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International Journal of Thermal Sciences 58 (2012) 20e28

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International Journal of Thermal Sciences

journal homepage: www.elsevier .com/locate/ i j ts

The harmonic method: A new procedure to obtain wall periodic cross responsefactors

Fernando Varela a,*, Francisco J. Rey b, Eloy Velasco b, Santiago Aroca a

aDepartment of Energy Engineering, Escuela Técnica Superior de Ingenieros Industriales, Universidad Nacional de Educación a Distancia, Calle de Juan del Rosal, 12, 28040 Madrid,SpainbDepartment of Energy Engineering and Fluid Mechanics, Escuela Técnica Superior de Ingenieros Industriales, Universidad de Valladolid, Paseo del Cauce, s/n 47011 Valladolid, Spain

a r t i c l e i n f o

Article history:Received 10 December 2010Received in revised form13 August 2011Accepted 4 March 2012Available online 11 April 2012

Keywords:Wall heat transferPeriodic response factorsTrigonometric interpolation

* Corresponding author.E-mail address: fvarela@ind.uned.es (F. Varela).

1290-0729/$ e see front matter � 2012 Elsevier Masdoi:10.1016/j.ijthermalsci.2012.03.005

a b s t r a c t

Thermal accumulation phenomena are one of the hardest problems to solve in thermal load and energycalculations in buildings. The evolution in the mathematical modeling of building envelopes wants toprovide a response to the growing need for accuracy in equipment dimensioning and energy certificationcalculation methods.

Among all existing methods, the response factors method and the transfer function method, developedby Mitalas (1967) and Stephenson (1971), stand out due to their simplicity, accuracy and widespread use.

After that, with the release of the RTS (Radiant Time Series) method, developed by J.D. Spitler, D.E.Fisher and C.O. Pedersen in 1997, the concept of periodic response factors is introduced, adding thehypothesis of 24-h periodicity of the input conditions.

The main difficulty of the method lies in obtaining these so called response factors or weightingfactors, which requires an approximate, complex, and high computing cost procedure.

In this paper, a new method to obtain 24 h periodic wall response factors called the Harmonic Methodis presented, which constitutes an improvement over usual methods used to obtain wall responsefactors. It will be shown that this method is able to reduce computational effort and increase accuracycompared to previous calculation algorithms.

� 2012 Elsevier Masson SAS. All rights reserved.

1. Introduction. Presentation of the problem

The problem to be solved is the transient heat conduction inmulti-layered walls present in buildings, in order to estimate heatloads due to wall conduction. In this context, the heat load ofa space in a building is essentially the heat power needed tomaintain a certain constant indoor temperature in the space, andcoincides with the sum of the inside face heat flows of the limitingwalls. To estimate the maximum load, is common to consider anoutdoor periodic temperature function, which represents thecoldest day possible for the climate where the building is located.The reason to be periodic is that we suppose that not only a cold dayis possible, but also a sequence of cold days can occur. This periodicfunction is given through hourly temperature data from climaticrecords, this is, 24 temperature values. The searched variable, heatflow in the inside face of the walls, or generally the heat flux (flowper area unit), allows us to choose an adequate size of the heatgenerator (boiler, heat pump.) for the building and heatingterminal units (radiator, radiant slab, fan-coils.) for the space.

son SAS. All rights reserved.

Traditionally, themost commonmethod used to obtain this heatflux is through the calculation of a set of values called wall responsefactors, which represent the hourly evolution of the heat fluxresponse in the inside surface of the wall to a triangular pulse.These are commonly obtained through the Laplace’s transformmethod. It roughly consists of transforming the one-dimensionalHeat Equation, taking it to a space where its solution is mucheasier to attain than in time domain: Laplace’s domain. Once this“easy” solution is obtained in Laplace’s domain, wemust proceed tofind its counterpart in the original space, which is the desiredfunction [1,2].

This is precisely the critical point of this procedure: invertingLaplace’s transform to obtain the solution in the time domain. Thispoint is dealt with in detail later, and as can be seen in [3] and [4],this method requires a search for the infinite roots of a complexfunction B(s), this complexity growing with the number of layers inthe wall. A series with the roots found must also be evaluated.

