Heat storage efficiency and effective thermal output: Indicators of thermal response and output of radiant heating and cooling systems

Heat storage efficiency and effective thermal output: Indicators of thermal response and output of radiant heating and cooling systems KRAJČÍK, M.; ŠIKULA, O. Energy and Buildings Volume 229, 15 December 2020, 110524, Pages 1-14 ISSN: 0378-7788 DOI: https://doi.org/10.1016/j.enbuild.2020.110524 Accepted manuscript © 2021. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ dspace.vutbr.cz 1 HEAT TRANSFER EFFICIENCY AND HEAT STORAGE EFFICIENCY: INDICATORS OF THERMAL 1 RESPONSE OF RADIANT HEATING AND COOLING SYSTEMS 2 3 Michal Krajčík1,*, Ondřej Šikula2 4 5 1Slovak University of Technology, Faculty of Civil Engineering, Radlinského 11, 81005 Bratislava 6 2Brno University of Technology, Faculty of Civil Engineering, Veveří 331/95, 60200 Brno 7 8 *Corresponding e-mail: [email protected] 9 10 Abstract 11 Alternative indicators of thermal response of radiant heating and cooling systems called heat transfer 12 efficiency (HTE) and heat storage efficiency (HSE) have been proposed. The HTE and HSE were 13 compared with established indicators represented by the time constant τ63, response time τ95, and 14 thermal energy stored (TES). The comparison was performed for three wall cooling systems with 15 various combinations of pipe location, configuration of material layers, and materials of the thermal 16 core. Taking into account the whole response curve (HTE, HSE) instead of focusing on one specific 17 point on the curve (τ63, τ95) allowed comparing the thermal response of systems with complex thermal 18 behaviour by a single value. It also permitted predicting thermal response consistently regardless of 19 the system and core material. TES predicted the thermal response of certain systems, but it may not 20 be suitable for comparing the thermal response of radiant systems with different thermal admittance. 21 Besides, a composite indicator called effective thermal output was proposed to consider both the 22 steady-state and dynamic thermal performance. For thermally active building systems (TABS), using 23 effective thermal output led to differences in thermal performance between the two core materials 24 lower than indicated by the maximum (nominal) thermal output. 25 26 Keywords 27 Radiant heating; radiant cooling; thermal response; thermally activated building systems (TABS); 28 thermal dynamics; response time 29 30 2 1. Introduction 1 2 In well-insulated modern and retrofitted existing buildings, installation of radiant heating and cooling 3 systems can be preferable due to their benefits such as the suitability for combination with low-grade 4 renewable energy sources [1,2,3], high sensible heating and cooling capacity [4], and comfortable 5 thermal environment [5,6,7]. The response time of these systems is higher than that of convective 6 systems and may be as high as tens of hours [8,9,10]. This, in combination with frequent changes in 7 weather conditions and internal heat gains, requires using precise control strategies to fulfil the criteria 8 on thermal comfort [11,12,13,14]. Knowing the thermal response of the radiant systems is therefore 9 crucial to determine an efficient control strategy. 10 The existing body of research describes various methods to evaluate the thermal response of 11 radiant heating and cooling systems (Table 1). The most common indicator is the time constant, which 12 is defined as 63% of the final value of the variable observed (τ63) such as the surface temperature of 13 the radiant system, thermal output, or room temperature [15,16,17,18]. Alternatively, the thermal 14 response can be expressed as response time defined as 80% (τ80) [19], 95% (τ95) [9] or other 15 percentages of the final value. A single value of the time constant or response time sufficiently 16 describes the thermal response in case that the temperature of the body is nearly uniform, i.e. that 17 internal conduction can be neglected. For such systems, the response time τ95 equals three times the 18 time constant τ63 [20]. Ning et al. [9] have shown that for radiant systems with pipes embedded in a 19 layer of thermal mass, using a single value of the time constant or response time is not suitable 20 because the ratio of τ95/τ63 varies and the thermal behaviour of some of the systems may be too 21 complex. They, therefore, recommended using several response times such as e.g., 25%, 50%, 63%, 22 or 80% of the difference between the final and initial values. 