4 E0020 Icmsn 2018

International Journal of Structural and Civil Engineering Research Vol. 8, No. 1, February 2019

The Modern Approach for Materials Construction Design

Elhadj. Benachour a Energetic laboratory in Arid Zones (ENERGARID), Departement of Sciences, Faculty of Science, University of Tahri Mohamed, Bechar, 08000, Algeria Email: benachour_elhadj @yahoo.fr

Belkacem. Draoui b, Bachir. Imine c and Khadidja. Asnoune d b Energetic laboratory in Arid Zones (ENERGARID), Departement of Sciences, Faculty of Science, University of Tahri Mohamed, Bechar, 08000, Algeria cLASP Laboratory, Faculty of Mechanical Engineering, USTOran, 31000, Algeria dLaboratory of Smart Grids and Renewable Energies, TAHRI Mohammed University of Bechar, BP. 417 Bechar (08000), Algeria Email: [email protected],{ imine_b, kh_asnoune}@yahoo.fr

Abstract—The simulation grant a deep insight into the construction problems but what are the most common and quantum mechanical and thermal heat transfer effects how can you avoid them?. Each building design team which determine material properties, and therefore, must complete its own due diligence on the system that computational materials design is traditionally used to works best for its building and the end user of the improve and further develop already existing materials. building, as well as the design of the exterior walls Especially when it comes to thermal transfer within building materials because it directly affects the quality thermal without forgetting the building material because the comfort. In this work the evolution of conduction convection building sector today is known to be consuming 40% of coupling through this material which represents the raw the world energy [3]. The building sector is experiencing material of the construction of the walls. Also, in this paper significant challenges in relation to the consumption of we have developed a new correlation for Controller the energy, climate change and energy poverty issues [4], average Nusselt number which gives us a prediction before there are even simulations and analyzes of vapor bubble conception walls and with Newton polynomial interpolation bundles to show the potentially erosive impact loads on method and the material optimization for a average  the walls [5]. Also, it has become common in recent years Rayleigh number equal to 1.E-4. to gather data on human attitudes and behavior in

Index Terms—materials, advanced materials, optimized building energy, research examples of this kind of design, convection in buildings research are found in the main journals which deal with technical aspects of energy in buildings, including Building and Environment [6–8], International review of I. INTRODUCTION mechanical engineering [9], Energy Storage [10-11], Construction and Building Materials [12-13], Energy and Cooling is one of the major concerns in building Buildings [14]. In numerical analysis, polynomial tropical houses. This problem is exacerbated by the heat interpolation is the interpolation of a given data set by the gain of the roof and the external walls, which constitutes polynomial of lowest possible degree that passes through more than 70% of the total heat gain [1]. The range of the points of the dataset. A new mathematical method is applications where heat sinks are specified as an integral developed for interpolation from a given set of data part of electronics components and circuitry has increased points in a plane and for fitting a smooth curve to the significantly in recent years. Heat sinks have typically points. This method is devised in such a way that the been used in high powered, electronic devices, such as resultant curve will pass through the given points and will stereo equipment, computing devices and appear smooth and natural. It is based on newton communications equipment, however, newer applications interpolation method. So, firstly we vary the thickness of in the automotive industry and consumer products are the building material of the outer wall four times and reshaping the way we design and manufacture heat sinks. calculate the Nusselt number aims to find a cloud point. In addition to traditional design requirements such as After, we developed new formula that helps us to know thermal performance and structural integrity, the high the exchange ratio and the Nusselt number for each volume, low cost requirements associated with newer thickness e by the Newton polynomial interpolation heat sink markets, necessitates a re-examination of the method for mean Rayleigh number equal 10.E4. Thus, it materials and methods used to manufacture heat sinks [2]. can be applicable for use in building simulators The last thing you wish to encounter are building concerning heat transfer phenomena in building physics.

Manuscript received April 9, 2018; revised August 15, 2018.

