This paper presents the pertinence of the radial footing map web for anticipation of natural convection heat transportation from a confined horizontal elliptic tubing. The RBF construction is developed and trained with the aid of informations obtained by a Mach-Zehnder interferometer. It is expensive and clip devouring to make experimental work with altering all variables. The radial footing map web is developed with tubing axis ratio, distance from the centre of tubing and rayliegh figure as inputs and mean nusselt figure as coveted end product. We used the radial footing map web to imitate the steady status of heat transportation rate distribution in described geometry. The consequences of web have an first-class understanding with experimental informations. Therefore, the web can be used to foretell the unobserved information points within the scope of experimental consequences.
Artificial nervous webs ( ANNs ) have seen an detonation of involvement over the last few old ages, and are being successfully applied across an extraordinary scope of job spheres, in countries such as diverse as finance medical specialty, technology, geology and natural philosophies. Indeed, anyplace that there are jobs of anticipation, categorization or control, nervous webs are being introduced. The survey of heat transportation is one of the most of import jobs in technology applications. For illustration, in electrical technology field, in topographic points like transmittal lines and wires transporting the current and are located compactly, if their distance are non chosen right, the heat generated by them will hold inauspicious consequence on their actions. It causes runing of their isolation and it is in bend may take to happen unwanted effects. Furthermore, a broad scope of practical applications involve the analysis of heat transportation, for illustration in heat money changers, reactors etc. [ 1 ] . For a better thermic design of such heat money changers, it is indispensable to analyze heat transportation around heated elliptic tubing of different cross subdivision confined between two walls. Heat transportation around round cylinder as a particular instance of elliptic tubing has been antecedently studied widely [ 2-7 ] . To heighten the heat transportation rate around elliptic tubing more research has to be done. Fieg and Roetzel in [ 8 ] showed that the egg-shaped distortion increases heat transportation coefficient during their analytical probe on laminar movie condensation on inclined egg-shaped tubing. A.O.Elsayed et. al. , studied free convection from a changeless heat flux elliptic tubing by experimentation [ 9 ] . To better heat transportation from the tubing surface, a technique was employed to restrict the tubing between two adiabatic walls [ 10 ] . In the work done so far [ 1 ] , an person ANN web for each entree ratio was developed, so an ANN is needed for each axis ratio, thereby this method is clip devouring and its computational velocity is being reduced. In order to get the better of such jobs, i.e. to hold merely one ANN for all axis ratio, a theoretical account was developed in [ 11 ] utilizing multilayer perceptron nervous web in such a manner that axis ratio will be considered as one of the input to the web. The purpose of this paper is to make an ANN theoretical account to foretell the norm nusselt figure for heat transportation from elliptic tubing cross subdivisions confined between two adiabatic walls utilizing radial footing map web.
a major axis ( m )
AAD Average Absolute Deviation
B minor axis ( m )
end product of each nerve cell
bias vector of each nerve cell
degree Fahrenheit transportation map
H wall length
H1 Distance from top to centre of tubing
H2 Distance from underside to centre of tubing
n transportation map input
figure of point
Nu mean Nusselt figure
P input vector
Ra Rayleigh figure
T wall spacing ( m )
a?†NU comparative mistake ( Nusselt difference )
% a?†NU mistake divergence ( Nusselt difference in per centum )
W weight vector
Y mark activation of the end product bed
R figure of elements in input
ARTIFICIAL NEURAL NETWORK
Artificial nervous webs are mathematical or computational theoretical accounts based on biological nervous webs. They are being used greatly to pattern complex relationships between inputs and end products or to happen forms in informations. By utilizing this capableness, an ANN theoretical account for our intent has been constructed. Fig. 1 shows a schematic of the proposed ANN model.In this theoretical account, the mean nusselt figure is adopted as a map of three variables viz. :
t/b, wall spacing to tube minor axis ration
Ra, Rayleigh figure
b/a, Axis ratio
Therefore an ANN theoretical account as shown in Fig. 1 is developed with tubing axis ratio, distance from the centre of tubing and rayliegh figure as inputs and mean nusselt figure as coveted end product.
Fig. 1. Input-output schematic of system.
Radial footing map
Radial footing maps ( RBFs ) can suit fickle informations. RBF webs have been widely applied in many scientific discipline and technology Fieldss due to their good estimate capablenesss, faster larning algorithms and simpler web constructions.
