Analytical expression of indoor temperature distribution in generally ventilated room with arbitrary boundary conditions

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Analytical Expression of Indoor Temperature Distribution in Generally Ventilated Room with Arbitrary Boundary Conditions Shuai Yan , Xianting Li PII: DOI: Reference:

S0378-7788(19)32286-8 https://doi.org/10.1016/j.enbuild.2019.109640 ENB 109640

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Energy & Buildings

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25 July 2019 31 October 2019 24 November 2019

Please cite this article as: Shuai Yan , Xianting Li , Analytical Expression of Indoor Temperature Distribution in Generally Ventilated Room with Arbitrary Boundary Conditions, Energy & Buildings (2019), doi: https://doi.org/10.1016/j.enbuild.2019.109640

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Analytical Expression of Indoor Temperature Distribution in Generally Ventilated Room with Arbitrary Boundary Conditions Shuai Yan, Xianting Li*

Department of Building Science, Beijing Key Laboratory of Indoor Air Quality Evaluation and Control, Tsinghua University, Beijing, People’s Republic of China * Corresponding author: Prof. Xianting Li Department of Building Science, School of Architecture Tsinghua University Beijing 100084, P.R. China Tel: +86-10-62785860 Fax: +86-10-62773461 E-mail: [email protected]

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Abstract The indoor temperature of a ventilated room is influenced by the airflow pattern, convective heat, and indoor heat sources. The influences of the thermal factors on indoor temperature are different, and vary with location. Though the influence of supply air and heat source can be estimated previously, the influence of convective heat is coupled with indoor temperature thus it has not been clearly investigated. However, the convective heat plays an important role in total building heat transfer. To determine the explicit influence of each thermal factor (especially the heat convection), the theoretical expression of the indoor temperature distribution of steady-state cases is established based on previous research. Three indices: the modified accessibilities of the supply air, ambient temperature, and heat source are used to reflect the explicit influence of different types. The distributions and influencing factors of the three modified indices are analyzed through case studies. The results indicate the following: 1) The influence from the convective boundary is of significance, e.g., the contribution of the water temperature of convective terminals exceeds 0.4 in most studied cases thus it is more reasonable and accurate to reflect the influence by the corresponding ambient temperature; 2) The influence degree of each thermal factor is determined by the flow field and heat transfer coefficient of the convective boundary, e.g., the contribution degree of the supply air is reduced from 0.43 to 0.35 when ventilation rate is reduced by half; and 3) Because the energy grades of the supply air and convective boundary are different, more low grade energy can be used to control the temperature in the target zone to save energy in a non-uniform indoor environment. Succinctly, the established expression can reveal the mechanism of the temperature distribution and provide guidance for the establishment of an energy-efficient non-uniform indoor environment. 2 / 53

Keywords

non-uniform indoor environment; temperature distribution; analytical expression; CFD; building energy efficiency

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Nomenclature 𝑎𝐶𝑃 𝑎𝐸𝑃 𝑎𝑆𝑃 𝑃 ̅𝑎̅̅̅ 𝐶 𝑃 ̅𝑎̅̅̅ 𝐸 𝑃 ̅𝑎̅̅̅ 𝑆 𝑎𝑣 ̅̅̅̅̅ 𝑎 𝐶 ̅̅̅̅̅ 𝑎𝐸𝑎𝑣 ̅̅̅̅̅ 𝑎𝑎𝑣 𝑆

A Cp D F ℎ k l 𝑚𝑠 𝑛𝑐 𝑛 ̂𝑐 𝑛𝑒 𝑛𝑓 𝑛 ̂𝑓 𝑛𝑠 Q 𝑡𝑎𝑚𝑏 𝑡𝑠 u V

accessibility of convective heat to location P accessibility of heat source with fixed intensity to location P accessibility of supply air to location P modified accessibility of ambient temperature to location P modified accessibility of heat source with fixed intensity to location P modified accessibility of supply air to location P volume-averaged modified accessibility of ambient temperature volume-averaged modified accessibility of heat source with fixed intensity volume-averaged modified accessibility of supply air matrix of accessibility indices specific heat of air, [J/(kg‧K)] matrix of modified ambient temperature area of convective boundary, [m2] convective heat transfer coefficient, [W/(m2∙K)] convective boundary number of nodes adjacent to convective boundary mass flow rate of supply air, [kg/s] number of convective boundaries number of discretized convective heat sources number of heat sources number of thermal factors of proposed method number of dependent thermal factors of traditional method number of supply air inlets intensity of heat source, [W] ambient temperature, [°C] supply air temperature, [°C] velocity, [m/s] target zone of ventilated room

Greek symbols 𝛤𝑒 δ 𝜌 λ

diffusion coefficient thickness, [m] density of air, [kg/m3] thermal conductivity, [W/(m∙K)]

Abbreviations

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ACH AHS ASA CFD MAAT MAHS MASA MV

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air change per hour accessibility of heat source accessibility of supply air computational fluid dynamics modified accessibility of ambient temperature modified accessibility of heat source modified accessibility of supply air mixing ventilation

1. Introduction Indoor thermal environments are generally affected by heat transfer through the building envelope, internal heat sources, and the infiltration of ambient air. The required indoor temperature is realized through either ventilation or radiantconvective terminals [1, 2]. Conventionally, uniform indoor environments are created through mixing ventilation (MV). However, temperature discrepancies have been even observed in with MV [3]. Owing to the increasing importance of thermal comfort and energy-saving heating and cooling methods, engineers have discovered that the thermal requirements of the local zone rather than the entire room must be satisfied. That is, a non-uniform thermal environment, which is created by advanced airflow patterns, has more energy saving potential than a uniform environment. For example, personalized ventilation (PV) focuses only on the local area, rather than on the entire room [4]. In addition, when ventilation and local heating/cooling devices are combined to create the desired local thermal environment, the temperature of the local zone differs significantly from the average temperature of the room [5, 6]. Therefore, research on the non-uniform indoor environment has attracted increasing attention [7, 8].

Over the past few decades, computational fluid dynamics (CFD) has become a popular approach for simulations of distributed indoor parameters [9, 10], because most indoor-airflow problems can be numerically solved owing to the continuous development of computational technology [11]. For instance, CFD simulations play an important role in airflow optimization and energy saving measures for residential and industrial buildings [12, 13]. However, CFD simulations have two main deficiencies: First, a theoretical model of how the airflow patterns influence the 6 / 53

distributed indoor parameters is lacking, although several cases with different settings could be numerically solved. For example, researchers have proposed empirical expressions to calculate the distributed indoor parameters of underfloor air distribution (UFAD) systems [14, 15]. However, the contribution of each heat source varies between the different cases and is based on several assumptions in the proposed empirical expression. Second, the direct use of the CFD results for guidance of operations in engineering processes is difficult. The CFD simulations are conducted based on typical cases with specific rooms and ventilation systems. However, room and ventilation systems vary between different cases. Thus, the results of typical cases are insufficient for guidance [9].