II. GEOMETRIC CONFIGURATION D. Energy The idea is illustrated in the configuration shown in T T T  1 (4) Figure 1. It is a square cavity where the outer wall is U V  2T   t X Y   concrete, the form factor equal to A = 1, the length L = 1 cp cp and an author H = 1 (dimensionless case). This cavity is The derived equation of motion Eq. (2) over Y and the differentially heated; in addition the upper and lower equation of motion Eq. (3) by contributing to x, then, walls are adiabatic. The thickness takes four values after subtracting the two equations obtained, we obtain respectively e = L / 0, e = L / 40, e = L / 20, e = L / 10. the equations dimensionless variables in writing Helmotz This is a necessary step to obtain the Nusselt numbers in terms of vorticity and stream function formulation are corresponding to the fluid flow. Inside the cavity. See Fig. as follows: 1.    T U V  Pr2  Ra Pr t X Y X (5)

 T T  2T  2T  L2 U V    (6) t X Y X 2 Y 2  Cp a T

2 2    (7) X 2 Y 2 The stream function and vorticity are related to the Figure 1. Schematic of the studied configuration. velocity components by the following expressions:   V U To simplify the problem, assume that: U  , V   and    (8) • The fluid is Newtonian and incompressible. Y X X Y • The Boussinesq approximation is considered The dimensionless parameters in the equations above III. OBJECTIVES are defined as follows:  x y u (9) We can express our aim for this study in the X  , Y  , U  L L ui following points:     Prediction of the wall design through numerical  P T Ti t V  , P  2 ,T*  ,t * 2 evaluation of the convection in buildings by the  ui  ui Tc Ti L / a Newton polynomial interpolation method for mean Rayleigh number equal 1.E4. For solid (Concrete), we are interested only in the following heat equation:  Modelling Study of the convection – conduction coupling. T 1 2T 2T  (  )  Study of the effect of the distribution of the heat t a X 2 Y 2 (10) inside buildings on the convection V. PROCEDURE OF SIMULATION IV. MATHEMATICAL MODEL First, the numerical calculation was conducted using The fluid is assumed incompressible and obeys the the GAMBIT and FLUENT software. The numerical Boussinesq approximation. In these cases, continuity in procedure used in this work is that of finite volume. It two dimensions and the equations governing the flow and involves the integration of differential equations of energy is given by: mathematical model on finite control volumes for the corresponding algebraic equations. In this context, an A. Continuity analogy was used for the functions which are discretized U V by the finite difference method and integrated in the CFD   0 X Y . (1) code which is based on the finite volume method. The SIMPLE algorithm [15] was chosen for the coupling B. X-momentum speed pressure in the Navier-Stokes equations on a staggered grid. The convective terms in all equations are U U U 1 P (2) evaluated using the schema apwind first order. The U V    2U t X Y  X discretization of the time term is made in a totally implicit scheme. The convergence of the solution is C. Y-momentum considered reached when the maximum relative change of all variables (u, v, w, p, t) between two successive time V V V 1 P T U V     2V  g (3) is not less than 1.e4. With an aim of following well any t X Y  X X variation of the fields thermal and hydrodynamic, we used a uniform grid of 14241 nodes and 14480 Elements

in non-stationary mode. Secondly, we developed a n Pn (x)  d j E j (x) FORTRAN program to control the linear equation of j0 (15) order three. This method of interpolation was used for Where: predicting exchange coefficient of convection to optimize the design of walls in buildings Pn (x)  d 0  d1 (x  x0 )  d 2 (x  x0 )(x  x1 )  .... (16)   .  VI. MATHEMATICAL MODEL FOR NEWTON d n (x  x0 )(x  x1 )...... (x  xn1 ). POLYNOMIAL INTERPOLATION METHOD

