Direct numerical simulation of contaminant removal in presence of underfloor air distribution system

Abstract

Indoor contaminant removal over 0.5 ≤ FrT ≤ 5.0, 0.5 ≤ N ≤ 5.0, and 50 ≤ Re ≤ 500 was investigated numerically, wherein Fr_T refers to the Froude number, N refers to the buoyancy ratio, and Re refers to the Reynolds number. As demonstrated, the ventilation effectiveness increased with increasing contaminant source intensity and air supply intensity at a constant air temperature, indicating that increase the fresh air can effectively eliminate contaminants in this case. At high air supply temperatures, the heat retention time and contaminant transport was extremely short, and the fresh air induced by strong natural convection floating lift was rapidly discharged. Additioanlly, the air supply intensity had significant effects on contaminant removal. Quantification of the ventilation effectiveness under the combined effects of air supply intensity, air supply temperature and contaminant source intensity was determined based on the results of direct numerical simulations.

1. Introduction

Indoor air pollution is hazardous to the human immune systems, causing the body to be more vulnerable to viral diseases in the era of COVID-19 pandemic. Therefore, appropriate indoor ventilation is essential to reduce concentrations of air pollutants and improve air quality [1,2]. The underfloor air distribution (UFAD) system generates fresh air via the floor air vent, thereby reducing the contaminant concentration and improve indoor air quality [3].

Nowadays, human typically spend 90 % of their time indoor, while natural ventilation is not sufficient to guarantee indoor air quality [4]. An appropriate ventilation system can not only facilitate the control of indoor temperature and humidity, but also remove indoor pollutants. The ventilation system mainly considers the rationality of indoor airflow distribution and different air distributions lead to different ventilation modes [5]. Meanwhile, the indoor pollutant concentrations can be several times higher than outdoor pollutant concentrations. Inappropriate indoor air flow would severely affect human health and may cause increased building energy consumption [[6], [7], [8]]. It has been demonstrated that indoor airflow play a key role in air quality and pollution control.

The UFAD systems can achieve rational indoor air quality at low costs. It has been the focus in the field of air conditioning over the past decades [9,10]. Indeed, the UFAD system injects fresh air into the room via the ventilation inlet on the floor, and the air pollutants are discharged from the air outlets [11].

The UFAD system was first used for cooling of computer rooms in the 1950s and subsequently introduced in the West German Foreign Office building in the 1970s [12]. Numerical simulation of heat and mass transport of indoor airflow is a critical to understand and evaluate the indoor air quality. By analyzing the indoor ventilation strategies (e.g., mixed ventilation, UFAD and displacement ventilation), indoor air quality and CO₂ concentration can be determined [[13], [14], [15]], as air velocity field, temperature field and pollutant distribution under different ventilation modes can be simulated. A series of working condition simulations based on different parameters and analysis of personal have been reported, including the correlation of air temperature and personal comfort [[16], [17], [18], [19], [20]]. Alajmi et al. [21] found that any inhabited room can be divided into the cold zone at the bottom, the narrower zone in the lower part, the transition zone and the upper warm zone. Meanwhile, the room height also affects air stratification. Tsai et al. [22] reported that the impacts of air temperature on the vertical distribution of temperature were negligible, and the vertical momentum had a positive correlation with the temperature in the lower part; as the airflow increased, temperature stratification induced by the increased supply of air momentum decreased. Ismail et al. [23] investigated the thermal performance of UFAD in rooms with one and two occupants using Computational Fluid Dynamics (CFD), and the results (e.g., distributions of airflow velocity and temperature, energy consumption and comfort) were compared with the top air distribution systems. Additionally, position and height of heat source and intensity of heat supply would greatly affect temperature stratification and personal comfort [24]. UFAD systems were numerically simulated in a large space with air supply intensity and the position of ventilation inlet as the variables [25,26]. The results showed that requirements on the personal comfort can be satisfied under all working conditions involved.

