# Efficient coupling of multizone and CFD indoor flow models through proper orthogonal decomposition.

INTRODUCTIONActive flow control in common rooms is becoming strategically important for critical applications involving the accidental or intentional release of chemical or biological agents in buildings. The control strategy will greatly depend on the speed and performance of the sensors used in the detection mechanism, the reaction speed of the occupants, and, more importantly, the airflow and contaminant dispersion in the affected zones. Knowledge of these flows permits the controllers to predict the contaminant dispersion rate, which is necessary information for effective evacuation planning and containment of the released contaminant. Active flow control may be put to good use also to optimize comfort and air quality in a building under normal operating conditions.

To compute the contaminant dispersion and spatial distribution within a complex, multi-room building, the airflow patterns must be known a priori or predicted in near real time. Furthermore, the time-varying airflow patterns that emerge once the control mechanism (e.g., the closing of a damper) is activated must be known. All these data must be available and must be processed at speeds faster than the contaminant dispersion rate. Typically, this is done by means of flow network zonal (FNZ) models (e.g., Dols et al. [2000]) that predict airflow and contaminant dispersion/distribution. Sensor placement and evacuation strategies based on such models have been reported by Arvelo et al. (2002) and Zhang et al. (2005). FNZ models, however, assume that all the rooms are well-mixed zones, i.e., the contaminant concentration is uniform inside each room (zone) and differs only from one zone to the other. This may be a suitable assumption for small rooms with mixing-type ventilation. The well-mixed assumption may also be utilized for corridors if a corridor is divided into subzones along its length to allow for contaminant front propagation. However, in large open spaces such as atria, auditoria, and theaters, the flow is highly nonuniform and three-dimensional. The advection of contaminants within such nonuniform open spaces (NOSs) will depend on accurate knowledge of the airflow, and it is unrealistic to model a NOS as a fully mixed zone. Alternatively, one may attempt to solve this problem totally by CFD; however, the execution time of CFD codes for realistic building configurations, even on powerful computers with relatively coarse meshes, far exceeds the time available for near-real-time control.

Several approaches for coupling multi-zone and CFD calculations have been proposed and employed to improve yearly energy evaluation in buildings (Negrao 1998; Zhai et al. 2002; Zhai and Chen 2003; Bartak et al. 2002; Beausoleile-Morrison 2000). CFD was mainly used to assess the airflow within zones of interest and the rest of the building was represented by the FNZ (Negrao 1995). For such applications, the coupling of both models at the interface openings and walls was needed. At the interface openings, the conservation of mass and energy is enforced, and at the walls the conservation of energy is enforced. Dynamic coupling (at the CFD pace) in these approaches was possible because the room air has a characteristic time scale of a few seconds, whereas the building envelope has a characteristic time scale of a few hours (Zhai et al. 2002; Zhai and Chen 2003). In our application, we expect that a contaminant release in the NOS will affect the office suites and vice versa, and, hence, there is a direct interaction between both models.

Zhai and Chen (2003) classify the coupling techniques as (1) static or (2) dynamic. In the static technique, the FNZ model is first exercised, then the CFD solution follows, and if needed, the FNZ is updated and exercised once more. In a dynamic coupling, the FNZ and CFD models are iteratively coupled. First the FNZ model is used to obtain uniform flow conditions at the interface for the CFD. The CFD model then computes the flow distribution within the space, and an average pressure is computed. Since the average pressure may be different from the one predicted by the FNZ model, which was used to predict the flow through the opening interfaces, the FNZ model is re-run with the new pressures to obtain modified flow rates. Such coupling can be time consuming; however, Zhai and Chen (2003) have shown that the scheme will converge to a unique solution for the case of thermal coupling. Another approach by Bartak et al. (2002) ensures accurate mass and momentum exchange between the CFD-and FNZ-modeled spaces by reducing or increasing the appropriate network area connections to maintain the correct mass flow rate. This is iteratively done until both the mass and momentum are satisfied at the interface for both the CFD-and the FNZ-modeled spaces. To speed up the process, the authors recommend the use of virtual dynamic coupling, where the CFD solutions would be known a priori and saved in a database. Djunaedy et al. (2004, 2005) further classified the coupling mechanisms as external and internal. Internal coupling involves computer code development to interface FNZ with CFD models, whereas external coupling sequentially feeds the results of one model to the other through an interface program.

