Analysis of adaptive-network-based fuzzy inference system (ANFIS) to estimate buoyancy-induced flow field in partially heated triangular enclosures

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Expert Systems with Applications Expert Systems with Applications 35 (2008) 1989–1997 www.elsevier.com/locate/eswa

Analysis of adaptive-network-based fuzzy inference system (ANFIS) to estimate buoyancy-induced flow field in partially heated triangular enclosures Yasin Varol

a,*

, Ahmet Koca a, Hakan F. Oztop b, Engin Avci

c

a

Department of Mechanical Education, Firat University, 23119 Elazig, Turkey Department of Mechanical Engineering, Firat University, 23119 Elazig, Turkey Department of Electronic and Computer Education, Firat University, 23119 Elazig, Turkey b

c

Abstract Adaptive-network-based Fuzzy Inference System (ANFIS) was used to predict temperature and flow field due to buoyancy-induced heat transfer in a partially heated right-angle triangular enclosure. Results are obtained for two-dimensional, steady and laminar flow. Data were generated from earlier studies which were obtained from a CFD-based computer code writing in Fortran. Equations were analyzed using finite difference method. Effective parameters on flow and temperature fields are Rayleigh number, location of partial heater and height-to-base ratio of triangular enclosures. Both instability and accuracy related results will be discarded by using ANFIS. Thanks to using of ANFIS, computation time, memory space and effort were reduced. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: CFD; Natural convection; Fuzzy system; Triangular enclosure

1. Introduction Buoyancy induced flow is formed inside enclosures due to buoyancy forces and temperature difference. This mode of heat transfer can be found in building applications as double pane windows, rooms or roofs, solar collectors, cooling of electronic equipments, heat exchangers, drying of foods etc. A wide review on application of buoyancyinduced flow was performed in earlier studies by Ostrach (1988), Yang (1987) and De Vahl Davis and Jones (1983). Analysis of flow and temperature field can be performed with experimental techniques but measurement of velocity and pressure is very difficult and expensive inside the enclosure due to low flow strength and velocity. Thus, computa-

*

Corresponding author. Tel.: +90 424 237 0000x4219; fax: +90 424 236 7064. E-mail address: [email protected] (Y. Varol). 0957-4174/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2007.08.073

tional fluid dynamics (CFD) is a powerful tool to obtain buoyancy-induced flow and temperature field in different shaped enclosures. But CFD has two important problems as accuracy and stability which are well known by CFD researchers. These problems occur especially at highly nonlinear cases and CFD analyze needs long calculation time and high computation cost. Thus, soft programming techniques (Artificial Neural Network, Fuzzy-logic, AdaptiveNetwork-Based Fuzzy Inference System etc.) can be used as a powerful tool to analyze and predict flow, heat and mass transfer problems as indicated by Mahmoud and Ben-Nakhi (2007). Some applications of soft programming methods in energy related areas are can be found in literature (Bishop, 1996; Avci, Turkoglu, & Poyraz, 2005; Guanghui et al., 2003; Kalogirou, 1999; Mahajan, Fichera, & Horst, 2005; Vaziri, Hojabri, Erfani, Monsefi, & Nilforooshan, 2007). Adaptive-Network-Based Fuzzy Inference System (ANFIS) is a newer method to predict new data using some

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Y. Varol et al. / Expert Systems with Applications 35 (2008) 1989–1997

Nomenclature c c0 g Gr L H Nu Pr p, r, q, Ra T u, v X, Y

Greek Letters t kinematic viscosity X vorticity h non-dimensional temperature b thermal expansion coefficient a thermal diffusivity W stream function

heater distance to bottom of enclosure dimensionless heater position, c/H gravitational acceleration Grashof number length of bottom wall maximum height of triangle Nusselt number Prandtl number rr linear output parameters Rayleigh number temperature velocities non-dimensional coordinates