Although improved ad-hoc root finding methods have beendeveloped [5], the computational cost of root finding is extremelyhigh, involving a great (and above all, indeterminate) number offunction evaluations [6,7]. In addition, root finding by means ofnumerical methods involves an error related to the convergence

Nomenclature

Ag (1,1) component characteristic matrix of the wallAj (1,1) component characteristic matrix of the layer jBg (1,2) component characteristic matrix of the wallBj (1,2) component characteristic matrix of the layer jCg (2,1) component characteristic matrix of the wallCj (2,1) component characteristic matrix of the layer jcn, ccn cosine coefficients [K]cp specific thermal capacity [J kg�1 K�1]Dg (2,2) component characteristic matrix of the wallDj (2,2) component characteristic matrix of the layer jG generic transfer functionk thermal conductivity [Wm�1 K�1]L layer length [m]mg charateristic matrix of the wallm j charateristic matrix of the layer jqe exterior heat flux [W m�2]qi interior heat flux [W m�2]q̂e Laplace transform of exterior heat flux [W m�2]q̂i Laplace transform of interior heat flux [W m�2]~qe Fourier transform of exterior heat flux [W m�2]~qi Fourier transform of interior heat flux [W m�2]R ramp function [K]s Laplace domain variable [s�1]sn, ssn sine coefficients [K]T temperature [K]T̂ Laplace transform of temperature [K]~T Fourier transform of temperature [K]T0 initial layer temperature [K]Te exterior surface temperature [K]Ti interior surface temperature [K]T̂e Laplace transform of exterior surface temperature [K]T̂ i Laplace transform of interior surface temperature [K]~Te Fourier transform of exterior surface temperature [K]~T i Fourier transform of interior surface temperature [K]t time [s]x spatial variable [m]X exterior triangle response function [Wm�2 K]

XR exterior ramp response function [Wm�2 K]X(j) j-th exterior response factor [Wm�2 K]XP(j) j-th periodic exterior response factor [Wm�2 K]Y cross triangle response function [Wm�2 K]YR cross ramp response function [Wm�2 K]Y(j) j-th cross response factor [Wm�2 K]YP(j) j-th periodic cross response factor [Wm�2 K]Z interior triangle response function [Wm�2 K]ZR interior ramp response function [Wm�2 K]Z(j) j-th interior response factor [Wm�2 K]ZP(j) j-th periodic interior response factor [Wm�2 K]a thermal diffusivity [m2 s-1]bk roots of Bg [s�1]c, cn exterior periodic response factors [Wm�2 K]D, Dn triangular pulse function [K]dk k-th trigonometric pulse function [K]f, fn interior phase function [rad]h, hn exterior phase function [rad]k, kn interior periodic response factors [Wm�2 K]r density [kg m�3]u, un frequency [rad s�1]j, jn cross phase function [rad]x, xn cross periodic response factors [Wm�2 K]

Subscriptsi interiore exterior0 initialg globalR rampH harmonicC common

List of abbreviationsASHRAE American Society of Heating, Refrigerating and Air

Conditioning EngineersFFT Fast Fourier TransformRQD Relative Quadratic DeviationTMY Typical Meteorological Year

F. Varela et al. / International Journal of Thermal Sciences 58 (2012) 20e28 21

tolerance of the method and maximum number of iterations [7,8].Another error source of this method is that we cannot obviouslycalculate all the infinite number of function roots, and are forced totruncate the series.

The alternative method proposed in this document, valid forperiodic boundary conditions (periodic response factors [9]),means a change in philosophy when solving the equation. Insteadof Laplace transform, Fourier transform is used, more suitable forperiodic problems, and, only trigonometric interpolation of inputfunctions will be needed, avoiding the process of root finding.

The use of this method involves certain advantages and,unavoidably, some disadvantages over the original method,disadvantages which we will try to avoid or at least minimize asefficiently as possible.

We will begin by briefly describing the classical solution to theproblem to stress in its key points and compare it with the oneobtained through the Harmonic Method.

2. Laplace’s transform method

Heat transfer by conduction is modeled in each of the layers ofa multi-layer wall by means of the partial differential equationknown as one-dimensional heat equation, which can be written as:

>>>>> vT ¼ av2T

8><>>>>>>:

vt vx2Tðx;0Þ ¼ T0ðxÞTð0; tÞ ¼ TeðtÞTðL; tÞ ¼ TiðtÞ

(1)

where Tðx; tÞ represents temperature as a function of position (x)and time (t), a is the thermal diffusivity of the layer, T0ðxÞ is theinitial temperature of the layer, and TeðtÞ and TiðtÞ are the outsideand inside surface temperature of the layer, respectively.

