A new transient method for determining thermal properties of wall sections

Accepted Manuscript Title: A New Transient Method for Determining Thermal Properties of Wall Sections Authors: A.J. Robinson, A.F.J. Lesage, A. Reilly, G. McGranaghan, G. Byrne, R. O’Hegarty, O. Kinnane PII: DOI: Reference:

S0378-7788(17)30491-7 http://dx.doi.org/doi:10.1016/j.enbuild.2017.02.029 ENB 7390

To appear in:

ENB

Received date: Revised date: Accepted date:

18-7-2016 20-12-2016 11-2-2017

Please cite this article as: A.J.Robinson, A.F.J.Lesage, A.Reilly, G.McGranaghan, G.Byrne, R.O’Hegarty, O.Kinnane, A New Transient Method for Determining Thermal Properties of Wall Sections, Energy and Buildings http://dx.doi.org/10.1016/j.enbuild.2017.02.029 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

A New Transient Method for Determining Thermal Properties of Wall Sections

A.J. Robinson1, A., F.J. Lesage2, A. Reilly3, G. McGranaghan4, G. Byrne1, R. O’Hegarty4, and O. Kinnane5,4,3

1

Department of Mechanical and Manufacturing Engineering, Trinity College Dublin, Ireland. Département d’informatique et d’ingénierie, Université du Québec en Outaouais, Gatineau ,QC,

2

Canada School of the Natural and Built Environment, Queen’s University Belfast, Northern Ireland

3

4

Department of Civil, Structural & Environmental Engineering, Trinity College Dublin, Ireland.

5

School of Architecture, Planning and Environmental Policy, University College Dublin, Ireland.

Abstract This investigation outlines a straight-forward and low cost methodology for determining thermal properties of wall structures. The method eliminates the need to produce a step change boundary condition, and the error inherent in the departure from a step change that finite properties necessarily impose. The transient technique involves an experimental component whereby a high temperature thermal ramp boundary condition is applied to one wall face with the other exposed to the cooler ambient surroundings. Temperature and heat flux sensors are installed to monitor the transient heating behaviour at the wall faces. The measured transient wall temperature profiles are subsequently imposed as boundary equations to Fourier’s equation in such a way that the analytic solution can provide a prediction of the transient surface heat flux. With this, the thermal diffusivity is estimated by using the effective thermal diffusivity of the wall material as a tuning parameter to regression fit the predicted and measured heat flux histories. Additionally, the steady solution facilitates the approximation of the effective thermal conductivity when used in conjunction with the steady surface temperature and heat flux measurements. To illustrate the technique, a test was performed on a 900 mm x 900 mm x 120 mm thick solid concrete wall section. The effective thermal diffusivity was determined to be 7.2 x 10-7 m2/s with corresponding effective thermal conductivity of 1.64 W/m.K and specific heat of 0.99 kJ/kg.K. Each property value is within the range of published literature for concrete.

Key Words: Building envelope, walls, thermal conductivity, thermal diffusivity, thermal mass, heat transfer

Introduction There is growing interest in accurately predicting the energy performance of building envelopes. This is largely driven by the fact that energy used in buildings accounts for over 30% of global energy consumption [1], with about half of this energy used for space heating/cooling and hot water [2]. The energy efficiency of buildings depends strongly on the magnitude of heat gains or losses across the building envelope structures, such as walls, since they in part determine the amount of energy required to maintain comfortable living conditions within the building. One major region of thermal energy transport to and from building envelopes is by diffusion across wall structures.

For given thermal loading conditions, the heat transfer characteristics of walls depend strongly on their effective thermal properties. The published thermophysical properties of well-known and

characterized materials, such as concrete, can vary by up to 80% [3]. Buildings constructed of these materials are proving difficult to characterise and hence problematic to retrofit. Prior to the 1930s in the UK and Ireland houses were commonly built using a variety of solid wall techniques including mass concrete. Retrofit of these buildings continues unabated even though the optimum strategy is still unclear. Inaccurate assumptions regarding their baseline performance is leading to inappropriate retrofit interventions, which in turn is leading to problems of moisture trapping, negatively affected indoor air quality and structural damage [4]. Greater knowledge of the thermal characteristics of solid walls, as an example, is necessary to avoid issues such as these.