23 The studies that use indicators other than time constant and response time include Price and 24 Smith [21], who studied the thermal response of various structures of the building envelope to thermal 25 load. The thermal response was expressed as the time in which the surface temperature reaches its 26 peak value. Peak values of the surface temperature were also reported. Sourbron et al. [22] used 27 thermal admittance (surface output/surface temperature) and thermal transmittance (pipe 28 output/surface temperature). The difference between the transmittance and the admittance curve is 29 proportional to the amount of energy stored in the thermal mass, which makes it a suitable indicator to 30 3 quantify the controllability of radiant systems. Chikh [23] used room temperature overshoot and 1 settling time after a step rise in the set point of the room temperature by 1°C in a room with radiant 2 floor heating. The heat output was provided through a heat source located in various depths within the 3 floor. 4 Kobayashi and Kohri [24] compared the thermal response of three types of floor heating systems 5 to a step-change in pipe surface temperature from 20°C to 40°C using three indicators: (i) total heat 6 transmission from pipe to the ambient space at upper and lower floor surfaces, (ii) heat absorption 7 from pipe to the floor, and (iii) ratio of heat transmission to heat absorption. The cumulative difference 8 between the heat transmission and absorption rates indicated the thermal storage. Other studies use 9 step-up or step-down curves of surface temperature and compare the curves visually to observe the 10 differences in the thermal response of various radiant systems [19,25,26] or various measurement 11 points within the same system [27], or count the number of operation cycles needed to keep the 12 cooling output between 90 and 63 % of its maximum value [8]. 13 Several studies have tested the effect of design on the thermal response of radiant heating and 14 cooling systems. Merabtine et al. [15] developed a simplified calculation model of the thermal 15 behaviour of a heating slab considering various design parameters. The heat capacity of the slab and 16 the water flowrate significantly affected the time constant τ63 as opposed to the insignificant effect of 17 the thermal conductivity and the pipe diameter. Ning et al. [9] tested the response time τ95 of radiant 18 heating and cooling, floor and ceiling systems as defined in ISO 11855 [28]. The concrete thickness, 19 pipe spacing, and concrete type had a significant impact on the response time of Type E of thermally 20 active building systems (TABS), whereas pipe diameter, room temperature, water temperature, and 21 water flow regime did not. The small effect of some of the design parameters in combination with 22 complex thermal behaviour may result in similar response times of radiant systems with different 23 thermal dynamics. For example, Krajčík and Šikula [29] used τ95 to test the thermal response of TABS 24 with pipes embedded in thermal insulation attached to the thermal core, TABS with pipes embedded in 25 the core, and a system with a thermally activated plaster decoupled from the core. In some cases, the 26 response times were similar despite the obvious differences in thermal dynamics. 27 The literature review has shown that (i) an indicator of thermal response may not reflect the 28 differences in thermal dynamics between various radiant systems, (ii) a single value of response time 29 is not enough to describe the thermal dynamics of systems with complex thermal behaviour, and (iii) 30 4 studies that would directly compare the performance of several indicators are lacking. We, therefore, 1 propose alternative indicators of thermal response called heat transfer efficiency (HTE) and heat 2 storage efficiency (HSE). Unlike time constant and response time, HTE and HSE present single 3 indicators of thermal response taking into account the evolution of surface temperature or thermal 4 output from the beginning until it reaches steady-state. These indicators help describe the thermal 5 response of radiant systems with complex thermal behaviour by a single indicator and thereby 6 facilitate direct comparison of their thermal dynamics. The calculation principle of HTE using a decline 7 curve of surface temperature was briefly outlined in Ref. [8]. This study provides a detailed explanation 8 of the concept and calculation principle of HTE and HSE, compares them with established indicators 9 of thermal response, and test their ability to overcome some of the limitations of the existing indicators. 10 Besides, a composite indicator called effective thermal output is proposed that allows taking into 11 account both the steady-state and dynamic thermal performance of radiant systems. 