An improvement to frequency-domain regression method for calculating conduction transfer functions of building walls

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Applied Thermal Engineering 28 (2008) 661–667 www.elsevier.com/locate/apthermeng

An improvement to frequency-domain regression method for calculating conduction transfer functions of building walls Xinhua Xu a, Shengwei Wang a

a,*

, Youming Chen

b

Department of Building Services Engineering, The Hong Kong Polytechnic University, Kowloon, Hong Kong b College of Civil Engineering, Hunan University, Changsha 410082, Hunan, China Received 8 March 2007; accepted 14 June 2007 Available online 22 June 2007

Abstract An improvement is proposed for the frequency-domain regression (FDR) method to calculate CTF (conduction transfer function) coefficients of multilayer constructions. This improvement aims at providing complete CTF coefficients with a unique set of d values. In the improvement, the external and internal polynomial s-transfer functions of heat conduction of a wall are identified based on equivalent frequency response characteristics by constraining their denominators equal to that of the cross polynomial s-transfer function while this cross polynomial s-transfer function of this wall is identified using FDR method in advance. The complete CTF coefficients of this wall (i.e., external CTF values a, cross CTF values b, internal CTF values c and a unique set of heat flow rate history coefficients d) can be easily calculated from these identified polynomial s-transfer functions. Examples and comparisons show that this improvement of the frequency-domain regression method works well. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: FDR method; CTF coefficients; Improvement; Frequency characteristics

1. Introduction Conduction transfer function (CTF) method, formerly introduced by [16], is commonly used to calculate space transient heat transfer for system design and system operations etc. To obtain the CTF coefficients of a multilayer construction for heat flow calculation, a computationally lengthy, tedious and inefficient process is involved in to find the poles of hyperbolic s-transfer function and to compute their residues [11]. This process may also occasionally lead to miscalculation due to missing a root, particularly in the case where two adjacent roots are close together [13]. Subsequently, state space methods [12,13,15] and the time-domain method [8] were developed to calculate these coefficients. Recently, frequency-domain regression (FDR) method [17,19] was developed to calculate response factors and conduction transfer function (CTF). This method can *

Corresponding author. Tel.: +852 27730345; fax: +852 27746146. E-mail address: [email protected] (S. Wang).

1359-4311/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2007.06.008

avoid the complicate root-finding process while providing accurate heat flow calculation. Furthermore, this method is easily understandable, simple, reliable, and has high computational efficiency [4–6,18]. In most currently used energy simulation packages such as HVACSIM+ [14], TRNSYS [10], and EnergyPlus [7], BLAST [9,3], a unique set of d values are used. These values are calculated using conventional ‘‘root-finding’’ method and/or state space method. In the published literature [18,19] using FDR method, there are three different sets of d values since these three approximate external, cross and internal polynomial s-transfer functions are regressed respectively and independently. Therefore, in this study, an improvement is proposed to this FDR method to obtain a unique set of d values to meet the conformability of the common understanding of CTF coefficients by conventional methods (i.e., the ‘‘so-called’’ feature of a unique set of d values). However, it is needed to point out that it is not the defect of this FDR method resulting in three different sets of d values. The difference definitely does not affect

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X. Xu et al. / Applied Thermal Engineering 28 (2008) 661–667

Nomenclature A, B transmission matrix element or polynomial D transmission matrix element a, b, c, d transfer function coefficients G transfer function H, g matrix J objective function j imaginary unit O, P, Q, R real number q heat flow (W m2) s Laplace variable or roots T temperature (°C or K) V, W real number

Superscripts ^ estimated T transpose 0 associated with regressed polynomial X associated with external heat conduction Y associated with cross heat conduction Z associated with internal heat conduction k, r, m, N integer count Subscripts in inside k integer count out outside X associated with external heat conduction Y associated with cross heat conduction Z associated with internal heat conduction

