Predicting discharge capacity of triangular labyrinth side weir located on a straight channel by using an adaptive neuro-fuzzy technique

Advances in Engineering Software 41 (2010) 154–160

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Predicting discharge capacity of triangular labyrinth side weir located on a straight channel by using an adaptive neuro-fuzzy technique M. Emin Emiroglu a,*, Ozgur Kisi b, Omer Bilhan a a b

Firat University, Civil Engineering Department, 23119 Elazig, Turkey Erciyes University, Civil Engineering Department, 38019 Kayseri, Turkey

a r t i c l e

i n f o

Article history: Received 27 July 2009 Received in revised form 11 September 2009 Accepted 14 September 2009 Available online 12 October 2009 Keywords: Side weir Discharge coefficient Intake Labyrinth weir Neuro-fuzzy

a b s t r a c t Side weirs are widely used for flow diversion in irrigation, land drainage, urban sewage systems and also in intake structures. It is essential to correctly predict the discharge coefficient for hydraulic engineers involved in the technical and economical design of side weirs. In this study, the discharge capacity of triangular labyrinth side weirs is estimated by using adaptive neuro-fuzzy inference system (ANFIS). Two thousand five hundred laboratory test results are used for determining discharge coefficient of triangular labyrinth side weirs. The performance of the ANFIS model is compared with multi nonlinear regression models. Root mean square errors (RMSE), mean absolute errors (MAE) and correlation coefficient (R) statistics are used as comparing criteria for the evaluation of the models’ performances. Based on the comparisons, it was found that the ANFIS technique could be employed successfully in modeling discharge coefficient from the available experimental data. There are good agreements between the measured values and the values obtained using the ANFIS model. It is found that the ANFIS model with RMSE of 0.0699 in validation stage is superior in estimation of discharge coefficient than the multiple nonlinear and linear regression models with RMSE of 0.1019 and 0.1507, respectively. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Flow over weirs a typical case of spatially varied flow. A complete analytical solution of the equations governing the flow over side weirs is complicated. Side weirs are extremely useful in irrigation and drainage systems as a means of diverting excess water into relief channels for flood protection works and as storm overflow from urban sewerage systems [14]. A review of previous studies indicated that rectangular sharp-crested side weirs have been investigated extensively by, for example, Ackers [1], Frazer [13], Collings [7], Subramanya and Awasthy [29], El-Khashab and Smith [11], Uyumaz and Muslu [30], Uyumaz and Smith [31], Helweg [15], Agaccioglu and Yüksel [2], Borghei et al. [5], Durga Rao and Pillai [14]. Kumar and Pathak [24] investigated the discharge coefficient of sharp and broad-crested triangular side weirs. Cosar and Agaccioglu [8] studied discharge coefficient of the triangular side weir on straight and curved channel. Aghayari et al. [3] have studied the spatial variation of discharge coefficient in broad-crested inclined side weirs. There are several types of the side weirs. They are commonly rectangular, triangular, circular, and labyrinth side weirs. A labyrinth weir is defined as a weir crest that is not straight in planform. The increased sill length provided by labyrinth weirs effectively re* Corresponding author. Tel.: +90 237 00 00; fax: +90 241 55 26. E-mail address: [email protected] (M.E. Emiroglu). 0965-9978/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.advengsoft.2009.09.006

duces upstream head to the particular discharge. They can therefore be used to a particular advantage where the width of a channel is restricted and a weir is required to pass a range of discharges with a limited variation in upstream water level. General type of side weirs is rectangular side weir as mentioned above. Emiroglu et al. [12] studied on the discharge coefficient of sharpcrested labyrinth side weirs on a straight channel. Emiroglu et al. [12] made an investigated for the first time on the labyrinth side weirs. Their results about labyrinth side weir discharge coefficient can be summarized the following: 1. Discharge coefficient of the labyrinth side weir is 1.5–4.5 times higher than rectangular side weir. 2. The discharge coefficient Cd increases when L/b ratio increases. Decrease in the labyrinth weir included angle h causes a considerable increase in Cd due to increasing the overflow length. The labyrinth side weir with h = 45° has the greatest Cd values among the weir included angle tested. 3. The following equation for Cd, De Marchi coefficient, for subcritical flow can be used reliable proposed:

"  0:012  0:112  4:024 L L p C d ¼ 18:6  23:535 þ 6:769  0:502 b ‘ h1 1:431 ð1Þ þ0:094  sin h  0:393F 2:155 1

M.E. Emiroglu et al. / Advances in Engineering Software 41 (2010) 154–160

155

Nomenclature b Cd E F1 g h h1 h2 J0 L l n p QL Q1

width of channel side weir discharge coefficient specific energy Froude number at upstream end of side weir acceleration due to gravity main channel depth flow depth at upstream end of side weir at channel center flow depth at downstream end of side weir at channel center slope of main channel length (width) of side weir the weir crest length roughness of the main channel height of weir crest discharge in the main channel at downstream end of side weir total discharge in the main channel at upstream end of side weir

The flow over a side weir falls under the category of spatially varied flow. The concept of constant specific energy [9] is often adopted for studying the flow characteristics of these weirs. Considering the discharge dQ through an elementary strip of length ds along the side weir in rectangular main channel as De Marchi equation, one gets



dQ 2 pffiffiffiffiffiffi ¼ C d 2g ½h  p3=2 ds 3

ð2Þ

where Q is discharge in the main channel, s is the distance from the beginning of the side weir, dQ/ds (or q) is discharge spilling for per unit length of the side weir, g is acceleration due to gravity, p is height of the side weir and h is depth of flow measured from the channel bottom along the channel centerline (Fig. 1). Neuro-fuzzy models have been successfully used in the hydrological sciences during recent years. Nayak et al. [26] evaluated the potential of neuro-fuzzy technique in forecasting river flow time series. Kisi [19] used a neuro-fuzzy model for daily suspended sediment estimation. Nayak et al. [27] used a neuro-fuzzy model for short-term flood forecasting. Kisi [20] investigated the accuracy of neuro-fuzzy computing technique in daily evaporation modeling. Chang and Chang [6] used a neuro-fuzzy approach to construct a water level forecasting system during flood period. Kisi and Ozturk [21] investigated the accuracy of adaptive neuro-fuzzy method for modeling reference evapotranspiration. Bae et al. [4] predicted the monthly dam inflows using a neuro-fuzzy technique. Kisi et al. [22] modeled the daily suspended sediment of rivers in Turkey using a neuro-fuzzy technique. Although the neuro-fuzzy approach applied for different hydrological processes have reported many exciting results, this method is still rarely mentioned in hydraulic research [32,23,28]. In this study, an adaptive neurofuzzy inference system (ANFIS) model was developed to determine discharge coefficient of triangular labyrinth side weirs.

Q Qw q dQ/ds R s z z/h V V1 Vs h

l q r w

discharge in the main channel total flow over side weir discharge per unit length over side weir discharge per unit length of side weir correlation coefficient distance along side weir measured from upstream end of side weir (m) local vertical coordinate dimensionless depth at any point mean velocity in any section of channel mean velocity of flow at upstream end of side weir velocity of flow dQs over the brink labyrinth side weir included angle dynamic viscosity of the fluid mass density of the fluid surface tension deviation angle of flow

the weir crest length (l), crest height (p), roughness of the main channel (n) and labyrinth side weir included angle (h)

C d ¼ f ðb; h1 ; V 1 ; L; g; J o ; w; l; r; q; ‘; p; n; hÞ

ð3Þ

The effect of Jo, n, r ve l on discharge coefficient for elementary flow particle is very small. Therefore they are negligible. Moreover, the water nape deviation or deflection angle w, is defined the deflection of the side weir nape from the water surface toward the weir side and it is given as follows [29]:

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2 V1 sin w ¼ 1  Vs

ð4Þ

In which, Vs is velocity of flow dQs over the brink. According to Eq. (3), w takes different values for each fluid particle and varies with Froude number, which changes along the side weir due to spilling over the side weir. The deviation angle increases towards the weir side when the Froude number in the main channel decreases towards the downstream direction. El-Khashab [11] also mentioned that the dimensionless length of the side weir (L/b) includes the effect of the deviation angle on the discharge coefficient. Therefore, the deviation angle w is not existed in the side weir discharge coefficient equations in the literature. Thus

