Heat Loss to Building Foundations:An Analysis Based on a Theory ofDynamic Thermal Networks

Eva-Lotta Wentzel Johan Claesson

ABSTRACT

The calculation of heat flow in the ground, in particular the time-dependent heat loss through the foundation, poses particularproblems due to three-dimensional flow and very long time scales. In this paper a new tool of analysis called “dynamic thermalnetworks” is presented briefly and used to analyze the time scales for the time-dependent heat loss for different types of foun-dations: a slab-on-ground, a cellar, and a split-level foundation.

The heat loss may be divided into a transmittive and an absorptive component. The transmittive and absorptive step-responsefluxes and the corresponding weighting functions provide a precise description of the time scales of the heat flow to the ground.It is shown that the transmittive component has time scales of a decade and even more, while the absorptive component has atime scale of days only.

INTRODUCTION

The modeling of heat flow in the thermal envelopecoupled to energy balances for buildings is a main task inbuilding physics. The calculation of heat flow in the ground,in particular the time-dependent heat loss through the founda-tion, is the part that is most difficult to calculate, depending onthree-dimensional heat flow and very long time scales.

In this paper a new tool of analysis called “dynamic ther-mal networks” is presented briefly and used to analyze thetime scales for the time-dependent or dynamic heat loss fordifferent types of foundations: a slab-on-ground, a cellar, anda split-level foundation.

The heat flux from the floor to the foundation depends onthe current indoor and outdoor temperatures and on thepreceding sequence of values for these boundary tempera-tures. A key question is how far back in time we have toaccount for these boundary temperatures. This paper presentsa methodology to answer this question within the conceptualframework of dynamic thermal networks.

The theory of dynamic thermal networks for heat flow insolid regions such as ground, walls, and roof is formulated tofit into energy balance models in a handy way. This is briefly

presented, and a way to minimize the storage and use ofpreceding boundary temperatures is indicated.

There is a vast literature with many publications about theproblem of the relations between boundary heat fluxes andboundary temperatures. The basic mathematics is found inCarslaw and Jaeger (1959). The response factor method, intro-duced by Mitalas in the late 1960s, is summarized togetherwith a large list of references in ASHRAE (1997). Otherimportant papers are Davies (1997), where the analysis isbased on ramp responses, and Kossecka and Kosny (1998),where so-called thermal structure factors for composite wallsare used.

DYNAMIC THERMAL NETWORKS

We consider heat conduction in a solid volume with twoboundary surfaces with the temperatures T1(t) and T2(t),respectively. An example is the thermal envelope of a buildingwith roof, walls, foundation, and surrounding ground betweenindoor and outdoor boundary temperatures. The theory, whichis briefly presented here without any derivations, is discussedin more detail in Claesson (2003, 2002).

©2004 ASHRAE.

Eva-Lotta Wentzel is a Ph.D. student and Johan Claesson is a professor in the Department of Building Technology, Chalmers University ofTechnology, Gothenburg, Sweden.

Basic Relations

The time-dependent or dynamic relations between bound-ary heat fluxes and boundary (air) temperatures may be writ-ten in the following way:

(1)

(2)

Here, K12 (W/K) is the (steady-state) thermal conduc-tance between the two surfaces. The factor K1 (W/K) is thesurface thermal conductance for surface 1. It is equal to thesurface area times the heat transfer coefficient between air andsolid surface. The conductances are multiplied by temperaturedifferences involving the boundary temperatures at the consid-ered time t and at preceding times t − τ, τ > 0, as describedbelow.

The second right-hand term in Equations 1 and 2 is thesame except for the sign. It may be called the transmittive heatflux between the surfaces. The first right-hand term may becalled the absorptive heat flux for surface 1 and surface 2,respectively. This separation of the dynamic boundary fluxesQ1(t) and Q2(t) into an absorptive component for each surfaceand a common transmittive component is quite intricate. It isdiscussed further in Claesson (2003). The basic equations 1and 2 are represented graphically as a dynamic thermalnetwork in Figure 1.

Transmittive and Absorptive Mean Temperatures

In Equations 1 and 2, we use mean values of boundarytemperatures backward in time, which are indicated by a bar:T. The transmittive mean temperatures for the two boundarytemperatures are

(3)

Here, κ12τ is the transmittive weighting function definedbelow. There is an absorptive weighting function for eachboundary surface and a corresponding absorptive meantemperature:

(4)

The integrals are extended backward in time, until theweighting function is sufficiently small so that the rest of theintegral (to infinity) is negligible. These backward integralsbecome sums in the discrete formulation. In the graphicalrepresentation of Figure 1, these backward averages are indi-cated by summation sign Σ.

