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Citation: Bobula, M.; Michalak, P.;

Wołoszyn, J. Inﬂuence of the TABS

Material, Design, and Operating

Factors on an Ofﬁce Room’s Thermal

Performance. Energies 2024,17, 1951.

https://doi.org/10.3390/en17081951

Academic Editor: Boris Igor Palella

Received: 18 March 2024

Revised: 16 April 2024

Accepted: 17 April 2024

Published: 19 April 2024

Copyright: © 2024 by the authors.

Licensee MDPI, Basel, Switzerland.

This article is an open access article

distributed under the terms and

conditions of the Creative Commons

Attribution (CC BY) license (https://

creativecommons.org/licenses/by/

4.0/).

energies

Article

Inﬂuence of the TABS Material, Design, and Operating Factors

on an Ofﬁce Room’s Thermal Performance

Mikołaj Bobula †, Piotr Michalak †and Jerzy Wołoszyn *,†

AGH University of Krakow, Faculty of Mechanical Engineering and Robotics, Department of Power Systems and

Environmental Protection Facilities, al. A. Mickiewicza 30, 30-059 Krakow, Poland; mbobula@agh.edu.pl (M.B.);

pmichal@agh.edu.pl (P.M.)

*Correspondence: jwoloszy@agh.edu.pl

†These authors contributed equally to this work.

Abstract: Reducing energy consumption in residential and commercial buildings is an important

research topic. Thermally activated building systems are a promising technology for signiﬁcantly

reducing energy consumption. The high thermal inertia, large surfaces, and radiative nature are

advantages of these systems, but, on the other hand, this makes the system control and design

complex. A transient simulation is also required to address the dynamic behavior of the system. The

inﬂuence of 19 factors (material, design, and operating parameters) on the air temperature and mean

radiant temperature inside the room as well as the required cooling equipment power were analyzed

to better understand the system. The screening experiment was conducted using the random balance

design method, and measurement data were used to validate the resistance–capacitance model. The

analysis was performed using the Plackett–Burman design and a design with randomly selected

points from a full factorial experiment. The results show that internal heat gains and the inlet water

temperature have a signiﬁcant inﬂuence on the system, and the inﬂuence of the screed’s properties is

insigniﬁcant compared to other parameters. It should be borne in mind that the obtained results and

conclusions are valid for the assumed range of factors’ variability.

Keywords: TABS; concrete core activation; resistance–capacitance model; sensitivity analysis; validation;

measurement

1. Introduction

Thermally activated building systems (TABS) are commonly used for heating and cool-

ing multi-story buildings. They cover a wide range of various constructions [

1

], including

one with pipes embedded in the concrete core of the slab. This type of TABS was intro-

duced in Switzerland in the early 1990s and, since then, it has been installed in numerous

buildings around the world [

2

]. TABS are known in German as Betonkerntemperierung

(BKT) systems, or in English as Concrete Core Activation (CCA) or Concrete Core Cooling

(CCC) [

3

–

5

]. During periods of high thermal load, heat is transferred through the ﬂoor

and ceiling and stored in concrete; it can then be removed by water ﬂowing in pipes when

the occupants are absent [

6

]. Spreading the removal of heat over a longer period results in

lower peak power requirements for cooling systems, which can lead to energy savings up

to 50% [7].

The high thermal inertia, large surfaces, and radiative nature are advantages of TABS;

however, on the other hand, this makes the system control and design complex. A transient

simulation is also required to address the dynamic behavior of the system [

8

,

9

]. In [

10

],

thermal comfort issues in a room located in an ofﬁce building with TABS were experi-

mentally investigated. Measurements performed during the everyday operation of the

room revealed very low differences in indoor air temperature (between 22.5

◦

C and 23.1

◦

C

during the working days) in its vertical proﬁle and good thermal conditions provided by

TABS cooperating with a balanced ventilation system. The average temperature of the

Energies 2024,17, 1951. https://doi.org/10.3390/en17081951 https://www.mdpi.com/journal/energies

Energies 2024,17, 1951 2 of 22

ﬂoor’s surface ranged from 20.6

◦

C to 26.2

◦

C. However, this study was not oriented to

dynamic thermal analysis of TABS. Such works are very rare because multi-ﬂoor buildings

with thermally activated slabs exchanging heat between upper and bottom adjacent zones

are, in general, large commercial facilities. Hence, they are normally very difﬁcult to access,

and measurements are very troublesome for occupants during their normal operation. That

is why most of the experimental studies on TABS concern control-oriented applications in

terms of energy use [

11

–

14

] or were performed under laboratory conditions in simple test

rooms [15,16].

One way of estimating system performance is through computational ﬂuid dynamics

(CFD). It can provide detailed information about the system, but it is time-consuming

and therefore impractical [

17

,

18

]. Simpliﬁed methods based on ordinary differential equa-

tions can be an alternative to detailed simulations because they require less time to solve.

Nageler et al. [18]

compared the results of CFD simulations and simpliﬁed physical models

with measurement data. They found that simpliﬁed models can provide satisfying results.

Shariﬁ et al.

[9]

used a simpliﬁed resistance–capacitance model of TABS to develop the con-

trol algorithm for optimal load splitting between TABS and the secondary cooling system.

The utilization of the simpliﬁed model allowed authors to run the optimization algorithm

for a one-year period (8760 h long time steps) within 2 h, which would not be possible with

detailed simulations. To make the calculations easier, the method for simulating a room

with TABS was standardized in ISO 11855-4 [

19

]. This standard proposes a one-dimensional

simpliﬁed resistance–capacitance mathematical model of the room and slab, which is solved

with the ﬁnite difference method (FDM). The method has been utilized by Behrendt

[20]

to develop a computer program for simulating TABS. Catalano

[6]

has modiﬁed the ISO

11855-4 method to deal with unrealistic behavior of the system in the case of high thermal

load and low cooling power, which resulted in internal temperature exceeding 40

◦

C. The

author has proposed a correction factor that was calculated based on air temperature inside

the room and was used to modify heat gain value. However, simpliﬁed methods still

require detailed information on room design as well as on internal and external conditions.

Therefore, identifying parameters with the highest inﬂuence on the performance of TABS

would simplify the calculations and result in a better understanding of the system.

Due to the complexity of TABS, many studies have been conducted to establish

the inﬂuence of various parameters on conditions in the TABS-equipped building and

control strategy. Lu et al.

[21]

examined building energy ﬂexibility under the inﬂuence of

various parameters that included thermal transmittance of the external wall (U-value), total

structural thermal mass, internal heat gains, and methods of cooling. They found that the U-

value of the external wall had a slight effect on cooling performance when compared to other

parameters. It should be kept in mind that this is valid for the assumed input parameters’

variability range. Saelens et al.