Indoor airflow structure is one of the most important physical parameters of the indoor air. Indeed, personal comfort is directly dependent on the rationality of indoor airflow structure. The indoor air is mainly reflected by a mixed process of air flow and heat/pollutant transmission, including air convection heat transfer and mass transfer. Therefore, improvement of indoor air quality can only be achieved based on full understanding of the transport and diffusion of indoor heat and pollutants under different conditions. In presence of both temperature gradients and concentration gradients, natural convection with double buoyancy may be observed. Concentration and temperature gradients generate concentration and thermal buoyancy, respectively. The magnitude, direction, and relative position of temperature and concentration gradients have direct impacts on natural convective heat transfer and mass transfer. Additionally, the interaction (coupling effect) of heat and mass transfer was also observed for indoor air. The concentration field (or the temperature field) affects the flow field, which in turn affects the temperature field (or the concentration field), indicating an indirect correlation of the concentration field and the temperature field; complex fluid flow conditions tend to be observed due to variations of thermal diffusion rate and solute diffusion rate. With the widespread of UFAD systems in recent years, both heat and pollutant buoyancy shall be taken into consideration. Although this phenomenon has been observed for a long time, various problems about the fluid transportation mechanism in presence of both temperature and concentration gradients remain to be solved. Theoretical analysis, as well as numerical simulation and experimental methods, has been widely employed for such problems. Lin et al. [27] proposed a model of UFAD systems comprising one heat source and one ventilation outlet and verified by experiments. The results showed that parameters influencing airflow include thermal buoyancy flux, volume energy and momentum. This model was established base on the flow fountain model and the heat source plume. Liu et al. [28] proposed models with multiple ventilation inlets and one heat source floor at different levels. The experimental results showed that the fluid flow is mainly determined by the entrainment and negative buoyancy jet entering the plume. A theoretical model based on stratified ventilation was established and the results obtained by this model were compared with the experimental results. Lin et al. [29] demonstrated that as the ventilation volume of the UFAD system increased, the gradient slope of vertical temperature and the stratified height increased. Kanaan [30] proposed simulation of UFAD systems with multiple ventilation inlets and multiple heat sources from both theoretical and CFD perspectives. The results showed that the CO₂ concentration predicted by theoretical analysis was consistent with that obtained by the CFD.

Ventilation effectiveness is a critical evaluation index reflecting the indoor discharging effect [31,32]. It is usually employed as an indoor air quality indicator for the design of heating ventilation, especially in cases with known pollutant concentrations [[33], [34], [35]]. In recent years, studies of thermal comfort focused on parameters such as predicted mean vote (PMV), percentage of dissatisfaction (PPD), ventilation efficiency and the feeling of blowing. CFD is a feasible method to evaluate indoor air quality from a microscopic perspective and facilitate the control of pollutant distribution and diffusion. Indeed, a comprehensive analysis of the indoor temperature field, velocity field and PMV/PPD is essential for development of optimized indoor air quality [36,37]. Cheong et al. [10] investigated the effects of underfloor ventilation system and traditional mixed ventilation system in a primary school on elimination of airborne pathogens. The results demonstrated that the vent shall be installed away from the breathing area. Fan et al. [38] thoroughly investigated different ventilation modes during the outbreak of COVID-19, aiming to improve thermal comfort, air quality and virus elimination, the thermal comfort, ventilation efficiency and discharging efficiency under different ventilation modes in different situations were assessed. Gao et al. [39] explored the influences of surface temperature of ceiling and air velocity on indoor air pollutant distribution. The vertical distribution of indoor air temperature, wind speed, pollutant concentration and discharging effect were investigated, respectively. The results showed that the surface temperature of ceiling had negligible effects on the vertical distribution of air pollutants, while the air supply intensity had significant influences on indoor air pollutant distribution and the discharging effect. Zhang et al. [40] focused on the thermal stratification of UFAD system, and obtained characteristic parameters of the horizontal, dimensionless temperature distribution between the ankle and head by using multivariable regression analysis.

