Acta Polytechnica doi:10.14311/AP.2019.59.0211 Acta Polytechnica 59(3):211–223, 2019 © Czech Technical University in Prague, 2019 available online at http://ojs.cvut.cz/ojs/index.php/ap GENERAL MODEL OF RADIATIVE AND CONVECTIVE HEAT TRANSFER IN BUILDINGS: PART I: ALGEBRAIC MODEL OF RADIATIVE HEAT TRANSFER Tomáš Ficker Brno University of Technology, Faculty of Civil Engineering, Department of Physics, Veveří 95, 602 00 Brno, Czech Republic correspondence: ficker.t@fce.vutbr.cz Abstract. Radiative heat transfer is the most effective mechanism of energy transport inside buildings. One of the methods capable of computing the radiative heat transport is based on the system of algebraic equations. The algebraic method has been initially developed by mechanical engineers for a wide range of thermal engineering problems. The first part of the present serial paper describes the basic features of the algebraic model and illustrates its applicability in the field of building physics. The computations of radiative heat transfer both in building enclosures and also in open building envelopes are discussed and their differences explained. The present paper serves as a preparation stage for the development of a more general model evaluating heat losses of buildings. The general model comprises both the radiative and convective heat transfers and is presented in the second part of this serial contribution. Keywords: Radiative heat transfer, view factor, radiosity, room envelope, radiative heat in interiors, heat loss. 1. Introduction Heat radiation represents a dominant transfer mechanism of heat energy inside buildings. Estimating this transfer is, therefore, a useful indicator of effectiveness of heating systems, especially those based on radiant panels. So far, several algebraic methods for determining radiative heat transfer have been published. Probably, the pioneering work in the field of algebraic models can be ascribed to Hottel [1], who introduced the concept of the total-view factor Fij. Soon afterwards, Hotell and Sarofim [2] improved the method by introducing the total-exchange area SiSj. Their model is based on the so-called radiosity heat flux and, thus, it is often referred to as a radiosity method. Besides the radiosity method, there are some other modifications of the algebraic approach. Gebhart [3, 4] introduced a method utilizing the so-called absorption factor. Among other methods dealing with radiative heat transfer in enclosures, it is possible to mention the methods by Sparrow [5], Sparrow and Cess [6], Oppenheim [7], Ecker and Drake [8], Love [9], Wiebelt [10], and Siegel and Howell [11]. A comprehensive analysis of the methods that were introduced by Hottel and Sarofim [1, 2] and Gebhart [3] was published by Clark and Korybalski [12]. These two authors showed that the methods of Hottel, Sarofim and Gebhart, although written in different forms, were mathematically equivalent. Liesen and Pedersen [13] published an overview of many radiant exchange models that range from the exact models using uniform radiosity networks and exact view factors to mean radiant temperature and area-weighted view factors. These authors compared the radiant exchange models to each other for a simple zone with varying aspect ratios. Since that time, many various applications of algebraic methods have been published in the field of mechanical engineering, but in the field of building thermal technology, the algebraic methods has not actually been used. The reason may be related to the fact that a complete solution of the combined radiative-convective transport leads to the system of transcendent non-linear equations whose solutions seem to be problematic in some cases. Recent studies of radiative heat transport in the inner spaces of buildings (i.e., in enclosures) avoid algebraic methods and prefer differential transport equations [14–18]. The main interest is focused on floor heating systems [14–17], but radiant panel systems situated on walls or ceilings are also investigated [18–21]. Coefficients of heat transfers at interiors or exteriors along with heat losses are often measured or calculated as well [20, 22–25]. As has been mentioned above, the radiative heat transfer is the most effective mechanism of energy transport as compared to the free convection or conduction. This fact has been verified many times both theoretically and experimentally. For example, Rahimi and Sabernaeemi [21] have experimentally investigated these transports between the radiant ceiling surface and other internal surfaces of a room and have found that more than 90 % of the heat is transferred by radiation. 