Here we propose a virtual dynamic coupling approach based on the use of proper orthogonal decomposition (POD) of CFD snapshots of the flow field in the NOS, which efficiently couples the CFD to the FNZ models for near-real-time control of air and contaminant flow in a given building. POD has been used in a wide variety of applications to reduce the order of complex flow phenomena (Holmes et al. 1996; Lumley 1981; Efe and Ozbay 2003) or to facilitate face recognition (Everson and Sirovich 1995). More recently, the method was applied to indoor airflows (Elhadidi and Khalifa 2005). The proposed approach differs in the way by which the coupling is achieved.

In this paper, we first develop the basic methodology and assumptions then apply the POD-based coupling approach to investigate different flow conditions in a typical NOS connected to a suite of offices through a corridor. A brief overview of the POD modes is presented and the coupling between the FNZ and POD models is then described. Finally, we present an investigation of two scenarios in which a contaminant is released from the office suite. Conclusions and future recommendations then follow.

METHODOLOGY AND ASSUMPTIONS

Consider an NOS, 0, connected through a single interface opening to each of several office suites (i = 1, 2, ... [n.sub.s]), as shown in Figure 1. The NOS has a number of ceiling supply diffusers and exhaust vents. Each of the offices (j = 1, 2, ... [N.sub.i]) is also equipped with a supply diffuser and an exhaust vent. The offices are treated as well-mixed zones characterized by a set of scalars such as concentration [C.sub.ij], temperature [T.sub.ij], etc. All exhausts are connected to a single, well-mixed plenum, maintained at a pressure [P.sub.e]. The flow through the interface opening, [m.sub.i0], is considered positive when it is from i to 0 (the NOS), i.e,. [m.sub.i0] = [m.sub.0i], with scalar properties [C.sub.i0], [T.sub.i0], ... at the corridor side of the interface and [C.sub.0i], [T.sub.0i], ... at the NOS side of the interface. All walls are considered adiabatic, and any of the offices, the corridors, or the NOS could have contaminant or energy sources of known strengths.

In this paper we use POD reconstruction to accelerate and improve the efficiency of coupling CFD solutions in an NOS with FNZ solutions in one office suite. We assume that the flow in the small offices and corridor is well represented by well-mixed, lumped-parameter models and ignore thermal effects. The corridor, however, could be subzoned lengthwise to account for contaminant propagation. The well-mixed assumption is satisfactory since (1) mixing-type ventilation systems are typically employed in such rooms and (2) for small rooms, given typical human reaction times, there is no reasonable action that can be taken in near real time based on a detailed knowledge of the concentration field resulting from an accidental or intentional release of a harmful substance (such is not the case in a large NOS). Further, with the assumption that each office suite is connected to the NOS through one opening only (a door to the corridor), we will be able to examine the effect of the flow going from the NOS to the office suite or vice versa, without iteration or with one additional FNZ solution iteration. The general case can be easily addressed later. The solution steps are outlined in the flow chart of Figure 2 and are summarized below.

[FIGURE 2 OMITTED]

1. Use the FNZ model to compute the interzonal flows and zone concentrations in the office suite and the NOS, treated in this step as an extra well-mixed zone. From this model we compute the mass flow rate and concentration at the interfaces. If the contaminant is released in the office suite, the value of the concentration at the interface is needed as a boundary condition for CFD only if the flow is entering from the corridor into the NOS, i.e., [m.sub.i0] > 0 (upwind scheme).

2. With a suitable CFD code, compute the three-dimensional flow and concentration field in the NOS using the interface mass flow rate and concentrations and following the upwind convention at the interface. If [m.sub.i0] < 0, the boundary conditions are the interface opening mass flow rate only. This will be done a priori for a range of boundary conditions to create a suitable number of snapshots of the flow and concentration fields to extract from them the POD modes (Sirovich 1987).

