Zonal Modeling for simulating indoor environment of buildings: review, recent developments, and applications.

INTRODUCTION

Various modeling approaches have been used for modeling the thermal distribution and airflow movement. The type of approach used depends on the complexity of the phenomena observed, the results expected, the parameters investigated, and the degree of required accuracy. The following four levels of modeling have been developed:

1. Nodal Method (One-Node Model). This method considers a zone to be perfectly homogenous. The applications of this technique are limited because of its oversimplified approach. For instance, this method is mostly used to study the energy consumption of a building by representing each room, or sometimes each apartment, by a single node. The outdoor climate is also reduced to a single node, and the physical parameters are determined from weather conditions. To make a more detailed study by considering the wind direction and the accurate prediction of solar radiation, a node is associated with each facade. This approach is suitable for design and system sizing since it provides a rapid solution.

2. Multi-Node Model. The multiroom models treat heat transfer and airflow between different zones of a building. Multiroom models do not treat the local phenomena within a room, such as stratification. In a thermal multiroom modeling approach, a zone is, in general, a specific room or even a collection of rooms. The whole apartment unit in a residential high-rise building can be represented by a single zone to study the interaction between units.

In an airflow multiroom approach, a building and its plant are treated as a collection of nodes representing rooms and HVAC system components, with internodal connections representing the distributed flow paths associated with cracks, doors, windows, HVAC ducts, fans, etc. The network nodes represent the zones, each of which are modeled at a uniform temperature and pollutant concentration. The pressures vary hydrostatically, so the zone pressure values are constant only at a specific elevation within the zone (Megri 1993).

3. Computational Fluid Dynamics Modeling. Another approach to studying the airflow and temperature and contaminant distributions in an environment incorporates computational fluid dynamics (CFD) analysis, which subdivides the volume of a single room into thousands, if not millions, of nodes (Haghighat et al. 1992). Despite the richness of the results in terms of detailed information regarding the flow and temperature fields within a room, CFD suffers from a need for significant problem definition and computation effort by the user. Thus, it is difficult to apply to situations involving a number of rooms in a building (Clarke et al. 1995). Further, the CFD models are very sensitive to the variation of boundary conditions and the quality of the data used for prediction. However, much of the building data previously used for energy simulation is not accurate enough to be used for CFD predictions. This detailed method is very demanding in terms of power and computer processors due to the fact that a very fine grid is necessary to study the heat and mass transfer processes at the boundaries. The design of innovative HVAC systems, as well as the evaluation of the comfort of occupants, requires a detailed estimation of airflows and heat transfers within building zones. The CFD method can, in principle, provide such details, but in practice it is difficult to apply to a whole building over long periods of time.

4. Zonal Method. This method is based upon the partitioning of a room into a number of subzones; it is an intermediate approach between a one-node (single-zone) and a CFD model. The zonal approach allows us to accurately define the physical parameters within each zone of a given room to provide us with a tool that could be used for a detailed investigation of thermal comfort, indoor air quality, and energy analysis. Zonal models provide an improvement over the well-mixed assumption used for nodal models. However, this approach needs complementary information and models to define flows. The CFD modeling approach could provide us with flow in a steady-state situation without previous knowledge of the dominant flow phenomena. Zonal models are a promising way to predict air movement in a room with respect to comfort conditions and gradient of temperature because they require extremely little computer time and could easily be included in multizone air movement models (Jiru and Haghighat 2004; Megri et al. 2005).

The zonal model was developed to study the interaction between the terminal unit and the rest of the room, including the floor and the roof (Lebrun 1970; Laret 1980; Howarth 1980, 1985; Sandberg and Lindstrom 1987; Ngendakumana 1988; Allard et al. 1990). Later, the zonal model was used to study the temperature stratification within nonindustrial buildings (Grelat 1987; Allard and Inard 1992; Togari et al. 1993; Arai et al. 1994; Inard et al. 1995, 1996; Cron et al. 2000; Voeltzel 1999; Gao et al. 2006). It has also been used to investigate contaminant and contaminant distributions (Huang and Haghighat 2005) and to solve some indoor and outdoor design problems (Bozonnet et al. 2005). An interesting application for modeling solar domestic hot-water systems also was developed (Kenjo et al. 2007).

This paper begins with a review of zonal models that have been developed over the past three decades and experimental studies that have been carried out to validate a model or provide one with input data. Discussions of specific zonal models and various improvements proposed over the years will be explored. Recent developments, as well as various applications, will be also presented.

EXPERIMENTALLY BASED (SPECIFIC) ZONAL MODELS

This category of models has been driven by experimental studies. The majority of these studies are motivated initially by the quantification of the temperature stratification within a single room under a heating system. The goal has been extended to study the performance of various systems and the level of thermal comfort they produce.