Today walls are more commonly constructed from multiple materials in numerous layers. In such cases, defining the effective properties of the composite structure is not trivial since it requires knowledge of the individual material properties and how they interact thermally. In other instances, such as when new materials are introduced into building structures, the properties are not known and need to be approximated [5]. Further to these, complex composite wall structures with threedimensional internal geometries have associated with them complex heat flow paths which cannot be approximated using simple lumped analysis or thermal network models [6]. All of the scenarios above can introduce significant errors when estimating the energy performance of buildings.

Accurate prediction of the thermal characteristics of building envelopes not only hinges on having correct information regarding the wall thermophysical properties, it also requires correct formulation of the thermal energy transport mechanisms. Historically, heat transfer formulations utilized steady state thermal network models which require only effective thermal conductivities or associated U- and R-values of the wall material [7]. The effective U- and R-values are determined by several techniques including thermal network analysis of the composite structure [8,9,10], laboratory-type measurements [8,11,12,13] and in-situ measurements [14,15,16].

In reality, external thermal loading conditions on building wall structures are not steady. This can be due to changing outdoor conditions such as wind speed, temperature, precipitation and/or solar radiation. It can also be due to indoor factors such as control of heating and air conditioning systems. When the external thermal loading conditions change over a time interval that is notably less than the effective thermal time scale of the wall structure, transient thermal storage effects can become significant with regard to the heat flow within, and ultimately across the wall. This was recently highlighted by Byrne et al. [17] who investigated the steady and transient thermal behaviour of retrofitted cavity wall insulation. Their in-situ measurements showed a remarkably different behaviour of the transient heat flux of the retrofitted wall due to the change in the effective thermal mass of the wall with the addition of sprayed-foam insulation in the cavity. The importance of

characterising both the effective U-value together with the thermal mass has also been emphasized recently by Biddulph et al. [2].

The physical property which dictates the transient behaviour of matter is the thermal diffusivity, defined as the ratio of thermal conductivity (k) to the product of density (ρ) and specific heat (Cp). Thermal diffusivity quantifies a material’s capacity to conduct heat by diffusion relative to its capacity to store heat. For a given wall structure, if the thermal diffusivity is low enough the storage of energy will influence the rate of change of the temperature field within the structure in such a way that correct formulation of the flow of heat within the structure, and across its faces, must include transient thermal storage effects. Ultimately, this influences the rate of energy exchange with the surroundings which of course influences the building’s energy performance.

Early investigations that aimed to estimate the thermal diffusivity of walls in buildings used approximate solutions to the one-dimensional Fourier equation. Pratt and Lacy [18] obtained analytic solutions for periodic heat flow through isotropic wall structures and estimated the thermal diffusivity from measurements of the time delay of the periodic temperature variation across the walls. Agarwal and Verma [19] estimated the thermal diffusivity of wall structures by simplifying the solution of Fourier’s equation and observing the rise time of a thermal response function at the midway point of a sandwich-type sample arrangement subsequent to one surface being subject to a step function heat input. Albeit an approximation, the technique did not require information about the effective heat transfer coefficients acting on the wall surfaces nor did it require the heat capacity. Raychaudhuri [20] proposed a transient in-situ technique whereby the measured surface temperatures and heat flow rates over a 24 hour period were represented by a series of complex harmonic terms which were related to the material properties in such a way that the thermal diffusivity could be approximated. Hoffman [21] later developed a technique whereby the wall structure was evenly cooled, insulated, and then immersed into a heated room. By removing the insulation, a symmetric thermal transient within the structure was induced. The classical solution to Fourier’s equation with identical convective boundary conditions allowed for the estimation of the thermal diffusivity as well as other parameters, including the effective air-side heat transfer coefficient which was assumed constant during the entire transient.

