Frequency domain and finite difference modeling of ventilated concrete slabs and comparison with field measurements: Part 1, modeling methodology

International Journal of Heat and Mass Transfer xxx (2013) xxx–xxx

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International Journal of Heat and Mass Transfer

journal homepage: www.elsevier .com/locate / i jhmt

Frequency domain and finite difference modeling of ventilated concreteslabs and comparison with field measurements: Part 1, modelingmethodology

0017-9310/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.05.076

⇑ Corresponding author. Address: Dept. BCEE, Rm: EV-6.139, Concordia Univer-sity, 1455 de Maisonneuve Blvd. West. Montréal, Québec H3G 1M8, Canada. Tel.: +1514 848 2424x7244; fax: +1 514 848 7965.

E-mail addresses: [email protected] (Y. Chen), [email protected](A.K. Athienitis), [email protected] (K.E. Galal).

Please cite this article in press as: Y. Chen et al., Frequency domain and finite difference modeling of ventilated concrete slabs and comparison wimeasurements: Part 1, modeling methodology, Int. J. Heat Mass Transfer (2013), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.05.076

Yuxiang Chen ⇑, Andreas K. Athienitis, Khaled E. GalalConcordia University, 1455 De Maisonneuve Blvd. W., Montreal, Québec H3G 1M8, Canada

a r t i c l e i n f o a b s t r a c t

Article history:Available online xxxx

Keywords:Frequency responseLumped-parameter finite differenceTransfer functionActive thermal energy storageVentilated concrete slabBuilding-integrated thermal energy storage

This paper is the first of two papers that focus on the thermal modeling of building-integrated thermalenergy storage (BITES) systems using frequency response (FR) and lumped-parameter finite difference(LPFD) techniques. Structural/non-structural building fabric components, such as ventilated concreteslabs (VCS) can actively store and release thermal energy effectively by passing air through their embed-ded air channels. These building components can be described as ventilated BITES systems. To assist thethermal analysis and control of BITES systems, modeling techniques and guidelines for FR and LPFD mod-els of VCS are presented in this two-part paper. In this first part, modeling techniques for FR and LPFDapproaches based on network theory are presented. A method for calculating the heat transfer betweenflowing air and ventilated components is developed for these two approaches. Discretization criteria forexplicit LPFD models are discussed. For the FR approach, discrete Fourier series in complex frequencyform are used to represent the boundary excitations. In the treatment of heat injection from the flowingair as internal source in the VCS, network techniques such as Thévenin theorem, heat flow division, andY-diakoptic transform are employed. The techniques presented in this paper are applicable to other BITESwith hydronic or electric charging/discharging systems. With the FR techniques, model-based controlstrategies based on transfer functions can be readily developed.

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1. Introduction

Thermal energy storage (TES) systems that are parts of buildingfabric and are exposed to room air (e.g. walls and slabs) can bedescribed as building-integrated thermal energy storage (BITES)systems. BITES systems with appropriate control strategies cansignificantly improve the thermal performance of buildings.Improvements include enhanced thermal comfort by reducingroom temperature fluctuations, decreasing total space cooling/heating energy consumption, shifting and shaving peak energy de-mands, offsetting the mismatch of energy supply/availability anddemand, and higher efficiency and smaller required capacity ofmechanical heating/cooling equipment [1–5]. Furthermore,making effective use of building fabric as thermal storage systemswill avoid extra occupation of living space and material cost. Thecritical factor influencing the potential savings is the effective stor-age utilization. The benefit of passive BITES systems (e.g. solid

slabs) is limited by their poor effectiveness–only passive heat ex-change with living space on their exposed surfaces [6]. On the otherhand, active BITES systems can be actively charged and/or dis-charged by exchanging heat with their internal heat sources/sinks,additional to the passive measures. Furthermore, low temperatureheat source for heating or high temperature source for cooling canbe adopted to achieve low exergy space conditioning. The effectivestorage utilization is significantly increased, as well as the potentialbeneficial impact on space heating/cooling load reduction.

Active BITES systems include hydronic, ventilated (or air-based),electric, and capillary systems [7–11] (Fig. 1). Indirect evaporativecooling systems [12] are not considered in this study. Electricsystems can only be used in heating applications and cannot beactively discharged. Meanwhile, the other three types can be usedfor both heating and cooling applications and can be actively dis-charged. Hydronic systems are often designated as thermo-active(or thermally active) building systems (TABS) when significantthermal storage is integrated (generally thicker than 5 cm of con-crete). Examples for ventilated systems include ventilated concreteslabs (VCS) [13] and ventilated masonry block walls [14]. The airflow can be open-loop (flowing air mixes with room air) orclosed-loop, and is possible to switch based on applications (e.g.open-loop in slab cooling and closed-loop in slab heating).

th field

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Nomenclature

Symbols^ DFS or complex frequency form– mean value/response� oscillatory value/responseffi approximately equal to order of layers in an assembly. 1 N means the assem-

bly contains layers from 1 to N, and the excitations areon surface l of layer N

Greeka diffusivity (m2/sec), a = k/q � cq density (kg/m3)x angular frequency (rad/s), xf = 2p/P and xh = h �xf

