Computational Fluid Dynamic Modelling of Thermal Periodic ... - MDPI

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Computational Fluid Dynamic Modelling of Thermal Periodic Stabilized Regime in Passive Buildings Fabio Nardecchia *, Benedetta Mattoni, Francesca Pagliaro, Lucia Cellucci, Fabio Bisegna and Franco Gugliermetti Department of Astronautical, Electrical and Energy Engineering (DIAEE), Sapienza University of Rome, Via Eudossiana 18, 00184 Rome, Italy; [email protected] (B.M.); [email protected] (F.P.); [email protected] (L.C.); [email protected] (F.B.); [email protected] (F.G.) * Correspondence: [email protected]; Tel.: +39-06-4458-5685 Academic Editors: Francesco Asdrubali and Pietro Buzzini Received: 30 September 2016; Accepted: 10 November 2016; Published: 12 November 2016

Abstract: The periodic stabilized regime is the condition where the temperature of each point of a certain environment varies following a periodic law. This phenomenon occurs in many practical applications, such as passive or ancient buildings not equipped with Heating, Ventilating and Air Conditioning HVAC systems and located in latitudes where the temperature greatly varies with Earth’s daily cycles. Despite that, the study of transient phenomena is often simplified, i.e., considering negligible the thermal response of the indoor microclimate. An exact solution to enclosures whose microclimate is free to evolve under a periodic stabilized regime does not exist nowadays, also from an analytical point of view. The aim of this study is to parametrically analyze the thermal variations inside a room when a transient periodic temperature is applied on one side. The phenomenon has been numerically studied through Computational Fluid Dynamics (CFD) and analytically validated using a function that reproduces the daily variation of the outdoor temperature. The results of this research would lay the groundwork to develop analytical correlations to solve and predict the thermal behavior of environments subject to a periodic stabilized regime. Keywords: periodic stabilized regime; CFD; combined heat transfer; time-dependent boundary conditions; passive buildings

1. Introduction The indoor microclimate of buildings is greatly influenced by outdoor climate conditions and by the ability of the architectural envelope to dampen them. These physical phenomena involve different heat transfer processes (coupling convection/conduction) which can affect several aspects related to sustainable architecture and design of buildings and cities, such as the reduction of energy demand [1–6], the decrease of the environmental impact [7,8] and the analysis of the urban canyon effect [9,10]. Differentially heated enclosures have been widely studied in the past [11–15]. However, the complexity of the problem has led the researchers to apply several simplifications in the analysis of these phenomena, such as setting the outdoor conditions as steady or considering indoor microclimate as a controlled environment (where the interior conditions are set, and, if required, dynamically controlled by an HVAC system). Furthermore, the ISO 13786:2007(E) [16] supplies the procedure for the calculation of the dynamic thermal behavior of building components imposing the temperature on one side of the element as constant. Time-periodic thermal conditions should be taken into account especially when the outdoor temperature greatly varies during the day [17–21]. One application is the so-called periodic stabilized regime, which is the condition where the average temperature varies with a periodic law in each point

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of a certain environment. This phenomenon finds practical applications in buildings not characterized by HVAC systems, as passive, ancient and/or historical buildings, where the temperature is free to evolve without thermal constraints. The problem has been analytically studied taking into account the case of the semi-infinite solid [22–24]. An analytical solution for the transient temperature distribution in a semi-infinite solid is known only for three surface conditions: constant surface temperature, constant surface heat flux, and surface convection. To the best of our knowledge, there is not an exact analytical law to predict the indoor thermal behavior in enclosures where a varying temperature is applied at least on one external surface and the indoor temperature is free to evolve. Time-periodic natural convection in 2D enclosures has been also numerically examined in the past literature [25–32]. However, these studies are more focused on the analysis of the fluid field inside the enclosure rather than supplying new analytical relations to solve the physical phenomenon of the periodic stabilized regime. This problem could be studied by coupling experimental measures and numerical simulations and comparing the results. Despite that, an experimental approach is not always applicable, for example when the case study is too complex to be experimentally duplicated or when no more existing buildings or sites are analyzed [33,34]. The aim of this study is to parametrically analyze the thermal variations inside a room when a transient periodic temperature is applied on one side. The results of this research would lay the groundwork to develop analytical correlations to solve and predict the thermal behavior of environments subject to time-dependent thermal forcing and where the indoor temperature is free to evolve, matching together all the approaches used up to now. 2. Materials and Methods Nowadays, Computational Fluid Dynamic (CFD) is one of the main tools used in several studies to analyze heat transfer processes involving convection and conduction. However, these studies did not supply analytical correlations to predict the thermal behavior of enclosures where the temperature is free to evolve under a periodic stabilized regime. In this work, the indoor temperature trends inside the model of a simple room has been numerically investigated through CFD. Parametric analyses have been carried out by changing the thermo-physical and boundary conditions of the case study. Several sets of simulations have been, indeed, taken into account in order to understand what the thermal behavior of the indoor environment is when geometry and thermal inputs vary. Four sensitivity tests have been carried out changing one parameter at a time. All the case studies analyzed are summarized in Table 1. Table 1. Case studies summary. Case Name

Sensitivity Test

Case 1 Case 2 Case 3 Case 4

Material of the solid domain Thickness of the solid domain External sinusoidal temperature Tsin External fixed temperature Twall

In the first two cases (Case 1 and Case 2), the thermo-physical properties of the envelope have been varied, while in the last two cases (Case 3 and Case 4), different thermal inputs have been applied on the sidewalls of the model. The varying profile of the external daily temperature has been reproduced through a sinusoidal trend on the external surface of the enclosure. 2.1. Analytical Statement In the study of heat transfer by conduction, the Fourier’s equation admits analytical solutions only for well-defined problems, usually characterized by simple geometries, simple boundary conditions, materials with uniform properties and independent temperature [22–24].


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Many heat transfer problems are time-dependent. It is possible to analyze transient conduction Many heat transfer problems are time-dependent. It is possible to analyze transient conduction phenomena referring to a semi-infinite solid that extends its dimensions to infinity with the exception phenomena referring to a semi-infinite solid that extends its dimensions to infinity with the of one of them. exception of one of them. A typical case of transient conduction is described by a body of a wide thickness delimited by A typical case of transient conduction is described by a body of a wide thickness delimited by a a flat surface where a transient periodic temperature is imposed on. The other surfaces of the solid are flat surface where a transient periodic temperature is imposed on. The other surfaces of the solid are at a distance large enough not to consider their influence. The temperature in all the points of the body at a distance large enough not to consider their influence. The temperature in all the points of the varies with the same periodic law of the imposed temperature after a certain time. This condition is body varies with the same periodic law of the imposed temperature after a certain time. This called periodic stabilized regime. condition is called periodic stabilized regime. Suppose to analyze a semi-infinite wall, whose material is homogeneous and isotropic, delimited Suppose to analyze a semi-infinite wall, whose material is homogeneous and isotropic, by a plan surface. The thermo-physical characteristics of the medium (cp , λ, ρ) are independent from delimited by a plan surface. The thermo-physical characteristics of the medium (cp, λ, ρ) are temperature and time. Imagine to impose on its surfaces a sinusoidal temperature in the normal independent from temperature and time. Imagine to impose on its surfaces a sinusoidal temperature x direction, which follows the analytical Equation (1): in the normal x direction, which follows the analytical Equation (1): TT( x, (ωτωτ ) ) ( xτ,τ) )==TTmm++Θsin Θ sin(

(1) (1)

aninitial initial transient period, the temperature periodicregime. stabilized regime. After an transient period, the temperature reaches areaches periodicastabilized Considering a time τ in the periodic regime, stabilized regime, since the temperature not depend on the aConsidering time τ in the periodic stabilized since the temperature does not does depend on the spatial spatial coordinates andsolution z, the solution to the Fourier’s equation coordinates y and z,ythe to the Fourier’s equation is (2): is (2): − βx

( xτ,τ) )==TTmm++Θe Θ−e βx sin sin( TT( x, (ωτωτ −− βxβ)x)

(2) (2)

Transient conduction in a semi-infinite solid can be solved also in three different conditions on Transient conduction in a semi-infinite solid can be solved also in three different 'conditions on the 00 the surface a constant temperature ≠ Ti (Figure a constant heat q 0' 1b); (Figure 1b); an surface [24]: [24]: a constant temperature Ts 6= TTi s(Figure 1a); a 1a); constant heat flux q0 flux (Figure an exposure exposure to a fluid whose temperature T∞ ≠a T i and a convective coefficient (Figure 1c). to a fluid whose temperature is T∞ 6= Tiisand convective coefficient h (Figureh1c).

