Long-term prediction of dynamic distribution of passive contaminant in complex recirculating ventilation system.

Recirculating ventilation systems may act as carriers of hazardous substances. The long-term prediction of the dynamic distribution of contaminants in this type of system is crucial for the evaluation of pollution and further design of more efficient ventilation systems. However, few convenient methods can predict the dynamic distribution of contaminants, because the dynamic supply air concentrations resulting from air recirculation are unknown, especially over long time periods, such as months or years. In this study, a novel method is proposed to predict the dynamic distribution of contaminants over a long time period in a complex recirculating ventilation system, where an algebraic expression based on the indices of the response coefficient is applied to account for the relationship between the contaminant distribution inside the room and various boundary conditions. The method is established by obtaining comprehensive mathematical descriptions of the relationships between concentrations of contaminants in the air handling units, supply air inlets, return air outlets, and fresh air. Hourly supply air concentrations can be easily obtained by solving a matrix, and the dynamic distribution of contaminants is then calculated using an expression based on the response coefficient. The reliability of the proposed method is analyzed by both experimental and numerical methods. A simplified method is suggested to accelerate the time-consuming calculation of the response coefficient. The proposed method is beneficial for predicting three-dimensional dynamic distribution of contaminants in complex ventilation systems with an acceptable accuracy and time cost.

average concentration of the initial contaminant [kg/m3]

contaminant concentration of point p at moment 0 [kg/m3]

contaminant concentration of point p at the jth time step [kg/m3]

average concentration from all of the outlets at the steady state, when the emission rate from the n th contaminant source is [kg/m3]

concentration in the th fresh air inlet in the th room [kg/m3]

outdoor air concentration at the th time step [kg/m3]

return air concentration in the th GAHU from the th room at the jth time step [kg/m3]

total return air concentration in the th GAHU at the jth time step [kg/m3]

concentration in the th outlet for the th GAHU in the th room [kg/m3]

contaminant concentration in supply air from the n th inlet at moment [kg/m3]

contaminant concentration in supply air from the n th inlet at the 0th time step [kg/m3]

concentration in the th inlet for the th GAHU in the th room [kg/m3]

supply air concentration of the th GAHU at the jth time step [kg/m3]

known item caused by the concentrations in supply air and fresh air, and emission rate of contaminant source at the previous time steps, considering the time delay in part of the air ducts [kg/m3]

fresh air ratio of the th GAHU [-]

delayed time step for the transport of contaminant from the outdoor air opening to the th inlet [-]

delayed time step for the transport of contaminant from the th outlet to the th GAHU [-]

delayed time step for the transport of contaminant from the th GAHU to the th inlet [-]

number of contaminant sources [-]

number of contaminant sources in the th room [-]

number of direct fresh air inlets in the th room [-]

number of independent rooms [-]

number of return air outlets for the th GAHU in the th room [-]

number of return air outlets with neglected time delay [-]

number of supply air inlets [-]

number of supply air inlets for the th GAHU in the th room [-]

number of supply air inlets with neglected time delay [-]

number of GAHUs [-]

ventilation rate of the room [m3/s]

ventilation rate of the th room [m3/s]

fresh air flow rate of the th GAHU [m3/s]

return air flow rate of the th GAHU [m3/s]

supply air flow rate of the th GAHU [m3/s]

ratio of return air flow rate in the th outlet of the th GAHU from the th room to the total return air flow rate of the th GAHU from the th room [-]

ratio of return air flow rate of the th GAHU from the th room to the total return air flow rate of the th GAHU [-]

emission rate from the n th contaminant source at moment [kg/s]

emission rate from the n th contaminant source at the 0th time step [kg/s]

emission rate from the th contaminant source at the ith time step [kg/s]

volume of the room [m3]

RCCS at an arbitrary point p from the n th contaminant source at the jth time step [-]

RCSA at the th outlet from the th fresh air inlet at the th time step [-]

RCID at an arbitrary point p from the initial concentration distribution at the jth time step [-]

RCSA at an arbitrary point p from the n th inlet at the jth time step [-]

purification efficiency of contaminant in the th GAHU [-]

purification efficiency of contaminant in fresh air in the th room [-]

transient time [s]

time interval [s]

Air Handling Unit

Computational Fluid Dynamics

Fan Coil Unit

Generalized Air Handling Unit

Room Air Conditioner

Response Coefficient to Contaminant Source

Response Coefficient to Initial Distribution

Response Coefficient to Supply Air

Introduction

People spend more than 90% of their time in enclosed environments, including vehicles and rooms of buildings [1], [2]. Therefore, effectively controlling the pollution from various types of indoor and outdoor sources of hazardous substances is crucial for the health and safety of human beings [3]. Buildings are vulnerable to pollutants such as chemical and biological agents [4], [5], [6], [7]. Moreover, events in the last decade, such as the outbreaks of Severe Acute Respiratory Syndrome (SARS, 2003) and the H1N1 Type A influenza (2009), have revealed the importance of taking more active measures to protect the environment in buildings [8], [9]. Furthermore, the increasingly serious hazy weather (i.e., fine particulate matter), especially in China, in recent years has greatly raised awareness of the need to maintain an acceptable indoor air quality [10].