Subscript c cold h hot

experimental and numerical data (Avci & Akpolat, 2006; Jang, 1993; Kosko, 1991). However, its application to the energy related studies are very limited. For instance, Ryoo, Dragojlovic, and Kaminski (2005) used the ANFIS to control of convergence in a Computational Fluid Dynamics Simulation. They tested benchmark problems as lid-driven cavity, differentially heated enclosure, backward-facing step and conjugate buoyancy-induced cavity. Lu, Cai, Xie, Li, and Soh (2005) used ANFIS to optimize in-building section of centralized heating, ventilation and air-conditioning (HVAC) systems. Their results indicated that the system energy usage can be minimized by operating components at the optimal set points calculated in real time with the changing cooling load. The optimal set points include chilled water supply temperature, chilled water Insulated wall u=0, v=0, ∂T/∂n=0

pump heat, air differential pressure in duct networks, and sequencing of pumps and chillers. Recently, Varol, Avci, Koca, and Oztop (2007) compared the prediction results from ANFIS and artificial neural network (ANN) with of flow and temperature field in a triangular enclosure due to natural convection. As well known that analysis of natural convection in triangular enclosure is important problem due to its wide application area in engineering (Akinsete & Coleman, 1982; Asan & Namli, 2000; Holtzman, Hill, & Ball, 2000; Ridouane, Campo, & McGarry, 2005; Tzeng, Liou, & Jou, 2005; Varol, Koca, & Oztop, 2006). Thus, they analyzed the natural convection for this problem. Finally, they observed that ANFIS gives better results than that of ANN especially at higher Rayleigh numbers.

y

TC, u=0, v=0

TH, u=0, v=0 H

h

c

g x

Insulated wall u=0, v=0, ∂T/∂n=0

L

Fig. 1. Treated physical model.

Y. Varol et al. / Expert Systems with Applications 35 (2008) 1989–1997

The main objective of this study is to estimate the flow and temperature distribution in a partially heated air-filled right-angle triangular enclosure using adaptive-networkbased fuzzy inference system (ANFIS). Data generated from the CFD code were used to compare ANFIS results using different governing parameters as Rayleigh number, aspect ratio and position of heater. Triangular shaped enclosure. 2. Treated physical model Considered physical model, which is right-angle enclosure, is depicted in Fig. 1 with boundary conditions. It has a heater with finite length on its left vertical. Sloping wall of enclosure is isothermally cooled. Location of heater can be change along the vertical wall. Remaining walls are adiabatic. Height to base ratio of the triangle is defined with aspect ratio (AR = H/L) which is important parameter on flow and temperature field. 3. Data generation with CFD code Steady, laminar, two-dimensional Navier-Stokes equations in streamfunction-vorticity form are written for constant thermo physical properties, incompressible, and Newtonian fluid with Boussinesq approximation. Gravity acts in vertical direction and radiation mode of heat transfer is neglected. Based on these assumptions, the governing equations in dimensionless form become as o2 W o2 W þ oX 2 oY 2     o2 X o2 X 1 oW oX oW oX oh  þ ¼  Ra 2 2 Pr oY oX oX oY oX oX oY X¼

ð5Þ

Nuy dy 0

4. Adaptive-network-based fuzzy inference system (ANFIS) Both artificial neural network and fuzzy logic are used in ANFIS’s architecture. ANFIS is consisted of if-then rules and couples of input-output. ANFIS training is used learning algorithms of neural network (Akinsete & Coleman, 1982; Holtzman et al., 2000; Lu et al., 2005; Varol et al., 2007). For simplicity, we assume the fuzzy inference system under consideration has three inputs (x, y and z) and one output (t) in the present study. For a first order Sugeno fuzzy model, a typical rule set with base fuzzy if-then rules can be expressed as Layer-4