Another sub-category of transient techniques for estimation of wall thermal diffusivity are those based on optical techniques. The laser-flash method [22, 23] utilises a laser pulse to instigate the thermal transient within the structure. The temperature response of the opposing surface is then measured with an infrared camera and the diffusivity is then determined by regression fitting the response to the known solution to the heat equation given prescribed initial and boundary conditions [24]. Cobîrzan et al. [25] proposed an alternative optical technique based on non-contact infrared lock-in thermography. Here the laser was intensity-modulated at a low frequency and the synchronized infrared camera

determined the phase of the modulated temperature which was related to the thermal diffusivity in a similar, yet much more sophisticated fashion to the early work of Pratt and Lacy [18].

The above thermal diffusivity measurement techniques suffer from a variety of shortcomings. For the contact-type techniques they often require assumptions which may or may not be physically realistic in order to simplify the solution of Fourier’s equation to the extent that the thermal diffusivity can be estimated. This could include requiring an estimation of the effective convective and radiative heat transfer coefficients acting at the surfaces, which are typically a non-linear function of surface temperature, or simply assuming that it is constant in space and time, for example. Also, the contacttype techniques require that the experiments impose an idealized thermal shock-type condition such as a step change in heat flux or temperature in order to satisfy the theoretical boundary conditions imposed when acquiring the analytic solution to Fourier’s equation. Accurate experimental thermal shock conditions on low conductivity-high thermal mass systems are notoriously difficult to realise in practice due to parasitic heat losses or thermal inertia of the heater systems which can add to the error in the property estimation. The non-contact measurements are expensive and are better suited for relatively thin isotropic materials.

The overarching objective of the present investigation is to develop a low-cost and simple to use method of estimating the effective thermal diffusivity of building wall structures. The method described is a transient contact-type technique whereby a quickly ramped thermal boundary condition is applied to an initially uniform temperature wall section. Measured temperature histories of the front and rear face are then used as time dependent boundary conditions which are implemented in the solution to Fourier’s equation. An inverse-type technique is then employed whereby the thermal diffusivity is adjusted until the predicted heat flux history from the exposed cold face of the wall is regression fit to that measured by the heat flux sensor. In this way, the solution does not require a precise thermal-shock to be realized experimentally nor does it require any approximations or assumptions with regard to the effective convective and radiative heat transfer from the exposed face.

Experimental Procedure Wall Section Preperation Example solid wall sections for testing were cast using a well-developed and reliable concrete mix. Concrete is commonly produced from a mix of cement binder, fine and course aggregate; the mix used in this study was composed of Portland cement, gravel sand and limestone aggregate in a ratio by weight of 0.15:0.35:0.45. Free potable water was mixed with cement in a ratio of 0.43. The gravel sand constituted the fine aggregate with particles of size 0-4mm. The limestone aggregate constituted the course aggregate with particle sizes of 6-14mm. A superplasticiser was added to increase the

workability, thereby reducing the total amount of water needed. The constituents of the concrete mix used in this study are listed in Table 1.