Models Based on Experimental Observation

The first zonal model scheme was developed during the 1970s (Lebrun 1970) from experimental observations realized in a test cell. Lebrun proposed an airflow and heat transfer for a six-zone configuration with a heating system source. In his model, the author selected a principal flow direction and also a recirculation at the level of the ceiling. Later on, Inard and Buty (1991) authenticated this model using an extensive experimental study measuring the distribution of temperature within a controlled environment: a Minibat cell.

Later, Ngendakumana (1988) applied a five-zone scheme to determine the temperature distribution within the same volume. In his scheme, he suggested seven airflow paths between the five zones (total of 12 unknowns). The flows were oriented from experimental observation. Based on this scheme, he developed an analytical model, which was able to predict two of the five zones' temperatures. The remaining three unknown temperatures at the floor and roof levels and within the volume were measured. Consequently, the number of unknowns was reduced to nine. This model is based on the conservation of energy equation and complementary equations. The complementary equations used were the convective heat flow equations assumed to be at various surfaces; the power heat flow equation of the heating system, which represents the summation of the convective heat flow; and, finally, the equation that represents the relationshp between the power heat flow and two zones temperatures. Later, the author used the computed temperatures and the airflow rates to determine the convective heat transfer coefficients. Ngendakumana repeated this process experimentally with different heating power for both steady and transient states. The values obtained are only valid for the experimental conditions imposed and cannot be generalized. However, this work developed a methodology to analyze the interior convective flux. Laret (1980) proposed a new scheme, slightly different than the configuration developed by Lebrun and Ngendakumana (1987). This model respected the same general flow with assumptions that allowed the analytical determination of the air temperatures within the volume. Based on these schemes and others developed later, simplified models were developed to study temperature stratification in the presence of a radiator (Howarth 1980, 1985) and a convector heating system (Laret 1980; Allard et al. 1990a).

Sandberg and Lindstrom (1987) proposed a transient model with only two zones to study the interaction between a building envelope and the displacement ventilation system. This model included an axisymmetric thermal plume generated from a point source. The characteristics of the thermal plume were obtained from integral equations, while the conservation of mass and energy equations were established for various elevations. This model was similar to the one developed by Howarth (1980); it is robust and allows the prediction of the concentration of a pollutant within a room. However, the quality of model prediction strongly depends on the description of the direction of the driven flow and, more specifically, the thermal plume.

Integration of Heating Systems

Over the years, this category of model has been extended through the integration of a branch of commercial heating systems in order to compare their performance. The integrated models include the following:

* Hydronic radiator (or baseboard heating) (Howarth 1985; Lebrun and Ngendakumana 1987; Inard and Molle 1989; Inard and Buty 1991; Musy et al. 2001; During 1994)

* Electric convector (During 1994; Inard et al. 1997a, 1997b; Musy et al. 2001)

* Radiant panel (During 1994)

* Heating ceiling (During 1994)

* Floor heating (Inard and Buty 1991; During 1994)

* Heat pump (Gschwind et al. 1995)

In particular, the results of the studies performed by During (1994) and Inard et al. (1997a, 1997b) have been in the form of recommendations to manufacturers for improving their products and to designers. For example, local heating systems (hydronic radiator, electric convector, and radiant panel) are more suited to new insulated commercial and residential buildings. For old buildings without insulation or with inappropriate insulation and high floor to roof height, it is not recommended to use heating systems such as baseboard locally or even at the opposite side of the room. Distributed heating systems (heated ceiling and floor heating) can be used for both building situations with and without insulation. Some of the conclusions are confidential and are directed principally to manufactures' quality control.

Integration of Displacement Ventilation

Different zonal models have been developed to study the displacement ventilation (Mundt 1996; Rees 1998; Rees and Haves 1999).

Rees (1998) modeled a displacement ventilation system without a chilled ceiling using a two-dimensional thermal resistance model. The authors studied models using three nodes for the air near the floor and ceiling and two nodes representing the ceiling and floor surfaces. Results near the floor and extraction points varied from measured results of temperature gradients and wall surface temperatures. This model was extended to model a plume that develops in a room with displacement ventilation. It provides eight additional nodes to model the plume flow as it moves toward the ceiling. The additional nodes reflect the four walls and the flow in and out of the plume. Additional assumptions were applied fixing the stationary front mixed conditions near the ceiling and flow parameters that were derived from CFD modeling. Airflow pattern is fixed in this model.

Results of the modeling reflected the shape of the temperature profile in the modeled room, matched measured temperature near the floor, and were lower by one degree Celsius near the ceiling. Temperature values away from the ceiling and floor varied from measurements by a degree or more depending on the amount of recirculation specified in the model assumptions. The exact path recirculation takes could not be pinpointed from the model analyses in order to improve the results.