C d ¼ f ðb; h1 ; V 1 ; L; g; q; ‘; p; hÞ

ð5Þ

Using the Buckingham P theorem, nondimensional equations in functional forms can be obtained as below

  L L C d ¼ f F 1 ; ; ; p=h1 ; h b ‘

ð6Þ

The above functional relationships have been worked out in the present study, in which F1 is Froude number at upstream end of side weir. 3. Experimental set-up

2. Dimensional analysis Referring to Fig. 1 the discharge coefficient (Cd) can be written as a function of width of channel (b), flow depth in the main channel (h1), mean velocity of flow at upstream end of side weir (V1), length of side weir (L), acceleration due to gravity (g), slope of main channel bed (Jo), deviation angle of flow (w), dynamic viscosity of the fluid (l), surface tension (r), mass density of the fluid (q),

The data used in this study were taken from study conducted by Emiroglu et al. [12] on a large model. A schematic representation of the experimental set-up is shown in Fig. 2. Experimental set-up consisted of a main channel and a discharge collection channel. The main channel was 12 m long and the bed rectangular crosssection. The main channel was 0.50 m wide, 0.50 deep and a 0.001 bed slope. The channel consisted of a smooth horizontal

156

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Downstream Section

Upstream Spatial Weir crest Section a Strip b

Downstream Section

Spatial Weir crest a Strip b

Upstream Section

ds Normal depth corresponding h1 to Q1

b

s

h

p

Normal depth E corresponding hL to Q L

Normal depth corresponding h1 to Q1

p

L

L

(a)

(c)

Q1 Flow V1 Main channel

QL VL

QL

Q1 Flow V1 Main channel

b

Qw

VL

L

Qw

L

Normal depth E corresponding to QL

hL

Qw

(d)

(b)

Fig. 1. Definition sketch for flow over side weir in a rectangular channel. (a) Longitudinal cross-section for the rectangular side weir, (b) plan for the rectangular side weir, (c) longitudinal cross-section for the labyrinth side weir and (d) plan for the labyrinth side weir.

0.50

Grid

1.0 m 1.0 m

0.50

Grid

Basin 0.50

o

90 triangular weir

Gate Approaching channel

0.94

Main channel 0.5 m

Basin

Grids

0.9 m 1.0 m

3.5 m

0.80 m

Basin 0.46

Side Weir 0.5 m Grids

0.73 Collection channel

Rectangular weir

12.0 m

1.5 m

0.50

1.0 m

Vane

1.5 m

Water supply Electromagnetic pipe (D=0.254 m) flowmeter 2.0 m

Outlet channel

1.0 m

Fig. 2. Experimental arrangement.

Channel Centerline

Upstream 1 2 3 4 5 C

D

Side Weir E

Flow Downstream

(a) L

p

C

D

E

θ

well-painted steel bed of vertical glass sidewall. A sluice gate was fitted at the end of the main channel to control flow depth. Collection channel was 0.50 m wide and 0.70 m deep, and situated parallel to the main channel. The width of collection channel across the side weir was 1.3 m and constructed as a circular shape to provide free overflow conditions. A rectangular weir was placed at the end of collection channel to measure the discharge of the side weir. An electronic point gauge, Mitutoyo brand name and with ±0.01 mm sensitivity, was fixed further 0.40 m from the weir. Labyrinth side weirs were fabricated from steel plates, which were sharp edged and fully aerated and installed flush with the main channel wall (Fig. 3). Water supplied to the main channel, through a supply pipe, from a sump and the flow was controlled by a gate valve. The discharge is ±0.01 L/s sensitivity measured by means of a Siemens brand name electromagnetic flow-meter installed in the supply line. Additionally, the results are compared by a calibrated 90° Vnotched weir. The overflow rate was measured by calibrated standard rectangular weir, located at the downstream end of the collection channel. Water depth measurements had been conducted by using the point gauge at the side weir region, along the channel centerline and the side weir of the main channel. Water surface measurements had been done by using a special type measurement car which can move in both directions on a rail. Velocities were measured by using Nortek Acoustic Doppler velocity-meter with high sensitivity. Experiments were conducted for subcritical flow, stable flow conditions and, free overflow conditions. The experiments were

(b) Fig. 3. Labyrinth side weir.