Greek symbols a, b polynomial s-transfer function coefficients b coefficient vector U, W matrix x frequency (s1) the validity and applicability of FDR method for heat flow calculation although a more step is required to calculate cross heat flow and internal heat flow separately resulting in the net heat flow through a wall eventually. This improvement to this FDR method is to provide an alternative approach to calculate CTF coefficients of multilayer constructions for heat flow calculation for current application programs. Examples and comparisons show that this improvement is feasible and works well. 2. Brief review of frequency-domain regression method The transmission equation relating temperatures to heat flows on both sides (inside and outside) of a multilayer construction in s-domain is given by Eq. (1). GX(s), GY(s) and GZ(s) are the external, cross and internal transfer functions of this construction respectively. They are expressed as Eqs. (2)–(4).      qout ðsÞ GX ðsÞ GY ðsÞ T out ðsÞ ¼ ð1Þ qin ðsÞ GY ðsÞ GZ ðsÞ T in ðsÞ ð2Þ GX ðsÞ ¼ AðsÞ=BðsÞ GY ðsÞ ¼ 1=BðsÞ GZ ðsÞ ¼ DðsÞ=BðsÞ;

ð3Þ ð4Þ

where, q is heat flow, T is temperature. A(s), B(s) and D(s) are the elements of the transmission matrix of the multilayer construction [19]. In the literature [17,19] using FDR method, three exact complex hyperbolic transfer functions GX(s), GY(s) and GZ(s) are approximately represented using G0X ðsÞ, G0Y ðsÞ and G0Z ðsÞ, respectively. These approximate transfer functions are regressed independently in term of their frequency responses equivalent to their coincident theoretical

frequency responses GX(s), GY(s), GZ(s) respectively, i.e., frequency-domain regression (FDR) method. These approximate s-transfer functions are expressed as the form of a ratio of two polynomials as Eqs. (5)–(7). BX ðsÞ 1 þ AX ðsÞ BY ðsÞ G0Y ðsÞ ¼ 1 þ AY ðsÞ BZ ðsÞ G0Z ðsÞ ¼ ; 1 þ AZ ðsÞ

G0X ðsÞ ¼

ð5Þ ð6Þ ð7Þ

where, BX(s), AX(s), BY(s), AY(s), BZ(s), AZ(s) are polynomials. These approximate transfer functions are easy to find the zeros and thus produce response factors and conduction transfer conduction (CTF) coefficients. It is obvious that AX(s) 5 AY(s) 5 AZ(s) because they are regressed independently. Three sets of dk in CTF coefficients resulted from these three approximate transfer functions are different. With conventional methods, such as direct root-finding method and time-domain method, dk resulted from these three exact transfer functions GX(s), GY(s) andGZ(s) are identical. Although three different sets of dk using FDR method do not affect the validity and applicability of FDR method, an improvement is needed for this FDR method to meet the requirement of a unique set of dk in current calculation programs for heat flow calculation and energy analysis. 3. Improvement to frequency-domain regression method In the improvement of this FDR method, the regressed denominator of G0Y ðsÞ is used as benchmark, and the denominators of G0X ðsÞ and G0Z ðsÞ to be regressed are

X. Xu et al. / Applied Thermal Engineering 28 (2008) 661–667

constrained to be equal to the denominator of the regressed G0Y ðsÞ. G0Y ðsÞ is regressed as Eq. (8) using the FDR method as described in the literature [17,19]. The approximate external and internal s-transfer functions G0X ðsÞ and G0Z ðsÞ to be regressed are expressed as Eqs. (9) and (10). After the coefficients f1; a1 ; a2 ;    ; am g of G0Y ðsÞ are identified using FDR method, the coefficients of G0X ðsÞ and the coefficients of G0Z ðsÞ to be identified are only fbX0 ; bX1 ; bX2 ;    ; bXr g and fbZ0 ; bZ1 ; bZ2 ;    ; bZr g, respectively. G0Y ðsÞ ¼ G0X ðsÞ ¼