Later, the authors (Rees et al. 1999) extended their work to include modeling of a chilled ceiling used with displacement ventilation. The ten-node model developed relies on CFD models to develop a fixed airflow pattern and parameters expected during the experiment for two different heat load cases. The CFD results for this study pinpointed recirculation to be very large between the ceiling-level flows and at intermediate levels near the floor. This information, not available in the previous study (Rees 1998) was incorporated in the ten-node model. Airflow parameters were calculated using heat and mass balance equations at the nodes of the model using a general-purpose equation-solver code to give values for various test cases. Convection coefficients were taken from design handbooks (Rees et al. 1999). The model was generalized using rules to define convection coefficient and capacity rate parameters for other experimental setups.

A comparison of measured and calculated results for displacement ventilation with and without a chilled ceiling showed reasonable comparison. The temperature profiles matched well with variations of less than one degree Celsius. The model results matched expected qualitative behavior of airflow and temperature profiles in the room as the supply air temperature was increased and the ceiling temperature was reduced.

GENERALIZED ZONAL MODELS

After the development of airflow modeling approaches for single-zone, multizone (Haghighat 1989), and, more specifically, the COMIS project (Feustel et al. 1990; Megri 1993, 1995), which incorporated the latest developments from an international research project in the framework of the International Energy Agency (Annex 23), zonal models have been known for their integration of airflow and thermal modeling. The actual models are based on the corresponding mass (Equation 1) and the energy conservations (Equation 2) in different cells, and no assumptions are needed for airflow direction.

[summation over (j)][m.sub.j[right arrow]i] = 0 (1)

[summation over (j)][[PHI].sub.j[right arrow]i] + [[PHI].sub.source] = [[rho].sub.i][V.sub.i][c.sub.p][[[partial derivative][T.sub.i]]/[[partial derivative]t]] (2)

[P.sub.i] = [[rho].sub.i]r[T.sub.i] (3)

[m.sub.j[right arrow]i] = [[rho].sub.j,i]S.[C.sub.d][([P.sub.j] [P.sub.i]).sup.n] (4)

[m.sub.j[right arrow]i] = [[rho].sub.j,i]S.[C.sub.d][[P.sub.j] - [P.sub.i] - [1/2]([[rho].sub.j][h.sub.i] + [[rho].sub.j][h.sub.i])].sup.n] (5)

[[PHI].sub.j[right arrow]i] = [c.sub.p]([m.sub.[j[right arrow]i].sup.+][T.sub.j] + [m.sub.j[right arrow]i].sup.-][T.sub.i]) - [[[lambda]S]/l]([T.sub.j] - [T.sub.i]) (6)

Other balance equations can be added to these two equations, such as the moisture mass balance (Mendonca et al. 2002; Wurtz et al. 2006b) and contaminant mass balance (Huang and Haghighat 2005) equations. Other complementary equations are usually needed, such as Equations 3 through 6. Equation 3 represents the ideal gas equation and Equation 4 assumes that the mass flow rate is a function of the pressure difference across the vertical face. Also, Equation 5 assumes that, for the horizontal faces, the hydrostatic variation of pressure is taken into account. The overall heat exchange fluxes are represented by Equation 6. Jets, plumes, and boundary layers are induced in various cells (Figure 1). The application of the airflow, determined as a function of pressure distribution using a reduced form of the Navier-Stokes equations (Bouia 1993; Dalicieux and Bouia 1993; Wurtz 1995; Gagneau et al. 1997; Haghighat et al. 2001), is obviously limited to nondriving flows (zones with relatively low velocity) in the absence of thermal plumes and jets.

New zonal models are developed with the capability to use both types of conservation equations for airflow and energy in the same cell. They use the specific conservation law for the part of the cell affected by thermal plumes, jets, and boundary conditions, as well as pressure distribution equations in the another region (part) of the same cell (Inard et al. 1996; Bouia 1998; Musy 1999; Wurtz et al. 1999; Lin et al. 1999; Haghighat et al. 2001). The balance equations describe the state of the subzones (or cells), and the equations of transfer describe the phenomena between two neighboring cells through their interface.

Efforts have also been made to couple zonal and CFD. Mora (2003) and Mora et al. (2003a, 2003b) developed an approach in which the zonal model uses the airflow structure from the results of a CFD model in the same volume. Bellivier (2004) defined the conditions under which a CFD model can be simplified with enlarging meshes to reach the zonal model's level.

Integrated Airflow and Energy Conservation-Based Models

Based on the fact that all of the previously cited models are valid for very limited configurations, various flexible models have been developed to study the impact of location and type of air supply and return diffusers on the indoor air quality and thermal comfort. These models require considering three categories of irregular cells (or meshes) that can be distinguished within a room: low velocity cells that represent the part of the room not affected directly by the presence of jets and/or thermal plumes (Bouia 1993); driving flow cells that represent the zones affected by dominant flows, such as jet, plume, and boundary layers (Allard et al. 1990; During 1994); and mixed cells composed of low velocity and driving flow cells (Inard et al. 1996) (Figure 1). Bouia (1993) and Wurtz (1995) initiated the development of zonal methods based on solving the pressure field to predict airflow and temperatures in large indoor spaces.

