:4 0 for M L 1 ð15Þ
:
for 1\Miþ12;j 0 ð18Þ
When the advection Mach number Miþ12;j tends to zero, the dissipation term in Eq. (8) will approach zero, too. Thus, there will be some disturbances which cannot be damped by the scheme. In order to solve this problem, it is proposed that the scaling of the dissipation term of the AUSM method be modified as follows 8 > 1 1 [d M for M > iþ2;j < iþ2;j 2 ; ð19Þ /AUSM 2 iþ12;j ¼ > M 1 iþ2;j þd > : for Miþ12;j d 2d where d is a small value ð0 d\5Þ. Hence, there will always be a sufficient amount of numerical dissipation. The weighting factor in Eq. (17) is defined as
i1;j
The pressure at the east face of the control volume is obtained from the splitting: þ
Eq. (18), 8 > 1 M > iþ2;j > > 1 < 2 R 1 /VL iþ12;j ¼ > Miþ2;j þ 2 ðM 1Þ > > > : M 1 þ 1 ðM L þ 1Þ2 iþ2;j 2
w ¼ minðmi;j ; miþ1;j Þ; Pi1;j 2Pi;j þ Piþ1;j ; 0 ; mi;j ¼ max 1 a P þ 2P þ P
b2 ¼ 3:0304 105
and
P ¼
for M R 1 R
is defined by where w is the weighting factor and uVL iþ1;j
and
8 > < 01
i;j
a ¼ 5:
iþ1;j
ð20Þ Thus according to Eqs. (17) and (20), it is observed that the hybrid method switches to the Van Leer scheme at high gradient regions.
4 Group method of data handling neural networks By means of group method of data handling (GMDH) type of ANNs a model can be represented as set of neurons in which different pairs of them in each layer are connected through a quadratic polynomial and thus produce new
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Neural Comput & Applic
neurons in the next layer. Such representation can be used in modeling to map inputs to outputs. The formal definition of the identification problem is to find a function f^ so that can be approximately used instead of actual one, f in order to predict output y^ for a given input vector X ¼ ðx1 ; x2 ; x3 ; . . .; xn Þ as close as possible to its actual output y. Therefore, given M observation of multi-input–single-output data pairs so that yi ¼ f ðxi1 ; xi2 ; xi3 ; . . .; xin Þ ði ¼ 1; 2. . .M Þ
ð21Þ
It is now possible to train a GMDH-type neural network to predict the output values y^i for any given input vector X ¼ ðxi1 ; xi2 ; xi3 ; . . .; xin Þ, that is y^i ¼ f^ðxi1 ; xi2 ; xi3 ; . . .; xin Þ ði ¼ 1; 2. . .M Þ
ð22Þ
The problem is now to determine a GMDH-type neural network so that the square of difference between the actual output and the predicted one is minimized, that is M
X 2 f^ðxi1 ; xi2 ; xi3 ; . . .; xin Þ yi ! min
ð23Þ
i¼1
General connection between inputs and output variables can be expressed by a complicated discrete form of the Volterra functional series in the form of y ¼ a0 þ þ
n X
ai x i þ
i¼1 n X n X n X
n X n X
aij xi xj
i¼1 j¼1
ð24Þ
aijk xi xj xk þ
i¼1 j¼1 k¼1
where is known as Kolmogorov–Gabor polynomial [27]. This full form of mathematical description can be represented by a system of partial quadratic polynomials consisting of only two variables (neurons) in the form of y^ ¼ Gðxi ;xj Þ ¼ a0 þ a1 xi þ a2 xj þ a3 xi xj þ a4 x2i þ a5 x2j
Fig. 1 Geometry of the rotor tip section and the computational domain [51]
has a very large stagger ðCÞ of 63.27. Chord length of the blade (C) and pitch length of the cascade (p) are equal to 43 mm and 37 mm, respectively. Also, the inlet flow angle (b) is -38. Moreover, inflow stagnation pressure and temperature are set to 99 kPa and 382 K, respectively. As shown in Fig. 2, size of the grid is specified as 498 9 65
ð25Þ
There are two main concepts involved within GMDH-type neural networks, namely, the parametric and the structural identification problems. In this way, some authors presented a hybrid GA and singular value decomposition (SVD) method to optimally design such polynomial neural networks [28, 29]. The methodology in these references has been successfully used in this paper to obtain the polynomial models of deviation angle and performance losses in wet steam turbines.