[22]

evaluated the inﬂuence of occupants’ behavior on

internal heat gains. They concluded that changes in various aspects of occupants’ behavior,

such as their mobility, the probability that they appear in the ofﬁce during the day, or

manual operation of the lights and shading devices, have a signiﬁcant inﬂuence on cooling

demand and thermal comfort in the ofﬁce. Their results show that switching off the lights

reduces cooling demand by 8% and shortens the time that the temperature is outside the

thermal comfort zone by 15%. Rijksen et al.

[23]

examined the inﬂuence of internal heat

gains and window area on the peak-shaving performance of cooling in an ofﬁce building

with TABS and without it. The authors discovered that TABS combined with a smaller area

of the windows resulted in a reduction of required cooling power by 50% when compared

to a system without energy buffering. Samuel et al.

[24]

experimentally assessed the effect

of parameters on thermal comfort in a room with TABS installed. The authors concluded

that increasing the number of cooling surfaces had a positive effect on thermal comfort

and that, with all surfaces cooled (ceiling, ﬂoor, and four walls), thermal comfort was

maintained throughout the day. They also found out that the use of ventilation could

result in rapid temperature changes as external air was mixed with internal.

Ning et al. [25]

examined the inﬂuence of geometric and thermal parameters on the system response

Energies 2024,17, 1951 3 of 22

time. They found that concrete thickness and pipe spacing have a signiﬁcant impact on

TABS response time.

Samuel et al. [26]

performed a sensitivity analysis based on CFD

simulations to investigate the inﬂuence of pipe diameter, pipe thermal conductivity, and

slab thickness on thermal comfort in a room with TABS. They found that increasing pipe

thermal conductivity from 0.14 to 1.4

W

m·K

and increasing its inner diameter from 9 to 17 mm

reduced the operative temperature by 2.8

◦

C and 1.8

◦

C, respectively. However, increasing

the ﬂoor and ceiling slab thickness from 0.1 to 0.2 m reduced the operative temperature

by only 0.3

◦

C.

Chandrashekar and Kumar [27]

experimentally studied the inﬂuence of

various ﬂoor covering materials on TABS performance. They found that using a granite

ﬂoor resulted in lowering air temperature in the room by 1.5

◦

C and reduced cooling load

by 10% compared to the vinyl ﬂoor, which had lower thermal conductivity than granite.

Rakesh et al.

[28]

experimentally evaluated the inﬂuence of inlet water velocity on the

temperature in a room with TABS. They found that increasing the water velocity from 0.35

to 1

m

s

reduced internal air temperature by 1.5

◦

C; however, further a increase in the water

velocity to 1.5

m

s

reduced air temperature by only 0.2

◦

C. The review presented in this

section has been summarized in Table 1.

Table 1. Literature review summary.

Reference Method 1Examined Parameters Findings

[21] S

external wall thermal transmittance,

total thermal mass,

internal heat gains,

cooling method

little effect of thermal transmittance of the wall

[22] S internal heat gains (occupants’ behavior)

signiﬁcant inﬂuence on internal conditions,

encouraging occupants to switch off lights may

reduce cooling demand by 8%

[23] E, S window area,

internal heat gains

internal heat gains have a signiﬁcant effect on

cooling power

[24] E number of cooled surfaces,

use of other cooling system

cooling all surfaces allows for maintenance of

thermal comfort in the room throughout the day

(tropical climate)

[25] S

concrete thickness,

pipe spacing,

concrete type,

pipe diameter,

water ﬂow pattern,

water temperature,

room operating temperature

concrete thickness and pipe spacing have the most

signiﬁcant inﬂuence on TABS response time

[26] S

pipe diameter,

pipe thermal conductivity,

slab thickness

increasing pipes’ thermal conductivity and

diameter is more inﬂuential on operative

temperature than increasing slab thickness

[27] E ﬂoor covering properties

granite ﬂoor with higher thermal conductivity

reduces cooling load by 10% and air temperature

in the room by 1.5 ◦C

[28] E water inlet velocity

increasing the water velocity from 0.35 to 1 m

s

reduced internal air temperature by 1.5 ◦C, but

further velocity increase has little effect on the

temperature

1E—Experiment, S—Simulation

As presented above, materials, design, and operating parameters have important

effects on the performance of TABS. However, the inﬂuence of these parameters has not

been widely discussed in terms of the internal air and mean radiant temperature, which

Energies 2024,17, 1951 4 of 22

is very important in terms of thermal comfort. Moreover, the researchers examined the

inﬂuence of a limited number of factors at the same time (all factors at a time analysis).

Therefore, the possibility of comparing the inﬂuence of the parameters is limited. To ﬁll

those gaps, we conducted a sensitivity analysis based on a random balance design and

found which parameters signiﬁcantly affect the temperature inside a room with TABS and

required cooling power. We examined the inﬂuence of 19 parameters, including the slab

material properties, properties of walls, circuit operating parameters, and heat gains. The

calculations were performed using the modiﬁed method presented in ISO 11855-4 [

19

] and

validated using measurement data.

2. Details of Research Objective

The experimental part of this study was performed in a single ofﬁce room in the

passive ofﬁce building (Figure 1) located in Katowice in south Poland. This facility has been

described recently [

10

,

29

]. It has a total and usable area of 8100 m

2

and 7500 m

2

, respectively.

Figure 1. General view of the building from the south.

Ceiling slabs (Figure 2b) and the basement ﬂoor were made from 30 cm thick reinforced

concrete with embedded modules and with polymer pipes with a total area of 4500 m

2

and

1870 m

2

, respectively. This is the main heating and cooling system in the building. It is

supported by ﬁve air handling units with heating and cooling coils, creating a complete

heating, ventilation, and air conditioning (HVAC) system. Four units have a common

intake and exhaust collector on the roof, and they supply fresh air to ofﬁce and social rooms.

The ﬁfth unit supplies fresh air to sanitary facilities and toilets and is equipped with a

separate air intake and exhaust on the roof. The HVAC system is supplied with water from

the set of six water/water heat pumps with a total heating and cooling capacity of 244 kW

and 187 kW, respectively. Additionally, 10 vacuum solar collectors are used to support the

heating and tap water systems. As a peak cooling source, two chillers can be used.

The ofﬁce rooms are located around the outer perimeter of the building, facilitating the

use of daylight. The large area of triple-glazed external windows (U = 0.7

W

m2·K

) increases

the share of solar energy in the thermal balance of the building. External blinds, on the other

hand, make it possible to reduce excessive solar gains during the summer. The corridors

are located on the inner perimeter of the building and, thanks to the glazing in its central

part, they are naturally illuminated. Photovoltaic modules with a total power of 107 kW

p

mounted on the roof and on the facade, as well as the three trackers in front of the building,

reduce electricity consumption in the building. For effective energy management and data

analysis, a building management system (BMS) is used. An ofﬁce room (Figure 2a), located

on the west side of the second ﬂoor of the building was chosen for the research. It has a

ﬂoor area of 46.11 m2and a height of 3.10 m. During measurements, it was not occupied.