Pn (xi)  f (xi) i  0,....,n.. Recall that for distinct points x0, x1, ..., xn, and a real (17) function f defined at these points, there is a unique Where, we shall see how to determine the coefficients polynomial interpolant pn ∈ πn. The idea of Newton (dj) 0 ≤ j ≤ n in the following section entitled the interpolation is to build up pn from the interpolant pn−1 divided differences. Divided differences for Newton for n ≥ 1. Defining the polynomial. Just as two points polynomial interpolation method determine a line, three (non-collinear) points determine a d0 Is called a zero-order divided difference. Where, unique quadratic function, four points that do not lie on a lower degree polynomial curve determine a cubic d0  f [xc ]  f (xc ) : (18) function and, in general, points uniquely determine a polynomial of degree n, presuming that they do not fall onto a polynomial of lower degree. The process of d1 is called 1st -order divided difference. Where, finding such a polynomial is called interpolation and the f [x ]- f [x ] (19) d  f [x , x ]  1 c two most important approaches used are Newton’s 1 1 c x1 - x 0 interpolating formula and Lagrange’s interpolating formula [16]. Each has its own advantages, as we will d2 is called 1nd -order divided difference. Where, discuss. In this article, we show how Newton approaches f [x , x ]- f [x , x ] d  f [x , x , x ]  1 2 c 1 can be introduced and developed at the precalculus level 2 c 1 2 x 2 - x0 in the context of fitting polynomials to data. This brings (20) some of the most powerful and useful tools of numerical By recurrence, we obtain: analysis to the attention of students who are still at the f [x ,..., x ] - f [x ,...., x ] (21) introductory level while simultaneously building on and d  f [x ,...., x ]  1 2 c 1 k c k x - x reinforcing many of the fundamental ideas in algebra and k 0 precalculus mathematics. f[x0, …, xk] is thus called a k th -order divided The polynomials of Newton’s basis, Ej, are defined by: difference Newton’s interpolation polynomial of degree n is obtained via the successive divided differences (see (11) equations:18-21) where: n j1  (12) P (x)  f [x ]  f [x ,..., x ] E (x) (22) E j (x)  (x  x0 ) n c  0 j j  .  j  0 j1   j 1,....,n

With the following convention: VII. RESULTS AND DISCUSSION We wish to find the polynomial interpolating the (13) points. So , In this table we have grouped the Nusselt Moreover numbers calculate after each change of wall thickness facing outwards and that represents the only source of E (x)  (x  x ) 1 0 heat, as an example for e = 0 represents the benchmark E (x)  (x  x )(x  x )  2 0 1 (14) case [15] for values Rayleigh number average Ra = 104, E (x)  (x  x )(x  x )(x  x )  3 0 1 2 same technique was used for the other thicknesses, the E4 (x)  (x  x0 )(x  x1 )(x  x0 )(x  x3 )  results are shown in Table I. .. ..  TABLE I. NUSSELT NUMBER FOR DIFFERENT VALUES Of .. THICKNESS AND FOR Ra=104 E (x)  (x  x )(x  x )...... (x  x )  n 0 1 n1 Ra i Xi=e Yi=Nu (e) :   j  1,....,n Average The set of polynomials (Ej) 0 ≤ j ≤ n (Newton’s Nusselt basis) are a basis of Pn, the space of polynomials of number 104 0 0 2.24 degree at most equal to n. Indeed, they constitute an 104 1 e=L/40 1.59 echelon-degree set of (n + 1) polynomials. 104 2 e=L/20 1.35 Newton’s interpolation polynomial of degree n related 104 3 e=L/10 0.98 to the subdivision {(x0, y0), (x1, y1), …, (xn, yn)} is:

 1 Where, the points of Nu (e) are obtained by numerical P (x)  2.24  26(x  0)  328(x  0)(x  )  ....  n 40  simulation of the FLUENT software, for the Rayleigh  . 4  1 1 number equal 10  2986.7(x  0)(x  ).(x  ). If (0, 2.24),(L/40, 1.59), (L/20, 1.35), (L/10, 0.98), are  40 20 (32) given data points, then the cubic polynomial passing P (x)  2986.66 x3  551.98 x2 37.9227 x  2.24 through these points can be expressed as, 3 (33)

P (x)  d  d (x  x )  d (x  x )(x  x )  .... (23) 3 2 n 0 1 0 2 0 1 e e e  Nu(e)  2986.6  551.98  37.92  2.24  . 3 2  L L L (34) d3 (x  x0 )(x  x1 ).(x  x2 ).

Where, d1 is called 1st -order divided difference is defined by VIII. CONCLUSION In this paper, concrete wall for different thicknesses is d  f [x ]  f (x )  2.24 (24) 0 c c viewed with 0 ≤ e ≤ L/10. We are interested in

