# Multizone airflow modeling in buildings: history and theory.

INTRODUCTIONMethods 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 flow element as follows, for example, for control volume of Figure 3:

[FIGURE 3 OMITTED]

[m.sup.c][equivalent to][m.sub.i] = [m.sub.j] (5)

The use of subscript indices to associate variables with port planes and superscript indices to associate variables with control volumes will be used consistently in this paper. These notational conventions and the use of the hat embellishment (e.g., as for [^.p]) to distinguish spatially averaged variables from others allow a unified treatment of the commonly used nodal approach and the less commonly used port-plane and loop approaches to macroscopic airflow analysis.

System Idealization

Consider the representative building idealization shown in Figure 4--an assemblage of nine control volumes, a to i, plus the outdoor control volume, o, with a total of twelve port planes numbered from 1 to 12. In this figure, the conventional flow variables are explicitly identified; for example, flow-element air-mass flow rates,[m.sup.a], [m.sup.c], and [m.sup.e]; zone-node pressures, [p.sup.b] and [p.sup.d]; and duct junction node pressures, [p.sup.i] and [p.sup.g]. The flow variables are complemented by control volume air temperatures, which may vary from zone to zone but, nevertheless, are typically specified as uniform within each zone. Port planes are indicated by dotted lines, but the variables associated with them are not explicitly represented in Figure 4. A line linking the centers of one port plane to the next along any possible airflow path and continuing back to the first port plane is a flow loop. For example, Loop 1, shown in Figure 4, links port planes 1-6 and returns to port plane 1. Finally, an approach-wind profile characterized by a reference time-smoothed. averaged approach wind speed, [[bar].v.sub.ref]; an outdoor air temperature, [T.sup.o], which is commonly but not necessarily assumed to be spatially uniform yet vary with time, [T.sup.o](t); and an outdoor ambient air pressure, [p.sup.o], associated with an outdoor node at elevation [z.sup.o]together define the environmental conditions governing the analysis.

[FIGURE 4 OMITTED]

With a building idealization established, the system variables can be defined. Given the two alternative building idealizations presented in Figure 2 and the fundamental principles of mass and mechanical energy conservation and loop consistency, a number of alternative strategies for defining the system variables and forming the system equations may be considered.

Nodal Approach. If system variables are defined in terms of node pressures, here collected into a single vector,{p}, then the system equations may be formed by imposing mass conservation at each node of the system idealization. For example, for the building in Figure 4, the nodal system variables include four unknown pressures, {p} = [{[p.sup.b], [p.sup.d], [p.sup.g], [p.sup.i]}.sup.T]

Port Plane Approach. Alternatively, if the system variables are defined in terms of the port-plane velocities, [[^.v].sub.i], and pressures, [[^.p].sub.i], (collected into a partitioned vector,[{[^.v]|[^.P]}.sup.T]), then the system equations may be formed by imposing both mass and mechanical energy conservation. For example, for the building in Figure 4, the port-plane system variables will include 24 unknown variables, [{[^.v]| [^.p]}.sup.T] = [{[^.v].sub.1] [[^.v].sub.2] ... [[^.v].sub.12]| [[^.p].sub.1] [[^.p].sub.2] ... [[^.p].sub.12]}.sup.T]

Loop Approach. Finally, system variables may be defined in terms of the flow-element mass airflow rates and collected into a single vector,{m}, and mass conservation coupled with loop consistency may be imposed to form the system equations. For example, for the building in Figure 4, the loop-system variables would include seven unknown mass airflow rates, defined as {m} = [{[m.sup.a], [m.sup.c], [m.sup.e], [m.sup.f], [m.sup.g], [m.sup.h], [m.sup.i]}.sup.T]

From these definitions, it may be tempting to conclude that the nodal approach (the basis of conventional analysis) is preferable, as it can be defined in general terms of far fewer system variables. However, as the details of the methods, based on each of these building idealizations, are studied, it will become clear that each approach offers certain advantages.

Conservation of Mass

At the most fundamental level, air mass flowing into and out of each control volume must be conserved. More precisely, the air mass flow difference must equal the rate at which the mass of air,[m.sup.l], within a control volume, l, is accumulated. For ports i and j of a two-port control volume, l, [[rho].sup.i]with being the air density distribution over port i (assumed here to be uniform), we demand the following:

[F.sub.m.sup.l][equivalent to]([[rho].sub.i][A.sub.i][^.v.sub.i] [[rho].sub.j][A.sub.j][[^.v].sub.j]) - [[d[m.sup.l]]/[dt]] = 0 (6)

This and subsequent control volume mass balance functions will be identified as [F.sub.m.sup.l]

For multiport control volumes, mass flow rates are simply summed over the inflow and outflow ports, but now the conservation relation is expressed in terms of either port-plane velocities or, if only two-port flow elements are involved, in terms of the linking mass flow rates as follows:

[F.sub.m.sup.l][equivalent to][summation over (inflowi i = i1,i2,..)][[rho].sub.i][A.sub.i][[[^.v].sub.i] - [summation over (outflow j = j1,j2,..)][[rho].sub.j][A.sub.j][[^.v].sub.j] - [[d[m.sup.l]]/[dt]] = 0 (7)

[F.sub.m.sup.l][equivalent to][summation over (inflow i = i1,i2,..)][[m.sup.i] - [summation over (outflow j - j1,j2,..)][m.sup.j] = [[d[m.sup.l]]/[dt]] (8)

Commonly, airflow is modeled as a steady phenomenon and, thus, the accumulation term d[m.sup.l]/dt is assumed to be zero. However, two programs, CONTAMW 2.0 and CONTAM97, allow accumulation (e.g., for smoke generation and dispersal studies) (Walton 1998; Dols 2001a).

In forming the system equations, mass balance functions [F.sub.m.sup.l] would be formed for each control volume l of the system idealization. For a system of control volumes, these may then be collected into a system mass balance vector as {[F.sub.m]} = [{[F.sub.m.sup.1], [F.sub.m.sup.2], ... [F.sub.m.sup.n]}.sup.T}]

Conservation of Mechanical Energy

For flow systems, one may unambiguously account for flow dissipation through the application of the conservation of mechanical energy (Bird et al. 1960, 2002), yet this conservation principle has been largely ignored by the building ventilation community. Instead, dissipation in flow-limiting paths has been indirectly modeled using semiempirical relations derived using the Bernoulli equation. Here we will begin with the former and demonstrate that the latter may be derived as special cases of the former.

The detailed derivation of the macroscopic mechanical-energy balance is subtle (e.g., see Section 7.8 of Bird et al. [2002]), yet the individual terms of both the microscopic and macroscopic mechanical-power balances have direct physical meaning. For example, consider the microscopic unsteady conservation of mechanical energy for a two-port control volume with unequal port-plane cross-sectional areas, [A.sub.i] [not equal to] [A.sub.j], to add generality (e.g., control volume of Figure 3) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The first term accounts for the difference in the rate at which pressure work is done on the flow, accounting here for density changes through the buoyancy term,[[rho].sub.i] g[z.sub.i], with port-plane elevation,[z.sub.i], and acceleration of gravity, g. The second term accounts for kinetic energy rate contributions, where [[upsilon].sub.i] is the resultant airflow velocity and [v.sub.i]is the normal velocity at port plane i. The third term,[E.sub.d.sup.l], accounts for the rate of viscous energy dissipation in the control volume, l, an unambiguous quantitative measure of the power dissipated. The fourth term accounts for the rate of accumulation of total kinetic energy,[K.sub.tot.sup.l], and geopotential energy,[[PHI].sub.tot.sup.l], within the control volume l (i.e., evaluated, in principle, via integration of 0.5 [rho][[upsilon].sup.2]and [rho]gz, respectively, over the control volume). Importantly, this fourth term models dynamic phenomena associated primarily with the inertia of airflow that, together with the mass accumulation dynamics noted above, accounts for macroscopic airflow dynamics. Yet it appears that no multizone analysis program has included this dynamic contribution. (A broad review of dynamic macroscopic airflow analysis is found in Axley [2006b].)

This detailed power balance may be recast in terms of the port-plane variables through the introduction of two terms, [[alpha].sub.i] and [[beta].sub.i], which account for the nonuniformity of pressure and velocity distributions at the ports, assuming that air density is uniform over port sections as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where

[[alpha].sub.i][equivalent to][[integral].[A.sub.i]][[upsilon].sub.i.sup.2][v.sub.i]dA/([[^.v].sub.i.sup.3][A.sub.i]) (11)

[[beta].sub.i][equivalent to][[integral].[A.sub.i]][p.sub.i][v.sub.i]dA/([[^.p].sub.i][A.sub.i]) (12)

For nearly uniform pressure and velocity distributions, [[beta].sub.i][right arrow]1.0. If, in addition, the port velocity is normal to the port section, then [[alpha].sub.i][right arrow] 1.0 For fully developed laminar flow in circular pipes, [[alpha].sub.i] = 2.0, while it is close to 1.0 for fully developed turbulent flow. It should be noted, that the velocity distribution correction factor, [[alpha].sub.i], is sometimes included in the published literature (e.g., see Bird et al. [2002] and ASHRAE [2005]) but is most often assumed to be close to unity for practical applications, while the pressure-work correction factor, [[beta].sub.i], is not commonly presented. For flow-limiting resistances found in building ventilation systems, there appears to be little justification to assume these factors are close to unity. Indeed, some preliminary studies have indicated quite the opposite (Axley and Chung 2005a, 2005b, 2006).