[FIGURE 1 OMITTED]

Wurtz (1995) developed a new zonal model where mass and energy balances are written in each subvolume, while the mass flows in the interfaces are calculated by power-pressure laws. However, the power law suggested uses constant coefficients (K and n). Wurtz took advantage of the modularity of the zonal method to implement it in an object-oriented environment using SPARK (Sowell and Haves 2001) environment, since SPARK fits the resolution of large nonlinear equations systems. The properties and advantages of the object-oriented environment were used to couple the zonal method developed with a thermal comfort model, a conductive model, and a mass transport model. Results were validated by comparison with various experimental and numerical references. A sensitivity analysis was then performed to determine the appropriate empirical coefficients, as well as the characteristics of an optimal mesh. The simulation of the influence of a heat source yields results consistent with experimental data in the whole domain studied. The combined natural and forced convection case has been treated by adding a model for kinetic energy conservation in the subzones, and the calculated mass flows computed with the CFD results.

Voeltzel (1999) applied the zonal model approach to predict airflow patterns and temperature fields in atria where she incorporated accurate solutions of radiative exchanges between indoor surfaces and solar gains. For airflow modeling, she used a standard set of power-law flow equations with constant coefficients. She obtained good agreement between time-dependent predictions and measurements of temperature. For experimental study, she used a 5.1 m high, highly glazed room (SunCell at Ecole Nationale des Travaux Publics de l'Etat, Lyon, France) to validate her zonal model. Temperatures were measured every minute along the vertical centerline of the room at four different heights for 56 hours. Time-dependent temperature predictions demonstrated satisfactory agreement with the measurements at these four locations.

Musy (1999) showed that it is possible to automatically build zonal models that allow the prediction of air movement, temperature distribution, and indoor air quality parameters not only in a specific zone, as can be done with CFD model, but in the whole building. She entirely reformulates the zonal model developed by Wurtz (1995) as a connection of small sets of equations in a way that all the equations describing the building behavior are grouped together into subsystems of equations. This assemblage represents the geometric representation of the rooms. Haghighat et al. (2001) put forth a zonal model in the framework of Annex 35 of the International Energy Agency (IEA). Specific differences occur in the treatment of mass flow rates using a pressure power law versus temperature and heat power laws. They provided a comprehensive background and validation of the pressurized zonal model with the air diffuser (POMA) model. POMA made use of a modeling of mass flow across normal boundaries (without the presence of jets), across a jet boundary to air not entrained in the jet, and across the jet. POMA is able to predict the airflow patterns and thermal distributions within a room. The POMA model is based on the conservation of mass and energy. Jet characteristic equations were introduced in the model to generalize its application to mechanically ventilated buildings. The POMA model is a simplified numerical model and uses pressure-driven power laws.

Results from the POMA model were compared with some experimental results and with predictions made by another zonal model, as well as with the CFD model (i.e., FLOVENT [Flometrics 2004]). The results showed that the POMA model tracks the shape, airflow pattern, and quantitative values for FLOVENT well for the cases examined. It is interesting to note, though, that the FLOVENT results did not track the experimental results in the region of a single wall of the test cell. The POMA model with a large mesh and a fine mesh showed good agreement with EXACT3 (Kurabuchi et al. 1990) results. However, the proper conditions that allow an extensive comparison between CFD and zonal models have not been defined.

Computational Fluid Dynamic-Based Models

Mora (2003) proposed a new simulation platform based on the object-oriented simulation environment, SPARK, to treat most of building zones using the nodal approach. This modeling method considers each zone as a fully and instantaneously well-mixed volume. In this case, each zone can be characterized by a unique computational node where temperature, pressure, and contaminant concentrations are determined. Then, some specific rooms are studied in more detail. In order to see the impact of these details on the entire building model, the author proposed different coupling methods depending on the model's associations between the nodal approach and zonal or CFD room models. The author also attempted to demonstrate the usefulness of applying one method instead of another, depending on the room characteristics or the modeler's objectives. A new platform was developed in which both nodal and zonal models can be solved, and which allows detailed models to be integrated with simplified ones. Through case studies, the author showed that the developed platform has the capability to adjust the level of detail needed for each volume.

Based on the fact that the CFD modeling approach is time consuming, that it is necessary to reduce computing time, and the need of building engineers to study increasingly larger volumes, the author proposes an integrated approach based on both CFD and zonal approaches and investigates the transition between both approaches. Basically, the objective is to come up with a zonal model from a simplified CFD model. First, the author developed a simplified CFD model adequate to be used under a number of conditions, including the use of coarse grids (large meshes), a constant effective viscosity law, and adapted coefficients for heat transfer adapted to thermal flow and airflow within buildings (Bellivier 2004).