Ventilation systems play an important role in controlling indoor environment, specifically air temperature, humidity, and pollutant concentration. However, in most buildings with central air conditioning and ventilating systems, a large portion of old indoor air is recirculated to the air handling unit (AHU) to save energy [11], [12], [13]. The ventilation systems in these situations, therefore, act as carriers of hazardous substances [14]. The contaminants released in one room may be transported with return air into neighboring rooms or even distant rooms through the ventilation system, causing widespread exposure to a hazardous environment. Therefore, accurately predicting the dynamic distribution of contaminants in buildings with complex ventilation systems that utilize air recirculation during long time periods, such as months and years, can contribute to the overall evaluation of the exposure level of occupants to pollution, and inform the future design of more efficient ventilation systems.

Computational fluid dynamics (CFD) has been widely utilized to predict the spatial and temporal distribution of indoor contaminants. Numerous studies have used CFD simulations to study ventilation and air quality [15], [16], [17], [18], [19], [20]. Other attempts have been conducted to improve the computing speed of CFD [21], [22], [23]. Most of the existing work related to CFD has focused on one room, where all of the boundary conditions are known. However, in many actual buildings, different rooms are always connected with each other through ventilation systems that recirculate air. As the concentration of contaminant in return air is unknown, the concentration of contaminant in supply air is unknown as well; therefore, the predictions of the CFD simulations are usually inaccurate [24].

Multizone airflow network models can solve the issue of contaminant variations in the entire building, as they assume that the concentration of contaminants in each room is uniform [25], [26], [27]. For a specific condition of contaminant release, the concentration of the contaminant at an arbitrary position in the room is uniform for ideal mixing ventilation; however, the concentrations of the contaminant at different indoor positions and in return air or exhaust air outlets are actually different for different airflow patterns, such as actual mixing ventilation [28], [29], displacement ventilation [30], [31], [32], [33], under floor air distribution [34], [35] and personalized ventilation [36], [37], [38], [39]. Therefore, the exposure levels in local areas of rooms (especially the occupied zones) cannot be predicted and evaluated by the multizone model.

Some prediction methods have been proposed for the distribution of contaminants in recirculating ventilation system. Waters and Simon [40] presented a method to consider the effect of air recirculation in a ventilated space; however, it can only be utilized in a simple case. Li et al. [24], [41] defined a general ventilation system covering all-air system with air recirculation, a fan coil unit (FCU), a room air conditioner (RAC), an air cleaner system, etc., and proposed two methods to predict the age of air and the contaminant distribution, respectively, in complex ventilation systems with air recirculation. Both methods are focused on the steady state and cannot be used for unsteady prediction. Hiyama et al. [42] proposed a method that predicts the transient contaminant distribution with air recirculation by coupling three-dimensional transient pollutant transport into a flow network model; however, this method is mainly used for predictions over a short period of time, like 1 h, and is not suitable for a long period of time because of the limited computing time.

In this paper, an algorithm to calculate the distribution of contaminant over a long period of time is proposed based on the general ventilation system described by Li et al. [41]. The reliability of the proposed method is comprehensively analyzed by an experiment and several numerical cases.

Algorithm to predict the distribution of contaminant over a long period of time in a recirculating ventilation system

In order to calculate the dynamic distribution of contaminant in a recirculating ventilation system, the following assumptions are made to simplify the problem: (1) The airflow field are at steady state, and the density of air is constant; (2) the contaminant is a passive gas that has no impact on the flow field; and (3) There is no air leakage in the ductwork, and the flow in the ductwork is fully mixed [24].

General ventilation system

Numerous types of ventilation and air conditioning devices have air recirculation characteristics, such as all-air systems with air recirculation, fan coil units (FCUs), room air conditioners (RACs), and air cleaners. Li et al. [41] constructed a general ventilation system to cover all of the mentioned devices, as shown in Fig. 1 .

Fig. 1

Schematic representation of the general ventilation system [41].

The general ventilation system consists of three parts: the ventilated rooms, the generalized air handling units (GAHUs) and air openings, and the ductwork that connects rooms with GAHUs, where a GAHU is an air handling unit in which return air is handled with or without fresh air mixing [24]. Each air handling unit (AHU), FCU, RAC, or air cleaner can be treated as a GAHU. In each room, air is supplied from one or more GAHUs and returned to different GAHUs after diluting indoor contaminants. As shown in Fig. 1, there are GAHUs and rooms that are connected with each other through the ductwork. In this study, a general algorithm to calculate the dynamic distribution of contaminant is proposed based on the general ventilation system in Fig. 1.