A 5-sided mold with interior dimensions 0.9m x 0.9m and 0.12m was made in birch plywood and varnished. The mold was laid horizontally, on a vibrating table, with one large square side left exposed into which the concrete was cast. To enhance movement of entrapped air to the surface the panel was vibrated during casting. Lifting eyes were cast into the concrete so that the panels could be moved from the manufacturing facility to transport, into the laboratory and onto the testing rig using cranes. Test cubes were prepared at the same time as the concrete was poured to check consistency and compressive strength. All panels reached a minimum strength of 45 N/mm2 after 28 days with a density of 2300±10 kg/m3. After casting the panel was allowed to set in the mold for 16 hours before the mold was removed. It was then moved to the factory floor and stored indoors for a week before being delivered for testing after 28 days. The panel was tested 24 months after casting. Apparatus and Procedure Figure 1 depicts the experimental apparatus used in this investigation. The system conmprised a heated water flow loop and the instrumented wall section under test, hereafter referred to as the wall section. The flow loop consisted of a Grundfos UP 20-45 circulation pump which drew water from a 60 L insulated water tank fitted with a 3 kW electrical immersion heating element. The temperature of the water in the storage tank was controlled by a Tecnologic TDF 11PID temperature controller to a nominal value of 50 oC for these experiments. The water exiting the pump was routed to two 0.9 m wide x 0.9 m high x 0.01 m thick flat aluminium plate heat exchangers. As depicted in Figure 2, the heat exchangers consisted of three aluminium sections within which serpentine channels were machined. The three sections were fastened together using dowel pins and screwed to a single backing plate of PVC measuring 0.9 m x 0.9 m x 0.012 m. Silicon sealant ensured water-tightness and unidirectional flow through the channels. The three sections were connected in series by external hose connections (not shown). The front heat exchanger provided the heat source for the transient and steady heat transfer across the wall section whilst the rear heat exchanger acted as a guard heater. The two heat exchangers were separated by 70 mm of Kooltherm K5 insulation.

The wall section was 0.9 m wide x 0.9 m high x 0.12 m thick solid concrete. A dimensioned schematic is shown in Figure 3. With the rear face heated by the front heat exchanger, the front face was exposed to the ambient environment in order to establish the temperature differential and subsequent heat transfer across the wall section. The room was approximately 10m x 21m x 6m which

can be considered a thermal reservoir, in the sense that the energy transfer across the wall section did not noticeably heat the room. In order to characterize the transient thermal behaviour of the wall section it was instrumented with two 80mm diameter Hukseflux HFP01 heat flux sensors and associated Hukseflux TRSYS01 Campbell Scientific CR1000 data loggers. The system accuracy for these sensor systems is ±5% of reading. The location of the heat flux sensors are depicted in Figure 3. The temperature of the front and rear faces were determined from four T-Type surface-mounted thermocouples, two on the front face and two on the rear face. The thermocouples signals were measured with a Datascan 7221 logger connected to a laboratory PC. Earlier tests using many more surface-mounted thermocouples showed that the heat flow across the wall is very close to onedimensional, with temperature variations across the wall faces typically within the thermocouple system accuracy of ±0.5 oC. It must be noted that standard low cost equipment used in building thermal evaluation have been used in this study to demonstrate the usefulness and accuracy of the technique for a broad range of practitioners. Of course more sophisticated DAQ hardware and software would improve the accuracy of the results, though as will be discussed, the added cost and complexity may be unwarranted.

The one dimensional transient conduction was aided by ensuring that there was a high enough flow rate of heated water through the front heat exchanger such that there was a negligible temperature difference between the inlet and outlet water streams. Furthermore, as shown in Figure 4, a thermally conductive gap filler pad was pressed between the front heat exchanger and the rear face of the wall. Importantly, the gap filler pad was sufficiently thick and compliant that it provided even thermal contact between the heat exchanger and the wall, which was not perfectly flat. In order to reduce heat flow at the perimeter of the wall section which could cause transverse temperature gradients, the edges were fitted with 70 mm of Kooltherm K5 insulation. Finally, all but the exposed front face were encased in a rigid wooden frame. Experiments were performed by first opening a control valve which flooded the heat exchangers with pre-heated water from the storage tank, and this defined the initiation time of the transient heat transfer process. The system was then allowed to heat until steady state conditions were achieved, as indicated by temperature and heat flux measurements not varying with time. All measurements were acquired at a sampling period of 10 minutes. Sample Results Figure 5 shows an example of the transient heating profile of the neat concrete wall section. In as much as the heated surface temperature increases quickly to a steady value of 48.7oC, Figure 5a shows that it is not a step function. Temperature step function boundary conditions are mathematical idealizations, which can be difficult to realize when testing large wall sections. Heat flux step function boundary conditions are also problematic for testing walls, especially those made from low-diffusivity