As initial condition for thewall T0ðxÞ is not a known function, weassume that initial temperature of the layer is null. This can be donedue to the properties of heat equation: is easy to prove [10] that thedifference between the solutions of the original problem Tðx; tÞ andthat of null initial condition T*ðx; tÞ8>>>>>>><>>>>>>>:

vT*

vt¼ a

v2T*

vx2

T*ðx;0Þ ¼ 0

T*ð0; tÞ ¼ TeðtÞT*ðL; tÞ ¼ TiðtÞ

35Dry Bulb Temperature 12th August. TMY Valladolid (Spain)

F. Varela et al. / International Journal of Thermal Sciences 58 (2012) 20e2822

diminishes exponentially with time:

���Tðx; tÞ � T*ðx; tÞ��� � Ke

��pL

�2

t; x˛½0; L�

0 5 10 15 20 250

5

10

15

20

25

30

solar hour

Tem

pera

ture

(ºC

)

Δ15

Fig. 1. Decomposition of a function as a triangle summation.

0.5

1

1.5

2

2.5

3Δn−2·RnRn+1Rn−1

Thus, influence of T0ðxÞ is negligible after a short period of timeand can be neglected. In a practical way, the temperature of thewalldoes not depend on the initial condition but on the boundaryconditions. To ensure that the influence of T0ðxÞ can be neglected,usually in thermal simulations in buildings the first day is repeatedperiodically.

Applying Laplace’s transform in the time domain, equation (1) istransformed into the ordinary differential equation8>>>>><>>>>>:

s$T̂ ¼ a$v2T̂vx2

T̂ð0Þ ¼ T̂e

T̂ðLÞ ¼ T̂ i

(2)

where T̂ðx; sÞ is Laplace’s Transform of function Tðx; tÞ.As can be found in the literature [3] [4], applying Fourier’s Law to

the solution of (2) in every single layer of thewall, after some simplealgebraic transformations it is possible to express heat flux in bothextreme surfaces of the complete multi-layer wall as a function ofinside and outside surface temperatures (in Laplace domain):

�q̂eq̂i

�¼

2664DgðsÞBgðsÞ �

1BgðsÞ

1BgðsÞ �

AgðsÞBgðsÞ

3775�T̂eT̂ i

�(3)

where AgðsÞ, BgðsÞ and DgðsÞ are elements of the characteristicmatrix of the wall MgðsÞ obtained by successive multiplication ofindividual characteristic matrices of each layer:

MgðsÞ ¼�AgðsÞ BgðsÞCgðsÞ DgðsÞ

�¼Ynj¼1

�AjðsÞ BjðsÞCjðsÞ DjðsÞ

�¼Ynj¼1

MjðsÞ (4)

The components of the characteristic matrix of each layer aredefined from the thermophysical properties of the material whichconstitutes the layer:

�AjðsÞ BjðsÞCjðsÞ DjðsÞ

�¼

26666666664

ch

Lj

ffiffiffiffiffisaj

r ! sh

Lj

ffiffiffiffiffisaj

r !

kj

ffiffiffiffiffisaj

r

kj

ffiffiffiffiffisaj

rsh

Li

ffiffiffiffiffisaj

r !ch

Li

ffiffiffiffiffisaj

r !

37777777775

(5)

where Lj is the width of layer j, aj is the thermal diffusivity of layer j,kj is the thermal conductivity of layer j.

Response in Laplace’s domain is thus obtained. The Laplacetransform must be inverted to attain the response in time domain,and it is precisely at this point where classical and harmonicmethods diverge.

n−3 n−2 n−1 n n+1 n+2 n+3−2

−1.5

−1

−0.5

0

Fig. 2. Decomposition of a triangle function into 3 ramp functions.

2.1. The classical method: linear interpolation of input functions

The traditional approach to the problem leads to a piecewiselinear interpolation of hourly temperature input data. As can beseen in [3] and [4], this is equivalent to describe the input functionas a linear combination of triangle functions Dn with height 1 unitof temperature and base width 2 time units (Fig. 1), centered intime unit n (usually hours):

TðtÞ ¼Xn

j¼�N

TðjÞ$DjðtÞ (6)

for t˛ðm� 1;mÞ, where TðjÞ are the available hourly temperaturedata.

