Int. Journal for Housing Science, Vol.38, No.1 pp.1-12, 2014 Published in the United States

0146-6518/01/1-12, 2014 Copyright©2013 IAHS

IN-SITU THERMAL RESISTANCE EVALUATION OF WALLS USING AN ITERATIVE DYNAMIC MODEL

António Jóse Barreto TADEU and Nuno SIMÕES University of Coimbra, Department of Civil Engineering (DEC)

Coimbra, Portugal

Inês SIMÕES and Filipe PEDRO Instituto de Investigação em Ciências da Construção (ITeCons)

Coimbra, Portugal

Leopold ŠKERGET University of Maribor, Faculty of Mechanical Engineering

Maribor, Slovenia

ABSTRACT This paper proposes and validates a numerical iterative model to evaluate the thermal resistance of multilayer systems (walls) when in a dynamic state. First, the validation is performed numerically, then the second step of validation uses the temperature and heat flux values recorded during experimental tests performed in a hot box chamber. These validations involve comparing the results obtained with those expected, given the thermal properties of each material and thickness of each wall layer. The paper first presents the analytical solution for simulating heat transfer by conduction in the frequency domain, through the multilayer system. This is generated by imposing temperatures on the external surfaces, when the thermal properties of the materials are known. The model is then modified by assuming the wall is composed of a single layer with unknown thermal properties. The temperatures and heat fluxes, provided earlier by the analytical model and imposed on the external surfaces, lead to a nonlinear system that can be solved for the unknown thermal properties. It is solved by implementing an iterative approach based on the Newton-Raphson method.

2 Tadeu, N. Simões, I. Simões, Pedro and Škerget

After the validation of the proposed model, this is used to evaluate the thermal resistance of a multilayered wall subjected to real conditions. Key words: Green’s Function Formulation, Frequency Domain, Multilayer Walls, Thermal Resistance, Iterative Dynamic Model.

Introduction The number of studies on estimating energy use for heating and cooling spaces in existing buildings has been steadily increasing as a result of people's growing expectations regarding energy consumption reduction [1]. In EU countries, the required energy certifications [Directive 2010/31/UE, Directive 2002/91/EC] have played a part in the recent increases observed in the number of energy audits. Since the energy performance of a building tends to depend on the envelope performance, it is very important to have accurate information about its thermal performance to define measures to increase the energy performance of existing buildings. Overall heat transmission is frequently based on the thermal transmittance, or U-value, of each element in the building envelope [ISO 13789:2007, ISO 13790:2008]. Most of the time there is no information about the composition of the envelope of existing buildings. Two possibilities may be explored in these circumstances: to apply destructive tests to identify the materials and their thickness, or to perform direct measurements of the heat flow [3]. The U-value can be found by measuring both the heat flow (in accordance with ISO 9869:19974) through the building element and the temperature on both sides of it. If the system is under steady-state conditions, the U-value can be very accurate. However, since outdoor conditions are always changing, is not usual to find steady-state conditions during in-situ measurements. Two main approaches can then be applied: record the heat flow rate and temperatures over a long period in such way that allows the good estimation of equivalent steady-state behavior, or apply a dynamic model to take into account the surface temperature and heat flow rate variations. In the first, simpler, approach, the results may be quite inaccurate unless the storage effect caused by thermal mass (inertia) is negligible for the heat flow rates in question. Changes in heat flow direction will also lead to imprecise measurements. Laurenti et al. [4] present a mathematical model for calculating the thermal resistance of a wall through the dynamic analysis of in-situ data, both heat flux and surface temperature measurements. The proposed method models the transient thermal response of the wall through a linear relation with constant parameters that links the instantaneous heat flux at the inner surface of the wall to the temperature difference between the surfaces of the same wall at the same instant. Cucumo et al. [5] proposed a method for the experimental determination of the in situ building wall conductance, based on both inside and outside heat fluxes and surface temperatures. This method was applied to a test wall of an external testing station at the University of Calabria at different times; the results agreed with values obtained by means of the progressive method [6]. The proposed method provides an equivalent conductivity. Wang et al.