211 http://dx.doi.org/10.14311/AP.2019.59.0211 http://ojs.cvut.cz/ojs/index.php/ap Tomáš Ficker Acta Polytechnica Figure 1. Heat exchange by radiation between two small surface elements dS1 and dS2. As is well-known, the radiative heat is formed by electromagnetic waves containing many different wavelengths. When the room under investigation contains glazed windows, it is necessary to consider the capability of glass to transmit these waves. Glass is transparent for wavelengths between 0 and 4µm but above this range, it behaves as an opaque matter. The wavelengths of solar radiation near the surface of the Earth assume values less than 2.5µm and thus they may pass through the windows into interiors. Temperatures of inner furnishings usually do not exceed 30 °C, i.e. 300 K, and, according to Wien’s displacement law, the wavelength of maximum radiation of such furnishings is λmax = 2898 µmK 300 K = 9.66 µm. This value is sufficiently above the critical transmittance range, which makes the glazed window an opaque barrier for inner radiation similarly as a wall [14]. However, like with walls, heat is transferred by convection and radiation in the vicinity of the internal and external sides of windows, but inside the glass panes, solely by conduction. When the windows are double or triple glazed, the cavities between glass panes filled with an inert gas represent a narrow space where the heat is transferred by convection, radiation and often the conduction may participate as well. Since the heat radiation within opaque enclosures represents a dominant heat transfer between inner surfaces, it is desirable to have a reliable computational tool for its quantification. The algebraic method is certainly such a tool. Its theoretical formalism is presented in the following section. 2. Basics of radiosity method The first step in performing the radiosity method consists in forming the matrix of view factors. These factors facilitate redistributing the radiative energy among the surfaces of the enclosure under study. 2.1. View factors A view factor F reduces the total energy irradiated by a surface to that part of energy that reaches the neighboring surface (Figure 1). View factors ares dimensionless, assume only positive values and are restricted to the interval F ∈ 〈0, 1〉. They are defined as follows [26–28]: Fij = 1 Si ∫ (Si) [∫ (Sj ) cos ϕi cos ϕj πR2 dSj ] dSi (1) Fji = 1 Sj ∫ (Sj ) [∫ (Si) cos ϕj cos ϕi πR2 dSi ] dSj (2) From Equations (1) and (2), the symmetry relation between Fij and Fji can be immediately deduced: Si ·Fij = Sj ·Fji (3) This symmetry rule holds quite generally regardless of the types of surfaces and their geometrical positions. There is another important property of view factors. If a radiant surface is incapable of irradiating itself, its view factor is zero: Fii = 0 (4) For example, the perfect planes or the external surfaces of spheres belong to this class of surfaces. It should be highlighted that zero rule (4) does not hold generally, but it is restricted to special surfaces. 212 vol. 59 no. 3/2019 Model of radiative and convective heat transfer in buildings: Part I Figure 2. Scheme of surface energy exchange. The third property concerns the closed envelopes that consist of different surfaces numbered as 1, 2, 3, . . . ,n: n∑ j=1 Fij = 1 (5) Relation (5) may be called the summation rule. It should be stressed that its validity is restricted solely to the closed envelopes. These may be, for example, inner spaces of rooms. The view factors can be ordered into a matrix whose elements have to fulfil the basic rules (3) - (5):  F11 F12 . . . F1n F21 F22 . . . F2n ... ... ... ... Fn1 Fn2 . . . Fnn   (6) 2.2. Algebraic equations of radiative heat transfer The radiosity method relies on diffuse and grey surfaces. These surfaces emit or reflect radiation that is directional and wavelength independent. So the term ‘grey’ has little to do with real colours of surfaces. The two assumptions (diffuse and grey) are reasonable for most engineering applications. The emissivities ε of grey diffuse Lambert’s surfaces assume the values lying between zero and one. The radiative heat transfer between such surfaces is based on two laws, namely the Stefan-Boltzmann law and the Kirchhoff law [26–28]. The grey diffuse solid surfaces fulfil two conditions: i) Transmittance is zero τ = 0. ii) Reflectance ρ and absorbance α (or emissivity ε) are independent of wavelengths and their sum equals one, i.e. ρ + ε = 1. In Figure 2, there is a scheme of energy exchange occurring on the surface of a radiant body. The symbol H represents the total radiative heat flux (W/m2) coming from all neighbouring surfaces whereas the symbol W represents radiosity, which is the total heat flux emitted from the surface (W/m2). Radiosity is the sum of the Stefan-Boltzmann radiation (εEb = εσT 4, σ = 5.67 · 10−8 W/(m2K4)) and the reflected heat flux (ρH), i.e. W = εEb + ρH (7) H = W −εEb ρ . (8) The resulted (net) radiative heat flux q related to the investigated