DYNAMI EAT TRANFER IN WALL: LIMITATION OF EAT FLUX METER DINAMIČKI PRENO TOPLOTE U ZIDOVIMA: OGRANIČENJA MERAČA TOPLOTNOG PROTOKA (TOPLOTNOG FLUKA) 1 I. Naveros a, b,. Ghiaus a a ETIL UMR58, INA-Lyon, 9 rue de la physique, F-69621 Villeurbanne, France b Department of ivil Engineering, University of Granada, ampus de Fuentenueva, E-1871 Granada, pain eat transfer in walls can be modelled by thermal networks, which represent the energy balance in the system formed by the wall and its surroundings. The continuous model may be expressed as an infinite set of differential algebraic equations. The infinite set of equations may be arranged in the state-space representation from which it may be obtained the matrix transfer function. In virtually every application, a reduced model order is used, i.e., a finite set of differential algebraic equations. The matrix transfer function may be used to study the characteristics of the heat transfer flows through the wall measured by a heat flux meter. The transfer function, related to the heat flux through a wall, is not strictly proper and the use of the heat flux meter is limited by the frequency of the input signal. Keywords: eat transfer, tate space, Transfer function, Frequency domain, Reduced model order Prenos toplote u zidovima može se modelovati toplotnim mrežama, koje predstavljaju energetski bilans u sistemu koga formiraju zid i njegovo okruženje. Kontinualni model se može izraziti kao beskonačan niz diferencijalnih algebarskih jednačina. Beskonačan skup jednačina može se poređati u prikazu prostora stanja i na osnovu toga se može dobiti funkcija prenosa matrice. U praktično svakoj primeni, ovaj model se mora diskretizovati u prostoru, to jest, red modela se mora smanjiti odabirom konačnog skupa diferencijalnih algebarskih jednačina. Funkcija prenosa matrice može se iskoristiti za dobijanje kriterijuma za određivanje modela manjeg (nižeg) reda, koji se može koristiti za modelovanje tokova prenosa toplote kroz zid izmerenih pomoću merača toplotnog protoka (toplotnog fluksa). Funkcija prenosa, koja se odnosi na toplotni fluks kroz zid, nije potpuno odgovarajuća i korišćenje merača toplotnog fluksa jeste ograničeno frekvencijom ulaznog signala. Ključne reči: prenos toplote; prostor stanja; funkcija prenosa; domen frekvencije; model manjeg (nižeg) reda I. Introduction The modelling of dynamic heat transfer through a wall needs to use a mathematical expression that appropriately relates the physical variables [1]. The problem of determining the mathematical model is essentially a problem of approximating the thermal behaviour of a wall using a mathematical expression, i.e., a mathematical model [2]. The mathematical model may be built from first principles, keeping as much physical information as possible. In this way, it is possible to use a reduced order model, formed by a finite set of differential algebraic equations (DAEs) [3], which does not need the use of the thermal-electrical analogy as it is usual in the engineering context [4-8]. As the heat energy transfer is a dissipative process, it is expected to obtain a proper transfer function relating the input signal, going into the wall by one of its faces, to the output signal, going out the wall by the opposite face of the wall. From the model, the transfer function may be obtained considering different output signals, as temperatures or heat flux densities [9-1]. Then, it may be possible to study the characteristics of the different transfer functions obtained for different output signal. 1 orresponding author: Tel.: +3347243881; Fax: +33472438811 E-mail address: iban.naveros-mesa@insa-lyon.fr 1
II. Methodology onsidering the principle of energy conservation and the Fourier s law, the general model in continuous time and space is expressed as an infinite set of differential algebraic equations. The model is written in state-space representation and the Laplace s transform, for zero initial conditions, is used to obtain the matrix transfer function. A. eat equation as a set of differential algebraic equations The thermal model may be expressed as an infinite set of differential algebraic equations (DAEs), supposed linear and time invariant [11]: θ T T = A GAθ + A Gb + f (1) where is the diagonal matrix of heat capacities (J/K), is the vector of temperatures in the nodes (K), is