5 Results and discussion The geometry under study is a rotor tip section of a steam turbine, taken from [48]. Figure 1 shows this geometry and computational domain between blades. The blade section
123
Fig. 2 Grid independency test; distribution of pressure ratio (P/P0i) on the blade surface
Neural Comput & Applic
after grid-independency test. This grid is used for numerical simulation and illustrated in Fig. 3 with the close-up near the leading and trailing edges. In this paper GMDH-type neural networks are used to model deviation angle (h), total pressure loss coefficient (x), and performance loss coefficient (n) in wet steam turbines with respect to four input variables, i.e., stagnation pressure (Pz), stagnation temperature (Tz), back pressure (Pb), and inflow angle (b). These variables are shown in Fig. 1. Deviation angle (h) shows the difference between flow angle and angle of camber line in the trailing edge, as depicted in Fig. 1. It is caused by the difference between the stagnation pressures over the suction and pressure surfaces around the trailing edge. The flow over the suction side experiences further losses with respect to that of the pressure side, as it passes a sequence of condensation- and aerodynamic-shocks over the suction side. In turbomachinery science, deviation angle shows deviation of flow direction in the trailing edge of blade with respect to on-design case. This parameter is increased due to viscous effects, condensation phenomenon, and formation of shock waves in the steam turbine blades.
Also x and n are computed using the following equations: x¼
DP0 P0i P0e ¼ ; P0i Pe P0i Pe
ð26Þ
he he;is V2 ¼ 1 2e ; h0i he;is Ve;is
ð27Þ
and n¼
where subscripts ‘‘i,’’ ‘‘e,’’ and ‘‘is’’ denote to inlet section, exit section, and isentropic state, respectively. As shown in Fig. 2, size of the grid is specified as 498 9 65 after grid-independency test. This grid is used for numerical simulation and illustrated in Fig. 3 with the close-up near the leading and trailing edges. 5.1 Validation To validate the present in-house CFD code, the experimental data reported by Bakhtar and Mahpeykar [49] have been used. The boundary conditions for the current numerical simulation have been chosen identical to those in the experimental test such as, the inflow direction (b) is
Fig. 3 Grid used for numerical simulation (498 9 65) with the close-up near the leading and trailing edges
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Neural Comput & Applic
Fig. 4 Distribution of pressure ratio on the pressure and suction surfaces using numerical and experimental results [48]
-38 and the inflow stagnation pressure and temperature are set to 99.9 kPa and 360.8 K, respectively. At the inlet section, the wetness fraction v = 0 and at the outlet section, the back pressure Pb = 42.7 kPa. Comparison of the pressure ratio on the suction and pressure surfaces between numerical and experimental results is shown in Fig. 4. As observed in this figure, a good agreement exists between the results and position of the condensation shock is captured well. 5.2 Effects of outlet pressure variations on the flow field Figure 5 presents Mach number and wetness fraction contours in the passage for different outlet pressures. The outflow regime is supersonic for Pb = 30, 45 kPa and subsonic for Pb = 60 kPa. As shown in Fig. 5a, an oblique shock is formed at the trailing edge of the blade to match the outlet pressure in supersonic outflow cases. It can be seen in Fig. 5b that the wetness fraction increases with the outlet pressure reduction. For example, the outlet wetness fraction increases by 223 % when the outlet pressure decreases from 60 to 30 kPa. Also the wetness fraction decreases across the oblique shock due to increase in vapor temperature. Sequences of condensation and evaporation phenomena over the suction side of the blade in wet cases are shown in Fig. 5b. 5.3 Modeling of deviation angle and performance losses using GMDH-type neural networks In this paper the GMDH-type neural networks are used for modeling of h, x and n with respect to four input variables:
123
Fig. 5 Comparison of contours in the passage for different outlet pressures, a Mach number contours, b wetness fraction contours; (P0)i = 99 kPa, (T0)i = 382 K