To the best of our knowledge, few studies on UFAD system using Direct Numerical Simulation (DNS) have been reported, and the influences of the air supply temperature, air supply intensity and contaminant source intensity remain unclear. This study aims to deepen the understanding of the effectiveness of pollutant removal in a room with a UFAD system.

This paper is organized as follows: Section 2 briefly describes the physical model, the numerical methodology, mesh generation and the results of time-step independence test, as well as the DNS results were introduced in Section 3.Section 4 presents a description of the evolution of typical thermal-solutal contours based on the DNS results, along with the pollutant source intensity, the air supply temperature and quantitative analysis of the air supply intensity. The conclusions are given in Section 5.

2. Numerical methodology and numerical simulations

Fig. 1 shows a 2D rectangular domain (height = H, width = W) representing a physical system and corresponding numerical. Herein, the origin of the 2D cylindrical coordinate system (X, Y) is located at the floor center. The working fluid is an air-pollutant mixture. Meanwhile, the ambient fluid is a Newtonian one, which is initially stationary under a uniform temperature (T_a) and a uniform concentration (Ca). At the center of the floor (Y = 0), the radius of the fluid is X₀. The remaining floor is an essentially rigid, insulated and stationary boundary. At t = 0, a fresh air jet with temperature of T₀ (T₀ < T_a) and pollutant concentration of C₀ is injected into the domain at a uniform rate (V₀) to trigger the air flow and exits naturally from the air outlet port at the top of the side walls.

Fig. 1.

The physical and numerical models.

The flow of air-pollutant in a room is governed by the 2D incompressible Navier-Stokes (N–S) equations. The buoyancy was determined by the Oberbeck-Boussinesq approximation, with the gravity along the negative Y (vertical) direction.

The density of fluid at T and C, $\rho \left(T,C\right)$, can be determined by:

$$\rho \left(T,C\right)={\rho }_a\left(T_a,C_a\right)\left[1-{\beta }_T\left(T-T_a\right)-{\beta }_C\left(C-C_a\right)\right]$$ (1)

where ${\rho }_a\left(T_a,C_a\right)$ refers to the initial density of the ambient fluid at Ta and Ca, and ${\beta }_T$ and ${\beta }_C$ are the volumetric thermal and mass expansion coefficients of the fluid, respectively.

$Re$, $F{r}_T$, $Sc$ , and N are defined as (equation (2) and (3)) follows:

$$Re=\frac{V_0{X}_0}{\nu },F{r}_T=\frac{V_0}{\sqrt{g{\beta }_T\left(T_a-T_0\right)X_0}},Sc=\frac{\nu }{D}$$ (2)

$$N=\frac{{\beta }_C\left(C_a-C_0\right)}{{\beta }_T\left(T_a-T_0\right)}$$ (3)

N denotes the influences of concentration on density in comparison with temperature, $\nu$ is kinematic viscosity, and D is the diffusivity.

The overall ventilation effectiveness for temperature distribution ${\epsilon }_t$ serves as a quantitative indicator associated with the heat distribution inside the cavity, and the effectiveness for pollutant removal, ${\epsilon }_c$, serves as a quantitative indicator associated with the pollutant distribution inside the cavity [41]. Herein, the two parameters are defined as follows (equation (4) and (5)):

$${\epsilon }_t=\frac{T_{out}-T_0}{T_{average}-T_0}$$ (4)

$${\epsilon }_c=\frac{C_{out}-C_0}{C_{average}-C_0}$$ (5)

where T_out and C_out are the average temperature and average pollutant concentration at the outlet, respectively, T_average and C_average are the average temperature and average pollutant concentration inside the cavity, respectively, T₀ and C₀ are the average temperature and average pollutant concentration at the inlet, respectively.