the incidence matrix, is the transpose of the incidence matrix, is the diagonal matrix of thermal conductances (W/K), is the vector of temperatures sources on the branches (K), and is the vector of heat rate sources (W) [11]. The model can be represented as an infinite thermal network, supposing fix boundary conditions, as it is shown in Figure 1. Figure 1. Thermal network for a wall modelled by N nodes In Figure 1, T o and T i are the outside and inside air temperatures, I sv is the solar irradiance, R so and R si are the outside and inside conductances, θ so and θ si are the outside and inside surface temperatures, α is the wall absorptivity and is the wall surface. B. tate-space representation of the thermal model The differential algebraic equations (DAEs) can be put in state-space representation. The DAEs, Eq. (1), can be expressed as: θ Kθ + K b + f (2) = b where, by notation and. If in Eq. (1) the differential equations are separated from algebraic equations, Eq. (2) can be expressed in blocks as: θ θ K = K θ θ K + K b2 I b + f f K b 11 12 1 11 21 K 22 I 22 (3) By eliminating the algebraic equations from DAEs represented by Eq. (3), the state-space representation can be obtained: θ = A θ B u (4) + 2
Once Eq. (4) is solved for the state variables, the other outputs can be obtained by using an algebraic equation: θ = θ + D u (5) 1) Differential equations The relation between blocks and the state matrix,, is: 6) and the relation between blocks and the input matrix,, is: 7) The input vector is given by: 2) Algebraic equations From Eq. (3), we can also obtain the set of algebraic equations which complete the state-space model: 8) (9) The relation between blocks and the output matrix, through matrix,, are:, and the relation between blocks and the feed (1) (11). Transfer function representation of thermal model: Laplace s transform The relation between the inputs, u, and the outputs,, can be expressed as a set of transfer functions or a transfer matrix. Applying Laplace transform, for zero initial conditions, Eqs. (4)-(5) can be represented as: (12) (13) where is the complex variable. From Eqs. (12)- (13) it is possible to obtain: (14) The matrix: (15) from Eq. (14) is composed of transfer functions relating each output, of the output vector particular input from the input vector u. For a wall, Figure 1, the transfer function can be written as:, with a 3
T 11 12 13 = [ 1 2 ] = (16) 21 22 23 If the surface temperatures are used as outputs (they are nodes with negligible heat capacity), then: ;. If the heat flux densities are used as outputs, then: ;. The transfer functions can be obtained from state-space matrices. III. Frequency study of not strictly proper transfer functions A. tate-space for a first order model For the case, is the state matrix is the input matrix, and is the input vector The state vector is, by noting, we obtain the differential equation: 17) is the output matrix is the feed through matrix. The two measurement equations for the surface temperatures are: (18) (19) A measurement equation, related to the use of a heat flux meter placed in the inner face of the wall, may be derived [9]: 4
(2) In Eq. (2) the heat flux density is defined as positive when it leaves the room from inside to outside. The matrices and are the same as those obtained in ection 3.1.1. and the new output matrix and the feed through matrix are:, the output matrix, the feed through matrix B. Bode diagrams of the wall model: inside surface temperature and heat flux density The transfer function for the first order model, related to the inside surface temperature and the heat flux density, can be derived from Eq. (15) using the state-space matrices obtained in ection 3.1. The procedure to obtain the transfer functions for higher order models is analogous. In Figure 2, Bode magnitude plots are represented for two outputs: the inside surface temperature, θ si (º), and the inside heat flux density, Q i (W/m 2 ), relating to two inputs: the inside air temperature, T i (º), and the outside air temperature, T o (º), as inputs. Three reduced order model are shown for three number of nodes, N=3, 4 and 1 (Figure 1). It can be seen that the transfer functions for the input T o are strictly proper and the output signal tends to be totally damped to high frequencies. Nonetheless, the transfer