Establishment and experimental validation of a ...

Energy and Buildings 82 (2014) 330–340

Contents lists available at ScienceDirect

Energy and Buildings journal homepage: www.elsevier.com/locate/enbuild

Establishment and experimental validation of a dynamic heat transfer model for concrete radiant cooling slab based on reaction coefficient method Zhe Tian a,∗ , Baodong Duan a , Xiaolei Niu b , Qi Hu a , Jide Niu a a School of Environmental Science and Engineering, Key Laboratory of Efficient Utilization of Low and Medium Grade Energy, MOE, Tianjin University, Tianjin 300072, China b Jinan Urban Planning and Design Institute, Jinan 250000, China

a r t i c l e

i n f o

Article history: Received 26 March 2014 Received in revised form 13 July 2014 Accepted 14 July 2014 Available online 22 July 2014 Keywords: Concrete radiation cooling Reaction coefficient method Dynamic heat transfer

a b s t r a c t To study the dynamic thermal response performance of concrete radiant cooling slab, this paper introduces the concept of core temperature layer. Heat transfer process in the concrete slab is divided into three sub-processes, correspondingly, and three heat transfer models are built by reaction coefficient method. Two-dimensional unsteady heat transfer model of concrete cooling slab is established ultimately. Experiments are conducted to test the slab thermal performance in steady and unsteady conditions. Error analysis and consistency verification are presented between the experimental value and calculated results of the model. The relative error between calculation and experimental value is within 2% in steady conditions and within 7% in unsteady conditions. The application scope of the heat transfer model for practical engineering is defined in this paper. © 2014 Elsevier B.V. All rights reserved.

1. Introduction Unlike other radiant cooling technologies, in concrete radiant cooling system, building itself with a large heat capacity acts as a part of air conditioning terminal, which is directly involved in the cooling process of air conditioning system. Active intervention of building regenerator not only causes the cooling transfer process of the terminal trend to be complex, but also changes the coupling process between heat transfer within the building and cooling by air conditioning system. In order to ensure the realization of energy saving effect of concrete radiant cooling technology and form a complete design and operation technology system, clear understandings need to be got about the coupling relationships of the two processes. One of the problems is how to find out the dynamic response of the thermal properties of concrete radiant cooling system. It is the dynamic changes of cooling load that determines that the concrete radiant cooling technology research must be built on the basis of unsteady heat transfer process. Therefore, studies on heat transfer process in the concrete slab have been the major research directions in this field. Numerical simulations and

∗ Corresponding author. Tel.: +86 22 27407800; fax: +86 22 27407800. E-mail address: [email protected] (Z. Tian). http://dx.doi.org/10.1016/j.enbuild.2014.07.031 0378-7788/© 2014 Elsevier B.V. All rights reserved.

simplified heat transfer models become two main technology roadmaps as a result of the difficulty to obtain analytical solutions. Finite difference method (FDM) and finite element method (FEM) are typically used in numerical methods. Fort establishes a numerical model of heat transfer process within concrete radiant cooling terminal by using FDM method. This method has high accuracy. However, it is not well balanced with circular pipe geometry in the slab for the use of square meshes. It also causes shocks in calculation and requires high quality grids as well as strong performance computers [1]. Holopainen et al. establish an uneven nodal network model with finite difference heat balance method, by which show the benefits of placing the densest gridding in steepest curvature of the temperature gradient [2]. FEM method is also used in calculation, but it is not widely used in engineering for the complexity in modeling and long time spent in computing. Babiak studies the dynamic response of 24-h-period sine wave signals on several concrete radiant cooling terminals. The author believes that when the load or cooling capacity changes, not the whole slab but 10–15 cm thick slab is involved in regenerative regulation. Due to the complexity in modeling and long time spent in computing of the FEM method, the author ignores the coupling of passive thermal storage of other building envelopes. Further analysis on how other forms of fluctuation influences the regenerative regulation is not carried out in the article [3]. Simplified heat transfer models are relatively simple to build, and they take little computational effort but can

Z. Tian et al. / Energy and Buildings 82 (2014) 330–340

Nomenclature L l d T Tk Re Pr y1 y2 q t s a n T¯ q¯ h T(t) AUST

spacing between adjacent pipes (mm) pipe length (mm) diameter of pipes (mm) temperature (◦ C) constant temperature = 273.15 ◦ C thermal conductivity [W/(m K)] Reynolds number Prandtl number spacing between pipes and the upper floor surface (mm) spacing between pipes and the underside floor surface (mm) heat flux (W/m2 ) time complex number of Laplace transform thermal diffusivity in the concrete slab (m2 /s) duration of unit triangular wave function reaction coefficient number temperature of the Laplace transform heat flux of the Laplace transform heat transfer coefficient [W/(m2 K)] temperature disturbance (◦ C) average unheated surface temperature (◦ C)