Energies 2024,17, 1951 5 of 22

(a) (b)

Figure 2. Experimental room: (a) General view; (b) Cross-section of the ﬂoor and the ceiling slabs

(dimensions in mm, 1-carpet, 2-cement screed, 3-reinforced concrete, 4-gypsum plaster).

3. Research Strategy

A sensitivity analysis for 19 (material, design, and operating) parameters for the

real TABS is impossible due to the high cost and the long time needed to perform each

experiment. One solution is to perform a series of simulation experiments. There are basic

CFD-based calculations that are detailed and accurate but time-consuming, so reduced-

order methods for the calculation of such complex systems are required. However, even

the use of supercomputers and distributed computing to perform sensitivity studies for

19 parameters is time-consuming. Therefore, there is a need to utilize an efﬁcient research

strategy by using effective and time-reducing methods.

3.1. Research Algorithm

According to the research algorithm outlined in Figure 3, the initial phase of the

study involved meticulous identiﬁcation of design speciﬁcations, material properties, and

operational parameters of the building room with TABS (Section 2). Simultaneously, a

decision was made to adopt a reduced computational model, aligning with the established

methodology outlined in ISO 11855-4 [19] (Section 5). The assumed computational model

of the ofﬁce room with TABS was veriﬁed and validated (Section 5.2) using measurement

results (Section 4). The mean air temperature and mean radiant temperatures of the room

were compared in the validation analysis. For a full factorial design experiment, there was

a need to conduct 2

19

= 524,288 computational experiments. It is still impossible to conduct

the full experiment due to time-consuming calculations.

Figure 3. The research algorithm.

It was decided to conduct research using the random balance design method, and

experiments were generated by the use of the Plackett–Burman (PB) design. The PB designs

are mainly used for screening research. This design is efﬁcient, but the main effects are, in

general, heavily confounded with two-factor interactions. For this reason, we decided to

Energies 2024,17, 1951 6 of 22

extend this research and conduct 512 randomly selected experiments from a full factorial

design. This allowed the results to be conﬁrmed or rejected. Consequently, we assess

the impact of 19 factors on ﬁve output parameters. This screening experiment has been

performed using scripts written in Python 3.10, and the pyDOE3 version 1.0.1 [

30

] library

has been used to prepare the design of the experiments.

3.2. Assumptions

Using data from diverse sources, an extensive array of cases within the TABS across

varied geographical locations can be analyzed. The ranges of variability (Table 2) in the

parameters under analysis were determined by synthesizing information based on existing

literature. Consequently, we assumed parameters such as a cooling surface area (

P

1

=AF

)

equal to 50% and 100% of the total ﬂoor area on low and high levels, respectively, and the

material parameters’ variability values. The thermal conductivity of the concrete layer

(

P

2

=λcon

) and pipe spacing (

P

13

=L

) are limited by the assumptions of the method

described in [

19

]; the thermal conductivity, the speciﬁc heat, the density of the screed layer

(

P

3

=λsc

,

P

5

=csc

,

P

7

=ρsc

), the speciﬁc heat, and the density of the concrete layer

(P4=ccon ,P6=ρcon) are based on [31].

The construction of the slab is based on Figure 2b, and properties of the 2nd layer

(screed) and the 3rd layer (concrete) have been considered as parameters in the sensitivity

analysis. The water pipes were always located in the middle of the concrete layer. The 1st

layer (carpet) has been ignored due to its negligible thermal capacity; hence, it has not been

considered a slab layer in the numerical model (there is no node representing the carpet

layer). However, the inﬂuence of the thermal resistance of the carpet has been examined

by introducing additional thermal resistance (

P

11

=RAddF

) at the ﬂoor. We have assumed

that, at the high level, the value of

RAddF

is equal to the thermal resistance of the carpet in

the considered room, and the low level of the parameter represents the situation where the

carpet is not present and therefore the additional thermal resistance is equal to 0

W

m2·K

. The

properties of the 4th layer (plaster gypsum) have been considered constant in the sensitivity

analysis, and they are presented in Table 3.

The internal building walls consist of two layers of plasterboard and a layer of min-

eral wool between them. Their structure is the same as in the building for which the

measurements were made. The high level for the internal wall thickness parameter

P

10

=dIW

decomposed into layers equaled 25 mm for plasterboard and 200 mm for

the mineral wool layer (

dIW =

25

mm +

200

mm +

25

mm =

250

mm

) and, for the low

value,

dIW =

8

mm +

50

mm +

8

mm =

66

mm

. To calculate internal walls’ thermal re-

sistance and speciﬁc heat capacity, which are the parameters in the mathematical model,

we have evaluated the substitute values of walls’ thermal conductivity (

λIW

) and speciﬁc

thermal capacity (CIW ) according to the following equations:

λIW =∑di

∑di

λi

(1)

CIW =∑di·ci·ρi

∑di

(2)

where

di

is thickness of the

i

th layer [m],

λi

is the thermal conductivity of the

i

th layer

[

W

m·K

],

ci

is the speciﬁc heat of the

i

th layer [

J

kg·K

], and

ρ

is the density of the

i

th layer [

kg

m3

].

The thermal properties of the walls’ materials were determined according to [

31

]. The

values of λIW and CIW are presented in Table 3.

The values of water ﬂow (

P

12

=˙

mH

) and inlet water temperature (

P

19

=θWaterIn

)

have been estimated according to measurement results. The values of the parameters at

low and high levels denote the minimum and maximum values provided by the building

management system (BMG), respectively.

Thermal transmittance of window

P

15

=UW

variability is based on values provided

by [

32

]. We have assumed that the values of the window area (

P

14

=AW

) equaled 30%

Energies 2024,17, 1951 7 of 22

and 70% of the external wall area at the low and high levels, respectively. Additionally, we

have assumed that the area of glazing (

Ag

) used in the process of calculating solar heat

gains equals 90% of the total window area (AW).

Thermal transmittance of the opaque part of the external wall (

P

16

=UEW

) is calcu-

lated according to [

33

]. The wall consists of four layers: (i) plaster, (ii) styrofoam with a

thickness of 300 mm, (iii) hollow bricks, and (iv) plaster. The values of the parameter were

determined by changing the thickness of the styrofoam layer, and for low and high values,

the thickness was equal to 50 mm and 300 mm, respectively.