In-Situ Thermal Resistance Evaluation of Walls 3

[7] established and validated a transient heat transfer model of a wall by applying the finite difference method. Additionally, an in-situ measurement program using a heat flow meter was undertaken, paying special attention to the wind velocity. The proposed method, the mean method and the dynamic analysis method proposed by ISO 9869:1994 were implemented on the test wall under different wind velocities. The wall thermal resistance value obtained by the proposed method was shown to be in better agreement with that obtained for a steady state. Our paper proposes and validates a numerical iterative model to evaluate the thermal resistance of multilayer systems (walls) when in a dynamic state. It first describes a dynamic model for simulating heat diffusion through multilayer systems in the frequency domain, generated by imposing temperature variation on the external surfaces. Then, a dynamic iterative model is implemented for evaluating the thermal resistance of a multilayer system, based on the Newton-Raphson method, where the external heat fluxes and temperatures are prescribed. After the validation of these models, the dynamic iterative model is used to evaluate the thermal resistance of multilayer systems subjected to heat diffusion in a controlled environment, using a hot-box, and in an in-situ environment. Different experimental measurements are presented and the expected thermal resistance is compared with numerically computed values.

Dynamic Model for a Multilayer System Consider a multi-layered system built from a set of m plane layers of infinite extent, as shown in Figure 1. This system is subjected to temperatures 0tt and 0bt at the top and bottom external surfaces. The layers are assumed to be infinite in the x and z directions. The thermal material properties and thickness of various layers may differ. The transient heat transfer by conduction in each layer is expressed by the equation ( ) ( )2 2 ( , ) ( , )j j jy T t y c T t y tλ ρ∂ ∂ = ∂ ∂ (1)

in which t is time, ( , )T t y is temperature, j identifies the layer, jλ is the thermal conductivity, jρ is the density and jc is the specific heat. The solution is defined in the frequency domain after a Fourier transformation is applied to equation [1]:

( )( )2

2 2 ˆi ( , ) 0jy T yω α ω⎛ ⎞∂ ∂ + − =⎜ ⎟⎝ ⎠

(2)

where i 1= − , ( )j j j jcα λ ρ= is the thermal diffusivity of the layer j , and ω is the frequency.

4 Tadeu, N. Simões, I. Simões, Pedro and Škerget

The total heat field is achieved by adding the surface terms arising within each layer and at each interface, as required satisfying the boundary conditions at the interfaces, i.e. continuity of temperatures and normal flows between layers. For the layer j , the heat surface terms on the upper and lower interfaces can be expressed as

( )1 0 1 0 0( , ) tj j j j jT y E E Aω ν=%

(3)

( )2 0 2 0 0( , ) bj j j j jT y E E Aω ν=%

(4)

where 0 1j jE λ= , 1

01

i

1 e

j

j ll

y h

jEν

−

=

− −∑= ,

01

i

2 e

j

j ll

y h

jEν

=

− −∑= and lh is the thickness of the layer l . 0

tjA and

0bjA are a priori unknown potential amplitudes. A system of 2m equations is derived,

ensuring the continuity of temperatures and heat flows along the 1m − interfaces between layers, and by imposing the temperatures 0tt$ and 0bt$ at the top and bottom external surfaces. 0tt$ and 0bt$ are obtained by Fourier transformation of 0tt and 0bt in the time domain. All the terms are organized according to the form

01 1

01 1

01 1

0

0

0

i

1 01 1 01i

i1

11 01 1 01

i

i

3 0 0i

0 0

1 e ... 0 0

e 1 ... 0 0e 1 ... 0 0

... ... ... ... ... ...0 0 ... 1 e

1 e0 0 ...

e 10 0 ...