Englisha coefficient or intermediate variableadm admittanceA time-series valuesb index of air channel bounding surfaceB total number of bounding surfacesc/cp capacitance (J/kg/K)chn air channelcnc concretecnv convective/convectioncrt criticald mathematical symbol for differentiale exponential baseeqv equivalentE excitationsG() response/transfer functionl thickness of a layer (m), also used to indicated the sur-

face at x = lL length (m)h convective heat transfer coefficient or harmonic indexH total number of harmonicsi index of sampling time for time-series valuesI total number of sampled valuesj imaginary unit; j ¼

ffiffiffiffiffiffiffi�1p

k conductivity (W/m/K)[M] matrix of transfer functionsn layer index, layer nN total number of layers in one assemblyp heat flux or power per unit area (W/m2)P heat power (W) or period in secondsQ flow rate (m2/sec)r thermal resistance per unit area (m2 K/W)Re{} real part of the complex numbersc sourceslb slabt time (second/s)Dt simulation time step or sampling time interval (sec)trs transmissionttl totalT temperature (�C)DT temperature difference or potential (�C)

u conductance per unit area (W/m2/K)x distance to an outer surface of a layer (m)Y self-/transfer-admittance

AcronymsBITES building-integrated thermal energy storageCHTC convective heat transfer coefficientCV/CVs control volume (s)DFS discrete Fourier seriesFD finite differenceFR frequency responseLPFD lumped-parameter finite differenceMPC model-based predictive controlTES thermal energy storageVCS ventilated concrete slab

Variableshxa heat exchange coefficientedge_CVc capacitance per square meter of the edge control vol-

ume (J/m2/K)ttlL length of one air channelBiotNo Biot number1 nadm ½M�h admittance matrix of the hth harmonic for an assembly

composed of layers 1 to N.heat_fluxp heat flux on a surface, such as solar radiation (W/m2)scpi heat flux from the heat source (W/m2)10~pi;h oscillatory heat flow response on surface at x = 0 of layer

1 at time i and of the 0h0th harmonic1sc 0~pi;h oscillatory heat flow response on surface at x = 0 of layer

1 caused by internal source at time i and of the 0h0th har-monic

1 Nt12h the element at the 1st row and 2nd column of transmis-sion matrix of assembly 1 N of the 0h0th harmonic.

chnT channel temperature weighted by the channel surfaces’respective heat transfer coefficients

chn_srfTb temperature of channel bounding surface ‘‘b’’inlet_airT inlet air flow temperaturemean_airT mean air flow temperaturenl T�

i;h oscillatory temperature response on surface at x = l oflayer n at time i and of the 0h0th harmonic

1eqv 0

eT i;h equivalent oscillatory temperature response on surfaceat x = 0 of layer 1 at time i and of the 0h0th harmonic

cnvDT temperature difference during convective heat transfer1eqv sc 0D

eT i;h equivalent oscillatory temperature potential on sur-face at x = 0 of layer 1 caused by internal source at timei and of the 0h0th harmonic

edge_CVTh thickness of edge control volume (m)chn_srfUb conductance per meter channel length of channel sub-

surface b (W/m/K)nodeu conductance per square meter between two capacitance

nodes (W/m2/K)1 sc1eqv slf yh equivalent self-admittance of assembly 1 sc1 of the

0h0th harmonic

2 Y. Chen et al. / International Journal of Heat and Mass Transfer xxx (2013) xxx–xxx

Although active BITES systems potentially offer greater benefitsover passive ones, realizing these benefits to an ideal extent re-quires cautious design and control [15,16]. For design purposes,simple yet accurate models are needed to compare and evaluatethe thermal response characteristics of different alternatives onrelative bases. Detailed models require more detailed knowledge

Please cite this article in press as: Y. Chen et al., Frequency domain and finite dmeasurements: Part 1, modeling methodology, Int. J. Heat Mass Transfer (201

of geometry, which may not be known in the early design stages.Simplified models are also needed for whole building simulationsover long simulation periods (e.g. monthly or yearly). For thedevelopment and deployment of control strategies, simple modelssuch as those based on transfer functions are needed, especially formodel-based predictive control (MPC) [17]. The weather forecast

ifference modeling of ventilated concrete slabs and comparison with field3), http://dx.doi.org/10.1016/j.ijheatmasstransfer.2013.05.076


Storage/structural support layerAir cavity

Storage/structural support layerInsulation/false panel and air gap

Air

flo

w

Heat exchangewith living space

(a)

Electric cable/capillary tube

Warm floor (heat goes up)Cold floor (heat goes down)

Storage/structural support layerInsulationStorage/structural support layer

(b)

Storage/structural support layerInsulationStorage/structural support layerTube

(c)

Fig. 1. Schematic of the typical cross sections of (a) ventilated, (b) electric or capillary, (c) hydronic systems (each system can be a floor, wall, or roof; insulation is optionaland can be placed on the other sides of the tube to orient heat flow in the opposite direction).

Y. Chen et al. / International Journal of Heat and Mass Transfer xxx (2013) xxx–xxx 3

information and corresponding space heating/cooling load pre-dicted from building models are needed as inputs for the controlof BITES systems. Cooperman et al. showed that predictive controlusing weather forecast can result in substantial savings in bothcommercial and residential buildings [18].

Laplace transform has been traditionally used in deriving trans-fer functions for transient heat conduction in solids, especially forsolid bounded by two parallel planes, such as wall and slab assem-blies [19,20]. After Laplace transforming, time-series (discrete) re-sponse coefficients, such as thermal response factors [21],conduction transfer functions (CTF) [22], and radiant time series[23,24], are derived with different methods and used in time do-main modeling. Other than using Laplace transform formulation,CTF can also be obtained with state space formulation [25,26].For assemblies with internal heat sources, Strand [27] incorporated‘‘source transfer functions’’ into CTF using either Laplace transformand state space formulations. The advantage of using time-seriesresponse coefficients is high computational efficiency for transientthermal simulations. A possible disadvantage is that it assumes alinear or linearized system. Besides time series coefficients, quasi-analytical algorithm was also developed to approximate the heattransfer among the assemblies, the heat transfer fluid, and room air[28].