CFD simulations are necessary because the complex equations of the indoor turbulent flow (rather than the energy balance equation) require comprehensive numerical simulations [16]. That is, it is easier to determine the temperature distribution inside the room once the flow field has been obtained. In a specific room, the indoor air movement is dominated by the supply air, and the airflow does not change significantly with changing heat source and supply air temperature [17]. Based on the fixed flow field assumption, studies have been conducted to reveal the influences of the supply air and heat source. Kato et al. proposed the contribution ratio of indoor climate (CRI) index to describe the influences of different thermal factors on the formation of the temperature field for steady-state cases [18]. To determine the transient impact of the supply air from each supply air inlet on a specific space within a finite time period, Li and Zhao proposed the accessibility of supply air (ASA) index, which was applied to an emergency ventilation system [19, 20]. Similarly, the accessibility of heat source (AHS) index was proposed to describe 7 / 53

the transient effect of a heat source in a specific zone [21, 22]. With the ASA and AHS, the indoor temperature can be predicted by superposing the influences of the supply air and heat source. In general, the proposed ASA predicts the indoor temperature distribution and provides an understanding of how the indoor temperature distribution is influenced by each thermal factor [23].

In previous research studies, the influence of supply air and the influence of heat source on indoor temperature has been investigated by ASA and AHS. Consequently, the thermal factors influencing the indoor temperature were classified into two categories: the influences of the supply air and heat source with a fixed heat intensity [24]. However, in real ventilated rooms, some of the heat sources have fixed intensities (such as lamps and computers), and some depend on the indoor temperature (such as heat transfer through the building envelope and chilled beams) [25]. Heat sources with fixed intensity are normally regarded as a second-type thermal boundary referring to [26]; those related to the indoor temperature are often regarded as convective thermal boundary (with known heat transfer coefficient and ambient temperature) while in previous research, they were regarded as heat sources with estimated heat intensity. However, it is difficult to analyze the influence of the convective thermal boundary with AHS because the transferred convective heat is coupled with the indoor temperature [27]. In fact, the convective heat is of significance and using indoor convective terminals to create the energy-efficient indoor environment is becoming more and more popular. Although the implicit heat transfer has been solved through the iteration method [21, 24], an intuitive understanding of the contributions of the convective boundary is lacking. Thus, a

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method for explicitly determining the influence of each thermal factor in the formation of the indoor temperature is essential.

To clarify how the indoor temperature distribution is explicitly affected by each thermal factor, an expression for the prediction of the temperature distribution of steady-state cases based on the superposition theorem is derived in this study. In addition, modified accessibility indices are proposed to describe the contributions of the supply air, heat source, and convective boundaries. The contribution of each thermal factor is analyzed and discussed based on the proposed method and typical cases.

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2. Explicit expression of indoor temperature distribution The steady-state cases are adopted in this study and the reasons are as following: 1) The thermal mass of indoor air is small compared with building envelope. Consequently, the heat transfer of indoor air can be regarded as several steady-state cases where only the thermal inertia of building envelope is considered; 2) In transient cases, the results are influenced by not only the thermal factor but also the thermal mass of building envelope thus it is case-sensitive. That is, the explicit influence of each thermal factor in transient cases are not easy to be understood compared with steady-state cases; and 3) The transient heat transfer of a ventilated room can be obtained based on steady-state cases [1, 28, 29] thus the significance of the expression of transient cases is the simple method for engineering calculation rather than the intuitive understanding between indoor temperature and the influencing factor. In a word, because it is much easier to understand how an indoor temperature field is affected by different thermal factors through steady-state cases, this study focuses only on steady-state cases. In addition, the longwave radiation between different surfaces is neglected in this study. This is because the temperature difference between different surfaces is limited. Besides, though longwave radiation exists in the ventilated room, the transferred radiative heat would finally be absorbed by indoor air. Generally, previous studies indicate that it is possible to be neglected and the accuracy is promising [17, 21].

To acquire the explicit analytical expression of the temperature distribution, the implicit expression of the temperature distribution, supply air temperature, and heat source intensity is introduced first. Next, the explicit expression is derived based on the implicit expression. The modified accessibility indices are proposed to determine 10 / 53

the degree of influence of each thermal factor. Finally, the determination of the established expression for general ventilated rooms is presented.

2.1 Implicit expression of indoor temperature distribution The well-known energy conservation equation of sensible thermal heat in steadystate cases is shown in Eq. (1). 𝜕𝜌𝑢𝑗 𝑡 𝜕 𝜕𝑡 = (𝛤𝑒 ) + 𝑆𝑡 𝜕𝑥𝑗 𝜕𝑥𝑗 𝜕𝑥𝑗

(1)

where 𝑡 is the temperature [°C], 𝜌 the density of air [kg/m3], 𝑢𝑗 the velocity in the j direction [m/s], 𝛤𝑒 the diffusion coefficient, and 𝑆𝑡 the heat source intensity [W].

As previously mentioned, owing to thermal buoyancy, the energy conservation equation is coupled with the momentum equation and is difficult to solve explicitly. However, when the flow field can be considered fixed, the transport equation of the sensible heat of Eq. (1) becomes linear. As a result, the effects of different boundaries on the arbitrary location P can be superposed.

After analyzing all boundaries in the ventilated space, the factors that influence the indoor temperature are the supply air, fixed heat sources, and convective boundaries. Thus, the presented approach is suitable for arbitrary boundary conditions.