Subscript 0 chilled water 1 upper slab surface 2 underside slab surface air in the test room air c core temperature layer air boundary upon upper concrete slab surface u w air boundary below lower concrete slab surface tot total cov convention rad radiation from air boundary upon upper concrete slab surface wc to core temperature layer cw from core temperature layer to air boundary upon upper concrete slab surface uc from air boundary below lower concrete slab surface to core temperature layer cu from core temperature layer to air boundary below lower concrete slab surface 0c from chilled water to core temperature layer c0 from core temperature layer to chilled water

ensure high accuracy. Therefore, many of the current studies focus on them. Thermal resistance and heat capacity network method (TRHC) and heat conduction transfer function method (HCTF) are two representative practices and they have already been used in Trnsys and Energy Plus current version, respectively. TRHC method is based on the similarity between heat transfer in building materials and electrical circuit. Methods of electrical circuit calculation are introduced to convert unsteady heat transfer in time domain to frequency domain, greatly reducing model complexity and computational workload. HCTF method is based on transfer function method, which is the basic analysis method of control theory. Transfer function is typically used in single-input and single-output analog circuit, which is mainly applied in signal processing, communication theory and control theory. For simple continuous-time input and output signals, the linear mapping between Laplace transform of the input signal and the output signal under conditions

331

of zero state is reflected by transfer function. Stephonson et al. apply transfer function to calculate air conditioning load of the building envelope, which is its beginning of the application in heat transfer problems [4]. Koschenz and Lehmann employ HCTF model to calculate heat transfer from water pipe node to core temperature layer and use TRHC algorithm to compute heat transfer of other part, thus forming a joint TRHC-HCTF model. This simplified model is adopted by the radiant cooling module in Trnsys Ver.16 and subsequent Ver.17 [5–7]. As a result of large calculation errors, this model is only suitable for lightweight construction and relatively constant temperature conditions, it is less applicable for simulation conditions with thin concrete radiant cooling terminals and temperature fluctuation period less than 10 h [8,9]. Weber develops a numerical model to obtain equivalent thermal resistance and heat capacity of nodes and then uses star TRHC method to compute. Although this approach expands the scope of the model, it complicates computing process at the same time [8]. Strand builds a concrete cooling heat transfer model based on the HCTF method, which is used in EnergyPlus energy simulation software to calculate heat transfer through building envelope, the author adds heat source and heat convergence in HCTR method and then solves the model via Laplace transform or state space method. This algorithm was adopted by EnergyPlus software in its radiant cooling module and has been used till now [10,11]. Compared with TRHC method, HCTF method can only handle linear equations because it can’t associate with nonlinear processes such as changes in water flow. When calculating heat transfer between pipes and concrete slab, HCTF method can only assume that fluid is still and its temperature does not change along pipe length. Consequently, HCTF model may not do very well in simulating unsteady heat transfer process in concrete slab under variable flow conditions [8]. Rijksen believes that regenerative effect is bound to reduce the peak load of concrete radiant cooling system. For the sake of achieving universal design guidelines, the maximum room heat gain handled by concrete radiant cooling system in Netherlands is simulated, and the value is about 60 w/m2 [12]. Zhang et al. found a two-dimension simplified calculation model for concrete cooling slab. They put forward a simplified method to calculate the cooling capacity of the concrete slab and temperature distribution of floor surface, pointing out that thickness and heat transfer coefficient of materials have great impact on cooling performance [13]. Liu et al. use the result of TRHC method to establish a simplified TRHC model for unsteady heat transfer process in concrete radiant cooling system. The authors define the equivalent heat capacity and thermal resistance of core temperature layer according to the geometrical and thermal parameters, making modeling process be separated from numerical simulation. The model further simplifies calculation process but brings big error since Fourier decomposition is introduced to deal with step changes of input variables [14]. Previous models have their own limitations in handling heat transfer process of concrete radiant cooling system, further studies are still necessary to improve and develop concrete radiation heat transfer models. This paper establishes a heat transfer model which takes unsteady heat transfer process in flat wall as theoretical foundation and processes input disturbance using response coefficient method (RC). Compared with simplified calculation methods proposed in Refs. [2,13,14], the model in this paper has better applicability for actual slab heat transfer process.

2. Heat transfer model 2.1. Heat transfer in the concrete slab In order to make it easy to study and analyze heat transfer process in concrete slab, core temperature layer concept is introduced

332


Fig. 3. Heat transfer process of lower heat transfer subsystem.

Fig. 1. Determination of core temperature layer.

to simplify heat transfer model in this paper [15]. On the center line of adjacent pipes, the concrete temperature is not uniform, the closer to the pipe the lower the temperature. Temperature distribution in the slab is shown as the curve in Fig. 1. The determination of thermal resistance and heat capacity between cool water and core temperature can refer to Ref. [14]. Heat transfer process in the concrete slab is greatly simplified thanks to the introduction of core temperature layer concept. Heat transfer in the slab can be considered to transfer as the following process: firstly, heat transfer through slab surfaces to core temperature layer driven by corresponding temperature difference. Then heat accumulates in core temperature layer, after that, it transfers form core temperature layer to chilled water ultimately. Heat transfer from core temperature layer to chilled water can be regarded as heat transfer from point C in Fig. 1 to chilled water. Thus, heat transfer process in the slab can be divided into three processes among four nodes, as shown in Fig. 2. Heat transfer system is divided into three sub-systems correspondingly which named upper heat transfer subsystem, lower heat transfer subsystem and core temperature layer heat transfer subsystem, respectively. 2.2. Simplified assumptions Heat transfer process in the concrete slab is complex, it is necessary to simplify the heat transfer process and ensure the accuracy and applicability at the same time. The three-dimensional concrete slab heat transfer process is simplified to two-dimensional heat transfer process based on the following assumptions: As the temperature difference between supply and return water is relatively small, temperature change is ignored along water flow direction. The mean water temperature is supposed to be equal to the water temperature. Each slab material layer is assumed as homogeneous.