2.1. Mesh generation and bench marking of numerical simulation

To keep the numerical simulation independent from time step and grid, we conducted a series of independent tests. Fig. 2 shows the results of grid and time-step independent tests with $\mathit{Pr}=0.71$, $R{a}_T=3.0\times {10}^8$, $Le=1.0$, $N=-1$, $Re=300$, and $F{r}_T=0.5$. Herein, the vertical profile of velocity, overall ventilation effectiveness in terms of temperature distribution and the pollutant removal were demonstrated. Three different meshes (1.05695 million meshes, 1.62104 million meshes, and 2.18026 million meshes) (Fig. 2a–c) and three different time steps (0.01s, 0.025s, and 0.05s) (Fig. 2d–f) were involved. As indicated, the DNS results with different meshes but constant time step (0.025 s) were identical. The DNS results with different time steps but constant mesh (1.62104 million meshes) were also identical. It was found that 1.62104 million meshes and a time step of 0.025 s were employed for all DNS cycles in this study as they can deliver numerical simulation results with sufficiently high accuracy.

Fig. 2.

DNS results in cases of different meshes (left column) and different steps (right column): ventilation effectiveness for temperature distribution (top row), ventilation effectiveness for pollutant removal (middle row), and vertical profiles of velocity (bottom row).

The ANSYS Fluent code used for DNS cycles were bench marked to guarantee that the accuracy was sufficiently high. Indeed, a numerical code requires verification and validation using data available. Specifically, the subject of several works of numerical validations was selected for code benchmarking [[42], [43], [44], [45]]. Additionally, the cavity is heated differentially with adiabatic horizontal walls. To achieve that, it was set that $\mathit{Pr}=0.71$, $R{a}_T={10}^7$, $Le=1$. Table 1 lists the average Nusselt number $Num$, which is defined by:

$$Num=\underset{0}{\overset{1}{\int }}{\left(N{u}_L\right)}_{X=0}dY=\underset{0}{\overset{1}{\int }}{\left(-\partial \theta /\partial X\right)}_{X=0}dY$$ (6)

Table 1.

Num in the case of Ra_T = 7.0 × 10⁹, Pr = 0.71, Le = 1.0.

N	Present study	Sezai [46]	Koufi L [47]	Xamán [48]	Béghein [49]
−0.01	16.6	16.3	13.4	16.6	16.4
−0.1	16.0	15.9	16.0	15.9	16.0
−0.2	15.5	15.4	15.3	15.4	15.5
−0.5	13.6	13.5	13.6	13.6	13.6
−0.8	10.7	10.5	10.6	10.6	10.6
−0.9	8.9	8.6	8.8	8.8	8.8
−1.5	13.7	13.5	13.5	13.6	13.6
−5.0	23.7	23.7	23.7	23.7	23.7

As demonstrated, the accuracy of ANSYS Fluent code is sufficiently high for direct numerical simulation of pollutant removal in a room with underfloor air distribution systems.

2.2. Numerical simulations

A total of 22 DNS runs (see Table 2 for details) were executed using ANSYS Fluent 2020 R2. Air mixture was employed as the fluid in DNS, wherein ${\rho }_a=1.16kg/m^3$, kinematic viscosity $\nu =1.85\times {10}^{-5}$ m²/s, ${\beta }_T=3.33\times {10}^{-3}$ 1/K, ${\beta }_C=-0.34$ 1/(%), Pr = 0.71, and Le = 1.0. As observed, parameters reflecting the air intensity, the air temperature and the pollutant intensity were Fr_T, Re and N, which varied in the ranges of 0.5–5.0, 50–500 and 0.1–10.0, respectively.

Table 2.

Key data of the DNS runs.