The transmittive component is represented by the conven-tional resistance symbol with the thermal conductance K12written above the component. To the resistance or conduc-tance symbol, we add summation signs on both sides. Thesigns signify that we take an average of the node temperaturesbackward in time according to Equation 3. The left-handsummation sign is reversed to indicate the symmetric charac-ter of the flux and that summation concerns the values at theleft-hand node. The weighting function κ12τ may be writtenbelow the resistance symbol.

The two absorptive components are represented by resis-tance signs with the surface thermal conductance (K1 and K2)written above. A summation sign is added at the free end afterthe surface conductance to indicate averages backward in timeaccording to Equation 4. There is not any summation sign onthe node side since the present node temperature T1(τ) is to beused at this side in accordance with the first right-hand termsof Equations 1 and 2.

Step-Response Heat Fluxes

The weighting functions are derivatives of certain basicstep-response heat fluxes as described below. There are twostep-response problems with a unit temperature step atsurfaces 1 and 2, respectively. We use τ and not t as the timevariable in step-response problems. Then the weighting func-tions have τ as an independent variable, while the backwardtemperatures are taken for t − τ. In the first step-response prob-lem, the temperature outside surface 1 is raised from 0 to 1atzero time and kept at this value for τ > 0. All temperatures arezero for τ < 0. There is an admittive heat flux Q11(τ) intosurface 1 and a cross-flux or transmittive flux Q12(t) outthrough surface 2. The corresponding step-response fluxes fora step at surface 2 are Q22(τ) and Q21(τ). The cross-fluxes areequal due to a general symmetry principle. We have three basicstep-response fluxes:

(5)

The general character of the two admittive fluxes and thetransmittive flux are shown in Figure 2. They all approach thesteady-state flux K12 as τ tends to infinity. The admittive fluxesstart from the surfaces conductances K1 and K2, respectively,

Q1 t( ) K1 T1 t( ) T1a t( )–[ ]⋅ K12+ T1t t( ) T2t t( )–[ ]⋅=

Q2 t( ) K2 T2 t( ) T2a t( )–[ ]⋅ K12+ T2t t( ) T1t t( )–[ ]⋅=

T1t t( ) κ12 τ( ) T1 t τ–( )⋅ τd

0

∞

∫= T2t t( ) κ12 τ( ) T2 t τ–( )⋅ τd

0

∞

∫=

T1a t( ) κ1a τ( ) T1 t τ–( )⋅ τd

0

∞

∫= T2a t( ) κ2a τ( ) T2 t τ–( )⋅ τd

0

∞

∫=

Figure 1 Dynamic thermal network to represent therelations in Equations 1 and 2.

Q11 τ( ) Q22 τ( ) Q21 τ( ) Q12 τ( )=

2 Buildings IX

and decrease to the steady-state value. The transmittive fluxstarts from zero and increases steadily to steady state.

The absorptive step-response fluxes for the two unitsteps are given by the absorbed heat, i.e., the differencebetween the admittive and transmittive fluxes. We have

(6)

These two fluxes start from the surface conductance anddecrease steadily to zero for a long time as shown in Figure 2.

Weighting Functions

The weighting functions are time derivatives of the step-response functions. The transmittive weighting function,κ12(τ), is the derivative of Q12(τ) multiplied by 1/K12, so thatthe integral (Equation 9) becomes equal to 1. We have

(7)

The adsorptive weighting function, κ1a(τ), is given by thefactor −1/K1 multiplied by the time derivative of Q1a(τ). Wehave for the two absorptive weighting functions,

(8)

The character of the weighting functions is shown inFigure 9. They are all non-negative, and the integrals are allequal to 1 due to the factor involving conductivity:

(9)

The transmittive weighting functions have a bell-shapedform (Figure 9, top and bottom). They are zero during a firstperiod (around an hour) until the temperature step is felt at theother side. They increase to a maximum at a certain time, andthen they slowly decrease to zero for a long time. The absorp-tive weighting functions decrease monotonously from a veryhigh value (Figure 9, middle).