12 5 Table 1 Research studies pertaining to thermal response of radiant heating and cooling systems, and their comparison with the present study Study Radiant system Determination method Evaluation of thermal response Athienitis [18] (1993) Floor heating (not specified) Step up curve of room temperature over time Time constant, defined as the time to reach 63% of the steady state room temperature Athienitis and Chen [30] (2000) Floor heating (not specified) Profiles of surface temperature over time Delay time, defined as the time duration between the time that the room temperature falls below the set point and the time when it reaches its minimum value Kobayashi and Kohri [24] (2003) Floor heating (hydronic) Step up or decay curve of: (1) heat transmission from pipe to room (2) heat absorption from pipe to floor (3) ratio of heat transmission to heat absorption Visual evaluation and comparison of the curves for the various cases investigated. The heat transmission and heat absorption are not defined Chikh [23] (2005) Floor heating (not specified) (1) Step up curve of room temperature over time (2) Response of room temperature to rise in set point (1) Time constant, defined as the time to reach 63% of the steady state room temperature (2a) Overshoot of room temperature after a step change in set point of the room temperature (2b) Settling time after a step change in set point of room temperature Sourbron et al. [22] (2009) Combined floor and ceiling, heating and cooling (hydronic) Profiles of thermal output and surface temperatures over time (1) Admittance, defined as the ratio of thermal output from the surface to surface temperature (2) Transmittance, defined as the ratio of thermal output from pipes to surface temperature Thomas et al. [19] (2011) Floor heating (hydronic) Step up curves of heat output over time Response time, defined as the time to reach 80% of the nominal heat output Zhao et al. [17] (2014) Floor cooling (hydronic) Step up curve of surface temperature. Decay curve of cooling output Time constant, defined as the time to reach 63% of the steady state value Ning et al. [10] (2015) Floor and ceiling, heating and cooling (hydronic) Step up curve of surface temperature over time Time constant, defined as the time when the average surface temperature reaches 63.2% of the steady state value Yu and Yao [16] (2015) Floor heating (hydronic) Step up curves of surface temp. and heating capacity Time constant, defined as the time to reach 63% of the steady state value Ning et al. [9] (2017) Floor and ceiling, heating and cooling (hydronic) Decay curve of surface temperature over time Response time, defined as the time it takes for the surface temperature of a radiant system to reach 95% of the difference between its final and initial values when a step change in control of the system is applied as input Shen et al. [27] (2017) Generic heating and cooling panel Step up curve of surface temperature over time Visual evaluation and comparison of the curves for various measurement locations on the panel Ferrarini et al. [31] (2018) Generic heating panel (electric) Step up curve of surface temperature over time Time constant, calculated as 𝜏=𝜌𝑐p𝐿x ℎCONV+ℎRAD Sun et al. [25] (2018) Generic heating panel with flat heat pipes (electric) Step up curve of surface temperature of the flat heat pipe over time (1) Visual evaluation and comparison of the curves for various measurement locations on the flat heat pipe (2) Duration of rising stage of the surface temperature Merabtine et al. [15] (2019) Floor heating (hydronic) Profiles of surface temperature over time (1) Time constant, defined as the time to reach 63% of the steady state surface temperature. (2) Time delay, determined from the equation: 𝑇s(𝑡)=�𝑇 s,0 𝑡< 𝑡 d 𝑇s,∞+�𝑇s,0−𝑇s,∞�𝑒−�𝑡−𝑡d� 𝜏 𝑡≥ 𝑡d Krajčík and Šikula [8] (2020) Wall cooling (hydronic) (1) Step-down curve of surface temperature over time. (2) Curve of cooling output (1) Heat transfer efficiency calculated from a step-down curve of surface temperature (2) Number of operation cycles to keep the cooling output between 90 and 63% its maximum Present study Wall cooling (hydronic) Step-up and step-down curves of surface temperature over time Heat transfer efficiency, heat storage efficiency, time constant τ63, response time τ95, energy stored 6 2. Indicators of thermal response used 1 2 Referring to Table 1, time constant and response time are the most frequent indicators to evaluate the 3 thermal response of radiant systems and have been therefore selected as representatives of the 4 established indicators. Thermal energy stored in the thermal mass has also been considered. Despite 5 not being widely used as an indicator, it can provide useful insights into the thermal dynamics and is a 6 suitable indicator to quantify the controllability of radiant systems of radiant systems [22]. Heat transfer 7 efficiency (HTE) and heat storage efficiency (HSE) present novel indicators that may provide certain 8 benefits but have not yet been properly described and compared with the established indicators. 