2. Thermal Diffusivity and Thermal Effusivity

Abstract: The work questions the significance of the steady-state R-value for high mass wall systems through experimental analysis of the thermal behaviour of two stabilized rammed earth (SRE) building components representing a building envelope subjected to variable temperatures in the form of 24 hour cyclic sinusoidal inputs. It shows that when the environment temperature is used to quantify the thermal fluctuation of a zone the response of the walls surfaces to energy cycles can be determined by dynamic thermal transfer properties expressed in accordance with EN ISO 13786: 2007 relating cyclic heat flux to cyclic temperature variations.

Keywords: Admittance, Decrement, Surface Factor, Thermal Mass,

1. Introduction

“Earth building is an appropriate, renewable, sustainable technology. Earth is possibly the ultimate Green Building material. It is a technology that is ancient yet still most relevant. In many cases earth building is based on sound design principles as a product of vernacular building traditions that have been be used for hundreds, sometimes thousands, of years to design comfortable buildings appropriate for their climate. In Australia the decline of vernacular architecture is relatively recent. It started with prefabricated housing in the 1970s and has become fully entrenched with the advent of affordable air-conditioning which is a technological fix for poor building design. Australian building laws accept that bulk and reflectivity can be used to insulate, yet deny the fact that mass too can insulate.”- Peter Hickson (president of the EBAA). If this was true, then why does the air temperature of many caves stay constant regardless of season? Or more applicably, why is it considered best practice in many parts of the world to couple a house directly to the ground without added thermal insulation? The truth is that mass insulation offers thermal lag, thermal inertia and capacitance. Mass is more than an insulator, it stores, balances and releases energy [1].

Pollack Periodica, 2013

2. Thermal diffusivity and thermal effusivity

Three primary parameters, density, ρ [kg/m3]; specific heat capacity, c, [J/kg.K]; and thermal conductivity, λ [W/m.K], that affect thermal performance of an opaque material may be combined to obtain thermal diffusivity, α = (λ / ρ.c) [m2.s-1], which is an indicator of how rapidly heat is conducted in a material [2]. The depth that the daily changes in temperature reach within the material will depend on thermal diffusivity. The thermal diffusivity of common materials ranges from 1.4 x 10-5 for steel to 8 x 10-7 for concrete. Materials with higher thermal diffusivity values can be more effective for cyclic heat storage at a greater depth than materials with lower values. Thermal effusivity, β = (λ.ρ.c.0,5) [J/m2.K.s0.5], is used to represent the capacity of a material to absorb and release heat. This relationship is also known as ‘thermal inertia’ [2]. Materials with high thermal effusivity will more readily dissipate heat from their surface, and so will be suitable for heat or coolth storage.

3. The admittance procedure

Admittance values provide a practical means of assessing the approximate in-use heat absorption performance of building elements such as walls. Both U-values and Admittance is measured in [W/m2.K]. However ‘K’ represents something different. For admittance, it is the difference between the mean temperature and the actual temperature at a specific time of day [3]. It is this dynamic temperature difference that drives heat in and out of the envelope. In contrast, the ‘K’ in U-values is the difference between internal and external temperature, which is assumed to be constant, which is why U-values are steady-state. Another difference is that high admittance values are desirable from a thermal mass perspective, whilst low U-values will minimise heat loss.

4. Principal factors used in the admittance procedure

Admittance and decrement are two principal factors which describe the amplitude of the response and associated time lead or lag (phase change) of a construction element to dynamic effects based on the construction elements’ thermo-physical properties i.e. thermal conductivity, specific heat capacity and density. Although admittance by itself is sometimes taken to be a measure of thermal storage, this is not completely true, since admittance only describes heat flow into the surface, whereas it is the difference between the heat flow into the interior surface (admittance) of the element and that leaving the exterior surface (decrement) that relates to the heat storage achieved [4]. To calculate effective thermal storage, therefore, requires both the admittance and decrement factor to be used with their associated time delays. This is done in the calculation of the ‘areal heat capacity’ which describes the effective heat storage achieved over a half-cycle [4]. The third parameter calculated is the surface factor, F, which is the ratio of the swing in heat flow from the internal surface of the element to the swing in heat flow received at the internal surface of the element (such as gains from localised sunshine through a window), and an associated time factor, ψ [h], that defines the time delay. For multilayer components admittance is determined primarily by the surface layer so that a 300 [mm] component with 30 [mm] of insulation on the inside surface would respond more as a lightweight material than a heavy one. The theoretical scenarios forming the basis of these three parameters are illustrated schematically in figure 1.