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M.E. Emiroglu et al. / Advances in Engineering Software 41 (2010) 154–160 Table 1 Range of variables tested. h (°)

L (cm)

p (cm)

L/l ()

p/h1 ()

Q (L/s)

F1 ()

Number of runs

45

25.0 50.0 75.0

12, 16 and 20 12, 16 and 20 12, 16 and 20

0.383

0.30–0.91

10–145

0.07–0.93

352

60

25.0 50.0 75.0

12, 16 and 20 12, 16 and 20 12, 16 and 20

0.500

0.30–0.91

10–145

0.07–0.93

392

90

25.0 50.0 75.0

12, 16 and 20 12, 16 and 20 12, 16 and 20

0.707

0.30–0.91

10–145

0.07–0.93

447

120

25.0 50.0 75.0

12, 16 and 20 12, 16 and 20 12, 16 and 20

0.866

0.30–0.91

10–145

0.07–0.93

415

150

25.0 50.0 75.0

12, 16 and 20 12, 16 and 20 12, 16 and 20

0.966

0.30–0.91

10–145

0.07–0.93

425

180 (Linear weir)

25.0 50.0 75.0

12, 16 and 20 12, 16 and 20 12, 16 and 20

1.000

0.30–0.91

10–145

0.07–0.93

469

conducted for lengths of the weir (0.25 m, 0.50 m and 0.75 m), heights of the weir (0.12 m, 0.16 m and, 0.20 m) and, labyrinth side weir included angles with h = 45°, 60°, 90°, 120° and 150° (Fig. 3). Notations and location of the labyrinth side weir and range of test variables were given respectively in Table 1. After completion of a good physical description of the labyrinth side weir flow at the straight rectangular channel, labyrinth side weir flow rates were tested for different Froude number, different p/h1 ratio, different L/b ratio, different L/l ratio and, different weir included angles in order to obtain the variation of the discharge coefficient. A total of 2500 test runs for discharge coefficient measurements were performed in the study. 4. Experimental results Discharge coefficient was computed using De Marchi’s equation (Eq. (2)). Dimensionless parameters for the triangular labyrinth side weir discharge coefficient were used in Eq. (6). Cd increases with the increase in p/h1. For h = 45°, this increasing is so significant. Cd values corresponding the same p/h1 values are so different each others. Scatter of the data is attributed the effect of Froude number. The effect of p/h1 on the discharge coefficient is very significant for the same Froude number on all the side weir dimensions [12]. Therefore, the effect of p/h1 on Cd has been considered for discharge coefficient of the labyrinth side weirs. Discharge coefficients of labyrinth side weirs have much higher values than those rectangular side weirs. Especially, labyrinth side weir with h = 45° has greater Cd values. The crest length of labyrinth side weir is always longer than that of classical rectangular side weir. The main reason of labyrinth side weir is to occur higher length of the crest and more severe secondary flow. The intensity of secondary motion created by lateral flow increases with an increasing the overflow length. An increase on the secondary flow causes the growth of the deviation angle and kinetic energy towards the side weir when the relative side weir length increases. Therefore, an increase at the F1 values also increases Cd values. L/b ratio increases Cd values also increase. On the other words, it is obtained higher Cd values at the high L/b ratios due to increase at the intensity of secondary flow created by lateral flow. 5. Adaptive Neuro-Fuzzy Inference System (ANFIS) Adaptive Neuro-Fuzzy Inference System (ANFIS), first introduced by Jang [16], is a universal approximator and as such is capa-

ble of approximating any real continuous function on a compact set to any degree of accuracy [17]. ANFIS is functionally equivalent to fuzzy inference systems [17]. Specifically the ANFIS system of interest here is functionally equivalent to the Sugeno first-order fuzzy model [17,10]. Below, the hybrid learning algorithm, which combines gradient descent and the least-squares method, is introduced. As a simple example we assume a fuzzy inference system with two inputs x and y and one output z. The first-order Sugeno fuzzy model, a typical rule set with two fuzzy If-Then rules can be expressed as:

Rule 1 : If x is A1 and y is B1 ; then f 1 ¼ p1 x þ q1 y þ r1

ð7Þ

Rule 2 : If x is A2 and y is B2 ; then f 2 ¼ p2 x þ q2 y þ r2

ð8Þ

The resulting Sugeno fuzzy reasoning system is shown in Fig. 4. Here the output z is the weighted average of the individual rule outputs and is itself a crisp value. The corresponding equivalent ANFIS architecture is shown in Fig. 5. Nodes at the same layer have similar functions. The node function is described next. The output of the ith node in layer l is denoted as Ol,i. Layer 1: Every node i in this layer is an adaptive node with node function

Ol;i ¼ uAi ðxÞ; Ol;i ¼ uBi2 ðyÞ;

for i ¼ 1; 2; or for i ¼ 3; 4

Min or Product

μ

μ

A1

B1 w1 f1 = p1 x + q1 y + r1

μ

X

μ

A2

Y B2 w2

x

X

y

f 2 = p2 x + q2 y + r2

weighted average

Y

f =

w1 f1 + w2 f 2 w1 + w2

Fig. 4. Two inputs first-order Sugeno fuzzy model with two rules.

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M.E. Emiroglu et al. / Advances in Engineering Software 41 (2010) 154–160 Layer 1

Layer 2

Layer 3

Layer 4

x

Layer 5

y

A1

x

w1

A2

__

__

w1

w1 f1

∑

f

__

B1

w2

y

B2

w2 f 2

__

w2 x

y

Fig. 5. Equivalent ANFIS architecture.

where x (or y) is the input to the ith node and Ai (or Bi2) is a linguistic label (such as ‘‘low’’ or ‘‘high’’) associated with this node. In words, Ol,i is the membership grade of a fuzzy set A (= A1, A2, B1, or B2) and it specifies the degree to which the given input x (or y) satisfies the quantifier A. The membership functions for A and B are generally described by generalized bell functions, e.g.

uAi ðxÞ ¼

1 1 þ ½ðx  ci Þ=ai 2bi

ð9Þ

where {ai, bi, ci} is the parameter set. As the values of these parameters change, the bell-shaped function varies accordingly, thus exhibiting various forms of membership functions on linguistic label Ai. In fact, any continuous and piecewise differentiable functions, such as commonly used triangular-shaped membership functions, are also qualified candidates for node functions in this layer [16]. Parameters in this layer are referred to as premise parameters. The outputs of this layer are the membership values of the premise part. Layer 2: This layer consists of the nodes labelled P which multiply incoming signals and sending the product out. For instance,

O2;i ¼ wi ¼ uAi ðxÞuBi ;

i ¼ 1; 2:

ð10Þ

Each node output represents the firing strength of a rule. Layer 3: In this layer, the nodes labelled N calculates the ratio of the ith rule’s firing strength to the sum of all rules’ firing strengths

i ¼ O3;i ¼ w

wi ; w1 þ w2

i ¼ 1; 2:

ð11Þ

The outputs of this layer are called normalized firing strengths. Layer 4: This layer’s nodes are adaptive with node functions

 i fi ¼ w  i ðpi x þ qi y þ r i Þ O4;i ¼ w

O5;i

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X RMSE ¼ t ðYiobserv ed  Yiestimate Þ2 N i¼1 MAE ¼

ð14Þ

N 1X jYiobserv ed  Yiestimate j N i¼1

ð15Þ

in which N is the number of data set, Yi is the discharge coefficient. Seven different ANFIS models are established to estimate predict discharge coefficient. Two or three membership functions to the ANFIS models were considered sufficient for modeling discharge coefficient. The triangle, trapezoidal, gaussian and gumbell membership functions are tried for each ANFIS model. The training and test statistics of the ANFIS models are given in Table 2. As can be obviously seen from Table 2, the ANFIS(3,trimf) model comprising three membership functions for the inputs, L/b, L/l, p/h1, h and F1 has the lowest RMSE (0.0637), MAE (0.0426) and the highest R (0.978) values in test period. Regression analyses were performed for 2500 experimental data sets by Emiroglu et al. [12] using the nonlinear regression module. Empirical correlations predicting the discharge coefficient for the triangular side weirs are given in Eq. (1).