All equations in this brief outline of the theory ofdynamic thermal networks are mathematically exact, or asexact as the numerically calculated step-response functionsand their derivatives, the weighting functions. The heat flowproblem must be linear so that superposition is applicable.

The heat flow problems considered above have twoboundary surfaces with different temperatures. The heat flowproblem may in a more general case have N boundary surfaceswith different temperatures. Then the dynamic thermalnetwork has N nodes. Each node or boundary surface has anabsorptive component. There are transmittive componentsbetween all pairs of nodes just as in the corresponding steady-state network (see Claesson 2003). The theory may also beapplied for a subsurface of the indoor area. An example isWentzel (2002), where the heat loss dynamics of differentparts of the floor is analyzed. Another application for compos-ite walls is presented in Wentzel and Claesson (2003), wherethe analyses are extended to the annual heating cost for a vari-able energy price.

A BUILDING WITH THREE TYPES OF FOUNDATION

We will consider a particular building with heat flowthrough ceiling, walls with window openings, and floor. Threetypes of foundation are studied: slab-on-ground, cellar, andsplit-level. The outdoor temperature, T2(t), follows a naturalclimate, while the indoor temperature T1(t) depends on theheating conditions.

Basic Formulas for the Heat Loss

Our main interest is the heat loss Q1(t) from the buildingthrough ceiling, walls, and floor (see Figure 3). We are notinterested in the heat flux through the outdoor surface. Weneed Equation 1 only:

(10)

The thermal conductance for the whole building is K12(W/K). The factor K1 (W/K) is the surface thermal conduc-tance at the building’s inside. It is equal to the surface area A1times the surface heat transfer coefficient α1: K1=A1⋅α1.

Figure 2 Character of the basic step-response heat fluxes.

Q1a τ( ) Q11 τ( ) Q12 τ( )–= Q2a τ( ) Q22 τ( ) Q21 τ( ) .–=

κ12 τ( )1

K12

--------dQ12

dτ------------ .⋅=

κ1a τ( )1

K1

------–dQ1a

dτ------------⋅= κ2a τ( )

1

K2

------–dQ2a

dτ------------⋅=

κ1 τ( ) 1 ,=

0

∞

∫ κ1 τ( ) 0 ,≥ I 12, 1a, 2a=

Figure 3 The heat loss from a building.

Q1 t( ) K1 T1 t( ) T1a t( )–[ ]⋅ K12+ T1t t( ) T2t t( )–[ ]⋅=

Buildings IX 3

The right-hand terms of the dynamic relations involvethermal conductances multiplied by temperature differences.The temperatures that are used are the present indoor temper-ature T1(t) and average temperatures backward in time. Thetemperature averages are given by Equations 2 and 4. We have:

(11)

Here, τ assumes values from zero to infinity or suffi-ciently far back in time. The transmittive weighting functionκ12(τ) and the absorptive weighting function κ1a(τ) arediscussed below.

Figure 4 shows the dynamic thermal network according toEquation 10 for the building’s heat loss. We have an absorptiveand a transmittive component.

The weighting functions κ1a(τ) and κ12(τ) in Equation 11are obtained from a basic step-response solution. We havefrom Equations 7 and 8,

(12)

Here, we need the basic step-response heat fluxes for aunit step at the inside surface 1:

(13)

See Figure 5. The admittive heat flux Q11(τ) at the insideboundary and the transmittive flux Q12(τ) out through theoutside boundary have to be determined from the calculationof the step-response problem. The absorptive heat flux, Q1a(τ),which is used to calculate the absorptive weighting function,is the difference between the admittive and transmittive fluxfor a unit step at surface 1.

Building and Foundations

A building with three different foundations is studied.The foundations are chosen to represent three common typesin Sweden: slab-on-ground, cellar, and a split-level. The build-ing’s floor area is 12.5 m × 8 m.

The building above the foundation has 0.2-m-thicklight expanded clay aggregate (Leca) walls with an externalinsulation layer of 0.1 m. The roof is made of light concretewith an external insulation layer of 0.1 m. The windows havea total area of 24 m2.

The studied slab-on-ground foundation consists of a0.1-m-thick concrete slab on a 0.2-m-thick insulation layer.The edge beams are 0.2 m thick and 0.3 m wide. The exteriorwalls, which are placed on the slab, are made of 0.2-m-thicklight expanded clay aggregate with a 0.1-m-thick exteriorinsulation layer. (see Figure 6, top).