The values of primary air gains (

P

17

=QPrim Air

) have been based on the results

of measurements conducted in the considered room. The ventilation system provided

constant airﬂow of 95

m3

h

between 3 a.m. and 7 p.m.; however, the difference between

supply and exhaust air temperature was greatest between 3 a.m. and 9 a.m. After 9 a.m.,

the difference between the supply and exhaust air temperature was reduced signiﬁcantly;

hence, the heat gain was considered to be 0 W despite the ventilation system still working.

To simplify the model, we have assumed that the ventilation system provides constant

airﬂow of 240

m3

h

, which is the maximum value that can be provided by the HVAC system.

To estimate the value of the parameters, we have assumed that the difference between

supply and exhaust air temperature equals

−

4.5

◦

C and 0

◦

C at low and high levels,

respectively, which results in the value of QPri mAir equal to −350 W and 0 W, respectively.

Internal heat gains (

P

18

=QInt

) originate from occupants, ofﬁce equipment, and

lighting in the room. In a sensitivity analysis, we have assumed that the occupants are

present and that they use the equipment between 7 a.m. and 5 p.m.; otherwise, the internal

heat gains are equal to 0 W. We have determined that the value of the parameter at a low

level, assuming that the room is not used, is 0 W. For the high level, the value of internal

heat gains was estimated according to [

34

]. To carry out the calculations, we have assumed

that ﬁve people work in the room and that each person uses a personal computer with two

monitors; additionally, there is one printer in the room. The internal heat gains estimated

according to those assumptions are equal to 1500 W. We have assumed that half of the

internal heat gain is transferred through radiation (

QInt Rad

) and the other half is through

convection (

QIntConv

) [

22

]. All parameters considered in the sensitivity analysis along with

their variability ranges are presented in Table 2.

Table 2. Variability ranges of the input parameters.

No Parameter Symbol Value Unit

Low High

P1 Cooling surface area AF23 46 m2

P2 Concrete thermal conductivity λcon 1.15 2 W/(m·K)

P3 Screed thermal conductivity λsc 1.4 1.8 W/(m·K)

P4 Concrete-speciﬁc heat ccon 900 1100 J/(kg·K)

P5 Screed-speciﬁc heat csc 900 1100 J/(kg·K)

P6 Concrete density ρcon 1800 2400 kg/m3

P7 Screed density ρsc 1800 2000 kg/m3

P8 Concrete layer thickness dcon 0.2 0.4 m

P9 Screed layer thickness dsc 0.005 0.03 m

P10 Internal wall thickness dIW 0.066 0.25 m

P11 Floor additional resistance RAdd F 0 0.032 m2·K/W

P12 Water ﬂow ˙

mH0.028 0.08 kg/s

P13 Pipe spacing L0.15 0.3 m

P14 Window area AW5.4 12.5 m2

P15 Thermal transmittance of window UW0.5 3.3 W/(m2·K)

P16 Thermal transmittance of external wall (opaque) UEW 0.088 0.203 W/(m2·K)

P17 Primary air heat gain QPrim Air −350 10 W

P18 Internal heat gains QI nt 01500 2W

P19 Inlet water temperature θWaterIn 16 22 ◦C

1Between 3 a.m. and 9 a.m.; otherwise 0 W. 2Between 7 a.m. and 5 p.m.; otherwise 0 W.

Energies 2024,17, 1951 8 of 22

To provide information about thermal comfort inside the room and the cooling power

necessary for maintaining constant water temperature in the circuit, we have selected the

following output parameters:

• Maximum and minimum air temperature (θMax

A,θMin

A),

• Maximum and minimum mean radiant temperature (θMax

MR ,θMin

MR ),

• Maximum power received by the circuit (QMax

Circuit).

The maximum/minimum air temperature (

θMax

A

,

θMin

A

) and maximum/minimum

mean radiant temperature (

θMax

MR

,

θMin

MR

) have been considered only between 7 a.m. and

5 p.m. when occupants are present, while power (

QMax

Circuit

) received by the circuit has been

considered throughout the day.

Other parameters that have not been considered in the sensitivity analysis but that are

necessary for calculations are presented in Table 3.

Table 3. Other system parameters.

Parameter Symbol Value Unit

Room width X5.8 m

Room length Y7.95 m

Room height Z3.1 m

Convection coefﬁcient on the ﬂoor hAF 2.37 W/(m2·K)

Convection coefﬁcient on the ceiling hAC 0.54 W/(m2·K)

Convection coefﬁcient on the internal walls hAW 1.79 W/(m2·K)

Radiant heat transfer coeff. between ﬂoor and ceiling hFC 2.79 W/(m2·K)

Radiant heat transfer coeff. between ﬂoor and internal walls hFW 2.07 W/(m2·K)

Ceiling additional resistance RAddC 0m2·K/W

Water-speciﬁc heat cw4183 J/(kg·K)

Plaster layer thickness dpl 0.005 m

Plaster thermal conductivity λpl 0.43 W/(m·K)

Plaster-speciﬁc heat cpl 1000 J/(kg·K)

Plaster density ρpl 1200 kg/m3

Internal wall thermal conductivity λIW 0.06 W/(m·K)

Substitute internal wall-speciﬁc thermal capacity CI W 321, 710 J/(m3·K)

Pipe external diameter da0.02 m

Pipe wall thickness s0.002 m

Pipe wall thermal conductivity λr0.35 W/(m·K)

Time step ∆t3600 s

Maximum iteration error allowed ξMax 0.00001 ◦C

Maximum number of iterations allowed nMax 10, 000 -

4. Details of Measurements

The measurements were carried out in a specially prepared ofﬁce room with 46.11 m

2

(Figure 2a). All sensors were placed in the room according to the layout presented in

Figure 4a,b. For the sake of clarity, supply and exhaust channels of the ventilation system

were indicated only for the ceiling.

All air and surface temperature measurements were performed using Pt100 platinum

resistance sensors. Heat ﬂux of the ceiling and the ﬂoor was measured using the HFP01

sensors of Hukseﬂux [

35

]. Global solar irradiance entering the room was measured by the

LP PYRA03 pyranometer of DeltaOhm [

36

]. At the center of the room, a tripod was placed

with a globe temperature sensor (diameter of 150 mm) at a height of 1.5 m. Based on [

37

,

38

],

it was assumed that the mean radiant temperature is the same as the measured globe

temperature. However, as noted in [

39

,

40

], this assumption may result in underestimation

of the mean radiant temperature. As the room was unoccupied during measurements, this

effect was minimized, but its precise assessment requires further analysis. Additionally,

three air temperature sensors were attached to a tripod at heights of 0.2 m, 0.6 m, and 1 m

above ﬂoor level. They were covered with low-emissivity (

ε=

0.1) coating to minimize

Energies 2024,17, 1951 9 of 22

the impact of heat transfer by radiation. The wall temperature sensors were placed 1.5 m

above the ﬂoor. All sensors are listed in Table 4. All sensors were connected directly to the

Fluke 2638A data logger [41].