d1 is called 1st -order divided difference, is defined by convection conduction coupling. Vary the thickness of the building material of the outer f [x ] - f [x ] 1 c (25) wall four times and calculate the Nusselt number and d1  f [x0 , x1]   25.99 x1 - x0 exchange coefficient of heat transfer aims to find a cloud point respectively for the thicknesses e = 0, L / 40, L / 20 and L / 10. After, we developed a relationship (Eq 34) f [x2 ]- f [x1] that helps us to know and for the estimation of Nusselt f [x1, x2 ]   9.6 (26) x2 - x1 number and exchange ratio for each thickness. Following a break and a lack of information, particularly painful for d2 is called 2nd -order divided difference, is defined by optimization in building materials. This technique and equation minimizes the challenges in building f [x1, x2 ]- f [xc , x1 ] (27) d2  f [xc , x1, x2 ]   327.99 construction based on the prediction of heat transfer from x - x 2 0 the outside to the inside of the houses once the wall thickness has been compensated for its value. f [x ]- f [x ] f [x , x ]  3 2  7.4 (28) 2 3 REFERENCES x3 - x2 [1] K. M. Al-Obaidi, M. Ismail, A. M. A. Rahman, “Passive cooling f [x , x ]- f [x , x ] techniques through reflective and radiative roofs in tropical houses f [x , x , x ]  2 3 1 2  29.333 in Southeast Asia: A literature review,” Frontiers of Architectural 1 2 3 x - x 3 1 (29) Research, vol. 3, no. 3, pp. 283-297, September 2014. [2] J. R. Culham, W. A. Khan, and M. M. Yovanovich, “The influence of material properties and spreading resistance in the d3 is called 3rd -order divided difference, is defined by thermal design of plate fin heat sinks,” in Proc. ASME NHTC’01 (30) 35th National Heat Transfer Conference Anaheim, California, f [x1 , x2 , x3 ]- f [xc , x1 , x2 ] d 3  f [xc , x1 , x2 , x3 ]   2986.6 June 10-12, 2001. x - x 3 0 [3] A. P. O. Obafemi and S. Kurtb, “Environmental impacts of adobe Therefore, the following table can be filled in Table II: as a building material: The north cyprus traditional building case,” Case Studies in Construction Materials, Rev., 2016, vol. 4, pp. 32- 41. TABLE II. PRACTICAL CALCULATION OF THE COEFFICIENTS FOR [4] M. Santamouris, “Innovating to zero the building sector in Europe: OUR CASE minimising the energy consumption, eradication of the energy i xi F[ xi ] F[xi,xi+1 ] F[ xi,xi+1, F[ xi,xi+1, poverty and mitigating the local climate change,” Solar Energy, xi+2 ] xi+2, xi+3 ] vol. 128, pp. 61-94, 2016. [5] D. Ogloblina, S. J. Schmidt, and N. A. Adams, “Simulation and analysis of collapsing vapor-bubble clusters with special emphasis on potentially erosive impact loads at walls,” in Proc. the 0 0 d0=2.24 International conference Experimental Fluid Mechanics 2017, d1= -25,99 November 21.-24., 017,Mikulov, Czech Republic 1 1/40 1.59 d2=327,99 -9.6 d = -2986.6 [6] R. Galvin, “How many interviews are enough? Do qualitative 3 interviews in building energy consumption research produce 2 1/20 1.35 29.333 reliable knowledge? Rev., 2015, vol. 1, pp. 2-12. [7] L. Bellia, F. Fragliasso, “New parameters to evaluate the -7.4 capability of a daylight-linked control system in complementing 3 1/10 0.98 daylight,” Rev., 2017, vol. 123, October 2017, pp. 223-242. [8] N. L. Lyng, L. Gunnarsen, H. V. Andersen, V. K. Sørensen, P. A. So, the cubic polynomial passing through these points Clausen, “Measurement of PCB emissions from building surfaces can be expressed as, using a novel portable emission test cell,” Building and Environment, Rev., 2016, vol. 101, pp. 77-84. [9] E. Benachour, B. Draoui, B. Imine. L. Rahmani. B. Mebarki. K. Pn (x)  d 0  d1 (x  x0 )  d 2 (x  x0 )(x  x1 )  .... (31)  Asnoune, M. Hasnat, “Study of the influence of the mural heating .  on the convection in the building in arid area,” International d (x  x )(x  x ).(x  x ).  3 0 1 2 Review of Mechanical Engineering (IREME), Rev., 2015, vol. 9, no. 1.

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Benachour El Hadj, born on November 20, 1973 in Bechar, Algeria. He received his B.S. degree in mechanical engineering from the Institute of Mechanics at the University of Bechar 1998, the MS Degree in Physics and heat the building of the Institute of Mechanics at the USTOran in 2010. He is currently a PhD candidate at the doctoral School. He is currently working in the convection and thermal comfort in the building.