Relationship between contaminant distribution, air supply inlet conditions, and contaminant sources in rooms

Li and Zhu [43] proposed an index of response coefficient (RC) and a corresponding formula to correlate transient contaminant distribution in ventilated rooms with variable supply air concentrations and emission rates of contaminant sources, which forms the basis of the method proposed in this study.

Index of response coefficient to supply air (RCSA)

For a steady flow field, assume that the initial contaminant concentration is 0, the emission rates from all of the contaminant sources are 0, and the contaminant concentration in supply air from the n th inlet is (Eq. (1)), and those of the other inlets are 0. Then, the RCSA at an arbitrary point p from the n th inlet at the jth time step, , is defined as Eq. (2) [43]: where is the contaminant concentration of point p at the jth time step (kg/m3); is the contaminant concentration in supply air from the n th inlet at the 0th time step (kg/m3); and is the time interval (s).

RCSA is a non-dimensional quantity reflecting the effect of ‘pulse’ contaminants in supply air on the indoor air quality at an arbitrary position p [43]. For a room with supply air inlets, the contribution from all of supply air inlets to the formation of the transient concentration at an arbitrary point p is expressed aswhere is the number of supply air inlets; is the contaminant concentration in supply air from the n th inlet at the ith time step.

Index of response coefficient to contaminant source (RCCS)

For a steady flow field, assume that the initial contaminant concentration is 0, the contaminant concentration in supply air from all of the inlets are 0, and the emission rate from the n th contaminant source is (Eq. (4)), and rates from the other sources are 0. Then, the RCCS at an arbitrary point p from the n th contaminant source at the jth time step, , is defined as Eq. (5) [43]: where is the emission rate from the n th contaminant source at the 0th time step (kg/s); Q is the ventilation rate of the room (m3/s); and is a nominal contaminant concentration, which is equal to the average concentration from all of the outlets at the steady state when the emission rate from the n th contaminant is (kg/m3).

RCCS is also a non-dimensional quantity, and is independent of the emission rate of the contaminant from its source when the flow field is fixed [43]. If the contaminant is released from a wall at a dynamic emission rate, RCCS can also be utilized to quantify the effect of the wall contaminant source. For a room with contaminant sources, the contribution from all of the sources to the formation of the transient concentration at an arbitrary point p is expressed aswhere is the number of the contaminant sources; is the emission rate from the n th contaminant source at the ith time step.

Index of response coefficient to initial distribution (RCID)

For a steady flow field, assume that the contaminant concentrations in supply air from all of the inlets are 0, and the emission rates from all of the contaminant sources are 0. Then, the RCID at an arbitrary point p from the initial concentration distribution at the jth time step, , is defined aswhere is the average concentration of the initial contaminant distribution (kg/m3); is the contaminant concentration of point p at moment 0 (kg/m3); and V is the volume of the room (m3).

For a room with an initial condition of contaminant, the contribution of the initial condition to the formation of the transient concentration at an arbitrary point p is expressed as

Expression for the transient distribution of contaminant concentration

The transient concentration of contaminant at an arbitrary point p under dynamic boundary conditions and a steady flow field is expressed as

Eq. (9) establishes a quantitative relationship between the resulting concentration at an arbitrary point p and multiple dynamic supply air concentrations, emission rates of contaminant sources, and the initial distribution of contaminant. In Eq. (9), the RC indices, i.e., , and , can be obtained in advance by limited rounds of CFD simulations. The number of simulations is equal to the total number of the boundary conditions of supply air, contaminant sources, and the initial condition. Once the RC indices are determined, the transient concentration can be quickly predicted when the concentration in supply air, the emission rate of the contaminant source, and the initial distribution of contaminant are given.

Mathematical descriptions of contaminant transport in the different components of the ventilation system

By analyzing the transport processes of the contaminant in different sections of the general ventilation system, four relationships are described through mathematical expressions: (1) The relationship between supply air concentrations in the GAHUs and in the inlets; (2) the relationship between supply air concentrations and return air concentrations in rooms; (3) the relationship between return air concentrations in the outlets and in the GAHUs; and (4) the relationship between the concentrations in fresh air, return air, and supply air in the GAHUs. It is assumed that the initial concentration of contaminant in the ventilation system (i.e., GAHU, air duct, and room) is 0, and therefore the contribution from the initial condition of contaminant (Eq. (8)) is 0.

Relationship between supply air concentrations in the GAHUs and in the inlets

Assuming that there are supply air inlets for the th GAHU in the th room (see Fig. 1), the concentration in the th inlet for the th GAHU in the th room, , is expressed aswhere is supply air concentration in the th GAHU at the th time step (kg/m3); is the delayed time step for the transport of contaminant from the th GAHU to the th inlet.

Similarly, assuming that there are direct fresh air inlets in the th room, the concentration in the th inlet in the th room, , is expressed aswhere is the outdoor air concentration at the th time step (kg/m3); is the purification efficiency of the contaminant in fresh air to the th room; and is the delayed time step for the transport of contaminant from the outdoor air opening to the th inlet.