To obtain a practical and general approach, 𝑛𝑠 supply air outlets with different supply air temperatures, 𝑛𝑒 heat sources with different heat intensities, and 𝑛𝑐 convective boundaries with different ambient temperatures in a ventilated room are considered, as shown in Fig. 1 a). Furthermore, the 𝑙𝑘 nodes are adjacent to the kth convective 11 / 53

boundary, and the convective heat is affected by the corresponding 𝑙𝑘 heat sources, as shown in Fig. 1 b). Consequently, the indoor temperature is the superposition of the contributions of all thermal factors (Eq. (2)). Furthermore, the ASA and AHS are used to present the degree of influence [19]. The third term of Eq. (2) represents the non-uniformly distributed convective heat of the convective boundaries. The convective heat can be expressed as 𝑛𝑠

𝑛𝑒

𝑛𝑐

𝑙𝑘

𝑄𝑘𝑟 𝑃 𝑄𝑗 𝑃 𝑃 𝑡 𝑃 = ∑ 𝑡𝑆,𝑖 𝑎𝑆,𝑖 +∑ 𝑎𝐸,𝑗 + ∑ ∑ 𝑎 𝑚𝑠 𝐶𝑝 𝑚𝑠 𝐶𝑝 𝐶,𝑘𝑟

(2)

𝑄𝑘𝑟 = 𝑈𝑘𝑟 𝐹𝑘𝑟 (𝑡𝑎𝑚𝑏,𝑘 − 𝑡𝑘𝑟 )

(3)

𝑖=1

𝑗=1

𝑘=1 𝑟=1

where 𝑡 𝑃 is the temperature at location P [°C], 𝑡𝑆,𝑖 the supply air temperature of the ith inlet [°C], 𝑄𝑗 the intensity of the jth heat source [W], 𝑄𝑘𝑟 the convective heat of the rth node of the kth convective boundary [W], 𝑈𝑘𝑟 the heat transfer coefficient of the rth node of the kth convective boundary [W/(m2·K)], 𝐹𝑘𝑟 the area of the rth node adjacent to the kth convective boundary [m2], 𝑡𝑘𝑟 the temperature of the rth node [°C], 𝑡𝑎𝑚𝑏,𝑘 the ambient temperature of the kth convective boundary [°C], 𝑚𝑠 the total mass flow rate of the supply air [kg/s], and 𝐶𝑝 the specific heat of air [J/(kg‧K)].

In addition, the 𝑛 ̂𝑐 discretized heat sources correspond to the 𝑛𝑐 convective boundaries (𝑛 ̂𝑐 > 𝑛𝑐 ), as shown in Eq. (4). To simplify the method, Eq. (2) can be expressed as a matrix, as shown in Eq. (5). Consequently, there are 𝑛 ̂𝑓 thermal factors that have an impact on the indoor temperature distribution, as shown in Eq. (6).

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𝑛𝑐

𝑛 ̂𝑐 = ∑ 𝑙𝑘

(4)

𝑘=1

⃗⃗⃗⃗𝑃 )𝑇 × ⃗⃗⃗ 𝑡 𝑃 = (𝑎 𝑡𝑆 + 𝑆

1 ⃗⃗⃗⃗𝑃 𝑇 1 ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗𝑃 𝑇 ⃗ + ̂ (𝑎𝐸 ) × 𝑄 (𝑈𝐹𝑎𝐶 ) × (𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡𝑘 ) 𝑎𝑚𝑏 − ⃗⃗⃗ 𝑚𝑠 𝐶𝑝 𝑚𝑠 𝐶𝑝

𝑛 ̂𝑓 = 𝑛𝑠 + 𝑛𝑒 + 𝑛 ̂𝑐

(5) (6)

where ⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡̂ ̂𝑐 𝑎𝑚𝑏 is the vector of the ambient temperature and corresponds to the 𝑛 adjacent nodes.

To analyze the common principle of the heat transfer in the convective boundaries, the heat convection in the boundaries in Fig. 2 can be simplified. The overall thermal resistance equals three serially connected thermal resistances, as shown in Eq. (7); ℎ𝑖𝑛 , which represents the convective heat transfer coefficient of the air and convective boundary, is determined by the Reynolds and Prandtl numbers. These can be determined for a fixed flow field. The other two thermal resistances are determined only by the thermal property of the convective boundary. Taking the building envelope as an example, the heat conduction is determined by the envelope material. Furthermore, the outdoor convective heat transfer coefficient is influenced by the wind speed. The two resistances are usually provided [29]. Succinctly, when the flow field and thermal property of the convective boundary are known, 𝑈𝑘𝑟 is fixed and can be determined. Furthermore, if the convective thermal resistance of the ambient and conductive thermal resistances are zero, the thermal boundary can be regarded as a that with fixed temperature. 𝑈𝑘𝑟 =

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1 1 𝛿 1 + + ℎ𝑎𝑚𝑏 𝜆 ℎ𝑖𝑛,𝑟

(7)

where ℎ𝑎𝑚𝑏 and ℎ𝑖𝑛,𝑟 are the ambient and indoor heat transfer coefficients, respectively [W/(m2·K)]; 𝛿 is the thickness [m], and 𝜆 is the heat conductivity [W/(m·K)].

2.2 Derivation of explicit expression and modified accessibility indices As previously mentioned, the convective heat 𝑄𝑘 is non-uniformly distributed and coupled with the indoor temperature. The temperature of the 𝑛 ̂𝑐 nodes adjacent to the 𝑛𝑐 convective boundaries can be obtained as follows: first, ⃗⃗⃗ 𝑡𝑘 is denoted as the temperature of the 𝑛 ̂𝑐 nodes adjacent to the 𝑛𝑐 convective boundaries, as shown in Eq. (8). The ambient temperature of the 𝑛𝑐 convective boundaries is denoted as ⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡𝑎𝑚𝑏 , as shown in Eq. (9). The ambient temperature of the 𝑛 ̂𝑐 adjacent nodes is denoted as ⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡̂ 𝑎𝑚𝑏 . Because the nodes adjacent to the same convective boundary experience the same ambient temperature, a relationship between ⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡̂ 𝑡𝑎𝑚𝑏 can be established 𝑎𝑚𝑏 and ⃗⃗⃗⃗⃗⃗⃗⃗⃗ through a matrix D, as shown in Eq. (10). Specifically, if the ith node of ⃗⃗⃗ 𝑡𝑘 is adjacent to the jth convective boundary 𝐶𝑗 , 𝐷𝑖,𝑗 equals one; otherwise, 𝐷𝑖,𝑗 equals zero, as shown in Eq. (11).