m m

m m

m m

h

h

h tnbn

h tnm

h bnm

m m mh

m m m m

AA

AA

ν

ν

ν

ν

ν

ν

λν λν

λν λν

λν λ ν

λ ν λ ν

−

−

−

−

−

−

⎡ ⎤− −⎢ ⎥⎢ ⎥

−⎢ ⎥⎢ ⎥ ⎡⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥− ⎢⎢ ⎥

⎣⎢ ⎥− −⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

0

0

00...00

t

b

t

t

⎡ ⎤⎢ ⎥⎤ ⎢ ⎥⎥ ⎢ ⎥⎥ ⎢ ⎥=⎥ ⎢ ⎥⎥ ⎢ ⎥⎢ ⎥⎦ ⎢ ⎥⎢ ⎥⎣ ⎦

$

$

(5)

The resolution of this system gives the amplitude of the surface terms at each interface, leading to the following temperature and heat flux fields at layer j ,

( ) ( )( )0 1 0 0 2 0 0( , ) t bj j j j j j jT y E E A E Aω ν ν= +%

if

1

1 1

j j

l ll lh y h

−

= =

< <∑ ∑

(6)

( )1 0 2 0( , ) t b

j j j j jT y i E A E Ayω

λ∂

= − +∂

%

if

1

1 1

j j

l ll lh y h

−

= =

< <∑ ∑.

(7)

In-Situ Thermal Resistance Evaluation of Walls 5

FIG 1. Geometry of the problem.

Iterative Dynamic Model to Evaluate Thermal Resistance

This section describes how the thermal resistance of a multilayer system is evaluated. For this purpose, the model described above is replaced by a single-layer wall bounded by two thin air layers to account for the surfaces’ thermal resistance (see Figure 2). The thickness of the wall is the same as the overall thickness of the multilayer system. Its thermal properties ( 2λ and 2 2 2v cρ= ) are a priori unknown. However, it assumed that in addition to 0tt$ and 0bt$ on the external surfaces (interfaces 1 and 4) being known in the frequency domain, the heat fluxes ( tq and bq ) on the wall surfaces are also known (interfaces 2 and 3). This results in a nonlinear system

01 1

01 1 02 2

01 1 02 2

02 2 03 3

02 2 03 3

03 3

01 1

i

1 01 1 01i i

i i

1 01 1 01 2 02 2 02i i

i i

2 02 2 02 3 03 3 03

i

3 03 3 03i

1 e 0 0 0 0

e 1 1 e 0 0e 1 1 0 0

0 0 1 1e 1 10 0

10 0 0 0

i i 0 0 0 00 0 0 0 i i

h

h h

h h

h h

h h

h

h

e

e ee

e

ee

ν

ν ν

ν ν

ν ν

ν ν

ν

ν

λν λν

λν λν λ ν λ ν

λ ν λ ν λν λν

λν λν

−

− −

− −

− −

− −

−

−

−

− −

− −

− −

− −

− −

−

− 03 3

0

1

1

2

2

03

3

i

0000

t

t

b

t

b

tb

bt

b

h

tAAAAA tA q

q

ν

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥ ⎡ ⎤⎢ ⎥ ⎢ ⎥

⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥ =⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦

⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎣ ⎦

⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

$

$

[8]

This system is solved by employing an iterative approach based on the Newton-Raphson method (Newton’s method for the common eigenvector problem). This requires defining a matrix of the first derivatives of the system of equations in equation [8], that is not presented in this paper.

 

h1

hm

Medium 1

Medium 2

Medium m

Interface 2

Interface 1

Interface m

Interface m+1

•  •  •  

xx

yx

0ttθ =

0btθ =

Interface 3

6 Tadeu, N. Simões, I. Simões, Pedro and Škerget

 

obtθ = $ y

x ottθ = $ Interface 1 Medium 1

Medium 2

Medium 3

Interface 2

Interface 3

Interface 4

1h

2h

3h

The iterative process starts with an initial guess that is updated in every iteration until convergence is reached. If the solution starts being unstable a new guess is introduced and the process continues until convergence is reached. The thermal resistance of the system corresponds to that to the static response, that is, for null frequency. FIG 2. Geometry of the problem used for the application of iterative dynamic model.