Figure 1. 1. Transient semi-infinite solid solid depending surface conditions: conditions: Figure Transient conduction conduction in in aa semi-infinite depending on on different different surface '00' i (a); (b); exposure exposure to to aa fluid fluid whose whose temperature temperature is constant temperature Tss ≠6=TT constantheat heatflux fluxqq00 (b); i (a);constant T 6 = T and a convective coefficient h (c). Original figure from [24]. ∞ i a convective coefficient h (c). Original figure from [24]. T∞ ≠ Ti and

be solved solved respectively respectively through through the the following following equations equations [24]: [24]: These three cases can be τ )T−s Ts  xx  T ( x,Tτ(x ) ,− erf = = er f  √  Ti −TiT− ατ  s Ts  22 ατ

(3)


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00 00 2q0 (ατ/π )1/2 q0 x − x2 x √ T ( x, τ ) − Ti = exp − er f c λ 4ατ λ 2 ατ √ T ( x, τ ) − Ti x h2 · ατ x hx h ατ √ √ = er f c + · er f c − exp + T∞ − Ti λ λ λ2 2 ατ 2 ατ

(4) (5)

The mathematical calculations to obtain the Equations (3)–(5) can be found in [22–24]. The thermo-physical characteristics of the solid body, such as the diffusivity, the phase lag and the thermal damping factor, are noticeable in all the equations. It means that the initial sinusoidal temperature is attenuated and delayed proportionally to the thickness and the physical and thermal properties of the wall. To the best of our knowledge, an analytical solution to enclosures subject to a periodic stabilized regime does not exist nowadays. 2.2. Numerical Approach The commercial software Ansys Fluent v. 14.5 [35] has been used for the transient simulations. The governing equations used for the gas phase are the unsteady form of the continuity, compressible Navier–Stokes, and energy equations for a perfect gas with the Sutherland model for the viscosity and temperature-dependent properties. The equations that govern the phenomenon (momentum (6), continuity (7) and energy conservation (8)) can be expressed as follows: uj

1 ∂p µ ∂2 u i ∂ 0 0 ∂ui =− + − u i u j + gi ∂x j ρ ∂xi ρ ∂xi ∂x j ∂x j ∂ui =0 ∂xi ∂ ∂T ∂T + ui k =0 ∂xi ∂xi ∂xi

(6)

(7) (8)

where i = 1, 2, 3 for the three-dimensional (3D) case and i = 1, 2 for the two-dimensional (2D) case. The Fourier law has been used for the analysis of the solid body. Time-accurate solutions have been obtained using the implicit coupled scheme with second-order accuracy in time with the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) segregated scheme, used for the gas phase until convergence was achieved. For time-accurate solutions, the fluxes for all equations at the cell faces have been interpolated by using the second-order upwind scheme. Pressure equation and the Laplace equation for the conduction have been also computed by using second-order accuracy. The Boussinesq approach has been applied to the model and RNG k-ε turbulence model [36,37] has been chosen to study the fluid turbulence after the calculation of the dimensionless groups for the natural convection (Ra = 2.54 × 1010 ). Basing on convergence and computational stability criteria, the RNG k-ε model has been considered the most suitable for this case in terms of indoor flow field characterization [38]. The simulations have run with the unsteady regime with an initial time step size between 10−7 and 10−5 s and it has been increased at 20 s after the residuals have reached the stability. This time step has been obtained after several tests based on numerical stability and numerical residual convergence. Convergence criteria have been reached when all the residuals were below 10−6 . 3. Validation Cases Heat conduction problems are often solved through many assumptions and simplifications, such as the steady state condition. Despite that, several applications involve time-dependent variation of the temperature, especially when it changes in a periodic manner. As previously said, there is not an exact analytical law to predict the indoor thermal behavior in enclosures where the indoor temperature is free to evolve subject to a periodic stabilized regime.

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variation of the temperature, especially when it changes in a periodic manner. As previously said, variation of the temperature, especially when it changes in a periodic manner. As previously said, there is not an exact analytical law to predict the indoor thermal behavior in enclosures where the numerical carried out in thisthe research beenbehavior validatedinthrough the simulation thereThe is not an exactanalyses analytical law to predict indoorhave thermal enclosures where the indoor temperature is free to evolve subject to a periodic stabilized regime. of simple cases where thetoanalytical solution known.stabilized Two case studies have been taken into indoor temperature is free evolve subject to a is periodic regime. The numerical analyses carried out in this research have been validated through the simulation of account The[34]: numerical analyses carried out in this research have been validated through the simulation of simple cases where the analytical solution is known. Two case studies have been taken into account [34]: simple cases where the analytical solution is known. Two case studies have been taken into account [34]: •• Case where a a time-dependent time-dependent sinusoidal Case A: A: heat heat transfer transfer through through aa solid solid where sinusoidal temperature temperature is is applied applied • Case A: heat transfer through a solid where a time-dependent sinusoidal temperature is applied on one side; on one side; on one side; transfer through a solid and an adjacent fluid domain. •• Case Case B: B: heat heat transfer through a solid and an adjacent fluid domain. • Case B: heat transfer through a solid and an adjacent fluid domain. The solid body has been assumed isotropic and homogeneous. The material of the solid is the The solid body has been assumed isotropic and33homogeneous. The material of the solid is the aluminum, which is characterized by: ρ = 2719 kg/m kg/m, ,cpcp= =871 871J/(kgK), J/(kgK), = 202.4 W/(mK). λ =λ 202.4 W/(mK). aluminum, which is characterized by: ρ = 2719 kg/m3, cp = 871 J/(kgK), λ = 202.4 W/(mK). The geometry of Case A is represented by a rectangular domain having dimensions Case A is represented by a rectangular domain having dimensionsofofLL1 1==5.0 5.0 m The geometry of Case A is represented by a rectangular domain having dimensions of L1 = 5.0 m and L22 = 3.0 m has been been applied applied on one m (Figure (Figure 2). 2). A A time-dependent time-dependent sinusoidal sinusoidal temperature Tsin sin has and L2 = 3.0 m (Figure 2). A time-dependent sinusoidal temperature Tsin has been applied on one sidewall = 298.15 298.15 K, sidewall in the normal x direction, direction, which which follows follows the analytical analytical Equation (1), where Tm m = sidewall in the normal x−5 direction, which follows the analytical Equation (1), where Tm = 298.15 K, − 5 Θ andωω==7.3 7.3×× rad/s. Θ00 ==55KKand 1010 rad/s. Θ0 = 5 K and ω = 7.3 × 10−5 rad/s. A fixed applied onon thethe opposite sidewall, while the top fixed temperature temperatureTT wall == 298.15 298.15KKhas hasbeen been applied opposite sidewall, while the and top wall A fixed temperature Twall = 298.15 K has been applied on the opposite sidewall, while the top the bottom walls have been considered as adiabatic. and the bottom walls have been considered as adiabatic. and the bottom walls have been considered as adiabatic. Since the geometry is very simple, a mesh with a structured grid, composed by 10,500 cells Since the geometry is very simple, a mesh with a structured grid, composed by 10,500 cells (interval size 0.035 m), has been been chosen chosen for for the the discretization. discretization. (interval size 0.035 m), has been chosen for the discretization.

Figure 2. Case A—Geometry, mesh and boundary boundary conditions. Figure 2. 2. Case Case A—Geometry, A—Geometry, mesh Figure mesh and and boundary conditions. conditions.