Relationship between supply air concentrations and return air concentrations in the rooms

Assume that there are return air outlets for the th GAHU in the th room. According to Eq. (9), the concentration in the th outlet for the th GAHU in the th room, , is expressed aswhere is the RCSA at the th outlet from the th fresh air inlet at the th time step; is the number of the contaminant sources in the th room; is the emission rate from the th contaminant source at the ith time step (kg/s); and is the ventilation rate of the th room (m3/s).

Considering the delayed time step for the transport of contaminant in supply air and fresh air, return air concentration becomes Eq. (13) by taking Eqs. (10), (11) into Eq. (12):

Relationship between the return air concentration in the outlets and in the GAHUs

Return air from different outlets will be recirculated back to the GAHUs. Return air concentration in the th GAHU from the th room at the jth time step, is expressed aswhere is the ratio of return air flow rate in the th outlet of the th GAHU from the th room to the total return air flow rate of the th GAHU from the th room; is the delayed time steps for the transport of contaminant from the th outlet to the th GAHU. When , equals 0.

Return air of the th GAHU comes from rooms, and therefore the total return air concentration in the th GAHU at the jth time step, , iswhere is the ratio of return air flow rate of the th GAHU from the th room to the total return air flow rate of the th GAHU.

Relationship between the concentrations in fresh air, return air, and supply air in the GAHUs

For the th GAHU, fresh air ratio, , is expressed aswhere , , are fresh air, return air, and supply air flow rates of the th GAHU, respectively (kg/m3). As return air exists for each GAHU, the range of fresh air ratio is . Supply air concentration of the th GAHU at the jth time step, , is obtained by the mass balance of the contaminant:where is the purification efficiency of contaminant in the th GAHU, .

Algorithm for the dynamic distribution of contaminant in a general recirculating ventilation system

Prediction of dynamic contaminant distribution with time step in seconds

When the dynamic prediction is based on a time step in seconds, the time delay in each air duct should be considered, and the equations in Section 2.3 are applicable. By combining Eqs. (13), (15), (17), supply air concentration in the th GAHU, , is expressed as

As a period of time in several seconds or minutes is needed for the transport of contaminant in return air, fresh air, and supply air ducts, the delayed time steps , , and are generally more than 1 when the time step of 1 s is adopted to calculate the RC indices and the dynamic concentration. In this case, Eq. (18) becomes an explicit scheme, and therefore the at each time step can be directly calculated based on the available concentrations at previous time steps. After the dynamic supply air concentrations in all of the GAHUs are obtained by Eq. (18), the dynamic distribution of contaminant in each room can be calculated by Eq. (9).

Prediction of dynamic contaminant distribution over a long period of time

When dynamic pollution during weeks', months', or years' operation needs to be predicted, the adopted time steps should be in minutes or hours like those adopted in building load prediction software, such as DeST and Energy Plus (always in hours). In this situation, the actual time required for the transport of contaminant in some air ducts may be less than the length of the adopted time step, and therefore the time delays along the air ducts can be neglected. Accordingly, the method used to predict the dynamic contaminant distribution over a long time period is proposed based on Eq. (18), as follows:

Calculate the delayed time steps (rounded off) for transport of contaminant in all of the air ducts (supply air, return air, and fresh air) based on the specified time step during prediction. Assume that the supply air inlets for the th GAHU in the th room are numbered as 1, 2, …, for the inlets with a neglected time delay, and for the inlets with a time delay. The return air outlets for the th GAHU in the th room are numbered as for the outlets with a neglected time delay, and for the outlets with a time delay. According to the number of the air openings, Eq. (18) becomes where is a known item caused by the concentrations in supply air and fresh air, and the emission rate of the contaminant from its source at the previous time steps, considering the time delay in part of the air ducts (kg/m3).

Let. Then, Eq. (19) becomes

At each time step, there are unknown concentrations in supply air of the GAHUs, and there are also equations based on Eq. (21). Therefore, a matrix can be constructed to solve the concentrations in supply air of GAHUs at each time step:

After the dynamic supply air concentrations in all of the GAHUs are obtained by Eq. (22), the dynamic distribution of contaminant in each room can be calculated by Eq. (9).

Generally, the process used to solve for the concentration of contaminant over a long period of time is as follows:

Prepare all the basic parameter information about the GAHUs, direct fresh air and exhaust air, and lengths of air ducts and rooms. Calculate the delayed time steps in each air duct.

Build the geometrical model of each room and simulate the fixed flow fields.

Calculate RCSA, RCCS and RCID in each room with Eqs. (2), (5), (7).

Calculate the dynamic concentrations in supply air of each GAHU with Eq. (22).

Calculate the dynamic contaminant distribution in each room with Eq. (9).

In the process above, step (3) is time intensive, consuming as much time as the traditional CFD simulation. However, as only one round of simulation is needed to obtain the RC distribution in advance, the subsequent prediction for contaminant distribution is less time intensive through simple matrix solution and algebraic operation regardless of the number of simulation cases.