functions considering the input T i are proper, but not strictly proper. Therefore, the output signal does not tend to be totally damped to high frequencies; furthermore, the output signal is amplified in the case of the heat flux density. This fact should be taken into account when experimental measurements are conducted. The outputs that have a proper but not strictly proper transfer function may be very sensitive to the high frequencies present in the input signal. This sensitivity explains why heat flux meters located on the outdoor surface of the walls need to be protected when they are exposed to solar irradiance or wind speed. The main problem is not the amplitude of the input signal but the high frequencies present in the input signal. a) Output θ si / Input T o b) Output θ si / Input T i c) Output Q i / Input T o d) Output Qi/ Input T i Figure 2. Bode magnitude plots for a wall 5
The values used to build the transfer functions, shown in Figure 2, using the methodology described above, are: R -1 total = 2W/K, R so = 2W/K, R si = 8W/K and = 2.1e6J/K. IV. onclusions The transfer of the signal from input to output implies a change of amplitude as a function of frequency. ince any heat transfer process is dissipative, the amplitude of the output signal will be lower than the amplitude of the input which generated this variation. It is the case of the variation of the inside surface temperature. Nonetheless, in the case of the inside heat flux density, though the heat flux density is limited by the resistances of the system, it is observed that the high frequencies are amplified. The domain of validity of models using the heat flux density as output is limited by this fact for signal inputs with frequencies higher than the cut-off frequency of the thermal system. The heat flux meter can also amplify the noise present in the signal inputs, when its frequency is high. Furthermore, the use of devices to measure the heat flux density through walls should be avoided in outside surfaces without a protection for inputs as solar irradiance or wind speed, which signals have high frequency components. AKNOWLEDGEMENT The research was financially supported by the LABEX IMU (ANR-1-LABX-88) of Université de Lyon, within the program "Investissements d'avenir" (ANR-11-IDEX-7) operated by the French National Research Agency (ANR) through the project IDeffE: Estimation of energy performance through experimental identification. BIBLIOGRAPY [1]. arslaw and J. Jaeger, onduction of eat in olids, larendo Press, 1986. [2] J. rassidis and J. Junkins, Optimal Estimation of Dynamic ystems, econd Edition, Taylor & Francis, 211. [3] G. trang, omputational cience and Engineering, Wellesley-ambridge Press, 27. [4] G. Fraisse,. Viardot, O. Lafabrie and G. Achard, "Development of a simplified and accurate building model based on electrical analogy," Energy and Buildings, vol. 34, no. 1, pp. 117-131, 22. [5] A. Ramallo-González, M. Eames and D. oley, "Lumped parameter models for building thermal modelling: An analytic approach to simplifying complex multi-layered constructions," Energy and Buildings, vol. 6, pp. 174-184, 213. [6] A. Rabl, "Parameter Estimation in Buildings: Methods for Dynamic Analysis of Measured Energy Use," Journal of olar Energy Engineering, vol. 11, no. 1, pp. 52-66, 1988. [7] A. Foucquier,. Robert, F. uard, L. téphan and A. Jay, "tate of the art in building modelling and energy performances prediction: A review," Renewable and ustainable Energy Reviews, vol. 23, pp. 272-288, 213. [8] I. azyuk,. Ghiaus and D. Penhouet, "Optimal temperature control of intermittently heated buildings using Model Predictive ontrol: Part I - Building modeling," Building and Environment, vol. 51, pp. 379-387, 212. [9] I. Naveros, P. Bacher, D. Ruiz, M. Jiménez and. Madsen, "etting up and validating a complex model for a simple homogeneous wall," Energy and Buildings, vol. 7, pp. 33-317, 214. [1] I. Naveros, M. J. Jiménez and M. R. eras, "Analysis of capabilities and limitations of the regression method based in averages, applied to the estimation of the U value of building component tested in Mediterranean weather," Energy and Buildings, vol. 55, pp. 854-872, 212. [11]. Ghiaus, "ausality issue in the heat balance method for calculating the design heating and cooling load," Energy, vol. 5, pp. 292-31, 213. 6