The studied cellar foundation consists of 0.2-m-thickcellar walls of concrete with a 0.1-m-thick exterior insulationlayer. The walls are 2.4 m high. The cellar floor consists of a0.1-m-thick concrete slab on a 0.2-m-thick insulation layer.The edge beams are 0.2 meter thick and are 0.3 m wide. Thefloor between the cellar and the upper building consist of 0.15m concrete. The upper exterior walls are the same as for theslab-on-ground foundation. The ground level lies 0.5 m belowthe cellar ceiling (see Figure 6, bottom, left).

The split-level foundation has the same construction asthe cellar above. The highest ground level is 0.5 m below theceiling and the lower ground level is at the same level as thefloor (see Figure 6, bottom, right).

Figure 4 Dynamic thermal network for the heat loss.

T1t t( ) T2t t( )– κ12 τ( )

0

∞

∫ T1 t τ–( ) T2 t τ–( )–[ ]dτ⋅=

T1a t( ) κ1a τ( )

0

∞

∫ T1 t τ–( )dτ .⋅=

κ12 τ( )1

K12

--------dQ12

dτ------------⋅= κ1a τ( )

1–

K1

------dQ1a

dτ------------⋅=

Q11 τ( ) Q12 τ( ) Q1a τ( ) Q11 τ( ) Q12 τ( )–=

Figure 5 Basic step-response heat fluxes for a unit step atthe inside boundary surface.

Figure 6 Three types of foundation: slab-on-ground,cellar, and split-level foundation.

4 Buildings IX

STEP-RESPONSE AND WEIGHTING FUNCTIONS

To calculate the dynamic heat loss (Equation 10) throughthe building’s foundation, we use the foundation’s transmittiveand absorptive response functions. These functions give infor-mation about how preceding temperature variations influenceon the current heat loss.

Numerical Solution

The step-response thermal problem is solved with thethree-dimensional numerical code Heat 3 (Blomberg 1998).The mesh involves up to 50 × 50 × 50 (=125,000) nodes. Thestudied foundations are symmetric, and only one- forth ofthem is considered in the numerical program, except for thesplit-level case where one-half of the foundation is considered.The ground depth is five times the building’s length. The Euro-pean Standards EN ISO 10211-1 (1995) and EN ISO 10211-2 (2001) recommend a ground depth of 2.5 times the build-ing’s width, but greater depth gives better precision.

Figure 7 shows parts of the mesh around the cellar foun-dation. The left-hand figure illustrates the different materiallayers. The white areas show the insulation outside theconcrete. The dark gray area outside the insulation is the soil.The light gray part in the top of the figure is the Leca wall. Theright-hand figure shows the boundary conditions. The whitepart indicates total insulation due to symmetry. The gray areaat the cellar’s inside (left) has a surface heat transfer coeffi-cient α1=7.7 W/(m2K) and a temperature T1=1 for τ > 0. Theother gray area outside the building (right) has a surface heattransfer coefficient α2=25 W/(m2K) and a temperature T2= 0.

Analytical Expressions forLonger and Shorter Times

The step-response fluxes for the heat flow through theground have a very long tail as they approach steady state. It

takes some time to calculate this part from, say, 5 to 100 yearsin a numerical model. To handle this problem, we havecombined the numerical solution with asymptotic analyticalexpressions (not yet published). The beginning, the first fourto five years, of the response fluxes is calculated numerically.Then analytical expressions for τ > τla (large) are fitted to thenumerical results. For the transmittive response and admittivefluxes, we use these expressions:

(14)

(15)

In the considered cases, tla is five years. The responsefluxes Q12(tla) and Q11(tla) are the last values after five yearsin the numerical calculation. The coefficients β1, β2, and β arechosen so that the expressions fit the numerical solutions aswell as possible during a year’s time before tla. The differencebetween the above expression and a continued numericalcalculation after τ = τla is very small.