ð2Þ

∏ x

A1 A2 A3

2

xðLÞ Pr ; t

Layer-3

Layer-2 Layer-1

where the following dimensionless variables have been used:

y

B1

∏

3

z

N N N

∏

N

∏

N

N

w2

w1 * f1

w3 Layer-5

w5

f

w6 w7

N

w8

N

w9

C1

x y z

w1

w4

N

B3

bgðT h  T c ÞL Pr ; t2

w1

∏

B2

Ra ¼

h

Governing equations in streamline-vorticity form (Eqs. (1)–(3)) are solved using finite difference method. Algebraic equations are obtained via Taylor series and they solved via successive under relaxation (SUR) technique, iteratively. Central difference method is used for discretization procedure. The detailed solution technique is well described in the literature (Varol, Koca, & Oztop, 2007). A database was generated by using finite difference method by writing a FORTRAN code. As convergence criteria, 104 is chosen for all depended variables and value of 0.1 is taken for under-relaxation parameter. To check accuracy of the code, the computational results are validated against the earlier studies and results can be found our earlier study (Varol et al., 2007).

ð3Þ

X¼

Nu ¼

Z

ð1Þ

o2 h o2 h oW oh oW oh  þ ¼ oX 2 oY 2 oY oX oX oY

x y wPr X ¼ ; Y ¼ ; W¼ ; L L t T  Tc ow ; h¼ ; u¼ oy Th  Tc   ow ov ou v¼ ; x¼  ; ox ox oy t Pr ¼ : a

  oh ; Nuy ¼  oX X ¼0

1991

C2

ð4Þ The hot wall is maintained at a temperature h = 1, and cold wall has a temperature h = 0. Remaining of vertical wall is adiabatic (oh/ox = 0). No-slip conditions (U = V = 0) are used on all walls. Calculation of local and mean Nusselt number are given as follow

C3

∏ ∏

N

∏

N

∏

N

w 25 w 26 w 27

w27* f 27

Fig. 2. Block diagram of ANFIS used in this study.

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Y. Varol et al. / Expert Systems with Applications 35 (2008) 1989–1997

If x is A1 and y is B1 and z is C 1 then f 1 ¼ p1 x þ q1 y þ r1 z þ rr1

ð6Þ

The ANFIS’s architecture is shown in Fig. 2a that it has three inputs and one output. This architecture is formed by using five layer and 27 if–then rules as follows: Layer-1: Each ‘‘i’’ node in this layer is a square node with a node function as O1;i ¼ lAi ðxÞ; for i ¼ 1; 2; 3 for i ¼ 4; 5; 6

O1;i ¼ lBi 3 ðyÞ;

O1;i ¼ lCi 6 ðzÞ; for i ¼ 7; 8; 9

ð7Þ

where x, y and z are inputs to node i, and Ai, Bi, and Ci are linguistic label associated with this node function. In order words, O1,i is the membership function of Ai, Bi, and Ci. We choose lAi(x), lBi(y), and lCi(z) to be bell-shaped with maximum (1) and minimum (0), such as li ðxÞ ¼ expðððxi  ci Þ=ðai ÞÞ2 Þ

Table 1 ANFIS architecture and training parameters used in this study

Training parameters Learning rule

Momentum constant Sum-squared error

O2;i ¼ wi ¼ lAi ðxÞ  lBi 3 ðyÞ  l i ¼ 1; 2; 3 . . . ; 27

5 Input: 3, Rules number: 27, Output: 1

ð9Þ

Each output nodes represent the firing strength of a rule. (In fact, other T-norm operators that performs generalized AND can be used as the node function in this layer). Layer-3: Every node in this layer is a circle node depicted by N. The ith node calculates the ratio of the ith rules firing strength to the sum of all rule’s firing strengths:  i ¼ wi =ðw1 þ w2 þ . . . þ w27 Þ; O3;i ¼ w i ¼ 1; 2; 3 . . . ; 27

ð10Þ

Layer-4: Every node i in this layer is a square node with a node function Table 2 R2 values Contour

Isotherm

AR (=H/L)

Dimensionless heater position (c 0 )