Figure 3 shows the temperature trends in the center of the solid domain (x = 2.5 m, y = 1.5 m) in Figure 3 shows the temperature trends in the center center of of the the solid solid domain domain (x (x == 2.5 m, y = 1.5 m) in both numerical and analytical analyses. The analytical data are derived (2). both numerical and analytical analyses. The analytical data are derived (2).

Figure 3. Case A: Temperature trends comparison between CFD and analytic analyses. Figure comparison between between CFD CFD and and analytic analytic analyses. analyses. Figure 3. 3. Case Case A: A: Temperature Temperature trends trends comparison

The results show that the two temperatures have the same trends and almost the same values. The results show that the two temperatures have the same trends and almost the same values. The comparison between and numerical data been performed also through root The results show thatanalytical the two temperatures have thehas same trends and almost the samethe values. The comparison between analytical and numerical data has been performed also through the root mean square error (RMSE) in a time of 24 h has after theperformed stabilization the simulation. The The comparison between analytical and period numerical data been alsoofthrough the root mean mean square error (RMSE) in a time period of 24 h after the stabilization of the simulation. The


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6 of 18 in a time period of 24 h after the stabilization of the simulation. The obtained RMSE value is 0.027 K. Therefore, CFD simulations can be considered accurate enough for the analysis obtained RMSE value is 0.027 K. Therefore, CFD simulations can be considered accurate enough for obtained RMSE heat value is 0.027 K.through Therefore, CFD simulations can be considered accurate enough for of the transient conduction a solid. the analysis of the transient heat conduction through a solid. the analysis of the transient heat conduction a solid. with a rectangular shape and a thickness The geometry of Case B is represented represented bythrough a solid domain rectangular geometry of Case B is represented by a solid domain with a rectangular shape and a thickness of ss =The 1 m, while the fluid The heat of fluid domain domain dimensions dimensions are L1 = = 5.0 m and L L22 = 3.0 m (Figure 4). The of s = 1 m, while the fluid domain dimensions are L 1 = 5.0 m and L 2 = 3.0 m (Figure 4). The heat 2 2K).K). exchangecoefficient coefficienthas hasbeen beenimposed imposedequal equaltoto1010 W/(m temperature of the domain exchange W/(m TheThe temperature of the fluidfluid domain has 2K). The temperature of the fluid domain has exchange coefficient has and beenequal imposed equal to 10 W/(m has been kept constant to T = 293.15 K, while the solid temperature T has been been kept constant and equal to Tfluid = fluid 293.15 K, while the solid temperature Tin has been left freeleft to in been kept constant and equal to T fluid = 293.15 K, while the solid temperature Tin has been left free to free to evolve, from a temperature Thethe top and the bottom have been evolve, starting starting from a temperature of 215.15 of K. 215.15 The topK.and bottom walls have walls been considered evolve, starting from a temperature of 215.15 K. The top and the bottom walls have been considered considered adiabatic. adiabatic. adiabatic. As the previous very simple. ForFor thisthis reason, a mesh withwith a structured grid, previouscase, case,the thegeometry geometryis is very simple. reason, a mesh a structured As the previous case, the geometry is very simple. For this reason, a mesh with a structured composed by 12,700 cells (interval size 0.035 m), has for the grid, composed by 12,700 cells (interval size 0.035 m),been has chosen been chosen fordiscretization. the discretization. grid, composed by 12,700 cells (interval size 0.035 m), has been chosen for the discretization.

Figure Figure 4. 4. Case Case B: B: Geometry, Geometry, mesh mesh and and boundary boundary conditions. conditions. Figure 4. Case B: Geometry, mesh and boundary conditions.

Figure 5 shows the temperature trends in the center of the solid domain (x = 0.5 m, y = 1.5 m) in Figure 5 shows the temperature trends in the center of the solid domain domain (x (x == 0.5 m, y = 1.5 m) in both numerical and analytical analyses. In this case, it is possible to use the Equation (5), where λ, α, α,α, h, both numerical numerical and and analytical analyticalanalyses. analyses.In Inthis thiscase, case,ititisispossible possibletotouse usethe theEquation Equation(5), (5),where whereλ,λ, h, and T∞ are assumed constant. The assumption of constant h and T is almost never encountered in and T are assumed constant. The assumption of constant h and T is almost never encountered in real h, and∞T∞ are assumed constant. The assumption of constant h and T is almost never encountered in real cases, as for the one-dimensional application. cases, as for application. real cases, asthe forone-dimensional the one-dimensional application.

Figure 5. Case B: Temperature trends comparison between CFD and analytic analysis. Figure between CFD CFD and and analytic analytic analysis. analysis. Figure 5. 5. Case Case B: B: Temperature Temperature trends trends comparison comparison between

Analytical and numerical temperature trends greatly differ only before the stabilization of the Analytical and numerical temperature trends greatly differ only before the stabilization of the simulation (around h). Thetemperature root mean square (RMSE) been evaluated in a time period Analytical and2000 numerical trendserror greatly differhas only before the stabilization of the simulation (around 2000 h). The root mean square error (RMSE) has been evaluated in a time period of 400 h after 2000 h of simulated time and the obtained valuehas is 0.0001 K. It is possible assume simulation (around 2000 h). The root mean square error (RMSE) been evaluated in a timetoperiod of of 400 h after 2000 h of simulated time and the obtained value is 0.0001 K. It is possible to assume that the numerical model approximates accurately the analytical one also in this case. 400 h after 2000 h of simulated time and the obtained value is 0.0001 K. It is possible to assume that the that the numerical model approximates accurately the analytical one also in this case. The CFD model has been considered accepting that the obtained values are affected numerical model approximates accurately validated, the analytical one also in this case. The CFD model has been considered validated, accepting that the obtained values are affected by a minimum error. Therefore, it has been applied to a similar real model where a solid envelope by a minimum error. Therefore, it has been applied to a similar real model where a solid envelope adjacent to the flow domains is subject to thermal sinusoidal boundary conditions varying with adjacent to the flow domains is subject to thermal sinusoidal boundary conditions varying with time. time.


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The CFD model has been considered validated, accepting that the obtained values are affected by a minimum error. Therefore, it has been applied to a similar real model where a solid envelope Sustainability 1172 domains is subject to thermal sinusoidal boundary conditions varying with 7time. of 18 adjacent to2016, the flow 4. 3D-2D 3D-2DSimplification Simplification The periodic stabilized regime regime has been numerically numerically applied applied to a model of a simple closed room with D)and andH H==33m. m.The Theroom roommodel model is is influenced influenced by an outdoor with aa rectangular rectangularplan planof of55×× 5 m (W × × D) thermal forcing applied applied on on aa solid solid wall, wall, while whilethe theopposite oppositewall wallisisadjacent adjacenttotoananenvironment environmentatata afixed fixed temperature. other walls have considered adjacent to environments having the temperature. TheThe other walls have beenbeen considered adjacent to environments having the same same temperature the room; therefore, it has been possible assumethat thatthere thereisisnot not heat heat transfer temperature of theofroom; therefore, it has been possible to to assume through those walls (adiabatic (adiabatic process). process). The 3D simplified in in a 2D computational domain representing the 3D geometry geometryofofthe theroom roomhas hasbeen been simplified a 2D computational domain representing longitudinal section on the xy plane (Figure 6). 6). the longitudinal section on central the central xy plane (Figure

Figure 6. 6. Geometry: Geometry: 3D 3D model model (a); (a); 2D 2D model model (b). (b). Figure