Pz, Tz, Pb and b. Variation range of input variables are given in Table 1. These data are used for CFD simulations and GMDH-type neural networks modeling. Response surface methodology (RSM) which is a submethod of design of experiments (DOE) is used to design number of input–output data [50, 51]. There has been a total number of 31 input–output CFD data considering four input variables and one objective function. Samples of numerical results that are used to learn the neural networks are shown in Table 2. The GMDH-type neural network is
Neural Comput & Applic Table 1 Variation range of input variables to learn the neural networks
Input variables
From
To
Selected values for GMDH-type neural network modeling
Pz (kPa)
90
110
90, 95, 100, 105, 110
350
400
350, 362.5, 375, 387.5, 400
Pb (kPa)
30
60
30, 37.5, 45, 52.5, 60
b ()
30
45
30, 33.75, 37.5, 41.25, 45
Tz (K)
Table 2 Samples of numerical results used to learn and test the neural networks
(Pz) 1 (Tz) 2 (Pb) 3 (β) 4
23
Number
Pz (kPa)
Output data Tz (K)
Pb (kPa)
b ()
h ()
x
n
1
95
362.5
37.5
33.75
4.42
0.218
0.0718
2
105
362.5
37.5
33.75
5.31
0.244
0.0732
3
95
387.5
37.5
33.75
3.87
0.171
0.0548
4
105
387.5
37.5
33.75
5.1
0.197
0.0623
5
95
362.5
52.5
33.75
1.33
0.151
0.0699
6
105
362.5
52.5
33.75
1.7
0.169
0.073
7
95
387.5
52.5
33.75
0.87
0.094
0.048
8
105
387.5
52.5
33.75
1.54
0.125
0.062
9
95
362.5
37.5
41.25
4.49
0.217
0.0717
10
105
362.5
37.5
41.25
5.39
0.243
0.0733
… 31
100
375
45
37.5
3.03
0.174
0.0762
2314
14 34
Input data
23143412 (θ) 3412
12
Fig. 6 Evolved structure of generalized GMDH-type neural network for deviation angle (h)
now used to find polynomial models for h, x and n with respect to input parameters. The structure of the evolved two hidden layer GMDHtype neural network is shown in Fig. 6 corresponding to the genome representation of h (23143412) in which 1, 2, 3, 4 stand for Pz, Tz, Pb and b, respectively. Different new parameters are made at inner layers using two parameters from the previous layer. Finally the output parameter is obtained from the inner variables. In order to demonstrate the prediction ability of evolved GMDH-type neural networks, CFD data have been divided into two different sets, namely, training and testing sets. The training set which consists of 25 out of 31 inputs– output data pairs is used for training the GMDH models. The testing set which consists of 6 unforeseen input–output
data samples during the training process is merely used for testing to show the prediction ability of such evolved GMDH-type neural network models. The corresponding polynomial representation of GMDH model for deviation angle h is as follows: y23 ¼ 0:001 þ 0:207Tz 1:285Pb 0:0003Tz2 þ 0:007P2b þ 0:001Tz Pb
ð28Þ
y14 ¼ 22:174 þ 0:345Pz þ 0:184b 0:001P2z 0:002b2 þ 0:0001Pz b
ð29Þ
y34 ¼ 27:582 0:883Pb 0:003b þ 0:007P2b þ 0:0003b2 0:0003Pb b
ð30Þ
y12 ¼ 0:008 1:530Pz þ 0:412Tz þ 0:0006P2z 0:001Tz2 þ 0:004Pz Tz
ð31Þ
y2314 ¼ 0:444 þ 0:498y23 0:227y14 0:001y223 þ 0:117y214 þ 0:165y23 y14
ð32Þ
y3412 ¼ 1:257 þ 0:621y34 þ 0:188y12 þ 0:0003y234 þ 0:071y212 þ 0:121y34 y12
ð33Þ
h ¼ 0:011 þ 0:047y2314 þ 1:013y3412 2:478y22314 2:636y23412 þ 5:111y2314 y3412
ð34Þ
The good behavior of such GMDH-type neural network model is depicted in Fig. 7, both for the training and testing
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Neural Comput & Applic
y1423 ¼ 0:231 þ 1:137y14 þ 1:565y23 þ 0:327y214 0:899y223 1:453y14 y23
ð40Þ
x ¼ 0:005 1:352y2413 þ 2:311y1423 83:109y22413 93:453y21423 þ 176:669y2413 y1423
ð41Þ
Figure 9 presents comparison of total pressure loss coefficient x predicted by GMDH model and simulated by CFD. As can be seen in this figure, the presented GMDHtype neural network model has enough accuracy to predict total pressure loss coefficient x, because GMDH model and CFD results coincide with each other in the test process. Figure 10 shows evolved structure of generalized GMDH-type neural network for performance loss coefficient n. The corresponding polynomial representation of n is as follows: Fig. 7 Comparison of deviation angle (h) modeled by GMDH and simulated by CFD
(Pz) 1
24 2413 13
(Tz) 2
14
(Pb) 3
24131423 (ω) 1423
23
(β) 4
Fig. 8 Evolved structure of generalized GMDH-type neural network for total pressure loss coefficient (x)
y13 ¼ 1:104 þ 0:025Pz 0:003Pb 0:0001P2z 0:000004P2b þ 0:00003Pz Pb
ð42Þ
y12 ¼ 0:00004 0:009Pz þ 0:003Tz 0:0001P2z 0:00002Tz2 þ 0:00008Pz Tz
ð43Þ