Rayleigh number Ra = 104

Ra = 106

1.0

0.15 0.50 0.85 0.15 0.50 0.85

0.9984 0.9981 0.9992 0.9974 0.9991 0.9990

0.9950 0.9782 0.9939 0.9898 0.9940 0.9792

0.15 0.50 0.85 0.15 0.50 0.85

0.9435 0.9426 0.9536 0.9476 0.9225 0.9746

0.9746 0.9367 0.9547 0.9467 0.9547 0.9856

Bell-shaped 0.5 Log-sigmoid Hybrid Learning Algorithm (Back-propagation for nonlinear parameters (ai, ci) and Least square errors for linear parameters (pi, qi, ri, rri)) 0.98 0.000001

a

Ci 6 ðzÞ;

ð8Þ

where ai and ci are the parameter set which referred as premise parameters.

Architecture The number of layers The number of inputs, rules and output Type of input membership functions Activation functions

Layer-2: Every note in this layer is a circle node depicted by P which multiplies the incoming signals and sends the product out. For instance,

CFD

Streamline

1.0

0.5

ANFIS

Ψmax= 0 Ψmin=-0.56

Ψmax= 0 Ψmin=-0.55

b Ψmax= 0 Ψmin=-13.1

Ψmax= 0 Ψmin=-13.08

Fig. 3. Streamlines obtained from CFD (on the left) and ANFIS (on the right) for AR = 0.5 and c 0 = 0.15, (a) Ra = 104, (b) Ra = 106.

Y. Varol et al. / Expert Systems with Applications 35 (2008) 1989–1997

 i  fi ¼ wi ðpi x þ qi y þ ri z þ rri Þ; O4;i ¼ w i ¼ 1; 2; 3 . . . ; 27

5. Results and discussion ð11Þ

where, wi is the output of layer 3, and {pi, qi, ri, rri} is the parameter set. Parameters in this layer will be referred to as consequent parameters. Layer-5: The P single node in this layer is a circle node depicted by that computes the overall output as the summation of all incoming signals: P X wi fi  i fi ¼ Pi Oi;5 ¼ Overall output ¼ ð12Þ w i wi i Table 1 shows the parameters for ANFIS which are used in this study.

a

ANFIS method has been used to predict the temperature and flow field in a partially heated triangular enclosure in this study. Data were generated by a CFD code which is written in Fortran platform. These are accepted as actual values. The code is also used in our earlier study (Varol et al., 2007). New data generated for governing parameters as Rayleigh number, aspect ratio and different location of heater. Value of Prandtl number of working fluid is taken as 0.7. Dimensionless streamfunction and temperature values were used as dependent variables. 61 by 61 matrix, in other words, 3721 nodal points were used. Rayleigh number was changed between 103 and 106. Streamfunction

CFD

ANFIS

0 .1

0.7 0.9

0.5

0 .3

0 .1

0 .5

0 .3

0 .7 0.9

b

0 .5

0 .1

0 .1

0 .5

0 .3

0.3

Fig. 4. Isotherms obtained from CFD (on the left) and ANFIS (on the right) for AR = 0.5 and c 0 = 0.15, (a) Ra = 104, (b) Ra = 106.

a

1993

CFD

ANFIS

Ψmax= 0 Ψmin=-0.678

Ψmax= 0 Ψmin=-0.676

b Ψmax=0.478 Ψmin=-9.79

Ψmax=0.499 Ψmin=-9.710

Fig. 5. Streamlines obtained from CFD (on the left) and ANFIS (on the right) for AR = 0.5 and c 0 = 0.50, (a) Ra = 104, (b) Ra = 106.