(a) (b)

Figure 4. Location of the measurement equipment in the room (all dimensions in meters): (a) Floor;

(b) Ceiling.

Table 4. Measurement sensors.

Symbol Type, Class Measured Variable

QF1,QF2HFP01 Floor heat ﬂux

QC1,QC2HFP01 Ceiling heat ﬂux

TF1,TF2,TF3,TF4Pt100, class AA Floor surface temperature

TF5,TF6Pt1000, class A Floor surface temperature

TC1,TC2Pt1000, class A Ceiling surface temperature

TIWS1,TIWS2Pt1000, class A Internal wall surface temperature

Texh Pt1000, class A Exhaust ventilation air temperature

Tsup Pt1000, class A Supply ventilation air temperature

Tglobe Pt100, class A Globe temperature

T0.2 Pt100, class AA Indoor air temperature at 0.2 m above the ﬂoor

T0.6 Pt100, class AA Indoor air temperature at 0.6 m above the ﬂoor

T1.0 Pt100, class AA Indoor air temperature at 1.0 m above the ﬂoor

Isol LP PYRA03 Global solar irradiance entering the room

Accuracy classes of Pt100 and Pt1000 follow IEC 60751 [42].

According to [

43

], combined uncertainty of measured variable

x

, when using a mea-

surement sensor and a signal processing and display unit (device), is given by:

u(x) = qu2

sens +u2

dev. (3)

Both sensor and device uncertainties can be obtained from the relevant standards or

manufacturer documentation. Following IEC 60751 [

42

] for the known temperature (

T

), the

tolerances of the A and AA class sensors are given by the following relationships, respectively:

∆T= (0.15 +0.002|T|), (4)

∆T= (0.01 +0.0017|T|). (5)

At 25

◦

C for the A class sensor, we obtain from Equation (4):

usens =∆T

√3

= 0.116

◦

C.

According to the manufacturer [

41

], for a 4-wire Pt100 sensor, uncertainty

udev

= 0.024

◦

C

Energies 2024,17, 1951 10 of 22

at 25

◦

C. Hence, from Equation (3):

u(x) =

0.12

◦

C. Apart from variables listed in Table 4,

inlet water temperature and ﬂow rate were obtained from the heat meter connected to the

BMS. For the second class heat meter [

44

] the maximum permissible error (MPE), expressed

in %, is deﬁned for the ﬂow rate (

Ef

) and for the temperature difference measured by the

pair of sensors (ET) by the following relationships:

Ef= (2+0.02 ˙

mper /˙

m), (6)

ET= (0.5 +3∆T/∆Tmin), (7)

where

˙

m

means ﬂow rate,

˙

mper

is the maximum permissible ﬂow rate,

∆T

is the current

inlet/outlet temperature difference, and

∆Tmin

is the minimum temperature difference

(

∆Tmin

= 5 K in the considered case). Based on available data, it was established that

both

ET

and

Ef

do not exceed 3%. Finally, according to the manufacturer’s data, it was

assumed that the ventilation ﬂow rate was measured with an accuracy of

±

5%. The most

troublesome to assess is the accuracy of heat ﬂux sensors. Following the manufacturer’s

recommendations [

35

] in buildings physics, it varies from

±

6% in ideal conditions to

±

20% in typical on-site applications. In the presented case, the second value seems to be

reasonable in the analysis of the measurement results.

5. Computational Model Description

5.1. Mathematical Model Description

The mathematical model of the room is based on resistance–capacitance simpliﬁcation

introduced in ISO 11855-4 [

19

]. The mathematical model includes the slab, internal walls

(walls between the considered room and adjacent ones), and air ﬁlling the room, which are

divided into thermal nodes and, in each node, the heat balance is calculated according to

the following equation:

Cdθ

dt =∑Q, (8)

where

C

is the thermal capacity of the node [

J

K

], and the right-hand side of the equation is

the sum of heat transfer rates received or delivered to the considered node in [W].

The RC representations of the slab and the room are presented in Figure 5a and

Figure 5b, respectively. The heat transport in the slab is simpliﬁed by the use of the thermal

resistance method provided in ISO 11855-2 [

45

], where

Rt

is the thermal resistance between

inlet water and concrete at the pipe level. It is introduced to calculate heat transfer between

water ﬂowing in pipes and concrete. In the part of the mathematical model representing

the room, nodes representing the ﬂoor, ceiling, and internal wall surface are connected,

which represents radiant heat transfer between them. The convective heat transfer is also

present, and it is represented by connections between surface nodes and air nodes. Several

simplifying assumptions were also made:

•

The heat transport in the perpendicular direction to the ﬂoor is taken into considera-

tion,

• Water mass ﬂow in the pipes is constant throughout the simulation time,

• Radiant heat gains are distributed evenly to all surfaces,

• Conditions in rooms above and below are the same as in the considered room,

• All thermophysical properties are constant and independent of temperature.

The mathematical model is solved through the implicit scheme of the ﬁnite difference

method, and the equation used for calculating temperature in the considered node can be

written as follows:

θh

p=

HA·θh

A+HIWS ·θh

IWS +HF·θh

F+HC·θh

C+HRad ·Qh

Rad +HCo nv ·Qh

Conv +

+HCondU p ·θh

p−1+HCond Down ·θh

p+1+HInertia ·θh−1

p+HCircuit ·θh

WaterIn

HA+HIWS +HF+HC+HCondU p +HCondDown +HInertia +HCircuit

, (9)

Energies 2024,17, 1951 11 of 22

where

θh−1

p

is the temperature of the considered node in the previous time step [

◦

C],

θh

p−1

is the temperature of the previous node [

◦

C],

θh

p+1

is the temperature of the next node [

◦

C],

Qh

Rad

is total radiant heat gain [W],

Qh

Conv

is total convective heat gain [W], and

θh

Water,In

is inlet water temperature [

◦

C].

H

represents the heat transfer coefﬁcients, which depend

on the type of node that is currently being considered. An iterative method is used for

determining temperatures in the nodes, and with each iteration, the error is calculated

as follows:

ξ=∑

p|θh

p−θh′

p|, (10)

where

θh′

p

is the temperature calculated in the previous iteration. Calculations are carried

out until the iteration error

ξ

is smaller than the maximum allowed error (

ξMax

) or until the

maximum number of iterations (

nMax

) is achieved. The values of maximum iteration error

and maximum iterations allowed are presented in Table 3.

(a) (b)

Figure 5. Scheme of the model (based on [

19

]): (a) Resistance network representing the slab; (b) Resis-

tance network representing the room.