y23 ¼ 0:000003 þ 0:001Tz 0:002Pb 0:000003Tz2 þ 0:000003P2b þ 0:000005Tz Pb ð44Þ y14 ¼ 1:267 þ 0:026Pz þ 0:001b 0:0001P2z 0:00003b2 0:000001Pz b
ð45Þ
y1312 ¼ 0:616 21:010y13 þ 2:521y12 þ 166:378y213 11:016y212 2:409y13 y12
ð46Þ
data. It is evident from Fig. 7 that the evolved GMDH neural network in terms of simple polynomial equations successfully model and predict outputs of the testing data. Figure 8 shows evolved structure of generalized GMDH-type neural network for total pressure loss coefficient x. The corresponding polynomial representation of x is as follows: y24 ¼ 0:0001 þ 0:005Tz 0:023b 0:00001Tz2 þ 0:00004b2 þ 0:00005Tz b
ð35Þ
y13 ¼ 0:064 þ 0:009Pz 0:010Pb 0:00003P2z þ 0:00006P2b 0:000003Pz Pb
ð36Þ
y14 ¼ 0:429 þ 0:011Pz 0:003b 0:00004P2z þ 0:00002b2 þ 0:00001Pz b
ð37Þ
y23 ¼ 0:000003 þ 0:005Tz 0:013Pb 0:000009Tz2 þ 0:00007P2b þ 0:000005Tz Pb y2413 ¼ 0:377 þ 3:157y24 þ 1:322y13 5:668y224 0:190y213 1:493y24 y13
123
ð38Þ ð39Þ
Fig. 9 Comparison of total pressure loss coefficient (x) modeled by GMDH and simulated by CFD
Neural Comput & Applic (Pz) 1 (Tz) 2 (Pb) 3 (β) 4
13 1312 12 23
13122314 (ξ) 2314
14
Fig. 10 Evolved structure of generalized GMDH-type neural network for performance loss coefficient (n)
y2314 ¼ 0:177 þ 13:545y23 9:030y14 53:105y223 þ 117:780y214 77:939y23 y14 n ¼ 0:026 þ 0:055y1312 0:018y2314 þ 8:533y21312 þ 11:139y22314 11:444y1312 y2314
ð49Þ ð48Þ
Figure 11 compares values of performance loss coefficient n modeled by GMDH-type neural network and simulated by CFD. As observed in this figure, the presented GMDH model has suitable accuracy to predict performance loss coefficient n in wet steam turbines. Figure 12 shows variations of different output variables (h, x, n) with back pressure using GMDH-type models for the conditions P0i ¼ 99 kPa; T0i ¼ 382 K; b ¼ 38 . As it is clear in this figure, deviation angle, pressure loss coefficient and performance loss coefficient decrease with back pressure. To assess the accuracy of GMDH-type neural network models, some statistical parameters are given in Table 3. These statistical parameters are R2 as absolute fraction of variance, RMSE as root-mean-squared error and MAPE as
Fig. 11 Comparison of performance loss coefficient (n) modeled by GMDH and simulated by CFD
Fig. 12 Variations of different output variables with back pressure using GMDH-type models for the conditions P0i ¼ 99 kPa; T0i ¼ 382 K; b ¼ 38 , a deviation angle h, b pressure loss coefficient x, c performance loss coefficient n
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Neural Comput & Applic Table 3 Amount of absolute fraction of variance (R2), root-meansquared error (RMSE) and mean absolute percentage of error (MAPE) for total pressure loss coefficient (x), performance loss coefficient (n), and deviation angle (h) Parameter
x
n
h
R2
0.992
0.946
0.816
RMSE
0.002
0.002
0.142
MAPE
0.037
0.170
0.020
mean absolute percentage of error which are defined as follows: R2 ¼ 1
n X ðYi ANN Yi CFD Þ2
Yi2CFD sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 ðYi ANN Yi CFD Þ RMSE ¼ n ! n 1X ðYi ANN Yi CFD Þ MAPE ¼ 100 n i¼1 Yi CFD
ð49Þ
i¼1
ð50Þ ð51Þ
where Y and n represent each of the outputs and number of input–output data, respectively. Also the subscripts ANN and CFD denote to artificial neural networks and computational fluid dynamics, respectively. It is evident from Table 3 that the parameters of RMSE and MAPE are close to zero and R2 is close to one, thus GMDH-type neural network models coincide with the CFD results.
6 Conclusions In this paper quadratic models were presented based upon numerical results, using evolved GMDH-type neural networks for modelling of deviation angle, total pressure loss coefficient and performance loss coefficient in wet steam turbines. The GMDH-type predicted values were compared with CFD results and excellent agreement was observed. It is clearly evident that the evolved GMDH-type neural networks in terms of simple polynomial equations can successfully model and predict deviation angle and performance losses in wet steam turbines. To assess the accuracy of neural network models, some statistical parameters were calculated. Absolute fraction of variance (R2) and root-meansquared error (RMSE) related to total pressure loss coefficient (x) are equal to 0.992 and 0.002, respectively. Thus the presented GMDH models have enough accuracy.
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