3. If the mass flow is leaving the NOS and entering the office suite ([m.sub.i0] < 0), the average concentration at the NOS side of the interface opening is computed, and the FNZ model is exercised once more for the office suite only (without including the NOS) to obtain improved concentration values for each office.

Flow Network Zonal (FNZ) Models

The basis for the FNZ model is the CONTAM code developed by Dols et al. (2000) at the US National Institute of Standards and Technology (NIST). The CONTAM code assumes that a building can be modeled as a set of interconnected wellmixed zones, where each zone typically represents either a room or a closely coupled set of rooms. The purpose of CONTAM is to predict the airflows between the zones, the resulting contaminant distributions, and ultimately the exposure of building occupants to airborne contaminants for risk assessment and exposure reduction.

In order to predict contaminant distribution, CONTAM includes models for infiltration, exfiltration, room-to-room airflows and pressure difference in building airflows driven by mechanical means, wind pressures acting on the exterior of the building, and buoyancy effects induced by temperature differences between the building zones or between interior and exterior conditions. CONTAM also contains models for the dispersal of airborne contaminants transported by these airflows and transformed by a variety of processes including chemical and radio-chemical, adsorption and desorption by building materials, filtration, and deposition on building surfaces (Dols and Walton 2002).

Clearly, CONTAM is a very capable tool for predicting the exposure of individuals to contaminants based upon a very rich physical model. But its sheer size makes it difficult to modify in order to couple with the NOS model in this early stage of development of the present method, which requires only a limited portion of CONTAM's capabilities.

Hence, for this analysis, we used a simplified CONTAMlike code, referred to here as C-Lite. C-Lite is a program for computing the time-dependent concentration and temperature distributions in a cluster of well-mixed zones that are linked together by advective paths. For this purpose, we assumed one-way coupling between the flow and thermal effects: the flow affects the temperature but the temperature does not affect the flow, i.e., there is no buoyancy. This simplifying assumption may be valid for fast events but is generally inaccurate. In any case, the energy balance equations in the model can be turned off entirely, leading to a formulation similar to that from CONTAM.

A zone is a region in space in which the air is assumed to be well mixed; that is, each zone has a single pressure, temperature, and contaminant concentration at every instant of time. Zones are characterized by their volume as well as the mass, heat, and contaminant sources (each of which can be a function of time). Zones are connected by paths, through which mass, energy, and contaminants are transferred advectively.

At each time step, the pressure in each of the zones is adjusted so that the net advection out of (or into) each zone is equal to the imposed mass source, which can be specified as a function of time. The advected mass flow from zone i to zone j is given by

[m.sub.ij] = sign([DELTA][P.sub.ij])[C.sub.f][a.sub.ij][(2[rho][approximately equal to[DELTA][P.sub.ij]).sup.n] (1)

where [C.sub.f] is an empirical flow coefficient (dimensional), [a.sub.ij] is the area of the opening between i and j, [rho] is the air density, [DELTA][P.sub.ij] = [P.sub.i][P.sub.j] and is the pressure difference between zones i and j, and n is an exponent ranging from 0.5 to 1.0. Implicit in this calculation is the assumption that the air is incompressible and that the density in each zone is equal to the outside air density. For the analysis presented in this paper, n was taken as 0.5, rendering [C.sub.f] nondimensional (the same as an orifice discharge coefficient). The zone pressures are computed via Newton's method, and the flows through the paths are linearized when the pressure difference is close to zero.

Subsequently, the temperature and contaminant concentration in each of the zones are updated based upon the heat/contaminant generation in each zone and the advection through each path. Heat is transferred by two mechanisms: by advection (given the above airflows) and by conduction:

[q.sub.i] + [SIGMA][c.sub.p][[F.sub.ji][m.sub.ji][T.sub.j] + (1 - [F.sub.ji])[m.sub.ji][T.sub.i]] = [SIGMA][U.sub.ij][A.sub.ij]([T.sub.i] - [T.sub.j]) (2)

where

[c.sub.p] = the specific heat of air

q = the prescribed heat generation rate in zone i

[m.sub.ji]= the mass flow rate from j to i (from Equation 1)