For multiport control volumes, flow contributions are simply summed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

For any two-port control volume, l, the viscous dissipation rate may be related to a characteristic kinetic-energy rate of the flow (taken here as that at the inflow port) and a dimensionless friction loss factor [[zeta].sup.l] (Bird et al. 2002), as follows:

[E.sub.d.sup.l] = 1/2 [[rho].sub.i][[^.v].sub.i.sup.3][A.sub.i][[zeta].sup.l] (14)

Friction loss factors have been published for a large variety of two-port flow components, most obviously for HVAC duct-network components, but also for simple and operable openings similar in geometry to common building ventilation devices (Fried and Idelchik 1989; Idelchik 1994; Blevins 2003; ASHRAE 2005). Although friction loss factors are not commonly used for such openings, the recent attention directed to wind-driven airflow through porous buildings (Sandberg 2002, 2004; True et al. 2003; Kato 2004; Seifert et al. 2004; Axley and Chung 2006; Kobayashi et al. 2006) and to evaluating the efficacy of stack ventilation devices (Hunt and Syrios 2004) has fostered interest in mechanical power balances and, consequently, the use of friction-loss factors.

Assuming steady flow conditions and nearly equal densities in the kinetic energy terms,[[rho].sub.i][approximately equal to][[rho].sub.j] = [[rho].sup.l], Equation 13 simplifies to a general form of the Bernoulli equation as follows:

([[beta].sub.i][[^.p].sub.i] + [[rho].sub.i]g[[^.z].sub.i] + [1/2][[alpha].sub.i][[rho].sup.l][[^.v].sub.i.sup.2]) - ([[beta].sub.j][[^.p].sub.j] + [[rho].sub.j]g[[^.z].sub.j] + [1/2][[alpha].sub.j][[rho].sup.l][[^.v].sub.j.sup.2]) - [1/2][[rho].sup.l][[^.v].sub.i.sup.2][[zeta].sup.l] = 0 (15)

Note that the actual inflow and outflow densities are retained for the geopotential terms (i.e., to account for buoyancy effects), while the mean zone density, [[rho].sup.l] is employed for the kinetic energy terms--a macroscopic form of the microscopic Boussinesq assumption.

The standard Bernoulli equation is obtained when port pressure and velocity distributions are uniform, and these velocities are normal to the opening (i.e., [alpha] = [beta] - 1.0) as follows:

([[^.p].sub.i] + [[rho].sub.i]g[[^.z].sub.i]) - ([[^.p].sub.j] + [[rho].sub.j]g[[^.z].sub.j]) - [[zeta].sup.l][1/2][[rho].sup.l][[^.v].sub.i.sup.2] = 0 (16)

This relation, in more familiar terms, defines a total pressure balance. While seldom used in building ventilation analysis, total pressure balances are favored by the piping and duct network analysis communities (Jeppson 1976; Wood and Funk 1993; Saleh 2002).

The standard Bernoulli relation may be simplified further for two-port control volumes with equal inflow and outflow port areas, [A.sub.i] = [A.sub.j], to obtain the following:

([[^.P].sub.i] + [[rho].sub.i]g[[^.z].sub.i]) - ([[^.P].sub.j] + [[rho].sub.j]) - [[zeta].aup.l][1/2][[^.v].sub.i.sup.2] = 0 (17)

This simplified relation, defined in terms of modified pressure differences and accounting for both hydrostatic and static differences,[DELTA][P.sup.l][equivalent to]([^.p.sub.i]+[[rho].sub.i]g[z.sub.i]-([^.p.sub.j]+[[rho].sub.j]g[z.sub.j]), is essentially the basis of conventional multizone airflow analysis--not the Bernoulli equation, as is often claimed. That is to say, in conventional multizone analysis, airflow rates are assumed to be directly related to (modified) static pressure differences. For example, following the tradition establish a century ago by Sir Napier Shaw (Shaw 1907), building openings are often modelled with a simplified version of the (steady isothermal) orifice equation defined in terms of the familiar discharge coefficient, [C.sub.d] as follows:

[DELTA][P.sup.l] - [1/2] [rho][[^.v].sub.i.sup.2](1/[C.sub.d.sup.2]) = 0 (18)

At extremely low-flow Reynolds numbers, airflow velocities are linearly related to static pressure differences (e.g., as modeled by the classic Hagen Poiseuille equation [Bird et al. 2002]) as follows:

[DELTA][P.sup.l] - [32[mu]L/[D.sub.h.sup.2]][[^.v].sub.i] = 0 (19)

where [mu] is the viscosity of air, L is the effective length from inlet to outlet, and [D.sub.h]is the effective hydraulic diameter of the two-port control volume under consideration. While this flow dissipation model seldom appears in the literature, it or a similar variant is invariably used internally in multizone airflow analysis programs for low-flow conditions (e.g., Re = [rho][^v.sub.i][D.sub.h]/[mu][less than]100) to provide greater accuracy and to assure convergence of iterative solvers. It also provides a useful relation to provide an initial estimate (iterate) of airflows needed for iterative solutions of the otherwise nonlinear systems of equations that govern multizone airflows.