The influence of pipe regenerative effect to heat transfer process is disregarded considering that thermal resistance and heat capacity of pipes are relatively small in comparison with concrete cooling slab. Heat transfer among pipes is not considered for the small temperature difference along the pipe. 2.3. Establishment of heat transfer equations Heat transfer processes are similar among the three heat transfer subsystems, so this paper only takes the lower heat transfer subsystem for example to analyze heat transfer process. When the lower heat transfer subsystem is formed of single homogeneous concrete slab, heat transfer in the slab is shown in Fig. 3. According to the thermal conduction differential equations, Fourier’s law and boundary conditions, heat transfer governing equations of the slab can be described in the following equation:

⎧ 2 ∂T (y, t) ∂ T (y, t) ⎪ ⎪ =a ⎪ ⎪ ∂t ∂ y2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ q (y, t) = − ∂T (y, t) ∂y

(1)

⎪ ⎪ ⎪ ⎪ T (y, t) = T2 (t) ⎪ y=0 ⎪ ⎪ ⎪ ⎪ ⎩ q (y, t)

y=0

= q2 (t)

For Eq. (1), Laplace transform is carried out with t as the independent variable and y as the reference variable:

⎧ d2 T¯ ⎪ sT¯ (y, s) = a ⎪ ⎪ ⎪ ∂ y2 ⎪ ⎪ ⎪ ⎪ ⎪ ∂T¯ ⎨ q¯ (y, s) = − ∂y

(2)

⎪ ⎪ ⎪ ⎪ T¯ = T¯ 2 (s) ⎪ y=0 ⎪ ⎪ ⎪ ⎪ ⎩ = q¯ 2 (s) q¯ y=0

By solving Eq. (2), when y = y2 , heat transfer equation in the slab is as show in the following equation:

T¯ c q¯ c

⎡

⎢ ch ⎢ =⎢ ⎣ =

−

s sh a

A(1)

B(1)

C(1)

D(1)

Fig. 2. Simplified heat transfer process in the slab.

where Tr(1) =

s y2 a − s a s y2 ch a sh

s y2 a

A(1) C(1)

s y2 a

¯ T2

q¯ 2 B(1) D(1)

= Tr(1)

⎤

⎥ T¯ 2 ⎥ ⎥ ⎦ q¯ 2

T¯ 2

q¯ 2

is transfer matrix.

(3)


333

Heat transfer equation will be given by solving Eqs. (3), (4) and (7):

T¯ c q¯ c

⎡

⎢ =⎣

A

B−

A hw

C

D−

C hw

⎤

¯ Tw E F ⎥ T¯ w = ⎦ G H q¯ w

(8)

q¯ w

when the values of Tw and Tc are both known, heat flux through the air boundary (qw ) and the core temperature layer (q2 ) can be expressed from Eq. (8):

Fig. 4. Lower heat transfer subsystem with multilayers.

q¯ w

⎡

⎢ =⎣

q¯ c For the multilayer concrete slab, it is assumed that the neighboring layers bond very closely to each other as shown in Fig. 4. For a given interface y = Li (1 < i < n + 1), it acts as not only the upper surface of the lower layer but also the lower surface of the upper layer. Heat transfer manner of one layer in multilayer structure is as the same with single homogeneous concrete slab. Heat transfers from lower surface of lower heat transfer subsystem to core temperature layer through material layers. The transfer matrix can be expressed as: Tr(n) =

n A(i) B(i) C(i) D(i)

i=1

=

A

B

C

D

(4)

Taking the air boundary layer upon slab surface into consideration, the boundary shown in Fig. 5 can be assumed as a homogeneous layer. Air capacity is ignored for that the boundary layer is thin and is composed of air with small heat quantity. By the Newton law, heat flux through air boundary is presented by the following equation: qw = hw (Tw − T2 )

(5)

where hw is the total heat transfer coefficient of the slab surface. When Tw stands for temperatures of different heights in the room, the determination of hw can refer to Ref. [16]. In this paper, Tw is the temperature of air which is 1.1 m above floor surface, hw is chosen as 5.7 W/(m2 K) and 7.67 (T1 − Tw )◦.1 W/(m2 K) for cooling and heating. Heat capacity of the air boundary is ignored, this leads to: q2 = qw