BY ðsÞ bY þ bY1 s þ bY2 s2 þ    þ bYr sr ¼ 0 1 þ a1 s þ a2 s2 þ    þ am sm 1 þ AY ðsÞ

ð8Þ

BX ðsÞ BX ðsÞ bX þ bX1 s þ bX2 s2 þ    þ bXr sr ¼ ¼ 0 1 þ a1 s þ a2 s2 þ    þ am sm 1 þ AX ðsÞ 1 þ AY ðsÞ

ð9Þ BZ ðsÞ BZ ðsÞ bZ þ bZ1 s þ bZ2 s2 þ    þ bZr sr ¼ ¼ 0 G0Z ðsÞ ¼ 1 þ AZ ðsÞ 1 þ AY ðsÞ 1 þ a1 s þ a2 s2 þ    þ am sm

ð10Þ

To identify the unknown parameters (coefficients) of G0X ðsÞ and G0Z ðsÞ, the same philosophy is used by equaling the frequency responses of G0X ðsÞ andG0Z ðsÞ to the coincident theoretical frequency responses of GX(s) and GZ(s), respectively. When s = jx, the frequency response characteristic of G0Z ðsÞ is given as Eq. (11). Choosing N frequency points, (x1, x2, . . . xN) within the frequency range of concern, the complex function GZ(jx) can be written in two parts: real component (Ok) and imaginary component (jPk) at the kth frequency point xk as shown in Eq. (12). An objective function (J) to be minimized is introduced as shown in Eq. (13). Substituting G(jxk), BZ(jxk), AZ(jxk) in Eq. (13) and letting 1 + AZ(jxk) = Qk + jRk, this equation can be rewritten as Eq. (14). H and g are expressed as Eqs. (15) and (16). G0Z ðjxÞ ¼

bZ0 þ bZ1 ðjxÞ þ bZ2 ðjxÞ2 þ    þ bZr ðjxÞr 1 þ a1 ðjxÞ þ a2 ðjxÞ2 þ    þ am ðjxÞm

¼

BZ ðjxÞ 1 þ AZ ðjxÞ

GZ ðjxk Þ ¼ GZk ¼ Ok þ jP k minðJ ðbZ ÞÞ ¼ ^Z b

N X

jBZ ðjxk Þ  ðOk þ jP k Þð1 þ AZ ðjxk ÞÞj2

ðON QN  P N RN Þ þ jðON RN þ P N QN Þ



ðjx1 Þ

.

 3 7 7 7 7 7 5 N 1

;

k

2

k

k

k

ð18Þ

k

k¼1

V0

6 6 0 6 6 6 V 2 6 6 0 U¼6 6 6 V 6 4 6 6 0 6 4 .. .

WT ¼ ½ W 0

0

V 2

0

V4

0

V2

0

V 4

0

V6

0

V4

0

V 6

0

V 4

0

V6

0

V 8

0

V 6

0

V8

0

V6 .. .

0 .. .

V 8 .. .

0 .. .

V 10 .. .

U1

W 2

U 3

W4

U5



3

7 7 7 7 7 7 7 7 ð19Þ 7 7 7 7 7 7 5 .. . ðrþ1Þðrþ1Þ

   1ðrþ1Þ

ð20Þ

^Z

Therefore, the coefficients b of the polynomial numerator of G0Z ðsÞ can be obtained when constraining the polynomial denominator of G0Z ðsÞ equaling to the known polynomial denominator of G0Y ðsÞ. With the same procedure, the coef^X of the polynomial numerator of the approximate ficients b polynomial s-transfer function G0X ðsÞ, i.e., bX ¼ external  X X X X T b0 b1 b2    br can also be deduced. After generating these three polynomial s-transfer functions, the formulae to calculate response factors and CTF coefficients are totally identical to those given in the literature [19].