surface is determined as the difference between the emitted (W) and coming (H) fluxes: q = W −H = εEb −εH = { > 0 =⇒ surface emits energy < 0 =⇒ surface absorbs energy (9) By replacing H in Eq. (9) by the fraction from Eq. (8), we obtain: q = W − W −εEb ρ = (ρ− 1)W + εEb ρ = −εW + εEb ρ = ε ρ (Eb −W) (10) Thus, the density heat flux qi of the i-th surface is given as follows qi = εi ρi (Ebi −Wi) (W/m2) (11) From this equation, it is clear that the density heat flux qi can be positive or negative. The positive value indicates that the surface emits the heat energy whereas negative value means that the surface absorbs the energy. To determine the density heat flux qi from Eq. (11), it is necessary to know radiosity Wi. The equations for radiosities are specified in the following paragraphs. 213 Tomáš Ficker Acta Polytechnica 2.3. Radiosity The i-th surface of area Si is supplied by the energy Hi coming from all the n surfaces (including the i-th surface itself if it is curved): SiHi = n∑ j=1 SjFjiWj (W) (12) Taking into account the rule of symmetry Sj ·Fji = Si ·Fij, the following equations may be obtained SiHi = n∑ j=1 SiFijWj (W) (13) Hi = n∑ j=1 FijWj (W/m2) (14) By considering Equations (7) and (14), the system of n linear algebraic equations emerges: Wi = εiEbi + ρi n∑ j=1 FijWj i,j = 1, 2, 3, . . . ,n (W/m2) (15) The solution of this system offers n values of radiosities {Wi} n i=1 by means of which the total radiative heat flux qi for each of the n surfaces may be calculated according to Eq. (11). If the heat flux qi is determined, it is easy to calculate the net heat flow φi corresponding to i-th surface: φi = Siqi (W) (16) If this quantity is positive, the surface emits energy but negative quantity determines the absorption of the energy. Equations (11), (15), and (16) are the basic algebraic relations that specify a radiative heat transfer between grey surfaces not only within the closed systems of surfaces, i.e. enclosures, but also in the open systems of surfaces. 2.4. Heat flux of absolutely black surface In previous section 2.3, the basic relation for heat flux (11) associated with grey surfaces has been introduced. By applying Eq. (11) to absolutely black surfaces (τ = 0, ρ = 0, ε = 1), it leads to an uncertain expression 0/0 and thus it is necessary to derive another expression that does not suffer from such drawback. From Eqs. (9) and (14), it follows qi = Wi − n∑ j=1 FijWj (17) which is a convenient alternative for calculating heat fluxes associated with both the black and grey surfaces. Relation (17) does not lead to any uncertain expression but in comparison with (11), it requires more numerical work and, for this reason, expression (11) is a preferable choice when dealing with grey surfaces. 2.5. Energy exchange between couples of surfaces Heat exchange between two surfaces i and j, i.e. φi↔j, will be calculated as a difference between those portions of heat that are absorbed by these surfaces φi↔j = φi←j −φi→j (18) The absorbed heat φi←j will be calculated under the condition requiring that the surface j may emit energy while other surfaces emit nothing, i.e. they are ascribed by zero temperatures. The absorbed heat φi←j may be determined by means of relations (11) and (16) in which Ebj = 0: φi←j = Siεi ρi (−W (j)i ) i 6= j (19) where W (j)i is a radiosity when the surface j irradiates energy while others emit nothing because they are ascribed by zero temperatures. 214 vol. 59 no. 3/2019 Model of radiative and convective heat transfer in buildings: Part I Similarly, φi→j will be calculated as the difference between absorbed portions of heat when the surface i irradiates energy while others do not emit any energy, i.e. they have zero temperatures: φi→j = Sjεj ρj (−W (i)j ) i 6= j (20) The expressions for φi←i (= φi→i) assume a bit different forms. Their derivation can be started from a general definition (9): q = W −H q = εEb + ρH −H q = εEb + (ρ− 1)H q = εEb −εH −εH = q −εEb (21) The term −εEb is actually the energy (W/m2) absorbed by the surface under the investigation. To obtain power in Watts, this term has to be multiplied by the area S of the surface: −SεH = Sq −SεE −SεH = φ−SεE −SεH = Sε ρ (E −W) −SεE −SεH = Sε ρ [E −W −ρE] −SεH = Sε ρ [(1 −ρ)E −W ] −SεH = Sε ρ [εE −W ] (22) As seen from Eqs. (22), the expressions φi←i (= φi→i) assume the following form φi←i = φi→i = −SiεiHi = Siεi ρi (εiEi −Wi) (23) Combining Eqs. (19), (20) and (23) more general expressions emerge: φi←j = Siεi ρi (εiEiδij −W (j) i ) =   Siεi ρi (−W (j)i ) for i 6= j Siεi ρi (εiEbi −W (j) i ) for i = j (24) φi→j = Sjεj ρj (εjEjδij −W (i) j ) =   Sjεj ρj (−W (i)j ) for i 6= j Sjεj ρj (εjEbj −W (i) j ) for i = j (25) where δij is the Kronecker symbol δij = { 1 for i = j 0 for i 6= j . A general expression for the energy exchange between couples of surfaces may be obtained by using Eqs. (18) and (24)/(25) φi↔j = φi←j −φi→j =   0 for i = j Sjεj ρj W (i) j − Siεi ρi W (j) i for i 6= j (W) (26) Hottel and Sarofim [2] published a different expression for φi↔j: Q̇j↔i = Siεi ρi ( jWi Ej − δijεj ) (Ej −Ei) = Sjεj ρj ( iWj Ei −δijεi ) (Ej −Ei) (27) In fact, Eqs. (26) and (27) are only two equivalent alternatives since they both yield the same numerical results. However, expression (26) seems to be more instructive. 