Surface-Drag Flow Relations for Zonal Modeling

Axley (2001) has developed an alternative formulation to model airflow resistance in zonal models. Axley points out that current zonal models use well-understood jet-momentum relations for regimes involved in forced airflow (by jets or buoyancy), but the use of power-law viscosity relations away from jet-driven areas does not have a strong physical basis, since viscous losses related to surface drag may dominate airflow. Axley examines the use of current viscous relations in models and determines that shear stress related to the air viscosity and time-smoothed air velocity profile more accurately reflects the physical mechanisms of airflow away from forcibly driven air.

Applying Newtonian laminar flow and using Prandtl's eddy viscosity approximation or mixing length approximation for turbulent flow produces relations that can be solved using empirical fits for the velocity profiles across a room. These relations yield linear pressure flow relations that can be solved iteratively. Specifically, they can be applied to zonal model cells to produce new pressure flow relations. These relations were used on CONTAM96 (Walton 1997) and compared to results using a cell boundary power law (i.e., pressure to the 0.5 power) and CFD results reported by Chen and Xu (1998). Results showed accurate reproduction of the CFD results, but the conventional cell boundary power law did vary dramatically from CFD results away from driven flow areas. The author concludes that the new surface-drag relations more faithfully reproduce the CFD results than do conventional power law relations.

Implementing a Subzonal Model into the Airflow Model COMIS

In order to simulate airflow, temperature (Ren and Stewart 2003), and concentration (Stewart and Ren 2003) distribution inside buildings using a modified version of COMIS (Allard et al. 1990b; Megri 1993) with subzonal divisions, the authors applied zonal model processes to the COMIS computer program to produce COMIS with subzones (CWSZ) (Stewart and Ren 2006). The physics methodology is similar to what was performed by Inard et al. (1996) in producing a zonal model where zones are divided out as standard or flow-driven zones. Each type of zone is driven by specific flow relations. Mass and energy balance is maintained between zones, and flow is governed by a pressure power law as used by Inard. Comparison to Inard's experimental results shows similar agreement. The authors note that increasing the number of zones did not change the temperature or airflows initially found with CWSZ using fewer zones. The authors also applied CWSZ to a more practical modeling of a room with a stove, windows, and ventilation hood. CWSZ provided detailed isotherms showing the use of CWSZ in practical applications.

Implementing a Zonal Model in Building Load and Energy Calculation Procedures

Building load and energy simulation programs based on the complete-mixing air model fail to consider the impact of nonuniform air temperature distributions. A momentum zonal model based on the Euler equation has been developed to enhance building load and energy simulations by predicting indoor airflows and temperatures (Chen and Xu 1998; Griffith and Chen 2003). This work shows some validation exercises by comparing model results to measurements and CFD. The model was found to predict thermal stratification conditions reasonably well and to err on the side of complete mixing. The model has been coupled to the heat balance model and tested on load calculations. Results for cooling and heating loads are compared to the traditional complete-mixing model with minor effects on total load but important differences in air-system flow rate and control options. Total computation times for load calculations were two orders of magnitude higher using the momentum zonal model compared to traditional complete mixing.

Octree Partitioning Method

Recently, Guernouti et al. (2004) analyzed boundary conditions applied to a room to deduce the occurring airflows. It consists of a dynamic partition of rooms that fits with the corresponding airflow pattern that can be modified in case the boundary conditions evolve. The analysis of boundary conditions consists of relating standard boundary conditions with the elementary airflow models. For instance, the linear diffuser boundary condition, applied on a wall, implies the use of a horizontal-plane jet model. Knowing the sort of elementary models that have to be used allows the definition of the spatial coverage of each specific airflow. The partitioning of the room can then be done according to the airflow pattern. For this stage, the authors use the octree method that allows refining the meshing locally. For each time step, there is a corresponding octree that represents the room's partitioning. This partitioning method eases the communication of information from one time step to the next (considering the mean value in case the cells are grouped together or transmitting values to child cells per inheritance if a cell is divided into smaller ones). Cells with driving flows are no longer divided into subcells. They either contain specific laws governing the flow expected in air-driven cells or use power law equations.

Regarding the airflow movement and heat transfer calculation process, the authors used the SPARK environment. Note that the use of this partitioning modifies one basis of zonal models: a zone does not necessarily have six neighbors but a number of neighbors that are not fixed from the beginning. For this reason, even if the authors had taken advantage of the previous work, they had to completely revise the zonal model construction processes to take this feature into account.

The first calculation carried out for static configuration (Guernoutti et al. 2004) shows that the partitioning method is well appropriated for this kind of airflow representation. The next stage of this work will be to perform the simulation of dynamic phenomena (sun patches, room heating, or ventilation), which would offer new prospects for using zonal models.