[T.sub.i] = the temperature in zone i

[T.sub.j] = the temperature in zone j

[F.sub.ji] = 1 if the flow is from j to i; 0 if the flow is from i to j (upwind convention)

[U.sub.ij] = the overall heat transfer coefficient between zones i and j

[A.sub.ij] = the heat transfer interface area between zones i and j

For each room, the summations are taken over all paths to the neighboring rooms. Contaminant is transferred solely by advection. The characteristics of the paths (area, discharge coefficient, and conductivity) can be specified as a function of time. The update is done via a Runge-Kutta scheme applied to the time-dependent governing equations.

For configurations involving 17 zones and 50 paths, such as the one used in this study, C-Lite can predict the evolution of the mass, heat, and contaminant fluxes to steady state in less than one second on a personal computer. The inputs to C-Lite are provided in an ASCII data file and output is generated in a series of ASCII files. In addition, C-Lite offers the user the ability to plot time evolutions as well as time-varying contour plots. The program is written in FORTRAN and uses the QuickWin interface for graphics.

Although any other FNZ model, such as CONTAM, could be used, we employed C-Lite in this study to demonstrate the coupling of an office suite and an NOS.

CFD/POD Models

The specific NOS used in this work simulates a large room with a long desk (a counter) and a single opening to an office suite (Figure 3). The x, y, z dimensions are 15 X 4 X 10 m (~49.2 X 13.1 X 32.8 ft), respectively. There are six supply air inlets, each 0.4 X 0.4 m (~1.3 X 1.3 ft) and two outlets. The opening to the office suite measures 1 X 2 m (~3.3 X 6.6 ft) and is located at one end of the NOS, next to the counter. The CFD results are computed using a commercial code. The flow is assumed incompressible, steady, and turbulent. The turbulence model is the indoor zero-equation model (Srebric et al. 1999). For the model presented in Figure 3, the typical number of grid points is 54,000; more points are placed in the inlet and exhaust vents to ensure that the gradients and mixing in the jet are well captured.

[FIGURE 3 OMITTED]

To construct the POD modes, we consider an ensemble of flow "snapshots" (Sirovich 1987), [U.sub.i](x), i = (1), (2), ... N, where [U.sub.i] represents a solution set (velocity, temperature, concentration, etc.), x represents the spatial coordinates, and N is the number of snapshots.1 Each snapshot [U.sub.i] represents the flow field inside the indoor space for a parameter value pi corresponding to a different interface or supply condition (e.g., [m.sub.i0], [C.sub.i0];[m.sub.j.sup.s],[C.sub.j.sup.s] ...). In the present work, these snapshots are obtained by running CFD simulations for each parameter value. Our goal is to represent any one of those snapshots as

[U.sub.i](x) = [bar.U](x) + [u.sub.i](x)[approximately equal to][bar.U](x) + [[N.sub.s].summation over (k = 1)][a.sub.k.sup.i][phi].sub.k](x) (3)

where the top bar designates the ensemble average of the spatial field and [u.sub.i](x) is the deviation of a given spatial field from the ensemble average, which can be expanded in terms of the eigenmodes (modes), [[phi].sub.k](x), with[a.sub.k.sup.i] representing the amplitude of each mode, which depends on the value of the parameter [p.sub.i] for snapshot solution [U.sub.i]. We would like to find the [N.sub.m] modes, [[phi].sub.k](x), that minimize the squared deviations of the reconstructed field and the original snapshots ([N.sub.m] < N). These modes are the solution of the classical Fredholm eigenvalue problem (Holmes et al. 1996). To obtain the POD modes using the method of snapshots, we first construct the eigenvalue problem:

R[psi] = [lambda][psi] (4)

where [PSI] and [lambda] are the eigenmodes and eigenvalues and R is the correlation matrix with entries,