In conventional multizone analysis-zone pressures, p(z) are assumed to vary hydrostatically within zones. Thus, for each pair of port planes i and j of a given zone or, alternatively, for the zone node, l, and any given port plane, i the hydrostatic field assumption is imposed as follows:

([[^.p].sub.i] + [[rho].sup.l]g[[^.z].sub.i]) + ([[^.p].sub.j] + [[rho].sup.l]g[[^.z].sub.i]) = 0 (20a)

([p.sup.l] + [[[rho].sup.l]g[z.sup.l]]) - ([[^.p].sup.i] + [[rho].sup.l]g[^.z.sub.i] = 0 (20b)

This is yet another simplification, this time assuming there is no dissipation.

Each of the control volume dissipation relations presented above (i.e., Equations 10, 13, and 15-21) have the form [F.sub.d.sup.l] (for control volume l). The set of the dissipation relations for all n control volumes of a building idealization {[F.sub.d]} = {[F.sub.d.sup.1], [F.sub.d.sup.2] ... [F.sub.d.sup.n]}.sup.T] defines the system dissipation vector that will be used for port-plane analysis.

Many of the dissipation relations used for practical multizone airflow analysis may be derived as special semiempirical cases of the relations presented above, yet other more empirical relations have also proven useful for practical multizone airflow analysis. The vast majority of these dissipation relations are limited to modeling two-port control volumes, as until recently multiport control volumes (e.g., building zones and duct plenums) have been modeled as being inviscid. When using these models for loop analysis, it is convenient to use functional notation and express them in terms of the modified pressure as [DELTA][P.sup.l] = [F.sub.d.sup.'l]([[^.v].sup.'l]).or [F.sub.d.sup.'l](m.sup.l) On the other hand, for conventional nodal analysis, two-port dissipation models are commonly expressed in terms of air mass flow rate as [m.sup.l] = [G.sub.d.sup.l]([DELTA][P.sup.l]). Detailed data needed to use these models and additional details relating to such matters as accuracy, application suitability, and limiting assumptions are published in a number of handbooks and program users' manuals (Liddament 1986a, 1986b; Pelletret and Keilholz 1997; Orme et al. 1998; Orme 1999; Dols et al. 2000; Haas 2000; Dols 2001a, 2001b; Persily and Ivy 2001; ASHRAE 2005; Axley 2006b).

Many of the available dissipation relations are valid for unidirectional flow, yet in larger building openings, bi-and multi-directional airflows commonly occur. A number of approaches have been proposed to model these large-opening conditions. Most of these large-opening models have been based on applying an infinitesimal form of the simplified orifice model to infinitesimal lamina over the height of the opening (van der Maas 1992; van der Maas et al. 1994; Dascalaki and Santamouris 1996; Etheridge and Sandberg 1996; Feustel and Smith 1997; Etheridge 2004). While these approaches may be intuitively satisfying, they lack physical or theoretical justification and may well overestimate the dissipation they seek to model (Axley 2000c, 2000d, 2001b).

Boundary Conditions

Equations governing the airflow in building systems may be formed using any one of the three approaches introduced above. Alone, however, these equations would be indeterminate and must be complemented by boundary conditions and, in general, zone-field assumptions to establish a determinate (closed) system of equations that may then be solved.

In general, the boundary conditions may be established in terms of any of the defined system variables used. While pressure conditions acting at envelope ports of a building system often define these boundary conditions, one may also specify internal zone or port-plane pressures (e.g., to investigate building pressurization strategies for smoke control) or components or portplane airflow rates (e.g., to model well-controlled mechanically induced airflows).

Port-Plane Approach. For the port-plane approach, one can directly specify numerical values for static pressure, [^.p*], total pressure, [[^.p*].sub.tot], or velocity [^.v*], as boundary conditions at any of the port planes of the system. While the algebraic specification of these three boundary conditions may be achieved in a number of ways, here they will be represented as additional boundary condition functions [F.sub.bi] for each port plane i involved, as follows:

[^.p.sub.i] - [[^.p*].sub.i] = 0 (21)

[[^.p].sub.i] + [1/2] [[rho].sup.l][[^.v].sub.i.sup.2] - [[^.p*].sub.itot] = 0 (22)

[[^.v].sub.i] - [[^.v].sub.i.sup.*] = 0 (23)

where [[rho].sup.l] is the air density of the control volume associated with the specification.