T¯ c q¯ c

⎡ =⎣

1

−

0

1

1 hw

⎤ ⎦

T¯ w

E F

−

−

1 F

H F

1 F

⎤

⎥ T¯ w ⎦

q¯ w

⎡

⎢

=⎣

q¯ c

−

E F

−

1 F

1 F

H F

⎤ ⎥ ⎦

T¯ w

(7)

⎡

⎢

=⎣

0

−

E F

1 − F

⎤

⎥¯ ⎦ Tw

(10)

The equation of heat flux response to temperature disturbance can be written as: E q¯ w (s) = − T¯ w (s) F

(11)

1 q¯ c (s) = − T¯ w (s) F

(12)

where −E/F and −1/F are endothermic reaction function and heat transfer reaction function, respectively. The impact of temperature changes of air boundary layer on heat fluxes of both sides can be known by Eqs. (11) and (12). The corresponding endothermic reaction function and heat transfer reaction function of upper heat transfer subsystem and core temperature layer heat transfer subsystem can be obtained using the same method above. In particular, heat transfer coefficient between pipe and water in core temperature layer subsystem (hs ) can be calculated by the following Eqs. (13)–(15) [17]:

hs =

×

3.66 +

0.0668RePr d/L

d d

1 + 0.04RePr d/L

Re < 2200,

RePr

q¯ w

(9)

T¯ c

To analyze the impact of unilateral temperature disturbance upon the heat flux of upper and lower surface, Tc is set to zero. This changes Eq. (9) to become:

(6)

Eqs. (5) and (6) can be written as Eq. (7) by Laplace transform:

−

hs = 1.86 RePr

d L

1/3 d

d

d ≤ 10 L

(13)

Re < 2200,

hs = 0.116 Re2/3 − 125 Pr 1/3

1+

RePr

d > 10 L

(14)

d 2/3

d

L

× (2200 ≤ Re ≤ 10, 000)

d (15)

2.4. Decomposition of temperature disturbance

Fig. 5. Heat transfer process in lower heat transfer subsystem considering air boundary layer.

Unit rectangular wave function or unit triangular wave function is commonly used by reaction coefficient method to decompose temperature disturbance. Both functions above can be derived from unit step function. Consequently, most of the turbulent motions can be derived by unit step functions.

334

Z. Tian et al. / Energy and Buildings 82 (2014) 330–340 Table 1 Wall classification standard [20].

Fig. 6. Temperature disturbance transfer process of lower heat transfer subsystem.

As for the lower heat transfer subsystem shown in Fig. 6, heat flux, which transfers through upper and lower surface of the subsystem, will respectively change when the air boundary forces a unit temperature disturbance. Function xu (t) stands for the variation of heat flux transfers through the lower surface, which called basic endothermic reaction function. Function yu (t) stands for the variation of heat flux transfers through the upper surface, which called basic heat transfer reaction function. When unit temperature disturbance is decomposed by unit triangle wave function, the corresponding functions which express the lower and upper surface heat flux variation are xT (t) and yT (t). Unit step disturbance U(t) can be expressed as Eq. (16) by Laplace transform: L[u(t)] =

(16)

Inserting Eq. (16) into Eqs. (11) and (12) gives: ∞

Kxi e−si t

(17)

i=1

yu (t) = K +

∞

Kyi e−si t

where Kxi = −E/

(18)

d/ds sF

s=−si

,

Kyi = −1/

d/ds sF

yT (t) =

∞ Kx

i

si

∞ Ky

i

i=1

s=−si

si

e

−si (t−)

(1 − e

−si 2

)

∞ Kx

i

i=1

YTk =

si

∞ Ky

i

i=1

≥600

amount of computation comes out at the same time, the value is 200 in this paper. The corresponding endothermic reaction coefficient and heat transfer reaction coefficient of upper heat transfer subsystem and core temperature layer heat transfer subsystem can be obtained by using the same method above. Then heat fluxes transfer through the subsystem within a period of time can be calculated by utilizing concrete slab stability and heat flux superposition. Tw (t) stands for temperature turbulence of air boundary layer and Tc (t) is temperature turbulence of core temperature layer. By decomposing temperature turbulence by unit triangular wave function, the total heat flux transfers through core temperature layer and air boundary layer of lower heat transfer subsystem can be obtained, respectively: n

qwc =

Tw [(n − k) ] YTkw −

k=0

e

(1 − e

−si 2

)

si

e−si (k−1) (1 − e−si )2

quc =

(23)

Tc [(n − k) ] YTkw −

n

Tw ≤ [(n − k) ] XTkw

(24)

k=0

n

qcu =

n

Tu [(n − k) ] YTku −

n

Tc [(n − k) ] XTku

(25)

Tu [(n − k) ] XTku

(26)

k=0 n

Tc [(n − k) ] YTku −

k=0

The total heat flux transfers through core temperature layer and air boundary layer of core temperature layer subsystem can be written, respectively, as: n

q0u =

T0 [(n − k) ] YTk0 −

k=0 n

n

Tc [(n − k) ] XTk0

(27)

T0 [(n − k) ] XTk0

(28)

k=0

Tc [(n − k)] YTk0 −

k=0

(20)