Once again, the triangle function can be decomposed into threeramp functions (Fig. 2), of which Laplace’s transform is known:

DnðtÞ ¼ Rn�1ðtÞ � 2RnðtÞ þ Rnþ1ðtÞ (7)

Combining equation (3) with expression (6) for temperatureinputs, evaluating the expression hourly, and taking advantage ofthe linearity and autonomous nature of the process, we reach anexpression for heat fluxes in extreme surfaces:8>>><>>>:

qeðnÞ ¼ PNj¼0

Teðn� jÞ$XðjÞ � Tiðn� jÞ$YðjÞ

qiðnÞ ¼ PNj¼0

Teðn� jÞ$YðjÞ þ Tiðn� jÞ$ZðjÞ(8)

where values XðjÞ, YðjÞ, ZðjÞ, j ¼ 1;2; ::: are known as wall responsefactors.

0 5 10 15 20 2512

14

16

18

20

22

24

26

28

30

32

solar hour

Tem

pera

ture

(ºC

)

Dry Bulb Temperature 12th August. TMY Valladolid (Spain)

Fig. 3. Trigonometric interpolation of outside temperature data.12th august city ofValladolid (Spain).

F. Varela et al. / International Journal of Thermal Sciences 58 (2012) 20e28 23

This is to say, it is possible to express incoming and outgoinghourly heat flux in a wall as a weighted summation of the outsideand inside surface temperatures preceding the current calculationhour.

Having described the process, it should be pointed out that themost difficult and computer time-consuming step is obtaining theso called response factors, by means of the inversion of Laplace’stransform for responses to ramp input functions in a proceduredescribed in [3] and [4].

Thus, to summarize, the series must be evaluated.

XRðtÞ ¼ t�DgðsÞBgðsÞ

�s¼0

þ dds

�DgðsÞBgðsÞ

�s¼0

þXNk¼1

1

b2k

DgðsÞB0gðsÞ

�����s¼�bk

e�bkt

YRðtÞ ¼ t�

1BgðsÞ

�s¼0

þ dds

�1

BgðsÞ�s¼0

þXNk¼1

1

b2k

1B0gðsÞ

�����s¼�bk

e�bkt

ZRðtÞ ¼ t�AgðsÞBgðsÞ

�s¼0

þ dds

�AgðsÞBgðsÞ

�s¼0

þXNk¼1

1

b2k

AgðsÞB0gðsÞ

�����s¼�bk

e�bkt

(9)

To do this, we must first find a reasonably high amount of bk(which are Bg roots) by an iterative process, and evaluate the wallcharacteristic matrix and its derivative to attain a certain level ofaccuracy.

Equation (9) shows that high frequencies present in tempera-ture signal, due, for example, to measurement noise, are neglectedwhen truncating the series, filtering the response. This method isnot discussed here because is not the aim of the present paper, butother choices of inversion of Laplace transform can be chosenwhich allow avoiding this circumstance [11].

Furthermore, we have to carry out the required operations toadd up the three summations and obtain the response factors as inequation (10).

8<:

XðjÞ ¼ XRðjþ 1Þ � 2$XRðjÞ þ XRðj� 1ÞYðjÞ ¼ YRðjþ 1Þ � 2$YRðjÞ þ YRðj� 1ÞZðjÞ ¼ ZRðjþ 1Þ � 2$ZRðjÞ þ ZRðj� 1Þ

(10)

To obtain the response to a 24 h periodic signal, the periodicresponse factors are as the sum the response factors in thefollowing fashion:8<:

XPðjÞ ¼ XðjÞ þ Xðjþ 24Þ þ Xðjþ 48Þ þ.YPðjÞ ¼ YðjÞ þ Yðjþ 24Þ þ Yðjþ 48Þ þ.ZPðjÞ ¼ ZðjÞ þ Zðjþ 24Þ þ Zðjþ 48Þ þ.