Among other modeling approaches, finite difference (FD) mod-els [29,30] for active BITES systems have been widely used. Themain advantages of FD approach are accurate treatment of non-linearization and easy formulation (e.g. heat transfer and control)[30]. Detailed models reflecting the actual heat transfer process,such as 2-or 3-dimension spatial discretization, can provide moreaccurate results; however, they require more computational effort.Simpler lumped-parameter finite difference (LPFD) models withacceptable accuracy are needed for long-period simulations, espe-cially for incorporation into whole building simulations and mod-el-based control.

On the other hand, frequency response (FR) approach facilitatesthe integration of design and model-based control. FR approachcan provide additional information, particularly for design optimi-zation and comparison of design alternatives on relative bassesand, without tedious simulation [31]. Furthermore, FR approachprovides analytical solution and does not require spatial discretiza-tion. However, its main disadvantage is that it cannot directlymodel non-linear components. Another important application ofFR approach is in the development of model-based control strate-gies for building HVAC systems [17,31]. FR models for ventilatedBITES systems for design and model-based control purposes arethus presented in this paper.

FR approach is based on transfer functions derived from Laplacetransform. Instead of inversing solutions back to time domain afterLaplace transforming, FR approach generates solutions in the com-plex frequency domain by simply replacing s with j �x[32], wherej �x is the imaginary angular frequency and j ¼

ffiffiffiffiffiffiffi�1p

. When exci-


tations are also represented in complex frequency form, the ther-mal responses of assemblies can be readily obtained in complexfrequency domain. In the analysis of multi-layer assemblies (e.g.wall/floor/ceiling), the temperatures and heat flux at nodes of nointerest do not need to be calculated. The magnitude and phaseangle of admittance or impedance obtained from FR approach ofan assembly provides substantial insight into its thermal behavior[31,33,34]. These variables can be readily used for parametric anal-ysis and design optimization. Athienitis [34], Davies [35,36], Athie-nitis et al. [37], and Hittle [38] conducted studies on the thermalbehavior of building components and thermal zones using thisapproach.

Optimized RC (resistance and capacitance) network theory hasbeen used to optimize the space-discretization of systems [39].After obtaining the optimal discretization based on the frequencyof the excitations of interest, thermal behavior is then obtainedwith finite difference method in time domain [39]. Schmidt andJóhannesson [40] described a modeling method for active TESsystems with heat transfer fluid flowing through. RC networktechnique was applied to optimize the space discretization. The studywas later extended to quasi-two dimensional models [41]. Weberand Jóhannesson [42] and Weber et al. [43] presented their studieson applying RC-network in modeling hydronic BITES systems.

The purpose of this part of the paper is to present network-based modeling methodologies for FR and explicit LPFD modelsof ventilated BITES systems. In the modeling of different activeBITES systems (i.e. hydronic or ventilated, walls or slabs), eventhough they have different configurations (i.e. different heat trans-fer fluids with different fluid paths), the modeling concepts areuniversal and the techniques are similar. In this paper, FR and LPFDmodeling concepts and techniques are presented using VCS sys-tems for demonstration. The developed modeling methodologiesare applicable to other active BITES systems. In the companion pa-per Part II, the presented modeling techniques are applied to twotypes of VCS. The two approaches are compared, and modelingguidelines are withdrawn.

2. Modeling methodology

This section describes the techniques adopted or developed inthis paper for FR and explicit LPFD modeling of VCS systems. Onetypical cross section of VCS is shown in Fig. 2-a. This cross sectioncan represent either a slab on-grade or an intermediate floor. Theinsulation is optional for intermediate floors. It will be used if occu-pants would like to orientate the heat to one direction. The insula-tion layer can also represent the false ceiling and the air layerbetween the slab and the ceiling, from the point of heat transfer.Lumped-parameter models of VCS and treatments of the effectivecapacitance around radial air channels were discussed by Bartonet al. [44] and Ren and Wright [45]. Chen et al. [13] studied



Concrete

Insulation

Room airnode

Soil node

Airchannel

Convectiveconductance

Slab bottomsurface node

Heat fluxfrom air flow

Concrete

InsulationSoil

Internalnodes

Slab topsurface node

Short/longwaveradiation

Room air

Fig. 2. Schematics of a VCS-b (‘‘-b’’ indicates the air channels at the bottom of theconcrete slab), its equivalent cross section, and one-dimensional thermal networkafter transformation (there is no need for ‘‘internal nodes’’ in FR thermal network).


experimentally and numerically the typical three dimensionaltemperature distribution in a VCS system. These studies showedthat one dimensional (normal to the room-side surface of the slab)thermal model can approximate the thermal behavior of the VCSwell in cases of practical interest. Considerations in substitutingtwo dimensional models with one dimensional ones were furtherdiscussed by Strand [27]. An approximation approach in simplify-ing three dimensional heat transfer calculation is presented byKoschenz and Lehmann [28].