[R.sub.ik] = [1/N][u.sub.i],[u.sub.k] = [1/N][integral][u.sub.i][u.sub.k.sup.T]dx (5)

in which the superscript T denotes the transpose of the vector. We note that the correlation matrix R is symmetric and positive definite, which means that all the eigenvalues will be real and that the eigenmodes will be orthogonal and represent a complete set of all solutions within the ensemble space. Hence, any indoor airflow solution within the given ensemble space can be optimally reconstructed from the expansion (Equation 3) given the correct value of the amplitudes, [a.sub.k.sup.i] Thus, we can reconstruct the spatial distributions of the NOS

velocity and concentration fields corresponding to any boundary condition (within the range considered) relatively quickly using linear combinations of previously stored POD modes (Equation 3). This may be viewed as an interpolation process; the basis modes for the interpolation are the eigenmodes [[phi].sub.k](x), and the interpolation coefficients are [a.sub.k.sup.i] The eigenmodes [[phi].sub.k](x) are computed from

[phi].sub.k](x) = [N.summation over (i = 1)][[psi].sub.i.sup.k][u.sub.i](x) (6)

and [a.sub.k.sup.i] is computed by applying the inner product of [u.sub.i](x) in expansion (Equation 3),

[a.sub.k.sup.i] = [([u.sub.i][[phi].sub.k])/([[phi].sub.k][[phi].sub.k])] = [[integral][u.sub.i][[phi].sub.k.sup.T]dx/[integral][[phi].sub.k][[phi].sub.k.sup.T]dx] (7)

As mentioned previously, the coefficients [a.sub.k.sup.i] depend on the interface and supply variables [m.sub.i0], [C.sub.i0], ... obtained from the FNZ model and others ([m.sub.j.sup.s], [C.sub.j.s] ...) that are either measured or otherwise given. The values [a.sub.k.sup.i] of are subsequently used for reconstructing NOS airflow and concentration fields that are not included in the original ensemble.

The relative magnitude of the POD eigenvalues is used as an indicator to determine the number of POD eigenmodes that must be included in the expansion (Equation 3) to achieve the desired accuracy. Typically, one will include [N.sub.s] basis vectors such that the relative energy,

[E.sub.r] = [[N.sub.s].summation over (i = 1)][[lambda].sub.i]/[N.summation over (j = 1)][[lambda].sub.j] (8)

captured by eigenmodes is greater than some convergence criterion. This measure estimates how accurate the reconstruction of a snapshot from the original ensemble is.

Figure 4 compares typical original and reconstructed concentration fields for the NOS (Figure 3) for a typical case. Figure 4a is the original CFD "snapshot" and Figure 4b is the reconstructed field from the POD eigenmodes. The agreement between the two is remarkable, even though we have included only the first three modes. For the case presented, the reconstruction calculations can be executed in less than one second on a typical personal computer.

[FIGURE 4 OMITTED]

RESULTS AND DISCUSSION

To demonstrate the capabilities of our approach, we will only consider isothermal flow and, hence, the coupling of the FNZ and CFD/POD models will only occur at the corridor openings (interface), with no heat transfer through the walls. To this end, we considered an office suite such as Office Suite 3 in Figure 1 connected to an NOS like that shown in Figure 3. The total air supply to the entire complex (office suite plus large room) was maintained at 4 ach. A contaminant was released in Office 6 (see Figure 5) and two scenarios were considered: (1) office doors are closed except for a small leak into the corridor and (2) office doors are open to the corridor. In both cases the flow at the interface opening between the office suite and the large room were from the corridor into the large room. The FNZ model was computationally coupled to the POD reconstruction of the flow and concentration fields in the large room (the NOS). Figures 5 and 6 show the change in the concentration field at a height of 1.5 m (~4.9 ft) in both the office suite and the large room for the closed door and open door scenarios, respectively. The results were generated in less than 1 second, using three POD modes. No iterations were required in this case because the interface flow is from the corridor into the large room.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

To obtain these results, we first computed the mass flows through the openings (vents, doors, interfaces) using the FNZ, including the NOS as an additional zone in the flow network. Then we computed the CFD/POD solution for the NOS using the flows at the openings from the FNZ model as boundary conditions. With contaminant release and advection from the office suite to the NOS, we applied a flow and concentration boundary conditions for the CFD/POD simulation and obtained the distribution in the large room. The method is readily applicable to the case of contaminant release and advection from the NOS into the office suite. Under these conditions, the average concentration at the interface opening is obtained from the CFD/POD calculations, and the FNZ model is exercised once more for just the office suite (without the NOS) to obtain the concentration in each office and along the corridor.