The RBF has a feed forward construction and in its most basic signifier consists of three separate beds called input bed, hidden bed and end product bed as shown in Fig. 2. The transmutation from input to conceal bed is nonlinear and from hidden to end product bed is additive. All preparation informations points are represented to the web and the interpolating surface has to go through through all of them.
The end product from jth nerve cell of the concealed bed is given by:
j=1,2, .. , K
K is a purely positive radially symmetric map ( meat ) . It has a alone upper limit at its centre, i.e. , and decreases to zero off from the centre. is the breadth of the receptive field in the input infinite from unit J and K is the figure of nerve cells in the concealed bed. This indirectly indicate that has an desired value merely when the distance is smaller than the.
Fig. 2. Radial footing map nervous web.
The end product bed consists of a set of summing up units and provides the response of the web. For an input vector, the end product of the mth nerve cell in the end product is defined by:
m=1,2, .. , M
Fig. 3 shows a conventional of elliptic tubing confined between two adiabatic walls [ 11 ] . In this figure, H1 and H2 are fixed and equal to 32 millimeters and 24 millimeter severally. The tried tubings are made up of aluminium with the length of 160 millimeters and major. Their minor axes are chosen to obtain the same fringe. The dimensions of walls are 56A-20A-160mm and the tubing is placed symmetrically between the two walls. Heating wire is placed inside the tubing and it is heated utilizing a variable power supply. The Mach-Zehnder Interferometer used in our experiment is the same as the one used by Ashjaee et. Al. [ 10 ] . The inside informations of the Mach-Zehnder Interferometer apparatus and the informations decrease process are to the full explained in [ 10 ] .
We get the experimental information points from [ 10 ] and a good understanding between the experimental consequences and consequences of the other researches have been observed, therefore comparings are non reexpressed here.
Fig. 3 Schematic of the job.
simulation with ann
In this survey, a RBF nervous web theoretical account was implemented to foretell mean nusselt figure from an isothermal horizontal cylinder of egg-shaped cross subdivision confined between two adiabatic walls. As mentioned before the experiment needs a considerable sum of clip to acquire accurate consequences. The purpose is to build a RBF theoretical account which is capable of qualified anticipation of mean nusselt figure.
To measure the anticipation truth of the proposed theoretical account, mistake difference ( a?†NU ) and error divergence ( a?†NU % ) for the mean nusselt figure are calculated as:
( 1 )
( 2 )
exp and pred represent experimental and predicted values, severally. Besides, the Average Absolute Deviation ( AAD % ) is defined as:
( 3 )
where is the figure of points. The AAD % for the RBF theoretical account implemented in the present work and for the MLP theoretical account which we created antecedently in [ 11 ] are shown in Table 1.
The Average Absolute Deviation ( AAD % )
As it can be observed from Table 1, there is a good understanding between experimental and predicted informations. The Comparison of experimental mean nusselt figure ( Nu exp. ) with predicted mean nusselt figure ( Nu pre. ) , mistake difference, mistake divergence for trial informations ( unobserved informations ) of the RBF and MLP theoretical accounts are shown in Table 2. The consequence show that the predicted values are in good understanding with experimental informations. The comparing between experimental and predicted values for preparation and proving sets in RBF theoretical account are shown in Fig. 4 and 5 severally. These figures besides shows the predicted values are really near to experimental values with least mistake.
Fig. 4. Comparison between experimental and predicted values for preparation set.
Fig. 5. Comparison between experimental and predicted values for proving set.
Besides, the comparing between mean nusselt Numberss obtained from experiment and those predicted with nervous web for tested informations, as a map of t/b ratio for some selected Rayleigh Numberss for each b/a=0.53, 0.67, 0.8, are shown in Fig. 6. This figure besides compares the truth of RBF and MLP [ 11 ] theoretical accounts.
Fig. 6a. Comparison between experimental and predicted values for b/a=0.53.
Comparison of experimental mean nusselt ( Nu exp. ) with predicted mean nusselt figure ( Nu pre. ) , mistake difference, mistake divergence.
Fig. 6b. Comparison between experimental and predicted values for b/a=0.67.
Fig. 6c. Comparison between experimental and predicted values for b/a= 0.8.