Runs	N	FrT	Re	Fr	V0(m/s)	Qs (L/s.m2)	ΔT (K)	ΔC (%)
1	3.0	2.0	50	1.0	0.02658	7.51 × 10−5	0.180231	0.005296
2	3.0	2.0	100	1.0	0.053161	1.50 × 10−4	0.720926	0.021182
3	3.0	2.0	200	1.0	0.106322	3.00 × 10−4	2.883703	0.08473
4	3.0	2.0	300	1.0	0.159483	4.51 × 10−4	6.488331	0.190642
5	3.0	2.0	500	1.0	0.265805	7.51 × 10−4	18.02314	0.529562
6	3.0	0.5	100	0.3	0.053161	1.50 × 10−4	11.53481	0.33892
7	3.0	1.0	100	0.5	0.053161	1.50 × 10−4	2.883703	0.08473
8	3.0	1.5	100	0.8	0.053161	1.50 × 10−4	1.281646	0.037658
9	3.0	3.0	100	1.5	0.053161	1.50 × 10−4	0.320411	0.009414
10	3.0	5.0	100	2.5	0.053161	1.50 × 10−4	0.115348	0.003389
11	0.5	0.5	100	0.4	0.053161	1.50 × 10−4	11.53481	0.056487
12	0.5	1.0	100	0.8	0.053161	1.50 × 10−4	2.883703	0.014122
13	0.5	1.5	100	1.2	0.053161	1.50 × 10−4	1.281646	0.006276
14	0.5	2.0	100	1.6	0.053161	1.50 × 10−4	0.720926	0.00353
15	0.5	3.0	100	2.4	0.053161	1.50 × 10−4	0.320411	0.001569
16	0.5	5.0	100	4.1	0.053161	1.50 × 10−4	0.115348	0.000565
17	0.1	2.0	100	1.9	0.053161	1.50 × 10−4	0.720926	0.000706
18	1.0	2.0	100	1.4	0.053161	1.50 × 10−4	0.720926	0.007061
19	1.5	2.0	100	1.3	0.053161	1.50 × 10−4	0.720926	0.010591
20	2.0	2.0	100	1.2	0.053161	1.50 × 10−4	0.720926	0.014122
21	5.0	2.0	100	0.8	0.053161	1.50 × 10−4	0.720926	0.035304
22	10.0	2.0	100	0.6	0.053161	1.50 × 10−4	0.720926	0.070608

3. Results and discussion

3.1. Qualitative analysis

Fig. 3 shows the contour evolution of transient temperature, pollutant concentration, and streamlines of a typical DNS run with Fr_T = 2.0, Re = 200, and N = 3.0 at six moments. As demonstrated, heat and pollutant concentration at the bottom, especially around the vent, decreased as the low-temperature fresh air passed the inlet port. The temperature and pollutant concentration profiles can be divided into three section: (1) the high temperature and pollutant concentration stage (top layer), (2) the intermediate stage where temperature and concentration decreases linearly to the bottom layer, and (3) the low temperature and pollutant concentration stage (bottom layer). The sequences of temperature and pollutant concentration profiles indicated that the height of the bottom layer increased and the cavity was filled by fresh air until the steady state, with pollutant discharged.

Fig. 3.

Transient contours of temperature (left column), pollutant concentration (middle column), and streamlines (right column), wherein Fr_T = 2.0, Re = 200, and N = 3.0. Herein, temperature (θ) and pollutant concentration (φ) were non-dimensionalized with (T − T₀)/(T_a − T₀) and (C–C₀)/(C_a– C₀), while τ was non-dimensionalized with X₀/V₀.

Fig. 4 shows the isotherms and iso-concentrations in the case of Fr_T = 2.0, Re = 200, and N = 3.0 at different heights, y = Y/H (0.03, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, and 0.9).

Fig. 4.

Time series of temperature (a) and pollutant concentration (b) at different heights with Fr_T = 2.0, Re = 200, and N = 3.0, and the time series of temperature difference (c) and pollutant concentration difference (d) at indoor inlet port and outlet port with temperature (θ), pollutant concentration (φ) being non-dimensionalized with (T − T₀)/(T_a − T₀), (C–C₀)/(C_a– C₀), and τ being non-dimensionalized with X₀/V₀.

