PERTURBATION NUMBERS EVALUATION OF THE INFLUENCE OF THE UNCERTAINTY OF PARAMETERS ON UNIDIRECTIONAL TEMPERATURE DISTRIBUTION IN A PLATE

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1 A R C H I T E C T U R E C I V I L E N G I N E E R I N G E N V I R O N M E N T The Silesian University of Technology No. 1/ 2016 PERTURBATION NUMBERS EVALUATION OF THE INFLUENCE OF THE UNCERTAINTY OF PARAMETERS ON UNIDIRECTIONAL TEMPERATURE DISTRIBUTION IN A PLATE Agnieszka WINKLER-SKALNA* * PhD Eng.; Building Research Institute, Korfantego 191, Katowice, Poland address: a.winkler-skalna@itb.pl Received: ; Revised: ; Accepted: A b s t r a c t This paper describes the issue of unidirectional temperature distribution in a plate with the assumption of perturbations in the problem parameters. The new algebraic perturbation system is proposed as a possible alternative to the traditional I-order perturbation methods. A calculation example is presented to compare the classical solution of a problem with that using perturbation numbers. The investigation of a possibility to apply the new algebraic system using a simple example represents an introduction to the further development of the proposed methodology which can be used to solve more complicated problems, taking into account among other properties, inhomogeneity of material, its porosity or the occurrence of thermal bridges. S t r e s z c z e n i e W artykule opisano zagadnienie dotyczące jednokierunkowego rozkładu temperatury w płycie przy założeniu zaburzeń parametrów zadania. Zaproponowano wykorzystanie nowej algebry perturbacyjnej, która może stanowić alternatywę dla tradycyjnych metod perturbacyjnych I rzędu. Przedstawiony został przykład obliczeniowy, w którym porównane zostało rozwiązanie zadania metodą klasyczną z rozwiązaniem wykorzystującym liczby perturbacyjne. Przetestowanie możliwości zastosowania nowej algebry na prostym przykładzie stanowi wstęp do dalszego rozwijania proponowanej metodologii, która będzie mogła być stosowana w rozwiązywaniu bardziej skomplikowanych zadań, uwzględniających m.in. niejednorodność materiału, jego porowatość lub występowanie mostków termicznych. K e y w o r d s : Heat transfer; Perturbation numbers; Perturbed parameters. 1. INTRODUCTION The continuous progress in the field of numerical calculations which has been observed through the decades gives better opportunity to use mathematical models in many areas of science today, including those connected with the heat transfer. Analyses concerning the calculation of temperature fields are most frequently carried out by means of the finite element method (FEM), the boundary element method (BEM), the finite difference method and elementary energy balances method [1, 3]. The two latter methods are based on discretization of the considered space and preparation of energy balances for the elements; in the FEM and BEM the so called shape function used to approximate the temperature curve is determined after dividing the space and identifying the nodes [3]. The possibility to solve problems associated with heat conduction in structures of complicated geometry is an advantage of methods based on the division of a space, a surface or a boundary into small elements [10]. Neural networks, fractal theory or perturbation methods are more and more frequently used in various engineering fields. The artificial neural networks were also used to solve the inverse problem of 1/2016 A R C H I T E C T U R E C I V I L E N G I N E E R I N G E N V I R O N M E N T 95