⃗⃗⃗ 𝑡𝑘 = [𝑡𝑘,1 , 𝑡𝑘,2 , … , 𝑡𝑘,𝑥 , … , 𝑡𝑘,𝑛̂𝑐 ]𝑇

(8)

⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡𝑎𝑚𝑏 = [𝑡𝑎𝑚𝑏,1 , 𝑡𝑎𝑚𝑏,2 , … , 𝑡𝑎𝑚𝑏,𝑛̂𝑐 ]𝑇

(9)

⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡̂ × ⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡𝑎𝑚𝑏 ̂×𝑛 𝑎𝑚𝑏 = (𝐷)𝑛 𝑐 𝑐 {

𝐷𝑖,𝑗 = 1 𝐷𝑖,𝑗 = 0

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(𝑖 ∈ 𝐶𝑗 ) (𝑖 ∉ 𝐶𝑗 )

(10) (11)

Based on Eq. (5), the temperatures of the nodes adjacent to the convective boundary are expressed in Eq. (12). Correspondingly, the matrices of the influences of the supply air, heat source, and heat convection of the convective boundary are presented in Eqs. (13), (14), and (15), respectively. In addition, Eq. (12) can be transformed into Eq. (16). (𝑡⃗⃗⃗𝑘 )𝑛̂×1 = (𝐴𝑠 )𝑛̂×𝑛 × (𝑡⃗⃗⃗𝑆 )𝑛𝑠 ×1 + 𝑐 𝑐 𝑠

1 1 ⃗ )𝑛 ×1 + (𝐴𝐸 )𝑛̂×𝑛 × (𝑄 (𝐴 ) 𝑐 ̂𝑐 𝑐 𝑒 𝑒 𝑚𝑠 𝐶𝑝 𝑚𝑠 𝐶𝑝 𝐶 𝑛̂×𝑛

(12)

̂ × (𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡𝑘 )𝑛̂×1 𝑎𝑚𝑏 − ⃗⃗⃗ 𝑐 𝑏 (𝐴𝑠 )𝑖,𝑏 = 𝑎𝑆,𝑖

(13)

𝑏 (𝐴𝐸 )𝑗,𝑏 = 𝑎𝐸,𝑗

(14)

(𝐴𝐶 )𝑘,𝑏 = 𝑈𝑘 𝐹𝑘 𝑎𝐶𝑏

(15)

1 ⃗⃗⃗ 𝑡𝑘 = (𝐸 + 𝐴 × 𝐷) 𝑚𝑠 𝐶𝑝 𝐶

−1

× (𝐴𝑠 × ⃗⃗⃗ 𝑡𝑆 +

1 1 ⃗ + 𝐴𝐸 × 𝑄 𝐴 × 𝐷 × ⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑡𝑎𝑚𝑏 ) 𝑚𝑠 𝐶𝑝 𝑚𝑠 𝐶𝑝 𝐶

(16)

where 𝐸 is the identity matrix.

According to Eq. (16), ⃗⃗⃗ 𝑡𝑘 is influenced by each supply air temperature, heat source intensity, and ambient temperature. With ⃗⃗⃗ 𝑡𝑘 of Eqs. (16) and (5), Eq. (17) can be derived. Similarly to ⃗⃗⃗ 𝑡𝑘 , the indoor temperature at an arbitrary location P is a superposition of the contributions of the supply air, heat source, and ambient temperature. Their contributions are defined in Eqs. (18), (19), and (20), respectively. ⃗⃗⃗⃗ 𝑃 ⃗⃗⃗⃗ 𝑃 ̅𝑎̅̅̅ ̅̅̅̅ ̅̅̅𝑃̅ Based on [19], ⃗⃗⃗⃗ 𝑆 , 𝑎𝐸 , and 𝑎𝐶 are defined as the modified accessibility of the supply air (MASA) to location P, modified accessibility of the heat source (MAHS) to location P, and modified accessibility of the ambient temperature (MAAT) to location P, respectively. These indices are used to present the degree of influence of each

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thermal factor. According to Eqs. (18), (19), and (20), they depend on the flow field and thermal properties of the convective boundary.

⃗⃗⃗⃗ ⃗⃗⃗⃗ 𝑃 𝑇 𝑃 𝑇 ̅̅̅̅ ̅̅̅̅ ⃗⃗⃗ 𝑡 𝑃 = (𝑎 𝑆 ) × 𝑡𝑆 + (𝑎𝐸 ) ×

⃗ 𝑄 ⃗⃗⃗⃗ 𝑃 𝑇 ̅̅̅̅ ⃗⃗⃗⃗⃗⃗⃗⃗⃗ + (𝑎 𝐶 ) × 𝑡𝑎𝑚𝑏 𝑚𝑠 𝐶𝑝 −1

⃗⃗⃗⃗ 𝑃 ̅𝑎̅̅̅ ⃗⃗⃗⃗𝑃 𝑆 = 𝑎𝑆 −

1 1 ((𝐸 + 𝐴 × 𝐷) 𝑚𝑠 𝐶𝑝 𝑚𝑠 𝐶𝑝 𝐶

⃗⃗⃗⃗ 𝑃 ̅𝑎̅̅̅ ⃗⃗⃗⃗𝑃 𝐸 = 𝑎𝐸 −

1 1 × 𝐴𝐶 × 𝐷 × ((𝐸 − 𝐴 × 𝐷) 𝑚𝑠 𝐶𝑝 𝑚𝑠 𝐶𝑝 𝐶

⃗⃗⃗⃗ 𝑃 ̅𝑎̅̅̅ 𝐶 =

(17)

× 𝐴𝑠 )𝑇 × ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑈𝐹𝑎𝐶𝑃 −1

(18)

× 𝐴𝐸 )𝑇 × ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑈𝐹𝑎𝐶𝑃

1 1 1 × 𝐴𝐶 × 𝐷 × (𝐸 − (𝐸 − 𝐴 × 𝐷) 𝑚𝑠 𝐶𝑝 𝑚𝑠 𝐶𝑝 𝑚𝑠 𝐶𝑝 𝐶

−1

× 𝐴𝐶 × 𝐷) × ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑈𝐹𝑎𝐶𝑃

(19)

(20)

where E is the identity matrix.

Consequently, for a general ventilated room with 𝑛𝑠 supply air inlets, 𝑛𝑒 heat sources, and 𝑛𝑐 convective boundaries, there are totally 𝑛𝑓 thermal factors influence the indoor temperature distribution, as shown in Eq. (21). By comparing Eqs. (6) and (21), it can be concluded that the influence of the 𝑛𝑐 convective boundaries is substituted by 𝑛𝑐 ambient temperatures rather than 𝑛 ̂𝑐 convective heat sources (𝑛 ̂𝑐 > 𝑛𝑐 ). More importantly, the 𝑛𝑐 ambient temperatures are independent variables, whereas the 𝑛 ̂𝑐 heat sources are implicit variables and coupled with the indoor temperature. 𝑛𝑓 = 𝑛𝑠 + 𝑛𝑒 + 𝑛𝑐

(21)

In most ventilated rooms, the volume-averaged temperature of the target zone should be artificially controlled. The volume-averaged temperature of target zone V is expressed in Eq. (22). The corresponding indices (i.e., 𝑎𝑆𝑎𝑣 , 𝑎𝐸𝑎𝑣 , and 𝑎𝐶𝑎𝑣 ) are presented in Eqs. (23)–(25) to present the influences of the supply air temperature, 16 / 53

heat source intensity, and ambient temperature on 𝑡𝑎𝑣 . In addition, the indices depend on the flow field and thermal properties of the convective boundary.