materials, due to difficulties ensuring all of the applied heat passes through the wall, rather than into surrounding insulation. Figure 5a also shows that there is an approximate 30 minute delay between the time at which heating was initiated at the rear face, and the time that the front face began heating. This is corroborated by Figure 5b which shows the same time interval between the initiation of the transient and the time it takes for the front surface heat flux to start increasing due to convective and radiative heat transfer to the room. This delay is the time it takes for the thermal penetration front to move from the back surface to the front surface, and is of course related to the nature of the heated surface boundary condition, the thermal diffusivity of the structure and its thickness. For the time interval of the entire test, Figure 5a also shows that the room temperature did not vary significantly.

Mathematical Model For the scenario under consideration the temperature evolution is considered to be one-dimensional and transient, T(x, t), for a plate of thickness ℒ. For the case here where there is no internal heat generation and the effective thermal conductivity can be assumed homogeneous and constant, Fourier’s equation can be expressed as, ∂T ∂2 T (1) (x, t) = α 2 (x, t) . ∂t ∂x The thermal diffusivity, α, is a tuning parameter which will be varied in such a way that the solution of Eq. 1 matches the measured data. The general solution of Eq. 1 is, ∞

T(x, t) = ∑ an e−αλn t cos √λn x + bn e−αλn t sin √λn x

(2)

n=0

for which an , bn and λn are coefficients. The initial condition considers the function ϵ as an incremental increase in temperature value. It serves to simulate a thermal shock on the heat load side of the wall; T + ϵ, x=0 T(x, 0) = { ∞ T∞ , 0
(3)

The initial condition can be approximated by the Fourier series expansion of, T∞ + 2ϵ x < 0 x=0 f(x) = {T∞ + ϵ T∞ x>0

(4)

over the interval [−2𝓅, 2𝓅] in which 2𝓅 is the period of the Fourier series expansion. It is arbitrarily chosen and must be 𝓅 > 12ℒ in order to ensure that the heat load temperature only appears once in the solution. Here it will be used as a second tuning parameter in order to achieve the best possible fit of the analytical solution with the measured transient profiles.

The Fourier series expansion of the initial condition is: ∞

2ϵ nπx f(x) ≈ T∞ + ϵ + ∑ ζn sin π 2𝓅

(5)

n=1

in which ζn =

(−1)n −1 n

.

Comparing Eq. (5) with Eq. (2) at t = 0 yields: ∞

2ϵ T(x, 0) = T∞ + ϵ + ∑ ζn e−αλn t sin √λn x π

(6)

n=1

nπ

in which √λn = 2𝓅. There are well known solutions to Fourier’s equation for step changes in heat flux or surface temperature. However, as mentioned, these idealizations are difficult to realize experimentally for realistic wall sections due to the relatively high thermal resistance and thermal mass. Here a new approach is adopted whereby the measured hot wall temperature history is used as a transient boundary condition, as opposed to a thermal shock-type boundary condition as is conventional. At position 𝐱 = 𝟎 the temperature evolution is found to have the appriximate form, T(0, t) = Ao −

Bo Co + t

(7)

Assuming that ϵ is linear with respect to x, such that ϵ = εx + θ(t), combining with Eq. (6) at T(0, t) = T∞ + ϵ shows, ϵ(0, t) = θ(t) = Ao − T∞ −

Bo Co + t

(8)

and ϵ(x, t) = εx + Ao − T∞ −

Bo Co + t

(9)

such that, lim T(ℒ, t) = T∞ + εℒ + Ao − T∞

t→∞

(10)

= TC where TC is the steady state exposed surface temperature. The above shows that ε =

TC −Ao ℒ

yielding,

ϵ=

TC − Ao Bo x + Ao − T∞ − ℒ Co + t

(11)