2 A. W i n k l e r - S k a l n a heat conduction, with the assumption of functional relationship between thermal conductivity and temperature, as well as to predict the effective thermal conductivity in porous materials [2, 5]. The fractal theory is mainly applied in evaluation of the effective thermal conductivity which is of particular importance in the case of porous and fibrous materials [4, 13, 14]. The basis of all models and analyses is the presentation of physical phenomena using a mathematical model. Equations describing the phenomena connected with heat conduction are based on Fourier s law which assumes an infinitely high heat transfer speed, which in turns means that any disturbances at the boundary of the structure cause an immediate change of the temperature within the whole volume of the structure. Thus, the knowledge of initial and boundary conditions, respectively, is necessary to solve the considered heat conduction problem. The theory of perturbations is now commonly applied in the theoretical mechanics. Since the determination of the exact solution of a given problem is often difficult, or its analytical solution is unknown, it is easier to use a known solution of a similar (simple solvable) problem and then to determine an approximate solution to the problem in question using the perturbation theory. The perturbation theory is also applied in the cases where the basic physical conditions are disturbed. In such situation, it is important to determine the influence of perturbations in the problem parameters on its solution [6-9, 12]. The perturbation methods are also applied to obtain an approximate analytical solution to the problems connected with unidirectional heat conduction through porous materials. Such materials have low thermal conductivity, which can play the role of a small parameter [1, 11]. Thus, the application of approximate analytical methods, which include the perturbation techniques, can be useful in solving the heat transport problems. According to the basic assumptions of the traditional theory of perturbations the model should be transformed to a dimensionless form so that the parameters and variables relating to the system behaviour can be established. As a next step, a parameter is selected (called also a small parameter or a perturbation value). In comparison with the other parameters it is small and is denoted by ε [1]. An attempt is made in this study to apply the alternative perturbation method using the perturbed numbers as defined by Skrzypczyk. The method has already been successfully used to solve some mechanic and acoustic problems [6-9, 12]. In many thermal analyses coefficients of a considered problem are not known exactly, but with only some approximation. The new perturbation method is the simplest way to introduce uncertainty into mathematical analysis by introducing disturbances to the classical solution of the problem. 2. DEFINITION OF A PERTURBATION NUMBER The perturbation number is defined as an ordered pair of real numbers (x,y) R 2. The set of perturbation numbers is denoted as R ε. If z 1, z 2 R ε denote any perturbation numbers and z 1 =(x 1,y 1 ), z 2 =(x 2,y 2 ), x i,y i R,i=1,2 we can say that two perturbation numbers z 1 z 2 are equal if x 1 =x 2 and y 1 = y 2. The addition (+ ε ) and multiplication ( ε ) are introduced to the set R ε, and neutral elements of addition 0 ε :=(0,0) and multiplication 1 ε :=(1,0) are determined. In such defined set R ε, which is called a field of perturbation numbers, the commutativity and associativity of addition and multiplication, as well as distributive property are maintained [6-9]. The above defined field R ε does not contain the field of real numbers R. It can be shown, however, that the real numbers can be considered as some elements of the field R ε with algebraic operations and neutral elements of addition and multiplication. The properties of the perturbation numbers and notions of the perturbation analysis such as convergence, continuity, etc., are also described in the studies listed below [6-9]. 3. NUMERICAL EXAMPLE Unidirectional heat flow through an infinite singlelayer plate of the thickness g = 0.2 m, made of material with thermal conductivity λ = 0.15 W/(mK), is considered. The similar example was discussed in [10]. The air temperatures of the heat inflow and outflow are: T 1 =20 C and T 2 =-20 C. Coefficients of heat transfer on the heated and cooled surface are α 1 =7.69 W/(m 2 K) and α 2 =25.00 W/(m 2 K), respectively. The scheme of the assumptions is shown in Fig. 1. The determination of unidirectional temperature distribution T(x) in a plate at the distance of 0.00, 0.02,, 0.18, 0.20 m from the heated plate surface constitutes the solution to the problem. The results, presented in Table 1, can be obtained using 96 A R C H I T E C T U R E C I V I L E N G I N E E R I N G E N V I R O N M E N T 1/2016

3 P E R T U R B AT I O N N U M B E R S E VA L U AT I O N O F T H E I N F L U E N C E O F T H E U N C E R TA I N T Y O F PA R A M E T E R S... the classical method based on Fourier s law in the following form:, (1) where q is the heat flux density in W/m 2. The boundary conditions are:, (2) Table 1. Solution to the problem without perturbations x, m T(x), C x, m T(x), C ec C I V I L E N G I N E E R I N G After the transformation and integration the equation (1) we get: The integration constant C can be obtained from the following condition: Introducing equation (4) into equation (3) the solution of the problem takes the form: where:. (3). (4), (5) and is the heat transfer coefficient [10]. It is then assumed that the values g and λ are burdened with uncertainties and are therefore written in the form of perturbation numbers g ε = g 0 +εg 1 and λ ε =λ 0 +ελ 1, where the symbols denoted by 0 index are equal to the previously assumed values, called main values, whereas g 1 = m and λ 1 = W/(mK) are disturbed values. The source of uncertainty can be, for example, local inhomogeneity of the plate, discontinuities of materials, small variations in the thickness, etc. Introduction of disturbance affects other values. The heat transfer coefficient through a plate and the heat flux density will also be perturbation values written as u ε = u 0 +εu 1 and q ε = q 0 +εq 1. Using the algebra of perturbation numbers we obtain u ε = ε W/(m 2 K) and q ε = ε W/m 2. Finally, the perturbed temperature distribution was determined from the equation: (6) The calculations were performed by means of the original program written in Fortran. The results are presented in Table 2; the graphs of temperature distributions with and without disturbances are shown in Fig. 2. Figure 1. Geometry of the problem Table 2. Solution to the problem with disturbances of input parameters x, m T 0 (x) main part, C T 1 (x) perturbation part, C /2016 A R C H I T E C T U R E C I V I L E N G I N E E R I N G E N V I R O N M E N T 97