𝑡𝑎𝑣

1 ∭ 𝑡 𝑃 𝑑𝑉 ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ 𝑎𝑣 𝑇 𝑎𝑣 𝑇 𝑎𝑣 𝑇 ̅̅̅̅̅ ̅̅̅̅̅ ̅̅̅̅̅ ⃗ ⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗⃗⃗ = = (𝑎 × (𝑎 𝑆 ) × 𝑡𝑆 + 𝐸 ) × 𝑄 + (𝑎𝐶 ) × 𝑡𝑎𝑚𝑏 𝑚𝑠 𝐶𝑝 ∭ 𝑑𝑉

𝑃 ̅̅̅̅ ∭𝑎 𝑆,𝑖 𝑑𝑉 𝑎𝑣 ̅̅̅̅̅ 𝑎𝑆,𝑖 = ∭ 𝑑𝑉

(22)

(23)

𝑎𝑣 ̅̅̅̅̅ 𝑎𝐸,𝑗 =

𝑃 𝑎𝐸,𝑗 𝑑𝑉 ∭ ̅̅̅̅̅ ∭ 𝑑𝑉

(24)

𝑎𝑣 ̅̅̅̅̅ 𝑎 𝐶,𝑘 =

𝑃 ̅̅̅̅̅ ∭𝑎 𝐶,𝑘 𝑑𝑉 ∭ 𝑑𝑉

(25)

where V is the target zone.

Based on Eqs. (5) and (17), the convective heat is regarded as a given heat source with traditional accessibility indices. However, for modified accessibility indices, the influence of the convective boundary is computed based on the ambient temperature. If there are 𝑛𝑐 convective boundaries with different ambient temperatures, the 𝑛𝑐 ambient temperatures contribute independently to the indoor space.

In general, the accessibility and modified indices exhibit differences: 1) the modified accessibility indices depend on the flow field and thermal properties of the convective boundary; 2) for modified accessibility indices, the indoor temperature is explicitly influenced by all corresponding thermal factors. Succinctly, the modified accessibility indices can effectively reveal the relationship between the indoor temperature and influencing factors.

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2.3 Determination of explicit expression for general ventilated room The theoretical expressions of MASA, MAHS, and MAAT presented in Section 2.2 are complex, and several matrices should be prepared in advance for their determination. To simplify the calculations, according to Eqs. (18)-(20), once the flow field and heat transfer coefficient of the convective boundary are known, the indoor temperature depends only on 𝑛𝑓 independent thermal factors (Eq. (21)). Thus, it is possible to obtain the distributed MASA, MAHS, and MAAT based on the 𝑛𝑓 cases.

The MASA, MAHS, and MAAT at an arbitrary location P can be obtained as follows: first, the airflow field of a typical thermal scenario should be calculated and prepared. The thermal scenario should be typical, i.e., the Grashof number, Reynolds number, and Archimedes number should be representative: the relative error should be within 15% [21]. In a word, one should make sure that the airflow is simulated based on the typical supply air temperature, heat source intensity, and ambient temperature thus it is accurate enough to be used in further studies. Second, the temperature at location P of the 𝑛𝑓 cases, denoted as (𝑡⃗⃗⃗𝑃 )

𝑛𝑓 ×1

, is calculated based on the prepared flow

field, as shown in Eq. (26). Third, the MASA, MAHS, and MAAT can be obtained through the results of the 𝑛𝑓 cases, as shown in Eq. (27). Because the MASA, MAHS, and MAAT are uniquely determined for a given flow field and thermal properties of the convective boundary and are not related to the thermal factor, they can be acquired with 𝑛𝑓 different cases.

(𝑡⃗⃗⃗𝑃 )

𝑛𝑓 ×1

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(𝑡⃗⃗⃗⃗⃗⃗ 𝑆,1 ) =[ …

𝑇 𝑇

(𝑡⃗⃗⃗⃗⃗⃗⃗⃗ 𝑆,𝑛𝑓 )

𝑇

⃗⃗⃗⃗1 ) (𝑄 … 𝑇 ⃗⃗⃗⃗⃗⃗⃗ (𝑄 𝑛 ) 𝑓

𝑇

(𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑎𝑚𝑏,1 ) … ] 𝑇 (𝑡⃗⃗⃗⃗⃗ 𝑛𝑓 )

𝑛𝑓 ×𝑛𝑓

⃗⃗⃗⃗ ̅𝑎̅̅𝑃̅ 𝑆 ̅𝑎̅̅𝑃̅ × ⃗⃗⃗⃗ 𝐸

,

⃗⃗⃗⃗ [̅𝑎̅̅𝐶𝑃̅]𝑛

𝑓 ×1

(26)

⃗⃗⃗⃗ 𝑃 ̅𝑎̅̅̅ 𝑆 ⃗⃗⃗⃗ 𝑃 ̅𝑎̅̅̅

(𝑡⃗⃗⃗⃗⃗⃗ 𝑆,1 ) =[ …

𝐸

⃗⃗⃗⃗𝑃 [̅𝑎̅̅̅ 𝐶 ]𝑛

𝑓 ×1

𝑇

(𝑡⃗⃗⃗⃗⃗⃗⃗⃗ 𝑆,𝑛𝑓 )

𝑇

𝑇

⃗⃗⃗⃗1 ) (𝑄 … 𝑇 ⃗⃗⃗⃗⃗⃗⃗ (𝑄 𝑛 ) 𝑓

𝑇

−1

(𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑎𝑚𝑏,1 ) … ] 𝑇 (𝑡⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ 𝑎𝑚𝑏,𝑛𝑓 )

× (𝑡⃗⃗⃗𝑃 )

𝑛𝑓 ×1

,

(27)

𝑛𝑓 ×𝑛𝑓

⃗⃗⃗ where 𝑡⃗⃗⃗⃗⃗ 𝑡𝑎𝑚𝑏,𝑖 are the vectors of the supply air temperature, heat source, 𝑆,𝑖 , 𝑄𝑖 , and ⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗⃗ and ambient temperature of the ith Case, respectively.