The admittive response flux decreases rapidly in thebeginning and the numerical solution, in particular the deriv-ative to get the absorptive weighting function, becomes inac-curate. Therefore, an analytical expression is used during thefirst period. We use the solution for a semi-infinite concreteslab with a surface heat transfer coefficient α1 with a unit stepat the boundary. During a first period, the admittive responseflux becomes (Carslaw and Jaeger 1959)

(16)

Here, erfc(s) is the complementary error function, λ is thethermal conductivity (W/(mK)), ρc the volumetric heat capac-ity (J/m3), d = λ/α an equivalent length of the surface layer,and α = λ/(ρc) the thermal diffusivity (m2/s) of the concreteindoor surface. The time τsm (small) is chosen as one hour.

Response Functions for the Three Foundations

Table 1 gives the thermal conductance, indoor area, andU-factor for the building above ground (walls and ceiling) and

Figure 7 Part of the mesh in Heat 3. Left: The differentmaterials have different shades of gray. Right:The different boundary conditions have differentshades of gray.

Table 1. The Foundations’ ThermalConductances and U-Factors

ConstructionK12

(W/K)Area(m2)

U-Factor(W/m2K)

Walls and ceiling 40.13 184.3 0.218

Slab-on-ground 18.27 100.0 0.183

Cellar 45.07 198.4 0.227

Split-level 48.57 198.4 0.245

Q12

τ( ) K12

K12

Q12

τ1a

( )–( )–τ1a

τ---------

1 β1

– τ1a

τ⁄( )⋅ β2

+ τ1a

τ⁄( )2

⋅

1 β1

– β2

+----------------------------------------------------------------------------------⋅ ⋅=

Q11

τ( ) K12

Q11

τ1a

( ) K12

–( )+τ1a

τ---------⎝ ⎠⎛ ⎞

1.5 1 β– τ1a

τ⁄( )⋅

1 β–-------------------------------------⋅ ⋅= for τ τ

1a>

Q11 τ( ) A α1 ea τ⋅ d

2⁄erfc a τ⋅ d⁄( )⋅ ⋅ ⋅= for 0 τ τsm< <

Buildings IX 5

for each foundation. The transmittive response functions,Q12(τ), for the three foundations and for the building above thefoundation are shown in Figure 8 (top). The curves are normal-ized to one by division with the thermal conductance K12. Theleft-hand graphs show the first week (168 hours). The buildingabove ground reaches steady state after, say, four days. Theresponse is very different for the foundations. We see that only20% to 50% of the steady-state value is reached during the firstweek. In the right-hand graphs, the time axis is shown in log-scale in order to represent both the long tail and the first periodproperly.

The transmittive response fluxes in Figure 8, top, arevirtually zero before one hour (τ = 1). The transmittiveresponse functions have reached 90% of their steady-statevalues after 18 years, 17 years, and 7 years for slab, cellar, andsplit-level, respectively.

The absorptive response functions Q1a(τ) are shown inFigure 8, bottom. All foundations have the same indoorsurface (concrete), which means that the normalized responsefunctions are virtually identical during the first hour in accor-dance with Equation 16.

Figure 9 shows the weighting functions correspondingto the response functions of Figure 8. The maximum of thetransmittive weighting functions is the time backward whenprevious temperature variations have the largest influence onthe current heat flow. We see in Figure 9, top, that the maxi-mum values for the different foundations occurs after 5.3(split-level), 5.7 (cellar), and 22 (slab) hours. The values of theweighting functions are very small after the first week, but thetails of the transmittive weighting functions are very long, andthe integrals over later parts involve most of the influence aswe have seen above. The bottom curve for the weighting func-tion of the building above ground does not have this long tail.All influence after, say, τ = 4 days is negligible.

Figure 8 Transmittive (top) and absorptive (bottom) step-response functions for the building above ground and for the slab,cellar, and split-level foundations. Left: The first week. Right: Logarithmic time from 1 hour to 106 hours or 114years.

6 Buildings IX

RELATIVE INFLUENCE OFBACKWARD TIME INTERVALS

The weighting functions κ(τ) give the relative influenceof boundary temperatures T (t−τ) backward in time from theconsidered time t, (τ >0). A wall, or a corner in two or threedimensions, will have a response time scale of up to a fewdays. After this time, the weighting factor tends to zero expo-nentially. The integrals (Equations 3 and 4) in τmay be termi-nated after a few days or a week. The basic reason for thislimited time scale is that the heat flow region has limitedextensions.

The heat flow region in the ground below and aroundthe building has very large extensions downward and outward.This leads to much longer time scales. The tail of the weight-ing must, in a precise description, be accounted for duringseveral years.