1994

Y. Varol et al. / Expert Systems with Applications 35 (2008) 1989–1997

and temperature values for Ra = 103 and 105 were used to predict further values as 104 and 106. A whole domain was used for estimating new values. Fig. 3 shows the comparison of streamlines of CFD and ANFIS for Ra = 104 (on the top) and Ra = 106 (on the bottom) at c 0 = 0.15. As can be observed from the flow field that single cell formed in clockwise direction due to increasing heated air from the heater. The heated air moves towards to the inclined cold wall. Thus, a circulation cell is formed. Values of streamfunction are given on the top right corner of the figure. Flow field shows good agreement between CFD results and treated ANFIS results. Difference between values of streamfunction of ANFIS and CFD are very small. This difference can be seen from the relaxation values which are listed in Table 2.

a

Specifically, a statistical indicator (R2) was used to evaluate the closeness of fit of network architecture as follows (Avci & Akpolat, 2006). P ðx  xp Þ2 2 R ¼1P ð13Þ 2 ðx  x m Þ While x (streamfunction or temperature) is the actual value obtained from CFD code, xp is the predicted value of x, and xm is the mean value of x values. As can be seen from the Table 2, R2 values are closer to 1. This means that there is a good fit between soft programming codes and CFD code especially for ANFIS method for both isotherms and stream functions. Isotherms are presented in Fig. 4 to compare results between ANFIS and CFD code. It can be seen from the

CFD

0.1

0.3

1 0.

0.3

0.5 0.7 0.9

0.9 0 0.5 .7

ANFIS

b

0.3

0.3

0.7

0.5

0.7

0.5

0.

1

1 0.

Fig. 6. Isotherms obtained from CFD (on the left) and ANFIS (on the right) for AR = 0.5 and c 0 = 0.50, (a) Ra = 104, (b) Ra = 106.

a

CFD

ANFIS

Ψmax= 0 Ψmin= -0.406

Ψmax= 0 Ψmin= -0.404

b Ψmax= 0 Ψmin= -4.35

Ψmax= 0 Ψmin= -4.31

Fig. 7. Streamlines obtained from CFD (on the left) and ANFIS (on the right) for AR = 0.5 and c 0 = 0.85, (a) Ra = 104, (b) Ra = 106.

Y. Varol et al. / Expert Systems with Applications 35 (2008) 1989–1997

figures that similar temperature distribution was obtained using ANFIS. For lower Rayleigh number, conduction

a

mode of heat transfer is effective. On the contrary, convection mode of heat transfer is dominant to conduction for

ANFIS

0.5

0 0.5 .7

CFD

0.3

0.3 0.1

0.1

3 0.

0. 1

0. 1

0. 5

0. 0. 5 3

b

Fig. 8. Isotherms obtained from CFD (on the left) and ANFIS (on the right) for AR = 0.5 and c 0 = 0.85, (a) Ra = 104, (b) Ra = 106.

a

CFD

ANFIS Ψmax= 0 Ψmin=-2.43

Ψmax= 0 Ψmin=-2.40

b Ψmax= 0 Ψmin=-20.0

1995

Ψmax= 0 Ψmin=-19.95

Fig. 9. Streamlines obtained from CFD (on the left) and ANFIS (on the right) for AR = 1 and c 0 = 0.15, (a) Ra = 104, (b) Ra = 106.

1996

Y. Varol et al. / Expert Systems with Applications 35 (2008) 1989–1997

a

CFD

ANFIS

0.3

0.1

0.3

0.5

0.1

0.7

0.7

0.5

0.9

0.9

b

0.5

0.5

0.3 0.1

0.1

0.3

Fig. 10. Isotherms obtained from CFD (on the left) and ANFIS (on the right) for AR = 1 and c 0 = 0.15, (a) Ra = 104, (b) Ra = 106.