Pressure-Based Model and Variable K Values

The pressurized zonal models use the power law model (PLM) with a constant and identical flow coefficient (K) for each cell (Bouia 1993; Wurtz 1995; Inard et al. 1996; Musy 1999; Voeltzel 1999; Haghighat et al. 1999, 2001; Lin et al. 1999). The value used was 0.83, but Wurtz (1995) showed that the PLM's prediction does not depend on this K value. Jiru and Haghighat (2006) showed discrepancies from applying the PLM with a constant flow coefficient for predicting indoor airflow distribution. To improve the quality of PLM predictions, they proposed using a given K value for each cell. The K value for each cell was estimated using a CFD model. This new model provided an appropriate and variable flow coefficient in each cell for the case of forced convection and reasonably predicted the recirculation in the standard zone. Jiru and Haghighat (2006) further improved the quality of PLM predictions by integrating the PLM equation with the surface-drag flow model (SDM). This integration greatly improves prediction of the recirculation air in the standard zone. The PLMK is obtained with variable K values and the models resulting from direct and indirect combinations of the SDM and the PLM. Such combinations as surface-drag power law type 1 (SD-PLM1), surface-drag power law type 2 (SD-PLM2), surface-drag power law type 3 (SD-PLM3), and modified power law model (MPLM) have been compared extensively. Comparison of the predictions of all of these models with each other and with experimental data showed that the MPLM (modified power law model) provides the best predictions of the recirculation in the standard zone, followed by the PLMK. All of these zonal models, except the MPLM, have similar quadratic general relationships linking the pressure difference between cells i and j and mass flow rate from cells i to j, with different a and b coefficients as expressed in the following equation:

[DELTA][P.sub.i,j] = a * [m.sub.i,j.sup.2] + b * [m.sub.i, j] (7)

The MPLM is given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Figures 2a, 2b, and 3 compare the results of the PLM, MPLM, and PLMK for different distances from the inlet side (Jiru and Haghighat 2006). The flow field predicted by the PLM using Nielsen's (1998) geometry is represented in Figure 2c. For Figures 2 and 3, [U.sub.0] is the inlet velocity, and Y is the characteristic room dimension equal to 0.5 H

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

An Integrated Zonal Model to Predict Transient Indoor Humidity Distribution

In order to investigate the impact of building material moisture adsorption and desorption processes on indoor air humidity and to predict the humidity distribution in a room, a zonal model was integrated with a building material moisture transfer model based on the conservation of energy, dry air, and water-vapor mass. The model was applied to a room conditioned by a fan-coil unit. The results suggested that this model was able to give a satisfactory prediction of transient humidity distribution in the room as well as to provide meaningful information on cooling loads (Wurtz et al. 2006b).

Integrated Zonal Model (IZM) to Predict Volatile Organic Compound (VOC)

Huang and Haghighat (2005) developed a three-dimensional IZM to predict airflow, temperature, and VOC concentration distributions within a room. The IZM integrated a three-dimensional zonal model with an air-jet model and a three-dimensional building material VOC emission/sink model. The IZM was validated at three levels: the airflow distribution in a mechanically ventilated room predicted by the IZM was compared with that of the standard k-[epsilon] CFD model; the temperature distribution for a natural convection case predicted by the IZM was validated with experimental data; and the predictions of the total VOC distribution were compared with CFD model predictions. It was found that the IZM, with quite coarse grids, could provide some global information regarding airflow pattern and thermal and VOC distributions within a room. The physical system considered (Huang and Haghighat 2005) is a room with a mechanical ventilation system in a nonisothermal condition. The room is subdivided into a number of three-dimensional small cells. The room configuration and partition are shown in Figure 4a.

[FIGURE 4 OMITTED]

The airflow was symmetric to the plane Y/W = 0.5. A further detailed mean air-velocity profile comparison was also conducted at this section, as shown in Figure 4b. It indicates that the air velocity predicted by the IZM was slightly stronger than that of the k-[epsilon] CFD model. This could be due to different jet models integrated in these two methods. However, the air velocity at the two locations predicted by both models closely followed a similar trend, even though there was a small discrepancy. As can be seen from the results, the IZM is able to provide global information about the airflow pattern in the ventilated room, which is quite helpful for the design of the mechanical ventilation system.

The discrepancy between the experimental results and the IZM predictions was less than 5%, and there was hardly any difference between the POMA predictions and the IZM predictions (Figure 5). In both Figures 4 and 5, [U.sub.0] = 0.75 [m.s.sup.-1], the dimensions of the room, are L x W x H = 3.0 x 3.0 x 2.7 [m.sup.3], and X is the distance from outlet (m).

[FIGURE 5 OMITTED]

An Equation-Based Simulation Environment to Investigate Fast Building Simulation

Two software tools, SimSPARK and SimZonal, were created to develop and test new zonal models. In addition, these tools were used to analyze airflow and temperature distributions in a cavity under mixed convection conditions and to evaluate the thermal coupling between an electric heater and the indoor environment (Wurtz et al. 2006a).