The initial and speciﬁc conditions have to be determined before calculating the heat

delivered or received from the considered room. Speciﬁc conditions can be divided into

three categories:

•

Radiant heat gains delivered to surface nodes (ﬂoor, ceiling, and internal wall surface

(Figure 5b)),

• Convective heat gains delivered to the air node,

• Heat received by the circuit from the pipe-level node.

Both radiant and convective heat gains depend on internal and external conditions and

are calculated for every time step of the simulation according to the following equations:

Qh

Rad =0.85 ·Qh

Transm +Qh

Sun +Qh

Int Rad (11)

Qh

Conv =0.15 ·Qh

Transm +Qh

Prim Air +Qh

IntConv (12)

where

Qh

Transm

is the transmission heat gain [W],

Qh

Sun

is the solar heat gain [W],

Qh

Prim Air

is

the primary air heat gain [W], and

Qh

Int Rad

and

Qh

IntConv

are internal radiant and convective

heat gains [W], respectively. The value of each term in Equations (11) and (12) has to be

known for every time step of the simulation. Primary air and internal heat gains have been

Energies 2024,17, 1951 12 of 22

determined according to measurement data and the literature (see Section 3.2). The solar

heat gains have been determined according to the following:

Qh

Sun =Isol ·Ag, (13)

where Isol is total solar irradiance entering the room [ W

m2], and Agis the area of glazing [m2].

According to Catalano

[6]

, adopting the value of transmission heat gain independent

of temperature may lead to errors. Hence, the value of the transmission heat gain has been

determined using the difference between external and internal temperatures according to

PN-EN 12831-1 [46], and it is equal to the following:

Qh

Transm = (UEW ·AEW +UW·AW)(θh

Ex −θh

A)(14)

where

UEW

is the thermal transmittance of the opaque part of the external wall [

W

m2·K

],

UW

is the thermal transmittance of the window [

W

m·K

],

AEW

is the area of the opaque

surface [m

2

],

AW

is the area of the window [m

2

], and

θh

Ex

is the external temperature

[

◦

C]. Since transmission heat gain is dependent on air temperature, it has to be calculated

with every iteration. The external temperature has been obtained from weather data. To

examine the inﬂuence of external conditions on the room, the weather data chosen for the

experiment contained outside temperatures differing from measured room temperatures.

The maximum value of the external temperature considered in the experiment was equal

to 18 ◦C.

Convective heat transfer coefﬁcients on the surfaces in the room were calculated

according to equations developed by Awbi

[47]

, presented in Table 5. It is assumed that the

difference between the surface and air temperatures (

∆T

) in each case is constant and equal

to 1.5 K, so the coefﬁcients are constant.

Table 5. Convective heat transfer coefﬁcient for the surfaces [47].

Surface Convective Coefﬁcient Range

Floor h=2.175

D0.076 ∆T0.308 9·108<Gr <7·1010

Ceiling h=0.704

D0.601 ∆T0.133 9·108<Gr <1·1011

Walls h=1.823

D0.121 ∆T0.293 9·108<Gr <6·1010

The initial temperature of the nodes has been set to 22

◦

C by default. However, due to

the high thermal inertia of the system, the time required by the node temperatures to reach

their ﬁnal distribution may exceed the simulation time. For that reason, a startup period

has been added to the simulation [

20

]. During the startup period, node temperatures can

reach their ﬁnal values without altering the solution. To verify whether the ﬁnal results

are achieved, the difference between the maximum temperatures of the considered and

the previous day is calculated. If the difference is smaller than 0.005 K, the solution is

considered valid.

5.2. Validation and Veriﬁcation

The mathematical model was validated using experimental data to assess whether

the model parameters and the speciﬁc conditions are properly set. During the process, a

six-day-long period was simulated, and the results were compared with data collected

during measurements. The speciﬁc conditions were based on measurements of solar

irradiation, external temperature, and air temperature delivered by an air conditioning

system. Since there were no occupants in the room during the measurements, the internal

heat gains resulted from measurement equipment only. They were assumed to have a

constant value of 10 W throughout the simulation time. The speciﬁc conditions used in the

simulation have been doubled to form a startup period; therefore, the total simulation time

was equal to 12 days, but only the second half of it was considered valid and not affected

Energies 2024,17, 1951 13 of 22

by initial conditions. The simulation was performed with a time step (

∆t

) equal to 1 h and

a maximum iteration error (ξMax ) equal to 10−5K.

A comparison of simulation results and measurements for air temperatures is pre-

sented in Figure 6a. The mean absolute error is equal to 0.16

◦

C, and the maximum error is

0.65

◦

C, which are acceptable values. In the case of mean radiant temperature, the error

values were higher when compared to the air temperature mean, and the maximum abso-

lute errors are equal to 0.18

◦

C and 2.02

◦

C, respectively. The high value of the maximum

error may result from high sun radiation entering the room and heating the thermometer

responsible for collecting the data. Despite the relatively high maximum error for the

mean radiant temperatures, both air and radiant temperatures show good agreement with

experimental data.

(a) (b)

Figure 6. Comparison of simulation and experimental results. (a) Air temperature. (b) Mean radiant

temperature.

To assess that the results would converge, a grid independence study was conducted.

A calculation with the number of nodes varying from 13 to 55 was considered. The grid

independence study was performed for the air temperature at 12 a.m. and at 11 a.m. on

the ﬁrst day of the simulation time (1st and 12th time steps, respectively, after rejecting the

startup period). The results are shown in Figure 7a and Figure 7b, respectively. In both

cases, temperatures reach a steady value, with the number of nodes equal to 35. Beyond

this value, no signiﬁcant change in the solution can be reported; thus, we have chosen

the model with approximately 35 nodes for sensitivity analysis. However, the number of

nodes may differ depending on the experiment, since the geometry of the slab varies, as

the thickness of the screed and concrete layer are parameters in the sensitivity analysis.

(a) (b)

Figure 7. Results of grid independence analysis: (a) Results for the beginning of the simulation;

(b) Results for the 12th hour of the simulation.

Energies 2024,17, 1951 14 of 22

6. Results of Sensitivity Analysis

The sensitivity analysis was conducted using two experimental design methods—

Plackett–Burman and full factorial design—from which 512 experiments were randomly

selected. The medians of the appropriate output parameters were calculated for a series of

design points when the speciﬁc factor assumed low or high levels. Scatter diagrams with

marked medians (dashed lines) for all design points and medians for low and high levels

(point) for both methods are presented in Figures 8–12. The inﬂuence of the parameters’

changes on maximum air temperature (

θMax

Air

) is presented in Figure 8. According to both

experiment designs, the area of the cooling surface (

P

1), internal heat gains (

P

18), and

inlet water temperature (

P

19) have a signiﬁcant impact on

θMax

A

, as the differences in their

medians have values greater than 2.2

◦

C. In the case of water ﬂow (

P

12), the difference in

the medians obtained with the full factorial design is at the same level as the difference

in the area of the cooling surface (

P

1). However, the value obtained with the PB design

is lower and equals 0.6

◦

C. Since the full factorial design contains more experiments, it is

considered to be more informative in the case of uncertainty; therefore, parameter

P

12 is

also considered to have an impact on

θMax

A

. The inﬂuence of the speciﬁc heat of concrete

(

P

4) cannot be conﬁrmed, since its inﬂuence should be considered according to PB design.