Analytical Verification of the Model

Two construction system walls are studied, as shown in Figure 3. These systems are composed of different materials, viz., concrete, traditional mortar and traditional plaster. The thermal insulation is provided by extruded polystyrene (XPS). Table 1 lists the thermal properties of these materials. In all computations thin air layers are simulated on the outer surfaces to account for convection and radiation phenomena. It was imposed a sinusoidal variation in temperature on the outer surface (the surface left of each building system shown in Figure 3): the initial temperature is assumed to be 20ºC and it fluctuates by 20ºC in each 24-hour period. The other surface is subjected to a constant temperature of 20ºC. The analytical computations were performed in the frequency domain for frequencies ranging from 0.0 Hz to -35.734 10 Hz× , with a frequency increment of -77.0 10 Hz× , in a full analysis window of approximately 16 days and 13 hours. Table 2 lists the imposed temperatures 0tt$ and 0bt$ , the computed tq and bq using the dynamic model for the multilayer system (the direct problem) that are used as input data in the iterative dynamic model for thermal resistance evaluation for a null frequency (static response). In addition, it includes equivalent thermal conductivity given by the iterative model modelλ and that expected assuming the existence of a permanent heat flow rate (static response),

1 1

2 2

m mi

equiv ii i i

hhλλ

− −

= =

=∑ ∑.

In-Situ Thermal Resistance Evaluation of Walls 7

Table 1. Materials’ thermal properties.

Material Conductivity λ

-1 -1(W.m .ºC )

Mass Density ρ -3(kg.m )

Specific Heat

c -1 -1( J.kg .ºC )

Thermal Diffusivity K

2 1(m . )s− (1) Concrete 1.4 2300.0 880.0 6.92e-07

(3) Extruded polystyrene (XPS)

0.035 35.0 1400.0 7.14e-07

(4) Traditional mortar 0.72 1860.0 780.0 4.96e-08

(5) Traditional plaster 0.50 1200.0 840.0 4.94e-07

Air (outer exposed surface) 0.075 1.29 1000.0 5.81e-05

Air (inner exposed surface) 0.0231 1.29 1000.0 1.79e-5

Example 1 Example 2

1 – Concrete. 3 - Thermal insulation (extruded polystyrene). 4 – Traditional mortar. 5 – Traditional plaster. Note: thicknesses in meters.

FIG 3. Wall systems studied: composition and dimensional characteristics of

the solutions.

Table 2. Thermal resistance evaluation for a null frequency (static response).

Case 00 (ºC)textt t= $

00 int (ºC)bt t= $ -2 (W.m )tq -2 (W.m )bq -1 -1(W.m .ºC )equivλ

-1 -1

model (W.m .ºC )λ

1 10.2 20.0 41.701 41.701 1.4 1.4 2 10.2 20.0 5.600 5.600 0.206 0.206

The results show a very good agreement between the thermal resistance evaluation given by the iterative model and the expected value.

8 Tadeu, N. Simões, I. Simões, Pedro and Škerget

Experimental Validation of the Model Using a Hot Box (Unsteady State Heat Flow) The proposed iterative dynamic model is verified using both the numerical results and the data recorded in a hot box. The test specimen is placed between two climatic chambers that simulate the indoor and outdoor environments. The numerical verification uses the results obtained from one calibration panel used as test specimens. This calibration panel was chosen as test samples because their thermal performances had been previously evaluated by the National Physical Laboratory, England. The test sample is subjected to both steady and unsteady heat flow conditions. The unsteady heat flow was generated by keeping a constant temperature in one of the climatic chambers (20°C) while the temperature in the other varied cyclically. Each cycle has a period of 24 h and an amplitude temperature of 10ºC. Thus, the temperature fluctuates between -10ºC and 10 ºC (see Figure 4) to simulate the outdoor environment.

FIG 4. Temperature variation simulating an outdoor environment during a dynamic hot box test.