Here, two examples are presented to show the correctness of CTF coefficients given by the improvement of this FDR method (i.e., FDR method (Improved) thereafter) The first wall is a brick/cavity wall as described in Table 1, which is often used to validate various methods for heat transfer calculation. The second wall is Wall group 4, which is a multilayer construction consisting of 5 layers of homogeneous materials including 50 mm insulation between 50 mm high density concrete and 20 mm plaster with outside and inside air films [1]. The detailed physical properties of this wall do not duplicate here for conciseness. Table 2 presents the complete CTF coefficients of the brick/cavity wall generated using the time-domain method [8]. The polynomial s-transfer function of the cross heat conduction ðG0Y ðsÞÞ was regressed using the FDR method as shown in Eq. (21). The polynomial s-transfer functions for external and internal heat transfer are generated by FDR method(Improved) as Eqs. (22) and (23). The CTF

ð14Þ

ð16Þ

 T  and g are the complex where, b ¼ bZ0 bZ1 bZ2    bZr , H conjugate matrices of H and g, respectively. Z

i

ð17Þ

ð12Þ ð13Þ 3

^Z ¼ U 1W b 8 N P > > V i ¼ ðxk Þi > > > k¼1 > > > < N P i W i ¼ ðxk Þ ðOk Qk  P k Rk Þ > k¼1 > > > > > N P > > : U ¼ ðx Þi ðO R þ P Q Þ

4. Examples

7 r ðjx2 Þ 7 7 7 ð15Þ .. 7 . 7 5 r ðjxN Þ Nðrþ1Þ

 ..

r

As the derivation in the FDR method [19], the estimated ^Z is calculated as Eq. (17) with some intermediate values b presented as Eqs. (18)–(20).

ð11Þ

k¼1

 Z  gÞT ðHbZ  gÞ J ðbZ Þ ¼ ðHb 2 1 jx1 ðjx1 Þ2 ðjx1 Þ3 ðjx1 Þ4 6 2 3 4 6 1 jx ðjx2 Þ ðjx2 Þ ðjx2 Þ 2 6 H¼6 6. 6 .. 4 2 3 4 1 jxN ðjxN Þ ðjxN Þ ðjxN Þ 2 ðO1 Q1  P 1 R1 Þ þ jðO1 R1 þ P 1 Q1 Þ 6 ðO 6 2 Q2  P 2 R2 Þ þ jðO2 R2 þ P 2 Q2 Þ 6 g¼6 .. 6 . 4

663

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X. Xu et al. / Applied Thermal Engineering 28 (2008) 661–667

Table 1 Details of physical properties of a brick/cavity wall Description

Thickness and thermal properties

Outside surface film Brickwork Cavity Heavyweight concrete Inside surface film

L (mm)

k (W m1 K1)

q (kg m3)

Cp (J kg1 K1)

105

0.840

1700

800

100

1.630

2300

1000

R (m2 KW1) 0.060 0.125 0.180 0.06135 0.120

Table 2 CTF coefficients of the brick/cavity wall using different methods k

0

1

2

3

4

5

R

Time-domain method

ak bk ck dk

9.548397 0.000179 6.953625 1.000000

18.528113 0.013915 12.223156 1.620834

10.569717 0.043460 5.985915 0.726131

1.575632 0.018036 0.660046 0.065025

0.062597 0.001034 0.020334 0.001594

0.000339 0.000005 0.000044 0.000000

0.076628 0.076628 0.076628 0.041866

FDR method (Improved)

ak bk ck dk

9.589008 0.000178 6.959635 1.000000

18.586934 0.013914 12.226366 1.619844

10.586818 0.043475 5.979019 0.724520

1.560687 0.018078 0.653456 0.064306

0.049680 0.001052 0.018078 0.001543

0.001181 0.000006 0.000207 0.000006

0.076703 0.076703 0.076703 0.041906

coefficients of this wall based on these three approximate polynomial transfer functions using FDR method(Improved) are also presented in Table 2 for comparison. These results show that FDR method (Improved) provides almost the same CTF coefficients to those generated by the time-domain method.

nal, cross and internal heat conduction of this wall were identified as Eqs. (24)–(26). With these polynomials, the complete CTF coefficients were also generated as presented in Table 3. These coefficients basically agreed with the results using the conventional method.