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3. Explicit expressions of typical cases The theoretical expressions of the temperature distribution and modified indices are presented in Section 2. In this section, the intuitive understanding of the modified indices is presented based on case studies. Specifically, the settings of the studied cases and the validation of the numerical method are illustrated first. Next, the distribution of the modified indices is analyzed. To determine how each index is influenced, the results of different cases are compared.

3.1 Case description A ventilated room with dimensions of 4 m (length, X), 2.5 m (height, Y), and 3 m (width, Z) is adopted. The ceiling has two supply air inlets, the side wall has two exhaust air outlets, and one heat source is installed at the room center. A sketch of the room is shown in Fig. 3 a). The top and side views of the ventilated room are presented in Fig. 3 b) and c), respectively.

To analyze how the modified indices are influenced by the flow field and thermal properties of the convective boundary, four typical cases are adopted. The settings of each case are listed in Table 1. In the studied room, the external building envelope is adjacent to the outdoor environment, while the other internal envelope is equipped with convective terminals. The convective terminals can be controlled artificially by adjusting the supply water temperature; 𝑈𝑖𝑛 , which represents the overall heat transfer coefficient of the convective terminals, is set to 2.5 W/(m2·K) [30]. The south wall is an external envelope in Cases 1, 3, and 4. In Case 2, both south and east walls are external envelopes.

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3.2 Numerical method and model validation The indoor airflow fields are simulated with the ANSYS 14.5 software. The RNG k-e turbulence model is used for the turbulent-flow calculations. In addition, the Boussinesq assumption is applied to include the buoyancy term. The momentum equation is solved with the SIMPLE algorithm, and the difference scheme is QUICK [21]. The room is discretized into 210,000, 430,000, and 880,000 cells for lowdensity, mid-density, and high-density meshes, respectively. After the grid independence tests, the mid-density mesh is adopted for further studies owing to its efficiency and accuracy.

The established expression of indoor temperature is theoretical thus the results by Eq. (17) is the same as that of CFD. However, it is based on a given flow field thus the numerical model for airflow under the typical thermal scenario should be validated. To determine the accuracy, the simulated results of CFD are compared with the experimental results of Tian et al. [31]. In the experiment, the air was supplied by a supply air inlet located at the center of the right wall and exhausted through the exhaust outlet in the ceiling. The ventilation rate was 5.5 air change per hour (ACH), and the total indoor heat source intensity is 399 W. The temperatures along a typical vertical line (x = 2.15 m, y = 1.45 m) is selected for comparison (Fig. 4). According to the results in Fig. 4, the trend of simulated temperature distribution fits well with that of experimental results and the standard derivation is only 0.45 °C, indicating that the accuracy of the numerical results is satisfactory. Therefore, the presented numerical method can be used for further studies.

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As mentioned in Section 2.3, the flow fields of typical thermal scenarios should be prepared in advance. In this study, two flow fields (for the high and low ventilation rates, respectively) of typical thermal scenarios are simulated first. The basic thermal settings of the two flow fields are listed in Table 2. As a result, two fields are obtained with the CFD. The distribution of the velocity vector in the central XZ plane is illustrated in Fig. 5.

3.3 Distribution of modified accessibility indices In the studied ventilated room, four thermal factors affect the temperature distribution: the supply air temperature, water temperature of the convective terminals, outdoor 𝑃 𝑃 𝑃 temperature, and heat source intensity. Based on Eq. (17), ̅𝑎̅̅𝑆𝑃̅, ̅̅̅̅̅̅̅ 𝑎𝐶,𝑡𝑒𝑟 , ̅̅̅̅̅̅ 𝑎𝐶,𝑒𝑥 , and ̅𝑎̅̅̅ 𝐸

are used to determine their influences on 𝑡 𝑃 (as Eq. (28)). Next, the distributions of 𝑃 ̅̅̅̅̅̅̅ 𝑃 𝑃 ̅𝑎̅̅̅ ̅̅̅̅̅̅ ̅̅̅𝑃̅ 𝑆 , 𝑎𝐶,𝑡𝑒𝑟 , 𝑎𝐶,𝑒𝑥 , and 𝑎𝐸 in the central XZ plane (y = 1.5 m) of Case 1 are analyzed.

Since there are four independent thermal factors, four cases with different thermal boundaries, whose detailed information is shown as Table 3, are used to calculate the modified accessibilities. 𝑄 𝑃 𝑃 𝑃 ̅̅̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅𝑃̅ 𝑡 𝑃 = ̅𝑎̅̅̅ 𝑆 × 𝑡𝑆 + 𝑎𝐶,𝑡𝑒𝑟 × 𝑡𝑡𝑒𝑟 + 𝑎𝐶,𝑒𝑥 × 𝑡𝑜𝑢𝑡 + 𝑎𝐸 × 𝑚𝑠 𝐶𝑝

(28)

𝑃 The distribution of ̅𝑎̅̅̅ 𝑆 is shown in Fig. 6. The MASA is higher in the core region of the

supply air jets: the MASA is approximately one near the supply air inlets. Thus, the supply air has a greater contribution to the regions in which the airflow is dominated mainly by the supply air. In other words, a higher ̅𝑎̅̅𝑆𝑃̅ indicates that the temperature is 22 / 53

more sensitive to the supply air temperature. However, 𝑎𝑆𝑃 equals one everywhere in the room according to the ASA [19]. This proves that the MASA reflects the contribution of the supply air in a more intuitive manner. 𝑃 The contour of ̅̅̅̅̅̅̅ 𝑎𝐶,𝑡𝑒𝑟 , which reflects the contribution of the water temperature of the

convective terminals to the indoor temperature, is shown in Fig. 7. In contrast to ̅𝑎̅̅𝑆𝑃̅, 𝑃 ̅̅̅̅̅̅̅ 𝑎𝐶,𝑡𝑒𝑟 is lower in the supply air jet zone. Thus, the water temperature has a greater

influence on the area around the convective terminals than on the jet core zone. 𝑃 Similarly, the contour of ̅̅̅̅̅̅ 𝑎𝐶,𝑒𝑥 , which reflects the contribution of the outdoor 𝑃 temperature to the indoor space, is shown in Fig. 8. According to Fig. 8, ̅̅̅̅̅̅ 𝑎𝐶,𝑒𝑥

exceeds 0.2 only around the ceiling. The contribution of the outdoor temperature is very limited compared with that of the water temperature of the convective terminals: 𝑃 ̅̅̅̅̅̅ the area-averaged 𝑎 𝐶,𝑒𝑥 is only approximately 0.15. Although the heat transfer

coefficients are approximately equal, the total area of the convective terminals is much larger than the external-building envelope in Case 1. Thus, the contributions of the convective terminals are greater.

