Development Of An Integrated

Energy and Buildings 35 (2003) 375–382

Development of an integrated dynamic thermal bridging assessment environment Abdullatif E. Ben-Nakhi* College of Technological Studies, P.O. Box 3665, 22037 Salmiya, Kuwait Received 10 September 2001; accepted 12 July 2002

Abstract Accurate thermal bridging assessment is becoming more important not only to predict the peak thermal load and the year round heat flow, but also to estimate the potential for condensation and mould growth in the heating season. This paper presents a new dynamic thermal bridging assessment module that is integrated within a state-of-the-art, whole building simulation environment in order to have more realistic boundary conditions. It integrates all inter-related energy subsystems that occur in buildings. From another point of view, it is a variable resolution whole building simulation program that allows efficient assessment of thermal bridging. In order to encourage its employment, the integrated environment had undergone rigorous validation tests and was furnished with a user-friendly interface and other user-friendly features, such as the online help and exemplars. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Built environment; Thermal bridging; Numerical modeling; Simulation; Energy; Validation

1. Introduction In general, heat flow through building construction is a 1D (i.e. in the direction perpendicular to the surface) because thermal conductance and temperature differential in the perpendicular direction are much greater than that in the lateral directions. However, localized multi-dimensional heat conduction through building envelope is common. In some buildings, up to 50% of the elevation area is subjected to multi-dimensional heat conduction [1,2]. Recent publications have shown that thermal resistance of building envelope can be significantly reduced by thermal bridging through framing at corners and junctions [3–6]. Thermal bridge is the part of building envelope through which heat conduction is multi-dimensional. The multi-dimensional character of heat conduction affects the local temperature distribution and heat flow rate. In other words, thermal bridging will make the internal surface temperature nearer to the other side environment temperature and causes a higher rate of heat flow between the two environments. While the internal surface temperature should be considered in mould growth and condensation risk assessments during the heating season, higher heat flow rate should be taken into * Tel.: þ965-562-2927; fax: þ965-561-8866. E-mail address: [email protected] (A.E. Ben-Nakhi).

account during the design of buildings and their year round environmental control systems. In general, buildings have several thermal bridges, which occur due to one or more of the following reasons. 1. Change in thermal properties of building envelope in the lateral direction (e.g. interface between concrete beam and cement block). 2. Change in construction thickness (e.g. a window within a wall). 3. Difference between internal and external surfaces areas (e.g. edges and corners). 4. Heat generation within building construction (e.g. hot water pipe). In addition, the legislation and energy awareness have led to increase insulation levels in buildings, which implies increased thermal resistance in the perpendicular direction. Of course, the addition of insulation layers in building constructions does not affect the lateral thermal resistance for the remaining layers in the construction. Consequently, the difference between thermal conductance in the perpendicular and lateral directions is reduced. Hence, the potential for multi-dimensional heat conduction in building constructions is increased. Furthermore, the severity of the thermal bridging due to the four reasons mentioned above has increased.

0378-7788/02/$ – see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 7 7 8 8 ( 0 2 ) 0 0 1 0 6 - 8

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Nomenclature A cp  g hc hr Nb Nm t T T V x, y, z

area (m2) specific heat (kJ/(kg K)) heat source term (W/m3) convective heat transfer coefficient (W/(m2 K)) radiative heat transfer coefficient (W/(m2 K)) number of homogeneous boundary conditions number of homogeneous materials in a control volume time (s) temperature (8C) initial temperature (8C) volume (m3) locations in the x, y and z directions (m)

Greek symbols b, g, Z eigenvalues l thermal conductivity (W/(m K)) r density (kg/m3) Traditionally, facilities available to designers for assessing thermal bridging have involved either the use of guidebooks or general purpose numerical analysis tools [7], such as TRNSYS [8]. The former suffers from the handicap that the design details in question do not necessarily match the details in the guidebooks. The latter can be time consuming to set up, unable or difficult to model multiple heterogeneous dynamic boundary conditions for a domain and are not dynamically integrated with other common mass and heat transfer processes in the building. There are several recent initiatives aimed at improving this situation. These approaches can be divided into two main groups. The first group encompasses attempts to computerize thermal catalogues and guidebooks. An example of the first group is the KOBRU86 software [9], which allows for 2D steady state thermal bridge analysis. KOBRU86 combined with the EUROKOBRA database, consisting of more than 3000 thermal bridge details, provide a user-friendly way to analyze building thermal bridges [7]. Although this approach simplified domain definition, the problem of boundary conditions was not completely tackled, as the boundaries and thermal sources and sinks have to be explicitly defined before performing simulation. This can be a major source of error since buildings and their environmental control systems are inherently complex (multi-dimensional, dynamic, highly interactive and non-linear) [10]. The second group includes endeavors to add 3D conduction capabilities for existing building energy simulation packages [11–16]. They differ in the employed numerical scheme (thermal response factor, finite element method (FEM), finite difference method (FDM), control volume, etc.) and extent of integration between 3D conduction module and the associated whole building energy simulation package.