It is observed from Fig. 6, that the mean nusselt figure additions with the addition of the Rayleigh figure for each wall spacing. There is an optimal wall spacing for a changeless Rayleigh figure, where the heat transportation from the elliptic cylinder is maximal. When the wall spacing additions from its optimal value, the mean nusselt figure lessenings and attacks to the value of mean nusselt figure for a tubing in infinite medium. Besides, for each Rayleigh figure, lessening of the wall spacing from its optimal value makes a crisp lessening in the mean nusselt figure. On the other manus, Fig. 6 shows that consequences obtained from nervous web have fitted good with the consequences of experiment [ 1 ] , [ 10 ] .
The consequences showed that the theoretical account could be used in this job for anticipation of mean nusselt figure, which is of import in free convection application.
In this work, an accurate RBF theoretical account was constructed to foretell the norm nusselt figure for heat transportation from elliptic tubing cross subdivisions confined between two adiabatic walls. The web was trained utilizing experimental informations. The maximal absolute mistake for trained and tried values are 0.265 % and 2.101 % severally and Average Absolute Deviation ( AAD % ) are 0.0361 % and 0.2580 % for train and tested informations severally. A comparative survey of soft calculating theoretical accounts for burden prediction shows that RBF is more accurate and effectual as compared to MLP. The consequences shows predicted values are really near to experimental values.
The consequences obtained clearly demonstrate that RBF is more accurate and dependable for the anticipation of free coefficient of convection heat transportation.
M. Ashjaee, R. Afzali, M. Niknami, M. Amiri. , and T. Yousefi, “ Nervous Network Analysis of Free Connection Around Isothermal Elliptic Tube ” , 8th Biennial ASME Conference on Engineering Systems Design and Analysis, Torino, Italy, July 4-7, 2006.
T.H. Kuehn, and R.J. Goldstein, “ Numeric Solution to the Navier-Stokes Equations for Laminar Natural Convection about a Horizontal Isothermal Circular Cylinder ” , Intl. J. Heat Mass Transfer, vol. 23, 1980, pp. 971-979.
T. Saitoh, T. Sajiki, and K. Maruhara, “ Bench Mark Solution to Natural Convection Heat Transfer Problem Around the Horizontal Circular Cylinder ” , Intl. J. Heat Mass Transfer, vol. 36, 1993, pp. 1251- 1259.
V.T. Morgan, “ The Overall Convection Heat Transfer from Smooth Circular Cylinders ” , Advances in Heat Transfer, vol. 11, 1975, pp.199-264.
S.W. Churchill, H.H.S. Chu, “ Correlating Equations for Laminar and Turbulent Free Convection from a Horizontal Cylinder ” , Intl. J. Heat Mass Transfer, vol. 18, 1975, pp. 1049-1053.
H.M. Badr, “ Heat Transfer Buoyancy Driven Flow Adjacent to a Horizontal Rod ” , Intl. J. Heat Mass Transfer, vol. 30, 1987, pp. 1997-2012.
R. Diez, M. Dolz, R. Belda, J.V. Herraez, and M. Buendia, “ Free Convection Around a Horizontal Circular Cylinder. A Dimensional Empirical Equation ” , Applied Scientific Research, vol.46, 1989, pp. 365-378.
G.P. Fieg, and W. Roetzel, “ Calculation of Laminar Film Condensation in/on Inclined Elliptical Tubes ” , Intl. J. Heat Mass Transfer, vol. 37, 1994, pp. 619-624.
A.O. Elsayed, E.Z. Ibrahim, and S.A. Elsayed, “ Free From a Changeless Heat Flux Elliptic Tube ” , Energy Conservation and Management, vol. 44, 2003, pp. 2445-2453, 2003.
M. Ashjaee, T. Yousefi, and M. Amiri, “ Free Convection Heat Transfer from a Confined Horizontal Elliptic Cylinder ” , accepted for publication in the Journal of Heat Transfer Engineering. ‘
M. Hayati, T. Yousefi, M. Asgjaee, A. Hamidi, and Y. Shirvani, “ Application of Artificial Neural Networks for Prediction of Natural Convection Heat Transfer from a Confined Horizontal Eliptic Tube ” , International Journal of Applied Science, Engineering and Technology, vol. 4, pp. 157-162, 2007.