In this work, this simplification has been possible since the considered room has a preferential In this work, this simplification has been possible since the considered room has a preferential symmetry plan along the z axis. The choice of a 2D model allows lower computational costs without symmetry plan along the z axis. The choice of a 2D model allows lower computational costs without compromising the accuracy of the results. compromising the accuracy of the results. In order to demonstrate the effectiveness of this choice, both the 3D and 2D models have been In order to demonstrate the effectiveness of this choice, both the 3D and 2D models have been simulated with the same boundary conditions and simulation settings. The results have been shown simulated with the same boundary conditions and simulation settings. The results have been shown in terms of temperature trends at the end of the solid domain (probe_wall) and in the center of the in terms of temperature trends at the end of the solid domain (probe_wall) and in the center of the fluid domain (probe_fluid) and in terms of contours of the velocity magnitude in the central plane xy fluid domain (probe_fluid) and in terms of contours of the velocity magnitude in the central plane xy (Figure 6). (Figure 6). The two computational models have been discretized using an orthogonal grid (Figure 7). The two computational models have been discretized using an orthogonal grid (Figure 7). A grid independence test has been performed for the 2D model in order to demonstrate the A grid independence test has been performed for the 2D model in order to demonstrate the independence of the results from the mesh spacing. A fixed temperature has been applied on the independence of the results from the mesh spacing. A fixed temperature has been applied on the external wall of the solid domain and the static temperature in the middle of the fluid domain external wall of the solid domain and the static temperature in the middle of the fluid domain (probe_fluid) has been chosen as a parameter for the test. Assuming the finest mesh considered, Mesh (probe_fluid) has been chosen as a parameter for the test. Assuming the finest mesh considered, D (mesh spacing equal to 0.0175 m), as the pivot case, it is possible to observe that the differences Mesh D (mesh spacing equal to 0.0175 m), as the pivot case, it is possible to observe that the differences percentage Δ% with the other three meshes are always very low (Table 2). The mesh with the lower percentage ∆% with the other three meshes are always very low (Table 2). The mesh with the lower error, Mesh A (mesh spacing equal to 0.035 m), has been used for these CFD analyses and the 3D error, Mesh A (mesh spacing equal to 0.035 m), has been used for these CFD analyses and the 3D model model has been discretized considering the same interval size of the mesh, with an amount of 155,344 has been discretized considering the same interval size of the mesh, with an amount of 155,344 cells. cells.

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Figure 7. 7. Mesh: Figure Mesh: 3D 3D model model (a); (a); 2D 2D model model (b). (b). Figure 7. Mesh: 3D model (a); 2D model (b). Table 2. Mesh sensitivity test for the 2D model. Table 2. Mesh sensitivity test for the 2D model. Table 2. Mesh sensitivity test for the 2D model.

Mesh A Mesh B Mesh C Mesh D A A Mesh B B Mesh CC Mesh D Mesh Mesh Mesh Mesh D Cells of the mesh Mesh 116,80 2920 730 46,720 Cells ofof the mesh 2920 730 46,720 Cells the mesh 116,80 116,80 2920 730 46,720 Interval size 0.035 0.07 0.14 0.0175 Interval size 0.035 0.07 0.14 0.0175 Interval 0.035 0.07 0.14 0.0175 Δ%size 0.02% 0.17% 0.30% ∆% 0.02% 0.17% 0.30% Δ% 0.02% 0.17% 0.30% Both models are composed by two computational domains, one for the solid wall and one for Bothinside models are composed by two two computational computational domains, one forthe the solid wall(Figure andone one for Both models are composed by domains, one for solid wall and the fluid the room. The same boundary conditions have been taken into account 8).for A thefluid fluidinside insidethe the room.The The same boundary conditions have been taken into account 8). the room. same boundary conditions have been taken into account (Figure A time-dependent sinusoidal temperature Tsin has been applied on the external surface of(Figure the 8). solid A time-dependent sinusoidal temperature has been applied theexternal external surface theon solid time-dependent sinusoidal temperature sinTsin has been applied ononthe ofofthe solid domain following the analytical EquationT(1), while a fixed temperature Twall hassurface been applied the domain following the analytical Equation (1), while a fixed temperature T has been applied on domain following Equation (1), and whilethe a fixed temperature Twall hasconsidered been applied on the right sidewall of the analytical fluid domain. The top bottom walls have been adiabatic wall the right sidewall of the fluid domain. The and the bottom walls havebeen been considered adiabatic right sidewall ofsurfaces the fluid domain. toptop and the bottom have adiabatic assuming those adjacent toThe environments at the samewalls temperature, asconsidered usually happens in a assuming those surfaces adjacent to environments at the same temperature, as usually happens assuming surfaces adjacent to environments the3D same temperature, as considered usually happens in in a room of athose building. Also, the other two sides ofatthe model have been adiabatic. a room building. Also, other two sidesofofthe the3D 3Dmodel model have been considered adiabatic. room of of aa a building. Also, thethe other sides have been considered adiabatic. Neither fixed temperature nor atwo convection coefficient have been applied to the indoor Neither aafixed temperature nortemperature a convection have been applied the indoor Neither fixed temperature nor a convection have been applied to environment. the external indoor environment. Therefore, the iscoefficient free coefficient to evolve under the to influence of the Therefore, the temperature is free to evolve under the influence of the external temperature. environment. temperature. Therefore, the temperature is free to evolve under the influence of the external temperature.

Figure 8. Geometry, Geometry, mesh and boundary conditions of the 2D model. Figure 8. Geometry, mesh and boundary conditions of the 2D model.

Figure shows the the temperature temperature trends trends in in probe_wall probe_wall and and probe_fluid probe_fluid for for the the two two models. models. Figure 99 shows Figure 9 shows the temperature trends in probe_wall and probe_fluid for the two models.

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Figure 9. Temperature trends for the 3D model (symbols) and the 2D model (lines).

Figure 9. 9. Temperature Temperature trends Figure trends for for the the 3D 3D model model (symbols) (symbols) and and the the 2D 2D model model (lines). (lines). Figure 9. Temperature trends for the 3D model (symbols) and the 2D model (lines). The results show that the temperatures have almost the same values and trends. The root mean The results that the Ktemperatures have theprobe_wall. same values and trends. The root mean square error show (RMSE) is 0.2 for probe_fluid andalmost 0.1 K for Therefore, it is possible The results show that the temperatures have almost the same values and trends. The roottomean assume the values obtained from the 2D numerical model describe the thermal phenomena with to K for for probe_fluid probe_fluid and 0.1 0.1 K K for probe_wall. probe_wall. square errorthat (RMSE) is 0.2 K and for Therefore, it is possible square error (RMSE) is 0.2 K for probe_fluid and 0.1 K for probe_wall. Therefore, it is possible to the that samethe accuracy the 3D model. assume valuesofobtained from the 2D numerical model describe the thermal phenomena with assume that the values obtained from the 2D numerical model describe the thermal phenomena with assumption has model. been supported also by the analysis of the fluid field and the convective model. the same This accuracy of the 3D the same accuracy of10 the 3D11). model. motions (Figures and convective This assumption has been supported also by the analysis of the fluid field and the convective

This assumption has been supported also by the analysis of the fluid field and the convective 11). motions (Figures 10 and 11). motions (Figures 10 and 11).

Figure 10. Contours of the velocity magnitude expressed in m/s in the highest peak of the sinusoidal temperature Tsin: 3D model (a); 2D model (b).

Figure 10. 10. Contours Contours of of the the velocity velocity magnitude magnitude expressed expressed in m/s m/s in the of the the sinusoidal sinusoidal Figure in the the highest highest peak peak of expressed in in m/s in model (a); 2D model (b). temperature Tsin:: 3D 3Dmodel model(a); (a);2D 2Dmodel model(b). (b). sin: 3D temperature Tsin

Figure 11. Contours of the velocity magnitude expressed in m/s in the lowest peak of the sinusoidal temperature Tsin: 3D model (a); 2D model (b).