Nodal Approach. For the nodal approach, one may specify node pressures directly as follows:

[p.sup.l] - [p*.sup.l] = 0 (24)

One may also specify the airflow rate for a given limiting flow element, l, simply by summing its specified numerical value, [m*.sup.l], when applying the mass conservation relation.

Finally, for all approaches, wind-induced envelope port-plane pressures are commonly assumed to be linked to the ambient outdoor zone-node pressure, [p.sup.o], via sealed building wind-pressure coefficients, [C.sub.p,i], for each envelope port plane, i as follows:

[[^.p].sub.i] - [C.sub.p,i][1/2][[rho].sup.o][[bar.v].sub.ref.sup.2] + [[rho].sup.o]g([z.sub.i][z.sup.o]) - [p.sup.o] = 0 (25)

(The hydrostatic increment and ambient outdoor node pressures,[p.sup.o], of this relation are often not included in the literature, yet are necessarily included in multizone airflow analysis programs.)

Each of the boundary conditions presented above (i.e., Equations 21-25) has the form [F.sub.b,i] = 0 (i.e., for port plane i). The m specified port planes i, j, ... k may then be collected into a system boundary condition vector {[F.sub.b]} = [{[F.sub.b,i], [F.sub.b,j], ... [F.sub.b,k]}.sup.T]for port-plane analysis.

Wind-pressure coefficients are available in a number of handbooks (Dascalaki and Santamouris 1996; Allard 1998; Orme et al. 1998; Santamouris and Dascalaki 1998; Persily and Ivy 2001; ASHRAE 2005). Increasingly, CFD is also used to predict wind-pressure coefficients (Murakami 1993; Holmes and McGowan 1997; Kurabuchi et al. 2000; Jensen et al. 2002a, 2002b). In general, wind-pressure coefficients vary with wind direction. A number of empirical correlations and computational tools have been developed that account for this variation (Walker and Wilson 1994; Dascalaki and Santamouris 1996; Knoll and Phaff 1996; Knoll et al. 1997; Orme et al. 1998). The handbooks cited above also provide guidance for adjusting airport wind data for building height, location, site topography, and shielding of nearby buildings.

The uncertainty of wind-pressure coefficients combined with that associated with estimating the reference wind conditions are thought to introduce the greater part of the uncertainty of computed results in multizone ventilation analysis (Bassett 1990; Furbringer et al. 1993, 1996a, 1996b; Roulet et al. 1996). Nevertheless, two additional sources of error may, in some instances, be equally important: (1) sealed-building wind-pressure coefficients may not properly represent driving forces for porous buildings when, for example, opening areas exceed approximately 20% of any given building surface (Aynsley et al. 1977a; Aynsley 1999) and (2) wind-induced turbulence may significantly contribute to ventilation flow rates, yet this effect is commonly ignored (Etheridge and Sandberg 1996; Siren 1997; Girault and Spennato 1999; Saraiva and Marques da Silva 1999; Haghighat et al. 2000; Etheridge 2002). Recent research into modeling porous buildings suggests that windward envelope pressures may be better modeled through the specification of total pressure rather than static pressure (Aynsley et al. 1977b; Seifert et al. 2000, 2004; Sandberg 2002, 2004; True et al. 2003; Karava et al. 2004; Ohba et al. 2004), yet the issue remains unresolved.

Zone Field Assumptions

The zone field assumptions introduced above in Equations 20a and 20b are specific cases of the assumption of hydrostatic conditions within a flow regime: p + [rho]gz = constant. Two additional possibilities can also be identified--a uniform pressure field and a uniform total pressure field. The latter, p + [rho]gz + (1/2)[rho][v.sup.2] = constant, is associated with irrotational flow in (nearly) inviscid fluids (Chorin and Marsden 1993). In conventional multizone analysis, the uniform pressure field assumption,p = constant, is commonly applied to duct junctions, while the hydrostatic field assumption is preferred for zones; both are quite reasonable approximations, but approximations nevertheless. Use of the irrotational field assumption has been investigated for crossventilation airflows characterized by a dominant stream tube (Axley et al. 2002a; Axley and Chung 2005a, 2005b). In the few cases investigated, however, the hydrostatic field assumption appears to provide a better approximation than the irrotational field assumption and avoids the complexity of the added nonlinearity introduced.

The hydrostatic field assumptions presented above (i.e., Equations 20a and 20b), and similar functions based on assumed uniform pressure or irrotational fields, have the form [F.sub.f,k] = 0 (i.e., for a pair of port planes identified as pair k). For f specified pairs of port planes, these may be collected into a system field assumption vector,{[F.sub.f]} = [{[F.sub.f,i], [F.sub.f,j] ... [F.sub.f,k]}.sup.T]for port-plane analysis.