215 Tomáš Ficker Acta Polytechnica Figure 3. A simple room with three surfaces: heated floor no. 1, side walls no. 2, and ceiling no. 3. Parameters Surface 1 Surface 2 Surface 3 ε 0.95 0.80 0.75 ρ 0.05 0.20 0.25 T (K) 300 295 290 S (m2) 72 108 72 εEij (W/m2) 436.3065 343.5272 300.7712 Table 1. Input data for the simple room shown in Fig. 3. As seen from Eq. (26), the expression φi↔j represents a quasi-symmetric matrix (φi↔j = −φj↔j ). Each row of the matrix describes the energy exchange between a particular surface and the remaining neighbouring surfaces. For example, the i-th row of the matrix contains energies that are exchanged between the i-th surface and all the remaining ones. The matrix φi↔j of energy exchange assumes the following form  φ1↔1 φ1↔2 . . . φ1↔n φ2↔1 φ2↔2 . . . φ2↔n ... ... ... ... φn↔1 φn↔2 . . . φn↔n   (28) 3. Application of radiosity method In this section, the algebraic radiosity method is applied to three particular cases. The first application presented in sub-section 3.1 illustrates the functionality of the method within an enclosure, which is represented by a simple room. The second application described in sub-sections 3.2 is focused on an open system created paradoxically as a closed room envelope in which some parts of the envelope completely transmit heat radiation (quasi-enclosure). The third application in sub-section 3.3 concerns a real open system in which some constructional parts of the room envelope are completely missing. In open systems, the radiative heat energy is not conserved as in the closed systems but a large portion of the heat disappears in the open space. The radiosity method is capable of determining the escaped energy independently whether the system of surfaces is closed with some transmitting parts or some completely missing parts. 3.1. Application to enclosure To illustrate the functionality of the radiosity method, a simple room with heated floor has been chosen as shown in Fig. 3. The matrix of view factors Fij determined by means of the graphs published in the technical literature [26–28] and the rules specified by Eqs. (3), (4), and (5) reads  0 0.5 0.50.3 0.3 0.3 0.5 0.5 0   (29) 216 vol. 59 no. 3/2019 Model of radiative and convective heat transfer in buildings: Part I Radiosities (Eq. (15)): W1 = 436.3065 + 0.05 (0.5W2 + 0.5W3) W2 = 343.5727 + 0.2(0.3W1 + 0.3W2 + 0.3W3) W3 = 300.7712 + 0.25(0.5W1 + 0.5W2) (30) W1 = 457.352710, W2 = 430.140549, W3 = 411.707857 W/m2 (31) Heat flows (Eqs. (11) and (16)): φ1 = S1q1 = +2 622.85272 W φ2 = S2q2 = −316.029168 W φ3 = S3q3 = −2 306.791512 W (32) 3∑ i=1 φi = 0.032040 ≈ 0 W (33) The power φ1 = +2 622.85272 W represents the energy flow from the heated floor. The floor has the highest temperature and thus emits the heat power from its surface into the room. The walls and the ceiling have lower temperatures as compared to the floor and from this reason, they assume the heat powers φ2 = −316.029168 W and φ3 = −2 306.791512 W. These energies are absorbed into the volumes of the walls and the ceiling. In enclosures, the emitted energy is redistributed among the cooler surfaces and thus no portion of radiative energy can be lost ( ∑n i=1 φi = 0). In our case, the sum of heat flows shows a tiny deviation from zero ( ∑n i=1 φi = 0.032), but this is caused by rounding errors during the computations and inaccurately reading the values of view factors from their published graphs. The zero sum of radiative heat energies is the general property of closed systems and follows from the first and second laws of thermodynamics. However, since the algebraic radiosity model is not a ’product of nature’ but a product of human creativity, it is desirable to verify whether this artificial model satisfies the laws of thermodynamics. For this reason, the property ∑ (i) φi = 0 (which may be termed as compensation theorem) will be verified mathematically: The proof is based on three properties related to view factors and specified by definition (1), symmetry property (3), and summation property (5): n∑ i=1 φi = n∑ i=1 Si  Wi − n∑ j=1 FijWj   = = n∑ i=1 SiWi − n∑ i=1 Si   n∑ j=1 FijWj   = = n∑ i=1 SiWi − n∑ i=1 n∑ j=1 SiFijWj = = n∑ i=1 SiWi − n∑ i=1 n∑ j=1 SjFjiWj = = n∑ i=1 SiWi − n∑ j=1 n∑ i=1 SjFjiWj = = n∑ j=1 SjWj − n∑ j=1 SjWj · ( n∑ i=1 Fji ) = = n∑ j=1 [ (SjWj ) − (SjWj ) ( n∑ i=1 Fji )] = = n∑ j=1 (SjWj ) · [ 1 − n∑ i=1 