Figure 6 presents air velocity predictions made by two zonal models with a jet model along with a prediction made by a CFD model using RANS k-[epsilon]. In this case, air entering the test room through the diffuser at velocity Uref = 1.71 [m.s.sup.-1] and a temperature Td of 23.1[degrees]C. In all three sections, CFD predictions give a satisfactory estimate when compared with experimental data. The jet section is well characterized and the recirculation slightly underestimated, especially in the vertical section located at x = 0.75 W. All zonal models give a satisfactory speed estimate in the jet region, while the recirculation (speed in the lower section of the room) is underestimated (Wurtz et al. 2006a).

[FIGURE 6 OMITTED]

A New Approach on Zonal Modeling of Indoor Environment with Mechanical Ventilation

Song et al. (2006) developed a new zoning approach based on room air age, a parameter that indicates the mixing condition of the air. Zoning criteria are developed based on the deviation ratio of air age, as well as location of the key source that is of concern (e.g., temperature, air pollutant, etc.). This study was limited to mechanically ventilated rooms. The principal issue of the new approach is that zoning should be based on the distributions of flow field, and the zoning results should indicate the uniformity of zones in the space. In this research, air age is adopted as a basis for zoning in mechanically ventilated rooms. A case study was presented for a displacement ventilated room to demonstrate the applicability of the new approach for predicting indoor temperature, and simulation results using the new zonal model were compared with those using a CFD model and a conventional zonal model. The authors state that this model is more accurate in predicting the zonal temperature distributions than the conventional zonal model. The model is suitable for dynamic simulations (e.g., whole-year) of indoor environmental parameters.

VALIDATION AND APPLICATIONS

Zonal models become a viable tool to analyze, assess, and design a system. They have been validated experimentally and through comparison with CFD models and many applications that have been developed to study the indoor and outdoor environment, since it is inappropriate to use the assumption of instantaneously well-mixed zones to model airflows and pollutant transport in indoor spaces.

Evaluation and Validation of Zonal Models

Inard and Buty (1991) proposed a validation of zonal models using a CFD modeling approach; the main objective of this exercise was to study the ability of the zonal models to predict the thermal behavior of airflow in the case of natural convection coupled with the presence of a hydronic radiator. A comparison between simplified two-and five-zone models and the results obtained with a low Reynolds number k-[epsilon] model showed that the five-zone model gave indoor air temperature profiles consistent with the low Reynolds number k-[epsilon] model. Concerning the convective heat fluxes, except for the two zone model, the values computed by the models are of the same order of magnitude with lower values for the low Reynolds number k-[epsilon] model.

Mora et al. (2003b) studied detailed airflow distribution in large indoor spaces and performed a comparison of velocity predictions using different formulations of zonal methods and coarse-grid k-[epsilon] CFD models for measurements in a two-dimensional, mechanically ventilated isothermal room. The results suggested that when airflow details are required, coarse-grid CFD is a method better suited to predict airflows in large indoor spaces coupled with complex multizone buildings than are the zonal methods. Based on the comparison of pressure predictions from different models, the authors provided guidance regarding the coupling of a model of detailed airflow in large spaces to algebraic multizone infiltration models.

General Application of Zonal Model

A Zonal Model for Large Enclosures with Combined Stratification Cooling and Natural Ventilation. An airflow and thermal zonal model approach was applied to design a combined mechanical air-conditioning and natural ventilation system for cooling a large enclosure (Gao et al. 2006). This technique uses stratified air conditioning and natural ventilation to cool the occupied and upper parts of a space, respectively, to reduce heat penetration into the lower air-conditioned part. The vertical temperature profiles of large enclosures under such a combined system were predicted using a zonal model, by cutting the space into horizontal settled zones. It introduces some particular flow dynamics and thermal effects into the predictions of mean airflows and temperature distributions. Different from those pressure-based zonal models applied generally to small rooms, it is termed a temperature-based zonal model, which uses correlations based on temperature differences in combination with submodels for modeling of mass flow and heat transfer in the large enclosures. The authors provided a calculation procedure for the model developed and demonstrated performance by analyzing the impacts of some influential factors, such as height above floor and ventilation flow rate on the space air-temperature profiles.

Effects of Sensor Location on Thermal Control. Models currently used for control studies are either oversimplified (one-node models) or not generic enough to provide a flexible and usable testing tool for room controllers. In fact, the temperature measured by the sensor of a room temperature controller depends on its position in the zone. The measured sensor temperature depends on the airflow, and more specifically on the convective coupling of the zone and its heater, and may be different from the mean air temperature. The experimental observation of the physical thermal and airflow phenomena have been used to develop a zonal model adapted to study the effect of the sensor position for a thermal control system. The proposed model has the ability to distinguish between air temperature and sensor temperature in a transient state situation. This model was used for a specific situation where a fan-coil unit is used for cooling and heating (Riederer et al. 2001).