One-dimensional lumped-parameter modeling is adopted inthis study. The original cross section is transformed into an equiv-alent cross section (Fig. 2-b) by replacing the air channels with animaginary layer without thickness (in this case at the bottom of themass which has the same transformed cross sectional area as ori-ginal). This transformation allows the heat transfer to be treatedas one dimensional. Fig. 2 also shows the thermal network of theVCS. Modeling results will be compared, in the companion paperPart II, with field measurements from a solar demonstration housewith a VCS system in the basement slab [46]. In the followingsections, formulations and calculations are developed to suitone-dimensional lumped-parameter modeling. The thermal char-acteristics of all material are assumed to be linear and time-invariant (e.g. conductivity and specific heat capacity do notdepend on temperature or time).

Fig. 3. A differential length of an air channel bounded by 2 surfaces (B = 2; onesurface contains the top, front, and back surfaces; the other one is the bottomsurface, which is in a different color).

2.1. Heat exchange between VCS and flowing air

In FR and LPFD models, heat transfer calculation is conductedon unit room-side surface area of the ventilated component. To cal-culate the average heat exchange per unit surface area between theflowing air and the slab, a representative (or mean) temperature ofthe flowing air, mean_airT is needed. When the thermal capacitanceof the control volumes (CV’s) surrounding the flowing air is much


higher than that of air, the air channel surface temperature can beassumed to be uniform without causing significant errors [45].Chen et al. [13] showed that the temperature gradient of a VCS-balong the air flow direction is not significant (less than 1 �C). Renand Wright [45], Charron and Athienitis [47], and Fraisse et al.[48] presented methods for obtaining the local air temperaturealong an air path, as well as the mean air temperature. Based ontheir work, a method applicable to FR and LPFD models isdeveloped.

Applying an energy balance to a differential length of the airchannel being bounded by a number of B surfaces (Fig. 3), weobtain

dðairTÞ � ðairQ �air q �air cpÞ ¼XB

b¼1

dðchnsrfTb �air TÞ �chn srf Ub

h i� dðchnLÞ

ð1Þ

where airQ is the volumetric air flow rate. chn_srfUb is the conductance(convective and/or conductive) per meter channel length betweenthe surface ‘‘b’’ and the air (unit: W/m/K). For example, Fig. 3 is usedto exemplify the application of Eq. (1) to VCS-b shown inFig. 3. chn_srfU2 in Fig. 3 represents the conductance between soilnode and air flow. It combines the conductance of the insulationand the convective heat transfer coefficient (CHTC) between airflow and the insulation surface. The CHTC can be calculated usingempirical equations [49] or obtained by experiments.

Integrating over the total length of the air channel ttlL, the outletair temperature can be obtained with Eq. (2):

outlet airT ¼eqv chn T þ ðinlet airT �eqv chn TÞ � expð�hxa �ttl LÞ ð2Þ

where inlet_airT is the air temperature at the inlet. chnT is the channeltemperature weighted by the surfaces’ respective heat transfercoefficients. When the air channel is bounded by only one surface,chnT = chn_srfT, where chn_srfT is the surface temperature of the airchannel. In the case of being bounded by B number surfaces

chnT ¼PB

b¼1ðchn srf Tb � chn srf UbÞPBb¼1ðchn srf UbÞ

hxa ¼XB

b¼1

ðchn srf UbÞ=ðairQ � airq � aircpÞ

mean_airT can be calculated using Eq. (3):

mean airT ¼chn T þ ðinlet airT �chn TÞ � ð1� expð�hxa �ttl LÞÞhxa �ttl L

ð3Þ

The total heat flow from the air to the channel can be calculatedas follows:

air chnP ¼ ðmean airT �chn TÞ �XB

b¼1 chn srf

Ub �ttl L



Fig. 4. Schematic of an assembly consisting of N layers, and the labeling of itsexcitations and its indices.


air_chnP calculated in this way is numerically equal to the heatloss/gain rate of the air flow (i.e. (inlet_airT� outlet_airT) � (air Q � airq � aircp)),but slightly different from calculating it by another approachair chnP ¼ttl L �

PBb¼1ððmean airT �chn srf TbÞ �chn srf UbÞ. The difference is

within 3% in practical situations (e.g. expected inlet air tempera-ture in the range 10 to 50 �C, flow rates of 5–20 L/s per air channel,and boundary temperature of 10 to 30 �C).

The analytical results obtained using Eqs. (2) and (3) are com-pared with those from a finite difference model with fine mesh –the air channel is divided into significant small CV’s along the airpath to approach accurate results. Upwind differencing scheme[50] is used for modeling the heat transfer between the flowingair and the air channel surface. This comparison validates themethod adopted in this paper for the calculation of heat flow fromthe flowing air to the slab. This method can be used for air channelswith any path configuration, such as ‘‘U’’ (inlet and outlet on thesame perimetric side of the slab) or ‘‘S’’ (inlet and outlet on theopposite sides after two or more turns) shapes. There may be sev-eral paths in one ventilated system. Furthermore, the air flowboundary can consist of more than one surfaces (e.g. insulationand concrete surfaces as shown in Fig. 2-a). This method can alsobe applied to hydronic BITES systems if the piping configuration re-sults in a nearly uniform temperature across the BITES.