Thus, the model requires as input an emperical regression fit of the hot wall temperature evolution with time along with the initial/ambient temperature and the steady state cold wall temperature, each easily measured with simple and low cost thermocouples and data loggers. Applying the above, the temperature profile evolution within the domain 0≤x≤ ℒ is therefore, ∞

2ϵ T(x, t) = T∞ + ϵ + ∑ ζn e−αλn t sin √λn x π

(12)

n=1

where ζn =

(−1)n −1 n

nπ

, √λn = 2𝓅, ϵ =

TC −Ao x+ ℒ

Bo . o +t

Ao − T∞ − C

From Fourier’s law of heat conduction the heat flux is given as, q′′(x, t) = −k

∂T ∂x ∞

∞

n=1

n=1

TC − Ao 2 TC − Ao 2 =− − ∑ ζn e−αλn t sin √λn x − ϵ ∑ √λn ζn e−αλn t cos √λn x ℒ π ℒ π

(13)

In implementing the above, it is first requires that the regression coefficients Ao , Bo , and Co be determined from a curve fit of hot-side temperature evolution to Eq. 7. Also, the initial/ambient temperature T∞ and steady state temperature at position x = ℒ , here defined as TC , are determined from the measured data set.

The effective thermal conductivity is evaluated simply as,

k = q"(ℒ, t → ∞)

ℒ Ao − TC

(14)

in which q"(L, t → ∞) is the steady state heat flux at position x = ℒ. To determine the thermal diffusivity, 𝓅 is initially selected to satisfy 𝓅 > 12ℒ and α is tuned until a least-squared regression fit of Eq. 13 to the measured cold-side heat flux data results. The parameter 𝓅 is then tuned, each time determining a new value of α, until the measured and calculated delay times are equal.

Results and Discussion Results Figure 6 illustrates the salient features of the model results and solution procedure. Specifically, Figure 6 shows that the measured hot wall temperature evolution is fit reasonably well by the simple hyperbolic function with regression coefficients Ao= 48.9, Bo= 18161 and Co= 667.8. Importantly, only the initial/ambient wall temperature and the steady-state cold-side temperatures are required as additional input, and the resulting agreement with the theoretical solution is quite good. This is a practical advantage over other techniques which require thermal shock-type boundary conditions and/or approximations of the convective and radiative thermal resistances at the exposed face(s). It is worth noting that the predicted cold-side temperature evolution also agrees with the measured profile which verifies the efficacy of the model. The thermal properties that are extracted from the solution procedure outlined above are the thermal conductivity and thermal diffusivity. These are summarized in Table 2 along with the density and specific heat capacity. The density was measured from a core sample taken from the wall section and the specific heat capacity was calculated from the definition of thermal diffusivity, Cp=k/ρα.

Comparison With Values from Other Studies The values of the thermal properties listed in Table 2 lie within the range of values reported in the literature for concrete. Most studies do not measure diffusivity directly, as done here, but calculate it from measurements of conductivity and heat capacity. Specific heat capacity is a difficult property to experimentally measure with accuracy [26]. Clarke and Yaneske [26], in a survey of available building material thermal properties, and their means of measurement, report a paucity of specific heat measurements undertaken by recognised laboratories. Following from this the thermal diffusivity of concrete is hence not widely reported in the literature. It can also be measured directly from small material samples by measuring the temperature difference between the interior and surface of a heated cylinder as it cools in a bath of water at constant temperature [27]. Using this test method Van Geem et al. report a thermal diffusivity of 9.5x10-7 m2/s for structural normal weight concrete of density 2290 kg/m3 [28]. Using a guarded hot plate they report a conductivity 2.32 W/m.K [29] for concrete of the same density [29]. These values are higher than reported in this study however it is well recognised that the thermophyscial properties of concrete vary considerably for different mixes, moisture contents, aggregate type and fraction [26,30.31]. Other values for thermal diffusivity of concrete that are reported in the literature and design guides are both directly measured and calculated from conductivity and specific heat. Documented values range between 5.5x10-7 and 15.5x10-7 m2/s depending on the aggregate used [30,31,32] with lower values reported for lightweight, aerated and CMU concretes [32]. More specifically, the diffusivity of concretes described in ISO 10456 [33] can