9 10 2.1 Time constant τ63 and response time τ95 11 Time constant τ63 is the most frequently used indicator of thermal response (Table 1). On the other 12 hand, Ning et al. [9] found it preferable to use the response time τ95 to classify radiant heating and 13 cooling systems according to their thermal response. Response time τ95 can be deduced from the time 14 constant τ63 in case that the temperature of the body is nearly uniform and internal conduction can be 15 neglected. The relationship between τ95 and τ63 may vary for various system types and designs, or it 16 may even be impossible to deduce if the thermal behaviour is too complex [9,20]. Both τ63 and τ95 have 17 been considered in the present study. They are defined as the time it takes for the surface 18 temperature of a radiant system to reach 63% (τ63) or 95% (τ95) of the difference between its final and 19 initial values when a step change in control of the system is applied as input. 20 21 2.2 Thermal energy stored 22 The rate of thermal energy stored in the heating/cooling structure is a function of its thermal 23 admittance and transmittance. Changing the water temperature with frequencies at which the 24 difference between the thermal energy admitted to the structure and transmitted to the interior is large 25 means that energy is exchanged between the pipe and the structure without any meaningful effect on 26 the indoor temperature. The amount of the thermal energy stored quantifies the thermal response and 27 controllability of the thermally active structure [22,32,33]. To calculate the amount of thermal energy 28 stored, the energy flows admitted to the structure and transmitted to the surfaces are integrated over 29 time as follows: 30 7 1 𝑇𝑇𝑇= ��𝑞𝑠𝑠𝑠𝑠𝑠𝑠,𝑖∗Δ𝑡𝑖� i=n i=1 =���𝑞p−𝑞int−𝑞ext�∗Δ𝑡𝑖� i=n i=1 (Wh/m2) (1) 2 where qstored,i is the thermal energy stored in the time step Δti (W/m2), Δti is the calculation time step 3 (h), qp is the heating or cooling power supplied to the structure through the pipe (W/m2), qint is the 4 thermal output of the structure (W/m2), and qext are thermal losses to the exterior (W/m2). 5 6 2.3 Heat transfer efficiency (HTE) and heat storage efficiency (HSE) 7 HTE shows how efficiently the heat is transferred from the pipes to the interior. Conversely, HSE 8 indicates how efficiently the heat flux is distributed over the heating or cooling structure and thereby 9 stored in the structure. A higher HTE (lower HSE) means a greater tendency of the heat to be 10 transferred to the inner surface, whereas at a lower HTE (higher HSE) the thermal energy tends to be 11 stored in the structure. 12 13 3 Concept and calculation principle of HTE and HSE 14 15 HTE and HSE are based on the mean age of heat flux concept. The mean age of heat flux is an 16 analogy to the mean age of air used in buildings ventilation to characterize the ability of the ventilation 17 system to efficiently distribute the fresh supply air in a ventilated room [6,34,35,36]. The analogy 18 between the mean age of air and mean age of heat flux is illustrated in Fig. 1. In buildings ventilation, 19 the age of air can be obtained by several methods using tracer gas, from which the most typical are 20 the tracer gas step-up and step-down method. In the step-up method, the tracer gas is dosed at a 21 constant rate throughout the whole measurement. The dosage starts at the time τ = 0 s, and its 22 concentration starts to continuously increase until it reaches steady state. Conversely, in the step-23 down method, the tracer gas is dosed in the room before the measurement. The measurement starts 24 when the supply of tracer gas stops and its concentration starts to decay due to ventilation. 