2:40729E  4s5  1:24894E  6s4  4:03896E  9s3  8:82551E  12s2  1:25852E  14s  9:10930E  18 s5  3:32927E  3s4 þ 3:71932E  6s3  1:57185E  9s2 þ 1:83226E  13s  4:97697E  18 8:02757s5  2:54250E  2s4 þ 2:69560E  5s3  1:04813E  9s2 þ 9:76241E  13s  9:06345E  18 G0Z ðsÞ ¼ s5  3:32927E  3s4 þ 3:71932E  6s3  1:57185E  9s2 þ 1:83226E  13s  4:97697E  18 1:45071E þ 1s5  4:11038E  2s4 þ 3:94287E  5s3  1:31387E  8s2 þ 8:57483E  13s  8:79758E  18 G0X ðsÞ ¼ s5  3:32927E  3s4 þ 3:71932E  6s3  1:57185E  9s2 þ 1:83226E  13s  4:97697E  18 G0Y ðsÞ ¼

For the second wall, i.e., Wall group 4, the complete CTF coefficients were produced using Fortran code of statespace method in ASHRAE Loads Toolkit [2]. The complete CTF coefficients of the wall using this conventional method are list in Table 3. Using FDR method (Improved), three approximate polynomial s-transfer functions of exter-

ð21Þ ð22Þ ð23Þ

To evaluate the dynamic thermal behaviors of this wall using both methods, numerical comparisons were made within the frequency range of normal concern (108– 103 rad s1) among the frequency responses of approximate polynomial s-transfer functions using FDR method (Improved), the frequency responses of CTF models using

Table 3 CTF coefficients of Wall group 4 using different methods k

0

1

2

3

4

5

R

Conventional methods

ak bk ck dk

11.237246 0.015822 4.944606 1.000000

14.893030 0.124793 7.284118 0.908805

3.885357 0.059605 2.551513 0.195469

0.028047 0.001300 0.010471 0.000014

0.000007 0.000000 0.000010 0.000000

0.000000 0.000000 0.000000 0.000000

0.201520 0.201520 0.201520 0.286650

FDR method (Improved)

ak bk ck dk

11.252526 0.016668 4.948458 1.000000

14.814006 0.125828 7.242681 0.900008

3.790569 0.059465 2.502143 0.189426

0.025714 0.001424 0.004536 0.000101

0.000011 0.000000 0.000001 0.000000

0.000000 0.000000 0.000000 0.000000

0.203385 0.203385 0.203385 0.289317

X. Xu et al. / Applied Thermal Engineering 28 (2008) 661–667

conventional method and these theoretical frequency response characteristics. These frequency responses are shown in term of amplitude and phase lag in Figs. 1–3. Among these frequency responses, the external, cross and internal frequency responses using FDR method (Improved) agrees very well with the coincident theoretical responses. These frequency responses almost overlap in term of amplitude and phase lag. Therefore, these regressed polynomials s-transfer functions using FDR method (Improved) can really and exactly represent the dynamic thermal behaviors of Wall group 4.

665

As for the frequency characteristics of CTF models using the conventional method, Fig. 2 shows the amplitudes and phase lags of the cross CTF model deviate slightly from the coincident theoretical values at the high frequency regions. It is also obvious that the phase lags of the external and internal CTF models using the conventional method are far from their theoretical values as shown in Figs. 1 and 3 in high frequency regions although the amplitudes of both models basically agree with their theoretical values. Therefore, these CTF models of Wall group 4 using the conventional method cannot well and