The air motions have been analyzed in two different time steps which represent the highest and Figure 11. Contours velocity magnitude expressed m/s inare the almost lowest the peaksame of thefor sinusoidal the lowest peaks of of thethe sinusoidal temperature Tsin. Theinresults the two Figure 11. Contours Contours of of the the velocity velocity magnitude magnitude expressed expressed in m/s m/s in peak sinusoidal Figure in the the lowest lowest peak of of the the sinusoidal models.11. The fluid fields show both time(b). steps two areasincharacterized by an acceleration of the air. temperature Tsin: 3D model (a);in2D model temperature Tsin::3D model (b). temperature T model(a); (a);2D 2Dmodel modelthe (b).two models is the air velocity of the acceleration zones, sin 3D difference The most noticeable between which highest in the 2Din model. This phenomenon due to represent an increasethe of highest kinetic and The airreaches motions have values been analyzed two different time stepsis which The air motions have been analyzed in two different time steps which represent the highest and energy in the 2D model, where the air in motions are only inalmost tworepresent directions. However, The airpeaks motions been analyzed two different steps which the highest the lowest of have the sinusoidal temperature T sin. distributed Thetime results are the same for the and two the lowest peaks of have the sinusoidal temperature Tsin.toThe results Therefore, are almost the same forshow the two these differences been considered small enough be ignored. the two models the lowest peaks the sinusoidal temperature Tsintwo . The results are almost by thean same for the two models. The fluidoffields show in both time steps areas characterized acceleration ofmodels. the air. models. The fluid fieldsalso show inaboth steps twoofareas comparable results from fluidtime dynamic point view.characterized by an acceleration of the air.

The show difference in both time steps two characterized acceleration the air. Thezones, most The fluid mostfields noticeable between theareas two models is the by air an velocity of the of acceleration The most noticeable difference between the two models is the air velocity of the acceleration zones, noticeable difference between the two models is the air velocity of the acceleration zones, which which reaches highest values in the 2D model. This phenomenon is due to an increase of kinetic which reaches highest values in the 2D model. This phenomenon is due to an increase of kinetic reaches highest values in the 2D model. phenomenon is due only to anin increase of kinetic However, energy in energy in the 2D model, where the air This motions are distributed two directions. energy in the 2D model, where the air motions are distributed only in two directions. However, the 2Ddifferences model, where the air motions are small distributed only directions. However, these differences these have been considered enough to in be two ignored. Therefore, the two models show these differences have been considered small enough to be ignored. Therefore, the two models show have been considered small enough to be ignored. Therefore, the two models show comparable results comparable results also from a fluid dynamic point of view. comparable results also from a fluid dynamic point of view. also from a fluid dynamic point of view.


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5. Analytical Evaluation of the Solid Envelope The analysis of building materials and building elements is a crucial aspect in the evaluation of the Sustainability 2016, 1172 10 of 18 effects of outdoor microclimatic conditions on indoor environment. For this reason, the temperature trends at the endEvaluation of the different analyzed walls have been analytically calculated considering 5. Analytical of the Solid Envelope a semi-infinite solid and applying Equation (2). This calculation has been performed for the walls in The analysis of building materials and building elements is a crucial aspect in the evaluation of Case 1 and Case 2 in order to compare the variation between an ideal geometry and a more realistic the effects of outdoor microclimatic conditions on indoor environment. For this reason, the one. In both cases, a sinusoidal temperature Tsin given by the Equation (1), where Tm = 298.15 K, temperature trends at − the end of the different analyzed walls have been analytically calculated Θ0 = considering 5 K and ω = 7.3 × 10 5 rad/s, hasapplying been applied on (2). theThis sidewall of thehas solid domain. a semi-infinite solid and Equation calculation been performed for Three different materials have been taken into account: aluminum, masonry the walls in Case 1 and Case 2 in order to compare the variation between an ideal geometryand and wood. a Thermo-physical of cases, the materials (density, specificTsin heat capacity, have been more realisticproperties one. In both a sinusoidal temperature given by theconductivity) Equation (1), where −5 rad/s, derived UNI [39] and UNI EN has ISObeen 10456:2007 [40]the (Table 3). of the solid domain. Tm =from 298.15 K, Θ10351:1994 0 = 5 K and ω = 7.3 × 10 applied on sidewall Three different materials have been taken into account: aluminum, masonry and wood. Table 3. of Thermo-physical properties of theheat chosen materials. Thermo-physical properties the materials (density, specific capacity, conductivity) have been derived from UNI 10351:1994 [39] and UNI EN ISO 10456:2007 [40] (Table 3). Aluminum Masonry Wood Table 3. Thermo-physical properties of the chosen materials.

1700 700 0.58 Masonry Wood 0.17 850 2310 ρ 1700 700 λ 202.4 0.58 0.17 c p 871 850 2310 These materials have been chosen because of their different capability in transmitting, damping ρ λ cp

2719 202.4 Aluminum 871 2719

and delaying a thermal input. Considering a fixed thickness of the solid wall s = 0.5 m, the properties These materials have been chosen because of their different capability in transmitting, damping of theand wall vary as shown in Table 4. delaying a thermal input. Considering a fixed thickness of the solid wall s = 0.5 m, the properties of the wall vary as shown in Table 4. Table 4. Thermo-physical properties of the solid wall as the material changes. Table 4. Thermo-physical properties of the solid wall as the material changes.

U σ τr θ

Aluminum Aluminum 5.80 U 5.80 72.18 σ 72.18 1.25 τr 3.611.25 ϴ 3.61

Masonry Masonry 0.97 0.97 0.86 0.86 18.18 18.18 0.04 0.04

Wood Wood 0.33 0.33 0.01 0.01 35.22 35.22 0.00 0.00

The The temperature trends atatthe havebeen beenshown shown Figure 12 the for three the three temperature trends theend end of of the the wall wall have in in Figure 12 for considered materials. considered materials.

Figure 12. Analytical temperature trends at the end of the wall (s = 0.5 m) for three different materials.

Figure 12. Analytical temperature trends at the end of the wall (s = 0.5 m) for three different materials.

Another important aspect that influences the thermal behavior of the building envelope is the

Another important aspect that influences the thicknesses thermal behavior of the building thickness of the building elements. Three different s of the solid domain haveenvelope been takenis the into account m, 0.5 elements. m and 1.0 m), while the material has been ssetofasthe aluminum (Table 3). The been thickness of the (0.1 building Three different thicknesses solid domain have thermo-physical properties of the solid wall vary as shown in Table 5. taken into account (0.1 m, 0.5 m and 1.0 m), while the material has been set as aluminum (Table 3). The temperature trends at of the foras the three considered The thermo-physical properties ofthe theend solid wallwall vary shown in Table 5.thicknesses s are shown in Figure 13. The temperature trends at the end of the wall for the three considered thicknesses s are shown in Figure 13.


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Table 5. Thermo-physical properties of the solid wall as the thicknesses changes. Table 5. Thermo-physical properties of the solid wall as the thicknesses changes. s = 0.1 m s = 0.5 m s = 1.0 m


s =5.86 0.1 m s = 0.5 s = 1.05.71 m U 5.80m 5.86 5.80 5.71 σ U 93.69 72.18 52.09 Table 5. Thermo-physical properties of the solid wall as the thicknesses changes. τr σ 0.25 1.25 93.69 72.18 52.092.49 θ τr 3.61m s =4.68 0.1 m s =1.25 0.5 s =2.49 1.02.60 m 0.25 U ϴ σ τr ϴ

5.86 4.68 93.69 0.25 4.68

5.80 3.61 72.18 1.25 3.61

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5.71 2.60 52.09 2.49 2.60

Figure 13. Analytical temperature trends at the end of the wall for three different thicknesses s.

Figure 13. Analytical temperature trends at the end of the wall for three different thicknesses s.