In order to analyze this problem, we divide the backwardtime 0< τ < ∞ in J intervals:

(17)

The integral of the weighting function over interval j,, gives the relative importance of the temperatures T(t −

τ) in that interval τj-1 < τ < τj on the transmittive mean temper-ature. Using Equation 7, we have

(18)

This means that the increase of Q12(τ)/K12 over any timeinterval represents the relative importance of that particularinterval.

The mean temperature in interval j (with the weightingfunction ) is

(19)

The transmittive mean temperature is then

(20)

This mathematically exact formula shows very clearly theinfluence from any time interval. There is a correspondingformula for T2t(t). The formulas for the absorptive meantemperature are of the same type as the ones above except forthe minus sign in Equation 8. We have

(21)

This means again that the decrease of Q1a(τ)/K1 over anytime interval represents the relative importance of that inter-val.

We may choose the times τj so that the J intervals allhave the same relative importance, i.e., 1/J. We get a set ofintervals for the transmittive and for the absorptive meantemperatures:

(22)

Figure 9 Transmittive (top) and absorptive (middle)weighting functions for the slab, cellar, and split-level foundations. Bottom: Transmittiveweighting function for the building above ground.

0 τ0 τ1 … τj … τJ< < < < < ∞= =

κ12,j

κ12,j κ12 τ( ) τd

τj 1–

τj

∫Q12 τj( ) Q12 τj 1–( )–

K12

--------------------------------------------------= = j 1,…, j .=

κ12 τ( ) κ12,j⁄

T1t,j1

κ12,j

----------- κ12 τ( )

τj 1–

τj

∫ T1 t τ–( )dτ⋅ ⋅= j 1,…, j .=

T1t t( ) κ12

τ( ) T1t τ–( )⋅ τd

τj 1–

τj

∫j 1=

J

∑Q

12τj

( ) Q12

τj 1–

( )–

K12

------------------------------------------------------- T1t ,j .⋅

j 1=

J

∑= =

T1a t( )Q1a τj 1–( ) Q1a τj( )–

K1

-------------------------------------------------- T1a,j .⋅

j 1=

J

∑=

Q12 τj ,t( )

K12

---------------------j

J-- ,=

Q1a τj,a( )

K1

----------------------- 1j

J-- ,–= j 0, 1, …, J=

Buildings IX 7

Then we have

(23)

Figure 10 shows these time intervals for J =10 for theslab-on-ground foundation. We see that the first 10% of thetransmittive heat loss is due to outdoor temperatures from 0(rather than 3) to 52 hours. The last 10% is due to temperaturesmore than 163,000 hours = 17 years backward. The limits forall ten intervals are

0, 52 h, 12 d, 44 d, 108 d, 217 d, 1.1 y, 2.1 y, 4.8 y, 17 y, ∞

We see that the transmittive heat loss involves precedingtemperatures during more than a decade.

The behavior for the absorptive heat flux is very differ-ent. The first 10% of the absorptive heat loss is due to thepreceding indoor temperatures from 0 to 0.15 hours (9minutes). The last 10% is due to temperatures more than 23hours backward. The limits for all ten intervals are

0, 0.15 h, 0.74 h, 1.8 h, 3.2 h, 4.9 h, 7.0 h, 9.9 h, 14 h, 23 h, ∞

We see that the absorptive heat loss involves precedingtemperatures during a few days only.

CALCULATION OF THE HEAT LOSS

In a numerical solution, we must use a discrete approxi-mation of the backward temperatures and the integrals (Equa-tion 11). Our approach is reported in Claesson (2003). We usea time step h, and an indoor and an outdoor temperature at eachtime step. We assume a linear variation of the temperaturesduring each time step.

The heat flux is obtained for each time step from thecorresponding discrete form of Equation 10. The integrals(Equation 11) are replaced by a sum of backward tempera-tures. The transmittive and absorptive weighting functions arereplaced by weighting factors, which are obtained directlyfrom integrals of the response functions.

A particular problem is the long time scale for the heatloss through the foundation. We have seen that we shouldaccount for boundary temperatures more than ten years backin time for the transmittive heat loss. With a time step of onehour, this means that we must consider more than 105 temper-atures at preceding time steps. But the variation of the weight-ing factors become smaller and smaller as the backward timeincreases. This means that we only need average backwardtemperature values for increasing time intervals. This isclearly illustrated in Equation 20. But these averages should beas correct as possible.