25 In a ventilated room (Fig. 1a) the air containing tracer gas is transferred by convection from the 26 supply, through the room, to the exhaust where the tracer gas concentration is measured. Thereby, 27 the step-up or step-down curve is obtained. In the heating or cooling element (radiant wall, floor or 28 8 ceiling, Fig. 1b), the heat is supplied through pipes and transferred by conduction to the surface where 1 it is emitted to the room. Similarly to the tracer gas which spends a certain time in the room before it is 2 exhausted, the heat or cool in the structure is not being transferred directly to the surface, but 3 distributed and stored throughout the structure. In a step-up test, heat is supplied to the structure at 4 constant water temperature and the surface temperature and thermal output start to increase until they 5 reach steady state (Fig. 2). Conversely, in a step-down test, the heat is extracted from the structure at 6 constant water temperature and the surface temperature and thermal output start plummeting until 7 they reach equilibrium. Although in Fig. 2 and Eqs. 4 to 7 the calculation procedure is described using 8 surface temperature, heat flux at the surface (thermal output) can be used for the calculation instead 9 of the surface temperature providing that the heat transfer coefficient at the inner wall surface is 10 constant. 11 12 Fig. 1 The analogy between mean age of air and mean age of heat flux 13 14 Mean age of air is a statistical concept based on the age distribution of the air components in a 15 point [34,35,36]. The age of an element of air is the time elapsed since the element of air entered the 16 room. The mean age of air is the mean age of all the elements at a certain point or in the whole room. 17 As an analogy to buildings ventilation, the age of an element of heat flux is the time elapsed until the 18 element of heat flux supplied to the structure through pipe reaches a point in the structure where it is 19 stored. For a point within the structure, the local mean age of heat flux is the time it takes for the heat 20 flux to reach and be stored in the actual point after entering the element. The structure mean age of 21 heat flux is the mean age of all the heat flux in the structure. Using the mean age of heat flux concept, 22 it is possible to determine the heat storage efficiency: 23 24 15 1 Fig. 6 The relationship between a) simulation time step and root mean squared error (RMSE), b) 2 simulation time and Heat Storage Efficiency (HSE). Key: A – aerated concrete, R – reinforced 3 concrete, c – core thickness in cm, TI – insulation thickness in cm, sp – pipe spacing in cm. 4 5 4.3 Thermal response curves 6 All the simulations have been done for nine realistic combinations of the concrete thickness (20, 30, 7 and 40 cm) and pipe spacing (15, 20, and 25 cm). Insulation thickness was always 20 cm. The thermal 8 response curves obtained for the three systems W1, W2, and W3 are shown in Fig. 7 for the two core 9 materials, aerated concrete (A) and reinforced concrete (R). 10 11 12 13 14 15 16 17 18 19 20 21 22 0 2 4 6 8 10 12 14 16 18 12345678910 RMSE * 10-3 Time step (min) W1(A)_c 30_TI 20_sp 15 W1(R)_c 30_TI 20_sp 15 W2(A)_c 30_TI 20_sp 15 W2(R)_c 30_TI 20_sp 15 W3(A)_c 30_TI 20_sp 15 W3(R)_c 30_TI 20_sp 15 (a) 2 4 6 8 10 12 14 16 18 20 22 52 53 54 55 56 57 58 59 60 61 62 HSE W2, W3 (%) HSE W1 (%) Simulation time (h) W1(A)_c 30_TI 20_sp 15 W1(R)_c 30_TI 20_sp 15 W2(A)_c 30_TI 20_sp 15 W2(R)_c 30_TI 20_sp 15 W3(A)_c 30_TI 20_sp 15 W3(R)_c 30_TI 20_sp 15 (b) 16 1 2 3 Fig. 7 Thermal response curves of the three wall cooling systems for two core materials: aerated (A) 4 and reinforced (R) concrete. Key: c – core thickness in cm, TI – insulation thickness in cm, sp – pipe 5 spacing in cm. 6 25,0 25,2 25,4 25,6 25,8 26,0 Surface temperature (°C) Time elapsed (h) W1(A) λ = 0.19 W/(m.K) W1(A)_c 20_TI 20_sp 15 W1(A)_c 20_TI 20_sp 20 W1(A)_c 20_TI 20_sp 25 W1(A)_c 30_TI 20_sp 15 W1(A)_c 30_TI 20_sp 20 W1(A)_c 30_TI 20_sp 25 W1(A)_c 40_TI 20_sp 15 W1(A)_c 40_TI 20_sp 20 W1(A)_c 40_TI 20_sp 25 22,0 22,4 22,8 23,2 23,6 24,0 24,4 24,8 25,2 25,6 26,0 Surface temperature (°C) Time elapsed (h) W1(R) λ = 1.58 W/(m.K) W1(R)_c 20_TI 20_sp 15 W1(R)_c 20_TI 20_sp 20 W1(R)_c 20_TI 20_sp 25 W1(R)_c 30_TI 20_sp 15 W1(R)_c 30_TI 20_sp 20 W1(R)_c 30_TI 20_sp 25 W1(R)_c 40_TI 20_sp 15 W1(R)_c 40_TI 20_sp 20 W1(R)_c 40_TI 20_sp 25 22,0 22,4 22,8 23,2 23,6 24,0 24,4 24,8 25,2 25,6 26,0 Surface temperature (°C) Time elapsed (h) W2(A) λ = 0.19 W/(m.K) W2(A)_c 20_TI 20_sp 15 W2(A)_c 20_TI 20_sp 20 W2(A)_c 20_TI 20_sp 25 W2(A)_c 30_TI 20_sp 15 W2(A)_c 30_TI 20_sp 20 W2(A)_c 30_TI 20_sp 25 W2(A)_c 40_TI 20_sp 15 W2(A)_c 40_TI 20_sp 20 W2(A)_c 40_TI 20_sp 25 22,0 22,4 22,8 23,2 23,6 24,0 24,4 24,8 25,2 25,6 26,0 Surface temperature (°C) Time elapsed (h) W2(R) λ = 1.58 W/(m.K) W2(R)_c 20_TI 20_sp 15 W2(R)_c 20_TI 20_sp 20 W2(R)_c 20_TI 20_sp 25 W2(R)_c 30_TI 20_sp 15 W2(R)_c 30_TI 20_sp 20 W2(R)_c 30_TI 20_sp 25 W2(R)_c 40_TI 20_sp 15 W2(R)_c 40_TI 20_sp 20 W2(R)_c 40_TI 20_sp 25 22,4 22,8 23,2 23,6 24,0 24,4 24,8 25,2 25,6 26,0 Surface temperature (°C) Time elapsed (h) W3(A) λ = 0.19 W/(m.K) W3(A)_c 20_TI 20_sp 15 W3(A)_c 20_TI 20_sp 20 W3(A)_c 20_TI 20_sp 25 W3(A)_c 30_TI 20_sp 15 W3(A)_c 30_TI 20_sp 20 W3(A)_c 30_TI 20_sp 25 W3(A)_c 40_TI 20_sp 15 W3(A)_c 40_TI 20_sp 20 W3(A)_c 40_TI 