The reported analytical data underline how and to what extent the external temperature is Figure Analytical trends athow the end of the forextent three thicknesses perceived on 13. the internal temperature surface of the wall in the semi-infinite solid different case. Θ 0 s.and the The reported analytical data underline and to wall what the Knowing external temperature is thermal factor σ of of each element, it has been to calculate the the perceived on damping the internal surface thebuilding wall in the semi-infinite solidpossible case. Knowing Θ0 and The reported underline how what surface extent the external temperature is semi-amplitude of analytical the sinusoidal wave applied onitand the external it is perceived on the thermal damping factor σ of eachdata building element, hasto been possible toonce calculate the semi-amplitude perceived on the internal surface of the wall in the semi-infinite solid case. Knowing Θ0 and the internal surface. of the sinusoidal wave applied on the external surface once it is perceived on the internal surface. thermal σ of each element, been possible The damping solid wall factor of aluminum delaysbuilding and damps a little itthehas temperature Tsin andtoit calculate is possiblethe to The solid wall of aluminum delays and damps a little the temperature and it is possible to sinperceived semi-amplitude oftemperature the sinusoidal wave applied the external once itTis on the expect that indoor largely follows theon sinusoidal trendsurface of the outdoor thermal forcing. On expect that indoor temperature largely follows the sinusoidal trend of the outdoor thermal forcing. internal surface. the contrary, the solid walls composed by masonry and wood greatly impact on the outdoor On the contrary, the solid wallsit composed masonry and greatly impact the outdoor The solidFor wall ofreason, aluminum andby damps the wood temperature TsinIn and ison possible to temperature. this is delays perceived almost asa alittle constant temperature. theitsame way, the temperature. thistemperature reason, islargely perceived almost as varies a constant In theforcing. same way, expect indoor follows the sinusoidal trend oftemperature. thethe outdoor thermal On the higher that theFor thickness s is, the it less the indoor temperature following external sinusoidal trend. thethe contrary, thes is, solid composed by masonry varies and wood greatlythe impact on sinusoidal the outdoortrend. higher thickness thewalls less the indoor temperature following external

temperature. ForSensitivity this reason, it is perceived almost as a constant temperature. In the same way, the 6. CFD Results: Analysis for Different Geometrical Parameters and Boundary Conditions

6. CFD Results: Sensitivity for Different Geometrical Parameters andsinusoidal trend. higher the thickness s is, theAnalysis less the indoor temperature varies following the external Boundary Conditions 6.1. Materials Sensitivity Test 6. CFD Results: Sensitivity Analysis for Different Geometrical Parameters and Boundary Conditions The materials described in Table 3 have been used for the solid wall of Case 1. The sinusoidal temperature Tsin described in the previous chapter has been applied on the external side of the solid 6.1. Materials Sensitivity Test wall;materials the temperature of theinright wall of the been fluid environment Twall =wall 293.15 while and The described Table 3 have used for theissolid of K, Case 1. the Thetop sinusoidal The materials described in Table 3 have been used for the solid wall of Case 1. The sinusoidal the bottom walls have been set adiabatic. temperature Tsin described in the previous chapter has been applied on the external side of the solid temperature Tsinshows described in the previous chapter hasend been onand thein external side of of the the solid Figure 14 the right temperature at the of applied the wall the center wall; the temperature of the wall of trends the fluid environment is Twall = 293.15 K, while thefluid top and wall; thefor temperature of the right wall of compared the fluid environment is Twallsinusoidal = 293.15 K, while the top and domain different solid wall materials with the external temperature. the bottom walls have been set adiabatic. the bottom walls have been set adiabatic. Figure 14 shows the temperature trends at the endend of the wallwall andand in the center of the fluid domain Figure 14 shows the temperature trends at the of the in the center of the fluid for different solid wall materials compared with the external sinusoidal temperature. domain for different solid wall materials compared with the external sinusoidal temperature.

6.1. Materials Sensitivity Test

Figure 14. Case 1: Temperature trends in probe_wall (a) and probe_fluid (b).




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The numerical results confirm the analytical data shown in Tables 3 and 4, underlining the relationship between thermo-physical properties of building materials and architectural elements. The characteristics of the sinusoidal wave at the end of the wall and in the center of the fluid domain have been derived from the numerical results and analyzed in-depth in Table 6. Table 6. Case 1: Characteristics of the sinusoidal wave in probe_wall and probe_fluid. Aluminum Tsin σ τr θ Tm

5.0 298.10

Masonry

Wood

probe_wall probe_fluid probe_wall probe_fluid probe_wall probe_fluid 99.15 0.55 4.96 298.13

49.14 1.23 2.46 295.67

1.52 18.05 0.08 296.39

0.76 18.60 0.04 294.76

0.01 31.14 0.00 294.96

0.01 31.26 0.00 294.04

As expected, the semi-amplitude θ of the thermal wave decreases between x = 0.0 m and x = 2.5 m. This decrement is easily shown in the aluminum and the masonry cases, while it is very slight in the wood wall. After passing through the masonry and wood walls, the sinusoidal wave is almost totally damped. On the contrary, in the case of the aluminum, the fluid environment damps the sinusoidal wave more than the wall. In this case, the thermal wave Tsin quickly passes through the solid domain and it is subject to a small damping, because of the high thermal conductivity of the material. Therefore, the resulting thermal wave dissipates a great part of its energy in the fluid environment, which is characterized by a lower thermal conductivity. For the other two materials, Tsin arrives into the fluid environment with almost completely lowered oscillations. This behavior is noticeable also in the variation of Tm for the three materials, which decreases with the decrease of the thermal conductivity. Higher energy levels correspond to higher Tm , as it happens for aluminum. Finally, regarding the phase lag, masonry and wood have similar behavior. The phase lag, indeed, remains almost the same between the end of the wall and the center of the fluid domain in both cases. On the contrary, the phase lag increases for aluminum. This phenomenon is probably due to the influence of Tsin on the indoor environment. In the cases where masonry and wood have been used, the entering temperature is almost constant and has almost the same values. Therefore, it is easy to suppose that the fluid field is almost the same. In the aluminum case, indeed, the great oscillation of the entering temperature causes different fluid motions, which take to different thermo-fluid dynamic behavior. The comparison between the numerical results in probe_wall and the analytical data (Table 4 and Figure 12) shows how the fluid environment is able to limit the energy losses from the solid wall. In the comparison with the analytical results, the phase lag is lower and the thermal damping factor and the semi-amplitude are higher in the numerical simulations. This phenomenon is emphasized by the decrease of the thermal conductivity of the material, which is related to a lower energy level dissipated by the wall and, consequently, to lower heat levels transferred to the fluid environment. This suggests that the analytical solution for the semi-infinite solid is too approximated for the case of a wall adjacent to a fluid environment. 6.2. Thickness Sensitivity Test Three different wall thicknesses s have been modeled in Case 2. The applied thermal boundary conditions are the same used in Case 1, while the material has been set as aluminum (Table 3); this choice has been made since the other two materials greatly damp and delay the external sinusoidal wave, approximating it to a constant temperature. For this reason, the indoor thermal variations are too small to be perceived. The obtained temperature trends are shown in Figure 15.

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Figure Case Temperaturetrends trends in in probe_wall probe_fluid (b). (b). Figure 15. 15. Case 2: 2: Temperature probe_wall(a)(a)and and probe_fluid

The characteristics of the sinusoidal wave at the end of the wall and in the center of the fluid

The characteristics of thefrom sinusoidal waveresults at the(Table end of7).the wall and in the center of the fluid domain have been derived the numerical domain have been derived from the numerical results (Table 7). Table 7. Case 2: Characteristics of the sinusoidal wave in probe_wall and probe_fluid.