ð12Þ

 i is the output of layer 3, and fpi ; qi ; r i g are the parameter where w set. Parameters of this layer are referred to as consequent parameters. Layer 5: This layer’s single fixed node labelled R computes the final output as the summation of all incoming signals

P X wf  i fi ¼ Pi i i w ¼ i wi i¼1

sionless weir height p/h1, triangular labyrinth side weir included angle h, and upstream Froude number F1. The dimensionless parameters, L/b, L/l, p/h1, h and F1 were used as inputs to the ANFIS model to predict discharge coefficient of the triangular side weirs, respectively. 2500 experimental data sets were used for the ANFIS simulations. The data were randomized and divided into three parts, training, testing and validation. After randomizing, the first 1500 data were used for training, the second 500 data were used for testing and the remaining 500 data were used for validation. Before applying the ANFIS to the data, the training input and output values were normalized to zero mean and unit variance following the suggestion of Lawrence et al. [25]. The root mean square errors (RMSE), mean absolute errors (MAE) and correlation coefficient (R) statistics are used to evaluate the model accuracies. The R shows the degree which two variables are linearly related to. Different types of information about the predictive capabilities of the model are measured through RMSE and MAE. The RMSE sizes the goodness of the fit related to high discharge coefficient values whereas the MAE measures a more balanced perspective of the goodness of the fit at moderate discharge coefficients [18]. The RMSE and MAE are defined as:

ð13Þ

Thus, an adaptive network which is functionally equivalent to a Sugeno first-order fuzzy inference system. More information for ANFIS can be found in Jang [16]. 6. ANFIS and regression analyses A program code including fuzzy toolbox, were written in MATLAB language for the ANFIS simulation. Different ANFIS architectures were tried using this code and the appropriate model structure was determined. The parameters considered in the study are the dimensionless weir length L/b, the dimensionless effective length L/l, the dimen-

Table 2 The RMSE, MAE and R statistics of the ANFIS models. Models

ANFIS(2,trimf) ANFIS(2,trapmf) ANFIS(2,gaussmf ANFIS(2,gbellmf) ANFIS(3,trimf) ANFIS(3,trapmf) ANFIS(3,gaussmf) ANFIS(3,gbellmf)

Training period

Test period

RMSE

MAE

R

RMSE

MAE

R

0.0670 0.0722 0.0658 0.0667 0.0690 0.0814 0.0684 0.0716

0.0377 0.0417 0.0367 0.0375 0.0392 0.0477 0.0390 0.0409

0.9768 0.9730 0.9775 0.9769 0.9753 0.9654 0.9757 0.9733

0.1096 0.0830 0.0745 0.0699 0.0637 0.0742 0.0644 0.0664

0.0749 0.0554 0.0536 0.0463 0.0426 0.0504 0.0428 0.0440

0.9367 0.9651 0.9700 0.9738 0.9782 0.9703 0.9777 0.9762

Table 3 The RMSE, MAE and R statistics of each model in validation period. Model

RMSE

MAE

R

ANFIS(3,trimf) Emiroglu et al. [12] NLR MLR

0.0699 0.0802 0.1019 0.1507

0.0411 0.0516 0.0664 0.1043

0.977 0.969 0.950 0.886

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M.E. Emiroglu et al. / Advances in Engineering Software 41 (2010) 154–160

3

3

Emiroglu et al. (2009)

y = 0.973x + 0.0206 R = 0.977

ANFIS

2

1

y = 0.9376x + 0.0485 R = 0.969 2

1

0

0

0

1

2

3

0

1

2

3

2

3

observed

observed 3

3

y = 0.9244x + 0.0552 R = 0.950

y = 0.7573x + 0.1682 R = 0.886 2

NLR

MLR

2

1

1

0

0

0

0.5

1 1.5 observed

2

2.5

0

1 observed

Fig. 6. The scatterplots of the ANFIS, Emiroglu et al. [12], NLR and MLR models in validation period.