Although the thermal response factor method is free from numerical instabilities and is fast [17], the control volume approach is used in the present work because of its simplicity of formulation and physical elegance. In addition, the numerical instabilities associated with control volume technique are controlled by using fully implicit schemes and the computation time is tackled by the advances in computing power and speed. Furthermore, because of its physical significance, control volume approach facilitates high level of integration between the 3D conduction module and the associated energy simulation tool. On the other hand, the analytical determination of thermal response factors for the 2D/3D heat conduction system is extremely difficult, instead they are determined based on the comprehensive 2D/3D FEM or FDM heat conduction calculations or based on measured heat conduction data by a calibrated hot box [12,14–17]. The 3D conduction module introduced by Ben-Nakhi [11] mainly differs from those offered by others in its degree of integration with whole building simulation environment. It is based on unstructured1 mesh, which inherits gridding flexibility, but requires high CPU effort and space. However, based on several years’ experience, gridding flexibility is usually not required in building’s thermal bridging assessment. In addition, the control of truncation error in that module is difficult. Because the truncation errors for a changing mesh spacing is related to the rate of space step change [18], which may be coarse in that module. Furthermore, the adopted scheme may have high round-off errors because of the great relative difference in the magnitudes of the system matrix coefficients. Accordingly, the present work is invoked in order to develop an adaptive thermal bridging assessment tool. The adaptive tool should be able to accurately model building elements and involve minimal user exertion, and CPU effort and space. This is mainly achieved by incorporating structured 3D dynamic conduction capabilities into an existing building energy simulation environment. In order to reduce CPU effort and space, sparse matrix storage and solution techniques are used. The developed module is considered to be of the second group of attempts to improve thermal bridging assessment tools. It has the same level of integration with the other mass and heat flow paths as that introduced by Ben-Nakhi [11]. The developed module prevails over that introduced by Ben-Nakhi [11] in some points, such as the truncation and round-off errors are lower and easier to control, and it requires less CPU effort and space. After introducing the theoretical background for 3D dynamic heat flow through multi-layered construction, the integration of the 3D module into a state-of-the-art building simulation package is presented. Then, the numerical and analytical validations of the developed tool are performed. Finally, conclusions are offered. 1

In unstructured mesh, the identification of grid points should be individually specified and they are not associated with an orderly defined grid lines.

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2. Theoretical background Usually, heat conduction occurring in buildings is transient. That is because of the changing boundary conditions, which are affected by the outside climate, plant operation, occupants’ activity, etc. The differential equation of heat conduction can be written as rcp

Nb X @Tð~ r; tÞ ¼  r ~ qi þ gð~ r; tÞ @t i¼1

(1)

Several investigators offer many methods for the numerical formulation of Eq. (1). The control volume approach is adopted in the present work because of its physical elegance, formulation simplicity and flexibility. There are several possible schemes for the positioning of control volumes and their associated grid points [18]. The approach adopted in the current work is based on putting one node on each material or boundary interface. Then positioning additional nodes in between according to the required resolution. After that, the control volume surfaces are located midway between grid points. This approach ensures continuity in the boundary conditions throughout each control volume surface and continuity in the thermal conductance between nodes. The control volume formulation is achieved by integrating the associated partial differential equation, i.e. Eq. (1), over a small control volume. Accordingly, for a rectangular parallelepiped control volume, with heterogeneous material and uniform boundary at each surface we have Nm X

ri ci Vi

i¼1

6 @T X ¼ As qs þ V g @t s¼1

(2)

where the heat fluxes (qs ) can be due to heat conduction, convection or radiation, which are defined by qcond ¼

lj!i ðTj  Ti Þ Dxj!i

(3)

qconv ¼ hc ðTj  Ti Þ

(4)

qrad ¼ hr ðTj4  Ti4 Þ

(5)

Based on the foregoing theory, a structured 3D gridding module has been developed. The main advantage of this module is that it enables localized 3D modeling. That is, after integrating the module into a whole building simulation environment, a building can be modeled as a 1D problem, except for parts of it that are represented by 3D model. The integration of the new module within a state-of-the-art simulation environment in presented in Section 3.