higher Rayleigh number. Thus, obtained contours from the ANFIS are not smooth especially for lower temperature values. In Figs. 5 and 6, the heater is located on the middle of the vertical (c 0 = 0.50) to see the effect of location of heater on flow and temperature distribution. A circle shaped main circulation cell was obtained for the case of Ra = 104. When Rayleigh number increased, it shows diagonally elongated shape. Predicted values show again good agreement with calculated values. Detailed results on heat transfer and flow field for this case are given in our earlier study (Varol et al., 2007). Here, we can compare the minimum and maximum streamfunction values obtained from ANFIS and CFD. Table 2 shows that R2 value is small (0.9225) for Ra = 104 for the middle position of the heater. However, R2 values are almost unity. It means that ANFIS shows acceptable results. When heater locates the top of the vertical wall (c 0 = 0.85), flow distributed from top corner into the cavity due to pressing flow between vertical and sloping wall. Single cell was formed in both Rayleigh number. All results showed that the location of heater is the most important parameter on flow and temperature

field. It can be seen that minimum and maximum values of streamfunctions are very close to each other (Fig. 7). Namely, Wmin = 0.406 (CFD) and Wmin = 0.404 (ANFIS) for Ra = 104 and Wmin = 4.35 (CFD) and Wmin = 4.31 (ANFIS) for Ra = 106. Isotherms show interesting distribution due to adjacent hot and cold wall (Fig. 8). Temperature different has the highest value in this case. Thus, most of the fluid inside the cavity becomes cold due to motionless fluid. Effects of aspect ratio (AR) on flow and temperature are given in Fig. 9 (streamlines) and 10 (isotherms) for the cases of c 0 = 0.15. To predict flow and temperature field using ANFIS at different aspect ratio, each aspect ratio is evaluated in the same class of aspect ratio Fig. 10. In other words, Ra = 103 and 105 are used to obtain values for Ra = 104 and 106 at AR = 1. In this case, circle shaped main cell was formed for lower aspect ratios and egg shaped cell was obtained for higher Rayleigh number due to increasing of flow velocity. Flow strength also increases with increasing of aspect ratio. ANFIS gives almost same minimum and maximum streamfunction values with CFD results.

Y. Varol et al. / Expert Systems with Applications 35 (2008) 1989–1997

6. Conclusion A comparison study between computational fluid dynamics (CFD) and adaptive-network-based fuzzy inference system (ANFIS) was performed for natural convection in partially heated triangular enclosure. CFD results were obtained from a computer code and they used to predict new flow and temperature field for different parameters using ANFIS. Ra = 103 and 105 values are used to predict new flow and temperature field for higher Rayleigh number. Regression coefficient results are acceptable. Both flow and temperature distribution is resembled to each other. Almost same minimum and maximum streamfunction values are obtained between CFD and ANFIS for all values of aspect ratios. Results showed that ANFIS can be used to predict new values from generated data. Thus, computational time and computational cost will be reduced. References Akinsete, V., & Coleman, T. A. (1982). Heat transfer by steady laminar free convection in triangular enclosures. International Journal of Heat Mass Transfer, 25, 991–998. Asan, H., & Namli, L. (2000). Laminar natural convection in a pitched roof of triangular cross-section: Summer day boundary conditions. Energy and Buildings, 33, 69–73. Avci, E., & Akpolat, Z. H. (2006). Speech recognition using a wavelet packet adaptive network based fuzzy inference system. Expert Systems with Applications, 31, 495–503. Avci, E., Turkoglu, I., & Poyraz, M. (2005). Intelligent target recognition based on wavelet packet neural network. Expert Systems with Applications, 29, 175–182. Bishop, C. M. (1996). Neural networks for pattern recognition. Oxford: Clarendon Press. De Vahl Davis, G., & Jones, I. P. (1983). Natural convection in a square cavity: A comparison exercise. Intenational Journal for Numerical Methods in Fluids, 3, 227–248. Guanghui, S., Morita, K., Fukuda, K., Pidduck, M., Dounan, J., & Miettinen, J. (2003). Analysis of the critical heat flux in round vertical tubes under low pressure and flow oscillation conditions: applications of artificial neural network. Nuclear Engineering and Design, 220, 17–35. Holtzman, G. A., Hill, R. W., & Ball, K. S. (2000). Laminar natural convection in isosceles triangular enclosures heated from below and symmetrically cooled from above. Jouranl of Heat Transfer, 122, 485–491.

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