7:47632E  5s5  1:43806E  6s4  1:82842E  8s3  1:28831E  10s2  5:90163E  13s  1:35035E  15 s5  1:07535E  2s4 þ 4:03705E  5s3  5:74017E  8s2 þ 2:01604E  11s  1:92079E  15 1:70043E þ 1s5  1:63653E  1s4 þ 5:82123E  4s3  7:50038E  7s2 þ 1:79312E  10s  1:35799E  15 G0X ðsÞ ¼ s5  1:07535E  2s4 þ 4:03705E  5s3  5:74017E  8s2 þ 2:01604E  11s  1:92079E  15 9:01391s5  8:03607E  2s4 þ 2:90017E  4s3  3:59375E  7s2 þ 5:88220E  11s  1:35597E  15 G0Z ðsÞ ¼ s5  1:07535E  2s4 þ 4:03705E  5s3  5:74017E  8s2 þ 2:01604E  11s  1:92079E  15

16

0.8

14

0.7

12

Amplitude (Wm-2K-1)

Amplitude (Wm-2K-1)

G0Y ðsÞ ¼

Theoretical model Conventional method FDR method (Improved)

10 8 6 4 2

ð26Þ

0.6 0.5 0.4

Theoretical model Conventional method FDR method (Improved)

0.3 0.2

10

-7

10

-6

10

-5

10

-4

10

0 -8 10

-3

10

-7

0.2

1

0

0

Phase Lag (rad)

-0.2 -0.4

Theoretical model Conventional method FDR method (Improved)

-0.6 -0.8

10

-6

10

Frequency (rad

Frequency (rad s-1)

Phase Lag (rad)

ð25Þ

0.1

0 -8 10

-5

10

-4

10

-3

s-1)

-1

-2

Theoretical model Conventional method FDR method (Improved)

-3

-4

-1

10

ð24Þ

-8

10

-7

10

-6

10

-5

10

-4

10

-3

-1

Frequency (rad s ) Fig. 1. Frequency response of external heat conduction of Wall group 4 (a) amplitude; (b) phase lag.

-5 -8 10

10

-7

10

-6

10

Frequency (rad

-5

10

-4

10

-3

s-1)

Fig. 2. Frequency response of cross heat conduction of Wall group 4 (a) amplitude; (b) phase lag.

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X. Xu et al. / Applied Thermal Engineering 28 (2008) 661–667

cations of current programs for heat flow calculation and energy analysis etc. Examples demonstrate that FDR method (Improved) can generate a unique set dk for a building wall and that all the generated CTF coefficients are consistent to those by other conventional method. These examples further show this improvement to this FDR method is valid and applicable for practical applications.

8

Amplitude (Wm-2K-1)

7 6 5 4

Theoretical model Conventional method FDR method (Improved)

3

Acknowledgements

2 1 0 -8 10

10

-7

10

-6

10

Frequency (rad

-5

10

-4

10

-3

s-1)

The research work presented in this paper is financially supported by a grant (PolyU 5283/05E) from the Research Grants Council (RGC) of the Hong Kong SAR and the National Nature Sciences Foundation of China (No. 50378033).

0.2

References

Phase Lag (rad)

0

-0.2

-0.4

Theoretical model Conventional method FDR method (Improved)

-0.6

-0.8

-1 -8 10

10

-7

10

-6

10

-5

10

-4

10

-3

Frequency (rad s-1) Fig. 3. Frequency response of internal heat conduction of Wall group 4 (a) amplitude; (b) phase lag.

exactly represent the dynamic thermal behaviors of this wall. However, FDR method (Improved) can generate CTF coefficients representing the real and exact frequency response characteristics of this wall. 5. Conclusion Frequency-domain regression (FDR) method is an easily understandable, simple and reliable approach to generate response factors and CTF coefficients for heat transfer calculation of multilayer constructions with high accuracy. This study presents an improvement made to this FDR method to produce a unique set of heat flow rate history coefficients dk conformable to the common understanding of CTF methods. This improved FDR method (i.e., FDR method (Improved)) aims at providing an alternative approach to generate CTF coefficients for practical appli-

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