3. Implementation The developed multi-dimensional conduction modeling module is based on the control volume technique which is established according to the energy conservation law. For

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building energy simulation, the energy conservation law should be combined with Fourier’s law of heat conduction in solids, Newton’s law for convection between the internal and external air and the wall, and Stefan–Boltzmann’s law for heat exchange by radiation between wall surfaces and the surrounding surfaces. In addition, several heat generation (source and sink) modes, such as plant interaction and radiation absorption should be coupled. It is out of the scope of this paper to present the theory behind the building energy simulation represented here by the ESP-r, which is well established and reported in details elsewhere [19]. However, a brief description of the ESP-r environment is necessary to present the implementation of the developed module within ESP-r. ESP-r is a tool for the transient simulation of heat and fluid flow within combined building/plant systems with control imposed. The structure of ESP-r is based on several integrated modules of which three are fundamental [20]. The three fundamental modules in ESP-r are Project Manager, Simulator and Results Analyzer. By means of Project Manager, a simulation problem is defined by a set of data files whose names and locations are saved in a single system configuration file. By defining the system configuration file name to the Simulator, it will represent the problem by its equivalent network of time-dependent thermal resistances and capacitances subjected to dynamic potential differences. By performing a simulation, the Simulator creates a result file that is analyzed by the Results Analyzer. Accordingly, three levels of integration between the developed module and the ESP-r environment are considered. These levels are problem definition, simulation and results analysis (Fig. 1). Since ESP-r is equipped with an advanced gridding module called grd, the required data for the structured 3D module are defined within grd. However, the 1D problem should be defined within the ESP-r environment first. This is to minimize data entry for the 3D domain and to ensure data resemblance between the 1D and 3D models. The developed module deals with one zone at a time. Each zone is divided into local components and one imported domain. The local components are the default ESP-r constructions composing the zone, such as walls. The imported domain is the 3D model. These two domains are linked during the simulation process within a single system matrix. For the purpose of clearness, the default ESP-r components and schemes are named 1D, and 3D is used to identify the 3D module constituent. Here, it is important to point out that the default ESP-r schemes are 3D, except heat flow through building constructions is locally 1D in the direction perpendicular to each construction. However, the default ESP-r schemes are denoted 1D in this paper in order to distinguish them from their counterpart schemes in the 3D conduction module. When the 3D module is invoked, the associated schemes for all inter-related energy processes within buildings should be modified to provide the proper boundary conditions and

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Fig. 1. A flow chart showing the integration of the new module within the ESP-r simulation environment.

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heat source or sink terms. For example, the default ESP-r’s scheme for internal long-wave radiation can be put in the form q2!1 ¼

e1 e2 sðA2 f2!1 T24  A1 f1!2 T14 Þ 1  ð1  e1 Þð1  e2 Þf1!2 f2!1 N X e1 e2 ð1  ei ÞsA2 f2!i fi!1 T24 þ 1  ð1  e1 Þð1  e2 Þð1  ei Þf1!2 f2!i fi!1 n¼1 

N X

e1 e2 ð1  ei ÞsA2 f1!i fi!2 T14 1  ð1  e1 Þð1  e2 Þð1  ei Þf2!1 f1!i fi!2 n¼1

where f1!2 is the geometric view factor between surface nodes 1 and 2, and N the total number of internal surface nodes. Therefore, its counterpart in the 3D module, which is based on the same theory, should be capable of dealing with internal surfaces represented by multiple nodes. Accordingly, the internal nodes topology (its internal surface shape, location and emissivity) should be saved for the 3D module, while for the 1D case the required topology is obtained directly from the details of zone constructions. The local (i.e. 1D module) components can be set as defined or not defined. The undefined components are those components that will be replaced by the imported domain. Therefore, in order to model 3D heat flow through, say, the east and south walls, they should be set as undefined local components and a 3D model of the two walls and the edge in between should be defined and imported. The imported domain is defined with respect to a Cartesian coordinate system. The definition of the imported domain requires three groups of data: grid data, material geometry and boundary conditions. The required grid data for the imported domain are the employed length unit (e.g. mm or cm), number of gridding lines and the distance between each two successive grid lines in each dimension. The group of grid data is used to facilitate a high level of gridding flexibility. The internal and external boundaries are referenced to existing dynamic boundaries in the 1D problem. Similarly, the thermophysical properties of the imported domain are defined by referencing them to existing layers within the 1D building constructions. The simulation of a problem within the Simulator is performed in a three-stage process: discretization of the problem, derivation of the simulation equation for the nodal system and simultaneous solution of the derived characteristic equations. The default ESP-r space discretization approach is based on 1D heat conduction through building constructions. Accordingly, each inter-constructional node has two heat conduction connections. However, construction surface nodes have only one conduction connection. Depending on the boundary conditions, the other connections for the construction surface node are defined. For example, the internal surface nodes have one convection connection with the zone air node and radiation connection with other internal surface nodes. Furthermore, a climate data file defines the building external boundary variables.