In contrast, according to full factorial design, its inﬂuence is negligible. The discrepancy

between the two methods suggests that further study may be necessary to evaluate the

impact of the concrete-speciﬁc heat.

Figure 8. Results of sensitivity analysis for maximum air temperature.

In the case of minimum air temperature (

θMin

A

) presented in Figure 9, the assumed

DOEs show more inconsistency compared to the case of

θMax

A

. According to the PB design

(dark lines), only the inlet water temperature (

P

19) has a signiﬁcant impact on

θMin

A

, with

the difference in the medians equal to 4.1

◦

C, while for other parameters, the difference in

the medians is smaller than 1

◦

C. However, according to the full factorial design, the area

of the cooling surface (

P

1), internal wall thickness (

P

10), ﬂoor additional resistance (

P

11),

water ﬂow (

P

12), thermal transmittance of the window (

P

15), primary air heat gains (

P

17),

internal heat gains (

P

18), and inlet water temperature (

P

19) have considerable inﬂuence on

θMin

A

, with the difference in the medians above 1

◦

C. Due to the more informative nature of

the full factorial design, we consider parameters

P

1,

P

10,

P

11,

P

12,

P

15,

P

17,

P

18,

P

19 as

inﬂuences on θMin

A.

The result of sensitivity analysis for the maximum value of mean radiant temperature

(

θMax

MR

) is presented in Figure 10. Both methods indicate that the cooling surface area (

P

1),

Energies 2024,17, 1951 15 of 22

internal heat gains (

P

18), and inlet water temperature (

P

19) have a signiﬁcant inﬂuence

on the results, with differences in the medians greater than 1

◦

C. The results of the PB

design indicate that the speciﬁc heat of concrete (

P

4), screed density (

P

7), and the window

area (

P

14) also have considerable inﬂuence on the maximum value of the mean radiant

temperature. However, this is not conﬁrmed by the results of the full factorial design,

according to which the differences in the medians for parameters

P

4,

P

7, and

P

14 are

smaller than 1

◦

C. Therefore, the inﬂuence of parameters

P

4 and

P

7 is not considered

signiﬁcant. The signiﬁcance of the window area (

P

14) cannot be conﬁrmed. According to

the PB design, the difference in the medians is equal to 2.3

◦

C; however, the value obtained

with a randomly selected design from a full factorial experiment equals 0.7

◦

C. Due to the

inconsistency between the results of both methods, further study is necessary to conﬁrm

the inﬂuence of parameter P14.

Figure 9. Results of sensitivity analysis for minimum air temperature.

Figure 10. Results of sensitivity analysis for maximum mean radiant temperature.

Energies 2024,17, 1951 16 of 22

The results for minimum mean radiant temperature (

θMin

MR

) are presented in Figure 11.

Similarly to the case of minimum air temperature (

θMin

A

), the results obtained with the

PB design indicate that only the inﬂuence of the inlet water temperature (

P

19) can be

considered signiﬁcant, with the difference in the medians equal to 5

◦

C. The signiﬁcance of

parameter

P

19 is conﬁrmed by the results of the full factorial design, with the difference in

the medians equal to 5.7

◦

C. The results obtained with the full factorial design indicate that

the parameters

P

1,

P

8,

P

10,

P

12,

P

15,

P

17, and

P

18 should also be considered signiﬁcant.

However, further study may be necessary to establish the inﬂuence of these parameters.

According to both methods, the differences in the medians for thermal conductivity of

concrete (

P

2), concrete-speciﬁc heat (

P

4), and distance between pipes (

P

13) are smaller

than 0.6 ◦C, which means that the inﬂuence of these parameters on θMin

MR is negligible.

Figure 11. Results of sensitivity analysis for minimum mean radiant temperature.

The last output parameter examined in this study was the maximum power received

by the circuit (

QMax

Circuit

). Its value corresponds with the power required by water cooling

equipment to maintain the constant inlet water temperature. According to the results

presented in Figure 12, both DOEs indicate that the cooling surface area (

P

1), thermal

transmittance of the window (

P

15), and internal heat gains (

P

18) have a signiﬁcant impact

on the value of

QMax

Circuit

, with the differences in the medians greater than 120 W. The

results of both methods show that the properties of the screed layer (parameters

P

3,

P

5,

and

P

7) and thermal transmittance of the opaque part of the external wall (

P

16) have

insigniﬁcant inﬂuence on the cooling power demand. In the case of the window area (

P

14)

and primary air heat gains (

P

17), the PB design indicates a signiﬁcant inﬂuence on the

results, with a difference in the medians greater than 110 W. However, according to the

results of the full factorial design, parameters

P

14 and

P

17 have no impact on the value of

QMax

Circuit

, as differences in the medians are smaller than 15 W. The inconsistency between

the two methods indicates that further study may be necessary to evaluate the inﬂuence of

parameters P14 and P17.

A comparison of the results shows that the internal heat gains (

P

18) have the most sig-

niﬁcant inﬂuence on

θMax

A

,

θMax

MR

, and

QMax

Circuit

, which is consistent with the results presented

in [

22

]. However, in this study, we have assumed that the internal heat gains are constant

within working hours and that their value is independent of the temperature inside the

room, which is a considerable simpliﬁcation. According to [

22

], even minor changes in

occupants’ presence and behavior (such as switching off the lights) might notably inﬂuence

Energies 2024,17, 1951 17 of 22

the temperature inside the room and the demand for cooling power. Therefore, designers

should pay particular attention to properly evaluate the value of internal heat gains.

The results show that the inlet water temperature (

P

19) has a signiﬁcant impact on

both air and mean radiant temperature in the room, while the impact of water ﬂow has

been considered signiﬁcant only in the case of

θMin

A

. The amount of water delivered to the

pipes and its temperature is controlled by the cooling system. Therefore, developing the

proper control strategy for the cooling system is essential for maintaining the temperature

inside the room at comfortable levels. However, keeping the appropriate value of the inlet

water temperature is more important than the water ﬂow, since the water temperature has

a greater inﬂuence on the internal room temperature than does the water ﬂow.

Figure 12. Results of sensitivity analysis for maximum power received by the circuit.