(21)

Tc [(n − k) ] XTkw

Similarly, the total heat flux transferring through core temperature layer and air boundary layer of upper heat transfer subsystem can be written, respectively as:

qc0 = −si (t−)

n k=0

n

qcw =

(19)

In order to calculate heat transfer capacity at any time, the values of Eqs. (19) and (20) at any time need to be obtained. When k ≥ 1, inserting t = k into Eqs. (19) and (20) gives expressions of reaction coefficient: XTk =

Heavy wall

300–600

k=0

and K is heat transfer coefficient between air boundary and core temperature layer. Through decomposing unit temperature disturbance by unit triangle wave function, the result can be written as:

i=1

Medium wall

≤300

k=0

i=1

xT (t) =

Lightweight wall

Weight (kg/m2 )

k=0

1 s

xu (t) = K +

Wall type

n k=0

The reaction coefficient number n can be divided into three categories. As for lightweight wall, generally the value is 48. For medium wall, n is set to 72 and the value is 96 for heavy wall [20]. Wall classification standard is shown in Table 1. By the energy conservation law, at point C the sum of the heat flows has to equal zero, this leads to: qwc + quc + q0c = 0

(29)

Inserting Eqs. (23), (25) and (27) into Eq. (29) gives: e−si (k−1) (1 − e−si )2

n

(22)

For the value of the infinity (∞) in above equations, some scholars have not reached an agreement. Yan et al. recommend that the value should not be less than 40 [18], while Chen et al. suggest that the value should not be less than 200 [19]. The bigger the value, the more accurate the calculation result. However, because large

Tc [(n − k) ] YTku +

k=0

=

n

k=0

n

Tc [(n − k) ] XTkw +

k=0

Tu [(n − k) ] YTku +

n

k=0

n

Tc [(n − k) ] XTk0

k=0

Tw [(n − k) ] XTkw +

n

(30) T0 [(n − k) ] XTk0

k=0

Inserting Tu , Tw and T0 together with corresponding endothermic reaction coefficient and heat transfer reaction coefficient into


335

Fig. 7. Test room layout and concrete structure.

Eq. (30) gives the temperature of core temperature layer (Tc ) at any time. Then, heat flux variation with time of upper slab surface, lower slab surface and the chilled water can be got by inserting the three input parameters (Tu , Tw and Tc ) into Eqs. (24), (26) and (28), respectively. 3. Validation of RC heat transfer model 3.1. Experimental set-up Experimental research of concrete slab radiant cooling system was done in a full scale test room (L × W × H, 4 m × 2 m × 3 m) with good performance of insulation and air tightness. Inner heat source controlled by input voltage was used to adjust surface temperature and heat load. Test room and concrete structure are shown in Fig. 7. Dynamic thermal response performance of concrete radiant cooling slab was measured in this paper. The accuracy of RC model is verified by comparing calculation result with experimental value.

are diameter and buried depth. Three nodes temperatures are air boundary layer temperatures upon upper and lower slab surfaces and the mean water temperature. The dynamic response performance of concrete radiant cooling slab can be reflected by the changes of temperature field in it. This paper selects floor surface heat flux as target parameter to compare model results with experimental values. The experimental values can be calculated by testing floor surface temperature and air temperature above the surface. Heat flux can be calculated by Eqs. (31)–(33) [21–23]. The calculation method of AUST can refer to Ref. [13]. qtot = qcov + qrad

(31)

qcov = 0.314(Tair − T1 )

1.25

−8

qrad = 5 × 10

2 (AUST + Tk ) + (T1 + Tk )

(32)

2

[(AUST + Tk )

+ (T1 + Tk )] (AUST − T1 )

(33)

3.2. Boundary conditions and validation parameter of experiment 3.3. Validation of steady heat transfer conditions Boundary conditions of experiment are as follows: physical parameters of every concrete slab layer such as density, heat capacity, thermal conductivity and so on. Physical parameters of pipes

There is a significant regenerative effect of concrete radiant cooling slab, which causes slow changes in parameter values during

336


Table 2 The maximum standard deviation of single measurement.

Table 6 Comparisons of RC model with simulation results in literatures [2,14].

Variations

The maximum standard deviation (◦ C)

Heat flux of upper slab surface (W/m2 )

Simulation

RC model

Error (%)

Chilled water temperature Room air temperature Concrete slab temperature Inner wall temperature

0.05 0.05 0.1 0.5

Condition 5 [2] Condition 6 [14] Condition 7 [14]

54.5 37.65 43.94

55.2 37.65 43.93

1.28 0.00 0.02

slab. Dynamic thermal response in the slab is complex and slow for the strong heat storage capacity of concrete slab. In this paper, unsteady conditions with changes in water temperature and the indoor heat gain are adopted to validate the RC model. Temperature and heat gain disturbance are decomposed into the form of triangular waves [19]. Step changes and sinusoidal variation are as input for supply water temperature and heat gain and the corresponding experiments are conducted to test the dynamic thermal response performance of concrete radiant cooling slab. Experimental results are compared with the calculated values by RC model. In order to reflect data consistency accurately and comprehensively by different methods, Bland–Altman method is used in this paper to analyze the consistency between simulation and experimental data.