Simplified method for the long-term prediction of dynamic contaminant distribution

The established model (Eq. (22)) can quickly predict the distribution of contaminant over a long period of time if the key indices of the RCs are obtained in advance. That is, the actual time cost depends on the RC calculation process. If a long-term prediction, such as a one-year or several-year prediction, is needed, the time cost for one RC calculation is still higher because of numerous time steps of simulation, especially when the building ventilation system is complex and the total grid number is large. Considering that the RC value for a “pulse” contaminant source is notable at the preliminary stage before decreasing sharply with time, as shown in Fig. 11, a simplified method for RC calculation and the proposed model was proposed as follows:

Fig. 11

RCSA variations at different positions.

For each RC simulation for a specific boundary condition, only limited hours (e.g., 1 h, 2 h or 3 h) of contaminant variation and corresponding RC values at the preliminary stage are simulated. RC values close to 0 in the following hours are not simulated. As a limited number of simulations are needed for RC calculation, the total time cost for a long-time period prediction is significantly lower with respect to a traditional CFD simulation, even if only one case prediction is conducted. The reliability of the simplified method is analyzed in Section 4.

Verification of the proposed method

The accuracy of the proposed method, i.e., Eqs. (18), (22), depends on the accuracy of the expression for the transient concentration of contaminant (Eq. (9)) and the mathematical descriptions of contaminant transport in the air ducts and the GAHUs. In Section 3, the experimental validation of the expression for transient concentration is described first; then, a description of the numerical verification of a general recirculating ventilation system follows. The time step adopted in each process was 1 s. Further exploration of the reliability of the proposed method for a longer time step (minutes or hours) is discussed in Section 4.

Experimental validation of the expression for transient concentration

The dimensions of the test chamber were 4 m (X) × 2.5 m (Y) × 3 m (Z), as shown in Fig. 2 . One inlet (0.2 m × 0.2 m) and one outlet (0.3 m × 0.2 m) were installed in the chamber. The coordinates of the center points of the inlet and outlet were (0, 2.3, 1.5) and (4, 0.3, 1.5), respectively. There was no heat source inside the chamber, and all the walls were well insulated during the experiment.

Fig. 2

Schematic of the test chamber.

A hot-bulb anemometer (range 0–20 m/s; accuracy ±3%) was used to measure the air velocity. Supply air inlet was uniformly divided into 16 sections, and the measured velocity in each section is illustrated in Fig. 3 . Therefore, the room air change rate was 8 ACH. CO2 was selected as the contaminant and released into supply air duct at a controlled constant rate of 25 L/min to create a constant CO2 concentration in supply air. Nine CO2 sensors (range 0–5000 ppm; accuracy ±3%) were arranged in the chamber (No.2–10) to record the transient concentration variation at different positions, as shown in Fig. 4 . An extra sensor (No.1) was used to monitor the background CO2 concentration. During the release of CO2 from supply air inlet, there was no CO2 source in the chamber. CO2 was continuously released for 3000 s; however, the total duration of measurement was 4530 s.

Fig. 3

Measured air supply velocity.

Fig. 4

Arrangement of CO2 sensors (Z = 1.5 m).

STACH–3, a well-validated CFD program, was used as the simulation tool for the RC index and the concentration of the contaminant [44]. An indoor zero-equation turbulence model was used to account for the turbulent flow in the room [45]. The Reynolds-averaged Navier-Stokes (RANS) equations, together with mass conservation equations, were discretized by a finite volume method (FVM). A power law scheme was used as the difference scheme. A SIMPLE algorithm was adopted while momentum equations were solved on non-uniform staggered grids [46]. After the flow field was obtained, the concentration of the passive gas was evaluated using the conventional transport equation for contaminants.

Based on the experiment parameters, the flow field and RCSA were simulated by STACH–3. The contaminant distribution was then calculated based on the proposed expression, i.e., Eq. (9). Comparisons of the experimental and predicted results are shown in Fig. 5 .

Fig. 5

Comparisons of the experimental results and the prediction by the proposed expression.

The overall variation tendencies of CO2 concentrations for the prediction and the experiment were in accordance with each other. There were some discrepancies between the prediction and the experimental data, which might be because the sensors cannot respond to the change of CO2 concentrations in a timely manner, resulting in the delayed readings of the sensors. Moreover, the transient concentrations at different positions have different variation characteristics because of the non-uniform distribution of indoor air parameters. Compared with the rapid rise of concentrations at positions close to the inlet (e.g., P9), the concentrations at positions close to the outlet (e.g., P6) change more slowly; therefore, no significant rise of the measured concentration occurred in the first few seconds because of the delayed response of the sensors. Although a certain discrepancy in CO2 concentration existed, no large discrepancy was observed for most of the positions. Therefore, the predicted results agreed well with the experimental results, showing that the expression Eq. (9) could be used to describe the relationship between the transient concentration of contaminant and different influencing factors.