To present the influence of the heat source, the contour of ̅𝑎̅̅𝐸𝑃̅ is shown in Fig. 9. ̅𝑎̅̅𝐸𝑃̅ is higher next to the heat source and low far away from the heat source. For example, 𝑃 in the jet core zone, ̅𝑎̅̅̅ 𝐸 is low because the air movement is dominated by the

momentum of the supply air. By contrast, the air movement around the heat source is determined by thermal buoyancy. Thus, ̅𝑎̅̅𝐸𝑃̅ is higher in this region.

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3.4 Analysis of influencing factors As previously mentioned, the MASA, MAAT, and MAHS are determined by the flow field and thermal properties of the building envelope. To determine how these indices 𝑎𝑣 ̅̅̅̅̅̅̅ 𝑎𝑣 𝑎𝑣 𝑎𝑣 ̅̅̅̅̅ ̅̅̅̅̅̅ ̅̅̅̅̅ are affected, the volume-averaged indices (𝑎 𝑆 , 𝑎𝐶,𝑡𝑒𝑟 , 𝑎𝐶,𝑒𝑥 , and 𝑎𝐸 ) of the four

cases, which can be calculated with Eqs. (23)-(25), are compared.

The contribution degree of the supply air with respect to the volume-averaged indoor temperature is shown in Fig. 10. According to Eqs. (18) and (23), a higher ventilation rate and better-insulated envelope should lead to a higher ̅̅̅̅̅ 𝑎𝑆𝑎𝑣 . Cases 1, 2, and 3 exhibit high ventilation rates, whereas Case 4 has a low ventilation rate. Consequently, ̅̅̅̅̅ 𝑎𝑆𝑎𝑣 of Case 4 is the lowest owing to the lower ACH. Regarding the insulation of the external building envelope, Case 3 has a better insulation, whereas Case 1 has a normal insulation. Although Cases 1, 2, and 3 have the same ventilation rates, ̅̅̅̅̅ 𝑎𝑆𝑎𝑣 of Case 3 is the highest. This is due to the better insulation. That is, the influence of the outdoor temperature is reduced, and the influence of the supply air increases. In a word, it is possible to adjust the contribution of the supply air by changing the flow field and heat transfer coefficient of the convective boundaries.

The contributions of the water temperatures of the convective terminals are shown in 𝑎𝑣 Fig. 11. It can be seen that ̅̅̅̅̅̅̅ 𝑎𝐶,𝑡𝑒𝑟 of Case 2 is the lowest because the total area of the

internal envelope is the smallest. Nevertheless, MAAT is also influenced by the flow field. Although Cases 1 and 4 have identical external and internal envelopes, the ACH of Case 4 is lower. In other words, the water temperatures of the convective

24 / 53

𝑎𝑣 ̅̅̅̅̅̅ terminals have a greater impact in Case 4. Correspondingly, 𝑎 𝐶,𝑒𝑥 of the four cases is 𝑎𝑣 illustrated in Fig. 12; ̅̅̅̅̅̅ 𝑎𝐶,𝑒𝑥 of Cases 2 and 4 are higher than those of Cases 1 and 3.

The total area of the external building envelope of Case 2 is the largest. As for Case 4, the contribution of supply air is limited because of low ACH thus that from convective terminals and outdoor temperature becomes larger. In addition, the contribution degrees of the supply air temperature and ambient temperature of the convective terminals are approximately the same in Case 1. Reducing the water temperature of the convective terminals is easier than supplying air because mostly chilled water (approximately 7 °C) is provided for the supply air, whereas the water temperatures of the radiant terminals are always higher (approximately 18 °C). 𝑎𝑣 ̅̅̅̅̅ Fig. 13 presents 𝑎 𝐸 of the four cases. According to the analytical expression of

MAHS in Eqs. (19) and (24), a higher supply air rate and better-insulated envelope should lead to a higher ̅̅̅̅̅ 𝑎𝐸𝑎𝑣 . Consequently, ̅̅̅̅̅ 𝑎𝐸𝑎𝑣 of Case 4 is the lowest; owing to the limited ventilation rate, the heat dissipated to the ambient has a larger proportion.

In general, it can be concluded that the airflow field and heat transfer coefficient of the convective boundaries should affect the modified accessibility indices. As a result, their influences on the indoor temperature should be adjusted. For example, a higher ventilation rate and lower heat transfer coefficient of the convective boundaries should lead to a greater influence of the supply air. In addition, because the influence of the different thermal factors are not equal, different grades of energy can be used to maintain the temperature in the target zone by changing the contributions of the supply air, heat source, and convective boundary.

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4. Discussion According to the study results, the indoor temperature is explicitly affected by the supply air, heat source, and ambient temperature of the convective boundary. As mentioned in Section 3.4, the different airflow fields and heat transfer coefficients of convective boundaries result in different contribution degrees of the thermal factors. Thus, the contribution of each thermal factor should be adjusted to save energy by using appropriate insulation and airflow patterns. Taking the insulation of the external building envelope as an example, in the cooling-dominated region, the outdoor temperature is mostly higher than the indoor set-point temperature. As a result, the corresponding MAAT should be lower. Thus, it is recommended to use a better insulated envelope and to reduce the infiltration. If the outdoor temperature is mild, the MAAT should be increased. Consequently, the building envelope and infiltration should be designed according to the climate characteristics to save energy [32]. In a word, the modified indices can be used to adjust the influence of each thermal factor to reduce the total energy consumption.

In addition, the energy grades of the supply air and convective terminals are different. Traditionally, the indoor heat source and heat convection of the building envelope are regarded as disturbances, and the supply air is used to compensate for the disturbance. According to the study results, the disturbance and supply air affect the indoor temperature. However, the energy grades of the supply air and disturbances are different. Thus, different energy grades can be applied to control the temperature in the target zone. For example, pre-handling technologies can be employed with high-efficiency low-grade energy to control the disturbance. Taking the double-skin facade as an example, the surface temperature of the inner glazing is very high 26 / 53

owing to strong solar radiation under summer conditions. This leads to a significant indoor heat gain. However, with ventilation in the cavity, the inner glazing surface temperature decreases, and the indoor heat gain decreases. For example, based on Eq. (29) and Eq.(30), if the outdoor temperature is 35 °C and solar intensity is 200 W/m2, the solar-air temperature of conventional opaque glass (𝛼𝑐 = 0.3) is 59 °C while that of conventional DSF is 43 °C (𝛼𝐷𝑆𝐹 = 0.1) [33]. As a result, the heat gain of DSF is 45 W/m2 per square window while that of conventional opaque window is 87 W/m2. Although the MAAT of DSF case equal to that of conventional glass, the solar-air temperature decreases with pre-handling in DSF thus less heat penetrated and less energy consumed [34].