As observed, the fresh air supply at the early stage, the indoor temperature and the pollutant concentration decreased monotonously, while the decreasing rates decreased. As the airflow reached full development, the dimensionless temperature and pollutant concentration remained basically unchanged, indicating that heat and pollutant had been completely removed at this height, as illustrated in Fig. 4 (a) and (b). As illustrated in Fig. 4 (c) and (d), the time series of temperature and pollutant concentration of indoor inlet port and outlet port. Such results confirmed the results shown in Fig. 3.

3.2. Quantitative analysis

As air pollutant and heat are closely related to each other owing to the shared source, the ventilation effectiveness for temperature distribution ε_t is equal to that for pollutant removal ε_c, hereinafter referred to as the ventilation effectiveness ε.

For each DNS run, a time series of ventilation effectiveness could be determined based on the results of direct numerical simulation. Fig. 5 shows four examples of such time series in cases of different Re, N, and Fr_T. At the early stage of inlet airflow, the ventilation effectiveness increased monotonously, while the increasing rate decreased gradually until the maximum ventilation effectiveness was reached at the dimensionless moment τ. As the airflow reached full development, the ventilation effectiveness remained unchanged, indicating that heat and pollutant of the whole room, at the inlet port, and at the outlet port were consistent, suggesting that heat and pollutants were completely removed at this height. Additionally, Fig. 5 (a)–(b) and (c)-(d) shows that the ventilation effectiveness in all DNS runs exceeded 1.0 as the temperature and pollutant concentration at the outlet were higher than those at the inlet, resulting in mixing and entrainment of heat and pollutant between the fresh airflow supply from inlet, the room airflow and the outlet airflow.

Fig. 5.

Time series of ventilation effectiveness at Fr_T = 2.0, N = 3.0, 50 ≤ Re ≤ 500 (a), Fr_T = 2.0, Re = 100, 0.1 ≤ N ≤ 10.0 (b), Re = 100, N = 0.5, 0.5 ≤ Fr_T ≤ 5.0 (c), Re = 100, N = 3.0, 0.5 ≤ Fr_T ≤ 5.0 (d), wherein τ was non-dimensionalized with X₀/V₀.

3.2.1. Effect of N (contaminant source intensity)

Fig. 6 shows the numerical results at 0.1 ≤ N ≤ 10.0, Fr_T = 2.0, and Re = 100. As observed, the indoor airflow moved by following a double vortex pattern (one counterclockwise and one clockwise). At the full development stage, the thermal buoyancy force was greater than the solutal buoyancy force, wherein the indoor airflow was driven by thermal natural convection. The gradient of pollutant concentration decreased, while the temperature gradient increased. As a result, the indoor air velocity decreased as the buoyancy negative ratio N increased. The maximum temperature and pollutant concentration were observed at the roof, while temperature and pollutant concentration at the floor were more uniformly distributed. When N > 1.0, the solutal buoyancy force was greater than the thermal buoyancy force, the movement of air-pollutant mixture was induced by the solutal buoyancy force. This can be attributed to the fact that as the buoyancy ratio N increased, the pollutant buoyancy increased and the fluid movement was accelerated. Additionally, the isotherms and iso-concentrations were linearly stratified from top down. Overall, the indoor air quality can be improved by rationally increasing the pollutant concentration (within a reasonable range).

Fig. 6.

Dimensionless isothermal (left), iso-concentrations (middle), and streamlines (right) at 0.1 ≤ N ≤ 10.0, Fr_T = 2.0, and Re = 100..

Fig. 9 (a) shows the ventilation effectiveness ε vs. N in all DNS runs. As observed, thet ventilation effectiveness increased as N increased, which can be attributed to the increase of negative buoyancy. The power-law fashion can be quantified by:

$$\mathrm{\epsilon }=3.27303N^{0.12585}$$ (7)

wherein the regression coefficient was 0.9962.

