is more sensible than inlet angle and exit angle due to its high slope in the surface in Figure 16(a), Figure 16(b), Figure 17(a) and Figure 17(b). As an important outcome, exit angle has the most significant effect on peak efficiency, while bias angle and inlet angle may have optimum values that maximize the peak efficiency (Figure 16(c), Figure 16(d), Figure 17(c), and Figure 17(d)). It is of note that the minimum value of peak efficiency at 20% area factor is lower than that at 17% area factor as shown in Figure 16(c) and Figure 17(c). This phenomenon can be explained by the fact that the curvature of the core increases with the increase of flow area of a torque converter, which may result in larger flow losses.

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 [23] . In this paper, two responses are defined as stall torque ratio and peak efficiency. The goal of optimization is to find a set of conditions that meet all the goals, not to get a desirability value of 1.0. Desirability reflects the preferred ranges for each response ( d ′ i ). The simultaneous objective function is a geometric mean of all transformed responses:

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 (Table 8). Optimized cases 1 and 4 have the maximum stall torque ratio while optimized cases 2 and 5 have the best peak efficiency. Optimized cases 3 and 6 have the largest desirability when the weights of responses are considered equal. It is clear that torque converters with 20% area factor have better performance characteristics than torque converters with 17% area factor.

In the present study, optimized case 6 is selected as the final optimized case which is designed and calculated. Table 9 compares the theoretical and calculation results. The CFD calculation results at the optimum parameter combination confirm the effectiveness of the response surface model for optimum design parameters. Compared to the original model, the stall torque ratio and peak efficiency of the optimized one have increased by 1.62% and 1.1%, respectively. Figure 18 shows the performance comparison of the optimized design with the original one which illustrates the improvement of performance.