3. The harmonic method

3.1. Trigonometric interpolation. An initial approach

We will first take advantage of the periodic character of theproblem considering the appropriate version of heat equation fora single layer that characterizes it:8>><>>:

vTvt

¼ av2Tvx2

; ðx; tÞ˛½0; L� � R

Tð0; tÞ ¼ TeðtÞ ¼ Teðt þ 24Þct˛RTðL; tÞ ¼ TiðtÞ ¼ Tiðt þ 24Þct˛R

(11)

Initial condition T0ðxÞ disappears, as the solution for the problem(11) has a unique solutionwhen considering the complete problemwith time coming from �N to N. This eliminates the necessity of

estimating or neglecting the initial condition, function that in heatload estimation problems in building walls is never known.

The second difference respect to the original method is that ismore convenient to apply Fourier transformation instead of Lap-lace’s, due to the periodic character of the problem and the fact thatexcitation is not null before time 0, but it is always present from -N.Applying Fourier transform to (11) and following an equivalentprocess to that of section 2, leads us to characteristic matrices inlayers like this:

�AjðuÞ BjðuÞCjðuÞ DjðuÞ

�¼

26666666664ch

Lj

ffiffiffiffiffiiuaj

s ! sh

Lj

ffiffiffiffiffiiuaj

s !

kj

ffiffiffiffiffisaj

r

kj

ffiffiffiffiffiiuaj

ssh

Li

ffiffiffiffiffiiuaj

s !ch

Li

ffiffiffiffiffiiuaj

s !

37777777775

and characteristic matrix of the wall results to be

"~qe~qi

#¼

26664DgðiuÞBgðiuÞ �

1BgðiuÞ

1BgðiuÞ �

AgðiuÞBgðiuÞ

37775"~Te~T i

#

identical to that obtained by Laplace’s transform substituting s byiu, and denoting by ~qe the Fourier transform of exterior heat fluxand so on.

The key to the difference in both methods is the choice of theinterpolant function of input data. In this case, as we have a set ofequally spaced temperature data (usually hourly data), inside andoutside temperature functions can be described as a finite sum ofsine and cosine functions by means of trigonometric interpolation(Fig. 3):

TeðtÞ ¼ P12n¼0

cn$cosðuntÞ þ sn$sinðuntÞ

TiðtÞ ¼ P12n¼0

ccn$cosðuntÞ þ ssn$sinðuntÞ(12)

where un ¼ 2pn24

and cn; sn; ccn; ssn are coefficients obtained from

Discrete Fourier’s Transform of the vectors of inside and outside

F. Varela et al. / International Journal of Thermal Sciences 58 (2012) 20e2824

temperature data. The analytical expression of these coefficientscan be found in [9]. The reason to look for such an interpolant isthat by considering sines and cosines as excitation functions inlinear systems (e.g. heat equation) it is very simple to find out itsresponse in time domain.

Due to the properties of the transfer functions present in thecharacteristic matrix, it is easy to prove that the response ofequation

~YðuÞ ¼ GðuÞ$~XðuÞto an excitation XðtÞ ¼ cosðutÞ, when G(u) is any of this threementioned transfer functions Dg(iu)/Bg(iu), 1/Bg(iu), Ag(iu)/Bg(iu) ismerely a change in wave amplitude (decrement factor) A anda phase delay (time lag) f [12] (Fig. 4):

YðtÞ ¼ AðuÞ$cosðut þ fðuÞÞ (13)

where AðuÞ ¼ jGðiuÞj and fðuÞ ¼ argðGðiuÞÞ.The response in time domain for equation (3) considering input

functions described by trigonometric interpolation (equation (12))will be

qeðtÞ ¼P12n¼0

An$½cn$cosðuntþfnÞþ sn$sinðuntþfnÞ�

� P12n¼0

Bn$½ccn$cosðuntþjnÞþ ssn$sinðuntþjnÞ�

qiðtÞ ¼P12n¼0

Bn$½cn$cosðuntþjnÞþ sn$sinðuntþjnÞ�

� P12n¼0

Cn$½ccn$cosðuntþhnÞþssn$sinðuntþhnÞ�

(14)

where

An ¼����DgðiunÞBgðiunÞ

����;Bn ¼���� 1BgðiunÞ

����;Cn ¼����AgðiunÞBgðiunÞ

����fn ¼ arg

�DgðiunÞBgðiunÞ

;jn ¼ arg

�1

BgðiunÞ;hn ¼ arg

�AgðiunÞBgðiunÞ

(15)