20_sp 25 22,4 22,8 23,2 23,6 24,0 24,4 24,8 25,2 25,6 26,0 Surface temperature (°C) Time elapsed (h) W3(R) λ = 1.58 W/(m.K) W3(R)_c 20_TI 20_sp 15 W3(R)_c 20_TI 20_sp 20 W3(R)_c 20_TI 20_sp 25 W3(R)_c 30_TI 20_sp 15 W3(R)_c 30_TI 20_sp 20 W3(R)_c 30_TI 20_sp 25 W3(R)_c 40_TI 20_sp 15 W3(R)_c 40_TI 20_sp 20 W3(R)_c 40_TI 20_sp 25 17 1 4.4 Time constant τ63 and response time τ95 2 The time constant τ63 (Fig. 8a) indicates that the thermal response is slowest for TABS with pipes 3 embedded in the thermal core W1(A) and W1(R), followed by TABS with a thermally conductive core 4 and pipes underneath the surface W2(R). The response time is fast when pipes are attached to core 5 with low thermal conductivity W2(A) and when the pipes are insulated from the thermal core (W3). 6 This does not entirely match the trend when response time τ95 is used to evaluate the thermal 7 response (Fig. 8b). For example, when using τ95 as an indicator, the response is slowest for the 8 system with pipes attached to a thermally conductive core, W2(R). In general, τ63 and τ95 of the 9 systems that accumulate thermal energy can vary over a wide interval depending on the thickness of 10 the core and spacing of the pipes. 11 12 Fig. 8 Boxplots of the nine cases investigated for each cooling system: a) time constant τ63 and b) 13 response time τ95. Key: A – aerated concrete, R – reinforced concrete, 14 15 Fig. 9 shows the relationship between the time constant τ63 and response time τ95 for individual 16 cases. The corresponding coefficients of determination are shown in the legend. A strong correlation 17 and large variation in the values was found for the two TABS with pipes embedded in the thermal core 18 (W1). For the systems with pipes insulated from the thermal core (W3) the correlation is weaker and 19 the variation is much lower. The correlation is weakest for system W2(R). For systems W2(A) and 20 W2(R), τ95 varies over a much wider interval than τ63 meaning that the difference in thermal response 21 between the systems and individual cases depends on the definition of response time. This is also 22 0 2 4 6 8 10 12 14 16 18 Time constant, τ63 (h) Type of wall, material 0 10 20 30 40 50 60 70 80 Response time, τ95 (h) Type of wall, material (b) (a) 18 illustrated by the outlying value of τ95 in system W3(R) (see Fig. 8b where the maximum value is much 1 higher than the rest). This is caused by the response curve of system W3 composed of three parts 2 (Fig. 7): (i) the steep decline, (ii) the sharp bend, and (iii) the slow gradual decline. The surface 3 temperature corresponding to τ63 lies on the steep part of the curve and τ63 is therefore always close to 4 zero. The surface temperature corresponding to τ95 lies on the slowly declining part of the curve. The 5 slow decline means that a small difference in surface temperature can lead to substantial differences 6 in τ95. These findings underline the fact that calculating several response times may be needed to 7 describe the thermal dynamics of radiant systems with complex thermal behaviour. 8 9 Fig. 9 Correlation between τ63 and τ95 for the three radiant systems and two core materials. Key: A – 10 aerated concrete; R – reinforced concrete. 11 12 4.5 Thermal energy stored (TES) 13 The simulations have shown that for the system W3 with pipes separated from the thermal core, TES 14 does not reach the peak even after the whole simulation period but keeps increasing. For W1 and W2, 15 the increase is maximal during the first hours. As expected, TES increased with the core thickness of 16 and density of pipe spacing. The values of TES differ substantially for various combinations of the 17 cooling system and core material, as illustrated in Fig. 10a for the concrete thickness of 30 cm, 18 insulation thickness of 20 cm, and pipe spacing of 15 cm. A smaller amount of TES indicates a smaller 19 difference between energy admitted to the structure and energy transmitted to the interior. This means 20 faster transmission of thermal energy from pipe to the interior and consequently more rapid thermal 21 response. Conversely, higher TES indicates a slower thermal response. 22 0 10 20 30 40 50 60 70 80 Response time τ95 Time constant τ 63 W1(A), R^2=0.99 W2(A), R^2=0.94 W3(A), R^2=0.84 W1(R), R^2=1.00 W2(R), R^2=0.17 W3(R), R^2=0.42 19 Fig. 10b shows boxplots of the nine combinations of concrete thickness and pipe spacing for each 1 cooling system. The two systems with pipes attached to a thermally conductive core, W1(R) and 2 W2(R), store significantly more energy than the other systems. TES is low for the two systems with 3 pipes attached to a core with low thermal conductivity, W1(A) and W2(A), because of their low thermal 4 admittance resulting from the low thermal conductivity of the core. TES is also low for the two systems 5 with pipes insulated from the thermal core (W3) due to the low thermal mass of the thermally active 6 plaster, which indicates a fast thermal response. 