𝑐 𝑡𝑠𝑜𝑙𝑎𝑟−𝑎𝑖𝑟 = 𝑡𝑜𝑢𝑡 +

𝛼𝑐 𝐼 ℎ𝑎𝑚𝑏

(29)

𝐷𝑆𝐹 𝑡𝑠𝑜𝑙𝑎𝑟−𝑎𝑖𝑟 = 𝑡𝑜𝑢𝑡 +

𝛼𝐷𝑆𝐹 𝐼 ℎ𝑎𝑚𝑏

(30)

27 / 53

5. Conclusions In this study, an explicit theoretical expression of the indoor temperature in a general ventilated room is established. Three indices (MASA, MAAT, and MAHS) are used to determine their degrees of influence on the indoor temperature. Their distributions and influencing factors are analyzed and discussed through case studies. Compared with previous studies, the established method could explicitly reveal the influence of each independent thermal factor, especially the heat convection from the convective boundaries. According to the results, the established method provides results efficiently and serves as guidance to save energy in non-uniform indoor environments. In the following, the research steps and conclusions are summarized: 1) The influence from the convective boundary is of significance, for example, the contribution of the water temperature of convective terminals exceeds 0.4 in most cases, similar to that of supply air. Thus, it is reasonable and accurate to reflect the influence by the corresponding ambient temperature; 2) The influence degree of each thermal factor is determined by the flow field and heat transfer coefficient of the convective boundary. Hence, less energy can be supplied to the room by optimizing the flow field and thermal properties of the convective boundaries. For example, the contribution degree of the supply air is reduced from 0.43 to 0.35 when ventilation rate is reduced by half; 3) The indoor temperature is influenced by both the supply air temperature and ambient temperature of convective boundaries, thus it is appropriate to use different grades of energy for the supply air and convective boundaries to control the temperature in the target zone and save energy.

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Acknowledgments This study was supported by the National Natural Science Foundation of China (Grant Nos. 51578306 and 51638010).

29 / 53

conflict of interest No conflict of interest exists in the submission of this manuscript. All the authors listed have read and approved this version of the article, and due care has been taken to ensure the integrity of the work. No part of this paper has been published or submitted elsewhere.

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a) Sketch of supply air inlet and heat source

b) Nodes adjacent to the kth convective boundary Fig. 1. Sketch of general ventilated room

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Fig. 2. Heat transfer through convective boundary

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a) Sketch of ventilated room

b) Top view of supply air inlets and heat source

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c) Side view of exhaust air outlets and heat source Fig. 3. Sketch and geometric parameters of ventilated room

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2.8 2.4

Height (m)

2 1.6 1.2 0.8

simulation 0.4

experiment 0 21

22

23

24

25

26

27

28

temperature (°C)

Fig. 4. Comparison between experimental and numerical results (x = 2.15 m, y = 1.45 m)

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a) Flow field for 4.8 ACH

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b) Flow field for 9.6 ACH Fig. 5. Velocity vector in a typical plane of Case 1

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Fig. 6. Distribution of MASA.

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Fig. 7. Distribution of modified accessibility of water temperature of convective terminals

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Fig. 8. Distribution of modified accessibility of outdoor temperature

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Fig. 9. Distribution of modified accessibility of heat source (MAHS)

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0.6 0.5 0.4 0.3 0.2 0.1 0 Case1 (LV-SE-NI)

Case2 (LV-LE-NI)

Case3 (LV-SE-BI)

Case4 (SV-SE-NI)

Fig. 10. Volume-averaged modified accessibility of supply air (MASA)

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0.6 0.5 0.4 0.3 0.2 0.1 0 Case1 (LV-SE-NI)

Case2 (LV-LE-NI)

Case3 (LV-SE-BI)

Case4 (SV-SE-NI)

Fig. 11. Volume-averaged modified accessibility of water temperature of convective terminals

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0.3

0.2

0.1

0 Case1 (LV-SE-NI)

Case2 (LV-LE-NI)

Case3 (LV-SE-BI)

Case4 (SV-SE-NI)

Fig. 12. Volume-averaged modified accessibility of outdoor temperature

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0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 Case1 (LV-SE-NI)

Case2 (LV-LE-NI)

Case3 (LV-SE-BI)

Case4 (SV-SE-NI)

Fig. 13. Volume-averaged modified accessibility of heat source (MAHS)

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Table 1. Basic settings of four cases Case

ACH

𝐹𝑒𝑥

𝐹𝑡𝑒𝑟

𝑈𝑒𝑥

2

2

2

Description No.

-1

(h )

Case 1 Case 2

9.6

Case 3 Case 4

4.8

(m )

(m )

(W/(m ∙K))

10

49

2.5

LV-SE-NI

17.5

41.5

2.5

LV-LE-NI

10

49

1

LV-SE-BI

10

49

2.5

SV-SE-NI

Note: 𝐹𝑒𝑥 and 𝐹𝑡𝑒𝑟 represent the area of the external building envelope and convective terminals, respectively; 𝑈𝑒𝑥 is the overall heat transfer coefficient of the external envelope [W/(m2·K)]; LV and SV represent high and low ventilation rates, respectively; LE and SE represent large and small external building envelopes, respectively; NI and BI represent normal and better insulations, respectively.

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Table 2. Basic settings of two flow fields

LV SV

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𝑡𝑠

𝑄

𝑈𝑒𝑥

𝑈𝑖𝑛

𝐹𝑡𝑒𝑟

𝐹𝑖𝑛

𝑡𝑎𝑚𝑏

𝑡𝑤𝑎𝑡

°C

W

W/(m2‧K)

W/(m2‧K)

m2

m2

°C

°C

18

150 2.5

2.5

10

49

35

27

18

75

Table 3. Thermal settings of four cases to access the modified accessibility Supply air

Water

Ambient

Intensity of

temperature

temperature

temperature

heat source

°C

°C

°C

W

a

17

27

35

150

b

20

20

30

100

c

17

27

30

100

d

20

20

35

150

Case No

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