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With respect to the employed discretization, a system matrix is created. In this matrix each node (i.e. 1D and 3D) is represented by one equation. These equations are then solved simultaneously with respect to the invoked control law. Two solution methods are available: direct and iterative. The iterative method is the default one, as it requires less storage space and it produces less round-off errors. The adopted direct and iterative solution methods are Gauss’s elimination method and the Gauss–Seidel method, respectively. The adopted Gauss–Seidel method incorporates linear under-relaxation factor. For the defined local components, the default ESP-r space discretization will be employed and their associated characteristic equations are created in the system matrix. The undefined local components will not be directly represented in the system matrix. The imported 3D domain replaces them. As in the 1D gridding, the internal surface nodes are connected with the space air node by convection and connected with each other by radiation. For the internal radiation calculations, the 1D view factors are employed after area weighting the 1D values and setting to zero the view factors between nodes within the same surface. The fully implicit discretization scheme is employed when the 3D modeling is invoked. This is because the fully implicit scheme is unconditionally stable, the coefficient generation process requires less CPU effort as compared with other implicit schemes, and the amplification factor is always positive, hence prevents oscillation in the results, as shown in Fig. 4. Based on the results obtained from the 3D module, the default 1D temperatures for the undefined local components are estimated either directly for the 1D constructions or by volume weighting for the 3D constructions. Because the 1D temperature distribution is required in the calculation of other thermal processes, such as short-wave radiation, heat absorption by transparent materials and convective heat transfer coefficient. At the result analyses level, the default options within the ESP-r environment, such as heat fluxes and temperature distribution can be used after 3D simulation. In addition, the new module allows monitoring the temperature profiles for several predefined nodes within the imported domain. This is an important option for condensation risk assessment. Because of the generality of the 3D module, it can be used to model heat transfer through the ground. This is a major error source in building energy simulation packages, which usually incorporate empirical ground temperatures at specified depths to approximate heat flow through the ground [21]. Furthermore, the 3D module is capable of dealing with heterogeneous thermophysical properties and boundary conditions for the ground. In order to encourage the usage of the developed module, it is furnished with the default ESP-r interface for the definition of the imported domain. In addition, online help and exemplar are also provided. Beside that, the validation of the developed module is performed and presented in Section 4.

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Fig. 2. A schematic diagram of the 3D multi-layered corner used in inter-model validation process.

4. Validation In general, validation processes fall largely into three categories: analytical verification, inter-model comparison and empirical validation [22]. In analytic tests, the predictions of programs are compared with the exact analytical solutions. In empirical validation the results from thermal programs are compared with the measurements made in buildings. In inter-model comparisons, the predictions of the program are compared with those of other programs, which usually, are of similar sophistication. The developed module was only validated by inter-model and analytical verifications because there was no empirical data available for validation. In inter-model comparison, the results of the developed 3D module, which was integrated within the ESP-r, were compared with the ESP-r’s default 1D analysis and VOLTRA2 [23] packages. First, the developed module was used to model the transient 3D heat flow through a building construction similar to the wall construction shown in Fig. 2. Since ESP-r deals with a whole building, the current wall was modeled by defining a single zoned building whose surfaces are adiabatic except for one wall. The boundaries at the lateral directions were set to be adiabatic. For the perpendicular direction, the internal ambient temperature was set to 24 8C and the external boundary was defined by the climate file of a typical meteorological year for Kuwait [24]. Therefore, the defined problem was of 1D nature even though a 3D gridding was employed. The main purpose of this task is to examine the validity of the integration of the 3D module within ESP-r. The heat flow rate for the 3D model was compared with that of an equivalent 1D model by ESP-r. The results matched up to two decimal digits. Regarding to the second inter-model comparison, the developed module within the ESP-r was invoked to compare with VOLTRA modeling accuracy. The problem modeled by VOLTRA was 3D transient heat conduction through the building corner as shown in Fig. 2, which is common in 2 VOLTRA is a tool developed by the Belgian Company, Physibel, for 3D transient heat conduction modeling.