be calculated as 6.39 x 10-7 to 8.33 x 10-7 m2/s for medium (1800 kg/m3) to high (2400 kg/m3) density concrete. Corresponding values of thermal conductivity range from 1.15 W/m.K (medium density) to 2 W/m.K (high density) [33], although again wide variability exists and the variable nature of the constituents of concrete has a strong influence on the thermal conductivity. Given this variability in concrete’s properties, it will be of significant advantage to have a comparatively low-cost method that is able to give accurate results. There is an increasing interest in modelling thermal performance at the design stage, and if this is to be accurate, it must include dynamic thermal modelling with accurate input data. The results from this experiment suggest that the procedure could be employed widely, reducing the reliance on a small selection of values that cannot cover all permutations of mix components, moisture contents and aggregate types. Error and Uncertainty In this study the uncertainty on the thermal conductivity was determined using the standard method outlined by Kline and McClintock [34],

 n  Z  2  wz     w   i 1  xi i    

(15)

For the purpose of uncertainty calculation, xi is the individual steady state measurement and the associated uncertainty in the measurement is wi. The result of a calculation using these measurements is denoted as Z and the uncertainty in the calculated result is denoted by wz. Using the average of 100 steady state measurements (i.e. 1000 minutes) the estimated uncertainty on the thermal conductivity is determined to be ±0.13 W/m/K. This represents ±8% of the measured value which is quite reasonable considering the low cost equipment used. Of course this can be improved significantly by using calibrated equipment and more sophisticated DAQ hardware and software. However, this can involve significant additional cost, complexity and know-how which may not always be warranted depending on the level of accuracy demanded.

Estimating the uncertainty associated with the effective thermal diffusivity is not as straight forward, largely due to the curve-fitting of the hot-side surface temperature as subsequent least-square fitting of the measured cold-side heat flux evolution. In order to gain a realistic sense of the propagation of the measured uncertainty on the estimated α, a technique described by Kempers et al. [35] was adopted. Here, the temperature and heat flux uncertainty is assumed to have a normal distribution where the stated uncertainty is equal to two standard deviations. A Monte Carlo method was then used to create over 2000 temperature and heat flux curves where the values are constrained by the respective uncertainty distribution at each point. Each randomly generated curve was then curve fit in such a way

that a unique estimation of α results. The resulting set of effective thermal diffusivity estimations is normally distributed and the uncertainty is taken as two standard deviations of this data set. In this way the magnitude of the uncertainty on α is related to the associated uncertainty on the primary measurement variables. For the case under consideration here the uncertainty is estimated to be ±0.27 x 10-7 m2/s, representing 3.8% of the estimated thermal diffusivity. This is quite reasonable considering the low cost equipment used in this study and, in the same way as the thermal conductivity, can be improved if necessary by using more sophisticated measurement equipment and performing instrument and equipment calibrations.

Conclusions In this work a straight forward and potentially low-cost method for estimating the thermal properties of wall structures is outlined. In particular, the thermal diffusivity is targeted since it is recognized that thermal storage effects of wall structures can play a non-negligible role in the energy performance of building envelopes.

Similar to other transient techniques, the method involves one face of the wall section to be in contact with a heat source with the other exposed to a heat sink, with appropriate temperature and heat flux meters installed at the faces. Unlike other transient techniques of this type, an alternative solution of Fourier’s equation is put forth whereby the hot side boundary condition represents a time-varying function which is regression fit to the measured profile. With the current formulation the only other input required are the initial/ambient temperature and the steady state cold side temperature, both easily obtained from the measurements. This is a practical advantage over other formulations which require thermal shock-type boundary conditions and/or knowledge of the convective and radiative thermal resistances at the exposed face. The effective thermal diffusivity of the wall structure is determined by simply tuning its value in such a way that the analytic cold side heat flux profile matches the measured profile.