In order to minimize the storage of preceding temper-atures, we proceed in the following way. We use each preced-ing time step until the weighting factor has decreased wellbelow half of its maximum value. Then we use the mean valueof two consecutive temperatures. When the weighting factor iswell below one-fourth of its maximum, we use the average of

22 = 4 consecutive temperatures. We double the intervals inthis systematic way. The number of preceding temperaturesthat we have to store and use decreases drastically; in a typicalcase from 105-106 to 100. The calculation of the heat lossduring a year may involve 104 time steps, where each time steprequires a few hundred additions and multiplications. Thecomputer time requirement is then a few seconds only. Itshould be noted that we retain then full three-dimensionalaccuracy via the step-response fluxes. The method requiresthat the two step-response fluxes be calculated separately andgiven as an input.

The original sequence of preceding temperatures isshifted one step backward for each time step. In the condensedrepresentation, with mean values of 2, 4, 8, etc., originaltemperatures, this backward shift must be done in a particularway in order to avoid what we call time-shift dispersion. Thisproblem and our solution will be reported in another publica-tion.

HEAT BALANCE MODELS

The above theory of dynamic thermal networks is partic-ularly designed to incorporate dynamic heat flow in solidcomponents in heat balance models. We will illustrate thiswith a simple example.

T1t t( )1

J--- T1t ,j ,

j 1=

J

∑⋅= T1a t( )1

J--- T1a,j .

j 1=

J

∑⋅=

Figure 10 Time intervals of equal influence (10%) on theheat fluxes for the slab-on-grade foundation. Top:Transmittive heat flux. Bottom: Absorptive heatflux.

8 Buildings IX

We consider a building with an indoor temperatureT1(t) and a given outdoor temperature T2(t). There is a givenheat input Qh(t) (W) and a given heat gain from solar radia-tion Qr(t). The heat input due to ventilation is determinedfrom a prescribed time-dependent ventilation conductanceKv(t) (W/K). The sum of these heat inputs is equal to the heatflux Q1(t) at the inside boundary of the building. Using Equa-tion 10, we have the heat balance

(24)

The corresponding dynamic network is shown in Figure 11.The indoor temperature T1(t) is readily solved from the

heat balance (Equation 24). We have

(25)

The indoor temperature is equal to the heat influx to theindoor node of Figure 11 from heating, solar radiation, venti-lation with the outdoor temperature, absorptive influx with theabsorptive backward mean T1a(t), and a transmittive influxwith the temperature difference T2t(t) – T1t(t), all divided bythe sum of conductances K1+Kv(τ) to the node.

The above simple case with prescribed heating mayreadily be extended to more complex relations for Qh (t) andto more complex networks. The dynamic thermal networks forheat flow in solid regions may in the above way be added andincorporated within the conceptual framework of steady-statenetworks. The level of complexity is higher than that of thecorresponding steady-state network. We have to add anabsorptive component at each node and store and use preced-ing boundary temperatures for the absorptive and transmittivecomponents.

CONCLUSION

The calculation methodology “Dynamic ThermalNetwork” is described and used to illustrate the time-depen-dent or dynamic heat loss for three different foundations, slab,cellar, and split-level. The dynamic heat loss consists of twoparts: absorptive and transmittive. The theory of dynamic ther-mal networks is based on step-response functions and theirderivatives (weighting functions).

The response functions for the three-dimensional foun-dations are calculated by a combination of a numerical solu-tion and analytical expressions for short and long times. Thiscombination saves computer time.

The response functions show the different thermalbehavior of the foundations. The absorptive heat loss is similarfor all foundations because they have the same inside material,concrete (Figure 8, bottom). The absorptive component has atime scale of days only. The transmittive response functionsshow that the foundations have reached 90% of their steady-

state value after 18, 17, and 7 years for the slab, cellar, andsplit-level foundations, respectively. This means that the last10% of the heat flux is determined by outdoor temperaturesmore that a decade ago.