Multizone System Equations

With fundamental mass and mechanical energy-conservation relations, semiempirical and fully empirical dissipation relations, boundary conditions, and zone-field assumptions in hand, one can assemble these equations to form system equations governing the airflows in multizone building systems. Here we will consider the conventional nodal approach, the more general port-plane approach, and the often overlooked loop approach, but for simplicity of presentation, consideration will be limited to steady conditions.

Nodal Approach. Given a nodal building idealization, one can implement the nodal approach by simply demanding zone mass airflow rates be conserved for each zone (Equation 8). Element mass airflow relations are then applied for each limiting flow resistance,, by substituting appropriate dissipation relations expressed in functional notation as [m.sup.i] = [G.sub.d.sup.i]([DELTA][P.sup.i]As each of the element-flow relations is defined in terms of port-plane pressures of the adjacent zones they link, zone-field assumptions must then be imposed (Equation 20b) and, typically, wind-pressure boundary conditions are imposed for envelope ports (Equation 25) to establish the primary dependency on the zone node pressures,[p.sup.z]. When formed for each of the zones of the building idealization, one will have equations defined in terms of the zone-node pressures.

Although the details of this equation assembly process may be presented in concise matrix notation (Axley 2006b), here we will resort to functional notation instead. The substitution of the specific numeric-model parameters outlined above has the net effect of mapping the dissipation relations from functions that depend on modified pressure differences,[DELTA][P.sup.i], to ones that depend on the system zone pressure vector,{p}, as, [G.sub.d.sup.i]([DELTA][p.sup.i])[right arrow] [G.sub.d.sup.i]({p}); hence, the mass conservation function for each zone, z, will also be dependent on the zone-node pressures as follows:

[F.sub.m.sup.z]({p}) = [summation over (elements linked to z i = i1,i2,...)][G.sub.d.sup.i]({p}) = 0 (26)

The equations governing ventilation airflows in the system as a whole (i.e., the system equations) are defined by the set of mass conservation equations for all zones, collected here into a single vector of functions as {[F.sub.m]({p}) = [{[F.sub.m.sup.1]({p})[F.sub.m.sup.2]({p}) ... }.sup.T] = {0} As each of these functions are coupled through shared dependencies on select zone-node pressures and, furthermore, these couplings are nonlinear, the solution of these coupled system equations is generally difficult.

Most multizone programs use the Newton-Raphson method or one of several variants of it to solve the system equations. The Newton-Raphson method is a well-known method based on a truncated Taylor-series approximation of the system equations in an iterative manner where given an initial iterate for the zone pressure vector,[{p}.sub.[0]], an improved estimate is formed, [{p}.sub.[1]] = [{p}.sub.[0]] + [{[DELTA]p}.sub.[1]], by solving the truncated Taylor series approximation (Press et al. 1992; Kelley 1995, 2003; Lorenzetti and Sohn 2000). Using this improved estimate, the process is repeated until the solution converges to sufficient accuracy.

More specifically, for the typical case of steady flow d{m}/dt = {0}at iterate k of the iterative process, one

1. forms the system Jacobian matrix: [MATHAMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (27)

2. solves the truncated Taylor series approximation: [[J].sub.[k]][{[DELTA]p}.sub.[k + 1]][approximately equal to] - {[F.sub.m]([{p}.sub.[k]])} (28)

3. updates the solution estimate: [{p}.sub.[k + 1]] = [{p}.sub.[k]] + [{[DELTA]p}.sub.[k + 1]] (29)

4. forms the current system residual: {[F.sub.m]([{p}.sub.[k + 1]])} (30)

5. evaluates convergence: {[F.sub.m]([{p}.sub.[k + 1]])}[less than or equal to][[tau].sub.rel] {[F.sub.m]([{p}.sub.[k]])} + [[tau].sub.abs] (31)

where both relative,[[tau].sub.rel],and an absolute,[[tau].sub.abs], error tolerances may be considered. If the convergence criterion is met, the approximate solution is deemed satisfactory.

The convergence of this iterative process depends in part on the nearness of the initial iterate to the actual solution. Therefore, computation of this initial iterate is an important step in the process, yet no method is available to obtain a completely reliable initial iterate. However, one may obtain an initial iterate by assembling the system equations using the linear Hagen-Pouiselle relation for all flow elements, as the resulting system equations will be linear and, thus, easily solved.

Port Plane Approach. The port plane approach is superficially similar to the nodal approach in that one assembles mass conservation and dissipation relations and imposes boundary conditions and zone-field assumptions to form the system equations. However, the relations used are expressed in terms of the port-plane variables of velocity,[^.v.sub.i]., and pressure, [^.p.sub.i], of each port plane i of the building model--i.e., in terms of the port-plane system variable vector, [{^.v|[^.p]}.sup.T].