The experiment used in this investigation used a constant temperature hot water loop as the heat source and a relatively constant temperature room as the heat sink together with standard thermocouple and heat flux measurement devices and data loggers. The idea was to illustrate the utility of the technique without relying on unnecessarily complicated and/or expensive experimental facilities. In fact, the straight forward and low cost experiment combined with the new mathematical formulation results in an effective thermal diffusivity of a concrete wall of 7.2 m2/s±0.27 m2/s, which is sufficiently accurate for most practical applications.

The wall section tested in this experiment was of a thickness typically found in buildings, and the area (0.9 x 0.9 m) was also substantial. It would be possible to adapt the method, with very similar

equipment, to measure the thermal diffusivity of wall sections in situ. That this could be done with portable equipment and, importantly, in a non-destructive manner, would offer substantial advantages over methods that rely on destructive techniques to measure the heat capacity. The biggest challenge to the method in this case, would be the uncontrolled heat loss to the side boundaries of the section under test. In this experiment, the side boundaries were insulated; for in situ measurements, an alternative must be sought, which may be an adaptation of the solution to the heat equation to account for this loss.

Acknowledgements Dr. Aidan Reilly was funded by the EU H2020 Framework Programme working on the IMPRESS project on which Dr. Oliver Kinnane is PI ( http://www.project-impress.eu/ ). Mr. Richard O’Hegarty was funded by the Irish Research Council Enterprise Partnership scheme with support from Firebird Heating Solutions Ltd.

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Front Plate Guard Plate Heater

Gap filler

PC

Heat Flux Gauges

Tank

Data Logger

Thermocouples PID Controller Wall Section Pump Figure 1: Schematic of experimental facility

Water inlet

Water feedthrough

Bolt holes

Water exit Figure 2: Aluminium water-heated flat plate heat exchanger showing grooved serpentine channels. Threaded bolt holes shown are for fixing the PVC backing plate.

Figure 3: Schematic of the wall section including placement of heat flux sensors and thermocouples.

Wooden Frame

Insulation

Wall Section

Front Heated Plate Rear Heated (guard) Plate

Figure 4: Cross section showing insulating and supporting frame.

Gap Filler

80

(a) T Hot 70

T Cold T Ambient

60

Temperature (oC)

50

40

30

20

10

0 0

100

200

300 Time (min)

400

500

600

400

500

600

300

(b) Heat Flux 250

Heat Flux (W/m2)

200

150

100

50

0 0

100

200

300 Time (min)

Figure 5: Transient heating curves showing (a) the heated surface, cooled surface and ambient temperature histories, and (b) the surface heat flux history from the exposed surface

80

200 T Hot 180

T Cold

70

T Cold, Theory 160

T Hot, Curve Fit 60

Heat Flux

Temperature (oC)

50

120

40

100 80

30

Heat Flux (W/m2)

140

Heat Flux, Regression Fit

60 20 40 10

20

0

0 0

100

200

300 Time (min)

400

500

600

Figure 6: Measured and curve fit functions of the wall surface temperatures and associated agreement of solution to Fouriers equation with tuned thermal diffusivity for least squarred regression fit to measured heat flux.

Table 1: Constituents of concrete mix used. Concrete constituent

Weight /m3

Rapid hardening grey cement

360

Crushed limestone aggregate, 6-14mm

1025

Concreting gravel sand, 0-4 mm

790

Potable water

155 liters

Superplasticiser

2.3 liters

Table 2: Thermo-physical properties of tested concrete wall section. k

Concrete

α x 107 2

ρ

Cp 3

(W/mK)

(m /s)

(kg/m )

(kJ/kgK)

1.64

7.2

2295

0.99