2.2. Frequency response approach

This section describes the techniques adopted or developed inthis paper for FR models. FR analysis is conducted in the complexfrequency domain, typically with periodic excitations. Time do-main equations have to be converted to frequency domain equa-tions. These equations include heat transfer and excitation (e.g.boundary temperatures and heat flux, and internal heat sources)representation equations. These equations will be expressed inexponential complex frequency form as explained in later sub-sec-tions or Appendix. Detailed information for representation of exci-tations and calculation of transfer functions in complex frequencydomain is provided in appendices. Some important points are gi-ven below:

� The analysis period, P, can be of any time span, from a few hoursto one year, depending on study objectives [51].� Simulation time step in does not have a significant impact on

the results, except in situations that there are changes/controlsof excitations. For example, on/off control of auxiliary heatingwould be on multi-minute time interval instead on hourlyinterval. Hence, hourly time step is not appropriate.� The equations for FR approach in this paper are given for unit

surface area.

2.2.1. Discrete frequency representation of excitationsIn time domain, time dependent excitations are generally given

in two forms: continuous form (i.e. a function of time) and discreteform (i.e. sampled values, discrete values in forms of time series,such as in Eq. (A 1)). Time-series discrete values are most commonin building thermal simulations. Values given in continuous formcan also be converted to discrete form by sampling at desired timeintervals. In the analysis for rough estimation, periodic excitationswith simple profiles, such as diurnal outdoor temperature and so-lar incident on a surface on a clear day, can be represented approx-imately with simple sine-or cosine-wave function [34]. For morecomplicate excitation profiles, they can be approximated by thesuperposition of several simple profiles, using the idea from seriesexpansions theorems such as Fourier series. In this paper, discreteFourier series (DFS) in exponential complex frequency form areused to represent the boundary excitations, such as the surfacetemperature variation and heat flux from the air flow to the VCS.


2.2.2. Influence of number of harmonicsThe desired number of harmonics in a DFS representation is

determined by the profile of the excitation (mainly its frequency)and the desired accuracy of its representation in frequency domain.Take the representation of a 24-h external solar radiation profilefor example, since its frequency is one (i.e. one cycle per 24 h),one harmonic can be used; however, in order to better representthe sharp change at the sunrise and sunset moments, three har-monics are more desirable [34]. Another example, if an excitationwith an irregular profile that has a shortest fluctuation time (e.g.the shortest time between two adjacent concave or convex points)of 3 h, 8 harmonics should be used for a 24-h simulation period(24/3 = 8), or 16 harmonics for a 48-h period (48/3 = 16).

Furthermore, the Nyquist–Shannon sampling theorem showsthat the excitation frequency or the highest frequency componentmust not be more than half the sampling frequency, to prevent ali-asing [52] (e.g. a different profile will appear). Since the highestfrequency component in a DFS representation is determined bythe number of harmonics, the number of harmonics shall not bemore than half of the number of the sampled (discrete) data. Forexample, hourly value in a 48-h period means 48 discrete values,and hence maximum of 24 harmonics are allowed.

2.2.3. Thermal response in complex frequency domainIn FR analysis, heat fluxes and temperatures on the two opposite

surfaces of an assembly are the four variables of interest. If any twoof the four variables are given, the total thermal responses (meanand oscillatory) of the other two variables can be readily obtainedwithout discretizing the solid (also called two-port network method[53]). The two given variables are considered as excitations, and canbe represented with discrete Fourier series (DFS) in exponentialcomplex frequency form as described in Appendix A.

Eq. (4) from references [19,20] shows the matrix expression forthe oscillatory response at time t (t = Dt � i, Dt is the time interval ofdata sampling). The oscillatory heat flux n

0~pi;h and temperature n0T�

i;h

on surface x = 0 are due to excitations on surface x = l of layer n(Fig. 4). n

trs½M�h is the transmission matrix of layer n. See AppendixB for more explanation. In this study, surfaces 0 and l are the oppo-site outer surfaces of any layer, including the room air (air filmlayer) and soil nodes as shown in Fig. 2-b. It is important to notethat the values in the excitation vectors are of various harmonicsh, as in their DFS representations.

n0eT i;h

n0~pi;h

" #¼ n

trs½M�hnleT i;h

nl~pi;h

" #ð4Þ

The overall transmission matrix for a N-layer assembly




1 ntrs ½M�h ¼ 1

trs½M�h � 2trs½M�h � 3

trs½M�h . . . � Ntrs½M�h ð5Þ

where left-hand-side superscript ‘‘1 N’’ indicates this transmis-sion matrix is of the assembly containing layers from 1 to N(Fig. 4). As indicated by Eqs. (4) and (5), the temperature and heatflux on surface l of layer N have to be the excitations for thisformulation.

If the oscillatory temperatures on the two surfaces of one layerare given as excitations, Eq. (6) can be used to obtain the oscillatoryheat fluxes. Matrix adm½M�h is referred to as the admittance matrix.See Appendix B for more explanation.

n0~pi;hnl~pi;h

" #¼ n

adm½M�hn0eT i;h

nleT i;h

" #ð6Þ

For any single layer, ntrs½M�h and n

adm½M�h can be derived from eachother by mathematical manipulation (i.e. rearrangement of thevariables). If the assembly of interest consists of more than one lay-ers, the overall admittance matrix can only be obtained by rewritingthe overall transmission matrix as follows, not by multiplying indi-vidual admittance matrices.

Let trs½M�h ¼t11h t12h

t21h t22h

� �and adm½M�h ¼

a11h a12h

a21h a22h

� �then

a11h = t22h/t12h; a12h = �1/t12h; a21h = 1/t12h; a22h = �t11h/t12h.For an assembly consisting of N layers of material, the matrix

expression for calculating the oscillatory heat flux and temperatureat surface 0 of layer 1 due to excitations on surface l of layer N isEq. (7). If the oscillatory temperature excitations on two oppositesurfaces are given, the oscillatory heat flow responses on thesetwo surfaces can be obtained with Eq. (8). For other combinationsof excitations, different matrices can be derived [22].