[W/m2.K] (1)

[-] (2)

[-](3)

Fig. 1Illustration of how dynamic thermal characteristics are calculated

Fig. 1 shows homogeneous wall components separating two thermal zones from which dynamic thermal characteristics are calculated. Qi is the swing in heat flux emanating from the internal surface of the element into the room, θi is the swing in the internal environmental temperature, θe is the swing in external environmental temperature, U is the U-value of the element and Qabsorb is the heat absorbed by the surface (heat flux is shown in orange and temperatures in green) [4].Two ‘surface heat capacities’, κ and κ30 are calculated even though they are not directly related to the admittance framework but to a simpler framework for calculating the dynamic thermal response of building elements used in ISO 13790:2004 [5]. These quantities relate to the volumetric heat capacity of different parts of the construction.

[kJ/K.m²](4)

Where

ρi - Density of layer i in the construction [kg/m³]

ci - Specific heat capacity of layer i [J/kg.K]

di - Thickness of layer i [mm].

4. Finite element input data

Since the shapes of the test specimens were symmetrical, it was possible to use the alternate solution for the finite element method. Time variable inputs were the only boundary condition for the exterior surface and were based upon data derived from the Australian Climate Data Base as used by the NatHERS scheme – Climate file 13, Perth (BCA 5) [6]. The suggested data is based upon the average of two consecutive hot days to which a sin function is fitted to smooth the values and to produce a symmetrical diurnal range enabling cycling through several days without a discontinuity. A rectangular finite element mesh was created for time-dependant-variable boundary conditions and then solved by computational methods.

Fig. 2. Comparison of summer temperatures & humidities

5. Results and analysis

Both wall specimens had an average density of ρ ~ 1800 [kg/m3], specific heat of c ~ 840 [J/kg.K] and steady state thermal conductivity dependant on the building standard boundary conditions used. Tab.1 reveals major incongruences amongst the standards.

Tab. 1Steady –state thermal conductivity based on boundary conditions

As a sanity check, the common materials listed in the STN 730540 norm were reviewed to see which materials have similar conductivities to those obtained by measurement. Table 16 part 20.2.1 of that norm revealed that sandstone has almost identical thermal physical-properties with a stated conductivity of 0,90 [W/m.K], density of 1800 [kg/m3] and specific heat of 840 [J/kg.K] [7].

Tab. 2Results of admittance method for an external envelope where the period is 24 [h]

*Rsi = 0,13; Rse = 0,04 [m2.K/W]

To obtain the volumetric heat capacities of different parts of the construction surface heat capacities were calculated.

Tab. 3. Surface heat capacities

The thermal admittance method is particularly useful for designers less experienced in building modelling, as they can gain an understanding of the sensitivity of the proposed designs to variations in the basic thermal properties of the construction [1]. Published values of thermal admittance are calculated on a basis of sinusoidal variation of heat input and temperature. In practice, these conditions rarely occur. Theoretically, it would be possible to calculate thermal admittance of periodic conductance for any pattern of heat gain, but to retain the simplicity of the thermal admittance method, standard thermal admittance values are generally used. For example, when assessing cooling loads, it treats the outdoor daily temperature profile as being constant over a repeating number of consecutive days and, for heating loads, it assumes a constant outside air temperature and no solar gain. Also, the basic implementation of the method maintains a constant infiltration/ventilation rate (outdoor air passing directly into the room). However all calculation techniques have inbuilt assumptions, and thus restrictions, and the admittance procedure should be seen as one of a number of calculation techniques which give realistic answers to the problem of cyclic temperature predictions in buildings based on these restrictions.

The results of the finite element method figures 3; 4 show that there is a tendency to not only attenuate, but also stabilize fluctuations within the internal surfaces of rammed earth when the outside temperature cycled above and below the set indoor temperature, which in this case was 24 [ºC].

Looking at θex (blue line) which is the same for figures 3; 4 we can see that the time lag between the peak and valley of the sin curve which is based on measured values only is 12 [h]. The decrement delay between θex (blue line) and θin (green line) for figure 3 (calculated) and figure 4. (measured) both show values of 8:45 [h] which is analogous to the Admittance procedure decrement delay based on STN 730540 boundary conditions (15 min variation). θc (red line) and θin (green line) attenuation (decrement factor) varied considerably between figures 3 and 4. The admittance procedure produced a decrement ranging from 0,28 - 0,38 depending on the boundary conditions used. Measured data produced a decrement of approximately 0,05 while (FEM) revealed a value of 0,275 which was close to the lower limit of the Admittance procedure using AS 4859.1 app. k sec. 3.1(summer) boundary conditions. U and R values are most favourable when subjected to AS 4859.1 app. k sec. 3.1(winter) since the temperature gradient set out in the boundary conditions is only six degrees between the inside and outside conditions.