Fig. 9.

(a) and (b) ε vs. N and Fr, respectively, (c) and (d) ε vs. Re and Fr-0.08Re0.39N0.0024, respectively.

3.2.2. Effect of Fr (air supply temperature)

Fig. 7 shows the numerical results at 0.5 ≤ Fr_T ≤ 5.0, N = 3.0, and Re = 100. As Fr decreased, the ventilation effectiveness increased, and the magnitude of the stratified effect increased. This can be attributed to the fact that the negative buoyancy induced by both temperature and pollutant concentration increased as Fr decreased, resulting in reduced Fr and enhanced mixing and entrainment of the inlet airflow.

Fig. 7.

Dimensionless isothermal (left column), iso-concentrations (middle column), and streamlines (right column) at 0.5 ≤ Fr_T ≤ 5.0, N = 3.0, and Re = 100.

The dependence of ε on Fr can be quantified by:

$$\mathrm{\epsilon }=3.58963{\mathit{Fr}}^{‐0.36}$$ (8)

wherein the regression coefficient was 0.96, as shown in Fig. 9(b).

3.2.3. Effect of Re (air supply intensity)

Fig. 8 illustrates the numerical results at 50 ≤ Re ≤ 500, Fr_T = 2.0, and N = 3.0. In cases of 0.1 ≤ N ≤ 10.0, the isotherms and isoconcentrations exhibited linear staratification from top down, with negligible entrainment of the inlet airflow into the core region of the room airflow. As a result, air-upflow and air-downflow were indistinguishable. When Re ≤ 50, the indoor airflow was essentially double diffusion natural convection with coexistence of temperature and pollutant concentration effects. Additionally, indoor heat and pollutant stayed in the room for a long time, as shown in Fig. 8 (a), (g), and (l). At relatively large Re, as the air supply intensity increased, the core region of room was separated from the upflow, and the air jet height of floor air supply outlet also increased with the air supply intensity.

Fig. 8.

Dimensionless isothermal (left column), iso-concentrations (middle column), and streamlines (right column) at 50≤ Re ≤ 500, N = 3.0, and Fr_T = 2.0.

The air supply velocity linearly decreased as room height increased, and the temperature difference would not cause the indoor wind speed to exceed the upper limit of 0.25 m/s recommended by ASHRAE.

As shown in Fig. 9(c), the influences of ε on Re can be obtained by:

$$\epsilon =\{\begin{array}{l}0.0075Re+2.934250\le Re\le 200\\ 0.00002R{e}^2-0.0009Re+4.511200<Re\le 500\end{array}$$ (9)

where the coefficient was 0.9875 and 1.0, respectively.

As the dependence of ε on N, Fr, and Re can be obtained by the relations (7), (8) and (9), respectively. The combined effect of N, Fr and Re can be quantified by multivariable regression analysis:

$$\mathrm{\epsilon }=0.36496{\mathit{Fr}}^{‐0.08}R{e}^{0.39}{N}^{0.0024}+1.28578$$ (10)

where the regression coefficient was 0.98, as shown in Fig. 9 (d).

4. Conclusions

Direct numerical simulations were conducted to investigate the UFAD systems, and explore the effects of Re, Fr and N on indoor air quality (IAQ) and ventilation effectiveness both qualitatively and quantitatively. As demonstrated, ε increased with Re and N, but decreased as Fr increased, ε was mainly influenced by Re. Stratifications of temperature and pollutant concentration were observed in all direct numerical simulations (DNS). Additionally, these feature parameters were significantly influenced by Fr and Re, although the effect of N was negligible and weaker than those of Fr and Re. This study provides references for preliminary design of UFAD systems.

Funding statement

The support from the National Natural Science Foundation of China (Grant No. 12362023) is gratefully acknowledged.

Data availability statement

Data will be made available on request.

CRediT authorship contribution statement

Yaowen Xia: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Saidong Lyu: Writing – original draft, Validation.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.