The test sample comprises two glass panels, each 4 mm thick sandwiching a layer of expanded polystyrene (EPS) about 50 mm thick. Thus, this sample has a total thickness of 58 mm. According the calibration report of National Physical Laboratory, the thermal conductance of the test sample is defined by equation [9].

1 0.0025 0.6203K T= × + (9) In this equation, T is the temperature in ºC. The equivalent thermal conductivity of the specimen ( ).equiv realλ is determined by equation [10].

( ). = 1equiv real ie Kλ (10)

where e is the total thickness of the specimen, in meters.

-10

-5

0

5

10

0 12 24 36 48 60 72

Time (h)

Tem

pera

ture

(ºC

)

In-Situ Thermal Resistance Evaluation of Walls 9

To account for convection and radiation phenomena the surfaces’ thermal resistance is modelled as thin air layers on the outer surfaces. A layer of air 1 mm thick is assumed on each outer surface (layers 1 and 3 in Figure 2) whose thermal properties are presented in Table 1. The equipment used to record the temperatures and heat fluxes on the test sample surfaces inside the hot box, is a transverse gradient heat flux sensor (TRSYS01 from Hukseflux). The temperatures and heat fluxes recorded on the external wall surfaces were used in the iterative dynamic model to evaluate the equivalent thermal conductivity ( )mod elλ of the test specimen, assuming the existence of a single-layer wall. The obtained results were compared with the actual thermal conductivity ( ).equiv realλ . The equivalent thermal conductivity was evaluated over three periods: 24 h, 36 h and 72 h of measurements. It was thus possible to see how important the duration of the data acquisition is for the estimation of the thermal conductivity of the test specimen. The analytical computations were performed in the frequency domain for different frequency ranges and different frequency increments, for the full analysis window. The frequency increments used were -65.787 10 Hz× , -63.858 10 Hz× and -61.929 10 Hz× , respectively. Figure 5 shows the temperature and heat flux recorded over three days when test samples were subjected to unsteady state conditions. The heat fluxes exhibit similar cycles for the unsteady state heat flow conditions.

-10

0

10

20

0 24 48 72

Time (h)

Te

mp

era

ture

(ºC

)

-5

0

5

10

15

20

25

0 24 48 72

Time (h)

He

at

fluxe

s (W

.m-2

)

a) b)

FIG 5. Results of unsteady state test: a) temperatures; b) heat fluxes.

The thermal conductivities, mod elλ and .equiv realλ , and the relative error ( )( ). mod .equiv real el equiv realλ λ λ− are given in Table 3, for three different measurement durations: 24 h, 36 h and 72 h. The results show the existence of a good agreement between the values determined by the proposed methodology and those given in the calibration report.

10 Tadeu, N. Simões, I. Simões, Pedro and Škerget

Table 3. Equivalent thermal conductivity obtained under unsteady state test conditions.

Test Sample Duration of Recorded

Data

model

-1 -1(W.m .ºC )

λ

-1 -1(W.m .ºC )equiv realλ

Relative Error (%)

1 24 h 0.0367 0.0374 1.87 1 36 h 0.0350 0.0372 5.90 1 72 h 0.0366 0.0374 2.14

In-Situ Thermal Resistance Evaluation

This section illustrates the application of the proposed algorithm to data recorded using a real dynamic heat transfer phenomenon throughout a wall. The wall studied faces the west direction. The wall is composed, from the outside to the inside, of these layers: 0.25 m concrete, 0.04 m air cavity, 0.03 m extruded polystyrene (XPS), horizontally perforated clay bricks, 0.11 m thick, and 0.02 m traditional mortar. According to the thermal conductivity/resistance and thickness of each layer, indicated in Table 4, the equivalent thermal conductivity of this wall when subjected to steady state conditions is -1 -10.31W.m .ºC .