Fji ] = { = 0 (for closed envelopes only) 6= 0 (for opened envelopes only) (34) The expression [1 − ∑n i=1 Fji] in Eq. (34) is zero only if ∑n i=1 Fji = 1, that holds solely for enclosures (closed envelopes), as follows from summation rule (5). However, in open envelopes, summation rule (5) does not hold, 217 Tomáš Ficker Acta Polytechnica Surface 1 radiates others not - W (1)i (W/m 2) W (1)1 = 438.677644 W (1) 2 = 35.565381 W (1) 3 = 59.280378 Surface 2 radiates others not - W (2)i (W/m 2) W (2)1 = 10.501990 W (2) 2 = 372.237215 W (2) 3 = 47.842401 Surface 3 radiates others not - W (3)i (W/m 2) W (3)1 = 8.173076 W (3) 2 = 22.337953 W (3) 3 = 304.585078 Table 2. Special radiosities W (j)i for computing the matrix φi↔j. i.e. ∑n i=1 Fji 6= 1, and thus the sum of radiative heat flows ∑n i=1 φi assumes non-zero values, as shown in Eq. (34). At first sight, it might seem that the values of radiosities, heat fluxes and heat flows shown in various places of the present paper include a too large number of figures after the decimal points. This is because the computations are performed in the regime of double precision using the input data containing more figures after the decimal points in order to suppress rounding errors and meet the requirement of the compensation theorem ( ∑ (i) φi = 0) as accurate as possible. The tested room shown in Fig. 3 has been chosen as a very simple room possessing only three surfaces with different temperatures and emissivities. In reality, rooms may have a much larger number of surfaces with different geometries, temperatures and emissivities (windows, doors, furnishings, wooden or artificial decorations, carpets, textiles, etc.). The possible geometric complexity of rooms concerns solely the matrix of view factors. Although there are many tables and formulae for determining view factors in the literature, e.g. in Ref. [26–28], many general cases are missing. Since the analytical derivation of view factors in these cases may be difficult, the numerical computations of double integrals (Eq. (1)) seem to be the only way to overcome this problem. An interesting method for the numerical evaluation of view factors has been presented only recently [29]. However, for common geometries of internal rooms, there is a sufficient number of formulae, graphs or tables to easily determine the corresponding view factors. As soon as the matrix of view factors is formed, other computational steps related to radiosites, heat flux and heat flows are the matter of routine numerical operations for which a large number of various surfaces with different temperatures or emissivities does not represent a larger problem. In addition, it is clear that quite small surfaces compared to the area of the room envelope have small influences on results and thus neglecting some of them will not introduce an essential inaccuracy. Finally, as the heat losses of the room under investigation are concerned, some parts of the absorbed energies φ2 = −316.029168 W and φ3 = −2 306.791512 W may be propagated by conduction through the solid constructions and at the external sides they may be transferred by convection and radiation into the exterior. But the real amount of heat loss depends on the quality of thermal insulation, the temperature difference between interior and exterior, external coefficients of heat transfer and external emissivities. The complete computations of heat losses that include inner convective and radiative transfers along with external convective and radiative transfers are accomplished in the second serial paper [30], which thematically continues the present paper. 3.1.1. Matrix of heat exchange In the following paragraphs, a detail analysis of radiative heat exchange between couples of surfaces of the investigated simple room (Fig. 3) is presented. Although such an analysis is not required for determining heat losses, it could be useful for understanding the mechanism of energy exchange between surfaces. For this purpose, the matrix of energy exchange φi↔j will be computed according to relation (26). This relation requires special radiosities W (j)i determined when some surfaces (i-surfaces) have zero temperatures and only one surface (j-surface) has its own correct non-zero temperature. These radiosities have been computed and their values are gathered in Tab. 2. The matrix φi↔j of energy exchange (Eq. (26)):  01↔1 +997.5222721↔2 +1 623.7936801↔3−997.5222722↔1 02↔2 +683.9629202↔3 −1 623.7936803↔1 −683.9629203↔2 03↔3   =⇒   ∑ 1 = +2 621.315952 ≈ φ1∑ 2 = −313.559352 ≈ φ2∑ 3 = −2 307.756600 ≈ φ3   W (35) ∑ 1 + ∑ 2 + ∑ 3 = 0 (Precisely) As seen from matrix (35), the sum of numbers in each row approximately equals the net heat power φ computed previously (see Eqs. (32)). 