Numerical verification of the proposed method for transient concentration in a general ventilation system

The accuracy of the proposed model for a general recirculating ventilation system (Eq. (18)) was further verified by comparing the simulated results obtained by CFD and the proposed model based on a numerical case. The geometry model of the ventilation system for CFD simulation is illustrated in Fig. 6 . The dimensions of the room were 4 m (X) × 2.5 m (Y) × 3 m (Z). There was one supply air inlet (0.2 m × 0.2 m) and one return air outlet (0.2 m × 0.2 m) in the room. The coordinates of the center points of the inlet and outlet were (0, 2.3, 1.5) and (0, 0.3, 1.5), respectively. Part of return air flowed back to an AHU along an air return duct (length: 16 m), whereas the rest was exhausted to the outdoor air. Fresh air was introduced into the AHU and mixed with return air to form supply air of the AHU. Supply air was transported into the room along an air supply duct (length: 14 m). The air velocity of supply air inlet was 2.412 m/s. Fresh air ratio was 0.21. There was a passive contaminant source at position (2, 1.5, 1.475). The dimensions of the source were 0.05 m (X) × 0.05 m (Y) × 0.05 m (Z). The constant emission rate from the passive source was 5 mg/s. The initial concentration of contaminant in the ventilation systems was 0.

Fig. 6

Geometrical model of the recirculating ventilation system.

Direct CFD simulation of the transient concentration of contaminant was conducted based on the geometrical model in Fig. 6. The CFD modeling approach was the same as that described in Section 3.1. The room model was built separately, and the RCSA and RCCS indices were simulated and calculated. The transient concentration of contaminant was predicted subsequently by using the proposed model (Eq. (18)). The predicted concentrations at different positions (Fig. 4) are shown in Fig. 7 .

Fig. 7

Comparisons of CFD and the proposed method (Eq. (18)) predictions.

The concentration variations at all of the positions for the proposed model agreed well with those for the CFD simulation. Therefore, it is plausible that the proposed model had essentially the same accuracy as the CFD simulation, indicating an acceptable reliability of prediction.

Reliability analysis of the proposed method

The accuracy of the proposed method with a time step of 1 s was verified in Section 3. However, the reliability of the model needs to be determined for larger time steps that are adopted for the prediction of contaminant distribution over a long period of time. We considered a year-round prediction and analyzed the reliability of the prediction based on the time step of 1 h. Generally, the reliability of the prediction depends on three aspects: the simulation accuracy with the adopted time step, whether the influence of contaminant in return air on contaminant in supply air is considered, and whether the time delay of contaminant transport is considered. In this section, the accuracy with different time steps is first analyzed, and the influences of return air concentration and time delay are then investigated.

Case setup

A recirculating ventilation system is established, as shown in Fig. 8 . There were two AHUs (AHU 1 and AHU 2) and two rooms (Room 1 and Room 2). The geometry of the room is shown in Fig. 9 .

Fig. 8

Schematic of the ventilation system with two AHUs.

Fig. 9

Geometry of the room.

There were two supply air inlets (S1 and S2) and two return air outlets (R1 and R2) in each room. Each AHU was connected with one room through one inlet, one outlet, and the corresponding air ducts. The dimensions of each room were 4 m (X) × 2.5 m (Y) × 3 m (Z). The two rooms did not contain a heat source, and all the walls were adiabatic. The temperature of the supplied air was 20 °C. A contaminant source was placed at position (1, 1.05, 1.5) in Room 1 and at position (3, 1.05, 1.5) in Room 2. The concentration in fresh air was 0 ppm. The air supply velocity was 1 m/s, and the corresponding air change rate was 9.6 ACH. Detailed information about the locations of the air openings is listed in Table 1 .

Table 1

Locations of the air openings.

Opening	Start point			End point
	XS (m)	YS (m)	ZS (m)	XE (m)	YE (m)	ZE (m)
S1	0.90	2.50	1.40	1.10	2.50	1.60
S2	2.90	2.50	1.40	3.10	2.50	1.60
R1	0.00	0.20	1.40	0.00	0.40	1.60
R2	4.00	0.20	1.40	4.00	0.40	1.60

The concentrations of contaminant at nine positions were monitored and recorded for analysis, as shown in Fig. 10 . The CFD modeling approach was also the same as that described in Section 3.1.

Fig. 10

Locations of monitoring points (Z = 0.5 m and Z = 1.5 m).

Response coefficient for a time scale of 1 h

For each room, the response coefficient indices, i.e., RCSA and RCCS, were simulated by CFD and calculated by Eqs. (2), (5). A time scale of 1 h was used to calculate RC and predict the dynamic concentration. However, when the transient concentration in Eq. (2) or (5) was simulated to calculate RC, a smaller time step could be used, such as 1 min, 10 min, or 30 min. As a representative, the RCSAs at 4 positions (P1, P5, P7, and P9) from inlet S1 in Room 1 are shown in Fig. 11 . The time steps adopted in the simulations of RCSA were 1 min, 10 min, 30 min, and 1 h.