10eT i;h

10~pi;h

" #¼ 1 n

trs ½M�hNleT i;h

Nl

~pi;h

" #ð7Þ

10~pi;h

Nl

~pi;h

" #¼ 1 n

adm ½M�h10eT i;h

NleT i;h

" #ð8Þ

2.2.4. Treatment of internal heat sourcesTo facilitate the FR analysis of a N-layer assembly (e.g. wall/

floor/ceiling), heat flow sources that are not located at the two out-ermost nodes can be transformed to equivalent temperaturepotentials, and then added to the original temperatures of thetwo nodes (Fig. 5). Alternatively, the assembly (e.g. floor system)and the heat source can be split into two parts at the level of theheat source. These two approaches will be presented here. Exam-ples of heat sources/sinks include heat flux from internal heattransfer fluids or from electric wires that are embedded in the ac-tive BITES components, and transmitted solar/long-wave radiationabsorbed by the top surface of these components.

Fig. 5. Transformation of thermal network (node 0 is the outermost node of assembly 1 in-between the two assemblies).


2.2.4.1. Thévenin theorem for heat sources transformation. Heat fluxcan be transformed to equivalent temperature potential usingThévenin theorem from network theory [54] as shown in Fig. 5.This transformation is similar to the process of obtaining thesolar-air temperature [55], which is commonly used in buildingthermal calculations. Eq. (9) can be used to calculate the oscillatoryvalue of the potential due to heat flux at the source. See Eq. (5) forthe definition of 1 sc1t12h. The mean potential can be obtained ina similar manner with normal resistances (e.g. r). If the assembly1 sc1 is purely resistive, 1 sc1t12h equals to 1 sc1r with 1 sc1rbeing the thermal resistance of assembly 1 sc1.

1eqv sc 0D T

�

i;h ¼1 sc1 t12h �sc ~pi;h ð9Þ

where 1 sc1 indicates the assembly consists of layers from 1 tosc1 (to the left of source node).

The total equivalent temperature at node 0 becomes

1eqv 0T

�

i;h ¼ 1eqv sc 0D T

�

i;h þ 10T�

i;h

2.2.4.2. Heat flow division. Instead of transforming the internal heatflow and transporting it to one of the outermost nodes, the heatflow can be divided into two portions, one for each node (e.g. nodes0 and l in Fig. 5), using current division method [54]. Take the ther-mal network from Fig. 5 for demonstration. The oscillatory portioninto node 0 will be

1sc 0~pi;h ¼ �

scN Nt12h1 Nt12h

�sc ~pi;h ð10Þ

where scN N indicates the assembly consists of layers from scN(to the right of source node) to N. The negative sign indicates theheat flow direction.

Thévenin theorem transformation and heat flow division areactually equivalent to each other (i.e. the same amount of heatflux from the internal source to the outermost nodes). Theirequivalence can be proved analytically. From Eq. (8), 1

sc0~pi;h ¼

1 Na12h �Neqv sc l DT�

i;h. Similar to Eq. (9), Neqv sc lD T

�

i;h ¼N scNt12h�sc~pi;h, and N scNt12h = scN Nt12h. Hence

1sc 0~pi;h ¼1 N a12h �N scN t12h �sc ~pi;h ¼1 N a12h �scN N t12h �sc ~pi;h

ð11Þ

As presented previously, 1 Na12h and �1/1 Nt12h are numeri-cally equal. Therefore, Eqs. (10) and (11) result in the same values,and hence the equivalency of these two techniques is proved.

2.2.5. Y-diakoptic methodIn ventilated BITES systems, the activation of the air flow (i.e.

fan on/off) depends on the temperatures of the available inlet airand the system. Generally, the air flow will be activated if the tem-perature of the inlet air is certain degrees higher that of theBITES system and the BITES system needs to be conditioned.

sc1, while node l is the outermost node of assembly scN N. Heat source node sc is




Furthermore, the heat exchange between the air flow and the sys-tem is affected by the temperature difference between the air flowand the air channel surface. Therefore, the temperature of thechannel surface is desired for these two reasons–activation of airflow and calculation of heat transfer.

In order to obtain the temperature at the internal source level(i.e. air channel) in FR approach, the VCS needs be split into twoparts at the source level and their transmission matrices need tobe calculated. Furthermore, considering the heat flow can be di-vided into two portions using heat flow division technique dis-cussed previously, if the two parts of the VCS can be treatedseparately, splitting into two parts will facilitate the solution forcomplex thermal network, resulting from integrating the VCS mod-el into a whole building model. The difficulty is that the capaci-tance of the floor assembly is common (i.e. linked) to its twooutermost nodes that have time-varying temperatures. The splitneeds special treatment. Athienitis et al. [53] used Y-diakopticmethod (Y represents admittance) to represent a two-zone com-mon wall with two self-admittances and one transfer-admittance.Then, the common wall is then split into two parts-each originaloutermost node is connected to one equivalent self-admittanceand an equivalent heat source.