218 vol. 59 no. 3/2019 Model of radiative and convective heat transfer in buildings: Part I Parameters Floor no. 1 Side walls no. 2 Ceiling no. 3 ε 0.95 1 0.75 ρ 0.05 0 0.25 T (K) 300 0 290 S (m2) 72 108 72 εEb (W/m2) 436.3065 0 300.7712 Table 3. Input data for the quasi-enclosure of the simple room. From the first row of matrix (35), it is obvious that surface no. 1 (floor) has emitted ∼ 998 W towards surface 2 (side walls) and ∼ 1 624 W towards surface no. 3 (ceiling). Both these transfers result in ∼ 2 621 W of the net heat power emitted by the floor (no. 1). From the second row of matrix (35), it follows that surface no. 2 (side walls) has absorbed ∼ 998ť W coming from the floor (no. 1) but, simultaneously, surface no. 2 has sent ∼ 684 W towards the ceiling (no. 3). By summing these two transfers, the net heat power absorbed by the side walls (no. 2) has amounted to ∼ 314 W. Similarly, the third row of matrix (35) has let us know that the ceiling (no. 3) has absorbed ∼ 1 624 W coming from the floor (no. 1) and also ∼ 684 W that has been sent from the walls (no. 2). Thus, the net absorbed power of the ceiling (no. 3) is ∼ 2 308 W. The radiative heat energies redistributed within the room enclosure and specified by matrix (35) fulfil precisely the compensation theorem in accordance with the first and the second laws of thermodynamics, namely, each transfer of energy is directed from a warmer surface to cooler one and all the transferred radiative energies are conserved inside the enclosure without any possibility to escape, i.e. their sum is precisely zero ( ∑3 i=1 φi = 0). Since the majority of basic monographs, e.g. [26–28], do not discuss behaviour of the radiosity method within open room envelopes, the following two sections 3.2 and 3.3 are devoted to this problem. Section 3.2 illustrates the behaviour of the radiosity method within the quasi-enclosure whereas section 3.3 explores the properties of this method within a real open envelope. We would like to mention that these two open structures do not serve as the prototypes of window openings since the glazed windows under the acting of long-wave room thermal radiation can be treated as non-transparent structures undertaking only surface absorption without surface transparency just as with solid walls. This fact has been explained in the introduction. Sections 3.2 and 3.3 that discuss open systems along with section 3.1 that explores closed systems provide a complete treatise concerning the functionality of the radiosity method applied to a variety of surface arrangements. 3.2. Application to quasi- enclosure The quasi-enclosure is realized like a simple room shown in Fig. 3 whose four side walls no. 2 are completely transparent by heat radiation. These side walls are considered as ideally black bodies with zero temperatures, i.e. ε = 1, ρ = 0, T2 = 0. In fact, such an enclosure behaves as an open system in which heat escapes through the ideally absorptive side walls. The input data are summarized in Tab. 3. The matrix of view factors Fij is the same as in section 3.1:  0 0.5 0.50.3 0.3 0.3 0.5 0.5 0   (36) Radiosities (Eq. 15): W1 = 436.3065 + 0.05 (0.5W2 + 0.5W3) W2 = 0 W3 = 300.7712 + 0.25 (0.5W1 + 0.5W2) (37) W1 = 445.2170, W2 = 0, W3 = 356.4233 (W/m2) (38) Heat flows (Eqs. (11), (16) and (17)): φ1 = S1q1 = 72 · 1 0.05 · (436.3065 − 0.95 · 445.21708) = +19 224 W (see Eq. (11)) φ2 = S2q2 = 108 · [ 0 − ( 0.3W1 + 0.3W3 )] = −28 859 W (see Eq. (17)) (39) φ3 = S3q3 = 72 · 1 0.25 · (300.7712 − 0.75 · 356.4233) = +9 635 W (see Eq. (11)) 219 Tomáš Ficker Acta Polytechnica Surface 1 radiates others not - W (1)i (W/m 2) W (1)1 = 437.67423 W (1) 2 = 0 W (1) 3 = 54.70928 Surface 2 “radiates” others not - W (2)i (W/m 2) W (2)1 = 0 W (2) 2 = 0 W (2) 3 = 0 Surface 3 radiates others not - W (3)i (W/m 2) W (3)1 = 7.54285 W (3) 2 = 0 W (3) 3 = 301.71410 Table 4. Special radiosities W (j)i for computing the matrix φi↔j of the quasi-enclosure. 3∑ i=1 φi = 0 W (40) In the case of the quasi-enclosure, the radiosity method provides results that resemble the results achieved with the real closed envelope, i.e. the heat powers transferred are ‘conserved’ since their sum is zero ( ∑3 i=1 φi = 0) in a complete accordance with the compensation theorem. In addition, the transfers of energies are directed from wormer surfaces (floor and ceilings) to cooler surface (side walls). Yet, there is some difference. The side walls (no. 2) are completely transparent for heat radiation and thus their total net absorbed power 28 859 W represents the heat that escapes into the open space beyond the quasi-enclosure. This result corresponds to that of a real open enclosure of a room whose side walls are completely removed. The question is whether the radiosity method will determine the same heat loss (28 859 W) if the real open enclosure consisting only of the heated floor and the ceiling is considered. The