Fig. 3.Surface temperature profile at points in a 300 [mm] (SRE) wall subjected to quasi-stationary conditions (FEM Calculation)

Fig. 4.Surface temperature profile at points in a 300 [mm] (SRE) wall subjected to quasi-stationary conditions (Measured Data)

The (FEM) method is useful when quasi-stationary conditions are considered and although more complex and climate dependant, it has the ability to offset false assumptions relating to thermal comfort and energy needed for heating. The decrement delays were accurate within 15 [min] (fig 3; 4) which shows a good cohesion with measured data and verifies the suitability of the method. Although attenuation (amplitude) was not the same, this was not due to inaccurate calculations, but inaccurate initial assumptions. Further, thermal conductivity was again assumed constant; the density had an accuracy variation of 0.3 % and the heat flux plate was accurate to within 5 [%]. Figure 5 used interpolation to express what constant thermal conductivity would be required by numerical methods (FEM) to obtain an indoor surface temperature that is analogous to those achieved by analytical means i.e. (measurements).

Fig. 5. Comparison between numerical and analytical data

The graph showed that for numerical methods where the inside surface θi was determined by (FEM) from values where λ=0,292 [W/m.K] the amplitude still did not adequately match the attenuation evident in high mass walls obtained by analytical means. This would mean that on average the wall was functioning as if it had a thermal conductivity better than 0,292 [W/m.K] as opposed to it steady state 0,892 [W/m.K] which is a considerable difference.

6. Discussion

True thermal conductivity is a key parameter for (FEM) and the Admittance procedure. Having varying thermal conductivities greatly affects the output of all the results. In general, steady-state techniques are useful when the temperature of the material does not change with time as it is primarily dependent on the medium's phase, temperature, density, and molecular bonding. The initial contribution of this work did not foresee that thermal conductivity of the same material in the same position subjected to different boundary temperatures would produce a maximum variation in thermal conductivity of 0,393 [W/m.K] for CS0108. This is a ~ 36 [%] variation. The guarded hot plate method for determining thermal conductivity, considered to be more accurate, was not a prerequisite of the study since an assumed accuracy of 0,1 [W/m.K] from the mean axis was deemed sufficient for a material with low insulating properties 0,7 [W/m.K] or higher (less than 14,28 [%] error;). The author noted when processing the results that a reduction in temperature increased thermal conductivity incrementally (column A), but that when the temperature went from a cooling to heating phase spikes in thermal conductivity were noticed (column B). See table 4.

Tab. 4. Temperature dependant changes in thermal conductivity

These variations have created an unexpected enigma and its effect can only be hypothesized since more accurate test methods for thermal conductivity were not implemented. In general, during any period in which temperatures are changing in time at any place within an object, the mode of thermal energy flow is termed transient conduction. Another term is "non-steady-state" conduction, referring to time-dependence of temperature fields in an object. Non-steady-state situations appear after an imposed change in temperature at a boundary of an element. They may also occur with temperature changes inside an element, as a result of a new source or sink of heat suddenly introduced within an object, causing temperatures near the source or sink to change in time. When a new perturbation of temperature occurs, temperatures within the system will tend toward a new equilibrium with new boundary conditions, provided that these also do not change. After equilibrium is achieved, heat flow into the system will again equal the heat flow out, and the temperatures at each point inside the system remain constant. Once this occurs, transient conduction ends, although steady-state conduction can continue if there continues to be heat flow. If external temperatures or internal heat generation change to quickly for equilibrium of temperatures in space to occur, then the system never reaches a state of unchanging temperature distribution in time, and the system remains in a transient state.

Conclusion

Although the analysis of non-steady-state conduction systems is more complex than steady-state systems, and (except for simple shapes) calls for the application of approximation theories, and/or numerical analysis by computer, ignoring its affects unfairly targets mass systems that, (based on climate region, design etc.), are comfortable to live in, have a small carbon footprint and are sustainable in every context of the word. Neither of the numerical methods applied adequately reflected the attenuation and stability of rammed earth and it is presumed that computer simulations will fare no better unless they use non-steady-state conduction. Since rammed earth is hygric in nature, it is assumed that moisture content can vary based on specific humidity, and evaporative effects in the material. Relative humidity values are deceptive in that a relative humidity of 50 % at 0 [ºC] has a specific humidity of 2 [g/kg] while a relative humidity of 50 % at 30 ºC has a specific humidity of 13,5 [g/kg] when consulting a typical h-x diagram. This may also affect the results which would explain the variation in thermal conductivity. Further research in the thermal conductivity of rammed earth is needed to verify whether or not it should be considered constant when subjected to real conditions based on the extent to which it influences the overall performance of rammed earth envelopes. Facilities at the newly built Vukonze laboratory can do this and once and for all validate the correctness of simulations should the need arise.

Acknowledgements

Supported by VEGA 1/1060/11: Monitoring changes in physical parameters of building envelope structures during quasi stationary states in regards to the dynamic changes of the external environment. This article, was created and realized through ITMS 26220120037 Centre for excellent research of progressive building construction, materials and technologies, with the support of the research and development program financed by the European Regional Development Fund.