The proposed method of coupling an FNZ model with a CFD/POD model is more beneficial because (1) it can be executed very rapidly to obtain the different spatial airflow and concentration patterns under different interface conditions, yielding a near-real-time active flow control scenario, and (2) it can be used to develop an external, dynamic, virtual coupling as suggested by Zhai et al. (2002) and Zhai and Chen (2003). The distribution of the contaminant in the NOS can be quickly obtained either from the POD modes or, possibly, by solving in near real time a simplified version of the species conservation differential equation.

While the execution time scales of the FNZ and the CFDbased POD models are comparable and allow virtual dynamic coupling, in this work we used a static external coupling technique. External coupling at this stage would facilitate future developments of the scheme and addition of more physics. Static coupling was chosen rather than dynamic coupling because we needed only to match the mass flowing from the office suite to the NOS and vice versa. The pressures are expected to be different in the CFD/POD and FNZ sides of the interface because they are modeled differently. The discrepancy in the pressure (average pressure in CFD) can arguably be modified by correcting the interface area (or flow coefficient) in the FNZ as was proposed by Bartak et al. (2002). In the CFD/POD model, the full Navier Stokes equations of motion are solved, whereas in the FNZ model, the continuity equation is satisfied assuming that the mass flux through the interface opening is proportional to the pressure difference across the opening raised to some exponent (Equation 1). The constant of proportionality represents an empirically determined flow coefficient, whose value is relatively uncertain. Hence, accuracy of the FNZ model will depend on the selected flow coefficients and exponents for each opening, and adjusting those within reason to secure better agreement with experimental or more detailed CFD data is acceptable.

CONCLUSIONS AND RECOMMENDATIONS

An efficient coupling technique between CFD and FNZ models has been developed incorporating POD solutions to extract the CFD results quickly with modest computing resources. The new technique resolves the timing dilemma in coupling two types of simulations that execute at radically different time scales (seconds for FNZ models vs. hours or days for CFD models), preventing their coupled use in near-real-time control applications.

Since it is possible for the contaminant to be released at any location within a large open space, the model must be further developed to include a fast unsteady, coarse grid solver for scalar transport equations (contaminant concentrations). The coarse grid solver would compute the contaminant spread rate sufficiently rapidly to permit near-real-time building flow control. The speed of execution also makes this method especially suitable for scenario assessments, design optimization, and optimal sensor placement. Continued development of the method is underway to allow rapid unsteady calculations of contaminant dispersion in multizone buildings having a combination of well-mixed zones and large open spaces where spatial gradients must be considered. Further work is also underway to generalize the method to more complex building configurations and to develop coupling schemes for popular FNZ codes such as CONTAM.

ACKNOWLEDGMENTS

The work described here was conducted with partial support from the New York Office of Science, Technology and Academic Research (NYSTAR) and the U.S. Environmental Protection Agency. Computing resources were provided by the NYSTAR-designated STAR Center for Environmental Quality Systems.

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1. We follow the methodology described by Elhadidi and Khalifa (2005) for indoor airflows.

H. Ezzat Khalifa is a New York Office of Science, Technology and Academic Research (NYSTAR) Distinguished Professor of Mechanical and Aerospace Engineering and director of the STAR Center for Environmental Quality Systems and John F. Dannenhoffer, III, is associate professor of mechanical and aerospace engineering at Syracuse University, Syracuse, NY, USA. Basman Elhadidi is a lecturer in aerospace engineering at Cairo University, Giza, Egypt.

Ezzat Khalifa, PhD Member ASHRAE

Basman Elhadidi, PhD

John F.Dannenhoffer,III, ScD

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Title Annotation: | Computational Fluid Dynamics |
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Author: | Khalifa, H. Ezzat; Elhadidi, Basman; Dannenhoffer, John F. |

Publication: | ASHRAE Transactions |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jul 1, 2007 |

Words: | 5174 |

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