Table 7. Case 2: Characteristics of the sinusoidal wave in probe_wall and probe_fluid. s = 0.1 m s = 0.5 m s = 1.0 m probe_wall probe_fluid probe_wall probe_fluid probe_wall probe_fluid Tsin σ 99.66 s = 0.1 m49.41 99.15 s = 0.5 m 49.14 93.84s = 1.0 m 46.45 probe_wall probe_fluid probe_wall probe_fluid probe_wall probe_fluid τr - Tsin 0.04 0.74 0.55 1.23 1.66 2.43 ϴ σ 5 4.98 2.47 4.96 2.46 4.69 2.32 99.66 49.41 99.15 49.14 93.84 46.45 298.13 295.67 298.13 295.67 298.12 295.66 Tm τ r 298.150.04 0.74 0.55 1.23 1.66 2.43 θ 5 4.98 2.47 4.96 2.46 4.69 2.32 TComparing 298.15 the numerical 298.13 295.67and the analytical 298.13 295.67 298.12 295.66 m results data (Table 5 and Figure 13), the trend of

thermal damping factor at thickness variation is the most evident phenomenon. Based on the analytical data, the sinusoidal wave is expected to be highly damped with the increase of thickness. Comparing the numerical results and the analytical data (Table 5 and Figure 13), the trend of Despite that, the thermal damping factor is almost the same in the numerical results in both thermal damping factor at thickness variation is the most evident phenomenon. Based on the analytical probe_wall (90%–100%) and probe_fluid (45%–50%). Furthermore, unlike Case 1, the numerical data, values the sinusoidal expected to be of highly damped with the increase of thickness. Despite of phasewave lag inisthe same point the wall are higher than the analytical ones, and this that, the thermal damping factor is almost the same in the numerical results in both probe_wall (90%–100%) difference is emphasized consequently to the increase of the thicknesses s, passing from an increase and probe_fluid (45%–50%). 1, the values of phase lag in the of thermal lag of 6% (s = Furthermore, 0.1 m) to 80% unlike (s = 1.0Case m). On thenumerical contrary, thermal damping factor andsame pointsemi-amplitude of the wall areare higher the analytical ones, and difference is emphasized lowerthan compared to the analytical ones.this Also in this case, the difference consequently increases the increase thickness s. These differences byof thethermal convective them) to to thewith increase of theofthicknesses s, passing from are an caused increase lagphenomena of 6% (s =in0.1 fluid environment, which are not taken into account in the analytical calculation. 80% (s = 1.0 m). On the contrary, thermal damping factor and semi-amplitude are lower compared The numerical suggest higher the increases thickness is,with the lower the energy of to the analytical ones. results Also also in this case,that thethedifference the increase oflevels thickness s. the entering wave are. This phenomenon causes lower heat exchanges among the fluid particles of These differences are caused by the convective phenomena in the fluid environment, which are not the enclosure, damping less the temperature. On the contrary, basing on the same assumption, a taken into account in the analytical calculation. lower thickness allows higher energy levels of the entering wave and a consequent higher heat The numerical resultsdamping also suggest the higher the thickness is, the lower the energy levels of exchange and a higher of thethat temperature. the entering wave are. This phenomenon causes lower heat exchanges fluid particles Also in this case, it is possible to assume that the analytical solutionamong for the the semi-infinite solid isof the enclosure, damping less thecase temperature. On thetocontrary, basing on the same assumption, a lower too approximated for the of a wall adjacent a fluid environment.

thickness allows higher energy levels of the entering wave and a consequent higher heat exchange and 6.3. External Sinusoidal Temperature Tsin Sensitivity Test a higher damping of the temperature. AlsoThe in this case, possible to assume that concern the analytical solutionoffor semi-infinite solid is third andit is fourth sensitivity analyses the variation thethe thermal boundary conditions, while properties the solid wall have been left unaltered. In the too approximated for the the thermo-physical case of a wall adjacent to of a fluid environment. third case the sinusoidal wave Tsin applied on the external surface of the solid wall has been changed

6.3. External varying Sinusoidal the initial TTemperature m,0, while Θ0T =sin5 Sensitivity K and ω = Test 7.3 × 10−5 rad/s have been left unaltered in all the simulations. Three Tm,0 have been considered: 295.15 K, 298.15 K and 301.15 K. The temperature on

The third and fourth sensitivity analyses concern the variation of the thermal boundary conditions, the right wall of the fluid environment is Twall = 293.15 K, while the top and the bottom walls have whilebeen the thermo-physical properties of the wall have been3)left Inof the set adiabatic. The material has been setsolid as aluminum (Table andunaltered. the thickness thethird wall case s is the sinusoidal wave Tsin applied on the external surface of the solid wall has been changed varying the equal to 1.0 m. −5 rad/s have been left unaltered in all the simulations. initial Tm,0The , while Θ0 =temperature 5 K and ωtrends = 7.3 × obtained are10 shown in Figure 16. Three Tm,0 have been considered: 295.15 K, 298.15 K and 301.15 K. The temperature on the right wall of the fluid environment is Twall = 293.15 K, while the top and the bottom walls have been set adiabatic. The material has been set as aluminum (Table 3) and the thickness of the wall s is equal to 1.0 m. The obtained temperature trends are shown in Figure 16.

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Figure 16. Case 3: Temperature trends in probe_wall (a) and probe_fluid (b). Figure 16. Case 3: Temperature trends in probe_wall (a) and probe_fluid (b).

The characteristics the3:sinusoidal wave at in theprobe_wall end of the(a)wall in the center Figure 16. of Case Temperature trends and and probe_fluid (b). of the fluid The characteristics of the sinusoidal wave at the end of the wall and in the center of the fluid domain have been derived from the numerical results (Table 8). domain have been derived from the numerical results (Table 8). The characteristics of the sinusoidal wave at the end of the wall and in the center of the fluid Case 3: Characteristics of the sinusoidal wave in8). probe_wall and probe_fluid. domain haveTable been8.derived from the numerical results (Table

Table 8. Case 3: Characteristics of the sinusoidal wave in probe_wall and probe_fluid. Tm,0 = 298.15 Tm,0 = 301.15 Tm,0 = 295.15 Table Case 3: Characteristics of theprobe_wall sinusoidal wave in probe_wall and probe_fluid. probe_fluid probe_fluid probe_wall probe_fluid Tsin 8. probe_wall Tm,0 = 295.15 Tm,0 = 298.15 Tm,0 = 301.15 σ 93.86 46.29 93.84 46.45 93.84 46.48 Tm,0 = 298.15 Tm,0 = 301.15 Tm,0 = 295.15 probe_wall probe_fluid probe_wall probe_fluid probe_wall probe_fluid τr Tsin1.59 2.35 1.66 2.43 1.59 2.36 probe_wall probe_fluid probe_wall probe_fluid probe_wall probe_fluid Tsin 4.69 2.32 4.69 2.32 4.6993.84 2.32 46.48 σ ϴ -5 93.86 46.29 93.84 46.45 σ 93.86 46.29 93.84 46.45 93.84 46.48 295.14 294.17 298.12 295.66 301.10 297.172.36 τ r Tm 298.15 1.59 2.35 1.66 2.43 1.59 τr 1.59 2.35 1.66 2.43 1.59 2.36 θ 5 4.69 2.32 4.69 2.32 4.69 2.32 ϴ 5 4.69 2.32 4.69 2.32 4.69 2.32 Tm Varying 298.15Tm,0 of the 295.14 295.66 in probe_wall 301.10 and probe_fluid 297.17 sinusoidal294.17 wave Tsin, the298.12 temperature trends Tm 298.15 295.14 294.17 298.12 295.66 301.10 297.17