It is clear that the only unknown parameters in equation (14) arethe coefficients of sines and cosines, which can easily be obtainedbymeans of the Fast Fourier Transform (FFT) of the input data [9]. Inthis method, the maximum frequency of the temperature signal isdetermined by the number n of nodes (temperature samples) per

0 2 4 6 8 10

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1arg(G(iω))

|G(iω)|

Excitation X(t)Response Y(t)

Fig. 4. Decrement factor and time lag in a wall due to sinusoidal excitation.

period (a day, for periodic response factors) and therefore themaximum frequency of the signal is n$p/12 rad/h. Very commonlythis quantity of samples is set to 24, due to data disposal in climaticrecords and accuracy reasons, which leads to a maximumfrequency in temperature signal of p/12 rad/h. As measurementerror in temperature probes is lower 0.2 K, with a sampling rate ofone sample per hour, measurement noise is of low amplitude andundetectable and therefore not considered in this method. To applythis method for other purposes different from the one which hasbeen designed for, where signals with high frequencies are present,some kind of filtering is recommended.

The simplicity of the offered solution compared to the classicalmethod is clear, avoiding the root finding procedure and thesubsequent series evaluation.

However, the need to make the FFT, which has an order ofN$logN operations (with an efficient algorithm, e.g. CooleyeTukeyor similar [13]) every time a heat flux is calculated, makes thismethod as time-consuming as the classical method. The“trigono-metric pulse” function.

As concluded in last paragraph, although the simplicity of theproposed method is evident compared to the original, the amountof calculation to be done in each heat flux evaluation slows theprocess down and eliminates the achieved advantage. However,using a simple mathematical transformation we will be able toavoid this problem.

The key to the simplification lies in the constructing method ofthe interpolant. The goal is to make the calculation of the FFTindependent of temperature input data, and reorder the terms sothat the functions which appear in the expression of temperatureare actually one single function delayed in time.

These two things are achieved by means of a function which wecall k-th trigonometric pulse (Fig. 5), constructed to be the trigono-

metric interpolant of the dataset ½0;0; :::;0; 1ðkÞ

;0; :::;0;0�, this is tosay, null data except the k-th, which is 1.

This way any discrete function can be written as a linearcombination of trigonometric pulses (Fig. 6), enabling us to re-write equation (12) as,

TeðtÞ ¼ P23n¼0

TeðnÞ$dnðtÞ

TiðtÞ ¼ P23n¼0

TiðnÞ$dnðtÞ(16)

0 5 10 15 20−0.5

0

0.5

1

1.5

solar hour

Tem

pera

ture

(ºC

)

Trigonometric Pulse k=7

Fig. 5. Trigonometric pulse for k ¼ 7.

Fig. 6. Function shown as sum of trigonometric pulses.

Table 2Deviation of cross periodic response factors.

Wall RQD Wall RQD Wall RQD

1 16.46% 15 1.01% 29 0.12%2 6.44% 16 0.78% 30 0.29%3 7.40% 17 0.76% 31 0.26%4 7.015% 18 0.67% 32 0.18%5 3.50% 19 0.53% 33 0.16%6 3.60% 20 0.40% 34 0.12%7 4.18% 21 0.24% 35 0.40%

F. Varela et al. / International Journal of Thermal Sciences 58 (2012) 20e28 25

Taking again into account the linear and autonomous nature ofthe response, by evaluating heat fluxes in the calculation hours, weobtain

qeðmÞ ¼ P23n¼0

Teðm� nÞ$cðnÞ � P23n¼0

Tiðm� nÞ$xðnÞ

qiðmÞ ¼ P23n¼0

Teðm� nÞ$xðnÞ � P23n¼0

Tiðm� nÞ$kðnÞ(17)

where

cðtÞ ¼�DgðsÞBgðsÞ$d̂0ðsÞ

ˇ

�1xðtÞ ¼

�1

BgðsÞ$d̂0ðsÞ ˇ

�1

kðtÞ ¼�AgðsÞBgðsÞ$d̂0ðsÞ

ˇ

�1

(18)

are the system responses to the 0-th trigonometric pulse, theanalogous to X, Y and Z functions in piecewise linear interpolation.

Using this technique, the FFT of temperature data does not needto be calculated each evaluation time, and only needs to be per-formed for the dataset ½1;0;0:::;0�. This can be pre-calculated andstored, with null computational cost.