In the present study, a nonlinear regression (NLR) and multiple linear regression (MLR) were also established for the 2000 experimental data and tested using the remaining 500 data. The obtained NLR and MLR are

"  0:600  1:519  9:230 L L p C d ¼ 0:507 þ 0:179 þ 0:072 þ 0:233 b ‘ h1 i3:242 0:003 sin h þ 0:245F 3:845 ð16Þ 1 C d ¼ ½1:0855 þ 0:2473 ðL=bÞ  1:0703 ðL=‘Þ þ 0:4455 ðp=h1 Þ  0:4226 sin h þ 0:2986 F 1 

ð17Þ

The validation results of the ANFIS (3,trimf) model was compared with the regression proposed by Emiroglu et al. [12], NLR given in Eq. (16) and MLR (Eq. (17)) Table 3. It can be seen from the table that the ANFIS model performs better than the regression models. The Emiroglu et al. [12] seems to be better than the NLR. The reason behind this may be the fact that the Emiroglu et al. [12] was obtained using the whole 2500 experimental data. The MLR has the lowest accuracy. The ANFIS and regression models were compared in Fig. 6 for the validation data. As seen from the fit line equations (assume that the equation is y = aox + a1) in the scatterplots that the ao and a1 coefficients for the ANFIS model is respectively closer to the 1 and 0 with a higher R value than those of the other models. The MLR estimates are far from the exact fit line. This can be clearly observed from its fit line equation coefficients. This implies the nonlinearity of the investigated phenomenon. 7. Conclusions In this study, an ANFIS model has been developed to determine the discharge coefficient of the triangular labyrinth weirs. 2500 experimental data sets were used for the ANFIS simulations. Input parameters used for the ANFIS simulations are the dimensionless

weir length, the dimensionless effective length, the dimensionless weir height, triangular labyrinth side weir included angle and upstream Froude number. The optimum ANFIS model was obtained after trying different structures in terms of membership function type and number. The ANFIS estimates were compared with those of the multiple nonlinear and linear regression models. It was found that the ANFIS model performs better than the regression models. The ANFIS could be successfully used in computation of discharge coefficient of labyrinth side weirs. In the present study, the ANFIS model was used for the determination of discharge coefficient of the triangular labyrinth weirs. The other data-driven techniques (e.g. radial basis function) could be used for the discharge coefficient estimation. This may be a subject of another study. Acknowledgment This work was financially supported by the Scientific and Technological Research Council of Turkey (TUBITAK). References [1] Ackers P. A theoretical consideration of side weirs as storm water overflows. Proc Ice, London 1957;6:250–69. [2] Agaccioglu H, Yüksel Y. Side weir flow in curved channels. J Irrigat Drain Eng 1998;124(3):163–75. [3] Aghayari F, Honar T, Keshavarzi A. A study of spatial variation of discharge coefficient in broad-crested inclined side weirs. Irrigat Drain 2009;58:246–54. [4] Bae D-H, Jeong DM, Kim G. Monthly dam inflow forecasts using weather forecasting information and neuro-fuzzy technique. Hydrol Sci J 2007;52(1): 99–113. [5] Borghei M, Jalili MR, Ghodsian M. Discaharge coefficient for sharp-crested side weir in subcritical flow. J Hydraul Eng 1999;125(10):1051–6. [6] Chang FJ, Chang YT. Adaptive neuro-fuzzy inference system for prediction of water level in reservoir. Adv Water Res 2006;29:1–10. [7] Collings VK. Discharge capacity of side weirs. In: Proceedings of the Institute Civil Engineerings, vol. 6, London, England: 1957, p. 288–304. [8] Cosar A, Agaccioglu H. Discharge coefficient of a triangular side weir located on a curved channel. J Irrigat Drain Eng 2004;130(5):321–33. [9] De Marchi G. Saggio di teoria de funzionamente degli stramazzi letarali. Energ Elettr 1934:849–60.

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