For each position, the RCSA was larger in the first hour and then decreased quickly to a value close to 0. Position P1 had the largest RCSA in the first hour among the 4 positions, indicating the fastest response to supply air from inlet S1. The RCSA values based on the time steps of 1 min, 10 min, and 30 min were close to each other, whereas there was a certain discrepancy in the RCSA values when the time step of 1 h was adopted.

Effect of the adopted time steps for all fresh air ventilation system

To obtain the long-term prediction of contaminant distribution by the proposed model (Eq. (22)) with an acceptable time expense, a time scale of 1 h was adopted. The given dynamic boundary conditions (supply air concentration and emission rate of contaminant source) will change hourly accordingly. The effect of the time step was first investigated based on the all-fresh air ventilation system illustrated in Fig. 8. Fresh air ratio of both AHUs was set as 100%. The dynamic supply air concentration was known. We assumed that the concentration in supply air from S1 complied with a sinusoidal function, i.e., (ppm). The time step was 1 h. The concentration in supply air from S2 was 0 ppm. Time steps of 1 s, 1 min, 10 min, 30 min, and 1 h were used to simulate the transient concentration for the RC calculation, while the time scale of 1 h was utilized for the RC indices and the final concentration distribution. The predicted results for 50 h are illustrated in Fig. 12 for Room 1. As a large calculation load is needed for the simulation when a time step of 1 s is applied, only 4 h of simulation (14400 steps) were conducted for this case.

Fig. 12

Dynamic concentrations at different positions in all fresh air ventilation system.

In the first hour, the predicted concentrations based on the time steps of 1 s, 1 min, 10 min and 30 min were consistent with each other for an arbitrary position (Fig. 12). There was a certain discrepancy in the prediction results for the time step of 1 h and the other time steps. However, from the second hour all the results from the different time steps were consistent with each other. Therefore, it was reliable to conduct a long-term prediction based on the time step of 1 h for an all-fresh air ventilation system.

Effect of the adopted time steps for a recirculating ventilation system when the time delay in the air ducts is neglected

The effect of the adopted time steps was further investigated for a recirculating ventilation system. We assumed that the time delay for contaminant transport in the air ducts was much shorter than the time step adopted in the prediction. In this case, the time delay had to be neglected, and the contaminant concentration in return air was treated to affect the contaminant concentration in supply air instantaneously during the process of calculating the transient concentration. The demonstration case in this section helped to reveal the reliability of the adopted time steps when considering the influence of the concentration in the return and supply air, which was the key characteristic of the recirculating ventilation system.

Fresh air ratio of AHU 1 and AHU 2 were set as 10% and 85%, respectively (Fig. 8). We assumed that the emission rates from the contaminant source in Room 1 and Room 2 both complied with a sinusoidal function, i.e., (10−6 m3/s). The time step was 1 h. During the prediction process, the concentration in supply air would change objectively during the 1 h period owing to the influence of return air, even if the rate of emission from the contaminant source was assumed to be constant during the same period. Therefore, the variation of supply air concentration could be captured when a time step smaller than 1 h was adopted in the calculation of RC. The time steps of 10 min, 30 min, and 1 h were adopted for the calculations of RCSA, RCCS, and the dynamic prediction of the concentration distribution by the proposed model, respectively. The time steps of the variation of supply air concentration within 1 h could be considered when time steps of 10 min and 30 min were adopted.

The time delay information is listed in Table 2 . The time delay in Table 2 was far less than the length of the adopted time steps above. Therefore, the time delay was neglected during the prediction process.

Table 2

Time delay for contaminant transport in air ducts (Case 1).

Number of air ducts	Path of air ducts	Time delay (s)
1	AHU 1 → S1 (Room1)	20
2	AHU 1 → S1 (Room2)	50
3	AHU 2 → S2 (Room1)	60
4	AHU 2 → S2 (Room2)	30
5	AHU 1 → R1 (Room1)	10
6	AHU 1 → R1 (Room2)	40
7	AHU 2 → R2 (Room1)	70
8	AHU 2 → R2 (Room2)	30

The predicted results at different positions are illustrated in Fig. 13 .

Fig. 13

Dynamic concentrations at different positions based on different time steps in recirculating ventilation system (without considering the time delay).

The dynamic concentrations in the supply air of each AHU and at different positions (P1 and P5) in each room were basically the same for the predictions based on the time steps of 10 min, 30 min, and 1 h, indicating that predictions of the RC index and the corresponding transient concentrations based on the 1 h time step were reliable, even if the influence of return air concentration on supply air concentration was considered, and supply air concentration changed objectively during a period of 1 h.