Sine and cosine coefficients for function d0 are shown in Table 1.Using the coefficients in Table 1, expression (14) becomes:

cðmÞ ¼ 112

X12n¼0

An½cosðunmþ fnÞ�

xðmÞ ¼ 112

X12n¼0

Bn½cosðunmþ jnÞ�

kðmÞ ¼ 112

X12n¼0

Cn½cosðunmþ hnÞ�

(19)

where A0;B0;C0;A12;B12;C12 must be divided by two.

Table 1Function d0 coefficients.

n 0 1 2 3 4 5 6 7 8 9 10 11 12

cn 1/24 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/12 1/24

sn 0 0 0 0 0 0 0 0 0 0 0 0 0

3.2. Validation of the method

The obtained heat flux response corresponds to the analyticalsolution of heat equation in multi-layered walls for periodicboundary conditions which can be written as trigonometricpolynomials. The aim of this section is in fact to compare periodicresponse factors obtained with both (harmonic and classical)methods.

4. ASHRAE reference walls. Y factors

To test the similarity between both sets of response factors, the41 ASHRAE reference walls [14] developed by Harris and McQuis-ton [15] will be used as a comparison dataset. These walls wereintended to span the complete range of constructions used in NorthAmerica, and are the result of studying some 2600 walls, classifiedinto 41 categories according to these four characteristics:

1. Insulation (global termal resistance)2. Material with higher thermal mass3. Second material with higher thermal mass4. Place of predominant thermal mass

a. Interiorb. Exteriorc. Essentially homogeneous

Any construction made with the ASHRAE layers dataset can beincluded in one of these 41 subsets and its response factorsdeduced [14]. That is why this set of walls has been considered anappropriate reference.

Cross response factors (Y factors) will be used to make thecomparison. The difference between two sets of factors will bemeasured with the Relative Quadratic Deviation (RQD) of the twovectors:

RQD ¼ k v!H � v!Ck2k v!Ck2

where v!H, v!C are the 24 position vectors of periodic responsefactors obtained with the harmonic and classical method, respec-tively. RQD is shown in Table 2 for the complete ASHRAE dataset.

Table 2 show deviations decrease inversely with wall number. Infact, as can be seen in Fig. 7, the logarithm of relative difference(RQD) between the two series of factors is strongly correlated withthe time lag and decrement factor of walls.

The difference increases exponentially with decrement factorand decreases exponentially with time lag. The reason for thisphenomenon will be explained in next subsection.

8 2.34% 22 0.64% 36 0.098%9 1.71% 23 0.53% 37 0.23%10 1.46% 24 0.35% 38 0.15%11 1.42% 25 0.27% 39 0.53%12 1.36% 26 0.23% 40 0.27%13 1.22% 27 0.36% 41 0.35%14 0.86% 28 0.17%

0 5 10 15

10−1

100

101

Wall Time Lag (h)

RQ

D (%

)

0 0.2 0.4 0.6 0.8 1

10−1

100

101

Wall Decrement Factor (adim.)

RQ

D (%

)

Fig. 7. Time lag and decrement factor of ASHRAE wall set vs. RQD.

0 5 10 15 20−0.5

0

0.5

1

1.5

solar hour

Tem

pera

ture

(ºC

)

Trigonometric pulse vs. Triangular pulse

Triangular pulseTrigonometric pulse

Fig. 8. Triangular and trigonometric pulse.

Fig. 9. Amplitude attenuation vs. frequency for different thicknesses (left), thermal co

F. Varela et al. / International Journal of Thermal Sciences 58 (2012) 20e2826

4.1. Triangular pulse vs. trigonometric pulse

Periodic response factors obtained with the two methodscannot be compared directly, because they are the system responseof two different input data: excitation for the classical method ispiecewise linear (triangle pulse), and for harmonic method issinusoidal (trigonometric pulse), being equal only in grid points, asshown in Fig. 8.

As can be seen in Table 2, lightest walls (the first) show a higherdifference between the two sets of response factors. This is becausethe wall acts as a low-pass filter when calculating heat flux in theopposite side to the excitation, decreasing high-frequency waveamplitude.

This way, high harmonics are attenuated and system responsesto triangular pulse and trigonometric pulse converge. The heavierthe wall, the higher the convergence, because the filter is moredemanding, as shown in Fig. 9.

