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Description
INTRODUCTION
Methods to model transport phenomena in physical systems fall within one of two broad categories--macroscopic or microscopic. Macroscopic methods are based on idealizing systems as collections of finite-sized control volumes within which mass, momentum, or energy transport is described in terms of ordinary differential conservation equations. They provide the means to predict the bulk response characteristics of this transport, including spatially averaged temperatures, concentrations, and mass flow rates. Microscopic methods, on the other hand, are based on continuum descriptions of mass, momentum, and energy transport defined in terms of partial differential conservation equations that are most often applied to selected portions of a physical system. In principle, they provide the means to predict the spatial details of response, but for systems involving turbulent or quasi turbulent fluid flow, the analyst must be satisfied with approximate solutions for often spatially limited flow cases.
As buildings are invariably designed as collections of thermally conditioned zones, it was quite natural for building researchers to turn to macroscopic methods. Sir Napier Shaw first idealized buildings as single control volumes linked to the outdoor environment via flow-limiting orifices (Shaw 1907). Building on Shaw's work, by the mid-twentieth century Dick was able to lay out the key principles of the macroscopic building airflow analysis used today (Dick 1949, 1950, 1951; Thomas and Dick 1953). Specifically, he correlated building envelope pressure boundary conditions with the approach-wind velocity, he presumed airflows through flow-limiting openings were driven by static-pressure differences, and he imposed the conservation of airflow into the (single) building control volume.
Dick's single-zone infiltration models (e.g., see the left-hand portion of Figure 1) led others to develop serial multizone infiltration models that were similar to serial electric-resistance models (e.g., Aynsley et al. [1977a]), although the airflow resistances were assumed to be nonlinearly dependent on the driving potential (i.e., static pressure differences) rather than linearly dependent. For buildings of more general configuration, electric-network circuit-analysis methods were adapted, but this demanded digital computation. For isothermal wind-driven airflows, network airflow models were introduced in the 1970s in the British LEAKS, SWIFIB, and VENT programs and were later integrated with macroscopic building thermal-analysis models in the thermal-analysis research program, TARP, by Walton in 1984 (de Gids 1978; Liddament and Allen 1983; Walton 1984). By the mid 1990s, methods to systematically account for buoyancy effects, which had no analog in electric resistance network analysis, were presented in multizone models (Walton 1988; Feustel 1990; Feustel and Raynor-Hoosen 1990; Wray 1990; Li 1993).
[FIGURE 1 OMITTED]
In addition, element assembly methods borrowed from the finite element analysis community (e.g., Bathe [1982]) and dynamic memory-management methods developed by computer scientists were introduced to facilitate continued program development and to offer the analyst a greater variety of flow-limiting models and the ability to model building systems of practically arbitrary complexity and scale (Axley 1987; Walton 1989). Immediately thereafter, these models were further modified to account for the interaction of mechanically ducted air-distribution systems (sometimes called duct-network models; see center portion of Figure 1) (Walton 1994, 1997; Feustel and Smith 1997; Pelletret and Keilholz 1997; Dols and Walton 2000; Dols et al. 2000).
The general nonlinearity of multizone building airflow modeling, the increasing complexity of the building systems considered, and, most recently, the shear size of building ventilation systems modeled in practice (e.g., with thousands of zones and tens of thousands of flow elements) has demanded continual improvements in equation-solving methods, computational strategies, and user-interface features of the leading multizone analysis programs. Lorenzetti has addressed the former problem (Lorenzetti and Sohn 2000; Lorenzetti 2002a) and, importantly, has outlined a number of fundamental theoretical limitations of conventional multizone airflow analysis that have largely escaped notice (Lorenzetti 2002b). Of these, he notes that conventional multizone approaches impose mass conservation but not momentum conservation.
While momentum conservation is not easily imposed, its close cousin, mechanical energy conservation (mechanical power balances), may be (Axley et al. 2002a; Axley and Chung 2005a, 2005b; Axley 2006a, 2006b). Although the practical application of power-balance methods may be limited for lack of measured model parameters, this research has demonstrated that the conventional multizone method is simply a special case of the more general power-balance method with another special case being that based on the generalized Bernoulli equation that is used in the piping network analysis community.