For a specific building idealization, the mass balance functions would be formed for each control volume and collected into a vector of functions, {[F.sub.m]([{^.v|^.p}.sup.T])} = {[F.sub.m.sup.1] [F.sub.m.sup.2] ...}.sup.T]; the dissipation conservation functions would be formed for each control volume and collected into a vector of functions, {[F.sub.d]([{^.v|[^.p]}.sup.T])} = {[F.sub.d.sup.1] [F.sub.d.sup.2] ...}.sup.T]; the boundary conditions functions would be formed for each specified port plane, m, n, ... (e.g., envelope port planes) and collected into a vector of functions, {[F.sub.f]([{^.v|^.p}.sup.T])} = {[F.sub.f,m] [F.sub.f,n] ...}.sup.T]; and, finally, zone-field assumptions would be applied to each pair of port planes, m, n, as needed and collected into a vector of functions {[F.sub.f]([{^.v|^.p}.supT])} = {[F.sub.f,m] [F.sub.f,n] ...}.sup.T]. The system equations {F} would then be defined by the collection of these functions: {F([{^.v|^.p}.sup.T])} = [{[F.sup.m]|[F.sup.d]|[F.sup.b]| [F.sup.f]}.sup.T] This system of equations may then be solved using the Newton-Raphson method or a variant, given an initial iterate for the port-plane system variables,{^.v|^.p}[0], by following the schematic algorithm presented above where the Jacobian matrix would now be formed in terms of the port-plane system variables, {^.v|^.p}.

Although the port plane approach may seem rather formidable, it is a general method that reveals the limiting assumptions of other approaches used to model flow networks. If, for a given building model, one models dissipation using power-balance dissipation relations for all control volumes (or, equivalently, Bernoulli dissipation for two-port control volumes), then the building model will be theoretically most complete, especially if one is able to include the velocity and pressure distribution correction factors [alpha]and [beta] For serial assemblies of two-port control volumes, zone-field assumptions will not be required for closure and, therefore, the system equations will be exact in the macroscopic sense. Furthermore, the imposition of total pressure boundary conditions is inherently direct in the port-plane approach, while it is not so for the nodal and loop approaches. Finally, as a full power-balance analysis allows the possibility of accounting for the dynamic contribution of both mass accumulation and flow inertia, it provides the only means presently available to properly account for realistic flow dynamics.

If, on the other hand, one utilizes the Bernoulli dissipation relation for all two-port control volumes (e.g., the flow-limiting resistances) and applies either the hydrostatic or the uniform field assumptions to multiport control volumes (e.g., zones), then the analytical model will correspond to those preferred by the piping network analysis community (Jeppson 1976; Wood and Funk 1993; Saleh 2002). Finally, if one models dissipation in two-port control volumes (conventional flow elements) using any of the conventional semi-empirical or fully empirical flow-element models and applies either the hydrostatic (for zones) or the uniform pressure field (for duct junctions) assumptions to multiport control volumes, then the analytical model will correspond to the conventional multizone method (although without being formulated in terms of nodal pressures). Again, however, the conventional formulation remains theoretically incomplete and approximate as mechanical energy conservation is ignored and field assumption approximations are required. While research is under way to evaluate the significance of these shortcomings, no general conclusion yet can be made (Guffey and Fraser 1989; Murakami et al. 1991; Kato 2004; Axley and Chung 2005a, 2005b).

Loop Approach. In this approach, one forms loop consistency functions demanding that port-to-port pressure changes, [DELTA][^.P.sub.k] while traversing flow loops in the airflow system simply sum to zero as follows:

[F.sub.l,i] = [Summation over (loop i)] [DELTA][[^.p].sup.k] = 0 (32)

where k is permuted through linked control volumes (including the outdoor zone) as one proceeds around a given loop, l. To properly account for wind-induced effects, one must include an outdoor ambient pressure node in the loop for loops that traverse the outdoor zone. The individual loop equations (functions) may then be collected into a vector of loop functions {[F.sub.l]} = [{[F.sub.l,1] [F.sub.l.2] ...}.sup.T], although one must be sure that the collection of these loop functions are an independent set of all the loop equations that could conceivably be formed. This last condition has been the principle barrier to the use of the loop method. Recently, however, graph theoretic algorithms have been employed to automatically identify the independent loops needed to achieve this critical objective (Jensen 2005).