To make both responses directly comparable, the excitationmust be a function whose piecewise linear and trigonometricinterpolations are similar (smooth functions). With that aim, an

nductivities (center) and thermal capacities (right) in a sample single layer wall.

0 5 10 15 20 2510

15

20

25

30

35

40

45

solar hour

Tem

pera

ture

(ºC

)

Sol−Air and Dry Bulb Temperature 12th August. TMY Valladolid (Spain)

Dry Bulb TemperatureSol−Air Temperature

Fig. 10. Soleair temperature daily evolution for south-oriented medium colored wall.12th August synthetic TMY of Valladolid (Spain).

Table 3Relative quadratic deviation between ordinary and harmonic periodic responsefactors for test walls.

Wall RQDa Wall RQD Wall RQD

1 1.01% 15 0.86% 29 0.10%2 0.83% 16 0.68% 30 0.44%3 0.65% 17 0.60% 31 0.43%4 0.82% 18 0.81% 32 0.29%5 0.80% 19 0.31% 33 0.28%6 0.95% 20 0.27% 34 0.24%7 0.42% 21 0.18% 35 1.10%8 0.38% 22 0.84% 36 0.17%9 0.76% 23 0.68% 37 0.26%10 0.80% 24 0.58% 38 0.17%11 0.42% 25 0.27% 39 1.47%12 0.81% 26 0.20% 40 0.66%13 0.38% 27 0.46% 41 1.06%14 0.29% 28 0.19%

a Relative Quadratic Deviation.

F. Varela et al. / International Journal of Thermal Sciences 58 (2012) 20e28 27

arbitrary soleair outside temperature function will be chosen ful-filling that condition, and the response of the 41 reference wallswill be obtained with both methods.

4.2. Comparison of responses to a smooth excitation

Radiation and temperature TMY data in the city of Valladolid(Spain) corresponding to 12th August have been used for the test,1

being expressed through the soleair temperature function shownin Fig. 10.

In Table 3, relative quadratic deviation between harmonic andclassical periodic wall response factors is shown for ASHRAE’sreference walls, using the above-described soleair test function asinput.

As it was predicted, considering a smooth input, the differencebetween both sets of periodic response factors along ASHRAEdataset and becomes independent of the wall mass.

1 The data used for this comparison have been obtained from the TypicalMeteorological Year (TMY), generated by the University of Seville (Spain). ThisMeteorological database was developed to be used in energy certifications in Spainwith the standard tool for energy Certifications in buildings CALENER.

5. Conclusions

A new periodic response factors calculation method has beendeveloped, derived from the ordinary response factors calculationmethod. A validation of the results offered by this method has beenperformed, contrasting them with those obtained with ASHRAEresponse factors in the 41 reference walls included in ASHRAEFundamentals: Handbook. Differences are below 1.5% in each testedwall and about 0.56% in average.

The presented method offers the following advantages:

1. Simpler:a. Based on insulation and thermal lag concepts. Using this

Fourier-like approachwe preserve this simple physical viewof wall behavior.

b. By avoiding the root finding algorithm, this method provesmuch easier to implement in a computer program.Response factor calculation is limited to the evaluation ofequation (19), where all the coefficients can be directlyobtained from the properties of the wall’s layers: thermalconductivity, specific heat, density and thickness.

2. Faster:a. Avoids the root finding procedure: As mentioned before, it

is not necessary to calculate the inverse of Laplace’s trans-form and therefore the computationally heavy root findingalgorithms.

b. Speeds up the RTS method: It obtains wall periodicresponse factors directly, which do not need to be obtainedfrom ordinary response factors, thus benefitting the RTSheat load calculation procedure.

3. More accurate:a. The response obtained is the analytical solution of heat

equation for boundary conditions described by trigono-metric polynomials, no numerical method being required tosolve it.

b. Avoids numerical errors arising when approximate rootfinding procedures are used.

c. Avoids truncating errors when series given by equation (9)are summed.

Acknowledgments

This work is included in the research lines funded by theNational RþD Plan. The funded project is entitled: “Reduction ofenergy consumption and carbon dioxide emission in buildingscombining evaporative cooling, free cooling and energy recovery inall-air systems”. Ref.: ENE2008-02274/CON.

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