heat transfer process. It is important to determine when the airconditioning system has reached the steady state. During the test, steady conditions can be reached when all the following temperature variations meet their standard deviation, respectively, and last for at least 1 h, as shown in Table 2. It is in the steady condition that the test data is recognized to be real and effective in subsequent calculations. Steady experimental conditions are shown in Table 3. Heat fluxes of upper concrete slab surface calculated by RC model are compared with experimental results and simulated values conducted by FEM method in Matlab PDE Toolbox, as shown in Table 4. It can be seen from Table 4, that heat flux differences of upper slab surface between RC model calculations and experimental values are less than 0.73 W/m2 , with relative errors within 1.5%. Yet the relative errors of FEM model results are within 4% compared with experimental values. The RC model is also compared with other models proposed in literatures [2,14]. The comparison conditions are shown in Table 5 and the results are shown in Table 6. Compared with the numerical value in Ref. [2], the error of RC model is within 2%, as is shown in Table 6. Particularly, the results of RC model and numerical model in Ref. [14] are in remarkably close agreement. RC model has a smaller error compared with the simplified calculation model in Ref. [13]. Therefore, RC model has better accuracy and applicability when dealing with the steadystate heat transfer problems.

3.4.1. Step changes in water temperature During the experiment, supply water temperature steps from 16 ◦ C to 18 ◦ C, and then from 18 ◦ C to 16 ◦ C by step. Polynomial functions fitted by experimental data are as model input parameters for the reason that the three nodes temperatures (mean water temperature, upper and lower surface air boundary temperatures) are changing with time in unsteady conditions. Heat fluxes of concrete slab surface are calculated by RC model in water temperature stepping up and stepping down conditions, and comparison and consistency analysis are done with experimental data, as shown in Figs. 8 and 9. Bland–Altman method is used to analyze the consistency between simulation and experimental data.

3.4. Validation of unsteady heat transfer conditions

3.4.2. Sinusoidal variation in water temperature During the experiment, water temperature is adjusted to change according to the sine law and the fluctuation range is 17 ± 1 ◦ C

In unsteady heat transfer conditions, regenerative process is accompanied by heat transfer process in concrete radiant cooling Table 3 Steady experimental conditions.

◦

Supply water temperature ( C) Return water temperature (◦ C) Mean water temperature (◦ C) Temperature of air at height 1.1 m (◦ C)

Condition 1

Condition 2

Condition 3

Condition 4

16 17 16.5 27.4

18 19 18.5 28.8

16.2 17.1 16.65 30.0

16.2 16.8 16.5 27.1

Table 4 Comparisons of upper concrete slab surface heat fluxes calculated by RC model with experimental values and FEM results. Experiment (W/m2 )

Conditions

RC model

FEM model 2

Condition 1 Condition 2 Condition 3 Condition 4

42.4 41 51.9 41.5

Calculation (W/m )

Error (%)

Simulation (W/m2 )

Error (%)

42.97 40.61 52.63 41.79

1.34 0.95 1.41 0.70

43 42.6 53.1 41.6

1.42 3.90 2.31 0.24

Table 5 Model conditions in literatures [2,14]. Condition 5 [2] ◦

Mean water temperature ( C) Mean air temperature (◦ C) Total heat transfer coefficient of upper slab surface (W/m2 K) Total heat transfer coefficient of lower slab surface (W/m2 K)

40 21 10.6 100,000

Condition 6 [14]

Condition 7 [14]

17 26 10 10

17 26 10 10

Condition 6 is the concrete core cooling slab condition while condition 7 is the capillary cooling system condition in literature [14].


Fig. 8. Experimental results and calculated values of heat flux in water temperature stepping up condition.

with the period of 4 h. Comparison and consistency analysis are done with experimental data, as shown in Fig. 10. Bland–Altman method is used to analyze the consistency between simulation and experimental data. 3.4.3. Step changes in heat gain In this case, supply water temperature is kept constant at 16.2 ◦ C, water flow rate is 0.65 m3 /h. Heat gain steps up from 360 W to 480 W, and then steps down from 480 W to 360 W. Slab surface heat fluxes are calculated by RC model in heat gain stepping up and stepping down conditions, and comparison and consistency analysis are done with experimental data, as shown in Figs. 11 and 12. Bland–Altman method is used to analyze the consistency between simulation and experimental data.

Fig. 9. Experimental results and calculated values of heat flux in water temperature stepping down condition.

Table 7 shows the errors of heat fluxes calculated by RC model compared with experimental results in 6 unsteady conditions. The maximum mean absolute error of RC model compared with experiment is 3.21 W/m2 and the maximum mean relative error is 6.79% among the 6 conditions. As shown in Table 8, the maximum mean standard deviation between RC model values and experimental results is 1.14 W/ m2 . Most values are located within the confidence interval and good consistency was verified between values calculated by RC model and experimental results.