Take the node 0 from Fig. 5 as one of the outermost nodes fordemonstration. The equivalent self-admittance 1 sc1

eqv slf Yh of theassembly 1 sc1 is obtained with Eq. (12). The equivalent heatsource is the product of the equivalent transfer-admittance1 sc1eqv trf Yh (Eq. (13)) and the temperature difference between thetwo outermost nodes. See paper Part 2 for further demonstration.1 sc1eqv slf Yh ¼1 N a11h þ1 N a12h ð12Þ

1 sc1eqv trf Yh ¼1 N a12h ð13Þ

2.2.6. Calculating the total responseThe final discrete response to any excitation is the summation

of the system’s mean response and the responses to all the har-monics (i.e. oscillatory response), as given in Eq. (14). The calcula-tion of the mean response is similar to that of the oscillatoryresponses, but using mean values of the excitations and steady-state conductance. If there are more than one excitations, the totalresponse will be the summation of individual responses to therespective excitations by superposition (based on the attributesof a linear time-invariant system).

GðEiÞ ¼ G�ðE�Þ þ

XH

h¼1

ðG�ðE�

i;hÞÞ ð14Þ

where G�ðÞ represents the steady state response function. G

�ðÞ is the

oscillatory response functions, and it is the transmission or admit-tance matrix in this paper.

Considering an example, if the excitations are the temperatureson two outermost surfaces–surface 0 of layer 1 and surface l oflayer N of a N-layer assembly, the steady-state response of heat

flow at surface 0 is 10�p ¼ ð10T

�� N

l T�Þ=1 Nr with 1 Nr being the total

thermal resistance of all layers. Oscillatory heat flow (i.e. response)at surface 0; 1

0~pi;h can be obtained using Eq. (8). The final response

can then be obtained using Eq. (14): 10p̂i ¼ 1

0�p þPH

h¼110~pi;h

� �, and

time domain (i.e. real value) heat flux is 10pi ¼ Re 1

0p̂ii

n o.

2.3. Explicit lumped-parameter finite difference approach

Techniques adopted in this study for the discretization and timestep selections in explicit LPFD models are presented and discussedin this section.


2.3.1. DiscretizationIn the multi-layer discretization schemes to be studied for

one-dimensional LPFD models, each layer of the concrete slab isrepresented by one CV. The optimal thickness of each CV has tobe determined for desirable simulation accuracy. The thickness ofthe outermost CV can be made as the thinnest to take into accountthe relatively higher heat transfer on the top surface. Paper Part 2discusses effects of different discretization schemes.

Dimensionless Biot number (ratio of the internal thermal resis-tance of a solid to the boundary layer thermal resistance) [29] canbe used to decide the thickness of CV’s. Ideally, Biot number shouldbe less than 0.1 in order to minimize the error introduced byassuming uniform temperature distribution inside one CV [29].However in LPFD models, coarse discretizations will result in Biotnumbers larger than 0.1. The recommendation for proper Biotnumber and its resulting errors for different types of VCS are pre-sented in paper Part 2. Once the Biot number is chosen, the thick-ness of the edge CVs (e.g. slab top surface node in Fig. 2-b) can bedetermined as follow

edge CV Th ¼ BiotNo �cnc k

crthð15Þ

where cnck is the conductivity of the concrete, BiotNo is the Biot num-ber of the CV, and crth is the critical (i.e. largest) combined convec-tive and radiative conductance on the concrete surface, either theroom side or the air flow side.

When direct heat flux, heat_fluxp, on the boundary (e.g. solar radi-ation) is significant enough to be considered, converting radiativeheat flux to equivalent temperature potential (e.g. solar-air tem-perature) is not correct here. The heat flux needs be converted intoan equivalent CHTC using

eqvh ¼heat flux p=cnvDT ð16Þ

and then superimposed on the CHTC, cnvh. Hence, crth = cnvh + eqvh.cnvD T is the air film temperature difference. This treatment willbe exemplified in paper Part 2.

2.3.2. Time stepIn explicit FD approach, Fourier number (Fo = a � D t/l2, the ratio

of the heat conduction rate to the rate of thermal energy storage)needs to be less than 0.5 to stabilize the marching of the FD equa-tions forward in time [30]. Therefore in our case of one-dimen-sional heat transfer, the maximum allowable time step can becalculated as follow:

Dt < edge CV c

nodeuþcrt hð17Þ

where edge_CVc is the capacitance of the edge CV, and nodeu is the con-ductance between the edge capacitance node and the adjacent in-ner capacitance node.

One of the advantages of using LPFD models is their less criticaltime step requirement in explicit formulation, extending from lessthan five minutes for a 1-cm thick layer to more than an hour for a5-cm layer in one dimensional model. However, large time stepmay not be adequate to obtain a sufficiently accurate solutionreflecting practical operating situations. Effects of time step selec-tion are discussed in the companion paper Part 2.

3. Conclusion

Modeling techniques for frequency response (FR) and lumped-parameter finite difference (LPFD) approaches are presented in thispaper for active building-integrated thermal energy storage(BITES) systems. The methodology is applied to ventilated concreteslabs (VCS) system. The techniques are applicable to other venti-lated, electric, and hydronic BITES systems.




Network modeling techniques, such as heat source transforma-tion with Thévenin theorem, heat flow division, and Y-diakopticmethod are presented as means to develop transfer function mod-els in frequency domain. Thévenin transformation and heat flowdivision are equivalent in the treatment of the heat flow fromthe flowing air. Y-diakoptic method can split the BITES system intotwo parts at the internal heat source level, and hence facilitates theformulation and calculation. Discrete Fourier series (DFS) repre-sentation in complex frequency form are used to represent theboundary excitations, such as the surface temperature variation,solar radiation and heat flux associated with flowing air. Sincethe heat transfer equations are also solved and represented in com-plex frequency domain, simple and efficient solutions can be read-ily obtained. Frequency domain results can be easily transformedback to the time domain. The criteria for choosing the number ofharmonics have also been discussed. Furthermore, equations forone-dimensional discretization and time step selection are dis-cussed, taking into account convective and radiative heat transferon the boundaries. A method used in simplified models for calcu-lating the heat transfer between flowing air and ventilated BITESsystems is developed. Application of these techniques is presentedin the second part of the paper.