7 TES logically predicts a slow thermal response for TABS with thermally conductive core, W1(R) 8 and W2(R), and fast thermal response for systems with pipes insulated from the core (W3). The 9 discrepancy between TES and the other indicators is largest for system W1(A). In this case, TES 10 indicates a fast thermal response similar to W2(A) and W3(A). This contradicts the visual comparison 11 of response curves in Fig. 7, and a slow thermal response predicted by τ63 and τ95 (Fig. 8) and HTE 12 (see Fig. 11, Section 4.7). This is because TES reflects the low thermal admittance of structure W1(A) 13 rather than the shape of the response curve. It means that TES can be used to compare the thermal 14 response of systems with similar thermal admittance. Thus, TES has limited use as an indicator of 15 thermal response. 16 17 Fig. 10 Thermal energy stored (TES) over 24 h: a) Representative curves, b) Boxplots of the nine 18 cases investigated for each cooling system. Key: A – aerated concrete; R – reinforced concrete. 19 20 4.6 Heat transfer efficiency (HTE) and heat storage efficiency (HSE) 21 0 100 200 300 400 500 600 700 800 900 1000 Thermal Energy Stored, TES (Wh/m2) Time elapsed (h) W1(A)_c 30_TI 20_sp 15 W1(R)_c 30_TI 20_sp 15 W2(A)_c 30_TI 20_sp 15 W2(R)_c 30_TI 20_sp 15 W3(A)_c 30_TI 20_sp 15 W3(R)_c 30_TI 20_sp 15 0 200 400 600 800 1000 1200 1400 Thermal Energy Stored, TES (Wh/m2) Type of wall, material (a) (b) 20 HTE and HSE are complementary indicators meaning that their sum is always 100%. The boxplots in 1 Fig. 11 visualize the relationship between HTE and HSE. The results show the differences in the 2 thermal response of the cooling systems. HTE and HSE are ranging from 40 to 60% for the systems 3 with pipes embedded in the thermal core (W1). HTE is substantially lower for W1 with pipes in the core 4 than for W2 and W3 with pipes underneath the surface because locating the pipes in the core causes 5 a more even distribution of heat flux in the structure. Thereby, the energy is stored at a higher rate and 6 consequently, its transfer to the surface is less efficient. 7 HTE is about 90% for the system with pipes attached to a core with low thermal conductivity, 8 W2(A), because the high thermal resistance of the core results in less efficient thermal storage. The 9 HTE is lower for W2(R) with a thermally conductive core because the low thermal resistance of the 10 core allows more efficient heat flux distribution and thermal storage. HTE is highest for system W3 11 regardless of the core material. Locating the pipes close to the surface and insulating them from the 12 thermal core causes inefficient thermal storage. 13 14 15 Fig. 11 Boxplots of the nine cases investigated for each cooling system: a) Heat transfer efficiency 16 (HTE), b) Heat storage efficiency (HSE). Key: A – aerated concrete, λ = 0.19 W/(m.K); R – reinforced 17 concrete, λ = 1.58 W/(m.K). 18 19 The age-of-heat-flux concept considers the whole response curve whereas the response-time 20 concept focuses on one point on the curve. The information provided about thermal response can, 21 therefore, differ depending on the method as illustrated in Figs. 12 and 13 where HSE is compared 22 40 50 60 70 80 90 100 Heat Transfer Efficiency, HTE (%) Type of wall, material (a) 0 10 20 30 40 50 60 Heat Storage Efficiency, HSE (%) Type of wall, material (b) 21 with τ63 and τ95 for the individual cases. Time constant τ63 shows a fast thermal response of W2(A) and 1 W3, but does not show any differences in thermal response between the systems and the individual 2 cases. Response time τ95 may lead to differences between the individual cases much wider than τ63, 3 e.g. in case of W2(R) and W3(R). This confirms the need for calculating several response times (e.g. 4 τ25, τ50, τ63, τ95) as pointed out by Ning et al. [9]. On the other hand, HSE (HTE) reflects even subtle 5 differences between the individual cases and systems but the values are more consistent as 6 compared to τ95 whereby excessive scatter of the values is prevented. Thus, the potential advantage 7 of using HSE (HTE) as compared to the response-time concept is that it allows comparing radiant 8 systems with complex thermal behaviour using a single value. 