Energy Consumption and Thermal Comfort in Dwelling Units. A simplified zonal model treating the thermal behavior of dwelling units was developed (Inard et al. 1998; Wurtz et al. 2006a). The objective was to evaluate the performance of various heating systems commonly used in dwelling units. Two types of heating sources--localized (hydronic radiator and an electrical convector) and distributed (a hot-water- heated floor and an electrical heated ceiling)--were integrated into the zonal model. The model was used to predict the loads and the distribution of indoor temperature specific to each system. The authors concluded that the distributed heat sources presented a slight advantage over the localized sources with regard to energy consumption and thermal comfort.

Thermal and Airflow Modeling of Passive Cooling. A thermal and airflow zonal model was developed (ZAER) to assess thermal comfort within unconditioned buildings in Mediterranean and North African regions under transient conditions (Gharbi et al. 2004). A power-pressure law is used with a constant flow coefficient. In the context of thermal comfort, thermal and airflow aspects have a strong interdependence. This approach is based on a temperature- and pressuredriven zonal airflow model based on the coupling of reduced-order state models. The integration between zonal and thermal comfort models allows a study of the influence of night natural cross-ventilation strategy upon the summer thermal comfort. The model validation was performed by comparing its predictions with the experimental data obtained from measurements on the experimental cell Minibat (CETHIL, INSA Lyon Laboratory), for different configurations.

Urban Microclimates Impact on the Building Energy Demand. Bozonnet et al. (2005) used a pressure-based zonal model to study the heat and mass transfer within an urban street canyon. The goal was to investigate the increase of air-conditioning energy demand from the heat island effects, as well as to demonstrate that a combined airflow and thermal modeling is necessary to study the heat island effects. The zonal model selected provides an acceptable level of accuracy of the building energy demand as well as the temperature and wind velocity distributions within a street canyon. Further, this zonal model was integrated with the model of natural convection, which takes into account the effect of the multiple reflections of solar radiation developed with the street canyon. The dominant airflows due to wind are determined from experimental isothermal airflow measurements. Coupled effects of forced (wind) and natural convections in a street canyon are simulated for a 28-day period and compared with experimental data.

CONCLUSIONS

Current one-zone or multizone models are inadequate to obtain the information used for many design applications, since the fundamental assumption of heat balance models is that the air in each thermal zone is considered to be well stirred with uniform temperature throughout. Zonal models are intermediate models between one-zone and CFD models. They have the ability to take into account different phenomena, ignored by one-node and multizone models, such as temperature stratification, thermal integration with cold facade, draft, asymmetric thermal radiation, and cold or hot floor surfaces.

In this paper, we presented the evolution of the zonal model from simple scheme to detailed and thermal and airflow models integrated with CFD, capable of predicting the distribution of temperature, airflow, and moisture. In addition, recent zonal model development used to solve a number of engineering and design applications has been presented.

To date, there is no commercial program or software based on the zonal modeling approach. The treatment of case studies where several particular flows are involved, in particular flow conflicts, as well as the application of a zonal modeling approach to natural ventilation, have never been done. Consequently, the development of a new generation of zonal models for particular applications, including the development of databases for these specific applications, should be considered for future developments.

NOMENCLATURE

Variables

[C.sub.d] = coefficient of power law, [ms.sup.-1].[Pa.sup.-n]

[C.sub.p] = heat capacity of air, [J.kg.sup.-1].[C.sup.-1]

[h.sub.i] = height of airflow element i, m

H = height of room/gap, m

l = distance between node to, m

[m.sub.j[right arrow]i] = airflow rate from cell to, [kg.s.sup.-1])

n = airflow model exponent, dimensionless

[P.sub.i]= pressure in cell i, Pa

S = surface area, [m.sup.2]

t = time, s

[T.sub.i] = air temperature in cell i, [degrees]C

U = velocity in the room, [m.s.sup.-1]

[V.sub.i] = volume of cell i, [m.sup.3]

y= distance of a cell from floor, m

X, Y, Z = coordinates

Greek Symbols

[DELTA]P = pressure difference, Pa

[lambda] = conductivity of air, [W.m sup.-1].[degrees][C.sup.-1]

[[rho].sub.i] = density of air in cell i, [kg.m.sup.-3]

[[rho].sub.i, j] = air density depending on sign ([m.sub.j[right arrow]i]), [kg.m.sup.-3]

[[PHI].sub.j[right arrow]i] = heat flux from cell j to I, W

[[PHI].sub.source] = heat source in cell i, W

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Ahmed Cherif Megri, PhD

Fariborz Haghighat, PhD

Fellow ASHRAE

Ahmed Cherif Megri is an assistant professor at the Illinois Institute of Technology, Chicago, and Fariborz Haghighat is a professor in the Department of Building, Civil and Environmental Engineering at Concordia University, Montreal, Quebec.

Received April 4, 2007; accepted June 29, 2007
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