Acknowledgements

The work is funded by Postgraduate Scholarship from NaturalSciences and Engineering Research Council (NSERC) of Canada,the NSERC Smart Net-zero Energy Buildings Strategic ResearchNetwork (SNEBRN), and Graduate Student Support Program fromthe Faculty of Engineering and Computer Science of ConcordiaUniversity, Montreal, Canada.

Appendix A. Discrete frequency representation of excitations

Given a time series of discrete values, in Eq. (A-1),

½A�I ¼ ½A1;A2;A3 . . . Ai . . . AI� ðA-1Þ

where i = 1,2 . . . I, is the time-series index, indicating the time (i.e.t = Dt � i) at which the value is sampled. Dt is the time interval ofdata sampling. I indicates how many values are given.

The function of any excitation can be assumed to be even (i.e.symmetric about time origin) since only one period (i.e. cycle) isof interest. Its complex DFS representation can be approximatedusing Eq. (A-2). See Chen [56] and Kreyszig [57] for more detailson Fourier series and discrete Fourier series.

Ai ffi a0 þ 2XH

h¼1

ðah � ejhxf iDtÞ ¼ a0 þ 2 �XH

h¼1

ah � ejh2pI i

� �ðA-2Þ

where accent ‘‘ ^ ’’ means DFS or complex frequency form, andxf = 2p/P is the fundamental angular frequency. P is the analysisperiod in seconds (P = Dt � I). H is a positive integer indicating thenumber of harmonics are used in the approximation.

a0 ¼1I

XI

i¼1

Ai � e�j02pI i

� �¼ 1

I

XI

i¼1

Ai

ah ffi1I

XI

i¼1

ðAi � e�jhxf iDtÞ ¼ 1I

XI

i¼1

Ai � e�jh2pI i

� �Coefficient a0 is the mean of all the given time-domain values

[A]I. Each ah calculates the magnitude (also called modulus) ofthe 0h0th harmonic, which has an angular frequency of xh = h �xf.The summation term in (A-2) gives the oscillation about the meanvalue. It is the superposition of all the harmonics of differentfrequencies. When h equals to zero, a0 numerically equals to ah.(A-2) shows that when a time-series excitation is represented with


a DFS (e.g. Ai in (A-2)), each value in the DFS consists of a mean va-lue A

�¼ a0, and an oscillating value approximated by the summa-

tion of different harmonics, each being A�

i;h ffi 2 � ah � ejh2pI i. Accent

‘‘-’’ represents mean value, and ‘‘ � ’’ the oscillatory values. To con-vert the complex values back to real values, Ai ffi RefAig, where Re{}takes the real part of the complex number.

Appendix B. Thermal response in complex frequency domain

ntrs½M�h ¼

coshðnl � chÞsinhðnl�chÞ

nk�ch

nk � chsinhðnl � chÞ coshðnl � chÞ

" #ðB-1Þ

where nl is the thickness of layer n, and nk is the thermal conductiv-

ity (W/m/K). ch ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijxf h=

naq

, and na = nk/nq � n c is the thermal diffu-

sivity of the material (m2/sec) of layer n.For a layer that can be considered as purely resistive/conductive

(e.g. insulation, air film), the transmission matrix becomes

trs½M�h ¼1 r0 1

� �. r is the thermal resistance of the corresponding

layer. For an exterior air film, r = 1/cnvh, with cnvh being the exteriorCHTC.

The matrix trs[M]h is referred to as the transmission matrix [22](sometimes as cascade matrix). Pipes [32] provided an approachfor direct derivation of the transmission matrix based on the anal-ogy between thermal and electrical circuits. The use of transmis-sion matrix simplifies the calculation of the heat transfer inmulti-layered assemblies (e.g. walls and slabs) – the overall trans-mission matrix simply equals to the product of the individualtransmission matrix in the order corresponding to their locationsin the assembly (Eq. (5)). The theory behind the direct multiplica-tion is that the temperature of the common surface of two adjacentlayers is obvious the same, and the heat flux through the commonsurface conserves [19,32].

nadm½M�h ¼

nk � chcothðnl � chÞ�nk�ch

sinhðnl�chÞnk�ch

sinhðnl�chÞ�nk � chcothðnl � chÞ

24 35 ðB-2Þ

In admittance matrices, elements a11h or a22h is referred to as theself-admittance that relates the heat flow response to the tempera-ture excitation at the same surface. a12h or a12h is transfer-admit-tance that relates the heat flow response to the temperatureexcitation at the opposite surface [58]. In some literatures, self-admittance is simply referred to as admittance, and transfer-admit-tance as dynamic transmittance [39,59]. Multiplying individualadmittance matrices like that for transmission matrices is notcorrect.

Note the sign conventions in the transmission and admittancematrices vary in the literature. In this paper, the formulation ofadmittance matrices defines that heat flux is positive if it flowsin the direction pointing from surface 0 to surface l of the samelayer.

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Documents

Frequency domain and finite difference modeling of ventilated concrete slabs and comparison with field measurements: Part 1, modeling methodology