The loop consistency functions are then complemented by mass conservation for each zone using Equation 8, dissipation relations for each flow element of the form [DELTA][[^.p].sup.k] = [F.sub.d.sup.'k](m.sup.k] - [[rho].sup.k]g[DELTA][z.sup.k], the hydrostatic relation for pressure changes encountered within zones [DELTA][[^.p].sup.k] = [[rho].sup.k]g[DELTA][z.sup.k], and the relation that accounts for wind-pressure increments at envelope ports [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Substituting these relations, the loop consistency relation may be simplified to the intuitively satisfying result that flow-element pressure losses simply balance the sum of wind-induced [DELTA][P.sub.w]and buoyancy induced [DELTA][P.sub.b]pressure changes as follows:

MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII (33)

where the envelope wind pressure coefficients,[C.sub.p,i] are summed as positive contributions for pressure increases as one moves along the loop in the forward direction. Note that zone-node pressures need not be considered at all in forming the loop equations.

These loop consistency relations would then be augmented with mass conservation relations,[F.sub.m.sup.l], for each zone l of the building idealization using Equation 8. Finally, recognizing the mass-conservation and loop-consistency relations are defined in terms of the flow-element mass airflow rates, one may take these airflow rates as the system variables, {m} = [{[m.sup.1] [m.sup.2] ...}.sup.T] collect the mass conservation relations into a single vector, {[F.sub.m]({m})} = [{[F.sub.m.sup.1] [F.sub.m.sup.2] ...}.sup.T] and define the loop system equations as the set of the mass conservation and loop equations {F({m})} = [{[F.sub.m]|[F.sub.l]}.sup.T] As before, this system of equations may then be solved using the Newton-Raphson method, given an initial iterate for the loop system variables, [{m}.sub.[0]] by following the schematic algorithm presented above where the Jacobian matrix now would be formed in terms of the loop system variables, {m}.

The loop approach, while overlooked by all but a few members of the building simulation community (Wray and Yuill 1993; Nitta 1994), provides an approach for sizing ventilation system components (Axley 1998, 1999, 2000a, 2000b, 2001a; Axley et al. 2002c; Ghiaus et al.

2003) and has recently been employed for modeling the coupled thermal/airflow interactions in buildings (Jensen 2005).

CONCLUSION

Multizone building airflow analysis may be based on either nodal or port-plane idealizations of integrated building/HVAC systems. The nodal approach, which evolved through an adaptation of nodal methods of electric resistance network analysis, idealizes the building system as collections of zones and duct junctions and associated node pressures within them linked by discrete flow-limiting elements. With the nodal model in hand, commonly available multizone analysis programs form equations governing system airflows by (1) demanding that mass airflow rates are conserved in zone and duct junctions, (2) assuming dissipation in the flow-limiting elements is solely dependent on static pressure differences, and (3) assuming hydrostatic conditions approximate the conditions within zones. This conventional approach to multizone airflow analysis has proven to be reasonably reliable, practically accurate, and computationally effective. Yet it ignores the fundamental principle that, in addition to mass, mechanical energy must be conserved; it presumes that airflow in zones is inviscid; it limits flow elements to two-port control volumes of constant cross section; and while in principle it allows consideration of the dynamic accumulation of air mass within zones, it does not support the consideration of the dynamic accumulation of kinetic and geopotential energy.

The alternative port-plane approach, based on established methods of macroscopic analysis used in the chemical engineering community (Bird et al. 2002), idealizes the building system as a collection of control volumes (i.e., of zones, the outdoor environment, flow elements, ducts, etc.) explicitly considering the airflow entering and exiting through the port planes separating these control volumes. By defining the primary system variables in terms of spatial averages of pressures and velocities at each of the port planes, the port-plane approach allows both mass and mechanical energy conservation principles to be imposed in forming the system equations and in principle can accommodate flow elements of nonconstant cross section, dissipation within zones, and the dynamic accumulation of kinetic and potential energy. Yet, when used for such a full power-balance analysis, it produces larger systems of equations of greater nonlinearity than the conventional approach that appear to be generally more difficult to solve. However, the commonly used theory, as well as a more general Bernoulli approach (i.e., one that models dissipation in flow elements with the Bernoulli equations), may also be implemented using a port-plane model of the building system. Thus, the port-plane approach is a general approach that includes the conventional, Bernoulli, and hybrid combinations of these two and the power-balance approach as special cases.

At this moment in the development of multizone airflow analysis methods, the port-plane approach has opened up new avenues for more fundamental investigations of the governing theory, while, simultaneously, promising developments have emerged in coupling these methods to multizone thermal and air-quality models of whole-building systems and to CFD models of individual zones that may significantly extend the utility of multizone airflow analysis. Thus, if we are to realize the full potential of the field, additional research is needed at both the fundamental and more applied levels, even though the past three decades of development have proven to be so fruitful.

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James Axley, PhD

James Axley is a professor at the School of Architecture and School of Forestry and Environmental Studies, Yale University, New Haven, CT.

Received June 29, 2007; accepted August 28, 2007

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Author: | Axley, James |
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Publication: | HVAC & R Research |

Geographic Code: | 1USA |

Date: | Nov 1, 2007 |

Words: | 11354 |

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