answer can be found in section 3.3. Prior to starting section 3.3, it would be instructive to explore the energy exchanges between surfaces within this quasi-enclosure. The corresponding results are presented in Tab. 4 and within the matrix φi↔j. The matrix φi↔j of energy exchange (Eq. (26)):  01↔1 +17 725.81↔2 +1 498.61↔3−17 725.82↔1 02↔2 −11 133.22↔3 −1 498.63↔1 +11 133.23↔2 03↔3   =⇒   ∑ 1 = +19 223.4 = φ1∑ 2 = −28 859.0 = φ2∑ 3 = +9 634.6 = φ3   W (41) From the first row of matrix (41) it is obvious that floor (no. 1) has sent 17 725.8 W to the transmittable side walls (no. 2) and this energy escapes into the open space. In addition, the floor (no. 1) has also sent 1 498.6 W to the ceiling (no. 3). The second row of matrix (41) summarizes the total energy coming from the floor (17 725.8 W) and the ceiling (11 133.2 W). These energies are absorbed by the transmittable side walls (no. 2) and represent the total heat losses going into the open space (28 859.0 W). The third row of matrix (41) describes the energy exchanges between the ceiling (no. 3) and the remaining surfaces. The ceiling has absorbed 1 498.6 W from the floor (no. 1) but it has emitted 11 133.2 W towards the transmittable side walls (no. 2) and this energy represents the heat loss escaping into the open space. The net power of the ceiling is 9 634.6 W. By comparing heat exchanges within the quasi-open enclosure (matrix (41)) and the regular enclosure (matrix (35)), it is obvious that the ‘quasi-open’ envelopes of rooms enormously increase heat losses. This is due to the energy that escapes through the transparent (open) parts of envelopes. In the next section, the case of the quasi-open enclosure will be replaced by the real open system and the problem will be recalculated. This will enable us to compare results of the radiosity method applied to differently arranged surfaces. 3.3. Application to regularly open envelope Let us consider the arrangement shown in Fig. 3, in which the side walls have been completely removed. The corresponding input data can be found in Tab. 1. The remaining surfaces are associated with the floor (no. 1) and the ceiling (no. 2). - the ceiling has been renumbered. This arrangement corresponds to the quasi- enclosure in which the side walls have really been removed. The matrix of view factors of this two-dimensional problem may be easily derived from matrix (36): ( 0 0.5 0.5 0 ) (42) Radiosities (Eq. 15): W1 = 436.3065 + 0.05 · (0.5W2) W2 = 300.7712 + 0.25 · (0.5W1) (43) 220 vol. 59 no. 3/2019 Model of radiative and convective heat transfer in buildings: Part I Surface 1 radiates other not - W (1)i (W/m 2) W (1)1 = 437.67423 W (1) 2 = 54.70928 Surface 2 radiates other not - W (2)i (W/m 2) W (2)1 = 7.54285 W (2) 2 = 301.71406 Table 5. Special radiosities W (j)i for computing the matrix φi↔j of the regularly open envelope. W1 = 445.2170, W2 = 0, W3 = 356.4233 (W/m2) (44) Heat flows (Eqs. (11), (16) and (17)): φ1 = S1q1 = 72 · 1 0.05 · (436.3065 − 0.95 · 445.21708) = +19 224.4 W (see Eq. (11)) φ2 = S2q2 = 72 · 1 0.25 · (300.7712 − 0.75 · 356.4233) = +9 634.6 W (see Eq. (11)) (45) 3∑ i=1 φi = +28 859 W (46) As seen from results (45) and (46), the floor and the ceiling emit energies (they provide only positive values), i.e. no portion of energy is absorbed by these surfaces (no negative values are present). This means that the emitted energies (28 859 W) represent heat losses directed into the open space. This conclusion is in a full agreement with the foregoing computations performed within the quasi-open enclosure presented in section 3.2. The total heat flow (46) of the regularly open system cannot be zero as in the case of the enclosure (40) or (33) since a large portion of energy escapes into the open space. Let us now investigate the matrix of energy exchange associated with the two-dimensional radiosity method (see Tab. 5). The matrix φi↔j of energy exchange (Eq. (26)):( 01↔1 +1 498.61↔2 −1 498.62↔1 02↔2 ) =⇒ (∑ 1 = +1 498.6∑ 2 = −1 498.6 ) W (47) Matrix (47) specifies the energy transferred between the floor and the ceiling. The energy of 1 498.6 W has been emitted by the floor (no. 1) and the same energy has been absorbed by the ceiling (no. 2). This is in agreement with the computations performed within the quasi-enclosure in sub-section 3.2. This energy exchange obeys the principle of the second law of thermodynamics that declares that, during natural processes, the heat always flows from the wormer body to the cooler one. The elements φ1↔2 and φ2↔1 of matrix (47) are in agreement with the elements φ1↔3 and φ3↔1 of matrix (41). However, the two-dimensional modification of radiosity model is not capable of providing such detailed information of energy exchanges between surfaces as in the case of thre-dimensional radiosity method. Although numerically more demanding, the three-dimensional radiosity model used in the previous section is more informative. 