9 10 Fig. 12 Comparison of HSE, τ63, and τ95 for thermal core made of aerated concrete. Key: c – core 11 thickness in cm, TI – insulation thickness in cm, sp – pipe spacing in cm. 12 13 0 10 20 30 40 50 60 70 0 10 20 30 40 50 60 70 τ63 , τ95 (h) Heat Storage Efficiency, HSE (%) Aerated concrete, λ = 0.19 W/(m.K) HSE τ63 τ95 22 1 Fig. 13 Comparison of HSE, τ63, and τ95 for thermal core made of reinforced concrete. Key: c – core 2 thickness in cm, TI – insulation thickness in cm, sp – pipe spacing in cm. 3 4 4.7 Effective thermal output 5 In addition to HTE and HSE, we propose a composite indicator of steady-state and dynamic thermal 6 performance called effective thermal output (qint,eff). The effective thermal output reflects both the 7 steady-state thermal performance represented by the heating/cooling capacity and the dynamic 8 thermal performance represented by HTE. In other words, it considers the magnitude of thermal output 9 as well as the speed with which the output is attained. It is defined as: 10 11 qint,eff = (qint * HTE) / 100 (W/(m2.K)) (12) 12 13 where qint is the maximum value of thermal output which corresponds to the nominal heating/cooling 14 capacity, expressed in W/m2 per temperature difference between water and room temperature 15 (W/(m2.K)). In the present study, the maximum (nominal) cooling output was used, and the 16 temperature difference between water and room temperature was 6 K in all the cases. HTE is heat 17 transfer efficiency (%). 18 Figs. 14 and 15 show the differences in effective thermal output (qint,eff) between some of the 19 systems and cases. For system W1, HTE is similar for aerated and reinforced concrete but the 20 nominal cooling output is substantially higher for reinforced concrete because of its high thermal 21 0 10 20 30 40 50 60 70 80 0 10 20 30 40 50 60 70 τ63 , τ95 (h) Heat Storage Efficiency, HSE (%) Reinforced concrete, λ = 1.58 W/(m.K) HSE τ63 τ95 23 conductivity. The effective thermal output shows that the difference between the two core materials is 1 considerably lower than indicated by the nominal cooling output. Nevertheless, the thermally 2 conductive core is still preferable. 3 Effective thermal output also equalizes the performance of the two systems with pipes attached to 4 the thermal core, W2(A) and W2(R). The nominal cooling output is lower, and HTE is higher for the 5 core made of aerated concrete, W2(A). However, looking at the effective thermal output, the 6 performance is similar regardless of the core material. For systems W3(A) and W3(R) the nominal and 7 effective thermal output are always similar because the HTE is very close to 100% in all cases. These 8 findings suggest that using effective thermal output can be useful especially when comparing TABS 9 (W1, W2). 10 11 Fig. 14 Comparison of nominal and effective thermal output for thermal core made of aerated 12 concrete. Key: c – core thickness in cm, TI – insulation thickness in cm, sp – pipe spacing in cm. 13 14 0 1 2 3 4 5 6 0 10 20 30 40 50 60 70 80 90 100 Thermal output (W/(m2.K)) Heat Transfer Efficiency, HTE (%) Aerated concrete, λ = 0.19 W/(m.K) HTE Nominal output Effective output 24 1 Fig. 15 Comparison of nominal and effective thermal output for thermal core made of reinforced 2 concrete. Key: c – core thickness in cm, TI – insulation thickness in cm, sp – pipe spacing in cm. 3 4 5 Conclusion 5 6 Novel indicators of thermal response of radiant heating and cooling systems called heat transfer 7 efficiency (HTE) and heat storage efficiency (HSE) have been proposed. Unlike the commonly used 8 time constant and response time that focus on a single point on the response curve, HTE and HSE 9 take into account the evolution of the response curve from the beginning until it reaches steady-state. 10 The HTE and HSE have been compared with established indicators of thermal response represented 11 by the time constant τ63, response time τ95, and thermal energy stored (TES). Besides, a composite 12 indicator called effective thermal output has been proposed for a combined evaluation of the steady-13 state and dynamic thermal performance of radiant systems. The novel indicators can serve to evaluate 14 the thermal performance of various types of radiant systems as well as to develop a control strategy in 15 buildings with radiant systems. The conclusions that may be drawn from this study are: 16 • Using the whole response curve (HTE, HSE) rather than one specific point on the curve (τ63, 17 τ95) helps predict the thermal response consistently regardless of the system and core 18 material. It clearly showed the differences between individual systems and materials of the 19 thermal core while preventing excessive scatter of the values. 20 0 1 2 3 4 5 6 0 10 20 30 40 50 60 70 80 90 100 Thermal output (W/(m2.K)) Heat Transfer Efficiency, HTE (%) Reinforced concrete, λ = 1.58 W/(m.K) HTE Nominal output Effective output