Comparisons of predicted results with and without considering the time delay of contaminant transport in air ducts

Another case was designed to explore the reliability of the prediction based on the time step of 1 h without considering the time delay. The results were then compared with those of the prediction that considered the time delay in the air ducts. The newly designed time delays in the air ducts are listed in Table 3 . The time step of 10 min (600 s) was used for prediction. During the prediction, the time delays for air ducts 2–4, and 6–8 could be considered with respect to the length of the time step (10 min).

Table 3

Time delay for contaminant transport in air ducts (Case 2).

Number of air ducts	Path of air ducts	Time delay (s)
1	AHU 1 → S1 (Room1)	200
2	AHU 1 → S1 (Room2)	500
3	AHU 2 → S2 (Room1)	600
4	AHU 2 → S2 (Room2)	300
5	AHU 1 → R1 (Room1)	100
6	AHU 1 → R1 (Room2)	400
7	AHU 2 → R2 (Room1)	700
8	AHU 2 → R2 (Room2)	300

The concentrations with and without considering the time delay through the air ducts were predicted based on the time step of 10 min, and the results are illustrated in Fig. 14 .

Fig. 14

Dynamic concentration at different positions in a recirculating ventilation system with and without considering time delay.

The larger discrepancy in the predicted concentrations with and without considering the time delay only occurred in the first hour for the different positions. From the second hour the predicted concentrations with and without considering the time delay were close to each other. From the view point of long-time period prediction, a low effect might be caused by neglecting the time delay in the air ducts. As the predictions with time steps of 10 min and 1 h were almost the same, as illustrated in Fig. 13, the overall discrepancy in the prediction based on the time step of 1 h without considering the time delay and the prediction based on the time step of 10 min with consideration of the time delay was believed to be small, indicating an acceptable reliability of prediction.

Furthermore, the time delay of up to 700 s (11.7 min) was designed for air ducts in the demonstration case; therefore, a certain effect on the prediction was observed if the time delay was neglected. However, for conventional building ventilation systems, the magnitude of the time delay in the air ducts is always tens of seconds or a few minutes, which is much less than the set time in the study Thus, a smaller effect on the prediction is expected to be observed when the time delay in the air ducts is neglected. This alternative method can be used to determine the prediction based on the time step of 1 h when the actual structure of the air ducts is complex, and there is a heavy workload in calculating the time delay in each air duct.

Reliability of the simplified method

In Section 4.3, 4.4, 4.5, the proposed model exhibited an acceptable reliability (with the time step 1 h) based on numerical cases of both fresh air ventilation systems and recirculating ventilation systems. In Section 4.6, the adaptability of the proposed simplified method was analyzed based on the case in Table 2, Section 4.4 (time step of 1 h). The predicted results by the simplified method (i.e., 1 h RC-simplified) and the original proposed method (i.e., 1 h RC-original) are illustrated and compared in Fig. 15 . For the prediction by the simplified method, only the RC value in the first hour was calculated.

Fig. 15

Comparisons between simplified and original proposed method.

All of the predicted concentrations over time were kept consistent between the original method and the simplified method. The simplified method, in which 1 h of RC was simulated, could achieve a result that was approximately equal to that achieved by the original method. Therefore, the simplified method was reliable in predicting the distribution of contaminant in a recirculating ventilation system over a long period of time. In this section, the 1 h RC simulation suitably predicted the distribution of contaminant when the simplified method was used under the given building ventilation situation. For other practical building ventilation systems, the required time steps of RC calculation for the simplified method should be determined individually according to different ventilation and air exchange conditions.

Conclusion

A numerical method is proposed to predict the dynamic contaminant distribution in a complex recirculating ventilation system over a long period of time. An algebraic expression based on the indices of response coefficient is applied to account for the relationship between contaminant distribution inside the room and various boundary conditions. The method is established by mathematical descriptions of the relationships between supply air concentrations in GAHUs and in inlets, return air concentration in outlets and in GAHUs, and fresh air concentrations in the total ventilation systems. Hourly supply air concentrations in the GAHUs can be easily obtained by solving a matrix, and the dynamic distribution of contaminant is then quickly calculated with an expression based on the response coefficient. The proposed method can provide long-term predictions almost without time expense, once the response coefficient is prepared in advance.

Both experimental and numerical methods are utilized to verify the reliability of the proposed method. It is shown that the proposed method has an acceptable accuracy in predicting the contaminant distribution based on the time step of 1 s. The prediction based on the time step of 1 h is reliable for long-term contaminant distribution when compared with smaller time steps. There is a certain prediction discrepancy if the time delay in the air ducts is directly neglected when the actual time delay is notable in part of the air ducts; however, the larger discrepancy only occurs in the first hour.

A simplified form of the established numerical method is further proposed to accelerate the calculation of the response coefficient, which alone is as time intensive as the traditional CFD simulation. The response coefficient at the preliminary stage, rather than for the entire time period, should be calculated. The demonstration case shows that only calculating 1 h of the response coefficient using the simplified method also yields reliable results.