4. Conclusions The discussed computational model for the estimation of radiative heat transfer is applicable both to the open and closed systems of surfaces. In the first computational step, the matrix of view factors is formed on the basis of the graphs or formulae published in the technical literature and by means of the three auxiliary rules termed as the symmetry rule (Eq. 3), the zero rule (Eq. 4) and the summation rule (Eq. 5). The radiosities are then determined from the system of linear algebraic equations (Eqs. 15). The radiosities enable to compute heat fluxes qi (Eqs. 11/17) and heat flows φi (Eq. 16) associated with particular surfaces. In addition, the radiative heat portions that are exchanged between couples of surfaces may be identified as the elements of the heat exchange matrix φi↔j. The radiative portions of heat that are transferred in the open and closed systems of surfaces differ as to the estimation of energy losses. In the open system, the large portion of the heat is emitted irreversibly into the open space whereas in the closed system, some portion of the heat is reflected back into the interior, which ensures a more economical performance of enclosures. In addition, the stationary thermal state in the interior of the enclosure is characterized by the equilibrium between emitted and absorbed portions of heat, i.e. the sum of heat flows is zero ( ∑ (i) φi = 0). This is in an agreement with the compensation theorem and the basic laws of 221 Tomáš Ficker Acta Polytechnica thermodynamics. On the contrary, the total heat flow of the regularly open system cannot be zero as in the case of the closed system since a large portion of energy escapes into the open space. The open system of surfaces may be investigated either as the quasi-enclosure or the regularly open envelopes. Although the radiosity method applied to both these arrangements provides equivalent results, the concept of quasi-enclosure seems to be more informative. The present paper has summarized the algebraic computational model based on radiosity. Different properties and behaviours of that model when applied to various systems of room envelopes have been analysed and discussed. All this should serve as a preparation stage for the development of a more general model for the estimation of heat losses of inner spaces of buildings, in which the heat transfer is realized by radiation and convection. The general model of combined radiative and convective heat transfer is formulated and applied in the separate serial paper [30] thematically related to the present contribution. The performed analysis of the algebraic radiosity model enables to draw several summarizing conclusions: (1.) The algebraic radiosisty model is capable of a correct functioning not only in closed systems but also in open enclosures. The analysis of its properties in open enclosures is usually missing in monographs related to this model. (2.) The validity of the compensation theorem related to the overall radiative heat flow in enclosures has been proven on the rigorous mathematical basis (Eq. 34). (3.) A new alternative formula for heat exchange between the couples of surfaces has been derived (Eq. (26)). (4.) So far, the algebraic radiosity model has been preferably used for thermal applications in the field of mechanical engineering where it was originally formed. The present paper has illustrated that this model may also be useful in the field of thermal building technology and building physics since it is capable of straightforwardly computing the radiative heat energy produced by heating systems. (5.) 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Acta Polytechnica 59(3):224 – 237, 2019. doi:10.14311/AP.2019.59.0224. 223 http://dx.doi.org/10.1016/j.enbuild.2017.06.052 http://dx.doi.org/10.1016/j.enbuild.2013.08.037 http://dx.doi.org/10.1016/j.enbuild.2017.05.027 http://dx.doi.org/10.1016/j.enbuild.2010.06.017 http://dx.doi.org/10.1016/j.enbuild.2014.06.045 http://dx.doi.org/10.1016/j.enbuild.2012.08.020 http://dx.doi.org/10.1016/j.enbuild.2017.03.043 http://dx.doi.org/10.1016/j.enbuild.2017.07.004 http://dx.doi.org/10.1016/B978-0-08-017787-8.50007-8 http://dx.doi.org/10.1016/j.ijthermalsci.2015.02.008 http://dx.doi.org/10.14311/AP.2019.59.0224 Acta Polytechnica 59(3):211–223, 2019 1 Introduction 2 Basics of radiosity method 2.1 View factors 2.2 Algebraic equations of radiative heat transfer 2.3 Radiosity 2.4 Heat flux of absolutely black surface 2.5 Energy exchange between couples of surfaces 3 Application of radiosity method 3.1 Application to enclosure 3.1.1 Matrix of heat exchange 3.2 Application to quasi- enclosure 3.3 Application to regularly open envelope 4 Conclusions References