4 A. W i n k l e r - S k a l n a Figure 2. Distribution of temperature in a plate taking into account the perturbations Comparing the results obtained from the calculations performed using the real numbers with those for perturbed values it has been found that the main parts of the perturbation solution are equal to the solution for real numbers. Thus, using the perturbation numbers we obtain the results which allow the evaluation of the influence of uncertainty on the final solution of the problem. 4. CONCLUSIONS For the issues that require determination of influence of perturbations or uncertainty of the input parameters of the problem on its solution, the application of the perturbation numbers provides the possibility to obtain results in a relatively simple way. Undoubted advantages of the proposed method, such as avoidance of complicated analytical calculations involving expanding approximated values of solutions in infinite series or easiness to adopt known algorithms to the new algebraic system provide great opportunities. Particularly, they allow us to obtain solutions equivalent to the I-order perturbation methods of the classical perturbation theory. 98 A R C H I T E C T U R E C I V I L E N G I N E E R I N G E N V I R O N M E N T 1/2016

5 P E R T U R B AT I O N N U M B E R S E VA L U AT I O N O F T H E I N F L U E N C E O F T H E U N C E R TA I N T Y O F PA R A M E T E R S... REFERENCES [1] Aziz A., Na T. Y.; Perturbation Methods in Heat Transfer. Hemisphere Publishing Corporation, USA; 1984 [2] Czél B., Woodbury K. A., Gróf G.; Simultaneous estimation of temperature-dependent volumetric heat capacity and thermal conductivity functions via neural networks. International Journal of Heat and Mass Transfer, Vol.68, 2014, p.1-13 [3] Modelowanie numeryczne pól temperatury (Numerical modelling of thermal fields). Ed.: Szargut J., Wydawnictwo Naukowo-Techniczne, Warszawa, 1992 (in Polish) [4] Shi M., Li X., Chen Y.; Determination of effective thermal conductivity for polyurethane foam by use of fractal method. Science in China Series E: Technological Sciences, Vol.49, No.4, 2006; p [5] Singh R., Bhoopal R.S., Kumar S.; Prediction of effective thermal conductivity of moist porous materials using artificial neural network approach. Building and Environment, Vol.46, 2011; p [6] Skrzypczyk J.; Perturbation methods New arithmetic. Zeszyty Naukowe Politechniki Śląskiej Budownictwo, Gliwice, 2003; p [7] Skrzypczyk J.; Metody perturbacyjne Nowa arytmetyka (Perturbation methods New arithmetic). Zeszyty Naukowe Katedry Mechaniki Stosowanej Politechniki Śląskiej, Vol.23, Gliwice, 2004; p (in Polish) [8] Skrzypczyk J.; Winkler-Skalna A.; Sound Wave Propagation Problems New Perturbation Methodology. Archives of Acoustic, Vol.31, No.4 Supl., 2006; p [9] Skrzypczyk J.; Metody perturbacyjne I, Nowa metodologia algebraiczna, Zastosowania w mechanice i akustyce (Perturbation methods I, New algebraic methodology. Applications in mechanics and acoustics). Wydawnictwo Politechniki Śląskiej, Gliwice, 2010 (in Polish) [10] Taler J., Duda P.; Rozwiązywanie prostych i odwrotnych zagadnień przewodzenia ciepła (Solution of direct and inverse problems of heat conduction). Wydawnictwa Naukowo-Techniczne, Warsaw, 2003 (in Polish) [11] Villatoro F.R., Perez J., Santander J.L.G., Borovsky M.A., Ratisf Yu.L., Izzheurov E.A., Fernandez de Cordoba P.; Perturbation analysis of the heat transfer in porous media with small thermal conductivity. Journal of Mathematical Analysis and Applications, Vol.374, 2011; p [12] Winkler-Skalna A.; Propagation of sound waves in uncertain environment new interval perturbation methodology. Proc. 55 th Open Seminar on Acoustics, Wrocław-Piechowice, 2008; p [13] Xia D., Guo S., Ren L.; Fractal structure reconstruction for alumina silicate refractory fiber and simulation of the thermal conductivity. Journal of Thermal Science, Vol.19, No.1, 2010; p [14] Zhu F., Cui S., Gu B.; Fractal analysis for effective thermal conductivity of random fibrous porous materials. Physics Letters A, Vol.374, 2010, p c e C I V I L E N G I N E E R I N G 1/2016 A R C H I T E C T U R E C I V I L E N G I N E E R I N G E N V I R O N M E N T 99

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MULTI-SCALE PERTURBATION METHODS IN MECHANICS

006/3 4 PAGES 13 17 RECEIVED 18 9 006 ACCEPTED 7 11 006 J SKRZYPCZYK MULTI-SCALE PERTURBATION METHODS IN MECHANICS Dr Hab Eng J Skrzypczyk, Prof ul Krzywoustego 7, 44-100 Gliwice, Pol, Phone:+48 604 540