Recognizing that building airflow dynamics interacts with building thermal and, in some instances, air contaminant-dispersal dynamics, methods to integrate multizone airflow analysis with building thermal and contaminant-dispersal analysis have been proposed (Axley and Grot 1989; Chen and Van Der Kooi 1990; Clarke and Hensen 1990; Hensen and Clarke 1990; Klobut et al. 1991; Woloszyn et al. 2000a, 2000b; Axley et al. 2002b, 2002c; Li 2002; Seifert et al. 2002; Mora et al. 2004). Similarly, as multizone methods cannot predict the details of airflow within zones, there is growing interest in embedding detailed microscopic models (e.g., see the right-hand portion of Figure 1)--most commonly computational fluid dynamics (CFD) models--within larger whole-building multizone models to predict intrazonal details while accounting for their interaction with the larger building system (Li and Holmberg 1993; Schaelin et al. 1993; Albrecht et al. 2002; Gao 2002; Mora et al. 2002, 2003; Lorenzetti et al. 2003; Chen and Wang 2004; Malkawi 2004; Axley 2006b).
The next section will first outline the fundamental principles and assumptions of macroscopic airflow analysis and apply these principles to develop a general theory of multizone building airflow modeling. More detailed presentations may be found in Axley (2006a, 2006b).
MULITZONE MODELING THEORY
In conventional multizone airflow analysis, building systems are idealized as collections of zones and duct junctions linked by discrete (flow-limiting) airflow paths, envelope wind-pressure boundary conditions and temperatures within the zones and duct junctions are specified (typically but not necessarily as uniform), and specific flow relations are assigned to each of the discrete airflow paths or flow elements of the building idealizations as shown in the left-hand portion of Figure 2. Then, equations governing the behavior of the system as a whole are formed by demanding that zone mass airflow rates be conserved. Finally, these equations are complemented by assuming hydrostatic conditions exist in each of the modeled zones to achieve closure. The resulting nonlinear algebraic system equations, defined in terms of zone and duct junction node pressures, are then solved and the solution is back-substituted into the flow-element equations to determine the airflow rates within these elements. Given the central role of the zone and duct nodes in this building idealization and its historical association with electric circuit analysis, this modeling technique is sometimes called a nodal approach to multizone airflow analysis.
[FIGURE 2 OMITTED]
Alternatively, building systems may be idealized, as shown in the right-hand portion of Figure 2--here, both flow paths and zones are treated as finite-size control volumes separated by distinct port planes. In this more general approach, airflow variables associated with each port plane include as primary variables pressures and airflow velocities and, as related secondary variables, volumetric and mass airflow rates. With the port-plane variables in hand, now the conservation of both mass and mechanical energy may be used to form the system equations, with boundary conditions and zone-field assumptions imposed to effect closure. The resulting nonlinear equations may then be solved to directly determine port-plane pressures and velocities.
Short of imposing full mechanical power balances or, equivalently, the Bernoulli relation for two-port control volumes, one may instead use the flow-element relations from the conventional approach to account for approximate dissipation within the multizone flow system. Thus, the conventional approach to multizone analysis may be formulated in terms of port-plane variables rather than zone-node pressures and is therefore a special case of port-plane analysis.
Finally, limiting consideration to conditions of steady flow, one may form system equations demanding that pressure changes encountered as one progresses from port plane to port plane around a continuous flow loop in a building system sum to zero. While not immediately evident from this introduction, the resulting system equations share the same theoretical basis of the nodal approach but are defined in terms of flow-element mass airflow rates instead of zone pressures. This less popular but useful approach is identified as loop analysis.
Flow Variable Representation and Notation
The detailed (time-averaged) velocity, [v.sub.i](r,s), and pressure, [p.sub.i](r,s), distributions associated with a port will, in general, vary across the section (r,s) of the port, as shown in Figure 3. The spatial average of these distributions, [^.v.sub.i]and [^.p.sub.i] (i.e., over the port-plane cross section,[A.sub.i]), will be identified as the port-plane variables for macroscopic analysis. By definition, these averages are as follows:
[^.v.sub.i] = [[integral].[A.sub.i]][v.sub.i]dA/[A.sub.i] (1)
[^.p.sub.i] = [[integral].[A.sub.i]][p.sub.i]dA/[A.sub.i] (2)
Two secondary flow quantities may then be defined in terms of the fundamental port-plane velocities-the volumetric flow rate, [V.sub.i] and the air mass flow rate,[m.sub.i], through port as follows:
[V.sub.i] = [[^.v].sub.i][A.sub.i] (3)
[m.sub.i] = [rho][V.sub.i] = [[rho].sub.i][[^.v].sub.i][A.sub.i] (4)
where the density of the airflow, [[rho].sub.i], is assumed uniform across the port plane.
In conventional analysis, system equations are formulated in terms of node pressures associated with zones or duct junctions, where it is tacitly assumed that these node pressures are time-averaged values. Thus, in control volume of Figure 3, which is nominally a zone, pressure [p.sup.b] is associated with a specific location (node) at a specified elevation, [z.sup.b] In addition, as conventional analysis typically limits consideration to steady flow in two-port flow elements of constant cross section, one may define a single mass airflow rate for each... |
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