Table 7 Error analyses of RC model values and experimental results in unsteady conditions. Unsteady conditions

3.4.4. Sinusoidal variation in heat gain In this case, heat gain is adjusted to change according to the sine law and the fluctuation range is 350 ± 150 W with the period of 4 h. Supply water temperature is kept constant at 17 ◦ C, and water flow rate is 0.39 m3 /h. Comparison and consistency analysis are done with experimental data, as shown in Fig. 13. Bland–Altman method is used to analyze the consistency between simulation and experimental data.

337

Stepping up in water temperature Stepping down in water temperature Sinusoidal variation in water temperature Stepping up in heat gain Stepping down in heat gain Sinusoidal variation in heat gain

Heat flux (W/m2 ) Mean absolute errors

Mean relative errors (%)

1.10 1.12 1.67 1.63 2.46 3.21

2.59 2.78 3.65 3.38 5.50 6.79

338


Fig. 10. Experimental results and calculated values of heat flux in water temperature sinusoidal variation condition.

4. Utilizations of RC model to different pipe spacing and buried depths in steady conditions In the actual radiation cooling engineering, plastic pipes with the diameters of DN16, DN20 and DN25 are commonly used and their buried depths are between 50 mm and 100 mm in the slab. The pipe spacing is usually from 100 mm to 500 mm [24]. Due to limited conditions, it is difficult to carry out experiments to test the full operating conditions. So FEM model is used to verify the utilizations

Fig. 11. Experimental results and calculated values of heat flux in heat gain stepping up condition.

of RC model to different pipe spacing and buried depths in steady conditions. The results calculated by RC model are compared with numerical values by FEM models, as shown in Tables 9 and 10. Input parameters of both models are as follows: mean water temperature is 17 ◦ C, room air temperature is 26 ◦ C and the total heat flux of concrete slab surface is 10 W/(m2 K). Table 9 shows the relative errors of upper slab surface heat flux calculated by RC model compared with FEM model when pipe spacing is constant. Most of the errors are less than 5% for pipe

Table 8 Consistency verifications of RC model values and experimental results in unsteady conditions. Unsteady conditions

Average differences between calculated and experimental values (W/m2 )

Average standard deviations between calculated and experimental values (W/m2 )

The 95% confidence interval (W/m2 )

Outside the 95% confidence interval (%)

Stepping up in water temperature Stepping down in water temperature Sinusoidal variation in water temperature Stepping up in heat gain Stepping down in heat gain Sinusoidal variation in heat gain

1.10 1.12 1.67 1.64 2.46 3.21

1.14 0.68 0.71 0.49 1.03 0.83

(−1.87, 3.49) (−0.23, 2.47) (0.27, 3.07) (0.68, 2.60) (0.45, 4.48) (1.57, 4.85)

5.15 2.55 3.33 2.83 0.53 0.42


339

Fig. 12. Experimental results and calculated values of heat flux in heat gain stepping down condition.

diameters and buried depths commonly used in engineering. In such circumstance when pipe buried depth is constant, as shown in Table 10, relative errors of upper slab surface heat flux calculated by RC model compared with FEM model is within 10% for pipe diameters and spacing commonly used in engineering. Thus, RC model has an error less than 10% compared with FEM model when applied in practical engineering in common conditions present in Tables 9 and 10. However, the validations of RC model in unsteady conditions are not given in this paper due to limited conditions, which need to be done in further studies.

Fig. 13. Experimental results and calculated values of heat flux in heat gain sinusoidal variation condition.

5. Conclusions This paper analyzes the heat transfer performance of radiant cooling system by model analysis and experiments, the conclusions can be summarized as: (1) Unsteady heat transfer model of concrete radiant cooling slab based on core temperature layer concept and reaction coefficient method can better describe the two-dimensional dynamic heat transfer process.

Table 9 Comparison of RC model with FEM model on different pipe buried depths in steady conditions (pipe spacing = 200 mm). Pipe buried depth (mm)

Upper slab surface heat flux (W/m2 ) Pipe diameter (mm)

Pipe diameter (mm)

16

25 50 80 100 120

Pipe diameter (mm)

20

25

FEM model

RC model

Error (%)

FEM model

RC model

Error (%)

FEM mode

RC model

Error (%)

57.93 50.14 44.17 40.57 38.89

56.31 51.09 45.95 43.07 40.53

2.80 1.89 4.03 6.17 4.22

55.90 51.35 45.31 42.59 40.00

58.50 51.77 46.54 43.61 41.03

4.64 0.82 2.71 2.39 2.57

59.25 53.53 48.45 44.54 41.86

58.12 52.66 47.31 44.31 41.67

1.91 1.62 2.35 0.51 0.47

340


Table 10 Comparison of RC model with FEM model on different pipe buried depths in steady conditions (pipe buried depth = 50 mm). Pipe spacing (mm)

Upper slab surface heat flux (W/m2 ) Pipe diameter (mm)

Pipe diameter (mm)

16

100 200 300 400 500

Pipe diameter (mm)

20

25

FEM model

RC model

Error (%)

FEM model

RC model

Error (%)

FEM model

RC model

Error (%)