the Middle East countries. As shown in Figs. 3 and 4, there are good agreement between the results of the two models. The minor differences in the results were due to the difference in the discretization schemes employed. While ESP-r is based on fully implicit scheme, VOLTRA incorporates Crank–Nicolson discretization scheme. The oscillation in the VOLTRA results is due to the nature of stability error associated with the Crank–Nicolson discretization scheme [25]. For 10 min time step, the oscillations were significantly dampened and better agreement was obtained between the

Fig. 3. A comparison between VOLTRA and ESP-r results for external corner temperature profile.

Fig. 4. A comparison between VOLTRA and ESP-r results for internal corner temperature profile.

A.E. Ben-Nakhi / Energy and Buildings 35 (2003) 375–382

381

ESP-r and the VOLTRA results. The results were not shown since they almost overlap each other. In the analytical verification, the results of ESP-r were compared with the exact solution of transient 3D heat conduction through a homogeneous slab. For a rectangular parallelepiped domain (0 x a, 0 y b and 0 z c) that is initially at 50 8C and for times t > 0, the boundaries are defined by l l

@T þ hout T ¼ 0 @x

@T þ hin T ¼ 0 @x

at x ¼ 0

(6a)

at x ¼ a

(6b)

@T ¼ 0 at y ¼ 0 @y l

@T þ hy T ¼ 0 @y

(6c) at y ¼ b

Fig. 5. A comparison between analytical and ESP-r results.

external ambient temperature was also intended to amplify the errors.

(6d) 5. Conclusions

@T ¼ 0 at z ¼ 0 @z l

(6e)

@T þ hz T ¼ 0 at z ¼ c @z

(6f)

Ozisik [26] has presented the general solution for multidimensional homogeneous heat conduction problems. For the current problem, the solution can be written as Tðx; y; z; tÞ ¼ 8T   

XXX

bm cosðbm xÞ þ ðhout =lÞ sinðbm xÞ e 2 2 ðbm þ ðhout =l2 ÞÞða þ ðhin =ðlðb2m þ ðh2in =l2 ÞÞÞÞÞ þ ðhout =lÞ ! cosðgn yÞ cosðZp zÞðg2n þ ðh2y =l2 ÞÞðZ2p þ ðh2z =l2 ÞÞ

ðbðg2n þ ðh2y =l2 ÞÞ þ ðhy =lÞÞðcðZ2p þ ðh2z =l2 ÞÞ þ ðhz =lÞÞ ! ðhout =lÞ þ bm sinðbm aÞ  ðhout =lÞ cosðbm aÞ sinðgn bÞ sinðZp cÞ bm gn Zp

bm ðhout þ hin Þ lðb2m  ðhout hin =l2 ÞÞ

gn tanðgn bÞ ¼

hy l

(8) (9)

hz (10) l The results of the analytical validation based on 500 eigenvalues of each of b, g and Z are shown in Fig. 5. A subroutine was developed for automatic estimation of the eigenvalues based on the sign count method [27]. The temperature profiles shown are for the center node of the external surface. The reason for selecting the external surface is because of the expected highest truncation error due to the highest temperature gradient in the time direction. In addition, the stepwise excitation from 50 to 0 8C in the Zp tanðZp cÞ ¼

!

aðb2m þg2n þZ2p Þt

where the eigenvalues bm , gn and Zp are the positive roots of the following equations tanðbm aÞ ¼

A new thermal bridging assessment module was developed and integrated within a state-of-the-art, whole building simulation environment (ESP-r). The developed tool is distinguished by the flexibility in domain definition, the lower numerical errors and the level of conflation with whole building simulation package, which facilitated more pragmatic assessment of thermal bridging through building constructions.

(7)

The developed thermal bridging assessment tool is furnished with effective CPU-related features, such as the capability of variable resolution simulation, employment of sparse storage and matrix solution techniques, automatic importing of all required 1D data from the default 1D configuration. In order to encourage the employment of the developed tool in practice, it was furnished with a user-friendly interface that is compatible with the whole building simulation package. In addition, both inter-model and analytical validations were performed to verify the adopted domain definition procedure. The tool was equipped with further user-friendly features, such as the online help and exemplars.

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