In every experiment, the inﬂuence of the screed properties (parameters

P

3,

P

5,

P

7,

and

P

9), the thermal conductivity of concrete (parameter

P

2), and pipe spacing (

P

13)

is insigniﬁcant when compared to other parameters. This means that, in the process of

dimensioning the room with TABS, designers do not have to pay particular attention to

those parameters, since their inﬂuence on internal conditions is negligible; instead, they

should focus on more inﬂuential ones.

The results show that the primary air gain (

P

17) does not inﬂuence

θMax

A

or

θMax

MR

, which

may be an effect of the assumption that the ventilation cools the room only in the hours

following its activation. In the case of high thermal load, the air temperature delivered

by the ventilation system and the air temperature in the room may differ signiﬁcantly.

Therefore, the primary air gain would have a non-zero value throughout the time the

ventilation system is on. To consider the inﬂuence of the internal temperature on the

primary air gain, a further modiﬁcation of the numerical model of the room is necessary.

The results of both the PB and full factorial design methods show that the transmittance

of the opaque part of the external wall (

P

16) has no signiﬁcant inﬂuence on temperatures

inside the room nor on the power received by the circuit. The insigniﬁcant inﬂuence of the

parameter

P

16 may be caused by the fact that its values are approximately one order of

magnitude lower than the values of the thermal transmittance of the window (

P

15). This

resulted in low heat ﬂux transmitted by the opaque part of the external wall and low heat

loss to the environment, which is consistent with the results obtained by Lu et al.

[21]

. It

should be borne in mind that the obtained results and conclusions are valid for the assumed

range of factors’ (P1–P19) variability.

Energies 2024,17, 1951 18 of 22

7. Conclusions

In this study, we performed a sensitivity analysis to ﬁnd which of the selected 19

factors (material, design, and operating parameters) affect the maximum/minimum air

temperature, maximum/minimum mean radiant temperature inside the room, and the

maximum required cooling equipment power. The analysis was performed using the PB

design and a randomly selected design from a full factorial experiment. The resistance–

capacitance model introduced in ISO 11855-4 was used for simulation research. The

speciﬁc conditions considered in the mathematical model and parameters assumed in the

sensitivity analysis were based on the measurement results collected in the ofﬁce building.

The measurement data were also used to validate the mathematical model. Below, we

provide a summary of the key ﬁndings from the research:

•

The internal heat gains (

P

18) and the inlet water temperature (

P

19) signiﬁcantly inﬂu-

ence the maximum/minimum air temperature, maximum/minimum mean radiant

temperature, and the maximum required cooling equipment power. Therefore, special

attention should be paid to the internal heat gain model and control strategy for TABS

while it is being designed.

•

The impact of the analyzed factors on the mean radiant temperature, however, is

inﬂuenced by simpliﬁcations resulting from the set of the experimental equipment

used. Hence, in the study, the globe temperature was assumed to be equal to

θMR

.

Precise measurement of air velocity near the globe sensor would be of special interest

in terms of obtaining the correct value of θMR.

•

The primary air heat gains (

P

17) signiﬁcantly inﬂuence the minimum air temperature.

•

The inﬂuence of the screed’s properties (

P

3,

P

5,

P

7) is insigniﬁcant compared to other

parameters, and it does not need to be considered in the process of dimensioning the

cooling system or in the cooling control strategy. However, it should be borne in mind

that this is valid for the assumed range of factors’ (P1–P19) variability.

•

Thermal transmittance of the external wall does not have a signiﬁcant effect on internal

conditions because, even at a high level, its value is small compared to the thermal

transmittance of the window.

•

The water inlet temperature (

P

19) has a considerably larger inﬂuence on the tempera-

ture in the room than does the water mass ﬂow rate (P12) in the pipes.

The simulation conditions have been simpliﬁed and are based on measurement data,

which may limit the results to speciﬁc situations. The authors of this study propose to ex-

tend the research, such as by making primary air heat gains dependent on air temperature.

Author Contributions: Conceptualization, J.W. and P.M; investigation, M.B., J.W. and P.M.; methodol-

ogy, M.B., P.M. and J.W.; software, M.B.; supervision, J.W.; validation, M.B. and P.M.; writing—original

draft, M.B., J.W. and P.M.; writing—review & editing, M.B., J.W. and P.M. All authors have read and

agreed to the published version of the manuscript.

Funding: This research project was partly supported by the program “Excellence initiative—research

university” for the AGH University of Science and Technology and by Poland national subvention,

Poland no. 16.16.130.942.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Data Availability Statement: The original contributions presented in the study are included in the

article, further inquiries can be directed to the corresponding authors.

Conﬂicts of Interest: The authors declare no conﬂicts of interest.

Energies 2024,17, 1951 19 of 22

Abbreviations

The following abbreviations are used in this manuscript:

BMS Building management system

DOE Design of experiment

FDM Finite difference method

HVAC Heating, ventilation, and air conditioning

MPE Maximum permissible error

PB Plackett–Burman

TABS Thermally activated building systems

Nomenclature

ASurface area, m2

cSpeciﬁc heat, J

kg·K

CSpeciﬁc thermal capacity, J

m2·K; Substitute speciﬁc thermal capacity, J

m3·K

dThickness, m

daPipe outer diameter, m

DCharacteristic dimension, m

EMaximum permissible error, [-]

Gr Grashof number, [-]

h,HHeat transfer coefﬁcient, W

m2·K

Isol Solar irradiance, W

m2

LPipe spacing, m

˙

mFlow rate, [kg

s]

˙

mHWater ﬂow in pipes, [kg

s]

nNumber of iterations, [-]

PParameter

QHeat gain, W

RThermal resistance, m·K

W

RAddC Ceiling additional thermal resistance, m·K

W

RAddF Floor additional thermal resistance, m·K

W

RtThermal resistance between inlet water and concrete at pipe level, m·K

W

sPipe wall thickness, m

tTime, [s]

TTemperature, ◦C

uUncertainty

UThermal transmittance, W

m2·K

X,Y,ZRoom dimensions, m

Greek symbols

εEmissivity, [-]

θTemperature, ◦C

λThermal conductivity, W

m·K

λrPipe wall thermal conductivity, W

m·K

ξIteration error, K

ρDensity, kg

m3

Subscripts

AAir

CCeiling

con Concrete layer

CondDown Conduction to next node

CondUp Conduction to previous node

Conv Convective

dev Device

EW External wall (opaque)

Ex External

exh Ventilation exhaust

fFlow

FFloor

Energies 2024,17, 1951 20 of 22

gGlazing

hTime step number

Int Internal

IntConv Internal convective

IntRad Internal radiant

IW Internal walls

IWS Internal wall surface

MR Medium radiant

pNode number

per Permissible

pl Plaster gypsum layer

Rad Radiant

sens Sensor

sc Screed layer

sup Ventilation supply

TTemperature

Transm Transmission

WWater

WWindow

WaterI n Water inlet

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