are damped and delayed in the same way, since the thermo-physical properties of the building materials and elements are the same. T The, only noticeable difference is the variation of Tm whichare Varying inin probe_wall and probe_fluid VaryingTm,0 Tm,0of ofthe thesinusoidal sinusoidalwave wave sin Tsin,the thetemperature temperaturetrends trends probe_wall and probe_fluid increases the Tm,0 external temperature. Furthermore, Tproperties m does not increase proportionally damped andwith delayed in of thethe same thermo-physical of the building materials are damped and delayed in the way, samesince way,the since the thermo-physical properties of the building to Tm,0 since it is influenced also by Twall. The results underline an interesting relation applicable in the and elements are the same. The only noticeable difference is the variation of T which increases with m materials and elements are the same. The only noticeable difference is the variation of Tm which analyzed case: the temperature Tm of the environment can be calculated, as the average between the the T of the external temperature. Furthermore, T does not increase proportionally to T since it increases with the Tm,0 of the external temperature. Furthermore, Tm does not increase proportionally m m,0 m,0 external Tm,0 and the fixed Twall basing on the distance from the considered walls. This phenomenon is istoinfluenced also by Twall . The results underline an interesting relation applicable in the analyzed Tm,0 since it is influenced also by Twall . The results underline an interesting relation applicable in case: the not computable when the analytical solution is applied. the temperature Tm temperature of the environment canenvironment be calculated, asbe thecalculated, average between the external Tm,0 the and analyzed case: the Tm of the can as the average between the fixed TTwall basing on theTwall distance from thedistance considered This phenomenon is phenomenon not computable external m,0 and theTemperature fixed basing on the fromwalls. the considered walls. This is 6.4. External Fixed Twall Sensitivity Test when the analytical solution is applied. not computable when the analytical solution is applied. Finally, the influence of the temperature Twall on the interior environment has been investigated. Thermo-physical properties T of materials andTest building elements are the same as Case 3, while the 6.4. Sensitivity Test 6.4.External ExternalFixed FixedTemperature Temperature Twall wall Sensitivity sinusoidal temperature Tsin follows Equation (2), where Tm = 295.15 K, Θ0 = 5 K and ω = 7.3 × 10−5 Finally, the influence of on the theKinterior interior environment hasbeen beeninvestigated. investigated. Finally, the influence ofthe thetemperature temperature wall 303.15 on environment has rad/s. Three Twall have been considered: 293.15TTwall K, and 323.15 K. Thermo-physical properties of materials and building elements are the same as Case 3, while Thermo-physical properties of materials and building elements are the same as Case 3, while the The obtained temperature trends are shown in Figure 17. −5 the sinusoidal temperature T follows Equation (2), where T = 295.15 K, Θ = 5 K and sinusoidal temperature Tsin follows Equation (2), where Tm = 295.15m K, Θ0 = 5 K and ω 0 = 7.3 × 10 sin − 5 ωrad/s. = 7.3 Three × 10 Twall rad/s. Twall have been considered: 293.15 K, 303.15 haveThree been considered: 293.15 K, 303.15 K and 323.15 K. K and 323.15 K. The Theobtained obtainedtemperature temperaturetrends trends are are shown shown in in Figure Figure 17.


The characteristics of the sinusoidal wave at the end of the wall and in the center of the fluid domain have been derived from the numerical results (Table 9). Figure 17. Case 4: Temperature trends in probe_wall (a) and probe_fluid (b). Figure 17. Case 4: Temperature trends in probe_wall (a) and probe_fluid (b).

The characteristics of the sinusoidal wave at the end of the wall and in the center of the fluid domain have been derived from the numerical results (Table 9).


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The characteristics of the sinusoidal wave at the end of the wall and in the center of the fluid domain have been derived from the numerical results (Table 9). Table 9. Case 4: Characteristics of the sinusoidal wave in probe_wall and probe_fluid. Twall = 293.15 K

σ τr θ Tm

Twall = 303.15 K

Twall = 323.15 K

Tsin

probe_wall

probe_fluid

probe_wall

probe_fluid

probe_wall

probe_fluid

5 298.15

93.86 1.59 4.69 295.14

46.29 2.35 2.31 294.17

93.84 1.60 4.69 295.20

45.99 2.40 2.30 299.19

93.83 1.61 4.69 295.32

51.02 2.24 2.55 309.20

The results show the same physical phenomena observed in Case 3. The indoor temperatures, indeed, are characterized by the same thermal damping factors and phase lags, and their Tm is function of the average temperature of Tsin and Twall . This consideration is helpful when materials with low thermal damping factors and high phase lags are taken into account (such as the previously analyzed wood and masonry). In that case, indeed, the external sinusoidal temperature is perceived inside as a constant temperature equal to Tm and the indoor temperature is equal to the average of the all the external temperatures considered. As for the previous cases, this phenomenon is not computable when the analytical solution is applied. 7. Conclusions The indoor microclimate of buildings is influenced by several parameters. Especially when the indoor conditions are subject to an outdoor transient thermal forcing and they are free to evolve in a periodic stabilized regime, the heat transfer phenomena are very complex. For this reason, several studies use some simplifications in the investigation of these physical problems. To the best of our knowledge, an exact solution to solve this phenomenon does not exist also from the analytical point of view. The aim of this research is to propose a first approach to solving the thermal phenomena inside environments where the temperature is free to evolve, such as buildings not equipped with HVAC systems. The coupling of conduction and convection inside a simplified room has been investigated through Computational Fluid Dynamics. Different preliminary analyses have been taken into account in order to validate the numerical results and to evaluate possible discrepancies between the 3D geometry of the room and its simplification in a 2D numerical model. The numerical model has been studied taking into account four configurations and performing parametric analysis in order to understand the response of the indoor environment when thermal inputs and thermo-physical properties of the envelope change. In the first and the second configurations (Case 1 and Case 2), the thermo-physical properties of the solid domain have been varied considering three different construction materials (aluminum, masonry and wood) and three different thicknesses of the wall (0.1 m, 0.5 m and 1 m.) The results of these analyses underline a strong dependence of the indoor temperature on thermo-physical properties of the solid envelope. Furthermore, the entering thermal wave is also greatly influenced by the energy exchanges with the fluid particles in the indoor environment. These heat exchanges are shown in terms of decrease of the thermal damping factor related to the variation of thermal conductivity. In the third and fourth sensitivity analyses (Cases 3 and Case 4), the thermal boundary conditions have been varied considering three Tm,0 of the sinusoidal temperature (295.15 K, 298.15 K and 301.15 K) and three different Twall (293.15 K, 303.15 K and 323.15 K). In these cases, the results underline that the thermal damping factor and the phase lag do not change with these variations, while the Tm in the middle of the fluid environment (probe_fluid) changes proportionally to the average of Tm,0 and Twall . The comparison between the obtained numerical data and the analytical ones for the four cases have shown that the case of the semi-infinite solid is too approximate to correctly describe


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the temperature trend inside a fluid environment adjacent to a wall in a periodic stabilized regime. More in detail, the analytical calculation does not take into account the convective phenomena in the fluid environment resulting from the heat levels exchanged with the wall. This situation is clear when the thermo-physical characteristics of the solid vary, especially in Case 2, where the phase lag and thermal damping factor do not change as expected from the analytical calculations. For this reason, further developments of the research are needed in order to formulate more accurate predictions of the periodic stabilized regime. This work is a first step to better understanding these physical phenomena and it would lay the groundwork to develop analytical correlations to predict the trend of the temperature inside an environment free from thermal control and subject only to the varying external thermal conditions. Author Contributions: The study has been conceived and designed by Fabio Nardecchia. Fabio Nardecchia, Francesca Pagliaro and Benedetta Mattoni have elaborated the case studies, performed the numerical simulations and analyzed analytical and numerical data. Lucia Cellucci has done the graphical works. Fabio Nardecchia, Francesca Pagliaro and Benedetta Mattoni have written the paper. English corrections have been undertaken by Francesca Pagliaro and Benedetta Mattoni. Franco Gugliermetti and Fabio Bisegna have revised the research work and the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

Nomenclature cp erf erfc D g h H

specific heat at constant pressure (J/kgK) Gaussian error function complementary error function, defined as er f c ≡ 1 − er f depth (m) gravitational acceleration (m/s2 ) convection heat transfer coefficient (W/m2 K) height (m)

k

turbulent kinetic energy (m2 /s2 ), defined as k = pressure (Pa) Rayleigh number thickness (m) temperature (K) fluid velocity component (m/s) thermal transmittance (W/m2 K), defined as U =

p Ra s T u U

W width (m) x,y,z spatial coordinates Greek letters thermal diffusivity (m2 /s), defined as α = α q π phase shift (rad), defined as β = ατ βx 0 Θ λ µ ρ σ τ

λ ρc p

semi-amplitude of a sinusoidal temperature thermal conductivity (W/mK) dynamic viscosity (kg/s m) mass density (kg/m3 ) thermal damping factor, defined as σ = e− βx time (s) q

τr

phase lag (h), defined as τr =

ω

angular velocity (rad/s)

βx ω

=

x 2

τ0 απ

ui0 u0j 2

1

1 s 1 h1 + λ + h2


Subscript 0 E i s m ∞ Overbar ¯

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initial condition external index; internal surface mean value free stream condition average conditions

References 1. 2. 3.

4.

5.

6.

7.

8.

9. 10. 11. 12. 13. 14. 15. 16.

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