Table 4. Thermal properties of the wall studied

Material Thickness (m)

Conductivity λ

-1 -1(W.m .ºC )

Thermal Resistance

2 1(m .ºC.W )−

Concrete 0.25 2.0 0.125 Air cavity 0.04 - 0.180 Extruded polystyrene (XPS) 0.03 0.035 0.857 Horizontally perforated clay bricks 0.11 - 0.270

Traditional mortar 0.02 1.8 0.011 TOTAL 0.45 - 1.46

Figure 6 illustrates the temperature and heat flux measurements during the real dynamic heat transfer. The lines with marks represent the outdoor data while the solid line illustrates the indoor measurements. The thermal conductivities obtained with the proposed iterative model, considering different periods of data acquisition, are presented in Table 5. Comparing the theoretical and numerical values, the smallest error is obtained for a last measurement day, while the largest error is given by the model that uses 58 h of measurement. Analysis of Table 5 indicates that the equivalent thermal conductivity

of the wall is ( ) -1 -10.298 0.014 W.m .ºC± . It can be concluded that in general the obtained values are acceptable even when relatively short periods of measurement are considered.

In-Situ Thermal Resistance Evaluation of Walls 11

15

17

19

21

23

0 6 12 18 24 30 36 42 48 54

Time (h)

Tem

pera

ture

(ºC

)

0

15

30

45

0 6 12 18 24 30 36 42 48 54

Time (h)

Hea

t flu

xes

(W/m

2 )

a) b)

FIG 6. Results of in-situ measurements over 7 days: a) temperatures; b) heat fluxes.

Table 5. Equivalent thermal conductivity obtained under real dynamic test conditions. Duration of

Recorded Data model

-1 -1(W.m .ºC )

λ

-1 -1(W.m .ºC )equiv realλ

Relative Error

(%) Last 12 h 0.312

0.308 -1.28

Last 24 h 0.284 7.79 58 h 0.312 -1.28

Conclusion

This paper has proposed an analytical formulation to simulate heat transfer through multilayer systems. The proposed method is based on a dynamic iterative model for evaluating the thermal resistance of a multilayer system, based on the Newton-Raphson method, when the external heat fluxes and temperatures are known. First, the method was presented and verified numerically using four wall systems. Then, the model was validated by using temperatures and heat fluxes recorded during experimental tests in a hot box. The thermal performance samples used in validations were known beforehand. The comparison has revealed a good agreement between the results. Finally, the thermal resistance of an external wall exposed to real climatic conditions is evaluated using the method, and the equivalent thermal conductivity was calculated successfully.

References

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[2] Desogus, G.; Mura, S. and Ricciu, R. – Comparing Different Approaches to in Situ Measurement of Building Components Thermal Resistance, in Energy and Buildings, Vol. 43/10 (2011), pp.2613-2620.

12 Tadeu, N. Simões, I. Simões, Pedro and Škerget

[3] Albatici, R., and Tonelli, A.M. – Infrared Thermovision Technique for the Assessment of Thermal Transmittance Value of Opaque Building Elements on Site, in Energy and Buildings, Vol. 42/11 (2010), pp.2177-2183.

[4] Roulet, C.; Gass, J. and Marcus, I. – In Situ U-Value Measurement: Reliable Results in Shorter Time by Dynamic Interpretation of the Measured Data, in ASHRAE Trans, Vol. 108 (1987), 1371–1379.

[5] Laurenti, L.; Marcotullio, F. and De Monte, F. – Determination of the Thermal Resistance of Walls Through a Dynamic Analysis of in-Situ Data, in International Journal of Thermal Sciences, Vol. 43/3 (2004), pp.297-306.

[6] Cucumo, M.; De Rosa, A.; Ferraro, V.; Kaliakatsos, D. and Marinelli, V. – A Method for the Experimental Evaluation in Situ of the Wall Conductance, in Energy and Buildings, Vol. 38/3 (2006), pp.238-244.

[7] Report on support to CEN TC89/WG8 Update of ISO 9869 to EN12494 ‘In-Situ Measurement of the Thermal Resistance and Thermal Transmittance’ by PASLINK EEIG; edited (1995) by J.J. BLOEM. Joint Research Centre Institute for Systems Engineering and Informatics.

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