The weighting functions κ (τ) give the relative influenceof boundary temperatures T(t−τ ) backward in time from theconsidered time t, (τ > 0). The heat flow region in the groundbelow and around the building has very large extensionsdownward and outward. This leads to the long time scales. Thetail of the weighting must, in a precise description, beaccounted for during several years. A way to illustrate this isto divide the backward time 0 < τ < ∞ in, say, 10 intervals. Eachinterval accounts for 10% of the backward influence. For theslab foundation, the transmittive heat loss involves precedingtemperatures during more than a decade. The behavior for theabsorptive heat flux is very different. The first 10% of theabsorptive heat loss is due to preceding indoor temperaturesfrom 0 to 0.15 hours (9 minutes). The last 10% is due totemperatures more than 23 hours backward.

NOMENCLATURE

A = area (m2)

α = surface heat transfer coefficient (W/m2K)

K12 = thermal conductance (W/K) between boundary surfaces 1 and 2

K1 = thermal surface conductance (W/K) of surfaces 1 (=A1α1)

κ/τ = weighting function (1/s)

κ12(τ) = transmittive weighting function (1/s) between 1 and 2

κ1a(τ) = absorptive weighting function (1/s) for surface 1

Q(t) = heat loss (W)

Q11(τ) = admittive step-response function (W/K)

Q12(τ) = transmittive step-response function between 1 and 2 (W/K)

Q1a(τ) = absorptive step-response function for surface 1 (W/K)

Q1 t( ) K1 T1 t( ) T1a t( )–[ ] K12+ T1t t( ) T2t t( )–[ ]⋅ ⋅=

Qh t( ) Qr t( ) Kv t( )+ + T2 t( ) T1 t( )–[ ] .⋅=

T1 t( )Qh t( ) Qr t( ) Kv t( )+ + T2 t( )⋅ QA+

K1 Kv t( )+------------------------------------------------------------------------------------=

QA K1 T1a t( )⋅ K12+ T2t t( ) T1t t( )–[ ] .⋅=

Figure 11 Dynamic thermal network for the indoortemperature with heating, solar radiation, andventilation.

Buildings IX 9

T(t) = temperature (°C)

T1t(τ) = transmittive mean boundary temperature for surface 1 (°C)

T1a(τ) = absorptive mean boundary temperature for surface 1 (°C)

t = time (s)

τ = preceding time (s)

Indices

1 = surface 1

2 = surface 2

12 = between surface 1 and 2 (transmittive)

11 = step at surface 1, flux at surface 1 (admittive)

t = transmittive

a = absorptive

REFERENCES

ASHRAE. 1997. 1997 ASHRAE Handbook—Fundamen-tals, chapters 28 and 30. Atlanta: American Society ofHeating, Refrigerating and Air-Conditioning Engineers,Inc.

Blomberg, T. 1998. HEAT 3, A PC program for heat transferin three dimensions. Manual with brief theory andexamples. Lund University, Sweden, Department ofBuilding Technology, Building Physics.

Carslaw, H.S., and J.C. Jaeger. 1959. Conduction of Heat inSolids. Oxford: Clarendon Press.

Claesson, J. 2003. Dynamic thermal networks. A methodol-ogy to account for time-dependent heat conduction. Pro-

ceedings of the 2d International Conference onResearch in Building Physics, Leuven, Belgium.

Claesson, J. 2002. Dynamic thermal networks. Outlines of ageneral theory. Proceedings of the 6th Symposium onBuilding Physics in the Nordic Countries, Trondheim,Norway.

Davies, M.G. 1997. Wall transient heat flow using time-domain analysis. J. of Building and Environment 32:427-446.

EN ISO 10211-1. 1995. Thermal bridges in building con-struction – Heat flows and surface temperatures – Part 1:General calculation methods. European Standard.

EN ISO 10211-2. 2001. Thermal bridges in building con-struction – Calculations of heat flows and surface tem-peratures – Part 2: Linear thermal bridges. EuropeanStandard.

Kossecka, E., and J. Kosny. 1998. Effect of insulation andmass distribution in exterior walls on dynamic thermalperformance of whole buildings. Proc. Thermal Perfor-mance of Exterior Envelope of Buildings VII, Dec.1998,Clearwater, Florida.

Wentzel, E.-L., and J. Claesson. 2003. Heat loss of walls.Analysis and optimizing based on the theory of dynamicthermal networks. Proceedings of the 2d InternationalConference on Research in Building Physics, Leuven,Belgium.

Wentzel, E.-L. 2002. Application of the theory of dynamicthermal networks for energy balances of a building withthree-dimensional heat conduction. Proceedings of the2d International Conference on Research in BuildingPhysics, Leuven, Belgium.

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