66.72 50.14 40.18 35.96 37.28

61.22 51.09 43.83 38.38 34.13

8.25 1.89 9.10 6.72 8.46

68.95 51.35 42.23 35.29 33.15

62.21 51.77 44.33 38.76 34.44

9.78 0.82 9.83 3.89 3.89

66.72 53.53 43.43 42.49 38.70

63.49 52.66 44.98 39.26 34.83

4.83 1.62 3.57 7.60 10.00

(2) In four different steady conditions, the relative error of calculated values by RC model which is developed in this paper keeps within 1.5% compared with experimental results. Compared with numerical results in Refs. [2,14], the relative error of RC model is less than 2%. (3) In six different unsteady conditions presented in this paper, the relative error of calculated values by RC model is within 7% compared with experimental results. The calculated values agree well with the experimental results. Acknowledgement The research described in this paper is financially supported by National Natural Science Foundation of China (No. 51178298). References [1] K. Fort, Floor Heating and Hypocaust Stuttgart, Transsolar Energietechnik GmbH, Germany, 1999. [2] R. Holopainen, P. Tuomaala, J. Piippo, Uneven gridding of thermal nodal networks in floor heating simulations, Energy and Buildings 39 (10) (2007) 1107–1114. [3] Jan Babiak, Maria Minarova, Bjarne W. Olesen, What is the effective thickness of a thermally activated concrete slab, in: Proceedings of Clima 2007, Well Being Indoors, 2007. [4] D.G. Stephonson, G.P. Mitalas, Calculation of heat conduction transfer functions for multi-layer slabs, ASHRAE Transaction 77 (2) (1971) 117–126. [5] TRNSYS16, Transient System Simulation Program, University of Wisconsin Madison, USA, 2006. [6] TRNSYS17, Transient System Simulation Program, University of Wisconsin Madison, USA, 2010. [7] M. Koschenz, B. Lehmann, Thermoaktive Bauteilsysteme tabs, EMPA Energiesysteme/Haustechnik, Duebendorf, Switzerland, 2000. [8] T. Weber, G. Jóhannesson, An optimized RC-network for thermally activated building components, Building and Environment 40 (1) (2005) 1–14.

[9] T. Weber, G. Jóhannesson, M. Koschenz, B. Lehmann, T. Baumgartner, Validation of a FEM-program (frequency-domain) and a simplified RC-model (time-domain) for thermally activated building component systems (TABS) using measurement data, Energy and Buildings 37 (7) (2005) 707–724. [10] R. Strand, C. Pederson, Modeling radiant systems in an integrated heat balance based energy simulation program, ASHRAE Transactions 108 (2) (2002) 1–9. [11] EnergyPlus, The Reference to EnergyPlus Calculations, EnergyPlus, California, USA, 2010. [12] D.O. Rijksen, C.J. Wisse, A.W.M. van Schijndel, Reducing peak requirements for cooling by using thermally activated building systems, Energy and Buildings 42 (3) (2010) 298–304. [13] L. Zhang, X. Liu, Y. Jiang, Simplified calculation for cooling/heating capacity, surface temperature distribution of radiant floor, Energy and Buildings 55 (2012) 397–404. [14] K. Liu, Z. Tian, C. Zhang, Y. Ding, W. Wang, Establishment and validation of modified star-type RC model for concrete core cooling slab, Energy and Buildings 43 (9) (2011) 2378–2384. [15] M. Koschenz, V. Dorer, Interaction of an air system with concrete core conditioning, Energy and Buildings 30 (2) (1999) 139–145. ´ ´ [16] T. Cholewa, M. Rosinski, Z. Spik, M.R. Dudzinska, A. Siuta-Olcha, On the heat transfer coefficients between heated/cooled radiant floor and room, Energy and Buildings 66 (2013) 599–606. [17] G. Dai, Heat Transfer, Higher Education Press, Beijing, 1999. [18] Q. Yan, Q. Zhao, Heat Process of Buildings, China Architecture & Building Press, Beijing, 1986. [19] P. Chen, S. Cao, Computational Theory and Methods of Air Conditioning Load, Tongji University Press, Shanghai, 1987. [20] J. He, H. Liu, X. Fang, D.F. Wang, The study for the number of reaction coefficients in method of reaction coefficient in unsteady heat transmission of wall, Journal of Applied Science 17 (1) (1999) 92–97. [21] T.C. Min, L.F. Schutrum, G.V. Parmelee, J.D. Vouris, Natural convection and radiation in a panel heated room, ASHRAE Transactions 62 (1) (1956) 337–358. [22] J. Jeong, S.A. Mumma, Practical cooling capacity estimation model for a suspended metal ceiling radiant cooling panel, Building and Environment 42 (9) (2007) 3176–3185. [23] Í.B. Kilkis, S.S. Sager, M. Uludag, A simplified model for radiant heating and cooling panels, Simulation Practice and Theory 2 (2) (1994) 61–76. [24] JGJ 142-2012, Technical Specification for Radiant Heating and Cooling, China Architecture & Building Press, Beijing, 2012.